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Wikiversity:Request custodian action
4
75745
2694058
2693968
2025-01-02T00:54:03Z
OhanaUnited
18921
/* IP block exempt request */ thank you
2694058
wikitext
text/x-wiki
{{/Header}}[[cs:Wikiverzita:Nástěnka správců]][[fr:Wikiversité:Requêtes aux bibliothécaires]][[pt:Wikiversidade:Pedidos a administradores]]
== Request to move image files to Commons ==
I got [[User_talk:Guy_vandegrift#Files_on_Commons|'''this request''']] to move files from [[:Category:NowCommons]] and [[:Category:Files from USGS]]. I delete lots of files, but usually let others delete image files because of my ignorance of copyright laws. I also have contributed a lot of files to Commons, but almost all of it is my own work. So I am out of my comfort zone on this. I don't even understand why these files should be moved.
{{ping|User:MGA73}} Maybe we can find someone with more expertise on file transfers here on [[Wikiversity:Request custodian action|Request custodian action]].--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 22:47, 7 January 2024 (UTC)
In a related vein, due to my inexperience with copyright regulations, perhaps it would be better if someone else processed the following files. All are up for speedy deletion. And all seem like quality images and/or on potentially high quality WV resources.
<gallery widths=50>
File:Merged fig1.png
File:Merged matrix2.png
File:Rps all hsa.png
File:Selected domfams fix.png
File:Service-pnp-fsa-8b32000-8b32000-8b32095r.jpg
File:Summary.svg
File:Transtree.png
File:Untitled-91274a-1024.jpg
</gallery>
: My request was primary to delete files that was moved to Commons allready. But if anyone have checked files they are of course very welcome to move files to Commons too. Same with [[:Category:Files from Flickr]]. --[[User:MGA73|MGA73]] ([[User talk:MGA73|discuss]] • [[Special:Contributions/MGA73|contribs]]) 16:32, 9 January 2024 (UTC)
::Thanks for the info. My ignorance of copyright law makes me very hesitant to delete image files.--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 19:50, 26 February 2024 (UTC)
::: I noticed [[User:Koavf]] just deleted a file moved to Commons. So perhaps Koavf could have a look at the files in [[:Category:NowCommons]] once there is a little time to spare? :-) --[[User:MGA73|MGA73]] ([[User talk:MGA73|discuss]] • [[Special:Contributions/MGA73|contribs]]) 19:14, 27 February 2024 (UTC)
::::lol@"time to spare", but sure. <3 —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 21:17, 27 February 2024 (UTC)
:::::Sometimes dirty tricks work ;-) --[[User:MGA73|MGA73]] ([[User talk:MGA73|discuss]] • [[Special:Contributions/MGA73|contribs]]) 08:00, 28 February 2024 (UTC)
== [[Special:Contributions/Hooglimkt]] (again) ==
{{Archive top|User is blocked so I guess were are {{Done}}. [[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 19:52, 26 February 2024 (UTC)}}
{{ping|Koavf}} After the last report ([[Wikiversity:Request_custodian_action/Archive/25#Special:Contributions/Hooglimkt]]), the user has restarted same types of edits. [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 13:25, 9 January 2024 (UTC)
:{{not done}} But what is the action here? He just wrote a bunch of Portuguese stuff on his userpage. What needs to be done? —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 13:30, 9 January 2024 (UTC)
:: They are writing non-English advertisements on someone else's userpage, how can this be allowed? [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 13:33, 9 January 2024 (UTC)
:: Please compare the reported user and [[Special:CentralAuth/Hoogli]] (user whose userpage is targeted), they don't look like the same user. [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 13:35, 9 January 2024 (UTC)
:::Ah, sorry--I got the usernames confused. Yes, that is inappropriate and he's not here for constructive purposes. Sorry. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 13:37, 9 January 2024 (UTC){{Archive bottom}}
== [[Special:Contributions/NotAReetBot]] ==
According to [[WV:IU]], this username is not acceptable (implying bot), should this account be blocked? [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 01:28, 10 January 2024 (UTC)
: I already sent a welcome and {{tl|uw-username}} (imported from enwiki). [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 01:42, 10 January 2024 (UTC)
:I think explicitly saying that you're not a bot is acceptable, but I agree that it's probably not ideal. E.g. someone could have the username "NotAReet" and run a bot under this name. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 03:46, 10 January 2024 (UTC)
== Call for rewriting [[WV:UNC]] ==
This agenda is suggested at [[Wikiversity_talk:Username#WV:UNC needs updates]], since this is related to policy documentation, I would like to have the attention of our custodians. [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 02:49, 10 January 2024 (UTC)
== [[Special:Contributions/2409:4064:810:DA39:FA73:D928:2C4D:B401]] ==
Possible vandalism (Massive enwiki copies with MOS issues), seems to be related to the recently reported IP, please consider range block. All targeted pages are semi-protected. Reverted revisions seem to be enwiki copies, please also consider revision deletion if needed. [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 03:38, 20 January 2024 (UTC)
: (Note) Currently stale, will report again if they come back. [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 09:04, 28 January 2024 (UTC)
== Scope of talk page usage for blocked users ==
I understand that the scope of talk page usage for blocked users is aimed at unblocking requests and relevant discussions. I would like to ask if Wikiversity has more exceptions accepted by the community. I'm asking this because I recently found [[special:diff/2602322]], and this does not seem to be related to an unblocking request. If unacceptable, custodians may need to remove talk page access from the user. [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 02:53, 30 January 2024 (UTC)
== Please review recent edits at [[Wikiversity:Verifiability]] ==
{{cot|long discussion}}
Recently we had many changes to this documentation. Reverting undiscussed changes would be non-controversial, but I'm not sure about the others. What would our custodians think about these edits? [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 15:03, 31 January 2024 (UTC)
: Each of my edit has an explanation/rationale in the edit summary. Here a summarization: I above all removed sentences that presented a contradiction within the same page. I also switched the page to policy proposal away from policy since I could not find a discussion establishing the page as a policy and since, given the contradictions before my edits, the page could not have been taken seriously as a policy, that is, a set of rigid rules contrasting to guidelines. I could have discussed the changes somewhere first, but since the changes are well documented in their edit summaries, I hoped they could remain. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 17:56, 31 January 2024 (UTC)
::For the record, the original version (before recent efforts) can be found at [[Special:Permalink/1375824]]. Regarding my thoughts about these edits, I think we should distinguish between top pages and subpages. If an instructor is inviting students to submit work in subspace, the instructor should have considerable flexibility regarding those subpages.--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 00:03, 1 February 2024 (UTC)
::: While I'm not sure about what type of flexibility is being mentioned, I generally believe that teachers should have enough privileges to complete their projects. If our policies (and related proposals) restrict legitimate educational activities, then we are no longer a place for education. [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 12:54, 1 February 2024 (UTC)
:: Thank you very much for the explanation and the summary, but I cannot guarantee that everyone will accept it. Removing contradictions sounds good. If the content was obvious nonsense or conflict with the entire Wikiversity, then your decision (blanking/removal) would be the most reasonable one. In this case, I think there were other options (such as rewriting to resolve contradictions), and that is why I'm calling for a review. For example, at [[special:diff/2602692]], you said that "The obligation to use verifiable and reliable sources lies with the editors wishing to include information on Wikiversity page, not on those seeking to question it or remove it" contradicts the option of scholarly research at Wikiversity. I don't understand how this becomes a contradiction (have you already explained that?). Even if it was a contradiction, I think blanking was not the only one option. We could have restricted the obligation to non-research content (such as educational resources) or downgraded the obligation to a recommendation, and avoid potential conflict with Wikiversity research content. The summary of my question is, "Why have you decided to remove instead of suggesting a rewrite?". [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 13:20, 1 February 2024 (UTC)
::: I see an obvious contradiction, as mentioned in the edit summary: if original research and original user-written essays are allowed, there is no "obligation to use verifiable and reliable sources".
::: As for dropping text vs. rewrite: a rewrite creates an opportunity to introduce new mistakes and non-consensualities, a bad thing. By contrast, removal of problematic sentences removes defects. After removal of problematic sentences, we may focus on whether the text that remained after removal is really accurate and fully fit for purpose, which I do not think to be the case either; more corrective work is required. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 11:12, 5 February 2024 (UTC)
:::: Thank you for additional explanations. If somebody is going to produce their own research where anything similar was never published elsewhere, there would be no other independent secondary sources, so the Wikipedia-like verifiability is no longer reasonable at here. On the other hand, I believe that authors should work hard to avoid errors (calculation errors, uploading wrong images etc., I was talking about this type of verifiablity for research content), if they want to pass Wikijournal peer reviews then they need to do so. In addition, I expect many type of research comes out from previous research history, and I think it is reasonable to expect the Wikipedia-like verifiablity when explaining research background and related history. What would you think about this? I'm not demanding the Wikipedia-like verifiability to research itself, I'm recommending this to things before entering research. [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 01:59, 6 February 2024 (UTC)
::::: As for "If somebody is going to produce their own research where anything similar was never published elsewhere", one may well publish result of research such that something similar ''was'' already published elsewhere; it is still ''original research'' in Wikipedia terminology.
::::: Wikiversity is great for articles that combine original research/element of originality with referenced material. For such articles, there is no duty to reference things but I would see inline referencing as recommended for consideration (not enforced) and adding great further reading/external links as recommended (not enforced). I fully agree that "authors should work hard to avoid errors". As for Wikijournals, that is a separate class of Wikiversity content, with its own rules and processes. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 07:22, 9 February 2024 (UTC)
::::: About "explaining research background": I know of no duty to explain research background (or is there one?) and therefore, there is no duty to explain the background and then reference it using Wikipedia-style inline referencing. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 08:13, 9 February 2024 (UTC)
{{cob}}
Would somebody like to vote between keeping page ''as is'' or returning it to [[Special:Permalink/1375824]]? If so, write "I move that we foobar" as vote yes or no.--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 17:54, 26 February 2024 (UTC)
== Can [[User:Ciphiorg/sandbox]] be an acceptable sandbox? ==
The sandbox was made by using talk page namespace so I moved it into userspace. After the page moved, I noticed that the sandbox was about physical geography but also aimed to promote a single website (physicalgeography.org) and its subpages. I checked the author's enwiki history, all edits were reverted and their enwiki sandbox was deleted per CSD U5. Could this be a xwiki spam case? [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 06:33, 2 February 2024 (UTC)
:{{done}} Deleted. He can ask for undeletion if he wants to remove self-promotion/spam links. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 06:36, 2 February 2024 (UTC)
:: Recent abuse filter logs suggests that the user came back to do something similar. You may need to take action to stop them. [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 05:43, 7 February 2024 (UTC)
::: (Update) Currently stale. [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 09:18, 9 February 2024 (UTC)
== Concern about an IP range starting from 165.199.181 ==
IP editors from this range ([[Special:Contributions/165.199.181.3]], [[Special:Contributions/165.199.181.9]], [[Special:Contributions/165.199.181.15]]) have done a lot of unhelpful actions in our project for months. I think our custodians should consider a range block for a reasonable amount of time. [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 02:06, 6 February 2024 (UTC)
: (Note) All IPs in this report are blocked in minimal range. [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 02:11, 7 February 2024 (UTC)
== Please consider blacklisting of physicalgeography.org ==
Dear custodians, I have reported about editors trying to get physicalgeography.org to appear in Wikiversity at [[special:permalink/2603578#Can_User:Ciphiorg/sandbox_be_an_acceptable_sandbox?]], and now we have another editor trying to get the link visible ([[Special:diff/2603646]]). Please consider the blacklisting of this URL. Thank you for your attention. [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 02:11, 7 February 2024 (UTC)
== [[Special:Contributions/103.150.214.192]] ==
Too many test edits at sandbox (RC flooding), possible proxy, already blocked at zhwiki. [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 09:00, 9 February 2024 (UTC)
:{{ping|MathXplore}} I blocked for 3 hours and then Googled {RC flooding}. I have no experience with these things. How long should I block for?----[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 13:01, 10 February 2024 (UTC)
:: When I reported the IP, they were violent, and at least a short-term block (perhaps several hours) may have been needed at that time. Currently, the IP editor is stale, so there may be no significant meaning to block them at this moment. On the other hand, GetIPIntel Prediction is 100% at [https://ipcheck.toolforge.org/index.php?ip=103.150.214.192 IPcheck information], this means that this IP might be a [[:m:No open proxies|proxy]] (and I guess that is why zhwiki blocked this IP, I don't know well about zhwiki proxy block policy), though the other parameters are negative. I think we need someone who knows more about proxies to choose the right range and terms. {{ping|Koavf}} can you take a look at this IP? [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 13:18, 10 February 2024 (UTC)
:: (Note) After my reply, another IP ([[Special:Contributions/103.150.214.135]], close to the one above) appeared with similar behavior (targeting sandbox). This IP is blocked at zhwikivoyage as an open proxy (1 year), also blocked at enwiki as a web host. [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 14:15, 10 February 2024 (UTC)
:::I am not a range block pro, but doing a little range block hacking, I see that both [[Special:Contributions/103.150.214.192/16]] and [[Special:Contributions/103.150.214.135/16]] contain all of the edits by the above IPs and ''only'' the edits by the above IPs. Both are globally blocked for a couple of months, but 1.) I take violent threats very seriously ({{Ping|MathXplore}}, did you write to legal@? If not, I will.) 2.) the sandbox is one of the only pages you really don't want to have escalated protection on, and 3.) oftentimes, rangeblocking open proxies is not going to harm the project. So, I'm willing to do a 12-month range block. Great work as always. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 19:17, 10 February 2024 (UTC)
:::: Sorry, I didn't write to legal. I was checking the edit frequencies and their global contributions rather than the context. Please go ahead for the report to legal. Thank you for the reactions and information. [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 06:28, 11 February 2024 (UTC)
:::::Hey, no worries MX. You do a ''lot'' across ''many'' wikis. It's a team effort, friend. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 07:27, 11 February 2024 (UTC)
::::::Wait--I actually ''looked'' at the diffs and some of them mention some weird violent content, but are not ''threats'', so it doesn't rise to that occasion. Sorry for my ignorance. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 07:29, 11 February 2024 (UTC)
== [[Special:Contributions/24.224.18.114]] ==
Vandalism from this IP, a targeted page is now semi-protected. [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 02:22, 16 February 2024 (UTC)
: (Note) Currently stale. [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 06:28, 20 February 2024 (UTC)
== [[special:permalink/2607000]] ==
Can this be considered as an academic profile, or should be handled as an advertisement? [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 06:27, 20 February 2024 (UTC)
:Tricky. I'm inclined to call it a valid profile ''if'' this user engages in actually editing and particularly in creating resources related to these kind of topics such as SEO, but call it just spam if this person is only here to say "I am so-and-so and I have [x] marketable skills". :/ So I could be persuaded either way, but it's not ''obviously'' spam as of now, as far as I can tell. I totally respect any other custodian or curator deleting it, tho. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 06:32, 20 February 2024 (UTC)
== [[Portal talk:Astronomy]] ==
This talk page is currently isolated but has a lot of things in here. Where can we move this page to save it as an archive? [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 13:39, 25 February 2024 (UTC)
:I created [[Draft:Archive]] without asking for a consensus. If nobody objects, we can all use it. The only open question in my mind is whether we need to nowikify the pages to avoid having titles appear on various lists and categories. I suggest the title [[Draft:Archive/2024/Portal talk-Astronomy]]. Personally, I am not very adept at undeleting pages, thought with a bit of practice I might find it more natural. With a small cleanup crew that tends to get bogged down in long discussions, it's easier if everybody can look at pages that have been removed in this fashion. Many years ago I remember an editor who annoyed administrators with frivolous requests to undelete for viewing purposes. If you want, I can move [[Portal talk:Astronomy]] right now.--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 14:37, 25 February 2024 (UTC)
:: What is wrong with [[Portal talk:Astronomy]] staying where it is? --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 14:41, 25 February 2024 (UTC)
:::Sorry! Again I read quickly but without accuracy. I didn't notice that it was a '''Talk''' page. I will archive it right now.[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 17:20, 25 February 2024 (UTC) {{Done}}[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 18:36, 25 February 2024 (UTC)
:::: You "archived" the page but not moved. Where should we move the talk page? That is my question. [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 00:22, 26 February 2024 (UTC)
::: According to [[WV:CSD]], isolated talk pages are subject to deletion. [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 00:21, 26 February 2024 (UTC)
::::I apparently just forgot to delete the talk page. Does anybody object to deleting the talk page and its archive?--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 01:50, 26 February 2024 (UTC)
::::: Why is this being deleted or archived? I guess it is because of [[WV:Deletions]], "Discussion about deleted resources where context is lost and becoming an independent resource is unlikely". But the resource was not deleted, it was moved: from looking at [[Portal:Astronomy]], one can see it was moved to [[Topic:Astronomy]], which is now a redirect to [[Astronomy]]. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 07:16, 26 February 2024 (UTC)
{{done}}[[file:Red question mark.svg|20px]] Taking Dan's lead, I assumed the hanging talk page [[Portal talk:Astronomy]] to have been attached to what is now [[Astronomy]], which already had a talk page. So I made the Archive a subpage with an explanatory note at [[Talk:Astronomy]]. I'm glad this is a hobby and not a serious effort to preserve the history of this ol wiki.--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 19:38, 26 February 2024 (UTC)
== Chronological order of [[Wikiversity:Request_custodian_action/Archive/23]] and [[Wikiversity:Request_custodian_action/Archive/24]] ==
I generally understand that archives are numbered in chronological order but I found an exception to this rule. [[special:permalink/2596291]] says that 23 is "January 2021 - June 2023" and 24 is "December 2021 - December 2022", this is breaking the chronological order. Should we fix this or keep it in the current state? [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 01:19, 26 February 2024 (UTC)
:I noticed that while archiving a while back. I think we should leave it alone. One problem is that we have two chronological orders: One is when the request was initiated, and the other is when the request is archived. To make matters worse, many topics get "archived" twice: First when <nowiki>{{Archive top}}..{{Archive bottom}}</nowiki> turns the background blue, and second when the conversation is moved. Also, these conversations are extremely chaotic. Reading them would make good reading for chatbots if and when humans ever decide to start punishing them for transgressions.--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 01:46, 26 February 2024 (UTC)
:: OK, thank you for your opinions. [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 01:49, 26 February 2024 (UTC)
== Can anybody explain how this turns into a proposed deletion? ==
I just deleted a lot of pages because I thought the author was confusing the prod template for speedy delete. [https://en.wikiversity.org/w/index.php?title=User:Ramosama/sandbox/Problem_Analysis_-_Provision&action=edit This is the source] for [[User:Ramosama/sandbox/Problem Analysis - Provision]]:
{{cot|Click to view the source code that triggers the prod}}
<code><nowiki>{{Problem analysis - measure|name=Reusing durables|identifier=reusing_durables
|definition= The reuse of durable goods in their original form.
|reasons=
|parents=
|instances=
* Design of equipment for reuse of their parts ("cradle to cradle").
* Prolonged storage of reusable goods in warehouses, such as deserted office buildings.
* Second-hand warehouses.
* Refund for returns of durables.
* Facilitation, for example, allowing customers to reuse packaging or containers.
|advantages=
|disadvantages= }}</nowiki></code> [[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 17:14, 26 February 2024 (UTC)
{{cob}}
Thankfully the user has been dormant for almost 4 years. See [[Special:Contributions/Ramosama]].[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 17:17, 26 February 2024 (UTC)
: I edited "[[:Template:Problem analysis - concept]]" to place its proposed deletion code into the noinclude tag. As a result, [[User:Ramosama/sandbox/Problem Analysis - Provision]]--which uses the template--no longer shows any proposed deletion tag. I hope it added some clarity and has no undesirable consequence. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 19:42, 26 February 2024 (UTC)
::Good news! I thought it was possible to accidentally make a prod. Thank's Dan.[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 19:48, 26 February 2024 (UTC)
== Does anybody know how to delete all pages by a single user? ==
We have a serial page creator. My hunch is that the pages were created in another language, translated using an auto-translator, and placed on en.wikiversity. I am currently trying to create a list from [https://en.wikiversity.org/w/index.php?title=Special%3AContributions&target=Saltrabook&namespace=all&tagfilter=&newOnly=1&start=&end=&limit=50 '''this list''']. If nobody knows how to do this, I will use a list under construction at '''[[Pre-diabetes diagnosis and remission]]'''.[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 16:43, 27 February 2024 (UTC)
:[[Special:Nuke]] can mass-delete, with some caveats. Oddly, it is only available to bureaucrats here. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 16:38, 27 February 2024 (UTC)
I don't know the answer. But let me list the pages created in 2024 (there are more from 2023):
* [[INVITATIONS TO SEAFARERS AND THE MARITIME MEDICAL CLINICS]]
* [[CONTENTS OF THE 16 WEEKS COACHING]]
* [[VIDEO PRESENTATION AND INVITATIONS]]
* [['''CONTENTS OF THE 16 WEEKS COACHING''']]
* [[DRAFT ARTICLE]]
* [[Maritime Diabetes-type 2 Intervention study/DRAFT PAHO PROTOCOL/CONTENTS OF THE 16 WEEKS COACHING]]
* [[Maritime Diabetes-type 2 Intervention study/DRAFT PAHO PROTOCOL/DRAFT PAHO PROTOCOLO EN ESPAÑOL]]
* [[Maritime Diabetes-type 2 Intervention study/DRAFT PAHO PROTOCOL/ESPAÑOL]]
--[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 16:43, 27 February 2024 (UTC)
:{{Done}} I deleted all the maritime health and diabetes pages made in the past several months. If more is needed, let me know. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 16:55, 27 February 2024 (UTC)
::Thanks Justin. You might want to change the parameters of my block of Saltrabook. I know little about blocking protocols. I will change my expiration date from one week to indefinite. I didn't know you could pagenuke. We need an active pagenuker on this wiki now that Dave is less active.[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 17:07, 27 February 2024 (UTC)
:::I have no perspective on an indefinite block, but it may be a good idea until/unless he can explain on his talk page what he's trying to do and where he is getting this information, etc. Note also that he has ''lots'' of pages going back to at least 2019. If we had consensus that [[Special:Nuke]] were available to admins (curators), then we could make the request on [[:phab:]] to change the local settings. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 17:58, 27 February 2024 (UTC)
::::Do you know whether Saltrabook can use his talk page? If so, there is no need to change the indefinite block.[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 19:19, 27 February 2024 (UTC)
::::: The latest block ([[special:redirect/logid/3389142]]) does not include edits, so I think they can. Generally, most blocked users can edit their own talk pages for unblock requests and related statements (unless revoked). [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 03:48, 28 February 2024 (UTC)
:::: Currently, curators cannot restore pages. I think allowing mass-delete without restoration permissions can be risky. Allowing mass-delete to our custodians should be enough. Why have we limited mass-delete to our bureaucrats? Are there any previous discussions in the past? [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 03:50, 28 February 2024 (UTC)
:::::No clue. That is very bizarre and atypical. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 04:48, 28 February 2024 (UTC)
:::::: I think we can ask to hear the community's opinion at [[Wikiversity:Colloquium]]. They may want to speak about what they think about this odd technical settings. [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 03:45, 1 March 2024 (UTC)
::::::: <s>(Note about this matter) I started a new thread over there.</s> [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 13:41, 1 March 2024 (UTC)
::::::: (Update) Per suggestion ([[special:diff/2610994]]), I started a proposal at [[Wikiversity_talk:Custodianship#Proposal_to_allow_custodians_to_use_mass-delete]]. [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 07:17, 8 March 2024 (UTC)
:: If possible, I suggest clarifying the deletion criteria (RFD? off-wiki request?). I'm sorry if I have missed anything. From my viewpoint, I only requested renaming without redirects, and now I see pages being deleted. Having more explanations would be better, I think. [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 03:53, 28 February 2024 (UTC)
:::{{ping|MathXplore}} Sorry, sometimes I act too swiftly. It turns out User:Saltrabook has been creating what looks like interesting pages for a long time, and he has created close to 100 such pages (probably much more.) He doesn't know English very well, so it is obvious that he is auto-translating the pages. I blocked his page creations, and he seems happy working on pages he already created (many of them were almost blank.) Personally, I would be happy if he works on the pages he has already created and left us alone. We get odd ones on WV. I should know; my family thinks I am one.-[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 03:37, 1 March 2024 (UTC)
:::: Thank you for the explanations. As can be seen in each page history, I'm one of the few editors handling the categorizations of their creations, but I didn't notice that there were auto-translations (has anyone identified which software has been used?), apologies for being late to notice such issues. I think we should clarify how to handle auto-translations via policy/guideline or previous discussions. [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 03:43, 1 March 2024 (UTC)
== [[Special:Contributions/Krutrimam]] ==
Lock evasion of [[User:Premaledu]], please see [[special:permalink/2609661#Offensive_username]]. [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 12:19, 1 March 2024 (UTC)
: Already {{done}}, globally locked. [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 13:23, 1 March 2024 (UTC)
== Explanation of edit ==
I was trying to link my pages and I got a notification to explain to a custodian. I hope I'm in the right place for that. [[User:An5189|An5189]] ([[User talk:An5189|discuss]] • [[Special:Contributions/An5189|contribs]]) 04:42, 4 March 2024 (UTC)
:Seems fine to me. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 04:49, 4 March 2024 (UTC)
::thanks [[User:An5189|An5189]] ([[User talk:An5189|discuss]] • [[Special:Contributions/An5189|contribs]]) 04:52, 4 March 2024 (UTC)
== create about user page ==
I was trying to create about User page
[[User:An5189|An5189]] ([[User talk:An5189|discuss]] • [[Special:Contributions/An5189|contribs]]) 05:17, 4 March 2024 (UTC)
:I'll create a blank one and you can modify it. Let me know if you have more problems. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 06:57, 4 March 2024 (UTC)
::thanks, I will[[User:An5189|An5189]] ([[User talk:An5189|discuss]] • [[Special:Contributions/An5189|contribs]]) 08:18, 4 March 2024 (UTC)
== [[Special:Contributions/39.50.199.52]] ==
Making bad pages (I already deleted them) and xwiki abuse (also reported at Wikiquote). [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 02:46, 10 March 2024 (UTC)
:{{done}} —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 05:49, 10 March 2024 (UTC)
== [[Special:Contributions/Precisiongroup]] ==
Spam-only account with promotional username (account named after company name). [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 06:49, 14 March 2024 (UTC)
:{{done}} —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 13:02, 14 March 2024 (UTC)
== [[Special:Contributions/Kroodham]] ==
Lock evasion of [[Special:CentralAuth/Premaledu]]. [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 07:45, 14 March 2024 (UTC)
: {{done}}, already locked. [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 08:57, 14 March 2024 (UTC)
== [[Special:Contributions/27.55.68.138]] ==
Vandalism and xwiki abuse. [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 10:07, 19 March 2024 (UTC)
:{{done}} Month-long rangeblock. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 10:36, 19 March 2024 (UTC)
== [[Special:Log/Cbtproxyus]] ==
The user has repeated user page spam, I already deleted it and set indefinite full protection. [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 05:56, 20 March 2024 (UTC)
:{{done}} indef block. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 15:42, 22 March 2024 (UTC)
== CAPTCHA Problem when creating an Account ==
I don't know how active Dave is at the moment, so I paste a message to Dave from [[User:Ireicher2]]:
{{quote|Hi Dave, Isabel here from Ohlone college. We've talked a couple of times before. Some of my students emailed me to let me know that they cannot create user accounts because of a CAPTCHA problem. I verified the information by attempting to create a new account and I received the same error message. Would you let me know how this can be resolved? Thank you!}}
{{ping|Ireicher2}} One thing you might try is having them create Wikipedia or Wikibooks accounts. I believe membership in one automatically creates membership on Wikiversity. Another thing to try is asking students to create the account from their homes. Does anybody else have any ideas????--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 01:49, 22 March 2024 (UTC)
: I think [[:w:Wikipedia:Request_an_account/Help_and_troubleshooting]] is related to this issue. It is a different project but shares the same technical basis. [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 01:55, 22 March 2024 (UTC)
:See [[meta: Mass account creation]]. I'll try adding Account creators to [[User:Ireicher2]] with an expiration of seven days and see if makes any difference. Yes, the suggestion that students create their accounts from home (or using their cell phones vs. school computers) should help. -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 03:44, 22 March 2024 (UTC)
::@[[User:Dave Braunschweig|Dave Braunschweig]] @[[User:Guy vandegrift|Guy vandegrift]] Of course. That makes sense. Thank you!<br> [[User:Ireicher2|Ireicher2]] ([[User talk:Ireicher2|discuss]] • [[Special:Contributions/Ireicher2|contribs]]) 04:46, 22 March 2024 (UTC)
== I need the custodians & curators to tell a user not to be involved with deletions. ==
{{Cot|Collapse as resolved}}
Here are two examples:
#He put a speedy delete on [[special:permalink/2617505]], saying among other things that there is "no clear explanation" of what ''ordinary'' differential equations are". This is a subpage, and the parent page at [[special:permalink/2483117]] gives a rather coherent explanation: "Differential equations serve as mathematical models of physical processes. This course is intended to be an introduction to ordinary differential equations and their solutions. <small>A '''differential equation''' (DE) is an equation relating a function to its derivatives. If the function is of only one variable, we call the equation an '''ordinary differential equation''' (ODE). ...</small><br> There is a movement to raise the standards regarding what should and should not be in namespace, but the the parent page at [[special:permalink/2483117]] has [[Special:PrefixIndex/Differential_equations/|13 subpages.]] If this resource is a problem, it has to be addressed from the top down, not one subpage at a time. As will be shown in the next example, I recently attempted to explain to him that it is inefficient to remove subpages without looking at the entire resource (via the parent page.)
#Days prior to the aforementioned effort to delete a subpage of [[Differential equations]], he proposed the deletion of one of some 300 subpages of [[Student Projects]] because it was unsourced. My reason for not deleting that page should have informed him that it would have been inappropriate to delete one subpage of [[Differential equations]], because it turns out that almost all subpages of [[Student Projects]] are unsourced, leaving us with the same issue involving the deletion of pages from the "bottom-up". For evidence that this user had been informed of the need for a "top-down" approach attempting to delete a subpage of [[Differential equations]], see [[special:permalink/2617342#Student_Projects/Major_rivers_in_India]]. This editor is a nice person with a lot of good ideas, but his stubbornness is making it difficult to moderate [[Wikiversity:Requests for Deletion]]--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 14:15, 31 March 2024 (UTC)
:I don't want this user blocked, or even banned from participating in discussions about deletion policy. He is not alone in advocating higher standard, and the community might want to do that. But there is a distinction between the nuts and bolts of deletion, and deletion as a policy. I am very conservative about deleting pages. So if the standards get tightened, there will be no need to revert anything I have done. I am asking the custodians/curators to encourage this user to go to [[Wikiversity:What-goes-where_2024#Personal_subpages_(with_visual_editing)|WV:WGW2024]] and create a subpage for sharing his ideas with the community.--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 14:41, 31 March 2024 (UTC)
-----
-----
-----
#: 1) The "unsourced" on [[Student Projects/Major rivers in India]] was only one reason; the other reason was that this page has nothing to add what is not in Wikipedia, a rationale previously recognized. 2) [[Student Projects/Major rivers in India]] is not integrated in any way to a [[Student Projects]] "project"; its being a subpage is just an attempt to escape deletion scrutiny. 3) I am not aware of any explanation to me that I should not nominate subpages; such an explanation has my talk page as a proper venue, and I am unaware of any such explanation, neither there or elsewhere. 4) Any disagreement about deletion can be resolved via RFD and via voting-cum-discussion there, as is usual in other projects, e.g. the English Wikipedia and the English Wiktionary; if I am mistaken in a particular nomination, it can be brought to RFD and quickly voted down. Even a single person opposing can prevent a deletion in which I am the sole, mistaken, deletion supporter. 5) I have a pretty good conversion rate between deletion nominations and actual deletions/moving out of mainspace, and therefore, I do not think that my nomination algorithm is too broad and too burdensome on those who have to oppose my nominations for deletions. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 14:26, 31 March 2024 (UTC)
#::All I am asking is that you stay out of active deletions and focus your talent on changing the policy. A great place to do that is at [[WV:WGW2024#Personal_subpages_(with_visual_editing)]] [[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 14:51, 31 March 2024 (UTC)
#::: I invite you to my talk page to make requests concerning change of behavior on my part. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 14:52, 31 March 2024 (UTC)
{{cob}}
[[File:Yes check.svg|18px]]'''Resolved''' We have corresponded in our talk pages and the problem has been resolved to my satisfaction.--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 18:15, 31 March 2024 (UTC)
== link on page looks possibly explicit to follow? ==
[[Other Free Learning Resources]] the univeristy of reddit link has a lot of very adult explicit words as links . I did not view other links from this page. Thanks
U - X
* [http://www.ureddit.com/ University of Reddit]
[[Special:Contributions/2001:8003:B120:8900:4D5:4E7A:36B2:58F3|2001:8003:B120:8900:4D5:4E7A:36B2:58F3]] ([[User talk:2001:8003:B120:8900:4D5:4E7A:36B2:58F3|discuss]]) 12:50, 1 April 2024 (UTC)
:Thanks, it's now spam, so I removed it. :/ —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 19:09, 1 April 2024 (UTC)
== [[Special:contribs/206.110.193.204]] ==
Vandalism [[User:Seawolf35|Seawolf35]] ([[User talk:Seawolf35|discuss]] • [[Special:Contributions/Seawolf35|contribs]]) 18:57, 16 April 2024 (UTC)
:{{done}} —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 19:32, 16 April 2024 (UTC)
== Induced stem cells copyright issues ==
[[Induced stem cells]] got imported to here from enWiki- which is fine, attribution was done correctly and everything- except for the fact that I'm just wrapping up a [[Wikipedia:Contributor copyright investigations/20240516|copyright investigation]]<nowiki> on the original contributor & his alternative account. Due to the fact that this contributor repeated and blatantly infringed on the copyright of multiple sources despite multiple warnings an even a block, I tagged the original page over on enWiki for presumptive deletion. I don't know what Wikiversity's process is for suspected copyright infringements without a clear source, but I figured you guys would want to know about the problems with this page anyways. -- ~~~~</nowiki> [[User:GreenLipstickLesbian|GreenLipstickLesbian]] ([[User talk:GreenLipstickLesbian|discuss]] • [[Special:Contributions/GreenLipstickLesbian|contribs]]) 20:34, 23 May 2024 (UTC)
:Very helpful, thanks. Do you have any relevant links to en.wp about the investigation or where he typically ripped off material? —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 23:48, 23 May 2024 (UTC)
::Okay- everthing I've gathered so far is going to be in this [[wikipedia:Wikipedia:Contributor_copyright_investigations/20240516|investigation page]]. ( tried to link it in the original post, but I failed spectacularly as you can see). They almost exclusively copied from scientific papers/ reviews, and news/blog reports. This user typically copied from the source they cited- or, at least, *a* source they cited. They'd regularly copy a paragraph of text from one source, then a cite a different source for each sentence. If a source was paywalled, they often would cite the source, but copy from a news report/blog report analyzing the source. One of the other investigators found a few instances where they copied another article in Wikipedia without attribution- but that was their rarest type of violation. They occasionally wrote their own material, but it was normally easily identifiable because English is not their first language.
::Sorry for not being more helpful on this article in particular-I saw they(and their alt) were essentially the sole author of this page, cited 300+ sources, made a noise somewhat akin to that of a distressed animal, and decided I was going to take advantage of enWiki's rule allowing us to delete articles written by serial copyright violators without any more evidence. --[[User:GreenLipstickLesbian|GreenLipstickLesbian]] ([[User talk:GreenLipstickLesbian|discuss]] • [[Special:Contributions/GreenLipstickLesbian|contribs]]) 00:09, 24 May 2024 (UTC)
:::That’s plenty to convince me that this should be assumed to be a copy II until proven otherwise. Merci. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 00:16, 24 May 2024 (UTC)
:::: Thank you for the responses, I think having a short intro, soft redirect to the CCI page, further readings section, and categories would be OK, what would you think about this? [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 01:56, 24 May 2024 (UTC)
:::::Sure. Do you want me to undelete and then redelete selected diffs? Or you’ll just create the redirect yourself? —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 02:10, 24 May 2024 (UTC)
:::::: I will create a soft redirect afterward. [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 02:29, 24 May 2024 (UTC)
::: Thank you for the information, do you think [[WikiJournal Preprints/Induced stem cells]] needs deletion? It is another page where the same editor has substantial involvement. [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 01:58, 24 May 2024 (UTC)
::::I did a brief check, and I found that [https://www.science.org/doi/10.1126/science.1248252 this source] (cited in the article) appears to have been partially copied. Specifically, the stuff about zebrafish has been copied word for word. It's not a promising sign. If this was on the English Wiki, I would ask for it to be presumptively deleted soley on the basis of the author and that confirmed instance of a copyright violation. I worked on the investigation for several days (and I was the one who asked for it to be opened), and I could confirm over half their writing to be blatant copy-and-paste jobs. [[User:GreenLipstickLesbian|GreenLipstickLesbian]] ([[User talk:GreenLipstickLesbian|discuss]] • [[Special:Contributions/GreenLipstickLesbian|contribs]]) 03:08, 24 May 2024 (UTC)
:::::Thank you for your service, hermana. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 03:38, 24 May 2024 (UTC)
::::: Thank you for the information, I have contacted an active Wikijournal contributor to learn about how this preprint should be handled. [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 02:05, 6 June 2024 (UTC)
::::::Thank you to [[User:GreenLipstickLesbian|GreenLipstickLesbian]] for informing us about the copyright violations and reference false attribution in this article, and [[User:Koavf|Koavf]] & [[User:MathXplore|MathXplore]] in participating in this conversation. Normally we would keep rejected articles in the preprint with the stated reason in the talk page for record purpose. However, since the induced stem cell contains copyright violation and may cause future accidental copyright violation by future text re-users under the assumption that the text is under Creative Commons license, I will request that the preprint be deleted while talk page remains undeleted to note the rationale. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;"><sup>Talk page</sup></span></b>]] 20:00, 18 June 2024 (UTC)
:::::::Good point. In addition to not deleting the talk page, I am redirecting the main page to the talk page and protecting it. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 20:07, 18 June 2024 (UTC)
::::::::@[[User:Koavf|Koavf]] Can you also delete [[WikiJournal Preprints/Induced stem cells]], redirecting it to talk page please? [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;"><sup>Talk page</sup></span></b>]] 04:03, 10 July 2024 (UTC)
:::::::::{{done}} —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 04:42, 10 July 2024 (UTC)
== Creating a section of my own talk page with a link to Wikipedia ==
I'm trying to set up my own talk page here at Wikiversity with my own example of trying separate the essence and accident of programming, as per <nowiki>[[w:No Silver Bullet|No Silver Bullet]]</nowiki> at Wikipedia, but it's rejected because of the external link (i.e. to Wikipedia). I'm doing this because most example code I see buries the essence in the accident and I wanted to show an example that there are better ways to write code. [[User:Philh-591|Philh-591]] ([[User talk:Philh-591|discuss]] • [[Special:Contributions/Philh-591|contribs]]) 10:33, 9 July 2024 (UTC)
:That's very weird: you can't create ''interwiki'' links? And to be clear, you're trying to put said links on your talk page at [[User talk:Philh-591]], not your userpage [[User:Philh-591]]? —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 12:05, 9 July 2024 (UTC)
::Well, your creation of the page with a welcome message has got it past that restriction, although I don't think it was the Wikipedia link. I'd not noticed that there are URL's in my example source referring to public information at the European Central Bank. However, it now insistently applies "nowiki" to what I insert. I guess I don't understand the formatting rules at Wikiversity; I'd assumed it was just like Wikipedia. I'll see if I can understand it more playing in the sandbox. [[User:Philh-591|Philh-591]] ([[User talk:Philh-591|discuss]] • [[Special:Contributions/Philh-591|contribs]]) 13:13, 9 July 2024 (UTC)
:::I figured that would fix the problem: sometimes, creating a new page (even your own user or user talk page) has restrictions. I forget the exact limitations per wiki, but they are usually very modest, like make at least five edits across two weeks or something. Re: formatting rules, they should be the same as Wikipedia, so I'm confused as to what you're trying to do again. :/ —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 13:25, 9 July 2024 (UTC)
== Won't let me publish "my about" page due to "New User Exceeded New Page Limit" ==
Unsure how to publish my about me page, is someone able to help me be able to publish it without it being disallowed? [[User:Lucywilson 546|Lucywilson 546]] ([[User talk:Lucywilson 546|discuss]] • [[Special:Contributions/Lucywilson 546|contribs]]) 03:20, 8 August 2024 (UTC)
: {{ping|Jtneill}} Can you grant confirmed status for this user? [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 03:41, 8 August 2024 (UTC)
:I made a blank page, which you can now edit. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 04:31, 8 August 2024 (UTC)
: {{ping|MathXplore}} Thankyou, I've confirmed the user. {{ping|Koavf}} Thankyou, a neat, instant solution :). {{ping|Lucywilson 546}} Thanks for letting us know. You should be good to go. Let us know if any other problems. Sincerely, James -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 10:24, 8 August 2024 (UTC)
== Delete revission ==
Could you delete [https://en.wikiversity.org/w/index.php?title=Wood_finishing&oldid=2651335 this revision], which is revealing my personal information, please? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 09:54, 9 September 2024 (UTC)
: Username is hidden, I have contacted the [[:m:stewards]]. [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 11:38, 9 September 2024 (UTC)
: {{done}} [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 12:00, 9 September 2024 (UTC)
== Spam filter exception request ==
I am prevented from creating [[Template:Vandal]] because a previous example in [[Template:Vandal/doc]] used an IP address, which is blocked by a spam filter. I removed that example, but am still blocked from creating that page. I have put the source code in [[Template:Vandal/sandbox]] in the interim. Perhaps allowing just <code>10.0.0.1</code> to avoid other IP spam? [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 04:55, 2 October 2024 (UTC)
:{{done}} I created a blank template, which you can now edit. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 06:11, 2 October 2024 (UTC)
::{{done}} again: I moved your sandbox to the template. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 06:12, 2 October 2024 (UTC)
:::It seems I still can't include the IP-user example (see [[Special:PermanentLink/2658932|an old version]] with the offending string) - [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 23:13, 2 October 2024 (UTC)
::::I don't understand the problem. What text are you trying to put where? —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 00:43, 3 October 2024 (UTC)
:::::See [https://en.wikiversity.org/w/index.php?title=Template:Vandal/doc&diff=prev&oldid=2658932 this diff] which shows the text and location {{--}} [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 00:48, 3 October 2024 (UTC)
::::::{{done}}. Longer-term issues with including IP addresses may still exist, but this particular edit at least is fixed. Thanks. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 00:53, 3 October 2024 (UTC)
== [[Special:Log/Tule-hog]] ==
As seen in the link above, [[User:Tule-hog]] has made various manual imports from WP to WV. Some may be OK, but others may be questionable. Despite various messages on their talk page ([[User talk:Tule-hog]]) from user:Dan Polansky, the user continues manual imports. Should we let this continue, keep talking with the user, or should we stop them? What would be the best option? ({{ping|Jtneill}} As Dan's mentor, your feedback is welcome here, and {{ping|Koavf}} since you previously communicated with the user in [[Special:Diff/2659041]], we would like to hear about your thoughts) [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 06:39, 7 October 2024 (UTC)
:{{Ping|Tule-hog}} From what I see on your talk, you are at least not doing this anymore. While copyright-wise, we can of course copy anything from en.wp to here, it is best to use [[Special:Import]] because it preserves edit histories, provides attribution, and can also import dependencies like another modules or templates. Can you explain what your goal is with this copying and what in general you want copied? —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 06:43, 7 October 2024 (UTC)
::I am motivated by updating Wikiversity template/module infrastructure in places where appropriate. Note I do not have the [[WV:Importer|importer]] role. I perform what I've been calling [[User:Tule-hog/Wikiversification|Wikiversification]] on docs and templates themselves, where much of the time the pages I come across are rough imports with raw Wikipedia links without modification, incorrect language for the project, bad category mapping, or are dependent on other undefined modules/templates.
::To be clear, I am ''not'' just going through picking out popular templates/modules and importing them. I approach a maintenance task, and where relevant spend the (not mindless) time to transform them to fit Wikiversity. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 06:50, 7 October 2024 (UTC)
::: I find "in places where appropriate" too non-specific. I do not see any specific need addressed. I find Colloquium a good forum for a proposal to copy (or import) a large number (how large?) of Wikipedia templates and categories; the approximate volume should be stated as part of the proposal. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 07:02, 7 October 2024 (UTC)
:::I undeleted and userfied to [[User:Tule-hog/Wikiversification]]. If you are thinking of making some large-scale change, then it's probably best to clarify your thoughts there, propose it (succinctly!) at the Colloquium, and then coordinate with a custodian who can import. This is kind of a [[:en:wikt:death by a thousand papercuts|death by a thousand papercuts]] situation: any one change is perfectly fine, but the volume may be systemic, so it's wise to get the community's input. Besides, we could help and many hands make for light lifting. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 07:08, 7 October 2024 (UTC)
::(Question due to unfamiliarity with importer mechanism:) Do we also submit requests to ''update'' already imported templates at [[WV:I]], or does that only happen once (and hence update requests should go to [[WV:RCA]])? Thanks, [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 23:25, 7 October 2024 (UTC)
:::Unfortunately, if you import a resource from another wiki and the original changes, the updates need to be imported again here manually and since [[WV:I]] is a dedicated space, it's probably best to put requests there. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 05:05, 8 October 2024 (UTC)
== [[:Category:Wikiversity policies and guidelines]] ==
Should this category finish being developed? (I could do so if desired.) [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 06:32, 8 October 2024 (UTC)
:Similarly with the list detailed at [[:Category:Wikiversity development]] (i.e. finishing up + deleting the list mentioned there) —[[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 06:55, 8 October 2024 (UTC)
::{{ping|Koavf}} double checking is alright for [[:Category:Wikiversity development]] as well (started by [[:User: McCormack|McCormack]]) —[[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 02:20, 9 October 2024 (UTC)
:Can you reword this question? I'm not sure what you're trying to do here. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 07:16, 8 October 2024 (UTC)
::The content of the category is "This category is being developed." so I believe it is in an unfinished state (i.e. adapting the categorization schema). [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 07:18, 8 October 2024 (UTC)
:::I think if you have some rational way of organizing the pages, that's fine. I don't know what
:::[User:CQ]][had in mind when he put that there, but he has basically not edited here in 4.5 years, so go for it. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 07:20, 8 October 2024 (UTC)
== [[:Category:Rejected policies]] ==
Should I use the list of tagged pages found in this category to update [[WV:POLICY#Rejected policies]]? Or is [[WV:IAR]] the only truly firmly rejected proposal? [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 22:59, 8 October 2024 (UTC)
:The category and that list should have the same items<ins> and at first glance, what is the category is in fact rejected proposals, therefore, the list should be updated.</ins> —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 02:04, 9 October 2024 (UTC)
::Just noting [[Wikiversity_talk:Policies#List_of_official_policies|this thread]] which suggests that another user made that list in the first place using tags, so it may have recursively snuck something in. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 02:12, 9 October 2024 (UTC)
== [[Wikiversity:Research guidelines]] ==
This page is listed in {{tlx|official policies}}. Should it be updated as adopted on [[WV:POLICY]], and if so, should it be considered a policy or guideline? [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 00:19, 11 October 2024 (UTC)
:(Also, should it link to the top-level [[Wikiversity:Research]] instead, which uses (the confusingly named) {{tlx|research policy}} navbox?) [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 00:46, 11 October 2024 (UTC)
:Good question. From what I can tell the beta Wikiversity research page is the official policy and the en.wv local version is a copy/fork that hasn't been officially endorsed.
:That leaves me wondering whether we want to pursue a local variation as an official policy or potentially remove the local variant and redirect to the beta version. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 03:12, 13 October 2024 (UTC)
:: What suggests that https://beta.wikiversity.org/wiki/Wikiversity:Research_guidelines is an official policy? And if it is, does the policy match the actual practice? For instance, it says "Original research at Wikiversity is subjected to ''peer review'' in order to allow the Wikiversity research community to strive for verifiability" (italics mine): is that really true outside of Wikijournals? Moreover, the putative policy states in a box: "This page contains summaries of discussions which have taken place in various languages." But this cannot be true since the policy reads like a monologue and a proper summary of discussions cannot be a monologue. A quick skimming of the page raises some red flags.
:: Be it as it may, I think keeping a local copy is vital since then we have the option to amend it without thereby requiring an international cross-language input to the changes. Of course, the amends will be unable to change some core features of Wikiversity (no metamorphosis allowed), but some amends should be possible.
:: As for the local [[Wikiversity:Research guidelines]], I propose to rank it as ''policy proposal'', given the misgivings.
:: In any case, this discussion does not belong to "Request custodian action" but rather to "Colloquium" since the outcome of the discussion can be implemented by anyone, not only custodians, and since input from non-custodians seems welcome. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 05:28, 13 October 2024 (UTC)
:::Agree with retaining local version and treating as ''policy proposal''.
:::I've hidden the note about the guidelines being a copy of the beta guidelines (it confused me at least into thinking that beta version was also the policy on en.wv).
:::Softened the peer review requirement to being "open" to peer review rather than being "subjected" to peer review.
:::Agree that further work e.g., on drafting and potentially making official should be followed through on Colloquium.
:::Thanks @[[User:Tule-hog|Tule-hog]] and @[[User:Dan Polansky|Dan Polansky]]. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 06:13, 13 October 2024 (UTC)
:: Oh, and I was not paying attention: [[Wikiversity:Research]] states "This page provides guidelines for research in Wikiversity" so there appears to be some redundancy/overlap between [[Wikiversity:Research]] and [[Wikiversity:Research guidelines]]. Confusing. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 05:56, 13 October 2024 (UTC)
== [[Wikiversity:Policies]] ➝ [[Wikiversity:Policies and guidelines]] ==
This is a proposal to move [[WV:Policies|Policies]] to a name matching the scope of the page, [[WV:Policies and guidelines|Policies and guidelines]].
The more descriptive title will make identifying the location of guidelines easier for newer participants. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 15:46, 25 October 2024 (UTC)
:See [[User_talk:Tule-hog#Wikiversity:Policies|more discussion]]. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 20:58, 25 October 2024 (UTC)
== Please fully protect... ==
[[Module:Message box/fmbox.css]]. It is used in 29 system messages. [[Special:Contributions/2604:3D08:9476:BE00:DCDC:4B47:21DA:D90E|2604:3D08:9476:BE00:DCDC:4B47:21DA:D90E]] ([[User talk:2604:3D08:9476:BE00:DCDC:4B47:21DA:D90E|discuss]]) 20:32, 25 October 2024 (UTC)
:{{done}} —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 06:26, 26 October 2024 (UTC)
== Uh Oh! ==
Hello. Something Went Wrong With Editing. My Dog And Me Is Editing The New Learning Resources. Dog Grooming (Learning Resources). So Help Me. Tanks. [[Special:Contributions/2603:9000:7AF0:5DA0:B940:EF5A:3D27:A8F0|2603:9000:7AF0:5DA0:B940:EF5A:3D27:A8F0]] ([[User talk:2603:9000:7AF0:5DA0:B940:EF5A:3D27:A8F0|discuss]]) 18:46, 11 December 2024 (UTC)
:Okay, it looks like you are editing [[Pomeranian]], which is a bit of a mess. I think that proper pet care could be a fine topic for this site or our sister site [[:b:|Wikibooks]], but the state this is in is pretty rough. I'd recommend you take a look at [[Wikiversity:Welcome]] and some of the pages linked there. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 18:57, 11 December 2024 (UTC)
== IP block exempt request ==
Can a custodian grant IP block exempt flag to {{u|Silver Dovelet}} please? Her account got into the crosshair of a very wide IP rangeblock and that rangeblock also prevented her from making the request directly here. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;"><sup>Talk page</sup></span></b>]] 09:04, 1 January 2025 (UTC)
:{{done}} for six months. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 11:12, 1 January 2025 (UTC)
::Thanks. Looks like she's back on track. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;"><sup>Talk page</sup></span></b>]] 00:54, 2 January 2025 (UTC)
oye7316finogi69lpl6on3ed5mcj6sw
2694086
2694058
2025-01-02T11:05:27Z
Koavf
147
/* IP block exempt request */ Reply
2694086
wikitext
text/x-wiki
{{/Header}}[[cs:Wikiverzita:Nástěnka správců]][[fr:Wikiversité:Requêtes aux bibliothécaires]][[pt:Wikiversidade:Pedidos a administradores]]
== Request to move image files to Commons ==
I got [[User_talk:Guy_vandegrift#Files_on_Commons|'''this request''']] to move files from [[:Category:NowCommons]] and [[:Category:Files from USGS]]. I delete lots of files, but usually let others delete image files because of my ignorance of copyright laws. I also have contributed a lot of files to Commons, but almost all of it is my own work. So I am out of my comfort zone on this. I don't even understand why these files should be moved.
{{ping|User:MGA73}} Maybe we can find someone with more expertise on file transfers here on [[Wikiversity:Request custodian action|Request custodian action]].--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 22:47, 7 January 2024 (UTC)
In a related vein, due to my inexperience with copyright regulations, perhaps it would be better if someone else processed the following files. All are up for speedy deletion. And all seem like quality images and/or on potentially high quality WV resources.
<gallery widths=50>
File:Merged fig1.png
File:Merged matrix2.png
File:Rps all hsa.png
File:Selected domfams fix.png
File:Service-pnp-fsa-8b32000-8b32000-8b32095r.jpg
File:Summary.svg
File:Transtree.png
File:Untitled-91274a-1024.jpg
</gallery>
: My request was primary to delete files that was moved to Commons allready. But if anyone have checked files they are of course very welcome to move files to Commons too. Same with [[:Category:Files from Flickr]]. --[[User:MGA73|MGA73]] ([[User talk:MGA73|discuss]] • [[Special:Contributions/MGA73|contribs]]) 16:32, 9 January 2024 (UTC)
::Thanks for the info. My ignorance of copyright law makes me very hesitant to delete image files.--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 19:50, 26 February 2024 (UTC)
::: I noticed [[User:Koavf]] just deleted a file moved to Commons. So perhaps Koavf could have a look at the files in [[:Category:NowCommons]] once there is a little time to spare? :-) --[[User:MGA73|MGA73]] ([[User talk:MGA73|discuss]] • [[Special:Contributions/MGA73|contribs]]) 19:14, 27 February 2024 (UTC)
::::lol@"time to spare", but sure. <3 —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 21:17, 27 February 2024 (UTC)
:::::Sometimes dirty tricks work ;-) --[[User:MGA73|MGA73]] ([[User talk:MGA73|discuss]] • [[Special:Contributions/MGA73|contribs]]) 08:00, 28 February 2024 (UTC)
== [[Special:Contributions/Hooglimkt]] (again) ==
{{Archive top|User is blocked so I guess were are {{Done}}. [[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 19:52, 26 February 2024 (UTC)}}
{{ping|Koavf}} After the last report ([[Wikiversity:Request_custodian_action/Archive/25#Special:Contributions/Hooglimkt]]), the user has restarted same types of edits. [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 13:25, 9 January 2024 (UTC)
:{{not done}} But what is the action here? He just wrote a bunch of Portuguese stuff on his userpage. What needs to be done? —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 13:30, 9 January 2024 (UTC)
:: They are writing non-English advertisements on someone else's userpage, how can this be allowed? [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 13:33, 9 January 2024 (UTC)
:: Please compare the reported user and [[Special:CentralAuth/Hoogli]] (user whose userpage is targeted), they don't look like the same user. [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 13:35, 9 January 2024 (UTC)
:::Ah, sorry--I got the usernames confused. Yes, that is inappropriate and he's not here for constructive purposes. Sorry. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 13:37, 9 January 2024 (UTC){{Archive bottom}}
== [[Special:Contributions/NotAReetBot]] ==
According to [[WV:IU]], this username is not acceptable (implying bot), should this account be blocked? [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 01:28, 10 January 2024 (UTC)
: I already sent a welcome and {{tl|uw-username}} (imported from enwiki). [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 01:42, 10 January 2024 (UTC)
:I think explicitly saying that you're not a bot is acceptable, but I agree that it's probably not ideal. E.g. someone could have the username "NotAReet" and run a bot under this name. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 03:46, 10 January 2024 (UTC)
== Call for rewriting [[WV:UNC]] ==
This agenda is suggested at [[Wikiversity_talk:Username#WV:UNC needs updates]], since this is related to policy documentation, I would like to have the attention of our custodians. [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 02:49, 10 January 2024 (UTC)
== [[Special:Contributions/2409:4064:810:DA39:FA73:D928:2C4D:B401]] ==
Possible vandalism (Massive enwiki copies with MOS issues), seems to be related to the recently reported IP, please consider range block. All targeted pages are semi-protected. Reverted revisions seem to be enwiki copies, please also consider revision deletion if needed. [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 03:38, 20 January 2024 (UTC)
: (Note) Currently stale, will report again if they come back. [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 09:04, 28 January 2024 (UTC)
== Scope of talk page usage for blocked users ==
I understand that the scope of talk page usage for blocked users is aimed at unblocking requests and relevant discussions. I would like to ask if Wikiversity has more exceptions accepted by the community. I'm asking this because I recently found [[special:diff/2602322]], and this does not seem to be related to an unblocking request. If unacceptable, custodians may need to remove talk page access from the user. [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 02:53, 30 January 2024 (UTC)
== Please review recent edits at [[Wikiversity:Verifiability]] ==
{{cot|long discussion}}
Recently we had many changes to this documentation. Reverting undiscussed changes would be non-controversial, but I'm not sure about the others. What would our custodians think about these edits? [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 15:03, 31 January 2024 (UTC)
: Each of my edit has an explanation/rationale in the edit summary. Here a summarization: I above all removed sentences that presented a contradiction within the same page. I also switched the page to policy proposal away from policy since I could not find a discussion establishing the page as a policy and since, given the contradictions before my edits, the page could not have been taken seriously as a policy, that is, a set of rigid rules contrasting to guidelines. I could have discussed the changes somewhere first, but since the changes are well documented in their edit summaries, I hoped they could remain. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 17:56, 31 January 2024 (UTC)
::For the record, the original version (before recent efforts) can be found at [[Special:Permalink/1375824]]. Regarding my thoughts about these edits, I think we should distinguish between top pages and subpages. If an instructor is inviting students to submit work in subspace, the instructor should have considerable flexibility regarding those subpages.--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 00:03, 1 February 2024 (UTC)
::: While I'm not sure about what type of flexibility is being mentioned, I generally believe that teachers should have enough privileges to complete their projects. If our policies (and related proposals) restrict legitimate educational activities, then we are no longer a place for education. [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 12:54, 1 February 2024 (UTC)
:: Thank you very much for the explanation and the summary, but I cannot guarantee that everyone will accept it. Removing contradictions sounds good. If the content was obvious nonsense or conflict with the entire Wikiversity, then your decision (blanking/removal) would be the most reasonable one. In this case, I think there were other options (such as rewriting to resolve contradictions), and that is why I'm calling for a review. For example, at [[special:diff/2602692]], you said that "The obligation to use verifiable and reliable sources lies with the editors wishing to include information on Wikiversity page, not on those seeking to question it or remove it" contradicts the option of scholarly research at Wikiversity. I don't understand how this becomes a contradiction (have you already explained that?). Even if it was a contradiction, I think blanking was not the only one option. We could have restricted the obligation to non-research content (such as educational resources) or downgraded the obligation to a recommendation, and avoid potential conflict with Wikiversity research content. The summary of my question is, "Why have you decided to remove instead of suggesting a rewrite?". [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 13:20, 1 February 2024 (UTC)
::: I see an obvious contradiction, as mentioned in the edit summary: if original research and original user-written essays are allowed, there is no "obligation to use verifiable and reliable sources".
::: As for dropping text vs. rewrite: a rewrite creates an opportunity to introduce new mistakes and non-consensualities, a bad thing. By contrast, removal of problematic sentences removes defects. After removal of problematic sentences, we may focus on whether the text that remained after removal is really accurate and fully fit for purpose, which I do not think to be the case either; more corrective work is required. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 11:12, 5 February 2024 (UTC)
:::: Thank you for additional explanations. If somebody is going to produce their own research where anything similar was never published elsewhere, there would be no other independent secondary sources, so the Wikipedia-like verifiability is no longer reasonable at here. On the other hand, I believe that authors should work hard to avoid errors (calculation errors, uploading wrong images etc., I was talking about this type of verifiablity for research content), if they want to pass Wikijournal peer reviews then they need to do so. In addition, I expect many type of research comes out from previous research history, and I think it is reasonable to expect the Wikipedia-like verifiablity when explaining research background and related history. What would you think about this? I'm not demanding the Wikipedia-like verifiability to research itself, I'm recommending this to things before entering research. [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 01:59, 6 February 2024 (UTC)
::::: As for "If somebody is going to produce their own research where anything similar was never published elsewhere", one may well publish result of research such that something similar ''was'' already published elsewhere; it is still ''original research'' in Wikipedia terminology.
::::: Wikiversity is great for articles that combine original research/element of originality with referenced material. For such articles, there is no duty to reference things but I would see inline referencing as recommended for consideration (not enforced) and adding great further reading/external links as recommended (not enforced). I fully agree that "authors should work hard to avoid errors". As for Wikijournals, that is a separate class of Wikiversity content, with its own rules and processes. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 07:22, 9 February 2024 (UTC)
::::: About "explaining research background": I know of no duty to explain research background (or is there one?) and therefore, there is no duty to explain the background and then reference it using Wikipedia-style inline referencing. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 08:13, 9 February 2024 (UTC)
{{cob}}
Would somebody like to vote between keeping page ''as is'' or returning it to [[Special:Permalink/1375824]]? If so, write "I move that we foobar" as vote yes or no.--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 17:54, 26 February 2024 (UTC)
== Can [[User:Ciphiorg/sandbox]] be an acceptable sandbox? ==
The sandbox was made by using talk page namespace so I moved it into userspace. After the page moved, I noticed that the sandbox was about physical geography but also aimed to promote a single website (physicalgeography.org) and its subpages. I checked the author's enwiki history, all edits were reverted and their enwiki sandbox was deleted per CSD U5. Could this be a xwiki spam case? [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 06:33, 2 February 2024 (UTC)
:{{done}} Deleted. He can ask for undeletion if he wants to remove self-promotion/spam links. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 06:36, 2 February 2024 (UTC)
:: Recent abuse filter logs suggests that the user came back to do something similar. You may need to take action to stop them. [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 05:43, 7 February 2024 (UTC)
::: (Update) Currently stale. [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 09:18, 9 February 2024 (UTC)
== Concern about an IP range starting from 165.199.181 ==
IP editors from this range ([[Special:Contributions/165.199.181.3]], [[Special:Contributions/165.199.181.9]], [[Special:Contributions/165.199.181.15]]) have done a lot of unhelpful actions in our project for months. I think our custodians should consider a range block for a reasonable amount of time. [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 02:06, 6 February 2024 (UTC)
: (Note) All IPs in this report are blocked in minimal range. [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 02:11, 7 February 2024 (UTC)
== Please consider blacklisting of physicalgeography.org ==
Dear custodians, I have reported about editors trying to get physicalgeography.org to appear in Wikiversity at [[special:permalink/2603578#Can_User:Ciphiorg/sandbox_be_an_acceptable_sandbox?]], and now we have another editor trying to get the link visible ([[Special:diff/2603646]]). Please consider the blacklisting of this URL. Thank you for your attention. [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 02:11, 7 February 2024 (UTC)
== [[Special:Contributions/103.150.214.192]] ==
Too many test edits at sandbox (RC flooding), possible proxy, already blocked at zhwiki. [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 09:00, 9 February 2024 (UTC)
:{{ping|MathXplore}} I blocked for 3 hours and then Googled {RC flooding}. I have no experience with these things. How long should I block for?----[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 13:01, 10 February 2024 (UTC)
:: When I reported the IP, they were violent, and at least a short-term block (perhaps several hours) may have been needed at that time. Currently, the IP editor is stale, so there may be no significant meaning to block them at this moment. On the other hand, GetIPIntel Prediction is 100% at [https://ipcheck.toolforge.org/index.php?ip=103.150.214.192 IPcheck information], this means that this IP might be a [[:m:No open proxies|proxy]] (and I guess that is why zhwiki blocked this IP, I don't know well about zhwiki proxy block policy), though the other parameters are negative. I think we need someone who knows more about proxies to choose the right range and terms. {{ping|Koavf}} can you take a look at this IP? [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 13:18, 10 February 2024 (UTC)
:: (Note) After my reply, another IP ([[Special:Contributions/103.150.214.135]], close to the one above) appeared with similar behavior (targeting sandbox). This IP is blocked at zhwikivoyage as an open proxy (1 year), also blocked at enwiki as a web host. [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 14:15, 10 February 2024 (UTC)
:::I am not a range block pro, but doing a little range block hacking, I see that both [[Special:Contributions/103.150.214.192/16]] and [[Special:Contributions/103.150.214.135/16]] contain all of the edits by the above IPs and ''only'' the edits by the above IPs. Both are globally blocked for a couple of months, but 1.) I take violent threats very seriously ({{Ping|MathXplore}}, did you write to legal@? If not, I will.) 2.) the sandbox is one of the only pages you really don't want to have escalated protection on, and 3.) oftentimes, rangeblocking open proxies is not going to harm the project. So, I'm willing to do a 12-month range block. Great work as always. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 19:17, 10 February 2024 (UTC)
:::: Sorry, I didn't write to legal. I was checking the edit frequencies and their global contributions rather than the context. Please go ahead for the report to legal. Thank you for the reactions and information. [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 06:28, 11 February 2024 (UTC)
:::::Hey, no worries MX. You do a ''lot'' across ''many'' wikis. It's a team effort, friend. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 07:27, 11 February 2024 (UTC)
::::::Wait--I actually ''looked'' at the diffs and some of them mention some weird violent content, but are not ''threats'', so it doesn't rise to that occasion. Sorry for my ignorance. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 07:29, 11 February 2024 (UTC)
== [[Special:Contributions/24.224.18.114]] ==
Vandalism from this IP, a targeted page is now semi-protected. [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 02:22, 16 February 2024 (UTC)
: (Note) Currently stale. [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 06:28, 20 February 2024 (UTC)
== [[special:permalink/2607000]] ==
Can this be considered as an academic profile, or should be handled as an advertisement? [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 06:27, 20 February 2024 (UTC)
:Tricky. I'm inclined to call it a valid profile ''if'' this user engages in actually editing and particularly in creating resources related to these kind of topics such as SEO, but call it just spam if this person is only here to say "I am so-and-so and I have [x] marketable skills". :/ So I could be persuaded either way, but it's not ''obviously'' spam as of now, as far as I can tell. I totally respect any other custodian or curator deleting it, tho. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 06:32, 20 February 2024 (UTC)
== [[Portal talk:Astronomy]] ==
This talk page is currently isolated but has a lot of things in here. Where can we move this page to save it as an archive? [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 13:39, 25 February 2024 (UTC)
:I created [[Draft:Archive]] without asking for a consensus. If nobody objects, we can all use it. The only open question in my mind is whether we need to nowikify the pages to avoid having titles appear on various lists and categories. I suggest the title [[Draft:Archive/2024/Portal talk-Astronomy]]. Personally, I am not very adept at undeleting pages, thought with a bit of practice I might find it more natural. With a small cleanup crew that tends to get bogged down in long discussions, it's easier if everybody can look at pages that have been removed in this fashion. Many years ago I remember an editor who annoyed administrators with frivolous requests to undelete for viewing purposes. If you want, I can move [[Portal talk:Astronomy]] right now.--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 14:37, 25 February 2024 (UTC)
:: What is wrong with [[Portal talk:Astronomy]] staying where it is? --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 14:41, 25 February 2024 (UTC)
:::Sorry! Again I read quickly but without accuracy. I didn't notice that it was a '''Talk''' page. I will archive it right now.[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 17:20, 25 February 2024 (UTC) {{Done}}[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 18:36, 25 February 2024 (UTC)
:::: You "archived" the page but not moved. Where should we move the talk page? That is my question. [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 00:22, 26 February 2024 (UTC)
::: According to [[WV:CSD]], isolated talk pages are subject to deletion. [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 00:21, 26 February 2024 (UTC)
::::I apparently just forgot to delete the talk page. Does anybody object to deleting the talk page and its archive?--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 01:50, 26 February 2024 (UTC)
::::: Why is this being deleted or archived? I guess it is because of [[WV:Deletions]], "Discussion about deleted resources where context is lost and becoming an independent resource is unlikely". But the resource was not deleted, it was moved: from looking at [[Portal:Astronomy]], one can see it was moved to [[Topic:Astronomy]], which is now a redirect to [[Astronomy]]. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 07:16, 26 February 2024 (UTC)
{{done}}[[file:Red question mark.svg|20px]] Taking Dan's lead, I assumed the hanging talk page [[Portal talk:Astronomy]] to have been attached to what is now [[Astronomy]], which already had a talk page. So I made the Archive a subpage with an explanatory note at [[Talk:Astronomy]]. I'm glad this is a hobby and not a serious effort to preserve the history of this ol wiki.--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 19:38, 26 February 2024 (UTC)
== Chronological order of [[Wikiversity:Request_custodian_action/Archive/23]] and [[Wikiversity:Request_custodian_action/Archive/24]] ==
I generally understand that archives are numbered in chronological order but I found an exception to this rule. [[special:permalink/2596291]] says that 23 is "January 2021 - June 2023" and 24 is "December 2021 - December 2022", this is breaking the chronological order. Should we fix this or keep it in the current state? [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 01:19, 26 February 2024 (UTC)
:I noticed that while archiving a while back. I think we should leave it alone. One problem is that we have two chronological orders: One is when the request was initiated, and the other is when the request is archived. To make matters worse, many topics get "archived" twice: First when <nowiki>{{Archive top}}..{{Archive bottom}}</nowiki> turns the background blue, and second when the conversation is moved. Also, these conversations are extremely chaotic. Reading them would make good reading for chatbots if and when humans ever decide to start punishing them for transgressions.--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 01:46, 26 February 2024 (UTC)
:: OK, thank you for your opinions. [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 01:49, 26 February 2024 (UTC)
== Can anybody explain how this turns into a proposed deletion? ==
I just deleted a lot of pages because I thought the author was confusing the prod template for speedy delete. [https://en.wikiversity.org/w/index.php?title=User:Ramosama/sandbox/Problem_Analysis_-_Provision&action=edit This is the source] for [[User:Ramosama/sandbox/Problem Analysis - Provision]]:
{{cot|Click to view the source code that triggers the prod}}
<code><nowiki>{{Problem analysis - measure|name=Reusing durables|identifier=reusing_durables
|definition= The reuse of durable goods in their original form.
|reasons=
|parents=
|instances=
* Design of equipment for reuse of their parts ("cradle to cradle").
* Prolonged storage of reusable goods in warehouses, such as deserted office buildings.
* Second-hand warehouses.
* Refund for returns of durables.
* Facilitation, for example, allowing customers to reuse packaging or containers.
|advantages=
|disadvantages= }}</nowiki></code> [[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 17:14, 26 February 2024 (UTC)
{{cob}}
Thankfully the user has been dormant for almost 4 years. See [[Special:Contributions/Ramosama]].[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 17:17, 26 February 2024 (UTC)
: I edited "[[:Template:Problem analysis - concept]]" to place its proposed deletion code into the noinclude tag. As a result, [[User:Ramosama/sandbox/Problem Analysis - Provision]]--which uses the template--no longer shows any proposed deletion tag. I hope it added some clarity and has no undesirable consequence. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 19:42, 26 February 2024 (UTC)
::Good news! I thought it was possible to accidentally make a prod. Thank's Dan.[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 19:48, 26 February 2024 (UTC)
== Does anybody know how to delete all pages by a single user? ==
We have a serial page creator. My hunch is that the pages were created in another language, translated using an auto-translator, and placed on en.wikiversity. I am currently trying to create a list from [https://en.wikiversity.org/w/index.php?title=Special%3AContributions&target=Saltrabook&namespace=all&tagfilter=&newOnly=1&start=&end=&limit=50 '''this list''']. If nobody knows how to do this, I will use a list under construction at '''[[Pre-diabetes diagnosis and remission]]'''.[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 16:43, 27 February 2024 (UTC)
:[[Special:Nuke]] can mass-delete, with some caveats. Oddly, it is only available to bureaucrats here. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 16:38, 27 February 2024 (UTC)
I don't know the answer. But let me list the pages created in 2024 (there are more from 2023):
* [[INVITATIONS TO SEAFARERS AND THE MARITIME MEDICAL CLINICS]]
* [[CONTENTS OF THE 16 WEEKS COACHING]]
* [[VIDEO PRESENTATION AND INVITATIONS]]
* [['''CONTENTS OF THE 16 WEEKS COACHING''']]
* [[DRAFT ARTICLE]]
* [[Maritime Diabetes-type 2 Intervention study/DRAFT PAHO PROTOCOL/CONTENTS OF THE 16 WEEKS COACHING]]
* [[Maritime Diabetes-type 2 Intervention study/DRAFT PAHO PROTOCOL/DRAFT PAHO PROTOCOLO EN ESPAÑOL]]
* [[Maritime Diabetes-type 2 Intervention study/DRAFT PAHO PROTOCOL/ESPAÑOL]]
--[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 16:43, 27 February 2024 (UTC)
:{{Done}} I deleted all the maritime health and diabetes pages made in the past several months. If more is needed, let me know. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 16:55, 27 February 2024 (UTC)
::Thanks Justin. You might want to change the parameters of my block of Saltrabook. I know little about blocking protocols. I will change my expiration date from one week to indefinite. I didn't know you could pagenuke. We need an active pagenuker on this wiki now that Dave is less active.[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 17:07, 27 February 2024 (UTC)
:::I have no perspective on an indefinite block, but it may be a good idea until/unless he can explain on his talk page what he's trying to do and where he is getting this information, etc. Note also that he has ''lots'' of pages going back to at least 2019. If we had consensus that [[Special:Nuke]] were available to admins (curators), then we could make the request on [[:phab:]] to change the local settings. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 17:58, 27 February 2024 (UTC)
::::Do you know whether Saltrabook can use his talk page? If so, there is no need to change the indefinite block.[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 19:19, 27 February 2024 (UTC)
::::: The latest block ([[special:redirect/logid/3389142]]) does not include edits, so I think they can. Generally, most blocked users can edit their own talk pages for unblock requests and related statements (unless revoked). [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 03:48, 28 February 2024 (UTC)
:::: Currently, curators cannot restore pages. I think allowing mass-delete without restoration permissions can be risky. Allowing mass-delete to our custodians should be enough. Why have we limited mass-delete to our bureaucrats? Are there any previous discussions in the past? [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 03:50, 28 February 2024 (UTC)
:::::No clue. That is very bizarre and atypical. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 04:48, 28 February 2024 (UTC)
:::::: I think we can ask to hear the community's opinion at [[Wikiversity:Colloquium]]. They may want to speak about what they think about this odd technical settings. [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 03:45, 1 March 2024 (UTC)
::::::: <s>(Note about this matter) I started a new thread over there.</s> [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 13:41, 1 March 2024 (UTC)
::::::: (Update) Per suggestion ([[special:diff/2610994]]), I started a proposal at [[Wikiversity_talk:Custodianship#Proposal_to_allow_custodians_to_use_mass-delete]]. [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 07:17, 8 March 2024 (UTC)
:: If possible, I suggest clarifying the deletion criteria (RFD? off-wiki request?). I'm sorry if I have missed anything. From my viewpoint, I only requested renaming without redirects, and now I see pages being deleted. Having more explanations would be better, I think. [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 03:53, 28 February 2024 (UTC)
:::{{ping|MathXplore}} Sorry, sometimes I act too swiftly. It turns out User:Saltrabook has been creating what looks like interesting pages for a long time, and he has created close to 100 such pages (probably much more.) He doesn't know English very well, so it is obvious that he is auto-translating the pages. I blocked his page creations, and he seems happy working on pages he already created (many of them were almost blank.) Personally, I would be happy if he works on the pages he has already created and left us alone. We get odd ones on WV. I should know; my family thinks I am one.-[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 03:37, 1 March 2024 (UTC)
:::: Thank you for the explanations. As can be seen in each page history, I'm one of the few editors handling the categorizations of their creations, but I didn't notice that there were auto-translations (has anyone identified which software has been used?), apologies for being late to notice such issues. I think we should clarify how to handle auto-translations via policy/guideline or previous discussions. [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 03:43, 1 March 2024 (UTC)
== [[Special:Contributions/Krutrimam]] ==
Lock evasion of [[User:Premaledu]], please see [[special:permalink/2609661#Offensive_username]]. [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 12:19, 1 March 2024 (UTC)
: Already {{done}}, globally locked. [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 13:23, 1 March 2024 (UTC)
== Explanation of edit ==
I was trying to link my pages and I got a notification to explain to a custodian. I hope I'm in the right place for that. [[User:An5189|An5189]] ([[User talk:An5189|discuss]] • [[Special:Contributions/An5189|contribs]]) 04:42, 4 March 2024 (UTC)
:Seems fine to me. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 04:49, 4 March 2024 (UTC)
::thanks [[User:An5189|An5189]] ([[User talk:An5189|discuss]] • [[Special:Contributions/An5189|contribs]]) 04:52, 4 March 2024 (UTC)
== create about user page ==
I was trying to create about User page
[[User:An5189|An5189]] ([[User talk:An5189|discuss]] • [[Special:Contributions/An5189|contribs]]) 05:17, 4 March 2024 (UTC)
:I'll create a blank one and you can modify it. Let me know if you have more problems. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 06:57, 4 March 2024 (UTC)
::thanks, I will[[User:An5189|An5189]] ([[User talk:An5189|discuss]] • [[Special:Contributions/An5189|contribs]]) 08:18, 4 March 2024 (UTC)
== [[Special:Contributions/39.50.199.52]] ==
Making bad pages (I already deleted them) and xwiki abuse (also reported at Wikiquote). [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 02:46, 10 March 2024 (UTC)
:{{done}} —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 05:49, 10 March 2024 (UTC)
== [[Special:Contributions/Precisiongroup]] ==
Spam-only account with promotional username (account named after company name). [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 06:49, 14 March 2024 (UTC)
:{{done}} —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 13:02, 14 March 2024 (UTC)
== [[Special:Contributions/Kroodham]] ==
Lock evasion of [[Special:CentralAuth/Premaledu]]. [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 07:45, 14 March 2024 (UTC)
: {{done}}, already locked. [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 08:57, 14 March 2024 (UTC)
== [[Special:Contributions/27.55.68.138]] ==
Vandalism and xwiki abuse. [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 10:07, 19 March 2024 (UTC)
:{{done}} Month-long rangeblock. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 10:36, 19 March 2024 (UTC)
== [[Special:Log/Cbtproxyus]] ==
The user has repeated user page spam, I already deleted it and set indefinite full protection. [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 05:56, 20 March 2024 (UTC)
:{{done}} indef block. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 15:42, 22 March 2024 (UTC)
== CAPTCHA Problem when creating an Account ==
I don't know how active Dave is at the moment, so I paste a message to Dave from [[User:Ireicher2]]:
{{quote|Hi Dave, Isabel here from Ohlone college. We've talked a couple of times before. Some of my students emailed me to let me know that they cannot create user accounts because of a CAPTCHA problem. I verified the information by attempting to create a new account and I received the same error message. Would you let me know how this can be resolved? Thank you!}}
{{ping|Ireicher2}} One thing you might try is having them create Wikipedia or Wikibooks accounts. I believe membership in one automatically creates membership on Wikiversity. Another thing to try is asking students to create the account from their homes. Does anybody else have any ideas????--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 01:49, 22 March 2024 (UTC)
: I think [[:w:Wikipedia:Request_an_account/Help_and_troubleshooting]] is related to this issue. It is a different project but shares the same technical basis. [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 01:55, 22 March 2024 (UTC)
:See [[meta: Mass account creation]]. I'll try adding Account creators to [[User:Ireicher2]] with an expiration of seven days and see if makes any difference. Yes, the suggestion that students create their accounts from home (or using their cell phones vs. school computers) should help. -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 03:44, 22 March 2024 (UTC)
::@[[User:Dave Braunschweig|Dave Braunschweig]] @[[User:Guy vandegrift|Guy vandegrift]] Of course. That makes sense. Thank you!<br> [[User:Ireicher2|Ireicher2]] ([[User talk:Ireicher2|discuss]] • [[Special:Contributions/Ireicher2|contribs]]) 04:46, 22 March 2024 (UTC)
== I need the custodians & curators to tell a user not to be involved with deletions. ==
{{Cot|Collapse as resolved}}
Here are two examples:
#He put a speedy delete on [[special:permalink/2617505]], saying among other things that there is "no clear explanation" of what ''ordinary'' differential equations are". This is a subpage, and the parent page at [[special:permalink/2483117]] gives a rather coherent explanation: "Differential equations serve as mathematical models of physical processes. This course is intended to be an introduction to ordinary differential equations and their solutions. <small>A '''differential equation''' (DE) is an equation relating a function to its derivatives. If the function is of only one variable, we call the equation an '''ordinary differential equation''' (ODE). ...</small><br> There is a movement to raise the standards regarding what should and should not be in namespace, but the the parent page at [[special:permalink/2483117]] has [[Special:PrefixIndex/Differential_equations/|13 subpages.]] If this resource is a problem, it has to be addressed from the top down, not one subpage at a time. As will be shown in the next example, I recently attempted to explain to him that it is inefficient to remove subpages without looking at the entire resource (via the parent page.)
#Days prior to the aforementioned effort to delete a subpage of [[Differential equations]], he proposed the deletion of one of some 300 subpages of [[Student Projects]] because it was unsourced. My reason for not deleting that page should have informed him that it would have been inappropriate to delete one subpage of [[Differential equations]], because it turns out that almost all subpages of [[Student Projects]] are unsourced, leaving us with the same issue involving the deletion of pages from the "bottom-up". For evidence that this user had been informed of the need for a "top-down" approach attempting to delete a subpage of [[Differential equations]], see [[special:permalink/2617342#Student_Projects/Major_rivers_in_India]]. This editor is a nice person with a lot of good ideas, but his stubbornness is making it difficult to moderate [[Wikiversity:Requests for Deletion]]--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 14:15, 31 March 2024 (UTC)
:I don't want this user blocked, or even banned from participating in discussions about deletion policy. He is not alone in advocating higher standard, and the community might want to do that. But there is a distinction between the nuts and bolts of deletion, and deletion as a policy. I am very conservative about deleting pages. So if the standards get tightened, there will be no need to revert anything I have done. I am asking the custodians/curators to encourage this user to go to [[Wikiversity:What-goes-where_2024#Personal_subpages_(with_visual_editing)|WV:WGW2024]] and create a subpage for sharing his ideas with the community.--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 14:41, 31 March 2024 (UTC)
-----
-----
-----
#: 1) The "unsourced" on [[Student Projects/Major rivers in India]] was only one reason; the other reason was that this page has nothing to add what is not in Wikipedia, a rationale previously recognized. 2) [[Student Projects/Major rivers in India]] is not integrated in any way to a [[Student Projects]] "project"; its being a subpage is just an attempt to escape deletion scrutiny. 3) I am not aware of any explanation to me that I should not nominate subpages; such an explanation has my talk page as a proper venue, and I am unaware of any such explanation, neither there or elsewhere. 4) Any disagreement about deletion can be resolved via RFD and via voting-cum-discussion there, as is usual in other projects, e.g. the English Wikipedia and the English Wiktionary; if I am mistaken in a particular nomination, it can be brought to RFD and quickly voted down. Even a single person opposing can prevent a deletion in which I am the sole, mistaken, deletion supporter. 5) I have a pretty good conversion rate between deletion nominations and actual deletions/moving out of mainspace, and therefore, I do not think that my nomination algorithm is too broad and too burdensome on those who have to oppose my nominations for deletions. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 14:26, 31 March 2024 (UTC)
#::All I am asking is that you stay out of active deletions and focus your talent on changing the policy. A great place to do that is at [[WV:WGW2024#Personal_subpages_(with_visual_editing)]] [[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 14:51, 31 March 2024 (UTC)
#::: I invite you to my talk page to make requests concerning change of behavior on my part. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 14:52, 31 March 2024 (UTC)
{{cob}}
[[File:Yes check.svg|18px]]'''Resolved''' We have corresponded in our talk pages and the problem has been resolved to my satisfaction.--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 18:15, 31 March 2024 (UTC)
== link on page looks possibly explicit to follow? ==
[[Other Free Learning Resources]] the univeristy of reddit link has a lot of very adult explicit words as links . I did not view other links from this page. Thanks
U - X
* [http://www.ureddit.com/ University of Reddit]
[[Special:Contributions/2001:8003:B120:8900:4D5:4E7A:36B2:58F3|2001:8003:B120:8900:4D5:4E7A:36B2:58F3]] ([[User talk:2001:8003:B120:8900:4D5:4E7A:36B2:58F3|discuss]]) 12:50, 1 April 2024 (UTC)
:Thanks, it's now spam, so I removed it. :/ —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 19:09, 1 April 2024 (UTC)
== [[Special:contribs/206.110.193.204]] ==
Vandalism [[User:Seawolf35|Seawolf35]] ([[User talk:Seawolf35|discuss]] • [[Special:Contributions/Seawolf35|contribs]]) 18:57, 16 April 2024 (UTC)
:{{done}} —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 19:32, 16 April 2024 (UTC)
== Induced stem cells copyright issues ==
[[Induced stem cells]] got imported to here from enWiki- which is fine, attribution was done correctly and everything- except for the fact that I'm just wrapping up a [[Wikipedia:Contributor copyright investigations/20240516|copyright investigation]]<nowiki> on the original contributor & his alternative account. Due to the fact that this contributor repeated and blatantly infringed on the copyright of multiple sources despite multiple warnings an even a block, I tagged the original page over on enWiki for presumptive deletion. I don't know what Wikiversity's process is for suspected copyright infringements without a clear source, but I figured you guys would want to know about the problems with this page anyways. -- ~~~~</nowiki> [[User:GreenLipstickLesbian|GreenLipstickLesbian]] ([[User talk:GreenLipstickLesbian|discuss]] • [[Special:Contributions/GreenLipstickLesbian|contribs]]) 20:34, 23 May 2024 (UTC)
:Very helpful, thanks. Do you have any relevant links to en.wp about the investigation or where he typically ripped off material? —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 23:48, 23 May 2024 (UTC)
::Okay- everthing I've gathered so far is going to be in this [[wikipedia:Wikipedia:Contributor_copyright_investigations/20240516|investigation page]]. ( tried to link it in the original post, but I failed spectacularly as you can see). They almost exclusively copied from scientific papers/ reviews, and news/blog reports. This user typically copied from the source they cited- or, at least, *a* source they cited. They'd regularly copy a paragraph of text from one source, then a cite a different source for each sentence. If a source was paywalled, they often would cite the source, but copy from a news report/blog report analyzing the source. One of the other investigators found a few instances where they copied another article in Wikipedia without attribution- but that was their rarest type of violation. They occasionally wrote their own material, but it was normally easily identifiable because English is not their first language.
::Sorry for not being more helpful on this article in particular-I saw they(and their alt) were essentially the sole author of this page, cited 300+ sources, made a noise somewhat akin to that of a distressed animal, and decided I was going to take advantage of enWiki's rule allowing us to delete articles written by serial copyright violators without any more evidence. --[[User:GreenLipstickLesbian|GreenLipstickLesbian]] ([[User talk:GreenLipstickLesbian|discuss]] • [[Special:Contributions/GreenLipstickLesbian|contribs]]) 00:09, 24 May 2024 (UTC)
:::That’s plenty to convince me that this should be assumed to be a copy II until proven otherwise. Merci. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 00:16, 24 May 2024 (UTC)
:::: Thank you for the responses, I think having a short intro, soft redirect to the CCI page, further readings section, and categories would be OK, what would you think about this? [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 01:56, 24 May 2024 (UTC)
:::::Sure. Do you want me to undelete and then redelete selected diffs? Or you’ll just create the redirect yourself? —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 02:10, 24 May 2024 (UTC)
:::::: I will create a soft redirect afterward. [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 02:29, 24 May 2024 (UTC)
::: Thank you for the information, do you think [[WikiJournal Preprints/Induced stem cells]] needs deletion? It is another page where the same editor has substantial involvement. [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 01:58, 24 May 2024 (UTC)
::::I did a brief check, and I found that [https://www.science.org/doi/10.1126/science.1248252 this source] (cited in the article) appears to have been partially copied. Specifically, the stuff about zebrafish has been copied word for word. It's not a promising sign. If this was on the English Wiki, I would ask for it to be presumptively deleted soley on the basis of the author and that confirmed instance of a copyright violation. I worked on the investigation for several days (and I was the one who asked for it to be opened), and I could confirm over half their writing to be blatant copy-and-paste jobs. [[User:GreenLipstickLesbian|GreenLipstickLesbian]] ([[User talk:GreenLipstickLesbian|discuss]] • [[Special:Contributions/GreenLipstickLesbian|contribs]]) 03:08, 24 May 2024 (UTC)
:::::Thank you for your service, hermana. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 03:38, 24 May 2024 (UTC)
::::: Thank you for the information, I have contacted an active Wikijournal contributor to learn about how this preprint should be handled. [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 02:05, 6 June 2024 (UTC)
::::::Thank you to [[User:GreenLipstickLesbian|GreenLipstickLesbian]] for informing us about the copyright violations and reference false attribution in this article, and [[User:Koavf|Koavf]] & [[User:MathXplore|MathXplore]] in participating in this conversation. Normally we would keep rejected articles in the preprint with the stated reason in the talk page for record purpose. However, since the induced stem cell contains copyright violation and may cause future accidental copyright violation by future text re-users under the assumption that the text is under Creative Commons license, I will request that the preprint be deleted while talk page remains undeleted to note the rationale. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;"><sup>Talk page</sup></span></b>]] 20:00, 18 June 2024 (UTC)
:::::::Good point. In addition to not deleting the talk page, I am redirecting the main page to the talk page and protecting it. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 20:07, 18 June 2024 (UTC)
::::::::@[[User:Koavf|Koavf]] Can you also delete [[WikiJournal Preprints/Induced stem cells]], redirecting it to talk page please? [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;"><sup>Talk page</sup></span></b>]] 04:03, 10 July 2024 (UTC)
:::::::::{{done}} —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 04:42, 10 July 2024 (UTC)
== Creating a section of my own talk page with a link to Wikipedia ==
I'm trying to set up my own talk page here at Wikiversity with my own example of trying separate the essence and accident of programming, as per <nowiki>[[w:No Silver Bullet|No Silver Bullet]]</nowiki> at Wikipedia, but it's rejected because of the external link (i.e. to Wikipedia). I'm doing this because most example code I see buries the essence in the accident and I wanted to show an example that there are better ways to write code. [[User:Philh-591|Philh-591]] ([[User talk:Philh-591|discuss]] • [[Special:Contributions/Philh-591|contribs]]) 10:33, 9 July 2024 (UTC)
:That's very weird: you can't create ''interwiki'' links? And to be clear, you're trying to put said links on your talk page at [[User talk:Philh-591]], not your userpage [[User:Philh-591]]? —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 12:05, 9 July 2024 (UTC)
::Well, your creation of the page with a welcome message has got it past that restriction, although I don't think it was the Wikipedia link. I'd not noticed that there are URL's in my example source referring to public information at the European Central Bank. However, it now insistently applies "nowiki" to what I insert. I guess I don't understand the formatting rules at Wikiversity; I'd assumed it was just like Wikipedia. I'll see if I can understand it more playing in the sandbox. [[User:Philh-591|Philh-591]] ([[User talk:Philh-591|discuss]] • [[Special:Contributions/Philh-591|contribs]]) 13:13, 9 July 2024 (UTC)
:::I figured that would fix the problem: sometimes, creating a new page (even your own user or user talk page) has restrictions. I forget the exact limitations per wiki, but they are usually very modest, like make at least five edits across two weeks or something. Re: formatting rules, they should be the same as Wikipedia, so I'm confused as to what you're trying to do again. :/ —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 13:25, 9 July 2024 (UTC)
== Won't let me publish "my about" page due to "New User Exceeded New Page Limit" ==
Unsure how to publish my about me page, is someone able to help me be able to publish it without it being disallowed? [[User:Lucywilson 546|Lucywilson 546]] ([[User talk:Lucywilson 546|discuss]] • [[Special:Contributions/Lucywilson 546|contribs]]) 03:20, 8 August 2024 (UTC)
: {{ping|Jtneill}} Can you grant confirmed status for this user? [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 03:41, 8 August 2024 (UTC)
:I made a blank page, which you can now edit. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 04:31, 8 August 2024 (UTC)
: {{ping|MathXplore}} Thankyou, I've confirmed the user. {{ping|Koavf}} Thankyou, a neat, instant solution :). {{ping|Lucywilson 546}} Thanks for letting us know. You should be good to go. Let us know if any other problems. Sincerely, James -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 10:24, 8 August 2024 (UTC)
== Delete revission ==
Could you delete [https://en.wikiversity.org/w/index.php?title=Wood_finishing&oldid=2651335 this revision], which is revealing my personal information, please? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 09:54, 9 September 2024 (UTC)
: Username is hidden, I have contacted the [[:m:stewards]]. [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 11:38, 9 September 2024 (UTC)
: {{done}} [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 12:00, 9 September 2024 (UTC)
== Spam filter exception request ==
I am prevented from creating [[Template:Vandal]] because a previous example in [[Template:Vandal/doc]] used an IP address, which is blocked by a spam filter. I removed that example, but am still blocked from creating that page. I have put the source code in [[Template:Vandal/sandbox]] in the interim. Perhaps allowing just <code>10.0.0.1</code> to avoid other IP spam? [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 04:55, 2 October 2024 (UTC)
:{{done}} I created a blank template, which you can now edit. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 06:11, 2 October 2024 (UTC)
::{{done}} again: I moved your sandbox to the template. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 06:12, 2 October 2024 (UTC)
:::It seems I still can't include the IP-user example (see [[Special:PermanentLink/2658932|an old version]] with the offending string) - [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 23:13, 2 October 2024 (UTC)
::::I don't understand the problem. What text are you trying to put where? —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 00:43, 3 October 2024 (UTC)
:::::See [https://en.wikiversity.org/w/index.php?title=Template:Vandal/doc&diff=prev&oldid=2658932 this diff] which shows the text and location {{--}} [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 00:48, 3 October 2024 (UTC)
::::::{{done}}. Longer-term issues with including IP addresses may still exist, but this particular edit at least is fixed. Thanks. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 00:53, 3 October 2024 (UTC)
== [[Special:Log/Tule-hog]] ==
As seen in the link above, [[User:Tule-hog]] has made various manual imports from WP to WV. Some may be OK, but others may be questionable. Despite various messages on their talk page ([[User talk:Tule-hog]]) from user:Dan Polansky, the user continues manual imports. Should we let this continue, keep talking with the user, or should we stop them? What would be the best option? ({{ping|Jtneill}} As Dan's mentor, your feedback is welcome here, and {{ping|Koavf}} since you previously communicated with the user in [[Special:Diff/2659041]], we would like to hear about your thoughts) [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 06:39, 7 October 2024 (UTC)
:{{Ping|Tule-hog}} From what I see on your talk, you are at least not doing this anymore. While copyright-wise, we can of course copy anything from en.wp to here, it is best to use [[Special:Import]] because it preserves edit histories, provides attribution, and can also import dependencies like another modules or templates. Can you explain what your goal is with this copying and what in general you want copied? —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 06:43, 7 October 2024 (UTC)
::I am motivated by updating Wikiversity template/module infrastructure in places where appropriate. Note I do not have the [[WV:Importer|importer]] role. I perform what I've been calling [[User:Tule-hog/Wikiversification|Wikiversification]] on docs and templates themselves, where much of the time the pages I come across are rough imports with raw Wikipedia links without modification, incorrect language for the project, bad category mapping, or are dependent on other undefined modules/templates.
::To be clear, I am ''not'' just going through picking out popular templates/modules and importing them. I approach a maintenance task, and where relevant spend the (not mindless) time to transform them to fit Wikiversity. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 06:50, 7 October 2024 (UTC)
::: I find "in places where appropriate" too non-specific. I do not see any specific need addressed. I find Colloquium a good forum for a proposal to copy (or import) a large number (how large?) of Wikipedia templates and categories; the approximate volume should be stated as part of the proposal. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 07:02, 7 October 2024 (UTC)
:::I undeleted and userfied to [[User:Tule-hog/Wikiversification]]. If you are thinking of making some large-scale change, then it's probably best to clarify your thoughts there, propose it (succinctly!) at the Colloquium, and then coordinate with a custodian who can import. This is kind of a [[:en:wikt:death by a thousand papercuts|death by a thousand papercuts]] situation: any one change is perfectly fine, but the volume may be systemic, so it's wise to get the community's input. Besides, we could help and many hands make for light lifting. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 07:08, 7 October 2024 (UTC)
::(Question due to unfamiliarity with importer mechanism:) Do we also submit requests to ''update'' already imported templates at [[WV:I]], or does that only happen once (and hence update requests should go to [[WV:RCA]])? Thanks, [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 23:25, 7 October 2024 (UTC)
:::Unfortunately, if you import a resource from another wiki and the original changes, the updates need to be imported again here manually and since [[WV:I]] is a dedicated space, it's probably best to put requests there. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 05:05, 8 October 2024 (UTC)
== [[:Category:Wikiversity policies and guidelines]] ==
Should this category finish being developed? (I could do so if desired.) [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 06:32, 8 October 2024 (UTC)
:Similarly with the list detailed at [[:Category:Wikiversity development]] (i.e. finishing up + deleting the list mentioned there) —[[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 06:55, 8 October 2024 (UTC)
::{{ping|Koavf}} double checking is alright for [[:Category:Wikiversity development]] as well (started by [[:User: McCormack|McCormack]]) —[[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 02:20, 9 October 2024 (UTC)
:Can you reword this question? I'm not sure what you're trying to do here. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 07:16, 8 October 2024 (UTC)
::The content of the category is "This category is being developed." so I believe it is in an unfinished state (i.e. adapting the categorization schema). [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 07:18, 8 October 2024 (UTC)
:::I think if you have some rational way of organizing the pages, that's fine. I don't know what
:::[User:CQ]][had in mind when he put that there, but he has basically not edited here in 4.5 years, so go for it. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 07:20, 8 October 2024 (UTC)
== [[:Category:Rejected policies]] ==
Should I use the list of tagged pages found in this category to update [[WV:POLICY#Rejected policies]]? Or is [[WV:IAR]] the only truly firmly rejected proposal? [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 22:59, 8 October 2024 (UTC)
:The category and that list should have the same items<ins> and at first glance, what is the category is in fact rejected proposals, therefore, the list should be updated.</ins> —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 02:04, 9 October 2024 (UTC)
::Just noting [[Wikiversity_talk:Policies#List_of_official_policies|this thread]] which suggests that another user made that list in the first place using tags, so it may have recursively snuck something in. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 02:12, 9 October 2024 (UTC)
== [[Wikiversity:Research guidelines]] ==
This page is listed in {{tlx|official policies}}. Should it be updated as adopted on [[WV:POLICY]], and if so, should it be considered a policy or guideline? [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 00:19, 11 October 2024 (UTC)
:(Also, should it link to the top-level [[Wikiversity:Research]] instead, which uses (the confusingly named) {{tlx|research policy}} navbox?) [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 00:46, 11 October 2024 (UTC)
:Good question. From what I can tell the beta Wikiversity research page is the official policy and the en.wv local version is a copy/fork that hasn't been officially endorsed.
:That leaves me wondering whether we want to pursue a local variation as an official policy or potentially remove the local variant and redirect to the beta version. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 03:12, 13 October 2024 (UTC)
:: What suggests that https://beta.wikiversity.org/wiki/Wikiversity:Research_guidelines is an official policy? And if it is, does the policy match the actual practice? For instance, it says "Original research at Wikiversity is subjected to ''peer review'' in order to allow the Wikiversity research community to strive for verifiability" (italics mine): is that really true outside of Wikijournals? Moreover, the putative policy states in a box: "This page contains summaries of discussions which have taken place in various languages." But this cannot be true since the policy reads like a monologue and a proper summary of discussions cannot be a monologue. A quick skimming of the page raises some red flags.
:: Be it as it may, I think keeping a local copy is vital since then we have the option to amend it without thereby requiring an international cross-language input to the changes. Of course, the amends will be unable to change some core features of Wikiversity (no metamorphosis allowed), but some amends should be possible.
:: As for the local [[Wikiversity:Research guidelines]], I propose to rank it as ''policy proposal'', given the misgivings.
:: In any case, this discussion does not belong to "Request custodian action" but rather to "Colloquium" since the outcome of the discussion can be implemented by anyone, not only custodians, and since input from non-custodians seems welcome. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 05:28, 13 October 2024 (UTC)
:::Agree with retaining local version and treating as ''policy proposal''.
:::I've hidden the note about the guidelines being a copy of the beta guidelines (it confused me at least into thinking that beta version was also the policy on en.wv).
:::Softened the peer review requirement to being "open" to peer review rather than being "subjected" to peer review.
:::Agree that further work e.g., on drafting and potentially making official should be followed through on Colloquium.
:::Thanks @[[User:Tule-hog|Tule-hog]] and @[[User:Dan Polansky|Dan Polansky]]. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 06:13, 13 October 2024 (UTC)
:: Oh, and I was not paying attention: [[Wikiversity:Research]] states "This page provides guidelines for research in Wikiversity" so there appears to be some redundancy/overlap between [[Wikiversity:Research]] and [[Wikiversity:Research guidelines]]. Confusing. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 05:56, 13 October 2024 (UTC)
== [[Wikiversity:Policies]] ➝ [[Wikiversity:Policies and guidelines]] ==
This is a proposal to move [[WV:Policies|Policies]] to a name matching the scope of the page, [[WV:Policies and guidelines|Policies and guidelines]].
The more descriptive title will make identifying the location of guidelines easier for newer participants. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 15:46, 25 October 2024 (UTC)
:See [[User_talk:Tule-hog#Wikiversity:Policies|more discussion]]. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 20:58, 25 October 2024 (UTC)
== Please fully protect... ==
[[Module:Message box/fmbox.css]]. It is used in 29 system messages. [[Special:Contributions/2604:3D08:9476:BE00:DCDC:4B47:21DA:D90E|2604:3D08:9476:BE00:DCDC:4B47:21DA:D90E]] ([[User talk:2604:3D08:9476:BE00:DCDC:4B47:21DA:D90E|discuss]]) 20:32, 25 October 2024 (UTC)
:{{done}} —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 06:26, 26 October 2024 (UTC)
== Uh Oh! ==
Hello. Something Went Wrong With Editing. My Dog And Me Is Editing The New Learning Resources. Dog Grooming (Learning Resources). So Help Me. Tanks. [[Special:Contributions/2603:9000:7AF0:5DA0:B940:EF5A:3D27:A8F0|2603:9000:7AF0:5DA0:B940:EF5A:3D27:A8F0]] ([[User talk:2603:9000:7AF0:5DA0:B940:EF5A:3D27:A8F0|discuss]]) 18:46, 11 December 2024 (UTC)
:Okay, it looks like you are editing [[Pomeranian]], which is a bit of a mess. I think that proper pet care could be a fine topic for this site or our sister site [[:b:|Wikibooks]], but the state this is in is pretty rough. I'd recommend you take a look at [[Wikiversity:Welcome]] and some of the pages linked there. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 18:57, 11 December 2024 (UTC)
== IP block exempt request ==
Can a custodian grant IP block exempt flag to {{u|Silver Dovelet}} please? Her account got into the crosshair of a very wide IP rangeblock and that rangeblock also prevented her from making the request directly here. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;"><sup>Talk page</sup></span></b>]] 09:04, 1 January 2025 (UTC)
:{{done}} for six months. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 11:12, 1 January 2025 (UTC)
::Thanks. Looks like she's back on track. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;"><sup>Talk page</sup></span></b>]] 00:54, 2 January 2025 (UTC)
:::Good deal. Let me know if it needs to be re-upped. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 11:05, 2 January 2025 (UTC)
klwshza6ro50ct0pj9wz7bin2z811aq
Complex Analysis
0
113381
2693970
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2025-01-01T14:03:24Z
Eshaa2024
2993595
/* Singularity and Residues - Part 3 */
2693970
wikitext
text/x-wiki
[[File:Wiki2Reveal Logo.png|146px|thumb|Course contains [[v:en:Wiki2Reveal|Wiki2Reveal]] Slides]]
[[File:Mapping f z equal 1 over z.gif|thumb|Moving the argument of function <math>f</math> in the complex number plane. The point <math>z</math> has a blue color and <math>f(z)= \frac{1}{z}</math> is marked in red color. <math>z</math> is moved on a curve with <math>\gamma(t)=t\cdot e^{it}</math>.]]
[[File:Image of path 1 over z.webm|thumb|Image of path in the complex numbers for the function <math>f(z)=\frac{1}{z}</math>]]
'''Complex analysis''' is a study of functions of a complex variable. This is a one quarter course in complex analysis at the undergraduate level.
==Articles==
* [[Algebra II]]
* [[Dummy variable]]
* [[Materials Science and Engineering/Equations/Quantum Mechanics]]
== Slides for Lectures ==
=== Chapter 1 - Intoduction ===
* '''[[Complex Numbers/From real to complex numbers|Complex Numbers]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Complex%20Numbers/From%20real%20to%20complex%20numbers&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Complex%20Numbers&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]
** [[w:en:Heine–Borel_theorem|Heine-Borel Theorem]]
* '''[[Riemann sphere|Riemann sphere]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Riemann%20sphere&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Riemann%20sphere&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]
* '''[[Complex_Analysis/Exponentiation_and_square_root|Exponentiation and roots]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Complex_Analysis/Exponentiation_and_square_root&author=Complex_Analysis&language=en&audioslide=yes&shorttitle=Exponentiation_and_square_root&coursetitle=Complex_Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]
=== Chapter 2 - Topological Foundations ===
* '''[[Complex Analysis/Sequences and series|Sequences and series]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Sequences%20and%20series&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Sequences%20and%20series&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]
* [[/Power series/]]
* '''[[Inverse-producing extensions of Topological Algebras/topological algebra|Topological algebra]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Inverse-producing%20extensions%20of%20Topological%20Algebras/topological%20algebra&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=topological%20algebra&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]
* [[w:en:Topological space|Topological space]] - Definition: [[Norms, metrics, topology#Definition:_topology|Topology]]
* '''[[Norms, metrics, topology|Norms, metrics, topology]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Norms,%20metrics,%20topology&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Norms,%20metrics,%20topology&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]
=== Chapter 3 - Complex Derivative ===
* '''[[Holomorphic function|Holomorphic function]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Holomorphic%20function&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Holomorphic%20function&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]
* '''[[Complex Analysis/Partial derivative|Partial Derivative]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Partial%20derivative&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Partial%20Derivative&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]
* '''[[Cauchy-Riemann-Differential equation|Cauchy-Riemann-Differential equation]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Cauchy-Riemann-Differential%20equation&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Cauchy-Riemann-Differential%20equation&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]
* '''[[Complex Analysis/Application of Cauchy-Riemann Equations|Application of Cauchy-Riemann Equations]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Application%20of%20Cauchy-Riemann%20Equations&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Application%20of%20Cauchy-Riemann%20Equations&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]
=== Chapter 4 - Curves and Line Integrals ===
* '''[[Line integral|Line integral in <math>\mathbb{R}^n</math>]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Line%20integral&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Line%20integral&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]
* '''[[/Curves/|Curves]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Complex%20Analysis/Curves&author=Complex_Analysis&language=en&audioslide=yes&shorttitle=Curves&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]
** [[w:en:Holomorphic function|Wikipedia: holomorphic function]]
** [[w:en:Integral|Wikipedia:Integral ]]
* '''[[Complex_Analysis/Paths|Paths]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Complex%20Analysis/Paths&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Paths&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]
* '''[[Path Integral|Path Integral]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Path%20Integral&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Path%20Integral&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]
* [[w:en:Curve integral |Wikipedia: Curve integral]]
* [[w:en:Continuity|Continuity]] and [[w:en:Limit of a sequence|Limit of a sequence]]
* '''[[Complex Analysis/Trace|Trace of Curve]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Trace&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Trace&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]
=== Chapter 5 - Holomorphic Functions ===
* '''[[Holomorphic function|Holomorphic function]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Holomorphic%20function&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Holomorphic%20function&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]
** [[Holomorphism/Criteria|Criteria]] - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Holomorphism/Criteria&author=Course:Complex_Analysis&language=en&audioslide=yes&shorttitle=Criteria&coursetitle=Complex_Analysis slideset]) [[File:Wiki2Reveal Logo.png|35px]]
** [[w:en:Holomorphic_function#.C3.84quivalent_properties_of_holomorphic_functions_of_one_variable|Wikipedia: Holomorphic function criteria]]
** [[/Differences from real differentiability/]]
** [[w:Conformal_mapping|conformal mappings]]<math>(\ast)</math>,
** [[Complex Analysis/Inequalities|Inequalities]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Inequalities&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Inequalities&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]
** [[Complex Analysis/rectifiable curve|rectifiable curve]] - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/rectifiable%20curve&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=rectifiable%20curve&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]
* '''[[Complex Analysis/Curve Integral|Curve Integral]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Curve%20Integral&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Curve%20Integral&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]
* '''[[Complex Analysis/Path of Integration|Path of Integration]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Path%20of%20Integration&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Path%20of%20Integration&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]
* '''[[Complex Analysis/Goursat's Lemma (Details)|Goursat's Lemma (Details)]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Goursat's%20Lemma%20(Details)&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Goursat's%20Lemma%20(Details)&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]
* '''[[Complex Analysis/Cauchy's Integral Theorem for Disks|Cauchy's Integral Theorem for Disks]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Cauchy's%20Integral%20Theorem%20for%20Disks&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Cauchy's%20Integral%20Theorem%20for%20Disks&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]
* '''[[Complex Analysis/Identity Theorem|Identity Theorem]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Identity%20Theorem&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Identity%20Theorem&coursetitle= Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]
* '''[[Complex Analysis/Liouville's Theorem|Liouville's Theorem]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Liouville's%20Theorem&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Liouville's%20Theorem&coursetitle= Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]
=== Complex Analysis Part 2 ===
* '''[[Complex Analysis/Chain|Chain]]''' - [https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Chain&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Chain&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]
* '''[[Complex Analysis/cycle|cycle]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/cycle&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=cycle&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]
* '''[[Laurent Series|Laurent Series]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Laurent%20Series&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Laurent%20Series&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]
* '''[[Cauchy Integral Theorem|Cauchy Integral Theorem]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Cauchy%20Integral%20Theorem&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Cauchy%20Integral%20Theorem&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]
* '''[[Cauchy's integral formula|Cauchy's integral formula]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Cauchy's%20integral%20formula&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Cauchy's%20integral%20formula&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]
*[[Complex Analysis/Example Computation with Laurent Series|Example Computation with Laurent Series]]
* '''[[Complex Analysics/Maximum Principle|Maximum Principle]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysics/Maximum%20Principle&author=Complex%20Analysics&language=en&audioslide=yes&shorttitle=Maximum%20Principle&coursetitle=Complex%20Analysics Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]
* '''[[Complex Analysis/Openness theorem theorem of territorial loyalty|Openness theorem theorem of territorial loyalty]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Openness%20theorem%20theorem%20of%20territorial%20loyalty&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Openness%20theorem%20theorem%20of%20territorial%20loyalty&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]
===Singularity and Residues - Part 3===
* '''[[Complex Analysis/Singularities|Singularities]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Singularities&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Singularities&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]
* '''[[Complex Analysis/Example - exp(1/z)|Example - exp(1/z)-essential singularity]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Example%20-%20exp(1/z)&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=z)&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]
* '''[[Complex Analysis/Residuals|Residuals]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Residuals&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Residuals&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]
* [[Complex Analysis/development in Laurent series|development in Laurent series]],
* [[Complex Analysis/Isolated singularity|Isolated singularity]],
* [[Complex Analysis/decomposition theorem|decomposition theorem]],
* [[Complex Analysis/Casorati-Weierstrass theorem|Casorati-Weierstrass theorem]],
==Lectures==
* [[/Cauchy-Riemann equations/]]
* [[Cauchy Theorem for a triangle]]
* [[Complex analytic function]]
* [[Complex Numbers]]
* [[Divergent series]]
* [[Estimation lemma]]
* [[Fourier series]]
* [[Fourier transform]]
* [[Fourier transforms]]
* [[Laplace transform]]
* [[Riemann hypothesis]]
* [[The Real and Complex Number System]]
* [[Warping functions]]
==Sample exams==
[[/Sample Midterm Exam 1/]]
[[/Sample Midterm Exam 2/]]
==See also==
* [[Boundary Value Problems]]
* [[Introduction to Elasticity]]
* [[The Prime Sequence Problem]]
* [[Wikipedia: Complex analysis]]
*[[Complex number]]
[[Category:Complex analysis| ]]
[[Category:Mathematics courses]]
[[Category:Mathematics]]
<noinclude>
[[de:Kurs:Funktionentheorie]]
</noinclude>
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/* Singularity and Residues - Part 3 */
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[[File:Wiki2Reveal Logo.png|146px|thumb|Course contains [[v:en:Wiki2Reveal|Wiki2Reveal]] Slides]]
[[File:Mapping f z equal 1 over z.gif|thumb|Moving the argument of function <math>f</math> in the complex number plane. The point <math>z</math> has a blue color and <math>f(z)= \frac{1}{z}</math> is marked in red color. <math>z</math> is moved on a curve with <math>\gamma(t)=t\cdot e^{it}</math>.]]
[[File:Image of path 1 over z.webm|thumb|Image of path in the complex numbers for the function <math>f(z)=\frac{1}{z}</math>]]
'''Complex analysis''' is a study of functions of a complex variable. This is a one quarter course in complex analysis at the undergraduate level.
==Articles==
* [[Algebra II]]
* [[Dummy variable]]
* [[Materials Science and Engineering/Equations/Quantum Mechanics]]
== Slides for Lectures ==
=== Chapter 1 - Intoduction ===
* '''[[Complex Numbers/From real to complex numbers|Complex Numbers]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Complex%20Numbers/From%20real%20to%20complex%20numbers&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Complex%20Numbers&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]
** [[w:en:Heine–Borel_theorem|Heine-Borel Theorem]]
* '''[[Riemann sphere|Riemann sphere]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Riemann%20sphere&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Riemann%20sphere&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]
* '''[[Complex_Analysis/Exponentiation_and_square_root|Exponentiation and roots]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Complex_Analysis/Exponentiation_and_square_root&author=Complex_Analysis&language=en&audioslide=yes&shorttitle=Exponentiation_and_square_root&coursetitle=Complex_Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]
=== Chapter 2 - Topological Foundations ===
* '''[[Complex Analysis/Sequences and series|Sequences and series]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Sequences%20and%20series&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Sequences%20and%20series&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]
* [[/Power series/]]
* '''[[Inverse-producing extensions of Topological Algebras/topological algebra|Topological algebra]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Inverse-producing%20extensions%20of%20Topological%20Algebras/topological%20algebra&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=topological%20algebra&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]
* [[w:en:Topological space|Topological space]] - Definition: [[Norms, metrics, topology#Definition:_topology|Topology]]
* '''[[Norms, metrics, topology|Norms, metrics, topology]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Norms,%20metrics,%20topology&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Norms,%20metrics,%20topology&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]
=== Chapter 3 - Complex Derivative ===
* '''[[Holomorphic function|Holomorphic function]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Holomorphic%20function&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Holomorphic%20function&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]
* '''[[Complex Analysis/Partial derivative|Partial Derivative]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Partial%20derivative&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Partial%20Derivative&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]
* '''[[Cauchy-Riemann-Differential equation|Cauchy-Riemann-Differential equation]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Cauchy-Riemann-Differential%20equation&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Cauchy-Riemann-Differential%20equation&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]
* '''[[Complex Analysis/Application of Cauchy-Riemann Equations|Application of Cauchy-Riemann Equations]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Application%20of%20Cauchy-Riemann%20Equations&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Application%20of%20Cauchy-Riemann%20Equations&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]
=== Chapter 4 - Curves and Line Integrals ===
* '''[[Line integral|Line integral in <math>\mathbb{R}^n</math>]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Line%20integral&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Line%20integral&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]
* '''[[/Curves/|Curves]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Complex%20Analysis/Curves&author=Complex_Analysis&language=en&audioslide=yes&shorttitle=Curves&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]
** [[w:en:Holomorphic function|Wikipedia: holomorphic function]]
** [[w:en:Integral|Wikipedia:Integral ]]
* '''[[Complex_Analysis/Paths|Paths]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Complex%20Analysis/Paths&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Paths&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]
* '''[[Path Integral|Path Integral]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Path%20Integral&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Path%20Integral&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]
* [[w:en:Curve integral |Wikipedia: Curve integral]]
* [[w:en:Continuity|Continuity]] and [[w:en:Limit of a sequence|Limit of a sequence]]
* '''[[Complex Analysis/Trace|Trace of Curve]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Trace&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Trace&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]
=== Chapter 5 - Holomorphic Functions ===
* '''[[Holomorphic function|Holomorphic function]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Holomorphic%20function&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Holomorphic%20function&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]
** [[Holomorphism/Criteria|Criteria]] - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Holomorphism/Criteria&author=Course:Complex_Analysis&language=en&audioslide=yes&shorttitle=Criteria&coursetitle=Complex_Analysis slideset]) [[File:Wiki2Reveal Logo.png|35px]]
** [[w:en:Holomorphic_function#.C3.84quivalent_properties_of_holomorphic_functions_of_one_variable|Wikipedia: Holomorphic function criteria]]
** [[/Differences from real differentiability/]]
** [[w:Conformal_mapping|conformal mappings]]<math>(\ast)</math>,
** [[Complex Analysis/Inequalities|Inequalities]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Inequalities&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Inequalities&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]
** [[Complex Analysis/rectifiable curve|rectifiable curve]] - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/rectifiable%20curve&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=rectifiable%20curve&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]
* '''[[Complex Analysis/Curve Integral|Curve Integral]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Curve%20Integral&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Curve%20Integral&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]
* '''[[Complex Analysis/Path of Integration|Path of Integration]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Path%20of%20Integration&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Path%20of%20Integration&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]
* '''[[Complex Analysis/Goursat's Lemma (Details)|Goursat's Lemma (Details)]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Goursat's%20Lemma%20(Details)&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Goursat's%20Lemma%20(Details)&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]
* '''[[Complex Analysis/Cauchy's Integral Theorem for Disks|Cauchy's Integral Theorem for Disks]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Cauchy's%20Integral%20Theorem%20for%20Disks&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Cauchy's%20Integral%20Theorem%20for%20Disks&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]
* '''[[Complex Analysis/Identity Theorem|Identity Theorem]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Identity%20Theorem&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Identity%20Theorem&coursetitle= Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]
* '''[[Complex Analysis/Liouville's Theorem|Liouville's Theorem]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Liouville's%20Theorem&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Liouville's%20Theorem&coursetitle= Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]
=== Complex Analysis Part 2 ===
* '''[[Complex Analysis/Chain|Chain]]''' - [https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Chain&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Chain&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]
* '''[[Complex Analysis/cycle|cycle]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/cycle&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=cycle&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]
* '''[[Laurent Series|Laurent Series]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Laurent%20Series&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Laurent%20Series&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]
* '''[[Cauchy Integral Theorem|Cauchy Integral Theorem]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Cauchy%20Integral%20Theorem&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Cauchy%20Integral%20Theorem&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]
* '''[[Cauchy's integral formula|Cauchy's integral formula]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Cauchy's%20integral%20formula&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Cauchy's%20integral%20formula&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]
*[[Complex Analysis/Example Computation with Laurent Series|Example Computation with Laurent Series]]
* '''[[Complex Analysics/Maximum Principle|Maximum Principle]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysics/Maximum%20Principle&author=Complex%20Analysics&language=en&audioslide=yes&shorttitle=Maximum%20Principle&coursetitle=Complex%20Analysics Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]
* '''[[Complex Analysis/Openness theorem theorem of territorial loyalty|Openness theorem theorem of territorial loyalty]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Openness%20theorem%20theorem%20of%20territorial%20loyalty&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Openness%20theorem%20theorem%20of%20territorial%20loyalty&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]
===Singularity and Residues - Part 3===
* '''[[Complex Analysis/Singularities|Singularities]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Singularities&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Singularities&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]
* '''[[Complex Analysis/Example - exp(1/z)|Example - exp(1/z)-essential singularity]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Example%20-%20exp(1/z)&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=z)&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]
* '''[[Complex Analysis/Residuals|Residuals]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Residuals&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Residuals&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]
* [[Complex Analysis/development in Laurent series|development in Laurent series]],
* [[Complex Analysis/Isolated singularity|Isolated singularity]],
* [[Complex Analysis/decomposition theorem|decomposition theorem]],
* [[Casorati-Weierstrass theorem|Casorati-Weierstrass theorem]],
==Lectures==
* [[/Cauchy-Riemann equations/]]
* [[Cauchy Theorem for a triangle]]
* [[Complex analytic function]]
* [[Complex Numbers]]
* [[Divergent series]]
* [[Estimation lemma]]
* [[Fourier series]]
* [[Fourier transform]]
* [[Fourier transforms]]
* [[Laplace transform]]
* [[Riemann hypothesis]]
* [[The Real and Complex Number System]]
* [[Warping functions]]
==Sample exams==
[[/Sample Midterm Exam 1/]]
[[/Sample Midterm Exam 2/]]
==See also==
* [[Boundary Value Problems]]
* [[Introduction to Elasticity]]
* [[The Prime Sequence Problem]]
* [[Wikipedia: Complex analysis]]
*[[Complex number]]
[[Category:Complex analysis| ]]
[[Category:Mathematics courses]]
[[Category:Mathematics]]
<noinclude>
[[de:Kurs:Funktionentheorie]]
</noinclude>
3zriehmhtolzyxhcd81aagdcljzla92
Caregiving and dementia/Topics/Nursing
0
132932
2694084
1042371
2025-01-02T10:46:04Z
202.165.232.168
/* References */
2694084
wikitext
text/x-wiki
<noinclude>
{{Caregiving and dementia/NPA}}
This NPA focuses on nursing in the [[caregiving and dementia|care of people with dementia]].
{{rightTOC}}
==Overview==
A review of 59 publications about nursing competencies and dementia synthesised 10 dementia care competencies (Traynor, Inoue, & Crookes, 2011):
# Understanding Dementia
# Recognising Dementia
# Effective Communication
# Assisting with Daily Living Activities;
# Promoting a Positive Environment
# Ethical and Person-Centred Care
# Therapeutic Work (Interventions)
# Responding the needs of Family Carers
# Preventative Work and Health Promotion and
#Special Needs Groups.
There were also five levels of practice:
# Novice
# Beginner
# Competent
# Proficient and
# Expert and no care setting
==Competency standards for dementia care==
As there is no dementia competency framework relevant across care settings or levels of practice, an empirical study will develop a multi-disciplinary dementia competency framework relevant across care settings and levels of practice to ensure the healthcare workforce can effectively deliver services to people with dementia and their carers (Traynor et al., 2011):
* [http://www.dementia.unsw.edu.au/images/dcrc/output-files/259-nc1_web_summary.pdf Competency standards for dementia care]
https://neurox.us/dementiaprogram/#government-organization
==References==
Traynor, V., Inoue, K., & Crookes, P. (2011). [http://onlinelibrary.wiley.com/doi/10.1111/j.1365-2702.2010.03511.x/abstract Literature review: understanding nursing competence in dementia care]. ''Journal of Clinical Nursing'', ''20'', 1948–1960. doi: 10.1111/j.1365-2702.2010.03511.x
[[Category:Caregiving and dementia/Topics]]
</noinclude>
7rmf2lkmziay9jmieqhy2iclf0oscc1
Acceleration field
0
164331
2694072
2286130
2025-01-02T06:33:24Z
Fedosin
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'''Acceleration field''' is a two-component vector field, describing in a covariant way the [[four-acceleration]] of individual particles and the [[four-force]] that occurs in systems with multiple closely interacting particles. The acceleration field is a component of the [[Physics/Essays/Fedosin/General field |general field]], which is represented in the Lagrangian and Hamiltonian of an arbitrary physical system by the term with the energy of particles’ motion and the term with the field energy. <ref name="ko"> Fedosin S.G. [http://www.oalib.com/paper/5263035#.VuFYxn2LQsY The Concept of the General Force Vector Field]. OALib Journal, Vol. 3, pp. 1-15 (2016), e2459. http://dx.doi.org/10.4236/oalib.1102459. </ref> <ref> Fedosin S.G. Two components of the macroscopic general field. Reports in Advances of Physical Sciences, Vol. 1, No. 2, 1750002, 9 pages (2017). http://dx.doi.org/10.1142/S2424942417500025. </ref>
The acceleration field is included in most [[equation of vector field |equations of vector field]]. Moreover, the acceleration field enters into the equation of motion through the [[acceleration tensor]] and into the equation for the metric through the [[acceleration stress-energy tensor]].
The acceleration field was presented by [[user:Fedosin |Sergey Fedosin]] within the framework of the [[Physics/Essays/Fedosin/Metric theory of relativity|metric theory of relativity]] and [[covariant theory of gravitation]], and the equations of this field were obtained as a consequence of the [[w:principle of least action |principle of least action]]. <ref name="pr"> Fedosin S.G. [http://vixra.org/abs/1406.0135 The procedure of finding the stress-energy tensor and vector field equations of any form]. Advanced Studies in Theoretical Physics, Vol. 8, No. 18, pp. 771-779 (2014). http://dx.doi.org/10.12988/astp.2014.47101. </ref> <ref name="ac"> [[user:Fedosin | Fedosin S.G.]] [http://journals.yu.edu.jo/jjp/Vol9No1Contents2016.html About the cosmological constant, acceleration field, pressure field and energy.] Jordan Journal of Physics. Vol. 9, No. 1, pp. 1-30 (2016). http://dx.doi.org/10.5281/zenodo.889304. </ref>
== Mathematical description ==
The 4-potential of the acceleration field is expressed in terms of the scalar <math>~ \vartheta </math> and vector <math>~ \mathbf {U} </math> potentials:
:<math>~U_\mu = \left(\frac {\vartheta }{c},- \mathbf {U} \right) .</math>
The antisymmetric [[acceleration tensor]] is calculated with the help of the 4-curl of the 4-potential:
:<math>~ u_{\mu \nu} = \nabla_\mu U_\nu - \nabla_\nu U_\mu = \partial_\mu U_\nu - \partial_\nu U_\mu . </math>
The acceleration tensor components are the components of the field strength <math>~\mathbf {S} </math> and the components of the solenoidal vector <math>~\mathbf {N} </math>:
:<math> ~ u_{\mu \nu}= \begin{vmatrix} 0 & \frac {S_x}{ c} & \frac {S_y}{ c} & \frac {S_z}{ c} \\ -\frac {S_x}{ c} & 0 & - N_{z} & N_{y} \\ -\frac {S_y}{ c} & N_{z} & 0 & -N_{x} \\ -\frac {S_z}{ c}& -N_{y} & N_{x} & 0 \end{vmatrix}. </math>
We obtain the following:
:<math>~ \mathbf {S} = - \nabla \vartheta - \frac {\partial \mathbf {U}}{\partial t},\qquad\qquad \mathbf {N} = \nabla \times \mathbf {U}.\qquad\qquad (1) </math>
In the general case the scalar and vector potentials are found by solving the wave equations for the acceleration field potentials.
=== Action, Lagrangian and energy ===
In the covariant theory of gravitation the 4-potential <math>~U_\mu </math> of the acceleration field is part of the 4-potential of the [[Physics/Essays/Fedosin/General field |general field]] <math>~ s_\mu</math>, which is the sum of the 4-potentials of particular fields, such as the electromagnetic and gravitational fields, acceleration field, [[pressure field]], [[dissipation field]], strong interaction field, weak interaction field and other vector fields, acting on the matter and its particles. All of these fields are somehow represented in the matter, so that the 4-potential <math>~ s_\mu</math> cannot consist of only one 4-potential <math>~U_\mu </math>.
The energy density of interaction of the general field and the matter is given by the product of the 4-potential of the general field and the mass 4-current: <math>~ s_\mu J^\mu </math>. We obtain the general field tensor from the 4-potential of the general field, using the 4-curl:
:<math>~ s_{\mu \nu} =\nabla_\mu s_\nu - \nabla_\nu s_\mu.</math>
The tensor invariant in the form <math>~ s_{\mu \nu} s^{\mu \nu} </math> is up to a constant factor proportional to the energy density of the general field. As a result, the action function, which contains the scalar curvature <math>~R</math> and the cosmological constant <math>~ \Lambda </math>, is given by the expression: <ref name="ko"/>
:<math>~S =\int {L dt}=\int (kR-2k \Lambda - \frac {1}{c}s_\mu J^\mu - \frac {c}{16 \pi \varpi} s_{\mu\nu}s^{\mu\nu} ) \sqrt {-g}d\Sigma,</math>
where <math>~L </math> is the Lagrange function or Lagrangian; <math>~dt </math> is the time differential of the coordinate reference system; <math>~k </math> and <math>~ \varpi </math> are the constants to be determined; <math>~c </math> is the speed of light as a measure of the propagation speed of the electromagnetic and gravitational interactions; <math>~\sqrt {-g}d\Sigma= \sqrt {-g} c dt dx^1 dx^2 dx^3</math> is the invariant 4-volume expressed in terms of the differential of the time coordinate <math>~ dx^0=cdt </math>, the product <math>~ dx^1 dx^2 dx^3 </math> of differentials of the space coordinates and the square root <math>~\sqrt {-g} </math> of the determinant <math>~g </math> of the metric tensor, taken with a negative sign.
The variation of the action function gives the general field equations, the four-dimensional equation of motion and the equation for the metric. Since the acceleration field is the general field component, then from the general field equations the corresponding equations of the acceleration field are derived.
Given the gauge condition of the cosmological constant in the form
:<math>~ c k \Lambda = - s_\mu J^\mu ,</math>
is met, the system energy does not depend on the term with the scalar curvature and is uniquely determined: <ref name="ac"/>
:<math>~E = \int {( s_0 J^0 + \frac {c^2 }{16 \pi \varpi } s_{ \mu\nu} s^{ \mu\nu} ) \sqrt {-g} dx^1 dx^2 dx^3}, </math>
where <math>~ s_0 </math> and <math>~ J^0</math> denote the time components of the 4-vectors <math>~ s_{\mu } </math> and <math>~ J^{\mu } </math>.
The system’s 4-momentum is given by the formula:
:<math>~p^\mu = \left( \frac {E}{c}{,} \mathbf {p}\right) = \left( \frac {E}{c}{,} \frac {E}{c^2}\mathbf {v} \right) , </math>
where <math>~ \mathbf {p}</math> and <math>~ \mathbf {v}</math> denote the system’s momentum and the velocity of the system’s center of momentum.
=== Equations ===
The four-dimensional equations of the acceleration field are similar in their form to Maxwell equations and are as follows:
:<math> \nabla_\sigma u_{\mu \nu}+\nabla_\mu u_{\nu \sigma}+\nabla_\nu u_{\sigma \mu}=\frac{\partial u_{\mu \nu}}{\partial x^\sigma} + \frac{\partial u_{\nu \sigma}}{\partial x^\mu} + \frac{\partial u_{\sigma \mu}}{\partial x^\nu} = 0. </math>
:<math>~ \nabla_\nu u^{\mu \nu} = - \frac{4 \pi \eta }{c^2} J^\mu, </math>
where <math>J^\mu = \rho_{0} u^\mu </math> is the mass 4-current, <math> \rho_{0}</math> is the mass density in the co-moving reference frame, <math> u^\mu </math> is the 4-velocity of the matter unit, <math>~ \eta </math> is a constant, which is determined in each problem, and it is supposed that there is an equilibrium between all fields in the observed physical system.
The gauge condition of the 4-potential of the acceleration field:
:<math>~ \nabla^\mu U_{\mu} =0 . </math>
If the second equation with the field source is written with the covariant index in the following form:
:<math>~ \nabla^\nu u_{\mu \nu} = - \frac{4 \pi \eta }{c^2} J_\mu, </math>
then after substituting here the expression for the acceleration tensor <math> u_{\mu \nu} </math> through the 4-potential <math> ~ U_\mu </math> of the acceleration field we obtain the wave equation for calculating the potentials of the acceleration field:
:<math>~ \nabla^\nu \nabla_\nu U_\mu + R_{\mu \nu} U^\nu = \frac{4 \pi \eta }{c^2} J_\mu, </math>
where <math>~ R_{\mu \nu} </math> is the Ricci tensor.
The continuity equation in curved space-time is:
:<math>~ R_{ \mu \alpha } u^{\mu \alpha }= \frac {4 \pi \eta }{c^2} \nabla_{\alpha}J^{\alpha}.</math>
In Minkowski space of the special theory of relativity, the Ricci tensor is set to zero, the form of the acceleration field equations is simplified and they can be expressed in terms of the field strength <math>~\mathbf {S} </math> and the solenoidal vector <math>~\mathbf {N} </math>:
:<math>~ \nabla \cdot \mathbf{S} = 4 \pi \eta \gamma \rho_0, \qquad\qquad \nabla \times \mathbf{N} = \frac{1}{c^2} \left( 4 \pi \eta \mathbf{J} + \frac{\partial \mathbf{S}} {\partial t} \right), </math>
:<math>~ \nabla \times \mathbf{S} = - \frac{\partial \mathbf{N} } {\partial t} , \qquad\qquad \nabla \cdot \mathbf{N} = 0 .</math>
where <math>~ \gamma = \frac {1}{\sqrt{1 - {v^2 \over c^2}}} </math> is the Lorentz factor, <math>~ \mathbf{J}= \gamma \rho_0 \mathbf{v }</math> is the mass current density, <math>~ \mathbf{v } </math> is the velocity of the matter unit.
The wave equation is also simplified and can be written separately for the scalar and vector potentials:
:<math>~ \partial^\nu \partial_\nu \vartheta = \frac {1}{c^2}\frac{\partial^2 \vartheta }{\partial t^2 } -\Delta \vartheta = 4 \pi \eta \gamma \rho_0, \qquad\qquad (2) </math>
:<math>~ \partial^\nu \partial_\nu \mathbf{U} =\frac {1}{c^2}\frac{\partial^2 \mathbf{U} }{\partial t^2 } -\Delta \mathbf{U}= \frac {4 \pi \eta}{c^2} \mathbf{J}. \qquad\qquad (3) </math>
The equation of motion of the matter unit in the general field is given by the formula:
:<math>~ s_{\mu \nu} J^\nu =0 </math>.
Since <math>~ J^\nu = \rho_0 u^\nu </math>, and the general field tensor is expressed in terms of the tensors of particular fields, then the equation of motion can be represented with the help of these tensors:
:<math>~ - u_{\mu \nu} J^\nu =F_{\mu \nu} j^\nu + \Phi_{\mu \nu} J^\nu + f_{\mu \nu} J^\nu + h_{\mu \nu} J^\nu + \gamma_{\mu \nu} J^\nu + w_{\mu \nu} J^\nu .</math>
Here <math>~ F_{\mu \nu}</math> is the [[w:electromagnetic tensor |electromagnetic tensor]], <math>~ j^\nu </math> is the charge [[4-current]], <math>~ \Phi_{\mu \nu}</math> is the [[Physics/Essays/Fedosin/Gravitational tensor |gravitational tensor]], <math>~ f_{\mu \nu}</math> is the [[pressure field tensor]], <math>~ h_{\mu \nu}</math> is the [[dissipation field tensor]], <math>~ \gamma_{\mu \nu}</math> is the strong interaction field tensor, <math>~ w_{\mu \nu}</math> is the weak interaction field tensor.
=== The stress-energy tensor ===
The [[acceleration stress-energy tensor]] is calculated with the help of the acceleration tensor:
:<math>~ B^{ik} = \frac{c^2} {4 \pi \eta }\left( -g^{im} u_{n m} u^{n k}+ \frac{1} {4} g^{ik} u_{m r} u^{m r}\right) </math>.
We find as part of the tensor <math>~ B^{ik}</math> the 3-vector of the energy-momentum flux <math>~\mathbf {K} </math>, which is similar in its meaning to the [[w:Poynting vector |Poynting vector]] and the [[Heaviside vector]]. The vector <math>~\mathbf {K} </math> can be represented through the vector product of the field strength <math>~ \mathbf {S} </math> and the solenoidal vector <math>~ \mathbf {N} </math>:
:<math>~ \mathbf {K}=c B^{0i} = \frac {c^2}{4 \pi \eta }[\mathbf {S}\times \mathbf {N}],</math>
here the index is <math>~ i=1,2,3.</math>
The covariant derivative of the stress-energy tensor of the acceleration field with mixed indices specifies the [[4-force]] density:
:<math> ~ f_\alpha = \nabla_\beta {B_\alpha}^\beta = - u_{\alpha k} J^k = - \rho_0 u_{\alpha k}u^k = \rho_0 \frac {DU_\alpha }{D \tau}- J^k \nabla_\alpha U_k = \rho_0 \frac {dU_\alpha }{d \tau} - J^k \partial_\alpha U_k ,\qquad \qquad (4)</math>
where <math>~ D \tau </math> denotes the proper time differential in the curved spacetime.
The stress-energy tensor of the acceleration field is part of the stress-energy tensor of the general field <math>~ T^{ik} </math>:
:<math>~ T^{ik}= W^{ik}+ U^{ik}+ B^{ik}+ P^{ik} + Q^{ik}+ L^{ik}+ A^{ik}, </math>
where <math>~ W^{ik} </math> is the [[w:electromagnetic stress–energy tensor |electromagnetic stress–energy tensor]], <math>~ U^{ik}</math> is the [[gravitational stress-energy tensor]], <math>~ P^{ik}</math> is the [[pressure stress-energy tensor]], <math>~ Q^{ik}</math> is the [[dissipation stress-energy tensor]], <math>~ L^{ik}</math> is the strong interaction stress-energy tensor, <math>~ A^{ik} </math> is the weak interaction stress-energy tensor.
Through the tensor <math>~ T^{ik} </math> the stress-energy tensor of the acceleration field enters into the equation for the metric:
:<math>~ R^{ik} - \frac{1} {4 }g^{ik}R = \frac{8 \pi G \beta }{ c^4} T^{ik}, </math>
where <math>~ R^{ik} </math> is the Ricci tensor, <math>~ G </math> is the [[gravitational constant]], <math>~ \beta </math> is a certain constant, and the gauge condition of the cosmological constant is used.
== Specific solutions for the acceleration field functions ==
The four-potential of any vector field, the global vector potential of which is equal to zero in the proper reference frame K', that is, in the center-of-momentum frame, in case of rectilinear motion in the laboratory reference frame K, can be presented as follows: <ref name="pr"/> <ref> Fedosin S.G. Equations of Motion in the Theory of Relativistic Vector Fields. International Letters of Chemistry, Physics and Astronomy, Vol. 83, pp. 12-30 (2019). https://doi.org/10.18052/www.scipress.com/ILCPA.83.12. </ref>
:<math>~ L_{\mu L} = \frac { k_f \varepsilon }{\rho_0 c^2} u_{\mu L},</math>
where <math>~ k_f = \frac {\rho_0}{\rho_{0q}}</math> is for the electromagnetic field and <math>~ k_f = 1</math> for the remaining fields; <math> ~ \rho_{0}</math> and <math> ~\rho_{0q}</math> are the invariant mass density and the charge density in the comoving reference frame, respectively; <math>~ \varepsilon </math> is the invariant energy density of the interaction, calculated as product of the four-potential of the field and the corresponding four-current; <math>~ u_{\mu L} </math> is the covariant four-velocity that determines the motion of the center of momentum of the physical system in K.
In the [[special relativity]] (SR), in the center-of-momentum frame K' the energy density is <math>~ \varepsilon = \gamma' \rho_0 c^2 </math>, where <math>~ \gamma' </math> is the Lorentz factor, and for the acceleration field, while the physical system is moving in K, the four-potential of the acceleration field will equal <math>~ U_{\mu L}= \gamma' u_{\mu L}</math>.
In case when the physical system is stationary in K, we will have <math>~ u_{\mu L} = (c,0,0,0) </math>, and consequently, the scalar potential will be <math>~ \vartheta = \gamma' c^2 </math>. If in the physical system, on the average, there are directed fluxes of matter or rotation of matter, the vector potential <math>~ \mathbf {U} </math> of the acceleration field is no longer equal to zero.
If the four-potential <math>~ U'_{\nu}</math> of acceleration field in K' is known, then in the laboratory reference frame K the four-potential is determined using the matrix <math>~ M_{\mu}^{\ \nu} </math> connecting the coordinates and time of both frames: <ref name="it"> Fedosin S.G. [http://dergipark.org.tr/gujs/issue/45480/435567 The Integral Theorem of the Field Energy.] Gazi University Journal of Science. Vol. 32, No. 2, pp. 686-703 (2019). http://dx.doi.org/10.5281/zenodo.3252783. </ref>
:<math>~ U_{\mu L}= M_{\mu}^{\ \nu} U'_{\nu}.</math>
In the special case of the system’s motion at the constant velocity <math>~ M_{\mu}^{\ \nu}</math> represents the Lorentz transformation matrix.
=== Ideally solid particle ===
In the approximation, when a particle is regarded as an ideally solid object, the matter inside the particle is motionless. It means that the Lorentz factor <math>~ \gamma' </math> of this matter in the center-of-momentum frame K' is equal to unity, so that the four-potential of the acceleration field becomes equal to the four-velocity of motion of the center of momentum:
:<math>~ U_\mu = u_\mu. </math>
In the SR, the expression for 4-velocity is simplified and we can write:
:<math>~U_\mu = \left( \frac {\vartheta }{c},- \mathbf {U} \right) = u_\mu = \left(\gamma c, - \gamma \mathbf {v} \right).</math>
The acceleration tensor components according to (1) will equal:
:<math>~ \mathbf {S} = - c^2 \nabla \gamma - \frac {\partial (\gamma \mathbf { v })}{\partial t},\qquad\qquad \mathbf {N} = \nabla \times (\gamma \mathbf { v }). </math>
Since in the solid-state motion equation for the four-acceleration with a covariant index <math>~ a_\mu </math> the relation holds
:<math>~ \rho_0 a_\mu = \rho_0 \frac {Du_\mu }{D \tau}= - u_{\mu \nu} J^\nu = - \rho_0 u_{\mu \nu} u^\nu, </math>
then in SR we obtain the following:
:<math>~ \frac {Du_\mu }{D \tau}= \frac {du_\mu }{d \tau} =\gamma \frac {du_\mu }{dt}, \qquad\qquad u^\nu =\left(\gamma c, \gamma \mathbf {v} \right), </math>
and the equations for the Lorentz factor <math>~ \gamma </math> and for the 3-acceleration <math>~ a= \frac {d \mathbf { v }}{dt} </math>:
:<math>~ \frac {d \gamma }{dt}= - \frac {1 }{c^2} \mathbf {S}\cdot \mathbf { v }, \qquad (5) \qquad \frac {d (\gamma \mathbf { v })}{dt}= \gamma \mathbf { a }+ \frac {d \gamma}{dt}\mathbf { v } = - \mathbf {S}- [\mathbf { v }\times \mathbf {N}]. \qquad (6) </math>
Multiplying equation (6) by the velocity <math>~ \mathbf { v }</math>, substituting the quantity <math>~ \mathbf {S}\cdot \mathbf { v } </math> from equation (5) to (6), taking into account relation <math>~\gamma^{-2}=1 - {v^2 \over c^2},</math> we find the well-known expression for the derivative of the Lorentz factor using the scalar product of the velocity and acceleration in SR:
:<math>~ \gamma^3 \mathbf {v}\cdot \mathbf { a }=c^2 \frac {d \gamma }{dt}.</math>
We can prove the validity of equation (6) by substituting in its right-hand side the expression for the strength and solenoidal vector:
:<math>~ \frac {d (\gamma \mathbf { v })}{dt}= c^2 \nabla \gamma + \frac {\partial (\gamma \mathbf { v })}{\partial t} - \mathbf { v }\times [ \nabla \times (\gamma \mathbf { v }) ] . \qquad\qquad (7) </math>
Indeed, the use of the [[w:material derivative |material derivative]] gives the following:
:<math>~ \frac {d (\gamma \mathbf { v })}{dt}= \frac {\partial (\gamma \mathbf { v })}{\partial t} + (\mathbf { v } \cdot \nabla) (\gamma \mathbf { v }) = \frac {\partial (\gamma \mathbf { v })}{\partial t}+\gamma (\mathbf { v } \cdot \nabla) \mathbf { v } + \mathbf { v } (\mathbf { v } \cdot \nabla\gamma) .</math>
In addition
:<math>~ - \mathbf { v }\times [ \nabla \times (\gamma \mathbf { v }) ] = - \gamma \mathbf { v }\times [ \nabla \times \mathbf { v } ] - \mathbf { v }\times [ \nabla \gamma \times \mathbf { v }] = -\frac {\gamma }{2} \nabla v^2 + \gamma (\mathbf { v } \cdot \nabla) \mathbf { v } - v^2 \nabla \gamma + \mathbf { v } (\mathbf { v } \cdot \nabla\gamma) .</math>
Substituting these relations in (7), taking into account the expression <math>~ \gamma^{-2}=1 - {v^2 \over c^2},</math> we obtain the identity:
:<math>~ c^2 \nabla \gamma - \frac {\gamma }{2} \nabla v^2 - v^2 \nabla \gamma =0 .</math>
If the components of the particle velocity are the functions of time and they do not directly depend on the space coordinates, then the solenoidal vector <math>~ \mathbf { N }</math> vanishes in such a motion.
In the SR <math>~ E = \gamma m c^2 </math> is the relativistic energy, <math>~ \mathbf p = \gamma m \mathbf v </math> is the 3-vector of relativistic momentum. If the mass <math>~ m </math> of a particle is constant, then multiplying (7) by the mass, we arrive to following equation for the force:
:<math>~ \mathbf F= \frac {d \mathbf p }{dt}= \nabla E + \frac {\partial \mathbf p }{\partial t} - \mathbf { v }\times [ \nabla \times \mathbf p ] . </math>
=== Rotation of a particle ===
For a small ideally solid particle, we can neglect the motion of the matter inside the particle and can assume that the four-potential of the acceleration field is equal to the four-velocity of the particle’s center of momentum.
Let us assume that the particle rotates about the axis OZ of the coordinate system at the distance <math>~ \rho = \sqrt {x^2 +y^2} </math> from the axis at the constant angular velocity <math>~ \omega</math> counterclockwise, as viewed from the side, in which the OZ axis is directed. Then we can assume that the linear velocity of the particle depends only on the coordinates <math>~ x</math> and <math>~ y</math>, and for the velocity’s projections on the axes of the coordinate system we can write: <math>~ \mathbf v = (-\omega y, \omega x, 0) </math>, while the square of the velocity equals <math>~ v^2 = \omega^2 (x^2 + y^2) </math>. For the Lorentz factor in the SR we obtain the following:
:<math>~ \gamma = \frac {1}{\sqrt {1- \frac { v^2}{ c^2}}} = \frac {1}{\sqrt {1- \frac { \omega^2 (x^2 + y^2)}{ c^2}}} . </math>
With this in mind, the potentials and field strengths of the acceleration field can be written as follows:
:<math>~ \vartheta = \gamma c^2, \qquad \mathbf {U} = \gamma \mathbf {v}. </math>
:<math>~ \mathbf {S} = - \nabla \vartheta - \frac {\partial \mathbf {U} }{\partial t}= \left( -\gamma^3 \omega^2 x, -\gamma^3 \omega^2 y, 0 \right). </math>
:<math>~ \mathbf {N} = \nabla \times \mathbf {U} = \left( 0, 0, \gamma \omega +\gamma^3 \omega \right). </math>
If we substitute <math>~ \gamma </math>, <math>~ \mathbf v </math>, <math>~\mathbf {S} </math> and <math>~\mathbf {N} </math> in (6), we can determine the acceleration components of the particle and the acceleration amplitude:
:<math>~ \mathbf {a} = \left( - \omega^2 x , -\omega^2 y, 0 \right). </math>
:<math>~ a = \sqrt {a^2_x +a^2_y +a^2_z} = \omega^2 \sqrt {x^2 +y^2}= \omega^2 \rho =\omega v = \frac {v^2} {\rho }. </math>
The acceleration is directed towards the center of rotation and represents [[centripetal acceleration]]. Using now the classic vector description, we have also for the time and coordinates of reference frame at the center of rotation:
:<math>~ \vec \rho = (x, y, 0) , \qquad \vec \omega = \frac {\vec {d\varphi} }{dt} =(0, 0, \omega) , </math>
:<math>~ \mathbf {v} = [\vec \omega \times \vec \rho] , \qquad \mathbf {a} = [\vec \omega \times \mathbf {v}] = [\vec \omega \times [\vec \omega \times \vec \rho]] = \vec \omega (\vec \omega \cdot \vec \rho) - \vec \rho (\vec \omega \cdot \vec \omega) = - \omega^2 \vec \rho , </math>
where <math>~ \rho </math> and <math>~ \varphi </math> are two coordinates of the [[Coordinate systems#Cylindrical coordinates.5B4.5D |cylindrical coordinate system]], <math>~ \vec \rho </math> is the vector from the center of rotation to the particle, <math>~ \vec {d\varphi}</math> is the axial vector of the differential of the rotation angle directed along the axis OZ.
As we can see, in case of such a motion with acceleration the vector product <math>~ [\mathbf {S}\times \mathbf {N}]</math> is not equal to zero, just as the three-vector <math>~ \mathbf {K}</math> of the energy-momentum flux of the acceleration field inside the particle.
=== The system of particles ===
Due to interaction of a number of particles with each other by means of various fields, including interaction at a distance without direct contact, the acceleration field in the matter changes and is different from the acceleration field of individual particles at the observation point. As a result, the density of the 4-force in the system of particles is given by the strength and the solenoidal vector, which represent the typical average characteristics of the matter motion. For example, in a gravitationally bound system there is a radial gradient of the vector <math>~ \mathbf { S },</math> and if the system is moving or rotating, there is a vector <math>~ \mathbf { N }.</math> From (4) there follows the general expression for the the density of the 4-force with covariant index:
:<math> ~ f_\nu = \rho_0 \frac {cdt}{ds}\left(-\frac {1}{c} \mathbf{S} \cdot \mathbf{v}{,} \qquad \mathbf{S}+[\mathbf{v} \times \mathbf{N}] \right),</math>
where <math> ~ ds </math> denotes a four-dimensional space-time interval.
For a stationary case, when the potentials of the acceleration field are independent of time, under the assumption that <math>~ \vartheta = \gamma c^2, </math> wave equation (2) for the scalar potential in the SR is transformed into the equation:
:<math>~ \Delta \gamma= - \frac {4 \pi \eta \gamma \rho_0}{c^2}. </math>
The solution of this equation for a fixed sphere with the particles randomly moving in it has the form: <ref name="ie"> Fedosin S.G. [http://vixra.org/abs/1403.0973 The Integral Energy-Momentum 4-Vector and Analysis of 4/3 Problem Based on the Pressure Field and Acceleration Field.] American Journal of Modern Physics. Vol. 3, No. 4, pp. 152-167 (2014). http://dx.doi.org/10.11648/j.ajmp.20140304.12.</ref>
:<math>~\gamma= \frac {c \gamma_c }{r \sqrt {4 \pi \eta \rho_0} } \sin \left(\frac {r}{c}\sqrt {4 \pi \eta \rho_0} \right) \approx \gamma_c - \frac {2 \pi \eta \rho_0 r^2 \gamma_c }{3 c^2}.</math>
where <math>~ \gamma_c = \frac {1}{\sqrt{1 - {v^2_c \over c^2}}} </math> is the Lorentz factor for the velocities <math>~ v_c</math> of the particles in the center of the sphere, and due to the smallness of the argument the sine is expanded to the second order terms. From the formula it follows that the average velocities of the particles are maximal in the center and decrease when approaching the surface of the sphere.
In such a system, the scalar potential <math>~ \vartheta</math> becomes the function of the radius, and the vector potential <math>~ \mathbf {U} </math> and the solenoidal vector <math>~ \mathbf { N }</math> are equal to zero. The acceleration field strength <math>~\mathbf {S} </math> is found with the help of (1). Then we can calculate all the functions of the acceleration field, including the energy of particles in this field and the energy of the acceleration field itself. <ref> Fedosin S.G. [http://journals.yu.edu.jo/jjp/Vol8No1Contents2015.html Relativistic Energy and Mass in the Weak Field Limit.] Jordan Journal of Physics. Vol. 8, No. 1, pp. 1-16 (2015). http://dx.doi.org/10.5281/zenodo.889210.</ref> For cosmic bodies the main contribution to the four-acceleration in the matter makes the gravitational force and the pressure field.
At the same time the relativistic rest energy of the system is automatically derived, taking into account the motion of particles inside the sphere. For the system of particles with the acceleration field, pressure field, gravitational and electromagnetic fields the given approach allowed solving the 4/3 problem and showed where and in what form the energy of the system is contained. The relation for the acceleration field constant in this problem was found:
:<math>~\eta = 3G- \frac {3q^2}{4 \pi \varepsilon_0 m^2 },</math>
where <math>~ \varepsilon_0</math> is the [[electric constant]], <math>~q </math> and <math>~m </math> are the total charge and mass of the system.
The solution of the wave equation for the acceleration field within the system results in temperature distribution according to the formula: <ref name="ie"/>
:<math>~ T=T_c - \frac {\eta M_p M(r)}{3kr} ,</math>
where <math>~ T_c </math> is the temperature in the center, <math>~ M_p </math> is the mass of the particle, for which the mass of the proton is taken (for systems which are based on hydrogen or nucleons in atomic nuclei), <math>~ M(r) </math> is the mass of the system within the current radius <math>~ r </math>, <math>~ k</math> is the Boltzmann constant.
This dependence is well satisfied for a variety of space objects, including gas clouds and Bok globules, the Earth, the Sun and neutron stars.
In articles <ref> Fedosin S.G. [http://www.nrcresearchpress.com/doi/10.1139/cjp-2015-0593#.Vv3piZyLQsY Estimation of the physical parameters of planets and stars in the gravitational equilibrium model]. Canadian Journal of Physics, Vol. 94, No. 4, pp. 370-379 (2016). http://dx.doi.org/10.1139/cjp-2015-0593.</ref> <ref>Fedosin S.G. The generalized Poynting theorem for the general field and solution of the 4/3 problem. International Frontier Science Letters, Vol. 14, pp. 19-40 (2019). https://doi.org/10.18052/www.scipress.com/IFSL.14.19. </ref> the ratio of the field’s coefficients for the fields was specified as follows:
:<math>~\eta + \sigma = G - \frac {\rho^2_{0q}}{4 \pi \varepsilon_0 \rho^2_{0}},</math>
where <math> ~ \sigma </math> is the pressure field constant.
If we introduce the parameter <math> ~ \mu </math> as the number of nucleons per ionized gas particle, then the acceleration field constant is expressed as follows:
:<math>~\eta = \frac {3\gamma_c \mu G}{2+ 3 \gamma_c \mu }.</math>
For the temperature inside the cosmic bodies in the gravitational equilibrium model we find the dependence on the current radius:
:<math>~ T=T_c - \frac {4 \pi \eta m_u \rho_{0c}\gamma_c r^2}{9k}+ \frac {2 \pi \eta A m_u \gamma_c r^3}{9k} + \frac {2 \pi \eta B m_u \gamma_c r^4}{15k} ,</math>
where <math> ~ m_u </math> is the mass of one gas particle, which is taken as the [[w:unified atomic mass unit |unified atomic mass unit]], and the coefficients <math> ~ A </math> and <math> ~ B </math> are included into the dependence of the mass density on the radius in the relation <math> ~ \rho_0 = \rho_{0c}- Ar - Br^2. </math>
Under the assumption that the system’s typical particles have the mass <math> ~\stackrel{-}{m } = \mu m_u </math>, and that it is typical particles that define the temperature and pressure, for the acceleration field constant we obtain the following: <ref>Fedosin S.G. The binding energy and the total energy of a macroscopic body in the relativistic uniform model. Middle East Journal of Science, Vol. 5, Issue 1, pp. 46-62 (2019). http://dx.doi.org/10.23884/mejs.2019.5.1.06. </ref>
:<math>~ \eta = \frac {3}{5} \left( G- \frac {\rho^2_{0q}}{ 4 \pi \varepsilon_0 \rho^2_0 } \right) .</math>
The Lorentz factor of the particles in the center of the system is also determined: <ref name="en"> Fedosin S.G. Energy and metric gauging in the covariant theory of gravitation. Aksaray University Journal of Science and Engineering, Vol. 2, Issue 2, pp. 127-143 (2018). http://dx.doi.org/10.29002/asujse.433947. </ref>
: <math>~\gamma_c = \frac {1}{\sqrt {1- \frac { v^2_c }{c^2}}} \approx 1+ \frac { v^2_c }{2c^2} +\frac {3 v^4_c }{8c^4} \approx 1+ \frac {3 \eta m}{10 a c^2} \left( 1+\frac {9}{2\sqrt {14}} \right) + \frac {27 \eta^2 m^2}{200 a^2 c^4} \left( 1+\frac {9}{2\sqrt {14}} \right)^2 . </math>
The wave equation (3) for the vector potential of the acceleration field was used to represent the relativistic equation of the fluid’s motion in the form of the [[w:Navier–Stokes equations |Navier–Stokes equations]] in hydrodynamics and to describe the motion of the viscous compressible and charged fluid. <ref> Fedosin S.G. Four-Dimensional Equation of Motion for Viscous Compressible and Charged Fluid with Regard to the Acceleration Field, Pressure Field and Dissipation Field. International Journal of Thermodynamics. Vol. 18, No. 1, pp. 13-24 (2015). http://dx.doi.org/10.5541/ijot.5000034003.</ref>
Taking into account the acceleration field and pressure field, within the framework of the [[relativistic uniform system]], it is possible to refine the [[w:virial theorem |virial theorem]], which in the relativistic form is written as follows: <ref>Fedosin S.G. The virial theorem and the kinetic energy of particles of a macroscopic system in the general field concept. Continuum Mechanics and Thermodynamics. Vol. 29, Issue 2, pp. 361-371 (2017). https://dx.doi.org/10.1007/s00161-016-0536-8. </ref>
: <math>~ \langle W_k \rangle \approx - 0.6 \sum_{k=1}^N\langle\mathbf{F}_k\cdot\mathbf{r}_k\rangle ,</math>
where the value <math>~ W_k \approx \gamma_c T </math> exceeds the kinetic energy of the particles <math>~ T </math> by a factor equal to the Lorentz factor <math>~ \gamma_c </math> of the particles at the center of the system. Under normal conditions we can assume that <math>~ \gamma_c \approx 1 </math>, then we can see that in the virial theorem the kinetic energy is related to the potential energy not by the coefficient 0.5, but rather by the coefficient close to 0.6. The difference from the classical case arises due to considering the pressure field and the acceleration field of particles inside the system, while the derivative of the virial scalar function <math>~ G_v </math> is not equal to zero and should be considered as the [[w:material derivative |material derivative]].
An analysis of the integral theorem of generalized virial makes it possible to find, on the basis of field theory, a formula for the root-mean-square speed of typical particles of a system without using the notion of temperature: <ref> Fedosin S.G. [http://em.rdcu.be/wf/click?upn=lMZy1lernSJ7apc5DgYM8f7AyOIJlVFO4uFv7zUQtzk-3D_DUeisO4Ue44lkDmCnrWVhK-2BAxKrUexyqlYtsmkyhvEp5zr527MDdThwbadScvhwZehXbanab8i5hqRa42b-2FKYwacOeM4LKDJeJuGA15M9FWvYOfBgfon7Bqg2f55NFYGJfVGaGhl0ghU-2BkIJ9Hz4M6SMBYS-2Fr-2FWWaj9eTxv23CKo9d8nFmYAbMtBBskFuW9fupsvIvN5eyv-2Fk-2BUc7hiS15rRISs1jpNnRQpDtk2OE9Hr6mYYe5Y-2B8lunO9GwVRw07Y1mdAqqtEZ-2BQjk5xUwPnA-3D-3D The integral theorem of generalized virial in the relativistic uniform model]. Continuum Mechanics and Thermodynamics, Vol. 31, Issue 3, pp. 627-638 (2019). https://dx.doi.org/10.1007/s00161-018-0715-x.</ref>
:<math> v_\mathrm{rms} = c \sqrt{1- \frac {4 \pi \eta \rho_0 r^2}{c^2 \gamma^2_c \sin^2 {\left( \frac {r}{c} \sqrt {4 \pi \eta \rho_0} \right) } } } .</math>
The integral [[field energy theorem]] for acceleration field in a curved space-time is as follows:<ref name="it"/>
:<math>~ - \int { \left( \frac {8 \pi \eta }{c^2} U_\alpha J^\alpha + u_{\alpha \beta} u^{\alpha \beta} \right) \sqrt {-g} dx^1 dx^2 dx^3 } = \frac {2}{c} \frac {d}{dt} \left( \int { U^\alpha u_\alpha ^{\ 0} \sqrt {-g} dx^1 dx^2 dx^3} \right) + 2 \iint \limits_S {U^\alpha u_\alpha ^{\ k} n_k \sqrt {-g} dS} . </math>
In the relativistic uniform system, the scalar potential <math>~\vartheta </math> of the acceleration field is related to the scalar potential <math>~\wp </math> of the pressure field:
<ref>Fedosin S.G. [https://rdcu.be/ccV9o The potentials of the acceleration field and pressure field in rotating relativistic uniform system]. Continuum Mechanics and Thermodynamics, Vol. 33, Issue 3, pp. 817-834 (2021). https://doi.org/10.1007/s00161-020-00960-7. </ref>
:<math>~ \wp = \frac {\sigma (\vartheta -c^2)}{ \eta } = \frac {2 (\vartheta -c^2)}{ 3 }. </math>
The relativistic expression for pressure is as follows:
<math> p = \frac{2\rho c^2 (\gamma - 1) }{3}= \frac {2 \rho c^2 }{3} \left( \frac {1}{\sqrt {1- v^2/ c^2 }}-1 \right) \approx \frac {\rho v^2}{3}, </math>
where <math>\rho </math> is the mass density of moving matter, <math> c </math> is the speed of light, <math> \gamma =\frac {1}{\sqrt {1- v^2/ c^2 }} </math> is the [[w:Lorentz factor |Lorentz factor]].
In the limit of low velocities, this relationship turns into the standard formula of the [[w:kinetic theory of gases |kinetic theory of gases]].
== Other approaches ==
Studying the Lorentz covariance of the 4-force, Friedman and Scarr found incomplete covariance of the expression for the 4-force in the form <math>~ F^\mu = \frac {d p^\mu }{d \tau } . </math> <ref> Yaakov Friedman and Tzvi Scarr. [http://iopscience.iop.org/1742-6596/437/1/012009 Covariant Uniform Acceleration]. Journal of Physics: Conference Series Vol. 437 (2013) 012009 doi:10.1088/1742-6596/437/1/012009. </ref>
This led them to conclude that the four-acceleration in SR must be expressed with the help of a certain antisymmetric tensor <math>~ {A^\mu}_\nu </math>:
:<math>~c \frac { d u^\mu }{d \tau } = {A^\mu}_\nu u^\nu . </math>
Based on the analysis of various types of motion, they estimated the required values of the acceleration tensor components, thereby giving indirect definition to this tensor.
From comparison with (4) it follows that the tensor <math>~ {A^\mu}_\nu </math> up to a sign and a constant multiplier coincides with the acceleration tensor <math> ~ {u^\alpha}_k </math> in case when rectilinear motion of a solid body without rotation is considered. Then indeed the four-potential of the acceleration field coincides with the four-velocity, <math>~ U_\mu = u_\mu </math>. As a result, the quantity <math>~ - J^k \partial_\alpha U_k =- \rho_0 u^k \partial_\alpha u_k </math> on the right-hand side of (4) vanishes, since the following relations hold true: <math>~ u^k u_k = c^2 </math>, <math>~ 2 u^k \partial_\alpha u_k = \partial_\alpha (u^k u_k) = \partial_\alpha c^2 =0 </math>. With this in mind, in (4) we can raise the index <math>~ \alpha </math> and cancel the mass density, which gives the following:
:<math> ~ - {u^\alpha}_k u^k =\frac {du^\alpha }{d \tau} .</math>
Mashhoon and Muench considered transformation of inertial reference frames, co-moving with the accelerated reference frame, and obtained the relation: <ref> Bahram Mashhoon and Uwe Muench. Length measurement in accelerated systems. Annalen der Physik. Vol. 11, Issue 7, P. 532–547, 2002. </ref>
:<math>~c \frac { d \lambda_\alpha }{d \tau } = {\Phi_\alpha}^\beta \lambda_\beta. </math>
The tensor <math>~ {\Phi_\alpha}^\beta </math> has the same properties as the acceleration tensor <math> ~ {u_\alpha}^\beta. </math>
== The use in the general theory of relativity ==
The action function in the [[general relativity]] (GR) can be represented as the sum of the four terms, which are responsible, respectively, for the spacetime metric, the matter in the form of substance, the electromagnetic field and the pressure field:
:<math>~ S = S_m + S_{mat} + S_{em} + S_p. </math>
Additional terms can be included in the action function, if other fields must be taken into account. The first, second and third terms of the action have the standard form: <ref> Fock, V. A. (1964). "The Theory of Space, Time and Gravitation". Macmillan. </ref>
:<math>~ S_m = \int (kR-2k \Lambda ) \sqrt {-g}d\Sigma.</math>
:<math>~ S_{mat} = \int ( - c \rho_0 ) \sqrt {-g}d\Sigma.</math>
:<math>~ S_{em} =\int ( - \frac {1}{c} A_\mu j^\mu - \frac {c \varepsilon_0}{4 } F_{\mu\nu}F^{\mu\nu} ) \sqrt {-g}d\Sigma,</math>
where <math> A_\mu </math> is the electromagnetic four-potential.
The term <math>~ S_p </math>, which is responsible for the contribution of pressure into the action function, is different in the works of different authors, depending on how the pressure is related to the elastic energy and whether the pressure field is considered to be a scalar field or a vector field. It should be noted that in the GR, the gravitational field is included in the action function not directly, but indirectly, by means of the metric tensor. In this case, as a rule, the pressure field is considered to be a scalar field.
In contrast, in the [[covariant theory of gravitation]] (CTG), the term <math>~ S_{ac} </math> associated with the acceleration field is used instead of the term <math>~ S_{mat} </math>, and the action function can be written as follows: <ref name="ac"/>
:<math>~ S = S_m + S_{ac} + S_{em} + S_p . </math>
Here
:<math>~ S_{ac} = \int ( - \frac {1}{c } U_\mu J^\mu - \frac {c}{ 16 \pi \eta } u_{\mu\nu}u^{\mu\nu} ) \sqrt {-g}d\Sigma , </math>
:<math>~ S_p =\int ( - \frac {1}{c } \pi_\mu J^\mu - \frac {c}{ 16 \pi \sigma } f_{\mu\nu}f^{\mu\nu} ) \sqrt {-g}d\Sigma,</math>
where <math> ~\pi_\mu </math> is the four-potential of the [[pressure field]], <math> ~ \sigma </math> is the coefficient of the pressure field, <math> ~ f_{\mu\nu}</math> is the [[pressure field tensor]], <math>J^\mu = \rho_{0} u^\mu </math>.
In the case of rectilinear motion of a rigid body without rotation, the following relations will hold: <math> U_\mu = u_\mu </math>, <math>~ u_\mu u^\mu = c^2 </math>, and in the term <math>~ S_{ac} </math> the relation <math>~ - \frac {1}{c } U_\mu J^\mu = - c \rho_0</math> is obtained.
In this particular case it is clear that the term <math>~ S_{ac} </math> differs from the term <math>~ S_{mat} </math> by an additional term associated with the energy of the acceleration field. This is due to the fact that in the CTG the acceleration field is considered to be a vector field, whereas as in the GR the acceleration field is actually used as a scalar field that does not depend on the particles’ velocities. In both theories, the acceleration field allows us to determine the contribution of the rest energy of the particles into the total energy of the system of particles and fields. However, the use of the acceleration field as a scalar field in the GR does not agree in its form with the vector nature of the electromagnetic field. Indeed, in the limiting case, when only the particles’ accelerations and electromagnetic forces are taken into account, the acceleration must be two-component, as is the case for the acceleration due to the action of the two-component [[Lorentz force]]. But this is possible only in the case, when the acceleration field is a vector field. The situation can be improved if, in addition to the gravitational field function, we ascribe to the metric field <math>~ g_{\mu \nu} </math> in the GR the function of the vector component of the acceleration field, but this makes the equations of the theory even more complex and complicated.
It should be noted that in the general case of arbitrary motion of the matter the relation <math>~ - \frac {1}{c } U_\mu J^\mu = - c \rho_0</math> is no longer satisfied and CTG does not coincide any more with GR in the method of describing the rest energy of a physical system. This means that in GR the motion of the matter is considered in a simplified way, as rectilinear motion of a solid body, whereas in CTG the use of the four-potential <math> U_\mu </math> of the acceleration field allows us to take into account the internal motion of the matter in each selected volume element. For example, when a particle moves round a circle, the four-potential <math> U_\mu </math> of the particle’s matter will depend on the location of this matter with respect to the circle line, since the velocity of the particle’s matter depends on the radius of rotation.
== See also ==
* [[General field]]
* [[Pressure field]]
* [[Dissipation field]]
* [[Covariant theory of gravitation]]
* [[Metric theory of relativity]]
* [[Acceleration tensor]]
* [[Acceleration stress-energy tensor]]
* [[Four-force]]
* [[Equation of vector field]]
== References ==
<references/>
==External links ==
* [http://www.wikiznanie.ru/ru-wz/index.php/%D0%9F%D0%BE%D0%BB%D0%B5_%D1%83%D1%81%D0%BA%D0%BE%D1%80%D0%B5%D0%BD%D0%B8%D0%B9 Acceleration field in Russian]
[[Category:Theoretical physics]]
[[Category:Concepts in physics]]
[[Category:Vector calculus]]
[[Category:Covariant theory of gravitation]]
[[Category: Metric theory of relativity]]
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'''Acceleration field''' is a two-component vector field, describing in a covariant way the [[four-acceleration]] of individual particles and the [[four-force]] that occurs in systems with multiple closely interacting particles. The acceleration field is a component of the [[Physics/Essays/Fedosin/General field |general field]], which is represented in the Lagrangian and Hamiltonian of an arbitrary physical system by the term with the energy of particles’ motion and the term with the field energy. <ref name="ko"> Fedosin S.G. [http://www.oalib.com/paper/5263035#.VuFYxn2LQsY The Concept of the General Force Vector Field]. OALib Journal, Vol. 3, pp. 1-15 (2016), e2459. http://dx.doi.org/10.4236/oalib.1102459. </ref> <ref> Fedosin S.G. Two components of the macroscopic general field. Reports in Advances of Physical Sciences, Vol. 1, No. 2, 1750002, 9 pages (2017). http://dx.doi.org/10.1142/S2424942417500025. </ref>
The acceleration field is included in most [[equation of vector field |equations of vector field]]. Moreover, the acceleration field enters into the equation of motion through the [[acceleration tensor]] and into the equation for the metric through the [[acceleration stress-energy tensor]].
The acceleration field was presented by [[user:Fedosin |Sergey Fedosin]] within the framework of the [[Physics/Essays/Fedosin/Metric theory of relativity|metric theory of relativity]] and [[covariant theory of gravitation]], and the equations of this field were obtained as a consequence of the [[w:principle of least action |principle of least action]]. <ref name="pr"> Fedosin S.G. [http://vixra.org/abs/1406.0135 The procedure of finding the stress-energy tensor and vector field equations of any form]. Advanced Studies in Theoretical Physics, Vol. 8, No. 18, pp. 771-779 (2014). http://dx.doi.org/10.12988/astp.2014.47101. </ref> <ref name="ac"> [[user:Fedosin | Fedosin S.G.]] [http://journals.yu.edu.jo/jjp/Vol9No1Contents2016.html About the cosmological constant, acceleration field, pressure field and energy.] Jordan Journal of Physics. Vol. 9, No. 1, pp. 1-30 (2016). http://dx.doi.org/10.5281/zenodo.889304. </ref>
== Mathematical description ==
The 4-potential of the acceleration field is expressed in terms of the scalar <math>~ \vartheta </math> and vector <math>~ \mathbf {U} </math> potentials:
:<math>~U_\mu = \left(\frac {\vartheta }{c},- \mathbf {U} \right) .</math>
The antisymmetric [[acceleration tensor]] is calculated with the help of the 4-curl of the 4-potential:
:<math>~ u_{\mu \nu} = \nabla_\mu U_\nu - \nabla_\nu U_\mu = \partial_\mu U_\nu - \partial_\nu U_\mu . </math>
The acceleration tensor components are the components of the field strength <math>~\mathbf {S} </math> and the components of the solenoidal vector <math>~\mathbf {N} </math>:
:<math> ~ u_{\mu \nu}= \begin{vmatrix} 0 & \frac {S_x}{ c} & \frac {S_y}{ c} & \frac {S_z}{ c} \\ -\frac {S_x}{ c} & 0 & - N_{z} & N_{y} \\ -\frac {S_y}{ c} & N_{z} & 0 & -N_{x} \\ -\frac {S_z}{ c}& -N_{y} & N_{x} & 0 \end{vmatrix}. </math>
We obtain the following:
:<math>~ \mathbf {S} = - \nabla \vartheta - \frac {\partial \mathbf {U}}{\partial t},\qquad\qquad \mathbf {N} = \nabla \times \mathbf {U}.\qquad\qquad (1) </math>
In the general case the scalar and vector potentials are found by solving the wave equations for the acceleration field potentials.
=== Action, Lagrangian and energy ===
In the covariant theory of gravitation the 4-potential <math>~U_\mu </math> of the acceleration field is part of the 4-potential of the [[Physics/Essays/Fedosin/General field |general field]] <math>~ s_\mu</math>, which is the sum of the 4-potentials of particular fields, such as the electromagnetic and gravitational fields, acceleration field, [[pressure field]], [[dissipation field]], strong interaction field, weak interaction field and other vector fields, acting on the matter and its particles. All of these fields are somehow represented in the matter, so that the 4-potential <math>~ s_\mu</math> cannot consist of only one 4-potential <math>~U_\mu </math>.
The energy density of interaction of the general field and the matter is given by the product of the 4-potential of the general field and the mass 4-current: <math>~ s_\mu J^\mu </math>. We obtain the general field tensor from the 4-potential of the general field, using the 4-curl:
:<math>~ s_{\mu \nu} =\nabla_\mu s_\nu - \nabla_\nu s_\mu.</math>
The tensor invariant in the form <math>~ s_{\mu \nu} s^{\mu \nu} </math> is up to a constant factor proportional to the energy density of the general field. As a result, the action function, which contains the scalar curvature <math>~R</math> and the cosmological constant <math>~ \Lambda </math>, is given by the expression: <ref name="ko"/>
:<math>~S =\int {L dt}=\int (kR-2k \Lambda - \frac {1}{c}s_\mu J^\mu - \frac {c}{16 \pi \varpi} s_{\mu\nu}s^{\mu\nu} ) \sqrt {-g}d\Sigma,</math>
where <math>~L </math> is the Lagrange function or Lagrangian; <math>~dt </math> is the time differential of the coordinate reference system; <math>~k </math> and <math>~ \varpi </math> are the constants to be determined; <math>~c </math> is the speed of light as a measure of the propagation speed of the electromagnetic and gravitational interactions; <math>~\sqrt {-g}d\Sigma= \sqrt {-g} c dt dx^1 dx^2 dx^3</math> is the invariant 4-volume expressed in terms of the differential of the time coordinate <math>~ dx^0=cdt </math>, the product <math>~ dx^1 dx^2 dx^3 </math> of differentials of the space coordinates and the square root <math>~\sqrt {-g} </math> of the determinant <math>~g </math> of the metric tensor, taken with a negative sign.
The variation of the action function gives the general field equations, the four-dimensional equation of motion and the equation for the metric. Since the acceleration field is the general field component, then from the general field equations the corresponding equations of the acceleration field are derived.
Given the gauge condition of the cosmological constant in the form
:<math>~ c k \Lambda = - s_\mu J^\mu ,</math>
is met, the system energy does not depend on the term with the scalar curvature and is uniquely determined: <ref name="ac"/>
:<math>~E = \int {( s_0 J^0 + \frac {c^2 }{16 \pi \varpi } s_{ \mu\nu} s^{ \mu\nu} ) \sqrt {-g} dx^1 dx^2 dx^3}, </math>
where <math>~ s_0 </math> and <math>~ J^0</math> denote the time components of the 4-vectors <math>~ s_{\mu } </math> and <math>~ J^{\mu } </math>.
The system’s 4-momentum is given by the formula:
:<math>~p^\mu = \left( \frac {E}{c}{,} \mathbf {p}\right) = \left( \frac {E}{c}{,} \frac {E}{c^2}\mathbf {v} \right) , </math>
where <math>~ \mathbf {p}</math> and <math>~ \mathbf {v}</math> denote the system’s momentum and the velocity of the system’s center of momentum.
=== Equations ===
The four-dimensional equations of the acceleration field are similar in their form to Maxwell equations and are as follows:
:<math> \nabla_\sigma u_{\mu \nu}+\nabla_\mu u_{\nu \sigma}+\nabla_\nu u_{\sigma \mu}=\frac{\partial u_{\mu \nu}}{\partial x^\sigma} + \frac{\partial u_{\nu \sigma}}{\partial x^\mu} + \frac{\partial u_{\sigma \mu}}{\partial x^\nu} = 0. </math>
:<math>~ \nabla_\nu u^{\mu \nu} = - \frac{4 \pi \eta }{c^2} J^\mu, </math>
where <math>J^\mu = \rho_{0} u^\mu </math> is the mass 4-current, <math> \rho_{0}</math> is the mass density in the co-moving reference frame, <math> u^\mu </math> is the 4-velocity of the matter unit, <math>~ \eta </math> is a constant, which is determined in each problem, and it is supposed that there is an equilibrium between all fields in the observed physical system.
The gauge condition of the 4-potential of the acceleration field:
:<math>~ \nabla^\mu U_{\mu} =0 . </math>
If the second equation with the field source is written with the covariant index in the following form:
:<math>~ \nabla^\nu u_{\mu \nu} = - \frac{4 \pi \eta }{c^2} J_\mu, </math>
then after substituting here the expression for the acceleration tensor <math> u_{\mu \nu} </math> through the 4-potential <math> ~ U_\mu </math> of the acceleration field we obtain the wave equation for calculating the potentials of the acceleration field:
:<math>~ \nabla^\nu \nabla_\nu U_\mu + R_{\mu \nu} U^\nu = \frac{4 \pi \eta }{c^2} J_\mu, </math>
where <math>~ R_{\mu \nu} </math> is the Ricci tensor.
The continuity equation in curved space-time is:
:<math>~ R_{ \mu \alpha } u^{\mu \alpha }= \frac {4 \pi \eta }{c^2} \nabla_{\alpha}J^{\alpha}.</math>
In Minkowski space of the special theory of relativity, the Ricci tensor is set to zero, the form of the acceleration field equations is simplified and they can be expressed in terms of the field strength <math>~\mathbf {S} </math> and the solenoidal vector <math>~\mathbf {N} </math>:
:<math>~ \nabla \cdot \mathbf{S} = 4 \pi \eta \gamma \rho_0, \qquad\qquad \nabla \times \mathbf{N} = \frac{1}{c^2} \left( 4 \pi \eta \mathbf{J} + \frac{\partial \mathbf{S}} {\partial t} \right), </math>
:<math>~ \nabla \times \mathbf{S} = - \frac{\partial \mathbf{N} } {\partial t} , \qquad\qquad \nabla \cdot \mathbf{N} = 0 .</math>
where <math>~ \gamma = \frac {1}{\sqrt{1 - {v^2 \over c^2}}} </math> is the Lorentz factor, <math>~ \mathbf{J}= \gamma \rho_0 \mathbf{v }</math> is the mass current density, <math>~ \mathbf{v } </math> is the velocity of the matter unit.
The wave equation is also simplified and can be written separately for the scalar and vector potentials:
:<math>~ \partial^\nu \partial_\nu \vartheta = \frac {1}{c^2}\frac{\partial^2 \vartheta }{\partial t^2 } -\Delta \vartheta = 4 \pi \eta \gamma \rho_0, \qquad\qquad (2) </math>
:<math>~ \partial^\nu \partial_\nu \mathbf{U} =\frac {1}{c^2}\frac{\partial^2 \mathbf{U} }{\partial t^2 } -\Delta \mathbf{U}= \frac {4 \pi \eta}{c^2} \mathbf{J}. \qquad\qquad (3) </math>
The equation of motion of the matter unit in the general field is given by the formula:
:<math>~ s_{\mu \nu} J^\nu =0 </math>.
Since <math>~ J^\nu = \rho_0 u^\nu </math>, and the general field tensor is expressed in terms of the tensors of particular fields, then the equation of motion can be represented with the help of these tensors:
:<math>~ - u_{\mu \nu} J^\nu =F_{\mu \nu} j^\nu + \Phi_{\mu \nu} J^\nu + f_{\mu \nu} J^\nu + h_{\mu \nu} J^\nu + \gamma_{\mu \nu} J^\nu + w_{\mu \nu} J^\nu .</math>
Here <math>~ F_{\mu \nu}</math> is the [[w:electromagnetic tensor |electromagnetic tensor]], <math>~ j^\nu </math> is the charge [[4-current]], <math>~ \Phi_{\mu \nu}</math> is the [[Physics/Essays/Fedosin/Gravitational tensor |gravitational tensor]], <math>~ f_{\mu \nu}</math> is the [[pressure field tensor]], <math>~ h_{\mu \nu}</math> is the [[dissipation field tensor]], <math>~ \gamma_{\mu \nu}</math> is the strong interaction field tensor, <math>~ w_{\mu \nu}</math> is the weak interaction field tensor.
=== The stress-energy tensor ===
The [[acceleration stress-energy tensor]] is calculated with the help of the acceleration tensor:
:<math>~ B^{ik} = \frac{c^2} {4 \pi \eta }\left( -g^{im} u_{n m} u^{n k}+ \frac{1} {4} g^{ik} u_{m r} u^{m r}\right) </math>.
We find as part of the tensor <math>~ B^{ik}</math> the 3-vector of the energy-momentum flux <math>~\mathbf {K} </math>, which is similar in its meaning to the [[w:Poynting vector |Poynting vector]] and the [[Physics/Essays/Fedosin/Heaviside vector |Heaviside vector]]. The vector <math>~\mathbf {K} </math> can be represented through the vector product of the field strength <math>~ \mathbf {S} </math> and the solenoidal vector <math>~ \mathbf {N} </math>:
:<math>~ \mathbf {K}=c B^{0i} = \frac {c^2}{4 \pi \eta }[\mathbf {S}\times \mathbf {N}],</math>
here the index is <math>~ i=1,2,3.</math>
The covariant derivative of the stress-energy tensor of the acceleration field with mixed indices specifies the [[4-force]] density:
:<math> ~ f_\alpha = \nabla_\beta {B_\alpha}^\beta = - u_{\alpha k} J^k = - \rho_0 u_{\alpha k}u^k = \rho_0 \frac {DU_\alpha }{D \tau}- J^k \nabla_\alpha U_k = \rho_0 \frac {dU_\alpha }{d \tau} - J^k \partial_\alpha U_k ,\qquad \qquad (4)</math>
where <math>~ D \tau </math> denotes the proper time differential in the curved spacetime.
The stress-energy tensor of the acceleration field is part of the stress-energy tensor of the general field <math>~ T^{ik} </math>:
:<math>~ T^{ik}= W^{ik}+ U^{ik}+ B^{ik}+ P^{ik} + Q^{ik}+ L^{ik}+ A^{ik}, </math>
where <math>~ W^{ik} </math> is the [[w:electromagnetic stress–energy tensor |electromagnetic stress–energy tensor]], <math>~ U^{ik}</math> is the [[gravitational stress-energy tensor]], <math>~ P^{ik}</math> is the [[pressure stress-energy tensor]], <math>~ Q^{ik}</math> is the [[dissipation stress-energy tensor]], <math>~ L^{ik}</math> is the strong interaction stress-energy tensor, <math>~ A^{ik} </math> is the weak interaction stress-energy tensor.
Through the tensor <math>~ T^{ik} </math> the stress-energy tensor of the acceleration field enters into the equation for the metric:
:<math>~ R^{ik} - \frac{1} {4 }g^{ik}R = \frac{8 \pi G \beta }{ c^4} T^{ik}, </math>
where <math>~ R^{ik} </math> is the Ricci tensor, <math>~ G </math> is the [[w:gravitational constant |gravitational constant]], <math>~ \beta </math> is a certain constant, and the gauge condition of the cosmological constant is used.
== Specific solutions for the acceleration field functions ==
The four-potential of any vector field, the global vector potential of which is equal to zero in the proper reference frame K', that is, in the center-of-momentum frame, in case of rectilinear motion in the laboratory reference frame K, can be presented as follows: <ref name="pr"/> <ref> Fedosin S.G. Equations of Motion in the Theory of Relativistic Vector Fields. International Letters of Chemistry, Physics and Astronomy, Vol. 83, pp. 12-30 (2019). https://doi.org/10.18052/www.scipress.com/ILCPA.83.12. </ref>
:<math>~ L_{\mu L} = \frac { k_f \varepsilon }{\rho_0 c^2} u_{\mu L},</math>
where <math>~ k_f = \frac {\rho_0}{\rho_{0q}}</math> is for the electromagnetic field and <math>~ k_f = 1</math> for the remaining fields; <math> ~ \rho_{0}</math> and <math> ~\rho_{0q}</math> are the invariant mass density and the charge density in the comoving reference frame, respectively; <math>~ \varepsilon </math> is the invariant energy density of the interaction, calculated as product of the four-potential of the field and the corresponding four-current; <math>~ u_{\mu L} </math> is the covariant four-velocity that determines the motion of the center of momentum of the physical system in K.
In the [[special relativity]] (SR), in the center-of-momentum frame K' the energy density is <math>~ \varepsilon = \gamma' \rho_0 c^2 </math>, where <math>~ \gamma' </math> is the Lorentz factor, and for the acceleration field, while the physical system is moving in K, the four-potential of the acceleration field will equal <math>~ U_{\mu L}= \gamma' u_{\mu L}</math>.
In case when the physical system is stationary in K, we will have <math>~ u_{\mu L} = (c,0,0,0) </math>, and consequently, the scalar potential will be <math>~ \vartheta = \gamma' c^2 </math>. If in the physical system, on the average, there are directed fluxes of matter or rotation of matter, the vector potential <math>~ \mathbf {U} </math> of the acceleration field is no longer equal to zero.
If the four-potential <math>~ U'_{\nu}</math> of acceleration field in K' is known, then in the laboratory reference frame K the four-potential is determined using the matrix <math>~ M_{\mu}^{\ \nu} </math> connecting the coordinates and time of both frames: <ref name="it"> Fedosin S.G. [http://dergipark.org.tr/gujs/issue/45480/435567 The Integral Theorem of the Field Energy.] Gazi University Journal of Science. Vol. 32, No. 2, pp. 686-703 (2019). http://dx.doi.org/10.5281/zenodo.3252783. </ref>
:<math>~ U_{\mu L}= M_{\mu}^{\ \nu} U'_{\nu}.</math>
In the special case of the system’s motion at the constant velocity <math>~ M_{\mu}^{\ \nu}</math> represents the Lorentz transformation matrix.
=== Ideally solid particle ===
In the approximation, when a particle is regarded as an ideally solid object, the matter inside the particle is motionless. It means that the Lorentz factor <math>~ \gamma' </math> of this matter in the center-of-momentum frame K' is equal to unity, so that the four-potential of the acceleration field becomes equal to the four-velocity of motion of the center of momentum:
:<math>~ U_\mu = u_\mu. </math>
In the SR, the expression for 4-velocity is simplified and we can write:
:<math>~U_\mu = \left( \frac {\vartheta }{c},- \mathbf {U} \right) = u_\mu = \left(\gamma c, - \gamma \mathbf {v} \right).</math>
The acceleration tensor components according to (1) will equal:
:<math>~ \mathbf {S} = - c^2 \nabla \gamma - \frac {\partial (\gamma \mathbf { v })}{\partial t},\qquad\qquad \mathbf {N} = \nabla \times (\gamma \mathbf { v }). </math>
Since in the solid-state motion equation for the four-acceleration with a covariant index <math>~ a_\mu </math> the relation holds
:<math>~ \rho_0 a_\mu = \rho_0 \frac {Du_\mu }{D \tau}= - u_{\mu \nu} J^\nu = - \rho_0 u_{\mu \nu} u^\nu, </math>
then in SR we obtain the following:
:<math>~ \frac {Du_\mu }{D \tau}= \frac {du_\mu }{d \tau} =\gamma \frac {du_\mu }{dt}, \qquad\qquad u^\nu =\left(\gamma c, \gamma \mathbf {v} \right), </math>
and the equations for the Lorentz factor <math>~ \gamma </math> and for the 3-acceleration <math>~ a= \frac {d \mathbf { v }}{dt} </math>:
:<math>~ \frac {d \gamma }{dt}= - \frac {1 }{c^2} \mathbf {S}\cdot \mathbf { v }, \qquad (5) \qquad \frac {d (\gamma \mathbf { v })}{dt}= \gamma \mathbf { a }+ \frac {d \gamma}{dt}\mathbf { v } = - \mathbf {S}- [\mathbf { v }\times \mathbf {N}]. \qquad (6) </math>
Multiplying equation (6) by the velocity <math>~ \mathbf { v }</math>, substituting the quantity <math>~ \mathbf {S}\cdot \mathbf { v } </math> from equation (5) to (6), taking into account relation <math>~\gamma^{-2}=1 - {v^2 \over c^2},</math> we find the well-known expression for the derivative of the Lorentz factor using the scalar product of the velocity and acceleration in SR:
:<math>~ \gamma^3 \mathbf {v}\cdot \mathbf { a }=c^2 \frac {d \gamma }{dt}.</math>
We can prove the validity of equation (6) by substituting in its right-hand side the expression for the strength and solenoidal vector:
:<math>~ \frac {d (\gamma \mathbf { v })}{dt}= c^2 \nabla \gamma + \frac {\partial (\gamma \mathbf { v })}{\partial t} - \mathbf { v }\times [ \nabla \times (\gamma \mathbf { v }) ] . \qquad\qquad (7) </math>
Indeed, the use of the [[w:material derivative |material derivative]] gives the following:
:<math>~ \frac {d (\gamma \mathbf { v })}{dt}= \frac {\partial (\gamma \mathbf { v })}{\partial t} + (\mathbf { v } \cdot \nabla) (\gamma \mathbf { v }) = \frac {\partial (\gamma \mathbf { v })}{\partial t}+\gamma (\mathbf { v } \cdot \nabla) \mathbf { v } + \mathbf { v } (\mathbf { v } \cdot \nabla\gamma) .</math>
In addition
:<math>~ - \mathbf { v }\times [ \nabla \times (\gamma \mathbf { v }) ] = - \gamma \mathbf { v }\times [ \nabla \times \mathbf { v } ] - \mathbf { v }\times [ \nabla \gamma \times \mathbf { v }] = -\frac {\gamma }{2} \nabla v^2 + \gamma (\mathbf { v } \cdot \nabla) \mathbf { v } - v^2 \nabla \gamma + \mathbf { v } (\mathbf { v } \cdot \nabla\gamma) .</math>
Substituting these relations in (7), taking into account the expression <math>~ \gamma^{-2}=1 - {v^2 \over c^2},</math> we obtain the identity:
:<math>~ c^2 \nabla \gamma - \frac {\gamma }{2} \nabla v^2 - v^2 \nabla \gamma =0 .</math>
If the components of the particle velocity are the functions of time and they do not directly depend on the space coordinates, then the solenoidal vector <math>~ \mathbf { N }</math> vanishes in such a motion.
In the SR <math>~ E = \gamma m c^2 </math> is the relativistic energy, <math>~ \mathbf p = \gamma m \mathbf v </math> is the 3-vector of relativistic momentum. If the mass <math>~ m </math> of a particle is constant, then multiplying (7) by the mass, we arrive to following equation for the force:
:<math>~ \mathbf F= \frac {d \mathbf p }{dt}= \nabla E + \frac {\partial \mathbf p }{\partial t} - \mathbf { v }\times [ \nabla \times \mathbf p ] . </math>
=== Rotation of a particle ===
For a small ideally solid particle, we can neglect the motion of the matter inside the particle and can assume that the four-potential of the acceleration field is equal to the four-velocity of the particle’s center of momentum.
Let us assume that the particle rotates about the axis OZ of the coordinate system at the distance <math>~ \rho = \sqrt {x^2 +y^2} </math> from the axis at the constant angular velocity <math>~ \omega</math> counterclockwise, as viewed from the side, in which the OZ axis is directed. Then we can assume that the linear velocity of the particle depends only on the coordinates <math>~ x</math> and <math>~ y</math>, and for the velocity’s projections on the axes of the coordinate system we can write: <math>~ \mathbf v = (-\omega y, \omega x, 0) </math>, while the square of the velocity equals <math>~ v^2 = \omega^2 (x^2 + y^2) </math>. For the Lorentz factor in the SR we obtain the following:
:<math>~ \gamma = \frac {1}{\sqrt {1- \frac { v^2}{ c^2}}} = \frac {1}{\sqrt {1- \frac { \omega^2 (x^2 + y^2)}{ c^2}}} . </math>
With this in mind, the potentials and field strengths of the acceleration field can be written as follows:
:<math>~ \vartheta = \gamma c^2, \qquad \mathbf {U} = \gamma \mathbf {v}. </math>
:<math>~ \mathbf {S} = - \nabla \vartheta - \frac {\partial \mathbf {U} }{\partial t}= \left( -\gamma^3 \omega^2 x, -\gamma^3 \omega^2 y, 0 \right). </math>
:<math>~ \mathbf {N} = \nabla \times \mathbf {U} = \left( 0, 0, \gamma \omega +\gamma^3 \omega \right). </math>
If we substitute <math>~ \gamma </math>, <math>~ \mathbf v </math>, <math>~\mathbf {S} </math> and <math>~\mathbf {N} </math> in (6), we can determine the acceleration components of the particle and the acceleration amplitude:
:<math>~ \mathbf {a} = \left( - \omega^2 x , -\omega^2 y, 0 \right). </math>
:<math>~ a = \sqrt {a^2_x +a^2_y +a^2_z} = \omega^2 \sqrt {x^2 +y^2}= \omega^2 \rho =\omega v = \frac {v^2} {\rho }. </math>
The acceleration is directed towards the center of rotation and represents [[centripetal acceleration]]. Using now the classic vector description, we have also for the time and coordinates of reference frame at the center of rotation:
:<math>~ \vec \rho = (x, y, 0) , \qquad \vec \omega = \frac {\vec {d\varphi} }{dt} =(0, 0, \omega) , </math>
:<math>~ \mathbf {v} = [\vec \omega \times \vec \rho] , \qquad \mathbf {a} = [\vec \omega \times \mathbf {v}] = [\vec \omega \times [\vec \omega \times \vec \rho]] = \vec \omega (\vec \omega \cdot \vec \rho) - \vec \rho (\vec \omega \cdot \vec \omega) = - \omega^2 \vec \rho , </math>
where <math>~ \rho </math> and <math>~ \varphi </math> are two coordinates of the [[Coordinate systems#Cylindrical coordinates.5B4.5D |cylindrical coordinate system]], <math>~ \vec \rho </math> is the vector from the center of rotation to the particle, <math>~ \vec {d\varphi}</math> is the axial vector of the differential of the rotation angle directed along the axis OZ.
As we can see, in case of such a motion with acceleration the vector product <math>~ [\mathbf {S}\times \mathbf {N}]</math> is not equal to zero, just as the three-vector <math>~ \mathbf {K}</math> of the energy-momentum flux of the acceleration field inside the particle.
=== The system of particles ===
Due to interaction of a number of particles with each other by means of various fields, including interaction at a distance without direct contact, the acceleration field in the matter changes and is different from the acceleration field of individual particles at the observation point. As a result, the density of the 4-force in the system of particles is given by the strength and the solenoidal vector, which represent the typical average characteristics of the matter motion. For example, in a gravitationally bound system there is a radial gradient of the vector <math>~ \mathbf { S },</math> and if the system is moving or rotating, there is a vector <math>~ \mathbf { N }.</math> From (4) there follows the general expression for the the density of the 4-force with covariant index:
:<math> ~ f_\nu = \rho_0 \frac {cdt}{ds}\left(-\frac {1}{c} \mathbf{S} \cdot \mathbf{v}{,} \qquad \mathbf{S}+[\mathbf{v} \times \mathbf{N}] \right),</math>
where <math> ~ ds </math> denotes a four-dimensional space-time interval.
For a stationary case, when the potentials of the acceleration field are independent of time, under the assumption that <math>~ \vartheta = \gamma c^2, </math> wave equation (2) for the scalar potential in the SR is transformed into the equation:
:<math>~ \Delta \gamma= - \frac {4 \pi \eta \gamma \rho_0}{c^2}. </math>
The solution of this equation for a fixed sphere with the particles randomly moving in it has the form: <ref name="ie"> Fedosin S.G. [http://vixra.org/abs/1403.0973 The Integral Energy-Momentum 4-Vector and Analysis of 4/3 Problem Based on the Pressure Field and Acceleration Field.] American Journal of Modern Physics. Vol. 3, No. 4, pp. 152-167 (2014). http://dx.doi.org/10.11648/j.ajmp.20140304.12.</ref>
:<math>~\gamma= \frac {c \gamma_c }{r \sqrt {4 \pi \eta \rho_0} } \sin \left(\frac {r}{c}\sqrt {4 \pi \eta \rho_0} \right) \approx \gamma_c - \frac {2 \pi \eta \rho_0 r^2 \gamma_c }{3 c^2}.</math>
where <math>~ \gamma_c = \frac {1}{\sqrt{1 - {v^2_c \over c^2}}} </math> is the Lorentz factor for the velocities <math>~ v_c</math> of the particles in the center of the sphere, and due to the smallness of the argument the sine is expanded to the second order terms. From the formula it follows that the average velocities of the particles are maximal in the center and decrease when approaching the surface of the sphere.
In such a system, the scalar potential <math>~ \vartheta</math> becomes the function of the radius, and the vector potential <math>~ \mathbf {U} </math> and the solenoidal vector <math>~ \mathbf { N }</math> are equal to zero. The acceleration field strength <math>~\mathbf {S} </math> is found with the help of (1). Then we can calculate all the functions of the acceleration field, including the energy of particles in this field and the energy of the acceleration field itself. <ref> Fedosin S.G. [http://journals.yu.edu.jo/jjp/Vol8No1Contents2015.html Relativistic Energy and Mass in the Weak Field Limit.] Jordan Journal of Physics. Vol. 8, No. 1, pp. 1-16 (2015). http://dx.doi.org/10.5281/zenodo.889210.</ref> For cosmic bodies the main contribution to the four-acceleration in the matter makes the gravitational force and the pressure field.
At the same time the relativistic rest energy of the system is automatically derived, taking into account the motion of particles inside the sphere. For the system of particles with the acceleration field, pressure field, gravitational and electromagnetic fields the given approach allowed solving the 4/3 problem and showed where and in what form the energy of the system is contained. The relation for the acceleration field constant in this problem was found:
:<math>~\eta = 3G- \frac {3q^2}{4 \pi \varepsilon_0 m^2 },</math>
where <math>~ \varepsilon_0</math> is the [[electric constant]], <math>~q </math> and <math>~m </math> are the total charge and mass of the system.
The solution of the wave equation for the acceleration field within the system results in temperature distribution according to the formula: <ref name="ie"/>
:<math>~ T=T_c - \frac {\eta M_p M(r)}{3kr} ,</math>
where <math>~ T_c </math> is the temperature in the center, <math>~ M_p </math> is the mass of the particle, for which the mass of the proton is taken (for systems which are based on hydrogen or nucleons in atomic nuclei), <math>~ M(r) </math> is the mass of the system within the current radius <math>~ r </math>, <math>~ k</math> is the Boltzmann constant.
This dependence is well satisfied for a variety of space objects, including gas clouds and Bok globules, the Earth, the Sun and neutron stars.
In articles <ref> Fedosin S.G. [http://www.nrcresearchpress.com/doi/10.1139/cjp-2015-0593#.Vv3piZyLQsY Estimation of the physical parameters of planets and stars in the gravitational equilibrium model]. Canadian Journal of Physics, Vol. 94, No. 4, pp. 370-379 (2016). http://dx.doi.org/10.1139/cjp-2015-0593.</ref> <ref>Fedosin S.G. The generalized Poynting theorem for the general field and solution of the 4/3 problem. International Frontier Science Letters, Vol. 14, pp. 19-40 (2019). https://doi.org/10.18052/www.scipress.com/IFSL.14.19. </ref> the ratio of the field’s coefficients for the fields was specified as follows:
:<math>~\eta + \sigma = G - \frac {\rho^2_{0q}}{4 \pi \varepsilon_0 \rho^2_{0}},</math>
where <math> ~ \sigma </math> is the pressure field constant.
If we introduce the parameter <math> ~ \mu </math> as the number of nucleons per ionized gas particle, then the acceleration field constant is expressed as follows:
:<math>~\eta = \frac {3\gamma_c \mu G}{2+ 3 \gamma_c \mu }.</math>
For the temperature inside the cosmic bodies in the gravitational equilibrium model we find the dependence on the current radius:
:<math>~ T=T_c - \frac {4 \pi \eta m_u \rho_{0c}\gamma_c r^2}{9k}+ \frac {2 \pi \eta A m_u \gamma_c r^3}{9k} + \frac {2 \pi \eta B m_u \gamma_c r^4}{15k} ,</math>
where <math> ~ m_u </math> is the mass of one gas particle, which is taken as the [[w:unified atomic mass unit |unified atomic mass unit]], and the coefficients <math> ~ A </math> and <math> ~ B </math> are included into the dependence of the mass density on the radius in the relation <math> ~ \rho_0 = \rho_{0c}- Ar - Br^2. </math>
Under the assumption that the system’s typical particles have the mass <math> ~\stackrel{-}{m } = \mu m_u </math>, and that it is typical particles that define the temperature and pressure, for the acceleration field constant we obtain the following: <ref>Fedosin S.G. The binding energy and the total energy of a macroscopic body in the relativistic uniform model. Middle East Journal of Science, Vol. 5, Issue 1, pp. 46-62 (2019). http://dx.doi.org/10.23884/mejs.2019.5.1.06. </ref>
:<math>~ \eta = \frac {3}{5} \left( G- \frac {\rho^2_{0q}}{ 4 \pi \varepsilon_0 \rho^2_0 } \right) .</math>
The Lorentz factor of the particles in the center of the system is also determined: <ref name="en"> Fedosin S.G. Energy and metric gauging in the covariant theory of gravitation. Aksaray University Journal of Science and Engineering, Vol. 2, Issue 2, pp. 127-143 (2018). http://dx.doi.org/10.29002/asujse.433947. </ref>
: <math>~\gamma_c = \frac {1}{\sqrt {1- \frac { v^2_c }{c^2}}} \approx 1+ \frac { v^2_c }{2c^2} +\frac {3 v^4_c }{8c^4} \approx 1+ \frac {3 \eta m}{10 a c^2} \left( 1+\frac {9}{2\sqrt {14}} \right) + \frac {27 \eta^2 m^2}{200 a^2 c^4} \left( 1+\frac {9}{2\sqrt {14}} \right)^2 . </math>
The wave equation (3) for the vector potential of the acceleration field was used to represent the relativistic equation of the fluid’s motion in the form of the [[w:Navier–Stokes equations |Navier–Stokes equations]] in hydrodynamics and to describe the motion of the viscous compressible and charged fluid. <ref> Fedosin S.G. Four-Dimensional Equation of Motion for Viscous Compressible and Charged Fluid with Regard to the Acceleration Field, Pressure Field and Dissipation Field. International Journal of Thermodynamics. Vol. 18, No. 1, pp. 13-24 (2015). http://dx.doi.org/10.5541/ijot.5000034003.</ref>
Taking into account the acceleration field and pressure field, within the framework of the [[relativistic uniform system]], it is possible to refine the [[w:virial theorem |virial theorem]], which in the relativistic form is written as follows: <ref>Fedosin S.G. The virial theorem and the kinetic energy of particles of a macroscopic system in the general field concept. Continuum Mechanics and Thermodynamics. Vol. 29, Issue 2, pp. 361-371 (2017). https://dx.doi.org/10.1007/s00161-016-0536-8. </ref>
: <math>~ \langle W_k \rangle \approx - 0.6 \sum_{k=1}^N\langle\mathbf{F}_k\cdot\mathbf{r}_k\rangle ,</math>
where the value <math>~ W_k \approx \gamma_c T </math> exceeds the kinetic energy of the particles <math>~ T </math> by a factor equal to the Lorentz factor <math>~ \gamma_c </math> of the particles at the center of the system. Under normal conditions we can assume that <math>~ \gamma_c \approx 1 </math>, then we can see that in the virial theorem the kinetic energy is related to the potential energy not by the coefficient 0.5, but rather by the coefficient close to 0.6. The difference from the classical case arises due to considering the pressure field and the acceleration field of particles inside the system, while the derivative of the virial scalar function <math>~ G_v </math> is not equal to zero and should be considered as the [[w:material derivative |material derivative]].
An analysis of the integral theorem of generalized virial makes it possible to find, on the basis of field theory, a formula for the root-mean-square speed of typical particles of a system without using the notion of temperature: <ref> Fedosin S.G. [http://em.rdcu.be/wf/click?upn=lMZy1lernSJ7apc5DgYM8f7AyOIJlVFO4uFv7zUQtzk-3D_DUeisO4Ue44lkDmCnrWVhK-2BAxKrUexyqlYtsmkyhvEp5zr527MDdThwbadScvhwZehXbanab8i5hqRa42b-2FKYwacOeM4LKDJeJuGA15M9FWvYOfBgfon7Bqg2f55NFYGJfVGaGhl0ghU-2BkIJ9Hz4M6SMBYS-2Fr-2FWWaj9eTxv23CKo9d8nFmYAbMtBBskFuW9fupsvIvN5eyv-2Fk-2BUc7hiS15rRISs1jpNnRQpDtk2OE9Hr6mYYe5Y-2B8lunO9GwVRw07Y1mdAqqtEZ-2BQjk5xUwPnA-3D-3D The integral theorem of generalized virial in the relativistic uniform model]. Continuum Mechanics and Thermodynamics, Vol. 31, Issue 3, pp. 627-638 (2019). https://dx.doi.org/10.1007/s00161-018-0715-x.</ref>
:<math> v_\mathrm{rms} = c \sqrt{1- \frac {4 \pi \eta \rho_0 r^2}{c^2 \gamma^2_c \sin^2 {\left( \frac {r}{c} \sqrt {4 \pi \eta \rho_0} \right) } } } .</math>
The integral [[field energy theorem]] for acceleration field in a curved space-time is as follows:<ref name="it"/>
:<math>~ - \int { \left( \frac {8 \pi \eta }{c^2} U_\alpha J^\alpha + u_{\alpha \beta} u^{\alpha \beta} \right) \sqrt {-g} dx^1 dx^2 dx^3 } = \frac {2}{c} \frac {d}{dt} \left( \int { U^\alpha u_\alpha ^{\ 0} \sqrt {-g} dx^1 dx^2 dx^3} \right) + 2 \iint \limits_S {U^\alpha u_\alpha ^{\ k} n_k \sqrt {-g} dS} . </math>
In the relativistic uniform system, the scalar potential <math>~\vartheta </math> of the acceleration field is related to the scalar potential <math>~\wp </math> of the pressure field:
<ref>Fedosin S.G. [https://rdcu.be/ccV9o The potentials of the acceleration field and pressure field in rotating relativistic uniform system]. Continuum Mechanics and Thermodynamics, Vol. 33, Issue 3, pp. 817-834 (2021). https://doi.org/10.1007/s00161-020-00960-7. </ref>
:<math>~ \wp = \frac {\sigma (\vartheta -c^2)}{ \eta } = \frac {2 (\vartheta -c^2)}{ 3 }. </math>
The relativistic expression for pressure is as follows:
<math> p = \frac{2\rho c^2 (\gamma - 1) }{3}= \frac {2 \rho c^2 }{3} \left( \frac {1}{\sqrt {1- v^2/ c^2 }}-1 \right) \approx \frac {\rho v^2}{3}, </math>
where <math>\rho </math> is the mass density of moving matter, <math> c </math> is the speed of light, <math> \gamma =\frac {1}{\sqrt {1- v^2/ c^2 }} </math> is the [[w:Lorentz factor |Lorentz factor]].
In the limit of low velocities, this relationship turns into the standard formula of the [[w:kinetic theory of gases |kinetic theory of gases]].
== Other approaches ==
Studying the Lorentz covariance of the 4-force, Friedman and Scarr found incomplete covariance of the expression for the 4-force in the form <math>~ F^\mu = \frac {d p^\mu }{d \tau } . </math> <ref> Yaakov Friedman and Tzvi Scarr. [http://iopscience.iop.org/1742-6596/437/1/012009 Covariant Uniform Acceleration]. Journal of Physics: Conference Series Vol. 437 (2013) 012009 doi:10.1088/1742-6596/437/1/012009. </ref>
This led them to conclude that the four-acceleration in SR must be expressed with the help of a certain antisymmetric tensor <math>~ {A^\mu}_\nu </math>:
:<math>~c \frac { d u^\mu }{d \tau } = {A^\mu}_\nu u^\nu . </math>
Based on the analysis of various types of motion, they estimated the required values of the acceleration tensor components, thereby giving indirect definition to this tensor.
From comparison with (4) it follows that the tensor <math>~ {A^\mu}_\nu </math> up to a sign and a constant multiplier coincides with the acceleration tensor <math> ~ {u^\alpha}_k </math> in case when rectilinear motion of a solid body without rotation is considered. Then indeed the four-potential of the acceleration field coincides with the four-velocity, <math>~ U_\mu = u_\mu </math>. As a result, the quantity <math>~ - J^k \partial_\alpha U_k =- \rho_0 u^k \partial_\alpha u_k </math> on the right-hand side of (4) vanishes, since the following relations hold true: <math>~ u^k u_k = c^2 </math>, <math>~ 2 u^k \partial_\alpha u_k = \partial_\alpha (u^k u_k) = \partial_\alpha c^2 =0 </math>. With this in mind, in (4) we can raise the index <math>~ \alpha </math> and cancel the mass density, which gives the following:
:<math> ~ - {u^\alpha}_k u^k =\frac {du^\alpha }{d \tau} .</math>
Mashhoon and Muench considered transformation of inertial reference frames, co-moving with the accelerated reference frame, and obtained the relation: <ref> Bahram Mashhoon and Uwe Muench. Length measurement in accelerated systems. Annalen der Physik. Vol. 11, Issue 7, P. 532–547, 2002. </ref>
:<math>~c \frac { d \lambda_\alpha }{d \tau } = {\Phi_\alpha}^\beta \lambda_\beta. </math>
The tensor <math>~ {\Phi_\alpha}^\beta </math> has the same properties as the acceleration tensor <math> ~ {u_\alpha}^\beta. </math>
== The use in the general theory of relativity ==
The action function in the [[general relativity]] (GR) can be represented as the sum of the four terms, which are responsible, respectively, for the spacetime metric, the matter in the form of substance, the electromagnetic field and the pressure field:
:<math>~ S = S_m + S_{mat} + S_{em} + S_p. </math>
Additional terms can be included in the action function, if other fields must be taken into account. The first, second and third terms of the action have the standard form: <ref> Fock, V. A. (1964). "The Theory of Space, Time and Gravitation". Macmillan. </ref>
:<math>~ S_m = \int (kR-2k \Lambda ) \sqrt {-g}d\Sigma.</math>
:<math>~ S_{mat} = \int ( - c \rho_0 ) \sqrt {-g}d\Sigma.</math>
:<math>~ S_{em} =\int ( - \frac {1}{c} A_\mu j^\mu - \frac {c \varepsilon_0}{4 } F_{\mu\nu}F^{\mu\nu} ) \sqrt {-g}d\Sigma,</math>
where <math> A_\mu </math> is the electromagnetic four-potential.
The term <math>~ S_p </math>, which is responsible for the contribution of pressure into the action function, is different in the works of different authors, depending on how the pressure is related to the elastic energy and whether the pressure field is considered to be a scalar field or a vector field. It should be noted that in the GR, the gravitational field is included in the action function not directly, but indirectly, by means of the metric tensor. In this case, as a rule, the pressure field is considered to be a scalar field.
In contrast, in the [[covariant theory of gravitation]] (CTG), the term <math>~ S_{ac} </math> associated with the acceleration field is used instead of the term <math>~ S_{mat} </math>, and the action function can be written as follows: <ref name="ac"/>
:<math>~ S = S_m + S_{ac} + S_{em} + S_p . </math>
Here
:<math>~ S_{ac} = \int ( - \frac {1}{c } U_\mu J^\mu - \frac {c}{ 16 \pi \eta } u_{\mu\nu}u^{\mu\nu} ) \sqrt {-g}d\Sigma , </math>
:<math>~ S_p =\int ( - \frac {1}{c } \pi_\mu J^\mu - \frac {c}{ 16 \pi \sigma } f_{\mu\nu}f^{\mu\nu} ) \sqrt {-g}d\Sigma,</math>
where <math> ~\pi_\mu </math> is the four-potential of the [[pressure field]], <math> ~ \sigma </math> is the coefficient of the pressure field, <math> ~ f_{\mu\nu}</math> is the [[pressure field tensor]], <math>J^\mu = \rho_{0} u^\mu </math>.
In the case of rectilinear motion of a rigid body without rotation, the following relations will hold: <math> U_\mu = u_\mu </math>, <math>~ u_\mu u^\mu = c^2 </math>, and in the term <math>~ S_{ac} </math> the relation <math>~ - \frac {1}{c } U_\mu J^\mu = - c \rho_0</math> is obtained.
In this particular case it is clear that the term <math>~ S_{ac} </math> differs from the term <math>~ S_{mat} </math> by an additional term associated with the energy of the acceleration field. This is due to the fact that in the CTG the acceleration field is considered to be a vector field, whereas as in the GR the acceleration field is actually used as a scalar field that does not depend on the particles’ velocities. In both theories, the acceleration field allows us to determine the contribution of the rest energy of the particles into the total energy of the system of particles and fields. However, the use of the acceleration field as a scalar field in the GR does not agree in its form with the vector nature of the electromagnetic field. Indeed, in the limiting case, when only the particles’ accelerations and electromagnetic forces are taken into account, the acceleration must be two-component, as is the case for the acceleration due to the action of the two-component [[Lorentz force]]. But this is possible only in the case, when the acceleration field is a vector field. The situation can be improved if, in addition to the gravitational field function, we ascribe to the metric field <math>~ g_{\mu \nu} </math> in the GR the function of the vector component of the acceleration field, but this makes the equations of the theory even more complex and complicated.
It should be noted that in the general case of arbitrary motion of the matter the relation <math>~ - \frac {1}{c } U_\mu J^\mu = - c \rho_0</math> is no longer satisfied and CTG does not coincide any more with GR in the method of describing the rest energy of a physical system. This means that in GR the motion of the matter is considered in a simplified way, as rectilinear motion of a solid body, whereas in CTG the use of the four-potential <math> U_\mu </math> of the acceleration field allows us to take into account the internal motion of the matter in each selected volume element. For example, when a particle moves round a circle, the four-potential <math> U_\mu </math> of the particle’s matter will depend on the location of this matter with respect to the circle line, since the velocity of the particle’s matter depends on the radius of rotation.
== See also ==
* [[General field]]
* [[Pressure field]]
* [[Dissipation field]]
* [[Covariant theory of gravitation]]
* [[Metric theory of relativity]]
* [[Acceleration tensor]]
* [[Acceleration stress-energy tensor]]
* [[Four-force]]
* [[Equation of vector field]]
== References ==
<references/>
==External links ==
* [http://www.wikiznanie.ru/ru-wz/index.php/%D0%9F%D0%BE%D0%BB%D0%B5_%D1%83%D1%81%D0%BA%D0%BE%D1%80%D0%B5%D0%BD%D0%B8%D0%B9 Acceleration field in Russian]
[[Category:Theoretical physics]]
[[Category:Concepts in physics]]
[[Category:Vector calculus]]
[[Category:Covariant theory of gravitation]]
[[Category: Metric theory of relativity]]
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'''Acceleration field''' is a two-component vector field, describing in a covariant way the [[four-acceleration]] of individual particles and the [[four-force]] that occurs in systems with multiple closely interacting particles. The acceleration field is a component of the [[Physics/Essays/Fedosin/General field |general field]], which is represented in the Lagrangian and Hamiltonian of an arbitrary physical system by the term with the energy of particles’ motion and the term with the field energy. <ref name="ko"> Fedosin S.G. [http://www.oalib.com/paper/5263035#.VuFYxn2LQsY The Concept of the General Force Vector Field]. OALib Journal, Vol. 3, pp. 1-15 (2016), e2459. http://dx.doi.org/10.4236/oalib.1102459. </ref> <ref> Fedosin S.G. Two components of the macroscopic general field. Reports in Advances of Physical Sciences, Vol. 1, No. 2, 1750002, 9 pages (2017). http://dx.doi.org/10.1142/S2424942417500025. </ref>
The acceleration field is included in most [[equation of vector field |equations of vector field]]. Moreover, the acceleration field enters into the equation of motion through the [[acceleration tensor]] and into the equation for the metric through the [[acceleration stress-energy tensor]].
The acceleration field was presented by [[user:Fedosin |Sergey Fedosin]] within the framework of the [[Physics/Essays/Fedosin/Metric theory of relativity|metric theory of relativity]] and [[covariant theory of gravitation]], and the equations of this field were obtained as a consequence of the [[w:principle of least action |principle of least action]]. <ref name="pr"> Fedosin S.G. [http://vixra.org/abs/1406.0135 The procedure of finding the stress-energy tensor and vector field equations of any form]. Advanced Studies in Theoretical Physics, Vol. 8, No. 18, pp. 771-779 (2014). http://dx.doi.org/10.12988/astp.2014.47101. </ref> <ref name="ac"> [[user:Fedosin | Fedosin S.G.]] [http://journals.yu.edu.jo/jjp/Vol9No1Contents2016.html About the cosmological constant, acceleration field, pressure field and energy.] Jordan Journal of Physics. Vol. 9, No. 1, pp. 1-30 (2016). http://dx.doi.org/10.5281/zenodo.889304. </ref>
== Mathematical description ==
The 4-potential of the acceleration field is expressed in terms of the scalar <math>~ \vartheta </math> and vector <math>~ \mathbf {U} </math> potentials:
:<math>~U_\mu = \left(\frac {\vartheta }{c},- \mathbf {U} \right) .</math>
The antisymmetric [[acceleration tensor]] is calculated with the help of the 4-curl of the 4-potential:
:<math>~ u_{\mu \nu} = \nabla_\mu U_\nu - \nabla_\nu U_\mu = \partial_\mu U_\nu - \partial_\nu U_\mu . </math>
The acceleration tensor components are the components of the field strength <math>~\mathbf {S} </math> and the components of the solenoidal vector <math>~\mathbf {N} </math>:
:<math> ~ u_{\mu \nu}= \begin{vmatrix} 0 & \frac {S_x}{ c} & \frac {S_y}{ c} & \frac {S_z}{ c} \\ -\frac {S_x}{ c} & 0 & - N_{z} & N_{y} \\ -\frac {S_y}{ c} & N_{z} & 0 & -N_{x} \\ -\frac {S_z}{ c}& -N_{y} & N_{x} & 0 \end{vmatrix}. </math>
We obtain the following:
:<math>~ \mathbf {S} = - \nabla \vartheta - \frac {\partial \mathbf {U}}{\partial t},\qquad\qquad \mathbf {N} = \nabla \times \mathbf {U}.\qquad\qquad (1) </math>
In the general case the scalar and vector potentials are found by solving the wave equations for the acceleration field potentials.
=== Action, Lagrangian and energy ===
In the covariant theory of gravitation the 4-potential <math>~U_\mu </math> of the acceleration field is part of the 4-potential of the [[Physics/Essays/Fedosin/General field |general field]] <math>~ s_\mu</math>, which is the sum of the 4-potentials of particular fields, such as the electromagnetic and gravitational fields, acceleration field, [[pressure field]], [[dissipation field]], strong interaction field, weak interaction field and other vector fields, acting on the matter and its particles. All of these fields are somehow represented in the matter, so that the 4-potential <math>~ s_\mu</math> cannot consist of only one 4-potential <math>~U_\mu </math>.
The energy density of interaction of the general field and the matter is given by the product of the 4-potential of the general field and the mass 4-current: <math>~ s_\mu J^\mu </math>. We obtain the general field tensor from the 4-potential of the general field, using the 4-curl:
:<math>~ s_{\mu \nu} =\nabla_\mu s_\nu - \nabla_\nu s_\mu.</math>
The tensor invariant in the form <math>~ s_{\mu \nu} s^{\mu \nu} </math> is up to a constant factor proportional to the energy density of the general field. As a result, the action function, which contains the scalar curvature <math>~R</math> and the cosmological constant <math>~ \Lambda </math>, is given by the expression: <ref name="ko"/>
:<math>~S =\int {L dt}=\int (kR-2k \Lambda - \frac {1}{c}s_\mu J^\mu - \frac {c}{16 \pi \varpi} s_{\mu\nu}s^{\mu\nu} ) \sqrt {-g}d\Sigma,</math>
where <math>~L </math> is the Lagrange function or Lagrangian; <math>~dt </math> is the time differential of the coordinate reference system; <math>~k </math> and <math>~ \varpi </math> are the constants to be determined; <math>~c </math> is the speed of light as a measure of the propagation speed of the electromagnetic and gravitational interactions; <math>~\sqrt {-g}d\Sigma= \sqrt {-g} c dt dx^1 dx^2 dx^3</math> is the invariant 4-volume expressed in terms of the differential of the time coordinate <math>~ dx^0=cdt </math>, the product <math>~ dx^1 dx^2 dx^3 </math> of differentials of the space coordinates and the square root <math>~\sqrt {-g} </math> of the determinant <math>~g </math> of the metric tensor, taken with a negative sign.
The variation of the action function gives the general field equations, the four-dimensional equation of motion and the equation for the metric. Since the acceleration field is the general field component, then from the general field equations the corresponding equations of the acceleration field are derived.
Given the gauge condition of the cosmological constant in the form
:<math>~ c k \Lambda = - s_\mu J^\mu ,</math>
is met, the system energy does not depend on the term with the scalar curvature and is uniquely determined: <ref name="ac"/>
:<math>~E = \int {( s_0 J^0 + \frac {c^2 }{16 \pi \varpi } s_{ \mu\nu} s^{ \mu\nu} ) \sqrt {-g} dx^1 dx^2 dx^3}, </math>
where <math>~ s_0 </math> and <math>~ J^0</math> denote the time components of the 4-vectors <math>~ s_{\mu } </math> and <math>~ J^{\mu } </math>.
The system’s 4-momentum is given by the formula:
:<math>~p^\mu = \left( \frac {E}{c}{,} \mathbf {p}\right) = \left( \frac {E}{c}{,} \frac {E}{c^2}\mathbf {v} \right) , </math>
where <math>~ \mathbf {p}</math> and <math>~ \mathbf {v}</math> denote the system’s momentum and the velocity of the system’s center of momentum.
=== Equations ===
The four-dimensional equations of the acceleration field are similar in their form to Maxwell equations and are as follows:
:<math> \nabla_\sigma u_{\mu \nu}+\nabla_\mu u_{\nu \sigma}+\nabla_\nu u_{\sigma \mu}=\frac{\partial u_{\mu \nu}}{\partial x^\sigma} + \frac{\partial u_{\nu \sigma}}{\partial x^\mu} + \frac{\partial u_{\sigma \mu}}{\partial x^\nu} = 0. </math>
:<math>~ \nabla_\nu u^{\mu \nu} = - \frac{4 \pi \eta }{c^2} J^\mu, </math>
where <math>J^\mu = \rho_{0} u^\mu </math> is the mass 4-current, <math> \rho_{0}</math> is the mass density in the co-moving reference frame, <math> u^\mu </math> is the 4-velocity of the matter unit, <math>~ \eta </math> is a constant, which is determined in each problem, and it is supposed that there is an equilibrium between all fields in the observed physical system.
The gauge condition of the 4-potential of the acceleration field:
:<math>~ \nabla^\mu U_{\mu} =0 . </math>
If the second equation with the field source is written with the covariant index in the following form:
:<math>~ \nabla^\nu u_{\mu \nu} = - \frac{4 \pi \eta }{c^2} J_\mu, </math>
then after substituting here the expression for the acceleration tensor <math> u_{\mu \nu} </math> through the 4-potential <math> ~ U_\mu </math> of the acceleration field we obtain the wave equation for calculating the potentials of the acceleration field:
:<math>~ \nabla^\nu \nabla_\nu U_\mu + R_{\mu \nu} U^\nu = \frac{4 \pi \eta }{c^2} J_\mu, </math>
where <math>~ R_{\mu \nu} </math> is the Ricci tensor.
The continuity equation in curved space-time is:
:<math>~ R_{ \mu \alpha } u^{\mu \alpha }= \frac {4 \pi \eta }{c^2} \nabla_{\alpha}J^{\alpha}.</math>
In Minkowski space of the special theory of relativity, the Ricci tensor is set to zero, the form of the acceleration field equations is simplified and they can be expressed in terms of the field strength <math>~\mathbf {S} </math> and the solenoidal vector <math>~\mathbf {N} </math>:
:<math>~ \nabla \cdot \mathbf{S} = 4 \pi \eta \gamma \rho_0, \qquad\qquad \nabla \times \mathbf{N} = \frac{1}{c^2} \left( 4 \pi \eta \mathbf{J} + \frac{\partial \mathbf{S}} {\partial t} \right), </math>
:<math>~ \nabla \times \mathbf{S} = - \frac{\partial \mathbf{N} } {\partial t} , \qquad\qquad \nabla \cdot \mathbf{N} = 0 .</math>
where <math>~ \gamma = \frac {1}{\sqrt{1 - {v^2 \over c^2}}} </math> is the Lorentz factor, <math>~ \mathbf{J}= \gamma \rho_0 \mathbf{v }</math> is the mass current density, <math>~ \mathbf{v } </math> is the velocity of the matter unit.
The wave equation is also simplified and can be written separately for the scalar and vector potentials:
:<math>~ \partial^\nu \partial_\nu \vartheta = \frac {1}{c^2}\frac{\partial^2 \vartheta }{\partial t^2 } -\Delta \vartheta = 4 \pi \eta \gamma \rho_0, \qquad\qquad (2) </math>
:<math>~ \partial^\nu \partial_\nu \mathbf{U} =\frac {1}{c^2}\frac{\partial^2 \mathbf{U} }{\partial t^2 } -\Delta \mathbf{U}= \frac {4 \pi \eta}{c^2} \mathbf{J}. \qquad\qquad (3) </math>
The equation of motion of the matter unit in the general field is given by the formula:
:<math>~ s_{\mu \nu} J^\nu =0 </math>.
Since <math>~ J^\nu = \rho_0 u^\nu </math>, and the general field tensor is expressed in terms of the tensors of particular fields, then the equation of motion can be represented with the help of these tensors:
:<math>~ - u_{\mu \nu} J^\nu =F_{\mu \nu} j^\nu + \Phi_{\mu \nu} J^\nu + f_{\mu \nu} J^\nu + h_{\mu \nu} J^\nu + \gamma_{\mu \nu} J^\nu + w_{\mu \nu} J^\nu .</math>
Here <math>~ F_{\mu \nu}</math> is the [[w:electromagnetic tensor |electromagnetic tensor]], <math>~ j^\nu </math> is the charge [[4-current]], <math>~ \Phi_{\mu \nu}</math> is the [[Physics/Essays/Fedosin/Gravitational tensor |gravitational tensor]], <math>~ f_{\mu \nu}</math> is the [[pressure field tensor]], <math>~ h_{\mu \nu}</math> is the [[dissipation field tensor]], <math>~ \gamma_{\mu \nu}</math> is the strong interaction field tensor, <math>~ w_{\mu \nu}</math> is the weak interaction field tensor.
=== The stress-energy tensor ===
The [[acceleration stress-energy tensor]] is calculated with the help of the acceleration tensor:
:<math>~ B^{ik} = \frac{c^2} {4 \pi \eta }\left( -g^{im} u_{n m} u^{n k}+ \frac{1} {4} g^{ik} u_{m r} u^{m r}\right) </math>.
We find as part of the tensor <math>~ B^{ik}</math> the 3-vector of the energy-momentum flux <math>~\mathbf {K} </math>, which is similar in its meaning to the [[w:Poynting vector |Poynting vector]] and the [[Physics/Essays/Fedosin/Heaviside vector |Heaviside vector]]. The vector <math>~\mathbf {K} </math> can be represented through the vector product of the field strength <math>~ \mathbf {S} </math> and the solenoidal vector <math>~ \mathbf {N} </math>:
:<math>~ \mathbf {K}=c B^{0i} = \frac {c^2}{4 \pi \eta }[\mathbf {S}\times \mathbf {N}],</math>
here the index is <math>~ i=1,2,3.</math>
The covariant derivative of the stress-energy tensor of the acceleration field with mixed indices specifies the [[4-force]] density:
:<math> ~ f_\alpha = \nabla_\beta {B_\alpha}^\beta = - u_{\alpha k} J^k = - \rho_0 u_{\alpha k}u^k = \rho_0 \frac {DU_\alpha }{D \tau}- J^k \nabla_\alpha U_k = \rho_0 \frac {dU_\alpha }{d \tau} - J^k \partial_\alpha U_k ,\qquad \qquad (4)</math>
where <math>~ D \tau </math> denotes the proper time differential in the curved spacetime.
The stress-energy tensor of the acceleration field is part of the stress-energy tensor of the general field <math>~ T^{ik} </math>:
:<math>~ T^{ik}= W^{ik}+ U^{ik}+ B^{ik}+ P^{ik} + Q^{ik}+ L^{ik}+ A^{ik}, </math>
where <math>~ W^{ik} </math> is the [[w:electromagnetic stress–energy tensor |electromagnetic stress–energy tensor]], <math>~ U^{ik}</math> is the [[gravitational stress-energy tensor]], <math>~ P^{ik}</math> is the [[pressure stress-energy tensor]], <math>~ Q^{ik}</math> is the [[dissipation stress-energy tensor]], <math>~ L^{ik}</math> is the strong interaction stress-energy tensor, <math>~ A^{ik} </math> is the weak interaction stress-energy tensor.
Through the tensor <math>~ T^{ik} </math> the stress-energy tensor of the acceleration field enters into the equation for the metric:
:<math>~ R^{ik} - \frac{1} {4 }g^{ik}R = \frac{8 \pi G \beta }{ c^4} T^{ik}, </math>
where <math>~ R^{ik} </math> is the Ricci tensor, <math>~ G </math> is the [[w:gravitational constant |gravitational constant]], <math>~ \beta </math> is a certain constant, and the gauge condition of the cosmological constant is used.
== Specific solutions for the acceleration field functions ==
The four-potential of any vector field, the global vector potential of which is equal to zero in the proper reference frame K', that is, in the center-of-momentum frame, in case of rectilinear motion in the laboratory reference frame K, can be presented as follows: <ref name="pr"/> <ref> Fedosin S.G. Equations of Motion in the Theory of Relativistic Vector Fields. International Letters of Chemistry, Physics and Astronomy, Vol. 83, pp. 12-30 (2019). https://doi.org/10.18052/www.scipress.com/ILCPA.83.12. </ref>
:<math>~ L_{\mu L} = \frac { k_f \varepsilon }{\rho_0 c^2} u_{\mu L},</math>
where <math>~ k_f = \frac {\rho_0}{\rho_{0q}}</math> is for the electromagnetic field and <math>~ k_f = 1</math> for the remaining fields; <math> ~ \rho_{0}</math> and <math> ~\rho_{0q}</math> are the invariant mass density and the charge density in the comoving reference frame, respectively; <math>~ \varepsilon </math> is the invariant energy density of the interaction, calculated as product of the four-potential of the field and the corresponding four-current; <math>~ u_{\mu L} </math> is the covariant four-velocity that determines the motion of the center of momentum of the physical system in K.
In the [[special relativity]] (SR), in the center-of-momentum frame K' the energy density is <math>~ \varepsilon = \gamma' \rho_0 c^2 </math>, where <math>~ \gamma' </math> is the Lorentz factor, and for the acceleration field, while the physical system is moving in K, the four-potential of the acceleration field will equal <math>~ U_{\mu L}= \gamma' u_{\mu L}</math>.
In case when the physical system is stationary in K, we will have <math>~ u_{\mu L} = (c,0,0,0) </math>, and consequently, the scalar potential will be <math>~ \vartheta = \gamma' c^2 </math>. If in the physical system, on the average, there are directed fluxes of matter or rotation of matter, the vector potential <math>~ \mathbf {U} </math> of the acceleration field is no longer equal to zero.
If the four-potential <math>~ U'_{\nu}</math> of acceleration field in K' is known, then in the laboratory reference frame K the four-potential is determined using the matrix <math>~ M_{\mu}^{\ \nu} </math> connecting the coordinates and time of both frames: <ref name="it"> Fedosin S.G. [http://dergipark.org.tr/gujs/issue/45480/435567 The Integral Theorem of the Field Energy.] Gazi University Journal of Science. Vol. 32, No. 2, pp. 686-703 (2019). http://dx.doi.org/10.5281/zenodo.3252783. </ref>
:<math>~ U_{\mu L}= M_{\mu}^{\ \nu} U'_{\nu}.</math>
In the special case of the system’s motion at the constant velocity <math>~ M_{\mu}^{\ \nu}</math> represents the Lorentz transformation matrix.
=== Ideally solid particle ===
In the approximation, when a particle is regarded as an ideally solid object, the matter inside the particle is motionless. It means that the Lorentz factor <math>~ \gamma' </math> of this matter in the center-of-momentum frame K' is equal to unity, so that the four-potential of the acceleration field becomes equal to the four-velocity of motion of the center of momentum:
:<math>~ U_\mu = u_\mu. </math>
In the SR, the expression for 4-velocity is simplified and we can write:
:<math>~U_\mu = \left( \frac {\vartheta }{c},- \mathbf {U} \right) = u_\mu = \left(\gamma c, - \gamma \mathbf {v} \right).</math>
The acceleration tensor components according to (1) will equal:
:<math>~ \mathbf {S} = - c^2 \nabla \gamma - \frac {\partial (\gamma \mathbf { v })}{\partial t},\qquad\qquad \mathbf {N} = \nabla \times (\gamma \mathbf { v }). </math>
Since in the solid-state motion equation for the four-acceleration with a covariant index <math>~ a_\mu </math> the relation holds
:<math>~ \rho_0 a_\mu = \rho_0 \frac {Du_\mu }{D \tau}= - u_{\mu \nu} J^\nu = - \rho_0 u_{\mu \nu} u^\nu, </math>
then in SR we obtain the following:
:<math>~ \frac {Du_\mu }{D \tau}= \frac {du_\mu }{d \tau} =\gamma \frac {du_\mu }{dt}, \qquad\qquad u^\nu =\left(\gamma c, \gamma \mathbf {v} \right), </math>
and the equations for the Lorentz factor <math>~ \gamma </math> and for the 3-acceleration <math>~ a= \frac {d \mathbf { v }}{dt} </math>:
:<math>~ \frac {d \gamma }{dt}= - \frac {1 }{c^2} \mathbf {S}\cdot \mathbf { v }, \qquad (5) \qquad \frac {d (\gamma \mathbf { v })}{dt}= \gamma \mathbf { a }+ \frac {d \gamma}{dt}\mathbf { v } = - \mathbf {S}- [\mathbf { v }\times \mathbf {N}]. \qquad (6) </math>
Multiplying equation (6) by the velocity <math>~ \mathbf { v }</math>, substituting the quantity <math>~ \mathbf {S}\cdot \mathbf { v } </math> from equation (5) to (6), taking into account relation <math>~\gamma^{-2}=1 - {v^2 \over c^2},</math> we find the well-known expression for the derivative of the Lorentz factor using the scalar product of the velocity and acceleration in SR:
:<math>~ \gamma^3 \mathbf {v}\cdot \mathbf { a }=c^2 \frac {d \gamma }{dt}.</math>
We can prove the validity of equation (6) by substituting in its right-hand side the expression for the strength and solenoidal vector:
:<math>~ \frac {d (\gamma \mathbf { v })}{dt}= c^2 \nabla \gamma + \frac {\partial (\gamma \mathbf { v })}{\partial t} - \mathbf { v }\times [ \nabla \times (\gamma \mathbf { v }) ] . \qquad\qquad (7) </math>
Indeed, the use of the [[w:material derivative |material derivative]] gives the following:
:<math>~ \frac {d (\gamma \mathbf { v })}{dt}= \frac {\partial (\gamma \mathbf { v })}{\partial t} + (\mathbf { v } \cdot \nabla) (\gamma \mathbf { v }) = \frac {\partial (\gamma \mathbf { v })}{\partial t}+\gamma (\mathbf { v } \cdot \nabla) \mathbf { v } + \mathbf { v } (\mathbf { v } \cdot \nabla\gamma) .</math>
In addition
:<math>~ - \mathbf { v }\times [ \nabla \times (\gamma \mathbf { v }) ] = - \gamma \mathbf { v }\times [ \nabla \times \mathbf { v } ] - \mathbf { v }\times [ \nabla \gamma \times \mathbf { v }] = -\frac {\gamma }{2} \nabla v^2 + \gamma (\mathbf { v } \cdot \nabla) \mathbf { v } - v^2 \nabla \gamma + \mathbf { v } (\mathbf { v } \cdot \nabla\gamma) .</math>
Substituting these relations in (7), taking into account the expression <math>~ \gamma^{-2}=1 - {v^2 \over c^2},</math> we obtain the identity:
:<math>~ c^2 \nabla \gamma - \frac {\gamma }{2} \nabla v^2 - v^2 \nabla \gamma =0 .</math>
If the components of the particle velocity are the functions of time and they do not directly depend on the space coordinates, then the solenoidal vector <math>~ \mathbf { N }</math> vanishes in such a motion.
In the SR <math>~ E = \gamma m c^2 </math> is the relativistic energy, <math>~ \mathbf p = \gamma m \mathbf v </math> is the 3-vector of relativistic momentum. If the mass <math>~ m </math> of a particle is constant, then multiplying (7) by the mass, we arrive to following equation for the force:
:<math>~ \mathbf F= \frac {d \mathbf p }{dt}= \nabla E + \frac {\partial \mathbf p }{\partial t} - \mathbf { v }\times [ \nabla \times \mathbf p ] . </math>
=== Rotation of a particle ===
For a small ideally solid particle, we can neglect the motion of the matter inside the particle and can assume that the four-potential of the acceleration field is equal to the four-velocity of the particle’s center of momentum.
Let us assume that the particle rotates about the axis OZ of the coordinate system at the distance <math>~ \rho = \sqrt {x^2 +y^2} </math> from the axis at the constant angular velocity <math>~ \omega</math> counterclockwise, as viewed from the side, in which the OZ axis is directed. Then we can assume that the linear velocity of the particle depends only on the coordinates <math>~ x</math> and <math>~ y</math>, and for the velocity’s projections on the axes of the coordinate system we can write: <math>~ \mathbf v = (-\omega y, \omega x, 0) </math>, while the square of the velocity equals <math>~ v^2 = \omega^2 (x^2 + y^2) </math>. For the Lorentz factor in the SR we obtain the following:
:<math>~ \gamma = \frac {1}{\sqrt {1- \frac { v^2}{ c^2}}} = \frac {1}{\sqrt {1- \frac { \omega^2 (x^2 + y^2)}{ c^2}}} . </math>
With this in mind, the potentials and field strengths of the acceleration field can be written as follows:
:<math>~ \vartheta = \gamma c^2, \qquad \mathbf {U} = \gamma \mathbf {v}. </math>
:<math>~ \mathbf {S} = - \nabla \vartheta - \frac {\partial \mathbf {U} }{\partial t}= \left( -\gamma^3 \omega^2 x, -\gamma^3 \omega^2 y, 0 \right). </math>
:<math>~ \mathbf {N} = \nabla \times \mathbf {U} = \left( 0, 0, \gamma \omega +\gamma^3 \omega \right). </math>
If we substitute <math>~ \gamma </math>, <math>~ \mathbf v </math>, <math>~\mathbf {S} </math> and <math>~\mathbf {N} </math> in (6), we can determine the acceleration components of the particle and the acceleration amplitude:
:<math>~ \mathbf {a} = \left( - \omega^2 x , -\omega^2 y, 0 \right). </math>
:<math>~ a = \sqrt {a^2_x +a^2_y +a^2_z} = \omega^2 \sqrt {x^2 +y^2}= \omega^2 \rho =\omega v = \frac {v^2} {\rho }. </math>
The acceleration is directed towards the center of rotation and represents [[centripetal acceleration]]. Using now the classic vector description, we have also for the time and coordinates of reference frame at the center of rotation:
:<math>~ \vec \rho = (x, y, 0) , \qquad \vec \omega = \frac {\vec {d\varphi} }{dt} =(0, 0, \omega) , </math>
:<math>~ \mathbf {v} = [\vec \omega \times \vec \rho] , \qquad \mathbf {a} = [\vec \omega \times \mathbf {v}] = [\vec \omega \times [\vec \omega \times \vec \rho]] = \vec \omega (\vec \omega \cdot \vec \rho) - \vec \rho (\vec \omega \cdot \vec \omega) = - \omega^2 \vec \rho , </math>
where <math>~ \rho </math> and <math>~ \varphi </math> are two coordinates of the [[Coordinate systems#Cylindrical coordinates.5B4.5D |cylindrical coordinate system]], <math>~ \vec \rho </math> is the vector from the center of rotation to the particle, <math>~ \vec {d\varphi}</math> is the axial vector of the differential of the rotation angle directed along the axis OZ.
As we can see, in case of such a motion with acceleration the vector product <math>~ [\mathbf {S}\times \mathbf {N}]</math> is not equal to zero, just as the three-vector <math>~ \mathbf {K}</math> of the energy-momentum flux of the acceleration field inside the particle.
=== The system of particles ===
Due to interaction of a number of particles with each other by means of various fields, including interaction at a distance without direct contact, the acceleration field in the matter changes and is different from the acceleration field of individual particles at the observation point. As a result, the density of the 4-force in the system of particles is given by the strength and the solenoidal vector, which represent the typical average characteristics of the matter motion. For example, in a gravitationally bound system there is a radial gradient of the vector <math>~ \mathbf { S },</math> and if the system is moving or rotating, there is a vector <math>~ \mathbf { N }.</math> From (4) there follows the general expression for the the density of the 4-force with covariant index:
:<math> ~ f_\nu = \rho_0 \frac {cdt}{ds}\left(-\frac {1}{c} \mathbf{S} \cdot \mathbf{v}{,} \qquad \mathbf{S}+[\mathbf{v} \times \mathbf{N}] \right),</math>
where <math> ~ ds </math> denotes a four-dimensional space-time interval.
For a stationary case, when the potentials of the acceleration field are independent of time, under the assumption that <math>~ \vartheta = \gamma c^2, </math> wave equation (2) for the scalar potential in the SR is transformed into the equation:
:<math>~ \Delta \gamma= - \frac {4 \pi \eta \gamma \rho_0}{c^2}. </math>
The solution of this equation for a fixed sphere with the particles randomly moving in it has the form: <ref name="ie"> Fedosin S.G. [http://vixra.org/abs/1403.0973 The Integral Energy-Momentum 4-Vector and Analysis of 4/3 Problem Based on the Pressure Field and Acceleration Field.] American Journal of Modern Physics. Vol. 3, No. 4, pp. 152-167 (2014). http://dx.doi.org/10.11648/j.ajmp.20140304.12.</ref>
:<math>~\gamma= \frac {c \gamma_c }{r \sqrt {4 \pi \eta \rho_0} } \sin \left(\frac {r}{c}\sqrt {4 \pi \eta \rho_0} \right) \approx \gamma_c - \frac {2 \pi \eta \rho_0 r^2 \gamma_c }{3 c^2}.</math>
where <math>~ \gamma_c = \frac {1}{\sqrt{1 - {v^2_c \over c^2}}} </math> is the Lorentz factor for the velocities <math>~ v_c</math> of the particles in the center of the sphere, and due to the smallness of the argument the sine is expanded to the second order terms. From the formula it follows that the average velocities of the particles are maximal in the center and decrease when approaching the surface of the sphere.
In such a system, the scalar potential <math>~ \vartheta</math> becomes the function of the radius, and the vector potential <math>~ \mathbf {U} </math> and the solenoidal vector <math>~ \mathbf { N }</math> are equal to zero. The acceleration field strength <math>~\mathbf {S} </math> is found with the help of (1). Then we can calculate all the functions of the acceleration field, including the energy of particles in this field and the energy of the acceleration field itself. <ref> Fedosin S.G. [http://journals.yu.edu.jo/jjp/Vol8No1Contents2015.html Relativistic Energy and Mass in the Weak Field Limit.] Jordan Journal of Physics. Vol. 8, No. 1, pp. 1-16 (2015). http://dx.doi.org/10.5281/zenodo.889210.</ref> For cosmic bodies the main contribution to the four-acceleration in the matter makes the gravitational force and the pressure field.
At the same time the relativistic rest energy of the system is automatically derived, taking into account the motion of particles inside the sphere. For the system of particles with the acceleration field, pressure field, gravitational and electromagnetic fields the given approach allowed solving the 4/3 problem and showed where and in what form the energy of the system is contained. The relation for the acceleration field constant in this problem was found:
:<math>~\eta = 3G- \frac {3q^2}{4 \pi \varepsilon_0 m^2 },</math>
where <math>~ \varepsilon_0</math> is the [[electric constant]], <math>~q </math> and <math>~m </math> are the total charge and mass of the system.
The solution of the wave equation for the acceleration field within the system results in temperature distribution according to the formula: <ref name="ie"/>
:<math>~ T=T_c - \frac {\eta M_p M(r)}{3kr} ,</math>
where <math>~ T_c </math> is the temperature in the center, <math>~ M_p </math> is the mass of the particle, for which the mass of the proton is taken (for systems which are based on hydrogen or nucleons in atomic nuclei), <math>~ M(r) </math> is the mass of the system within the current radius <math>~ r </math>, <math>~ k</math> is the Boltzmann constant.
This dependence is well satisfied for a variety of space objects, including gas clouds and Bok globules, the Earth, the Sun and neutron stars.
In articles <ref> Fedosin S.G. [http://www.nrcresearchpress.com/doi/10.1139/cjp-2015-0593#.Vv3piZyLQsY Estimation of the physical parameters of planets and stars in the gravitational equilibrium model]. Canadian Journal of Physics, Vol. 94, No. 4, pp. 370-379 (2016). http://dx.doi.org/10.1139/cjp-2015-0593.</ref> <ref>Fedosin S.G. The generalized Poynting theorem for the general field and solution of the 4/3 problem. International Frontier Science Letters, Vol. 14, pp. 19-40 (2019). https://doi.org/10.18052/www.scipress.com/IFSL.14.19. </ref> the ratio of the field’s coefficients for the fields was specified as follows:
:<math>~\eta + \sigma = G - \frac {\rho^2_{0q}}{4 \pi \varepsilon_0 \rho^2_{0}},</math>
where <math> ~ \sigma </math> is the pressure field constant.
If we introduce the parameter <math> ~ \mu </math> as the number of nucleons per ionized gas particle, then the acceleration field constant is expressed as follows:
:<math>~\eta = \frac {3\gamma_c \mu G}{2+ 3 \gamma_c \mu }.</math>
For the temperature inside the cosmic bodies in the gravitational equilibrium model we find the dependence on the current radius:
:<math>~ T=T_c - \frac {4 \pi \eta m_u \rho_{0c}\gamma_c r^2}{9k}+ \frac {2 \pi \eta A m_u \gamma_c r^3}{9k} + \frac {2 \pi \eta B m_u \gamma_c r^4}{15k} ,</math>
where <math> ~ m_u </math> is the mass of one gas particle, which is taken as the [[w:unified atomic mass unit |unified atomic mass unit]], and the coefficients <math> ~ A </math> and <math> ~ B </math> are included into the dependence of the mass density on the radius in the relation <math> ~ \rho_0 = \rho_{0c}- Ar - Br^2. </math>
Under the assumption that the system’s typical particles have the mass <math> ~\stackrel{-}{m } = \mu m_u </math>, and that it is typical particles that define the temperature and pressure, for the acceleration field constant we obtain the following: <ref>Fedosin S.G. The binding energy and the total energy of a macroscopic body in the relativistic uniform model. Middle East Journal of Science, Vol. 5, Issue 1, pp. 46-62 (2019). http://dx.doi.org/10.23884/mejs.2019.5.1.06. </ref>
:<math>~ \eta = \frac {3}{5} \left( G- \frac {\rho^2_{0q}}{ 4 \pi \varepsilon_0 \rho^2_0 } \right) .</math>
The Lorentz factor of the particles in the center of the system is also determined: <ref name="en"> Fedosin S.G. Energy and metric gauging in the covariant theory of gravitation. Aksaray University Journal of Science and Engineering, Vol. 2, Issue 2, pp. 127-143 (2018). http://dx.doi.org/10.29002/asujse.433947. </ref>
: <math>~\gamma_c = \frac {1}{\sqrt {1- \frac { v^2_c }{c^2}}} \approx 1+ \frac { v^2_c }{2c^2} +\frac {3 v^4_c }{8c^4} \approx 1+ \frac {3 \eta m}{10 a c^2} \left( 1+\frac {9}{2\sqrt {14}} \right) + \frac {27 \eta^2 m^2}{200 a^2 c^4} \left( 1+\frac {9}{2\sqrt {14}} \right)^2 . </math>
The wave equation (3) for the vector potential of the acceleration field was used to represent the relativistic equation of the fluid’s motion in the form of the [[w:Navier–Stokes equations |Navier–Stokes equations]] in hydrodynamics and to describe the motion of the viscous compressible and charged fluid. <ref> Fedosin S.G. Four-Dimensional Equation of Motion for Viscous Compressible and Charged Fluid with Regard to the Acceleration Field, Pressure Field and Dissipation Field. International Journal of Thermodynamics. Vol. 18, No. 1, pp. 13-24 (2015). http://dx.doi.org/10.5541/ijot.5000034003.</ref>
Taking into account the acceleration field and pressure field, within the framework of the [[Physics/Essays/Fedosin/Relativistic uniform system |relativistic uniform system]], it is possible to refine the [[w:virial theorem |virial theorem]], which in the relativistic form is written as follows: <ref>Fedosin S.G. The virial theorem and the kinetic energy of particles of a macroscopic system in the general field concept. Continuum Mechanics and Thermodynamics. Vol. 29, Issue 2, pp. 361-371 (2017). https://dx.doi.org/10.1007/s00161-016-0536-8. </ref>
: <math>~ \langle W_k \rangle \approx - 0.6 \sum_{k=1}^N\langle\mathbf{F}_k\cdot\mathbf{r}_k\rangle ,</math>
where the value <math>~ W_k \approx \gamma_c T </math> exceeds the kinetic energy of the particles <math>~ T </math> by a factor equal to the Lorentz factor <math>~ \gamma_c </math> of the particles at the center of the system. Under normal conditions we can assume that <math>~ \gamma_c \approx 1 </math>, then we can see that in the virial theorem the kinetic energy is related to the potential energy not by the coefficient 0.5, but rather by the coefficient close to 0.6. The difference from the classical case arises due to considering the pressure field and the acceleration field of particles inside the system, while the derivative of the virial scalar function <math>~ G_v </math> is not equal to zero and should be considered as the [[w:material derivative |material derivative]].
An analysis of the integral theorem of generalized virial makes it possible to find, on the basis of field theory, a formula for the root-mean-square speed of typical particles of a system without using the notion of temperature: <ref> Fedosin S.G. [http://em.rdcu.be/wf/click?upn=lMZy1lernSJ7apc5DgYM8f7AyOIJlVFO4uFv7zUQtzk-3D_DUeisO4Ue44lkDmCnrWVhK-2BAxKrUexyqlYtsmkyhvEp5zr527MDdThwbadScvhwZehXbanab8i5hqRa42b-2FKYwacOeM4LKDJeJuGA15M9FWvYOfBgfon7Bqg2f55NFYGJfVGaGhl0ghU-2BkIJ9Hz4M6SMBYS-2Fr-2FWWaj9eTxv23CKo9d8nFmYAbMtBBskFuW9fupsvIvN5eyv-2Fk-2BUc7hiS15rRISs1jpNnRQpDtk2OE9Hr6mYYe5Y-2B8lunO9GwVRw07Y1mdAqqtEZ-2BQjk5xUwPnA-3D-3D The integral theorem of generalized virial in the relativistic uniform model]. Continuum Mechanics and Thermodynamics, Vol. 31, Issue 3, pp. 627-638 (2019). https://dx.doi.org/10.1007/s00161-018-0715-x.</ref>
:<math> v_\mathrm{rms} = c \sqrt{1- \frac {4 \pi \eta \rho_0 r^2}{c^2 \gamma^2_c \sin^2 {\left( \frac {r}{c} \sqrt {4 \pi \eta \rho_0} \right) } } } .</math>
The integral [[field energy theorem]] for acceleration field in a curved space-time is as follows:<ref name="it"/>
:<math>~ - \int { \left( \frac {8 \pi \eta }{c^2} U_\alpha J^\alpha + u_{\alpha \beta} u^{\alpha \beta} \right) \sqrt {-g} dx^1 dx^2 dx^3 } = \frac {2}{c} \frac {d}{dt} \left( \int { U^\alpha u_\alpha ^{\ 0} \sqrt {-g} dx^1 dx^2 dx^3} \right) + 2 \iint \limits_S {U^\alpha u_\alpha ^{\ k} n_k \sqrt {-g} dS} . </math>
In the relativistic uniform system, the scalar potential <math>~\vartheta </math> of the acceleration field is related to the scalar potential <math>~\wp </math> of the pressure field:
<ref>Fedosin S.G. [https://rdcu.be/ccV9o The potentials of the acceleration field and pressure field in rotating relativistic uniform system]. Continuum Mechanics and Thermodynamics, Vol. 33, Issue 3, pp. 817-834 (2021). https://doi.org/10.1007/s00161-020-00960-7. </ref>
:<math>~ \wp = \frac {\sigma (\vartheta -c^2)}{ \eta } = \frac {2 (\vartheta -c^2)}{ 3 }. </math>
The relativistic expression for pressure is as follows:
<math> p = \frac{2\rho c^2 (\gamma - 1) }{3}= \frac {2 \rho c^2 }{3} \left( \frac {1}{\sqrt {1- v^2/ c^2 }}-1 \right) \approx \frac {\rho v^2}{3}, </math>
where <math>\rho </math> is the mass density of moving matter, <math> c </math> is the speed of light, <math> \gamma =\frac {1}{\sqrt {1- v^2/ c^2 }} </math> is the [[w:Lorentz factor |Lorentz factor]].
In the limit of low velocities, this relationship turns into the standard formula of the [[w:kinetic theory of gases |kinetic theory of gases]].
== Other approaches ==
Studying the Lorentz covariance of the 4-force, Friedman and Scarr found incomplete covariance of the expression for the 4-force in the form <math>~ F^\mu = \frac {d p^\mu }{d \tau } . </math> <ref> Yaakov Friedman and Tzvi Scarr. [http://iopscience.iop.org/1742-6596/437/1/012009 Covariant Uniform Acceleration]. Journal of Physics: Conference Series Vol. 437 (2013) 012009 doi:10.1088/1742-6596/437/1/012009. </ref>
This led them to conclude that the four-acceleration in SR must be expressed with the help of a certain antisymmetric tensor <math>~ {A^\mu}_\nu </math>:
:<math>~c \frac { d u^\mu }{d \tau } = {A^\mu}_\nu u^\nu . </math>
Based on the analysis of various types of motion, they estimated the required values of the acceleration tensor components, thereby giving indirect definition to this tensor.
From comparison with (4) it follows that the tensor <math>~ {A^\mu}_\nu </math> up to a sign and a constant multiplier coincides with the acceleration tensor <math> ~ {u^\alpha}_k </math> in case when rectilinear motion of a solid body without rotation is considered. Then indeed the four-potential of the acceleration field coincides with the four-velocity, <math>~ U_\mu = u_\mu </math>. As a result, the quantity <math>~ - J^k \partial_\alpha U_k =- \rho_0 u^k \partial_\alpha u_k </math> on the right-hand side of (4) vanishes, since the following relations hold true: <math>~ u^k u_k = c^2 </math>, <math>~ 2 u^k \partial_\alpha u_k = \partial_\alpha (u^k u_k) = \partial_\alpha c^2 =0 </math>. With this in mind, in (4) we can raise the index <math>~ \alpha </math> and cancel the mass density, which gives the following:
:<math> ~ - {u^\alpha}_k u^k =\frac {du^\alpha }{d \tau} .</math>
Mashhoon and Muench considered transformation of inertial reference frames, co-moving with the accelerated reference frame, and obtained the relation: <ref> Bahram Mashhoon and Uwe Muench. Length measurement in accelerated systems. Annalen der Physik. Vol. 11, Issue 7, P. 532–547, 2002. </ref>
:<math>~c \frac { d \lambda_\alpha }{d \tau } = {\Phi_\alpha}^\beta \lambda_\beta. </math>
The tensor <math>~ {\Phi_\alpha}^\beta </math> has the same properties as the acceleration tensor <math> ~ {u_\alpha}^\beta. </math>
== The use in the general theory of relativity ==
The action function in the [[general relativity]] (GR) can be represented as the sum of the four terms, which are responsible, respectively, for the spacetime metric, the matter in the form of substance, the electromagnetic field and the pressure field:
:<math>~ S = S_m + S_{mat} + S_{em} + S_p. </math>
Additional terms can be included in the action function, if other fields must be taken into account. The first, second and third terms of the action have the standard form: <ref> Fock, V. A. (1964). "The Theory of Space, Time and Gravitation". Macmillan. </ref>
:<math>~ S_m = \int (kR-2k \Lambda ) \sqrt {-g}d\Sigma.</math>
:<math>~ S_{mat} = \int ( - c \rho_0 ) \sqrt {-g}d\Sigma.</math>
:<math>~ S_{em} =\int ( - \frac {1}{c} A_\mu j^\mu - \frac {c \varepsilon_0}{4 } F_{\mu\nu}F^{\mu\nu} ) \sqrt {-g}d\Sigma,</math>
where <math> A_\mu </math> is the electromagnetic four-potential.
The term <math>~ S_p </math>, which is responsible for the contribution of pressure into the action function, is different in the works of different authors, depending on how the pressure is related to the elastic energy and whether the pressure field is considered to be a scalar field or a vector field. It should be noted that in the GR, the gravitational field is included in the action function not directly, but indirectly, by means of the metric tensor. In this case, as a rule, the pressure field is considered to be a scalar field.
In contrast, in the [[covariant theory of gravitation]] (CTG), the term <math>~ S_{ac} </math> associated with the acceleration field is used instead of the term <math>~ S_{mat} </math>, and the action function can be written as follows: <ref name="ac"/>
:<math>~ S = S_m + S_{ac} + S_{em} + S_p . </math>
Here
:<math>~ S_{ac} = \int ( - \frac {1}{c } U_\mu J^\mu - \frac {c}{ 16 \pi \eta } u_{\mu\nu}u^{\mu\nu} ) \sqrt {-g}d\Sigma , </math>
:<math>~ S_p =\int ( - \frac {1}{c } \pi_\mu J^\mu - \frac {c}{ 16 \pi \sigma } f_{\mu\nu}f^{\mu\nu} ) \sqrt {-g}d\Sigma,</math>
where <math> ~\pi_\mu </math> is the four-potential of the [[pressure field]], <math> ~ \sigma </math> is the coefficient of the pressure field, <math> ~ f_{\mu\nu}</math> is the [[pressure field tensor]], <math>J^\mu = \rho_{0} u^\mu </math>.
In the case of rectilinear motion of a rigid body without rotation, the following relations will hold: <math> U_\mu = u_\mu </math>, <math>~ u_\mu u^\mu = c^2 </math>, and in the term <math>~ S_{ac} </math> the relation <math>~ - \frac {1}{c } U_\mu J^\mu = - c \rho_0</math> is obtained.
In this particular case it is clear that the term <math>~ S_{ac} </math> differs from the term <math>~ S_{mat} </math> by an additional term associated with the energy of the acceleration field. This is due to the fact that in the CTG the acceleration field is considered to be a vector field, whereas as in the GR the acceleration field is actually used as a scalar field that does not depend on the particles’ velocities. In both theories, the acceleration field allows us to determine the contribution of the rest energy of the particles into the total energy of the system of particles and fields. However, the use of the acceleration field as a scalar field in the GR does not agree in its form with the vector nature of the electromagnetic field. Indeed, in the limiting case, when only the particles’ accelerations and electromagnetic forces are taken into account, the acceleration must be two-component, as is the case for the acceleration due to the action of the two-component [[Lorentz force]]. But this is possible only in the case, when the acceleration field is a vector field. The situation can be improved if, in addition to the gravitational field function, we ascribe to the metric field <math>~ g_{\mu \nu} </math> in the GR the function of the vector component of the acceleration field, but this makes the equations of the theory even more complex and complicated.
It should be noted that in the general case of arbitrary motion of the matter the relation <math>~ - \frac {1}{c } U_\mu J^\mu = - c \rho_0</math> is no longer satisfied and CTG does not coincide any more with GR in the method of describing the rest energy of a physical system. This means that in GR the motion of the matter is considered in a simplified way, as rectilinear motion of a solid body, whereas in CTG the use of the four-potential <math> U_\mu </math> of the acceleration field allows us to take into account the internal motion of the matter in each selected volume element. For example, when a particle moves round a circle, the four-potential <math> U_\mu </math> of the particle’s matter will depend on the location of this matter with respect to the circle line, since the velocity of the particle’s matter depends on the radius of rotation.
== See also ==
* [[General field]]
* [[Pressure field]]
* [[Dissipation field]]
* [[Covariant theory of gravitation]]
* [[Metric theory of relativity]]
* [[Acceleration tensor]]
* [[Acceleration stress-energy tensor]]
* [[Four-force]]
* [[Equation of vector field]]
== References ==
<references/>
==External links ==
* [http://www.wikiznanie.ru/ru-wz/index.php/%D0%9F%D0%BE%D0%BB%D0%B5_%D1%83%D1%81%D0%BA%D0%BE%D1%80%D0%B5%D0%BD%D0%B8%D0%B9 Acceleration field in Russian]
[[Category:Theoretical physics]]
[[Category:Concepts in physics]]
[[Category:Vector calculus]]
[[Category:Covariant theory of gravitation]]
[[Category: Metric theory of relativity]]
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'''Acceleration field''' is a two-component vector field, describing in a covariant way the [[four-acceleration]] of individual particles and the [[four-force]] that occurs in systems with multiple closely interacting particles. The acceleration field is a component of the [[Physics/Essays/Fedosin/General field |general field]], which is represented in the Lagrangian and Hamiltonian of an arbitrary physical system by the term with the energy of particles’ motion and the term with the field energy. <ref name="ko"> Fedosin S.G. [http://www.oalib.com/paper/5263035#.VuFYxn2LQsY The Concept of the General Force Vector Field]. OALib Journal, Vol. 3, pp. 1-15 (2016), e2459. http://dx.doi.org/10.4236/oalib.1102459. </ref> <ref> Fedosin S.G. Two components of the macroscopic general field. Reports in Advances of Physical Sciences, Vol. 1, No. 2, 1750002, 9 pages (2017). http://dx.doi.org/10.1142/S2424942417500025. </ref>
The acceleration field is included in most [[equation of vector field |equations of vector field]]. Moreover, the acceleration field enters into the equation of motion through the [[acceleration tensor]] and into the equation for the metric through the [[acceleration stress-energy tensor]].
The acceleration field was presented by [[user:Fedosin |Sergey Fedosin]] within the framework of the [[Physics/Essays/Fedosin/Metric theory of relativity|metric theory of relativity]] and [[covariant theory of gravitation]], and the equations of this field were obtained as a consequence of the [[w:principle of least action |principle of least action]]. <ref name="pr"> Fedosin S.G. [http://vixra.org/abs/1406.0135 The procedure of finding the stress-energy tensor and vector field equations of any form]. Advanced Studies in Theoretical Physics, Vol. 8, No. 18, pp. 771-779 (2014). http://dx.doi.org/10.12988/astp.2014.47101. </ref> <ref name="ac"> [[user:Fedosin | Fedosin S.G.]] [http://journals.yu.edu.jo/jjp/Vol9No1Contents2016.html About the cosmological constant, acceleration field, pressure field and energy.] Jordan Journal of Physics. Vol. 9, No. 1, pp. 1-30 (2016). http://dx.doi.org/10.5281/zenodo.889304. </ref>
== Mathematical description ==
The 4-potential of the acceleration field is expressed in terms of the scalar <math>~ \vartheta </math> and vector <math>~ \mathbf {U} </math> potentials:
:<math>~U_\mu = \left(\frac {\vartheta }{c},- \mathbf {U} \right) .</math>
The antisymmetric [[acceleration tensor]] is calculated with the help of the 4-curl of the 4-potential:
:<math>~ u_{\mu \nu} = \nabla_\mu U_\nu - \nabla_\nu U_\mu = \partial_\mu U_\nu - \partial_\nu U_\mu . </math>
The acceleration tensor components are the components of the field strength <math>~\mathbf {S} </math> and the components of the solenoidal vector <math>~\mathbf {N} </math>:
:<math> ~ u_{\mu \nu}= \begin{vmatrix} 0 & \frac {S_x}{ c} & \frac {S_y}{ c} & \frac {S_z}{ c} \\ -\frac {S_x}{ c} & 0 & - N_{z} & N_{y} \\ -\frac {S_y}{ c} & N_{z} & 0 & -N_{x} \\ -\frac {S_z}{ c}& -N_{y} & N_{x} & 0 \end{vmatrix}. </math>
We obtain the following:
:<math>~ \mathbf {S} = - \nabla \vartheta - \frac {\partial \mathbf {U}}{\partial t},\qquad\qquad \mathbf {N} = \nabla \times \mathbf {U}.\qquad\qquad (1) </math>
In the general case the scalar and vector potentials are found by solving the wave equations for the acceleration field potentials.
=== Action, Lagrangian and energy ===
In the covariant theory of gravitation the 4-potential <math>~U_\mu </math> of the acceleration field is part of the 4-potential of the [[Physics/Essays/Fedosin/General field |general field]] <math>~ s_\mu</math>, which is the sum of the 4-potentials of particular fields, such as the electromagnetic and gravitational fields, acceleration field, [[pressure field]], [[dissipation field]], strong interaction field, weak interaction field and other vector fields, acting on the matter and its particles. All of these fields are somehow represented in the matter, so that the 4-potential <math>~ s_\mu</math> cannot consist of only one 4-potential <math>~U_\mu </math>.
The energy density of interaction of the general field and the matter is given by the product of the 4-potential of the general field and the mass 4-current: <math>~ s_\mu J^\mu </math>. We obtain the general field tensor from the 4-potential of the general field, using the 4-curl:
:<math>~ s_{\mu \nu} =\nabla_\mu s_\nu - \nabla_\nu s_\mu.</math>
The tensor invariant in the form <math>~ s_{\mu \nu} s^{\mu \nu} </math> is up to a constant factor proportional to the energy density of the general field. As a result, the action function, which contains the scalar curvature <math>~R</math> and the cosmological constant <math>~ \Lambda </math>, is given by the expression: <ref name="ko"/>
:<math>~S =\int {L dt}=\int (kR-2k \Lambda - \frac {1}{c}s_\mu J^\mu - \frac {c}{16 \pi \varpi} s_{\mu\nu}s^{\mu\nu} ) \sqrt {-g}d\Sigma,</math>
where <math>~L </math> is the Lagrange function or Lagrangian; <math>~dt </math> is the time differential of the coordinate reference system; <math>~k </math> and <math>~ \varpi </math> are the constants to be determined; <math>~c </math> is the speed of light as a measure of the propagation speed of the electromagnetic and gravitational interactions; <math>~\sqrt {-g}d\Sigma= \sqrt {-g} c dt dx^1 dx^2 dx^3</math> is the invariant 4-volume expressed in terms of the differential of the time coordinate <math>~ dx^0=cdt </math>, the product <math>~ dx^1 dx^2 dx^3 </math> of differentials of the space coordinates and the square root <math>~\sqrt {-g} </math> of the determinant <math>~g </math> of the metric tensor, taken with a negative sign.
The variation of the action function gives the general field equations, the four-dimensional equation of motion and the equation for the metric. Since the acceleration field is the general field component, then from the general field equations the corresponding equations of the acceleration field are derived.
Given the gauge condition of the cosmological constant in the form
:<math>~ c k \Lambda = - s_\mu J^\mu ,</math>
is met, the system energy does not depend on the term with the scalar curvature and is uniquely determined: <ref name="ac"/>
:<math>~E = \int {( s_0 J^0 + \frac {c^2 }{16 \pi \varpi } s_{ \mu\nu} s^{ \mu\nu} ) \sqrt {-g} dx^1 dx^2 dx^3}, </math>
where <math>~ s_0 </math> and <math>~ J^0</math> denote the time components of the 4-vectors <math>~ s_{\mu } </math> and <math>~ J^{\mu } </math>.
The system’s 4-momentum is given by the formula:
:<math>~p^\mu = \left( \frac {E}{c}{,} \mathbf {p}\right) = \left( \frac {E}{c}{,} \frac {E}{c^2}\mathbf {v} \right) , </math>
where <math>~ \mathbf {p}</math> and <math>~ \mathbf {v}</math> denote the system’s momentum and the velocity of the system’s center of momentum.
=== Equations ===
The four-dimensional equations of the acceleration field are similar in their form to Maxwell equations and are as follows:
:<math> \nabla_\sigma u_{\mu \nu}+\nabla_\mu u_{\nu \sigma}+\nabla_\nu u_{\sigma \mu}=\frac{\partial u_{\mu \nu}}{\partial x^\sigma} + \frac{\partial u_{\nu \sigma}}{\partial x^\mu} + \frac{\partial u_{\sigma \mu}}{\partial x^\nu} = 0. </math>
:<math>~ \nabla_\nu u^{\mu \nu} = - \frac{4 \pi \eta }{c^2} J^\mu, </math>
where <math>J^\mu = \rho_{0} u^\mu </math> is the mass 4-current, <math> \rho_{0}</math> is the mass density in the co-moving reference frame, <math> u^\mu </math> is the 4-velocity of the matter unit, <math>~ \eta </math> is a constant, which is determined in each problem, and it is supposed that there is an equilibrium between all fields in the observed physical system.
The gauge condition of the 4-potential of the acceleration field:
:<math>~ \nabla^\mu U_{\mu} =0 . </math>
If the second equation with the field source is written with the covariant index in the following form:
:<math>~ \nabla^\nu u_{\mu \nu} = - \frac{4 \pi \eta }{c^2} J_\mu, </math>
then after substituting here the expression for the acceleration tensor <math> u_{\mu \nu} </math> through the 4-potential <math> ~ U_\mu </math> of the acceleration field we obtain the wave equation for calculating the potentials of the acceleration field:
:<math>~ \nabla^\nu \nabla_\nu U_\mu + R_{\mu \nu} U^\nu = \frac{4 \pi \eta }{c^2} J_\mu, </math>
where <math>~ R_{\mu \nu} </math> is the Ricci tensor.
The continuity equation in curved space-time is:
:<math>~ R_{ \mu \alpha } u^{\mu \alpha }= \frac {4 \pi \eta }{c^2} \nabla_{\alpha}J^{\alpha}.</math>
In Minkowski space of the special theory of relativity, the Ricci tensor is set to zero, the form of the acceleration field equations is simplified and they can be expressed in terms of the field strength <math>~\mathbf {S} </math> and the solenoidal vector <math>~\mathbf {N} </math>:
:<math>~ \nabla \cdot \mathbf{S} = 4 \pi \eta \gamma \rho_0, \qquad\qquad \nabla \times \mathbf{N} = \frac{1}{c^2} \left( 4 \pi \eta \mathbf{J} + \frac{\partial \mathbf{S}} {\partial t} \right), </math>
:<math>~ \nabla \times \mathbf{S} = - \frac{\partial \mathbf{N} } {\partial t} , \qquad\qquad \nabla \cdot \mathbf{N} = 0 .</math>
where <math>~ \gamma = \frac {1}{\sqrt{1 - {v^2 \over c^2}}} </math> is the Lorentz factor, <math>~ \mathbf{J}= \gamma \rho_0 \mathbf{v }</math> is the mass current density, <math>~ \mathbf{v } </math> is the velocity of the matter unit.
The wave equation is also simplified and can be written separately for the scalar and vector potentials:
:<math>~ \partial^\nu \partial_\nu \vartheta = \frac {1}{c^2}\frac{\partial^2 \vartheta }{\partial t^2 } -\Delta \vartheta = 4 \pi \eta \gamma \rho_0, \qquad\qquad (2) </math>
:<math>~ \partial^\nu \partial_\nu \mathbf{U} =\frac {1}{c^2}\frac{\partial^2 \mathbf{U} }{\partial t^2 } -\Delta \mathbf{U}= \frac {4 \pi \eta}{c^2} \mathbf{J}. \qquad\qquad (3) </math>
The equation of motion of the matter unit in the general field is given by the formula:
:<math>~ s_{\mu \nu} J^\nu =0 </math>.
Since <math>~ J^\nu = \rho_0 u^\nu </math>, and the general field tensor is expressed in terms of the tensors of particular fields, then the equation of motion can be represented with the help of these tensors:
:<math>~ - u_{\mu \nu} J^\nu =F_{\mu \nu} j^\nu + \Phi_{\mu \nu} J^\nu + f_{\mu \nu} J^\nu + h_{\mu \nu} J^\nu + \gamma_{\mu \nu} J^\nu + w_{\mu \nu} J^\nu .</math>
Here <math>~ F_{\mu \nu}</math> is the [[w:electromagnetic tensor |electromagnetic tensor]], <math>~ j^\nu </math> is the charge [[4-current]], <math>~ \Phi_{\mu \nu}</math> is the [[Physics/Essays/Fedosin/Gravitational tensor |gravitational tensor]], <math>~ f_{\mu \nu}</math> is the [[pressure field tensor]], <math>~ h_{\mu \nu}</math> is the [[dissipation field tensor]], <math>~ \gamma_{\mu \nu}</math> is the strong interaction field tensor, <math>~ w_{\mu \nu}</math> is the weak interaction field tensor.
=== The stress-energy tensor ===
The [[acceleration stress-energy tensor]] is calculated with the help of the acceleration tensor:
:<math>~ B^{ik} = \frac{c^2} {4 \pi \eta }\left( -g^{im} u_{n m} u^{n k}+ \frac{1} {4} g^{ik} u_{m r} u^{m r}\right) </math>.
We find as part of the tensor <math>~ B^{ik}</math> the 3-vector of the energy-momentum flux <math>~\mathbf {K} </math>, which is similar in its meaning to the [[w:Poynting vector |Poynting vector]] and the [[Physics/Essays/Fedosin/Heaviside vector |Heaviside vector]]. The vector <math>~\mathbf {K} </math> can be represented through the vector product of the field strength <math>~ \mathbf {S} </math> and the solenoidal vector <math>~ \mathbf {N} </math>:
:<math>~ \mathbf {K}=c B^{0i} = \frac {c^2}{4 \pi \eta }[\mathbf {S}\times \mathbf {N}],</math>
here the index is <math>~ i=1,2,3.</math>
The covariant derivative of the stress-energy tensor of the acceleration field with mixed indices specifies the [[4-force]] density:
:<math> ~ f_\alpha = \nabla_\beta {B_\alpha}^\beta = - u_{\alpha k} J^k = - \rho_0 u_{\alpha k}u^k = \rho_0 \frac {DU_\alpha }{D \tau}- J^k \nabla_\alpha U_k = \rho_0 \frac {dU_\alpha }{d \tau} - J^k \partial_\alpha U_k ,\qquad \qquad (4)</math>
where <math>~ D \tau </math> denotes the proper time differential in the curved spacetime.
The stress-energy tensor of the acceleration field is part of the stress-energy tensor of the general field <math>~ T^{ik} </math>:
:<math>~ T^{ik}= W^{ik}+ U^{ik}+ B^{ik}+ P^{ik} + Q^{ik}+ L^{ik}+ A^{ik}, </math>
where <math>~ W^{ik} </math> is the [[w:electromagnetic stress–energy tensor |electromagnetic stress–energy tensor]], <math>~ U^{ik}</math> is the [[gravitational stress-energy tensor]], <math>~ P^{ik}</math> is the [[pressure stress-energy tensor]], <math>~ Q^{ik}</math> is the [[dissipation stress-energy tensor]], <math>~ L^{ik}</math> is the strong interaction stress-energy tensor, <math>~ A^{ik} </math> is the weak interaction stress-energy tensor.
Through the tensor <math>~ T^{ik} </math> the stress-energy tensor of the acceleration field enters into the equation for the metric:
:<math>~ R^{ik} - \frac{1} {4 }g^{ik}R = \frac{8 \pi G \beta }{ c^4} T^{ik}, </math>
where <math>~ R^{ik} </math> is the Ricci tensor, <math>~ G </math> is the [[w:gravitational constant |gravitational constant]], <math>~ \beta </math> is a certain constant, and the gauge condition of the cosmological constant is used.
== Specific solutions for the acceleration field functions ==
The four-potential of any vector field, the global vector potential of which is equal to zero in the proper reference frame K', that is, in the center-of-momentum frame, in case of rectilinear motion in the laboratory reference frame K, can be presented as follows: <ref name="pr"/> <ref> Fedosin S.G. Equations of Motion in the Theory of Relativistic Vector Fields. International Letters of Chemistry, Physics and Astronomy, Vol. 83, pp. 12-30 (2019). https://doi.org/10.18052/www.scipress.com/ILCPA.83.12. </ref>
:<math>~ L_{\mu L} = \frac { k_f \varepsilon }{\rho_0 c^2} u_{\mu L},</math>
where <math>~ k_f = \frac {\rho_0}{\rho_{0q}}</math> is for the electromagnetic field and <math>~ k_f = 1</math> for the remaining fields; <math> ~ \rho_{0}</math> and <math> ~\rho_{0q}</math> are the invariant mass density and the charge density in the comoving reference frame, respectively; <math>~ \varepsilon </math> is the invariant energy density of the interaction, calculated as product of the four-potential of the field and the corresponding four-current; <math>~ u_{\mu L} </math> is the covariant four-velocity that determines the motion of the center of momentum of the physical system in K.
In the [[special relativity]] (SR), in the center-of-momentum frame K' the energy density is <math>~ \varepsilon = \gamma' \rho_0 c^2 </math>, where <math>~ \gamma' </math> is the Lorentz factor, and for the acceleration field, while the physical system is moving in K, the four-potential of the acceleration field will equal <math>~ U_{\mu L}= \gamma' u_{\mu L}</math>.
In case when the physical system is stationary in K, we will have <math>~ u_{\mu L} = (c,0,0,0) </math>, and consequently, the scalar potential will be <math>~ \vartheta = \gamma' c^2 </math>. If in the physical system, on the average, there are directed fluxes of matter or rotation of matter, the vector potential <math>~ \mathbf {U} </math> of the acceleration field is no longer equal to zero.
If the four-potential <math>~ U'_{\nu}</math> of acceleration field in K' is known, then in the laboratory reference frame K the four-potential is determined using the matrix <math>~ M_{\mu}^{\ \nu} </math> connecting the coordinates and time of both frames: <ref name="it"> Fedosin S.G. [http://dergipark.org.tr/gujs/issue/45480/435567 The Integral Theorem of the Field Energy.] Gazi University Journal of Science. Vol. 32, No. 2, pp. 686-703 (2019). http://dx.doi.org/10.5281/zenodo.3252783. </ref>
:<math>~ U_{\mu L}= M_{\mu}^{\ \nu} U'_{\nu}.</math>
In the special case of the system’s motion at the constant velocity <math>~ M_{\mu}^{\ \nu}</math> represents the Lorentz transformation matrix.
=== Ideally solid particle ===
In the approximation, when a particle is regarded as an ideally solid object, the matter inside the particle is motionless. It means that the Lorentz factor <math>~ \gamma' </math> of this matter in the center-of-momentum frame K' is equal to unity, so that the four-potential of the acceleration field becomes equal to the four-velocity of motion of the center of momentum:
:<math>~ U_\mu = u_\mu. </math>
In the SR, the expression for 4-velocity is simplified and we can write:
:<math>~U_\mu = \left( \frac {\vartheta }{c},- \mathbf {U} \right) = u_\mu = \left(\gamma c, - \gamma \mathbf {v} \right).</math>
The acceleration tensor components according to (1) will equal:
:<math>~ \mathbf {S} = - c^2 \nabla \gamma - \frac {\partial (\gamma \mathbf { v })}{\partial t},\qquad\qquad \mathbf {N} = \nabla \times (\gamma \mathbf { v }). </math>
Since in the solid-state motion equation for the four-acceleration with a covariant index <math>~ a_\mu </math> the relation holds
:<math>~ \rho_0 a_\mu = \rho_0 \frac {Du_\mu }{D \tau}= - u_{\mu \nu} J^\nu = - \rho_0 u_{\mu \nu} u^\nu, </math>
then in SR we obtain the following:
:<math>~ \frac {Du_\mu }{D \tau}= \frac {du_\mu }{d \tau} =\gamma \frac {du_\mu }{dt}, \qquad\qquad u^\nu =\left(\gamma c, \gamma \mathbf {v} \right), </math>
and the equations for the Lorentz factor <math>~ \gamma </math> and for the 3-acceleration <math>~ a= \frac {d \mathbf { v }}{dt} </math>:
:<math>~ \frac {d \gamma }{dt}= - \frac {1 }{c^2} \mathbf {S}\cdot \mathbf { v }, \qquad (5) \qquad \frac {d (\gamma \mathbf { v })}{dt}= \gamma \mathbf { a }+ \frac {d \gamma}{dt}\mathbf { v } = - \mathbf {S}- [\mathbf { v }\times \mathbf {N}]. \qquad (6) </math>
Multiplying equation (6) by the velocity <math>~ \mathbf { v }</math>, substituting the quantity <math>~ \mathbf {S}\cdot \mathbf { v } </math> from equation (5) to (6), taking into account relation <math>~\gamma^{-2}=1 - {v^2 \over c^2},</math> we find the well-known expression for the derivative of the Lorentz factor using the scalar product of the velocity and acceleration in SR:
:<math>~ \gamma^3 \mathbf {v}\cdot \mathbf { a }=c^2 \frac {d \gamma }{dt}.</math>
We can prove the validity of equation (6) by substituting in its right-hand side the expression for the strength and solenoidal vector:
:<math>~ \frac {d (\gamma \mathbf { v })}{dt}= c^2 \nabla \gamma + \frac {\partial (\gamma \mathbf { v })}{\partial t} - \mathbf { v }\times [ \nabla \times (\gamma \mathbf { v }) ] . \qquad\qquad (7) </math>
Indeed, the use of the [[w:material derivative |material derivative]] gives the following:
:<math>~ \frac {d (\gamma \mathbf { v })}{dt}= \frac {\partial (\gamma \mathbf { v })}{\partial t} + (\mathbf { v } \cdot \nabla) (\gamma \mathbf { v }) = \frac {\partial (\gamma \mathbf { v })}{\partial t}+\gamma (\mathbf { v } \cdot \nabla) \mathbf { v } + \mathbf { v } (\mathbf { v } \cdot \nabla\gamma) .</math>
In addition
:<math>~ - \mathbf { v }\times [ \nabla \times (\gamma \mathbf { v }) ] = - \gamma \mathbf { v }\times [ \nabla \times \mathbf { v } ] - \mathbf { v }\times [ \nabla \gamma \times \mathbf { v }] = -\frac {\gamma }{2} \nabla v^2 + \gamma (\mathbf { v } \cdot \nabla) \mathbf { v } - v^2 \nabla \gamma + \mathbf { v } (\mathbf { v } \cdot \nabla\gamma) .</math>
Substituting these relations in (7), taking into account the expression <math>~ \gamma^{-2}=1 - {v^2 \over c^2},</math> we obtain the identity:
:<math>~ c^2 \nabla \gamma - \frac {\gamma }{2} \nabla v^2 - v^2 \nabla \gamma =0 .</math>
If the components of the particle velocity are the functions of time and they do not directly depend on the space coordinates, then the solenoidal vector <math>~ \mathbf { N }</math> vanishes in such a motion.
In the SR <math>~ E = \gamma m c^2 </math> is the relativistic energy, <math>~ \mathbf p = \gamma m \mathbf v </math> is the 3-vector of relativistic momentum. If the mass <math>~ m </math> of a particle is constant, then multiplying (7) by the mass, we arrive to following equation for the force:
:<math>~ \mathbf F= \frac {d \mathbf p }{dt}= \nabla E + \frac {\partial \mathbf p }{\partial t} - \mathbf { v }\times [ \nabla \times \mathbf p ] . </math>
=== Rotation of a particle ===
For a small ideally solid particle, we can neglect the motion of the matter inside the particle and can assume that the four-potential of the acceleration field is equal to the four-velocity of the particle’s center of momentum.
Let us assume that the particle rotates about the axis OZ of the coordinate system at the distance <math>~ \rho = \sqrt {x^2 +y^2} </math> from the axis at the constant angular velocity <math>~ \omega</math> counterclockwise, as viewed from the side, in which the OZ axis is directed. Then we can assume that the linear velocity of the particle depends only on the coordinates <math>~ x</math> and <math>~ y</math>, and for the velocity’s projections on the axes of the coordinate system we can write: <math>~ \mathbf v = (-\omega y, \omega x, 0) </math>, while the square of the velocity equals <math>~ v^2 = \omega^2 (x^2 + y^2) </math>. For the Lorentz factor in the SR we obtain the following:
:<math>~ \gamma = \frac {1}{\sqrt {1- \frac { v^2}{ c^2}}} = \frac {1}{\sqrt {1- \frac { \omega^2 (x^2 + y^2)}{ c^2}}} . </math>
With this in mind, the potentials and field strengths of the acceleration field can be written as follows:
:<math>~ \vartheta = \gamma c^2, \qquad \mathbf {U} = \gamma \mathbf {v}. </math>
:<math>~ \mathbf {S} = - \nabla \vartheta - \frac {\partial \mathbf {U} }{\partial t}= \left( -\gamma^3 \omega^2 x, -\gamma^3 \omega^2 y, 0 \right). </math>
:<math>~ \mathbf {N} = \nabla \times \mathbf {U} = \left( 0, 0, \gamma \omega +\gamma^3 \omega \right). </math>
If we substitute <math>~ \gamma </math>, <math>~ \mathbf v </math>, <math>~\mathbf {S} </math> and <math>~\mathbf {N} </math> in (6), we can determine the acceleration components of the particle and the acceleration amplitude:
:<math>~ \mathbf {a} = \left( - \omega^2 x , -\omega^2 y, 0 \right). </math>
:<math>~ a = \sqrt {a^2_x +a^2_y +a^2_z} = \omega^2 \sqrt {x^2 +y^2}= \omega^2 \rho =\omega v = \frac {v^2} {\rho }. </math>
The acceleration is directed towards the center of rotation and represents [[centripetal acceleration]]. Using now the classic vector description, we have also for the time and coordinates of reference frame at the center of rotation:
:<math>~ \vec \rho = (x, y, 0) , \qquad \vec \omega = \frac {\vec {d\varphi} }{dt} =(0, 0, \omega) , </math>
:<math>~ \mathbf {v} = [\vec \omega \times \vec \rho] , \qquad \mathbf {a} = [\vec \omega \times \mathbf {v}] = [\vec \omega \times [\vec \omega \times \vec \rho]] = \vec \omega (\vec \omega \cdot \vec \rho) - \vec \rho (\vec \omega \cdot \vec \omega) = - \omega^2 \vec \rho , </math>
where <math>~ \rho </math> and <math>~ \varphi </math> are two coordinates of the [[Coordinate systems#Cylindrical coordinates.5B4.5D |cylindrical coordinate system]], <math>~ \vec \rho </math> is the vector from the center of rotation to the particle, <math>~ \vec {d\varphi}</math> is the axial vector of the differential of the rotation angle directed along the axis OZ.
As we can see, in case of such a motion with acceleration the vector product <math>~ [\mathbf {S}\times \mathbf {N}]</math> is not equal to zero, just as the three-vector <math>~ \mathbf {K}</math> of the energy-momentum flux of the acceleration field inside the particle.
=== The system of particles ===
Due to interaction of a number of particles with each other by means of various fields, including interaction at a distance without direct contact, the acceleration field in the matter changes and is different from the acceleration field of individual particles at the observation point. As a result, the density of the 4-force in the system of particles is given by the strength and the solenoidal vector, which represent the typical average characteristics of the matter motion. For example, in a gravitationally bound system there is a radial gradient of the vector <math>~ \mathbf { S },</math> and if the system is moving or rotating, there is a vector <math>~ \mathbf { N }.</math> From (4) there follows the general expression for the the density of the 4-force with covariant index:
:<math> ~ f_\nu = \rho_0 \frac {cdt}{ds}\left(-\frac {1}{c} \mathbf{S} \cdot \mathbf{v}{,} \qquad \mathbf{S}+[\mathbf{v} \times \mathbf{N}] \right),</math>
where <math> ~ ds </math> denotes a four-dimensional space-time interval.
For a stationary case, when the potentials of the acceleration field are independent of time, under the assumption that <math>~ \vartheta = \gamma c^2, </math> wave equation (2) for the scalar potential in the SR is transformed into the equation:
:<math>~ \Delta \gamma= - \frac {4 \pi \eta \gamma \rho_0}{c^2}. </math>
The solution of this equation for a fixed sphere with the particles randomly moving in it has the form: <ref name="ie"> Fedosin S.G. [http://vixra.org/abs/1403.0973 The Integral Energy-Momentum 4-Vector and Analysis of 4/3 Problem Based on the Pressure Field and Acceleration Field.] American Journal of Modern Physics. Vol. 3, No. 4, pp. 152-167 (2014). http://dx.doi.org/10.11648/j.ajmp.20140304.12.</ref>
:<math>~\gamma= \frac {c \gamma_c }{r \sqrt {4 \pi \eta \rho_0} } \sin \left(\frac {r}{c}\sqrt {4 \pi \eta \rho_0} \right) \approx \gamma_c - \frac {2 \pi \eta \rho_0 r^2 \gamma_c }{3 c^2}.</math>
where <math>~ \gamma_c = \frac {1}{\sqrt{1 - {v^2_c \over c^2}}} </math> is the Lorentz factor for the velocities <math>~ v_c</math> of the particles in the center of the sphere, and due to the smallness of the argument the sine is expanded to the second order terms. From the formula it follows that the average velocities of the particles are maximal in the center and decrease when approaching the surface of the sphere.
In such a system, the scalar potential <math>~ \vartheta</math> becomes the function of the radius, and the vector potential <math>~ \mathbf {U} </math> and the solenoidal vector <math>~ \mathbf { N }</math> are equal to zero. The acceleration field strength <math>~\mathbf {S} </math> is found with the help of (1). Then we can calculate all the functions of the acceleration field, including the energy of particles in this field and the energy of the acceleration field itself. <ref> Fedosin S.G. [http://journals.yu.edu.jo/jjp/Vol8No1Contents2015.html Relativistic Energy and Mass in the Weak Field Limit.] Jordan Journal of Physics. Vol. 8, No. 1, pp. 1-16 (2015). http://dx.doi.org/10.5281/zenodo.889210.</ref> For cosmic bodies the main contribution to the four-acceleration in the matter makes the gravitational force and the pressure field.
At the same time the relativistic rest energy of the system is automatically derived, taking into account the motion of particles inside the sphere. For the system of particles with the acceleration field, pressure field, gravitational and electromagnetic fields the given approach allowed solving the 4/3 problem and showed where and in what form the energy of the system is contained. The relation for the acceleration field constant in this problem was found:
:<math>~\eta = 3G- \frac {3q^2}{4 \pi \varepsilon_0 m^2 },</math>
where <math>~ \varepsilon_0</math> is the [[electric constant]], <math>~q </math> and <math>~m </math> are the total charge and mass of the system.
The solution of the wave equation for the acceleration field within the system results in temperature distribution according to the formula: <ref name="ie"/>
:<math>~ T=T_c - \frac {\eta M_p M(r)}{3kr} ,</math>
where <math>~ T_c </math> is the temperature in the center, <math>~ M_p </math> is the mass of the particle, for which the mass of the proton is taken (for systems which are based on hydrogen or nucleons in atomic nuclei), <math>~ M(r) </math> is the mass of the system within the current radius <math>~ r </math>, <math>~ k</math> is the Boltzmann constant.
This dependence is well satisfied for a variety of space objects, including gas clouds and Bok globules, the Earth, the Sun and neutron stars.
In articles <ref> Fedosin S.G. [http://www.nrcresearchpress.com/doi/10.1139/cjp-2015-0593#.Vv3piZyLQsY Estimation of the physical parameters of planets and stars in the gravitational equilibrium model]. Canadian Journal of Physics, Vol. 94, No. 4, pp. 370-379 (2016). http://dx.doi.org/10.1139/cjp-2015-0593.</ref> <ref>Fedosin S.G. The generalized Poynting theorem for the general field and solution of the 4/3 problem. International Frontier Science Letters, Vol. 14, pp. 19-40 (2019). https://doi.org/10.18052/www.scipress.com/IFSL.14.19. </ref> the ratio of the field’s coefficients for the fields was specified as follows:
:<math>~\eta + \sigma = G - \frac {\rho^2_{0q}}{4 \pi \varepsilon_0 \rho^2_{0}},</math>
where <math> ~ \sigma </math> is the pressure field constant.
If we introduce the parameter <math> ~ \mu </math> as the number of nucleons per ionized gas particle, then the acceleration field constant is expressed as follows:
:<math>~\eta = \frac {3\gamma_c \mu G}{2+ 3 \gamma_c \mu }.</math>
For the temperature inside the cosmic bodies in the gravitational equilibrium model we find the dependence on the current radius:
:<math>~ T=T_c - \frac {4 \pi \eta m_u \rho_{0c}\gamma_c r^2}{9k}+ \frac {2 \pi \eta A m_u \gamma_c r^3}{9k} + \frac {2 \pi \eta B m_u \gamma_c r^4}{15k} ,</math>
where <math> ~ m_u </math> is the mass of one gas particle, which is taken as the [[w:unified atomic mass unit |unified atomic mass unit]], and the coefficients <math> ~ A </math> and <math> ~ B </math> are included into the dependence of the mass density on the radius in the relation <math> ~ \rho_0 = \rho_{0c}- Ar - Br^2. </math>
Under the assumption that the system’s typical particles have the mass <math> ~\stackrel{-}{m } = \mu m_u </math>, and that it is typical particles that define the temperature and pressure, for the acceleration field constant we obtain the following: <ref>Fedosin S.G. The binding energy and the total energy of a macroscopic body in the relativistic uniform model. Middle East Journal of Science, Vol. 5, Issue 1, pp. 46-62 (2019). http://dx.doi.org/10.23884/mejs.2019.5.1.06. </ref>
:<math>~ \eta = \frac {3}{5} \left( G- \frac {\rho^2_{0q}}{ 4 \pi \varepsilon_0 \rho^2_0 } \right) .</math>
The Lorentz factor of the particles in the center of the system is also determined: <ref name="en"> Fedosin S.G. Energy and metric gauging in the covariant theory of gravitation. Aksaray University Journal of Science and Engineering, Vol. 2, Issue 2, pp. 127-143 (2018). http://dx.doi.org/10.29002/asujse.433947. </ref>
: <math>~\gamma_c = \frac {1}{\sqrt {1- \frac { v^2_c }{c^2}}} \approx 1+ \frac { v^2_c }{2c^2} +\frac {3 v^4_c }{8c^4} \approx 1+ \frac {3 \eta m}{10 a c^2} \left( 1+\frac {9}{2\sqrt {14}} \right) + \frac {27 \eta^2 m^2}{200 a^2 c^4} \left( 1+\frac {9}{2\sqrt {14}} \right)^2 . </math>
The wave equation (3) for the vector potential of the acceleration field was used to represent the relativistic equation of the fluid’s motion in the form of the [[w:Navier–Stokes equations |Navier–Stokes equations]] in hydrodynamics and to describe the motion of the viscous compressible and charged fluid. <ref> Fedosin S.G. Four-Dimensional Equation of Motion for Viscous Compressible and Charged Fluid with Regard to the Acceleration Field, Pressure Field and Dissipation Field. International Journal of Thermodynamics. Vol. 18, No. 1, pp. 13-24 (2015). http://dx.doi.org/10.5541/ijot.5000034003.</ref>
Taking into account the acceleration field and pressure field, within the framework of the [[Physics/Essays/Fedosin/Relativistic uniform system |relativistic uniform system]], it is possible to refine the [[w:virial theorem |virial theorem]], which in the relativistic form is written as follows: <ref>Fedosin S.G. The virial theorem and the kinetic energy of particles of a macroscopic system in the general field concept. Continuum Mechanics and Thermodynamics. Vol. 29, Issue 2, pp. 361-371 (2017). https://dx.doi.org/10.1007/s00161-016-0536-8. </ref>
: <math>~ \langle W_k \rangle \approx - 0.6 \sum_{k=1}^N\langle\mathbf{F}_k\cdot\mathbf{r}_k\rangle ,</math>
where the value <math>~ W_k \approx \gamma_c T </math> exceeds the kinetic energy of the particles <math>~ T </math> by a factor equal to the Lorentz factor <math>~ \gamma_c </math> of the particles at the center of the system. Under normal conditions we can assume that <math>~ \gamma_c \approx 1 </math>, then we can see that in the virial theorem the kinetic energy is related to the potential energy not by the coefficient 0.5, but rather by the coefficient close to 0.6. The difference from the classical case arises due to considering the pressure field and the acceleration field of particles inside the system, while the derivative of the virial scalar function <math>~ G_v </math> is not equal to zero and should be considered as the [[w:material derivative |material derivative]].
An analysis of the integral theorem of generalized virial makes it possible to find, on the basis of field theory, a formula for the root-mean-square speed of typical particles of a system without using the notion of temperature: <ref> Fedosin S.G. [http://em.rdcu.be/wf/click?upn=lMZy1lernSJ7apc5DgYM8f7AyOIJlVFO4uFv7zUQtzk-3D_DUeisO4Ue44lkDmCnrWVhK-2BAxKrUexyqlYtsmkyhvEp5zr527MDdThwbadScvhwZehXbanab8i5hqRa42b-2FKYwacOeM4LKDJeJuGA15M9FWvYOfBgfon7Bqg2f55NFYGJfVGaGhl0ghU-2BkIJ9Hz4M6SMBYS-2Fr-2FWWaj9eTxv23CKo9d8nFmYAbMtBBskFuW9fupsvIvN5eyv-2Fk-2BUc7hiS15rRISs1jpNnRQpDtk2OE9Hr6mYYe5Y-2B8lunO9GwVRw07Y1mdAqqtEZ-2BQjk5xUwPnA-3D-3D The integral theorem of generalized virial in the relativistic uniform model]. Continuum Mechanics and Thermodynamics, Vol. 31, Issue 3, pp. 627-638 (2019). https://dx.doi.org/10.1007/s00161-018-0715-x.</ref>
:<math> v_\mathrm{rms} = c \sqrt{1- \frac {4 \pi \eta \rho_0 r^2}{c^2 \gamma^2_c \sin^2 {\left( \frac {r}{c} \sqrt {4 \pi \eta \rho_0} \right) } } } .</math>
The integral [[field energy theorem]] for acceleration field in a curved space-time is as follows:<ref name="it"/>
:<math>~ - \int { \left( \frac {8 \pi \eta }{c^2} U_\alpha J^\alpha + u_{\alpha \beta} u^{\alpha \beta} \right) \sqrt {-g} dx^1 dx^2 dx^3 } = \frac {2}{c} \frac {d}{dt} \left( \int { U^\alpha u_\alpha ^{\ 0} \sqrt {-g} dx^1 dx^2 dx^3} \right) + 2 \iint \limits_S {U^\alpha u_\alpha ^{\ k} n_k \sqrt {-g} dS} . </math>
In the relativistic uniform system, the scalar potential <math>~\vartheta </math> of the acceleration field is related to the scalar potential <math>~\wp </math> of the pressure field:
<ref>Fedosin S.G. [https://rdcu.be/ccV9o The potentials of the acceleration field and pressure field in rotating relativistic uniform system]. Continuum Mechanics and Thermodynamics, Vol. 33, Issue 3, pp. 817-834 (2021). https://doi.org/10.1007/s00161-020-00960-7. </ref>
:<math>~ \wp = \frac {\sigma (\vartheta -c^2)}{ \eta } = \frac {2 (\vartheta -c^2)}{ 3 }. </math>
The relativistic expression for pressure is as follows:
<math> p = \frac{2\rho c^2 (\gamma - 1) }{3}= \frac {2 \rho c^2 }{3} \left( \frac {1}{\sqrt {1- v^2/ c^2 }}-1 \right) \approx \frac {\rho v^2}{3}, </math>
where <math>\rho </math> is the mass density of moving matter, <math> c </math> is the speed of light, <math> \gamma =\frac {1}{\sqrt {1- v^2/ c^2 }} </math> is the [[w:Lorentz factor |Lorentz factor]].
In the limit of low velocities, this relationship turns into the standard formula of the [[w:kinetic theory of gases |kinetic theory of gases]].
== Other approaches ==
Studying the Lorentz covariance of the 4-force, Friedman and Scarr found incomplete covariance of the expression for the 4-force in the form <math>~ F^\mu = \frac {d p^\mu }{d \tau } . </math> <ref> Yaakov Friedman and Tzvi Scarr. [http://iopscience.iop.org/1742-6596/437/1/012009 Covariant Uniform Acceleration]. Journal of Physics: Conference Series Vol. 437 (2013) 012009 doi:10.1088/1742-6596/437/1/012009. </ref>
This led them to conclude that the four-acceleration in SR must be expressed with the help of a certain antisymmetric tensor <math>~ {A^\mu}_\nu </math>:
:<math>~c \frac { d u^\mu }{d \tau } = {A^\mu}_\nu u^\nu . </math>
Based on the analysis of various types of motion, they estimated the required values of the acceleration tensor components, thereby giving indirect definition to this tensor.
From comparison with (4) it follows that the tensor <math>~ {A^\mu}_\nu </math> up to a sign and a constant multiplier coincides with the acceleration tensor <math> ~ {u^\alpha}_k </math> in case when rectilinear motion of a solid body without rotation is considered. Then indeed the four-potential of the acceleration field coincides with the four-velocity, <math>~ U_\mu = u_\mu </math>. As a result, the quantity <math>~ - J^k \partial_\alpha U_k =- \rho_0 u^k \partial_\alpha u_k </math> on the right-hand side of (4) vanishes, since the following relations hold true: <math>~ u^k u_k = c^2 </math>, <math>~ 2 u^k \partial_\alpha u_k = \partial_\alpha (u^k u_k) = \partial_\alpha c^2 =0 </math>. With this in mind, in (4) we can raise the index <math>~ \alpha </math> and cancel the mass density, which gives the following:
:<math> ~ - {u^\alpha}_k u^k =\frac {du^\alpha }{d \tau} .</math>
Mashhoon and Muench considered transformation of inertial reference frames, co-moving with the accelerated reference frame, and obtained the relation: <ref> Bahram Mashhoon and Uwe Muench. Length measurement in accelerated systems. Annalen der Physik. Vol. 11, Issue 7, P. 532–547, 2002. </ref>
:<math>~c \frac { d \lambda_\alpha }{d \tau } = {\Phi_\alpha}^\beta \lambda_\beta. </math>
The tensor <math>~ {\Phi_\alpha}^\beta </math> has the same properties as the acceleration tensor <math> ~ {u_\alpha}^\beta. </math>
== The use in the general theory of relativity ==
The action function in the [[general relativity]] (GR) can be represented as the sum of the four terms, which are responsible, respectively, for the spacetime metric, the matter in the form of substance, the electromagnetic field and the pressure field:
:<math>~ S = S_m + S_{mat} + S_{em} + S_p. </math>
Additional terms can be included in the action function, if other fields must be taken into account. The first, second and third terms of the action have the standard form: <ref> Fock, V. A. (1964). "The Theory of Space, Time and Gravitation". Macmillan. </ref>
:<math>~ S_m = \int (kR-2k \Lambda ) \sqrt {-g}d\Sigma.</math>
:<math>~ S_{mat} = \int ( - c \rho_0 ) \sqrt {-g}d\Sigma.</math>
:<math>~ S_{em} =\int ( - \frac {1}{c} A_\mu j^\mu - \frac {c \varepsilon_0}{4 } F_{\mu\nu}F^{\mu\nu} ) \sqrt {-g}d\Sigma,</math>
where <math> A_\mu </math> is the electromagnetic four-potential.
The term <math>~ S_p </math>, which is responsible for the contribution of pressure into the action function, is different in the works of different authors, depending on how the pressure is related to the elastic energy and whether the pressure field is considered to be a scalar field or a vector field. It should be noted that in the GR, the gravitational field is included in the action function not directly, but indirectly, by means of the metric tensor. In this case, as a rule, the pressure field is considered to be a scalar field.
In contrast, in the [[covariant theory of gravitation]] (CTG), the term <math>~ S_{ac} </math> associated with the acceleration field is used instead of the term <math>~ S_{mat} </math>, and the action function can be written as follows: <ref name="ac"/>
:<math>~ S = S_m + S_{ac} + S_{em} + S_p . </math>
Here
:<math>~ S_{ac} = \int ( - \frac {1}{c } U_\mu J^\mu - \frac {c}{ 16 \pi \eta } u_{\mu\nu}u^{\mu\nu} ) \sqrt {-g}d\Sigma , </math>
:<math>~ S_p =\int ( - \frac {1}{c } \pi_\mu J^\mu - \frac {c}{ 16 \pi \sigma } f_{\mu\nu}f^{\mu\nu} ) \sqrt {-g}d\Sigma,</math>
where <math> ~\pi_\mu </math> is the four-potential of the [[pressure field]], <math> ~ \sigma </math> is the coefficient of the pressure field, <math> ~ f_{\mu\nu}</math> is the [[pressure field tensor]], <math>J^\mu = \rho_{0} u^\mu </math>.
In the case of rectilinear motion of a rigid body without rotation, the following relations will hold: <math> U_\mu = u_\mu </math>, <math>~ u_\mu u^\mu = c^2 </math>, and in the term <math>~ S_{ac} </math> the relation <math>~ - \frac {1}{c } U_\mu J^\mu = - c \rho_0</math> is obtained.
In this particular case it is clear that the term <math>~ S_{ac} </math> differs from the term <math>~ S_{mat} </math> by an additional term associated with the energy of the acceleration field. This is due to the fact that in the CTG the acceleration field is considered to be a vector field, whereas as in the GR the acceleration field is actually used as a scalar field that does not depend on the particles’ velocities. In both theories, the acceleration field allows us to determine the contribution of the rest energy of the particles into the total energy of the system of particles and fields. However, the use of the acceleration field as a scalar field in the GR does not agree in its form with the vector nature of the electromagnetic field. Indeed, in the limiting case, when only the particles’ accelerations and electromagnetic forces are taken into account, the acceleration must be two-component, as is the case for the acceleration due to the action of the two-component [[Lorentz force]]. But this is possible only in the case, when the acceleration field is a vector field. The situation can be improved if, in addition to the gravitational field function, we ascribe to the metric field <math>~ g_{\mu \nu} </math> in the GR the function of the vector component of the acceleration field, but this makes the equations of the theory even more complex and complicated.
It should be noted that in the general case of arbitrary motion of the matter the relation <math>~ - \frac {1}{c } U_\mu J^\mu = - c \rho_0</math> is no longer satisfied and CTG does not coincide any more with GR in the method of describing the rest energy of a physical system. This means that in GR the motion of the matter is considered in a simplified way, as rectilinear motion of a solid body, whereas in CTG the use of the four-potential <math> U_\mu </math> of the acceleration field allows us to take into account the internal motion of the matter in each selected volume element. For example, when a particle moves round a circle, the four-potential <math> U_\mu </math> of the particle’s matter will depend on the location of this matter with respect to the circle line, since the velocity of the particle’s matter depends on the radius of rotation.
== See also ==
* [[Physics/Essays/Fedosin/General field |General field]]
* [[Pressure field]]
* [[Dissipation field]]
* [[Covariant theory of gravitation]]
* [[Physics/Essays/Fedosin/Metric theory of relativity |Metric theory of relativity]]
* [[Acceleration tensor]]
* [[Acceleration stress-energy tensor]]
* [[Four-force]]
* [[Equation of vector field]]
== References ==
<references/>
==External links ==
* [http://www.wikiznanie.ru/ru-wz/index.php/%D0%9F%D0%BE%D0%BB%D0%B5_%D1%83%D1%81%D0%BA%D0%BE%D1%80%D0%B5%D0%BD%D0%B8%D0%B9 Acceleration field in Russian]
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'''Acceleration field''' is a two-component vector field, describing in a covariant way the [[four-acceleration]] of individual particles and the [[four-force]] that occurs in systems with multiple closely interacting particles. The acceleration field is a component of the [[Physics/Essays/Fedosin/General field |general field]], which is represented in the Lagrangian and Hamiltonian of an arbitrary physical system by the term with the energy of particles’ motion and the term with the field energy. <ref name="ko"> Fedosin S.G. [http://www.oalib.com/paper/5263035#.VuFYxn2LQsY The Concept of the General Force Vector Field]. OALib Journal, Vol. 3, pp. 1-15 (2016), e2459. http://dx.doi.org/10.4236/oalib.1102459. </ref> <ref> Fedosin S.G. Two components of the macroscopic general field. Reports in Advances of Physical Sciences, Vol. 1, No. 2, 1750002, 9 pages (2017). http://dx.doi.org/10.1142/S2424942417500025. </ref>
The acceleration field is included in most [[equation of vector field |equations of vector field]]. Moreover, the acceleration field enters into the equation of motion through the [[acceleration tensor]] and into the equation for the metric through the [[acceleration stress-energy tensor]].
The acceleration field was presented by [[user:Fedosin |Sergey Fedosin]] within the framework of the [[Physics/Essays/Fedosin/Metric theory of relativity|metric theory of relativity]] and [[covariant theory of gravitation]], and the equations of this field were obtained as a consequence of the [[w:principle of least action |principle of least action]]. <ref name="pr"> Fedosin S.G. [http://vixra.org/abs/1406.0135 The procedure of finding the stress-energy tensor and vector field equations of any form]. Advanced Studies in Theoretical Physics, Vol. 8, No. 18, pp. 771-779 (2014). http://dx.doi.org/10.12988/astp.2014.47101. </ref> <ref name="ac"> [[user:Fedosin | Fedosin S.G.]] [http://journals.yu.edu.jo/jjp/Vol9No1Contents2016.html About the cosmological constant, acceleration field, pressure field and energy.] Jordan Journal of Physics. Vol. 9, No. 1, pp. 1-30 (2016). http://dx.doi.org/10.5281/zenodo.889304. </ref>
== Mathematical description ==
The 4-potential of the acceleration field is expressed in terms of the scalar <math>~ \vartheta </math> and vector <math>~ \mathbf {U} </math> potentials:
:<math>~U_\mu = \left(\frac {\vartheta }{c},- \mathbf {U} \right) .</math>
The antisymmetric [[acceleration tensor]] is calculated with the help of the 4-curl of the 4-potential:
:<math>~ u_{\mu \nu} = \nabla_\mu U_\nu - \nabla_\nu U_\mu = \partial_\mu U_\nu - \partial_\nu U_\mu . </math>
The acceleration tensor components are the components of the field strength <math>~\mathbf {S} </math> and the components of the solenoidal vector <math>~\mathbf {N} </math>:
:<math> ~ u_{\mu \nu}= \begin{vmatrix} 0 & \frac {S_x}{ c} & \frac {S_y}{ c} & \frac {S_z}{ c} \\ -\frac {S_x}{ c} & 0 & - N_{z} & N_{y} \\ -\frac {S_y}{ c} & N_{z} & 0 & -N_{x} \\ -\frac {S_z}{ c}& -N_{y} & N_{x} & 0 \end{vmatrix}. </math>
We obtain the following:
:<math>~ \mathbf {S} = - \nabla \vartheta - \frac {\partial \mathbf {U}}{\partial t},\qquad\qquad \mathbf {N} = \nabla \times \mathbf {U}.\qquad\qquad (1) </math>
In the general case the scalar and vector potentials are found by solving the wave equations for the acceleration field potentials.
=== Action, Lagrangian and energy ===
In the covariant theory of gravitation the 4-potential <math>~U_\mu </math> of the acceleration field is part of the 4-potential of the [[Physics/Essays/Fedosin/General field |general field]] <math>~ s_\mu</math>, which is the sum of the 4-potentials of particular fields, such as the electromagnetic and gravitational fields, acceleration field, [[pressure field]], [[dissipation field]], strong interaction field, weak interaction field and other vector fields, acting on the matter and its particles. All of these fields are somehow represented in the matter, so that the 4-potential <math>~ s_\mu</math> cannot consist of only one 4-potential <math>~U_\mu </math>.
The energy density of interaction of the general field and the matter is given by the product of the 4-potential of the general field and the mass 4-current: <math>~ s_\mu J^\mu </math>. We obtain the general field tensor from the 4-potential of the general field, using the 4-curl:
:<math>~ s_{\mu \nu} =\nabla_\mu s_\nu - \nabla_\nu s_\mu.</math>
The tensor invariant in the form <math>~ s_{\mu \nu} s^{\mu \nu} </math> is up to a constant factor proportional to the energy density of the general field. As a result, the action function, which contains the scalar curvature <math>~R</math> and the cosmological constant <math>~ \Lambda </math>, is given by the expression: <ref name="ko"/>
:<math>~S =\int {L dt}=\int (kR-2k \Lambda - \frac {1}{c}s_\mu J^\mu - \frac {c}{16 \pi \varpi} s_{\mu\nu}s^{\mu\nu} ) \sqrt {-g}d\Sigma,</math>
where <math>~L </math> is the Lagrange function or Lagrangian; <math>~dt </math> is the time differential of the coordinate reference system; <math>~k </math> and <math>~ \varpi </math> are the constants to be determined; <math>~c </math> is the speed of light as a measure of the propagation speed of the electromagnetic and gravitational interactions; <math>~\sqrt {-g}d\Sigma= \sqrt {-g} c dt dx^1 dx^2 dx^3</math> is the invariant 4-volume expressed in terms of the differential of the time coordinate <math>~ dx^0=cdt </math>, the product <math>~ dx^1 dx^2 dx^3 </math> of differentials of the space coordinates and the square root <math>~\sqrt {-g} </math> of the determinant <math>~g </math> of the metric tensor, taken with a negative sign.
The variation of the action function gives the general field equations, the four-dimensional equation of motion and the equation for the metric. Since the acceleration field is the general field component, then from the general field equations the corresponding equations of the acceleration field are derived.
Given the gauge condition of the cosmological constant in the form
:<math>~ c k \Lambda = - s_\mu J^\mu ,</math>
is met, the system energy does not depend on the term with the scalar curvature and is uniquely determined: <ref name="ac"/>
:<math>~E = \int {( s_0 J^0 + \frac {c^2 }{16 \pi \varpi } s_{ \mu\nu} s^{ \mu\nu} ) \sqrt {-g} dx^1 dx^2 dx^3}, </math>
where <math>~ s_0 </math> and <math>~ J^0</math> denote the time components of the 4-vectors <math>~ s_{\mu } </math> and <math>~ J^{\mu } </math>.
The system’s 4-momentum is given by the formula:
:<math>~p^\mu = \left( \frac {E}{c}{,} \mathbf {p}\right) = \left( \frac {E}{c}{,} \frac {E}{c^2}\mathbf {v} \right) , </math>
where <math>~ \mathbf {p}</math> and <math>~ \mathbf {v}</math> denote the system’s momentum and the velocity of the system’s center of momentum.
=== Equations ===
The four-dimensional equations of the acceleration field are similar in their form to Maxwell equations and are as follows:
:<math> \nabla_\sigma u_{\mu \nu}+\nabla_\mu u_{\nu \sigma}+\nabla_\nu u_{\sigma \mu}=\frac{\partial u_{\mu \nu}}{\partial x^\sigma} + \frac{\partial u_{\nu \sigma}}{\partial x^\mu} + \frac{\partial u_{\sigma \mu}}{\partial x^\nu} = 0. </math>
:<math>~ \nabla_\nu u^{\mu \nu} = - \frac{4 \pi \eta }{c^2} J^\mu, </math>
where <math>J^\mu = \rho_{0} u^\mu </math> is the mass 4-current, <math> \rho_{0}</math> is the mass density in the co-moving reference frame, <math> u^\mu </math> is the 4-velocity of the matter unit, <math>~ \eta </math> is a constant, which is determined in each problem, and it is supposed that there is an equilibrium between all fields in the observed physical system.
The gauge condition of the 4-potential of the acceleration field:
:<math>~ \nabla^\mu U_{\mu} =0 . </math>
If the second equation with the field source is written with the covariant index in the following form:
:<math>~ \nabla^\nu u_{\mu \nu} = - \frac{4 \pi \eta }{c^2} J_\mu, </math>
then after substituting here the expression for the acceleration tensor <math> u_{\mu \nu} </math> through the 4-potential <math> ~ U_\mu </math> of the acceleration field we obtain the wave equation for calculating the potentials of the acceleration field:
:<math>~ \nabla^\nu \nabla_\nu U_\mu + R_{\mu \nu} U^\nu = \frac{4 \pi \eta }{c^2} J_\mu, </math>
where <math>~ R_{\mu \nu} </math> is the Ricci tensor.
The continuity equation in curved space-time is:
:<math>~ R_{ \mu \alpha } u^{\mu \alpha }= \frac {4 \pi \eta }{c^2} \nabla_{\alpha}J^{\alpha}.</math>
In Minkowski space of the special theory of relativity, the Ricci tensor is set to zero, the form of the acceleration field equations is simplified and they can be expressed in terms of the field strength <math>~\mathbf {S} </math> and the solenoidal vector <math>~\mathbf {N} </math>:
:<math>~ \nabla \cdot \mathbf{S} = 4 \pi \eta \gamma \rho_0, \qquad\qquad \nabla \times \mathbf{N} = \frac{1}{c^2} \left( 4 \pi \eta \mathbf{J} + \frac{\partial \mathbf{S}} {\partial t} \right), </math>
:<math>~ \nabla \times \mathbf{S} = - \frac{\partial \mathbf{N} } {\partial t} , \qquad\qquad \nabla \cdot \mathbf{N} = 0 .</math>
where <math>~ \gamma = \frac {1}{\sqrt{1 - {v^2 \over c^2}}} </math> is the Lorentz factor, <math>~ \mathbf{J}= \gamma \rho_0 \mathbf{v }</math> is the mass current density, <math>~ \mathbf{v } </math> is the velocity of the matter unit.
The wave equation is also simplified and can be written separately for the scalar and vector potentials:
:<math>~ \partial^\nu \partial_\nu \vartheta = \frac {1}{c^2}\frac{\partial^2 \vartheta }{\partial t^2 } -\Delta \vartheta = 4 \pi \eta \gamma \rho_0, \qquad\qquad (2) </math>
:<math>~ \partial^\nu \partial_\nu \mathbf{U} =\frac {1}{c^2}\frac{\partial^2 \mathbf{U} }{\partial t^2 } -\Delta \mathbf{U}= \frac {4 \pi \eta}{c^2} \mathbf{J}. \qquad\qquad (3) </math>
The equation of motion of the matter unit in the general field is given by the formula:
:<math>~ s_{\mu \nu} J^\nu =0 </math>.
Since <math>~ J^\nu = \rho_0 u^\nu </math>, and the general field tensor is expressed in terms of the tensors of particular fields, then the equation of motion can be represented with the help of these tensors:
:<math>~ - u_{\mu \nu} J^\nu =F_{\mu \nu} j^\nu + \Phi_{\mu \nu} J^\nu + f_{\mu \nu} J^\nu + h_{\mu \nu} J^\nu + \gamma_{\mu \nu} J^\nu + w_{\mu \nu} J^\nu .</math>
Here <math>~ F_{\mu \nu}</math> is the [[w:electromagnetic tensor |electromagnetic tensor]], <math>~ j^\nu </math> is the charge [[4-current]], <math>~ \Phi_{\mu \nu}</math> is the [[Physics/Essays/Fedosin/Gravitational tensor |gravitational tensor]], <math>~ f_{\mu \nu}</math> is the [[pressure field tensor]], <math>~ h_{\mu \nu}</math> is the [[dissipation field tensor]], <math>~ \gamma_{\mu \nu}</math> is the strong interaction field tensor, <math>~ w_{\mu \nu}</math> is the weak interaction field tensor.
=== The stress-energy tensor ===
The [[acceleration stress-energy tensor]] is calculated with the help of the acceleration tensor:
:<math>~ B^{ik} = \frac{c^2} {4 \pi \eta }\left( -g^{im} u_{n m} u^{n k}+ \frac{1} {4} g^{ik} u_{m r} u^{m r}\right) </math>.
We find as part of the tensor <math>~ B^{ik}</math> the 3-vector of the energy-momentum flux <math>~\mathbf {K} </math>, which is similar in its meaning to the [[w:Poynting vector |Poynting vector]] and the [[Physics/Essays/Fedosin/Heaviside vector |Heaviside vector]]. The vector <math>~\mathbf {K} </math> can be represented through the vector product of the field strength <math>~ \mathbf {S} </math> and the solenoidal vector <math>~ \mathbf {N} </math>:
:<math>~ \mathbf {K}=c B^{0i} = \frac {c^2}{4 \pi \eta }[\mathbf {S}\times \mathbf {N}],</math>
here the index is <math>~ i=1,2,3.</math>
The covariant derivative of the stress-energy tensor of the acceleration field with mixed indices specifies the [[4-force]] density:
:<math> ~ f_\alpha = \nabla_\beta {B_\alpha}^\beta = - u_{\alpha k} J^k = - \rho_0 u_{\alpha k}u^k = \rho_0 \frac {DU_\alpha }{D \tau}- J^k \nabla_\alpha U_k = \rho_0 \frac {dU_\alpha }{d \tau} - J^k \partial_\alpha U_k ,\qquad \qquad (4)</math>
where <math>~ D \tau </math> denotes the proper time differential in the curved spacetime.
The stress-energy tensor of the acceleration field is part of the stress-energy tensor of the general field <math>~ T^{ik} </math>:
:<math>~ T^{ik}= W^{ik}+ U^{ik}+ B^{ik}+ P^{ik} + Q^{ik}+ L^{ik}+ A^{ik}, </math>
where <math>~ W^{ik} </math> is the [[w:electromagnetic stress–energy tensor |electromagnetic stress–energy tensor]], <math>~ U^{ik}</math> is the [[gravitational stress-energy tensor]], <math>~ P^{ik}</math> is the [[pressure stress-energy tensor]], <math>~ Q^{ik}</math> is the [[dissipation stress-energy tensor]], <math>~ L^{ik}</math> is the strong interaction stress-energy tensor, <math>~ A^{ik} </math> is the weak interaction stress-energy tensor.
Through the tensor <math>~ T^{ik} </math> the stress-energy tensor of the acceleration field enters into the equation for the metric:
:<math>~ R^{ik} - \frac{1} {4 }g^{ik}R = \frac{8 \pi G \beta }{ c^4} T^{ik}, </math>
where <math>~ R^{ik} </math> is the Ricci tensor, <math>~ G </math> is the [[w:gravitational constant |gravitational constant]], <math>~ \beta </math> is a certain constant, and the gauge condition of the cosmological constant is used.
== Specific solutions for the acceleration field functions ==
The four-potential of any vector field, the global vector potential of which is equal to zero in the proper reference frame K', that is, in the center-of-momentum frame, in case of rectilinear motion in the laboratory reference frame K, can be presented as follows: <ref name="pr"/> <ref> Fedosin S.G. Equations of Motion in the Theory of Relativistic Vector Fields. International Letters of Chemistry, Physics and Astronomy, Vol. 83, pp. 12-30 (2019). https://doi.org/10.18052/www.scipress.com/ILCPA.83.12. </ref>
:<math>~ L_{\mu L} = \frac { k_f \varepsilon }{\rho_0 c^2} u_{\mu L},</math>
where <math>~ k_f = \frac {\rho_0}{\rho_{0q}}</math> is for the electromagnetic field and <math>~ k_f = 1</math> for the remaining fields; <math> ~ \rho_{0}</math> and <math> ~\rho_{0q}</math> are the invariant mass density and the charge density in the comoving reference frame, respectively; <math>~ \varepsilon </math> is the invariant energy density of the interaction, calculated as product of the four-potential of the field and the corresponding four-current; <math>~ u_{\mu L} </math> is the covariant four-velocity that determines the motion of the center of momentum of the physical system in K.
In the [[special relativity]] (SR), in the center-of-momentum frame K' the energy density is <math>~ \varepsilon = \gamma' \rho_0 c^2 </math>, where <math>~ \gamma' </math> is the Lorentz factor, and for the acceleration field, while the physical system is moving in K, the four-potential of the acceleration field will equal <math>~ U_{\mu L}= \gamma' u_{\mu L}</math>.
In case when the physical system is stationary in K, we will have <math>~ u_{\mu L} = (c,0,0,0) </math>, and consequently, the scalar potential will be <math>~ \vartheta = \gamma' c^2 </math>. If in the physical system, on the average, there are directed fluxes of matter or rotation of matter, the vector potential <math>~ \mathbf {U} </math> of the acceleration field is no longer equal to zero.
If the four-potential <math>~ U'_{\nu}</math> of acceleration field in K' is known, then in the laboratory reference frame K the four-potential is determined using the matrix <math>~ M_{\mu}^{\ \nu} </math> connecting the coordinates and time of both frames: <ref name="it"> Fedosin S.G. [http://dergipark.org.tr/gujs/issue/45480/435567 The Integral Theorem of the Field Energy.] Gazi University Journal of Science. Vol. 32, No. 2, pp. 686-703 (2019). http://dx.doi.org/10.5281/zenodo.3252783. </ref>
:<math>~ U_{\mu L}= M_{\mu}^{\ \nu} U'_{\nu}.</math>
In the special case of the system’s motion at the constant velocity <math>~ M_{\mu}^{\ \nu}</math> represents the Lorentz transformation matrix.
=== Ideally solid particle ===
In the approximation, when a particle is regarded as an ideally solid object, the matter inside the particle is motionless. It means that the Lorentz factor <math>~ \gamma' </math> of this matter in the center-of-momentum frame K' is equal to unity, so that the four-potential of the acceleration field becomes equal to the four-velocity of motion of the center of momentum:
:<math>~ U_\mu = u_\mu. </math>
In the SR, the expression for 4-velocity is simplified and we can write:
:<math>~U_\mu = \left( \frac {\vartheta }{c},- \mathbf {U} \right) = u_\mu = \left(\gamma c, - \gamma \mathbf {v} \right).</math>
The acceleration tensor components according to (1) will equal:
:<math>~ \mathbf {S} = - c^2 \nabla \gamma - \frac {\partial (\gamma \mathbf { v })}{\partial t},\qquad\qquad \mathbf {N} = \nabla \times (\gamma \mathbf { v }). </math>
Since in the solid-state motion equation for the four-acceleration with a covariant index <math>~ a_\mu </math> the relation holds
:<math>~ \rho_0 a_\mu = \rho_0 \frac {Du_\mu }{D \tau}= - u_{\mu \nu} J^\nu = - \rho_0 u_{\mu \nu} u^\nu, </math>
then in SR we obtain the following:
:<math>~ \frac {Du_\mu }{D \tau}= \frac {du_\mu }{d \tau} =\gamma \frac {du_\mu }{dt}, \qquad\qquad u^\nu =\left(\gamma c, \gamma \mathbf {v} \right), </math>
and the equations for the Lorentz factor <math>~ \gamma </math> and for the 3-acceleration <math>~ a= \frac {d \mathbf { v }}{dt} </math>:
:<math>~ \frac {d \gamma }{dt}= - \frac {1 }{c^2} \mathbf {S}\cdot \mathbf { v }, \qquad (5) \qquad \frac {d (\gamma \mathbf { v })}{dt}= \gamma \mathbf { a }+ \frac {d \gamma}{dt}\mathbf { v } = - \mathbf {S}- [\mathbf { v }\times \mathbf {N}]. \qquad (6) </math>
Multiplying equation (6) by the velocity <math>~ \mathbf { v }</math>, substituting the quantity <math>~ \mathbf {S}\cdot \mathbf { v } </math> from equation (5) to (6), taking into account relation <math>~\gamma^{-2}=1 - {v^2 \over c^2},</math> we find the well-known expression for the derivative of the Lorentz factor using the scalar product of the velocity and acceleration in SR:
:<math>~ \gamma^3 \mathbf {v}\cdot \mathbf { a }=c^2 \frac {d \gamma }{dt}.</math>
We can prove the validity of equation (6) by substituting in its right-hand side the expression for the strength and solenoidal vector:
:<math>~ \frac {d (\gamma \mathbf { v })}{dt}= c^2 \nabla \gamma + \frac {\partial (\gamma \mathbf { v })}{\partial t} - \mathbf { v }\times [ \nabla \times (\gamma \mathbf { v }) ] . \qquad\qquad (7) </math>
Indeed, the use of the [[w:material derivative |material derivative]] gives the following:
:<math>~ \frac {d (\gamma \mathbf { v })}{dt}= \frac {\partial (\gamma \mathbf { v })}{\partial t} + (\mathbf { v } \cdot \nabla) (\gamma \mathbf { v }) = \frac {\partial (\gamma \mathbf { v })}{\partial t}+\gamma (\mathbf { v } \cdot \nabla) \mathbf { v } + \mathbf { v } (\mathbf { v } \cdot \nabla\gamma) .</math>
In addition
:<math>~ - \mathbf { v }\times [ \nabla \times (\gamma \mathbf { v }) ] = - \gamma \mathbf { v }\times [ \nabla \times \mathbf { v } ] - \mathbf { v }\times [ \nabla \gamma \times \mathbf { v }] = -\frac {\gamma }{2} \nabla v^2 + \gamma (\mathbf { v } \cdot \nabla) \mathbf { v } - v^2 \nabla \gamma + \mathbf { v } (\mathbf { v } \cdot \nabla\gamma) .</math>
Substituting these relations in (7), taking into account the expression <math>~ \gamma^{-2}=1 - {v^2 \over c^2},</math> we obtain the identity:
:<math>~ c^2 \nabla \gamma - \frac {\gamma }{2} \nabla v^2 - v^2 \nabla \gamma =0 .</math>
If the components of the particle velocity are the functions of time and they do not directly depend on the space coordinates, then the solenoidal vector <math>~ \mathbf { N }</math> vanishes in such a motion.
In the SR <math>~ E = \gamma m c^2 </math> is the relativistic energy, <math>~ \mathbf p = \gamma m \mathbf v </math> is the 3-vector of relativistic momentum. If the mass <math>~ m </math> of a particle is constant, then multiplying (7) by the mass, we arrive to following equation for the force:
:<math>~ \mathbf F= \frac {d \mathbf p }{dt}= \nabla E + \frac {\partial \mathbf p }{\partial t} - \mathbf { v }\times [ \nabla \times \mathbf p ] . </math>
=== Rotation of a particle ===
For a small ideally solid particle, we can neglect the motion of the matter inside the particle and can assume that the four-potential of the acceleration field is equal to the four-velocity of the particle’s center of momentum.
Let us assume that the particle rotates about the axis OZ of the coordinate system at the distance <math>~ \rho = \sqrt {x^2 +y^2} </math> from the axis at the constant angular velocity <math>~ \omega</math> counterclockwise, as viewed from the side, in which the OZ axis is directed. Then we can assume that the linear velocity of the particle depends only on the coordinates <math>~ x</math> and <math>~ y</math>, and for the velocity’s projections on the axes of the coordinate system we can write: <math>~ \mathbf v = (-\omega y, \omega x, 0) </math>, while the square of the velocity equals <math>~ v^2 = \omega^2 (x^2 + y^2) </math>. For the Lorentz factor in the SR we obtain the following:
:<math>~ \gamma = \frac {1}{\sqrt {1- \frac { v^2}{ c^2}}} = \frac {1}{\sqrt {1- \frac { \omega^2 (x^2 + y^2)}{ c^2}}} . </math>
With this in mind, the potentials and field strengths of the acceleration field can be written as follows:
:<math>~ \vartheta = \gamma c^2, \qquad \mathbf {U} = \gamma \mathbf {v}. </math>
:<math>~ \mathbf {S} = - \nabla \vartheta - \frac {\partial \mathbf {U} }{\partial t}= \left( -\gamma^3 \omega^2 x, -\gamma^3 \omega^2 y, 0 \right). </math>
:<math>~ \mathbf {N} = \nabla \times \mathbf {U} = \left( 0, 0, \gamma \omega +\gamma^3 \omega \right). </math>
If we substitute <math>~ \gamma </math>, <math>~ \mathbf v </math>, <math>~\mathbf {S} </math> and <math>~\mathbf {N} </math> in (6), we can determine the acceleration components of the particle and the acceleration amplitude:
:<math>~ \mathbf {a} = \left( - \omega^2 x , -\omega^2 y, 0 \right). </math>
:<math>~ a = \sqrt {a^2_x +a^2_y +a^2_z} = \omega^2 \sqrt {x^2 +y^2}= \omega^2 \rho =\omega v = \frac {v^2} {\rho }. </math>
The acceleration is directed towards the center of rotation and represents [[centripetal acceleration]]. Using now the classic vector description, we have also for the time and coordinates of reference frame at the center of rotation:
:<math>~ \vec \rho = (x, y, 0) , \qquad \vec \omega = \frac {\vec {d\varphi} }{dt} =(0, 0, \omega) , </math>
:<math>~ \mathbf {v} = [\vec \omega \times \vec \rho] , \qquad \mathbf {a} = [\vec \omega \times \mathbf {v}] = [\vec \omega \times [\vec \omega \times \vec \rho]] = \vec \omega (\vec \omega \cdot \vec \rho) - \vec \rho (\vec \omega \cdot \vec \omega) = - \omega^2 \vec \rho , </math>
where <math>~ \rho </math> and <math>~ \varphi </math> are two coordinates of the [[Coordinate systems#Cylindrical coordinates.5B4.5D |cylindrical coordinate system]], <math>~ \vec \rho </math> is the vector from the center of rotation to the particle, <math>~ \vec {d\varphi}</math> is the axial vector of the differential of the rotation angle directed along the axis OZ.
As we can see, in case of such a motion with acceleration the vector product <math>~ [\mathbf {S}\times \mathbf {N}]</math> is not equal to zero, just as the three-vector <math>~ \mathbf {K}</math> of the energy-momentum flux of the acceleration field inside the particle.
=== The system of particles ===
Due to interaction of a number of particles with each other by means of various fields, including interaction at a distance without direct contact, the acceleration field in the matter changes and is different from the acceleration field of individual particles at the observation point. As a result, the density of the 4-force in the system of particles is given by the strength and the solenoidal vector, which represent the typical average characteristics of the matter motion. For example, in a gravitationally bound system there is a radial gradient of the vector <math>~ \mathbf { S },</math> and if the system is moving or rotating, there is a vector <math>~ \mathbf { N }.</math> From (4) there follows the general expression for the the density of the 4-force with covariant index:
:<math> ~ f_\nu = \rho_0 \frac {cdt}{ds}\left(-\frac {1}{c} \mathbf{S} \cdot \mathbf{v}{,} \qquad \mathbf{S}+[\mathbf{v} \times \mathbf{N}] \right),</math>
where <math> ~ ds </math> denotes a four-dimensional space-time interval.
For a stationary case, when the potentials of the acceleration field are independent of time, under the assumption that <math>~ \vartheta = \gamma c^2, </math> wave equation (2) for the scalar potential in the SR is transformed into the equation:
:<math>~ \Delta \gamma= - \frac {4 \pi \eta \gamma \rho_0}{c^2}. </math>
The solution of this equation for a fixed sphere with the particles randomly moving in it has the form: <ref name="ie"> Fedosin S.G. [http://vixra.org/abs/1403.0973 The Integral Energy-Momentum 4-Vector and Analysis of 4/3 Problem Based on the Pressure Field and Acceleration Field.] American Journal of Modern Physics. Vol. 3, No. 4, pp. 152-167 (2014). http://dx.doi.org/10.11648/j.ajmp.20140304.12.</ref>
:<math>~\gamma= \frac {c \gamma_c }{r \sqrt {4 \pi \eta \rho_0} } \sin \left(\frac {r}{c}\sqrt {4 \pi \eta \rho_0} \right) \approx \gamma_c - \frac {2 \pi \eta \rho_0 r^2 \gamma_c }{3 c^2}.</math>
where <math>~ \gamma_c = \frac {1}{\sqrt{1 - {v^2_c \over c^2}}} </math> is the Lorentz factor for the velocities <math>~ v_c</math> of the particles in the center of the sphere, and due to the smallness of the argument the sine is expanded to the second order terms. From the formula it follows that the average velocities of the particles are maximal in the center and decrease when approaching the surface of the sphere.
In such a system, the scalar potential <math>~ \vartheta</math> becomes the function of the radius, and the vector potential <math>~ \mathbf {U} </math> and the solenoidal vector <math>~ \mathbf { N }</math> are equal to zero. The acceleration field strength <math>~\mathbf {S} </math> is found with the help of (1). Then we can calculate all the functions of the acceleration field, including the energy of particles in this field and the energy of the acceleration field itself. <ref> Fedosin S.G. [http://journals.yu.edu.jo/jjp/Vol8No1Contents2015.html Relativistic Energy and Mass in the Weak Field Limit.] Jordan Journal of Physics. Vol. 8, No. 1, pp. 1-16 (2015). http://dx.doi.org/10.5281/zenodo.889210.</ref> For cosmic bodies the main contribution to the four-acceleration in the matter makes the gravitational force and the pressure field.
At the same time the relativistic rest energy of the system is automatically derived, taking into account the motion of particles inside the sphere. For the system of particles with the acceleration field, pressure field, gravitational and electromagnetic fields the given approach allowed solving the 4/3 problem and showed where and in what form the energy of the system is contained. The relation for the acceleration field constant in this problem was found:
:<math>~\eta = 3G- \frac {3q^2}{4 \pi \varepsilon_0 m^2 },</math>
where <math>~ \varepsilon_0</math> is the [[electric constant]], <math>~q </math> and <math>~m </math> are the total charge and mass of the system.
The solution of the wave equation for the acceleration field within the system results in temperature distribution according to the formula: <ref name="ie"/>
:<math>~ T=T_c - \frac {\eta M_p M(r)}{3kr} ,</math>
where <math>~ T_c </math> is the temperature in the center, <math>~ M_p </math> is the mass of the particle, for which the mass of the proton is taken (for systems which are based on hydrogen or nucleons in atomic nuclei), <math>~ M(r) </math> is the mass of the system within the current radius <math>~ r </math>, <math>~ k</math> is the Boltzmann constant.
This dependence is well satisfied for a variety of space objects, including gas clouds and Bok globules, the Earth, the Sun and neutron stars.
In articles <ref> Fedosin S.G. [http://www.nrcresearchpress.com/doi/10.1139/cjp-2015-0593#.Vv3piZyLQsY Estimation of the physical parameters of planets and stars in the gravitational equilibrium model]. Canadian Journal of Physics, Vol. 94, No. 4, pp. 370-379 (2016). http://dx.doi.org/10.1139/cjp-2015-0593.</ref> <ref>Fedosin S.G. The generalized Poynting theorem for the general field and solution of the 4/3 problem. International Frontier Science Letters, Vol. 14, pp. 19-40 (2019). https://doi.org/10.18052/www.scipress.com/IFSL.14.19. </ref> the ratio of the field’s coefficients for the fields was specified as follows:
:<math>~\eta + \sigma = G - \frac {\rho^2_{0q}}{4 \pi \varepsilon_0 \rho^2_{0}},</math>
where <math> ~ \sigma </math> is the pressure field constant.
If we introduce the parameter <math> ~ \mu </math> as the number of nucleons per ionized gas particle, then the acceleration field constant is expressed as follows:
:<math>~\eta = \frac {3\gamma_c \mu G}{2+ 3 \gamma_c \mu }.</math>
For the temperature inside the cosmic bodies in the gravitational equilibrium model we find the dependence on the current radius:
:<math>~ T=T_c - \frac {4 \pi \eta m_u \rho_{0c}\gamma_c r^2}{9k}+ \frac {2 \pi \eta A m_u \gamma_c r^3}{9k} + \frac {2 \pi \eta B m_u \gamma_c r^4}{15k} ,</math>
where <math> ~ m_u </math> is the mass of one gas particle, which is taken as the [[w:unified atomic mass unit |unified atomic mass unit]], and the coefficients <math> ~ A </math> and <math> ~ B </math> are included into the dependence of the mass density on the radius in the relation <math> ~ \rho_0 = \rho_{0c}- Ar - Br^2. </math>
Under the assumption that the system’s typical particles have the mass <math> ~\stackrel{-}{m } = \mu m_u </math>, and that it is typical particles that define the temperature and pressure, for the acceleration field constant we obtain the following: <ref>Fedosin S.G. The binding energy and the total energy of a macroscopic body in the relativistic uniform model. Middle East Journal of Science, Vol. 5, Issue 1, pp. 46-62 (2019). http://dx.doi.org/10.23884/mejs.2019.5.1.06. </ref>
:<math>~ \eta = \frac {3}{5} \left( G- \frac {\rho^2_{0q}}{ 4 \pi \varepsilon_0 \rho^2_0 } \right) .</math>
The Lorentz factor of the particles in the center of the system is also determined: <ref name="en"> Fedosin S.G. Energy and metric gauging in the covariant theory of gravitation. Aksaray University Journal of Science and Engineering, Vol. 2, Issue 2, pp. 127-143 (2018). http://dx.doi.org/10.29002/asujse.433947. </ref>
: <math>~\gamma_c = \frac {1}{\sqrt {1- \frac { v^2_c }{c^2}}} \approx 1+ \frac { v^2_c }{2c^2} +\frac {3 v^4_c }{8c^4} \approx 1+ \frac {3 \eta m}{10 a c^2} \left( 1+\frac {9}{2\sqrt {14}} \right) + \frac {27 \eta^2 m^2}{200 a^2 c^4} \left( 1+\frac {9}{2\sqrt {14}} \right)^2 . </math>
The wave equation (3) for the vector potential of the acceleration field was used to represent the relativistic equation of the fluid’s motion in the form of the [[w:Navier–Stokes equations |Navier–Stokes equations]] in hydrodynamics and to describe the motion of the viscous compressible and charged fluid. <ref> Fedosin S.G. Four-Dimensional Equation of Motion for Viscous Compressible and Charged Fluid with Regard to the Acceleration Field, Pressure Field and Dissipation Field. International Journal of Thermodynamics. Vol. 18, No. 1, pp. 13-24 (2015). http://dx.doi.org/10.5541/ijot.5000034003.</ref>
Taking into account the acceleration field and pressure field, within the framework of the [[Physics/Essays/Fedosin/Relativistic uniform system |relativistic uniform system]], it is possible to refine the [[w:virial theorem |virial theorem]], which in the relativistic form is written as follows: <ref>Fedosin S.G. The virial theorem and the kinetic energy of particles of a macroscopic system in the general field concept. Continuum Mechanics and Thermodynamics. Vol. 29, Issue 2, pp. 361-371 (2017). https://dx.doi.org/10.1007/s00161-016-0536-8. </ref>
: <math>~ \langle W_k \rangle \approx - 0.6 \sum_{k=1}^N\langle\mathbf{F}_k\cdot\mathbf{r}_k\rangle ,</math>
where the value <math>~ W_k \approx \gamma_c T </math> exceeds the kinetic energy of the particles <math>~ T </math> by a factor equal to the Lorentz factor <math>~ \gamma_c </math> of the particles at the center of the system. Under normal conditions we can assume that <math>~ \gamma_c \approx 1 </math>, then we can see that in the virial theorem the kinetic energy is related to the potential energy not by the coefficient 0.5, but rather by the coefficient close to 0.6. The difference from the classical case arises due to considering the pressure field and the acceleration field of particles inside the system, while the derivative of the virial scalar function <math>~ G_v </math> is not equal to zero and should be considered as the [[w:material derivative |material derivative]].
An analysis of the integral theorem of generalized virial makes it possible to find, on the basis of field theory, a formula for the root-mean-square speed of typical particles of a system without using the notion of temperature: <ref> Fedosin S.G. [http://em.rdcu.be/wf/click?upn=lMZy1lernSJ7apc5DgYM8f7AyOIJlVFO4uFv7zUQtzk-3D_DUeisO4Ue44lkDmCnrWVhK-2BAxKrUexyqlYtsmkyhvEp5zr527MDdThwbadScvhwZehXbanab8i5hqRa42b-2FKYwacOeM4LKDJeJuGA15M9FWvYOfBgfon7Bqg2f55NFYGJfVGaGhl0ghU-2BkIJ9Hz4M6SMBYS-2Fr-2FWWaj9eTxv23CKo9d8nFmYAbMtBBskFuW9fupsvIvN5eyv-2Fk-2BUc7hiS15rRISs1jpNnRQpDtk2OE9Hr6mYYe5Y-2B8lunO9GwVRw07Y1mdAqqtEZ-2BQjk5xUwPnA-3D-3D The integral theorem of generalized virial in the relativistic uniform model]. Continuum Mechanics and Thermodynamics, Vol. 31, Issue 3, pp. 627-638 (2019). https://dx.doi.org/10.1007/s00161-018-0715-x.</ref>
:<math> v_\mathrm{rms} = c \sqrt{1- \frac {4 \pi \eta \rho_0 r^2}{c^2 \gamma^2_c \sin^2 {\left( \frac {r}{c} \sqrt {4 \pi \eta \rho_0} \right) } } } .</math>
The integral [[field energy theorem]] for acceleration field in a curved space-time is as follows:<ref name="it"/>
:<math>~ - \int { \left( \frac {8 \pi \eta }{c^2} U_\alpha J^\alpha + u_{\alpha \beta} u^{\alpha \beta} \right) \sqrt {-g} dx^1 dx^2 dx^3 } = \frac {2}{c} \frac {d}{dt} \left( \int { U^\alpha u_\alpha ^{\ 0} \sqrt {-g} dx^1 dx^2 dx^3} \right) + 2 \iint \limits_S {U^\alpha u_\alpha ^{\ k} n_k \sqrt {-g} dS} . </math>
In the relativistic uniform system, the scalar potential <math>~\vartheta </math> of the acceleration field is related to the scalar potential <math>~\wp </math> of the pressure field:
<ref>Fedosin S.G. [https://rdcu.be/ccV9o The potentials of the acceleration field and pressure field in rotating relativistic uniform system]. Continuum Mechanics and Thermodynamics, Vol. 33, Issue 3, pp. 817-834 (2021). https://doi.org/10.1007/s00161-020-00960-7. </ref>
:<math>~ \wp = \frac {\sigma (\vartheta -c^2)}{ \eta } = \frac {2 (\vartheta -c^2)}{ 3 }. </math>
The relativistic expression for pressure is as follows:
<math> p = \frac{2\rho c^2 (\gamma - 1) }{3}= \frac {2 \rho c^2 }{3} \left( \frac {1}{\sqrt {1- v^2/ c^2 }}-1 \right) \approx \frac {\rho v^2}{3}, </math>
where <math>\rho </math> is the mass density of moving matter, <math> c </math> is the speed of light, <math> \gamma =\frac {1}{\sqrt {1- v^2/ c^2 }} </math> is the [[w:Lorentz factor |Lorentz factor]].
In the limit of low velocities, this relationship turns into the standard formula of the [[w:kinetic theory of gases |kinetic theory of gases]].
In <ref> Fedosin S.G. The Mass Hierarchy in the Relativistic Uniform System. Bulletin of Pure and Applied Sciences, Vol. 38 D (Physics), No. 2, pp. 73-80 (2019). http://dx.doi.org/10.5958/2320-3218.2019.00012.5. </ref> it is shown how the acceleration field contributes to the mass of a physical system. Similarly, the acceleration field contributes to the space-time metric, both in the matter of the physical system and beyond it. <ref>Fedosin S.G. The relativistic uniform model: the metric of the covariant theory of gravitation inside a body. St. Petersburg Polytechnical State University Journal. Physics and Mathematics, Vol. 14, No. 3, pp.168-184 (2021). http://dx.doi.org/10.18721/JPM.14313. </ref>
The concept of the acceleration field allows one to define in a covariant form in curved space-time the generalized four-momentum, <ref>Fedosin S.G. Generalized Four-momentum for Continuously Distributed Materials. Gazi University Journal of Science, Vol. 37, Issue 3, pp. 1509-1538 (2024). https://doi.org/10.35378/gujs.1231793. </ref> the energy, momentum and total four-momentum of a physical system taking into account particles and fields. <ref>Fedosin S.G. What should we understand by the four-momentum of physical system? Physica Scripta, Vol. 99, No. 5, 055034 (2024). https://doi.org/10.1088/1402-4896/ad3b45. </ref>
== Other approaches ==
Studying the Lorentz covariance of the 4-force, Friedman and Scarr found incomplete covariance of the expression for the 4-force in the form <math>~ F^\mu = \frac {d p^\mu }{d \tau } . </math> <ref> Yaakov Friedman and Tzvi Scarr. [http://iopscience.iop.org/1742-6596/437/1/012009 Covariant Uniform Acceleration]. Journal of Physics: Conference Series Vol. 437 (2013) 012009 doi:10.1088/1742-6596/437/1/012009. </ref>
This led them to conclude that the four-acceleration in SR must be expressed with the help of a certain antisymmetric tensor <math>~ {A^\mu}_\nu </math>:
:<math>~c \frac { d u^\mu }{d \tau } = {A^\mu}_\nu u^\nu . </math>
Based on the analysis of various types of motion, they estimated the required values of the acceleration tensor components, thereby giving indirect definition to this tensor.
From comparison with (4) it follows that the tensor <math>~ {A^\mu}_\nu </math> up to a sign and a constant multiplier coincides with the acceleration tensor <math> ~ {u^\alpha}_k </math> in case when rectilinear motion of a solid body without rotation is considered. Then indeed the four-potential of the acceleration field coincides with the four-velocity, <math>~ U_\mu = u_\mu </math>. As a result, the quantity <math>~ - J^k \partial_\alpha U_k =- \rho_0 u^k \partial_\alpha u_k </math> on the right-hand side of (4) vanishes, since the following relations hold true: <math>~ u^k u_k = c^2 </math>, <math>~ 2 u^k \partial_\alpha u_k = \partial_\alpha (u^k u_k) = \partial_\alpha c^2 =0 </math>. With this in mind, in (4) we can raise the index <math>~ \alpha </math> and cancel the mass density, which gives the following:
:<math> ~ - {u^\alpha}_k u^k =\frac {du^\alpha }{d \tau} .</math>
Mashhoon and Muench considered transformation of inertial reference frames, co-moving with the accelerated reference frame, and obtained the relation: <ref> Bahram Mashhoon and Uwe Muench. Length measurement in accelerated systems. Annalen der Physik. Vol. 11, Issue 7, P. 532–547, 2002. </ref>
:<math>~c \frac { d \lambda_\alpha }{d \tau } = {\Phi_\alpha}^\beta \lambda_\beta. </math>
The tensor <math>~ {\Phi_\alpha}^\beta </math> has the same properties as the acceleration tensor <math> ~ {u_\alpha}^\beta. </math>
== The use in the general theory of relativity ==
The action function in the [[general relativity]] (GR) can be represented as the sum of the four terms, which are responsible, respectively, for the spacetime metric, the matter in the form of substance, the electromagnetic field and the pressure field:
:<math>~ S = S_m + S_{mat} + S_{em} + S_p. </math>
Additional terms can be included in the action function, if other fields must be taken into account. The first, second and third terms of the action have the standard form: <ref> Fock, V. A. (1964). "The Theory of Space, Time and Gravitation". Macmillan. </ref>
:<math>~ S_m = \int (kR-2k \Lambda ) \sqrt {-g}d\Sigma.</math>
:<math>~ S_{mat} = \int ( - c \rho_0 ) \sqrt {-g}d\Sigma.</math>
:<math>~ S_{em} =\int ( - \frac {1}{c} A_\mu j^\mu - \frac {c \varepsilon_0}{4 } F_{\mu\nu}F^{\mu\nu} ) \sqrt {-g}d\Sigma,</math>
where <math> A_\mu </math> is the electromagnetic four-potential.
The term <math>~ S_p </math>, which is responsible for the contribution of pressure into the action function, is different in the works of different authors, depending on how the pressure is related to the elastic energy and whether the pressure field is considered to be a scalar field or a vector field. It should be noted that in the GR, the gravitational field is included in the action function not directly, but indirectly, by means of the metric tensor. In this case, as a rule, the pressure field is considered to be a scalar field.
In contrast, in the [[covariant theory of gravitation]] (CTG), the term <math>~ S_{ac} </math> associated with the acceleration field is used instead of the term <math>~ S_{mat} </math>, and the action function can be written as follows: <ref name="ac"/>
:<math>~ S = S_m + S_{ac} + S_{em} + S_p . </math>
Here
:<math>~ S_{ac} = \int ( - \frac {1}{c } U_\mu J^\mu - \frac {c}{ 16 \pi \eta } u_{\mu\nu}u^{\mu\nu} ) \sqrt {-g}d\Sigma , </math>
:<math>~ S_p =\int ( - \frac {1}{c } \pi_\mu J^\mu - \frac {c}{ 16 \pi \sigma } f_{\mu\nu}f^{\mu\nu} ) \sqrt {-g}d\Sigma,</math>
where <math> ~\pi_\mu </math> is the four-potential of the [[pressure field]], <math> ~ \sigma </math> is the coefficient of the pressure field, <math> ~ f_{\mu\nu}</math> is the [[pressure field tensor]], <math>J^\mu = \rho_{0} u^\mu </math>.
In the case of rectilinear motion of a rigid body without rotation, the following relations will hold: <math> U_\mu = u_\mu </math>, <math>~ u_\mu u^\mu = c^2 </math>, and in the term <math>~ S_{ac} </math> the relation <math>~ - \frac {1}{c } U_\mu J^\mu = - c \rho_0</math> is obtained.
In this particular case it is clear that the term <math>~ S_{ac} </math> differs from the term <math>~ S_{mat} </math> by an additional term associated with the energy of the acceleration field. This is due to the fact that in the CTG the acceleration field is considered to be a vector field, whereas as in the GR the acceleration field is actually used as a scalar field that does not depend on the particles’ velocities. In both theories, the acceleration field allows us to determine the contribution of the rest energy of the particles into the total energy of the system of particles and fields. However, the use of the acceleration field as a scalar field in the GR does not agree in its form with the vector nature of the electromagnetic field. Indeed, in the limiting case, when only the particles’ accelerations and electromagnetic forces are taken into account, the acceleration must be two-component, as is the case for the acceleration due to the action of the two-component [[Lorentz force]]. But this is possible only in the case, when the acceleration field is a vector field. The situation can be improved if, in addition to the gravitational field function, we ascribe to the metric field <math>~ g_{\mu \nu} </math> in the GR the function of the vector component of the acceleration field, but this makes the equations of the theory even more complex and complicated.
It should be noted that in the general case of arbitrary motion of the matter the relation <math>~ - \frac {1}{c } U_\mu J^\mu = - c \rho_0</math> is no longer satisfied and CTG does not coincide any more with GR in the method of describing the rest energy of a physical system. This means that in GR the motion of the matter is considered in a simplified way, as rectilinear motion of a solid body, whereas in CTG the use of the four-potential <math> U_\mu </math> of the acceleration field allows us to take into account the internal motion of the matter in each selected volume element. For example, when a particle moves round a circle, the four-potential <math> U_\mu </math> of the particle’s matter will depend on the location of this matter with respect to the circle line, since the velocity of the particle’s matter depends on the radius of rotation.
== See also ==
* [[Physics/Essays/Fedosin/General field |General field]]
* [[Pressure field]]
* [[Dissipation field]]
* [[Covariant theory of gravitation]]
* [[Physics/Essays/Fedosin/Metric theory of relativity |Metric theory of relativity]]
* [[Acceleration tensor]]
* [[Acceleration stress-energy tensor]]
* [[Four-force]]
* [[Equation of vector field]]
== References ==
<references/>
==External links ==
* [http://www.wikiznanie.ru/ru-wz/index.php/%D0%9F%D0%BE%D0%BB%D0%B5_%D1%83%D1%81%D0%BA%D0%BE%D1%80%D0%B5%D0%BD%D0%B8%D0%B9 Acceleration field in Russian]
[[Category:Theoretical physics]]
[[Category:Concepts in physics]]
[[Category:Vector calculus]]
[[Category:Covariant theory of gravitation]]
[[Category: Metric theory of relativity]]
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Evidence-based assessment/Templates and Instructions/Creating A New Instrument Page
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<noinclude>{{Helping Give Away Psychological Science Banner}}</noinclude>
{{Wikipedia2|XXX}}
This is a shell that we use for building new Wikiversity instrument pages on Wikiversity.
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#Make sure to include a link to the "anchor citation"
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== Psychometrics ==
=== Steps for evaluating reliability and validity ===
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# Evaluate the instrument by referring to the rubrics for evaluating [[v:Evidence_based_assessment/Reliability|reliability]] and [[v:Evidence_based_assessment/Validity|validity]] (both '''external''' Wikiversity pages). For easy reference, open these pages in separate tabs.
## [[Evidence based assessment/Reliability|Reliability rubric]]
## [[Evidence based assessment/Validity|Validity rubric]]
# Refer to the relevant instrument rubric table. This is the table that you will be editing. Do not confuse this with the '''external pages''' on reliability and validity.
##[[OToPS/Measures/template#Rubric_table:_Reliability|Instrument rubric table: Reliability]]
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=== Instrument rubric table: Reliability===
'''Note:''' Not all of the different types of reliability apply to the way that questionnaires are typically used. Internal consistency (whether all of the items measure the same construct) is not usually reported in studies of questionnaires; nor is inter-rater reliability (which would measure how similar peoples' responses were if the interviews were repeated again, or different raters listened to the same interview). Therefore, make adjustments as needed.
{| class="wikitable"
|+
!Criterion
!Rating (adequate, good, excellent, too good)
!Explanation with Reference
|-
|Norms
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|-
|Internal Consistency
|Good
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|-
|Test-retest Reliability
|Good
|Intraclass correlation coefficients = 0.7-0.9<ref>{{Cite journal|last=BIRMAHER|first=BORIS|last2=KHETARPAL|first2=SUNEETA|last3=BRENT|first3=DAVID|last4=CULLY|first4=MARLANE|last5=BALACH|first5=LISA|last6=KAUFMAN|first6=JOAN|last7=NEER|first7=SANDRA MCKENZIE|title=The Screen for Child Anxiety Related Emotional Disorders (SCARED): Scale Construction and Psychometric Characteristics|url=https://doi.org/10.1097/00004583-199704000-00018|journal=Journal of the American Academy of Child & Adolescent Psychiatry|volume=36|issue=4|pages=545–553|doi=10.1097/00004583-199704000-00018}}</ref>
|-
|Repeatability
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|}
{{Collapse top|Click here for instrument reliability table}}
==== Reliability ====
Not all of the different types of reliability apply to the way that questionnaires are typically used. Internal consistency (whether all of the items measure the same construct) is not usually reported in studies of questionnaires; nor is inter-rater reliability (which would measure how similar peoples' responses were if the interviews were repeated again, or different raters listened to the same interview). Therefore, make adjustments as needed.
[[w:Reliability (statistics)|Reliability]] refers to whether the scores are reproducible. Unless otherwise specified, the reliability scores and values come from studies done with a United States population sample. Here is the rubric for evaluating the [[v:Evidence_based_assessment/Reliability#Evaluating_norms_and_reliability|reliability]] of scores on a measure for the purpose of evidence based assessment.
{| class="wikitable"
|+ Evaluation for norms and reliability for the XXX (table from Youngstrom et al., extending Hunsley & Mash, 2008; *indicates new construct or category)
! Criterion !! Rating (adequate, good, excellent, too good*) !! Explanation with references
|-
| Norms || Adequate || Multiple convenience samples and research studies, including both clinical and nonclinical samples{{citation needed|date=August 2016}}
|-
|[[Internal consistency]] (Cronbach’s alpha, split half, etc.) || Excellent; too good for some contexts || Alphas routinely over .94 for both scales, suggesting that scales could be shortened for many uses{{citation needed|date=August 2016}}
|-
|[[w:Inter-rater reliability|Interrater reliability]] ||Not applicable ||Designed originally as a self-report scale; parent and youth report correlate about the same as cross-informant scores correlate in general<ref>{{cite journal|last1=Achenbach|first1=TM|last2=McConaughy|first2=SH|last3=Howell|first3=CT|title=Child/adolescent behavioral and emotional problems: implications of cross-informant correlations for situational specificity.|journal=Psychological Bulletin|date=March 1987|volume=101|issue=2|pages=213–32|pmid=3562706}}</ref>
|-
|[[Test-retest reliability]] (stability || Good || r = .73 over 15 weeks. Evaluated in initial studies,<ref name=":0">{{cite journal|last1=Depue|first1=Richard A.|last2=Slater|first2=Judith F.|last3=Wolfstetter-Kausch|first3=Heidi|last4=Klein|first4=Daniel|last5=Goplerud|first5=Eric|last6=Farr|first6=David|title=A behavioral paradigm for identifying persons at risk for bipolar depressive disorder: A conceptual framework and five validation studies.|journal=Journal of Abnormal Psychology|date=1981|volume=90|issue=5|pages=381–437|doi=10.1037/0021-843X.90.5.381|accessdate=15 August 2016}}</ref> with data also show high stability in clinical trials{{citation needed|date=August 2016}}
|-
|Repeatability || Not published || No published studies formally checking repeatability
|}
{{collapse bottom}}
=== Instrument rubric table: Validity ===
{{collapse top|Click here for instrument validity table}}
====Validity====
[[w:Validity (statistics)|Validity]] describes the evidence that an assessment tool measures what it was supposed to measure. There are many different ways of checking validity. For screening measures, diagnostic accuracy and [[w:discriminative validity]] are probably the most useful ways of looking at validity. Unless otherwise specified, the validity scores and values come from studies done with a United States population sample. Here is a [[v:Evidence_based_assessment/Validity|rubric for describing validity]] of test scores in the context of evidence-based assessment.
{| class="wikitable"
|+ Evaluation of validity and utility for the XXX (table from Youngstrom et al., unpublished, extended from Hunsley & Mash, 2008; *indicates new construct or category)
! Criterion !! Rating (adequate, good, excellent, too good*) !! Explanation with references
|-
| [[w:Content validity|Content validity]] || Excellent || Covers both DSM diagnostic symptoms and a range of associated features<ref name=":0" />
|-
|[[w:Construct validity|Construct validity]] (e.g., predictive, concurrent, convergent, and discriminant validity) || Excellent || Shows [[w:convergent validity|Convergent validity]] with other symptom scales, longitudinal prediction of development of mood disorders,<ref>{{cite journal|last1=Klein|first1=DN|last2=Dickstein|first2=S|last3=Taylor|first3=EB|last4=Harding|first4=K|title=Identifying chronic affective disorders in outpatients: validation of the General Behavior Inventory.|journal=Journal of consulting and clinical psychology|date=February 1989|volume=57|issue=1|pages=106–11|pmid=2925959|accessdate=15 August 2016}}</ref><ref>{{cite journal|last1=Mesman|first1=Esther|last2=Nolen|first2=Willem A.|last3=Reichart|first3=Catrien G.|last4=Wals|first4=Marjolein|last5=Hillegers|first5=Manon H.J.|title=The Dutch Bipolar Offspring Study: 12-Year Follow-Up|journal=American Journal of Psychiatry|date=May 2013|volume=170|issue=5|pages=542–549|doi=10.1176/appi.ajp.2012.12030401|accessdate=15 August 2016}}</ref><ref>{{cite journal|last1=Reichart|first1=CG|last2=van der Ende|first2=J|last3=Wals|first3=M|last4=Hillegers|first4=MH|last5=Nolen|first5=WA|last6=Ormel|first6=J|last7=Verhulst|first7=FC|title=The use of the GBI as predictor of bipolar disorder in a population of adolescent offspring of parents with a bipolar disorder.|journal=Journal of affective disorders|date=December 2005|volume=89|issue=1-3|pages=147–55|pmid=16260043|accessdate=15 August 2016}}</ref> criterion validity via metabolic markers<ref name=":0" /><ref>{{cite journal|last1=Depue|first1=RA|last2=Kleiman|first2=RM|last3=Davis|first3=P|last4=Hutchinson|first4=M|last5=Krauss|first5=SP|title=The behavioral high-risk paradigm and bipolar affective disorder, VIII: Serum free cortisol in nonpatient cyclothymic subjects selected by the General Behavior Inventory.|journal=The American journal of psychiatry|date=February 1985|volume=142|issue=2|pages=175–81|pmid=3970242|accessdate=15 August 2016}}</ref> and associations with family history of mood disorder.<ref>{{cite journal|last1=Klein|first1=DN|last2=Depue|first2=RA|title=Continued impairment in persons at risk for bipolar affective disorder: results of a 19-month follow-up study.|journal=Journal of abnormal psychology|date=August 1984|volume=93|issue=3|pages=345–7|pmid=6470321|accessdate=15 August 2016}}</ref> Factor structure complicated;<ref name=":0" /><ref name=":1">{{cite journal|last1=Pendergast|first1=Laura L.|last2=Youngstrom|first2=Eric A.|last3=Brown|first3=Christopher|last4=Jensen|first4=Dane|last5=Abramson|first5=Lyn Y.|last6=Alloy|first6=Lauren B.|title=Structural invariance of General Behavior Inventory (GBI) scores in Black and White young adults.|journal=Psychological Assessment|date=2015|volume=27|issue=1|pages=21–30|doi=10.1037/pas0000020|accessdate=15 August 2016}}</ref> the inclusion of “biphasic” or “mixed” mood items creates a lot of cross-loading
|-
|Discriminative validity ||Excellent ||Multiple studies show that GBI scores discriminate cases with [[w:depression (mood)|unipolar]] and [[w:bipolar disorder|bipolar mood disorders]] from other clinical disorders<ref name=":0" /><ref name=":2">{{cite journal|last1=Danielson|first1=CK|last2=Youngstrom|first2=EA|last3=Findling|first3=RL|last4=Calabrese|first4=JR|title=Discriminative validity of the general behavior inventory using youth report.|journal=Journal of abnormal child psychology|date=February 2003|volume=31|issue=1|pages=29–39|pmid=12597697|accessdate=15 August 2016}}</ref><ref>{{cite journal|last1=Findling|first1=RL|last2=Youngstrom|first2=EA|last3=Danielson|first3=CK|last4=DelPorto-Bedoya|first4=D|last5=Papish-David|first5=R|last6=Townsend|first6=L|last7=Calabrese|first7=JR|title=Clinical decision-making using the General Behavior Inventory in juvenile bipolarity.|journal=Bipolar disorders|date=February 2002|volume=4|issue=1|pages=34–42|pmid=12047493|accessdate=15 August 2016}}</ref> [[effect size|effect sizes]] are among the largest of existing scales<ref>{{cite journal|last1=Youngstrom|first1=Eric A.|last2=Genzlinger|first2=Jacquelynne E.|last3=Egerton|first3=Gregory A.|last4=Van Meter|first4=Anna R.|title=Multivariate meta-analysis of the discriminative validity of caregiver, youth, and teacher rating scales for pediatric bipolar disorder: Mother knows best about mania.|journal=Archives of Scientific Psychology|date=2015|volume=3|issue=1|pages=112–137|doi=10.1037/arc0000024|accessdate=15 August 2016}}</ref>
|-
|Validity generalization || Good || Used both as self-report and caregiver report; used in college student<ref name=":1" /><ref>{{cite journal|last1=Alloy|first1=LB|last2=Abramson|first2=LY|last3=Hogan|first3=ME|last4=Whitehouse|first4=WG|last5=Rose|first5=DT|last6=Robinson|first6=MS|last7=Kim|first7=RS|last8=Lapkin|first8=JB|title=The Temple-Wisconsin Cognitive Vulnerability to Depression Project: lifetime history of axis I psychopathology in individuals at high and low cognitive risk for depression.|journal=Journal of abnormal psychology|date=August 2000|volume=109|issue=3|pages=403–18|pmid=11016110|accessdate=15 August 2016}}</ref> as well as outpatient<ref name=":2" /><ref>{{cite journal|last1=Klein|first1=Daniel N.|last2=Dickstein|first2=Susan|last3=Taylor|first3=Ellen B.|last4=Harding|first4=Kathryn|title=Identifying chronic affective disorders in outpatients: Validation of the General Behavior Inventory.|journal=Journal of Consulting and Clinical Psychology|date=1989|volume=57|issue=1|pages=106–111|doi=10.1037/0022-006X.57.1.106|accessdate=15 August 2016}}</ref><ref>{{cite journal|last1=Youngstrom|first1=EA|last2=Findling|first2=RL|last3=Danielson|first3=CK|last4=Calabrese|first4=JR|title=Discriminative validity of parent report of hypomanic and depressive symptoms on the General Behavior Inventory.|journal=Psychological assessment|date=June 2001|volume=13|issue=2|pages=267–76|pmid=11433802|accessdate=15 August 2016}}</ref> and inpatient clinical samples; translated into multiple languages with good reliability
|-
|Treatment sensitivity || Good || Multiple studies show sensitivity to treatment effects comparable to using interviews by trained raters, including placebo-controlled, masked assignment trials<ref>{{cite journal|last1=Findling|first1=RL|last2=Youngstrom|first2=EA|last3=McNamara|first3=NK|last4=Stansbrey|first4=RJ|last5=Wynbrandt|first5=JL|last6=Adegbite|first6=C|last7=Rowles|first7=BM|last8=Demeter|first8=CA|last9=Frazier|first9=TW|last10=Calabrese|first10=JR|title=Double-blind, randomized, placebo-controlled long-term maintenance study of aripiprazole in children with bipolar disorder.|journal=The Journal of clinical psychiatry|date=January 2012|volume=73|issue=1|pages=57–63|pmid=22152402|accessdate=15 August 2016}}</ref><ref name=":3">{{cite journal|last1=Youngstrom|first1=E|last2=Zhao|first2=J|last3=Mankoski|first3=R|last4=Forbes|first4=RA|last5=Marcus|first5=RM|last6=Carson|first6=W|last7=McQuade|first7=R|last8=Findling|first8=RL|title=Clinical significance of treatment effects with aripiprazole versus placebo in a study of manic or mixed episodes associated with pediatric bipolar I disorder.|journal=Journal of child and adolescent psychopharmacology|date=March 2013|volume=23|issue=2|pages=72–9|pmid=23480324|accessdate=15 August 2016}}</ref> Short forms appear to retain sensitivity to [[treatment effect|treatment effects]] while substantially reducing burden<ref name=":3" /><ref>{{cite journal|last1=Ong|first1=ML|last2=Youngstrom|first2=EA|last3=Chua|first3=JJ|last4=Halverson|first4=TF|last5=Horwitz|first5=SM|last6=Storfer-Isser|first6=A|last7=Frazier|first7=TW|last8=Fristad|first8=MA|last9=Arnold|first9=LE|last10=Phillips|first10=ML|last11=Birmaher|first11=B|last12=Kowatch|first12=RA|last13=Findling|first13=RL|last14=LAMS|first14=Group|title=Comparing the CASI-4R and the PGBI-10 M for Differentiating Bipolar Spectrum Disorders from Other Outpatient Diagnoses in Youth.|journal=Journal of abnormal child psychology|date=1 July 2016|pmid=27364346|accessdate=15 August 2016}}</ref>
|-
|Clinical utility || Good || Free (public domain), strong [[psychometrics]], extensive research base. Biggest concerns are length and reading level. Short forms have less research, but are appealing based on reduced burden and promising data
|}
{{collapse bottom}}
==Development and history==
{{Collapse top|Click here for instructions for development and history}}
*Why was this instrument developed? Why was there a need to do so? What need did it meet?
*What was the theoretical background behind this assessment? (e.g. addresses importance of 'negative cognitions', such as intrusions, inaccurate, sustained thoughts)
*How was the scale developed? What was the theoretical background behind it?
*If there were previous versions, when were they published?
*Discuss the theoretical ideas behind the changes.
{{collapse bottom}}
==Impact==
*What was the impact of this assessment? How did it affect assessment in psychiatry, psychology and health care professionals?
*What can the assessment be used for in clinical settings? Can it be used to measure symptoms longitudinally? Developmentally?
==Use in other populations==
*How widely has it been used? Has it been translated into different languages? Which languages?
==Scoring instructions and syntax ==
We have syntax in three major languages: '''R, SPSS, and SAS'''. All variable names are the same across all three, and all match the CSV shell that we provide as well as the Qualtrics export.
=== Hand scoring and general instructions ===
{{collapse top|Click here for hand scoring and general administration instructions}}
<Information about hand scoring and general instructions go here>
{{collapse bottom}}
If there are any hand scoring and general administration instructions, it should go here.
=== CSV shell for sharing ===
{{collapse top|Click here for CSV shell}}
*<Paste link to CSV shell here>
{{collapse bottom}}
Here is a shell data file that you could use in your own research. The variable names in the shell corresponds with the scoring code in the code for all three statistical programs.
Note that our CSV includes several demographic variables, which follow current conventions in most developmental and clinical psychology journals. You may want to modify them, depending on where you are working. Also pay attention to the possibility of '''"deductive identification"''' -- if we ask personal information in enough detail, then it may be possible to figure out the identity of a participant based on a combination of variables.
When different research projects and groups use the same variable names and syntax, it makes it easier to share the data and work together on integrative data analyses or "mega" analyses (which are different and better than meta-analysis in that they are combining the raw data, versus working with summary descriptive statistics).
=== R/SPSS/SAS syntax ===
{{collapse top|Click here for R code}}
R code goes here
{{collapse bottom}}
{{collapse top|Click here for SPSS code}}
SPSS code goes here
{{collapse bottom}}
{{collapse top|Click here for SAS code}}
SAS code goes here
{{collapse bottom}}
==See also==
Here, it would be good to link to any related articles on Wikipedia. For instance:
* [[Evidence based assessment/Obsessive-compulsive disorder (assessment portfolio)]]
==External links==
* [https://unc.az1.qualtrics.com/jfe/form/SV_cBlUQk8Y85LHF41 Depression and Bipolar Support Alliance: 7 Up 7 Down Online Screener]
==Example page==
* [[w:General Behavior Inventory|General Behavior Inventory]]
==References==
{{collapse top|Click here for references}}
{{reflist|3}}
{{collapse bottom}}
{{DEFAULTSORT:7 Up 7 Down Inventory}}
[[Category:Psychological measures]]
[[Category:OToPS measures]]
[[Category:OToPS Fall 2017 measures]]
[[Category:Assessment measures]]
{{DEFAULTSORT:7 Up 7 Down Inventory}}
cda867y164mxm2qo9weydpra59halxj
Evidence-based assessment/Instruments/Alcohol Use Disorders Identification Test
0
216861
2694012
2693953
2025-01-01T20:00:35Z
Yinuo May Liu
2995206
add validity table
2694012
wikitext
text/x-wiki
<noinclude>{{Helping Give Away Psychological Science Banner}}</noinclude>
{{wikipedia2| Alcohol Use Disorders Identification Test}}
The '''Alcohol Use Disorders Identification Test (AUDIT)''' is a ten-question test developed by a World Health Organization-sponsored collaborative project to determine if a person may be at risk for alcohol abuse problems <ref name=Saunders1993>{{cite journal|last1=Saunders|first1=JB|last2=Aasland|first2=OG|first3=Babor|last3=TF|last4=de la Fuente|first4=JR|last5=Grant|first5=M|title=Development of the Alcohol Use Disorders Identification Test (AUDIT):WHO Collaborative Project on Early Detection of Persons with Harmful Alcohol Consumption 1. |journal=Addiction |date=1993 |volume=88 |issue=12 |pages=791–804 |doi=10.1111/j.1360-0443.1993.tb02093.x|pmid=8329970}}</ref>. The test was designed to be used internationally, and was validated in a study drawing patients from six countries <ref name=Saunders1993 />. It performs well in medical settings like primary health care, outpatient settings, and hospital units <ref> Bohn, MJ; Babor, TF; Kranzler, HR. (July, 1995). "The Alcohol Use Disorders Identification Test (AUDIT): validation of a screening instrument for use in medical settings". ''Journal of studies on alcohol'', '''56''' (4), 423-432. [[doi:10.15288/jsa.1995.56.423]]</ref>.
The AUDIT alcohol consumption questions (AUDIT-C) is a 3-question screening test for problem drinking used frequently in primary care settings <ref> Bush, K; Kivlahan, DR; McDonell, MB; Fihn, SD; Bradley, KA; Ambulatory Care Quality Improvement Project. (September, 1998). "The AUDIT alcohol consumption questions (AUDIT-C): an effective brief screening test for problem drinking". ''Archives of internal medicine'', '''158''' (16):1789-1795. [[doi:10.1001/archinte.158.16.1789]]</ref>.
==Psychometrics==
===Reliability===
{| class="wikitable"
|+ Evaluation for norms and reliability for the AUDIT
! Criterion !! Rating (adequate, good, excellent) !! Explanation with references
|-
| Norms || Good || Multiple studies tested AUDIT for possible sex<ref>Cherpitel, CJ. (January, 1995). "Analysis of cut points for screening instruments for alcohol problems in the emergency room.". ''Journal of studies on alcohol'', '''56''' (6): 695-700.</ref>, ethnic<ref>Cherpitel, CJ. (1998). "Differences in performance of screening instruments for problem drinking among blacks, whites and Hispanics in an emergency room population.". ''Journal of Studies on Alcohol'', '''59''' (4): 420-426.</ref>, and age<ref>Foster, AI; Blondell, RD; Looney, SW. (March, 1997). "The practicality of using the SMAST and AUDIT to screen for alcoholism among adolescents in an urban private family practice.". ''The Journal of the Kentucky Medical Association'', '''95''' (3): 105-107.</ref> differences
|-
| Internal consistency || Good || A review of 18 studies reported a median Cronbach's alpha of 0.8<ref name=Reinert2002>Reinert, DF; Allen, JP. (April, 2002). "The alcohol use disorders identification test (AUDIT): a review of recent research.". ''Alcoholism: Clinical and Experimental Research'', '''26''' (2): 272-279.</ref>
|-
| Test-retest reliability || Good || r = 0.81 among 126 primary care patients over 6 weeks<ref name=Daeppen2000>Daeppen, JB; Yersin, B; Landry, U; Pécoud, A; Decrey, H. (April, 2000). "Reliability and validity of the Alcohol Use Disorders Identification Test (AUDIT) imbedded within a general health risk screening questionnaire: results of a survey in 332 primary care patients.". ''Alcoholism: Clinical and Experimental Research'', '''24''' (5): 659-665.</ref>; 69% to 89% of subjects fall into the same category after 6-week interval<ref name= Daeppen2000/>
|-
| Inter-rater reliability || Not applicable || Can be completed as a clinician-administered interview or a self-report questionnaire
|}
===Validity===
{| class="wikitable"
|+ Evaluation of validity and utility for the AUDIT
! Criterion !! Rating (adequate, good, excellent) !! Explanation with references
|-
|Content validity || Good || Covers drinking behaviors, adverse psychological reactions, alcohol-related problems, and alcohol consumption
|-
|Construct validity || Good || Studies generally supported a two-factor model with a consumption factor and an adverse consequences of drinking factor<ref>de Meneses-Gaya, C; Zuardi, AW; Loureiro, SR; Crippa, JAS. (2009). "Alcohol Use Disorders Identification Test (AUDIT): An updated systematic review of psychometric properties." ''Psychology & Neuroscience'', '''2''' (1): 83.</ref>
|-
|Validity generalization || Excellent || Used both as a clinician-administered interview and a self-report questionnaire; used in college students<ref>Kokotailo, PK; Egan, J; Gangnon, R; Brown, D; Mundt, M; Fleming, M. (May, 2004). "Validity of the alcohol use disorders identification test in college students.". ''Alcoholism: Clinical and Experimental Research'', '''28''' (6): 914-920.</ref> and psychiatric patients<ref name=Reinert2007>Reinert, DF; Allen, JP. (January, 2007). "The alcohol use disorders identification test: an update of research findings.". ''Alcoholism: Clinical and Experimental Research'', '''31''' (2): 185-199.</ref>; translated into multiple languages with satisfactory psychometric qualities<ref name=Reinert2007/>
|}
== Scoring and interpretation ==
Scoring the AUDIT is based on a 0-4 point scale. Six of the ten questions ask about the frequency of certain alcohol abuse behaviors and are scored by the following responses:
* '''0 points:''' "Never"
* '''1 point:''' "Less than monthly"
* '''2 points:''' "Monthly"
* '''3 points:''' "Weekly"
* '''4 points:''' "Daily, or almost daily"
The other four questions vary in participant response choice but are scored on a 0-4 point scale.
=== Item breakdown ===
The questions measure different domains of alcohol consumption problems. The breakdown is as follows:
* '''1-3''': Measure frequency in alcohol consumption
* '''4-6''': Measure alcohol dependence
* '''7-10''': Measure alcohol related problems
=== Interpretation of scores ===
In order to score the AUDIT, point values of each answer choice are summed together and then interpreted based on the following criteria.
* A score of 8 or more in men (7 in women) indicates a strong likelihood of hazardous or harmful alcohol consumption.
* A score of 20 or more is suggestive of alcohol dependence (although some authors quote scores of more than 13 in women and 15 in men as indicating likely dependence).<ref>[http://whqlibdoc.who.int/hq/2001/WHO_MSD_MSB_01.6a.pdf AUDIT: The Alcohol Use Disorders Identification Test: Guidelines for Use in Primary Care], second edition, by Thomas F. Babor, John C. Higgins-Biddle, John B. Saunders, and Maristela G. Monteiro. Retrieved June 24, 2006.</ref>
== External links ==
* [http://effectivechildtherapy.org/concerns-symptoms-disorders/disorders/drug-and-alcohol-abuse/ EffectiveChildTherapy.Org information on Substance Abuse]
* [https://sccap53.org Society of Clinical Child and Adolescent Psychology]
== References ==
{{Reflist}}
{{:{{BASEPAGENAME}}/Navbox}}
{{DEFAULTSORT:Alcohol Use Disorders Identification Test}}
[[Category:Psychological measures]]
8bbio11r9u8so5tqcrawimzh6q6dh40
2694013
2694012
2025-01-01T20:01:35Z
Yinuo May Liu
2995206
add citation
2694013
wikitext
text/x-wiki
<noinclude>{{Helping Give Away Psychological Science Banner}}</noinclude>
{{wikipedia2| Alcohol Use Disorders Identification Test}}
The '''Alcohol Use Disorders Identification Test (AUDIT)''' is a ten-question test developed by a World Health Organization-sponsored collaborative project to determine if a person may be at risk for alcohol abuse problems <ref name=Saunders1993>{{cite journal|last1=Saunders|first1=JB|last2=Aasland|first2=OG|first3=Babor|last3=TF|last4=de la Fuente|first4=JR|last5=Grant|first5=M|title=Development of the Alcohol Use Disorders Identification Test (AUDIT):WHO Collaborative Project on Early Detection of Persons with Harmful Alcohol Consumption 1. |journal=Addiction |date=1993 |volume=88 |issue=12 |pages=791–804 |doi=10.1111/j.1360-0443.1993.tb02093.x|pmid=8329970}}</ref>. The test was designed to be used internationally, and was validated in a study drawing patients from six countries <ref name=Saunders1993 />. It performs well in medical settings like primary health care, outpatient settings, and hospital units <ref> Bohn, MJ; Babor, TF; Kranzler, HR. (July, 1995). "The Alcohol Use Disorders Identification Test (AUDIT): validation of a screening instrument for use in medical settings". ''Journal of studies on alcohol'', '''56''' (4), 423-432. [[doi:10.15288/jsa.1995.56.423]]</ref>.
The AUDIT alcohol consumption questions (AUDIT-C) is a 3-question screening test for problem drinking used frequently in primary care settings <ref> Bush, K; Kivlahan, DR; McDonell, MB; Fihn, SD; Bradley, KA; Ambulatory Care Quality Improvement Project. (September, 1998). "The AUDIT alcohol consumption questions (AUDIT-C): an effective brief screening test for problem drinking". ''Archives of internal medicine'', '''158''' (16):1789-1795. [[doi:10.1001/archinte.158.16.1789]]</ref>.
==Psychometrics==
===Reliability===
{| class="wikitable"
|+ Evaluation for norms and reliability for the AUDIT
! Criterion !! Rating (adequate, good, excellent) !! Explanation with references
|-
| Norms || Good || Multiple studies tested AUDIT for possible sex<ref>Cherpitel, CJ. (January, 1995). "Analysis of cut points for screening instruments for alcohol problems in the emergency room.". ''Journal of studies on alcohol'', '''56''' (6): 695-700.</ref>, ethnic<ref>Cherpitel, CJ. (1998). "Differences in performance of screening instruments for problem drinking among blacks, whites and Hispanics in an emergency room population.". ''Journal of Studies on Alcohol'', '''59''' (4): 420-426.</ref>, and age<ref>Foster, AI; Blondell, RD; Looney, SW. (March, 1997). "The practicality of using the SMAST and AUDIT to screen for alcoholism among adolescents in an urban private family practice.". ''The Journal of the Kentucky Medical Association'', '''95''' (3): 105-107.</ref> differences
|-
| Internal consistency || Good || A review of 18 studies reported a median Cronbach's alpha of 0.8<ref name=Reinert2002>Reinert, DF; Allen, JP. (April, 2002). "The alcohol use disorders identification test (AUDIT): a review of recent research.". ''Alcoholism: Clinical and Experimental Research'', '''26''' (2): 272-279.</ref>
|-
| Test-retest reliability || Good || r = 0.81 among 126 primary care patients over 6 weeks<ref name=Daeppen2000>Daeppen, JB; Yersin, B; Landry, U; Pécoud, A; Decrey, H. (April, 2000). "Reliability and validity of the Alcohol Use Disorders Identification Test (AUDIT) imbedded within a general health risk screening questionnaire: results of a survey in 332 primary care patients.". ''Alcoholism: Clinical and Experimental Research'', '''24''' (5): 659-665.</ref>; 69% to 89% of subjects fall into the same category after 6-week interval<ref name= Daeppen2000/>
|-
| Inter-rater reliability || Not applicable || Can be completed as a clinician-administered interview or a self-report questionnaire
|}
===Validity===
{| class="wikitable"
|+ Evaluation of validity and utility for the AUDIT
! Criterion !! Rating (adequate, good, excellent) !! Explanation with references
|-
|Content validity || Good || Covers drinking behaviors, adverse psychological reactions, alcohol-related problems, and alcohol consumption<ref name=Saunders1993/>
|-
|Construct validity || Good || Studies generally supported a two-factor model with a consumption factor and an adverse consequences of drinking factor<ref>de Meneses-Gaya, C; Zuardi, AW; Loureiro, SR; Crippa, JAS. (2009). "Alcohol Use Disorders Identification Test (AUDIT): An updated systematic review of psychometric properties." ''Psychology & Neuroscience'', '''2''' (1): 83.</ref>
|-
|Validity generalization || Excellent || Used both as a clinician-administered interview and a self-report questionnaire; used in college students<ref>Kokotailo, PK; Egan, J; Gangnon, R; Brown, D; Mundt, M; Fleming, M. (May, 2004). "Validity of the alcohol use disorders identification test in college students.". ''Alcoholism: Clinical and Experimental Research'', '''28''' (6): 914-920.</ref> and psychiatric patients<ref name=Reinert2007>Reinert, DF; Allen, JP. (January, 2007). "The alcohol use disorders identification test: an update of research findings.". ''Alcoholism: Clinical and Experimental Research'', '''31''' (2): 185-199.</ref>; translated into multiple languages with satisfactory psychometric qualities<ref name=Reinert2007/>
|}
== Scoring and interpretation ==
Scoring the AUDIT is based on a 0-4 point scale. Six of the ten questions ask about the frequency of certain alcohol abuse behaviors and are scored by the following responses:
* '''0 points:''' "Never"
* '''1 point:''' "Less than monthly"
* '''2 points:''' "Monthly"
* '''3 points:''' "Weekly"
* '''4 points:''' "Daily, or almost daily"
The other four questions vary in participant response choice but are scored on a 0-4 point scale.
=== Item breakdown ===
The questions measure different domains of alcohol consumption problems. The breakdown is as follows:
* '''1-3''': Measure frequency in alcohol consumption
* '''4-6''': Measure alcohol dependence
* '''7-10''': Measure alcohol related problems
=== Interpretation of scores ===
In order to score the AUDIT, point values of each answer choice are summed together and then interpreted based on the following criteria.
* A score of 8 or more in men (7 in women) indicates a strong likelihood of hazardous or harmful alcohol consumption.
* A score of 20 or more is suggestive of alcohol dependence (although some authors quote scores of more than 13 in women and 15 in men as indicating likely dependence).<ref>[http://whqlibdoc.who.int/hq/2001/WHO_MSD_MSB_01.6a.pdf AUDIT: The Alcohol Use Disorders Identification Test: Guidelines for Use in Primary Care], second edition, by Thomas F. Babor, John C. Higgins-Biddle, John B. Saunders, and Maristela G. Monteiro. Retrieved June 24, 2006.</ref>
== External links ==
* [http://effectivechildtherapy.org/concerns-symptoms-disorders/disorders/drug-and-alcohol-abuse/ EffectiveChildTherapy.Org information on Substance Abuse]
* [https://sccap53.org Society of Clinical Child and Adolescent Psychology]
== References ==
{{Reflist}}
{{:{{BASEPAGENAME}}/Navbox}}
{{DEFAULTSORT:Alcohol Use Disorders Identification Test}}
[[Category:Psychological measures]]
e5h0zhp9jybvmywasxp13buw0gkc0qf
Appreciation of Failure
0
236663
2693985
1899712
2025-01-01T17:50:33Z
Nigerto Baldini
2995710
2693985
wikitext
text/x-wiki
This learning resource is about appreciation of failure. Failure has often a negative connotation. Therefore learners try to avoid failure. In the context of problem-solving in complex dynamic systems many people are involved to contribute to the solution. If the team tries to cover a large Domain of possible solutions, the workload of testing possible solution is distributed among the team members. The failure rate is very high if only a few appropriate solutions are among the domain of possible solutions.
Appreciation of failure convert the negatives annotation into a positive one. Failure this document well, so there are others do not have to repeat the testing of a possible solution, and the individuals understand, birthday Report over failure it Is necessary to cover systematically the domain of possible solutions. Well documented failure, which is accessible to the team members, and avoids repeating the same error all over again.
== Learning Task ==
* '''Fear of Failure:''' read the book chapter about [[Motivation and emotion/Book/2013/Fear_of_failure|fear failure]] and and explain, why this fear of failure is an obstacle for collaborative [[w:Problem solving|problem solving]]!
* '''Learning Environment in Schools:''' Create a learning environment for schools, in which students can identify and learn about the appreciation of failure (including benefits and obstacles)<ref>Johnson, D. W., Skon, L., & Johnson, R. (1980). Effects of cooperative, competitive, and individualistic conditions on children’s problem-solving performance. American Educational Research Journal, 17(1), 83-93.</ref>! Complex problems are associated with a large domain of test cases and only very few of the case lead to an acceptable solution. So if students fail testing one case, they should learn that this failure contributes to testing the large number of cases by a community. Students should learn about the requirement to document the failure transparent to others, so that
:* others do not repeat the test case and focus on other uncovered test cases or
:* learn from failure and improve the test case.
* '''Small and Medium Business Entprises (SME)''': SMEs can be very flexible and innovative with a small budget in comparison to large companies. Successful ideas of SMEs are bought from large companies (e.g.
:* [[w:Waze|Waze Community Driven Navigation]] bought by Google,
:* [[w:Skype|Skype]] bought by Microsoft,
:* [[w:WhatsApp|Facebook]] bought by Facebook,
:* [[w:GitHub|GitHub - as backbone of OpenSource Development]] bought by MircoSoft 2018, the OpenSource repositories could be accessed by anyone from GitHub including Mircosoft and the private software repositories cannot be sold by Mircosoft, because it is the intellectual property of the repository owners.
: Explain, what is the value of the SME (mentioned above), that justifies the price for buying the SME!
: Approach the assessment from either a collaborative and competetive angle!
: Describe the perspective from an individual SME (try and error) and from the community of SMEs with its diversity as an ecosystem of innovation that can be regarded as a Research and Development Unit where only the success stories of SMEs get paid.
: Explain the concept of diversity (e.g. bio diversity) as '''Richness of possible Answers and Expertise'''. How is biodiversity helpful to answers challenges to a ecosystem change? How can SME diversity and the appreciation of failure can help to understand the value of a diverse network of SME?
: Many SMEs fail with their business plan. Compare the costs for an own research and development unit of big companies with the costs for buying a SME for billion dollars. Link the concept to the appreciation of failure and the huge number of SMEs that invest money to explore a certain business idea. Do you think a reward system for well-documented failure makes sense for complex problems like [[climate change]] or issues related to the [[Sustainable Development Goals]]? What are the PROs and CONs of such a business idea that supports the appreciation of failure?
* '''[[Swarm intelligence]]''': explain similarities and differences between appreciation of failure and [[swarm intelligence]].
* '''Global Challenges:'''<ref>Ostrom, E., Burger, J., Field, C. B., Norgaard, R. B., & Policansky, D. (1999). Revisiting the commons: local lessons, global challenges. science, 284(5412), 278-282.</ref> Analyse global challenges like [[w:Climate Change|Climate Change]] and the requirement to identify appropriate response of humanity. [[Small and medium-sized enterprises]] can be regarded as flexible innovative units that contribute to [[w:Problem solving|problem solving]] in [[w:Complex dynamics|complex dynamic systems]]. If a large group of SMEs cover different areas with testbeds, pilots, case studies, .... the contribute to collaborative problem solving. Due to the complexity of global challenges a huge number of SMEs will fail because the will identify a deadend after analysis of certain approach in a domain of possible answers/responses. Nevertheless a well-documented report of failure contributes to collaborative problem solving. Explain the role of appreciation of well-documented failure for SME and create a draft of an economic reward system the takes the contributions into account.
* '''Failure analysis in health care systems:''' Failure occurs in all systems including the health care system with occasionally severe consequences for the patients<ref>DeRosier, J., Stalhandske, E., Bagian, J. P., & Nudell, T. (2002). Using health care failure mode and effect analysis™: the VA National Center for Patient Safety’s prospective risk analysis system. The Joint Commission journal on quality improvement, 28(5), 248-267.</ref>. Explain obstacles for reporting failure in health care system and explain the need for failure analysis and failure management in health care systems for avoiding a certain type of failure in the future. Include public perception of failure and trust in health care system in your consideration. Explain the role public appreciation of transparent failure management and a neutral point of view on failure and successes.
* '''Failure and Success<ref>Conchas, G. (2001). Structuring failure and success: Understanding the variability in Latino school engagement. Harvard Educational Review, 71(3), 475-505.</ref>:''' Failure can be regarded as a subjective assessment of an outcome/result. Provide examples in which a failure of one group member is a success of other group members. This leads to the competive problem solving. Explain differences and analogies between '''competitive'''<ref>Qin, Z., Johnson, D. W., & Johnson, R. T. (1995). Cooperative versus competitive efforts and problem solving. Review of educational Research, 65(2), 129-143.</ref> and '''collaborative'''<ref>Roschelle, J., & Teasley, S. D. (1995). The construction of shared knowledge in collaborative problem solving. In Computer supported collaborative learning (pp. 69-97). Springer, Berlin, Heidelberg.</ref> approaches to failure! Consider also the role of motivation for contribution to the problem solving approach.
* '''Failure and Predictive Models:''' Why do scientist combine failure analysis and predictive models<ref>Liang, Y., Zhang, Y., Sivasubramaniam, A., Jette, M., & Sahoo, R. (2006, June). Bluegene/l failure analysis and prediction models. In Dependable Systems and Networks, 2006. DSN 2006. International Conference on (pp. 425-434). IEEE.</ref>?
== See also ==
* [[Motivation and emotion/Book/2013/Fear of failure|Motivation and Emotion - Book]]
* [[Swarm Intelligence]]
== References ==
[[Category:Problem solving]]
[[Category:Systems Thinking]]
[[Category:Diversity]]
s6tbzlg5bzqwnnxdquao1d1zidltr7s
2693987
2693985
2025-01-01T17:56:41Z
Nigerto Baldini
2995710
Page's English is poor, fixed some, but still much more to do
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This learning resource is about appreciation of failure. Failure has often a negative connotation. Therefore learners try to avoid failure. In the context of problem-solving in complex dynamic systems many people are involved to contribute to the solution. If the team tries to cover a large Domain of possible solutions, the workload of testing possible solution is distributed among the team members. The failure rate is very high if only a few appropriate solutions are among the domain of possible solutions.
Appreciation of failure convert the negatives annotation into a positive one. Failure this document well, so others do not have to repeat the testing of a possible solution, and the individuals understand, birthday Report over failure it Is necessary to cover the domain of possible solutions systematically. Well documented failure, which is accessible to the team members, and avoids repeating the same error all over again.
== Learning Task ==
* '''Fear of Failure:''' read the book chapter about [[Motivation and emotion/Book/2013/Fear_of_failure|fear failure]] and and explain, why this fear of failure is an obstacle for collaborative [[w:Problem solving|problem solving]]
* '''Learning Environment in Schools:''' Create a learning environment for schools, in which students can identify and learn about the appreciation of failure (including benefits and obstacles)<ref>Johnson, D. W., Skon, L., & Johnson, R. (1980). Effects of cooperative, competitive, and individualistic conditions on children’s problem-solving performance. American Educational Research Journal, 17(1), 83-93.</ref>! Complex problems are associated with a large domain of test cases and only very few of the case lead to an acceptable solution. So i,f students fail testing one case, they should learn that this failure contributes to testing the large number of cases by a community. Students should learn about the requirement to document the failure transparent to others, so that
:* others do not repeat the test case and focus on other uncovered test cases or
:* learn from failure and improve the test case.
* '''Small and Medium Business Entprises (SME)''': SMEs can be very flexible and innovative with a small budget in comparison to large companies. Successful ideas of SMEs are bought from large companies (e.g.
:* [[w:Waze|Waze Community Driven Navigation]] bought by Google,
:* [[w:Skype|Skype]] bought by Microsoft,
:* [[w:WhatsApp|Facebook]] bought by Facebook,
:* [[w:GitHub|GitHub - as backbone of OpenSource Development]] bought by MircoSoft 2018, the OpenSource repositories could be accessed by anyone from GitHub including Microsoft and the private software repositories cannot be sold by Microsoft, because it is the intellectual property of the repository owners.
: Explain, what is the value of the SME (mentioned above), that justifies the price for buying the SME!
: Approach the assessment from either a collaborative and competitive angle!
: Describe the perspective from an individual SME (try and error) and from the community of SMEs with its diversity as an ecosystem of innovation that can be regarded as a Research and Development Unit where only the success stories of SMEs get paid.
: Explain the concept of diversity (e.g. bio diversity) as '''Richness of possible Answers and Expertise'''. How is biodiversity helpful to answers challenges to a ecosystem change? How can SME diversity and the appreciation of failure can help to understand the value of a diverse network of SME?
: Many SMEs fail with their business plan. Compare the costs for an own research and development unit of big companies with the costs for buying a SME for billion dollars. Link the concept to the appreciation of failure and the huge number of SMEs that invest money to explore a certain business idea. Do you think a reward system for well-documented failure makes sense for complex problems like [[climate change]] or issues related to the [[Sustainable Development Goals]]? What are the PROs and CONs of such a business idea that supports the appreciation of failure?
* '''[[Swarm intelligence]]''': explain similarities and differences between appreciation of failure and [[swarm intelligence]].
* '''Global Challenges:'''<ref>Ostrom, E., Burger, J., Field, C. B., Norgaard, R. B., & Policansky, D. (1999). Revisiting the commons: local lessons, global challenges. science, 284(5412), 278-282.</ref> Analyse global challenges like [[w:Climate Change|Climate Change]] and the requirement to identify appropriate response of humanity. [[Small and medium-sized enterprises]] can be regarded as flexible innovative units that contribute to [[w:Problem solving|problem solving]] in [[w:Complex dynamics|complex dynamic systems]]. If a large group of SMEs cover different areas with testbeds, pilots, case studies, .... the contribute to collaborative problem solving. Due to the complexity of global challenges a huge number of SMEs will fail because the will identify a deadend after analysis of certain approach in a domain of possible answers/responses. Nevertheless a well-documented report of failure contributes to collaborative problem solving. Explain the role of appreciation of well-documented failure for SME and create a draft of an economic reward system the takes the contributions into account.
* '''Failure analysis in health care systems:''' Failure occurs in all systems including the health care system with occasionally severe consequences for the patients<ref>DeRosier, J., Stalhandske, E., Bagian, J. P., & Nudell, T. (2002). Using health care failure mode and effect analysis™: the VA National Center for Patient Safety’s prospective risk analysis system. The Joint Commission journal on quality improvement, 28(5), 248-267.</ref>. Explain obstacles for reporting failure in health care system and explain the need for failure analysis and failure management in health care systems for avoiding a certain type of failure in the future. Include public perception of failure and trust in health care system in your consideration. Explain the role public appreciation of transparent failure management and a neutral point of view on failure and successes.
* '''Failure and Success<ref>Conchas, G. (2001). Structuring failure and success: Understanding the variability in Latino school engagement. Harvard Educational Review, 71(3), 475-505.</ref>:''' Failure can be regarded as a subjective assessment of an outcome/result. Provide examples in which a failure of one group member is a success of other group members. This leads to he competive problem solving. Explain differences and analogies between '''competitive'''<ref>Qin, Z., Johnson, D. W., & Johnson, R. T. (1995). Cooperative versus competitive efforts and problem solving. Review of educational Research, 65(2), 129-143.</ref> and '''collaborative'''<ref>Roschelle, J., & Teasley, S. D. (1995). The construction of shared knowledge in collaborative problem solving. In Computer supported collaborative learning (pp. 69-97). Springer, Berlin, Heidelberg.</ref> approaches to failure! Consider also the role of motivation for contribution to the problem solving approach.
* '''Failure and Predictive Models:''' Why do scientist combine failure analysis and predictive models<ref>Liang, Y., Zhang, Y., Sivasubramaniam, A., Jette, M., & Sahoo, R. (2006, June). Bluegene/l failure analysis and prediction models. In Dependable Systems and Networks, 2006. DSN 2006. International Conference on (pp. 425-434). IEEE.</ref>?
== See also ==
* [[Motivation and emotion/Book/2013/Fear of failure|Motivation and Emotion - Book]]
* [[Swarm Intelligence]]
== References ==
[[Category:Problem solving]]
[[Category:Systems Thinking]]
[[Category:Diversity]]
qja232x9htz30aa1n2qcprtdi41x04l
Sylheti language/Greetings and basic polite expressions
0
242402
2694077
2693954
2025-01-02T07:36:05Z
82.30.47.86
i made a few minor changes, couple translations were slightly off.
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text/x-wiki
[These brackets are for different pronunciations towards older people for respect]
{| class="wikitable"
|-
! To say !! Say
|-
| Greeting people || (assalamu alaikum) - used by Muslims, means peace be upon you. (nomoshkar) - used by Hindus
|-
| Are you good? || Bhala aso ni?
|-
| I am good/well || tik asi [asoin]
|-
| Congratulations || Mubarak
|-
| Blessed Wedding || Shaadi Mubarak
|-
| Goodbye || Allah hafiz
|-
| Good/Blessed morning || Biantubala Mubarak
|-
| Good/Blessed afternoon || Madan bala mubarak
|-
| Good/Blessed Evening || Hainja mubarak
|-
| Good/Blessed Night || Rait mubarak
|-
| See you again || Abar dekha oibo
|-
| Excuse me || Maaf karba
|-
| Forgive me || Maaf khor [khoro]
|-
| Thank you || Dhoinnobad
|-
| Thanks a lot || Shukriya
|-
| Please || Doya khori
|-
| Nice to meet you. || Afnar loge forisito oiya khushi oilam
|-
| I am good || Ami Bhala asi
|-
| Yes || [gee (pronounced like the letter g)] oy
|-
| No || [gee] naa
|-
| I don't understand [what you are saying] || (Afne kita khoisoin) ami buztam farsi na
|-
| Repeat please || Abar khoin(khoro)/ Arokhbar khoin(khoro)
|-
| Speak [a bit] slowly please || (Thura) aste kho [khoin]
|-
| I can't speak Sylheti || Ami Sylheti mattam fari na
|-
| I can't speak Sylheti very well || Ami bhala kori Sylheti mattam fari na
|-
| Do you speak English? || Afne Inglish matta faroin ni?
|}
What’s your name? Tumar nam kita?
{{subpage navbar}}
[[Category:Sylheti Language]]
52j38z4flv5ob0bhr283x7ek0ocd3dw
Digital Media Concepts/Color-blind lens/EnChroma
0
254380
2694033
2459546
2025-01-01T22:08:33Z
CommonsDelinker
9184
Removing [[:c:File:Enchroma_Lens.png|Enchroma_Lens.png]], it has been deleted from Commons by [[:c:User:DMacks|DMacks]] because: [[:c:COM:L|Copyright violation]]: Obvious license laundering, the image has a copyright watermark.
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= =
[[File:Red to Green Color Blindness.png|thumb|Red to Green Color Blindness]]
== '''How Human Eye work & Color-blindness''' ==
[[File:Contact Lenses.svg|Contact Lenses]]
The way the [https://en.wikipedia.org/wiki/Human_eye human eye] works is that [https://en.wikipedia.org/wiki/Retina retina] capture light waves from an object. There is the length of the light wave small, medium, and long-range that it is collected by the [https://en.wikipedia.org/wiki/Cone_cell cones]. The cones are more commonly referred to, and the cones differentiate as blue, green, and red cones. Cones may overlap at times which in turn affects the eyesight vision in color. That is the root of the color blindness.
<ref>Pappas, S. (2010, April 29). "How Do We See Color?" Retrieved from ttps://www.livescience.com/32559-why-do-we-see-in-color.html</ref>
=='''Types of colorblindness'''==
[[File:Speelgoed-tuin-dichroom.jpg|thumb|Speelgoed-tuin-dichroom]]
When people think [https://en.wikipedia.org/wiki/Color_blindness color-blind], they might think that one can only see in black, white or gray. Only seeing black and white, [https://en.wikipedia.org/wiki/Monochromacy monochromacy], is a rare form of color blindness and it is more common to be red-green color blindness. Within the red-green color blindness, there are four more distinctions such as the green looking more red and verse. The brightness of the colors would look less bright or green and red were color that one could not be able to tell the difference between. Rare-common color blindness is blue-yellow with two distinctions having a similar effect as the red. One would not be able to distinguish the “...difference between blue and green, and between yellow and red.”
<ref name=":0" />monochromacy, is a rare form of color blindness and it is more common to be red-green color blindness. Within the red-green color blindness, there are four more distinctions such as the green looking more red and verse. The brightness of the colors would look less bright or green and red were color that one could not be able to tell the difference between. Rare-common color blindness is blue-yellow with two distinctions having a similar effect as the red. One would not be able to distinguish the “...difference between blue and green, and between yellow and red.”
<ref name=":0">Types of Color Blindness. (2019, June). Retrieved from https://nei.nih.gov/learn-about-eye-health/eye-conditions-and-diseases/color-blindness/types-color-blindness.</ref>
==''''''How the lens work''''''==
Knowing how the human collects color, and how color-blindness is a result of cones overlaps the color-blind [https://en.wikipedia.org/wiki/Lens lens] can be explained. The lens from the EnChroma Labs corrects the overlap with filter the wavelength with its focus on the red and green cones. The main or big brands that the public knows are [https://en.wikipedia.org/wiki/Enchroma EnChroma], so many articles explain how the colorblind lens work from the EnChroma brand. The lens would not be exact of ‘regular’ vision, but similar to the vision.
<ref>Cross, R. (2016, June 29). This is how EnChroma's color-blindness glasses achieve their powerful effects. Retrieved from https://www.technologyreview.com/s/601782/how-enchromas-glasses-correct-color-blindness/.</ref>
=='''Dr. Donald McPherson'''==
{{Infobox
|name = Dr. Donald McPherson
|image = [[File:201705 Scientist desk M.svg|201705 Scientist desk M]]
|title = "Dr. Donald McPherson"
|headerstyle = background:#ccf;
|labelstyle = background:#ddf;
|label1 =
|data2 = {{{item_one| Ph.D. Chief Science Officer}}}
|data3 = {{{item_two|Co-Founder & Inventor of EnChroma}}}
}}
<div style="clear:both;"></div>
Glass scientist Dr. Donald McPherson one of the founders EnChroma. He studied at Alfred University with “.... an undergraduate degree in math and art, [and] a masters in ceramics”. Ph.D. in glass science. ”.<ref> Rehbock, B. (2015, June 16)."Color For The Colorblind Journalist: Exclusive Interview With EnChroma VP." Retrieved from http://techdrive.co/color-for-the-colorblind-journalist-exclusive-interview-with-enchroma-vp/.</ref>
<ref>Zhou, L. (2015, March 3). "A Scientist Accidentally Developed Sunglasses That Could Correct Color Blindness." Retrieved from https://www.smithsonianmag.com/innovation/scientist-accidentally-developed-sunglasses-that-could-correct-color-blindness-180954456/.</ref> He started with making glass lens for surgeons to protect their eyes from the laser. That lens that Dr.McPherson created the surgeons was the domino to soon create EnChroma Labs. McPherson would use the [https://en.wikipedia.org/wiki/Glasses glasses] he made for surgeons as normal sunglasses. In the [https://en.wikipedia.org/wiki/Frisbee frisbee] game, a friend of McPherson had borrowed his sunglasses and notice that he could see the difference in color tones between the grass and concrete.
<ref>Zhou, L. (2015, March 3). "A Scientist Accidentally Developed Sunglasses That Could Correct Color Blindness." Retrieved from https://www.smithsonianmag.com/innovation/scientist-accidentally-developed-sunglasses-that-could-correct-color-blindness-180954456/.</ref> Between 2004 to 2009 McPherson and his colleges, Andy and Tony Dykes would put the lens to scientific clinical trials. Later in 2012 EnChroma Labs had been in business.<ref> Rehbock, B. (2015, June 16)."Color For The Colorblind Journalist: Exclusive Interview With EnChroma VP." Retrieved from http://techdrive.co/color-for-the-colorblind-journalist-exclusive-interview-with-enchroma-vp/.</ref>
=='''Future product'''==
[[File:Contact lens 1.jpg|thumb|Contact lens 1]]
EnChroma Labs has worked on the sunglasses that helps with color blind, but could not work with clear glasses. Clear glasses would not work because the light wave would not be able to bound off the lens as sunglasses would. It is possible for contact lens. EnChroma has been in the process of creating correction colorblind len. The company does strive to improve the lens that they make now for a better experience. <ref> Rehbock, B. (2015, June 16)."Color For The Colorblind Journalist: Exclusive Interview With EnChroma VP." Retrieved from http://techdrive.co/color-for-the-colorblind-journalist-exclusive-interview-with-enchroma-vp/.</ref>
=='''Controversy'''==
There has been controversy on the EnChroma brand because of false marketing on the data they put out. The major attraction is a study EnChroma put that deals with 48 color-blind volunteers.<ref>Builder, M. (2019, February 20)."Do Color-blind Glasses Actually Work?" Retrieved from http://nymag.com/strategist/article/color-blind-glasses-enchroma.html.</ref> Especially the company brands in a way that made consumers think that they are ‘curing’ their color-blindness when the lens is glasses and can only work when one wears them.<ref>staff, S. X. (2018, October 29). Scientists debunk the effectiveness of EnChroma glasses for colorblind people. Retrieved from https://phys.org/news/2018-10-scientists-debunk-effectiveness-enchroma-glasses.html.</ref> As mentioned before there the companies branding would make it seem like one would see if they would not color blind, but the lens is a close second to that vision.
=='''Other Companies in the Industry'''==
The well-known production of this tech is EnChroma there are many more companies that make this lens. Some of these companies are the ColorCorrection System, Golden, Pilestone, and Vino. The brand ColorCorrection System is a prescription color-blind lens. The Golden brands focus more difficult colors like pink. There is red-green color blindness and blue-yellow color blindness that products by Pilestone.<ref>Can Glasses Really Fix Colorblindness? (EnChroma & Others). (2019). Retrieved from https://www.nvisioncenters.com/best-lenses/fix-colorblindness/.</ref>
==External Links==
https://enchroma.com/
==Reference==
[[Category:Digital Media Concepts]]
64bobtd1gdpysnmeowuplcjiwazejp9
Social Victorians/People/Bourke
0
263813
2693989
2693948
2025-01-01T18:54:18Z
Scogdill
1331941
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==Also Known As==
* Family name: Bourke [pronounced ''burk'']<ref name=":62">{{Cite journal|date=2024-05-07|title=Earl of Mayo|url=https://en.wikipedia.org/w/index.php?title=Earl_of_Mayo&oldid=1222668659|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Earl_of_Mayo.</ref>
* The Hon. Algernon Bourke
* Mrs. Guendoline Bourke
* Lady Florence Bourke
* See also the [[Social Victorians/People/Mayo|page for the Earl of Mayo]], the Hon. Algernon Bourke's father.
== Overview ==
Although the Hon. Algernon Henry Bourke was born in Dublin in 1854 and came from a family whose title is in the Peerage of Ireland,<ref name=":6">1911 England Census.</ref> he seems to have spent much of his adult life generally in England and especially in London. Because he was the son of the [[Social Victorians/People/Mayo|Earl of Mayo]], perhaps, or perhaps because he was so involved in projects that got reported on, he was mentioned a great deal in the newspapers, but after his bankruptcy, he seems to have receded in prominence, in part because he was living outside of the U.K.
Mrs. Guendoline Bourke was a noted horsewoman and an excellent shot, exhibited at dog shows successfully and was "an appreciative listener to good music."<ref>"Vanity Fair." ''Lady of the House'' 15 June 1899, Thursday: 4 [of 44], Col. 2c [of 2]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0004836/18990615/019/0004.</ref> She was reported as attending many social events without her husband, especially into the 20th century, usually with a quick description of what she wore. Unlike her husband's, Guendoline's social status seems to have risen as time passed, and she appears in stories associated with the Princess of Wales, and then later with Queen Alexandra.
The Hon. Algernon Bourke and Mr. Algernon Bourke, depending on the newspaper article, were the same person. Calling him Mr. Bourke in the newspapers, especially when considered as a businessman or (potential) member of Parliament, does not rule out the son of an earl, who would normally be accorded the honorific of ''Honorable''.
== Acquaintances, Friends and Enemies ==
=== Algernon Bourke ===
* [[Social Victorians/People/Montrose|Marcus Henry Milner]], "one of the zealous assistants of that well-known firm of stockbrokers, Messrs. Bourke and Sandys"<ref name=":8">"Metropolitan Notes." ''Nottingham Evening Post'' 31 July 1888, Tuesday: 4 [of 4], Col. 2a [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000321/18880731/025/0004.</ref>
* Caroline, Duchess of Montrose — her "legal advisor" on the day of her marriage to Marcus Henry Milner<ref>"Metropolitan Notes." ''Nottingham Evening Post'' 31 July 1888, Tuesday: 4 [of 4], Col. 1b [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000321/18880731/025/0004.</ref>
=== Guendoline Bourke ===
* Lord and Lady Alington, Belvedere House, Scarborough
== Organizations ==
=== Guendoline Bourke ===
* Member, the Ladies Committee for the Prince's Skating Club, which also included [[Social Victorians/People/Princess Louise|Princess Louise]] (Duchess of Argyll), the [[Social Victorians/People/Portland|Duchess of Portland]], [[Social Victorians/People/Londonderry|Lady Londonderry]], [[Social Victorians/People/Campbell|Lady Archibald Campbell]], [[Social Victorians/People/Ribblesdale|Lady Ribblesdale]], and [[Social Victorians/People/Asquith|Mrs. Asquith]]<ref name=":11">"What the 'World' Says." ''Northwich Guardian'' 01 November 1902, Saturday: 6 [of 8], Col. 8a [of 9]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001975/19021101/134/0006. Print title: The ''Guardian'', p. 6.</ref> (1902, at least)
=== Algernon Bourke ===
* Eton
* Cambridge University, Trinity College, 1873, Michaelmas term<ref name=":7">Cambridge University Alumni, 1261–1900. Via Ancestry.</ref>
* Conservative Party
* 1879: Appointed a Poor Law Inspector in Ireland, Relief of Distress Act
* 1885: Office of the 7th Surrey Rifles Regiment<ref>"7th Surrey Rifles." ''South London Press'' 08 August 1885, Saturday: 12 [of 16], Col. 4a [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000213/18850808/165/0012. Print p. 12.</ref>
* Special Correspondent of The ''Times'' for the Zulu War, accompanying Lord Chelmsford
* Head, Messrs. Bourke and Sandys, "that well-known firm of stockbrokers"<ref name=":8" /> ( – 1901 [at least])
* White's gentleman's club, St. James's,<ref>{{Cite journal|date=2024-10-09|title=White's|url=https://en.wikipedia.org/wiki/White's|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/White%27s.</ref> Manager (1897)<ref>"Side Lights on Drinking." ''Waterford Standard'' 28 April 1897, Wednesday: 3 [of 4], Col. 7a [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001678/18970428/053/0003.</ref>
* Willis's Rooms<blockquote>... the Hon. Algernon Burke [sic], son of the 6th Earl of Mayo, has turned the place into a smart restaurant where choice dinners are served and eaten while a stringed band discourses music. Willis's Rooms are now the favourite dining place for ladies who have no club of their own, or for gentlemen who are debarred by rules from inviting ladies to one of their own clubs. The same gentleman runs a hotel in Brighton, and has promoted several clubs. He has a special faculty for organising places of the kind, without which such projects end in failure.<ref>"Lenten Dullness." ''Cheltenham Looker-On'' 23 March 1895, Saturday: 11 [of 24], Col. 2c [of 2]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000226/18950323/004/0011. Print p. 275.</ref></blockquote>
==== Boards of Directors ====
*1883: One of the directors, the Franco-English Tunisian Esparto Fibre Supply Company, Ltd.<ref>''Money Market Review'', 20 Jan 1883 (Vol 46): 124.</ref>
*1891: One of the founders, the Discount Banking Company, Ltd., which says Algernon Bourke is a director of District Messenger Services and News Company, Ltd.<ref>"Public Company." ''Nottingham Journal'' 31 October 1891, Saturday: 4 [of 8], Col. 8a [of 8]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001896/18911031/099/0004. Print title: ''The Nottingham Daily Express'', p. 4.</ref>
*1894: One of the directors, the Frozen Lake, Ltd., with Admiral Maxse, Lord [[Social Victorians/People/Beresford|Marcus Beresford]], [[Social Victorians/People/Williams|Hwfa Williams]]<ref>"The Frozen Lake, Limited." ''St James's Gazette'' 08 June 1894, Friday: 15 [of 16], Col. 4a [of 4]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001485/18940608/085/0015. Print p. 15.</ref>
==== Committees ====
*Member, Men's Committee of the Prince's Skating Club, which also included Lord Edward Cecil, Lord Redesdale, Mr. [[Social Victorians/People/Lyttelton|Alfred Lyttelton]], Sir Edgar Vincent, Sir William Hart Dyke, and Mr. [[Social Victorians/People/Grenfell|W. H. Grenfell]]<ref name=":11" /> (1902, at least)
*[[Social Victorians/Timeline/1896#25 March 1896, Wednesday|The Sala Memorial Fund]], member of the committee (from 25 March 1896)
* Member of an "influential committee" headed by the Lord Mayor "to restore salmon to the Thames" (June 1899)<ref>"Salmon in the Thames." ''Berks and Oxon Advertiser'' 30 June 1899, Friday: 5 [of 8], Col. 4a [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0002298/18990630/079/0005. Print n.p.</ref>
== Timeline ==
'''1872 February 8''', Richard Bourke, 6th Earl of Mayo was assassinated while inspecting a "convict settlement at Port Blair in the Andaman Islands ... by Sher Ali Afridi, a former Afghan soldier."<ref>{{Cite journal|date=2024-12-01|title=Richard Bourke, 6th Earl of Mayo|url=https://en.wikipedia.org/wiki/Richard_Bourke,_6th_Earl_of_Mayo|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Richard_Bourke,_6th_Earl_of_Mayo.</ref> The Hon. Algernon's brother Dermot became the 7th Earl at 19 years old.
'''1876 November 24, Friday''', the Hon. Algernon Bourke was one of 6 men (2 students, one of whom was Bourke; 2 doctors; a tutor and another man) from Cambridge who gave evidence as witnesses in an inquest about the death from falling off a horse of a student.<ref>"The Fatal Accident to a Sheffield Student at Cambridge." ''Sheffield Independent'' 25 November 1876, Saturday: 7 [of 12], Col. 5a [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000181/18761125/040/0007. Print title: ''Sheffield and Rotherham Independent'', n. p.</ref>
'''1881 May 10, Tuesday''', Algernon Bourke attended the [[Social Victorians/Timeline/1881#1881 May 10, Tuesday|wedding of Marion Lascelles, eldest daughter of the Hon. Egremont W. Lascelles, brother of the Earl of Harewood, and Lieutenant Henry Dent Brocklehurst, of the Second Life Guards, nephew of Mr. Philip Brocklehurst, of Swithamley Park, Macclesfield]]. His gift was an "old enamelled watch set in pearls."<ref>"Nuptial Rejoicings at Middlethorpe Manor. Marriage of Miss Lascelles and Lieut. Brocklehurst." ''Yorkshire Gazette'' 14 May 1881, Saturday: 9 [of 12], Cols. 3a–4a [of 6]. ''British Newspaper Archive''https://www.britishnewspaperarchive.co.uk/viewer/bl/0000266/18810514/057/0009. Print same title and p.</ref>
'''1884 May 3, Saturday''', the "Rochester Conservatives" announced that they would "bring forward the Hon. Algernon Bourke, brother of Lord Mayo, as their second candidate,"<ref>"Election Intelligence." ''Yorkshire Gazette'' 03 May 1884, Saturday: 4 [of 12], Col. 6a [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000266/18840503/011/0004.</ref> but because he could not be the first candidate, Bourke declined.<ref>"Rochester." London ''Daily Chronicle'' 09 May 1884, Friday: 3 [of 8], Col. 8b [of 8]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0005049/18840509/049/0003.</ref>
'''1884 June 18, Wednesday''', Mr. Algernon Bourke was on a committee to watch a [[Social Victorians/Timeline/1884#18 June 1884, Wednesday|Mr. Bishop's "thought-reading" experiment]], which was based on a challenge by Henry Labourchere made the year before. This "experiment" took place before a fashionable audience.
'''1885 October 3, Saturday''', the Hon. Algernon Bourke was named as the Conservative candidate for Clapham in the Battersea and Clapham borough after the Redistribution Bill determined the electoral districts for South London.<ref>"South London Candidates." ''South London Press'' 03 October 1885, Saturday: 9 [of 16], Col. 5b [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000213/18851003/096/0009. Print p. 9.</ref> The Liberal candidate, who won, was Mr. J. F. Moulton.
'''1886 July 27, Tuesday''', Algernon Bourke attended a service honoring a memorial at St. Paul's for his father, who had been assassinated.<ref>"Memorial to the Late Earl of Mayo." ''Northern Whig'' 28 July 1886, Wednesday: 6 [of 8], Col. 6b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000434/18860728/143/0006. Print p. 6.</ref>
'''1886 September 2, Thursday''', Mr. Algernon Bourke was part of a group of mostly aristocratic men taking part in [[Social Victorians/Timeline/1886#8 September 1886, Wednesday|a "trial-rehearsal" as part of Augustus Harris's production]] ''A Run of Luck'', about sports.
'''1886 October 2, Saturday''', the Duke of Beaufort and the Hon. Algernon Bourke arrived in Yougal: "His grace has taken a residence at Lismore for a few weeks, to enjoy some salmon fishing on the Blackwater before the close of the season."<ref>"Chippenham." ''Wilts and Gloucestershire Standard'' 02 October 1886, Saturday: 8 [of 8], Col. 6a [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001955/18861002/142/0008. Print p. 8.</ref>
'''1886 December 30, Thursday''', Algernon Bourke was back in London and attending the [[Social Victorians/Timeline/1886#1886 December 30, Thursday|"Forty Thieves" pantomime at the Drury Lane Theatre]].
'''1887 December 15''', Hon. Algernon Bourke and Guendoline Stanley were married at St. Paul's, Knightsbridge, by Bourke's uncle the Hon. and Rev. George Bourke. Only family members attended because of "the recent death of a near relative of the bride."<ref>"Court Circular." ''Morning Post'' 16 December 1887, Friday: 5 [of 8], Col. 7c [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000174/18871216/066/0005.</ref>
'''1888 July 26''', [[Social Victorians/People/Montrose|Caroline Graham Stirling-Crawford]] (known as Mr. Manton for her horse-breeding and -racing operations) and Marcus Henry Milner married.<ref name=":12">"Hon. Caroline Agnes Horsley-Beresford." {{Cite web|url=https://thepeerage.com/p6863.htm#i68622|title=Person Page|website=thepeerage.com|access-date=2020-11-21}}</ref> According to the ''Nottingham Evening Post'' of 31 July 1888,<blockquote>LONDON GOSSIP.
(From the ''World''.)
The marriage of "Mr. Manton" was the surprise as well the sensation of last week. Although some wise people noticed a certain amount of youthful ardour in the attentions paid by Mr. Marcus Henry Milner to Caroline Duchess of Montrose at '''Mrs. Oppenheim's ball''', nobody was prepared for the sudden ''dénouement''; '''and it''' were not for the accidental and unseen presence [[Social Victorians/People/Mildmay|a well-known musical amateur]] who had received permission to practice on the organ, the ceremony performed at half-past nine on Thursday morning at St. Andrew's, Fulham, by the Rev. Mr. Propert, would possibly have remained a secret for some time to come. Although the evergreen Duchess attains this year the limit of age prescribed the Psalmist, the bridegroom was only born in 1864. Mr. "Harry" Milner (familiarly known in the City as "Millions") was one of the zealous assistants of that well-known firm of stockbrokers, Messrs. Bourke and Sandys, and Mr. Algernon Bourke, the head of the house (who, of course, takes a fatherly interest in the match) went down to Fulham to give away the Duchess. The ceremony was followed by a ''partie carrée'' luncheon at the Bristol, and the honeymoon began with a visit to the Jockey Club box at Sandown. Mr. Milner and the Duchess of Montrose have now gone to Newmarket. The marriage causes a curious reshuffling of the cards of affinity. Mr. Milner is now the stepfather of the [[Social Victorians/People/Montrose|Duke of Montrose]], his senior by twelve years; he is also the father-in-law of [[Social Victorians/People/Lady Violet Greville|Lord Greville]], Mr. Murray of Polnaise, and [[Social Victorians/People/Breadalbane|Lord Breadalbane]].<ref name=":8" /></blockquote>'''1888 December 1st week''', according to "Society Gossip" from the ''World'', the Hon. Algernon Bourke was suffering from malaria, presumably which he caught when he was in South Africa:<blockquote>I am sorry to hear that Mr. Algernon Bourke, who married Miss Sloane-Stanley a short time ago, has been very dangerously ill. Certain complications followed an attack of malarian fever, and last week his mother, the Dowager Lady Mayo, and his brother, Lord Mayo, were hastily summoned to Brighton. Since then a change for the better has taken place, and he is now out of danger.<ref>"Society Gossip. What the ''World'' Says." ''Hampshire Advertiser'' 08 December 1888, Saturday: 2 [of 8], Col. 5b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000495/18881208/037/0002. Print title: ''The Hampshire Advertiser County Newspaper''; print p. 2.</ref></blockquote>'''1889 – 1899 January 1''', the Hon. Algernon Bourke was "proprietor" of White's Club, St. James's Street.<ref name=":9">"The Hon. Algernon Bourke's Affairs." ''Eastern Morning News'' 19 October 1899, Thursday: 6 [of 8], Col. 7c [of7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001152/18991019/139/0006. Print p. 6.</ref>
'''1889 June 8, Saturday''', the Hon. Algernon Bourke contributed some art he owned to the collection of the Royal Institute of Painters in Water-Colours' [[Social Victorians/Timeline/1889#8 June 1889, Saturday|exhibition of "the works of the 'English Humourists in Art.'"]]
'''1892''', the Hon. Algernon Bourke privately published his ''The History of White's'', the exclusive gentleman's club.
'''1893 February 11, Tuesday''', Algernon Bourke opened Willis's Restaurant:<blockquote>Mr. Algernon Bourke has in his time done many things, and has generally done them well. His recently published history of White's Club is now a standard work. White's Club itself was a few years ago in its agony when Mr. Bourke stepped in and gave it a renewed lease of life. Under Mr. Bourke's auspices "Willis's Restaurant" opened its doors to the public on Tuesday last in a portion of the premises formerly so well known as Willis's Rooms. This new venture is to rival the Amphitryon in the matter of cuisine and wines; but it is not, like the Amphitryon, a club, but open to the public generally. Besides the restaurant proper, there are several ''cabinets particuliers'', and these are decorated with the very best of taste, and contain some fine portraits of the Georges.<ref>"Marmaduke." "Letter from the Linkman." ''Truth'' 20 April 1893, Thursday: 25 [of 56], Col. 1a [of 2]. ''British Newspaper Archive'' [https://www.britishnewspaperarchive.co.uk/viewer/bl/0002961/18930420/075/0025# https://www.britishnewspaperarchive.co.uk/viewer/bl/0002961/18930420/075/0025]. Print p. 855.</ref></blockquote>'''1893 April 1, Saturday''', Algernon Bourke published a letter to the editor of the ''Times'', reprinted in the ''Kildare Observer'', arguing against Gladstone's Home Rule bill on the grounds that Ireland would not be able to take out a loan on its own behalf because of its obligations to the U.K., including what was called its share of the national debt.<ref>"Irish Unionist Alliance." ''Kildare Observer and Eastern Counties Advertiser'' 01 April 1893, Saturday: 6 [of 8], Col. 4c [of 5]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001870/18930401/062/0006. Print: The ''Kildare Observer'', n.p.</ref>
'''1893 November 30, Thursday''', with Sir Walter Gilbey the Hon. Algernon Bourke "assisted" in "forming [a] collection" of engravings by George Morland that was exhibited at Messrs. J. and W. Vokins’s, Great Portland-street.<ref>"The George Morland Exhibition at Vokins's." ''Sporting Life'' 30 November 1893, Thursday: 4 [of 4], Col. 4c [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000893/18931130/058/0004.</ref>
'''1895 February 23, Saturday''', the Hon. Algernon Bourke attended the [[Social Victorians/Timeline/1895#23 February 1895, Saturday|fashionable wedding of Laurence Currie and Edith Sibyl Mary Finch]].
'''1895 August 24, Saturday''', "Marmaduke" in the Graphic says that Algernon Bourke "opened a cyclists' club in Chelsea."<ref>"Marmaduke." "Court and Club." The ''Graphic'' 24 August 1895, Saturday: 11 [of 32], Col. 3c [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/9000057/18950824/017/0011. Print p. 223.</ref>
'''1895 October''', the Hon. Algernon Bourke [[Social Victorians/Timeline/1900s#24 October 1902, Friday|opened the Prince's ice-skating rink for the season]].
'''1896 June 29, Monday''', Algernon Bourke published a letter to the editor of the ''Daily Telegraph'':<blockquote>To the Editor of “The Daily Telegraph.”
Sir — Permit me to make my bow to the public. I am the manager of the Summer Club, which on two occasions bas been the subject of Ministerial interpellation in Parliament. The Summer Club is a small combination, which conceived the idea of attempting to make life more pleasant in London by organising breakfast, luncheon, and teas in Kensington Gardens for its members. This appears to have given offence in some way to Dr. Tanner, with the result that the catering arrangements of the club are now "by order" thrown open to the public. No one is more pleased than I am at the result of the doctor's intervention, for it shows that the idea the Summer Club had of using the parks for something more than mere right of way bas been favourably received. In order, however, that the great British public may not be disappointed, should they all come to lunch at once, I think it necessary to explain that the kitchen, which by courtesy of the lessee of the kiosk our cook was permitted to use, is only 10ft by 5ft; it has also to serve as a scullery and pantry, and the larder, from which our luxurious viands are drawn, is a four-wheeled cab, which comes up every day with the food and returns after lunch with the scraps. Nevertheless, the Summer Club says to the British public — What we have we will share with you, though it don't amount to very much — I am, Sir, your obedient servant,
ALGERNON BOURKE.
White's Club, June 27<ref>"The Summer Club." ''Daily Telegraph & Courier'' (London) 29 June 1896, Monday: 8 [of 12], Col. 2b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001112/18960629/072/0008. Print title: ''Daily Telegraph'', p. 8.</ref></blockquote>'''1896 July 4, Saturday''', "Marmaduke" in the ''Graphic'' took Bourke's side on the Summer Club in Kensington Park:<blockquote>Most of us have noticed that if we read in the newspapers the account of some matter which we are personally acquainted with the account will generally contain several errors. I have also noticed that when a question is asked in the House of Commons regarding some matter about which I know all the facts the question and the official answer to it frequently contain serious errors. Last week Mr. Akers-Douglas was asked in the House to explain how it was that Mr. Algernon Bourke obtained permission to open the "Summer Club" in Kensington Gardens, and he was questioned upon other particulars connected with the same matter. Both the questions and the official reply showed considerable ignorance of the facts. There has been from time immemorial a refreshment kiosk in Kensington Gardens. Mr. Bourke obtained from the tenant of this permission to use the kitchen for the benefit of the "Summer Club," and to supply the members of the latter with refreshments. It was a purely commercial transaction. Mr. Bourke then established some wicker seats, a few tables, a tent, and a small hut upon a lawn in the neighbourhood of the kiosk. To do this he must have obtained the permission of Mr. Akers-Douglas, as obviously he would otherwise have been immediately ordered to remove them. Mr. Akers-Douglas equally obviously would not have given his sanction unless he had been previously informed of the objects which Mr. Bourke had in view — to wit, that the latter intended to establish a club there. That being the case, it is difficult to understand for what reason Mr. Akers-Douglas has now decided that any member of the public can use the chairs, tables, and tent belonging to the "Summer Club," can insist upon the club servants attending upon him, and can compel them to supply him with refreshments. Mr. Akers-Douglas should have thought of the consequences before he granted the permission.<ref>"Marmaduke." "Court and Club." The ''Graphic'' 04 July 1896, Saturday: 14 [of 32], Col. 1b [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/9000057/18960704/029/0014. Print p. 14.<blockquote></blockquote></ref></blockquote>'''1896 August 10, Monday''', the Morning Leader reported that the Hon. Algernon Bourke, for the Foreign Office, received Li Hung Chang at St. Paul's:<blockquote>At St. Paul's Li Hung was received by Field-Marshal Simmons, Colonel Lane, the Hon. Algernon Bourke, of the Foreign Office (who made the necessary arrangements for the visit) and Canon Newbolt, on behalf of the Dean and Chapter. A crowd greeted Li with a cheer as he drove up in Lord Lonsdale’s striking equipage, and his Excellency was carried up the steps in an invalid chair by two stalwart constables. He walked through the centre door with his suite, and was immediately conducted by Canon Newbolt to General Gordon’s tomb in the north aisle, where a detachment of boys from the Gordon Home received him as a guard of honor.
Li inspected the monument with marked interest, and drew the attention of his suite to the remarkable likeness to the dead hero. He laid a handsome wreath of royal purple asters, lilies, maidenhair fern, and laurel, tied with a broad band of purple silk, on the tomb.
The visit was not one of inspection of the building, but on passing the middle aisle the interpreter called the attention of His Excellency to the exquisite architecture and decoration of the chancel. Li shook hands in hearty English fashion with Canon Newbolt and the other gentlemen who had received him, and, assisted by his two sons, walked down the steps to his carriage. He returned with his suite to Carlton House-terrace by way of St. Paul’s Churchyard, Cannon-st., Queen Victoria-st., and the Embankment.<ref>"At St. Paul's." ''Morning Leader'' 10 August 1896, Monday: 7 [of 12], Col. 2b [of 5]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0004833/18960810/134/0007. Print p. 7.</ref></blockquote>'''1896 August 19, Wednesday''', the ''Edinburgh Evening News'' reported on the catering that White's Club and Mr Algernon Bourke arranged for the visiting Li Hung Chang:<blockquote>It is probably not generally known (says the "Chef") that Mr Algernon Bourke, manager of White's Club, London, has undertaken to the whole of the catering for our illustrious visitor front the Flowery Land. Li Hung Chang has five native cooks in his retinue, and the greatest good fellowship exists between them and their English ''confreres'', although considerable difficulty is experienced in conversation in understanding one another's meaning. There are between 40 and and 50 to cater for daily, besides a staff about 30; that Mr Lemaire finds his time fully occupied. The dishes for his Excellency are varied and miscellaneous, and from 14 to 20 courses are served at each meal. The bills of fare contain such items as bird's-nest soup, pigs' kidneys stewed in cream, boiled ducks and green ginger, sharks' fins, shrinips and prawns stewed with leeks and muscatel grapes, fat pork saute with peas and kidney beans. The meal usually winds with fruit and sponge cake, and freshly-picked green tea as liqueur.<ref>"Li Hung Chang's Diet." ''Edinburgh Evening News'' 19 August 1896, Wednesday: 3 [of 4], Col. 8b [of 8]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000452/18960819/057/0003.</ref></blockquote>
'''1896 November 6, Friday''', Algernon Bourke was on the committee for [[Social Victorians/Timeline/1896#1896 November 6, Friday|the Prince's Club ice-skating rink, which opened on this day]].<p>
'''1896 November 25, Wednesday''', Mr. and Mrs. Algernon Bouke attended [[Social Victorians/Timeline/1896#23 November 1896, Monday23 November 1896, Monday|Lord and Lady Burton's party for Derby Day]].<p>
'''1896 December 4, Friday''', the Orleans Club at Brighton was robbed:<blockquote>The old building of the Orleans Club at Brighton, which opens its new club house at 33, Brunswick-terrace to-day, was the scene of a very ingenious burglary during the small hours of yesterday morning. The greater portion of the club property had already been removed to the new premises, but Mr Algernon Bourke, his private secretary, and some of the officials of the club, still occupied bed-rooms at the house in the King’s-road. The corner shop of the street front is occupied by Mr. Marx, a jeweller in a large way of business, and upon his manager arriving at nine o'clock he discovered that the place had been entered through hole in the ceiling, and a great part of a very valuable stock of jewelry extracted. An examination of the morning rooms of the club, which runs over Mr. Marx's establishment reveal a singularly neat specimen of the burglar's art. A piece of the flooring about 15in square had been removed by a series of holes bored side by side with a centre-bit, at a spot where access to the lofty shop was rendered easy by a tall showcase which stood convemently near. A massive iron girder had been avoided by a quarter of an inch, and this circumstance and the general finish of the operation point to an artist in his profession, who had acquired an intimate knowledge of the premises. The club doors were all found locked yesterday morning, and the means of egress adopted by the thief are at present a mystery.<ref>"Burglary at Brighton." ''Daily Telegraph & Courier'' (London) 05 December 1896, Saturday: 5 [of 12], Col. 7a [of 7]. British Newspaper Archive https://www.britishnewspaperarchive.co.uk/viewer/bl/0001112/18961205/090/0005. Print title: ''Daily Telegraph''; p. 5.</ref></blockquote>
'''1896 December 10, Thursday''', Guendoline Bourke was present to help staff a stall at the [[Social Victorians/Timeline/1896#10 December 1896, Thursday|Irish Industries Exhibition and Sale, Brighton]].
'''1897 July 2, Friday''', the Hon. A. and Mrs. A. Bourke and Mr. and Mrs. Bourke attended the [[Social Victorians/1897 Fancy Dress Ball | Duchess of Devonshire's fancy-dress ball]] at Devonshire House.
'''1897 July 11–16, week of''', a dog of Guendoline Bourke's won a prize at the [[Social Victorians/Timeline/1897#11–16 July 1897, Week Of|Ladies' Kennel Association show in the Royal Botanic Gardens in Regent's Park]].<p>
'''23 July 1897 — or 30 July 1897 – Friday''', Guendonline Bourke attended [[Social Victorians/Timeline/1897#23 July 1897, Friday|Lady Burton's party at Chesterfield House]]. <blockquote>Far the prettiest women in the room were Lady Henry Bentinck (who looked perfectly lovely in pale yellow, with a Iong blue sash; and Mrs. Algernon Bourke, who was as smart as possible in pink, with pink and white ruchings on her sleeves and a tall pink feather in her hair.<ref>"Lady Burton's Party at Chesterfield House." ''Belper & Alfreton Chronicle'' 30 July 1897, Friday: 7 [of 8], Col. 1c [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0004151/18970730/162/0007. Print title: ''Belper and Alfreton Chronicle''; n.p.</ref></blockquote>
'''1897 October 30, Saturday''', ''Black and White'' published '''J.P.B.'''<nowiki/>'s "The Case of Mrs. Elliott,"<ref name=":13">J.P.B. "The Case of Mrs. Elliott." ''Black & White'' 30 October 1897, Saturday: 12 [of 34], Cols. 1a–2b [of 2]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0004617/18971030/036/0012. Print title ''Black and White'', p. 542.</ref> an odd short short story in which the Honourable Algernon Bourke Herriott is "rude to Mrs. Elliott,"<ref name=":13" />{{rp|Col. 2b}} presumably having proposed that they have sexual relations while her husband is out. J.P.B. links to the biographical Algernon Bourke's career in the stock market in the description of Mrs. Christine Elliott not even simulating interest in her husband's bicycling: "a soul is a grievous burthen for a stockbroker's wife,"<ref name=":13" />{{rp|Col. 2a}} suggesting that Mr. Elliott rather than Algernon Bourke Herriott is the stockbroker. The Hon. Algy<blockquote>was a senior member of several junior clubs. A woman had dubbed him once "a rip with a taste for verses." The description was severe, but not unwarranted. His was a pretty pagan sensualism, though, singing from a wine palate to Church music. For the rest, he had just imagination enough to despise mediocrity.<ref name=":13" />{{rp|Col. 2a}}</blockquote>
'''1898 January 5, Wednesday''', the ''Irish Independent'' reported that "Mr Algernon Bourke, the aristocratic stock broker ... was mainly responsible for the living pictures at the Blenheim Palace entertainment.<ref>"Mr Algernon Bourke ...." ''Irish Independent'' 05 January 1898, Wednesday: 6 [of 8], Col. 2c [of 8]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001985/18980105/115/0006.</ref><p>
'''1899 January 10, Tuesday''', the Brighton Championship Dog Show opened:<blockquote>Princess of Wales a Winner at the Ladies’ Kennel Club Show.
[Exclusive to "The Leader.")
The Brighton Championship Dog Show opened in the Dome and Corn Exchange yesterday, and was very well patronised by visitors and exhibitors. Among the latter was H.R.H. the Princess of Wales, who did very well; and others included Princess Sophie Duleep Singh, Countess De Grey, Sir Edgar Boehm, the Hon Mrs. Algernon Bourke, Lady Cathcart, Lady Reid, Mr. Shirley (chairman of the Kennel Club), and the Rev. Hans Hamiiton (president of the Kennel Club).
The entry of bloodhounds is one of the best seen for some time; the Great Danes are another stronyg lot; deerhounds are a fine entry, all good dogs, and most of the best kennels represented; borzois are another very stylish lot. The bigger dogs are, as usual, in the Corn Exchange and the "toy" dogs in the Dome. To everyone's satsfaction the Princess of Wales carried off two first prizes with Alex in the borzois class.<ref>"Dogs at Brighton." ''Morning Leader'' 11 January 1899, Wednesday: 8 [of 12], Col. 3b [of 5]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0004833/18990111/142/0008. Print p. 8.</ref></blockquote>'''1899 January 11, Wednesday''', Guendoline Bourke attended [[Social Victorians/Timeline/1899#11 January 1899, Wednesday|a luncheon at Stanfield-hall, home of Mr. and Mrs. Basil Montogomery, for Princess Henry of Battenberg]], that also included the Countess of Dudley (sister of Mrs. Montgomery), General Oliphant, and the Mayor and Mayoress of Romsey.
'''1899 February 7, Tuesday''', Guendoline Bourke was a member of the very high-ranking committee organizing a [[Social Victorians/Timeline/1899#1899 February 7, Tuesday|ball at the Hotel Cecil on 7 February 1899]].
'''1899 June 1, Thursday''', the Hon. Algernon and Guendoline Bourke attended the wedding of her brother, Sloane Stanley and Countess Cairns at Holy Trinity Church, Brompton.<ref>"Marriage of Mr. Sloane Stanley and Countess Cairns." ''Hampshire Advertiser'' 03 June 1899, Saturday: 6 [of 8], Col. 3b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000495/18990603/049/0006. Print p. 6.</ref>
'''1899 July 1, Saturday''', Algernon Bourke attended a [[Social Victorians/Timeline/1899#1 July 1899, Saturday|meeting in London at the Duke of Westminster's Grosvenor House]] about preserving Killarney as part of the National Trust and seems to have been acting for someone who wanted to purchase the Muckross Estate.<p>
'''1899 October 19, Thursday''', the Hon. Algernon Bourke had a bankruptcy hearing:<blockquote>The public examination of the Hon. Algernon Bourke was held before Mr Registrar Giffard yesterday, at the London Bankruptcy Court. The debtor, described as proprietor of a St. James's-street club, furnished a statement of affairs showing unsecured debts £13,694 and debts fully secured £12,800, with assets which are estimated at £4,489 [?]. He stated, in reply to the Official Receiver, that he was formerly a member of the Stock Exchange, but had nothing to do with the firm of which he was a member during the last ten years. He severed his connection with the firm in May last, and believed he was indebted to them to the extent of £2,000 or £3,000. He repudiated a claim which they now made for £37,300. In 1889 he became proprietor of White's Club, St. James's-street, and carried it on until January 1st last, when he transferred it to a company called Recreations, Limited. One of the objects of the company was to raise money on debentures. The examination was formally adjourned.<ref name=":9" /></blockquote>'''1899 November 8, Wednesday''', the Hon. Algernon Bourke's bankruptcy case came up again:<blockquote>At Bankruptcy Court, yesterday, the case the Hon. Algernon Bourke again came on for hearing before Mr. Registrar Giffard, and the examination was concluded. The debtor has at various times been proprietor of White’s Club, St. James’s-street, and the Orleans’ Club, Brighton, and also of Willis's Restaurant, King-street, St. James's. He attributed his failure to losses sustained by the conversion of White’s Club and the Orleans' Club into limited companies, to the payment of excessive Interest on borrowed money, and other causes. The liabilities amount to £26,590, of which £13,694 are stated to be unsecured, and assets £4,409.<ref>"Affairs of the Hon. A. Bourke." ''Globe'' 09 November 1899, Thursday: 2 [of 8], Col. 1c [of 5]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001652/18991109/020/0002. Print p. 2.</ref></blockquote>
'''1899 December 23, Saturday''', "Mr. Algernon Bourke has departed for a tour in Africa, being at present the guest of his brother in Tunis."<ref>"The Society Pages." ''Walsall Advertiser'' 23 December 1899, Saturday: 7 [of 8], Col. 7b [of 8]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001028/18991223/143/0007. Print p. 7.</ref>
'''1900 February 15, Thursday''', Miss Daphne Bourke, the four-year-old daughter of the Hon. Algernon and Mrs. Bourke was a bridesmaid in the wedding of Enid Wilson and the Earl of Chesterfield, so presumably her parents were present as well.<ref>"London Day by Day." ''Daily Telegraph'' 15 February 1900, Thursday: 8 [of 12], Col. 3b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001112/19000215/175/0008. Name in British Newspaper Archive: ''Daily Telegraph & Courier'' (London). Print p. 8.</ref>
'''1900 September 16''', the Hon. Algernon Bourke became the heir presumptive to the Earldom of Mayo when his older brother Captain Hon. Sir Maurice Archibald Bourke died.
'''1900 October 06, Saturday''', the ''Weekly Irish Times'' says that Mr. Algernon Bourke, now heir presumptive to the earldom of Mayo, "has been for some months lately staying with Mr. Terence Bourke in Morocco."<ref>"Society Gossip." ''Weekly Irish Times'' 06 October 1900, Saturday: 14 [of 20], Col. 3b [of 5]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001684/19001006/121/0014. Print p. 14.</ref><p>
'''1901 May 30, Thursday''', the Hon. Mrs. Algernon Bourke attended the fashionable [[Social Victorians/Timeline/1900s#1901 May 30, Thursday|Ladies' Kennel Association Dog Show at the Botanic Garden]].<p>
'''1901 July 4, Thursday''', Guendoline and Daphne Bourke attended a children's party hosted by the Countess of Yarborough:<blockquote>The Countess of Yarborough gave a charming children's party on Thursday (4th) afternoon at her beautiful house in Arlington Street. The spacious ballroom was quite filled with little guests and their mothers. Each little guest received a lovely present from their kind hostess. The Duchess of Beaufort, in grey, and with a large black picture hat, brought her two lovely baby girls, Lady Blanche and Lady Diana Somerset, both in filmy cream [Col. 2b–3a] lace frocks. Lady Gertrude Corbett came with her children, and Ellen Lady Inchiquin with hers. Lady Southampton, in black, with lovely gold embroideries on her bodice, brought her children, as also did Lady Heneage and Mr. and Lady Beatrice Kaye. Lady Blanche Conyngham, in écru lace, over silk, and small straw hat, was there; also Mrs. Smith Barry, in a lovely gown of black and white lace. The Countess of Kilmorey, in a smart grey and white muslin, brought little Lady Cynthia Needham, in white; Mrs. Arthur James, in black and white muslin; and the Countess of Powys, in mauve silk with much white lace; Lady Sassoon, in black and white foulard; Victoria Countess of Yarborough, came on from hearing Mdme. Réjane at Mrs. Wernher's party at Bath House; and there were also present Lord Henry Vane-Tempest, the Earl of Yarborough, Lady Naylor-Leyland's little boys; the pretty children of Lady Constance Combe, Lady Florence Astley and her children, and Lady Meysey Thompson (very smart in mauve and white muslin) with her children; also Hon. Mrs. Algernon Bourke, in pale grey, with her pretty little girl.<ref>"The Countess of Yarborough ...." ''Gentlewoman'' 13 July 1901, Saturday: 76 [of 84], Col. 2b, 3a [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0003340/19010713/381/0076. Print p. xxxvi.</ref></blockquote>'''1901 July 20, Saturday''', the ''Gentlewoman'' published the Hon. Mrs. Algernon Bourke's portrait (identified with "Perthshire") in its 3rd series of "The Great County Sale at Earl's Court. Portraits of Stallholders."<ref>"The Great County Sale at Earl's Court. Portraits of Stallholders." ''Gentlewoman'' 20 July 1901, Saturday: 31 [of 60], Col. 4b [of 5]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0003340/19010720/141/0031. Print n.p.</ref> Their daughter Daphne appears in the portrait as well.<p>
'''1901 September 12, Thursday''', Mrs. Guendoline Bourke's name is listed as Gwendolen Bourke, but the spelling is not what she objected to:<blockquote>Mr. Underhill, the Conservative agent, mentioned to the Revising Barrister (Mr. William F. Webster) that the name of the Hon. Mrs. Gwendolen Bourke was on the list in respect of the house, 75, Gloucester-place. The lady had written to him to say that she was the Hon. Mrs. Algernon Bourke and that she wished that name to appear on the register. In reply to the Revising Barrister, Mr. Underhill said that “Algernon” was the '''name the lady’s husband'''.
Mr. Cooke, the rate-collector, said that Mrs. Bourke had asked to be addressed Mrs. Algernon Bourke, but that the Town Clerk thought the address was not a correct one. The lady signed her cheques Gwendolen.”
Mr. Underhill said the agents frequently had indignant letters from ladies because they were not addressed by their husband’s Christian name.
The Revising Barrister — lf a lady gave me the name of Mrs. John Smith I should say I had not got the voter’s name. The name Gwendolen must remain.<ref>"Ladies’ Names." ''Morning Post'' 12 September 1901, Thursday: 7 [of 10], Col. 3a [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000174/19010912/130/0007. Print p. 7.</ref></blockquote>
'''1902 September 4, Thursday''', the ''Daily Express'' reported that "Mrs. Algernon Bourke is staying with Lord and Lady Alington at Scarborough."<ref>"Onlooker." "My Social Diary." "Where People Are." ''Daily Express'' 04 September 1902, Thursday: 5 [of 8], Col. 1b? [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0004848/19020904/099/0005. Print p. 4, Col. 7b [of 7].</ref><p>
'''1902 October 24, Friday''', the Hon. Algernon Bourke [[Social Victorians/Timeline/1900s#24 October 1902, Friday|opened the Prince's ice-skating rink for the season]], which he had been doing since 1895.<p>
'''1902 October 31, Friday''', the [[Social Victorians/Timeline/1900s#31 October 1902, Friday|7th opening of the Prince's Skating Club]]. Guendoline Bourke was on the Women's Committee and Algernon Bourke was on the Men's.<p>
'''1902 December 9, Tuesday''', Guendonline Bourke attended [[Social Victorians/Timeline/1900s#9 December 1902, Tuesday|Lady Eva Wyndham-Quin's "at home," held at the Welch Industrial depot]] for the sale Welsh-made Christmas gifts and cards. Bourke wore "a fur coat and a black picture hat."<ref>"A Lady Correspondent." "Society in London." ''South Wales Daily News'' 11 December 1902, Thursday: 4 [of 8], Col. 5a [of 8]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000919/19021211/082/0004. Print p. 4.</ref><p>
'''1903 March 17, Tuesday''', Guendoline Bourke staffed a booth at a [[Social Victorians/Timeline/1900s#1903 March 17, Tuesday|sale of the Irish Industries Association]] on St. Patrick's Day with [[Social Victorians/People/Mayo|Lady Mayo]], [[Social Victorians/People/Dudley|Georgina Lady Dudley]] and [[Social Victorians/People/Beresford|Miss Beresford]]. A number of other aristocratic women were also present at the sale in other booths, including [[Social Victorians/People/Londonderry|Lady Londonderry]] and [[Social Victorians/People/Lucan|Lady Lucan]].<p>
'''1903 June 23, Tuesday''', Guendoline and Daphne Bourke were invited to a [[Social Victorians/Timeline/1900s#1903 June 23, Tuesday|children's party at Buckingham Palace for Prince Eddie's birthday]].<p>
'''1905 February 17, Friday''', the Dundee ''Evening Post'' reported that Algernon Bourke "set up a shop in Venice for the sale of art treasures and old furniture."<ref>"Social News." Dundee ''Evening Post'' 17 February 1905, Friday: 6 [of 6], Col. 7b [of 8]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000582/19050217/105/0006. Print p. 6.</ref><p>
'''1905, last week of July''', Guendoline Bourke and daughter Daphne Bourke — who was 10 years old — attended [[Social Victorians/Timeline/1900s#Last week of July, 1905|Lady Cadogan's children's party at Chelsea House]]. Daphne was "One of loveliest little girls present."<ref>"Court and Social News." ''Belfast News-Letter'' 01 August 1905, Tuesday: 7 [of 10], Col. 6b [of 8]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000038/19050801/157/0007. Print p. 7.</ref><p>
'''1913 May 7, Wednesday''', Guendoline Bourke presented her daughter Daphne Bourke at court:<blockquote>Mrs. Algernon Bourke presented her daughter, and wore blue and gold broché with a gold lace train.<ref>"Social and Personal." London ''Daily Chronicle'' 08 May 1913, Thursday: 6 [of 12], Col. 6b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0005049/19130508/120/0006. Print p. 6.</ref></blockquote>
The ''Pall Mall Gazette'' has a description of Daphne Bourke's dress, but what exactly "chiffon [[Social Victorians/Terminology#Hoops|paniers]]" means in 1913 is not clear:<blockquote>Court dressmakers appear to have surpassed all previous records in their efforts to make the dresses for to-night’s Court as beautiful as possible. Noticeable among these is the dainty presentation gown to be worn by Miss Bourke, who will be presented by her mother, the Hon. Mrs. Algernon Bourke. This has a skirt of soft white satin draped with chiffon [[Social Victorians/Terminology#Hoops|paniers]] and a bodice veiled with chiffon and trimmed with diamanté and crystal embroidery. Miss Bourke’s train, gracefully hung from the shoulders, is of white satin lined with pale rose pink chiffon and embroidered with crystal and diamanté.<ref>"Fashion Day by Day. Lovely Gowns for To-night's Court." ''Pall Mall Gazette'' 07 May 1913, Wednesday: 13 [of 18], Col. 1a [of 5]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000098/19130507/199/0013. Print n.p.</ref></blockquote>
'''1904 September 15, Thursday''', according to what was at the time called the ''Irish Daily Independent and Nation'', Algernon Bourke was living in Venice and not in the UK at this point:<blockquote>Algernon Bourke, who usually lives in Venice, has spent some time in England during the present summer, and has now gone on a fishing expedition to Sweden, accompanied by his brother, Lord Mayo. Lady Mayo has been staying meanwhile in Ireland, and has had a visit from her mother, Lady Maria Ponsonby, who is a sister of Lend Obventry.<ref name=":10">"Society Notes." ''Irish Independent'' 15 September 1904, Thursday: 4 [of 8], Col. 5b [of 9]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001986/19040915/131/0004. Print title: ''Irish Daily Independent and Nation'', p. 4.</ref></blockquote>'''1909 May 22, Saturday''', Algernon Bourke appears to have been living in Pisa. A columnist for the ''Queen'' reported on the Royal School of Art Needlework:<blockquote>Lady Leconfield [?] was there, also her sister-in-law, the Dowager Lady Mayo, only just back from her winter on the Continent, when she spent most of the time at Pisa, where her son Mr Algernon Bourke has also been staying. The latter is a great connoisseur as regards [art?] notably in what is really good in the way of old Italian sculpture and carving. He and his handsome wife have a place near to Putney, and this winter again Mr Bourke, as the result of his Italian travels, has been sending home such relics of the old Italian palace gardens as as stone and marble carved vases, garden seats, and what-not of the kind — not all for himself and his own gardens by any means, I fancy; but his friends, relying on his knowledge in such matters, get him when abroad to choose for [them?] the adornment of their English terraces and gardens.<ref>"My Social Diary." The ''Queen'' 22 May 1909, Saturday: 31 [of 86], Col. 1b [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0002627/19090522/203/0031. Print p. 871.</ref></blockquote>
== Costume at the Duchess of Devonshire's 2 July 1897 Fancy-dress Ball ==
According to both the ''Morning Post'' and the ''Times'', the Hon. Algernon Bourke was among the Suite of Men in the [[Social Victorians/1897 Fancy Dress Ball/Quadrilles Courts#"Oriental" Procession|"Oriental" procession]] at the [[Social Victorians/1897 Fancy Dress Ball | Duchess of Devonshire's fancy-dress ball]].<ref name=":2" /><ref name=":3" /> Based on the people they were dressed as, Guendonine Bourke was probably in this procession but it seems unlikely that Algernone Bourke was.
[[File:Guendoline-Irene-Emily-Bourke-ne-Sloane-Stanley-as-Salammb.jpg|thumb|alt=Black-and-white photograph of a standing woman richly dressed in an historical costume with a headdress and a very large fan|Hon. Guendoline Bourke as Salammbô. ©National Portrait Gallery, London.]]
=== Hon. Guendoline Bourke ===
[[File:Alfons Mucha - 1896 - Salammbô.jpg|thumb|left|alt=Highly stylized orange-and-yellow painting of a bare-chested woman with a man playing a harp at her feet|Alfons Mucha's 1896 ''Salammbô''.]]
Lafayette's portrait (right) of "Guendoline Irene Emily Bourke (née Sloane-Stanley) as Salammbô" in costume is photogravure #128 in the album presented to the Duchess of Devonshire and now in the National Portrait Gallery.<ref name=":4">"Devonshire House Fancy Dress Ball (1897): photogravures by Walker & Boutall after various photographers." 1899. National Portrait Gallery https://www.npg.org.uk/collections/search/portrait-list.php?set=515.</ref> The printing on the portrait says, "The Hon. Mrs. Algernon Bourke as Salammbo."<ref>"Mrs. Algernon Bourke as Salammbo." ''Diamond Jubilee Fancy Dress Ball''. National Portrait Gallery https://www.npg.org.uk/collections/search/portrait/mw158491/Guendoline-Irene-Emily-Bourke-ne-Sloane-Stanley-as-Salammb.</ref>
==== Newspaper Accounts ====
The Hon. Mrs. A. Bourke was dressed as
* Salambo in the Oriental procession.<ref name=":2">"Fancy Dress Ball at Devonshire House." ''Morning Post'' Saturday 3 July 1897: 7 [of 12], Col. 4a–8 Col. 2b. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000174/18970703/054/0007.</ref><ref name=":3">"Ball at Devonshire House." The ''Times'' Saturday 3 July 1897: 12, Cols. 1a–4c ''The Times Digital Archive''. Web. 28 Nov. 2015.</ref>
* "(Egyptian Princess), drapery gown of white and silver gauze, covered with embroidery of lotus flowers; the top of gown appliqué with old green satin embroidered blue turquoise and gold, studded rubies; train of old green broché."<ref>“The Duchess of Devonshire’s Ball.” The ''Gentlewoman'' 10 July 1897 Saturday: 32–42 [of 76], Cols. 1a–3c [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0003340/18970710/155/0032.</ref>{{rp|p. 40, Col. 3a}}
*"Mrs. A. Bourke, as an Egyptian Princess, with the Salambo coiffure, wore a flowing gown of white and silver gauze covered with embroidery of lotus flowers. The top of the gown was ornamented with old green satin embroidered with blue turquoise and gold, and studded with rubies. The train was of old green broché with sides of orange and gold embroidery, and from the ceinture depended long bullion fringe and an embroidered ibis."<ref>“The Ball at Devonshire House. Magnificent Spectacle. Description of the Dresses.” London ''Evening Standard'' 3 July 1897 Saturday: 3 [of 12], Cols. 1a–5b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000183/18970703/015/0004.</ref>{{rp|p. 3, Col. 3b}}
==== Salammbô ====
Salammbô is the eponymous protagonist in Gustave Flaubert's 1862 novel.<ref name=":5">{{Cite journal|date=2024-04-29|title=Salammbô|url=https://en.wikipedia.org/w/index.php?title=Salammb%C3%B4&oldid=1221352216|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Salammb%C3%B4.</ref> Ernest Reyer's opera ''Salammbô'' was based on Flaubert's novel and published in Paris in 1890 and performed in 1892<ref>{{Cite journal|date=2024-04-11|title=Ernest Reyer|url=https://en.wikipedia.org/w/index.php?title=Ernest_Reyer&oldid=1218353215|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Ernest_Reyer.</ref> (both Modest Mussorgsky and Sergei Rachmaninoff had attempted but not completed operas based on the novel as well<ref name=":5" />). Alfons Mucha's 1896 lithograph of Salammbô was published in 1896, the year before the ball (above left).[[File:Algernon Henry Bourke Vanity Fair 20 January 1898.jpg|thumb|alt=Old colored drawing of an elegant elderly man dressed in a 19th-century tuxedo with a cloak, top hat and formal pointed shoes with bows, standing facing 1/4 to his right|''Algy'' — Algernon Henry Bourke — by "Spy," ''Vanity Fair'' 20 January 1898]]
=== Hon. Algernon Bourke ===
[[File:Hon-Algernon-Henry-Bourke-as-Izaak-Walton.jpg|thumb|left|alt=Black-and-white photograph of a man richly dressed in an historical costume sitting in a fireplace that does not have a fire and holding a tankard|Hon. Algernon Henry Bourke as Izaak Walton. ©National Portrait Gallery, London.]]
'''Lafayette's portrait''' (left) of "Hon. Algernon Henry Bourke as Izaak Walton" in costume is photogravure #129 in the album presented to the Duchess of Devonshire and now in the National Portrait Gallery.<ref name=":4" /> The printing on the portrait says, "The Hon. Algernon Bourke as Izaak Walton."<ref>"Hon. Algernon Bourke as Izaak Walton." ''Diamond Jubilee Fancy Dress Ball''. National Portrait Gallery https://www.npg.org.uk/collections/search/portrait/mw158492/Hon-Algernon-Henry-Bourke-as-Izaak-Walton.</ref>
This portrait is amazing and unusual: Algernon Bourke is not using a photographer's set with theatrical flats and props, certainly not one used by anyone else at the ball itself. Isaak Walton (baptised 21 September 1593 – 15 December 1683) wrote ''The Compleat Angler''.<ref>{{Cite journal|date=2021-09-15|title=Izaak Walton|url=https://en.wikipedia.org/w/index.php?title=Izaak_Walton&oldid=1044447858|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Izaak_Walton.</ref> A cottage Walton lived in and willed to the people of Stafford was photographed in 1888, suggesting that its relationship to Walton was known in 1897, raising a question about whether Bourke could have used the fireplace in the cottage for his portrait. (This same cottage still exists, as the [https://www.staffordbc.gov.uk/izaak-waltons-cottage Isaak Walton Cottage] museum.)
A caricature portrait (right) of the Hon. Algernon Bourke, called "Algy," by Leslie Ward ("Spy") was published in the 20 January 1898 issue of ''Vanity Fair'' as Number 702 in its "Men of the Day" series,<ref>{{Cite journal|date=2024-01-14|title=List of Vanity Fair (British magazine) caricatures (1895–1899)|url=https://en.wikipedia.org/w/index.php?title=List_of_Vanity_Fair_(British_magazine)_caricatures_(1895%E2%80%931899)&oldid=1195518024|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/List_of_Vanity_Fair_(British_magazine)_caricatures_(1895%E2%80%931899).</ref> giving an indication of what he looked like out of costume.
=== Mr. and Mrs. Bourke ===
The ''Times'' made a distinction between the Hon. Mr. and Mrs. A. Bourke and Mr. and Mrs. Bourke, including both in the article.<ref name=":3" /> Occasionally this same article mentions the same people more than once in different contexts and parts of the article, so they may be the same couple. (See [[Social Victorians/People/Bourke#Notes and Question|Notes and Question]] #2, below.)
== Demographics ==
*Nationality: Anglo-Irish<ref>{{Cite journal|date=2020-11-14|title=Richard Bourke, 6th Earl of Mayo|url=https://en.wikipedia.org/w/index.php?title=Richard_Bourke,_6th_Earl_of_Mayo&oldid=988654078|journal=Wikipedia|language=en}}</ref>
*Occupation: journalist. 1895: restaurant, hotel and club owner and manager<ref>''Cheltenham Looker-On'', 23 March 1895. Via Ancestry but taken from the BNA.</ref>
=== Residences ===
*Ireland: 1873: Palmerston House, Straffan, Co. Kildare.<ref name=":7" /> Not Co. Mayo?
*1888–1891: 33 Cadogan Terrace, S.W., Kensington and Chelsea, a dwelling house<ref>Kensington and Chelsea, London, England, Electoral Registers, 1889–1970, Register of Voters, 1891.</ref>
*1894: 181 Pavilion Road, Kensington and Chelsea<ref>Kensington and Chelsea, London, England, Electoral Registers, 1889–1970. Register of Voters, 1894. Via Ancestry.</ref>
*1900: 181 Pavilion Road, Kensington and Chelsea<ref>Kensington and Chelsea, London, England, Electoral Registers, 1889–1970. Register of Voters, 1900. Via Ancestry.</ref>
*1904: Algernon Bourke was "usually liv[ing] in Venice"<ref name=":10" />
*1911: 1911 Fulham, London<ref name=":6" />
*20 Eaton Square, S.W. (in 1897)<ref name=":0">{{Cite book|url=https://books.google.com/books?id=Pl0oAAAAYAAJ|title=Who's who|date=1897|publisher=A. & C. Black|language=en}} 712, Col. 1b.</ref> (London home of the [[Social Victorians/People/Mayo|Earl of Mayo]])
== Family ==
*Hon. Algernon Henry Bourke (31 December 1854 – 7 April 1922)<ref>"Hon. Algernon Henry Bourke." {{Cite web|url=https://www.thepeerage.com/p29657.htm#i296561|title=Person Page|website=www.thepeerage.com|access-date=2020-12-10}}</ref>
*Guendoline Irene Emily Sloane-Stanley Bourke (c. 1869 – 30 December 1967)<ref name=":1">"Guendoline Irene Emily Stanley." {{Cite web|url=https://www.thepeerage.com/p51525.htm#i515247|title=Person Page|website=www.thepeerage.com|access-date=2020-12-10}}</ref>
#Daphne Marjory Bourke (5 April 1895 – 22 May 1962)
=== Relations ===
*Hon. Algernon Henry Bourke (the 3rd son of the [[Social Victorians/People/Mayo|6th Earl of Mayo]]) was the older brother of Lady Florence Bourke.<ref name=":0" />
==== Other Bourkes ====
*Hubert Edward Madden Bourke (after 1925, Bourke-Borrowes)<ref>"Hubert Edward Madden Bourke-Borrowes." {{Cite web|url=https://www.thepeerage.com/p52401.htm#i524004|title=Person Page|website=www.thepeerage.com|access-date=2021-08-25}} https://www.thepeerage.com/p52401.htm#i524004.</ref>
*Lady Eva Constance Aline Bourke, who married [[Social Victorians/People/Dunraven|Windham Henry Wyndham-Quin]] on 7 July 1885;<ref>"Lady Eva Constance Aline Bourke." {{Cite web|url=https://www.thepeerage.com/p2575.htm#i25747|title=Person Page|website=www.thepeerage.com|access-date=2020-12-02}} https://www.thepeerage.com/p2575.htm#i25747.</ref> he became 5th Earl of Dunraven and Mount-Earl on 14 June 1926.
== Writings, Memoirs, Biographies, Papers ==
=== Writings ===
* Bourke, the Hon. Algernon. ''The History of White's''. London: Algernon Bourke [privately published], 1892.
* Bourke, the Hon. Algernon, ed., "with a brief Memoir." ''Correspondence of Mr Joseph Jekyll with His Sister-in-Law, Lady Gertrude Sloane Stanley, 1818–1838''. John Murray, 1893.
* Bourke, the Hon. Algernon, ed. ''Correspondence of Mr Joseph Jekyll''. John Murray, 1894.
=== Papers ===
* Where are the papers for the Earl of Mayo family? Are Algernon Bourke's papers with them?
== Notes and Questions ==
#The portrait of Algernon Bourke in costume as Isaac Walton is really an amazing portrait with a very interesting setting, far more specific than any of the other Lafayette portraits of these people in their costumes. Where was it shot? Lafayette is given credit, but it's not one of his usual backdrops. If this portrait was taken the night of the ball, then this fireplace was in Devonshire House; if not, then whose fireplace is it?
#The ''Times'' lists Hon. A. Bourke (at 325) and Hon. Mrs. A. Bourke (at 236) as members of a the "Oriental" procession, Mr. and Mrs. A. Bourke (in the general list of attendees), and then a small distance down Mr. and Mrs. Bourke (now at 511 and 512, respectively). This last couple with no honorifics is also mentioned in the report in the London ''Evening Standard'', which means the Hon. Mrs. A. Bourke, so the ''Times'' may have repeated the Bourkes, who otherwise are not obviously anyone recognizable. If they are not the Hon. Mr. and Mrs. A. Bourke, then they are unidentified. It seems likely that they are the same, however, as the newspapers were not perfectly consistent in naming people with their honorifics, even in a single story, especially a very long and detailed one in which people could be named more than once.
#Three slightly difficult-to-identify men were among the Suite of Men in the [[Social Victorians/1897 Fancy Dress Ball/Quadrilles Courts#"Oriental" Procession|"Oriental" procession]]: [[Social Victorians/People/Halifax|Gordon Wood]], [[Social Victorians/People/Portman|Arthur B. Portman]] and [[Social Victorians/People/Sarah Spencer-Churchill Wilson|Wilfred Wilson]]. The identification of Gordon Wood and Wilfred Wilson is high because of contemporary newspaper accounts. The Hon. Algernon Bourke, who was also in the Suite of Men, is not difficult to identify at all. Arthur Portman appears in a number of similar newspaper accounts, but none of them mentions his family of origin.
#[http://thepeerage.com The Peerage] has no other Algernon Bourkes.
#The Hon Algernon Bourke is #235 on the [[Social Victorians/1897 Fancy Dress Ball#List of People Who Attended|list of people who were present]]; the Hon. Guendoline Bourke is #236; a Mr. Bourke is #703; a Mrs. Bourke is #704.
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[[Social Victorians/Timeline/1850s | 1850s]] [[Social Victorians/Timeline/1860s | 1860s]] [[Social Victorians/Timeline/1870s | 1870s]] [[Social Victorians/Timeline/1880s | 1880s Headlines]] [[Social Victorians/Timeline/1880 | 1880]] 1881 [[Social Victorians/Timeline/1882 | 1882]] [[Social Victorians/Timeline/1883 | 1883]] [[Social Victorians/Timeline/1884 | 1884]] [[Social Victorians/Timeline/1885 | 1885]] [[Social Victorians/Timeline/1886 | 1886]] [[Social Victorians/Timeline/1887 | 1887]] [[Social Victorians/Timeline/1888 | 1888]] [[Social Victorians/Timeline/1889 | 1889]] [[Social Victorians/Timeline/1890s | 1890s Headlines]] [[Social Victorians/Timeline/1910s|1910s]] [[Social Victorians/Timeline/1920s-30s|1920s-30s]]
==January==
"Some time before the last illness he [Disraeli] gathered round him one evening some friends, who may pardon the mention of their names, as a proof that he never permitted politics to interfere with friendships. The Duke and Duchess of Sutherland, Lord and Lady Granville, Lord and Lady Spencer, Lady Chesterfield, Lady Dudley, Lady Lonsdale, Lord and Lady Barrington, Lord and Lady Cadogan, Lord Bradford, Mr. Alfred de Rothschild. Sir Frederick Leighton, and Mr. Henry Manners dined with him." (Ewald, quoting Lady John Manners, p. 581, Col B, fn)
===1 January, Thursday, New Year's Day===
===18 January 1881, Tuesday===
"Tuesday, Jan. 18, will ever be memorable in meteorological annals for the Snow Hurricane with bleached London and England generally" (Penny Illustrated Paper, "Our Illustrations," "The Reign of Jack Frost," p. 4). The Queen was on the Isle of Wight, and "The roads to Ryde and Newport were blocked and impassable. The drifts near Osborne were ten and twelve feet deep. The steamers were unable to cross the Solent; and, as at one period of the day the telegraph-poles were blown down, all communication between the island and the mainland was cut off" (Penny Illustrated Paper, "Our Illustrations," "Royalty in the Snow and on the Ice," p. 4.)
In London itself, "By eight in the morning a full gale was blowing, accompanied by sleet, the wind appearing at different times to blow from different quarters. Snow descended to heavily that by eleven the roads and paths were impassable, and the scene generally was of the wildest possible description. The comparatively few persons in the streets had the greatest difficulty in making progress through the blinding snow, now ankle-deep; horses with heavily-laden vehicles could not move; the tram-cars in all parts of the metropolis had to temporarily cease running, and even on some of the railways the snow-drift occasioned by the gale lay so deep – in parts two feet and even three feet – that trains were brought to a stand-still. Added to all this, the force of the 'circular wind' brought additional dangers in the shape of tiles, chimney-pots, &c., from the housetops, and it is computed that at least sixty persons were admitted to the London hospitals, which, since the present severe weather has set in, have had more cases of broken limbs, dislocations, and wounds of various kinds attended within their walls than have been known for some time.
"In the City especially, now crossed and recrossed over the housetops by telegraphic wires in every direction, which swayed violently with each rapidly succeeding gust of wind, the danger to pedestrians was greatly increased. Our picture of St. Paul's-churchyard, as viewed from the top of Ludgate-hill, gives a fair idea of the fury of the storm. The snow continued during the whole of the day. Attempts were made in many parts of London to remove it, but they were all futile, and night witnessed the remarkable spectacle of a metropolis comparatively deserted, theatres and other places of amusement being half empty. At the doors of the West-End club-houses commissionaires and porters during the night were constantly blowing their whistles for cabs, but without response. Not until the Wednesday morning could any estimate of the damage be arrived at. The first fact which impressed itself upon business men was the absense of letters either from the Continent or the provinces, not a mail of any description getting through to London in time for the early morning deliveries, which a good portion did not arrive before the Thursday. The breakage of the telegraph wires in many places also causes a serious delay to messages." (p. 5)
===19 January 1881, Wednesday===
Constance (de Rothschild) Battersea, in her Memoirs, says of the wedding between her cousin Leopold de Rothschild and Marie Perugia, the sister of Mrs. Arthur Sassoon, <quote>Many were the messages of congratulation that poured in upon the bridegroom, amongst them a letter from Lord Beaconsfield, who wrote: "I have always been of opinion that there cannot be too many Rothschilds." The marriage took place in 1881, and the day proved a memorable one. It was the 19th of January, following upon a terrific snowstorm and blizzard, which prevented some of the guests from attending the wedding. Amongst those, however, who did was the Prince of Wales, who seemed much impressed with the Jewish marriage ceremony, and had his place, I remember, opposite the Ark between two of my cousins. From this happy marriage sprang three sons: Lionel, Evelyn, and Anthony — the second son, most charming and well-beloved, doomed, alas, to the great grief of his family, to fall in Palestine during the Great War, in November 1917!</quote> (http://books.google.com/books?id=Z0gJAAAAIAAJ, p. 48).
===20 January 1881, Thursday===
The Prince of Wales and Lord Beaconsfield ("Dizzy") dine with Lord Ronald Gower at Stafford House (Gower 5).
===27 January 1881, Thursday===
Lord Ronald Gower has luncheon at Kensington Palace, where William Morris, "poet and paper manufacturer," was one of the speakers (Gower 5).
==February==
=== 5 February 1881, Saturday ===
Thomas Carlyle died.
===18 February 1881, Friday===
Bret Harte met the Van de Velde, dining at their house (accompanied by the Trübners?) (Gary Scharnhorst. Bret Harte: Opening the American Literary West.The Oklahoma Western Biographies. Vol. 17. Norman, OK: U of Oklahoma P, 2000. Page 163).
===21 February 1881, Monday===
In her ''Journals'', Lady Knightley says, "I came up [to London] for a lark, combined with G.F.S. Council, spent the whole day on the committee, dined with the old Duchess, and went to a charming party at Nora's" (Journals 346).
===22 February 1881, Tuesday===
In her ''Journals'', Lady Knightley says, "Again spent most of the day at the G.F.S. Council, dined with Lord Leven, and went to parties at Polly Ridley’s and Mrs. Brand’s. I sat between Lord Reay and Sir Stafford Northcote, who was particularly pleasant" (Journals 346). The House of Lords was debating Ireland, and this party and her journal discussed those debates.
==March==
Thomas E. Kebbel "saw him [Disraeli] for the last time at a London party one evening in March, and he then seemed to be quite as strong and well as a man of his age could be expected to be" (154). (Kebbel, T. [Thomas] E. [Edward] Life of Lord Beaconsfield. International Statesmen Series. Ed., Lloyd C. Sanders. Philadelphia: J.B. Lippincott, 1888. Google Books. Retrieved 12 February 2010.)
Disraeli's homeopathic physician, Dr. Joseph Kidd, says, “In the spring of 1881 he felt the cold most keenly, and seldom went out for a walk, his only exercise. Yet he could not deny himself the pleasure of going into society in the evening. He thought that with fur coats and shut carriage he might risk it. But on one of the worst nights in March he went out to dinner, and returning / home was caught for a minute by the deadly blast of the north-east wind laden with sleet. Bronchitis developed the next morning with distressing asthma, loss of appetite, fever, and congestion of the kidneys.” (68-69)
=== 2 March 1881 ===
Russian Emperor Alexander II was assassinated.
===19 March 1881, Saturday===
Disraeli dined with the Prince of Wales at Marlborough House (the last time Disraeli dined "from home"). ("Edward VII." Dictionary of National Biography. Ed., Sir Sidney Lee. Second Supplement, Vol. 1. London: Smith Elder, 1912. Page 533, Col. B.)
===26 March 1881, Saturday===
Disraeli "took a slight part in a political gathering held at his residence a week later" (Irving, Joseph. Annals of Our Time: A Diurnal of Events, Social and Political Home and Foreign: From February 24, 1871, to the Jubilee, June 20, 1887. London: Macmillan, 1889. Google Books. Retrieved 14 February 2010. [Google citation says Volume 2, but I don’t see it on the title page or anywhere around.]).
===30 March 1881, Wednesday===
First notice to the Times that Disraeli was ill, which begins with this: "It has been known throughout the country that Lord Beaconsfield for some days past was suffering from an attack of asthma, caused by exposure to the prevalent east winds. During the time he had been attended by his medical advisor, Dr. Kidd, who, however, entertained no anxiety. The symptoms of Lord Beaconsfield's indisposition fluctuated to some extent, but it was not thought that he was suffering from anything beyond an ordinary cold, which a few days' confinement to the house would eradicate. Within the last few days, however, symptoms of a graver character manifested themselves, and Dr. Kidd became more concerned on his patient's behalf. He attended him more closely, paying him frequent visits in the course of each day. On Sunday night still graver symptoms appeared, causing the doctor to remain all night. It was found that, in addition to a severe asthmatical cough, Lord Beaconsfield was suffering from an attack of suppressed gout, and the object which Dr. Kidd had in view was to develop the gout, and so relieve the asthma. During Monday Lord Beaconsfield was suffering greatly, and towards evening reports reached the House of Commons and the political clubs of his illness." ("The Earl of Beaconsfield," The Times. Wednesday, March 30, 1881, p. 7. Issue 30155, Col. E. Accessed online 7 February 2010. The Times Digital Archive 1785-1985.)
==April 1881==
===1 April 1881, Friday===
According to the ''Pall Mall Gazette'', <blockquote>Lord Beaconsfield passed a somewhat quieter night, and is said to be a little better to-day. Dr. Quain arrived at Curzon-street about half-past nine and after a consultation with Dr. Kidd had taken place, the following bulletin was issued at a quarter past ten o'clock: —
"Lord Beaconsfield has passed the night without any severe attack of spasms. His lordship is weak, but in other respects the symptoms have improved."
Dr. Quain, the Central News states, informed their reporter that his lordship has been able to take good nourishment; but though the case is more hopeful the patient will require continued and careful watching. The next bulletin will be issued at nine o'clock this evening, after the usual evening conference between Dr. Kidd and Dr. Quain.
The Press Associate is able to state upon authority that, although there is much to contend with in the state of Lord Beaconsfield's health, his medical advisers are of opinion that the crisis has been passed, and that there is every hope for the recovery of the noble earl.
The inquiries to-day have already been very numerous. Among the earliest inquirers were the Duke of Cambridge, Lord Lytton, Sir Moses Montefiore, Mr. A. de Rothschild, Mr. Beresford-Hope, Baron Ferdinand de Rothschild, and the Duchess of Sutherland. Mr. Gladstone has also sent to inquire.
It is stated that yesterday Lord Beaconsfield was employed in correcting for Hansard his last speech in the House of Lords. In the course of the afternoon Dr. Kidd superintended the removal of his patient from one room to another, and the change did not seem in any way to cause him distress. Dr. Kidd remained with his lordship again last night, it being deemed necessary that some one should be hourly in attendance.
The Lancet says: — "The noble earl suffers from a lack of nerve power — as distinguished from cerebral energy — which is by no means uncommon in men of Lord Beacsonsfield's type. The intensity of his lordship's vital force has for may years been remarkable, but it has been mainly due to mental energy, called forth in response to mental stimuli. With an organism so energetic and thus vitalized, there must needs be a perpetual liability to the suppression or metastasis of diseases which require a somewhat high grade of local disturbance to reach their normal type; meanwhile there are necessarily great irritability and weakness. The difficulties attending the management of such a case are obviously great, and its vicissitudes many and various." (The Pall Mall Gazette 1 April 1881 Issue 5025; page 8)</blockquote>
===2 April 1881, Saturday===
According to the ''Pall Mall Gazette'', <blockquote>The improvement in Lord Beaconsfield's condition yesterday morning was not maintained all day. In the evening he was more feverish and restless, and effect possibly due to the occurrene of gout in the foot not previously affected. Dr. Kidd again remained with his patient during the night, and his report htis morning was more favourable. His lordship passed a quiet night and took more nourishment. Dr. Quain arrived about half-past nine o'clock this morning, and after a consultation had taken palce between Dr. Quain and Dr. Kidd, which lasted an hour and ten minutes, the following bulleting was issued:—
"10.40 A.M. — Lord Beaconsfield has had some quiet sleep during the night. The gout in the right foot is rather more developed. The spasms have been relieved, but otherwise the chest symptoms continue the same."
The anxiety felt as to Lord Beaconsifield's conditions, owing to the somewhat unfavourabloe bulletin of last night, was shown by the large number of inquiries this morning. The central News says that the medical gentlemen in attendance consider his lordship to be better than he wa slast night, but not so well as he was yesterday morning. Rest and quietness are essential to recovery, and strict injunctions have been given that no visitors are to be admitted. Dr.s Quain and Kidd again both saw his lordship after the issue of this morning's bulletin, and they then left Curzon-street.
The callers yesterday were as numerous as on Wednesday and Thursday. Prince Leopold and the Duke of Cambridge paid personal visits to the house, and special telgrams were, by request, sent to the Prince of Wales and the Duke of Edinburgh.
A telegram was received by Lord Barrington last evening from Lord Rowton to the effect that he was absolutely detained at Marseilles on account of his sister's illness, and that the doctors forbade him to leave her. </blockquote>
===13 April 1881, Wednesday===
In its daily news item on the crisis in the health of Disraeli, Earl of Beaconsfield, the ''Times'' reported that, after it inquired "shortly before midnight," and was told the following:
<quote>Lord Beaconsfield has passed a day of restlessness, followed at 7 o'clock this evening by an attack of difficult respiration. The attack, which created considerable anxiety for a short time, passed over and later on his lordship was able to take some food and rest.
It is difficult to suppress the anxiety which these repeated attacks give rise to — for, though the patient rallies from day to day, each successive attack leaves him less equal to resist their recurrence.</quote> ("The Earl of Beaconsfield," The Times. Wednesday, April 13, 1881, p. 7. Issue 30167, Col. F. Accessed online 7 February 2010. The Times Digital Archive 1785-1985.)
===14 April 1881, Thursday===
People continued to gather in Curzon Street, where Disraeli lived. As always, the ''Times'' also lists who notable inquired or visited. ''The Lancet'' and the ''British Medical Journal'', which had been reporting on Disraeli's illness as well, are quoted. According to the ''Times'' report, the ''British Medical Journal'' says the following: "It is unnecessary to say that every variety of strong and digestible nourishment that can be suggested is at the command of his medical attendants; and it is a fact of the greatest interest and curiosity that from all parts of Europe, as well as of this country, different forms of food, wine, and other stimulants continue to be sent." ("The Earl of Beaconsfield," The Times. Thursday, April 14, 1881, p. 6. Issue 30168, Col. B. Accessed online 7 February 2010. The Times Digital Archive 1785-1985.)
===15 April 1881, Friday===
Good Friday.
According to the daily story on Disraeli's illness in the ''Times'', the usual physicians' report was issued at 10:15 a.m., and "Large numbers of persons, including members of the working classes, were assembled when this statement was published. A copy of it was forwarded to Her Majesty and members of the Royal Family." "The profound interest manifested by Her Majesty in the well-being of Lord Beaconsfield has been shared by Her Majesty's subjects of all degrees. The number of letters received offering suggestions or making inquiries of one kind or another would form a most curious volume, whilst the collection of articles of food or diet sent to Curzon-street would constitute a perfect cabinet of hygiene or physic." ("The Earl of Beaconsfield," The Times. Friday, April 15, 1881, p. 7. Issue 30169, Col. F. Accessed online 7 February 2010. The Times Digital Archive 1785-1985.)
===16 April 1881, Saturday===
According to the daily story on Disraeli's illness in the ''Times'', "favourable" reports on his condition were issued at 1:00 a.m., 5:00 a.m., and then it quotes from the 10:15 a.m. report. The story ends with a report from the 11:30 p.m. reponse to inquiries: "Lord Beaconsfield has passed the day in a condition satisfactory to his physicians. He has taken nourishment at longer intervals and readily, on one occasion even expressing a desire to have it. His Lordship's progress must be slow under the most favourable circumstances; but there is nothing in the present aspect of the case to show that it will not be sure." ("The Earl of Beaconsfield," The Times. Saturday, April 16, 1881, p. 5. Issue 30170, Col. F. Accessed online 7 February 2010. The Times Digital Archive 1785-1985.)
===17 April 1881, Sunday===
Easter Sunday.
===18 April 1881, Monday===
The ''Times'' reported "two bulletins" from Saturday that were "favourable" and that Dr. Kidd did not spend Saturday night at Disraeli's house in Curzon Street, for the first time in three weeks, but the story ends with this:
<quote>The following detailed account of Lord Beaconsfield's condition during the day was furnished to us in answer to our inquiries at 11 30 last night: —
Lord Beaconsfield has been more restless at intervals during the last 24 hours, and he has taken less nourishment. Rest and food being essential elements in the recovery of strength, deficiency in these effects must have an unfavourable effect. His Lordship consequently is found somewhat weaker, though in other respect unchanged to night. A failure of this kind, however slight, which would be immaterial in a younger person, naturally creates anxiety concerning the result in the case of one of advanced years who, like Lord Beaconsfield, as gone through so grave an illness. There is, however, happily, no material retrogression to-night.</quote> ("The Earl of Beaconsfield," The Times. Monday, April 18, 1881, p. 9. Issue 30171, Col. F. Accessed online 7 February 2010. The Times Digital Archive 1785-1985.)
===19 April 1881, Tuesday===
Benjamin Disraeli, Lord Beaconsfield, died. The ''Times'' report, moved up to page 3 from its usual page 6 or so, reports the previous day's events like this: "Our readers will bear with deep concern that, with the return of the east wind, Lord Beaconsfield's condition has materially changed for the worse." "During the day a very large number of persons visited Curzon-street to read the bulletin posted in front of Lord Beaconsfield's house." There was a report at 9:00 p.m.: "Lord Beaconsfield's condition has not been satisfactory during the day. He has been free from urgent symptoms, and taken more nourishment, yet he rather loses strength." Here is the 11:30 report: "The improvement which occurred in Lord Beaconsfield's health during last week is not maintained to-day. He is free from urgent or distressing symptoms, and during the afternoon he has been able to take nourishment sufficiently well. Still, he rather loses than gains strength, and he sleeps heavily at intervals. The physicians see in those symptoms grounds for more grave anxiety as to the result than at any period during his lordship's illness." Then, the next paragraph, which concludes the story, says, "At 1 o'clock this morning Lord Beaconsfield's condition had not changed since the issue of the last bulletin. No improvement was perceptible. Lord Rowton remained in the house." ("The Earl of Beaconsfield," The Times. Tuesday, April 19, 1881, p. 3. Issue 30172, Col. E. Accessed online 7 February 2010. The Times Digital Archive 1785-1985.)
“William Gladstone was at home in Flintshire, North Wales, when the news came early that morning [at 8:00 a.m.]. Benjamin Disraeli was dead. It was hardly unexpected, but Gladstone immediately recognized the implications for himself and the country. ‘It is a telling, touching event,’ he confided to his diary. ‘There is no more extraordinary man surviving in Englad, perhaps none in Europe. I must not say much, in the presence as it were of his Urn.’” (Aldous, Richard. The Lion and the Unicorn: Gladstone vs Disraeli. New York: W.W. Norton, 2006. Google Books, retrieved 13 February 2010. Page 1.)
"The sad tidings of the great statesman's death did not become generally known in the metropolis until the forenoon of Easter Tuesday, and as the railway trains kept bringing in their carriages full of passengers, returning from their Easter holiday-making in all parts of the country, intense became the excitement at the various metropolitan stations, on their seeing posted up in large letters at the newspaper-stands, 'Death Of Lord Beaconsfield.' Crowds collected, and groups of people of all classes, in grave and often tearful converse, gathered there all throughout the day, evidently absorbed in the one great and mournful subject of so melancholy yet memorable an occasion. And it was much the same in every other place of public resort. The clubs of every character and complexion, political and social, were agitated all day long by the exciting topic. It was one that, in all directions, threw every other into the shade. And it may be said with considerable truth that, with very partial, and by no means very respectable exceptions, the public mind of London — as, indeed, it soon appeared, of every other part of the kingdom — was sincerely and deeply affected by the mournful event" (Kebbel 300). Baron Redesdale says, "day after day the blasts, charged with all the filth of the great city, blew fiercely and yet more fiercely, bringing poison to those parched lungs. On the 19th of April he died, choked by London" (Redesdale 676).
===20 April 1881, Wednesday===
The article announcing Disraeli's death and summarizing his career appears to take up more than one entire page: 6 columns on one page and perhaps 1.25 on the next. Here is the main description of his Jewish heritage, and it begins the second paragraph of the article: "Lord Beaconsfield was of alien, although not obscure, extraction; he came of the separate people which, since it has been scattered from a land of its own, has been persecuted or ostracized by Christian intolerance. His family was ancient; allied, it is said, with that high Hebrew aristocracy of Spain that embraced individuals of the stamp of his own Sidonias, it traced its descent through merchant princes of Venice to a stem that had been transplanted from the East in very early days. But, like other privileges, such claims of blood came under the head of Jewish disabilities, and did less than nothing to help him in the struggle towards a position that seemed practically beyond his dreams. Now that he has pioneered the way for his people, blunting in 50 years the hard fighting the prejudices that every step imposed themselves to his own advance towards power; now that Jews sit as matter of right among the representatives of the country, legislating for interests in which they have a common concern with their fellows — it is difficult to measure the distance that then divided the young aspirant from the Premiership of England."
There is another discussion of his "Jewish descent" beginning at the bottom of Column C and continuing at the top of Column D: "We have remarked that, like a man of spirit and shrewdness, in his writings as in his speeches, [Col break] Disraeli boldly prided himself on his Jewish descent and the glories of his race. Jews rich in gifts as in gold are the mythical heroes of the Utopias in his fictions. But the most eloquent defence of his people against the prejudices of Christendom is to be found in that chapter of the 'Political Biography' which precedes the explanation of Lord George Bentinck's conduct with respect to the Jewish disabilities. In ingenious arguments, more sophistical than satisfactory, he seeks to demonstrate that these prejudices are neither historically true nor dogmatically sound, and urges characteristically that we owe the Jews a large debt of gratitude for becoming the instruments to carry out the great doctrine of the Atonement. That he felt more than natural sympathy, that he took a genuine pride in his people, there can be no doubt whatever, and as little that he had no bigoted prejudice against religious emancipation in the abstract. Yet, when Lord George Bentinck resigned the leadership of his party rather than countenance its intolerance on the question of Jewish disabilities; when he not only voted, but exerted himself, under great physical suffering, to address the House on behalf of the Jews, Mr. Disraeli took a different view of his duty. It is impossible not to suspect that here, as elsewhere, he sacrificed conscience and inclination alike to what he considered as the paramount claims of party." ("The Earl of Beaconsfield," The Times. Wednesday, April 20, 1881, p. 7. Issue 30173, Col. A. Accessed online 7 February 2010. The Times Digital Archive 1785-1985.)
There is another article addressing his death directly. ("The Death of the Earl of Beaconsfield," The Times. Wednesday, April 20, 1881, p. 9. Issue 30173, Col. E. Accessed online 7 February 2010. The Times Digital Archive 1785-1985.)
===21 April 1881, Thursday===
An article in the ''Times'' directly addresses the questions of the two funerals: "It is as yet undecided whether Lord Beaconsfield is to be buried in Westminster Abbey with a public funeral, or privately by the side of this wife in Hughenden churchyard. We stated yesterday that Mr. Gladstone, on receiving the sad news of Lord Beaconsfield's death, telegraphed at once to Lord Rowton and to the executors, Sir N. M. de Rothschild, M.P., and Sir Philip Rose, offering, on the part of the Government, a public funeral, and we are enabled this morning to give the continuation of the correspondence ...." The article quotes the will, providing for burial in Hughenden and a simple funeral, and says the Queen prefers he be buried in Westminster. It describes his lying in state in the room in which he died, though not on the couch on which he died, which has been removed from the room. There is a description of the moment of his death, when he seemed to the two men in the room to be about to rise to speak in Parliament. More stories from the foreign press are reported, as are descriptions of meetings in which people speak of Disraeli or of meetings that are cancelled. ("The Late Lord Beaconsfield," The Times. Thursday, April 21, 1881, p. 5. Issue 30174, Col. F [and continuing on to p. 6, Cols A-C]. Accessed online 7 February 2010. The Times Digital Archive 1785-1985.)
===22 April 1881, Friday===
The ''Times'' reports that the Queen has decided because of Disraeli's will that the funeral will be in Hughenden and Disraeli interred beside his late wife. The ''Times'' also reports stories from Vienna and ''The Lancet''. ("The Late Lord Beaconsfield," The Times. Friday, April 22, 1881, p. 5. Issue 30175, Col. F. Accessed online 7 February 2010. The Times Digital Archive 1785-1985.)
===23 April 1881, Saturday===
Leonora Braham opens in Gilbert and Sullivan's ''Patience, or Bunthorne's Bride'' (at the Opera Comique) to positive reviews.
===26 April 1881, Tuesday===
Benjamin Disraeli, Lord Beaconsfield's funeral.
===30 April 1881, Saturday===
Annual banquet for the Royal Academy. Gladstone attended: "Staring down at him was a magnificent four-foot oil portrait of Disraeli by the brilliant society artist Sir John Millais. The picture was a work in progress, which gave it an added poignancy. More to the point, it was an obvious partner for a portrait of Gladstone completed two years earlier. Gladstone had been painted in right profile. Now Disraeli, in left profile, would catch his eye in artistic perpetuity. / Speaker after speaker at the dinner referred to Millais’ painting, and to its pair. Gladstone, who often struggled with social off-the-cuff remarks, was not prepared to speak on Disraeli. When he rose, he astonished the audience by remarking coldly of the portrait that ‘it is, indeed an unfinished work. In this sense it was a premature death.’ / ‘Made my speech,’ Gladstone wrote in his diary when he returned home: ‘this year especially difficult.’” (Aldous 5) Also, Victoria went to Hughenden, to the vault where Disraeli was buried, to pay her respects.
Here is the ILN's version of it: "Last Saturday her Majesty and Princess Beatrice drove through Rayner's Park, the residence of Sir Philip Rose, to Hughenden church, where they were recieved by Lord Rowton and the Rev. Henry and Mrs. Blagden, who conducted them to the tomb of the late Earl of Beaconsfield, where they placed a wreath and cross of flowers. The Queen afterwards proceeded to Hughenden Manor, and drove back to Windsor through High Wycombe. Princess Louise of Lorne and the Duchess of Connaught arrived at the castle early in the evening; and the Duke of Connaught and Prince Leopold arrived after attending the royal Academy dinner" ("The Court." Illustrated London News, Saturday, May 07, 1881; pg. 450; Issue 2190).
==May 1881==
===1 May 1881, Sunday===
In her ''Journals'', Lady Knightley says, "I went to Westminster Abbey, which was crowded with a dense congregation, listening with rapt interest to an intellectual treat — a dissertation by Dean Stanley on Lord Beaconsfield. He selected a curious text: Judg. Xvi. 30: 'So the dead which he slew at his death were more than they which he slew in his life.' All the way through he coupled him with Gladstone, calling them the Great Twin Brethren. And I sat on the altar steps, at the foot of one of the columns, with the statues of the mightier rivals, Pitt and Fox, facing me, and listening to a magnificent anthem on King David, composed by Handel and finished by Goss, which had never been performed since the death of Wellington. …" (Journals 347).
“At a packed Westminster Abbey, Gladstone attended Disraeli’s [5] /[6] memorial service. The dean, Arthur Stanley, who had attended the same prep school as Gladstone, gave the address. His text was Judges 16:30: ‘So the dead which he slew at his death were more than they which he slew in his life.’ In the middle of the peroration, the dean unexpectedly combined Disraeli with (the still very much alive) Gladstone. They were, observed Stanley while the prime minister flushed and squirmed, the ‘Great Twin Brothers’ of British political life.” (Aldous 5-6) "So vivid was the impression made on that young mind [of Frederick Bridges, the Permanent Deputy-Organist of Westminster Abbey 1875-1882] by the dirge 'Know ye not that a Prince and a great man is fallen this day in Israel?' that when, twenty-nine years later, he had to prepare the music for the memorial service of Lord Beaconsfield, he suggested this anthem to Dean Stanley. Upon being told for what occasion it had been composed, the Dean observed that the death of Lord Beaconsfield had made a greater impression on the public mind than the death of any great man since Wellington. It is also interesting to record that this most appropriate dirge was sung at the Dean's own funeral in the Abbey, July 25, 1881." ("Sir Frederick Bridge," 514 Col A)
===4 May 1881, Wednesday===
In the 30 April 1881 issue of the ''Illustrated London News'', the "Home News" column says that "Mr. Russell Lowell, the United States Minister, will take the chair at the next anniversary dinner of the Literary Fund, which is to held [sic] at Willis's Rooms next Wednesday," which is probably Wednesday, 4 May 1881 ("Home News." Illustrated London News. Saturday, April 30, 1881; pg. 427; Issue 2189). George Augustus Sala, in his 14 May 1881 column "Echoes of the Week," says this: "I went, on Wednesday, the Fourth of May, to the annual festival, at Willis's Rooms, of the Royal Literary Fund. The American Minister, the Hon. James Russell Lowell, was in the chair. There was a large gathering; and I was glad to notice, among the usual assemblage of Peers, members of Parliament, dignified clerics, medical men, and publishers, a fair springling of working men of letters. Professional literature was represented by Mr. Leslie Stephen, Mr. Justin M'Carthy, Mr. Edund Yates, Mr. Blanchard Jerrold, Mr. Fraser Rae, and a few others; but I should have liked to see a great man more 'live authors' present. The Royal Literary Fund is a most admirable charity, generously, sagaciously, and delicately administered; and it is entitled to the support of every literary man. If he be a prosperous one, to help his less for-fortunate [sic; pb at hyphen] brethren, through the medium of this quietly beneficent institution, becomes a bounden duty. [new paragraph] Mr. James Russell Lowell made several speeches, full of tranquil humour adn refined scholarship. This Excellency, it is true, fathered on Swift a droll anecdote about a charity sermon, which anecdone, I believe, was first narrated in connection with the Rev. Rowland Hill; and again, from his interesting enumeration of American writers Mr. Lowell, oddly enough, omitted the names of Ralph Waldo Emerson and Edgar Allan Poe. Whether he mentioned Bret Harte, Mark Twain, adn George W. Curtis, I am not quite certain. If am rather deaf. Sir Garnet Wolseley made a capital speech, delievered with ringing emphasis; Mr. Justin M'Carthy returned thanks with equal elegance and eloquence for English Literature; but the finest oratorical display of the evening was unquestionably that made by Lord Coleridge. It was splendidly polished and sonorous, in matter and in manner alike unimpeachable; and to listen to it was a rich literary treat. [new paragraph] One of the noble speakers at the top table, in responding to the toast of the House of Peers, quoted the names of Lords Bacon, Bolingbroke, Derby, Macaulay, Lytton, adn Beaconsfield as exemplifying the close connection between literature and politics. The noble speaker might have added to his list Lord Shaftesbury of the 'Characteristics,' two Lord Strangfords — the translator of the Lusiad and the accomplished peer but lately among us: Lord Lyttelton, the historian; Lords Brougham, Campbell, Dorset, Roscommon, Stanhope, and Orford. For did not Horace Walpole die an Earl? Finally, the noble speaker might have known that there was never a 'Lord' Bacon. There was a wonderful genius by the name of Sir Francis Bacon, Baron Veralum and Viscount St. Albans. Those who study with love and reverence his immortal works would blush to call him 'Lord' Bacon. They speak of him as 'Bacon,' or as he, with simple dignity, was wont to speak of himsef, 'Francis of Veralum.'" (Sala, George Augustus. "Echoes of the Week." Illustrated London News, Saturday, May 14, 1881; pg. 467; Issue 2191)
=== 1881 May 10, Tuesday ===
This is the worst-written description of this kind of event I have read in a 19th-century British newspaper — wordy and bombastic even by Victorian standards. Perhaps a new reporter for this small regional paper? The wedding of Miss Marion Lascelles and Lieutenant Henry Dent Brocklehurst:<blockquote>NUPTIAL REJOICINGS AT MIDDLETHORPE MANOR.
MARRIAGE OF MISS LASCELLES AND LIEUT. BROCKLEHURST.
Tuesday last was without doubt a red letter day in the calendar associated with Middlethorpe Manor, the residence of the Hon. Egremont W. Lascelles, brother of the [[Social Victorians/People/Harewood|Earl of Harewood]], as on that day Miss Marion Lascelles, his eldest daughter, was united in the holy bands of matrimony to Lieutenant Henry Dent Brocklehurst, of the Second Life Guards, nephew of Mr. Philip Brocklehurst, of Swithamley Park, Macclesfield.
Before speaking of the marriage ceremonial, which took place at Bishopthorpe Church, we may state that in the immediate vicinity of Middlethorpe Manor bunting was displayed in token of rejoicing at the auspicious event about to take place, and along the approach to the church was a profuse display of national and other flags extending in rows across the road. At the entrance to the church yard was a triumphal arch composed of flowers and evergreens, and on one side was the motto "God bless them," whilst on the other was the inscription "Health and happiness." These mottoes were flanked with the monograms of the bride and bridegroom. The long and straight footpath leading from the archway we have named to the church porch was covered with scarlet cloth, and it was fringed on each side with a bordering of primroses and other flowers now in season. The aisle of the church from the porch to the altar rail was adorned precisely in a similar style, but with a little more elaboration in the details. A collection of beautiful flowering plants was displayed in the chancel, and it may be stated that the immense number of primroses and other flowers were the gift of Miss Steward, Bishopthorpe, and the Hon. and Rev. J. W. Lascelles, Goldsbro'. So much for the preliminaries, and before proceeding with the facts in connection with the marital ceremony to be celebrated in the church we will give a list of the numerous and valuable presents given to Miss Lascelles, as a token of esteem entertained for the young lady, and respect for her parents, the occasion of the marriage being the fitting opportunity for giving a tangible proof of the high regard in which the Lascelles family are held by their numerous friends. The following is a list of the presents, namely:—
The Hon. Alfred and Mrs. A. Gathorne Hardy, enamelled dish; Miss Violet Beckett Denison, pearl horseshoe brooch; Sir Frederick Milner, silver necklace; Mr. Walter Forbes, antique silver box; the Hon. and Rev. James Lascelles, set of pearl pins; Earl and Countess of Wharncliffe, diamond and pearl pig brooch; Mr. Dudley Milner, gold horse shoe bracelet; Mr. and Mrs. M'Intosh, pair of China plates; Miss Ogilvy, China plaque; Hon. Mrs. Arthur Lascelles, antique silver sugar bowl and sifter; Hon. Robert Lawley, embroidered pin-cushion cover; Servants of the house, stables, garden, and farm, inlaid black marble clock; Lord and Lady George Hamilton, pair of large silver-mounted scent bottles; Lord and Lady Wenlock, pearl and diamond spray brooch; Misses Edith, Rachel, Catharine, Emma, and Mabel Lascelles, pearl and diamond brocch; Hon. Mrs. James Lascelles, gold enamelled thimble; Major and Mrs. Prendergast, looking-glass in antique tortoiseshell frame; Mr. and Mrs. Jervoise Smith, turquoise and diamond ring; Sir Henry and Lady White, diamond daisy; Colonel Fairfax, travelling clock; Mrs. Fairfax, antique silver box; Colonel and Mrs. Malcolm, inlaid mother of pearl fan; Mr. H. O. Brocklehurst, five diamond stars; Miss Mills and Miss Mabel Mills, gold fan brooch; Hon. Hamilton and Lady Margaret Cuffe, Dresden china lamp; Sir Charles and Lady Louisa Mills, diamond bracelet; Miss F. and Miss H. Cole, antique silver pencil-case; the [[Social Victorians/People/Harewood|Countess of Harewood]], gold mounted travelling bag; Miss Milner, table cover; Miss Fairfax and Mr. Bryan Fairfax, pair of glass flower vases; Dowager Lady Middleton, pair of gold mounted scent bottles; Mr. and Mrs. Smallwood, inkstand; Miss Dent and Miss M. Dent, Dresden cup and saucer, mounted on bracket; Misses Winn (2), painted fan, mounted in mother of pearl; the Hon. Egremont and Mrs. Lascelles, dressing-case with gold mounts; Mrs. Oliver, antique silver pincushion; Captain and Miss Robertson, gilt tray; Mrs. W. Wickham, driving whip; Mr. John Malcolm, gold and intaglio bracelet; Mrs. Winn, pearl and diamond pin; Mr. Dudley Smith, scent bottle and vinaigrette; Colonel and Lady Florence Cust, old paste shoe buckles; Miss F. and Miss E. Duncombe, photograph screen; Mrs. Coke, hand-painted photograph frame; Capt. and Mrs. Slingsby, diamond frog brooch; Captain and Mrs. Starkey, plated spoon, fork, and salad bowl; Captain and Mrs. Palairet, antique silver salver; Mrs. Vyner, set of pearl pins; Miss Wickham, looking-glass in antique brass frame; Hon. Frederick and Mrs. Lascelles, umbrella with agate handle; Miss Brockleharst and Mr. E. Brocklehnrst, pair of diamond earrings; Mr. R. C. Vyner, 12 embroidered handkerchiefs; [[Social Victorians/People/Bourke|Hon. Algernon Bourke]], old enamelled watch set in pearls; Mrs. Scott, silver sugar basin: Miss M. Brocklehurst, old Flemish carved oak cupboard; Mr. P. Crutchley, six gold and silver bangles; Earl of Feversham, small travelling clock; Mrs. Dent, opal and diamond cross; the Hon. Mrs. Beckett Denison, travelling clock; Mr. Gordon Cunard, old Louis XVI. fan, painted by Boucher; Mr. T., Mr. N., Mr. H., and Miss Malcolm, six cut glass flower vases; Mr. and Mrs. George Thompson, pair of brass candlesticks; Mrs. Fairfax, a newspaper holder; Lady Sheffield, a pearl, diamond and ruby bee; Mr. A. Meysey-Thompson, a silver rose muffonier [sic s/b ''muffinier'']; Mrs. Malcolm, a revolving tea-table; Miss Shiffner, silver-mounted Russian leather card case; Mr. Robett Swann, 8 Worcester cups and saucers; Mr. and Mrs. A. Scott, tortoiseshell paper knife; Hon. Alethea Lawley, picture, copy of Carlo Dolci; Miss Maud White, antique silver buckle; Mr. A. Brocklehurst, pearl Marguerite necklace and earrings; Mr. A. Brocklcharst, pearl and diamond bracelet; Miss Liddell and Miss M. Liddell, brass flower stand; Mrs. Clayton, diamond pendant; the Archbishop of York and Mrs. Thomson, silver inkstand; Mr. Cecil, Walter, and Reginald Lascelles, pair of painted terra-cotta vases; Hon. Algernon Lawley, Indian shawl; Hon. Arthur Lawley, antique Florentine cross; Hon. George and Lady Louisa Lascelles, China tea service and tray; Mr. and Mrs. Hartley, pair of silver pig muffoniers [sic s/b muffiniers?]; Mr. and Mrs. Francis Johnstone, umbrella with silver handle; Mr. and Mrs. Johnson, two velvet tables; Miss Evelyn Lascelles, work basket; Mrs. Hewetson, Dresden cup and saucer; Mr. H. D. Brocklehurst, pearl and diamond ring; Mr. W. B. Denison, leather luncheon bag; Miss Maud Denison, pair brass candlesticks aud inkstand; Lord Burghersh, pair of Carltons; Dowager Lady Wenlock, florentine turquoise necklace; the Hon. Katherine Lawley, three turquoise arrows; the Hon. Caryl and Mrs. Molyneux, old silver etui [case]; the Hon. Gerald and Mrs. Lascelles, two Wedgwood bowls; Mrs. Whiteley and children, two Dresden cups and saucers; Mr. Geor[g]e Lane Fox, silver etui [case]; the Dean of York and Lady Emma Cust, pair of China plaques; Mr. Gerald Leatham, card case; Miss Alexander, letter weighing machine; Miss Clare Lascelles, two silver muffiniers; Miss Clare Lascelles, silver penholder; Mr. John Shafto, silver clothes brush; Lady Caroline Lascelles, black and white pearl brooch; Lady Susan Lascelles, silver clasp; the Earl of Harewood, pearl and diamond locket and earrings; Mr. and Mrs. Algernon Mills, silver cream jug; the Rev. Walter and Mrs. Hudson, Dresden china looking-glass; Lady Louisa Mills, pearl and enamel locket; the [[Social Victorians/People/Mayo|Countess of Mayo]], silver scent bottle; Mr. and Mrs. Worthington, silver tea and coffee set; Mr. Dent, old silver salt cellars, mustard pot, bowl, tankard, tray, and urn; Mr. Cosmo Little, silver claret jug and two silver cups; Mr G. Farquhar, pearl and diamond pin; Mr. and Mrs. Talbot, set of silver dessert spoons; the Countess of Shrewsbury, silver pincushion; Mr. Arthur Brocklehurst, silver tray; Mr. and Mrs. C. Brand, silver claret jug; Lord Burghersh, pearl diamond pin; Mr. Harter, spirit case; Hon. W. S. Hanbury, silver mustard pot; Mr. French, silver cream jug; Mr. Millward, silver salad bowl; Mr. Anstruther Thomson, silver bowl; Mrs. Harford, pair of silver gilt candlesticks; Mr. and Mrs. W. C. Brocklehurst, two florentine chairs; Mr. Spicer, spirit stand; Mr. H. Barclay, set of carvers; Mrs. Rodcliffe, silver teapot; Miss Violet Denison, ivory paper knife; Lady Milner, inkstand; Lord and Lady Downe, pair of silver candlesticks; Mr. Atkinson, two silver muffiniers; Captain Tennant, two silver dishes; Captain and Hon. Mrs. Neild, Dresden inkstand; Mr L. H. Jones, set of gold studs and links; Captain Fife, spirit case; Captain and Mrs. Benyon, silver teapot, cream jug, and sugar basin; Mr T. S. Cunningham, set of silver spoons; Lord G. Cains, silver cup; Mr. Fitz Brocklehurst, silver tankard.
Shortly before half-past eleven o'clock the bride and bridegroom, accompanied by the bridesmaids, relatives, and friends, proceeded in carriages from Middlethorpe Manor to Bishopthorpe Church. The day was beautifully fine, and there was a large concourse of spectators on the route, and especially near to the church. The bridal party walked along the footway we have named, first passing under the floral archway, and then entered the church, the bride being conducted by her father. The sacred edifice was crowded. The festal march, by Scotson Clark, was played by Miss Crosby, who presided at the organ, and immediately after the wedding party entered the church the hymn, "Thine for ever, God of love," was sung. The bride and bridegroom went to the front of the alter rail, and the marriage ceremony was commenced. The officiating clergyman was the Hon. and Rev. J. W. Lascelles, Goldsborough, uncle of the bride, and he was assisted by the Rev. Walter Hudson, vicar of Bishopthorpe. The bride was "given away" by her father, the Hon. E. W. Lascelles, and the bridegroom's "best man" was Lord Burghersh, eldest son of the [[Social Victorians/People/Westmorland|Earl of Westmoreland]] [sic]. The young ladies who officiated as bridesmaids were Miss Clara Lascelles, Lady Susan Lascelles, Miss A. Brocklehurst, Miss Edith Lascelles, Miss Rachel Lascelles, and Miss Violet Denison. They were all similarly attired, being habited in cream coloured dresses of gauze de Venise and satin merveilleux trimmed with lace. They had lace bonnets, diamond and pearl brooches, and elegant bouquets, the gift of the bridegroom. The bride was attired in an ivory satin and brocade dress, trimmed with Valenciennes lace. She also wore a lace tulle veil, and had diamond and pearl ornaments, and a large and magnificent wedding bouquet. Amongst those present on the interesting occasion we noticed the Earl Harewood, uncle of the bride; Lord Lascelles, Sir Frederick and Lady Milner, Lord and Lady Wenlock, the [[Social Victorians/People/Bourke|Hon. Algernon Bourke]], Capt., Mrs., and Miss Starkie, Mr. and Mrs. Worthington, Mr. A. Brocklehurst, Mr. E. Brocklehurst, Mr. F. Brocklehurst, Lord Burghersh, Mr. J. D., Mrs., and the Misses (2), Dent, Ribston Park; Mr. Forbes, Mr. and Mrs. Hewetson, Mrs. Oliver, Mrs. W. Wickham, Colonel and Mrs. Fairfax, Hon. and Rev. James and Mrs. Lascelles, the Misses Lascelles (5), Miss V. and Miss M. Nun Apploton; the Hon. Henry Boyle, Lady Susan Lascelles, Mr. and Mrs. G. S. Thompson, Moorlands; Lady Louisa and Miss Lascelles, Sion Hill, Thirsk; Colonel Cust, Miss Shiffner, Mr. J. Brocklehurst, Rev. Walter and Mrs. Hudson, and Miss Alexander, Miss A. Brocklehurst, Mr. S. Bateman, and Major Malcolm, and many other ladies and gentlemen who are named in the list of presents.
At the conclusion of the ceremony the newly married couple walked arm and arm up the church and left the sacred edifice. They were accompanied by the bridesmaids, two and two, and the relatives and friends, Miss Crosby playing Mendellsohn's [sic] Wedding March on the organ. On appearing outside the church the happy pair were most cordially greeted, as the crowd "rent the skies with loud [Col. 3c–4a] applause," whilst the school children strewed before them flowers as they passed along. Having re-entered the carriages, the bridal party returned to Middlethorpe Manor, where an elegant ''dejeuner'' [sic no accent] was served. In the afternoon the bride and bridegroom left Middlethorpe Manor for the railway station ''en route'' for the seat of Mr. Philip Brocklehurst, uncle of the bridegroom, Swithamley Park, Macclesfield, where they will pass the honeymoon.
We may state that the Archbishop of York and the Very Rev. the Dean were unavoidably prevented from being present the marriage. The employes and others at Middlethorpe Manor were hospitably entertained at Bishopthorpe by the Hon. E. W. Lascelles.<ref>"Nuptial Rejoicings at Middlethorpe Manor. Marriage of Miss Lascelles and Lieut. Brocklehurst." ''Yorkshire Gazette'' 14 May 1881, Saturday: 9 [of 12], Cols. 3a–4a [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000266/18810514/057/0009. Print same title and p.</ref> </blockquote>
==June 1881==
===1 June 1881, Wednesday===
Derby Day.
According to the ''Morning Post'', <quote>The Marchioness of Salisbury's assembly. / Lady Mary Windsor Clive's dance. / The Hon. Mrs. Magniac's dance, instead of the 31st inst. / Mrs. Millais's dance. / Art exhibition at River House, Chelsea Embankment (by kind permission of the Hon. Mrs. John Dundas), in behalf of recreation rooms for working girls in the East of London, and the following day. / Derby Day.</quote> ("Arrangements for This Day." The Morning Post Wednesday, 1 June 1881: p. 5 [of 8], Col. 6B).
In her ''Journals'', Lady Knightley says, "Went to Buckingham Palace to see Princess Christian, and with her and Lady Marian Alford to a Domestic Economy Congress meeting; brought her back to luncheon with Emmy Hamilton, Louisa Gordon, and Mr. Leveson, after which I drove with Nora and took her to a Derby tea at Mrs. Lloyd's. Iroquois, an American horse, won. We dined with Lady Lyveden, and went on to a pleasant party at Lady Salisbury's, where I was introduced to Sir Richard Temple, the author of India in 1880, and about the ugliest man whom I ever saw. But he is clever and agreeable, and I am pleased at the testimony he bears to the success of missions in India, which people are so ready to decry" (Journals 148).
===15 June 1881, Wednesday===
In her Journals, Lady Knightley says she and Sir Knightley "went to a party given by the Spencers at South Kensington Museum, where were all the world and his wife, including the King of the Sandwich Islands, who walked about arm in arm with the Princess of Wales. The courts were lighted with electric light, which has a peculiar and not very becoming effect" (Journals 348-49).
===16 June 1881, Thursday===
In her Journals, Lady Knightley says, "We had a most delightful drive to Wimbledon on Lord Tollemache's coach, taking with us Princess Mary's two nice boys. We came in for the Lords and Commons match, and had tea with 'my beautiful lady,' Lady Brownlow .... We met the Crown Prince of Germany to-night at Ishbel Aberdeen's, and Princess Frederica at Londonderry House. I was presented to her" (Journals 348-49).
==July==
===14 July 1881, Thursday===
Thursday afternoon, beginning about 2 p.m., Garden Party at Marlborough House for the Queen; [[Social Victorians/People/Arthur Collins|Arthur Collins]] is listed as having been invited, as are [[Social Victorians/People/Fanny Ronalds|Mrs. Ronalds]], [[Social Victorians/People/Arthur Sullivan|Arthur Sullivan]], and a number other familiar names (http://www.britishnewspaperarchive.co.uk/viewer/BL/0000174/18810716/051/0005). Ronald, Lord Gower says, "A garden party at Marlborough House. The Queen present, evidently suffering from the intense heat. I noticed Her Majesty talking much to John Bright" (8).
===18 July 1881, Monday===
Ronald, Lord Gower says, "I made the acquaintance, at a dinner at the Cardross's, of Howard Vincent, who, although but a little over thirty is already at the head of the detective force, and has published half a dozen books. He works all day, and dances half the night" (8).
===19 July 1881, Tuesday===
Ronald, Lord Gower says, "on waking, one heard the bells of the Abbey tolling, Dean Stanley having died during the night. He is a greater loss to the Queen than to the Church" (8).
Ronald, Lord Gower says, "To a fancy dress dance at Lowther Lodge that night. The dance, and Lord Houghton in a skull cap, were both successful and picturesque" (8).
==August 1881==
===29 August 1881, Monday===
Summer Bank Holiday.
==September==
==October 1881==
===10 October 1881, Monday===
"Savoy Theatre, erected for Mr. D'Oyly Carte by Mr. C. J. Phipps, opened 10 Oct. 1881" (Hayden's Dictionary 1104).
==November==
==December 1881==
===15 December 1881, Thursday===
The first performance was in the afternoon. The advertisement announcing the performance says this: <quote>THEATRE ROYAL, HAYMARKET.— A MORNING PERFORMANCE (for which Mr. and Mrs. Bancroft have given the use of their theatre) will take place on Thursday next, December 15, commencing at 3 o'clock, in AID of the ROYAL GENERAL THEATRICAL FUND, under the patronage of their Royal Highnesses the Prince and Princess of Wales. Goldsmith's comedy of SHE STOOPS TO CONQUER will be played. Mrs. Langtry has kindly consented to take the part of Miss Hardcastle. Messrs. Lionel Brough, Kyrle Bellow, George Barrett, J. Maclean, M. R. Crauford, F. Barsby, A. Bishop, Lestocq, Raiemond, Haines, &c; Mesdames Sophie Larkin, Helen Cresswell, and Mary Brough have generously proffered their valuable services (by permission of their respective managers).</quote> <cite>"The Times may be Purchased in Paris,." Times [London, England] 9 Dec. 1881: 6. The Times Digital Archive. Web. 3 May 2013.</cite>
The review, published on the 16th, says this: <quote>When we say that yesterday's representation was eminently successful, we are paying the highest compliment to the performers who principally contributed to this result. Foremost among these was Mrs. Langtry, who, it would be affectation to conceal, was the grand attraction of the piece — the attraction which brought together one of the most distinguished audiences that have recently assembled in a theatre. The house overflowed with rank, fashion, and celebrity, including the Prince and Princess of Wales, who are rarely absent when a praiseworthy purpose is to be forwarded or a kind action to be done. The proceeds of the representation, it will be remembered, were to go in aid of the funds of an excellent institution. High-raised as the general expectations might have been, they were not disappointed.</quote> <cite>"The Haymarket Theatre." Times [London, England] 16 Dec. 1881: 6. The Times Digital Archive. Web. 3 May 2013.</cite>.
==Works Cited==
*Ewald, Alexander Charles. The Right Hon. Benjamin Disraeli, Earl of Beaconsfield, K.G., and His Times. Vol. 2. London: William Mackenzie, 1884. Google Books. Retrieved 13 February 2010.
*Hayden's Dictionary of Dates and Universal Information Relating to All Ages. Ed., Benjamin Vincent. 23rd Edition, Containing the History of the World to the End of 1903. New York: G. P. Putnam's Sons, 1904. Page 1104. Google Books, retrieved 23 February 2010.
*Irving, Joseph. Annals of Our Time: A Diurnal of Events, Social and Political Home and Foreign: From February 24, 1871, to the Jubilee, June 20, 1887. London: Macmillan, 1889. Google Books. Retrieved 14 February 2010. [Google citation says Volume 2, but I don’t see it on the title page or anywhere around.]
*Kebbel, T. [Thomas] E. [Edward] Life of Lord Beaconsfield. International Statesmen Series. Ed., Lloyd C. Sanders. Philadelphia: J.B. Lippincott, 1888. Google Books. Retrieved 12 February 2010.
*Knightly.
*“Sir Frederick Bridge.” The Musical Times and Singing-Class Circular. Vol. XXXVIII. London: Novello, Ewer and Co., 1897. JSTOR. Google Books, retrieved 14 February 2010. [Get page numbers: 513 or so -]
*Redesdale, Algernon Bertram Freeman-Mitford, Baron. Memories. Vol. 2. 9th edition. London: Hutchinson, 1916. Google Books, retrieved 15 February 2010.
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[[Social Victorians/Timeline/1850s | 1850s]] [[Social Victorians/Timeline/1860s | 1860s]] [[Social Victorians/Timeline/1870s | 1870s]] [[Social Victorians/Timeline/1880s | 1880s Headlines]] [[Social Victorians/Timeline/1880 | 1880]] [[Social Victorians/Timeline/1881 | 1881]] [[Social Victorians/Timeline/1882 | 1882]] [[Social Victorians/Timeline/1883 | 1883]] [[Social Victorians/Timeline/1884 | 1884]] [[Social Victorians/Timeline/1885 | 1885]] 1886 [[Social Victorians/Timeline/1887 | 1887]] [[Social Victorians/Timeline/1888 | 1888]] [[Social Victorians/Timeline/1889 | 1889]] [[Social Victorians/Timeline/1890s | 1890s Headlines]] [[Social Victorians/Timeline/1910s|1910s]] [[Social Victorians/Timeline/1920s-30s|1920s-30s]]
Mary Cora (Urquhart) Brown-Potter and her husband (and daughter?) visited England in 1886 and met the Prince of Wales, who invited them to spend a weekend. (Wikipedia: Brown-Potter).
The Shelley Society mounted a production of ''The Cenci'', which lasted four hours. According to Neil Fraistat, "Wilde, Shaw, and Browning were all in the audience. It was a hard ticket to get. The audience gave it a rapturous reception. The newspaper critics, not so much. Wilde was wild about it. Shaw had reservations."<ref>Freistat, Neil. ''Twitter''. 14 October 2022 https://twitter.com/fraistat/status/1578404994021310465 (Retrieved 2022-10-14). In a comment that follows this tweet, Freistat says, "The Shelley Society Notebooks have great descriptions, but for an excellent overview, see Curran, Shelley’s Cenci, 183–92."</ref>
==January 1886==
Annie Eastty and Isabella Clemes gave their papers at the Men and Women's Club regular January meeting, talking about morality; Henrietta Muller and Mrs. Walters were not present at this meeting (Bland 22).
===1 January 1886, Friday, New Year's Day===
===15 January 1886, Friday===
Reading of ''A Doll's House'' "in a Bloomsbury drawing room in which all the participant were not only associated with the feminist cause but had achieved or would achieve prominence in the British socialist movment: Eleanor Marx, the daughter of Karl, in the role of Nora; her common-law husband Edward Aveling, who played Helmer; William Morris's daughter May, portraying Mrs Linde; and, as Krogstad, none other than [[Social Victorians/People/George Bernard Shaw|Bernard Shaw]]." (McFarlane 89) "'I feel I must do something to make people understand our Ibsen a little more than they do,' wrote Eleanor Marx to Havelock Ellis in Late December 1885.1 [fn 1] So invitations went out to a 'few people worth reading Nora to'; and on 15 January 1886, in their flat in Great Russell Street, Karl Marx's youngest daughter and her common-law husband, Edward Aveling, played host to one of the first readings in England of an Ibsen play — A Doll's House in the Henrietta Frances Lord translation. Bernard Shaw was a favoured invitee, playing the part of Krogstad to the Mrs Linde of William Morris's daughter, May." (McFarlane 233) [McFarlane, James Walter. The Cambridge companion to Ibsen. Cambridge: Cambridge UP, 1996.]
==February 1886==
Robert Parker gave a paper on "Sexual Relations among the Greeks of the Periclean Era" at the regular February 1886 meeting of the Men and Women's Club (Bland 31).
==March 1886==
==April 1886==
=== 15 April 1886, Thursday ===
Hon. [[Social Victorians/People/Mills|Charles W. Mills]] and Hon. Alice Marion Harbord married:<blockquote>The marriage of the Hon. Charles W. Mills, M.P., eldest son of Lord Hillingdon, with the Hon. Alice Marion Harbord, second daughter of Lord Suffield, took place yesterday afternoon at St Peter's Chapel, Vere-street. Among the relatives and friends who assembled were the Duchess of Leeds and Lady Harriet Godolphin Osborne, the Dowager Marchioness of Lansdowne and Lady Emily Fitzmaurice, the Earl and Countess of Wharnciiife, the Earl of Arran and Lady Alice Gore, Lord and Lady Suffield, [[Social Victorians/People/Mills|Lord and Lady Hillingdon]], Viscountess Bury and Hon. Misses Keppel, Lord and Lady George Hamilton, Lord and Lady Hastings and Hon. Miss Astley, Lord and Lady Claud Hamilton, Lord and Lady Revelstoke, Viscountess Downe, Sir E. and Hon. Lady Birkbeck, Captain and Lady Agneta Moutagu, Colonel Hon. C. Edgoumbe [sic], Hon. Assheton E. Harbord, Colonel and Hon. Mrs. Ellis, and Mrs. Windham Baring. Mr. Adolphus Liddell was the bridegroom's best man, and the eight bridesmaids were the Hon. Winifred, Hon. Eleanor, and Hon. Bridget Harbord, sisters of the bride; the Hon. Isabel, Hon. Mabel, and Hon. Violet Mills, sisters of the bridegroom; the Hon. Elizabeth Baring, and Miss Alexandra Ellis. The Hon. and Rev. James W. Lascelles, rector of Goldsborough, Yorkshire, uncle of the bridegroom, officiated, assisted by the Rev. P. Roberts, M.A., Lord Suffield giving his daughter away. After the ceremony Lord and Lady Suffield received the friends present at the wedding at their residence in Grosvenor-street. Early in the evening the bride and bridegroom left for Queen's Mede, near Windsor, lent to them by Colonel the Hon. Reginald and Mrs. Talbot for the honeymoon.<ref>"Arrangements for This Day." ''Morning Post'' 16 April 1886 Friday: 5 [of 8], Col. 6b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000174/18860416/041/0005.</ref></blockquote>
===23 April 1886, Friday===
Good Friday
===25 April 1886, Sunday===
Easter Sunday
==May 1886==
===7 May 1886, Friday===
[[Social Victorians/People/Muriel Wilson|Muriel Wilson]] taking part in the program for a benefit. Mr. Clive Wilson was a "sable attendant," probably in racist blackface. The copy of the newspaper the British Library digitized is sometimes quite difficult to read.
<blockquote>At the Anlaby Room on Friday evening, an entertainment, provided by Mrs Arthur Wilson of Tranby Croft, and house [?] party, was successfully presented. The performance was given in aid of the School Building Fund, which, as the room was crowded to excess, will materially benefited. The first part of the programme consisted of a concert, in which every item was encored, the quaint trio, "Three little maids from school," sang [sic?] by Miss Wilson, Mrs Travers, Messrs Whiting and Mahoney, and the recital of "The Stowaway" by Mr F. W. Wood, deservedly drawing forth hearty approval, the recitation being given with good dramatic feeling. A right merry finale was furnished in the exhibition of Mr Wood's famous “Waxworks," the figures in which included the "Grand Old Man” (Mr Reginald Voase), "Laughing Girl" (Miss Wilson), "The Walking Doll" (Miss Muriel Wilson), "The Duchess of Devonshire" (Mrs Reynard), "The Babes [?] in the Wood" (Miss Mabel Wilkinson and Mr Harry Sykes), and "Alonzo the Brave and the fair Imogene” (Lady Boynton and Mr Reynard). The showman found his humour heightened the able assistance rendered by Mr R. Hxxyard [?] as clown, and Master Clive Wilson as a picturesquely-attired sable attendant with a passion for the drum. The whole affair was in every way a complete success. (“Entertainment at Anlaby.” Hull Daily Mail 10 May 1886, Monday: 2 [of 4], Col. [of ]. British Newspaper Archive (accessed July 2019).</blockquote>
===16 May 1886, Sunday===
Albert Edward, Prince of Wales, recovering from his conflict with Randolph Churchill in 1876, agreed to attend a dinner party hosted by Jennie Churchill. Bertie had attended a dinner in March 1884 that included the Gladstones and Lord and Lady Randolph Churchill, his first meeting with them since 1876. <quote>...another two years elapsed and Lord Randolph had become Secretary for India before the Prince could bring himself to enter the Churchill home. On this occasion Blandford also attended, and Jennie, the bewitching Jennie, was thirty-two — approaching the age at which the Prince really appreciated his lady friends. The date was May 16, 1886, and it proved her most successful dinner party because so much was at stake</quote><cite>(Leslie 66)</cite>.
===26 May 1886, Wednesday===
Derby Day.
According to the ''Morning Post'', <quote>The Derby Day. / The Countess of Dalhousie's reception. / Mrs. Smith's second evening party. / Mrs. Charles Mills's dance, instead of Friday, ths 28th inst. / Mrs. Burton's first dance, at 6, Chesterfield-gardens. / Chevalier and Mrs. Desanges' at home, at 16, Stratford-place, from four to seven. No cards. / New Club Dance.</quote> ("Arrangements for This Day." The Morning Post Wednesday, 26 May 1886: p. 7 [of 12], Col. 7B).
==June 1886==
===13 June 1886, Sunday===
Whit Sunday
===26 June 1886, Saturday===
There was apparently a regular celebration of Arthur Collins' birthday, 26 June, by Bret Harte, George Du Maurier, Arthur Sullivan, Alfred Cellier, Arthur Blunt, and John Hare (Nissen, Axel. Brent Harte: Prince and Pauper: 239. [http://books.google.com/books?id=WEDewmUnapcC]). Choosing 1885–1902 as the dates because those apparently are the dates of the close relationship between Harte and Collins, ending in Harte's death in 1902.
===28 June 1886, Monday===
George, Duke of Cambridge: "Went to Kneller Hall. There met Sir Arthur Sullivan who had come down from London in my waggonette. Heard the band in the Chapel first and then a fine and powerful band, and was very much satisfied with the whole condition of things, as was Sullivan, who said he had no sort of suggestions to offer for improvements." (Sheppard, Edgar. George, Duke of Cambridge: a memoir of his private life based on the journals and correspondence of His Royal Highness. Volume 2. London: Longmans, Green, 1906: 153. Google Books, retrieved 23 February 2010.)
==July 1886==
===12 July 1886, Monday===
Bret "Harte became an official member" of the Beefsteak Club "on July 12, 1886" (Axel Nissen, Bret Harte: Prince & Pauper. Jackson, MS: U P of Mississippi: 2000: 300; from personal correspondence to Nissen by the Beefsteak Club).
==August 1886==
=== 23 August 1886 ===
From the 28 August 1886 ''Vanity Fair'':<blockquote>THINGS IN HOMBURG
H<small>OMBURG</small>, 23rd August.
MY DEAR VANITY,— Here I am once more in Homburg, and it seems hard to realise that it is a full year ago that I wrote to you from this place. Everything goes on just as when I left in August, 1885. The band plays the same music, nearly all the same people walk up and down the avenue of the springs, the water is served out by the same young women, the same "watery" shop is talked all over the place, and the restaurants have the same bills of fare. In fact, had I, like Rip Van Winkle, fallen asleep in August, 1885, and been suddenly awakened in August, 1886, I should have noticed no change to tell me that twelve months had elapsed.
His Royal Highness the Prince of Wales arrived about ten days ago, and whilst going steadily through the cure, seems to be enjoying himself with occasional visits to the Frankfort Opera and little picnics in the woods. The Prince's kind and genial manner makes him popular wherever he goes, and one hears nothing but words of praise about him from both English and Germans here. Princess Christian and the blind Grand Duke of Mecklenburg are also here, so that we have a fair share of Royalty in the town.
Frankfort Races, which came off on Sunday and Monday, the 15th and 16th, were very amusing, and drew many people away from Homburg. On the second day, Monday, the Prince of Wales went over with a small party of friends, and was met by his brother-in-law, the Grand Duke of Hesse. The Stand and Paddock are very prettily situated in the middle of a pine wood; and as the day was hot, the shade was delicious.
Betting is strictly forbidden now in Germany, but in spite of this, a little wagering was done when an English bookmaker put in an appearance; for, although he could not shout out the odds, / yet he was quite willing to walk to the wood with any would be backer for a few minutes' conversation, and once safely there, would explain what he could do on the next race.
The last event on the card, a steeplechase, was great fun. Two German officers rode in uniform, and went very well indeed, although it must have been a severe trial for both horses and riders in this almost tropical heat.
Lord Rendlesham gave a charming little dance on Wednesday evening to about forty friends. The Prince of Wales was present, and everyone enjoyed it. Dancing, which began at half-past nine, was stopped at half-past eleven. Then came a small supper, and to bed soon afterwards, for in Homburg no one dreams of sitting up late.
On Monday afternoon all the ''beau monde'' here were at a concert, at which Herr Hollman, violoncellist to the King of Holland, played most beautifully. Mrs. Burrowes sang divinely, and Mr. Alexander Yorke gave his impersonation of actors and actresses. The whole performance was so good that, in spite of the intense heat, everybody stayed till the end.
A good many people have left during the last two or three days, but fresh arrivals still pour in to take their places, so that it is difficult to find rooms, unless they are engaged a few days beforehand.
Homburg is quite celebrated for its lovely roses, and this year they seem to be finer than ever. One of the great occupations at the Elizabeth spring in the morning is to "bunch” the fair sex with bouquets and sprays. When leaving Homburg, the popularity of a fair lady may always be guessed by the amount of flowers which are sent by friends to the station. A certain beautiful and popular Countess left a morning or two ago, and her carriage looked like a florist's shop, in such numbers did her many friends come to wish her God speed.
Never have I seen a finer August in Homburg. The result is that picnics are quite the order of the day. Last week the Prince of Wales gave a charming one in the Taunus Woods, and ordered a photographer there, who did some groups of the party. Mr. Hargreaves also got up a most successful picnic, at which every pretty face in Homburg was to be seen, and the lunch being quite excellent was most thoroughly enjoyed.
Amongst the many well-known English who are here just now are the Duke of Manchester, Lord and Lady Erne, Lord Rendlesham, Lady Shrewsbury, Lady Conyers and Miss Lane-Fox, Mrs. and the Miss Verschoyles, Lord Dartmouth, Mr. [[Social Victorians/People/Webb|Godfrey Webb]], Colonel Larking, Mrs. and Miss Legh of Lyme, Lady Headfort, Mrs. Dixon, Mr. and Mrs. Cunard, Mr. and Miss Clarke Thornhill, Mrs. Francis Slone Stanley, Mr. and Mrs. Walter Campbell, the Hon. Mrs. Candy and Major Westenra, Lady and Miss De Bathe, Mrs. Rochford and Miss Crabbe, Colonel Colville, Colonel and Mrs. Chaine, Lord and Lady Cork, Mrs. Keith Fraser, Miss Ethel Cadogan, Mr. Alfred Montgomery, Mr. and Mrs. Ker Seymer, Mr. and Mrs. Hare, Mr. and Mrs. Grossmith, Lord Wolverton, Mrs. Maxey, Countess Tolstoi and Miss Helen Henniker, Mr. Alexander Yorke, Lady Sophia Macnamara, Mr. and Mrs. Powell, Mr. Hargreaves, Mr. Percy Barker, the Hon. Mrs. Roche and Miss Werke, Lady Macpherson Grant, Mrs. and the Miss Shaws, Admiral, Mrs., and the Miss Cochranes, and a heap more whose names do not come to me just now.<ref>"Things in Homburg." ''Vanity Fair'' 28 August 1886 (Vol. 36): 123–124. ''Google Books'' https://books.google.com/books?id=MGtHAQAAMAAJ.</ref></blockquote>
===30 August 1886, Monday===
Summer Bank Holiday
==September 1886==
"... in September 1886, the decision was taken to move [the head-centre organization of the Liberal party] to London, with the proviso that its annual meetings were always to be held in some provincial centre" (Spender 12). Robert Hudson, who was Assistant Secretary to Schnadhorst, moved to Palace Chambers, where he lived until May 1888 (Spender 15).
'''8 September 1886, Wednesday'''
The ''Taunton Courier'' reprinted an article from the ''World'' with some theatre and sports gossip. The Thursday of the trial rehearsal was probably 2 September 1886, and the Saturday of the production that Rosebery, Chetwynd and the Comptroller of the Household attended was probably 4 September 1886.<blockquote>DRURY LANE THEATRE.
Mr Augustus Harris has abundant reason to be satisfied with the advice and assistance he has received in the production of "A Run of Luck." Mr [[Social Victorians/People/Rothschild Family|Leopold de Rothschild]], Prince Soltykoff, "[[Social Victorians/People/Montrose|Mr Manton]]," the [[Social Victorians/People/Portland|Duke of Portland]], Mr Chaplin, and the [[Social Victorians/People/Beaufort|Duke of Beaufort]] all lent him their colours to be used in his race, and on Thursday his Grace of Beaufort came up expressly from Badmintont officiate as the foreman of a critical jury of sportsmen. Sir John Willoughby, Lord Baring (who missed a division in consequence), Colonel Vivian, Mr [[Social Victorians/People/Bourke|Algernon Bourke]], Mr Hume Webster, Captain Thurton, Mr James Selby (no relation to the squire of that ilk), and the proprietor of Kempton were all amongst the empanelled; the trial-rehearsal lasted for more than five hours, and the final verdict was pronounced in the small hours of Friday morning. At the production on Saturday [[Social Victorians/People/Rosebery|Lord Rosebery]] and Sir George Chetwynd were both present, the Comptroller of the Household dropped in on his way back from Covent Garden, and Sir John Gorst mused on the mysteries of the marriage-laws in the stalls. — ''The World''.<ref>"Drury Lane Theatre." ''Taunton Courier and Western Advertiser'' 08 September 1886, Wednesday: 7 [of 8], Col. 5b [of 5]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000526/18860908/026/0007. Print title ''Taunton Courier''; print p. 7.</ref></blockquote>
==October 1886==
===28 October 1886, Thursday===
In New York City, the Statue of Liberty was dedicated.
==November 1886==
===5 November 1886, Friday===
Guy Fawkes Day
===16 November 1886, Tuesday===
"On the 16th inst., at St. George's, Hanover-square, by the Rev. E. W. Pownall, M.A., Emslie John, only son of F. J. Horniman, Esq., of Surrey Mount, Forest Hill, to Laura Isabel, only daughter of Colonel A. G. Plomer, of 7, Chesterfield-street, Mayfair. (No cards.)" ("Marriages." Illustrated London News (London, England), Saturday, November 20, 1886; pg. 544; Issue 2483, Col. B)
==December 1886==
===25 December 1886, Saturday===
Christmas Day
===26 December 1886, Sunday===
Boxing Day
=== 1886 December 30, Thursday ===
On 30 December 1886, the Newcastle ''Evening Chronicle'' reprinted a story from the ''World'' about "The Forty Thieves" pantomime at the Drury Lane Theatre. This account contains essentializing generalizations about Asian people and cultures.<blockquote>THE DRURY LANE PANTOMIME.
(From ''The World''.)
Mr. August Harris has actually accomplished the apparently impossible task of eclipsing the pantomimes which year after year have been in turn pronounced unsurpassable. It seems almost as if Aladdin had left his lamp behind him at Drury Lane, to call forth, as if by magic, the marvellous magnificence which forms so sumptuous a setting for the frolic, fun, and drollery of this latest and most diverting edition of "The Forty Thieves." The time-honoured story is clearly and tersely told, and certainly loses none of its original interest by its skilful adaptation to the most absorbing topics of the day. Mr. Harris may well be somewhat embarrassed by the strength of the company he has collected. Miss Constance Gilchrist makes a charming and most ideal Morgiana; the unctuous humour of Mr. Harry Nicholls as a sort of Persian "Eccles," and of Mr. Herbert Campbell as a tradesman's wife, savouring equally of Ispahan and our own East-end; the mirth-provoking antics of Mr. Ali Sloper-Stevens, and the aggressive juvenility of Miss M. A. Victor as the elderly Mrs. Cassim (an Orientalised Miss Minnie Palmer), are simply irresistible. The go and vivacity of Miss Dot Marie and her sister; the dash of Miss Brereton; the graceful dancing of the D'Aubans; the delectable cheekiness of Miss Edith Bruce as Ganem; the brightness of Miss Marie Williams, and the amusing earnestness of Mr. Pateman as Cassim and Cassim's ghost, are one and all deftly used in giving new life to the old legend, in the picturesque treatment of which Paul Martinetti as the most agile of monkeys, Charles Laurie as the sagest of donkeys, and Madame Ænea with her daring flights (all of them great Continental stars), each play an important part. The costumes, the scenery, and the processions of "The Forty Thieves" mark an epoch in the history of pantomimic production. Mr. Harris has ransacked all London, all Paris, and all Bohemia for his matchless old brocades, his armour, his banners, and his gems. India and South Kensington have alike been laid under contribution, and in the great spectacle of the robbers' cave the poetry of Mr. Beverly's brush has, perhaps, at last found its limits. The realism of Mr. Emden in dealing with Eastern subjects, the good taste of Mr. Ryan, and the power of Mr. Telbin all find adequate expression in Ali Baba's courtyard, the New Club (which everybody would like to belong to), and the Jubilee Temple of Fame, in which patriotic playgoers will be perplexed and dazzled by the fanciful conception of the inventor and the artist, the glittering splendour of the details, and the rare beauty of the fair representatives of the various jewels of the British Crown. Madame Katti Lanner brings her usual artistic contribution to the delights of Boxing Night. The opening ballet of houris puts everybody at once in the best of humour; the dance of monkeys is very well done indeed, and so are the capers of the nimble little sailors, who, of course, appear in the final apotheosis of British loyalty. Signora Zanfretta and Signora Bettina de Sortis both distinguished themselves as ''premieres danseuses''; Harry Payne put his boys and policemen in quaint and characteristic masks; and amongst the comic "properties" the famous boots and shoes which drove Paris wild over ''Le Petit Poucci'' are introduced effectively as a subordinate incident. The singing of Mr. Scott's Jubilee Ode and the National Anthem, as beautiful Britannia and her comely companions defile majestically before Miss Telbin's statue of the Queen, is well calculated to provoke a display of enthusiasm which will not easily be forgotten by those who witnessed it. In the glories of "The Forty Thieves" the ex-Khedive Ismail Pasha and his sons might almost forget the grandeur of Aida, and amongst the audience on this memorable night were [[Social Victorians/People/Montrose|Caroline Duchess of Montrose]] and Sir George Arthur, [[Social Victorians/People/Londesborough|Lord and Lady Londesborough]], Lord and Lady De La Warr, Lord Cantelupe and the Ladies Sackville, Lord Alfred Paget, Sir John Gorst and Mr. T. P. O'Connor (if I am not mistaken, in one box), Colonel Herbert Eaton (who finds London much pleasanter than Richmond Barracks), Lord and Lady de Clifford, Lords Truro, Cairns, and Lurgan, Mr. [[Social Victorians/People/Bourke|Algernon Bourke]], Colonel Ralph Vivian, Sir George Chetwynd, Mr. Carl Rosa, Mr. Alfred Cooper and Lady Agnes Cooper, and Mr. Michael Sandys, while the 10th Hussars sent up their contingent from Aldershot.<ref>"The Drury Lane Pantomime." ''Newcastle Evening Chronicle'' 30 December 1886, Thursday: 4 [of 8], Col. 4b–c [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000726/18861230/092/0004. Print: The ''Evening Chronicle'', p. 4.</ref></blockquote>
==Works Cited==
*Gray, Eugene F. "Chronology of Events in the Life of Emma Nevada." Emma Nevada: An American Diva. https://www.msu.edu/~graye/emma/chronolo.html (retrieved 14 April 2010).
*Sheppard, Edgar. George, Duke of Cambridge: a memoir of his private life based on the journals and correspondence of His Royal Highness. Volume 2, 1871-1904. London: Longmans, Green, 1906. Google Books, retrieved 23 February 2010.
*Spender, J. A. Sir Robert Hudson: A Memoir. London: Cassell, 1930.
idvlrridx1o4lkdkzwx563uasp0szt9
Planck units (geometrical)
0
275012
2694068
2619127
2025-01-02T01:29:27Z
Platos Cave (physics)
2562653
2694068
wikitext
text/x-wiki
'''Natural Planck units as geometrical objects (the mathematical electron model)'''
In a geometrical [[w:Planck units |Planck unit]] theory, the dimensioned universe at the Planck scale is defined by discrete geometrical objects for the Planck units; [[w:Planck mass |Planck mass]], [[w:Planck units |Planck length]], [[w:Planck time |Planck time]] and Planck charge. The object embeds the attribute (mass, length, time, charge) of the unit, whereas for numerical based constants, the numerical values are dimensionless frequencies of the [[w:SI units |SI unit]] (kg, m, s, A), 3kg refers to 3 of the unit kg, the number 3 carries no mass-specific information.
=== Geometrical objects ===
The [[v:Electron_(mathematical) |mathematical electron]] <ref>Macleod, M.J. {{Cite journal |title= Programming Planck units from a mathematical electron; a Simulation Hypothesis |journal=Eur. Phys. J. Plus |volume=113 |pages=278 |date=22 March 2018 | doi=10.1140/epjp/i2018-12094-x }}</ref> is a Planck unit model where mass <math>M</math>, length <math>L</math>, time <math>T</math>, and ampere <math>A</math> are each assigned discrete geometrical objects from the geometry of 2 [[w:dimensionless physical constant | dimensionless physical constants]]; the (inverse) [[w:fine-structure constant | fine structure constant '''α''']] and [[v:Planck_units_(geometrical)#Omega | Omega '''Ω''']] (note: [[v:Planck_units_(geometrical)#Omega |Omega]] has an intriguing solution in terms of pi and [[w:Natural_logarithm |e]] and so may be a mathematical, not physical, constant).
Embedded into each object is the object function (attribute).
{| class="wikitable"
|+Table 1. Geometrical units
! Attribute
! Geometrical object
|-
| mass
| <math>M = (1)</math>
|-
| time
| <math>T = (\pi)</math>
|-
| [[v:Sqrt_Planck_momentum | sqrt(momentum)]]
| <math>P = (\Omega)</math>
|-
| velocity
| <math>V = (2\pi\Omega^2)</math>
|-
| length
| <math>L = (2\pi^2\Omega^2)</math>
|-
| ampere
| <math>A = \frac{16 V^3}{\alpha P^3} = (\frac{2^7 \pi^3 \Omega^3}{\alpha})</math>
|}
As the geometries of dimensionless constants, these objects are also dimensionless and so are independent of any system of units, and of any numerical system, and so could qualify as "natural units" (naturally occuring units);
{{bq|''...ihre Bedeutung für alle Zeiten und für alle, auch außerirdische und außermenschliche Kulturen notwendig behalten und welche daher als »natürliche Maßeinheiten« bezeichnet werden können...''
...These necessarily retain their meaning for all times and for all civilizations, even extraterrestrial and non-human ones, and can therefore be designated as "natural units"... -Max Planck <ref>Planck (1899), p. 479.</ref><ref name="TOM">*Tomilin, K. A., 1999, "[http://www.ihst.ru/personal/tomilin/papers/tomil.pdf Natural Systems of Units: To the Centenary Anniversary of the Planck System]", 287–296.</ref>}}
As geometrical objects, they may combine [[w:Lego |Lego-style]] to form more complex objects such as electrons (i.e.: by embedding ''mass'' and ''ampere'' objects into the geometry of the electron (the electron object), the electron can have wavelength and charge) <ref>[https://codingthecosmos.com/ A Planck scale mathematical universe model]</ref>. This requires a mathematical (unit number) relationship that defines how the objects interact with each other.
{| class="wikitable"
|+Table 2. Unit number
! Attribute
! Object
! Unit number θ
|-
| mass
| <math>M = (1)</math>
| <math>15</math>
|-
| time
| <math>T = (\pi)</math>
| <math>-30</math>
|-
| [[v:Sqrt_Planck_momentum | sqrt(momentum)]]
| <math>P = (\Omega)</math>
| <math>16</math>
|-
| velocity
| <math>V = (2\pi\Omega^2)</math>
| <math>17</math>
|-
| length
| <math>L = (2\pi^2\Omega^2)</math>
| <math>-13</math>
|-
| ampere
| <math>A = \frac{16 V^3}{\alpha P^3} = (\frac{2^7 \pi^3 \Omega^3}{\alpha})</math>
| <math>3</math>
|}
As alpha (α = 137.035 999 084) and Omega (Ω = 2.007 134 949 636) both have numerical solutions, we can assign to MLTA numerical values, i.e.: ''V'' = 2πΩ<sup>2</sup> = 25.3123819 and use to solve geometrical physical constant equivalents.
{| class="wikitable"
|+Table 3. Physical constant equivalents
! CODATA 2014 <ref>[http://www.codata.org/] | CODATA, The Committee on Data for Science and Technology | (2014)</ref>
! SI unit
! Geometrical constant
! unit u<sup>θ</sup>
|-
| ''c'' = 299 792 458 (exact)
| <math>\frac{m}{s}</math>
| ''c*'' = V = 25.312381933
| <math>u^{17}</math>
|-
| ''h'' = 6.626 070 040(81) e-34
| <math>\frac{kg \;m^2}{s}</math>
| ''h*'' = <math>2 \pi M V L</math> = 12647.2403
| <math>u^{15+17-13}</math> = <math>u^{19}</math>
|-
| ''G'' = 6.674 08(31) e-11
| <math>\frac{m^3}{kg \;s^2}</math>
| ''G*'' = <math>\frac{V^2 L}{M}</math> = 50950.55478
| <math>u^{34-13-15}</math> = <math>u^{6}</math>
|-
| ''e'' = 1.602 176 620 8(98) e-19
| <math>C = A s</math>
| ''e*'' = <math>A T</math> = 735.70635849
| <math>u^{3-30}</math> = <math>u^{-27}</math>
|-
| ''k<sub>B</sub>'' = 1.380 648 52(79) e-23
| <math>\frac{kg \;m^2}{s^2 \;K}</math>
| ''k<sub>B</sub>*'' = <math>\frac{2 \pi V M}{A}</math> = 0.679138336
| <math>u^{17+15-3}</math> = <math>u^{29}</math>
|}
We then find that where the unit numbers cancel, the numerical solutions agree (see Table 8).
{| class="wikitable"
|+Table 4. Dimensionless combinations
! CODATA 2014 (mean)
! (α, Ω)
! units u<sup>Θ</sup> = 1
|-
| <math>\frac{k_B e c}{h} =</math> {{font color|green|yellow|'''1.000 8254'''}}
| <math>\frac{(k_B^*) (e^*) (c^*)}{(h^*)}</math> = {{font color|green|yellow|'''1.0'''}}
| <math>\frac{ (u^{29}) (u^{-27}) (u^{17}) }{ (u^{19}) } = 1</math>
|-
| <math>\frac{h^3}{e^{13} c^{24}} =</math> {{font color|green|yellow| '''0.228 473 639... 10<sup>-58</sup>'''}}
| <math>\frac{(h^*)^3}{(e^*)^{13} (c^*)^{24}} = \frac{\alpha^{13}}{2^{106} \pi^{64} (\color{red}\Omega^{15})^5\color{black}} =</math> {{font color|green|yellow| '''0.228 473 759... 10<sup>-58</sup>'''}}
| <math>\frac{(u^{19})^{3}}{(u^{-27})^{13} (u^{17})^{24}} = 1</math>
|-
| <math>\frac{h c^2 e m_p}{G^2 k_B} =</math> {{font color|green|yellow| '''3.376 716'''}}
| <math>\frac{(h^*) (c^*)^2 (e^*) M}{(G^*)^2 (k_B^*)} = \frac{2^{11} \pi^3}{\alpha^2} =</math> {{font color|green|yellow| '''3.381 506'''}}
| <math>\frac{ (u^{19}) (u^{17})^2 (u^{-27}) (u^{15}) }{ (u^{6})^2 (u^{29}) } = 1</math>
|}
==== Scalars ====
To translate from geometrical objects to a numerical system of units requires system dependent scalars ('''kltpva'''). For example;
:If we use ''k'' to convert ''M'' to the SI Planck mass (M*''k''<sub>SI</sub> = <math>m_P</math>), then ''k''<sub>SI</sub> = 0.2176728e-7kg ([[w:SI_units |SI units]])
:Using ''v''<sub>SI</sub> = 11843707.905m/s gives ''c'' = V*''v''<sub>SI</sub> = 299792458m/s ([[w:SI_units |SI units]])
:Using ''v<sub>imp</sub>'' = 7359.3232155miles/s gives ''c'' = V*''v''<sub>imp</sub> = 186282miles/s ([[w:Imperial_units |imperial units]])
{| class="wikitable"
|+Table 5. Geometrical units
! Attribute
! Geometrical object
! Scalar
! Unit ''u''<sup>θ</sup>
|-
| mass
| <math>M = (1)</math>
| ''k''
| <math>u^{15}</math>
|-
| time
| <math>T = (\pi)</math>
| ''t''
| <math>u^{-30}</math>
|-
| [[v:Sqrt_Planck_momentum | sqrt(momentum)]]
| <math>P = (\Omega)</math>
| ''r''<sup>2</sup>
| <math>u^{16}</math>
|-
| velocity
| <math>V = (2\pi\Omega^2)</math>
| ''v''
| <math>u^{17}</math>
|-
| length
| <math>L = (2\pi^2\Omega^2)</math>
| ''l''
| <math>u^{-13}</math>
|-
| ampere
| <math>A = (\frac{2^7 \pi^3 \Omega^3}{\alpha})</math>
| ''a''
| <math>u^3</math>
|}
==== Scalar relationships ====
Because the scalars also include the SI unit, ''v'' = 11843707.905'''m/s''' ... they follow the unit number relationship ''u''<sup>θ</sup>. This means that we can find ratios where the scalars cancel. Here are examples (units = 1), as such ''only 2 scalars are required'', for example, if we know the numerical value for ''a'' and for ''l'' then we know the numerical value for ''t'' ('''t = a<sup>3</sup>l<sup>3</sup>'''), and from ''l'' and ''t'' we know the value for ''k''.
:<math>\frac{u^{3*3} u^{-13*3}}{u^{-30}}\;(\frac{a^3 l^3}{t}) = \frac{u^{-13*15}}{u^{15*9} u^{-30*11}} \;(\frac{l^{15}}{k^9 t^{11}}) = \;...\; =1</math>
This means that once any 2 scalars have been assigned values, the other scalars are then defined by default, consequently the CODATA 2014 values are used here as only 2 constants (c, [[w:premeability of vacuum|μ<sub>0</sub>]]) are assigned exact values, following the [[w:2019 redefinition of SI base units|2019 redefinition of SI base units]] 4 constants have been independently assigned exact values which is problematic in terms of this model.
Scalars ''r'' (θ = 8) and ''v'' (θ = 17) are chosen as they can be derived directly from the 2 constants with exact values; ''c'' and ''μ<sub>0</sub>''.
:<math>v = \frac{c}{2 \pi \Omega^2}= 11 843 707.905 ...,\; units = \frac{m}{s}</math>
:<math>r^7 = \frac{2^{11} \pi^5 \Omega^4 \mu_0}{\alpha};\; r = 0.712 562 514 304 ...,\; units = (\frac{kg.m}{s})^{1/4}</math>
{| class="wikitable"
|+Table 6. Geometrical objects
! attribute
! geometrical object
! unit number θ
! scalar r(8), v(17)
|-
| mass
| <math>M = (1)</math>
| 15 = 8*4-17
| <math>k = \frac{r^4}{v}</math>
|-
| time
| <math>T = (\pi)</math>
| -30 = 8*9-17*6
| <math>t = \frac{r^9}{v^6}</math>
|-
| velocity
| <math>V = (2\pi\Omega^2)</math>
| 17
| ''v''
|-
| length
| <math>L = (2\pi^2\Omega^2)</math>
| -13 = 8*9-17*5
| <math>l = \frac{r^9}{v^5}</math>
|-
| ampere
| <math>A = (\frac{2^7 \pi^3 \Omega^3}{\alpha})</math>
| 3 = 17*3-8*6
| <math>a = \frac{v^3}{r^6}</math>
|}
{| class="wikitable"
|+ Table 7. Comparison; SI and θ
! constant
! θ (SI unit)
! MLTVA
! scalar r(8), v(17)
|-
| ''c''
| <math>\frac{m}{s}</math> (-13+30 = {{font color|red|white|17}})
| ''c*'' = <math>V*v</math>
| {{font color|red|white|17}}
|-
| ''h''
| <math>\frac{kg \;m^2}{s}</math> (15-26+30={{font color|red|white|19}})
| ''h*'' = <math>2 \pi M V L * \frac{r^{13}}{v^5}</math>
| 8*13-17*5={{font color|red|white|19}}
|-
| ''G''
| <math>\frac{m^3}{kg \;s^2}</math> (-39-15+60={{font color|red|white|6}})
| ''G*'' = <math>\frac{V^2 L}{M} * \frac{r^5}{v^2}</math>
| 8*5-17*2={{font color|red|white|6}}
|-
| ''e''
| <math>C = A s</math> (3-30={{font color|red|white|-27}})
| ''e*'' = <math>A T * \frac{r^3}{v^3}</math>
| 8*3-17*3={{font color|red|white|-27}}
|-
| ''k<sub>B</sub>''
| <math>\frac{kg \;m^2}{s^2 \;K}</math> (15-26+60-20={{font color|red|white|29}})
| ''k<sub>B</sub>*'' = <math>\frac{2 \pi V M}{A} * \frac{r^{10}}{v^3}</math>
| 8*10-17*3={{font color|red|white|29}}
|-
| ''μ<sub>0</sub>''
| <math>\frac{kg \;m}{s^2 \;A^2}</math> (15-13+60-6={{font color|red|white|56}})
| ''μ<sub>0</sub>*'' = <math>\frac{4 \pi V^2 M}{\alpha L A^2} * r^7</math>
| 8*7={{font color|red|white|56}}
|}
==== Fine structure constant ====
The fine structure constant can be derived from this formula (units and scalars cancel).
:<math>\frac{2 (h^*)}{(\mu_0^*) (e^*)^2 (c^*)} = 2({2^3 \pi^4 \Omega^4})/(\frac{\alpha}{2^{11} \pi^5 \Omega^4})(\frac{2^{7} \pi^4 \Omega^3}{\alpha})^2(2 \pi \Omega^2) = \color{red}\alpha \color{black}</math>
:<math>units \;\frac{u^{19}}{u^{56} (u^{-27})^2 u^{17}} = 1</math>
:<math>scalars \;(\frac{r^{13}}{v^5})(\frac{1}{r^7})(\frac{v^6}{r^6})(\frac{1}{v}) = 1</math>
==== Electron formula ====
{{main|Electron (mathematical)}}
The ''electron object'' (formula ''f<sub>e</sub>'') is a mathematical particle (units and scalars cancel).
:<math>f_e = 4\pi^2(2^6 3 \pi^2 \alpha \Omega^5)^3 = .23895453...x10^{23}</math> units = 1
In this example, embedded within the electron are the objects for charge, length and time ALT. AL as an ampere-meter (ampere-length) are the units for a [[w:magnetic monopole | magnetic monopole]].
:<math>T = \pi \frac{r^9}{v^6},\; u^{-30}</math>
:<math>\sigma_{e} = \frac{3 \alpha^2 A L}{2\pi^2} = {2^7 3 \pi^3 \alpha \Omega^5}\frac{r^3}{v^2},\; u^{-10}</math>
:<math>f_e = \frac{\sigma_{e}^3}{2 T} = \frac{(2^7 3 \pi^3 \alpha \Omega^5)^3}{2\pi},\; units = \frac{(u^{-10})^3}{u^{-30}} = 1, scalars = (\frac{r^3}{v^2})^3 \frac{v^6}{r^9} = 1</math>
Associated with the electron are dimensioned parameters, these parameters however are a function of the MLTA units, the formula ''f<sub>e</sub>'' dictating the frequency of these units. By setting MLTA to their SI Planck unit equivalents (Table 6.);
[[w:electron mass | electron mass]] <math>m_e^* = \frac{M}{f_e}</math> (M = [[w:Planck mass | Planck mass]] = <math>\frac{r^4}{v})</math> = 0.910 938 232 11 e-30
[[w:Compton wavelength | electron wavelength]] <math>\lambda_e^* = 2\pi L f_e</math> (L = [[w:Planck length | Planck length]] = <math>2\pi\Omega^2\frac{r^9}{v^5})</math> = 0.242 631 023 86 e-11
[[w:elementary charge | elementary charge]] <math>e^* = A\;T</math> (T = [[w:Planck time | Planck time]]) = <math>\frac{2^7 \pi^4 \Omega^3}{\alpha}\frac{r^3}{v^3}</math> = 0.160 217 651 30 e-18
[[w:Rydberg constant | Rydberg constant]] <math>R^* = (\frac{m_e}{4 \pi L \alpha^2 M}) = \frac{1}{2^{23} 3^3 \pi^{11} \alpha^5 \Omega^{17}}\frac{v^5}{r^9}\;u^{13}</math> = 10 973 731.568 508
==== Omega ====
The most precise of the experimentally measured constants is the [[w:Rydberg constant | Rydberg constant]] ''R'' = 10973731.568508(65) 1/m. Here ''c'' (exact), [[w:Vacuum permeability | Vacuum permeability]] μ<sub>0</sub> = 4π/10^7 (exact) and ''R'' (12-13 digits) are combined into a unit-less ratio;
:<math>\mu_0^* = \frac{4 \pi V^2 M}{\alpha L A^2} = \frac{\alpha}{2^{11} \pi^5 \Omega^4} r^7,\; u^{56}</math>
:<math>R^* = (\frac{m_e}{4 \pi L \alpha^2 M}) = \frac{1}{2^{23} 3^3 \pi^{11} \alpha^5 \Omega^{17}} \frac{v^5}{r^9},\;u^{13}</math>
:<math>\frac{(c^*)^{35}}{(\mu_0^*)^9 (R^*)^7} = (2 \pi \Omega^2)^{35}/(\frac{\alpha}{2^{11} \pi^5 \Omega^4})^9 .(\frac{1}{2^{23} 3^3 \pi^{11} \alpha^5 \Omega^{17}})^7,\;units = \frac{(u^{17})^{35}}{(u^{56})^9 (u^{13})^7}</math>
:<math>\frac{(c^*)^{35}}{(\mu_0^*)^9 (R^*)^7} = 2^{295} \pi^{157} 3^{21} \alpha^{26} (\Omega^{15})^{15}</math>, units = 1
We can now define ''Ω'' using the geometries for (''c<sup>*</sup>, μ<sub>0</sub><sup>*</sup>, R<sup>*</sup>'') and then solve by replacing (''c<sup>*</sup>, μ<sub>0</sub><sup>*</sup>, R<sup>*</sup>'') with the numerical (''c, μ<sub>0</sub>, R'').
:<math>\Omega^{225}=\frac{(c^*)^{35}}{2^{295} 3^{21} \pi^{157} (\mu_0^*)^9 (R^*)^7 \alpha^{26}}, \;units = 1</math>
:<math>\Omega = 2.007\;134\;949\;636...,\; units = 1</math> (CODATA 2014 mean values)
:<math>\Omega = 2.007\;134\;949\;687...,\; units = 1</math> (CODATA 2018 mean values)
:<math>\Omega = 2.007\;134\;949\;584...,\; units = 1</math> (CODATA 2022 mean values)
There is a close natural number for Ω that is a square root
:<math>\Omega = \sqrt{ \left(\pi^e e^{(1-e)}\right)} = 2.007\;134\;9543... </math>
implying that Ω can have a plus or a minus solution, and this agrees with theory (in the mass domain Ω occurs as Ω<sup>2</sup> = plus only, in the charge domain Ω occurs as Ω<sup>3</sup> = can be plus or minus; see [[v:Sqrt_Planck_momentum | sqrt(momentum)]]). This solution however would re-classify Omega as a mathematical constant (as being derivable from other mathematical constants), leaving alpha as the only physical constant used in this model. From this mathematical Omega and reversing the above equation, we can solve α = 137.035996376...
This formula is of the form (where {{mvar|i}} is the [[w:imaginary unit | imaginary unit]])
:<math>\sqrt{ \left(i\; a^b\; b^{(1-b)}\right)}</math>
for which the real and imaginary parts will be equal and this implies a 45 degree angle in the complex plane (i.e.: this complex number lies exactly on the line y = x in the complex plane). This is relevant to particle rotations and symmetries, [[w:Quantum_superposition | Quantum_superposition]] ... <ref>Macleod, M.J., "[https://theprogrammergod.com The Programmer God, are we in a computer simulation?]", Chapter 10 Alpha and Omega (2024)</ref>
[[w:Euler%27s_formula | Euler's formula]]
:<math display="block">e^{i x} = \cos x + i \sin x</math>
Adding {{mvar|i}}
:<math>\sqrt{ \left(i \pi^e e^{(1-e)}\right)}</math>
Solves to 1.4192587369597... + 1.4192587369597...{{mvar|i}}.
==== Dimensionless combinations ====
Reference List of dimensionless combinations. These can be solved using only α, Ω (and the mathematical constants 2, 3, π) as the units and scalars have cancelled. The precision of the results depends on the precision of the SI constants; combinations with ''G'' and ''k''<sub>B</sub> return the least precise values. These combinations can be used to test the veracity of the MLTA geometries as natural Planck units. See also [[v:Planck_units_(geometrical)#Anomalies |Anomalies]] (below).
Example
:<math>\frac{(h^*)^3}{(e^*)^{13} (c^*)^{24}} = (2^3 \pi^4 \Omega^4 \frac{r^{13}}{v^5})^3/(\frac{2^7 \pi^4 \Omega^3 r^3}{\alpha v^3})^7.(2\pi\Omega^2 v)^{24} = \frac{\alpha^{13}}{2^{106} \pi^{64} (\color{red}\Omega^{15})^5\color{black}} = </math> {{font color|green|yellow| '''0.228 473 759... 10<sup>-58</sup>'''}}
Note: the geometry <math>\color{red}(\Omega^{15})^n\color{black}</math> (integer n ≥ 0) is common to all ratios where units and scalars cancel, suggesting a geometrical base-15.
{| class="wikitable"
|+Table 8. Dimensionless combinations
! CODATA 2014 mean
! (α, Ω) mean
! units = 1
! scalars = 1
|-
| <math>\frac{k_B e c}{h} =</math> {{font color|green|yellow|'''1.000 8254'''}}
| <math>\frac{(k_B^*) (e^*) (c^*)}{(h^*)}</math> = {{font color|green|yellow|'''1.0'''}}
| <math>\frac{ (u^{29}) (u^{-27}) (u^{17}) }{ (u^{19}) } = 1</math>
| <math>(\frac{r^{10}}{v^3}) (\frac{r^3}{v^3}) (v) / (\frac{r^{13}}{v^5}) = 1</math>
|-
| <math>\frac{h^3}{e^{13} c^{24}} =</math> {{font color|green|yellow| '''0.228 473 639... 10<sup>-58</sup>'''}}
| <math>\frac{(h^*)^3}{(e^*)^{13} (c^*)^{24}} = \frac{\alpha^{13}}{2^{106} \pi^{64} (\color{red}\Omega^{15})^5\color{black}} =</math> {{font color|green|yellow| '''0.228 473 759... 10<sup>-58</sup>'''}}
| <math>\frac{(u^{19})^{3}}{(u^{-27})^{13} (u^{17})^{24}} = 1</math>
| <math>(\frac{r^{13}}{v^5})^3 / (\frac{r^3}{v^3})^{13} (v^{24}) = 1</math>
|-
| <math>\frac{c^{35}}{\mu_0^9 R^7} =</math> {{font color|green|yellow| '''0.326 103 528 6170... 10<sup>301</sup>'''}}
| <math>\frac{(c^*)^{35}}{(\mu_0^*)^9 (R^*)^7} = 2^{295} \pi^{157} 3^{21} \alpha^{26} \color{red}(\Omega^{15})^{15}\color{black} =</math> {{font color|green|yellow| '''0.326 103 528 6170... 10<sup>301</sup>'''}}
| <math>\frac{(u^{17})^{35}}{(u^{56})^9 (u^{13})^7} = 1</math>
| <math>(v^{35})/(r^7)^9 (\frac{v^5}{r^9})^7 = 1</math>
|-
| <math>\frac{c^9 e^4}{m_e^3} =</math> {{font color|green|yellow| '''0.170 514 342... 10<sup>92</sup>'''}}
| <math>\frac{(c^*)^9 (e^*)^4}{(m_e^*)^3} = 2^{97} \pi^{49} 3^9 \alpha^5 (\color{red}\Omega^{15})^5\color{black}=</math> {{font color|green|yellow| '''0.170 514 368... 10<sup>92</sup>'''}}
| <math>\frac{ (u^{29}) (u^{-27}) (u^{17}) }{ (u^{19}) } = 1</math>
| <math>(v^9) (\frac{r^3}{v^3})^4 / (\frac{r^4}{v})^3 = 1</math>
|-
| <math>\frac{k_B}{e^2 m_e c^4} =</math> {{font color|green|yellow| '''73 095 507 858.'''}}
| <math>\frac{(k_B^*)}{(e^*)^2 (m_e^*) (c^*)^4} = \frac{3^3 \alpha^6}{2^3 \pi^5} =</math> {{font color|green|yellow| '''73 035 235 897.'''}}
| <math>\frac{(u^{29})}{(u^{-27})^2 (u^{15}) (u^{17})^4} = 1</math>
| <math>(\frac{r^{10}}{v^3}) / (\frac{r^3}{v^3})^2 (\frac{r^4}{v}) (v)^4 = 1</math>
|-
| <math>\frac{h c^2 e m_p}{G^2 k_B} =</math> {{font color|green|yellow| '''3.376 716'''}}
| <math>\frac{(h^*) (c^*)^2 (e^*) (m_p^*)}{(G^*)^2 (k_B^*)} = \frac{2^{11} \pi^3}{\alpha^2} =</math> {{font color|green|yellow| '''3.381 506'''}}
| <math>\frac{ (u^{19}) (u^{17})^2 (u^{-27}) (u^{15}) }{ (u^{6})^2 (u^{29}) } = 1</math>
| <math>(\frac{r^{13}}{v^5}) v^2 (\frac{r^{3}}{v^3})(\frac{r^{4}}{v^1}) / (\frac{r^5}{v^2})^2 (\frac{r^{10}}{v^3}) = 1</math>
|}
==== Table of Constants ====
We can construct a table of constants using these 3 geometries. Setting
:<math>f(x)\;units = (\frac{L^{15}}{M^9 T^{11}})^n = 1</math>
i.e.: unit number θ = (-13*15) - (15*9) - (-30*11) = 0
:<math>\color{red}i\color{black} = \pi^2 \Omega^{15}</math>, units = <math>\sqrt{f(x)}</math> = 1 (unit number = 0, no scalars)
:<math>\color{red}x\color{black} = \Omega \frac{v}{r^2}</math> , units = <math>\sqrt{\frac{L}{M T}}</math> = u<sup>1</sup> = u (unit number = -13 -15 +30 = 2/2 = 1, with scalars ''v'', ''r'')
:<math>\color{red}y\color{black} = \pi \frac{r^{17}}{v^8}</math> , units = <math>M^2 T</math> = 1, (unit number = 15*2 -30 = 0, with scalars ''v'', ''r'')
Note: The following suggests a numerical boundary to the values the SI constants can have.
:<math>\frac{v}{r^2} = a^{1/3} = \frac{1}{t^{2/15}k^{1/5}} = \frac{\sqrt{v}}{\sqrt{k}}</math> ... = 23326079.1...; unit = u
:<math>\frac{r^{17}}{v^8} = k^2 t = \frac{k^{17/4}}{v^{15/4}} = ... </math> gives a range from 0.812997... x10<sup>-59</sup> to 0.123... x10<sup>60</sup>
Note: Influence of <math>f(x)</math>, units = 1
:<math>\frac{r^{17}}{v^8} \;\;units \;(\frac{M^2 L^8}{T^7}) (\frac{T}{L})^8 = M^2 T</math>
:<math>r^{17} \;\;units \;(\frac{M\;L}{T})^{17/4} fx^{1/4} = \frac{M^2\;L^8}{T^7}</math>
:<math>r \;\;units \;(\frac{M\;L}{T})^{1/4} fx^{1/4} = \frac{L^4}{M^2 T^3}</math>
{| class="wikitable"
|+Table 9. Table of Constants
! Constant
! θ
! Geometrical object (α, Ω, v, r)
! Unit
! Calculated
! CODATA 2014
|-
| Time (Planck)
| <math>\color{red}-30\color{black}</math>
| <math>T = \color{red}\frac{x^\theta i^2}{y^3}\color{black} = \frac{\pi r^9}{v^6}</math>
| <math>T</math>
| T = 5.390 517 866 e-44
| ''t<sub>p</sub>'' = 5.391 247(60) e-44
|-
| [[w:Elementary charge |Elementary charge]]
| <math>\color{red}-27\color{black}</math>
| <math>e^* = (\frac{2^7 \pi^3}{\alpha}) \color{red}\frac{x^\theta i^2}{y^3}\color{black} = (\frac{2^7 \pi^3}{\alpha}) \;\frac{\pi \Omega^3 r^3}{v^3}</math>
| <math>\frac{L^{3/2}}{T^{1/2} M^{3/2}} = AT</math>
| ''e<sup>*</sup>'' = 1.602 176 511 30 e-19
| ''e'' = 1.602 176 620 8(98) e-19
|-
| Length (Planck)
| <math>\color{red}-13\color{black}</math>
| <math>L = (2\pi) \color{red}\frac{x^\theta i}{y}\color{black} = (2\pi) \;\frac{\pi \Omega^2 r^9}{v^5}</math>
| <math>L</math>
| L = 0.161 603 660 096 e-34
| ''l<sub>p</sub>'' = 0.161 622 9(38) e-34
|-
| Ampere
| <math>\color{red}3\color{black}</math>
| <math>A = (\frac{2^7 \pi^3}{\alpha}) \color{red}x^\theta\color{black} = (\frac{2^7 \pi^3}{\alpha}) \; \frac{\Omega^3 v^3}{r^6} </math>
| <math>A = \frac{L^{3/2}}{M^{3/2} T^{3/2}}</math>
| A = 0.297 221 e25
| ''e/t<sub>p</sub>'' = 0.297 181 e25
|-
| [[w:Gravitational constant |Gravitational constant]]
| <math>\color{red}6\color{black}</math>
| <math>G^* = (2^3 \pi^3) \color{red}\color{red}x^\theta y\color{black} = (2^3 \pi^3) \;\frac{\pi \Omega^6 r^5}{v^2}</math>
| <math>\frac{L^3}{M T^2}</math>
| ''G<sup>*</sup>'' = 6.672 497 192 29 e11
| ''G'' = 6.674 08(31) e-11
|-
|
| <math>\color{red}8\color{black}</math>
| <math>X = (2^4 \pi^4) \color{red}\color{red}x^\theta y\color{black} = (2^4 \pi^4) \pi \Omega^8 r</math>
| <math>\frac{L^4}{M^2 T^3}</math>
| ''X'' = 918 977.554 22
|
|-
| Mass (Planck)
| <math>\color{red}\color{red}15\color{black}</math>
| <math>M = \color{red}\color{red}\frac{x^\theta y^2}{i}\color{black} = \frac{r^4}{v}</math>
| <math>M</math>
| M = .217 672 817 580 e-7
| ''m<sub>P</sub>'' = .217 647 0(51) e-7
|-
| [[v:Sqrt_Planck_momentum | sqrt(momentum)]]
| <math>\color{red}16\color{black}</math>
| <math>P = \color{red}\color{red}\frac{x^\theta y^2}{i}\color{black} = \Omega r^2</math>
| <math>\frac{M^{1/2} L^{1/2}}{T^{1/2}}</math>
|
|
|-
| Velocity
| <math>\color{red}\color{red}17\color{black}</math>
| <math>V = (2\pi) \color{red}\color{red}\frac{x^\theta y^2}{i}\color{black} = (2\pi) \;\Omega^2 v</math>
| <math>V = \frac{L}{T}</math>
| V = 299 792 458
| ''c'' = 299 792 458
|-
| [[w:Planck constant |Planck constant]]
| <math>\color{red}19\color{black}</math>
| <math>h^* = (2^3 \pi^3) \color{red}\frac{x^\theta y^3}{i}\color{black} = (2^3 \pi^3) \;\frac{\pi \Omega^4 r^{13}}{v^5}</math>
| <math>\frac{L^2 M}{T}</math>
| ''h<sup>*</sup>'' = 6.626 069 134 e-34
| ''h'' = 6.626 070 040(81) e-34
|-
| [[w:Planck temperature |Planck temperature]]
| <math>\color{red}\color{red}20\color{black}</math>
| <math>{T_p}^* = (\frac{2^7 \pi^3}{\alpha}) \color{red}\color{red}\frac{x^\theta y^2}{i}\color{black} = (\frac{2^7 \pi^3 }{\alpha}) \; \frac{\Omega^5 v^4}{r^6}</math>
| <math>\frac{L^{5/2}}{M^{3/2} T^{5/2}} = AV</math>
| ''T<sub>p</sub><sup>*</sup>'' = 1.418 145 219 e32
| ''T<sub>p</sub>'' = 1.416 784(16) e32
|-
| [[w:Boltzmann constant |Boltzmann constant]]
| <math>\color{red}\color{red}29\color{black}</math>
| <math>{k_B}^* = (\frac{\alpha}{2^5 \pi}) \color{red}\frac{x^\theta y^4}{i^2}\color{black} = (\frac{\alpha}{2^5 \pi }) \;\frac{r^{10}}{\Omega v^3}</math>
| <math>\frac{M^{5/2} T^{1/2}}{L^{1/2}} = \frac{M L}{T A}</math>
| ''k<sub>B</sub><sup>*</sup>'' = 1.379 510 147 52 e-23
| ''k<sub>B</sub>'' = 1.380 648 52(79) e-23
|-
| [[w:Vacuum permeability |Vacuum permeability]]
| <math>\color{red}56\color{black}</math>
| <math>{\mu_0}^* = (\frac{\alpha}{2^{11} \pi^4}) \color{red}\frac{x^\theta y^7}{i^4}\color{black} = (\frac{\alpha}{2^{11} \pi^4})\; \frac{r^7}{\pi \Omega^4}</math>
| <math>\frac{M\;L}{T^2 A^2}</math>
| ''μ<sub>0</sub><sup>*</sup>'' = 4π/10^7
| ''μ<sub>0</sub>'' = 4π/10^7
|}
From the perspective of geometries
note: <math>\color{red}(u^{15})^n\color{black}</math> constants have no Omega term.
{| class="wikitable"
|+Table 10. Dimensioned constants; geometrical vs CODATA 2014
! Constant
! In Planck units
! Geometrical object
! SI calculated (r, v, Ω, α<sup>*</sup>)
! SI CODATA 2014 <ref>[http://www.codata.org/] | CODATA, The Committee on Data for Science and Technology | (2014)</ref>
|-
| [[w:Speed of light | Speed of light]]
| V
| <math>c^* = (2\pi\Omega^2)v,\;u^{17} </math>
| ''c<sup>*</sup>'' = 299 792 458, unit = u<sup>17</sup>
| ''c'' = 299 792 458 (exact)
|-
| [[w:Fine structure constant | Fine structure constant]]
|
|
| ''α<sup>*</sup>'' = 137.035 999 139 (mean)
| ''α'' = 137.035 999 139(31)
|-
| [[w:Rydberg constant | Rydberg constant]]
| <math>R^* = (\frac{m_e}{4 \pi L \alpha^2 M})</math>
| <math>R^* = \frac{1}{2^{23} 3^3 \pi^{11} \alpha^5 \Omega^{17}}\frac{v^5}{r^9},\;u^{13} </math>
| ''R<sup>*</sup>'' = 10 973 731.568 508, unit = u<sup>13</sup>
| ''R'' = 10 973 731.568 508(65)
|-
| [[w:Vacuum permeability | Vacuum permeability]]
| <math>\mu_0^* = \frac{4 \pi V^2 M}{\alpha L A^2}</math>
| <math>\mu_0^* = \frac{\alpha}{2^{11} \pi^5 \Omega^4} r^7,\; u^{56}</math>
| ''μ<sub>0</sub><sup>*</sup>'' = 4π/10^7, unit = u<sup>56</sup>
| ''μ<sub>0</sub>'' = 4π/10^7 (exact)
|-
| [[w:Vacuum permittivity | Vacuum permittivity]]
| <math>\epsilon_0^* = \frac{1}{\mu_0^* (c^*)^2}</math>
| <math>\epsilon_0^* = \frac{2^9 \pi^3}{\alpha}\frac{1}{r^7 v^2},\; \color{red}1/(u^{15})^6\color{black} = u^{-90}</math>
|
|
|-
| [[w:Planck constant | Planck constant]]
| <math>h^* = 2 \pi M V L</math>
| <math>h^* = 2^3 \pi^4 \Omega^4 \frac{r^{13}}{v^5},\; u^{19}</math>
| ''h<sup>*</sup>'' = 6.626 069 134 e-34, unit = u<sup>19</sup>
| ''h'' = 6.626 070 040(81) e-34
|-
| [[w:Gravitational constant | Gravitational constant]]
| <math>G^* = \frac{V^2 L}{M}</math>
| <math>G^* = 2^3 \pi^4 \Omega^6 \frac{r^5}{v^2},\; u^{6}</math>
| ''G<sup>*</sup>'' = 6.672 497 192 29 e11, unit = u<sup>6</sup>
| ''G'' = 6.674 08(31) e-11
|-
| [[w:Elementary charge | Elementary charge]]
| <math>e^* = A T</math>
| <math>e^* = \frac{2^7 \pi^4 \Omega^3}{\alpha}\frac{r^3}{v^3},\; u^{-27}</math>
| ''e<sup>*</sup>'' = 1.602 176 511 30 e-19, unit = u<sup>-27</sup>
| ''e'' = 1.602 176 620 8(98) e-19
|-
| [[w:Boltzmann constant | Boltzmann constant]]
| <math>k_B^* = \frac{2 \pi V M}{A}</math>
| <math>k_B^* = \frac{\alpha}{2^5 \pi \Omega} \frac{r^{10}}{v^3},\; u^{29}</math>
| ''k<sub>B</sub><sup>*</sup>'' = 1.379 510 147 52 e-23, unit = u<sup>29</sup>
| ''k<sub>B</sub>'' = 1.380 648 52(79) e-23
|-
| [[w:Electron mass | Electron mass]]
|
| <math>m_e^* = \frac{M}{f_e},\; u^{15}</math>
| ''m<sub>e</sub><sup>*</sup>'' = 9.109 382 312 56 e-31, unit = u<sup>15</sup>
| ''m<sub>e</sub>'' = 9.109 383 56(11) e-31
|-
| [[w:Classical electron radius | Classical electron radius]]
|
| <math>\lambda_e^* = 2\pi L f_e,\; u^{-13}</math>
| ''λ<sub>e</sub><sup>*</sup>'' = 2.426 310 2366 e-12, unit = u<sup>-13</sup>
| ''λ<sub>e</sub>'' = 2.426 310 236 7(11) e-12
|-
| [[w:Planck temperature | Planck temperature]]
| <math>T_p^* = \frac{A V}{\pi}</math>
| <math>T_p^* = \frac{2^7 \pi^3 \Omega^5}{\alpha} \frac{v^4}{r^6} ,\; u^{20} </math>
| ''T<sub>p</sub><sup>*</sup>'' = 1.418 145 219 e32, unit = u<sup>20</sup>
| ''T<sub>p</sub>'' = 1.416 784(16) e32
|-
| [[w:Planck mass | Planck mass]]
| M
| <math>m_P^* = (1)\frac{r^4}{v} ,\; \color{red}\color{red}(u^{15})^1\color{black}</math>
| ''m<sub>P</sub><sup>*</sup>'' = .217 672 817 580 e-7, unit = u<sup>15</sup>
| ''m<sub>P</sub>'' = .217 647 0(51) e-7
|-
| [[w:Planck length | Planck length]]
| L
| <math>l_p^* = (2\pi^2\Omega^2)\frac{r^9}{v^5},\;u^{-13} </math>
| ''l<sub>p</sub><sup>*</sup>'' = .161 603 660 096 e-34, unit = u<sup>-13</sup>
| ''l<sub>p</sub>'' = .161 622 9(38) e-34
|-
| [[w:Planck time | Planck time]]
| T
| <math>t_p^* = (\pi)\frac{r^9}{v^6} ,\; \color{red}\color{red}1/(u^{15})^2\color{black} </math>
| ''t<sub>p</sub><sup>*</sup>'' = 5.390 517 866 e-44, unit = u<sup>-30</sup>
| ''t<sub>p</sub>'' = 5.391 247(60) e-44
|-
| [[w:Ampere | Ampere]]
| <math>A = \frac{16 V^3}{\alpha P^3}</math>
| <math>A^* = \frac{2^7\pi^3\Omega^3}{\alpha}\frac{v^3}{r^6} ,\; u^3 </math>
| A<sup>*</sup> = 0.297 221 e25, unit = u<sup>3</sup>
| ''e/t<sub>p</sub>'' = 0.297 181 e25
|-
| [[w:Quantum Hall effect | Von Klitzing constant ]]
| <math>R_K^* = (\frac{h}{e^2})^*</math>
| <math>R_K^* = \frac{\alpha^2}{2^{11} \pi^4 \Omega^2} r^7 v ,\; u^{73}</math>
| ''R<sub>K</sub><sup>*</sup>'' = 25812.807 455 59, unit = u<sup>73</sup>
| ''R<sub>K</sub>'' = 25812.807 455 5(59)
|-
| [[w:Gyromagnetic ratio | Gyromagnetic ratio]]
|
| <math>\gamma_e/2\pi = \frac{g l_p^* m_P^*}{2 k_B^* m_e^*},\; unit = u^{-42}</math>
| ''γ<sub>e</sub>/2π<sup>*</sup>'' = 28024.953 55, unit = u<sup>-42</sup>
| ''γ<sub>e</sub>/2π'' = 28024.951 64(17)
|}
Note that ''r, v, Ω, α'' are dimensionless numbers, however when we replace ''u''<sup>θ</sup> with the SI unit equivalents (''u''<sup>15</sup> → kg, ''u''<sup>-13</sup> → m, ''u''<sup>-30</sup> → s, ...), the ''geometrical objects'' (i.e.: ''c<sup>*</sup>'' = 2πΩ<sup>2</sup>v = 299792458, units = u<sup>17</sup>) become '''indistinguishable''' from their respective ''physical constants'' (i.e.: ''c'' = 299792458, units = m/s).
==== 2019 SI unit revision ====
Following the 26th General Conference on Weights and Measures ([[w:2019 redefinition of SI base units|2019 redefinition of SI base units]]) are fixed the numerical values of the 4 physical constants (''h, c, e, k<sub>B</sub>''). In the context of this model however only 2 base units may be assigned by committee as the rest are then numerically fixed by default and so the revision may lead to unintended consequences.
{| class="wikitable"
|+Table 11. Physical constants
! Constant
! CODATA 2018 <ref>[http://www.codata.org/] | CODATA, The Committee on Data for Science and Technology | (2018)</ref>
|-
| [[w:Speed of light | Speed of light]]
| ''c'' = 299 792 458 (exact)
|-
| [[w:Planck constant | Planck constant]]
| ''h'' = 6.626 070 15 e-34 (exact)
|-
| [[w:Elementary charge | Elementary charge]]
| ''e'' = 1.602 176 634 e-19 (exact)
|-
| [[w:Boltzmann constant | Boltzmann constant]]
| ''k<sub>B</sub>'' = 1.380 649 e-23 (exact)
|-
| [[w:Fine structure constant | Fine structure constant]]
| ''α'' = 137.035 999 084(21)
|-
| [[w:Rydberg constant | Rydberg constant]]
| ''R'' = 10973 731.568 160(21)
|-
| [[w:Electron mass | Electron mass]]
| ''m<sub>e</sub>'' = 9.109 383 7015(28) e-31
|-
| [[w:Vacuum permeability | Vacuum permeability]]
| ''μ<sub>0</sub>'' = 1.256 637 062 12(19) e-6
|-
| [[w:Quantum_Hall_effect#Applications | Von Klitzing constant]]
| ''R<sub>K</sub>'' = 25812.807 45 (exact)
|}
For example, if we solve using the above formulas;
<math>R^* = \frac{4 \pi^5}{3^3 c^4 \alpha^8 e^3} = 10973\;729.082\;465</math>
<math>{(m_e^*)}^3 = \frac{2^4 \pi^{10} R \mu_0^3}{3^6 c^8 \alpha^7},\;m_e^* = 9.109\;382\;3259 \;10^{-31}</math>
<math>{(\mu_0^*)}^3 = \frac{3^6 h^3 c^5 \alpha^{13} R^2}{2 \pi^{10}},\;\mu_0^* = 1.256\;637\;251\;88\;10^{-6}</math>
<math>{(h^*)}^3 = \frac{2 \pi^{10} \mu_0^3}{3^6 c^5 \alpha^{13} R^2},\;h^* = 6.626\;069\;149\;10^{-34}</math>
<math>{(e^*)}^3 = \frac{4 \pi^5}{3^3 c^4 \alpha^8 R},\; e^* = 1.602\;176\;513\;10^{-19}</math>
=== Anomalies ===
{{main|Physical_constant_(anomaly)}}
The following are notes on the anomalies as evidence of a simulation universe source code <ref>Macleod, Malcolm J. {{Cite journal |title= Physical constant anomalies suggest a mathematical relationship linking SI units |journal=RG | doi=10.13140/RG.2.2.15874.15041/6 }}</ref>.
====== m<sub>P</sub>, l<sub>p</sub>, t<sub>p</sub> ======
In this ratio, the MLT units and ''klt'' scalars both cancel; units = scalars = 1, reverting to the base MLT objects. Setting the scalars ''klt'' for SI Planck units;
:k = 0.217 672 817 580... ''x'' 10<sup>-7</sup>kg
:l = 0.203 220 869 487... ''x'' 10<sup>-36</sup>m
:t = 0.171 585 512 841... ''x'' 10<sup>-43</sup>s
:<math>\frac{L^{15}}{M^{9} T^{11}} = \frac{(2\pi^2\Omega^2)^{15}}{(1)^{9} (\pi)^{11}} (\frac{l^{15}}{k^9 t^{11}}) = \frac{l_p^{15}}{m_P^{9} t_p^{11}} </math> (CODATA 2018 mean)
The ''klt'' scalars cancel, leaving;
:<math>\frac{L^{15}}{M^{9} T^{11}} = \frac{(2\pi^2\Omega^2)^{15}}{(1)^{9} (\pi)^{11}} (\frac{l^{15}}{k^9 t^{11}}) = 2^{15} \pi^{19} \color{red}(\Omega^{15})^2\color{black} = </math>{{font color|blue|yellow|'''0.109 293... 10<sup>24</sup> '''}}, <math>(\frac{l^{15}}{k^9 t^{11}}) = 1, \;\frac{u^{-13*15}}{u^{15*9} u^{-30*11}} = 1</math>
Solving for the SI units;
:<math>\frac{l_p^{15}}{m_P^{9} t_p^{11}} = \frac{(1.616255e-35)^{15}}{(2.176434e-8)^{9} (5.391247e-44)^{11}} = </math> {{font color|blue|yellow| '''0.109 485... 10<sup>24</sup>'''}}
====== A, l<sub>p</sub>, t<sub>p</sub> ======
:a = 0.126 918 588 592... ''x'' 10<sup>23</sup>A
:<math>\frac{A^3 L^3}{T} = (\frac{2^7 \pi^3 \Omega^3}{\alpha})^3 \frac{(2\pi^2\Omega^2)^3}{(\pi)} (\frac{a^3 l^3}{t}) = \frac{2^{24} \pi^{14} \color{red}(\Omega^{15})^1\color{black}}{\alpha^3} = </math> {{font color|green|yellow| '''0.205 571... 10<sup>13</sup>'''}}, <math>(\frac{a^3 l^3}{t}) = 1,\; \frac{u^{3*3} u^{-13*3}}{u^{-30}} = 1</math>
:<math>\frac{(e / t_p)^3 l_p^3}{t_p} = \frac{(1.602176634e-19/5.391247e-44)^3 (1.616255e-35)^3}{(5.391247e-44)} = </math> {{font color|green|yellow| '''0.205 543... 10<sup>13</sup>'''}}, <math>units = \frac{(C/s)^3 m^3}{s} </math>
The Planck units are known with low precision, and so by defining the 3 most accurately known dimensioned constants in [[v:Planck_units_(geometrical)#Physical_constants_(as_geometrical_formulas) |terms of these objects]] (c, R = Rydberg constant, <math>\mu_0</math>; CODATA 2014 mean values), we can test to greater precision;
====== c, μ<sub>0</sub>, R ======
:<math>\frac{(c^*)^{35}}{(\mu_0^*)^9 (R^*)^7} = (2 \pi \Omega^2 v)^{35}/(\frac{\alpha r^7}{2^{11} \pi^5 \Omega^4})^9 .(\frac{v^5}
{2^{23} 3^3 \pi^{11} \alpha^5 \Omega^{17} r^9})^7 = 2^{295} \pi^{157} 3^{21} \alpha^{26} \color{red}(\Omega^{15})^{15}\color{black} = </math> {{font color|red|yellow| '''0.326 103 528 6170... 10<sup>301</sup>'''}}, <math>\frac{(u^{17})^{35}}{(u^{56})^9 (u^{13})^7} = 1, \;(v^{35})/(r^7)^9 (\frac{v^5}{r^9})^7 = 1</math>
:<math>\frac{c^{35}}{\mu_0^9 R^7} = \frac{(299792458)^{35}}{(4 \pi/10^7)^9 (10973731.568160)^7} = </math> {{font color|red|yellow| '''0.326 103 528 6170... 10<sup>301</sup>'''}}, <math>units = \frac{m^{33}A^{18}}{s^{17}kg^9} == \frac{(u^{-13})^{33} (u^{3})^{18}}{(u^{-30})^{17} (u^{15})^9} = 1</math>
====== c, e, k<sub>B</sub>, h ======
:<math>\frac{(k_B^*) (e^*) (c^*)}{(h^*)} = (\frac{\alpha}{2^5 \pi \Omega} \frac{r^{10}}{v^3}) (\frac{2^7 \pi^4 \Omega^3}{\alpha} \frac{r^3}{v^3}) (2 \pi \Omega^2 v) / (2^3 \pi^4 \Omega^4 \frac{r^{13}}{v^5}) </math> = {{font color|blue|yellow|'''1.0'''}}, <math>\frac{ (u^{29}) (u^{-27}) (u^{17}) }{ (u^{19}) } = 1,\; (\frac{r^{10}}{v^3}) (\frac{r^3}{v^3}) (v) / (\frac{r^{13}}{v^5}) = 1</math>
:<math>\frac{k_B e c}{h} = </math> {{font color|blue|yellow|'''1.000 8254'''}}, <math>units = \frac{m C}{s^2 K} == \frac{(u^{-13}) (u^{-27})}{(u^{-30})^2 (u^{20})} = 1</math>
====== c, h, e ======
:<math>\frac{(h^*)^3}{(e^*)^{13} (c^*)^{24}} = (2^3 \pi^4 \Omega^4 \frac{r^{13}}{v^5})^3/(\frac{2^7 \pi^4 \Omega^3 r^3}{\alpha v^3})^7.(2\pi\Omega^2 v)^{24} = \frac{\alpha^{13}}{2^{106} \pi^{64} (\color{red}\Omega^{15})^5\color{black}} = </math> {{font color|green|yellow| '''0.228 473 759... 10<sup>-58</sup>'''}}, <math>\frac{(u^{19})^{3}}{(u^{-27})^{13} (u^{17})^{24}} = 1, \;(\frac{r^{13}}{v^5})^3 / (\frac{r^3}{v^3})^{13} (v^{24}) = 1</math>
:<math>\frac{h^3}{e^{13} c^{24}} = </math> {{font color|green|yellow| '''0.228 473 639... 10<sup>-58</sup>'''}}, <math>units = \frac{kg^3 s^{21}}{m^{18} C^{13}} == \frac{(u^{15})^3 (u^{-30})^{21}}{(u^{-13})^{18} (u^{-27})^{13}} = 1</math>
====== m<sub>e</sub>, λ<sub>e</sub> ======
:<math>\sigma_{e} = \frac{3 \alpha^2 A L}{2\pi^2} = {2^7 3 \pi^3 \alpha \Omega^5}\frac{r^3}{v^2},\; u^{-10}</math>
:<math>f_e = \frac{\sigma_{e}^3}{2 T} = 2^{20} 3^3 \pi^8 \alpha^3 (\color{red}\Omega^{15})\color{black},\;
\frac{(u^{-10})^3}{u^{-30}} =1,\; (\frac{r^3}{v^2})^3 \frac{v^6}{r^9} = 1</math>
:<math>(m_e^*) = \frac{M}{f_e} = \color{blue}9.109\;382\;3227 \;10^{-31}\color{black}\;u^{15}</math>
:<math>(m_e^*) = \frac{2^3 \pi^5 (h^*)}{3^3 \alpha^6 (e^*)^3 (c^*)^5} = \frac{1}{2^{20} \pi^8 3^3 \alpha^3 (\color{red}\Omega^{15})\color{black}} \frac{r^4 u^{15}}{v} = \color{blue}9.109\;382\;3227 \;10^{-31}\color{black}\;u^{15}</math>
:<math>m_e = \color{blue}9.109\;383\;7015... \;10^{-31}\color{black}\;kg</math>
:<math>(\lambda_e^*) = 2 \pi L f_e = \color{purple}2.426\;310\;238\;667 \;10^{-12}\color{black}\;u^{-13}</math>
:<math>\lambda_e = \frac{h}{m_e c} = \color{purple}2.426 \;310 \;238 \;67 \;10^{-12}\color{black}\;m</math>
====== c, e, m<sub>e</sub> ======
:<math>(m_e^*)= \frac{M}{f_e}, \;f_e = 2^{20} 3^3 \pi^8 \alpha^3 (\color{red}\Omega^{15})^1\color{black} </math>, units = scalars = 1 ([[v:Planck_units_(geometrical)#Electron_formula |m<sub>e</sub> formula]])
:<math>\frac{(c^*)^9 (e^*)^4}{(m_e^*)^3} = 2^{97} \pi^{49} 3^9 \alpha^5 (\color{red}\Omega^{15})^5\color{black} = </math> {{font color|red|yellow| '''0.170 514 368... 10<sup>92</sup>'''}}, <math>\frac{(u^{17})^9 (u^{-27})^4}{(u^{15})^3} = 1,\; (v^9) (\frac{r^3}{v^3})^4 / (\frac{r^4}{v})^3 = 1</math>
:<math>\frac{c^9 e^4}{m_e^3} = </math> {{font color|red|yellow| '''0.170 514 342... 10<sup>92</sup>'''}}, <math>units = \frac{m^9 C^4}{s^9 kg^3} == \frac{(u^{-13})^9 (u^{-27})^4}{(u^{-30})^9 (u^{15})^3} = 1</math>
====== k<sub>B</sub>, c, e, m<sub>e</sub> ======
:<math>\frac{(k_B^*)}{(e^*)^2 (m_e^*) (c^*)^4} = \frac{3^3 \alpha^6}{2^3 \pi^5} = </math> {{font color|blue|yellow| '''73 035 235 897.'''}}, <math>\frac{(u^{29})}{(u^{-27})^2 (u^{15}) (u^{17})^4} = 1,\; (\frac{r^{10}}{v^3}) / (\frac{r^3}{v^3})^2 (\frac{r^4}{v}) (v)^4 = 1</math>
:<math>\frac{k_B}{e^2 m_e c^4} = </math> {{font color|blue|yellow| '''73 095 507 858.'''}}, <math>units = \frac{s^2}{m^2 K C^2} == \frac{(u^{-30})^2}{(u^{-13})^2 (u^{20}) (u^{-27})^2} = 1</math>
====== m<sub>P</sub>, t<sub>p</sub>, ε<sub>0</sub> ======
These 3 constants, Planck mass, Planck time and the vacuum permittivity have no Omega term.
:<math>\frac{M^4 (\epsilon_0^*)}{T} = (1) (\frac{2^9 \pi^3}{\alpha}) / (\pi) = \frac{2^9 \pi^2}{\alpha} = </math> {{font color|green|yellow| '''36.875'''}}, <math>\frac{(u^{15})^4 (u^{-90})}{(u^{-30})} = 1,\; (\frac{r^4}{v})^4 (\frac{1}{r^7 v^2}) / (\frac{r^9}{v^6}) = 1</math>
:<math>\frac{m_p^4 (\epsilon_0)}{t_p} = </math> {{font color|green|yellow| '''36.850'''}}, <math>units = \frac{kg^4}{s} \frac{s^4 A^2}{m^3 kg} = \frac{kg^3 A^2 s^3}{m^3} == \frac{(u^{15})^3 (u^{3})^2 (u^{-30})^3}{(u^{-13})^3} = 1</math>
====== G, h, c, e, m<sub>e</sub>, K<sub>B</sub> ======
:<math>\frac{(h^*) (c^*)^2 (e^*) (m_e^*)}{(G^*)^2 (k_B^*)} = (m_e^*) (\frac{2^{11} \pi^3}{\alpha^2}) = </math> {{font color|red|yellow| '''0.1415... 10<sup>-21</sup>'''}}, <math>\frac{ (u^{19}) (u^{17})^2 (u^{-27}) (u^{15}) }{ (u^{6})^2 (u^{29}) } = 1,\; (\frac{r^{13}}{v^5}) v^2 (\frac{r^{3}}{v^3})(\frac{r^{4}}{v^1}) / (\frac{r^5}{v^2})^2 (\frac{r^{10}}{v^3}) = 1</math>
:<math>\frac{h c^2 e m_e}{G^2 k_B} = </math> {{font color|red|yellow| '''0.1413... 10<sup>-21</sup>'''}}, <math>units = \frac{kg^3 s^3 C K}{m^4} == \frac{(u^{15})^3 (u^{-30})^3 (u^{-27}) (u^{20}) }{(u^{-13})^4} = 1</math>
====== α ======
:<math>\frac{2 (h^*)}{(\mu_0^*) (e^*)^2 (c^*)} = 2({2^3 \pi^4 \Omega^4})/(\frac{\alpha}{2^{11} \pi^5 \Omega^4})(\frac{2^{7} \pi^4 \Omega^3}{\alpha})^2(2 \pi \Omega^2) = \color{blue}\alpha \color{black},\; \frac{u^{19}}{u^{56} (u^{-27})^2 u^{17}} = 1,\; (\frac{r^{13}}{v^5})(\frac{1}{r^7})(\frac{v^6}{r^6})(\frac{1}{v}) = 1</math>
Note: The above will apply to any combinations of constants (alien or terrestrial) where '''scalars = 1'''.
===== SI Planck unit scalars =====
:<math>M = m_P = (1)k;\; k = m_P = .217\;672\;817\;58... \;10^{-7},\; u^{15}\; (kg)</math>
:<math>T = t_p = {\pi}t;\; t = \frac{t_p}{\pi} = .171\;585\;512\;84... 10^{-43},\; u^{-30}\; (s)</math>
:<math>L = l_p = {2\pi^2\Omega^2}l;\; l = \frac{l_p}{2\pi^2\Omega^2} = .203\;220\;869\;48... 10^{-36},\; u^{-13}\; (m)</math>
:<math>V = c = {2\pi\Omega^2}v;\; v = \frac{c}{2\pi\Omega^2} = 11\;843\;707.905... ,\; u^{17}\; (m/s)</math>
:<math>A = e/t_p = (\frac{2^7 \pi^3 \Omega^3}{\alpha})a = .126\;918\;588\;59... 10^{23},\; u^{3}\; (A)</math>
====== MT to LPVA ======
In this example LPVA are derived from MT. The formulas for MT;
:<math>M = (1)k,\; unit = u^{15}</math>
:<math>T = (\pi) t,\; unit = u^{-30}</math>
Replacing scalars ''pvla'' with ''kt''
:<math>P = (\Omega)\;\frac{k^{12/15}}{t^{2/15}},\; unit = u^{12/15*15-2/15*(-30)=16}</math>
:<math>V = \frac{2 \pi P^2}{M} = (2 \pi \Omega^2)\; \frac{k^{9/15}}{t^{4/15}},\; unit = u^{9/15*15-4/15*(-30)=17} </math>
:<math>L = T V = (2 \pi^2 \Omega^2) \; k^{9/15} t^{11/15},\; unit = u^{9/15*15+11/15*(-30)=-13}</math>
:<math>A = \frac{2^4 V^3}{\alpha P^3} = \left(\frac{2^7 \pi^3 \Omega^3}{\alpha}\right)\; \frac{1}{k^{3/5} t^{2/5}},\; unit =
u^{9/15*(-15)+6/15*30=3} </math>
====== PV to MTLA ======
In this example MLTA are derived from PV. The formulas for PV;
:<math>P = (\Omega)p,\; unit = u^{16}</math>
:<math>V = (2\pi\Omega^2)v,\; unit = u^{17}</math>
Replacing scalars ''klta'' with ''pv''
:<math>M = \frac{2\pi P^2}{V} = (1)\frac{p^2}{v},\; unit = u^{16*2-17=15} </math>
:<math>T = (\pi) \frac{p^{9/2}}{v^6},\; unit = u^{16*9/2-17*6=-30} </math>
:<math>L = T V = (2\pi^2\Omega^2)\frac{p^{9/2}}{v^5},\; unit = u^{16*9/2-17*5=-13}</math>
:<math>A = \frac{2^4 V^3}{\alpha P^3} = (\frac{2^7 \pi^3 \Omega^3}{\alpha})\frac{v^3}{p^3},\; unit = u^{17*3-16*3=3}</math>
===== G, h, e, m<sub>e</sub>, k<sub>B</sub> =====
As geometrical objects, the physical constants (''G, h, e, m<sub>e</sub>, k<sub>B</sub>'') can also be defined using the geometrical formulas for (''c<sup>*</sup>, μ<sub>0</sub><sup>*</sup>, R<sup>*</sup>'') and solved using the numerical (mean) values for (''c, μ<sub>0</sub>, R, α''). For example;
:<math>{(h^*)}^3 = (2^3 \pi^4 \Omega^4 \frac{r^{13} u^{19}}{v^5})^3 = \frac{3^{19} \pi^{12} \Omega^{12} r^{39} u^{57}}{v^{15}},\; \theta = 57</math> ... '''and''' ...
:<math>\frac{2\pi^{10} {(\mu_0^*)}^3} {3^6 {(c^*)}^5 \alpha^{13} {(R^*)}^2} = \frac{3^{19} \pi^{12} \Omega^{12} r^{39} u^{57}}{v^{15}},\; \theta = 57</math>
{| class="wikitable"
|+Table 12. Calculated from (R, c, μ<sub>0</sub>, α) columns 2, 3, 4 vs CODATA 2014 columns 5, 6
! Constant
! Formula
! Units
! Calculated from (R, c, μ<sub>0</sub>, α)
! CODATA 2014 <ref>[http://www.codata.org/] | CODATA, The Committee on Data for Science and Technology | (2014)</ref>
! Units
|-
| [[w:Planck constant | Planck constant]]
| <math>{(h^*)}^3 = \frac{2\pi^{10} {\mu_0}^3} {3^6 {c}^5 \alpha^{13} {R}^2}</math>
| <math>\frac{kg^3}{A^6 s}</math>, θ = 57
| ''h<sup>*</sup>'' = 6.626 069 134 e-34, θ = 19
| ''h'' = 6.626 070 040(81) e-34
| <math>\frac{kg \;m^2}{s}</math>, θ = 19
|-
| [[w:Gravitational constant | Gravitational constant]]
| <math>{(G^*)}^5 = \frac{\pi^3 {\mu_0}}{2^{20} 3^6 \alpha^{11} {R}^2}</math>
| <math>\frac{kg\; m^3}{A^2 s^2}</math>, θ = 30
| ''G<sup>*</sup>'' = 6.672 497 192 29 e11, θ = 6
| ''G'' = 6.674 08(31) e-11
| <math>\frac{m^3}{kg \;s^2}</math>, θ = 6
|-
| [[w:Elementary charge | Elementary charge]]
| <math>{(e^*)}^3 = \frac{4 \pi^5}{3^3 {c}^4 \alpha^8 {R}}</math>
| <math>\frac{s^4}{A^3}</math>, θ = -81
| ''e<sup>*</sup>'' = 1.602 176 511 30 e-19, θ = -27
| ''e'' = 1.602 176 620 8(98) e-19
| <math>A s</math>, θ = -27
|-
| [[w:Boltzmann constant | Boltzmann constant]]
| <math>{(k_B^*)}^3 = \frac{\pi^5 {\mu_0}^3}{3^3 2 {c}^4 \alpha^5 {R}}</math>
| <math>\frac{kg^3}{s^2 A^6}</math>, θ = 87
| ''k<sub>B</sub><sup>*</sup>'' = 1.379 510 147 52 e-23, θ = 29
| ''k<sub>B</sub>'' = 1.380 648 52(79) e-23
| <math>\frac{kg \;m^2}{s^2 \;K}</math>, θ = 29
|-
| [[w:Electron mass | Electron mass]]
| <math>{(m_e^*)}^3 = \frac{16 \pi^{10} {R} {\mu_0}^3}{3^6 {c}^8 \alpha^7}</math>
| <math>\frac{kg^3 s^2}{m^6 A^6}</math>, θ = 45
| '' m<sub>e</sub><sup>*</sup>'' = 9.109 382 312 56 e-31, θ = 15
| ''m<sub>e</sub>'' = 9.109 383 56(11) e-31
| <math>kg</math>, θ = 15
|-
| [[w:Gyromagnetic ratio | Gyromagnetic ratio]]
| <math>({(\gamma_e^*)/2\pi})^3 = \frac{g_e^3 3^3 c^4}{2^8 \pi^8 \alpha \mu_0^3 R_\infty^2}</math>
| <math>\frac{m^3 s^2 A^6}{kg^3}</math>, θ = -126
| ''(γ<sub>e</sub><sup>*</sup>/2π)'' = 28024.953 55, θ = -42
| ''γ<sub>e</sub>/2π'' = 28024.951 64(17)
| <math>\frac{A\;s}{kg}</math>, θ = -42
|-
| [[w:Planck mass | Planck mass]]
| <math>({m_P^*})^{15} = \frac{2^{25} \pi^{13} {\mu_0}^6}{3^6 c^5 \alpha^{16} R^2}</math>
| <math>\frac{kg^6 m^3}{s^7 A^{12}}</math>, θ = 225
| ''m<sub>P</sub><sup>*</sup>'' = 0.217 672 817 580 e-7, θ = 15
| ''m<sub>P</sub>'' = 0.217 647 0(51) e-7
| <math>kg</math>, θ = 15
|-
| [[w:Planck length | Planck length]]
| <math>({l_p^*})^{15} = \frac{\pi^{22} {\mu_0}^9}{2^{35} 3^{24} \alpha^{49} c^{35} R^8}</math>
| <math>\frac{kg^9 s^{17}}{m^{18}A^{18}}</math>, θ = -195
| ''l<sub>p</sub><sup>*</sup>'' = 0.161 603 660 096 e-34, θ = -13
| ''l<sub>p</sub>'' = 0.161 622 9(38) e-34
| <math>m</math>, θ = -13
|}
==== External links ====
* [[v:electron_(mathematical) | Mathematical electron]]
* [[v:Physical_constant_(anomaly) | Physical constant anomalies]]
* [[v:Relativity_(Planck) | Programming relativity at the Planck scale]]
* [[v:Quantum_gravity_(Planck) | Programming gravity at the Planck scale]]
* [[v:Black-hole_(Planck) | Programming the cosmic microwave background at the Planck scale]]
* [[v:Sqrt_Planck_momentum | The sqrt of Planck momentum]]
* [[v:God_(programmer) | The Programmer God]]
* [[w:Simulation_hypothesis | The Simulation hypothesis]]
* [https://codingthecosmos.com/ Programming at the Planck scale using geometrical objects] -Malcolm Macleod's website
* [http://www.simulation-argument.com/ Simulation Argument] -Nick Bostrom's website
* [https://www.amazon.com/Our-Mathematical-Universe-Ultimate-Reality/dp/0307599809 Our Mathematical Universe: My Quest for the Ultimate Nature of Reality] -Max Tegmark
* [https://www.amazon.com/Programmer-God-Are-We-Simulation-ebook/dp/B0B5BC1PQK The Programmer God, an overview of the mathematical electron model] -ebook
* [https://link.springer.com/article/10.1134/S0202289308020011/ Dirac-Kerr-Newman black-hole electron] -Alexander Burinskii (article)
==== References ====
{{Reflist}}
[[Category: Physics]]
[[Category: Philosophy of science]]
9etnsh03zw8juwsoaujzzij17cd9foc
Workings of gcc and ld in plain view
0
285384
2693971
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Young1lim
21186
/* Linking Libraries */
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wikitext
text/x-wiki
=== Workings of the GNU Compiler for IA-32 ===
==== Overview ====
* Overview ([[Media:Overview.20200211.pdf |pdf]])
==== Data Processing ====
* Access ([[Media:Access.20200409.pdf |pdf]])
* Operators ([[Media:Operator.20200427.pdf |pdf]])
==== Control ====
* Conditions ([[Media:Condition.20230630.pdf |pdf]])
* Control ([[Media:Control.20220616.pdf |pdf]])
==== Function calls ====
* Procedure ([[Media:Procedure.20220412.pdf |pdf]])
* Recursion ([[Media:Recursion.20210824-2.pdf |pdf]])
==== Pointer and Aggregate Types ====
* Arrays ([[Media:Array.20211018.pdf |pdf]])
* Structures ([[Media:Structure.20220101.pdf |pdf]])
* Alignment ([[Media:Alignment.20201117.pdf |pdf]])
* Pointers ([[Media:Pointer.20201106.pdf |pdf]])
==== Integer Arithmetic ====
* Overview ([[Media:gcc.1.Overview.20240813.pdf |pdf]])
* Carry Flag ([[Media:gcc.2.Carry.20241204.pdf |pdf]])
* Overflow Flag ([[Media:gcc.3.Overflow.20241205.pdf |pdf]])
* Examples ([[Media:gcc.4.Examples.20240724.pdf |pdf]])
* Borrow ([[Media:Borrow.20241228.pdf |pdf]])
==== Floating point Arithmetic ====
</br>
=== Workings of the GNU Linker for IA-32 ===
==== Linking Libraries ====
* Static Libraries ([[Media:LIB.1A.Static.20241128.pdf |A.pdf]])
* Shared Libraries ([[Media:LIB.2A.Shared.20241231-1.pdf |A.pdf]], [[Media:LIB.2B.Shared.20250101-1.pdf |B.pdf]])
==== Dynamic Linking - Directories and Symbolic Links ====
* Shared Library Names ([[Media:DIR.1A.Names.20241230.pdf |pdf]])
* Managing Shared Libraries ([[Media:DIR.2A.Manage.20241230.pdf |pdf]])
==== Dynamic Loading - API Functions ====
* DL API ([[Media:API.1A.Functions.20241230.pdf |pdf]])
==== Library Search Path ====
* Using -L and -l only ([[Media:Link.4A.LibSearch-withLl.20240807.pdf |A.pdf]], [[Media:Link.4B.LibSearch-withLl.20240705.pdf |B.pdf]])
* Using RPATH ([[Media:Link.5A.LibSearch-RPATH.20241228.pdf |A.pdf]], [[Media:Link.5B.LibSearch-RPATH.20240705.pdf |B.pdf]])
==== Linking Process ====
* Object Files ([[Media:Link.3.A.Object.20190121.pdf |A.pdf]], [[Media:Link.3.B.Object.20190405.pdf |B.pdf]])
* Symbols ([[Media:Link.4.A.Symbol.20190312.pdf |A.pdf]], [[Media:Link.4.B.Symbol.20190312.pdf |B.pdf]])
* Relocation ([[Media:Link.5.A.Relocation.20190320.pdf |A.pdf]], [[Media:Link.5.B.Relocation.20190322.pdf |B.pdf]])
* Loading ([[Media:Link.6.A.Loading.20190501.pdf |A.pdf]], [[Media:Link.6.B.Loading.20190126.pdf |B.pdf]])
* Static Linking ([[Media:Link.7.A.StaticLink.20190122.pdf |A.pdf]], [[Media:Link.7.B.StaticLink.20190128.pdf |B.pdf]], [[Media:LNK.5C.StaticLinking.20241128.pdf |C.pdf]])
* Dynamic Linking ([[Media:Link.8.A.DynamicLink.20190207.pdf |A.pdf]], [[Media:Link.8.B.DynamicLink.20190209.pdf |B.pdf]], [[Media:LNK.6C.DynamicLinking.20241128.pdf |C.pdf]])
* Position Independent Code ([[Media:Link.9.A.PIC.20190304.pdf |A.pdf]], [[Media:Link.9.B.PIC.20190309.pdf |B.pdf]])
==== Example I ====
* Vector addition ([[Media:Eg1.1A.Vector.20190121.pdf |A.pdf]], [[Media:Eg1.1B.Vector.20190121.pdf |B.pdf]])
* Swapping array elements ([[Media:Eg1.2A.Swap.20190302.pdf |A.pdf]], [[Media:Eg1.2B.Swap.20190121.pdf |B.pdf]])
* Nested functions ([[Media:Eg1.3A.Nest.20190121.pdf |A.pdf]], [[Media:Eg1.3B.Nest.20190121.pdf |B.pdf]])
==== Examples II ====
* analysis of static linking ([[Media:Ex1.A.StaticLinkEx.20190121.pdf |A.pdf]], [[Media:Ex2.B.StaticLinkEx.20190121.pdf |B.pdf]])
* analysis of dynamic linking ([[Media:Ex2.A.DynamicLinkEx.20190121.pdf |A.pdf]])
* analysis of PIC ([[Media:Ex3.A.PICEx.20190121.pdf |A.pdf]])
</br>
go to [ [[C programming in plain view]] ]
[[Category:C programming language]]
fwcv8xhbknb56gib1cr1ilbqy9619cz
2694055
2693971
2025-01-02T00:22:54Z
Young1lim
21186
/* Linking Libraries */
2694055
wikitext
text/x-wiki
=== Workings of the GNU Compiler for IA-32 ===
==== Overview ====
* Overview ([[Media:Overview.20200211.pdf |pdf]])
==== Data Processing ====
* Access ([[Media:Access.20200409.pdf |pdf]])
* Operators ([[Media:Operator.20200427.pdf |pdf]])
==== Control ====
* Conditions ([[Media:Condition.20230630.pdf |pdf]])
* Control ([[Media:Control.20220616.pdf |pdf]])
==== Function calls ====
* Procedure ([[Media:Procedure.20220412.pdf |pdf]])
* Recursion ([[Media:Recursion.20210824-2.pdf |pdf]])
==== Pointer and Aggregate Types ====
* Arrays ([[Media:Array.20211018.pdf |pdf]])
* Structures ([[Media:Structure.20220101.pdf |pdf]])
* Alignment ([[Media:Alignment.20201117.pdf |pdf]])
* Pointers ([[Media:Pointer.20201106.pdf |pdf]])
==== Integer Arithmetic ====
* Overview ([[Media:gcc.1.Overview.20240813.pdf |pdf]])
* Carry Flag ([[Media:gcc.2.Carry.20241204.pdf |pdf]])
* Overflow Flag ([[Media:gcc.3.Overflow.20241205.pdf |pdf]])
* Examples ([[Media:gcc.4.Examples.20240724.pdf |pdf]])
* Borrow ([[Media:Borrow.20241228.pdf |pdf]])
==== Floating point Arithmetic ====
</br>
=== Workings of the GNU Linker for IA-32 ===
==== Linking Libraries ====
* Static Libraries ([[Media:LIB.1A.Static.20241128.pdf |A.pdf]])
* Shared Libraries ([[Media:LIB.2A.Shared.20241231-1.pdf |A.pdf]], [[Media:LIB.2B.Shared.20250102.pdf |B.pdf]])
==== Dynamic Linking - Directories and Symbolic Links ====
* Shared Library Names ([[Media:DIR.1A.Names.20241230.pdf |pdf]])
* Managing Shared Libraries ([[Media:DIR.2A.Manage.20241230.pdf |pdf]])
==== Dynamic Loading - API Functions ====
* DL API ([[Media:API.1A.Functions.20241230.pdf |pdf]])
==== Library Search Path ====
* Using -L and -l only ([[Media:Link.4A.LibSearch-withLl.20240807.pdf |A.pdf]], [[Media:Link.4B.LibSearch-withLl.20240705.pdf |B.pdf]])
* Using RPATH ([[Media:Link.5A.LibSearch-RPATH.20241228.pdf |A.pdf]], [[Media:Link.5B.LibSearch-RPATH.20240705.pdf |B.pdf]])
==== Linking Process ====
* Object Files ([[Media:Link.3.A.Object.20190121.pdf |A.pdf]], [[Media:Link.3.B.Object.20190405.pdf |B.pdf]])
* Symbols ([[Media:Link.4.A.Symbol.20190312.pdf |A.pdf]], [[Media:Link.4.B.Symbol.20190312.pdf |B.pdf]])
* Relocation ([[Media:Link.5.A.Relocation.20190320.pdf |A.pdf]], [[Media:Link.5.B.Relocation.20190322.pdf |B.pdf]])
* Loading ([[Media:Link.6.A.Loading.20190501.pdf |A.pdf]], [[Media:Link.6.B.Loading.20190126.pdf |B.pdf]])
* Static Linking ([[Media:Link.7.A.StaticLink.20190122.pdf |A.pdf]], [[Media:Link.7.B.StaticLink.20190128.pdf |B.pdf]], [[Media:LNK.5C.StaticLinking.20241128.pdf |C.pdf]])
* Dynamic Linking ([[Media:Link.8.A.DynamicLink.20190207.pdf |A.pdf]], [[Media:Link.8.B.DynamicLink.20190209.pdf |B.pdf]], [[Media:LNK.6C.DynamicLinking.20241128.pdf |C.pdf]])
* Position Independent Code ([[Media:Link.9.A.PIC.20190304.pdf |A.pdf]], [[Media:Link.9.B.PIC.20190309.pdf |B.pdf]])
==== Example I ====
* Vector addition ([[Media:Eg1.1A.Vector.20190121.pdf |A.pdf]], [[Media:Eg1.1B.Vector.20190121.pdf |B.pdf]])
* Swapping array elements ([[Media:Eg1.2A.Swap.20190302.pdf |A.pdf]], [[Media:Eg1.2B.Swap.20190121.pdf |B.pdf]])
* Nested functions ([[Media:Eg1.3A.Nest.20190121.pdf |A.pdf]], [[Media:Eg1.3B.Nest.20190121.pdf |B.pdf]])
==== Examples II ====
* analysis of static linking ([[Media:Ex1.A.StaticLinkEx.20190121.pdf |A.pdf]], [[Media:Ex2.B.StaticLinkEx.20190121.pdf |B.pdf]])
* analysis of dynamic linking ([[Media:Ex2.A.DynamicLinkEx.20190121.pdf |A.pdf]])
* analysis of PIC ([[Media:Ex3.A.PICEx.20190121.pdf |A.pdf]])
</br>
go to [ [[C programming in plain view]] ]
[[Category:C programming language]]
eiewkdoh87pgah7puvtb2qq3xsywdbh
Social Victorians/Terminology
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Scogdill
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Especially with respect to fashion, the newspapers at the end of the 19th century in the UK often used specialized terminology. The definitions on this page are to provide a sense of what someone in the late 19th century might have meant by the term rather than a definition of what we might mean by it today. In the absence of a specialized glossary from the end of the 19th century in the U.K., we use the ''Oxford English Dictionary'' because the senses of a word are illustrated with examples that have dates so we can be sure that the senses we pick are appropriate for when they are used in the quotations we have.
We also sometimes use the French ''Wikipédia'' to define a word because many technical terms of fashion were borrowings from the French. Also, often the French ''Wikipédia'' provides historical context for the uses of a word similar to the way the OED does.
== Articles or Parts of Clothing: Non-gender-specific ==
=== Mantle, Cloak, Cape ===
In 19th-century newspaper accounts, these terms are sometimes used without precision as synonyms. These are all outer garments.
'''Mantle'''
A mantle — often a long outer garment — might have elements like a train, sleeves, collars, revers, fur, and a cape. A late-19th-century writer making a distinction between a mantle and a cloak might use ''mantle'' if the garment is more voluminous.
'''Cloak'''
'''Cape'''
=== Peplum ===
According to the French ''Wiktionnaire'', a peplum is a "Short skirt or flared flounce layered at the waist of a jacket, blouse or dress" [translation by Google Translate].<ref>{{Cite journal|date=2021-07-02|title=péplum|url=https://fr.wiktionary.org/w/index.php?title=p%C3%A9plum&oldid=29547727|journal=Wiktionnaire, le dictionnaire libre|language=fr}} https://fr.wiktionary.org/wiki/p%C3%A9plum.</ref> The ''Oxford English Dictionary'' has a fuller definition, although, it focuses on women's clothing because the sense is written for the present day:<blockquote>''Fashion''. ... a kind of overskirt resembling the ancient peplos (''obsolete''). Hence (now usually) in modern use: a short flared, gathered, or pleated strip of fabric attached at the waist of a woman's jacket, dress, or blouse to create a hanging frill or flounce.<ref name=":5">“peplum, n.”. ''Oxford English Dictionary'', Oxford University Press, September 2023, <https://doi.org/10.1093/OED/1832614702>.</ref></blockquote>Men haven't worn peplums since the 18th century, except when wearing costumes based on historical portraits. The ''Daily News'' reported in 1896 that peplums had been revived as a fashion item for women.<ref name=":5" />
=== Revers ===
According to the ''Oxford English Dictionary'', ''revers'' are the "edge[s] of a garment turned back to reveal the undersurface (often at the lapel or cuff) (chiefly in ''plural''); the material covering such an edge."<ref>"revers, n." ''OED Online'', Oxford University Press, March 2023, www.oed.com/view/Entry/164777. Accessed 17 April 2023.</ref> The term is French and was used this way in the 19th century (according to the ''Wiktionnaire'').<ref>{{Cite journal|date=2023-03-07|title=revers|url=https://fr.wiktionary.org/w/index.php?title=revers&oldid=31706560|journal=Wiktionnaire|language=fr}} https://fr.wiktionary.org/wiki/revers.</ref>
== Articles or Parts of Clothing: Men's ==
[[Social Victorians/Terminology#Military|Men's military uniforms]] are discussed below.
=== À la Romaine ===
[[File:Johann Baptist Straub - Mars um 1772-1.jpg|thumb|left|alt=Old and damaged marble statue of a Roman god of war with flowing cloak, big helmet with a plume on top, and armor|Johann Baptist Straub's 1772 ''à la romaine'' ''Mars'']]
A few people who attended the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's fancy-dress ball in 1897]] personated Roman gods or people. They were dressed not as Romans, however, but ''à la romaine'', which was a standardized style of depicting Roman figures that was used in paintings, sculpture and the theatre for historical dress from the 17th until the 20th century. The codification of the style was developed in France in the 17th century for theatre and ballet, when it became popular for masked balls.
Women as well as men could be dressed ''à la romaine'', but much sculpture, portraiture and theatre offered opportunities for men to dress in Roman style — with armor and helmets — and so it was most common for men. In large part because of the codification of the style as well as the painting and sculpture, the style persisted and remained influential into the 20th century and can be found in museums and galleries and on monuments.
For example, Johann Baptist Straub's 1772 statue of Mars (left), now in the Bayerisches Nationalmuseum, Munich, missing part of an arm, shows Mars ''à la romaine''. In London, an early 17th-century example of a figure of Mars ''à la romaine'', with a helmet, '''was''' "at the foot of the Buckingham tomb in Henry VII's Chapel at Westminster Abbey."<ref>Webb, Geoffrey. “Notes on Hubert Le Sueur-II.” ''The Burlington Magazine for Connoisseurs'' 52, no. 299 (1928): 81–89. http://www.jstor.org/stable/863535.</ref>{{rp|81, Col. 2c}}
=== Cavalier ===
[[File:Sir-Anthony-van-Dyck-Lord-John-Stuart-and-His-Brother-Lord-Bernard-Stuart.jpg|thumb|alt=Old painting of 2 men flamboyantly and stylishly dressed in colorful silk, with white lace, high-heeled boots and long hair|Van Dyck's c. 1638 painting of cavaliers Lord John Stuart and his brother Lord Bernard Stuart]]
As a signifier in the form of clothing of a royalist political and social ideology begun in France in the early 17th century, the cavalier established France as the leader in fashion and taste. Adopted by [[Social Victorians/Terminology#Military|wealthy royalist British military officers]] during the time of the Restoration, the style signified a political and social position, both because of the loyalty to Charles I and II as well the wealth required to achieve the cavalier look. The style spread beyond the political, however, to become associated generally with dress as well as a style of poetry.<ref>{{Cite journal|date=2023-04-25|title=Cavalier poet|url=https://en.wikipedia.org/w/index.php?title=Cavalier_poet&oldid=1151690299|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Cavalier_poet.</ref>
Van Dyck's 1638 painting of two brothers (right) emphasizes the cavalier style of dress.
=== Coats ===
==== Doublet ====
* In the 19th-century newspaper accounts we have seen that use this word, doublet seems always to refer to a garment worn by a man, but historically women may have worn doublets. In fact, a doublet worn by Queen Elizabeth I exists and '''is somewhere'''.
* Technically doublets were long sleeved, although we cannot be certain what this or that Victorian tailor would have done for a costume. For example, the [[Social Victorians/People/Spencer Compton Cavendish#Costume at the Duchess of Devonshire's 2 July 1897 Fancy-dress Ball|Duke of Devonshire's costume as Charles V]] shows long sleeves that may be part of the surcoat but should be the long sleeves of the doublet.
==== Pourpoint ====
A padded doublet worn under armor to protect the warrior from the metal chafing. A pourpoint could also be worn without the armor.
==== Surcoat ====
Sometimes just called ''coat''.
[[File:Oscar Wilde by Sarony 1882 18.jpg|thumb|alt=Old photograph of a young man wearing a velvet jacket, knee breeches, silk hose and shiny pointed shoes with bows, seated on a sofa and leaning on his left hand and holding a book in his right| Oscar Wilde, 1882, by Napoleon Sarony]]
=== Hose, Stockings and Tights ===
Newspaper accounts from the late 19th century of men's clothing use the term ''hose'' for what we might call stockings or tights.
In fact, the terminology is specific. ''Stockings'' is the more general term and could refer to hose or tights. With knee breeches men wore hose, which ended above the knee, and women wore hose under their dresses.
The ''Oxford English Dictionary'' defines tights as "Tight-fitting breeches, worn by men in the 18th and early 19th centuries, and still forming part of court-dress."<ref>“Tights, N.” ''Oxford English Dictionary'', Oxford UP, July 2023, https://doi.org/10.1093/OED/2693287467.</ref> By 1897, the term was in use for women's stockings, which may have come up only to the knee. Tights were also worn by dancers and acrobats. This general sense of ''tights'' does not assume that they were knitted.
''Clocking'' is decorative embroidery on hose, usually, at the ankles on either the inside or the outside of the leg. It started at the ankle and went up the leg, sometimes as far as the knee. On women's hose, the clocking could be quite colorful and elaborate, while the clocking on men's hose was more inconspicuous.
In many photographs men's hose are wrinkled, especially at the ankles and the knees, because they were shaped from woven fabric. Silk hose were knitted instead of woven, which gave them elasticity and reduced the wrinkling.
The famous Sarony carte de visite photograph of Oscar Wilde (right) shows him in 1882 wearing knee breeches and silk hose, which are shiny and quite smoothly fitted although they show a few wrinkles at the ankles and knees. In the portraits of people in costume at the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's 1897 fancy-dress ball]], the men's hose are sometimes quite smooth, which means they were made of knitted silk and may have been smoothed for the portrait.
In painted portraits the hose are almost always depicted as smooth, part of the artist's improvement of the appearance of the subject.
=== Shoes and Boots ===
== Articles or Parts of Clothing: Women's ==
=== '''Chérusque''' ===
According to the French ''Wikipedia'', ''chérusque'' is a 19th-century term for the kind of standing collar like the ones worn by ladies in the Renaissance.<ref>{{Cite journal|date=2021-06-26|title=Collerette (costume)|url=https://fr.wikipedia.org/w/index.php?title=Collerette_(costume)&oldid=184136746|journal=Wikipédia|language=fr}} https://fr.wikipedia.org/wiki/Collerette_(costume)#Au+xixe+siècle+:+la+Chérusque.</ref>
=== Corsage ===
According to the ''Oxford English Dictionary'', the corsage is the "'body' of a woman's dress; a bodice."<ref>"corsage, n." ''OED Online'', Oxford University Press, December 2022, www.oed.com/view/Entry/42056. Accessed 7 February 2023.</ref> This sense is well documented in the ''OED'' for the mid and late 19th-century, used this way in fiction as well as in a publication like ''Godey's Lady's Book'', which would be expected to use appropriate terminology associated with fashion and dress making.
The sense of "a bouquet worn on the bodice" is, according to the ''OED'', American.
=== Décolletage ===
=== Girdle ===
=== Mancheron ===
According to the ''Oxford English Dictionary'', a ''mancheron'' is a "historical" word for "A piece of trimming on the upper part of a sleeve on a woman's dress."<ref>"mancheron, n." ''OED Online'', Oxford University Press, March 2023, www.oed.com/view/Entry/113251. Accessed 17 April 2023.</ref> At the present, in French, a ''mancheron'' is a cap sleeve "cut directly on the bodice."<ref>{{Cite journal|date=2022-11-28|title=Manche (vêtement)|url=https://fr.wikipedia.org/w/index.php?title=Manche_(v%C3%AAtement)&oldid=199054843|journal=Wikipédia|language=fr}} https://fr.wikipedia.org/wiki/Manche_(v%C3%AAtement).</ref>
=== Petticoat ===
According to the ''O.E.D.'', a petticoat is a <blockquote>skirt, as distinguished from a bodice, worn either externally or showing beneath a dress as part of the costume (often trimmed or ornamented); an outer skirt; a decorative underskirt. Frequently in ''plural'': a woman's or girl's upper skirts and underskirts collectively. Now ''archaic'' or ''historical''.<ref>“petticoat, n., sense 2.b”. ''Oxford English Dictionary'', Oxford University Press, September 2023, <https://doi.org/10.1093/OED/1021034245></ref> </blockquote>This sense is, according to the ''O.E.D.'', "The usual sense between the 17th and 19th centuries." However, while petticoats belong in both outer- and undergarments — that is, meant to be seen or hidden, like underwear — they were always under another garment, for example, underneath an open overskirt. The primary sense seems to have shifted through the 19th century so that, by the end, petticoats were underwear and the term ''underskirt'' was used to describe what showed under an open overskirt.
=== Stomacher ===
According to the ''O.E.D.'', a stomacher is "An ornamental covering for the chest (often covered with jewels) worn by women under the lacing of the bodice,"<ref>“stomacher, n.¹, sense 3.a”. ''Oxford English Dictionary'', Oxford University Press, September 2023, <https://doi.org/10.1093/OED/1169498955></ref> although by the end of the 19th century, the bodice did not often have visible laces. Some stomachers were so decorated that they were thought of as part of the jewelry.
=== Train ===
A train is
The Length of the Train
'''For the monarch [or a royal?]'''
According to Debrett's,<blockquote>A peeress's coronation robe is a long-trained crimson velvet mantle, edged with miniver pure, with a miniver pure cape. The length of the train varies with the rank of the wearer:
* Duchess: for rows of ermine; train to be six feet
* Marchioness: three and a half rows of ermine; train to be three and three-quarters feet
* Countess: three rows of ermine; train to be three and a half feet
* Viscountess: two and a half rows of ermine; train to be three and a quarter feet
* Baroness: two rows of ermine; train to be three feet<ref name=":2">{{Cite web|url=https://debretts.com/royal-family/dress-codes/|title=Dress Codes|website=debretts.com|language=en-US|access-date=2023-07-27}} https://debretts.com/royal-family/dress-codes/.</ref>
</blockquote>The pattern on the coronet worn was also quite specific, similar but not exactly the same for peers and peeresses. Debrett's also distinguishes between coronets and tiaras, which were classified more like jewelry, which was regulated only in very general terms.
Peeresses put on their coronets after the Queen or Queen Consort has been crowned. ['''peers?''']
=== Foundation Garments ===
Unlike undergarments, Victorian women's foundation garments created the distinctive silhouette. Victorian undergarments included the chemise, the bloomers, the corset cover — articles that are not structural.
The corset was an important element of the understructure of foundation garments — hoops, bustles, petticoats and so on — but it has never been the only important element.
=== Corset ===
[[File:Corset - MET 1972.209.49a, b.jpg|thumb|alt=Photograph of an old silk corset on a mannequin, showing the closure down the front, similar to a button, and channels in the fabric for the boning. It is wider at the top and bottom, creating smooth curves from the bust to the compressed waist to the hips, with a long point below the waist in front.|French 1890s corset, now in the Metropolitan Museum of Art, NYC]]
The understructure of Victorian women's clothing is what makes the costumes worn by the women at the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's 1897 fancy-dress ball]] so distinctly Victorian in appearance. An example of a corset that has the kind of structure often worn by fashionably dressed women in 1897 is the one at right.
This corset exaggerated the shape of the women's bodies and made possible a bodice that looked and was fitted in the way that is so distinctive of the time — very controlled and smooth. And, as a structural element, this foundation garment carried the weight of all those layers and all that fabric and decoration on the gowns, trains and mantles. (The trains and mantles could be attached directly to the corset itself.)
* This foundation emphasizes the waist and the bust in particular, in part because of the contrast between the very small waist and the rounded fullness of the bust and hips.
* The idealized waist is defined by its small span and the sexualizing point at the center-bottom of the bodice, which directs the eye downwards. Interestingly, the pointed waistline worn by Elizabethan men has become level in the Victorian age. Highly fashionable Victorian women wearing the traditional style, however, had extremely pointed waists.
* The busk (a kind of boning in the front of a corset that is less flexible than the rest) smoothed the bodice, flattened the abdomen and prevented the point on the bodice from curling up.
* The sharp definition of the waist was caused by
** length of the corset (especially on the sides)
** the stiffness of the boning
** the layers of fabric
** the lacing (especially if the woman used tightlacing)
** the over-all shape, which was so much wider at the top and the bottom
** the contrast between the waist and the wider top and bottom
* The late-19th-century corset was long, ending below the waist even on the sides and back.
* The boning and the top edge of the late 19th-century fashion corset pushed up the bust, rounding (rather than flattening, as in earlier styles) the breasts, drawing attention to their exposed curves and creating cleavage.
* The exaggerated bust was larger than the hips, whenever possible, an impression reinforced by the A-line of the skirt and the inverted Vs in the decorative trim near the waist and on the skirt.
* This corset made the bodice very smooth with a very precise fit, that had no wrinkles, folds or loose drapery. The bodice was also trimmed or decorated, but the base was always a smooth bodice. More formal gowns would still have the fitted bodice and more elaborate trim made from lace, embroidery, appliqué, beading and possibly even jewels.
The advantages and disadvantages of corseting and especially tight lacing were the subject of thousands of articles and opinions in the periodical press for a great part of the century, but the fetishistic and politicized tight lacing was practiced by very few women. And no single approach to corsetry was practiced by all women all the time. Most of the women at the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's 1897 ball]] were not tightly laced, but the progressive style does not dominate either, even though all the costumes are technically historical dress. Part of what gives most of the costumes their distinctive 19th-century "look" is the more traditional corset beneath them. Even though this highly fashionable look was widely present in the historical costumes at the ball, some women's waists were obviously very small and others were hardly '''emphasized''' at all. Women's waists are never mentioned in the newspaper coverage of the ball — or, indeed, of any of the social events attended by the network at the ball — so it is only in photographs that we can see the effects of how they used their corsets.
=== Hoops ===
'''This section is under construction right now'''.
''Hoops'' is a mid-19th-century term for a cage-like structure worn under a skirt to hold it away from the body. '''Striking''' for how long they lasted and '''the ways''' they evolved, hoops were the foundation undergarment for the bottom half of a woman's body, for a skirt and petticoat.
Women wore this cage-like structure from the '''15th century''' through the late 19th century. The 16th-century Katherine of Aragon is credited with making it fashionable outside Spain.
The cage caused the silhouette of skirts to change shape over time and enabled the extreme distortions of 17th-and-18th-century panniers and the late 19th-century bustle. Early hoops circled the body in a bell, cone or drum shape, then were moved to the sides with panniers, then ballooned around the body like the top half of a sphere, and finally were pulled to the rear with a bustle.
That is, the distorted shapes of high fashion were made possible by hoops. High fashion demanded these shapes, which disguised women's bodies, especially below the waist, while [[Social Victorians/Terminology#Corsets|corsets]] did their work above it.
Besides the shape, the structure used to construct hoops evolved — from cane and wood to whalebone, then steel '''bands''' and wire. Add fabric structural stuff: tabs, wires inserted into casings in a linen, muslin or, later, crinoline underskirt
[[File:Pedro García de Benabarre St John Retable Detail.jpg|thumb|alt=Old oil painting of a woman wearing a dress from the 1400s holding the decapitated head of a man with a halo before a table of people at a dinner party|Pedro García de Benabarre, Detail from St. John Altarpiece, c. 1470, Showing Visible Hoops]]
[[File:Alonso Sánchez Coello 011.jpg|thumb|alt=Old painting of a princess wearing a richly jeweled outfit|Infanta Isabel Clara Eugenia Wearing a Vertugado, or Spanish Farthingale]]
==== 15th Century ====
Hoops first appeared in Spain in the 15th century and influenced European fashion for '''many years'''.
A detail (right) from Pedro García de Benabarre's c. 1470 larger altarpiece painting shows women wearing a style of hoops that predates the farthingale but marks the beginning point of the development of that fashion. Salome (holding John the Baptist's head) is wearing a dress with what looks like visible wooden hoops attached to the outside of the skirt, which also appears to have padding at the hips underneath it.
De Benabarre was "active in Aragon and in Catalonia, between 1445–1496,"<ref>{{Cite web|url=https://www.mfab.hu/artworks/10528/|title=Saint Peter|website=Museum of Fine Arts, Budapest|language=en-US|access-date=2024-12-11}} https://www.mfab.hu/artworks/10528/.</ref> so perhaps he saw the styles worn by people like Katharine of Aragon.
==== 16th Century ====
Styles in personal adornment and architectural decoration: The "Golden Age" in '''England''', the Elizabethan Age.
[[File:Queen Elizabeth I ('The Ditchley portrait') by Marcus Gheeraerts the YoungerFXD.jpg|thumb|alt=Old oil painting of a queen in a white dress with shoulders and hips exaggerated by her dress|Queen Elizabeth I in a French Cartwheel Farthingale]]
In the 16th century, the garment we call ''hoops'' was called a farthingale.<blockquote>''"FARTHINGALE: Renaissance (1450-1550 C.E. to Elizabethan (1550-1625 C.E.). Linen underskirt with '''wire supports''' which, when shaped, produced a variety of dome, bell, and oblong shapes."<ref name=":7" />''{{rp|105}} ['''our emphasis''']</blockquote>''Vertugadin'' is a French term for ''farthingale'' — "un élément essentiel de la mode Tudor en Angleterre [an essential element of Tudor fashion in England]."<ref name=":0">{{Cite journal|date=2022-03-12|title=Vertugadin|url=https://fr.wikipedia.org/w/index.php?title=Vertugadin&oldid=191825729|journal=Wikipédia|language=fr}} https://fr.wikipedia.org/wiki/Vertugadin.</ref> ''Farthingale'' is the term in English; in French, it's ''vertugadin'', and in Spanish ''vertugado''. The hoops in the Pedro García de Benabarre painting (above right) predate what would technically be a vertugado.<p>
Blanche Payne says,<blockquote>Katherine of Aragon is reputed to have introduced the Spanish farthingale ... into England early in the century. The result was to convert the columnar skirt of the fifteenth century into the cone shape of the sixteenth. ...<p>
Spanish influence had introduced the hoop-supported skirt, smooth in contour, '''which was quite generally worn'''.<ref name=":11" />{{rp|291}} ['''our emphasis''']</blockquote>
In fact, "The Spanish princess Catherine of Aragon brought the fashion to England for her marriage to Prince Arthur, eldest son of Henry VII in 1501 [La princesse espagnole Catherine d'Aragon amena la mode en Angleterre pour son mariage avec le prince Arthur, fils aîné d'Henri VII en 1501]."<ref name=":0" /> Catherine of Aragon, of course, married Henry VIII after Arthur's death.
The vertugado was "quite generally worn" among the ruling and culturally elite classes in Spain, and not by working-class women, which was enforced by sumptuary laws.
By the end of the 16th century the French and Spanish farthingales were not identical.
The Spanish vertugado shaped the skirt into an A-line with a graduated series of hoops sewn to an undergarment. Alonso Sánchez Coello's c. 1584<ref name=":11" />{{rp|316}} portrait (right) shows infanta Isabel Clara Eugenia wearing a vertugado, with its "typically Spanish smooth cone-shaped contour."<ref name=":11" />{{rp|315–316}}
The French vertugadin was a flattish "cartwheel" '''in which a''' platter of hoops worn below the waist and above the hips held the skirt out more or less horizontally. Once past the vertugadin, the skirt then fell straight to the floor, shaping it into a kind of drum. Marcus Gheeraerts the Younger's portrait (right) of Queen Elizabeth I shows an English queen wearing a French drum-shaped farthingale. The skirt over a cartwheel farthingale did not touch the floor in front, so the dress flowed and the women's feet would show as they walked. Interestingly, shoes often appear in portraits of women wearing the vertugadin, as Elizabeth's do in Gheeraerts' image.
The shoes do not show in the portraits of women wearing the Spanish vertugado. The round hoops stayed in place in front, giving their feet enough room to take steps.
By the end of the 16th century France had become the arbiter of fashion for the western world, which it still is.
==== 17th Century ====
The 17th century encompassed movements in the arts and architecture as well as styling for personal adornment and the decoration of the home. The Cavalier, for example, was a style of men's dress but at base it was a political movement in support of England's King Charles II. The Baroque style centered on the courts of Europe. The speed with which fashion trends changed began to accelerate in the 17th century, making fashion more expensive and making it more difficult to keep up.
Hoops disappeared from high fashion for '''most of the''' century, reappearing flattened and widened as panniers and shifted to the back as bustles. Without structures like hoops, dresses draped loosely to the floor, shaped to some degree by padding, like bum rolls. And what had been starched and stiff in the 16th century became looser and flatter in the 17th.
People associate bustles with late-19th-century styles, but in fact the bustle existed in the 17th century, sometimes as padding rather than a structural cage. Panniers are associated with 18th-century styles, but they first appeared in the 17th century as well.
Payne says, "The bustle was a continuation of the 1690 mode."<ref name=":11" />{{rp|411}}
Generally, panniers were a kind of undergarment worn in the 17th and 18th centuries. Their design '''evolved during the century'''. Made of hoops of wood, they were "baskets" or cages worn on either side of the waist to broaden the skirts to the sides.
==== 18th Century ====
Styles in personal adornment and architectural decoration: Rococo, post French Revolution, Empire
Blanche Payne outlines the evolution of hoops, and thus the shape of the skirt, in the 18th century:<blockquote>SKIRT FASHIONS. Since skirts experienced the greatest alterations, a brief summary of the successive silhouettes should help to place individual costumes in their proper niches. Six basic forms appeared during the century, in the following order:
# The bustle was a continuation of the 1690 mode.
# The bell or dome shape resulted from the reintroduction of hoops; in England by 1710, in France by 1720.
# The ellipse, the second phase of the hoop skirt, was achieved by broadening the support from side to side and compressing it from front to back. It had a long run of popularity, from 1740 to 1770, the extreme width being retained in court costumes. In France it persisted until the revolution, except that skirts were allowed to curve outward in [the] back again. English court costume [411/413] followed this fashion well into the nineteenth century.
# The dairy maid, or polonaise, style could be achieved either by pulling the lower part of the overskirt through its own pocket holes, thus creating a bouffant effect, or by planned control of the overskirt, through the cut or by means of draw cords, ribbons, or loops and buttons, which were used to form the three great ‘poufs’ known as the polonaise .... These diversions appeared in the late sixties and became prevalent in the seventies. They were much like the familiar styles of our own [American] Revolutionary War period.
# The return of the bustle in the 1780s.
# The tubular form, drawn from classic art, in the 1790s.<ref name=":11" />{{rp|411, 413}}
</blockquote>While we think of the bustle as a 19th-century look, it can be found in the 18th century, as Payne says.
By the 18th century, the farthingale was called hoops, which were at this point made of wood.
The 1760–70 French panniers below are "a rare surviving example"<ref name=":15">{{Citation|title=Panniers|url=https://www.metmuseum.org/art/collection/search/139668|date=1760–70|accessdate=2025-01-01}}. The Costume Institute, Metropolitan Museum of Art. https://www.metmuseum.org/art/collection/search/139668.</ref> of the structure of this foundation garment, giving an idea of what they looked like. Almost no examples of panniers survive. The hoops are bent cane, held together with red velvet silk ribbon that looks pinked. The cane also appears to be covered with red velvet, and the hoops are hinged with metal "hinges that allow the hoops to be lifted, facilitating movement in tight spaces."<ref name=":15" /> This inventive hinging permitted the wearer to lift the bottom cane and her skirts, folding them like an accordion towards her underarms, greatly reducing the width.
[[File:Panniers 1.jpg|thumb|alt=Photograph of the wooden and fabric skeleton of an 18th-century women's foundation garment|Wooden and Fabric Structure for 18th-century Panniers|center]]Images:
* Hoop petticoat and corset England 1750-1780 LACMA.jpg
* The hoops are wooden, connected with fabric. This structure is not for the extremely wide panniers, so it probably represents panniers predating the '''1740s''' and certainly before Marie Antoinette exaggerated the width of panniers in court dress.
[[File:Hoop petticoat and corset England 1750-1780 LACMA.jpg|center|thumb|Hoops as Panniers, 1750–1780]]
* '''Johanna Gabriele of Habsburg Lorraine1 copy.jpg'''
* [[File:Johanna Gabriele of Habsburg Lorraine1 copy.jpg|center|thumb|Martin van Meytens, Johanna Gabriele of Habsburg Lorraine, c. 1760]]
*Thomas Gainsborough (workshop of) - Queen Charlotte of England, 1781 (Schwerin).jpg
[[File:Queen Charlotte, by studio of Thomas Gainsborough.jpg|center|thumb|Queen Charlotte of England, 1781]]
*<br />
=== Polonaise ===
* Thomas Gainsborough, Giovanna Baccelli. Oil on canvas, c.. 1782. Tate.jpg
[[File:Thomas Gainsborough, Giovanna Baccelli. Oil on canvas, c.. 1782. Tate.jpg|center|thumb|Polonaise]]
The Polonaise was a late-Georgian or late-18th-century style, dating in written English, according to the ''Oxford English Dictionary'', from 1773:<blockquote>A woman's dress consisting of a tight, unboned bodice and a skirt open from the waist downwards to reveal a decorative underskirt. Now historical.<ref name=":13">“Polonaise, N. & Adj.” ''Oxford English Dictionary'', Oxford UP, September 2024, https://doi.org/10.1093/OED/2555138986.</ref></blockquote>Even though it looks ''à la français'', the term itself does not appear as a term used to describe clothing by the French, either now or in the past.<p>
Payne says,<blockquote>The dairy maid, or polonaise, style could be achieved either by pulling the lower part of the overskirt through its own pocket holes, thus creating a bouffant effect, or by planned control of the overskirt, through the cut or by means of draw cords, ribbons, or loops and buttons, [or, later, buckles] which were used to form the three great ‘poufs’ known as the polonaise .... These diversions appeared in the late sixties and became prevalent in the seventies. They were much like the familiar styles of our own [American] Revolutionary War period.<ref name=":11" />{{rp|413}}</blockquote>
==== 19th Century ====
Possible images:
* File:Crinoline era3.gif
[[File:Crinoline era3.gif|center|thumb|Crinoline era3.gif]]
* Crinoline (6795291959).jpg
[[File:Crinoline (6795291959).jpg|center|thumb|Augusta Congreve, side entrance of Clonbrock House, Ahascragh, Co. Galway. 31 January 1866]]
* Elisabeth Franziska wearing a crinoline and feathered hat.jpg
[[File:Elisabeth Franziska wearing a crinoline and feathered hat.jpg|center|thumb|Archduchess Elisabeth Franziska (1831-1903) wearing a crinoline and feathered hat, 1860s]]
*HM Queen Victoria. Photograph by C. Clifford of Madrid, 1861 Wellcome V0027547.jpg
*Queen Victoria at Windsor Castle, in evening dress, with diadem & jewels. The Regalportrait photographed from live by C.Clifford of Madrid. 14 November 1861. Carte de visite[[File:Her Majesty the Queen Victoria.JPG|center|thumb|Queen Victoria at Windsor Castle, in evening dress, with diadem & jewels. The Regalportrait photographed from live by C.Clifford of Madrid. 14 November 1861]]
* Her Majesty the Queen Victoria.JPG
* Queen Victoria photographed by Mayall.JPG
[[File:Queen Victoria photographed by Mayall.JPG|center|thumb|Queen Victoria photographed by Mayall, 1860s, carte de visite]]
*
* File:The Secret of England's Greatness' (Queen Victoria presenting a Bible in the Audience Chamber at Windsor) by Thomas Jones Barker.jpg
[[File:The Secret of England's Greatness' (Queen Victoria presenting a Bible in the Audience Chamber at Windsor) by Thomas Jones Barker.jpg|center|thumb|The Secret of England's Greatness' (Queen Victoria presenting a Bible in the Audience Chamber at Windsor) by Thomas Jones Barker, c. 1863]]
* Bustle: Princess Victoria Mary of Teck.jpg
[[File:Princess Victoria Mary of Teck.jpg|center|thumb|Princess Victoria Mary of Teck, 1886. Bustle]]
* Development of the full-cage hoop to flatter in front, with fabric going to the back: Queen Emma of Hawaii, photograph by John & Charles Watkins, The Royal Collection Trust.jpg
[[File:Queen Emma of Hawaii, photograph by John & Charles Watkins, The Royal Collection Trust (crop).jpg|center|thumb|Queen Emma of Hawaii, photograph by John & Charles Watkins, The Royal Collection Trust, 1865]]
* Miss Victoria Stuart-Wortley, later Victoria, Lady Welby (1837-1912) 1859.jpg
[[File:Cutaway sketch of crinoline.gif|thumb|Cutaway sketch of crinoline]]
[[File:Paris voulant englober la banlieue.JPG|thumb|Paris voulant englober la banlieue]]
[[File:Vrouw moet haar hoepelrok uitdoen om de tram te betreden New Omnibus Regulation. Werry sorry'm, but yer l'av to leave yer Krinerline outside (Vide Punch) (titel op object), RP-F-F10723.jpg|thumb|Vrouw moet haar hoepelrok uitdoen om de tram te betreden New Omnibus Regulation. Werry sorry'm, but yer l'av to leave yer Krinerline outside (Vide Punch) (titel op object), RP-F-F10723]]
Styles in personal adornment and architectural decoration: Romantic, Victorian (at least in '''the UK'''), "New Woman," [[Social Victorians/Terminology#Traditional vs Progressive Style|Traditional vs Progressive Style]], Crinoline
In the 19th century, the hoops were made of wire and became lighter. By the 1860s, hoops caused skirts to be huge and round.
By the 19th century, fashion had begun to move down the social classes so that hoops (and, for example, top hats) were worn by the middle and sometimes working classes.
'''''1880s'''''
Laura Ingalls Wilder wrote about the hoops her fictionalized self wore the century before. In ''These Happy Golden Years'' (1943), she gives a detailed description of the clothing under her dress:<blockquote>
“Then carefully over her under-petticoats she put on her hoops. She liked these new hoops. They were the very latest style in the East, and these were the first of the kind that Miss Bell had got. Instead of wires, there were wide tapes across the front, almost to her knees, holding the petticoats so that her dress would lie flat. These tapes held the wire bustle in place at the back, and it was an adjustable bustle. Short lengths of tape were fastened either end of it; these could be buckled together underneath the bustle to puff it out, either large or small. Or they could be buckled together in front, drawing the bustle down close in back so that a dress rounded smoothly over it. Laura did not like a large bustle, so she buckled the tapes in front.
"Then carefully over all she buttoned her best petticoat, and over all the starched petticoats she put on the underskirt of her new dress. It was of brown cambric, fitting smoothly around the top over the bustle, and gored to flare smoothly down over the hoops. At the bottom, just missing the floor, was a twelve-inch-wide flounce of the brown poplin, bound with an inch-wide band of plain brown silk. The poplin was not plain poplin, but striped with an openwork silk stripe.
"Then over this underskirt and her starched white corset-cover, Laura put on the polonaise. Its smooth, long sleeves fitted her arms perfectly to the wrists, where a band of the plain silk ended them. The neck was high with a smooth band of the plain silk around the throat. The polonaise fitted tightly and buttoned all down the front with small round buttons covered with the plain brown silk. Below the smooth hips it flared and rippled down and covered the top of the flounce on the underskirt. A band of the plain silk finished the polonaise at the bottom."<ref>Wilder, Laura Ingalls. ''These Happy Golden Years.'' Harper & Row, Publishers, 1943. Pp. 161–163.</ref></blockquote>
When a 20th-century Laura Ingalls Wilder calls her character's late-19th-century dress a polonaise, she is probably referring to the "tight, unboned bodice"<ref name=":13" /> and perhaps the simple, modest look of a dairy maid.
In Wilder's 1941 ''Little Town on the Prairie'', she provides an interesting story about how the wind could affect hoops:<blockquote>“Well,” Laura began; then she stopped and spun round and round, for the strong wind blowing against her always made the wires of her hoop skirt creep slowly upward under her skirts until they bunched around her knees. Then she must whirl around and around until the wires shook loose and spiraled down to the bottom of her skirts where they should be.
“As she and Carrie hurried on she began again. “I think it was silly, the way they dressed when Ma was a girl, don’t you? Drat this wind!” she exclaimed as the hoops began creeping upward again.
“Quietly Carrie stood by while Laura whirled. “I’m glad I’m not old enough to have to wear hoops,” she said. “They’d make me dizzy.”
“They are rather a nuisance,” Laura admitted. “But they are stylish, and when you’re my age you’ll want to be in style.”<ref>Wilder, Laura Ingalls. ''Little Town on the Prairie.'' Harper and Row, 1941. Pp. 272–273.</ref></blockquote>This moment is set in 1883.<ref>Hill, Pamela Smith, ed. ''Pioneer Girl: The Annotated Autobiography''.</ref> The 16-year-old Laura makes the comment that she wants to be in style, but she lives on the prairie, far from a large city, and would not necessarily wear the latest Parisian style. This description of the way the wind could make hoops creep — and the solution of spinning to get the hoops to go back down — is very unusual. It must have been happening to other women wearing hoops at the time, but no other writer addresses this.
== '''Traditional vs Progressive Style''' ==
=== Progressive Style ===
The terms ''artistic dress'' and ''aesthetic dress'' are not synonymous and were in use at different times to refer to different groups of people in different contexts, but we recognize them as referring to a similar kind of personal style in clothing, a style we call progressive dress or the progressive style. Used in a very precise way, ''artistic dress'' is associated with the Pre-Raphaelite artists and the women in their circle beginning in the 1860s. Similarly, ''aesthetic dress'' is associated with the 1880s and 1890s and dress reform movements. In general, the progressive style is characterized by its resistance to the highly structured fashion of its day, especially corseting, aniline dyes and an extremely close fit.
=== Traditional Style ===
Images
* Smooth bodice, fabric draped to the back, bustle, laters: Victoria Hesse NPG 95941 crop.jpg
By the end of the century designs from the [[Social Victorians/People/Dressmakers and Costumiers#The House of Worth|House of Worth]] (or Maison Worth) define what we think of as the traditional Victorian look, which was very stylish and expensive. Blanche Payne describes an example of the 1895 "high style" in a gown by Worth with "the idiosyncrasies of the [1890s] full blown":<blockquote>The dress is white silk with wine-red stripes. Sleeves, collars, bows, bag, hat, and hem border match the stripes. The sleeve has reached its maximum volume; the bosom full and emphasized with added lace; the waistline is elongated, pointed, and laced to the point of distress; the skirt is smooth over the hips, gradually swinging out to sweep the floor. This is the much vaunted hourglass figure.<ref name=":11">Payne, Blanche. ''History of Costume from the Ancient Egyptians to the Twentieth Century''. Harper & Row, 1965.</ref>{{rp|530}}</blockquote>
The Victorian-looking gowns at the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's 1897 fancy-dress ball]] are stylish in a way that recalls the designs of the House of Worth. The elements that make their look so Victorian are anachronisms on the costumes representing fashion of earlier eras. The women wearing these gowns preferred the standards of beauty from their own day to a more-or-less historically accurate look. The style competing at the very end of the century with the Worth look was not the historical, however, but a progressive style called at the time ''artistic'' or ''aesthetic''.
William Powell Frith's 1883 painting ''A Private View at the Royal Academy, 1881'' (discussion below) pits this kind of traditional style against the progressive or artistic style.
=== The Styles ===
[[File:Frith A Private View.jpg|thumb|William Powell Frith, ''A Private View at the Royal Academy, 1881'']]
We typically think of the late-Victorian silhouette as universal but, in the periods in which corsets dominated women's dress, not all women wore corsets and not all corsets were the same, as William Powell Frith's 1883 ''A Private View at the Royal Academy, 1881'' (right) illustrates. Frith is clear in his memoir that this painting — "recording for posterity the aesthetic craze as regards dress" — deliberately contrasts what he calls the "folly" of the Artistic Dress movement and the look of the traditional corseted waist.<ref>Frith, William Powell. ''My Autobiography and Reminiscences''. 1887.</ref> Frith considered the Artistic Movement and Artistic Dress "ephemeral," but its rejection of corsetry looks far more consequential to us in hindsight than it did in the 19th century.
As Frith sees it, his painting critiques the "craze" associated with the women in this set of identifiable portraits who are not corseted, but his commitment to realism shows us a spectrum, a range, of conservatism and if not political then at least stylistic progressivism among the women. The progressives, oddly, are the women wearing artistic (that is, somewhat historical) dress, because they’re not corseted. It is a misreading to see the presentation of the women’s fashion as a simple opposition. Constance, Countess of Lonsdale — situated at the center of this painting with Frederick Leighton, president of the Royal Academy of Art — is the most conservatively dressed of the women depicted, with her narrow sleeves, tight waist and almost perfectly smooth bodice, which tells us that her corset has eyelets so that it can be laced precisely and tightly, and it has stays (or "bones") to prevent wrinkles or natural folds in the overclothing. Lillie Langtry, in the white dress, with her stylish narrow sleeves, does not have such a tightly bound waist or smooth bodice, suggesting she may not be corseted at all, as we know she sometimes was not.['''citation'''] Jenny Trip, a painter’s model, is the woman in the green dress in the aesthetic group being inspected by Anthony Trollope, who may be taking notes. She looks like she is not wearing a corset. Both Langtry and Trip are toward the middle of this spectrum: neither is dressed in the more extreme artistic dress of, say, the two figures between Trip and Trollope.
A lot has been written about the late-Victorian attraction to historical dress, especially in the context of fancy-dress balls and the Gothic revival in social events as well as art and music. Part of the appeal has to have been the way those costumes could just be beautiful clothing beautifully made. Historical dress provided an opportunity for some elite women to wear less-structured but still beautiful and influential clothing. ['''Calvert'''<ref>Calvert, Robyne Erica. ''Fashioning the Artist: Artistic Dress in Victorian Britain 1848-1900''. Ph.D. thesis, University of Glasgow, 2012. <nowiki>https://theses.gla.ac.uk/3279/</nowiki></ref>] The standards for beauty, then, with historical dress were Victorian, with the added benefit of possibly less structure. So, at the Duchess of Devonshire's ball, "while some attendees tried to hew closely to historical precedent, many rendered their historical or mythological personage in the sartorial vocabulary they knew best. The [photographs of people in their costumes at the ball offer] a glimpse into how Victorians understood history, not a glimpse into the costume of an authentic historical past."<ref>Mitchell, Rebecca N. "The Victorian Fancy Dress Ball, 1870–1900." ''Fashion Theory'' 2017 (21: 3): 291–315. DOI: 10.1080/1362704X.2016.1172817.</ref> (294)
* historical dress: beautiful clothing.
* the range at the ball, from Minnie Paget to Gwladys
* "In light of such efforts, the ball remains to this day one of the best documented outings of the period, and a quick glance at the album shows that ..."
Women had more choices about their waists than the simple opposition between no corset and tightlacing can accommodate. The range of choices is illustrated in Frith's painting, with a woman locating herself on it at a particular moment for particular reasons. Much analysis of 19th-century corsetry focuses on its sexualizing effects — corsets dominated Victorian photographic pornography ['''citations'''] and at the same time, the absence of a corset was sexual because it suggested nudity.['''citations'''] A great deal of analysis of 19th-century corsetry, on the other hand, assumes that women wore corsets for the male gaze ['''citations'''] or that they tightened their waists to compete with other women.['''citations''']
But as we can see in Frith's painting, the sexualizing effect was not universal or sweeping, and these analyses do not account for the choices women had in which corset to wear or how tightly to lace it. Especially given the way that some photographic portraits were mechanically altered to make the waist appear smaller, the size of a woman's waist had to do with how she was presenting herself to the world. That is, the fact that women made choices about the size of or emphasis on their waists suggests that they had agency that needs to be taken into account.
As they navigated the complex social world, women's fashion choices had meaning. Society or political hostesses had agency not only in their clothing but generally in that complex social world. They had roles managing social events of the upper classes, especially of the upper aristocracy and oligarchy, like the Duchess of Devonshire's ball. Their class and rank, then, were essential to their agency, including to some degree their freedom to choose what kind of corset to wear and how to wear it. Also, by the end of the century lots of different kinds of corsets were available for lots of different purposes. Special corsets existed for pregnancy, sports (like tennis, bicycling, horseback riding, golf, fencing, archery, stalking and hunting), theatre and dance and, of course, for these women corsets could be made to support the special dress worn over it.
Women's choices in how they presented themselves to the world included more than just their foundation garments, of course. "Every cap, bow, streamer, ruffle, fringe, bustle, glove," that is, the trim and decorations on their garments, their jewelry and accessories — which Davidoff calls "elaborations"<ref name=":1">Davidoff, Leonore. ''The Best Circles: Society Etiquette and the Season''. Intro., Victoria Glendinning. The Cressett Library (Century Hutchinson), 1986 (orig 1973).</ref>{{rp|93}} — pointed to a host of status categories, like class, rank, wealth, age, marital status, engagement with the empire, how sexual they wanted to seem, political alignment and purpose at the social event. For example, when women were being presented to the monarch, they were expected to wear three ostrich plumes, often called the [[Social Victorians/Terminology#Prince of Wales's Feathers or White Plumes|Prince of Wales's feathers]].
Like all fashions, the corset, which was quite long-lasting in all its various forms, eventually went out of style. Of the many factors that might have influenced its demise, perhaps most important was the women's movement, in which women's rights, freedom, employment and access to their own money and children were less slogan-worthy but at least as essential as votes for women. The activities of the animal-rights movements drew attention not only to the profligate use of the bodies and feathers of birds but also to the looming extinction of the baleen whale, which made whale bone scarce and expensive. Perhaps the century's debates over corseting and especially tightlacing were relevant to some decisions not to be corseted. And, of course, perhaps no other reason is required than that the nature of fashion is to change.
== Cinque Cento ==
According to the ''Oxford English Dictionary'', ''Cinque Cento'' is a shortening of ''mil cinque cento'', or 1500.<ref>"cinquecento, n." ''OED Online'', Oxford University Press, December 2022, www.oed.com/view/Entry/33143. Accessed 7 February 2023.</ref> The term, then would refer, perhaps informally, to the sixteenth century.
== Crevé ==
''Creve'', without the accent, is an old word in English (c. 1450) for burst or split.<ref>"creve, v." ''OED Online'', Oxford University Press, December 2022, www.oed.com/view/Entry/44339. Accessed 8 February 2023.</ref> ['''With the acute accent, it looks like a past participle in French.''']
== Elastic ==
Elastic had been invented and was in use by the end of the 19th century. For the sense of "Elastic cord or string, usually woven with india-rubber,"<ref name=":6">“elastic, adj. & n.”. ''Oxford English Dictionary'', Oxford University Press, September 2023, <https://doi.org/10.1093/OED/1199670313>.</ref> the ''Oxford English Dictionary'' has usage examples beginning in 1847. The example for 1886 is vivid: "The thorough-going prim man will always place a circle of elastic round his hair previous to putting on his college cap."<ref name=":6" />
== Elaborations ==
In her 1973 ''The Best Circles: Society, Etiquette and the Season'', Leonore Davidoff notes that women’s status was indicated by dress and especially ornament: “Every cap, bow, streamer, ruffle, fringe, bustle, glove and other elaboration,” she says, “symbolised some status category for the female wearer.”<ref name=":1" />{{rp|93}}
Looking at these elaborations as meaningful rather than dismissing them as failed attempts at "historical accuracy" reveals a great deal about the individual women who wore or carried them — and about the society women and political hostesses in their roles as managers of the social world. In her review of ''The House of Worth: Portrait of an Archive'', Mary Frances Gormally says,<blockquote>In a socially regulated year, garments custom made with a Worth label provided women with total reassurance, whatever the season, time of day or occasion, setting them apart as members of the “Best Circles” dressed in luxurious, fashionable and always appropriate attire (Davidoff 1973). The woman with a Worth wardrobe was a woman of elegance, lineage, status, extreme wealth and faultless taste.<ref>Gormally, Mary Frances. Review essay of ''The House of Worth: Portrait of an Archive'', by Amy de la Haye and Valerie D. Mendes (V&A Publishing, 2014). ''Fashion Theory'' 2017 (21, 1): 109–126. DOI: 10.1080/1362704X.2016.1179400.</ref> (117)</blockquote>
[[File:Aglets from Spanish portraits - collage by shakko.jpg|thumb|alt=A collage of 12 different ornaments typically worn by elite people from Spain in the 1500s and later|Aglets — Detail from Spanish Portraits]]
=== Aglet, Aiglet ===
Historically, an aglet is a "point or metal piece that capped a string [or ribbon] used to attach two pieces of the garment together, i.e., sleeve and bodice."<ref name=":7" />{{rp|4}} Although they were decorative, they were not always visible on the outside of the clothing. They were often stuffed inside the layers at the waist (for example, attaching the bodice to a skirt or breeches).
Alonso Sánchez Coello's c. 1584 (316) portrait (above right, in the [[Social Victorians/Terminology#16th Century|Hoops section]]) shows infanta Isabel Clara Eugenia wearing a vertugado, with its "typically Spanish smooth cone-shaped contour," with "handsome aiglets cascad[ing] down center front."<ref name=":11" /> (315)
=== Frou-frou ===
In French, ''frou-frou'' or, spelled as ''froufrou'', is the sound of the rustling of silk or sometimes of fabrics in general.<ref>{{Cite journal|date=2023-07-25|title=frou-frou|url=https://fr.wiktionary.org/w/index.php?title=frou-frou&oldid=32508509|journal=Wiktionnaire, le dictionnaire libre|language=fr}} https://fr.wiktionary.org/wiki/frou-frou.</ref> The first use the French ''Wiktionnaire'' lists is Honoré Balzac, ''La Cousine Bette'', 1846.<ref>{{Cite journal|date=2023-06-03|title=froufrou|url=https://fr.wiktionary.org/w/index.php?title=froufrou&oldid=32330124|journal=Wiktionnaire, le dictionnaire libre|language=fr}} https://fr.wiktionary.org/wiki/froufrou.</ref>
''Frou-frou'' is a term clothing historians use to describe decorative additions to an article of clothing; often the term has a slight negative connotation, suggesting that the additions are superficial.
=== Pouf, Puff, Poof ===
According to the French ''Wikipédia'', a pouf was, beginning in 1744, a "kind of women's hairstyle":<blockquote>The hairstyle in question, known as the “pouf”, had launched the reputation of the enterprising Rose Bertin, owner of the Grand Mogol, a very prominent fashion accessories boutique on Rue Saint-Honoré in Paris in 1774. Created in collaboration with the famous hairdresser, Monsieur Léonard, the pouf was built on a scaffolding of wire, fabric, gauze, horsehair, fake hair, and the client's own hair held up in an almost vertical position. — (Marie-Antoinette, ''Queen of Fashion'', translated from the American by Sylvie Lévy, in ''The Rules of the Game'', n° 40, 2009)</blockquote>''Puff'' and ''poof'' are used to describe clothing.
=== Shirring ===
''Shirring'' is the gathering of fabric to make poufs or puffs. The 19th century is known for its use of this decorative technique. Even men's clothing had shirring: at the shoulder seam.
=== Sequins ===
Sequins, paillettes, spangles
Sequins — or paillettes — are "small, scalelike glittering disks."<ref name=":7" />(216) The French ''Wiktionnaire'' defines ''paillette'' as "Lamelle de métal, brillante, mince, percée au milieu, ordinairement ronde, et qu’on applique sur une étoffe pour l’orner [A strip of metal, shiny, thin, pierced in the middle, usually round, and which is applied to a fabric in order to decorate it.]"<ref name=":8">{{Cite journal|date=2024-03-18|title=paillette|url=https://fr.wiktionary.org/w/index.php?title=paillette&oldid=33809572|journal=Wiktionnaire, le dictionnaire libre|language=fr}} https://fr.wiktionary.org/wiki/paillette.</ref>
According to the ''OED'', the use of ''sequin'' as a decorative device for clothing (as opposed to gold coins minted and used for international trade) goes back to the 1850s.<ref>“Sequin, N.” ''Oxford English Dictionary'', Oxford UP, September 2023, https://doi.org/10.1093/OED/4074851670.</ref> The first instance of ''spangle'' as "A small round thin piece of glittering metal (usually brass) with a hole in the centre to pass a thread through, used for the decoration of textile fabrics and other materials of various sorts" is from c. 1420.<ref>“Spangle, N. (1).” ''Oxford English Dictionary'', Oxford UP, July 2023, https://doi.org/10.1093/OED/4727197141.</ref> The first use of ''paillette'' listed in the French ''Wiktionnaire'' is in Jules Verne in 1873 to describe colored spots on icy walls.<ref name=":8" />
Currently many distinguish between sequins (which are smaller) and paillettes (which are larger).
Before the 20th century, sequins were metal discs or foil leaves, and so of course if they were silver or copper, they tarnished. It is not until well into the 20th century that plastics were invented and used for sequins.
=== Trim and Lace ===
''A History of Feminine Fashion'', published sometime before 1927 and probably commissioned by [[Social Victorians/People/Dressmakers and Costumiers#Worth, of Paris|the Maison Worth]], describes Charles Frederick Worth's contributions to the development of embroidery and [[Social Victorians/Terminology#Passementerie|passementerie]] (trim) from about the middle of the 19th century:<blockquote>For it must be remembered that one of M. Worth's most important and lasting contributions to the prosperity of those who cater for women's needs, as well as to the variety and elegance of his clients' garments, was his insistence on new fabrics, new trimmings, new materials of every description. In his endeavours to restore in Paris the splendours of the days of La Pompadour, and of Marie Antoinette, he found himself confronted at the outset with a grave difficulty, which would have proved unsurmountable to a man of less energy, resource and initiative. The magnificent materials of those days were no longer to be had! The Revolution had destroyed the market for beautiful materials of this, type, and the Restoration and regime of Louis Philippe had left a dour aspect in the City of Light. ... On parallel lines [to his development of better [[Social Victorians/Terminology#Satin|satin]]], [Worth] stimulated also the manufacture of embroidery and ''passementerie''. It was he who first started the manufacture of laces copied from the designs of the real old laces. He was the / first dressmaker to use fur in the trimming of light materials — but he employed only the richer furs, such as sable and ermine, and had no use whatever for the inferior varieties of skins.<ref name=":9" />{{rp|6–7}}</blockquote>
==== Gold and Silver Fabric and Lace ====
The ''Encyclopaedia Britannica'' (9th edition) has an article on gold and silver fabric, threads and lace attached to the article on gold. (This article is based on knowledge that would have been available toward the end of the 19th century and does not, obviously, reflect current knowledge or ways of talking.)<blockquote>GOLD AND SILVER LACE. Under this heading a general account may be given of the use of the precious metals in textiles of all descriptions into which they enter. That these metals were used largely in the sumptuous textiles of the earliest periods of civilization there is abundant testimony; and to this day, in the Oriental centres whence a knowledge and the use of fabrics inwoven, ornamented, and embroidered with gold and silver first spread, the passion for such brilliant and costly textiles is still most strongly and generally prevalent. The earliest mention of the use of gold in a woven fabric occurs in the description of the ephod made for Aaron (Exod. xxxix. 2, 3) — "And he made the ephod of gold, blue, and purple, and scarlet, and fine twined linen. And they did beat the gold into thin plates, and cut it into wires (strips), to work it in the blue, and in the purple, and in the scarlet, and in the fine linen, with cunning work." In both the ''Iliad'' and the ''Odyssey'' distinct allusion is frequently made to inwoven and embroidered golden textiles. Many circumstances point to the conclusion that the art of weaving and embroidering with gold and silver originated in India, where it is still principally prosecuted, and that from one great city to another the practice travelled westward, — Babylon, Tarsus, Baghdad, Damascus, the islands of Cyprus and Sicily, Con- / stantinople and Venice, all in the process of time becoming famous centres of these much prized manufactures. Alexander the Great found Indian kings and princes arrayed in robes of gold and purple; and the Persian monarch Darius, we are told, wore a war mantle of cloth of gold, on which were figured two golden hawks as if pecking at each other. There is reason, according to Josephus, to believe that the “royal apparel" worn by Herod on the day of his death (Acts xii. 21) was a tissue of silver. Agrippina, the wife of the emperor Claudius, had a robe woven entirely of gold, and from that period downwards royal personages and high ecclesiastical dignitaries used cloth and tissues of gold and silver for their state and ceremonial robes, as well as for costly hangings and decorations. In England, at different periods, various names were applied to cloths of gold, as ciclatoun, tartarium, naques or nac, baudekiu or baldachin, Cyprus damask, and twssewys or tissue. The thin flimsy paper known as tissue paper, is so called because it originally was placed between the folds of gold "tissue" to prevent the contiguous surfaces from fraying each other. At what time the drawing of gold wire for the preparation of these textiles was first practised is not accurately known. The art was probably introduced and applied in different localities at widely different dates, but down till mediaeval times the method graphically described in the Pentateuch continued to be practised with both gold and silver.
Fabrics woven with gold and silver continue to be used on the largest scale to this day in India; and there the preparation of the varieties of wire, and the working of the various forms of lace, brocade, and embroidery, is at once an important and peculiar art. The basis of all modern fabrics of this kind is wire, the "gold wire" of the manufacturer being in all cases silver gilt wire, and silver wire being, of course, composed of pure silver. In India the wire is drawn by means of simple draw-plates, with rude and simple appliances, from rounded bars of silver, or gold-plated silver, as the case may be. The wire is flattened into the strip or ribbon-like form it generally assumes by passing it, fourteen or fifteen strands simultaneously, over a fine, smooth, round-topped anvil, and beating it as it passes with a heavy hammer having a slightly convex surface. From wire so flattened there is made in India soniri, a tissue or cloth of gold, the web or warp being composed entirely of golden strips, and ruperi, a similar tissue of silver. Gold lace is also made on a warp of thick yellow silk with a weft of flat wire, and in the case of ribbons the warp or web is composed of the metal. The flattened wires are twisted around orange (in the case of silver, white) coloured silk thread, so as completely to cover the thread and present the appearance of a continuous wire; and in this form it is chiefly employed for weaving into the rich brocades known as kincobs or kinkhábs. Wires flattened, or partially flattened, are also twisted into exceedingly fine spirals, and in this form they are the basis of numerous ornamental applications. Such spirals drawn out till they present a waved appearance, and in that state flattened, are much used for rich heavy embroideries termed karchobs. Spangles for embroideries, &c., are made from spirals of comparatively stout wire, by cutting them down ring by ring, laying each C-like ring on an anvil, and by a smart blow with a hammer flattening it out into a thin round disk with a slit extending from the centre to one edge. Fine spirals are also used for general embroidery purposes. The demand for various kinds of loom-woven and embroidered gold and silver work in India is immense; and the variety of textiles so ornamented is also very great. "Gold and silver," says Dr Birdwood in his ''Handbook to the British-Indian Section, Paris Exhibition'', 1878, "are worked into the decoration of all the more costly loom-made garments and Indian piece goods, either on the borders only, or in stripes throughout, or in diapered figures. The gold-bordered loom embroideries are made chiefly at Sattara, and the gold or silver striped at Tanjore; the gold figured ''mashrus'' at Tanjore, Trichinopoly, and Hyderabad in the Deccau; and the highly ornamented gold-figured silks and gold and silver tissues principally at Ahmedabad, Benares, Murshedabad, and Trichinopoly."
Among the Western communities the demand for gold and silver lace and embroideries arises chiefly in connexion with naval and military uniforms, court costumes, public and private liveries, ecclesiastical robes and draperies, theatrical dresses, and the badges and insignia of various orders. To a limited extent there is a trade in gold wire and lace to India and China. The metallic basis of the various fabrics is wire round and flattened, the wire being of three kinds — 1st, gold wire, which is invariably silver gilt wire; 2d, copper gilt wire, used for common liveries and theatrical purposes; and 3d, silver wire. These wires are drawn by the ordinary processes, and the flattening, when done, is accomplished by passing the wire between a pair of revolving rollers of fine polished steel. The various qualities of wire are prepared and used in precisely the same way as in India, — round wire, flat wire, thread made of flat gold wire twisted round orange-coloured silk or cotton, known in the trade as "orris," fine spirals and spangles, all being in use in the West as in the East. The lace is woven in the same manner as ribbons, and there are very numerous varieties in richness, pattern, and quality. Cloth of gold, and brocades rich in gold and silver, are woven for ecclesiastical vestments and draperies.
The proportions of gold and silver in the gold thread for the lace trade varies, but in all cases the proportion of gold is exceedingly small. An ordinary gold lace wire is drawn from a bar containing 90 parts of silver and 7 of copper, coated with 3 parts of gold. On an average each ounce troy of a bar so plated is drawn into 1500 yards of wire; and therefore about 16 grains of gold cover a mile of wire. It is estimated that about 250,000 ounces of gold wire are made annually in Great Britain, of which about 20 per cent, is used for the headings of calico, muslin, &c., and the remainder is worked up in the gold lace trade.<ref>William Chandler Roberts-Austen and H. Bauerman [W.C.R. — H.B.]. "Gold and Silver Lace." In "Gold." ''Encyclopaedia Britannica'', 9th Edition (1875–1889). Vol. 10 (X). Adam and Charles Black (Publisher). https://archive.org/details/encyclopaedia-britannica-9ed-1875/Vol%2010%20%28G-GOT%29%20193592738.23/page/753/mode/1up (accessed January 2023): 753, Col. 2c – 754, Cols. 1a–b – 2a–b.</ref></blockquote>
==== Honiton Lace ====
Kate Stradsin says,<blockquote>Honiton lace was the finest English equivalent of Brussels bobbin lace and was constructed in small ‘sprigs, in the cottages of lacemakers[.'] These sprigs were then joined together and bleached to form the large white flounces that were so sought after in the mid-nineteenth century.<ref>Strasdin, Kate. "Rediscovering Queen Alexandra’s Wardrobe: The Challenges and Rewards of Object-Based Research." ''The Court Historian'' 24.2 (2019): 181-196. Rpt http://repository.falmouth.ac.uk/3762/15/Rediscovering%20Queen%20Alexandra%27s%20Wardrobe.pdf: 13, and (for the little quotation) n. 37, which reads "Margaret Tomlinson, ''Three Generations in the Honiton Lace Trade: A Family History'', self-published, 1983."</ref></blockquote>
[[File:Strook in Alençon naaldkant, 1750-1775.jpg|thumb|alt=A long piece of complex white lace with garlands, flowers and bows|Point d'Alençon lace, 1750-1775]]
==== Passementerie ====
''Passementerie'' is the French term for trim on clothing or furniture. The 19th century (especially during the First and Second Empire) was a time of great "''exubérance''" in passementerie in French design, including the development and widespread use of the Jacquard loom.<ref>{{Cite journal|date=2023-06-10|title=Passementerie|url=https://fr.wikipedia.org/w/index.php?title=Passementerie&oldid=205068926|journal=Wikipédia|language=fr}} https://fr.wikipedia.org/wiki/Passementerie.</ref>
==== Point d'Alençon Lace ====
A lace made by hand using a number of complex steps and layers. The lacemakers build the point d'Alençon design on some kind of mesh and sometimes leave some of the mesh in as part of the lace and perhaps to provide structure.
Elizabeth Lewandowski defines point d'Alençon lace and Alençon lace separately. Point lace is needlepoint lace,<ref name=":7">Lewandowski, Elizabeth J. ''The Complete Costume Dictionary''. Scarecrow Press, 2011.</ref>{{rp|233}} so Alençon point is "a two thread [needlepoint] lace."<ref name=":7" />{{rp|7}} Alençon lace has a "floral design on [a] fine net ground [and is] referred to as [the] queen of French handmade needlepoint laces. The original handmade Alençon was a fine needlepoint lace made of linen thread."<ref name=":7" />{{rp|7}}
The sample of point d'Alençon lace (right), from 1750–1775, shows the linen mesh that the lace was constructed on.<ref>{{Cite web|url=http://openfashion.momu.be/#9ce5f00e-8a06-4dab-a833-05c3371f3689|title=MoMu - Open Fashion|website=openfashion.momu.be|access-date=2024-02-26}} ModeMuseum Antwerpen. http://openfashion.momu.be/#9ce5f00e-8a06-4dab-a833-05c3371f3689.</ref> The consistency in this sample suggests it may have been made by machine.
== Fabric ==
=== Brocatelle ===
Brocatelle is a kind of brocade, more simple than most brocades because it uses fewer warp and weft threads and fewer colors to form the design. The article in the French ''Wikipédia'' defines it like this:<blockquote>La '''brocatelle''' est un type de tissu datant du <abbr>xvi<sup>e</sup></abbr> siècle qui comporte deux chaînes et deux trames, au minimum. Il est composé pour que le dessin ressorte avec un relief prononcé, grâce à la chaîne sur un fond en sergé. Les brocatelles les plus anciennes sont toujours fabriquées avec une des trames en lin.<ref>{{Cite journal|date=2023-06-01|title=Brocatelle|url=https://fr.wikipedia.org/w/index.php?title=Brocatelle&oldid=204796410|journal=Wikipédia|language=fr}} https://fr.wikipedia.org/wiki/Brocatelle.</ref></blockquote>Which translates to this:<blockquote>Brocatelle is a type of fabric dating from the 16th century that has two warps and two wefts, at a minimum. It is composed so that the design stands out with a pronounced relief, thanks to the weft threads on a twill background. The oldest brocades were always made with one of the wefts being linen.</blockquote>The ''Oxford English Dictionary'' says, brocatelle is an "imitation of brocade, usually made of silk or wool, used for tapestry, upholstery, etc., now also for dresses. Both the nature and the use of the stuff have changed" between the late 17th century and 1888, the last time this definition was revised.<ref>"brocatelle, n." ''OED Online'', Oxford University Press, March 2023, www.oed.com/view/Entry/23550. Accessed 4 July 2023.</ref>
=== Broché ===
=== Ciselé ===
=== Crépe de Chine ===
The ''Oxford English Dictionary'' distinguishes the use of ''crêpe'' (using a circumflex rather than an acute accent over the first ''e'') from ''crape'' in textiles, saying ''crêpe'' is "often borrowed [from the French] as a term for all crapy fabrics other than ordinary black mourning crape,"<ref>"crêpe, n." ''OED Online'', Oxford University Press, December 2022, www.oed.com/view/Entry/44242. Accessed 10 February 2023.</ref> with usage examples ranging from 1797 to the mid 20th century. Crêpe de chine, it says is "a white or other coloured crape made of raw silk."
=== Épinglé Velvet ===
Often spelled ''épingle'' rather than ''épinglé'', this term appears to have been used for a fabric made of wool, or at least wool along with linen or cotton, that was heavier and stiffer than silk velvet. It was associated with outer garments and men's clothing. Nowadays, épinglé velvet is an upholstery fabric in which the pile is cut into designs and patterns, and the portrait of [[Social Victorians/People/Douglas-Hamilton Duke of Hamilton|Mary, Duchess of Hamilton]] shows a mantle described as épinglé velvet that does seem to be a velvet with a woven pattern perhaps cut into the pile.
=== Lace ===
While lace also functioned sometimes as fabric — at the décolletage, for example, on the stomacher or as a veil — here we organize it as a [[Social Victorians/Terminology#Trim and Lace|part of the elaboration of clothing]].
=== Liberty Fabrics ===
=== Lisse ===
According to the ''Oxford English Dictionary'', the term ''lisse'' as a "kind of silk gauze" was used in the 19th-century UK and US.<ref>"lisse, n.1." ''OED Online'', Oxford University Press, March 2023, www.oed.com/view/Entry/108978. Accessed 4 July 2023.</ref>
=== Satin ===
The pre-1927 ''History of Feminine Fashion'', probably commissioned by Charles Frederick Worth's sons, describes Worth's "insistence on new fabrics, new trimmings, new materials of every description" at the beginning of his career in the mid 19th century:<blockquote>When Worth first entered the business of dressmaking, the only materials of the richer sort used for woman's dress were velvet, faille, and watered silk. Satin, for example, was never used. M. Worth desired to use satin very extensively in the gowns he designed, but he was not satisfied with what could be had at the time; he wanted something very much richer than was produced by the mills at Lyons. That his requirements entailed the reconstruction of mills mattered little — the mills were reconstructed under his directions, and the Lyons looms turned out a richer satin than ever, and the manufacturers prospered accordingly.<ref name=":9">[Worth, House of.] {{Cite book|url=http://archive.org/details/AHistoryOfFeminineFashion|title=A History Of Feminine Fashion (1800s to 1920s)}} Before 1927. [Likely commissioned by Worth. Link is to Archive.org; info from Wikimedia Commons: https://commons.wikimedia.org/wiki/File:Worth_Biarritz_salon.jpg.]</ref>{{rp|6 in printed, 26 in digital book}}</blockquote>
=== Selesia ===
According to the ''Oxford English Dictionary'', ''silesia'' is "A fine linen or cotton fabric originally manufactured in Silesia in what is now Germany (''Schlesien'').<ref>"Silesia, n." ''OED Online'', Oxford University Press, December 2022, www.oed.com/view/Entry/179664. Accessed 9 February 2023.</ref> It may have been used as a lining — for pockets, for example — in garments made of more luxurious or more expensive cloth. The word ''sleazy'' — "Of textile fabrics or materials: Thin or flimsy in texture; having little substance or body."<ref>"sleazy, adj." ''OED Online'', Oxford University Press, December 2022, www.oed.com/view/Entry/181563. Accessed 9 February 2023.</ref> — may be related.
=== Shot Fabric ===
According to the ''Oxford English Dictionary'', "Of a textile fabric: Woven with warp-threads of one colour and weft-threads of another, so that the fabric (usually silk) changes in tint when viewed from different points."<ref>“Shot, ''Adj.''” ''Oxford English Dictionary'', Oxford UP, July 2023, https://doi.org/10.1093/OED/2977164390.</ref> A shot fabric might also be made of silk and cotton fibers.
=== Tissue ===
A lightly woven fabric like gauze or chiffon. The light weave can make the fabric translucent and make pleating and gathering flatter and less bulky. Tissue can be woven to be shot, sheer, stiff or soft.
Historically, the term in English was used for a "rich kind of cloth, often interwoven with gold or silver" or "various rich or fine fabrics of delicate or gauzy texture."<ref>“Tissue, N.” ''Oxford English Dictionary'', Oxford UP, March 2024, https://doi.org/10.1093/OED/5896731814.</ref>
== Fan ==
The ''Encyclopaedia Britannica'' (9th edition) has an article on the fan. (This article is based on knowledge that would have been available toward the end of the 19th century and does not, obviously, reflect current knowledge or ways of talking.)<blockquote>FAN (Latin, ''vannus''; French, ''éventail''), a light implement used for giving motion to the air. ''Ventilabrum'' and ''flabellum'' are names under which ecclesiastical fans are mentioned in old inventories. Fans for cooling the face have been in use in hot climates from remote ages. A bas-relief in the British Museum represents Sennacherib with female figures carrying feather fans. They were attributes of royalty along with horse-hair fly-flappers and umbrellas. Examples may be seen in plates of the Egyptian sculptures at Thebes and other places, and also in the ruins of Persepolis. In the museum of Boulak, near Cairo, a wooden fan handle showing holes for feathers is still preserved. It is from the tomb of Amen-hotep, of the 18th dynasty, 17th century <small>B</small>.<small>C</small>. In India fans were also attributes of men in authority, and sometimes sacred emblems. A heartshaped fan, with an ivory handle, of unknown age, and held in great veneration by the Hindus, was given to the prince of Wales. Large punkahs or screens, moved by a servant who does nothing else, are in common use by Europeans in India at this day.
Fans were used in the early Middle Ages to keep flies from the sacred elements during the celebrations of the Christian mysteries. Sometimes they were round, with bells attached — of silver, or silver gilt. Notices of such fans in the ancient records of St Paul’s, London, Salisbury cathedral, and many other churches, exist still. For these purposes they are no longer used in the Western church, though they are retained in some Oriental rites. The large feather fans, however, are still carried in the state processions of the supreme pontiff in Rome, though not used during the celebration of the mass. The fan of Queen Theodolinda (7th century) is still preserved in the treasury of the cathedral of Monza. Fans made part of the bridal outfit, or ''mundus muliebris'', of ancient Roman ladies.
Folding fans had their origin in Japan, and were imported thence to China. They were in the shape still used—a segment of a circle of paper pasted on a light radiating frame-work of bamboo, and variously decorated, some in colours, others of white paper on which verses or sentences are written. It is a compliment in China to invite a friend or distinguished guest to write some sentiment on your fan as a memento of any special occasion, and this practice has continued. A fan that has some celebrity in France was presented by the Chinese ambassador to the Comtesse de Clauzel at the coronation of Napoleon I. in 1804. When a site was given in 1635, on an artificial island, for the settlement of Portuguese merchants in Nippo in Japan, the space was laid out in the form of a fan as emblematic of an object agreeable for general use. Men and women of every rank both in China and Japan carry fans, even artisans using them with one hand while working with the other. In China they are often made of carved ivory, the sticks being plates very thin and sometimes carved on both sides, the intervals between the carved parts pierced with astonishing delicacy, and the plates held together by a ribbon. The Japanese make the two outer guards of the stick, which cover the others, occasionally of beaten iron, extremely thin and light, damascened with gold and other metals.
Fans were used by Portuguese ladies in the 14th century, and were well known in England before the close of the reign of Richard II. In France the inventory of Charles V. at the end of the 14th century mentions a folding ivory fan. They were brought into general use in that country by Catherine de’ Medici, probably from Italy, then in advance of other countries in all matters of personal luxury. The court ladies of Henry VIII.’s reign in England were used to handling fans, A lady in the Dance of Death by Holbein holds a fan. Queen Elizabeth is painted with a round leather fan in her portrait at Gorhambury; and as many as twenty-seven are enumerated in her inventory (1606). Coryat, an English traveller, in 1608 describes them as common in Italy. They also became of general use from that time in Spain. In Italy, France, and Spain fans had special conventional uses, and various actions in handling them grew into a code of signals, by which ladies were supposed to convey hints or signals to admirers or to rivals in society. A paper in the ''Spectator'' humorously proposes to establish a regular drill for these purposes.
The chief seat of the European manufacture of fans during the 17th century was Paris, where the sticks or frames, whether of wood or ivory, were made, and the decorations painted on mounts of very carefully prepared vellum (called latterly ''chicken skin'', but not correctly), — a material stronger and tougher than paper, which breaks at the folds. Paris makers exported fans unpainted to Madrid and other Spanish cities, where they were decorated by native artists. Many were exported complete; of old fans called Spanish a great number were in fact made in France. Louis XIV. issued edicts at various times to regulate the manufacture. Besides fans mounted with parchment, Dutch fans of ivory were imported into Paris, and decorated by the heraldic painters in the process called “Vernis Martin,” after a famous carriage painter and inventor of colourless lac varnish. Fans of this kind belonging to the Queen and to the late baroness de Rothschild were exhibited in 1870 at Kensington. A fan of the date of 1660, representing sacred subjects, is attributed to Philippe de Champagne, another to Peter Oliver in England in the / 17th century. Cano de Arevalo, a Spanish painter of the 17th century devoted himself to fan painting. Some harsh expressions of Queen Christina to the young ladies of the French court are said to have caused an increased ostentation in the splendour of their fans, which were set with jewels and mounted in gold. Rosalba Carriera was the name of a fan painter of celebrity in the 17th century. Lebrun and Romanelli were much employed during the same period. Klingstet, a Dutch artist, enjoyed a considerable reputation for his fans from the latter part of the 17th and the first thirty years of the 18th century.
The revocation of the edict of Nantes drove many fan-makers out of France to Holland and England. The trade in England was well established under the Stuart sovereigns. Petitions were addressed by the fan-makers to Charles II. against the importation of fans from India, and a duty was levied upon such fans in consequence. This importation of Indian fans, according to Savary, extended also to France. During the reign of Louis XV. carved Indian and China fans displaced to some extent those formerly imported from Italy, which had been painted on swanskin parchment prepared with various perfumes.
During the 18th century all the luxurious ornamentation of the day was bestowed on fans as far as they could display it. The sticks were made of mother-of-pearl or ivory, carved with extraordinary skill in France, Italy, England, and other countries. They were painted from designs of Boucher, Watteau, Lancret, and other "genre" painters, Hébert, Rau, Chevalier, Jean Boquet, Mad. Verité, are known as fan painters. These fashions were followed in most countries of Europe, with certain national differences. Taffeta and silk, as well as fine parchment, were used for the mounts. Little circles of glass were let into the stick to be looked through, and small telescopic glasses were sometimes contrived at the pivot of the stick. They were occasionally mounted with the finest point lace. An interesting fan (belonging to Madame de Thiac in France), the work of Le Flamand, was presented by the municipality of Dieppe to Marie Antoinette on the birth of her son the dauphin. From the time of the Revolution the old luxury expended on fans died out. Fine examples ceased to be exported to England and other countries. The painting on them represented scenes or personages connected with political events. At a later period fan mounts were often prints coloured by hand. The events of the day mark the date of many examples found in modern collections. Amongst the fanmakers of the present time the names of Alexandre, Duvelleroy, Fayet, Vanier, may be mentioned as well known in Paris. The sticks are chiefly made in the department of Oise, at Le Déluge, Crèvecœur, Méry, Ste Geneviève, and other villages, where whole families are engaged in preparing them; ivory sticks are carved at Dieppe. Water-colour painters of distinction often design and paint the mounts, the best designs being figure subjects. A great impulse has been given to the manufacture and painting of fans in England since the exhibition which took place at South Kensington in 1870. Other exhibitions have since been held, and competitive prizes offered, one of which was gained by the Princess Louise. Modern collections of fans take their date from the emigration of many noble families from France at the time of the Revolution. Such objects were given as souvenirs and occasionally sold by families in straitened circumstances. A large number of fans of all sorts, principally those of the 18th century, French, English, German, Italian Spanish, &c., have been lately bequeathed to the South Kensington Museum.
Regarding the different parts of folding fans it may be well to state that the sticks are called in French ''brins'', the two outer guards ''panaches'', and the mount ''feuille''.<ref>J. H. Pollen [J.H.P.]. "Fan." ''Encyclopaedia Britannica'', 9th Edition (1875–1889). Vol. '''10''' ('''X'''). Adam and Charles Black (Publisher). https://archive.org/details/encyclopaedia-britannica-9ed-1875/Vol%209%20%28FAL-FYZ%29%20193323016.23/page/26/mode/2up (accessed January 2023): 27, Col. 1b – 28, Col. 1c.</ref></blockquote>
== Fancy-dress Ball ==
Fancy-dress (or costume) balls were popular and frequent in the U.K. and France as well as the rest of Europe during the 19th century. The themes and styles of the fancy-dress balls influenced those that followed.
At the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's 1897 fancy-dress ball]], the guests came dressed in costume from times before 1820, as instructed on '''the invitation''', but their clothing was much more about late-Victorian standards of beauty and fashion than the standards of whatever time period the portraits they were copying or basing their costumes on.
''The Queen'' published dress and fashion information and advice under the byline of Ardern Holt, who regularly answered questions from readers about fashion as well as about fancy dress. (More about Ardern Holt, which is almost certainly a pseudonym, can be found on the [[Social Victorians/People/Working in Publishing#Journalists|People Working in Publishing]] page.) Holt also ran wrote entire articles with suggestions for what might make an appealing fancy-dress costume as well as pointing readers away from costumes that had been worn too frequently. The suggestions for costumes are based on familiar types or portraits available to readers, similar to Holt's books on fancy dress, which ran through a number of editions in the 1880s and 1890s. Fancy-dress questions sometimes asked for details about costumes worn in theatrical or operatic productions, which Holt provides.
In November 1897, Holt refers to the Duchess of Devonshire's 2 July ball: "Since the famous fancy ball, given at Devonshire House during this year, historical fancy dresses have assumed a prominence that they had not hitherto known."<ref>Holt, Ardern. "Fancy Dress a la Mode." The ''Queen'' 27 November 1897, Saturday: 94 [of 145 in BNA; print p. 1026], Col. 1a [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0002627/18971127/459/0094.</ref> Holt goes on to provide a number of ideas for costumes for historical fancy dress, as always with a strong leaning toward Victorian standards of beauty and style and away from any concern for historical accuracy.
Ardern Holt published books on fancy dress as well as writing for the ''Queen'' and other periodicals, but not all of them were about fancy dress.
# ''Gentlemen's Fancy Dress: How to Choose It''. Wyman & Sons, 1882. (''Google Books'' has this: https://books.google.com/books/about/Gentlemen_s_Fancy_Dress.html?id=ED8CAAAAQAAJ.) Later editions: 1898 (HathiTrust)
# ''Fancy Dresses Described; Or, What to Wear at Fancy Balls''. Debenham & Freebody, 1882. Illustr., Lillian Young. (HathiTrust has this.) Later editions: 4th ed — 1884; 1887 (HathiTrust); 6th ed. — 1896 (HathiTrust)
As Leonore Davidoff says, "Every cap, bow, streamer, ruffle, fringe, bustle, glove and other elaboration symbolised some status category for the female wearer."<ref name=":1" />{{rp|93}} [handled under Elaborations]
=== Historical Accuracy ===
Many of the costumes at the ball were based on portraits, especially when the guest was dressed as a historical figure. If possible, we have found the portraits likely to have been the originals, or we have found, if possible, portraits that show the subjects from the two time periods at similar ages.
The way clothing was cut changed quite a bit between the 18th and 19th centuries. We think of Victorian clothing — particularly women's clothing, and particularly at the end of the century — as inflexible and restrictive, especially compared to 20th- and 21st-century customs permitting freedom of movement. The difference is generally evolutionary rather than absolute — that is, as time has passed since the 18th century, clothing has allowed an increasingly greater range of movement, especially for people who did not do manual labor.
By the end of the 19th century, garments like women's bodices and men's coats were made fitted and smooth by attention to the grain of the fabric and by the use of darts (rather than techniques that assembled many small, individual pieces of fabric).
* clothing construction and flat-pattern techniques
* Generally, the further back in time we go, the more 2-dimensional the clothing itself was.
==== Women's Versions of Historical Accuracy at the Ball ====
As always with this ball, whatever historical accuracy might be present in a woman's costume is altered so that the wearer is still a fashionable Victorian lady. What makes the costumes look "Victorian" to our eyes is the line of the silhouette caused by the foundation undergarments as well as the many "elaborations"<ref name=":1" />{{rp|93}}, mostly in the decorations, trim and accessories.
Also, the clothing hangs and drapes differently because the fabric was cut on grain and the shoulders were freed by the way the sleeves were set in.
==== Men's Versions of Historical Accuracy at the Ball ====
Because men were not wearing a Victorian foundation garment at the end of the century, the men's costumes at the ball are more historically accurate in some ways.
* Trim
* Mixing neck treatments
* Hair
* Breeches
* Shoes and boots
* Military uniforms, arms, gloves, boots
== Feathers and Plumes ==
=== Aigrette ===
Elizabeth Lewandowski defines ''aigrette'' as "France. Feather or plume from an egret or heron."<ref name=":7" />(5) Sometimes the newspapers use the term to refer to an accessory (like a fan or ornament on a hat) that includes such a feather or plume. The straight and tapered feathers in an aigrette are in a bundle.
=== Prince of Wales's Feathers or White Plumes ===
The feathers in an aigrette came from egrets and herons; Prince of Wales's feathers came from ostriches. A fuller discussion of Prince of Wales's feathers and the white ostrich plumes worn at court appears on [[Social Victorians/Victorian Things#Ostrich Feathers and Prince of Wales's Feathers|Victorian Things]].
For much of the late 18th and 19th centuries, white ostrich plumes were central to fashion at court, and at a certain point in the late 18th century they became required for women being presented to the monarch and for their sponsors. Our purpose here is to understand why women were wearing plumes at the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's 1897 fancy-dress ball]] as part of their costumes.
First published in 1893, [[Social Victorians/People/Lady Colin Campbell|Lady Colin Campbell]]'s ''Manners and Rules of Good Society'' (1911 edition) says that<blockquote>It was compulsory for both Married and Unmarried Ladies to Wear Plumes. The married lady’s Court plume consisted of three white feathers. An unmarried lady’s of two white feathers. The three white feathers should be mounted as a Prince of Wales plume and worn towards the left hand side of the head. Colored feathers may not be worn. In deep mourning, white feathers must be worn, black feathers are inadmissible.<p>
White veils or lace lappets must be worn with the feathers. The veils should not be longer than 45 inches.<ref>{{Cite web|url=https://www.edwardianpromenade.com/etiquette/the-court-presentation/|title=The Court Presentation|last=Holl|first=Evangeline|date=2007-12-07|website=Edwardian Promenade|language=en-US|access-date=2022-12-18}} https://www.edwardianpromenade.com/etiquette/the-court-presentation/.</ref></blockquote>[[Social Victorians/Victorian Things#Ostrich Feathers and Prince of Wales's Feathers|This fashion was imported from France]] in the mid 1770s.<ref>"Abstract" for Blackwell, Caitlin. "'<nowiki/>''The Feather'd Fair in a Fright''': The Emblem of the Feather in Graphic Satire of 1776." ''Journal for Eighteenth-Century Studies'' 20 January 2013 (Vol. 36, Issue 3): 353-376. ''Wiley Online'' DOI: https://doi.org/10.1111/j.1754-0208.2012.00550.x (accessed November 2022).</ref>
Separately, a secondary heraldic emblem of the Prince of Wales has been a specific arrangement of 3 ostrich feathers in a gold coronet<ref>{{Cite journal|date=2022-11-07|title=Prince of Wales's feathers|url=https://en.wikipedia.org/w/index.php?title=Prince_of_Wales%27s_feathers&oldid=1120556015|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Prince_of_Wales's_feathers.</ref> since King Edward III (1312–1377<ref>{{Cite journal|date=2022-12-14|title=Edward III of England|url=https://en.wikipedia.org/w/index.php?title=Edward_III_of_England&oldid=1127343221|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Edward_III_of_England.</ref>).
Some women at the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's 1897 fancy-dress ball]] wore white ostrich feathers in their hair, but most of them are not Prince of Wales's feathers. Most of the plumes in these portraits are arrangements of some kind of headdress to accompany the costume. A few, wearing what looks like the Princes of Wales's feathers, might be signaling that their character is royal or has royal ancestry. '''One of the women [which one?] was presented to the royals at this ball?'''
Here is the list of women who are wearing white ostrich plumes in their portraits in the [[Social Victorians/1897 Fancy Dress Ball/Photographs|''Diamond Jubilee Fancy Dress Ball'' album of 286 photogravure portraits]]:
# Kathleen Pelham-Clinton, the [[Social Victorians/People/Newcastle|Duchess of Newcastle]]
# [[Social Victorians/People/Louisa Montagu Cavendish|Luise Cavendish]], the Duchess of Devonshire
# Jesusa Murrieta del Campo Mello y Urritio (née Bellido), [[Social Victorians/People/Santurce|Marquisa de Santurce]]
# Lady [[Social Victorians/People/Farquhar|Emilie Farquhar]]
# Princess (Laura Williamina Seymour) Victor of [[Social Victorians/People/Gleichen#Laura%20Williamina%20Seymour%20of%20Hohenlohe-Langenburg|Hohenlohe Langenburg]]
# Louisa Acheson, [[Social Victorians/People/Gosford|Lady Gosford]]
# Alice Emily White Coke, [[Social Victorians/People/Leicester|Viscountess Coke]]
# Lady Mary Stewart, Helen Mary Theresa [[Social Victorians/People/Londonderry|Vane-Tempest-Stewart]]
#[[Social Victorians/People/Consuelo Vanderbilt Spencer-Churchill|Consuelo Vanderbilt Spencer-Churchill]], Duchess of [[Social Victorians/People/Marlborough|Marlborough]], dressed as the wife of the French Ambassador at the Court of Catherine of Russia (not white, but some color that reads dark in the black-and-white photograph)
#Mrs. Mary [[Social Victorians/People/Chamberlain|Chamberlain]] (at 491), wearing white plumes, as Madame d'Epinay
#Lady Clementine [[Social Victorians/People/Tweeddale|Hay]] (at 629), wearing white plumes, as St. Bris (''Les Huguenots'')
#[[Social Victorians/People/Meysey-Thompson|Lady Meysey-Thompson]] (at 391), wearing white plumes, as Elizabeth, Queen of Bohemia
#Mrs. [[Social Victorians/People/Grosvenor|Algernon (Catherine) Grosvenor]] (at 510), wearing white plumes, as Marie Louise
#Lady [[Social Victorians/People/Ancaster|Evelyn Ewart]], at 401), wearing white plumes, as the Duchess of Ancaster, Mistress of the Robes to Queen Charlotte, 1757, after a picture by Hudson
#[[Social Victorians/People/Lyttelton|Edith Sophy Balfour Lyttelton]] (at 580), wearing what might be white plumes on a large-brimmed white hat, after a picture by Romney
#[[Social Victorians/People/Yznaga|Emilia Yznaga]] (at 360), wearing what might be white plumes, as Cydalise of the Comedie Italienne from the time of Louis XV
#Lady [[Social Victorians/People/Ilchester|Muriel Fox Strangways]] (at 403), wearing what might be two smallish white plumes, as Lady Sarah Lennox, one of the bridesmaids of Queen Charlotte A.D. 1761
#Lady [[Social Victorians/People/Lucan|Violet Bingham]] (at 586), wearing perhaps one white plume in a headdress not related to the Prince of Wales's feathers
#Rosamond Fellowes, [[Social Victorians/People/de Ramsey|Lady de Ramsey]] (at 329), wearing a headdress that includes some white plumes, as Lady Burleigh
#[[Social Victorians/People/Dupplin|Agnes Blanche Marie Hay-Drummond]] (at 682), in a big headdress topped with white plumes, as Mademoiselle Andrée de Taverney A.D. 1775
#Florence Canning, [[Social Victorians/People/Garvagh|Lady Garvagh]] (at 336), wearing what looks like Prince of Wales's plumes
#[[Social Victorians/People/Suffolk|Marguerite Hyde "Daisy" Leiter]] (at 684), wearing what looks like Prince of Wales's plumes
#Lady [[Social Victorians/People/Spicer|Margaret Spicer]] (at 281), wearing one smallish white and one black plume, as Countess Zinotriff, Lady-in-Waiting to the Empress Catherine of Russia
#Mrs. [[Social Victorians/People/Cavendish Bentinck|Arthur James]] (at 318), wearing what looks like Prince of Wales's plumes, as Elizabeth Cavendish, daughter of Bess of Hardwick
#Nellie, [[Social Victorians/People/Kilmorey|Countess of Kilmorey]] (at 207), wearing three tall plumes, 2 white and one dark, as Comtesse du Barri
#Daisy, [[Social Victorians/People/Warwick|Countess of Warwick]] (at 53), wearing at least 1 white plume, as Marie Antoinette
More men than women were wearing plumes reminiscent of the Prince of Wales's feathers:
*
==== Bibliography for Plumes and Prince of Wales's Feathers ====
* Blackwell, Caitlin. "'''The Feather'd Fair in a Fright'<nowiki/>'': The Emblem of the Feather in Graphic Satire of 1776." Journal for ''Eighteenth-Century Studies'' 20 January 2013 (Vol. 36, Issue 3): 353-376. Wiley Online DOI: https://doi.org/10.1111/j.1754-0208.2012.00550.x.
* "Prince of Wales's feathers." ''Wikipedia'' https://en.wikipedia.org/wiki/Prince_of_Wales%27s_feathers (accessed November 2022). ['''Add women to this page''']
* Simpson, William. "On the Origin of the Prince of Wales' Feathers." ''Fraser's magazine'' 617 (1881): 637-649. Hathi Trust https://babel.hathitrust.org/cgi/pt?id=chi.79253140&view=1up&seq=643&q1=feathers (accessed December 2022). Deals mostly with use of feathers in other cultures and in antiquity; makes brief mention of feathers and plumes in signs and pub names that may not be associated with the Prince of Wales. No mention of the use of plumes in women's headdresses or court dress.
== Honors ==
=== The Bath ===
The Most Honourable Order of the Bath (GCB, Knight or Dame Grand Cross; KCB or DCB, Knight or Dame Commander; CB, Companion)
=== The Garter ===
The Most Noble Order of the Knights of the Garter (KG, Knight Companion; LG, Lady Companion)
[[File:The Golden Fleece - collar exhibited at MET, NYC.jpg|thumb|The Golden Fleece collar and pendant for the 2019 "Last Knight" exhibition at the MET, NYC.|alt=Recent photograph of a gold necklace on a wide band, with a gold skin of a sheep hanging from it as a pendant]]
=== The Golden Fleece ===
To wear the golden fleece is to wear the insignia of the Order of the Golden Fleece, said to be "the most prestigious and historic order of chivalry in the world" because of its long history and strict limitations on membership.<ref name=":10">{{Cite journal|date=2020-09-25|title=Order of the Golden Fleece|url=https://en.wikipedia.org/w/index.php?title=Order_of_the_Golden_Fleece&oldid=980340875|journal=Wikipedia|language=en}}</ref> The monarchs of the U.K. were members of the originally Spanish order, as were others who could afford it, like the Duke of Wellington,<ref name=":12">Thompson, R[obert]. H[ugh]. "The Golden Fleece in Britain." Publication of the ''British Numismatic Society''. 2009 https://www.britnumsoc.org/publications/Digital%20BNJ/pdfs/2009_BNJ_79_8.pdf (accessed January 2023).</ref> the first Protestant to be admitted to the order.<ref name=":10" /> Founded in 1429/30 by Philip the Good, Duke of Burgundy, the order separated into two branches in 1714, one Spanish and the other Austrian, still led by the House of Habsburg.<ref name=":10" />
[[File:Prince Albert - Franz Xaver Winterhalter 1842.jpg|thumb|1842 Winterhalter portrait of Prince Albert wearing the insignia of the Order of the Golden Fleece, 1842|left|alt=1842 Portrait of Prince Albert by Winterhalter, wearing the insignia of the Golden Fleece]]
The photograph (upper right) is of a Polish badge dating from the "turn of the XV and XVI centuries."<ref>{{Citation|title=Polski: Kolana orderowa orderu Złotego Runa, przełom XV i XVI wieku.|url=https://commons.wikimedia.org/wiki/File:The_Golden_Fleece_-_collar_exhibited_at_MET,_NYC.jpg|date=2019-11-10|accessdate=2023-01-10|last=Wulfstan}}. https://commons.wikimedia.org/wiki/File:The_Golden_Fleece_-_collar_exhibited_at_MET,_NYC.jpg.</ref> The collar to this Golden Fleece might be similar to the one the [[Social Victorians/People/Spencer Compton Cavendish#The Insignia of the Order of the Golden Fleece|Duke of Devonshire is wearing in the 1897 Lafayette portrait]].
The badges and collars that Knights of the Order actually wore vary quite a bit.
The 1842 Franz Xaver Winterhalter portrait (left) of Prince Consort Albert, Victoria's husband and father of the Prince of Wales, shows him wearing the Golden Fleece on a red ribbon around his neck and the star of the Garter on the front of his coat.<ref>Winterhalter, Franz Xaver. ''Prince Albert''. {{Cite web|url=https://www.rct.uk/collection/search#/16/collection/401412/prince-albert-1819-61|title=Explore the Royal Collection Online|website=www.rct.uk|access-date=2023-01-16}} https://www.rct.uk/collection/search#/16/collection/401412/prince-albert-1819-61.</ref>
=== Royal Victorian Order ===
(GCVO, Knight or Dame Grand Cross; KCVO or DCVO, Knight or Dame Commander; CVO, Commander; LVO, Lieutenant; MVO, Member)
=== St. John ===
The Order of the Knights of St. John
=== Star of India ===
Most Exalted Order of the Star of India (GCSI, Knight Grand Commander; KCSI, Knight Commander; CSI, Companion)
=== Thistle ===
The Most Ancient and Most Noble Order of the Thistle
== Jewelry and Stones ==
=== Cabochon ===
This term describes both the treatment and shape of a precious or semiprecious stone. A cabochon treatment does not facet the stone but merely polishes it, removing "the rough parts" and the parts that are not the right stone.<ref>"cabochon, n." ''OED Online'', Oxford University Press, December 2022, www.oed.com/view/Entry/25778. Accessed 7 February 2023.</ref> A cabochon shape is often flat on one side and oval or round, forming a mound in the setting.
=== Jet ===
=== ''Orfèvrerie'' ===
Sometimes misspelled in the newspapers as ''orvfèvrerie''. ''Orfèvrerie'' is the artistic work of a goldsmith, silversmith, or jeweler.
=== Turquoises ===
== Military ==
Several men from the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's 1897 fancy-dress ball at Devonshire House]] were dressed in military uniforms, some historical and some, possibly, not.
=== Baldric ===
According to the ''Oxford English Dictionary'', the primary sense of ''baldric'' is "A belt or girdle, usually of leather and richly ornamented, worn pendent from one shoulder across the breast and under the opposite arm, and used to support the wearer's sword, bugle, etc."<ref>"baldric, n." ''OED Online'', Oxford University Press, March 2023, www.oed.com/view/Entry/14849. Accessed 17 May 2023.</ref> This sense has been in existence since c. 1300.
=== Cuirass ===
According to the ''Oxford English Dictionary'', the primary sense of ''cuirass'' is "A piece of armour for the body (originally of leather); ''spec.'' a piece reaching down to the waist, and consisting of a breast-plate and a back-plate, buckled or otherwise fastened together ...."<ref>"cuirass, n." ''OED Online'', Oxford University Press, March 2023, www.oed.com/view/Entry/45604. Accessed 17 May 2023.</ref>
[[File:Knötel IV, 04.jpg|thumb|alt=An Old drawing in color of British soldiers on horses brandishing swords in 1815.|1890 illustration of the Household Cavalry (Life Guard, left; Horse Guard, right) at the Battle of Waterloo, 1815]]
=== Household Cavalry ===
The Royal Household contains the Household Cavalry, a corps of British Army units assigned to the monarch. It is made up of 2 regiments, the Life Guards and what is now called The Blues and Royals, which were formed around the time of "the Restoration of the Monarchy in 1660."<ref name=":3">Joll, Christopher. "Tales of the Household Cavalry, No. 1. Roles." The Household Cavalry Museum, https://householdcavalry.co.uk/app/uploads/sites/2/2021/06/Household-Cavalry-Museum-video-series-large-print-text-Tales-episode-01.pdf.</ref>{{rp|1}} Regimental Historian Christopher Joll says, "the original Life Guards were formed as a mounted bodyguard for the exiled King Charles II, The Blues were raised as Cromwellian cavalry and The Royals were established to defend Tangier."<ref name=":3" />{{rp|1–2}} The 1st and 2nd Life Guards were formed from "the Troops of Horse and Horse Grenadier Guards ... in 1788."<ref name=":3" />{{rp|3}} The Life Guards were and are still official bodyguards of the queen or king, but through history they have been required to do quite a bit more than serve as bodyguards for the monarch.
The Household Cavalry fought in the Battle of Waterloo on Sunday, 18 June 1815 as heavy cavalry.<ref name=":3" />{{rp|3}} Besides arresting the Cato Steet conspirators in 1820 "and guarding their subsequent execution," the Household Cavalry contributed to the "the expedition to rescue General Gordon, who was trapped in Khartoum by The Mahdi and his army of insurgents" in 1884.<ref name=":3" />{{rp|3}} In 1887 they "were involved ... in the suppression of rioters in Trafalgar Square on Bloody Sunday."<ref name=":3" />{{rp|3}}
==== Grenadier Guards ====
Three men — [[Social Victorians/People/Gordon-Lennox#Lord Algernon Gordon Lennox|Lord Algernon Gordon-Lennox]], [[Social Victorians/People/Stanley#Edward George Villiers Stanley, Lord Stanley|Lord Stanley]], and [[Social Victorians/People/Stanley#Hon. Ferdinand Charles Stanley|Hon. F. C. Stanley]] — attended the ball as officers of the Grenadier Guards, wearing "scarlet tunics, ... full blue breeches, scarlet hose and shoes, lappet wigs" as well as items associated with weapons and armor.<ref name=":14">“The Duchess of Devonshire’s Ball.” The ''Gentlewoman'' 10 July 1897 Saturday: 32–42 [of 76], Cols. 1a–3c [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0003340/18970710/155/0032.</ref>{{rp|p. 34, Col. 2a}}
Founded in England in 1656 as Foot Guards, this infantry regiment "was granted the 'Grenadier' designation by a Royal Proclamation" at the end of the Napoleonic Wars.<ref>{{Cite journal|date=2023-04-22|title=Grenadier Guards|url=https://en.wikipedia.org/w/index.php?title=Grenadier_Guards&oldid=1151238350|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Grenadier_Guards.</ref> They were not called Grenadier Guards, then, before about 1815. In 1660, the Stuart Restoration, they were called Lord Wentworth's Regiment, because they were under the command of Thomas Wentworth, 5th Baron Wentworth.<ref>{{Cite journal|date=2022-07-24|title=Lord Wentworth's Regiment|url=https://en.wikipedia.org/w/index.php?title=Lord_Wentworth%27s_Regiment&oldid=1100069077|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Lord_Wentworth%27s_Regiment.</ref>
At the time of Lord Wentworth's Regiment, the style of the French cavalier had begun to influence wealthy British royalists. In the British military, a Cavalier was a wealthy follower of Charles I and Charles II — a commander, perhaps, or a field officer, but probably not a soldier.<ref>{{Cite journal|date=2023-04-22|title=Cavalier|url=https://en.wikipedia.org/w/index.php?title=Cavalier&oldid=1151166569|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Cavalier.</ref>
The Guards were busy as infantry in the 17th century, engaging in a number of armed conflicts for Great Britain, but they also served the sovereign. According to the Guards Museum,<blockquote>In 1678 the Guards were ordered to form Grenadier Companies, these men were the strongest and tallest of the regiment, they carried axes, hatches and grenades, they were the shock troops of their day. Instead of wearing tri-corn hats they wore a mitre shaped cap.<ref>{{Cite web|url=https://theguardsmuseum.com/about-the-guards/history-of-the-foot-guards/history-page-2/|title=Service to the Crown|website=The Guards Museum|language=en-GB|access-date=2023-05-15}} https://theguardsmuseum.com/about-the-guards/history-of-the-foot-guards/history-page-2/.</ref></blockquote>The name comes from ''grenades'', then, and we are accustomed to seeing them in front of Buckingham Palace, with their tall mitre hats.
The Guard fought in the American Revolution, and in the 19th century, the Grenadier Guards fought in the Crimean War, Sudan and the Boer War. They have roles as front-line troops and as ceremonial for the sovereign, which makes them elite:<blockquote>Queen Victoria decreed that she did not want to see a single chevron soldier within her Guards. Other then [sic] the two senior Warrant Officers of the British Army, the senior Warrant Officers of the Foot Guards wear a large Sovereigns personal coat of arms badge on their upper arm. No other regiments of the British Army are allowed to do so; all the others wear a small coat of arms of their lower arms. Up until 1871 all officers in the Foot Guards had the privilege of having double rankings. An Ensign was ranked as an Ensign and Lieutenant, a Lieutenant as Lieutenant and Captain and a Captain as Captain and Lieutenant Colonel. This was because at the time officers purchased their own ranks and it cost more to purchase a commission in the Foot Guards than any other regiments in the British Army. For example if it cost an officer in the Foot Guards £1,000 for his first rank, in the rest of the Army it would be £500 so if he transferred to another regiment he would loose [sic] £500, hence the higher rank, if he was an Ensign in the Guards and he transferred to a Line Regiment he went in at the higher rank of Lieutenant.<ref>{{Cite web|url=https://theguardsmuseum.com/about-the-guards/history-of-the-foot-guards/history-page-1/|title=Formation and role of the Regiments|website=The Guards Museum|language=en-GB|access-date=2023-05-15}} https://theguardsmuseum.com/about-the-guards/history-of-the-foot-guards/history-page-1/.</ref></blockquote>
==== Life Guards ====
[[Social Victorians/People/Shrewsbury#Reginald Talbot's Costume|General the Hon. Reginald Talbot]], a member of the 1st Life Guards, attended the Duchess of Devonshire's ball dressed in the uniform of his regiment during the Battle of Waterloo.<ref name=":14" />{{rp|p. 36, Col. 3b}}
At the Battle of Waterloo the 1st Life Guards were part of the 1st Brigade — the Household Brigade — and were commanded by Major-General Lord Edward Somerset.<ref name=":4">{{Cite journal|date=2023-09-30|title=Battle of Waterloo|url=https://en.wikipedia.org/w/index.php?title=Battle_of_Waterloo&oldid=1177893566|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Battle_of_Waterloo.</ref> The 1st Life Guards were on "the extreme right" of a French countercharge and "kept their cohesion and consequently suffered significantly fewer casualties."<ref name=":4" />
== Photography ==
== Footnotes ==
{{reflist}}
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Nowadays pornography is more available than ever. It is free and people only needs internet to access billions of bites of pornography. However, is it good for human development to have that access? Is it bad for young people who are only starting to experience their sexual life? Does pornography opposes to sex education? Are there any ethical considerations to have in mind regarding pornography?
== It is correct to watch pornography ==
=== Arguments for ===
* {{Argument for}} There is plenty of porn made in an ethical and safe way<ref>{{Cite web|url=https://mysteryvibe.com/blogs/learn/guide-ethical-porn|title=Our Guide To: Ethical Porn|website=MysteryVibe|language=en|access-date=2023-03-10}}</ref>. There should not be any problem with watching this type of porn.
** {{Objection}} It is not always possible to know if the porn you are watching is ethical.
* {{Argument for}} Certifications allow consumers to choose porn with the consent of the participants.
** {{Objection}} If a person could be coerced into making porn, [[wikt:thon#Pronoun|wt:thon]] could coerced into signing a piece or paper or holding up a little sign.
** {{Objection}} Feminist and ethical porn spaces represent a small proportion of porn overall. Most and/or Mainstream porn depicts the subjugation of women, including activities such as "facials", gaging and fake female orgasms. Such has and still dominates the internet porn world: and is probably getting worse. Also, porn such as BBC, indicates an unhealthy obsession with race.
*** {{Objection}} The issue isn't the actions, but context. e.g. women watching facials, gang-bangs, and choking isn't problematic, or at least as problematic, as men watching it.
**** {{Objection}} This doesn't respond to the problem of men watching it: who are also likely in the majority of viewers of such.
*** {{Objection}} There is no "feminist and ethical porn." Porn is the objectification of people—mostly women—and there is nothing feminist nor ethical about that.
** {{Objection}} which is why countries such as the US, Canada, et al, get rid of their stupid laws against obscenity and pornography and allow people to produce porn more freely and officially.
*{{Argument for}} Ethical porn focuses on treating performers fairly, showing a diverse range of bodies and representing more realistic human interactions<ref>{{Cite web|url=https://www.dailydot.com/nsfw/guides/porn-ethical-premium/|title=The 8 best sites to watch ethical, fair trade porn|last=Danger|first=Danni|date=2017-12-16|website=The Daily Dot|language=en-US|access-date=2023-03-10}}</ref>.
*{{Argument for}} Porn is made to promote arousal, entertainment and even escapism.
* {{Argument for}} Masturbation is natural and beneficial<ref>{{Cite web|url=https://www.besthealthmag.ca/article/6-healthy-reasons-to-masturbate/|title=It’s Time for Some Self-Love—Here Are 6 Ways Masturbation Is Good for You|last=src="https://secure.gravatar.com/avatar/?s=96|first=<img class="avatar" alt="Best Health"|last2=#038;d=mm|date=2022-05-26|website=Best Health|language=en-US|access-date=2023-03-10|last3=May 25|first3=#038;r=g" width="50" height="50">Best HealthUpdated:|last4=2022}}</ref><ref>{{Cite web|url=https://www.medicalnewstoday.com/articles/319536|title=Frequent ejaculation and prostate cancer: What's the link?|date=2022-03-28|website=www.medicalnewstoday.com|language=en|access-date=2023-03-10}}</ref>, and porn is a useful masturbation aid.
** {{Objection}} It is not beneficial if it reduces people's motivation to connect who may find partners and sex otherwise.
** {{Objection}} Porn masturbation seems to reduce testosterone<ref>https://doi.org/10.1007/s003450100222</ref> and may reduce androgen receptor density<ref>https://doi.org/10.1159/000099250</ref> which can be problematic
** {{Objection}} You don't need porn to masturbate. Over consumption might lead to needing porn to masturbate.
* {{Argument for}} It’s the obligation of producers, platforms and law enforcement to ensure legal and ethical standards are met. On this basis (and also using your own judgment in addition), there’s nothing incorrect in consuming available material.
* {{Argument for}} Pornography is like a style of music, such as rap, heavy metal, or blue grass: each is harmless fun some people engage in. Some people enjoy watching ball games, others "vanilla" soap operas, and some enjoy watch some porn actress partaking in "gang bangs": to each [[wikt:thon#Pronoun|wt:thon]]s own.
* {{Argument for}} It has [[w:Miller v. California|serious literary, artistic, political, or scientific value]].
** {{Objection}} Such could be exempted: perhaps producers can try to have such reviewed by government boards.
* {{Argument for}} Most pornography is for and/or consumed by losers who can't find dates or interest in sex with them from neglected and hardworking wives, but at least it's not sexual assault or bothering women for sexual favors.
** {{Objection}} Maybe such "losers" should "get a life" and work on self-improvent, much like drug policies such as harm-reducting might be better than prohibition, but it would be even better if the addict worked on [[wikt:thon#Pronoun|wt:thon]]s addiction(s).
** {{Objection}} Porn is probably a major reason why people aren't reproducing and population rates are falling.
*** {{Objection}} Good! The Earth has too many people already. (See also: [[Should we aim to reduce the Earth population?]])
**** {{Objection}} National governments need more taxpayers, companies need more consumers, old people need young people to take care of them when they become too infirmed.
* {{Argument for}} Since the 1990s with the internet, nay, since the popularity of video cassettes and photocopying, supression of porn is infeasible: resources are better spent in mitigating and controlling its harm (if any).
** {{Objection}} The question is not whether it should be suppressed but whether people should make that personal choice (or make that choice often) or whether it should be as encouraged and promoted as it is in today's world with phenomena like for example OnlyFans.
** {{Objection}} Just because it's considered difficult, maybe even impossible, doesn't mean we shouldn't try.
* {{Argument for}} Sometimes porn can help individuals discover and explore their sexuality.
=== Arguments against ===
* {{Argument against}} People can become addicted to pornography, which is as problematic as drug or alcohol adiction. For example, porn acts affecting the reward circuitry<ref>{{Cite journal|last=Watts|first=Clark|last2=Hilton|first2=DonaldL|date=2011|title=Pornography addiction: A neuroscience perspective|url=https://surgicalneurologyint.com/surgicalint-articles/pornography-addiction-a-neuroscience-perspective/|journal=Surgical Neurology International|language=en|volume=2|issue=1|pages=19|doi=10.4103/2152-7806.76977|issn=2152-7806|pmc=PMC3050060|pmid=21427788}}</ref><ref>{{Cite journal|last=Love|first=Todd|last2=Laier|first2=Christian|last3=Brand|first3=Matthias|last4=Hatch|first4=Linda|last5=Hajela|first5=Raju|date=2015-09-18|title=Neuroscience of Internet Pornography Addiction: A Review and Update|url=http://www.mdpi.com/2076-328X/5/3/388|journal=Behavioral Sciences|language=en|volume=5|issue=3|pages=388–433|doi=10.3390/bs5030388|issn=2076-328X|pmc=PMC4600144|pmid=26393658}}</ref>, or causing the person to lose contact with the real world sexual situations<ref>{{Cite web|url=https://auresnotes.com/summary-your-brain-on-porn-gary-wilson/|title=Your Brain on Porn Summary - Gary Wilson - Aure's Notes|date=2022-04-06|language=en-US|access-date=2023-03-10}}</ref>.
** {{Objection}} People can become addicted to nearly everything, including sex.
** {{Objection}} Very few survey respondents in Australia reported that they were addicted to pornography (men 4%, women 1%), and of those who said they were addicted only about half also reported that using pornography had had a bad effect on them<ref>{{Cite journal|last=Rissel|first=Chris|last2=Richters|first2=Juliet|last3=de Visser|first3=Richard O.|last4=McKee|first4=Alan|last5=Yeung|first5=Anna|last6=Caruana|first6=Theresa|date=2017-02-12|title=A Profile of Pornography Users in Australia: Findings From the Second Australian Study of Health and Relationships|url=https://www.tandfonline.com/doi/full/10.1080/00224499.2016.1191597|journal=The Journal of Sex Research|language=en|volume=54|issue=2|pages=227–240|doi=10.1080/00224499.2016.1191597|issn=0022-4499}}</ref>.
*** {{Objection}} Reality sucks: my fantasy partner(s) never let me down. (Let Chad have his harem; I'll have mine.) ([https://www.youtube.com/watch?v=iyI542boYs0 "Goddess of Empathy" 00:00 to 00:44, and 01:44 to 02:13])
** {{Objection}} Things such as drug addictions are individualistic insofar that if a person consumes a drug, or any other sudstance, no one else is required to participate. Normal sex requires two people and the second person might not be willing, hence the first person resorts to porn to at least somewhat please [[wikt:thon#Pronoun|wt:thon]]self.
* {{Argument against}} Porn is damaging to relationships. One can get demotivated to find a real partner when they are satisfied by porn. It might be a problem, both personal and social.
** {{Objection}} Porn can be used to improve a couple's sex life by bringing new ideas. It can also inspire one to find a partner to share the experience.
** {{Objection}} Research suggests that there is no evidence of porn consumption being either adaptive or maladaptive when it comes to relationship satisfaction, closeness, and loneliness<ref>{{Cite journal|last=Hesse|first=Colin|last2=Floyd|first2=Kory|date=2019-11|title=Affection substitution: The effect of pornography consumption on close relationships|url=http://journals.sagepub.com/doi/10.1177/0265407519841719|journal=Journal of Social and Personal Relationships|language=en|volume=36|issue=11-12|pages=3887–3907|doi=10.1177/0265407519841719|issn=0265-4075}}</ref>.
*** {{Objection}} That study only assessed 357 adults, too unrepresentative of the average porn consumer.
*** {{Objection}} That study says porn comes to replace affection deprivation, and fight loneliness and depression. The same situations that leads a person to drugs or alcohol consumption. Thus, porn is replacing a deep job the person has to do with itself with the help of psychologists.
* {{Argument against}} Watching pornography contributes to human rights abuses, including sex trafficking and gender inequality<ref>{{Cite web|url=https://humantraffickingsearch.org/the-connection-between-sex-trafficking-and-pornography/|title=The Connection Between Sex Trafficking and Pornography|last=Search|first=Human Trafficking|date=2014-04-14|website=Human Trafficking Search|language=en-US|access-date=2023-03-10}}</ref>.
** {{Objection}} It is possible to choose a platform that ony features material with high ethical standards (see [[w:en:Feminist pornography]]).
** {{Objection}} no more than watching depictions of violence is being violent.
* {{Argument against}} People feel guilt and shame after watching pornography, which indicates that our conscience finds that the consumption of pornography is wrong<ref>{{Cite web|url=https://www.onlyyouforever.com/how-shame-perpetuates-porn-addiction/|title=How Shame Perpetuates Porn Addiction|last=Simonyi-Gindele|first=Caleb|date=2019-08-21|website=OnlyYouForever|language=en-US|access-date=2023-03-10}}</ref><ref>{{Cite web|url=https://www.therecoveryvillage.com/process-addiction/porn-addiction/pornography-statistics/|title=Pornography Facts and Statistics {{!}} Effects of Porn Addiction on Relationships|website=The Recovery Village Drug and Alcohol Rehab|language=en-US|access-date=2023-03-10}}</ref>.
** {{Objection}} many LGBTQI2S+ feel shame for being LGBTQI2S+: Christianity, Islam, Judaism, et al has a lot to answer for here.
* {{Argument against}} Mainstream pornography engenders and cultivates a subliminally misogynistic or otherwise violent way of men seeing women.
** {{Objection}} mainstream porn ≠ porn; mainstream porn ⊂ porn.
* {{Argument against}} [[wikisource:Bible (Tyndale)/Matthew#Chapter 5]] ''"27 Ye haue hearde howe it was sayde to the of olde tyme: Thou shalt not comitt advoutrie.<br/>28 But I say vnto you that whosoeuer looketh on a wyfe lustynge after her hathe comitted advoutrie with hir alredy in his hert."''<br/><br/>By looking at a woman with lust in your heart, you are essentially committing adultery with her.
* {{Argument against}} One will not die if one is deprived of it.
* {{Argument against}} Children might see it, be it a magazine on a rack, a late night television show (and/or cable or video), or someone's computer, and such might scar their tender minds.
** {{Objection}} Make sure they don't see it.
* {{Argument against}} Average consumption almost certainly involves parts of the psyche that are more vulnerable than the conscious intellect. Our minds are vulnerable beyond the scope of our conscious awareness which may lead to a confusion of what happens on screen with reality.
** {{Objection}} One should be mature enough.
* {{Argument against}} Amateur porn is the 11th most popular genre on Pornhub<ref>{{Cite web|url=https://www.esquire.com/lifestyle/sex/news/a52061/most-popular-porn-searches/|title=The Human Race Really Outdid Itself with Porn Searches in 2018|date=2018-12-12|website=Esquire|language=en-US|access-date=2023-03-10}}</ref>, and it is possible a lot of that porn was leaked without the consent of one of the participants, or even forcing one participant to act.
** {{Objection}} Maybe ban amateur, while keeping other forms of porn.
== Notes and references ==
{{Reflist}}
[[Category:Sexuality]]
[[Category:Pornography]]
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|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
| {{{4}}}
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
| {{{5}}}
!class="sum"| {{{71}}}
|-
! 2
| {{{6}}}
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
| {{{7}}}
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
| {{{8}}}
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
| {{{9}}}
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
| {{{10}}}
!class="sum"| {{{72}}}
|- style="font-size: {{{76|80}}}%;"
! 3
| {{{11}}}
|class="dummy"|
|class="dummy"|
|class="dummy"|
| {{{12}}}
|class="dummy"|
|class="dummy"|
|class="dummy"|
| {{{13}}}
|class="dummy"|
|class="dummy"|
|class="dummy"|
| {{{14}}}
|class="dummy"|
|class="dummy"|
|class="dummy"|
| {{{15}}}
|class="dummy"|
|class="dummy"|
|class="dummy"|
| {{{16}}}
|class="dummy"|
|class="dummy"|
|class="dummy"|
| {{{17}}}
|class="dummy"|
|class="dummy"|
|class="dummy"|
| {{{18}}}
|class="dummy"|
|class="dummy"|
|class="dummy"|
| {{{19}}}
!class="sum"| {{{73}}}
|- style="font-size: {{{77|70}}}%;"
! 4
| {{{20}}}
|class="dummy"|
| {{{21}}}
|class="dummy"|
| {{{22}}}
|class="dummy"|
| {{{23}}}
|class="dummy"|
| {{{24}}}
|class="dummy"|
| {{{25}}}
|class="dummy"|
| {{{26}}}
|class="dummy"|
| {{{27}}}
|class="dummy"|
| {{{28}}}
|class="dummy"|
| {{{29}}}
|class="dummy"|
| {{{30}}}
|class="dummy"|
| {{{31}}}
|class="dummy"|
| {{{32}}}
|class="dummy"|
| {{{33}}}
|class="dummy"|
| {{{34}}}
|class="dummy"|
| {{{35}}}
|class="dummy"|
| {{{36}}}
!class="sum"| {{{74}}}
|- style="font-size: {{{78|50}}}%;"
! 5
| {{{37}}}
| {{{38}}}
| {{{39}}}
| {{{40}}}
| {{{41}}}
| {{{42}}}
| {{{43}}}
| {{{44}}}
| {{{45}}}
| {{{46}}}
| {{{47}}}
| {{{48}}}
| {{{49}}}
| {{{50}}}
| {{{51}}}
| {{{52}}}
| {{{53}}}
| {{{54}}}
| {{{55}}}
| {{{56}}}
| {{{57}}}
| {{{58}}}
| {{{59}}}
| {{{60}}}
| {{{61}}}
| {{{62}}}
| {{{63}}}
| {{{64}}}
| {{{65}}}
| {{{66}}}
| {{{67}}}
| {{{68}}}
| {{{69}}}
!class="sum"| {{{75}}}
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<!-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------->
{{Collapsible START|integer weight|collapsed light wide followed}}
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! {{diagonal split header|''a''|''w''}}
! 0
! 1
! 2
! 3
! 4
! 5
! 6
! 7
! 8
! 9
! 10
! 11
! 12
! 13
! 14
! 15
! 16
! 17
! 18
! 19
! 20
! 21
! 22
! 23
! 24
! 25
! 26
! 27
! 28
! 29
! 30
! 31
! 32
!class="sum"| sums
|-
! 0
| {{{1}}}
| {{{2}}}
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
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|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
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|class="dummy"|
|class="dummy"|
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|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
!class="sum"| {{{70}}}
|-
! 1
| {{{3}}}
| {{{4}}}
| {{{5}}}
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
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|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
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|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
!class="sum"| {{{71}}}
|-
! 2
| {{{6}}}
| {{{7}}}
| {{{8}}}
| {{{9}}}
| {{{10}}}
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
!class="sum"| {{{72}}}
|- style="font-size: {{{76|80}}}%;"
! 3
| {{{11}}}
| {{{12}}}
| {{{13}}}
| {{{14}}}
| {{{15}}}
| {{{16}}}
| {{{17}}}
| {{{18}}}
| {{{19}}}
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
!class="sum"| {{{73}}}
|- style="font-size: {{{77|70}}}%;"
! 4
| {{{20}}}
| {{{21}}}
| {{{22}}}
| {{{23}}}
| {{{24}}}
| {{{25}}}
| {{{26}}}
| {{{27}}}
| {{{28}}}
| {{{29}}}
| {{{30}}}
| {{{31}}}
| {{{32}}}
| {{{33}}}
| {{{34}}}
| {{{35}}}
| {{{36}}}
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
!class="sum"| {{{74}}}
|- style="font-size: {{{78|50}}}%;"
! 5
| {{{37}}}
| {{{38}}}
| {{{39}}}
| {{{40}}}
| {{{41}}}
| {{{42}}}
| {{{43}}}
| {{{44}}}
| {{{45}}}
| {{{46}}}
| {{{47}}}
| {{{48}}}
| {{{49}}}
| {{{50}}}
| {{{51}}}
| {{{52}}}
| {{{53}}}
| {{{54}}}
| {{{55}}}
| {{{56}}}
| {{{57}}}
| {{{58}}}
| {{{59}}}
| {{{60}}}
| {{{61}}}
| {{{62}}}
| {{{63}}}
| {{{64}}}
| {{{65}}}
| {{{66}}}
| {{{67}}}
| {{{68}}}
| {{{69}}}
!class="sum"| {{{75}}}
|}
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<!-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------->
<templatestyles src="Template:Boolf triangles/style.css" />
{{Collapsible START|<small>(merged weights)</small>|collapsed wide light}}
{| class="wikitable boolf-triangle" style="margin: 0;"
|-
!style="color: gray;"| ''w''
!style="color: gray; font-size: 60%;"| 0
!style="color: gray;"| 1
!style="color: gray;"| 2
!style="color: gray;"| 3...4
!style="color: gray;"| 5...8
!style="color: gray;"| 9...16
!style="color: gray;"| 17...32
!rowspan="2" class="sum"| sums
|-
! {{diagonal split header|''a''|''k''}}
!style="font-size: 60%;"| -1
! 0
! 1
! 2
! 3
! 4
! 5
|-
! 0
|style="font-size: 60%;"| {{{1}}}
| {{{2}}}
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
!class="sum"| {{{70}}}
|-
! 1
|style="font-size: 60%;"| {{{3}}}
| {{{4}}}
| {{{5}}}
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
!class="sum"| {{{71}}}
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! 2
|style="font-size: 60%;"| {{{6}}}
| {{{7}}}
| {{{8}}}
| {{#expr: {{{9}}} + {{{10}}} }}
|class="dummy"|
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!class="sum"| {{{72}}}
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! 3
|style="font-size: 60%;"| {{{11}}}
| {{{12}}}
| {{{13}}}
| {{#expr: {{{14}}} + {{{15}}} }}
| {{#expr: {{{16}}} + {{{17}}} + {{{18}}} + {{{19}}} }}
|class="dummy"|
|class="dummy"|
!class="sum"| {{{73}}}
|-
! 4
|style="font-size: 60%;"| {{{20}}}
| {{{21}}}
| {{{22}}}
| {{#expr: {{{23}}} + {{{24}}} }}
| {{#expr: {{{25}}} + {{{26}}} + {{{27}}} + {{{28}}} }}
| {{#expr: {{{29}}} + {{{30}}} + {{{31}}} + {{{32}}} + {{{33}}} + {{{34}}} + {{{35}}} + {{{36}}} }}
|class="dummy"|
!class="sum"| {{{74}}}
|-
! 5
|style="font-size: 60%;"| {{{37}}}
| {{{38}}}
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| {{#expr: {{{40}}} + {{{41}}} }}
| {{#expr: {{{42}}} + {{{43}}} + {{{44}}} + {{{45}}} }}
| {{#expr: {{{46}}} + {{{47}}} + {{{48}}} + {{{49}}} + {{{50}}} + {{{51}}} + {{{52}}} + {{{53}}} }}
| {{#expr: {{{54}}} + {{{55}}} + {{{56}}} + {{{57}}} + {{{58}}} + {{{59}}} + {{{60}}} + {{{61}}} + {{{62}}} + {{{63}}} + {{{64}}} + {{{65}}} + {{{66}}} + {{{67}}} + {{{68}}} + {{{69}}} }}
!class="sum"| {{{75}}}
|}
{{Collapsible END}}<noinclude>
----
see e.g. {{tl|Boolf triangle Magnolia}}
[[Category:Boolf triangles with weight columns]]
[[Category:Some templates created by Watchduck]]
</noinclude>
hmcyug627451alyr3ai3vxksykthfm8
2694015
2694009
2025-01-01T20:15:31Z
Watchduck
137431
2694015
wikitext
text/x-wiki
<templatestyles src="Template:Boolf triangles/style.css" />
{{Collapsible START|rational weight|open wide light followed}}
{| class="wikitable boolf-triangle" style="margin: 0;"
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! <math>0</math>
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!class="sum"| sums
|-
! 0
| {{{1}}}
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
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|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
| {{{2}}}
!class="sum"| {{#expr: {{{1}}} + {{{2}}} }}
|-
! 1
| {{{3}}}
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
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|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
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| {{{4}}}
|class="dummy"|
|class="dummy"|
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|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
| {{{5}}}
!class="sum"| {{#expr: {{{3}}} + {{{4}}} + {{{5}}} }}
|-
! 2
| {{{6}}}
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
| {{{7}}}
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
| {{{8}}}
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
| {{{9}}}
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
| {{{10}}}
!class="sum"| {{#expr: {{{6}}} + {{{7}}} + {{{8}}} + {{{9}}} + {{{10}}} }}
|- style="font-size: {{{76|80}}}%;"
! 3
| {{{11}}}
|class="dummy"|
|class="dummy"|
|class="dummy"|
| {{{12}}}
|class="dummy"|
|class="dummy"|
|class="dummy"|
| {{{13}}}
|class="dummy"|
|class="dummy"|
|class="dummy"|
| {{{14}}}
|class="dummy"|
|class="dummy"|
|class="dummy"|
| {{{15}}}
|class="dummy"|
|class="dummy"|
|class="dummy"|
| {{{16}}}
|class="dummy"|
|class="dummy"|
|class="dummy"|
| {{{17}}}
|class="dummy"|
|class="dummy"|
|class="dummy"|
| {{{18}}}
|class="dummy"|
|class="dummy"|
|class="dummy"|
| {{{19}}}
!class="sum"| {{#expr: {{{11}}} + {{{12}}} + {{{13}}} + {{{14}}} + {{{15}}} + {{{16}}} + {{{17}}} + {{{18}}} + {{{19}}} }}
|- style="font-size: {{{77|70}}}%;"
! 4
| {{{20}}}
|class="dummy"|
| {{{21}}}
|class="dummy"|
| {{{22}}}
|class="dummy"|
| {{{23}}}
|class="dummy"|
| {{{24}}}
|class="dummy"|
| {{{25}}}
|class="dummy"|
| {{{26}}}
|class="dummy"|
| {{{27}}}
|class="dummy"|
| {{{28}}}
|class="dummy"|
| {{{29}}}
|class="dummy"|
| {{{30}}}
|class="dummy"|
| {{{31}}}
|class="dummy"|
| {{{32}}}
|class="dummy"|
| {{{33}}}
|class="dummy"|
| {{{34}}}
|class="dummy"|
| {{{35}}}
|class="dummy"|
| {{{36}}}
!class="sum"| {{#expr: {{{20}}} + {{{21}}} + {{{22}}} + {{{23}}} + {{{24}}} + {{{25}}} + {{{26}}} + {{{27}}} + {{{28}}} + {{{29}}} + {{{30}}} + {{{31}}} + {{{32}}} + {{{33}}} + {{{34}}} + {{{35}}} + {{{36}}} }}
|- style="font-size: {{{78|50}}}%;"
! 5
| {{{37}}}
| {{{38}}}
| {{{39}}}
| {{{40}}}
| {{{41}}}
| {{{42}}}
| {{{43}}}
| {{{44}}}
| {{{45}}}
| {{{46}}}
| {{{47}}}
| {{{48}}}
| {{{49}}}
| {{{50}}}
| {{{51}}}
| {{{52}}}
| {{{53}}}
| {{{54}}}
| {{{55}}}
| {{{56}}}
| {{{57}}}
| {{{58}}}
| {{{59}}}
| {{{60}}}
| {{{61}}}
| {{{62}}}
| {{{63}}}
| {{{64}}}
| {{{65}}}
| {{{66}}}
| {{{67}}}
| {{{68}}}
| {{{69}}}
!class="sum"| {{#expr: {{{37}}} + {{{38}}} + {{{39}}} + {{{40}}} + {{{41}}} + {{{42}}} + {{{43}}} + {{{44}}} + {{{45}}} + {{{46}}} + {{{47}}} + {{{48}}} + {{{49}}} + {{{50}}} + {{{51}}} + {{{52}}} + {{{53}}} + {{{54}}} + {{{55}}} + {{{56}}} + {{{57}}} + {{{58}}} + {{{59}}} + {{{60}}} + {{{61}}} + {{{62}}} + {{{63}}} + {{{64}}} + {{{65}}} + {{{66}}} + {{{67}}} + {{{68}}} + {{{69}}} }}
|}
{{Collapsible END}}
<!-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------->
{{Collapsible START|integer weight|collapsed light wide followed}}
{| class="wikitable boolf-triangle" style="margin: 0;"
! {{diagonal split header|''a''|''w''}}
! 0
! 1
! 2
! 3
! 4
! 5
! 6
! 7
! 8
! 9
! 10
! 11
! 12
! 13
! 14
! 15
! 16
! 17
! 18
! 19
! 20
! 21
! 22
! 23
! 24
! 25
! 26
! 27
! 28
! 29
! 30
! 31
! 32
!class="sum"| sums
|-
! 0
| {{{1}}}
| {{{2}}}
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
!class="sum"| {{#expr: {{{1}}} + {{{2}}} }}
|-
! 1
| {{{3}}}
| {{{4}}}
| {{{5}}}
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
!class="sum"| {{#expr: {{{3}}} + {{{4}}} + {{{5}}} }}
|-
! 2
| {{{6}}}
| {{{7}}}
| {{{8}}}
| {{{9}}}
| {{{10}}}
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
!class="sum"| {{#expr: {{{6}}} + {{{7}}} + {{{8}}} + {{{9}}} + {{{10}}} }}
|- style="font-size: {{{76|80}}}%;"
! 3
| {{{11}}}
| {{{12}}}
| {{{13}}}
| {{{14}}}
| {{{15}}}
| {{{16}}}
| {{{17}}}
| {{{18}}}
| {{{19}}}
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
!class="sum"| {{#expr: {{{11}}} + {{{12}}} + {{{13}}} + {{{14}}} + {{{15}}} + {{{16}}} + {{{17}}} + {{{18}}} + {{{19}}} }}
|- style="font-size: {{{77|70}}}%;"
! 4
| {{{20}}}
| {{{21}}}
| {{{22}}}
| {{{23}}}
| {{{24}}}
| {{{25}}}
| {{{26}}}
| {{{27}}}
| {{{28}}}
| {{{29}}}
| {{{30}}}
| {{{31}}}
| {{{32}}}
| {{{33}}}
| {{{34}}}
| {{{35}}}
| {{{36}}}
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
!class="sum"| {{#expr: {{{20}}} + {{{21}}} + {{{22}}} + {{{23}}} + {{{24}}} + {{{25}}} + {{{26}}} + {{{27}}} + {{{28}}} + {{{29}}} + {{{30}}} + {{{31}}} + {{{32}}} + {{{33}}} + {{{34}}} + {{{35}}} + {{{36}}} }}
|- style="font-size: {{{78|50}}}%;"
! 5
| {{{37}}}
| {{{38}}}
| {{{39}}}
| {{{40}}}
| {{{41}}}
| {{{42}}}
| {{{43}}}
| {{{44}}}
| {{{45}}}
| {{{46}}}
| {{{47}}}
| {{{48}}}
| {{{49}}}
| {{{50}}}
| {{{51}}}
| {{{52}}}
| {{{53}}}
| {{{54}}}
| {{{55}}}
| {{{56}}}
| {{{57}}}
| {{{58}}}
| {{{59}}}
| {{{60}}}
| {{{61}}}
| {{{62}}}
| {{{63}}}
| {{{64}}}
| {{{65}}}
| {{{66}}}
| {{{67}}}
| {{{68}}}
| {{{69}}}
!class="sum"| {{#expr: {{{37}}} + {{{38}}} + {{{39}}} + {{{40}}} + {{{41}}} + {{{42}}} + {{{43}}} + {{{44}}} + {{{45}}} + {{{46}}} + {{{47}}} + {{{48}}} + {{{49}}} + {{{50}}} + {{{51}}} + {{{52}}} + {{{53}}} + {{{54}}} + {{{55}}} + {{{56}}} + {{{57}}} + {{{58}}} + {{{59}}} + {{{60}}} + {{{61}}} + {{{62}}} + {{{63}}} + {{{64}}} + {{{65}}} + {{{66}}} + {{{67}}} + {{{68}}} + {{{69}}} }}
|}
{{Collapsible END}}
<!-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------->
<templatestyles src="Template:Boolf triangles/style.css" />
{{Collapsible START|<small>(merged weights)</small>|collapsed wide light}}
{| class="wikitable boolf-triangle" style="margin: 0;"
|-
!style="color: gray;"| ''w''
!style="color: gray; font-size: 60%;"| 0
!style="color: gray;"| 1
!style="color: gray;"| 2
!style="color: gray;"| 3...4
!style="color: gray;"| 5...8
!style="color: gray;"| 9...16
!style="color: gray;"| 17...32
!rowspan="2" class="sum"| sums
|-
! {{diagonal split header|''a''|''k''}}
!style="font-size: 60%;"| -1
! 0
! 1
! 2
! 3
! 4
! 5
|-
! 0
|style="font-size: 60%;"| {{{1}}}
| {{{2}}}
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
!class="sum"| {{#expr: {{{1}}} + {{{2}}} }}
|-
! 1
|style="font-size: 60%;"| {{{3}}}
| {{{4}}}
| {{{5}}}
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
!class="sum"| {{#expr: {{{3}}} + {{{4}}} + {{{5}}} }}
|-
! 2
|style="font-size: 60%;"| {{{6}}}
| {{{7}}}
| {{{8}}}
| {{#expr: {{{9}}} + {{{10}}} }}
|class="dummy"|
|class="dummy"|
|class="dummy"|
!class="sum"| {{#expr: {{{6}}} + {{{7}}} + {{{8}}} + {{{9}}} + {{{10}}} }}
|-
! 3
|style="font-size: 60%;"| {{{11}}}
| {{{12}}}
| {{{13}}}
| {{#expr: {{{14}}} + {{{15}}} }}
| {{#expr: {{{16}}} + {{{17}}} + {{{18}}} + {{{19}}} }}
|class="dummy"|
|class="dummy"|
!class="sum"| {{#expr: {{{11}}} + {{{12}}} + {{{13}}} + {{{14}}} + {{{15}}} + {{{16}}} + {{{17}}} + {{{18}}} + {{{19}}} }}
|-
! 4
|style="font-size: 60%;"| {{{20}}}
| {{{21}}}
| {{{22}}}
| {{#expr: {{{23}}} + {{{24}}} }}
| {{#expr: {{{25}}} + {{{26}}} + {{{27}}} + {{{28}}} }}
| {{#expr: {{{29}}} + {{{30}}} + {{{31}}} + {{{32}}} + {{{33}}} + {{{34}}} + {{{35}}} + {{{36}}} }}
|class="dummy"|
!class="sum"| {{#expr: {{{20}}} + {{{21}}} + {{{22}}} + {{{23}}} + {{{24}}} + {{{25}}} + {{{26}}} + {{{27}}} + {{{28}}} + {{{29}}} + {{{30}}} + {{{31}}} + {{{32}}} + {{{33}}} + {{{34}}} + {{{35}}} + {{{36}}} }}
|-
! 5
|style="font-size: 60%;"| {{{37}}}
| {{{38}}}
| {{{39}}}
| {{#expr: {{{40}}} + {{{41}}} }}
| {{#expr: {{{42}}} + {{{43}}} + {{{44}}} + {{{45}}} }}
| {{#expr: {{{46}}} + {{{47}}} + {{{48}}} + {{{49}}} + {{{50}}} + {{{51}}} + {{{52}}} + {{{53}}} }}
| {{#expr: {{{54}}} + {{{55}}} + {{{56}}} + {{{57}}} + {{{58}}} + {{{59}}} + {{{60}}} + {{{61}}} + {{{62}}} + {{{63}}} + {{{64}}} + {{{65}}} + {{{66}}} + {{{67}}} + {{{68}}} + {{{69}}} }}
!class="sum"| {{#expr: {{{37}}} + {{{38}}} + {{{39}}} + {{{40}}} + {{{41}}} + {{{42}}} + {{{43}}} + {{{44}}} + {{{45}}} + {{{46}}} + {{{47}}} + {{{48}}} + {{{49}}} + {{{50}}} + {{{51}}} + {{{52}}} + {{{53}}} + {{{54}}} + {{{55}}} + {{{56}}} + {{{57}}} + {{{58}}} + {{{59}}} + {{{60}}} + {{{61}}} + {{{62}}} + {{{63}}} + {{{64}}} + {{{65}}} + {{{66}}} + {{{67}}} + {{{68}}} + {{{69}}} }}
|}
{{Collapsible END}}<noinclude>
----
There is also {{tl|Boolf weight triangle 4}}. (For some cases, the calculation for arity 5 is not feasible.)
see e.g. {{tl|Boolf triangle Magnolia}}
[[Category:Boolf triangles with weight columns]]
[[Category:Some templates created by Watchduck]]
</noinclude>
9dn9usn9t3gjijqto3mxjf74uwpz475
2694023
2694015
2025-01-01T20:22:42Z
Watchduck
137431
2694023
wikitext
text/x-wiki
<templatestyles src="Template:Boolf triangles/style.css" />
{{Collapsible START|rational weight|open wide light followed}}
{| class="wikitable boolf-triangle" style="margin: 0;"
! {{diagonal split header|''a''|''w''}}
! <math>0</math>
! <math>\frac{1}{32}</math>
! <math>\frac{1}{16}</math>
! <math>\frac{3}{32}</math>
! <math>\frac{1}{8}</math>
! <math>\frac{5}{32}</math>
! <math>\frac{3}{16}</math>
! <math>\frac{7}{32}</math>
! <math>\frac{1}{4}</math>
! <math>\frac{9}{32}</math>
! <math>\frac{5}{16}</math>
! <math>\frac{11}{32}</math>
! <math>\frac{3}{8}</math>
! <math>\frac{13}{32}</math>
! <math>\frac{7}{16}</math>
! <math>\frac{15}{32}</math>
! <math>\frac{1}{2}</math>
! <math>\frac{17}{32}</math>
! <math>\frac{9}{16}</math>
! <math>\frac{19}{32}</math>
! <math>\frac{5}{8}</math>
! <math>\frac{21}{32}</math>
! <math>\frac{11}{16}</math>
! <math>\frac{23}{32}</math>
! <math>\frac{3}{4}</math>
! <math>\frac{25}{32}</math>
! <math>\frac{13}{16}</math>
! <math>\frac{27}{32}</math>
! <math>\frac{7}{8}</math>
! <math>\frac{29}{32}</math>
! <math>\frac{15}{16}</math>
! <math>\frac{31}{32}</math>
! <math>1</math>
!class="sum"| sums
|-
! 0
| {{{1}}}
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
| {{{2}}}
!class="sum"| {{#expr: {{{1}}} + {{{2}}} }}
|-
! 1
| {{{3}}}
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
| {{{4}}}
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
| {{{5}}}
!class="sum"| {{#expr: {{{3}}} + {{{4}}} + {{{5}}} }}
|-
! 2
| {{{6}}}
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
| {{{7}}}
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
| {{{8}}}
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
| {{{9}}}
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
| {{{10}}}
!class="sum"| {{#expr: {{{6}}} + {{{7}}} + {{{8}}} + {{{9}}} + {{{10}}} }}
|- style="font-size: {{{76|80}}}%;"
! 3
| {{{11}}}
|class="dummy"|
|class="dummy"|
|class="dummy"|
| {{{12}}}
|class="dummy"|
|class="dummy"|
|class="dummy"|
| {{{13}}}
|class="dummy"|
|class="dummy"|
|class="dummy"|
| {{{14}}}
|class="dummy"|
|class="dummy"|
|class="dummy"|
| {{{15}}}
|class="dummy"|
|class="dummy"|
|class="dummy"|
| {{{16}}}
|class="dummy"|
|class="dummy"|
|class="dummy"|
| {{{17}}}
|class="dummy"|
|class="dummy"|
|class="dummy"|
| {{{18}}}
|class="dummy"|
|class="dummy"|
|class="dummy"|
| {{{19}}}
!class="sum"| {{#expr: {{{11}}} + {{{12}}} + {{{13}}} + {{{14}}} + {{{15}}} + {{{16}}} + {{{17}}} + {{{18}}} + {{{19}}} }}
|- style="font-size: {{{77|70}}}%;"
! 4
| {{{20}}}
|class="dummy"|
| {{{21}}}
|class="dummy"|
| {{{22}}}
|class="dummy"|
| {{{23}}}
|class="dummy"|
| {{{24}}}
|class="dummy"|
| {{{25}}}
|class="dummy"|
| {{{26}}}
|class="dummy"|
| {{{27}}}
|class="dummy"|
| {{{28}}}
|class="dummy"|
| {{{29}}}
|class="dummy"|
| {{{30}}}
|class="dummy"|
| {{{31}}}
|class="dummy"|
| {{{32}}}
|class="dummy"|
| {{{33}}}
|class="dummy"|
| {{{34}}}
|class="dummy"|
| {{{35}}}
|class="dummy"|
| {{{36}}}
!class="sum"| {{#expr: {{{20}}} + {{{21}}} + {{{22}}} + {{{23}}} + {{{24}}} + {{{25}}} + {{{26}}} + {{{27}}} + {{{28}}} + {{{29}}} + {{{30}}} + {{{31}}} + {{{32}}} + {{{33}}} + {{{34}}} + {{{35}}} + {{{36}}} }}
|- style="font-size: {{{78|50}}}%;"
! 5
| {{{37}}}
| {{{38}}}
| {{{39}}}
| {{{40}}}
| {{{41}}}
| {{{42}}}
| {{{43}}}
| {{{44}}}
| {{{45}}}
| {{{46}}}
| {{{47}}}
| {{{48}}}
| {{{49}}}
| {{{50}}}
| {{{51}}}
| {{{52}}}
| {{{53}}}
| {{{54}}}
| {{{55}}}
| {{{56}}}
| {{{57}}}
| {{{58}}}
| {{{59}}}
| {{{60}}}
| {{{61}}}
| {{{62}}}
| {{{63}}}
| {{{64}}}
| {{{65}}}
| {{{66}}}
| {{{67}}}
| {{{68}}}
| {{{69}}}
!class="sum"| {{#expr: {{{37}}} + {{{38}}} + {{{39}}} + {{{40}}} + {{{41}}} + {{{42}}} + {{{43}}} + {{{44}}} + {{{45}}} + {{{46}}} + {{{47}}} + {{{48}}} + {{{49}}} + {{{50}}} + {{{51}}} + {{{52}}} + {{{53}}} + {{{54}}} + {{{55}}} + {{{56}}} + {{{57}}} + {{{58}}} + {{{59}}} + {{{60}}} + {{{61}}} + {{{62}}} + {{{63}}} + {{{64}}} + {{{65}}} + {{{66}}} + {{{67}}} + {{{68}}} + {{{69}}} }}
|}
{{Collapsible END}}
<!-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------->
{{Collapsible START|integer weight|collapsed light wide followed}}
{| class="wikitable boolf-triangle" style="margin: 0;"
! {{diagonal split header|''a''|''w''}}
! 0
! 1
! 2
! 3
! 4
! 5
! 6
! 7
! 8
! 9
! 10
! 11
! 12
! 13
! 14
! 15
! 16
! 17
! 18
! 19
! 20
! 21
! 22
! 23
! 24
! 25
! 26
! 27
! 28
! 29
! 30
! 31
! 32
!class="sum"| sums
|-
! 0
| {{{1}}}
| {{{2}}}
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
!class="sum"| {{#expr: {{{1}}} + {{{2}}} }}
|-
! 1
| {{{3}}}
| {{{4}}}
| {{{5}}}
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
!class="sum"| {{#expr: {{{3}}} + {{{4}}} + {{{5}}} }}
|-
! 2
| {{{6}}}
| {{{7}}}
| {{{8}}}
| {{{9}}}
| {{{10}}}
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
!class="sum"| {{#expr: {{{6}}} + {{{7}}} + {{{8}}} + {{{9}}} + {{{10}}} }}
|- style="font-size: {{{76|80}}}%;"
! 3
| {{{11}}}
| {{{12}}}
| {{{13}}}
| {{{14}}}
| {{{15}}}
| {{{16}}}
| {{{17}}}
| {{{18}}}
| {{{19}}}
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
!class="sum"| {{#expr: {{{11}}} + {{{12}}} + {{{13}}} + {{{14}}} + {{{15}}} + {{{16}}} + {{{17}}} + {{{18}}} + {{{19}}} }}
|- style="font-size: {{{77|70}}}%;"
! 4
| {{{20}}}
| {{{21}}}
| {{{22}}}
| {{{23}}}
| {{{24}}}
| {{{25}}}
| {{{26}}}
| {{{27}}}
| {{{28}}}
| {{{29}}}
| {{{30}}}
| {{{31}}}
| {{{32}}}
| {{{33}}}
| {{{34}}}
| {{{35}}}
| {{{36}}}
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
!class="sum"| {{#expr: {{{20}}} + {{{21}}} + {{{22}}} + {{{23}}} + {{{24}}} + {{{25}}} + {{{26}}} + {{{27}}} + {{{28}}} + {{{29}}} + {{{30}}} + {{{31}}} + {{{32}}} + {{{33}}} + {{{34}}} + {{{35}}} + {{{36}}} }}
|- style="font-size: {{{78|50}}}%;"
! 5
| {{{37}}}
| {{{38}}}
| {{{39}}}
| {{{40}}}
| {{{41}}}
| {{{42}}}
| {{{43}}}
| {{{44}}}
| {{{45}}}
| {{{46}}}
| {{{47}}}
| {{{48}}}
| {{{49}}}
| {{{50}}}
| {{{51}}}
| {{{52}}}
| {{{53}}}
| {{{54}}}
| {{{55}}}
| {{{56}}}
| {{{57}}}
| {{{58}}}
| {{{59}}}
| {{{60}}}
| {{{61}}}
| {{{62}}}
| {{{63}}}
| {{{64}}}
| {{{65}}}
| {{{66}}}
| {{{67}}}
| {{{68}}}
| {{{69}}}
!class="sum"| {{#expr: {{{37}}} + {{{38}}} + {{{39}}} + {{{40}}} + {{{41}}} + {{{42}}} + {{{43}}} + {{{44}}} + {{{45}}} + {{{46}}} + {{{47}}} + {{{48}}} + {{{49}}} + {{{50}}} + {{{51}}} + {{{52}}} + {{{53}}} + {{{54}}} + {{{55}}} + {{{56}}} + {{{57}}} + {{{58}}} + {{{59}}} + {{{60}}} + {{{61}}} + {{{62}}} + {{{63}}} + {{{64}}} + {{{65}}} + {{{66}}} + {{{67}}} + {{{68}}} + {{{69}}} }}
|}
{{Collapsible END}}
<!-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------->
<templatestyles src="Template:Boolf triangles/style.css" />
{{Collapsible START|<small>(merged weights)</small>|collapsed wide light}}
{| class="wikitable boolf-triangle" style="margin: 0;"
|-
!style="color: gray;"| ''w''
!style="color: gray; font-size: 60%;"| 0
!style="color: gray;"| 1
!style="color: gray;"| 2
!style="color: gray;"| 3...4
!style="color: gray;"| 5...8
!style="color: gray;"| 9...16
!style="color: gray;"| 17...32
!rowspan="2" class="sum"| sums
|-
! {{diagonal split header|''a''|''k''}}
!style="font-size: 60%;"| -1
! 0
! 1
! 2
! 3
! 4
! 5
|-
! 0
|style="font-size: 60%;"| {{{1}}}
| {{{2}}}
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
!class="sum"| {{#expr: {{{1}}} + {{{2}}} }}
|-
! 1
|style="font-size: 60%;"| {{{3}}}
| {{{4}}}
| {{{5}}}
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
!class="sum"| {{#expr: {{{3}}} + {{{4}}} + {{{5}}} }}
|-
! 2
|style="font-size: 60%;"| {{{6}}}
| {{{7}}}
| {{{8}}}
| {{#expr: {{{9}}} + {{{10}}} }}
|class="dummy"|
|class="dummy"|
|class="dummy"|
!class="sum"| {{#expr: {{{6}}} + {{{7}}} + {{{8}}} + {{{9}}} + {{{10}}} }}
|-
! 3
|style="font-size: 60%;"| {{{11}}}
| {{{12}}}
| {{{13}}}
| {{#expr: {{{14}}} + {{{15}}} }}
| {{#expr: {{{16}}} + {{{17}}} + {{{18}}} + {{{19}}} }}
|class="dummy"|
|class="dummy"|
!class="sum"| {{#expr: {{{11}}} + {{{12}}} + {{{13}}} + {{{14}}} + {{{15}}} + {{{16}}} + {{{17}}} + {{{18}}} + {{{19}}} }}
|-
! 4
|style="font-size: 60%;"| {{{20}}}
| {{{21}}}
| {{{22}}}
| {{#expr: {{{23}}} + {{{24}}} }}
| {{#expr: {{{25}}} + {{{26}}} + {{{27}}} + {{{28}}} }}
| {{#expr: {{{29}}} + {{{30}}} + {{{31}}} + {{{32}}} + {{{33}}} + {{{34}}} + {{{35}}} + {{{36}}} }}
|class="dummy"|
!class="sum"| {{#expr: {{{20}}} + {{{21}}} + {{{22}}} + {{{23}}} + {{{24}}} + {{{25}}} + {{{26}}} + {{{27}}} + {{{28}}} + {{{29}}} + {{{30}}} + {{{31}}} + {{{32}}} + {{{33}}} + {{{34}}} + {{{35}}} + {{{36}}} }}
|-
! 5
|style="font-size: 60%;"| {{{37}}}
| {{{38}}}
| {{{39}}}
| {{#expr: {{{40}}} + {{{41}}} }}
| {{#expr: {{{42}}} + {{{43}}} + {{{44}}} + {{{45}}} }}
| {{#expr: {{{46}}} + {{{47}}} + {{{48}}} + {{{49}}} + {{{50}}} + {{{51}}} + {{{52}}} + {{{53}}} }}
| {{#expr: {{{54}}} + {{{55}}} + {{{56}}} + {{{57}}} + {{{58}}} + {{{59}}} + {{{60}}} + {{{61}}} + {{{62}}} + {{{63}}} + {{{64}}} + {{{65}}} + {{{66}}} + {{{67}}} + {{{68}}} + {{{69}}} }}
!class="sum"| {{#expr: {{{37}}} + {{{38}}} + {{{39}}} + {{{40}}} + {{{41}}} + {{{42}}} + {{{43}}} + {{{44}}} + {{{45}}} + {{{46}}} + {{{47}}} + {{{48}}} + {{{49}}} + {{{50}}} + {{{51}}} + {{{52}}} + {{{53}}} + {{{54}}} + {{{55}}} + {{{56}}} + {{{57}}} + {{{58}}} + {{{59}}} + {{{60}}} + {{{61}}} + {{{62}}} + {{{63}}} + {{{64}}} + {{{65}}} + {{{66}}} + {{{67}}} + {{{68}}} + {{{69}}} }}
|}
{{Collapsible END}}<noinclude>
----
There is also {{tl|Boolf weight triangle 4}}. (For some cases, the calculation for arity 5 is not feasible.)
see e.g. {{tl|Boolf triangle Magnolia}}
[[Category:Boolf triangles with weight columns| ]]
[[Category:Some templates created by Watchduck]]
</noinclude>
6m60mw0hdq4zr4pey3qsjom7b6bosbl
2694024
2694023
2025-01-01T20:23:06Z
Watchduck
137431
2694024
wikitext
text/x-wiki
<templatestyles src="Template:Boolf triangles/style.css" />
{{Collapsible START|rational weight|open wide light followed}}
{| class="wikitable boolf-triangle" style="margin: 0;"
! {{diagonal split header|''a''|''w''}}
! <math>0</math>
! <math>\frac{1}{32}</math>
! <math>\frac{1}{16}</math>
! <math>\frac{3}{32}</math>
! <math>\frac{1}{8}</math>
! <math>\frac{5}{32}</math>
! <math>\frac{3}{16}</math>
! <math>\frac{7}{32}</math>
! <math>\frac{1}{4}</math>
! <math>\frac{9}{32}</math>
! <math>\frac{5}{16}</math>
! <math>\frac{11}{32}</math>
! <math>\frac{3}{8}</math>
! <math>\frac{13}{32}</math>
! <math>\frac{7}{16}</math>
! <math>\frac{15}{32}</math>
! <math>\frac{1}{2}</math>
! <math>\frac{17}{32}</math>
! <math>\frac{9}{16}</math>
! <math>\frac{19}{32}</math>
! <math>\frac{5}{8}</math>
! <math>\frac{21}{32}</math>
! <math>\frac{11}{16}</math>
! <math>\frac{23}{32}</math>
! <math>\frac{3}{4}</math>
! <math>\frac{25}{32}</math>
! <math>\frac{13}{16}</math>
! <math>\frac{27}{32}</math>
! <math>\frac{7}{8}</math>
! <math>\frac{29}{32}</math>
! <math>\frac{15}{16}</math>
! <math>\frac{31}{32}</math>
! <math>1</math>
!class="sum"| sums
|-
! 0
| {{{1}}}
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
| {{{2}}}
!class="sum"| {{#expr: {{{1}}} + {{{2}}} }}
|-
! 1
| {{{3}}}
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
| {{{4}}}
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
| {{{5}}}
!class="sum"| {{#expr: {{{3}}} + {{{4}}} + {{{5}}} }}
|-
! 2
| {{{6}}}
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
| {{{7}}}
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
| {{{8}}}
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
| {{{9}}}
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
| {{{10}}}
!class="sum"| {{#expr: {{{6}}} + {{{7}}} + {{{8}}} + {{{9}}} + {{{10}}} }}
|- style="font-size: {{{76|80}}}%;"
! 3
| {{{11}}}
|class="dummy"|
|class="dummy"|
|class="dummy"|
| {{{12}}}
|class="dummy"|
|class="dummy"|
|class="dummy"|
| {{{13}}}
|class="dummy"|
|class="dummy"|
|class="dummy"|
| {{{14}}}
|class="dummy"|
|class="dummy"|
|class="dummy"|
| {{{15}}}
|class="dummy"|
|class="dummy"|
|class="dummy"|
| {{{16}}}
|class="dummy"|
|class="dummy"|
|class="dummy"|
| {{{17}}}
|class="dummy"|
|class="dummy"|
|class="dummy"|
| {{{18}}}
|class="dummy"|
|class="dummy"|
|class="dummy"|
| {{{19}}}
!class="sum"| {{#expr: {{{11}}} + {{{12}}} + {{{13}}} + {{{14}}} + {{{15}}} + {{{16}}} + {{{17}}} + {{{18}}} + {{{19}}} }}
|- style="font-size: {{{77|70}}}%;"
! 4
| {{{20}}}
|class="dummy"|
| {{{21}}}
|class="dummy"|
| {{{22}}}
|class="dummy"|
| {{{23}}}
|class="dummy"|
| {{{24}}}
|class="dummy"|
| {{{25}}}
|class="dummy"|
| {{{26}}}
|class="dummy"|
| {{{27}}}
|class="dummy"|
| {{{28}}}
|class="dummy"|
| {{{29}}}
|class="dummy"|
| {{{30}}}
|class="dummy"|
| {{{31}}}
|class="dummy"|
| {{{32}}}
|class="dummy"|
| {{{33}}}
|class="dummy"|
| {{{34}}}
|class="dummy"|
| {{{35}}}
|class="dummy"|
| {{{36}}}
!class="sum"| {{#expr: {{{20}}} + {{{21}}} + {{{22}}} + {{{23}}} + {{{24}}} + {{{25}}} + {{{26}}} + {{{27}}} + {{{28}}} + {{{29}}} + {{{30}}} + {{{31}}} + {{{32}}} + {{{33}}} + {{{34}}} + {{{35}}} + {{{36}}} }}
|- style="font-size: {{{78|50}}}%;"
! 5
| {{{37}}}
| {{{38}}}
| {{{39}}}
| {{{40}}}
| {{{41}}}
| {{{42}}}
| {{{43}}}
| {{{44}}}
| {{{45}}}
| {{{46}}}
| {{{47}}}
| {{{48}}}
| {{{49}}}
| {{{50}}}
| {{{51}}}
| {{{52}}}
| {{{53}}}
| {{{54}}}
| {{{55}}}
| {{{56}}}
| {{{57}}}
| {{{58}}}
| {{{59}}}
| {{{60}}}
| {{{61}}}
| {{{62}}}
| {{{63}}}
| {{{64}}}
| {{{65}}}
| {{{66}}}
| {{{67}}}
| {{{68}}}
| {{{69}}}
!class="sum"| {{#expr: {{{37}}} + {{{38}}} + {{{39}}} + {{{40}}} + {{{41}}} + {{{42}}} + {{{43}}} + {{{44}}} + {{{45}}} + {{{46}}} + {{{47}}} + {{{48}}} + {{{49}}} + {{{50}}} + {{{51}}} + {{{52}}} + {{{53}}} + {{{54}}} + {{{55}}} + {{{56}}} + {{{57}}} + {{{58}}} + {{{59}}} + {{{60}}} + {{{61}}} + {{{62}}} + {{{63}}} + {{{64}}} + {{{65}}} + {{{66}}} + {{{67}}} + {{{68}}} + {{{69}}} }}
|}
{{Collapsible END}}
<!-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------->
{{Collapsible START|integer weight|collapsed light wide followed}}
{| class="wikitable boolf-triangle" style="margin: 0;"
! {{diagonal split header|''a''|''w''}}
! 0
! 1
! 2
! 3
! 4
! 5
! 6
! 7
! 8
! 9
! 10
! 11
! 12
! 13
! 14
! 15
! 16
! 17
! 18
! 19
! 20
! 21
! 22
! 23
! 24
! 25
! 26
! 27
! 28
! 29
! 30
! 31
! 32
!class="sum"| sums
|-
! 0
| {{{1}}}
| {{{2}}}
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
!class="sum"| {{#expr: {{{1}}} + {{{2}}} }}
|-
! 1
| {{{3}}}
| {{{4}}}
| {{{5}}}
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
!class="sum"| {{#expr: {{{3}}} + {{{4}}} + {{{5}}} }}
|-
! 2
| {{{6}}}
| {{{7}}}
| {{{8}}}
| {{{9}}}
| {{{10}}}
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
!class="sum"| {{#expr: {{{6}}} + {{{7}}} + {{{8}}} + {{{9}}} + {{{10}}} }}
|- style="font-size: {{{76|80}}}%;"
! 3
| {{{11}}}
| {{{12}}}
| {{{13}}}
| {{{14}}}
| {{{15}}}
| {{{16}}}
| {{{17}}}
| {{{18}}}
| {{{19}}}
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
!class="sum"| {{#expr: {{{11}}} + {{{12}}} + {{{13}}} + {{{14}}} + {{{15}}} + {{{16}}} + {{{17}}} + {{{18}}} + {{{19}}} }}
|- style="font-size: {{{77|70}}}%;"
! 4
| {{{20}}}
| {{{21}}}
| {{{22}}}
| {{{23}}}
| {{{24}}}
| {{{25}}}
| {{{26}}}
| {{{27}}}
| {{{28}}}
| {{{29}}}
| {{{30}}}
| {{{31}}}
| {{{32}}}
| {{{33}}}
| {{{34}}}
| {{{35}}}
| {{{36}}}
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
!class="sum"| {{#expr: {{{20}}} + {{{21}}} + {{{22}}} + {{{23}}} + {{{24}}} + {{{25}}} + {{{26}}} + {{{27}}} + {{{28}}} + {{{29}}} + {{{30}}} + {{{31}}} + {{{32}}} + {{{33}}} + {{{34}}} + {{{35}}} + {{{36}}} }}
|- style="font-size: {{{78|50}}}%;"
! 5
| {{{37}}}
| {{{38}}}
| {{{39}}}
| {{{40}}}
| {{{41}}}
| {{{42}}}
| {{{43}}}
| {{{44}}}
| {{{45}}}
| {{{46}}}
| {{{47}}}
| {{{48}}}
| {{{49}}}
| {{{50}}}
| {{{51}}}
| {{{52}}}
| {{{53}}}
| {{{54}}}
| {{{55}}}
| {{{56}}}
| {{{57}}}
| {{{58}}}
| {{{59}}}
| {{{60}}}
| {{{61}}}
| {{{62}}}
| {{{63}}}
| {{{64}}}
| {{{65}}}
| {{{66}}}
| {{{67}}}
| {{{68}}}
| {{{69}}}
!class="sum"| {{#expr: {{{37}}} + {{{38}}} + {{{39}}} + {{{40}}} + {{{41}}} + {{{42}}} + {{{43}}} + {{{44}}} + {{{45}}} + {{{46}}} + {{{47}}} + {{{48}}} + {{{49}}} + {{{50}}} + {{{51}}} + {{{52}}} + {{{53}}} + {{{54}}} + {{{55}}} + {{{56}}} + {{{57}}} + {{{58}}} + {{{59}}} + {{{60}}} + {{{61}}} + {{{62}}} + {{{63}}} + {{{64}}} + {{{65}}} + {{{66}}} + {{{67}}} + {{{68}}} + {{{69}}} }}
|}
{{Collapsible END}}
<!-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------->
<templatestyles src="Template:Boolf triangles/style.css" />
{{Collapsible START|<small>(merged weights)</small>|collapsed wide light}}
{| class="wikitable boolf-triangle" style="margin: 0;"
|-
!style="color: gray;"| ''w''
!style="color: gray; font-size: 60%;"| 0
!style="color: gray;"| 1
!style="color: gray;"| 2
!style="color: gray;"| 3...4
!style="color: gray;"| 5...8
!style="color: gray;"| 9...16
!style="color: gray;"| 17...32
!rowspan="2" class="sum"| sums
|-
! {{diagonal split header|''a''|''k''}}
!style="font-size: 60%;"| -1
! 0
! 1
! 2
! 3
! 4
! 5
|-
! 0
|style="font-size: 60%;"| {{{1}}}
| {{{2}}}
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
!class="sum"| {{#expr: {{{1}}} + {{{2}}} }}
|-
! 1
|style="font-size: 60%;"| {{{3}}}
| {{{4}}}
| {{{5}}}
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
!class="sum"| {{#expr: {{{3}}} + {{{4}}} + {{{5}}} }}
|-
! 2
|style="font-size: 60%;"| {{{6}}}
| {{{7}}}
| {{{8}}}
| {{#expr: {{{9}}} + {{{10}}} }}
|class="dummy"|
|class="dummy"|
|class="dummy"|
!class="sum"| {{#expr: {{{6}}} + {{{7}}} + {{{8}}} + {{{9}}} + {{{10}}} }}
|-
! 3
|style="font-size: 60%;"| {{{11}}}
| {{{12}}}
| {{{13}}}
| {{#expr: {{{14}}} + {{{15}}} }}
| {{#expr: {{{16}}} + {{{17}}} + {{{18}}} + {{{19}}} }}
|class="dummy"|
|class="dummy"|
!class="sum"| {{#expr: {{{11}}} + {{{12}}} + {{{13}}} + {{{14}}} + {{{15}}} + {{{16}}} + {{{17}}} + {{{18}}} + {{{19}}} }}
|-
! 4
|style="font-size: 60%;"| {{{20}}}
| {{{21}}}
| {{{22}}}
| {{#expr: {{{23}}} + {{{24}}} }}
| {{#expr: {{{25}}} + {{{26}}} + {{{27}}} + {{{28}}} }}
| {{#expr: {{{29}}} + {{{30}}} + {{{31}}} + {{{32}}} + {{{33}}} + {{{34}}} + {{{35}}} + {{{36}}} }}
|class="dummy"|
!class="sum"| {{#expr: {{{20}}} + {{{21}}} + {{{22}}} + {{{23}}} + {{{24}}} + {{{25}}} + {{{26}}} + {{{27}}} + {{{28}}} + {{{29}}} + {{{30}}} + {{{31}}} + {{{32}}} + {{{33}}} + {{{34}}} + {{{35}}} + {{{36}}} }}
|-
! 5
|style="font-size: 60%;"| {{{37}}}
| {{{38}}}
| {{{39}}}
| {{#expr: {{{40}}} + {{{41}}} }}
| {{#expr: {{{42}}} + {{{43}}} + {{{44}}} + {{{45}}} }}
| {{#expr: {{{46}}} + {{{47}}} + {{{48}}} + {{{49}}} + {{{50}}} + {{{51}}} + {{{52}}} + {{{53}}} }}
| {{#expr: {{{54}}} + {{{55}}} + {{{56}}} + {{{57}}} + {{{58}}} + {{{59}}} + {{{60}}} + {{{61}}} + {{{62}}} + {{{63}}} + {{{64}}} + {{{65}}} + {{{66}}} + {{{67}}} + {{{68}}} + {{{69}}} }}
!class="sum"| {{#expr: {{{37}}} + {{{38}}} + {{{39}}} + {{{40}}} + {{{41}}} + {{{42}}} + {{{43}}} + {{{44}}} + {{{45}}} + {{{46}}} + {{{47}}} + {{{48}}} + {{{49}}} + {{{50}}} + {{{51}}} + {{{52}}} + {{{53}}} + {{{54}}} + {{{55}}} + {{{56}}} + {{{57}}} + {{{58}}} + {{{59}}} + {{{60}}} + {{{61}}} + {{{62}}} + {{{63}}} + {{{64}}} + {{{65}}} + {{{66}}} + {{{67}}} + {{{68}}} + {{{69}}} }}
|}
{{Collapsible END}}<noinclude>
----
There is also {{tl|Boolf weight triangle 4}}. (For some cases the calculation for arity 5 is not feasible.)
see e.g. {{tl|Boolf triangle Magnolia}}
[[Category:Boolf triangles with weight columns| ]]
[[Category:Some templates created by Watchduck]]
</noinclude>
l3cx2vr72b0oxefryxq2o555sh60uu0
2694038
2694024
2025-01-01T22:59:26Z
Watchduck
137431
2694038
wikitext
text/x-wiki
<templatestyles src="Template:Boolf triangles/style.css" />
{{Collapsible START|rational weight|open wide light followed}}
{| class="wikitable boolf-triangle" style="margin: 0;"
! {{diagonal split header|''a''|''w''}}
! <math>0</math>
! <math>\frac{1}{32}</math>
! <math>\frac{1}{16}</math>
! <math>\frac{3}{32}</math>
! <math>\frac{1}{8}</math>
! <math>\frac{5}{32}</math>
! <math>\frac{3}{16}</math>
! <math>\frac{7}{32}</math>
! <math>\frac{1}{4}</math>
! <math>\frac{9}{32}</math>
! <math>\frac{5}{16}</math>
! <math>\frac{11}{32}</math>
! <math>\frac{3}{8}</math>
! <math>\frac{13}{32}</math>
! <math>\frac{7}{16}</math>
! <math>\frac{15}{32}</math>
! <math>\frac{1}{2}</math>
! <math>\frac{17}{32}</math>
! <math>\frac{9}{16}</math>
! <math>\frac{19}{32}</math>
! <math>\frac{5}{8}</math>
! <math>\frac{21}{32}</math>
! <math>\frac{11}{16}</math>
! <math>\frac{23}{32}</math>
! <math>\frac{3}{4}</math>
! <math>\frac{25}{32}</math>
! <math>\frac{13}{16}</math>
! <math>\frac{27}{32}</math>
! <math>\frac{7}{8}</math>
! <math>\frac{29}{32}</math>
! <math>\frac{15}{16}</math>
! <math>\frac{31}{32}</math>
! <math>1</math>
!class="sum"| sums
|-
! 0
| {{{1}}}
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
| {{{2}}}
!class="sum"| {{#expr: {{{1}}} + {{{2}}} }}
|-
! 1
| {{{3}}}
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
| {{{4}}}
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
| {{{5}}}
!class="sum"| {{#expr: {{{3}}} + {{{4}}} + {{{5}}} }}
|-
! 2
| {{{6}}}
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
| {{{7}}}
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
| {{{8}}}
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
| {{{9}}}
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
| {{{10}}}
!class="sum"| {{#expr: {{{6}}} + {{{7}}} + {{{8}}} + {{{9}}} + {{{10}}} }}
|- style="font-size: {{{76|80}}}%;"
! 3
| {{{11}}}
|class="dummy"|
|class="dummy"|
|class="dummy"|
| {{{12}}}
|class="dummy"|
|class="dummy"|
|class="dummy"|
| {{{13}}}
|class="dummy"|
|class="dummy"|
|class="dummy"|
| {{{14}}}
|class="dummy"|
|class="dummy"|
|class="dummy"|
| {{{15}}}
|class="dummy"|
|class="dummy"|
|class="dummy"|
| {{{16}}}
|class="dummy"|
|class="dummy"|
|class="dummy"|
| {{{17}}}
|class="dummy"|
|class="dummy"|
|class="dummy"|
| {{{18}}}
|class="dummy"|
|class="dummy"|
|class="dummy"|
| {{{19}}}
!class="sum"| {{#expr: {{{11}}} + {{{12}}} + {{{13}}} + {{{14}}} + {{{15}}} + {{{16}}} + {{{17}}} + {{{18}}} + {{{19}}} }}
|- style="font-size: {{{77|70}}}%;"
! 4
| {{{20}}}
|class="dummy"|
| {{{21}}}
|class="dummy"|
| {{{22}}}
|class="dummy"|
| {{{23}}}
|class="dummy"|
| {{{24}}}
|class="dummy"|
| {{{25}}}
|class="dummy"|
| {{{26}}}
|class="dummy"|
| {{{27}}}
|class="dummy"|
| {{{28}}}
|class="dummy"|
| {{{29}}}
|class="dummy"|
| {{{30}}}
|class="dummy"|
| {{{31}}}
|class="dummy"|
| {{{32}}}
|class="dummy"|
| {{{33}}}
|class="dummy"|
| {{{34}}}
|class="dummy"|
| {{{35}}}
|class="dummy"|
| {{{36}}}
!class="sum"| {{#expr: {{{20}}} + {{{21}}} + {{{22}}} + {{{23}}} + {{{24}}} + {{{25}}} + {{{26}}} + {{{27}}} + {{{28}}} + {{{29}}} + {{{30}}} + {{{31}}} + {{{32}}} + {{{33}}} + {{{34}}} + {{{35}}} + {{{36}}} }}
|- style="font-size: {{{78|50}}}%;"
! 5
| {{{37}}}
| {{{38}}}
| {{{39}}}
| {{{40}}}
| {{{41}}}
| {{{42}}}
| {{{43}}}
| {{{44}}}
| {{{45}}}
| {{{46}}}
| {{{47}}}
| {{{48}}}
| {{{49}}}
| {{{50}}}
| {{{51}}}
| {{{52}}}
| {{{53}}}
| {{{54}}}
| {{{55}}}
| {{{56}}}
| {{{57}}}
| {{{58}}}
| {{{59}}}
| {{{60}}}
| {{{61}}}
| {{{62}}}
| {{{63}}}
| {{{64}}}
| {{{65}}}
| {{{66}}}
| {{{67}}}
| {{{68}}}
| {{{69}}}
!class="sum"| {{#expr: {{{37}}} + {{{38}}} + {{{39}}} + {{{40}}} + {{{41}}} + {{{42}}} + {{{43}}} + {{{44}}} + {{{45}}} + {{{46}}} + {{{47}}} + {{{48}}} + {{{49}}} + {{{50}}} + {{{51}}} + {{{52}}} + {{{53}}} + {{{54}}} + {{{55}}} + {{{56}}} + {{{57}}} + {{{58}}} + {{{59}}} + {{{60}}} + {{{61}}} + {{{62}}} + {{{63}}} + {{{64}}} + {{{65}}} + {{{66}}} + {{{67}}} + {{{68}}} + {{{69}}} }}
|}
{{Collapsible END}}
<!-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------->
{{Collapsible START|integer weight|collapsed light wide followed}}
{| class="wikitable boolf-triangle" style="margin: 0;"
! {{diagonal split header|''a''|''w''}}
! 0
! 1
! 2
! 3
! 4
! 5
! 6
! 7
! 8
! 9
! 10
! 11
! 12
! 13
! 14
! 15
! 16
! 17
! 18
! 19
! 20
! 21
! 22
! 23
! 24
! 25
! 26
! 27
! 28
! 29
! 30
! 31
! 32
!class="sum"| sums
|-
! 0
| {{{1}}}
| {{{2}}}
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
!class="sum"| {{#expr: {{{1}}} + {{{2}}} }}
|-
! 1
| {{{3}}}
| {{{4}}}
| {{{5}}}
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
!class="sum"| {{#expr: {{{3}}} + {{{4}}} + {{{5}}} }}
|-
! 2
| {{{6}}}
| {{{7}}}
| {{{8}}}
| {{{9}}}
| {{{10}}}
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
!class="sum"| {{#expr: {{{6}}} + {{{7}}} + {{{8}}} + {{{9}}} + {{{10}}} }}
|- style="font-size: {{{76|80}}}%;"
! 3
| {{{11}}}
| {{{12}}}
| {{{13}}}
| {{{14}}}
| {{{15}}}
| {{{16}}}
| {{{17}}}
| {{{18}}}
| {{{19}}}
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
!class="sum"| {{#expr: {{{11}}} + {{{12}}} + {{{13}}} + {{{14}}} + {{{15}}} + {{{16}}} + {{{17}}} + {{{18}}} + {{{19}}} }}
|- style="font-size: {{{77|70}}}%;"
! 4
| {{{20}}}
| {{{21}}}
| {{{22}}}
| {{{23}}}
| {{{24}}}
| {{{25}}}
| {{{26}}}
| {{{27}}}
| {{{28}}}
| {{{29}}}
| {{{30}}}
| {{{31}}}
| {{{32}}}
| {{{33}}}
| {{{34}}}
| {{{35}}}
| {{{36}}}
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
!class="sum"| {{#expr: {{{20}}} + {{{21}}} + {{{22}}} + {{{23}}} + {{{24}}} + {{{25}}} + {{{26}}} + {{{27}}} + {{{28}}} + {{{29}}} + {{{30}}} + {{{31}}} + {{{32}}} + {{{33}}} + {{{34}}} + {{{35}}} + {{{36}}} }}
|- style="font-size: {{{78|50}}}%;"
! 5
| {{{37}}}
| {{{38}}}
| {{{39}}}
| {{{40}}}
| {{{41}}}
| {{{42}}}
| {{{43}}}
| {{{44}}}
| {{{45}}}
| {{{46}}}
| {{{47}}}
| {{{48}}}
| {{{49}}}
| {{{50}}}
| {{{51}}}
| {{{52}}}
| {{{53}}}
| {{{54}}}
| {{{55}}}
| {{{56}}}
| {{{57}}}
| {{{58}}}
| {{{59}}}
| {{{60}}}
| {{{61}}}
| {{{62}}}
| {{{63}}}
| {{{64}}}
| {{{65}}}
| {{{66}}}
| {{{67}}}
| {{{68}}}
| {{{69}}}
!class="sum"| {{#expr: {{{37}}} + {{{38}}} + {{{39}}} + {{{40}}} + {{{41}}} + {{{42}}} + {{{43}}} + {{{44}}} + {{{45}}} + {{{46}}} + {{{47}}} + {{{48}}} + {{{49}}} + {{{50}}} + {{{51}}} + {{{52}}} + {{{53}}} + {{{54}}} + {{{55}}} + {{{56}}} + {{{57}}} + {{{58}}} + {{{59}}} + {{{60}}} + {{{61}}} + {{{62}}} + {{{63}}} + {{{64}}} + {{{65}}} + {{{66}}} + {{{67}}} + {{{68}}} + {{{69}}} }}
|}
{{Collapsible END}}
<!-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------->
<templatestyles src="Template:Boolf triangles/style.css" />
{{Collapsible START|<small>(merged weights)</small>|collapsed wide light}}
{| class="wikitable boolf-triangle" style="margin: 0;"
|-
!style="color: gray;"| ''w''
!style="color: gray; font-size: 60%;"| 0
!style="color: gray;"| 1
!style="color: gray;"| 2
!style="color: gray;"| 3...4
!style="color: gray;"| 5...8
!style="color: gray;"| 9...16
!style="color: gray;"| 17...32
!rowspan="2" class="sum"| sums
|-
! {{diagonal split header|''a''|''k''}}
!style="font-size: 60%;"| -1
! 0
! 1
! 2
! 3
! 4
! 5
|-
! 0
|style="font-size: 60%;"| {{{1}}}
| {{{2}}}
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
!class="sum"| {{#expr: {{{1}}} + {{{2}}} }}
|-
! 1
|style="font-size: 60%;"| {{{3}}}
| {{{4}}}
| {{{5}}}
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
!class="sum"| {{#expr: {{{3}}} + {{{4}}} + {{{5}}} }}
|-
! 2
|style="font-size: 60%;"| {{{6}}}
| {{{7}}}
| {{{8}}}
| {{#expr: {{{9}}} + {{{10}}} }}
|class="dummy"|
|class="dummy"|
|class="dummy"|
!class="sum"| {{#expr: {{{6}}} + {{{7}}} + {{{8}}} + {{{9}}} + {{{10}}} }}
|-
! 3
|style="font-size: 60%;"| {{{11}}}
| {{{12}}}
| {{{13}}}
| {{#expr: {{{14}}} + {{{15}}} }}
| {{#expr: {{{16}}} + {{{17}}} + {{{18}}} + {{{19}}} }}
|class="dummy"|
|class="dummy"|
!class="sum"| {{#expr: {{{11}}} + {{{12}}} + {{{13}}} + {{{14}}} + {{{15}}} + {{{16}}} + {{{17}}} + {{{18}}} + {{{19}}} }}
|-
! 4
|style="font-size: 60%;"| {{{20}}}
| {{{21}}}
| {{{22}}}
| {{#expr: {{{23}}} + {{{24}}} }}
| {{#expr: {{{25}}} + {{{26}}} + {{{27}}} + {{{28}}} }}
| {{#expr: {{{29}}} + {{{30}}} + {{{31}}} + {{{32}}} + {{{33}}} + {{{34}}} + {{{35}}} + {{{36}}} }}
|class="dummy"|
!class="sum"| {{#expr: {{{20}}} + {{{21}}} + {{{22}}} + {{{23}}} + {{{24}}} + {{{25}}} + {{{26}}} + {{{27}}} + {{{28}}} + {{{29}}} + {{{30}}} + {{{31}}} + {{{32}}} + {{{33}}} + {{{34}}} + {{{35}}} + {{{36}}} }}
|-
! 5
|style="font-size: 60%;"| {{{37}}}
| {{{38}}}
| {{{39}}}
| {{#expr: {{{40}}} + {{{41}}} }}
| {{#expr: {{{42}}} + {{{43}}} + {{{44}}} + {{{45}}} }}
| {{#expr: {{{46}}} + {{{47}}} + {{{48}}} + {{{49}}} + {{{50}}} + {{{51}}} + {{{52}}} + {{{53}}} }}
| {{#expr: {{{54}}} + {{{55}}} + {{{56}}} + {{{57}}} + {{{58}}} + {{{59}}} + {{{60}}} + {{{61}}} + {{{62}}} + {{{63}}} + {{{64}}} + {{{65}}} + {{{66}}} + {{{67}}} + {{{68}}} + {{{69}}} }}
!class="sum"| {{#expr: {{{37}}} + {{{38}}} + {{{39}}} + {{{40}}} + {{{41}}} + {{{42}}} + {{{43}}} + {{{44}}} + {{{45}}} + {{{46}}} + {{{47}}} + {{{48}}} + {{{49}}} + {{{50}}} + {{{51}}} + {{{52}}} + {{{53}}} + {{{54}}} + {{{55}}} + {{{56}}} + {{{57}}} + {{{58}}} + {{{59}}} + {{{60}}} + {{{61}}} + {{{62}}} + {{{63}}} + {{{64}}} + {{{65}}} + {{{66}}} + {{{67}}} + {{{68}}} + {{{69}}} }}
|}
{{Collapsible END}}<noinclude>
----
There is also {{tl|Boolf weight triangle 4}}. (For some cases the calculation for arity 5 is not feasible.)
see e.g. {{tl|Boolf weight triangle; dense}}
[[Category:Boolf triangles with weight columns| ]]
[[Category:Some templates created by Watchduck]]
</noinclude>
jjsaock82e6fu7qw0b6sc76y5ryu9dy
Template:Boolf weight triangle; dense
10
310773
2694011
2669488
2025-01-01T19:51:35Z
Watchduck
137431
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{{Boolf weight triangle 5|
1|1|
0|1|1|
0|1|4|4|1|
0|1|13|44|67|56|28|8|1|
0|1|40|360|1546|4144|7896|11408|12866|11440|8008|4368|1820|560|120|16|1|
0|1|121|2680|27550|180096|866432|3308736|10453960|27991600|64472200|129002640|225783740|347370800|471435000|565722640|601080385|565722720|471435600|347373600|225792840|129024480|64512240|28048800|10518300|3365856|906192|201376|35960|4960|496|32|1|
2|2|10|218|64594|4294642034}}
<noinclude>
----
<div style="font-weight: bold;">
🌊 triangle ''Magnolia'' <small>({{oeislink|A163353}})</small> row sums ''Lotus'' <small>({{oeislink|A000371}})</small>
</div>
summation of {{tl|Boolf triangle Myrtle}}
'''rational weight:'''
Let ''C'' be the central column. Then ''C''(''a'') + ''a'' = 2 · {{oeislink|A069954}}(''a−1'') = {{oeislink|A000984}}(2<sup>''a''−1</sup>) for ''a'' > 0.
'''integer weight:'''
Diagonal is {{oeislink|A134174}}.
Column 2 is {{oeislink|A003462}}. a(n) = (3<sup>n</sup> − 1) / 2.
'''merged weights:'''
The diagonal 1, 5, 93, 26333... is half of {{oeislink|A037267}}(1...) = 2, 10, 186, 52666...
[[Category:Boolf triangles with weight columns]]
</noinclude>
9f1knf7539xsebkj7cnejvafiav1nfr
2694016
2694011
2025-01-01T20:16:14Z
Watchduck
137431
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wikitext
text/x-wiki
{{Boolf weight triangle 5|
1|1|
0|1|1|
0|1|4|4|1|
0|1|13|44|67|56|28|8|1|
0|1|40|360|1546|4144|7896|11408|12866|11440|8008|4368|1820|560|120|16|1|
0|1|121|2680|27550|180096|866432|3308736|10453960|27991600|64472200|129002640|225783740|347370800|471435000|565722640|601080385|565722720|471435600|347373600|225792840|129024480|64512240|28048800|10518300|3365856|906192|201376|35960|4960|496|32|1}}
<noinclude>
----
<div style="font-weight: bold;">
🌊 triangle ''Magnolia'' <small>({{oeislink|A163353}})</small> row sums ''Lotus'' <small>({{oeislink|A000371}})</small>
</div>
summation of {{tl|Boolf triangle Myrtle}}
'''rational weight:'''
Let ''C'' be the central column. Then ''C''(''a'') + ''a'' = 2 · {{oeislink|A069954}}(''a−1'') = {{oeislink|A000984}}(2<sup>''a''−1</sup>) for ''a'' > 0.
'''integer weight:'''
Diagonal is {{oeislink|A134174}}.
Column 2 is {{oeislink|A003462}}. a(n) = (3<sup>n</sup> − 1) / 2.
'''merged weights:'''
The diagonal 1, 5, 93, 26333... is half of {{oeislink|A037267}}(1...) = 2, 10, 186, 52666...
[[Category:Boolf triangles with weight columns]]
</noinclude>
4d2v67oqs2lp3g9xahk5u68bi42rzwv
2694018
2694016
2025-01-01T20:18:38Z
Watchduck
137431
Watchduck moved page [[Template:Boolf triangle Magnolia]] to [[Template:Boolf weight triangle; dense]]
2694016
wikitext
text/x-wiki
{{Boolf weight triangle 5|
1|1|
0|1|1|
0|1|4|4|1|
0|1|13|44|67|56|28|8|1|
0|1|40|360|1546|4144|7896|11408|12866|11440|8008|4368|1820|560|120|16|1|
0|1|121|2680|27550|180096|866432|3308736|10453960|27991600|64472200|129002640|225783740|347370800|471435000|565722640|601080385|565722720|471435600|347373600|225792840|129024480|64512240|28048800|10518300|3365856|906192|201376|35960|4960|496|32|1}}
<noinclude>
----
<div style="font-weight: bold;">
🌊 triangle ''Magnolia'' <small>({{oeislink|A163353}})</small> row sums ''Lotus'' <small>({{oeislink|A000371}})</small>
</div>
summation of {{tl|Boolf triangle Myrtle}}
'''rational weight:'''
Let ''C'' be the central column. Then ''C''(''a'') + ''a'' = 2 · {{oeislink|A069954}}(''a−1'') = {{oeislink|A000984}}(2<sup>''a''−1</sup>) for ''a'' > 0.
'''integer weight:'''
Diagonal is {{oeislink|A134174}}.
Column 2 is {{oeislink|A003462}}. a(n) = (3<sup>n</sup> − 1) / 2.
'''merged weights:'''
The diagonal 1, 5, 93, 26333... is half of {{oeislink|A037267}}(1...) = 2, 10, 186, 52666...
[[Category:Boolf triangles with weight columns]]
</noinclude>
4d2v67oqs2lp3g9xahk5u68bi42rzwv
2694020
2694018
2025-01-01T20:20:50Z
Watchduck
137431
2694020
wikitext
text/x-wiki
{{Boolf weight triangle 5|
1|1|
0|1|1|
0|1|4|4|1|
0|1|13|44|67|56|28|8|1|
0|1|40|360|1546|4144|7896|11408|12866|11440|8008|4368|1820|560|120|16|1|
0|1|121|2680|27550|180096|866432|3308736|10453960|27991600|64472200|129002640|225783740|347370800|471435000|565722640|601080385|565722720|471435600|347373600|225792840|129024480|64512240|28048800|10518300|3365856|906192|201376|35960|4960|496|32|1}}
<noinclude>
----
<div style="font-weight: bold;">
🌊 triangle ''Magnolia'' <small>({{oeislink|A163353}})</small> row sums ''Lotus'' <small>({{oeislink|A000371}})</small>
</div>
[[Boolf-term#dense|Dense]] <small>(non-degenerate)</small> Boolean functions by weight.
'''rational weight:'''
Let ''C'' be the central column. Then ''C''(''a'') + ''a'' = 2 · {{oeislink|A069954}}(''a−1'') = {{oeislink|A000984}}(2<sup>''a''−1</sup>) for ''a'' > 0.
'''integer weight:'''
Diagonal is {{oeislink|A134174}}.
Column 2 is {{oeislink|A003462}}. a(n) = (3<sup>n</sup> − 1) / 2.
'''merged weights:'''
The diagonal 1, 5, 93, 26333... is half of {{oeislink|A037267}}(1...) = 2, 10, 186, 52666...
[[Category:Boolf triangles with weight columns]]
</noinclude>
qx1pnjzwucv9g3o0w65z7xzvxl300b0
2694021
2694020
2025-01-01T20:21:28Z
Watchduck
137431
2694021
wikitext
text/x-wiki
{{Boolf weight triangle 5|
1|1|
0|1|1|
0|1|4|4|1|
0|1|13|44|67|56|28|8|1|
0|1|40|360|1546|4144|7896|11408|12866|11440|8008|4368|1820|560|120|16|1|
0|1|121|2680|27550|180096|866432|3308736|10453960|27991600|64472200|129002640|225783740|347370800|471435000|565722640|601080385|565722720|471435600|347373600|225792840|129024480|64512240|28048800|10518300|3365856|906192|201376|35960|4960|496|32|1}}
<noinclude>
----
<div style="font-weight: bold;">
🌊 triangle {{oeislink|A163353}} row sums ''Lotus'' <small>({{oeislink|A000371}})</small>
</div>
[[Boolf-term#dense|Dense]] <small>(non-degenerate)</small> Boolean functions by weight.
'''rational weight:'''
Let ''C'' be the central column. Then ''C''(''a'') + ''a'' = 2 · {{oeislink|A069954}}(''a−1'') = {{oeislink|A000984}}(2<sup>''a''−1</sup>) for ''a'' > 0.
'''integer weight:'''
Diagonal is {{oeislink|A134174}}.
Column 2 is {{oeislink|A003462}}. a(n) = (3<sup>n</sup> − 1) / 2.
'''merged weights:'''
The diagonal 1, 5, 93, 26333... is half of {{oeislink|A037267}}(1...) = 2, 10, 186, 52666...
[[Category:Boolf triangles with weight columns]]
</noinclude>
rb5miqq3pgce85ljwhvoq2m98aqkj1a
2694025
2694021
2025-01-01T20:23:25Z
Watchduck
137431
2694025
wikitext
text/x-wiki
{{Boolf weight triangle 5|
1|1|
0|1|1|
0|1|4|4|1|
0|1|13|44|67|56|28|8|1|
0|1|40|360|1546|4144|7896|11408|12866|11440|8008|4368|1820|560|120|16|1|
0|1|121|2680|27550|180096|866432|3308736|10453960|27991600|64472200|129002640|225783740|347370800|471435000|565722640|601080385|565722720|471435600|347373600|225792840|129024480|64512240|28048800|10518300|3365856|906192|201376|35960|4960|496|32|1}}
<noinclude>
----
<div style="font-weight: bold;">
🌊 triangle {{oeislink|A163353}} row sums ''Lotus'' <small>({{oeislink|A000371}})</small>
</div>
[[Boolf-term#dense|Dense]] <small>(non-degenerate)</small> Boolean functions by weight.
'''rational weight:'''
Let ''C'' be the central column. Then ''C''(''a'') + ''a'' = 2 · {{oeislink|A069954}}(''a−1'') = {{oeislink|A000984}}(2<sup>''a''−1</sup>) for ''a'' > 0.
'''integer weight:'''
Diagonal is {{oeislink|A134174}}.
Column 2 is {{oeislink|A003462}}. a(n) = (3<sup>n</sup> − 1) / 2.
'''merged weights:'''
The diagonal 1, 5, 93, 26333... is half of {{oeislink|A037267}}(1...) = 2, 10, 186, 52666...
[[Category:Boolf triangles with weight columns|dense]]
</noinclude>
8sr59o836bogbuzv48c770z11s8m4uz
Mentors of Boolean functions
0
313565
2694037
2693916
2025-01-01T22:42:37Z
Watchduck
137431
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wikitext
text/x-wiki
{{Boolf header}}
__NOTOC__
The mentor is a rather dubious [[Soft properties of Boolean functions|soft property]] of a BF. But it seems surprisingly interesting.<br>
It is found in three steps: Creating a [[Boolf-hard#family|family matrix]], getting the senior [[Noble Boolean functions|nobles]] of its rows, and getting their [[Boolf-hard#prefect|prefects]].<br>
The first digits of the prefects form the mentor.
{{Mentors of Boolean functions/illustrated examples 3-ary}}
The following images are the 4-ary equivalents of those above. The results are either the same as above, or the complement.
{{Mentors of Boolean functions/illustrated examples 4-ary}}
==Walsh permutations==
A Boolean function has a unique mentor for a given arity. The map between mentors can be expressed by four different [[Walsh permutation]]s.<br>
<small style="opacity: .7;">(That is, because a BF can be denoted by its truth table or by its Zhegalkin index. In the images above, they are shown red and green.)</small><br>
In all there are six Walsh permutations, which shall be denoted by Cyrillic letters:
Ж <small>(Zhe)</small>, Ч <small>(Che)</small>, Ш <small>(Sha)</small>, Ю <small>(Yu)</small>, Я <small>(Ya)</small>, Щ <small>(Shcha)</small><br>
Their degree is <math>d = 2^{arity}</math>, i.e. they correspond to invertible binary <math>d \times d</math> matrices.<br>
<small style="opacity: .7;">(The letter Ж is used in two different ways: On its own it represents the permutation. Followed by an integer it represents a BF, identified by its [[Zhegalkin matrix|Zhegalkin index]].)</small>
{{Mentors of Boolean functions/four WP relationships}}
Ч and Ш are both self-inverse. Ю and Я are inverse to each other.
The matrix of Щ is part of a top right Sierpinki triangle. Its diagonals follow a negated XOR pattern. <small style="opacity: .5;">(See [[c:File:Variadic5 antipode; ESAND (ESNOR twin).svg|image]].)</small><br>
The matrix of Ш is almost the same, but with the top right entry inverted.<br>
The matrix of Ч is a family matrix. Its top row is related to that of the Ш matrix. <small>The calculation involves the Zhegalkin twin and reversing the truth table.</small>
{{Mentors of Boolean functions/code Che matrix}}
{{Collapsible START|relationship between the matrix patterns|collapsed gap-above gap-below}}
The red family matrix has the pattern of Ч.<br>
The green matrix shows the twins of the red rows, and has the pattern of Ю and Я.<br>
The blue matrix shows the twins of the green columns, and has the pattern of Ш.<br>
[[File:Family of Zhe 38504.svg|400px|center]]
{{Collapsible END}}
{{Mentors of Boolean functions/code WP vectors}}
See the fixed points of Ч and Ш ordered by weight: [[Template:Boolf weight triangle; fixed points of Che|Ч]] [[Template:Boolf weight triangle; fixed points of Sha|Ш]]
===1-ary===
The permutations are all six Walsh permutations of degree 2. {{spaces|5}} Ч = (0, 2, 1, 3) {{spaces|5}} Ш = (0, 1, 3, 2) {{spaces|5}} Щ = I <small>(neutral permutation)</small>
{{Collapsible START|permutations|wide collapsed}}
{{Mentors of Boolean functions/illustrated WP 1-ary}}
{{Collapsible END}}
===2-ary===
Ч = Ш = I <small>(neutral permutation)</small> {{spaces|5}} Ю = Я = Ж {{spaces|5}} <small>Щ = (0, 1, 2, 3, 4, 5, 6, 7, 9, 8, 11, 10, 13, 12, 15, 14)</small>
===3-ary===
{{Mentors of Boolean functions/illustrated WP 3-ary}}
{{Collapsible START|code|collapsed light gap-above}}
{{Mentors of Boolean functions/code/3-ary}}
{{Collapsible END}}
{{Mentors of Boolean functions/small WP example 199}}
===4-ary===
{{Mentors of Boolean functions/illustrated WP 4-ary}}
{{Collapsible START|code|collapsed light gap-above gap-below}}
{{Mentors of Boolean functions/code/4-ary}}
{{Collapsible END}}
==seminars==
The mentor is not simply a bijection between Boolean functions, but between the truth tables for a given arity.<br>
The permutation from Zhegalkin indices to those of their ''n''-ary mentors is '''Ш<sub>''n''</sub>'''. The beginning of Ш<sub>''n''+1</sub> is '''Щ<sub>''n''</sub>'''. <small style="opacity: .5; font-size: 60%;">([[w:Shcha|This letter]] has a little hook on the right.)</small><br>
These two permutations are very similar. They are equal in the first half, and differ by exchanged neighbors in the second.
Neighboring Zhegalkin indices <small>(i.e. 2·''n'' and 2·''n''+1)</small> denote complements.<br>
So although there is no mentor bijection between Boolean functions, there is one between pairs of complements.<br>
Complement and mentor partition the set of all Boolean functions into blocks of size 4 or 2. Such a block shall be called (big or small) ''seminar''.<br>
The Zhegalkin indices in a big seminar are <math>\{ a, a+1, b, b+1 \}</math> with even <math>a</math> and <math>b</math>, so it can be represented by the pair <math>(a, b)</math>.<br>
<small>An example of a seminar is {138, 139, 156, 157}, represented as (138, 156). See [[c:File:Set of 3-ary Boolean functions 12855504354077768210885020350402125463028803369886765232947200.svg|image]]. <small style="opacity: .5;">In the permutation it is represented by the pair (69, 78).</small></small>
The pairs <math>\left( \frac{a}{2}, \frac{b}{2} \right)</math> are the cycles of a self-inverse Walsh permutation of degree <math>2^{arity} - 1</math>. <small style="opacity: .5;">(For arity 3 the degree is 7, and the permuted integers are 0...127.)</small><br>
For arities 1 and 2 this permutation is neutral. For arity 3 is has 64 fixed points <small>(of 128 places, i.e. 1/2)</small>. For arity 5 it has 1024 fixed points <small>(of 32768 places, i.e. 1/32)</small>.<br>
{{Collapsible START|mentor permutation|collapsed wide}}
[[File:Mentors 4; seminar permutation.svg|thumb|15×15 matrix corresponding to the Walsh permutation for arity 4.]]
The first 64 entries of the sequence are the fixed points. The next 64 entries form these 32 cycles:
<source lang="python" style="font-size: 60%;">
[ 64, 75], [ 65, 74], [ 66, 73], [ 67, 72], [ 68, 79], [ 69, 78], [ 70, 77], [ 71, 76],
[ 80, 91], [ 81, 90], [ 82, 89], [ 83, 88], [ 84, 95], [ 85, 94], [ 86, 93], [ 87, 92],
[ 96, 107], [ 97, 106], [ 98, 105], [ 99, 104], [100, 111], [101, 110], [102, 109], [103, 108],
[112, 123], [113, 122], [114, 121], [115, 120], [116, 127], [117, 126], [118, 125], [119, 124]
</source>
The permutation for arity 4 corresponds to the 15×15 matrix shown on the right.<br>
<small style="opacity: .5; font-size: 60%;">It is described by this vector: (1, 2, 4, 8, 16, 32, 75, 128, 256, 512, 1155, 2048, 4233, 8330, 19252)</small>
The matrix is always that of Ш or Щ without the top row and left column.
{{Collapsible END}}
For arity 3 the Boolean functions in big seminars are the sharp ones <small>(i.e. those with odd weight)</small>. See {{Boolf prop 3-ary|seminar|images}}.
<small>For a given arity, each seminar is part of a {{Boolf prop 3-ary|chunky seminar}}. For arity 3 they all have size 16. {{Mentors of Boolean functions/example chunky seminar}}</small>
[[Category:Mentors of Boolean functions]]
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{{Boolf header}}
__NOTOC__
The mentor is a rather dubious [[Soft properties of Boolean functions|soft property]] of a BF. But it seems surprisingly interesting.<br>
It is found in three steps: Creating a [[Boolf-hard#family|family matrix]], getting the senior [[Noble Boolean functions|nobles]] of its rows, and getting their [[Boolf-hard#prefect|prefects]].<br>
The first digits of the prefects form the mentor.
{{Mentors of Boolean functions/illustrated examples 3-ary}}
The following images are the 4-ary equivalents of those above. The results are either the same as above, or the complement.
{{Mentors of Boolean functions/illustrated examples 4-ary}}
==Walsh permutations==
A Boolean function has a unique mentor for a given arity. The map between mentors can be expressed by four different [[Walsh permutation]]s.<br>
<small style="opacity: .7;">(That is, because a BF can be denoted by its truth table or by its Zhegalkin index. In the images above, they are shown red and green.)</small><br>
In all there are six Walsh permutations, which shall be denoted by Cyrillic letters:
Ж <small>(Zhe)</small>, Ч <small>(Che)</small>, Ш <small>(Sha)</small>, Ю <small>(Yu)</small>, Я <small>(Ya)</small>, Щ <small>(Shcha)</small><br>
Their degree is <math>d = 2^{arity}</math>, i.e. they correspond to invertible binary <math>d \times d</math> matrices.<br>
<small style="opacity: .7;">(The letter Ж is used in two different ways: On its own it represents the permutation. Followed by an integer it represents a BF, identified by its [[Zhegalkin matrix|Zhegalkin index]].)</small>
{{Mentors of Boolean functions/four WP relationships}}
Ч and Ш are both self-inverse. Ю and Я are inverse to each other.
The matrix of Щ is part of a top right Sierpinki triangle. Its diagonals follow a negated XOR pattern. <small style="opacity: .5;">(See [[c:File:Variadic5 antipode; ESAND (ESNOR twin).svg|image]].)</small><br>
The matrix of Ш is almost the same, but with the top right entry inverted.<br>
The matrix of Ч is a family matrix. Its top row is related to that of the Ш matrix. <small>The calculation involves the Zhegalkin twin and reversing the truth table.</small>
{{Mentors of Boolean functions/code Che matrix}}
{{Collapsible START|relationship between the matrix patterns|collapsed gap-above gap-below}}
The red family matrix has the pattern of Ч.<br>
The green matrix shows the twins of the red rows, and has the pattern of Ю and Я.<br>
The blue matrix shows the twins of the green columns, and has the pattern of Ш.<br>
[[File:Family of Zhe 38504.svg|400px|center]]
{{Collapsible END}}
{{Mentors of Boolean functions/code WP vectors}}
See the fixed points of Ч and Ш ordered by weight: [[Template:Boolf weight triangle; fixed points of Che|Ч]], [[Template:Boolf weight triangle; fixed points of Sha|Ш]]
===1-ary===
The permutations are all six Walsh permutations of degree 2. {{spaces|5}} Ч = (0, 2, 1, 3) {{spaces|5}} Ш = (0, 1, 3, 2) {{spaces|5}} Щ = I <small>(neutral permutation)</small>
{{Collapsible START|permutations|wide collapsed}}
{{Mentors of Boolean functions/illustrated WP 1-ary}}
{{Collapsible END}}
===2-ary===
Ч = Ш = I <small>(neutral permutation)</small> {{spaces|5}} Ю = Я = Ж {{spaces|5}} <small>Щ = (0, 1, 2, 3, 4, 5, 6, 7, 9, 8, 11, 10, 13, 12, 15, 14)</small>
===3-ary===
{{Mentors of Boolean functions/illustrated WP 3-ary}}
{{Collapsible START|code|collapsed light gap-above}}
{{Mentors of Boolean functions/code/3-ary}}
{{Collapsible END}}
{{Mentors of Boolean functions/small WP example 199}}
===4-ary===
{{Mentors of Boolean functions/illustrated WP 4-ary}}
{{Collapsible START|code|collapsed light gap-above gap-below}}
{{Mentors of Boolean functions/code/4-ary}}
{{Collapsible END}}
==seminars==
The mentor is not simply a bijection between Boolean functions, but between the truth tables for a given arity.<br>
The permutation from Zhegalkin indices to those of their ''n''-ary mentors is '''Ш<sub>''n''</sub>'''. The beginning of Ш<sub>''n''+1</sub> is '''Щ<sub>''n''</sub>'''. <small style="opacity: .5; font-size: 60%;">([[w:Shcha|This letter]] has a little hook on the right.)</small><br>
These two permutations are very similar. They are equal in the first half, and differ by exchanged neighbors in the second.
Neighboring Zhegalkin indices <small>(i.e. 2·''n'' and 2·''n''+1)</small> denote complements.<br>
So although there is no mentor bijection between Boolean functions, there is one between pairs of complements.<br>
Complement and mentor partition the set of all Boolean functions into blocks of size 4 or 2. Such a block shall be called (big or small) ''seminar''.<br>
The Zhegalkin indices in a big seminar are <math>\{ a, a+1, b, b+1 \}</math> with even <math>a</math> and <math>b</math>, so it can be represented by the pair <math>(a, b)</math>.<br>
<small>An example of a seminar is {138, 139, 156, 157}, represented as (138, 156). See [[c:File:Set of 3-ary Boolean functions 12855504354077768210885020350402125463028803369886765232947200.svg|image]]. <small style="opacity: .5;">In the permutation it is represented by the pair (69, 78).</small></small>
The pairs <math>\left( \frac{a}{2}, \frac{b}{2} \right)</math> are the cycles of a self-inverse Walsh permutation of degree <math>2^{arity} - 1</math>. <small style="opacity: .5;">(For arity 3 the degree is 7, and the permuted integers are 0...127.)</small><br>
For arities 1 and 2 this permutation is neutral. For arity 3 is has 64 fixed points <small>(of 128 places, i.e. 1/2)</small>. For arity 5 it has 1024 fixed points <small>(of 32768 places, i.e. 1/32)</small>.<br>
{{Collapsible START|mentor permutation|collapsed wide}}
[[File:Mentors 4; seminar permutation.svg|thumb|15×15 matrix corresponding to the Walsh permutation for arity 4.]]
The first 64 entries of the sequence are the fixed points. The next 64 entries form these 32 cycles:
<source lang="python" style="font-size: 60%;">
[ 64, 75], [ 65, 74], [ 66, 73], [ 67, 72], [ 68, 79], [ 69, 78], [ 70, 77], [ 71, 76],
[ 80, 91], [ 81, 90], [ 82, 89], [ 83, 88], [ 84, 95], [ 85, 94], [ 86, 93], [ 87, 92],
[ 96, 107], [ 97, 106], [ 98, 105], [ 99, 104], [100, 111], [101, 110], [102, 109], [103, 108],
[112, 123], [113, 122], [114, 121], [115, 120], [116, 127], [117, 126], [118, 125], [119, 124]
</source>
The permutation for arity 4 corresponds to the 15×15 matrix shown on the right.<br>
<small style="opacity: .5; font-size: 60%;">It is described by this vector: (1, 2, 4, 8, 16, 32, 75, 128, 256, 512, 1155, 2048, 4233, 8330, 19252)</small>
The matrix is always that of Ш or Щ without the top row and left column.
{{Collapsible END}}
For arity 3 the Boolean functions in big seminars are the sharp ones <small>(i.e. those with odd weight)</small>. See {{Boolf prop 3-ary|seminar|images}}.
<small>For a given arity, each seminar is part of a {{Boolf prop 3-ary|chunky seminar}}. For arity 3 they all have size 16. {{Mentors of Boolean functions/example chunky seminar}}</small>
[[Category:Mentors of Boolean functions]]
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{{Boolf header}}
__NOTOC__
The mentor is a rather dubious [[Soft properties of Boolean functions|soft property]] of a BF. But it seems surprisingly interesting.<br>
It is found in three steps: Creating a [[Boolf-hard#family|family matrix]], getting the senior [[Noble Boolean functions|nobles]] of its rows, and getting their [[Boolf-hard#prefect|prefects]].<br>
The first digits of the prefects form the mentor.
{{Mentors of Boolean functions/illustrated examples 3-ary}}
The following images are the 4-ary equivalents of those above. The results are either the same as above, or the complement.
{{Mentors of Boolean functions/illustrated examples 4-ary}}
==Walsh permutations==
A Boolean function has a unique mentor for a given arity. The map between mentors can be expressed by four different [[Walsh permutation]]s.<br>
<small style="opacity: .7;">(That is, because a BF can be denoted by its truth table or by its Zhegalkin index. In the images above, they are shown red and green.)</small><br>
In all there are six Walsh permutations, which shall be denoted by Cyrillic letters:
Ж <small>(Zhe)</small>, Ч <small>(Che)</small>, Ш <small>(Sha)</small>, Ю <small>(Yu)</small>, Я <small>(Ya)</small>, Щ <small>(Shcha)</small><br>
Their degree is <math>d = 2^{arity}</math>, i.e. they correspond to invertible binary <math>d \times d</math> matrices.<br>
<small style="opacity: .7;">(The letter Ж is used in two different ways: On its own it represents the permutation. Followed by an integer it represents a BF, identified by its [[Zhegalkin matrix|Zhegalkin index]].)</small>
{{Mentors of Boolean functions/four WP relationships}}
Ч and Ш are both self-inverse. Ю and Я are inverse to each other.
The matrix of Щ is part of a top right Sierpinki triangle. Its diagonals follow a negated XOR pattern. <small style="opacity: .5;">(See [[c:File:Variadic5 antipode; ESAND (ESNOR twin).svg|image]].)</small><br>
The matrix of Ш is almost the same, but with the top right entry inverted.<br>
The matrix of Ч is a family matrix. Its top row is related to that of the Ш matrix. <small>The calculation involves the Zhegalkin twin and reversing the truth table.</small>
{{Mentors of Boolean functions/code Che matrix}}
{{Collapsible START|relationship between the matrix patterns|collapsed gap-above gap-below}}
The red family matrix has the pattern of Ч.<br>
The green matrix shows the twins of the red rows, and has the pattern of Ю and Я.<br>
The blue matrix shows the twins of the green columns, and has the pattern of Ш.<br>
[[File:Family of Zhe 38504.svg|400px|center]]
{{Collapsible END}}
{{Mentors of Boolean functions/code WP vectors}}
See the fixed points of Ч and Ш ordered by weight: [[Template:Boolf weight triangle; fixed points of Che|Ч]], [[Template:Boolf weight triangle; fixed points of Sha|Ш]]
===1-ary===
The permutations are all six Walsh permutations of degree 2. {{spaces|5}} Ч = (0, 2, 1, 3) {{spaces|5}} Ш = (0, 1, 3, 2) {{spaces|5}} Щ = I <small>(neutral permutation)</small>
{{Collapsible START|permutations|wide collapsed}}
{{Mentors of Boolean functions/illustrated WP 1-ary}}
{{Collapsible END}}
===2-ary===
Ч = Ш = I <small>(neutral permutation)</small> {{spaces|5}} Ю = Я = Ж {{spaces|5}} <small>Щ = (0, 1, 2, 3, 4, 5, 6, 7, 9, 8, 11, 10, 13, 12, 15, 14)</small>
===3-ary===
{{Mentors of Boolean functions/illustrated WP 3-ary}}
{{Collapsible START|code|collapsed light gap-above}}
{{Mentors of Boolean functions/code/3-ary}}
{{Collapsible END}}
{{Mentors of Boolean functions/small WP example 199}}
===4-ary===
{{Mentors of Boolean functions/illustrated WP 4-ary}}
{{Collapsible START|code|collapsed light gap-above gap-below}}
{{Mentors of Boolean functions/code/4-ary}}
{{Collapsible END}}
==seminars==
The mentor is not simply a bijection between Boolean functions, but between the truth tables for a given arity.<br>
The permutation from Zhegalkin indices to those of their ''n''-ary mentors is '''Ш<sub>''n''</sub>'''. The beginning of Ш<sub>''n''+1</sub> is '''Щ<sub>''n''</sub>'''. <small style="opacity: .5; font-size: 60%;">([[w:Shcha|This letter]] has a little hook on the right.)</small><br>
These two permutations are very similar. They are equal in the first half, and differ by exchanged neighbors in the second.
Neighboring Zhegalkin indices <small>(i.e. 2·''n'' and 2·''n''+1)</small> denote complements.<br>
So although there is no mentor bijection between Boolean functions, there is one between pairs of complements.<br>
Complement and mentor partition the set of all Boolean functions into blocks of size 4 or 2. Such a block shall be called (big or small) ''seminar''.<br>
The Zhegalkin indices in a big seminar are <math>\{ a, a+1, b, b+1 \}</math> with even <math>a</math> and <math>b</math>, so it can be represented by the pair <math>(a, b)</math>.<br>
<small>An example of a seminar is {138, 139, 156, 157}, represented as (138, 156). See [[c:File:Set of 3-ary Boolean functions 12855504354077768210885020350402125463028803369886765232947200.svg|image]]. <small style="opacity: .5;">In the permutation it is represented by the pair (69, 78).</small></small>
The pairs <math>\left( \frac{a}{2}, \frac{b}{2} \right)</math> are the cycles of a self-inverse Walsh permutation of degree <math>2^{arity} - 1</math>. <small style="opacity: .5;">(For arity 3 the degree is 7, and the permuted integers are 0...127.)</small><br>
For arities 1 and 2 this permutation is neutral. For arity 3 is has 64 fixed points <small>(of 128 places, i.e. 1/2)</small>. For arity 5 it has 1024 fixed points <small>(of 32768 places, i.e. 1/32)</small>.<br>
{{Collapsible START|mentor permutation|collapsed wide}}
[[File:Mentors 4; seminar permutation.svg|thumb|15×15 matrix corresponding to the Walsh permutation for arity 4.]]
The first 64 entries of the sequence are the fixed points. The next 64 entries form these 32 cycles:
<source lang="python" style="font-size: 60%;">
[ 64, 75], [ 65, 74], [ 66, 73], [ 67, 72], [ 68, 79], [ 69, 78], [ 70, 77], [ 71, 76],
[ 80, 91], [ 81, 90], [ 82, 89], [ 83, 88], [ 84, 95], [ 85, 94], [ 86, 93], [ 87, 92],
[ 96, 107], [ 97, 106], [ 98, 105], [ 99, 104], [100, 111], [101, 110], [102, 109], [103, 108],
[112, 123], [113, 122], [114, 121], [115, 120], [116, 127], [117, 126], [118, 125], [119, 124]
</source>
The permutation for arity 4 corresponds to the 15×15 matrix shown on the right.<br>
<small style="opacity: .5; font-size: 60%;">It is described by this vector: (1, 2, 4, 8, 16, 32, 75, 128, 256, 512, 1155, 2048, 4233, 8330, 19252)</small>
The matrix is always that of Ш or Щ without the top row and left column.
{{Collapsible END}}
For arity 3 the Boolean functions in big seminars are the sharp ones <small>(i.e. those with odd weight)</small>. See {{Boolf prop 3-ary|seminar|images}}.
<small>For a given arity, each seminar is part of a {{Boolf prop 3-ary|chunky seminar}}. For arity 3 they all have size 16. {{Mentors of Boolean functions/example chunky seminar}}</small>
==chains==
The permutations Ю and Я have fewer fixed points and longer cycles than Ч and Ш.<br>
A cycle of Ю shall be called '''chain'''. <small style="opacity: .5;">(Cycles of Я are the same, but reversed.)</small><br>
The XOR of all entries of a chain is one of the fixed points, and shall be called '''anchor'''. The fixed points are [[Noble Boolean functions|nobles]].
'''tables of chains:''' [[Template:Mentors of Boolean functions/chains/3-ary|3-ary]] <small>(4 fixed points)</small>, [[Template:Mentors of Boolean functions/chains/4-ary|4-ary]] <small>([[Template:Mentors of Boolean functions/chains/4-ary/fixed|8 fixed points]])</small>
'''3-ary partitions:'''
{{Boolf prop 3-ary|chain}},
{{Boolf prop 3-ary|chain length}},
{{Boolf prop 3-ary|chain quadrants}},
{{Boolf prop 3-ary|reduced chain quadrants}},
{{Boolf prop 3-ary|chunky chain}},
{{Boolf prop 3-ary|anchor}}
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<templatestyles src="Template:Mentors of Boolean functions/cycles/style.css" />
{| class="wikitable" style="text-align: center;"
|+ 4 fixed points
| [[File:Venn 0000 0000.svg|40px]]<br>0<br><small style="opacity: .5;">0</small>
| [[File:Venn 0001 0100.svg|40px]]<br>40<br><small style="opacity: .5;">2</small>
| [[File:Venn 0001 0010.svg|40px]]<br>72<br><small style="opacity: .5;">4</small>
| [[File:Venn 0000 0110.svg|40px]]<br>96<br><small style="opacity: .5;">6</small>
|}
{| class="wikitable sortable mentor-cycles"
|+ 6 cycles of length 2
!class="f"| F !!class="q"| Q !!class="g"| G !!class="w"| W !!colspan="3"| cycle
|-
|class="f"| 40 ||class="q"| (0, 0) ||class="g"| (0, 0) ||class="w"| (2, 4) ||class="c"| (20, 60) ||class="entry q0 g0"| 20<sub>2</sub> ||class="entry q0 g0"| 60<sub>4</sub>
|-
|class="f"| 40 ||class="q"| (0, 0) ||class="g"| (0, 0) ||class="w"| (4, 4) ||class="c"| (92, 116) ||class="entry q0 g0"| 92<sub>4</sub> ||class="entry q0 g0"| 116<sub>4</sub>
|-
|class="f"| 72 ||class="q"| (0, 0) ||class="g"| (0, 0) ||class="w"| (2, 4) ||class="c"| (18, 90) ||class="entry q0 g0"| 18<sub>2</sub> ||class="entry q0 g0"| 90<sub>4</sub>
|-
|class="f"| 72 ||class="q"| (0, 0) ||class="g"| (0, 0) ||class="w"| (4, 4) ||class="c"| (58, 114) ||class="entry q0 g0"| 58<sub>4</sub> ||class="entry q0 g0"| 114<sub>4</sub>
|-
|class="f"| 96 ||class="q"| (0, 0) ||class="g"| (0, 0) ||class="w"| (2, 4) ||class="c"| (6, 102) ||class="entry q0 g0"| 6<sub>2</sub> ||class="entry q0 g0"| 102<sub>4</sub>
|-
|class="f"| 96 ||class="q"| (0, 0) ||class="g"| (0, 0) ||class="w"| (4, 4) ||class="c"| (46, 78) ||class="entry q0 g0"| 46<sub>4</sub> ||class="entry q0 g0"| 78<sub>4</sub>
|}
{| class="wikitable sortable mentor-cycles"
|+ 12 cycles of length 5
!class="f"| F !!class="q"| Q !!class="g"| G !!class="w"| W !!colspan="6"| cycle
|-
|class="f"| 0 ||class="q"| (0, 1, 1, 2, 2) ||class="g"| (1, 0, 0, 0, 1) ||class="w"| (4, 4, 4, 3, 7) ||class="c"| (232, 105, 23, 104, 254) ||class="entry q0 g1"| 232<sub>4</sub> ||class="entry q1 g0"| 105<sub>4</sub> ||class="entry q1 g0"| 23<sub>4</sub> ||class="entry q2 g0"| 104<sub>3</sub> ||class="entry q2 g1"| 254<sub>7</sub>
|-
|class="f"| 0 ||class="q"| (0, 2, 1, 0, 3) ||class="g"| (1, 0, 1, 0, 0) ||class="w"| (4, 3, 2, 6, 7) ||class="c"| (150, 22, 129, 126, 127) ||class="entry q0 g1"| 150<sub>4</sub> ||class="entry q2 g0"| 22<sub>3</sub> ||class="entry q1 g1"| 129<sub>2</sub> ||class="entry q0 g0"| 126<sub>6</sub> ||class="entry q3 g0"| 127<sub>7</sub>
|-
|class="f"| 0 ||class="q"| (1, 3, 3, 2, 3) ||class="g"| (1, 1, 1, 1, 1) ||class="w"| (8, 1, 5, 1, 5) ||class="c"| (255, 1, 233, 128, 151) ||class="entry q1 g1"| 255<sub>8</sub> ||class="entry q3 g1"| 1<sub>1</sub> ||class="entry q3 g1"| 233<sub>5</sub> ||class="entry q2 g1"| 128<sub>1</sub> ||class="entry q3 g1"| 151<sub>5</sub>
|-
|class="f"| 40 ||class="q"| (0, 1, 1, 2, 2) ||class="g"| (1, 1, 0, 1, 1) ||class="w"| (2, 2, 6, 1, 5) ||class="c"| (192, 65, 63, 64, 214) ||class="entry q0 g1"| 192<sub>2</sub> ||class="entry q1 g1"| 65<sub>2</sub> ||class="entry q1 g0"| 63<sub>6</sub> ||class="entry q2 g1"| 64<sub>1</sub> ||class="entry q2 g1"| 214<sub>5</sub>
|-
|class="f"| 40 ||class="q"| (0, 2, 1, 0, 3) ||class="g"| (1, 0, 1, 0, 0) ||class="w"| (6, 5, 4, 4, 5) ||class="c"| (190, 62, 169, 86, 87) ||class="entry q0 g1"| 190<sub>6</sub> ||class="entry q2 g0"| 62<sub>5</sub> ||class="entry q1 g1"| 169<sub>4</sub> ||class="entry q0 g0"| 86<sub>4</sub> ||class="entry q3 g0"| 87<sub>5</sub>
|-
|class="f"| 40 ||class="q"| (1, 3, 3, 2, 3) ||class="g"| (1, 0, 1, 1, 1) ||class="w"| (6, 3, 3, 3, 7) ||class="c"| (215, 41, 193, 168, 191) ||class="entry q1 g1"| 215<sub>6</sub> ||class="entry q3 g0"| 41<sub>3</sub> ||class="entry q3 g1"| 193<sub>3</sub> ||class="entry q2 g1"| 168<sub>3</sub> ||class="entry q3 g1"| 191<sub>7</sub>
|-
|class="f"| 72 ||class="q"| (0, 1, 1, 2, 2) ||class="g"| (1, 1, 0, 1, 1) ||class="w"| (2, 2, 6, 1, 5) ||class="c"| (160, 33, 95, 32, 182) ||class="entry q0 g1"| 160<sub>2</sub> ||class="entry q1 g1"| 33<sub>2</sub> ||class="entry q1 g0"| 95<sub>6</sub> ||class="entry q2 g1"| 32<sub>1</sub> ||class="entry q2 g1"| 182<sub>5</sub>
|-
|class="f"| 72 ||class="q"| (0, 2, 1, 0, 3) ||class="g"| (1, 0, 1, 0, 0) ||class="w"| (6, 5, 4, 4, 5) ||class="c"| (222, 94, 201, 54, 55) ||class="entry q0 g1"| 222<sub>6</sub> ||class="entry q2 g0"| 94<sub>5</sub> ||class="entry q1 g1"| 201<sub>4</sub> ||class="entry q0 g0"| 54<sub>4</sub> ||class="entry q3 g0"| 55<sub>5</sub>
|-
|class="f"| 72 ||class="q"| (1, 3, 3, 2, 3) ||class="g"| (1, 0, 1, 1, 1) ||class="w"| (6, 3, 3, 3, 7) ||class="c"| (183, 73, 161, 200, 223) ||class="entry q1 g1"| 183<sub>6</sub> ||class="entry q3 g0"| 73<sub>3</sub> ||class="entry q3 g1"| 161<sub>3</sub> ||class="entry q2 g1"| 200<sub>3</sub> ||class="entry q3 g1"| 223<sub>7</sub>
|-
|class="f"| 96 ||class="q"| (0, 1, 1, 2, 2) ||class="g"| (1, 1, 0, 1, 1) ||class="w"| (2, 2, 6, 1, 5) ||class="c"| (136, 9, 119, 8, 158) ||class="entry q0 g1"| 136<sub>2</sub> ||class="entry q1 g1"| 9<sub>2</sub> ||class="entry q1 g0"| 119<sub>6</sub> ||class="entry q2 g1"| 8<sub>1</sub> ||class="entry q2 g1"| 158<sub>5</sub>
|-
|class="f"| 96 ||class="q"| (0, 2, 1, 0, 3) ||class="g"| (1, 0, 1, 0, 0) ||class="w"| (6, 5, 4, 4, 5) ||class="c"| (246, 118, 225, 30, 31) ||class="entry q0 g1"| 246<sub>6</sub> ||class="entry q2 g0"| 118<sub>5</sub> ||class="entry q1 g1"| 225<sub>4</sub> ||class="entry q0 g0"| 30<sub>4</sub> ||class="entry q3 g0"| 31<sub>5</sub>
|-
|class="f"| 96 ||class="q"| (1, 3, 3, 2, 3) ||class="g"| (1, 0, 1, 1, 1) ||class="w"| (6, 3, 3, 3, 7) ||class="c"| (159, 97, 137, 224, 247) ||class="entry q1 g1"| 159<sub>6</sub> ||class="entry q3 g0"| 97<sub>3</sub> ||class="entry q3 g1"| 137<sub>3</sub> ||class="entry q2 g1"| 224<sub>3</sub> ||class="entry q3 g1"| 247<sub>7</sub>
|}
{| class="wikitable sortable mentor-cycles"
|+ 18 cycles of length 10
!class="f"| F !!class="q"| Q !!class="g"| G !!class="w"| W !!colspan="11"| cycle
|-
|class="f"| 40 ||class="q"| (0, 1, 1, 2, 2, 0, 1, 1, 2, 2) ||class="g"| (1, 0, 0, 0, 1, 1, 0, 0, 0, 1) ||class="w"| (4, 4, 4, 3, 3, 4, 4, 4, 3, 3) ||class="c"| (156, 53, 99, 52, 138, 180, 29, 75, 28, 162) ||class="entry q0 g1"| 156<sub>4</sub> ||class="entry q1 g0"| 53<sub>4</sub> ||class="entry q1 g0"| 99<sub>4</sub> ||class="entry q2 g0"| 52<sub>3</sub> ||class="entry q2 g1"| 138<sub>3</sub> ||class="entry q0 g1"| 180<sub>4</sub> ||class="entry q1 g0"| 29<sub>4</sub> ||class="entry q1 g0"| 75<sub>4</sub> ||class="entry q2 g0"| 28<sub>3</sub> ||class="entry q2 g1"| 162<sub>3</sub>
|-
|class="f"| 40 ||class="q"| (0, 1, 1, 2, 2, 0, 1, 1, 2, 2) ||class="g"| (1, 0, 0, 0, 1, 1, 1, 1, 1, 1) ||class="w"| (4, 6, 4, 5, 3, 6, 4, 2, 3, 5) ||class="c"| (212, 125, 43, 124, 194, 252, 85, 3, 84, 234) ||class="entry q0 g1"| 212<sub>4</sub> ||class="entry q1 g0"| 125<sub>6</sub> ||class="entry q1 g0"| 43<sub>4</sub> ||class="entry q2 g0"| 124<sub>5</sub> ||class="entry q2 g1"| 194<sub>3</sub> ||class="entry q0 g1"| 252<sub>6</sub> ||class="entry q1 g1"| 85<sub>4</sub> ||class="entry q1 g1"| 3<sub>2</sub> ||class="entry q2 g1"| 84<sub>3</sub> ||class="entry q2 g1"| 234<sub>5</sub>
|-
|class="f"| 40 ||class="q"| (0, 2, 1, 0, 3, 0, 2, 1, 0, 3) ||class="g"| (1, 0, 1, 0, 0, 1, 1, 1, 0, 0) ||class="w"| (2, 3, 4, 2, 5, 4, 1, 6, 4, 3) ||class="c"| (130, 42, 149, 66, 107, 170, 2, 189, 106, 67) ||class="entry q0 g1"| 130<sub>2</sub> ||class="entry q2 g0"| 42<sub>3</sub> ||class="entry q1 g1"| 149<sub>4</sub> ||class="entry q0 g0"| 66<sub>2</sub> ||class="entry q3 g0"| 107<sub>5</sub> ||class="entry q0 g1"| 170<sub>4</sub> ||class="entry q2 g1"| 2<sub>1</sub> ||class="entry q1 g1"| 189<sub>6</sub> ||class="entry q0 g0"| 106<sub>4</sub> ||class="entry q3 g0"| 67<sub>3</sub>
|-
|class="f"| 40 ||class="q"| (0, 2, 1, 0, 3, 0, 2, 1, 0, 3) ||class="g"| (1, 0, 1, 1, 1, 1, 0, 1, 1, 1) ||class="w"| (4, 3, 6, 2, 3, 4, 3, 6, 2, 3) ||class="c"| (202, 98, 221, 10, 35, 226, 74, 245, 34, 11) ||class="entry q0 g1"| 202<sub>4</sub> ||class="entry q2 g0"| 98<sub>3</sub> ||class="entry q1 g1"| 221<sub>6</sub> ||class="entry q0 g1"| 10<sub>2</sub> ||class="entry q3 g1"| 35<sub>3</sub> ||class="entry q0 g1"| 226<sub>4</sub> ||class="entry q2 g0"| 74<sub>3</sub> ||class="entry q1 g1"| 245<sub>6</sub> ||class="entry q0 g1"| 34<sub>2</sub> ||class="entry q3 g1"| 11<sub>3</sub>
|-
|class="f"| 40 ||class="q"| (1, 3, 3, 2, 3, 1, 3, 3, 2, 3) ||class="g"| (1, 0, 1, 1, 1, 1, 0, 1, 1, 1) ||class="w"| (4, 3, 5, 3, 5, 6, 5, 7, 5, 3) ||class="c"| (195, 21, 213, 148, 171, 235, 61, 253, 188, 131) ||class="entry q1 g1"| 195<sub>4</sub> ||class="entry q3 g0"| 21<sub>3</sub> ||class="entry q3 g1"| 213<sub>5</sub> ||class="entry q2 g1"| 148<sub>3</sub> ||class="entry q3 g1"| 171<sub>5</sub> ||class="entry q1 g1"| 235<sub>6</sub> ||class="entry q3 g0"| 61<sub>5</sub> ||class="entry q3 g1"| 253<sub>7</sub> ||class="entry q2 g1"| 188<sub>5</sub> ||class="entry q3 g1"| 131<sub>3</sub>
|-
|class="f"| 40 ||class="q"| (1, 3, 3, 2, 3, 1, 3, 3, 2, 3) ||class="g"| (1, 0, 1, 1, 1, 1, 0, 1, 1, 1) ||class="w"| (4, 5, 5, 5, 5, 4, 5, 5, 5, 5) ||class="c"| (139, 93, 157, 220, 227, 163, 117, 181, 244, 203) ||class="entry q1 g1"| 139<sub>4</sub> ||class="entry q3 g0"| 93<sub>5</sub> ||class="entry q3 g1"| 157<sub>5</sub> ||class="entry q2 g1"| 220<sub>5</sub> ||class="entry q3 g1"| 227<sub>5</sub> ||class="entry q1 g1"| 163<sub>4</sub> ||class="entry q3 g0"| 117<sub>5</sub> ||class="entry q3 g1"| 181<sub>5</sub> ||class="entry q2 g1"| 244<sub>5</sub> ||class="entry q3 g1"| 203<sub>5</sub>
|-
|class="f"| 72 ||class="q"| (0, 1, 1, 2, 2, 0, 1, 1, 2, 2) ||class="g"| (1, 0, 0, 0, 1, 1, 0, 0, 0, 1) ||class="w"| (4, 4, 4, 3, 3, 4, 4, 4, 3, 3) ||class="c"| (154, 83, 101, 82, 140, 210, 27, 45, 26, 196) ||class="entry q0 g1"| 154<sub>4</sub> ||class="entry q1 g0"| 83<sub>4</sub> ||class="entry q1 g0"| 101<sub>4</sub> ||class="entry q2 g0"| 82<sub>3</sub> ||class="entry q2 g1"| 140<sub>3</sub> ||class="entry q0 g1"| 210<sub>4</sub> ||class="entry q1 g0"| 27<sub>4</sub> ||class="entry q1 g0"| 45<sub>4</sub> ||class="entry q2 g0"| 26<sub>3</sub> ||class="entry q2 g1"| 196<sub>3</sub>
|-
|class="f"| 72 ||class="q"| (0, 1, 1, 2, 2, 0, 1, 1, 2, 2) ||class="g"| (1, 0, 0, 0, 1, 1, 1, 1, 1, 1) ||class="w"| (4, 6, 4, 5, 3, 6, 4, 2, 3, 5) ||class="c"| (178, 123, 77, 122, 164, 250, 51, 5, 50, 236) ||class="entry q0 g1"| 178<sub>4</sub> ||class="entry q1 g0"| 123<sub>6</sub> ||class="entry q1 g0"| 77<sub>4</sub> ||class="entry q2 g0"| 122<sub>5</sub> ||class="entry q2 g1"| 164<sub>3</sub> ||class="entry q0 g1"| 250<sub>6</sub> ||class="entry q1 g1"| 51<sub>4</sub> ||class="entry q1 g1"| 5<sub>2</sub> ||class="entry q2 g1"| 50<sub>3</sub> ||class="entry q2 g1"| 236<sub>5</sub>
|-
|class="f"| 72 ||class="q"| (0, 2, 1, 0, 3, 0, 2, 1, 0, 3) ||class="g"| (1, 0, 1, 0, 0, 1, 1, 1, 0, 0) ||class="w"| (2, 3, 4, 2, 5, 4, 1, 6, 4, 3) ||class="c"| (132, 76, 147, 36, 109, 204, 4, 219, 108, 37) ||class="entry q0 g1"| 132<sub>2</sub> ||class="entry q2 g0"| 76<sub>3</sub> ||class="entry q1 g1"| 147<sub>4</sub> ||class="entry q0 g0"| 36<sub>2</sub> ||class="entry q3 g0"| 109<sub>5</sub> ||class="entry q0 g1"| 204<sub>4</sub> ||class="entry q2 g1"| 4<sub>1</sub> ||class="entry q1 g1"| 219<sub>6</sub> ||class="entry q0 g0"| 108<sub>4</sub> ||class="entry q3 g0"| 37<sub>3</sub>
|-
|class="f"| 72 ||class="q"| (0, 2, 1, 0, 3, 0, 2, 1, 0, 3) ||class="g"| (1, 0, 1, 1, 1, 1, 0, 1, 1, 1) ||class="w"| (4, 3, 6, 2, 3, 4, 3, 6, 2, 3) ||class="c"| (172, 100, 187, 12, 69, 228, 44, 243, 68, 13) ||class="entry q0 g1"| 172<sub>4</sub> ||class="entry q2 g0"| 100<sub>3</sub> ||class="entry q1 g1"| 187<sub>6</sub> ||class="entry q0 g1"| 12<sub>2</sub> ||class="entry q3 g1"| 69<sub>3</sub> ||class="entry q0 g1"| 228<sub>4</sub> ||class="entry q2 g0"| 44<sub>3</sub> ||class="entry q1 g1"| 243<sub>6</sub> ||class="entry q0 g1"| 68<sub>2</sub> ||class="entry q3 g1"| 13<sub>3</sub>
|-
|class="f"| 72 ||class="q"| (1, 3, 3, 2, 3, 1, 3, 3, 2, 3) ||class="g"| (1, 0, 1, 1, 1, 1, 0, 1, 1, 1) ||class="w"| (4, 3, 5, 3, 5, 6, 5, 7, 5, 3) ||class="c"| (165, 19, 179, 146, 205, 237, 91, 251, 218, 133) ||class="entry q1 g1"| 165<sub>4</sub> ||class="entry q3 g0"| 19<sub>3</sub> ||class="entry q3 g1"| 179<sub>5</sub> ||class="entry q2 g1"| 146<sub>3</sub> ||class="entry q3 g1"| 205<sub>5</sub> ||class="entry q1 g1"| 237<sub>6</sub> ||class="entry q3 g0"| 91<sub>5</sub> ||class="entry q3 g1"| 251<sub>7</sub> ||class="entry q2 g1"| 218<sub>5</sub> ||class="entry q3 g1"| 133<sub>3</sub>
|-
|class="f"| 72 ||class="q"| (1, 3, 3, 2, 3, 1, 3, 3, 2, 3) ||class="g"| (1, 0, 1, 1, 1, 1, 0, 1, 1, 1) ||class="w"| (4, 5, 5, 5, 5, 4, 5, 5, 5, 5) ||class="c"| (141, 59, 155, 186, 229, 197, 115, 211, 242, 173) ||class="entry q1 g1"| 141<sub>4</sub> ||class="entry q3 g0"| 59<sub>5</sub> ||class="entry q3 g1"| 155<sub>5</sub> ||class="entry q2 g1"| 186<sub>5</sub> ||class="entry q3 g1"| 229<sub>5</sub> ||class="entry q1 g1"| 197<sub>4</sub> ||class="entry q3 g0"| 115<sub>5</sub> ||class="entry q3 g1"| 211<sub>5</sub> ||class="entry q2 g1"| 242<sub>5</sub> ||class="entry q3 g1"| 173<sub>5</sub>
|-
|class="f"| 96 ||class="q"| (0, 1, 1, 2, 2, 0, 1, 1, 2, 2) ||class="g"| (1, 0, 0, 0, 1, 1, 0, 0, 0, 1) ||class="w"| (4, 4, 4, 3, 3, 4, 4, 4, 3, 3) ||class="c"| (166, 71, 89, 70, 176, 198, 39, 57, 38, 208) ||class="entry q0 g1"| 166<sub>4</sub> ||class="entry q1 g0"| 71<sub>4</sub> ||class="entry q1 g0"| 89<sub>4</sub> ||class="entry q2 g0"| 70<sub>3</sub> ||class="entry q2 g1"| 176<sub>3</sub> ||class="entry q0 g1"| 198<sub>4</sub> ||class="entry q1 g0"| 39<sub>4</sub> ||class="entry q1 g0"| 57<sub>4</sub> ||class="entry q2 g0"| 38<sub>3</sub> ||class="entry q2 g1"| 208<sub>3</sub>
|-
|class="f"| 96 ||class="q"| (0, 1, 1, 2, 2, 0, 1, 1, 2, 2) ||class="g"| (1, 0, 0, 0, 1, 1, 1, 1, 1, 1) ||class="w"| (4, 6, 4, 5, 3, 6, 4, 2, 3, 5) ||class="c"| (142, 111, 113, 110, 152, 238, 15, 17, 14, 248) ||class="entry q0 g1"| 142<sub>4</sub> ||class="entry q1 g0"| 111<sub>6</sub> ||class="entry q1 g0"| 113<sub>4</sub> ||class="entry q2 g0"| 110<sub>5</sub> ||class="entry q2 g1"| 152<sub>3</sub> ||class="entry q0 g1"| 238<sub>6</sub> ||class="entry q1 g1"| 15<sub>4</sub> ||class="entry q1 g1"| 17<sub>2</sub> ||class="entry q2 g1"| 14<sub>3</sub> ||class="entry q2 g1"| 248<sub>5</sub>
|-
|class="f"| 96 ||class="q"| (0, 2, 1, 0, 3, 0, 2, 1, 0, 3) ||class="g"| (1, 0, 1, 0, 0, 1, 1, 1, 0, 0) ||class="w"| (2, 3, 4, 2, 5, 4, 1, 6, 4, 3) ||class="c"| (144, 112, 135, 24, 121, 240, 16, 231, 120, 25) ||class="entry q0 g1"| 144<sub>2</sub> ||class="entry q2 g0"| 112<sub>3</sub> ||class="entry q1 g1"| 135<sub>4</sub> ||class="entry q0 g0"| 24<sub>2</sub> ||class="entry q3 g0"| 121<sub>5</sub> ||class="entry q0 g1"| 240<sub>4</sub> ||class="entry q2 g1"| 16<sub>1</sub> ||class="entry q1 g1"| 231<sub>6</sub> ||class="entry q0 g0"| 120<sub>4</sub> ||class="entry q3 g0"| 25<sub>3</sub>
|-
|class="f"| 96 ||class="q"| (0, 2, 1, 0, 3, 0, 2, 1, 0, 3) ||class="g"| (1, 0, 1, 1, 1, 1, 0, 1, 1, 1) ||class="w"| (4, 3, 6, 2, 3, 4, 3, 6, 2, 3) ||class="c"| (184, 88, 175, 48, 81, 216, 56, 207, 80, 49) ||class="entry q0 g1"| 184<sub>4</sub> ||class="entry q2 g0"| 88<sub>3</sub> ||class="entry q1 g1"| 175<sub>6</sub> ||class="entry q0 g1"| 48<sub>2</sub> ||class="entry q3 g1"| 81<sub>3</sub> ||class="entry q0 g1"| 216<sub>4</sub> ||class="entry q2 g0"| 56<sub>3</sub> ||class="entry q1 g1"| 207<sub>6</sub> ||class="entry q0 g1"| 80<sub>2</sub> ||class="entry q3 g1"| 49<sub>3</sub>
|-
|class="f"| 96 ||class="q"| (1, 3, 3, 2, 3, 1, 3, 3, 2, 3) ||class="g"| (1, 0, 1, 1, 1, 1, 0, 1, 1, 1) ||class="w"| (4, 3, 5, 3, 5, 6, 5, 7, 5, 3) ||class="c"| (153, 7, 143, 134, 241, 249, 103, 239, 230, 145) ||class="entry q1 g1"| 153<sub>4</sub> ||class="entry q3 g0"| 7<sub>3</sub> ||class="entry q3 g1"| 143<sub>5</sub> ||class="entry q2 g1"| 134<sub>3</sub> ||class="entry q3 g1"| 241<sub>5</sub> ||class="entry q1 g1"| 249<sub>6</sub> ||class="entry q3 g0"| 103<sub>5</sub> ||class="entry q3 g1"| 239<sub>7</sub> ||class="entry q2 g1"| 230<sub>5</sub> ||class="entry q3 g1"| 145<sub>3</sub>
|-
|class="f"| 96 ||class="q"| (1, 3, 3, 2, 3, 1, 3, 3, 2, 3) ||class="g"| (1, 0, 1, 1, 1, 1, 0, 1, 1, 1) ||class="w"| (4, 5, 5, 5, 5, 4, 5, 5, 5, 5) ||class="c"| (177, 47, 167, 174, 217, 209, 79, 199, 206, 185) ||class="entry q1 g1"| 177<sub>4</sub> ||class="entry q3 g0"| 47<sub>5</sub> ||class="entry q3 g1"| 167<sub>5</sub> ||class="entry q2 g1"| 174<sub>5</sub> ||class="entry q3 g1"| 217<sub>5</sub> ||class="entry q1 g1"| 209<sub>4</sub> ||class="entry q3 g0"| 79<sub>5</sub> ||class="entry q3 g1"| 199<sub>5</sub> ||class="entry q2 g1"| 206<sub>5</sub> ||class="entry q3 g1"| 185<sub>5</sub>
|}<noinclude>
[[Category:Mentors of Boolean functions; chains]]
</noinclude>
tod334059t57d0hwkybdsovecjl71sx
Template:Mentors of Boolean functions/chains/4-ary
10
313707
2693996
2693776
2025-01-01T19:23:31Z
Watchduck
137431
Watchduck moved page [[Template:Mentors of Boolean functions/cycles/4-ary]] to [[Template:Mentors of Boolean functions/chains/4-ary]]
2693776
wikitext
text/x-wiki
* {{tl|Mentors of Boolean functions/cycles/4-ary/fixed}}
* {{tl|Mentors of Boolean functions/cycles/4-ary/length 2}} (28)
* {{tl|Mentors of Boolean functions/cycles/4-ary/length 3}} (168)
* {{tl|Mentors of Boolean functions/cycles/4-ary/length 4}} (48)
* {{tl|Mentors of Boolean functions/cycles/4-ary/length 6}} (2636)
* There are 4080 of length 12, but that exceeds the size limit.
<noinclude>
[[Category:Mentors of Boolean functions; chains]]
</noinclude>
4c7dwq2et2f17j2w7ri4kuj7kk13zsp
2694049
2693996
2025-01-01T23:46:09Z
Watchduck
137431
2694049
wikitext
text/x-wiki
* {{tl|Mentors of Boolean functions/chains/4-ary/fixed}}
* {{tl|Mentors of Boolean functions/chains/4-ary/length 2}} (28)
* {{tl|Mentors of Boolean functions/chains/4-ary/length 3}} (168)
* {{tl|Mentors of Boolean functions/chains/4-ary/length 4}} (48)
* {{tl|Mentors of Boolean functions/chains/4-ary/length 6}} (2636)
* There are 4080 of length 12, but that exceeds the size limit.
<noinclude>
[[Category:Mentors of Boolean functions; chains]]
</noinclude>
4ngk26nhhu5ifx9rqj9vs6hhfyvyhxe
Template:Mentors of Boolean functions/chains/4-ary/length 2
10
313709
2694000
2693778
2025-01-01T19:24:00Z
Watchduck
137431
Watchduck moved page [[Template:Mentors of Boolean functions/cycles/4-ary/length 2]] to [[Template:Mentors of Boolean functions/chains/4-ary/length 2]]
2693778
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text/x-wiki
<templatestyles src="Template:Mentors of Boolean functions/cycles/style.css" />
{| class="wikitable sortable mentor-cycles"
|+ 28 cycles of length 2
!class="f"| F !!class="q"| Q !!class="g"| G !!class="w"| W !!colspan="3"| cycle
|-
|class="f"| 854 ||class="q"| (2, 2) ||class="g"| (1, 1) ||class="w"| (9, 11) ||class="c"| (59580, 60394) ||class="entry q2 g1"| 59580<sub>9</sub> ||class="entry q2 g1"| 60394<sub>11</sub>
|-
|class="f"| 854 ||class="q"| (2, 2) ||class="g"| (1, 1) ||class="w"| (9, 11) ||class="c"| (60810, 61148) ||class="entry q2 g1"| 60810<sub>9</sub> ||class="entry q2 g1"| 61148<sub>11</sub>
|-
|class="f"| 854 ||class="q"| (2, 2) ||class="g"| (1, 1) ||class="w"| (9, 11) ||class="c"| (63906, 64244) ||class="entry q2 g1"| 63906<sub>9</sub> ||class="entry q2 g1"| 64244<sub>11</sub>
|-
|class="f"| 854 ||class="q"| (2, 2) ||class="g"| (1, 1) ||class="w"| (9, 11) ||class="c"| (64660, 65474) ||class="entry q2 g1"| 64660<sub>9</sub> ||class="entry q2 g1"| 65474<sub>11</sub>
|-
|class="f"| 1334 ||class="q"| (2, 2) ||class="g"| (1, 1) ||class="w"| (9, 11) ||class="c"| (59610, 60908) ||class="entry q2 g1"| 59610<sub>9</sub> ||class="entry q2 g1"| 60908<sub>11</sub>
|-
|class="f"| 1334 ||class="q"| (2, 2) ||class="g"| (1, 1) ||class="w"| (9, 11) ||class="c"| (60300, 61114) ||class="entry q2 g1"| 60300<sub>9</sub> ||class="entry q2 g1"| 61114<sub>11</sub>
|-
|class="f"| 1334 ||class="q"| (2, 2) ||class="g"| (1, 1) ||class="w"| (9, 11) ||class="c"| (63940, 64754) ||class="entry q2 g1"| 63940<sub>9</sub> ||class="entry q2 g1"| 64754<sub>11</sub>
|-
|class="f"| 1334 ||class="q"| (2, 2) ||class="g"| (1, 1) ||class="w"| (9, 11) ||class="c"| (64146, 65444) ||class="entry q2 g1"| 64146<sub>9</sub> ||class="entry q2 g1"| 65444<sub>11</sub>
|-
|class="f"| 1632 ||class="q"| (0, 0) ||class="g"| (0, 0) ||class="w"| (4, 4) ||class="c"| (102, 1542) ||class="entry q0 g0"| 102<sub>4</sub> ||class="entry q0 g0"| 1542<sub>4</sub>
|-
|class="f"| 1632 ||class="q"| (0, 0) ||class="g"| (0, 0) ||class="w"| (4, 4) ||class="c"| (816, 1360) ||class="entry q0 g0"| 816<sub>4</sub> ||class="entry q0 g0"| 1360<sub>4</sub>
|-
|class="f"| 1632 ||class="q"| (0, 0) ||class="g"| (0, 0) ||class="w"| (6, 6) ||class="c"| (4472, 5912) ||class="entry q0 g0"| 4472<sub>6</sub> ||class="entry q0 g0"| 5912<sub>6</sub>
|-
|class="f"| 1632 ||class="q"| (0, 0) ||class="g"| (0, 0) ||class="w"| (6, 6) ||class="c"| (4654, 5198) ||class="entry q0 g0"| 4654<sub>6</sub> ||class="entry q0 g0"| 5198<sub>6</sub>
|-
|class="f"| 4382 ||class="q"| (2, 2) ||class="g"| (1, 1) ||class="w"| (9, 11) ||class="c"| (59622, 63992) ||class="entry q2 g1"| 59622<sub>9</sub> ||class="entry q2 g1"| 63992<sub>11</sub>
|-
|class="f"| 4382 ||class="q"| (2, 2) ||class="g"| (1, 1) ||class="w"| (9, 11) ||class="c"| (60336, 64174) ||class="entry q2 g1"| 60336<sub>9</sub> ||class="entry q2 g1"| 64174<sub>11</sub>
|-
|class="f"| 4382 ||class="q"| (2, 2) ||class="g"| (1, 1) ||class="w"| (9, 11) ||class="c"| (60880, 64718) ||class="entry q2 g1"| 60880<sub>9</sub> ||class="entry q2 g1"| 64718<sub>11</sub>
|-
|class="f"| 4382 ||class="q"| (2, 2) ||class="g"| (1, 1) ||class="w"| (9, 11) ||class="c"| (61062, 65432) ||class="entry q2 g1"| 61062<sub>9</sub> ||class="entry q2 g1"| 65432<sub>11</sub>
|-
|class="f"| 4680 ||class="q"| (0, 0) ||class="g"| (0, 0) ||class="w"| (4, 4) ||class="c"| (90, 4626) ||class="entry q0 g0"| 90<sub>4</sub> ||class="entry q0 g0"| 4626<sub>4</sub>
|-
|class="f"| 4680 ||class="q"| (0, 0) ||class="g"| (0, 0) ||class="w"| (4, 4) ||class="c"| (780, 4420) ||class="entry q0 g0"| 780<sub>4</sub> ||class="entry q0 g0"| 4420<sub>4</sub>
|-
|class="f"| 4680 ||class="q"| (0, 0) ||class="g"| (0, 0) ||class="w"| (6, 6) ||class="c"| (1388, 5924) ||class="entry q0 g0"| 1388<sub>6</sub> ||class="entry q0 g0"| 5924<sub>6</sub>
|-
|class="f"| 4680 ||class="q"| (0, 0) ||class="g"| (0, 0) ||class="w"| (6, 6) ||class="c"| (1594, 5234) ||class="entry q0 g0"| 1594<sub>6</sub> ||class="entry q0 g0"| 5234<sub>6</sub>
|-
|class="f"| 5160 ||class="q"| (0, 0) ||class="g"| (0, 0) ||class="w"| (4, 4) ||class="c"| (60, 5140) ||class="entry q0 g0"| 60<sub>4</sub> ||class="entry q0 g0"| 5140<sub>4</sub>
|-
|class="f"| 5160 ||class="q"| (0, 0) ||class="g"| (0, 0) ||class="w"| (4, 4) ||class="c"| (1290, 4386) ||class="entry q0 g0"| 1290<sub>4</sub> ||class="entry q0 g0"| 4386<sub>4</sub>
|-
|class="f"| 5160 ||class="q"| (0, 0) ||class="g"| (0, 0) ||class="w"| (6, 6) ||class="c"| (874, 5954) ||class="entry q0 g0"| 874<sub>6</sub> ||class="entry q0 g0"| 5954<sub>6</sub>
|-
|class="f"| 5160 ||class="q"| (0, 0) ||class="g"| (0, 0) ||class="w"| (6, 6) ||class="c"| (1628, 4724) ||class="entry q0 g0"| 1628<sub>6</sub> ||class="entry q0 g0"| 4724<sub>6</sub>
|-
|class="f"| 6014 ||class="q"| (2, 2) ||class="g"| (1, 1) ||class="w"| (5, 15) ||class="c"| (59520, 65534) ||class="entry q2 g1"| 59520<sub>5</sub> ||class="entry q2 g1"| 65534<sub>15</sub>
|-
|class="f"| 6014 ||class="q"| (2, 2) ||class="g"| (1, 1) ||class="w"| (9, 11) ||class="c"| (61152, 63902) ||class="entry q2 g1"| 61152<sub>9</sub> ||class="entry q2 g1"| 63902<sub>11</sub>
|-
|class="f"| 6014 ||class="q"| (2, 2) ||class="g"| (1, 1) ||class="w"| (9, 11) ||class="c"| (64200, 60854) ||class="entry q2 g1"| 64200<sub>9</sub> ||class="entry q2 g1"| 60854<sub>11</sub>
|-
|class="f"| 6014 ||class="q"| (2, 2) ||class="g"| (1, 1) ||class="w"| (9, 11) ||class="c"| (64680, 60374) ||class="entry q2 g1"| 64680<sub>9</sub> ||class="entry q2 g1"| 60374<sub>11</sub>
|}<noinclude>
[[Category:Mentors of Boolean functions; chains]]
</noinclude>
t1i6vkl312xr7wlg8013ktwf9zfoma1
Template:Mentors of Boolean functions/chains/4-ary/length 3
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Watchduck
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Watchduck moved page [[Template:Mentors of Boolean functions/cycles/4-ary/length 3]] to [[Template:Mentors of Boolean functions/chains/4-ary/length 3]]
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wikitext
text/x-wiki
<templatestyles src="Template:Mentors of Boolean functions/cycles/style.css" />
{| class="wikitable sortable mentor-cycles"
|+ 168 cycles of length 3
!class="f"| F !!class="q"| Q !!class="g"| G !!class="w"| W !!colspan="4"| cycle
|-
|class="f"| 0 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (2, 6, 8) ||class="c"| (6, 26208, 26214) ||class="entry q0 g0"| 6<sub>2</sub> ||class="entry q0 g0"| 26208<sub>6</sub> ||class="entry q0 g0"| 26214<sub>8</sub>
|-
|class="f"| 0 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (2, 6, 8) ||class="c"| (18, 23112, 23130) ||class="entry q0 g0"| 18<sub>2</sub> ||class="entry q0 g0"| 23112<sub>6</sub> ||class="entry q0 g0"| 23130<sub>8</sub>
|-
|class="f"| 0 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (2, 6, 8) ||class="c"| (20, 15400, 15420) ||class="entry q0 g0"| 20<sub>2</sub> ||class="entry q0 g0"| 15400<sub>6</sub> ||class="entry q0 g0"| 15420<sub>8</sub>
|-
|class="f"| 0 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (2, 6, 8) ||class="c"| (258, 21672, 21930) ||class="entry q0 g0"| 258<sub>2</sub> ||class="entry q0 g0"| 21672<sub>6</sub> ||class="entry q0 g0"| 21930<sub>8</sub>
|-
|class="f"| 0 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (2, 6, 8) ||class="c"| (260, 13000, 13260) ||class="entry q0 g0"| 260<sub>2</sub> ||class="entry q0 g0"| 13000<sub>6</sub> ||class="entry q0 g0"| 13260<sub>8</sub>
|-
|class="f"| 0 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (2, 6, 8) ||class="c"| (272, 3808, 4080) ||class="entry q0 g0"| 272<sub>2</sub> ||class="entry q0 g0"| 3808<sub>6</sub> ||class="entry q0 g0"| 4080<sub>8</sub>
|-
|class="f"| 0 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (4, 4, 8) ||class="c"| (278, 26752, 27030) ||class="entry q0 g0"| 278<sub>4</sub> ||class="entry q0 g0"| 26752<sub>4</sub> ||class="entry q0 g0"| 27030<sub>8</sub>
|-
|class="f"| 0 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (6, 10, 8) ||class="c"| (26758, 4086, 26480) ||class="entry q0 g0"| 26758<sub>6</sub> ||class="entry q0 g0"| 4086<sub>10</sub> ||class="entry q0 g0"| 26480<sub>8</sub>
|-
|class="f"| 0 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (6, 10, 8) ||class="c"| (26770, 13278, 23372) ||class="entry q0 g0"| 26770<sub>6</sub> ||class="entry q0 g0"| 13278<sub>10</sub> ||class="entry q0 g0"| 23372<sub>8</sub>
|-
|class="f"| 0 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (6, 10, 8) ||class="c"| (26772, 21950, 15658) ||class="entry q0 g0"| 26772<sub>6</sub> ||class="entry q0 g0"| 21950<sub>10</sub> ||class="entry q0 g0"| 15658<sub>8</sub>
|-
|class="f"| 0 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (6, 10, 8) ||class="c"| (27010, 15678, 21692) ||class="entry q0 g0"| 27010<sub>6</sub> ||class="entry q0 g0"| 15678<sub>10</sub> ||class="entry q0 g0"| 21692<sub>8</sub>
|-
|class="f"| 0 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (6, 10, 8) ||class="c"| (27012, 23390, 13018) ||class="entry q0 g0"| 27012<sub>6</sub> ||class="entry q0 g0"| 23390<sub>10</sub> ||class="entry q0 g0"| 13018<sub>8</sub>
|-
|class="f"| 0 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (6, 10, 8) ||class="c"| (27024, 26486, 3814) ||class="entry q0 g0"| 27024<sub>6</sub> ||class="entry q0 g0"| 26486<sub>10</sub> ||class="entry q0 g0"| 3814<sub>8</sub>
|-
|class="f"| 0 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (8, 8, 8) ||class="c"| (3826, 21944, 23370) ||class="entry q0 g0"| 3826<sub>8</sub> ||class="entry q0 g0"| 21944<sub>8</sub> ||class="entry q0 g0"| 23370<sub>8</sub>
|-
|class="f"| 0 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (8, 8, 8) ||class="c"| (3828, 13272, 15660) ||class="entry q0 g0"| 3828<sub>8</sub> ||class="entry q0 g0"| 13272<sub>8</sub> ||class="entry q0 g0"| 15660<sub>8</sub>
|-
|class="f"| 0 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (8, 8, 8) ||class="c"| (4066, 23384, 21690) ||class="entry q0 g0"| 4066<sub>8</sub> ||class="entry q0 g0"| 23384<sub>8</sub> ||class="entry q0 g0"| 21690<sub>8</sub>
|-
|class="f"| 0 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (8, 8, 8) ||class="c"| (4068, 15672, 13020) ||class="entry q0 g0"| 4068<sub>8</sub> ||class="entry q0 g0"| 15672<sub>8</sub> ||class="entry q0 g0"| 13020<sub>8</sub>
|-
|class="f"| 0 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (8, 8, 8) ||class="c"| (13006, 21932, 26466) ||class="entry q0 g0"| 13006<sub>8</sub> ||class="entry q0 g0"| 21932<sub>8</sub> ||class="entry q0 g0"| 26466<sub>8</sub>
|-
|class="f"| 0 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (8, 8, 8) ||class="c"| (13258, 26468, 21678) ||class="entry q0 g0"| 13258<sub>8</sub> ||class="entry q0 g0"| 26468<sub>8</sub> ||class="entry q0 g0"| 21678<sub>8</sub>
|-
|class="f"| 0 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (8, 8, 8) ||class="c"| (15406, 23132, 26226) ||class="entry q0 g0"| 15406<sub>8</sub> ||class="entry q0 g0"| 23132<sub>8</sub> ||class="entry q0 g0"| 26226<sub>8</sub>
|-
|class="f"| 0 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (8, 8, 8) ||class="c"| (15418, 26228, 23118) ||class="entry q0 g0"| 15418<sub>8</sub> ||class="entry q0 g0"| 26228<sub>8</sub> ||class="entry q0 g0"| 23118<sub>8</sub>
|-
|class="f"| 854 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (2, 10, 6) ||class="c"| (576, 27606, 27328) ||class="entry q0 g0"| 576<sub>2</sub> ||class="entry q0 g0"| 27606<sub>10</sub> ||class="entry q0 g0"| 27328<sub>6</sub>
|-
|class="f"| 854 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (4, 6, 8) ||class="c"| (3232, 25638, 27600) ||class="entry q0 g0"| 3232<sub>4</sub> ||class="entry q0 g0"| 25638<sub>6</sub> ||class="entry q0 g0"| 27600<sub>8</sub>
|-
|class="f"| 854 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (4, 6, 8) ||class="c"| (12424, 22554, 27588) ||class="entry q0 g0"| 12424<sub>4</sub> ||class="entry q0 g0"| 22554<sub>6</sub> ||class="entry q0 g0"| 27588<sub>8</sub>
|-
|class="f"| 854 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (4, 6, 8) ||class="c"| (22536, 12684, 27346) ||class="entry q0 g0"| 22536<sub>4</sub> ||class="entry q0 g0"| 12684<sub>6</sub> ||class="entry q0 g0"| 27346<sub>8</sub>
|-
|class="f"| 854 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (4, 6, 8) ||class="c"| (25632, 3504, 27334) ||class="entry q0 g0"| 25632<sub>4</sub> ||class="entry q0 g0"| 3504<sub>6</sub> ||class="entry q0 g0"| 27334<sub>8</sub>
|-
|class="f"| 854 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (4, 8, 6) ||class="c"| (582, 3510, 3238) ||class="entry q0 g0"| 582<sub>4</sub> ||class="entry q0 g0"| 3510<sub>8</sub> ||class="entry q0 g0"| 3238<sub>6</sub>
|-
|class="f"| 854 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (4, 8, 6) ||class="c"| (594, 12702, 12442) ||class="entry q0 g0"| 594<sub>4</sub> ||class="entry q0 g0"| 12702<sub>8</sub> ||class="entry q0 g0"| 12442<sub>6</sub>
|-
|class="f"| 854 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (4, 8, 6) ||class="c"| (836, 22814, 22796) ||class="entry q0 g0"| 836<sub>4</sub> ||class="entry q0 g0"| 22814<sub>8</sub> ||class="entry q0 g0"| 22796<sub>6</sub>
|-
|class="f"| 854 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (4, 8, 6) ||class="c"| (848, 25910, 25904) ||class="entry q0 g0"| 848<sub>4</sub> ||class="entry q0 g0"| 25910<sub>8</sub> ||class="entry q0 g0"| 25904<sub>6</sub>
|-
|class="f"| 854 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (4, 12, 10) ||class="c"| (596, 22526, 22268) ||class="entry q0 g0"| 596<sub>4</sub> ||class="entry q0 g0"| 22526<sub>12</sub> ||class="entry q0 g0"| 22268<sub>10</sub>
|-
|class="f"| 854 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (4, 12, 10) ||class="c"| (834, 16254, 16234) ||class="entry q0 g0"| 834<sub>4</sub> ||class="entry q0 g0"| 16254<sub>12</sub> ||class="entry q0 g0"| 16234<sub>10</sub>
|-
|class="f"| 854 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (6, 6, 10) ||class="c"| (3252, 22542, 22508) ||class="entry q0 g0"| 3252<sub>6</sub> ||class="entry q0 g0"| 22542<sub>6</sub> ||class="entry q0 g0"| 22508<sub>10</sub>
|-
|class="f"| 854 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (6, 6, 10) ||class="c"| (3490, 12430, 15994) ||class="entry q0 g0"| 3490<sub>6</sub> ||class="entry q0 g0"| 12430<sub>6</sub> ||class="entry q0 g0"| 15994<sub>10</sub>
|-
|class="f"| 854 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (6, 6, 10) ||class="c"| (12444, 25650, 22520) ||class="entry q0 g0"| 12444<sub>6</sub> ||class="entry q0 g0"| 25650<sub>6</sub> ||class="entry q0 g0"| 22520<sub>10</sub>
|-
|class="f"| 854 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (6, 6, 10) ||class="c"| (12682, 3250, 15982) ||class="entry q0 g0"| 12682<sub>6</sub> ||class="entry q0 g0"| 3250<sub>6</sub> ||class="entry q0 g0"| 15982<sub>10</sub>
|-
|class="f"| 854 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (6, 6, 10) ||class="c"| (22556, 3492, 22254) ||class="entry q0 g0"| 22556<sub>6</sub> ||class="entry q0 g0"| 3492<sub>6</sub> ||class="entry q0 g0"| 22254<sub>10</sub>
|-
|class="f"| 854 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (6, 6, 10) ||class="c"| (22794, 25892, 16248) ||class="entry q0 g0"| 22794<sub>6</sub> ||class="entry q0 g0"| 25892<sub>6</sub> ||class="entry q0 g0"| 16248<sub>10</sub>
|-
|class="f"| 854 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (6, 6, 10) ||class="c"| (25652, 12696, 22266) ||class="entry q0 g0"| 25652<sub>6</sub> ||class="entry q0 g0"| 12696<sub>6</sub> ||class="entry q0 g0"| 22266<sub>10</sub>
|-
|class="f"| 854 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (6, 6, 10) ||class="c"| (25890, 22808, 16236) ||class="entry q0 g0"| 25890<sub>6</sub> ||class="entry q0 g0"| 22808<sub>6</sub> ||class="entry q0 g0"| 16236<sub>10</sub>
|-
|class="f"| 854 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (8, 8, 10) ||class="c"| (27348, 15976, 22506) ||class="entry q0 g0"| 27348<sub>8</sub> ||class="entry q0 g0"| 15976<sub>8</sub> ||class="entry q0 g0"| 22506<sub>10</sub>
|-
|class="f"| 854 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (8, 8, 10) ||class="c"| (27586, 22248, 15996) ||class="entry q0 g0"| 27586<sub>8</sub> ||class="entry q0 g0"| 22248<sub>8</sub> ||class="entry q0 g0"| 15996<sub>10</sub>
|-
|class="f"| 1334 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (2, 10, 6) ||class="c"| (1056, 28086, 27808) ||class="entry q0 g0"| 1056<sub>2</sub> ||class="entry q0 g0"| 28086<sub>10</sub> ||class="entry q0 g0"| 27808<sub>6</sub>
|-
|class="f"| 1334 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (4, 6, 8) ||class="c"| (2752, 25158, 28080) ||class="entry q0 g0"| 2752<sub>4</sub> ||class="entry q0 g0"| 25158<sub>6</sub> ||class="entry q0 g0"| 28080<sub>8</sub>
|-
|class="f"| 1334 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (4, 6, 8) ||class="c"| (14344, 20874, 27828) ||class="entry q0 g0"| 14344<sub>4</sub> ||class="entry q0 g0"| 20874<sub>6</sub> ||class="entry q0 g0"| 27828<sub>8</sub>
|-
|class="f"| 1334 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (4, 6, 8) ||class="c"| (20616, 14364, 28066) ||class="entry q0 g0"| 20616<sub>4</sub> ||class="entry q0 g0"| 14364<sub>6</sub> ||class="entry q0 g0"| 28066<sub>8</sub>
|-
|class="f"| 1334 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (4, 6, 8) ||class="c"| (25152, 3024, 27814) ||class="entry q0 g0"| 25152<sub>4</sub> ||class="entry q0 g0"| 3024<sub>6</sub> ||class="entry q0 g0"| 27814<sub>8</sub>
|-
|class="f"| 1334 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (4, 8, 6) ||class="c"| (1062, 3030, 2758) ||class="entry q0 g0"| 1062<sub>4</sub> ||class="entry q0 g0"| 3030<sub>8</sub> ||class="entry q0 g0"| 2758<sub>6</sub>
|-
|class="f"| 1334 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (4, 8, 6) ||class="c"| (1076, 20894, 20636) ||class="entry q0 g0"| 1076<sub>4</sub> ||class="entry q0 g0"| 20894<sub>8</sub> ||class="entry q0 g0"| 20636<sub>6</sub>
|-
|class="f"| 1334 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (4, 8, 6) ||class="c"| (1314, 14622, 14602) ||class="entry q0 g0"| 1314<sub>4</sub> ||class="entry q0 g0"| 14622<sub>8</sub> ||class="entry q0 g0"| 14602<sub>6</sub>
|-
|class="f"| 1334 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (4, 8, 6) ||class="c"| (1328, 25430, 25424) ||class="entry q0 g0"| 1328<sub>4</sub> ||class="entry q0 g0"| 25430<sub>8</sub> ||class="entry q0 g0"| 25424<sub>6</sub>
|-
|class="f"| 1334 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (4, 12, 10) ||class="c"| (1074, 14334, 14074) ||class="entry q0 g0"| 1074<sub>4</sub> ||class="entry q0 g0"| 14334<sub>12</sub> ||class="entry q0 g0"| 14074<sub>10</sub>
|-
|class="f"| 1334 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (4, 12, 10) ||class="c"| (1316, 24446, 24428) ||class="entry q0 g0"| 1316<sub>4</sub> ||class="entry q0 g0"| 24446<sub>12</sub> ||class="entry q0 g0"| 24428<sub>10</sub>
|-
|class="f"| 1334 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (6, 6, 10) ||class="c"| (2770, 14350, 14314) ||class="entry q0 g0"| 2770<sub>6</sub> ||class="entry q0 g0"| 14350<sub>6</sub> ||class="entry q0 g0"| 14314<sub>10</sub>
|-
|class="f"| 1334 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (6, 6, 10) ||class="c"| (3012, 20622, 24188) ||class="entry q0 g0"| 3012<sub>6</sub> ||class="entry q0 g0"| 20622<sub>6</sub> ||class="entry q0 g0"| 24188<sub>10</sub>
|-
|class="f"| 1334 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (6, 6, 10) ||class="c"| (14362, 3010, 14062) ||class="entry q0 g0"| 14362<sub>6</sub> ||class="entry q0 g0"| 3010<sub>6</sub> ||class="entry q0 g0"| 14062<sub>10</sub>
|-
|class="f"| 1334 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (6, 6, 10) ||class="c"| (14604, 25410, 24440) ||class="entry q0 g0"| 14604<sub>6</sub> ||class="entry q0 g0"| 25410<sub>6</sub> ||class="entry q0 g0"| 24440<sub>10</sub>
|-
|class="f"| 1334 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (6, 6, 10) ||class="c"| (20634, 25172, 14328) ||class="entry q0 g0"| 20634<sub>6</sub> ||class="entry q0 g0"| 25172<sub>6</sub> ||class="entry q0 g0"| 14328<sub>10</sub>
|-
|class="f"| 1334 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (6, 6, 10) ||class="c"| (20876, 2772, 24174) ||class="entry q0 g0"| 20876<sub>6</sub> ||class="entry q0 g0"| 2772<sub>6</sub> ||class="entry q0 g0"| 24174<sub>10</sub>
|-
|class="f"| 1334 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (6, 6, 10) ||class="c"| (25170, 20888, 14076) ||class="entry q0 g0"| 25170<sub>6</sub> ||class="entry q0 g0"| 20888<sub>6</sub> ||class="entry q0 g0"| 14076<sub>10</sub>
|-
|class="f"| 1334 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (6, 6, 10) ||class="c"| (25412, 14616, 24426) ||class="entry q0 g0"| 25412<sub>6</sub> ||class="entry q0 g0"| 14616<sub>6</sub> ||class="entry q0 g0"| 24426<sub>10</sub>
|-
|class="f"| 1334 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (8, 8, 10) ||class="c"| (27826, 24168, 14316) ||class="entry q0 g0"| 27826<sub>8</sub> ||class="entry q0 g0"| 24168<sub>8</sub> ||class="entry q0 g0"| 14316<sub>10</sub>
|-
|class="f"| 1334 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (8, 8, 10) ||class="c"| (28068, 14056, 24186) ||class="entry q0 g0"| 28068<sub>8</sub> ||class="entry q0 g0"| 14056<sub>8</sub> ||class="entry q0 g0"| 24186<sub>10</sub>
|-
|class="f"| 1632 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (2, 4, 6) ||class="c"| (2176, 2448, 1904) ||class="entry q0 g0"| 2176<sub>2</sub> ||class="entry q0 g0"| 2448<sub>4</sub> ||class="entry q0 g0"| 1904<sub>6</sub>
|-
|class="f"| 1632 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (2, 4, 6) ||class="c"| (24576, 24582, 1638) ||class="entry q0 g0"| 24576<sub>2</sub> ||class="entry q0 g0"| 24582<sub>4</sub> ||class="entry q0 g0"| 1638<sub>6</sub>
|-
|class="f"| 1632 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (4, 8, 8) ||class="c"| (2194, 21464, 23850) ||class="entry q0 g0"| 2194<sub>4</sub> ||class="entry q0 g0"| 21464<sub>8</sub> ||class="entry q0 g0"| 23850<sub>8</sub>
|-
|class="f"| 1632 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (4, 8, 8) ||class="c"| (2196, 13752, 15180) ||class="entry q0 g0"| 2196<sub>4</sub> ||class="entry q0 g0"| 13752<sub>8</sub> ||class="entry q0 g0"| 15180<sub>8</sub>
|-
|class="f"| 1632 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (4, 8, 8) ||class="c"| (2434, 23864, 21210) ||class="entry q0 g0"| 2434<sub>4</sub> ||class="entry q0 g0"| 23864<sub>8</sub> ||class="entry q0 g0"| 21210<sub>8</sub>
|-
|class="f"| 1632 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (4, 8, 8) ||class="c"| (2436, 15192, 13500) ||class="entry q0 g0"| 2436<sub>4</sub> ||class="entry q0 g0"| 15192<sub>8</sub> ||class="entry q0 g0"| 13500<sub>8</sub>
|-
|class="f"| 1632 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (4, 8, 8) ||class="c"| (24594, 14926, 23612) ||class="entry q0 g0"| 24594<sub>4</sub> ||class="entry q0 g0"| 14926<sub>8</sub> ||class="entry q0 g0"| 23612<sub>8</sub>
|-
|class="f"| 1632 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (4, 8, 8) ||class="c"| (24596, 23598, 14938) ||class="entry q0 g0"| 24596<sub>4</sub> ||class="entry q0 g0"| 23598<sub>8</sub> ||class="entry q0 g0"| 14938<sub>8</sub>
|-
|class="f"| 1632 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (4, 8, 8) ||class="c"| (24834, 13486, 21452) ||class="entry q0 g0"| 24834<sub>4</sub> ||class="entry q0 g0"| 13486<sub>8</sub> ||class="entry q0 g0"| 21452<sub>8</sub>
|-
|class="f"| 1632 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (4, 8, 8) ||class="c"| (24836, 21198, 13738) ||class="entry q0 g0"| 24836<sub>4</sub> ||class="entry q0 g0"| 21198<sub>8</sub> ||class="entry q0 g0"| 13738<sub>8</sub>
|-
|class="f"| 1632 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (4, 10, 6) ||class="c"| (2182, 28656, 24854) ||class="entry q0 g0"| 2182<sub>4</sub> ||class="entry q0 g0"| 28656<sub>10</sub> ||class="entry q0 g0"| 24854<sub>6</sub>
|-
|class="f"| 1632 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (4, 10, 6) ||class="c"| (24848, 28390, 2454) ||class="entry q0 g0"| 24848<sub>4</sub> ||class="entry q0 g0"| 28390<sub>10</sub> ||class="entry q0 g0"| 2454<sub>6</sub>
|-
|class="f"| 1632 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (6, 6, 8) ||class="c"| (1650, 23592, 23610) ||class="entry q0 g0"| 1650<sub>6</sub> ||class="entry q0 g0"| 23592<sub>6</sub> ||class="entry q0 g0"| 23610<sub>8</sub>
|-
|class="f"| 1632 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (6, 6, 8) ||class="c"| (1652, 14920, 14940) ||class="entry q0 g0"| 1652<sub>6</sub> ||class="entry q0 g0"| 14920<sub>6</sub> ||class="entry q0 g0"| 14940<sub>8</sub>
|-
|class="f"| 1632 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (6, 6, 8) ||class="c"| (1890, 21192, 21450) ||class="entry q0 g0"| 1890<sub>6</sub> ||class="entry q0 g0"| 21192<sub>6</sub> ||class="entry q0 g0"| 21450<sub>8</sub>
|-
|class="f"| 1632 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (6, 6, 8) ||class="c"| (1892, 13480, 13740) ||class="entry q0 g0"| 1892<sub>6</sub> ||class="entry q0 g0"| 13480<sub>6</sub> ||class="entry q0 g0"| 13740<sub>8</sub>
|-
|class="f"| 1632 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (8, 8, 12) ||class="c"| (1910, 28384, 28662) ||class="entry q0 g0"| 1910<sub>8</sub> ||class="entry q0 g0"| 28384<sub>8</sub> ||class="entry q0 g0"| 28662<sub>12</sub>
|-
|class="f"| 1632 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (8, 10, 10) ||class="c"| (13498, 28644, 23870) ||class="entry q0 g0"| 13498<sub>8</sub> ||class="entry q0 g0"| 28644<sub>10</sub> ||class="entry q0 g0"| 23870<sub>10</sub>
|-
|class="f"| 1632 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (8, 10, 10) ||class="c"| (15178, 28404, 21470) ||class="entry q0 g0"| 15178<sub>8</sub> ||class="entry q0 g0"| 28404<sub>10</sub> ||class="entry q0 g0"| 21470<sub>10</sub>
|-
|class="f"| 1632 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (8, 10, 10) ||class="c"| (21212, 28642, 15198) ||class="entry q0 g0"| 21212<sub>8</sub> ||class="entry q0 g0"| 28642<sub>10</sub> ||class="entry q0 g0"| 15198<sub>10</sub>
|-
|class="f"| 1632 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (8, 10, 10) ||class="c"| (23852, 28402, 13758) ||class="entry q0 g0"| 23852<sub>8</sub> ||class="entry q0 g0"| 28402<sub>10</sub> ||class="entry q0 g0"| 13758<sub>10</sub>
|-
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|-
|class="f"| 4382 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (4, 6, 8) ||class="c"| (8896, 19026, 31116) ||class="entry q0 g0"| 8896<sub>4</sub> ||class="entry q0 g0"| 19026<sub>6</sub> ||class="entry q0 g0"| 31116<sub>8</sub>
|-
|class="f"| 4382 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (4, 6, 8) ||class="c"| (11296, 17826, 30876) ||class="entry q0 g0"| 11296<sub>4</sub> ||class="entry q0 g0"| 17826<sub>6</sub> ||class="entry q0 g0"| 30876<sub>8</sub>
|-
|class="f"| 4382 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (4, 6, 8) ||class="c"| (17568, 11316, 31114) ||class="entry q0 g0"| 17568<sub>4</sub> ||class="entry q0 g0"| 11316<sub>6</sub> ||class="entry q0 g0"| 31114<sub>8</sub>
|-
|class="f"| 4382 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (4, 6, 8) ||class="c"| (19008, 9156, 30874) ||class="entry q0 g0"| 19008<sub>4</sub> ||class="entry q0 g0"| 9156<sub>6</sub> ||class="entry q0 g0"| 30874<sub>8</sub>
|-
|class="f"| 4382 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (4, 8, 6) ||class="c"| (4122, 9174, 8914) ||class="entry q0 g0"| 4122<sub>4</sub> ||class="entry q0 g0"| 9174<sub>8</sub> ||class="entry q0 g0"| 8914<sub>6</sub>
|-
|class="f"| 4382 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (4, 8, 6) ||class="c"| (4124, 17846, 17588) ||class="entry q0 g0"| 4124<sub>4</sub> ||class="entry q0 g0"| 17846<sub>8</sub> ||class="entry q0 g0"| 17588<sub>6</sub>
|-
|class="f"| 4382 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (4, 8, 6) ||class="c"| (4362, 11574, 11554) ||class="entry q0 g0"| 4362<sub>4</sub> ||class="entry q0 g0"| 11574<sub>8</sub> ||class="entry q0 g0"| 11554<sub>6</sub>
|-
|class="f"| 4382 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (4, 8, 6) ||class="c"| (4364, 19286, 19268) ||class="entry q0 g0"| 4364<sub>4</sub> ||class="entry q0 g0"| 19286<sub>8</sub> ||class="entry q0 g0"| 19268<sub>6</sub>
|-
|class="f"| 4382 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (4, 12, 10) ||class="c"| (4110, 8190, 7918) ||class="entry q0 g0"| 4110<sub>4</sub> ||class="entry q0 g0"| 8190<sub>12</sub> ||class="entry q0 g0"| 7918<sub>10</sub>
|-
|class="f"| 4382 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (4, 12, 10) ||class="c"| (4376, 30590, 30584) ||class="entry q0 g0"| 4376<sub>4</sub> ||class="entry q0 g0"| 30590<sub>12</sub> ||class="entry q0 g0"| 30584<sub>10</sub>
|-
|class="f"| 4382 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (6, 6, 10) ||class="c"| (8902, 11314, 8170) ||class="entry q0 g0"| 8902<sub>6</sub> ||class="entry q0 g0"| 11314<sub>6</sub> ||class="entry q0 g0"| 8170<sub>10</sub>
|-
|class="f"| 4382 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (6, 6, 10) ||class="c"| (9168, 17586, 30332) ||class="entry q0 g0"| 9168<sub>6</sub> ||class="entry q0 g0"| 17586<sub>6</sub> ||class="entry q0 g0"| 30332<sub>10</sub>
|-
|class="f"| 4382 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (6, 6, 10) ||class="c"| (11302, 9154, 7930) ||class="entry q0 g0"| 11302<sub>6</sub> ||class="entry q0 g0"| 9154<sub>6</sub> ||class="entry q0 g0"| 7930<sub>10</sub>
|-
|class="f"| 4382 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (6, 6, 10) ||class="c"| (11568, 19266, 30572) ||class="entry q0 g0"| 11568<sub>6</sub> ||class="entry q0 g0"| 19266<sub>6</sub> ||class="entry q0 g0"| 30572<sub>10</sub>
|-
|class="f"| 4382 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (6, 6, 10) ||class="c"| (17574, 19028, 8172) ||class="entry q0 g0"| 17574<sub>6</sub> ||class="entry q0 g0"| 19028<sub>6</sub> ||class="entry q0 g0"| 8172<sub>10</sub>
|-
|class="f"| 4382 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (6, 6, 10) ||class="c"| (17840, 8916, 30330) ||class="entry q0 g0"| 17840<sub>6</sub> ||class="entry q0 g0"| 8916<sub>6</sub> ||class="entry q0 g0"| 30330<sub>10</sub>
|-
|class="f"| 4382 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (6, 6, 10) ||class="c"| (19014, 17828, 7932) ||class="entry q0 g0"| 19014<sub>6</sub> ||class="entry q0 g0"| 17828<sub>6</sub> ||class="entry q0 g0"| 7932<sub>10</sub>
|-
|class="f"| 4382 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (6, 6, 10) ||class="c"| (19280, 11556, 30570) ||class="entry q0 g0"| 19280<sub>6</sub> ||class="entry q0 g0"| 11556<sub>6</sub> ||class="entry q0 g0"| 30570<sub>10</sub>
|-
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|-
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|-
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|-
|class="f"| 4680 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (2, 4, 6) ||class="c"| (18432, 18450, 4698) ||class="entry q0 g0"| 18432<sub>2</sub> ||class="entry q0 g0"| 18450<sub>4</sub> ||class="entry q0 g0"| 4698<sub>6</sub>
|-
|class="f"| 4680 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (4, 8, 8) ||class="c"| (8326, 18404, 29994) ||class="entry q0 g0"| 8326<sub>4</sub> ||class="entry q0 g0"| 18404<sub>8</sub> ||class="entry q0 g0"| 29994<sub>8</sub>
|-
|class="f"| 4680 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (4, 8, 8) ||class="c"| (8340, 7596, 12144) ||class="entry q0 g0"| 8340<sub>4</sub> ||class="entry q0 g0"| 7596<sub>8</sub> ||class="entry q0 g0"| 12144<sub>8</sub>
|-
|class="f"| 4680 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (4, 8, 8) ||class="c"| (8578, 29996, 18150) ||class="entry q0 g0"| 8578<sub>4</sub> ||class="entry q0 g0"| 29996<sub>8</sub> ||class="entry q0 g0"| 18150<sub>8</sub>
|-
|class="f"| 4680 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (4, 8, 8) ||class="c"| (8592, 12132, 7356) ||class="entry q0 g0"| 8592<sub>4</sub> ||class="entry q0 g0"| 12132<sub>8</sub> ||class="entry q0 g0"| 7356<sub>8</sub>
|-
|class="f"| 4680 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (4, 8, 8) ||class="c"| (18438, 11890, 29756) ||class="entry q0 g0"| 18438<sub>4</sub> ||class="entry q0 g0"| 11890<sub>8</sub> ||class="entry q0 g0"| 29756<sub>8</sub>
|-
|class="f"| 4680 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (4, 8, 8) ||class="c"| (18452, 29754, 11878) ||class="entry q0 g0"| 18452<sub>4</sub> ||class="entry q0 g0"| 29754<sub>8</sub> ||class="entry q0 g0"| 11878<sub>8</sub>
|-
|class="f"| 4680 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (4, 8, 8) ||class="c"| (18690, 7354, 18416) ||class="entry q0 g0"| 18690<sub>4</sub> ||class="entry q0 g0"| 7354<sub>8</sub> ||class="entry q0 g0"| 18416<sub>8</sub>
|-
|class="f"| 4680 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (4, 8, 8) ||class="c"| (18704, 18162, 7594) ||class="entry q0 g0"| 18704<sub>4</sub> ||class="entry q0 g0"| 18162<sub>8</sub> ||class="entry q0 g0"| 7594<sub>8</sub>
|-
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|class="f"| 4680 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (4, 10, 6) ||class="c"| (18692, 31450, 8598) ||class="entry q0 g0"| 18692<sub>4</sub> ||class="entry q0 g0"| 31450<sub>10</sub> ||class="entry q0 g0"| 8598<sub>6</sub>
|-
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|-
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|-
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|-
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|-
|class="f"| 4680 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (8, 8, 12) ||class="c"| (4958, 31432, 31710) ||class="entry q0 g0"| 4958<sub>8</sub> ||class="entry q0 g0"| 31432<sub>8</sub> ||class="entry q0 g0"| 31710<sub>12</sub>
|-
|class="f"| 4680 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (8, 10, 10) ||class="c"| (7342, 31704, 30014) ||class="entry q0 g0"| 7342<sub>8</sub> ||class="entry q0 g0"| 31704<sub>10</sub> ||class="entry q0 g0"| 30014<sub>10</sub>
|-
|class="f"| 4680 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (8, 10, 10) ||class="c"| (12130, 31452, 18422) ||class="entry q0 g0"| 12130<sub>8</sub> ||class="entry q0 g0"| 31452<sub>10</sub> ||class="entry q0 g0"| 18422<sub>10</sub>
|-
|class="f"| 4680 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (8, 10, 10) ||class="c"| (18164, 31690, 12150) ||class="entry q0 g0"| 18164<sub>8</sub> ||class="entry q0 g0"| 31690<sub>10</sub> ||class="entry q0 g0"| 12150<sub>10</sub>
|-
|class="f"| 4680 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (8, 10, 10) ||class="c"| (30008, 31438, 7614) ||class="entry q0 g0"| 30008<sub>8</sub> ||class="entry q0 g0"| 31438<sub>10</sub> ||class="entry q0 g0"| 7614<sub>10</sub>
|-
|class="f"| 5160 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (2, 4, 6) ||class="c"| (10240, 10260, 5180) ||class="entry q0 g0"| 10240<sub>2</sub> ||class="entry q0 g0"| 10260<sub>4</sub> ||class="entry q0 g0"| 5180<sub>6</sub>
|-
|class="f"| 5160 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (2, 4, 6) ||class="c"| (16512, 16770, 5418) ||class="entry q0 g0"| 16512<sub>2</sub> ||class="entry q0 g0"| 16770<sub>4</sub> ||class="entry q0 g0"| 5418<sub>6</sub>
|-
|class="f"| 5160 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (4, 8, 8) ||class="c"| (10246, 20084, 29274) ||class="entry q0 g0"| 10246<sub>4</sub> ||class="entry q0 g0"| 20084<sub>8</sub> ||class="entry q0 g0"| 29274<sub>8</sub>
|-
|class="f"| 5160 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (4, 8, 8) ||class="c"| (10258, 29276, 20070) ||class="entry q0 g0"| 10258<sub>4</sub> ||class="entry q0 g0"| 29276<sub>8</sub> ||class="entry q0 g0"| 20070<sub>8</sub>
|-
|class="f"| 5160 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (4, 8, 8) ||class="c"| (10500, 6876, 10224) ||class="entry q0 g0"| 10500<sub>4</sub> ||class="entry q0 g0"| 6876<sub>8</sub> ||class="entry q0 g0"| 10224<sub>8</sub>
|-
|class="f"| 5160 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (4, 8, 8) ||class="c"| (10512, 9972, 7116) ||class="entry q0 g0"| 10512<sub>4</sub> ||class="entry q0 g0"| 9972<sub>8</sub> ||class="entry q0 g0"| 7116<sub>8</sub>
|-
|class="f"| 5160 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (4, 8, 8) ||class="c"| (16518, 10210, 29516) ||class="entry q0 g0"| 16518<sub>4</sub> ||class="entry q0 g0"| 10210<sub>8</sub> ||class="entry q0 g0"| 29516<sub>8</sub>
|-
|class="f"| 5160 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (4, 8, 8) ||class="c"| (16530, 7114, 20336) ||class="entry q0 g0"| 16530<sub>4</sub> ||class="entry q0 g0"| 7114<sub>8</sub> ||class="entry q0 g0"| 20336<sub>8</sub>
|-
|class="f"| 5160 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (4, 8, 8) ||class="c"| (16772, 29514, 9958) ||class="entry q0 g0"| 16772<sub>4</sub> ||class="entry q0 g0"| 29514<sub>8</sub> ||class="entry q0 g0"| 9958<sub>8</sub>
|-
|class="f"| 5160 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (4, 8, 8) ||class="c"| (16784, 20322, 6874) ||class="entry q0 g0"| 16784<sub>4</sub> ||class="entry q0 g0"| 20322<sub>8</sub> ||class="entry q0 g0"| 6874<sub>8</sub>
|-
|class="f"| 5160 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (4, 10, 6) ||class="c"| (10498, 31932, 16790) ||class="entry q0 g0"| 10498<sub>4</sub> ||class="entry q0 g0"| 31932<sub>10</sub> ||class="entry q0 g0"| 16790<sub>6</sub>
|-
|class="f"| 5160 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (4, 10, 6) ||class="c"| (16532, 32170, 10518) ||class="entry q0 g0"| 16532<sub>4</sub> ||class="entry q0 g0"| 32170<sub>10</sub> ||class="entry q0 g0"| 10518<sub>6</sub>
|-
|class="f"| 5160 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (6, 6, 8) ||class="c"| (5166, 29256, 29262) ||class="entry q0 g0"| 5166<sub>6</sub> ||class="entry q0 g0"| 29256<sub>6</sub> ||class="entry q0 g0"| 29262<sub>8</sub>
|-
|class="f"| 5160 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (6, 6, 8) ||class="c"| (5178, 20064, 20082) ||class="entry q0 g0"| 5178<sub>6</sub> ||class="entry q0 g0"| 20064<sub>6</sub> ||class="entry q0 g0"| 20082<sub>8</sub>
|-
|class="f"| 5160 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (6, 6, 8) ||class="c"| (5420, 9952, 10212) ||class="entry q0 g0"| 5420<sub>6</sub> ||class="entry q0 g0"| 9952<sub>6</sub> ||class="entry q0 g0"| 10212<sub>8</sub>
|-
|class="f"| 5160 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (6, 6, 8) ||class="c"| (5432, 6856, 7128) ||class="entry q0 g0"| 5432<sub>6</sub> ||class="entry q0 g0"| 6856<sub>6</sub> ||class="entry q0 g0"| 7128<sub>8</sub>
|-
|class="f"| 5160 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (8, 8, 12) ||class="c"| (5438, 31912, 32190) ||class="entry q0 g0"| 5438<sub>8</sub> ||class="entry q0 g0"| 31912<sub>8</sub> ||class="entry q0 g0"| 32190<sub>12</sub>
|-
|class="f"| 5160 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (8, 10, 10) ||class="c"| (6862, 32184, 29534) ||class="entry q0 g0"| 6862<sub>8</sub> ||class="entry q0 g0"| 32184<sub>10</sub> ||class="entry q0 g0"| 29534<sub>10</sub>
|-
|class="f"| 5160 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (8, 10, 10) ||class="c"| (9970, 32172, 20342) ||class="entry q0 g0"| 9970<sub>8</sub> ||class="entry q0 g0"| 32172<sub>10</sub> ||class="entry q0 g0"| 20342<sub>10</sub>
|-
|class="f"| 5160 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (8, 10, 10) ||class="c"| (20324, 31930, 10230) ||class="entry q0 g0"| 20324<sub>8</sub> ||class="entry q0 g0"| 31930<sub>10</sub> ||class="entry q0 g0"| 10230<sub>10</sub>
|-
|class="f"| 5160 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (8, 10, 10) ||class="c"| (29528, 31918, 7134) ||class="entry q0 g0"| 29528<sub>8</sub> ||class="entry q0 g0"| 31918<sub>10</sub> ||class="entry q0 g0"| 7134<sub>10</sub>
|-
|class="f"| 6014 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (4, 6, 12) ||class="c"| (6280, 28686, 32760) ||class="entry q0 g0"| 6280<sub>4</sub> ||class="entry q0 g0"| 28686<sub>6</sub> ||class="entry q0 g0"| 32760<sub>12</sub>
|-
|class="f"| 6014 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (4, 6, 12) ||class="c"| (9376, 19506, 32748) ||class="entry q0 g0"| 9376<sub>4</sub> ||class="entry q0 g0"| 19506<sub>6</sub> ||class="entry q0 g0"| 32748<sub>12</sub>
|-
|class="f"| 6014 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (4, 6, 12) ||class="c"| (10816, 17346, 32508) ||class="entry q0 g0"| 10816<sub>4</sub> ||class="entry q0 g0"| 17346<sub>6</sub> ||class="entry q0 g0"| 32508<sub>12</sub>
|-
|class="f"| 6014 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (4, 6, 12) ||class="c"| (17088, 10836, 32746) ||class="entry q0 g0"| 17088<sub>4</sub> ||class="entry q0 g0"| 10836<sub>6</sub> ||class="entry q0 g0"| 32746<sub>12</sub>
|-
|class="f"| 6014 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (4, 6, 12) ||class="c"| (19488, 9636, 32506) ||class="entry q0 g0"| 19488<sub>4</sub> ||class="entry q0 g0"| 9636<sub>6</sub> ||class="entry q0 g0"| 32506<sub>12</sub>
|-
|class="f"| 6014 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (4, 6, 12) ||class="c"| (28680, 6552, 32494) ||class="entry q0 g0"| 28680<sub>4</sub> ||class="entry q0 g0"| 6552<sub>6</sub> ||class="entry q0 g0"| 32494<sub>12</sub>
|-
|class="f"| 6014 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (6, 6, 6) ||class="c"| (6298, 10822, 9634) ||class="entry q0 g0"| 6298<sub>6</sub> ||class="entry q0 g0"| 10822<sub>6</sub> ||class="entry q0 g0"| 9634<sub>6</sub>
|-
|class="f"| 6014 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (6, 6, 6) ||class="c"| (6300, 19494, 17348) ||class="entry q0 g0"| 6300<sub>6</sub> ||class="entry q0 g0"| 19494<sub>6</sub> ||class="entry q0 g0"| 17348<sub>6</sub>
|-
|class="f"| 6014 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (6, 6, 6) ||class="c"| (6538, 9382, 10834) ||class="entry q0 g0"| 6538<sub>6</sub> ||class="entry q0 g0"| 9382<sub>6</sub> ||class="entry q0 g0"| 10834<sub>6</sub>
|-
|class="f"| 6014 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (6, 6, 6) ||class="c"| (6540, 17094, 19508) ||class="entry q0 g0"| 6540<sub>6</sub> ||class="entry q0 g0"| 17094<sub>6</sub> ||class="entry q0 g0"| 19508<sub>6</sub>
|-
|class="f"| 6014 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (6, 6, 6) ||class="c"| (9396, 28698, 17360) ||class="entry q0 g0"| 9396<sub>6</sub> ||class="entry q0 g0"| 28698<sub>6</sub> ||class="entry q0 g0"| 17360<sub>6</sub>
|-
|class="f"| 6014 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (6, 6, 6) ||class="c"| (9648, 17106, 28700) ||class="entry q0 g0"| 9648<sub>6</sub> ||class="entry q0 g0"| 17106<sub>6</sub> ||class="entry q0 g0"| 28700<sub>6</sub>
|-
|class="f"| 6014 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (6, 6, 6) ||class="c"| (11076, 28938, 19760) ||class="entry q0 g0"| 11076<sub>6</sub> ||class="entry q0 g0"| 28938<sub>6</sub> ||class="entry q0 g0"| 19760<sub>6</sub>
|-
|class="f"| 6014 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (6, 6, 6) ||class="c"| (11088, 19746, 28940) ||class="entry q0 g0"| 11088<sub>6</sub> ||class="entry q0 g0"| 19746<sub>6</sub> ||class="entry q0 g0"| 28940<sub>6</sub>
|-
|class="f"| 6014 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (6, 8, 8) ||class="c"| (6286, 5742, 6558) ||class="entry q0 g0"| 6286<sub>6</sub> ||class="entry q0 g0"| 5742<sub>8</sub> ||class="entry q0 g0"| 6558<sub>8</sub>
|-
|class="f"| 6014 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (6, 8, 8) ||class="c"| (9394, 5754, 9654) ||class="entry q0 g0"| 9394<sub>6</sub> ||class="entry q0 g0"| 5754<sub>8</sub> ||class="entry q0 g0"| 9654<sub>8</sub>
|-
|class="f"| 6014 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (6, 8, 8) ||class="c"| (11074, 5994, 11094) ||class="entry q0 g0"| 11074<sub>6</sub> ||class="entry q0 g0"| 5994<sub>8</sub> ||class="entry q0 g0"| 11094<sub>8</sub>
|-
|class="f"| 6014 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (6, 8, 8) ||class="c"| (17108, 5756, 17366) ||class="entry q0 g0"| 17108<sub>6</sub> ||class="entry q0 g0"| 5756<sub>8</sub> ||class="entry q0 g0"| 17366<sub>8</sub>
|-
|class="f"| 6014 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (6, 8, 8) ||class="c"| (19748, 5996, 19766) ||class="entry q0 g0"| 19748<sub>6</sub> ||class="entry q0 g0"| 5996<sub>8</sub> ||class="entry q0 g0"| 19766<sub>8</sub>
|-
|class="f"| 6014 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (6, 8, 8) ||class="c"| (28952, 6008, 28958) ||class="entry q0 g0"| 28952<sub>6</sub> ||class="entry q0 g0"| 6008<sub>8</sub> ||class="entry q0 g0"| 28958<sub>8</sub>
|-
|class="f"| 6014 ||class="q"| (0, 0, 0) ||class="g"| (0, 0, 0) ||class="w"| (6, 14, 10) ||class="c"| (5736, 32766, 32488) ||class="entry q0 g0"| 5736<sub>6</sub> ||class="entry q0 g0"| 32766<sub>14</sub> ||class="entry q0 g0"| 32488<sub>10</sub>
|}<noinclude>
[[Category:Mentors of Boolean functions; chains]]
</noinclude>
9xw2mil6kss94ikoton2phevc3y9t1v
Template:Mentors of Boolean functions/chains/4-ary/length 4
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Watchduck
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Watchduck moved page [[Template:Mentors of Boolean functions/cycles/4-ary/length 4]] to [[Template:Mentors of Boolean functions/chains/4-ary/length 4]]
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wikitext
text/x-wiki
<templatestyles src="Template:Mentors of Boolean functions/cycles/style.css" />
{| class="wikitable sortable mentor-cycles"
|+ 48 cycles of length 4
!class="f"| F !!class="q"| Q !!class="g"| G !!class="w"| W !!colspan="5"| cycle
|-
|class="f"| 0 ||class="q"| (1, 3, 1, 3) ||class="g"| (1, 0, 1, 0) ||class="w"| (10, 7, 12, 5) ||class="c"| (59581, 875, 65475, 5141) ||class="entry q1 g1"| 59581<sub>10</sub> ||class="entry q3 g0"| 875<sub>7</sub> ||class="entry q1 g1"| 65475<sub>12</sub> ||class="entry q3 g0"| 5141<sub>5</sub>
|-
|class="f"| 0 ||class="q"| (1, 3, 1, 3) ||class="g"| (1, 0, 1, 0) ||class="w"| (10, 7, 12, 5) ||class="c"| (59611, 1389, 65445, 4627) ||class="entry q1 g1"| 59611<sub>10</sub> ||class="entry q3 g0"| 1389<sub>7</sub> ||class="entry q1 g1"| 65445<sub>12</sub> ||class="entry q3 g0"| 4627<sub>5</sub>
|-
|class="f"| 0 ||class="q"| (1, 3, 1, 3) ||class="g"| (1, 0, 1, 0) ||class="w"| (10, 7, 12, 5) ||class="c"| (59623, 4473, 65433, 1543) ||class="entry q1 g1"| 59623<sub>10</sub> ||class="entry q3 g0"| 4473<sub>7</sub> ||class="entry q1 g1"| 65433<sub>12</sub> ||class="entry q3 g0"| 1543<sub>5</sub>
|-
|class="f"| 0 ||class="q"| (1, 3, 1, 3) ||class="g"| (1, 0, 1, 0) ||class="w"| (10, 7, 12, 5) ||class="c"| (60301, 1595, 64755, 4421) ||class="entry q1 g1"| 60301<sub>10</sub> ||class="entry q3 g0"| 1595<sub>7</sub> ||class="entry q1 g1"| 64755<sub>12</sub> ||class="entry q3 g0"| 4421<sub>5</sub>
|-
|class="f"| 0 ||class="q"| (1, 3, 1, 3) ||class="g"| (1, 0, 1, 0) ||class="w"| (10, 7, 12, 5) ||class="c"| (60337, 4655, 64719, 1361) ||class="entry q1 g1"| 60337<sub>10</sub> ||class="entry q3 g0"| 4655<sub>7</sub> ||class="entry q1 g1"| 64719<sub>12</sub> ||class="entry q3 g0"| 1361<sub>5</sub>
|-
|class="f"| 0 ||class="q"| (1, 3, 1, 3) ||class="g"| (1, 0, 1, 0) ||class="w"| (10, 7, 12, 5) ||class="c"| (60811, 1629, 64245, 4387) ||class="entry q1 g1"| 60811<sub>10</sub> ||class="entry q3 g0"| 1629<sub>7</sub> ||class="entry q1 g1"| 64245<sub>12</sub> ||class="entry q3 g0"| 4387<sub>5</sub>
|-
|class="f"| 0 ||class="q"| (1, 3, 1, 3) ||class="g"| (1, 0, 1, 0) ||class="w"| (10, 7, 12, 5) ||class="c"| (60881, 5199, 64175, 817) ||class="entry q1 g1"| 60881<sub>10</sub> ||class="entry q3 g0"| 5199<sub>7</sub> ||class="entry q1 g1"| 64175<sub>12</sub> ||class="entry q3 g0"| 817<sub>5</sub>
|-
|class="f"| 0 ||class="q"| (1, 3, 1, 3) ||class="g"| (1, 0, 1, 0) ||class="w"| (10, 7, 12, 5) ||class="c"| (61063, 5913, 63993, 103) ||class="entry q1 g1"| 61063<sub>10</sub> ||class="entry q3 g0"| 5913<sub>7</sub> ||class="entry q1 g1"| 63993<sub>12</sub> ||class="entry q3 g0"| 103<sub>5</sub>
|-
|class="f"| 0 ||class="q"| (1, 3, 1, 3) ||class="g"| (1, 0, 1, 0) ||class="w"| (10, 7, 12, 5) ||class="c"| (61153, 4383, 63903, 1633) ||class="entry q1 g1"| 61153<sub>10</sub> ||class="entry q3 g0"| 4383<sub>7</sub> ||class="entry q1 g1"| 63903<sub>12</sub> ||class="entry q3 g0"| 1633<sub>5</sub>
|-
|class="f"| 0 ||class="q"| (1, 3, 1, 3) ||class="g"| (1, 0, 1, 0) ||class="w"| (10, 7, 12, 5) ||class="c"| (63907, 4725, 61149, 1291) ||class="entry q1 g1"| 63907<sub>10</sub> ||class="entry q3 g0"| 4725<sub>7</sub> ||class="entry q1 g1"| 61149<sub>12</sub> ||class="entry q3 g0"| 1291<sub>5</sub>
|-
|class="f"| 0 ||class="q"| (1, 3, 1, 3) ||class="g"| (1, 0, 1, 0) ||class="w"| (10, 7, 12, 5) ||class="c"| (63941, 5235, 61115, 781) ||class="entry q1 g1"| 63941<sub>10</sub> ||class="entry q3 g0"| 5235<sub>7</sub> ||class="entry q1 g1"| 61115<sub>12</sub> ||class="entry q3 g0"| 781<sub>5</sub>
|-
|class="f"| 0 ||class="q"| (1, 3, 1, 3) ||class="g"| (1, 0, 1, 0) ||class="w"| (10, 7, 12, 5) ||class="c"| (64147, 5925, 60909, 91) ||class="entry q1 g1"| 64147<sub>10</sub> ||class="entry q3 g0"| 5925<sub>7</sub> ||class="entry q1 g1"| 60909<sub>12</sub> ||class="entry q3 g0"| 91<sub>5</sub>
|-
|class="f"| 0 ||class="q"| (1, 3, 1, 3) ||class="g"| (1, 0, 1, 0) ||class="w"| (10, 7, 12, 5) ||class="c"| (64201, 1335, 60855, 4681) ||class="entry q1 g1"| 64201<sub>10</sub> ||class="entry q3 g0"| 1335<sub>7</sub> ||class="entry q1 g1"| 60855<sub>12</sub> ||class="entry q3 g0"| 4681<sub>5</sub>
|-
|class="f"| 0 ||class="q"| (1, 3, 1, 3) ||class="g"| (1, 0, 1, 0) ||class="w"| (10, 7, 12, 5) ||class="c"| (64661, 5955, 60395, 61) ||class="entry q1 g1"| 64661<sub>10</sub> ||class="entry q3 g0"| 5955<sub>7</sub> ||class="entry q1 g1"| 60395<sub>12</sub> ||class="entry q3 g0"| 61<sub>5</sub>
|-
|class="f"| 0 ||class="q"| (1, 3, 1, 3) ||class="g"| (1, 0, 1, 0) ||class="w"| (10, 7, 12, 5) ||class="c"| (64681, 855, 60375, 5161) ||class="entry q1 g1"| 64681<sub>10</sub> ||class="entry q3 g0"| 855<sub>7</sub> ||class="entry q1 g1"| 60375<sub>12</sub> ||class="entry q3 g0"| 5161<sub>5</sub>
|-
|class="f"| 0 ||class="q"| (1, 3, 1, 3) ||class="g"| (1, 0, 1, 1) ||class="w"| (6, 11, 16, 1) ||class="c"| (59521, 6015, 65535, 1) ||class="entry q1 g1"| 59521<sub>6</sub> ||class="entry q3 g0"| 6015<sub>11</sub> ||class="entry q1 g1"| 65535<sub>16</sub> ||class="entry q3 g1"| 1<sub>1</sub>
|-
|class="f"| 6014 ||class="q"| (0, 1, 2, 3) ||class="g"| (1, 0, 0, 1) ||class="w"| (8, 4, 7, 9) ||class="c"| (59624, 105, 5910, 59625) ||class="entry q0 g1"| 59624<sub>8</sub> ||class="entry q1 g0"| 105<sub>4</sub> ||class="entry q2 g0"| 5910<sub>7</sub> ||class="entry q3 g1"| 59625<sub>9</sub>
|-
|class="f"| 6014 ||class="q"| (0, 1, 2, 3) ||class="g"| (1, 0, 0, 1) ||class="w"| (8, 4, 7, 9) ||class="c"| (61064, 1545, 4470, 61065) ||class="entry q0 g1"| 61064<sub>8</sub> ||class="entry q1 g0"| 1545<sub>4</sub> ||class="entry q2 g0"| 4470<sub>7</sub> ||class="entry q3 g1"| 61065<sub>9</sub>
|-
|class="f"| 6014 ||class="q"| (0, 1, 2, 3) ||class="g"| (1, 0, 0, 1) ||class="w"| (8, 4, 7, 9) ||class="c"| (64160, 4641, 1374, 64161) ||class="entry q0 g1"| 64160<sub>8</sub> ||class="entry q1 g0"| 4641<sub>4</sub> ||class="entry q2 g0"| 1374<sub>7</sub> ||class="entry q3 g1"| 64161<sub>9</sub>
|-
|class="f"| 6014 ||class="q"| (0, 1, 2, 3) ||class="g"| (1, 0, 0, 1) ||class="w"| (8, 4, 7, 9) ||class="c"| (64704, 5185, 830, 64705) ||class="entry q0 g1"| 64704<sub>8</sub> ||class="entry q1 g0"| 5185<sub>4</sub> ||class="entry q2 g0"| 830<sub>7</sub> ||class="entry q3 g1"| 64705<sub>9</sub>
|-
|class="f"| 6014 ||class="q"| (0, 1, 2, 3) ||class="g"| (1, 0, 0, 1) ||class="w"| (8, 8, 7, 13) ||class="c"| (59534, 1647, 6000, 61167) ||class="entry q0 g1"| 59534<sub>8</sub> ||class="entry q1 g0"| 1647<sub>8</sub> ||class="entry q2 g0"| 6000<sub>7</sub> ||class="entry q3 g1"| 61167<sub>13</sub>
|-
|class="f"| 6014 ||class="q"| (0, 1, 2, 3) ||class="g"| (1, 0, 0, 1) ||class="w"| (8, 8, 7, 13) ||class="c"| (59570, 4731, 5964, 64251) ||class="entry q0 g1"| 59570<sub>8</sub> ||class="entry q1 g0"| 4731<sub>8</sub> ||class="entry q2 g0"| 5964<sub>7</sub> ||class="entry q3 g1"| 64251<sub>13</sub>
|-
|class="f"| 6014 ||class="q"| (0, 1, 2, 3) ||class="g"| (1, 0, 0, 1) ||class="w"| (8, 8, 7, 13) ||class="c"| (59604, 5245, 5930, 64765) ||class="entry q0 g1"| 59604<sub>8</sub> ||class="entry q1 g0"| 5245<sub>8</sub> ||class="entry q2 g0"| 5930<sub>7</sub> ||class="entry q3 g1"| 64765<sub>13</sub>
|-
|class="f"| 6014 ||class="q"| (0, 1, 2, 3) ||class="g"| (1, 0, 0, 1) ||class="w"| (8, 8, 7, 13) ||class="c"| (60290, 5931, 5244, 65451) ||class="entry q0 g1"| 60290<sub>8</sub> ||class="entry q1 g0"| 5931<sub>8</sub> ||class="entry q2 g0"| 5244<sub>7</sub> ||class="entry q3 g1"| 65451<sub>13</sub>
|-
|class="f"| 6014 ||class="q"| (0, 1, 2, 3) ||class="g"| (1, 0, 0, 1) ||class="w"| (8, 8, 7, 13) ||class="c"| (60804, 5965, 4730, 65485) ||class="entry q0 g1"| 60804<sub>8</sub> ||class="entry q1 g0"| 5965<sub>8</sub> ||class="entry q2 g0"| 4730<sub>7</sub> ||class="entry q3 g1"| 65485<sub>13</sub>
|-
|class="f"| 6014 ||class="q"| (0, 1, 2, 3) ||class="g"| (1, 0, 0, 1) ||class="w"| (8, 8, 7, 13) ||class="c"| (63888, 6001, 1646, 65521) ||class="entry q0 g1"| 63888<sub>8</sub> ||class="entry q1 g0"| 6001<sub>8</sub> ||class="entry q2 g0"| 1646<sub>7</sub> ||class="entry q3 g1"| 65521<sub>13</sub>
|-
|class="f"| 6014 ||class="q"| (0, 1, 2, 3) ||class="g"| (1, 0, 0, 1) ||class="w"| (10, 6, 5, 11) ||class="c"| (60376, 1337, 5158, 60857) ||class="entry q0 g1"| 60376<sub>10</sub> ||class="entry q1 g0"| 1337<sub>6</sub> ||class="entry q2 g0"| 5158<sub>5</sub> ||class="entry q3 g1"| 60857<sub>11</sub>
|-
|class="f"| 6014 ||class="q"| (0, 1, 2, 3) ||class="g"| (1, 0, 0, 1) ||class="w"| (10, 6, 5, 11) ||class="c"| (60388, 4397, 5146, 63917) ||class="entry q0 g1"| 60388<sub>10</sub> ||class="entry q1 g0"| 4397<sub>6</sub> ||class="entry q2 g0"| 5146<sub>5</sub> ||class="entry q3 g1"| 63917<sub>11</sub>
|-
|class="f"| 6014 ||class="q"| (0, 1, 2, 3) ||class="g"| (1, 0, 0, 1) ||class="w"| (10, 6, 5, 11) ||class="c"| (60856, 857, 4678, 60377) ||class="entry q0 g1"| 60856<sub>10</sub> ||class="entry q1 g0"| 857<sub>6</sub> ||class="entry q2 g0"| 4678<sub>5</sub> ||class="entry q3 g1"| 60377<sub>11</sub>
|-
|class="f"| 6014 ||class="q"| (0, 1, 2, 3) ||class="g"| (1, 0, 0, 1) ||class="w"| (10, 6, 5, 11) ||class="c"| (60898, 4427, 4636, 63947) ||class="entry q0 g1"| 60898<sub>10</sub> ||class="entry q1 g0"| 4427<sub>6</sub> ||class="entry q2 g0"| 4636<sub>5</sub> ||class="entry q3 g1"| 63947<sub>11</sub>
|-
|class="f"| 6014 ||class="q"| (0, 1, 2, 3) ||class="g"| (1, 0, 0, 1) ||class="w"| (10, 6, 5, 11) ||class="c"| (61108, 4637, 4426, 64157) ||class="entry q0 g1"| 61108<sub>10</sub> ||class="entry q1 g0"| 4637<sub>6</sub> ||class="entry q2 g0"| 4426<sub>5</sub> ||class="entry q3 g1"| 64157<sub>11</sub>
|-
|class="f"| 6014 ||class="q"| (0, 1, 2, 3) ||class="g"| (1, 0, 0, 1) ||class="w"| (10, 6, 5, 11) ||class="c"| (61138, 5147, 4396, 64667) ||class="entry q0 g1"| 61138<sub>10</sub> ||class="entry q1 g0"| 5147<sub>6</sub> ||class="entry q2 g0"| 4396<sub>5</sub> ||class="entry q3 g1"| 64667<sub>11</sub>
|-
|class="f"| 6014 ||class="q"| (0, 1, 2, 3) ||class="g"| (1, 0, 0, 1) ||class="w"| (10, 6, 5, 11) ||class="c"| (63916, 869, 1618, 60389) ||class="entry q0 g1"| 63916<sub>10</sub> ||class="entry q1 g0"| 869<sub>6</sub> ||class="entry q2 g0"| 1618<sub>5</sub> ||class="entry q3 g1"| 60389<sub>11</sub>
|-
|class="f"| 6014 ||class="q"| (0, 1, 2, 3) ||class="g"| (1, 0, 0, 1) ||class="w"| (10, 6, 5, 11) ||class="c"| (63946, 1379, 1588, 60899) ||class="entry q0 g1"| 63946<sub>10</sub> ||class="entry q1 g0"| 1379<sub>6</sub> ||class="entry q2 g0"| 1588<sub>5</sub> ||class="entry q3 g1"| 60899<sub>11</sub>
|-
|class="f"| 6014 ||class="q"| (0, 1, 2, 3) ||class="g"| (1, 0, 0, 1) ||class="w"| (10, 6, 5, 11) ||class="c"| (64156, 1589, 1378, 61109) ||class="entry q0 g1"| 64156<sub>10</sub> ||class="entry q1 g0"| 1589<sub>6</sub> ||class="entry q2 g0"| 1378<sub>5</sub> ||class="entry q3 g1"| 61109<sub>11</sub>
|-
|class="f"| 6014 ||class="q"| (0, 1, 2, 3) ||class="g"| (1, 0, 0, 1) ||class="w"| (10, 6, 5, 11) ||class="c"| (64198, 5159, 1336, 64679) ||class="entry q0 g1"| 64198<sub>10</sub> ||class="entry q1 g0"| 5159<sub>6</sub> ||class="entry q2 g0"| 1336<sub>5</sub> ||class="entry q3 g1"| 64679<sub>11</sub>
|-
|class="f"| 6014 ||class="q"| (0, 1, 2, 3) ||class="g"| (1, 0, 0, 1) ||class="w"| (10, 6, 5, 11) ||class="c"| (64666, 1619, 868, 61139) ||class="entry q0 g1"| 64666<sub>10</sub> ||class="entry q1 g0"| 1619<sub>6</sub> ||class="entry q2 g0"| 868<sub>5</sub> ||class="entry q3 g1"| 61139<sub>11</sub>
|-
|class="f"| 6014 ||class="q"| (0, 1, 2, 3) ||class="g"| (1, 0, 0, 1) ||class="w"| (10, 6, 5, 11) ||class="c"| (64678, 4679, 856, 64199) ||class="entry q0 g1"| 64678<sub>10</sub> ||class="entry q1 g0"| 4679<sub>6</sub> ||class="entry q2 g0"| 856<sub>5</sub> ||class="entry q3 g1"| 64199<sub>11</sub>
|-
|class="f"| 6014 ||class="q"| (0, 1, 2, 3) ||class="g"| (1, 0, 0, 1) ||class="w"| (12, 8, 3, 13) ||class="c"| (60350, 831, 5184, 60351) ||class="entry q0 g1"| 60350<sub>12</sub> ||class="entry q1 g0"| 831<sub>8</sub> ||class="entry q2 g0"| 5184<sub>3</sub> ||class="entry q3 g1"| 60351<sub>13</sub>
|-
|class="f"| 6014 ||class="q"| (0, 1, 2, 3) ||class="g"| (1, 0, 0, 1) ||class="w"| (12, 8, 3, 13) ||class="c"| (60894, 1375, 4640, 60895) ||class="entry q0 g1"| 60894<sub>12</sub> ||class="entry q1 g0"| 1375<sub>8</sub> ||class="entry q2 g0"| 4640<sub>3</sub> ||class="entry q3 g1"| 60895<sub>13</sub>
|-
|class="f"| 6014 ||class="q"| (0, 1, 2, 3) ||class="g"| (1, 0, 0, 1) ||class="w"| (12, 8, 3, 13) ||class="c"| (63990, 4471, 1544, 63991) ||class="entry q0 g1"| 63990<sub>12</sub> ||class="entry q1 g0"| 4471<sub>8</sub> ||class="entry q2 g0"| 1544<sub>3</sub> ||class="entry q3 g1"| 63991<sub>13</sub>
|-
|class="f"| 6014 ||class="q"| (0, 1, 2, 3) ||class="g"| (1, 0, 0, 1) ||class="w"| (12, 8, 3, 13) ||class="c"| (65430, 5911, 104, 65431) ||class="entry q0 g1"| 65430<sub>12</sub> ||class="entry q1 g0"| 5911<sub>8</sub> ||class="entry q2 g0"| 104<sub>3</sub> ||class="entry q3 g1"| 65431<sub>13</sub>
|-
|class="f"| 6014 ||class="q"| (0, 1, 2, 3) ||class="g"| (1, 1, 1, 1) ||class="w"| (12, 4, 3, 9) ||class="c"| (61166, 15, 4368, 59535) ||class="entry q0 g1"| 61166<sub>12</sub> ||class="entry q1 g1"| 15<sub>4</sub> ||class="entry q2 g1"| 4368<sub>3</sub> ||class="entry q3 g1"| 59535<sub>9</sub>
|-
|class="f"| 6014 ||class="q"| (0, 1, 2, 3) ||class="g"| (1, 1, 1, 1) ||class="w"| (12, 4, 3, 9) ||class="c"| (64250, 51, 1284, 59571) ||class="entry q0 g1"| 64250<sub>12</sub> ||class="entry q1 g1"| 51<sub>4</sub> ||class="entry q2 g1"| 1284<sub>3</sub> ||class="entry q3 g1"| 59571<sub>9</sub>
|-
|class="f"| 6014 ||class="q"| (0, 1, 2, 3) ||class="g"| (1, 1, 1, 1) ||class="w"| (12, 4, 3, 9) ||class="c"| (64764, 85, 770, 59605) ||class="entry q0 g1"| 64764<sub>12</sub> ||class="entry q1 g1"| 85<sub>4</sub> ||class="entry q2 g1"| 770<sub>3</sub> ||class="entry q3 g1"| 59605<sub>9</sub>
|-
|class="f"| 6014 ||class="q"| (0, 1, 2, 3) ||class="g"| (1, 1, 1, 1) ||class="w"| (12, 4, 3, 9) ||class="c"| (65450, 771, 84, 60291) ||class="entry q0 g1"| 65450<sub>12</sub> ||class="entry q1 g1"| 771<sub>4</sub> ||class="entry q2 g1"| 84<sub>3</sub> ||class="entry q3 g1"| 60291<sub>9</sub>
|-
|class="f"| 6014 ||class="q"| (0, 1, 2, 3) ||class="g"| (1, 1, 1, 1) ||class="w"| (12, 4, 3, 9) ||class="c"| (65484, 1285, 50, 60805) ||class="entry q0 g1"| 65484<sub>12</sub> ||class="entry q1 g1"| 1285<sub>4</sub> ||class="entry q2 g1"| 50<sub>3</sub> ||class="entry q3 g1"| 60805<sub>9</sub>
|-
|class="f"| 6014 ||class="q"| (0, 1, 2, 3) ||class="g"| (1, 1, 1, 1) ||class="w"| (12, 4, 3, 9) ||class="c"| (65520, 4369, 14, 63889) ||class="entry q0 g1"| 65520<sub>12</sub> ||class="entry q1 g1"| 4369<sub>4</sub> ||class="entry q2 g1"| 14<sub>3</sub> ||class="entry q3 g1"| 63889<sub>9</sub>
|}<noinclude>
[[Category:Mentors of Boolean functions; chains]]
</noinclude>
d6vp0d4au8ea18eay9lyh0cu6txqbis
Template:Mentors of Boolean functions/chains/4-ary/length 6
10
313712
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2025-01-01T19:24:48Z
Watchduck
137431
Watchduck moved page [[Template:Mentors of Boolean functions/cycles/4-ary/length 6]] to [[Template:Mentors of Boolean functions/chains/4-ary/length 6]]
2693781
wikitext
text/x-wiki
<templatestyles src="Template:Mentors of Boolean functions/cycles/style.css" />
{| class="wikitable sortable mentor-cycles"
|+ 2636 cycles of length 6
!class="f"| F !!class="q"| Q !!class="g"| G !!class="w"| W !!colspan="7"| cycle
|-
|class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 8, 6, 8, 6) ||class="c"| (1672, 28425, 1655, 28168, 1695, 1889) ||class="entry q0 g0"| 1672<sub>4</sub> ||class="entry q1 g0"| 28425<sub>8</sub> ||class="entry q1 g0"| 1655<sub>8</sub> ||class="entry q0 g0"| 28168<sub>6</sub> ||class="entry q1 g0"| 1695<sub>8</sub> ||class="entry q1 g0"| 1889<sub>6</sub>
|-
|class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 8, 6, 8, 6) ||class="c"| (2152, 24825, 1895, 24808, 2415, 1649) ||class="entry q0 g0"| 2152<sub>4</sub> ||class="entry q1 g0"| 24825<sub>8</sub> ||class="entry q1 g0"| 1895<sub>8</sub> ||class="entry q0 g0"| 24808<sub>6</sub> ||class="entry q1 g0"| 2415<sub>8</sub> ||class="entry q1 g0"| 1649<sub>6</sub>
|-
|class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 8, 6, 8, 6) ||class="c"| (4768, 31521, 4703, 31264, 4791, 4937) ||class="entry q0 g0"| 4768<sub>4</sub> ||class="entry q1 g0"| 31521<sub>8</sub> ||class="entry q1 g0"| 4703<sub>8</sub> ||class="entry q0 g0"| 31264<sub>6</sub> ||class="entry q1 g0"| 4791<sub>8</sub> ||class="entry q1 g0"| 4937<sub>6</sub>
|-
|class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 8, 6, 8, 6) ||class="c"| (5312, 32065, 5183, 31808, 5335, 5417) ||class="entry q0 g0"| 5312<sub>4</sub> ||class="entry q1 g0"| 32065<sub>8</sub> ||class="entry q1 g0"| 5183<sub>8</sub> ||class="entry q0 g0"| 31808<sub>6</sub> ||class="entry q1 g0"| 5335<sub>8</sub> ||class="entry q1 g0"| 5417<sub>6</sub>
|-
|class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 8, 6, 8, 6) ||class="c"| (6688, 29361, 5423, 29344, 6951, 5177) ||class="entry q0 g0"| 6688<sub>4</sub> ||class="entry q1 g0"| 29361<sub>8</sub> ||class="entry q1 g0"| 5423<sub>8</sub> ||class="entry q0 g0"| 29344<sub>6</sub> ||class="entry q1 g0"| 6951<sub>8</sub> ||class="entry q1 g0"| 5177<sub>6</sub>
|-
|class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 8, 6, 8, 6) ||class="c"| (7232, 29905, 4943, 29888, 7495, 4697) ||class="entry q0 g0"| 7232<sub>4</sub> ||class="entry q1 g0"| 29905<sub>8</sub> ||class="entry q1 g0"| 4943<sub>8</sub> ||class="entry q0 g0"| 29888<sub>6</sub> ||class="entry q1 g0"| 7495<sub>8</sub> ||class="entry q1 g0"| 4697<sub>6</sub>
|-
|class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 8, 6, 8, 6) ||class="c"| (8296, 18669, 4955, 18664, 8571, 4685) ||class="entry q0 g0"| 8296<sub>4</sub> ||class="entry q1 g0"| 18669<sub>8</sub> ||class="entry q1 g0"| 4955<sub>8</sub> ||class="entry q0 g0"| 18664<sub>6</sub> ||class="entry q1 g0"| 8571<sub>8</sub> ||class="entry q1 g0"| 4685<sub>6</sub>
|-
|class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 8, 6, 8, 6) ||class="c"| (9736, 20109, 5435, 20104, 10011, 5165) ||class="entry q0 g0"| 9736<sub>4</sub> ||class="entry q1 g0"| 20109<sub>8</sub> ||class="entry q1 g0"| 5435<sub>8</sub> ||class="entry q0 g0"| 20104<sub>6</sub> ||class="entry q1 g0"| 10011<sub>8</sub> ||class="entry q1 g0"| 5165<sub>6</sub>
|-
|class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 8, 6, 8, 6) ||class="c"| (13376, 23749, 1907, 23744, 13651, 1637) ||class="entry q0 g0"| 13376<sub>4</sub> ||class="entry q1 g0"| 23749<sub>8</sub> ||class="entry q1 g0"| 1907<sub>8</sub> ||class="entry q0 g0"| 23744<sub>6</sub> ||class="entry q1 g0"| 13651<sub>8</sub> ||class="entry q1 g0"| 1637<sub>6</sub>
|-
|class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 8, 6, 8, 6) ||class="c"| (16488, 10475, 5437, 10472, 16765, 5163) ||class="entry q0 g0"| 16488<sub>4</sub> ||class="entry q1 g0"| 10475<sub>8</sub> ||class="entry q1 g0"| 5437<sub>8</sub> ||class="entry q0 g0"| 10472<sub>6</sub> ||class="entry q1 g0"| 16765<sub>8</sub> ||class="entry q1 g0"| 5163<sub>6</sub>
|-
|class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 8, 6, 8, 6) ||class="c"| (17928, 11915, 4957, 11912, 18205, 4683) ||class="entry q0 g0"| 17928<sub>4</sub> ||class="entry q1 g0"| 11915<sub>8</sub> ||class="entry q1 g0"| 4957<sub>8</sub> ||class="entry q0 g0"| 11912<sub>6</sub> ||class="entry q1 g0"| 18205<sub>8</sub> ||class="entry q1 g0"| 4683<sub>6</sub>
|-
|class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 8, 6, 8, 6) ||class="c"| (21024, 15011, 1909, 15008, 21301, 1635) ||class="entry q0 g0"| 21024<sub>4</sub> ||class="entry q1 g0"| 15011<sub>8</sub> ||class="entry q1 g0"| 1909<sub>8</sub> ||class="entry q0 g0"| 15008<sub>6</sub> ||class="entry q1 g0"| 21301<sub>8</sub> ||class="entry q1 g0"| 1635<sub>6</sub>
|-
|class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 10, 6, 8, 8) ||class="c"| (680, 703, 27351, 27176, 27433, 27585) ||class="entry q0 g0"| 680<sub>4</sub> ||class="entry q1 g0"| 703<sub>8</sub> ||class="entry q1 g0"| 27351<sub>10</sub> ||class="entry q0 g0"| 27176<sub>6</sub> ||class="entry q1 g0"| 27433<sub>8</sub> ||class="entry q1 g0"| 27585<sub>8</sub>
|-
|class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 10, 6, 8, 8) ||class="c"| (1224, 1247, 27831, 27720, 27977, 28065) ||class="entry q0 g0"| 1224<sub>4</sub> ||class="entry q1 g0"| 1247<sub>8</sub> ||class="entry q1 g0"| 27831<sub>10</sub> ||class="entry q0 g0"| 27720<sub>6</sub> ||class="entry q1 g0"| 27977<sub>8</sub> ||class="entry q1 g0"| 28065<sub>8</sub>
|-
|class="f"| 0 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 10, 6, 8, 8) ||class="c"| (2600, 2863, 28071, 25256, 25273, 27825) ||class="entry q0 g0"| 2600<sub>4</sub> ||class="entry q1 g0"| 2863<sub>8</sub> ||class="entry q1 g0"| 28071<sub>10</sub> ||class="entry q0 g0"| 25256<sub>6</sub> ||class="entry q1 g0"| 25273<sub>8</sub> ||class="entry q1 g0"| 27825<sub>8</sub>
|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
|class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 11, 5, 11, 13, 7) ||class="c"| (38378, 38845, 37891, 65084, 64893, 38467) ||class="entry q2 g1"| 38378<sub>9</sub> ||class="entry q3 g1"| 38845<sub>11</sub> ||class="entry q3 g1"| 37891<sub>5</sub> ||class="entry q2 g1"| 65084<sub>11</sub> ||class="entry q3 g1"| 64893<sub>13</sub> ||class="entry q3 g1"| 38467<sub>7</sub>
|-
|class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 11, 5, 11, 13, 7) ||class="c"| (40508, 40315, 36901, 62954, 63419, 37477) ||class="entry q2 g1"| 40508<sub>9</sub> ||class="entry q3 g1"| 40315<sub>11</sub> ||class="entry q3 g1"| 36901<sub>5</sub> ||class="entry q2 g1"| 62954<sub>11</sub> ||class="entry q3 g1"| 63419<sub>13</sub> ||class="entry q3 g1"| 37477<sub>7</sub>
|-
|class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 11, 5, 11, 13, 7) ||class="c"| (46652, 46447, 33817, 56810, 57263, 34393) ||class="entry q2 g1"| 46652<sub>9</sub> ||class="entry q3 g1"| 46447<sub>11</sub> ||class="entry q3 g1"| 33817<sub>5</sub> ||class="entry q2 g1"| 56810<sub>11</sub> ||class="entry q3 g1"| 57263<sub>13</sub> ||class="entry q3 g1"| 34393<sub>7</sub>
|-
|class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 11, 5, 11, 13, 7) ||class="c"| (54634, 54847, 33065, 48828, 48383, 33641) ||class="entry q2 g1"| 54634<sub>9</sub> ||class="entry q3 g1"| 54847<sub>11</sub> ||class="entry q3 g1"| 33065<sub>5</sub> ||class="entry q2 g1"| 48828<sub>11</sub> ||class="entry q3 g1"| 48383<sub>13</sub> ||class="entry q3 g1"| 33641<sub>7</sub>
|-
|class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 11, 9, 11, 9, 7) ||class="c"| (38380, 61917, 62053, 65082, 39709, 61477) ||class="entry q2 g1"| 38380<sub>9</sub> ||class="entry q3 g1"| 61917<sub>11</sub> ||class="entry q3 g1"| 62053<sub>9</sub> ||class="entry q2 g1"| 65082<sub>11</sub> ||class="entry q3 g1"| 39709<sub>9</sub> ||class="entry q3 g1"| 61477<sub>7</sub>
|-
|class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 11, 9, 11, 9, 7) ||class="c"| (38392, 52725, 52825, 65070, 42805, 52249) ||class="entry q2 g1"| 38392<sub>9</sub> ||class="entry q3 g1"| 52725<sub>11</sub> ||class="entry q3 g1"| 52825<sub>9</sub> ||class="entry q2 g1"| 65070<sub>11</sub> ||class="entry q3 g1"| 42805<sub>9</sub> ||class="entry q3 g1"| 52249<sub>7</sub>
|-
|class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 11, 9, 11, 9, 7) ||class="c"| (40506, 64283, 63043, 62956, 37339, 62467) ||class="entry q2 g1"| 40506<sub>9</sub> ||class="entry q3 g1"| 64283<sub>11</sub> ||class="entry q3 g1"| 63043<sub>9</sub> ||class="entry q2 g1"| 62956<sub>11</sub> ||class="entry q3 g1"| 37339<sub>9</sub> ||class="entry q3 g1"| 62467<sub>7</sub>
|-
|class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 11, 9, 11, 9, 7) ||class="c"| (40760, 44979, 41961, 62702, 50547, 41385) ||class="entry q2 g1"| 40760<sub>9</sub> ||class="entry q3 g1"| 44979<sub>11</sub> ||class="entry q3 g1"| 41961<sub>9</sub> ||class="entry q2 g1"| 62702<sub>11</sub> ||class="entry q3 g1"| 50547<sub>9</sub> ||class="entry q3 g1"| 41385<sub>7</sub>
|-
|class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 11, 9, 11, 9, 7) ||class="c"| (46638, 61223, 56899, 56824, 34279, 56323) ||class="entry q2 g1"| 46638<sub>9</sub> ||class="entry q3 g1"| 61223<sub>11</sub> ||class="entry q3 g1"| 56899<sub>9</sub> ||class="entry q2 g1"| 56824<sub>11</sub> ||class="entry q3 g1"| 34279<sub>9</sub> ||class="entry q3 g1"| 56323<sub>7</sub>
|-
|class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 11, 9, 11, 9, 7) ||class="c"| (46892, 48015, 35817, 56570, 53583, 35241) ||class="entry q2 g1"| 46892<sub>9</sub> ||class="entry q3 g1"| 48015<sub>11</sub> ||class="entry q3 g1"| 35817<sub>9</sub> ||class="entry q2 g1"| 56570<sub>11</sub> ||class="entry q3 g1"| 53583<sub>9</sub> ||class="entry q3 g1"| 35241<sub>7</sub>
|-
|class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 11, 9, 11, 9, 7) ||class="c"| (54382, 58615, 45797, 49080, 36407, 45221) ||class="entry q2 g1"| 54382<sub>9</sub> ||class="entry q3 g1"| 58615<sub>11</sub> ||class="entry q3 g1"| 45797<sub>9</sub> ||class="entry q2 g1"| 49080<sub>11</sub> ||class="entry q3 g1"| 36407<sub>9</sub> ||class="entry q3 g1"| 45221<sub>7</sub>
|-
|class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 11, 9, 11, 9, 7) ||class="c"| (54394, 55519, 36569, 49068, 45599, 35993) ||class="entry q2 g1"| 54394<sub>9</sub> ||class="entry q3 g1"| 55519<sub>11</sub> ||class="entry q3 g1"| 36569<sub>9</sub> ||class="entry q2 g1"| 49068<sub>11</sub> ||class="entry q3 g1"| 45599<sub>9</sub> ||class="entry q3 g1"| 35993<sub>7</sub>
|-
|class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 11, 11, 11, 9, 9) ||class="c"| (38826, 64619, 65219, 64636, 38571, 64643) ||class="entry q2 g1"| 38826<sub>9</sub> ||class="entry q3 g1"| 64619<sub>11</sub> ||class="entry q3 g1"| 65219<sub>11</sub> ||class="entry q2 g1"| 64636<sub>11</sub> ||class="entry q3 g1"| 38571<sub>9</sub> ||class="entry q3 g1"| 64643<sub>9</sub>
|-
|class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 11, 11, 11, 9, 9) ||class="c"| (40060, 63149, 64229, 63402, 40045, 63653) ||class="entry q2 g1"| 40060<sub>9</sub> ||class="entry q3 g1"| 63149<sub>11</sub> ||class="entry q3 g1"| 64229<sub>11</sub> ||class="entry q2 g1"| 63402<sub>11</sub> ||class="entry q3 g1"| 40045<sub>9</sub> ||class="entry q3 g1"| 63653<sub>9</sub>
|-
|class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 11, 11, 11, 9, 9) ||class="c"| (46204, 57017, 61145, 57258, 46201, 60569) ||class="entry q2 g1"| 46204<sub>9</sub> ||class="entry q3 g1"| 57017<sub>11</sub> ||class="entry q3 g1"| 61145<sub>11</sub> ||class="entry q2 g1"| 57258<sub>11</sub> ||class="entry q3 g1"| 46201<sub>9</sub> ||class="entry q3 g1"| 60569<sub>9</sub>
|-
|class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 11, 11, 11, 9, 9) ||class="c"| (55082, 48617, 60393, 48380, 55081, 59817) ||class="entry q2 g1"| 55082<sub>9</sub> ||class="entry q3 g1"| 48617<sub>11</sub> ||class="entry q3 g1"| 60393<sub>11</sub> ||class="entry q2 g1"| 48380<sub>11</sub> ||class="entry q3 g1"| 55081<sub>9</sub> ||class="entry q3 g1"| 59817<sub>9</sub>
|-
|class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 13, 11, 11, 11, 13) ||class="c"| (38140, 65341, 64917, 65322, 38397, 65493) ||class="entry q2 g1"| 38140<sub>9</sub> ||class="entry q3 g1"| 65341<sub>13</sub> ||class="entry q3 g1"| 64917<sub>11</sub> ||class="entry q2 g1"| 65322<sub>11</sub> ||class="entry q3 g1"| 38397<sub>11</sub> ||class="entry q3 g1"| 65493<sub>13</sub>
|-
|class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 13, 11, 11, 11, 13) ||class="c"| (40746, 62971, 63923, 62716, 40763, 64499) ||class="entry q2 g1"| 40746<sub>9</sub> ||class="entry q3 g1"| 62971<sub>13</sub> ||class="entry q3 g1"| 63923<sub>11</sub> ||class="entry q2 g1"| 62716<sub>11</sub> ||class="entry q3 g1"| 40763<sub>11</sub> ||class="entry q3 g1"| 64499<sub>13</sub>
|-
|class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 13, 11, 11, 11, 13) ||class="c"| (46890, 56815, 60815, 56572, 46895, 61391) ||class="entry q2 g1"| 46890<sub>9</sub> ||class="entry q3 g1"| 56815<sub>13</sub> ||class="entry q3 g1"| 60815<sub>11</sub> ||class="entry q2 g1"| 56572<sub>11</sub> ||class="entry q3 g1"| 46895<sub>11</sub> ||class="entry q3 g1"| 61391<sub>13</sub>
|-
|class="f"| 854 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 13, 11, 11, 11, 13) ||class="c"| (54396, 48831, 59583, 49066, 54399, 60159) ||class="entry q2 g1"| 54396<sub>9</sub> ||class="entry q3 g1"| 48831<sub>13</sub> ||class="entry q3 g1"| 59583<sub>11</sub> ||class="entry q2 g1"| 49066<sub>11</sub> ||class="entry q3 g1"| 54399<sub>11</sub> ||class="entry q3 g1"| 60159<sub>13</sub>
|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
|class="f"| 1334 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 5, 7, 11, 7, 9) ||class="c"| (55808, 33896, 45956, 57142, 33118, 46770) ||class="entry q2 g1"| 55808<sub>5</sub> ||class="entry q2 g1"| 33896<sub>5</sub> ||class="entry q2 g1"| 45956<sub>7</sub> ||class="entry q2 g1"| 57142<sub>11</sub> ||class="entry q2 g1"| 33118<sub>7</sub> ||class="entry q2 g1"| 46770<sub>9</sub>
|-
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|-
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|-
|class="f"| 1334 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 9, 7, 5, 11) ||class="c"| (37188, 38244, 63698, 38002, 36946, 64996) ||class="entry q2 g1"| 37188<sub>5</sub> ||class="entry q2 g1"| 38244<sub>7</sub> ||class="entry q2 g1"| 63698<sub>9</sub> ||class="entry q2 g1"| 38002<sub>7</sub> ||class="entry q2 g1"| 36946<sub>5</sub> ||class="entry q2 g1"| 64996<sub>11</sub>
|-
|class="f"| 1334 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 9, 7, 5, 11) ||class="c"| (37394, 38450, 64388, 38692, 37636, 65202) ||class="entry q2 g1"| 37394<sub>5</sub> ||class="entry q2 g1"| 38450<sub>7</sub> ||class="entry q2 g1"| 64388<sub>9</sub> ||class="entry q2 g1"| 38692<sub>7</sub> ||class="entry q2 g1"| 37636<sub>5</sub> ||class="entry q2 g1"| 65202<sub>11</sub>
|-
|class="f"| 1334 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 9, 7, 9, 7) ||class="c"| (32846, 47186, 54768, 34168, 48484, 53446) ||class="entry q2 g1"| 32846<sub>5</sub> ||class="entry q2 g1"| 47186<sub>7</sub> ||class="entry q2 g1"| 54768<sub>9</sub> ||class="entry q2 g1"| 34168<sub>7</sub> ||class="entry q2 g1"| 48484<sub>9</sub> ||class="entry q2 g1"| 53446<sub>7</sub>
|-
|class="f"| 1334 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 9, 7, 9, 7) ||class="c"| (32860, 57882, 36778, 34154, 59180, 35484) ||class="entry q2 g1"| 32860<sub>5</sub> ||class="entry q2 g1"| 57882<sub>7</sub> ||class="entry q2 g1"| 36778<sub>9</sub> ||class="entry q2 g1"| 34154<sub>7</sub> ||class="entry q2 g1"| 59180<sub>9</sub> ||class="entry q2 g1"| 35484<sub>7</sub>
|-
|class="f"| 1334 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 9, 7, 9, 7) ||class="c"| (33098, 35482, 58940, 33916, 36780, 58122) ||class="entry q2 g1"| 33098<sub>5</sub> ||class="entry q2 g1"| 35482<sub>7</sub> ||class="entry q2 g1"| 58940<sub>9</sub> ||class="entry q2 g1"| 33916<sub>7</sub> ||class="entry q2 g1"| 36780<sub>9</sub> ||class="entry q2 g1"| 58122<sub>7</sub>
|-
|class="f"| 1334 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 9, 7, 9, 7) ||class="c"| (33112, 53458, 48230, 33902, 54756, 47440) ||class="entry q2 g1"| 33112<sub>5</sub> ||class="entry q2 g1"| 53458<sub>7</sub> ||class="entry q2 g1"| 48230<sub>9</sub> ||class="entry q2 g1"| 33902<sub>7</sub> ||class="entry q2 g1"| 54756<sub>9</sub> ||class="entry q2 g1"| 47440<sub>7</sub>
|-
|class="f"| 1334 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 9, 7, 9, 7) ||class="c"| (33294, 54148, 48944, 34616, 54962, 47622) ||class="entry q2 g1"| 33294<sub>5</sub> ||class="entry q2 g1"| 54148<sub>7</sub> ||class="entry q2 g1"| 48944<sub>9</sub> ||class="entry q2 g1"| 34616<sub>7</sub> ||class="entry q2 g1"| 54962<sub>9</sub> ||class="entry q2 g1"| 47622<sub>7</sub>
|-
|class="f"| 1334 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 9, 7, 9, 7) ||class="c"| (33308, 35276, 58730, 34602, 36090, 57436) ||class="entry q2 g1"| 33308<sub>5</sub> ||class="entry q2 g1"| 35276<sub>7</sub> ||class="entry q2 g1"| 58730<sub>9</sub> ||class="entry q2 g1"| 34602<sub>7</sub> ||class="entry q2 g1"| 36090<sub>9</sub> ||class="entry q2 g1"| 57436<sub>7</sub>
|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
|class="f"| 1334 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 9, 7, 9, 7) ||class="c"| (37634, 39122, 62580, 38452, 40420, 61762) ||class="entry q2 g1"| 37634<sub>5</sub> ||class="entry q2 g1"| 39122<sub>7</sub> ||class="entry q2 g1"| 62580<sub>9</sub> ||class="entry q2 g1"| 38452<sub>7</sub> ||class="entry q2 g1"| 40420<sub>9</sub> ||class="entry q2 g1"| 61762<sub>7</sub>
|-
|class="f"| 1334 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 9, 7, 9, 7) ||class="c"| (37648, 49818, 44590, 38438, 51116, 43800) ||class="entry q2 g1"| 37648<sub>5</sub> ||class="entry q2 g1"| 49818<sub>7</sub> ||class="entry q2 g1"| 44590<sub>9</sub> ||class="entry q2 g1"| 38438<sub>7</sub> ||class="entry q2 g1"| 51116<sub>9</sub> ||class="entry q2 g1"| 43800<sub>7</sub>
|-
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|-
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|-
|class="f"| 1334 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 9, 11, 7, 7) ||class="c"| (39104, 44604, 52334, 40438, 43786, 51544) ||class="entry q2 g1"| 39104<sub>5</sub> ||class="entry q2 g1"| 44604<sub>9</sub> ||class="entry q2 g1"| 52334<sub>9</sub> ||class="entry q2 g1"| 40438<sub>11</sub> ||class="entry q2 g1"| 43786<sub>7</sub> ||class="entry q2 g1"| 51544<sub>7</sub>
|-
|class="f"| 1334 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 9, 11, 7, 7) ||class="c"| (39552, 50666, 42670, 40886, 49372, 41880) ||class="entry q2 g1"| 39552<sub>5</sub> ||class="entry q2 g1"| 50666<sub>9</sub> ||class="entry q2 g1"| 42670<sub>9</sub> ||class="entry q2 g1"| 40886<sub>11</sub> ||class="entry q2 g1"| 49372<sub>7</sub> ||class="entry q2 g1"| 41880<sub>7</sub>
|-
|class="f"| 1334 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 9, 11, 7, 7) ||class="c"| (43080, 63014, 42922, 44414, 62224, 41628) ||class="entry q2 g1"| 43080<sub>5</sub> ||class="entry q2 g1"| 63014<sub>9</sub> ||class="entry q2 g1"| 42922<sub>9</sub> ||class="entry q2 g1"| 44414<sub>11</sub> ||class="entry q2 g1"| 62224<sub>7</sub> ||class="entry q2 g1"| 41628<sub>7</sub>
|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
|class="f"| 1334 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 11, 9, 9, 9) ||class="c"| (51290, 52328, 64406, 52588, 51550, 65184) ||class="entry q2 g1"| 51290<sub>7</sub> ||class="entry q2 g1"| 52328<sub>7</sub> ||class="entry q2 g1"| 64406<sub>11</sub> ||class="entry q2 g1"| 52588<sub>9</sub> ||class="entry q2 g1"| 51550<sub>9</sub> ||class="entry q2 g1"| 65184<sub>9</sub>
|-
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|-
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|-
|class="f"| 1334 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 11, 9) ||class="c"| (35722, 59612, 35724, 36540, 60906, 36538) ||class="entry q2 g1"| 35722<sub>7</sub> ||class="entry q2 g1"| 59612<sub>9</sub> ||class="entry q2 g1"| 35724<sub>7</sub> ||class="entry q2 g1"| 36540<sub>9</sub> ||class="entry q2 g1"| 60906<sub>11</sub> ||class="entry q2 g1"| 36538<sub>9</sub>
|-
|class="f"| 1334 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 11, 9) ||class="c"| (39362, 64148, 39364, 40180, 65442, 40178) ||class="entry q2 g1"| 39362<sub>7</sub> ||class="entry q2 g1"| 64148<sub>9</sub> ||class="entry q2 g1"| 39364<sub>7</sub> ||class="entry q2 g1"| 40180<sub>9</sub> ||class="entry q2 g1"| 65442<sub>11</sub> ||class="entry q2 g1"| 40178<sub>9</sub>
|-
|class="f"| 1334 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 11, 9) ||class="c"| (39572, 63938, 39570, 40866, 64756, 40868) ||class="entry q2 g1"| 39572<sub>7</sub> ||class="entry q2 g1"| 63938<sub>9</sub> ||class="entry q2 g1"| 39570<sub>7</sub> ||class="entry q2 g1"| 40866<sub>9</sub> ||class="entry q2 g1"| 64756<sub>11</sub> ||class="entry q2 g1"| 40868<sub>9</sub>
|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 7, 7, 9, 9) ||class="c"| (34178, 36421, 35467, 59444, 58085, 36523) ||class="entry q2 g1"| 34178<sub>5</sub> ||class="entry q3 g1"| 36421<sub>7</sub> ||class="entry q3 g1"| 35467<sub>7</sub> ||class="entry q2 g1"| 59444<sub>7</sub> ||class="entry q3 g1"| 58085<sub>9</sub> ||class="entry q3 g1"| 36523<sub>9</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 7, 7, 9, 9) ||class="c"| (34192, 54285, 53457, 59430, 47277, 54513) ||class="entry q2 g1"| 34192<sub>5</sub> ||class="entry q3 g1"| 54285<sub>7</sub> ||class="entry q3 g1"| 53457<sub>7</sub> ||class="entry q2 g1"| 59430<sub>7</sub> ||class="entry q3 g1"| 47277<sub>9</sub> ||class="entry q3 g1"| 54513<sub>9</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 7, 7, 9, 9) ||class="c"| (34868, 33507, 35021, 58754, 60995, 36077) ||class="entry q2 g1"| 34868<sub>5</sub> ||class="entry q3 g1"| 33507<sub>7</sub> ||class="entry q3 g1"| 35021<sub>7</sub> ||class="entry q2 g1"| 58754<sub>7</sub> ||class="entry q3 g1"| 60995<sub>9</sub> ||class="entry q3 g1"| 36077<sub>9</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 7, 7, 9, 9) ||class="c"| (35120, 45099, 47873, 58502, 56459, 48929) ||class="entry q2 g1"| 35120<sub>5</sub> ||class="entry q3 g1"| 45099<sub>7</sub> ||class="entry q3 g1"| 47873<sub>7</sub> ||class="entry q2 g1"| 58502<sub>7</sub> ||class="entry q3 g1"| 56459<sub>9</sub> ||class="entry q3 g1"| 48929<sub>9</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 7, 7, 9, 9) ||class="c"| (41990, 40201, 43531, 51632, 61865, 44587) ||class="entry q2 g1"| 41990<sub>5</sub> ||class="entry q3 g1"| 40201<sub>7</sub> ||class="entry q3 g1"| 43531<sub>7</sub> ||class="entry q2 g1"| 51632<sub>7</sub> ||class="entry q3 g1"| 61865<sub>9</sub> ||class="entry q3 g1"| 44587<sub>9</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 7, 7, 9, 9) ||class="c"| (42004, 51009, 61521, 51618, 44001, 62577) ||class="entry q2 g1"| 42004<sub>5</sub> ||class="entry q3 g1"| 51009<sub>7</sub> ||class="entry q3 g1"| 61521<sub>7</sub> ||class="entry q2 g1"| 51618<sub>7</sub> ||class="entry q3 g1"| 44001<sub>9</sub> ||class="entry q3 g1"| 62577<sub>9</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 7, 7, 9, 9) ||class="c"| (49190, 37049, 49357, 44432, 64537, 50413) ||class="entry q2 g1"| 49190<sub>5</sub> ||class="entry q3 g1"| 37049<sub>7</sub> ||class="entry q3 g1"| 49357<sub>7</sub> ||class="entry q2 g1"| 44432<sub>7</sub> ||class="entry q3 g1"| 64537<sub>9</sub> ||class="entry q3 g1"| 50413<sub>9</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 7, 7, 9, 9) ||class="c"| (49442, 41585, 62209, 44180, 52945, 63265) ||class="entry q2 g1"| 49442<sub>5</sub> ||class="entry q3 g1"| 41585<sub>7</sub> ||class="entry q3 g1"| 62209<sub>7</sub> ||class="entry q2 g1"| 44180<sub>7</sub> ||class="entry q3 g1"| 52945<sub>9</sub> ||class="entry q3 g1"| 63265<sub>9</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 11, 7, 5, 9) ||class="c"| (34180, 59429, 60653, 59442, 33925, 59597) ||class="entry q2 g1"| 34180<sub>5</sub> ||class="entry q3 g1"| 59429<sub>7</sub> ||class="entry q3 g1"| 60653<sub>11</sub> ||class="entry q2 g1"| 59442<sub>7</sub> ||class="entry q3 g1"| 33925<sub>5</sub> ||class="entry q3 g1"| 59597<sub>9</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 11, 7, 5, 9) ||class="c"| (34866, 58499, 61099, 58756, 34851, 60043) ||class="entry q2 g1"| 34866<sub>5</sub> ||class="entry q3 g1"| 58499<sub>7</sub> ||class="entry q3 g1"| 61099<sub>11</sub> ||class="entry q2 g1"| 58756<sub>7</sub> ||class="entry q3 g1"| 34851<sub>5</sub> ||class="entry q3 g1"| 60043<sub>9</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 11, 7, 5, 9) ||class="c"| (42244, 51617, 65441, 51378, 42241, 64385) ||class="entry q2 g1"| 42244<sub>5</sub> ||class="entry q3 g1"| 51617<sub>7</sub> ||class="entry q3 g1"| 65441<sub>11</sub> ||class="entry q2 g1"| 51378<sub>7</sub> ||class="entry q3 g1"| 42241<sub>5</sub> ||class="entry q3 g1"| 64385<sub>9</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 11, 7, 5, 9) ||class="c"| (49202, 44177, 64753, 44420, 49201, 63697) ||class="entry q2 g1"| 49202<sub>5</sub> ||class="entry q3 g1"| 44177<sub>7</sub> ||class="entry q3 g1"| 64753<sub>11</sub> ||class="entry q2 g1"| 44420<sub>7</sub> ||class="entry q3 g1"| 49201<sub>5</sub> ||class="entry q3 g1"| 63697<sub>9</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 7, 11, 7, 9) ||class="c"| (33504, 56461, 55617, 61270, 45101, 56673) ||class="entry q2 g1"| 33504<sub>5</sub> ||class="entry q3 g1"| 56461<sub>9</sub> ||class="entry q3 g1"| 55617<sub>7</sub> ||class="entry q2 g1"| 61270<sub>11</sub> ||class="entry q3 g1"| 45101<sub>7</sub> ||class="entry q3 g1"| 56673<sub>9</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 7, 11, 7, 9) ||class="c"| (36416, 47275, 45713, 58358, 54283, 46769) ||class="entry q2 g1"| 36416<sub>5</sub> ||class="entry q3 g1"| 47275<sub>9</sub> ||class="entry q3 g1"| 45713<sub>7</sub> ||class="entry q2 g1"| 58358<sub>11</sub> ||class="entry q3 g1"| 54283<sub>7</sub> ||class="entry q3 g1"| 46769<sub>9</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 7, 11, 7, 9) ||class="c"| (37032, 52933, 51977, 64798, 41573, 53033) ||class="entry q2 g1"| 37032<sub>5</sub> ||class="entry q3 g1"| 52933<sub>9</sub> ||class="entry q3 g1"| 51977<sub>7</sub> ||class="entry q2 g1"| 64798<sub>11</sub> ||class="entry q3 g1"| 41573<sub>7</sub> ||class="entry q3 g1"| 53033<sub>9</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 7, 11, 7, 9) ||class="c"| (39944, 43747, 41177, 61886, 50755, 42233) ||class="entry q2 g1"| 39944<sub>5</sub> ||class="entry q3 g1"| 43747<sub>9</sub> ||class="entry q3 g1"| 41177<sub>7</sub> ||class="entry q2 g1"| 61886<sub>11</sub> ||class="entry q3 g1"| 50755<sub>7</sub> ||class="entry q3 g1"| 42233<sub>9</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 7, 11, 7, 9) ||class="c"| (41568, 64777, 51725, 53206, 37289, 52781) ||class="entry q2 g1"| 41568<sub>5</sub> ||class="entry q3 g1"| 64777<sub>9</sub> ||class="entry q3 g1"| 51725<sub>7</sub> ||class="entry q2 g1"| 53206<sub>11</sub> ||class="entry q3 g1"| 37289<sub>7</sub> ||class="entry q3 g1"| 52781<sub>9</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 7, 11, 7, 9) ||class="c"| (45096, 61249, 55365, 56734, 33761, 56421) ||class="entry q2 g1"| 45096<sub>5</sub> ||class="entry q3 g1"| 61249<sub>9</sub> ||class="entry q3 g1"| 55365<sub>7</sub> ||class="entry q2 g1"| 56734<sub>11</sub> ||class="entry q3 g1"| 33761<sub>7</sub> ||class="entry q3 g1"| 56421<sub>9</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 7, 11, 7, 9) ||class="c"| (50752, 61625, 41163, 44022, 39961, 42219) ||class="entry q2 g1"| 50752<sub>5</sub> ||class="entry q3 g1"| 61625<sub>9</sub> ||class="entry q3 g1"| 41163<sub>7</sub> ||class="entry q2 g1"| 44022<sub>11</sub> ||class="entry q3 g1"| 39961<sub>7</sub> ||class="entry q3 g1"| 42219<sub>9</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 7, 11, 7, 9) ||class="c"| (54280, 58097, 45699, 47550, 36433, 46755) ||class="entry q2 g1"| 54280<sub>5</sub> ||class="entry q3 g1"| 58097<sub>9</sub> ||class="entry q3 g1"| 45699<sub>7</sub> ||class="entry q2 g1"| 47550<sub>11</sub> ||class="entry q3 g1"| 36433<sub>7</sub> ||class="entry q3 g1"| 46755<sub>9</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 9, 7, 7, 11) ||class="c"| (33926, 48269, 47431, 59696, 53293, 48487) ||class="entry q2 g1"| 33926<sub>5</sub> ||class="entry q3 g1"| 48269<sub>9</sub> ||class="entry q3 g1"| 47431<sub>9</sub> ||class="entry q2 g1"| 59696<sub>7</sub> ||class="entry q3 g1"| 53293<sub>7</sub> ||class="entry q3 g1"| 48487<sub>11</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 9, 7, 7, 11) ||class="c"| (33940, 59077, 58141, 59682, 35429, 59197) ||class="entry q2 g1"| 33940<sub>5</sub> ||class="entry q3 g1"| 59077<sub>9</sub> ||class="entry q3 g1"| 58141<sub>9</sub> ||class="entry q2 g1"| 59682<sub>7</sub> ||class="entry q3 g1"| 35429<sub>7</sub> ||class="entry q3 g1"| 59197<sub>11</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 9, 7, 7, 11) ||class="c"| (34854, 55467, 53911, 58768, 46091, 54967) ||class="entry q2 g1"| 34854<sub>5</sub> ||class="entry q3 g1"| 55467<sub>9</sub> ||class="entry q3 g1"| 53911<sub>9</sub> ||class="entry q2 g1"| 58768<sub>7</sub> ||class="entry q3 g1"| 46091<sub>7</sub> ||class="entry q3 g1"| 54967<sub>11</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 9, 7, 7, 11) ||class="c"| (35106, 60003, 57691, 58516, 34499, 58747) ||class="entry q2 g1"| 35106<sub>5</sub> ||class="entry q3 g1"| 60003<sub>9</sub> ||class="entry q3 g1"| 57691<sub>9</sub> ||class="entry q2 g1"| 58516<sub>7</sub> ||class="entry q3 g1"| 34499<sub>7</sub> ||class="entry q3 g1"| 58747<sub>11</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 9, 7, 7, 11) ||class="c"| (42242, 44993, 39367, 51380, 50017, 40423) ||class="entry q2 g1"| 42242<sub>5</sub> ||class="entry q3 g1"| 44993<sub>9</sub> ||class="entry q3 g1"| 39367<sub>9</sub> ||class="entry q2 g1"| 51380<sub>7</sub> ||class="entry q3 g1"| 50017<sub>7</sub> ||class="entry q3 g1"| 40423<sub>11</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 9, 7, 7, 11) ||class="c"| (42256, 62857, 50077, 51366, 39209, 51133) ||class="entry q2 g1"| 42256<sub>5</sub> ||class="entry q3 g1"| 62857<sub>9</sub> ||class="entry q3 g1"| 50077<sub>9</sub> ||class="entry q2 g1"| 51366<sub>7</sub> ||class="entry q3 g1"| 39209<sub>7</sub> ||class="entry q3 g1"| 51133<sub>11</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 9, 7, 7, 11) ||class="c"| (49204, 51953, 39575, 44418, 42577, 40631) ||class="entry q2 g1"| 49204<sub>5</sub> ||class="entry q3 g1"| 51953<sub>9</sub> ||class="entry q3 g1"| 39575<sub>9</sub> ||class="entry q2 g1"| 44418<sub>7</sub> ||class="entry q3 g1"| 42577<sub>7</sub> ||class="entry q3 g1"| 40631<sub>11</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 9, 7, 7, 11) ||class="c"| (49456, 63545, 43355, 44166, 38041, 44411) ||class="entry q2 g1"| 49456<sub>5</sub> ||class="entry q3 g1"| 63545<sub>9</sub> ||class="entry q3 g1"| 43355<sub>9</sub> ||class="entry q2 g1"| 44166<sub>7</sub> ||class="entry q3 g1"| 38041<sub>7</sub> ||class="entry q3 g1"| 44411<sub>11</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 9, 11, 11, 7) ||class="c"| (34496, 45371, 46561, 60278, 56731, 45505) ||class="entry q2 g1"| 34496<sub>5</sub> ||class="entry q3 g1"| 45371<sub>9</sub> ||class="entry q3 g1"| 46561<sub>9</sub> ||class="entry q2 g1"| 60278<sub>11</sub> ||class="entry q3 g1"| 56731<sub>11</sub> ||class="entry q3 g1"| 45505<sub>7</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 9, 11, 11, 7) ||class="c"| (35424, 54557, 56881, 59350, 47549, 55825) ||class="entry q2 g1"| 35424<sub>5</sub> ||class="entry q3 g1"| 54557<sub>9</sub> ||class="entry q3 g1"| 56881<sub>9</sub> ||class="entry q2 g1"| 59350<sub>11</sub> ||class="entry q3 g1"| 47549<sub>11</sub> ||class="entry q3 g1"| 55825<sub>7</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 9, 11, 11, 7) ||class="c"| (38024, 41843, 42921, 63806, 53203, 41865) ||class="entry q2 g1"| 38024<sub>5</sub> ||class="entry q3 g1"| 41843<sub>9</sub> ||class="entry q3 g1"| 42921<sub>9</sub> ||class="entry q2 g1"| 63806<sub>11</sub> ||class="entry q3 g1"| 53203<sub>11</sub> ||class="entry q3 g1"| 41865<sub>7</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 9, 11, 11, 7) ||class="c"| (38952, 51029, 52345, 62878, 44021, 51289) ||class="entry q2 g1"| 38952<sub>5</sub> ||class="entry q3 g1"| 51029<sub>9</sub> ||class="entry q3 g1"| 52345<sub>9</sub> ||class="entry q2 g1"| 62878<sub>11</sub> ||class="entry q3 g1"| 44021<sub>11</sub> ||class="entry q3 g1"| 51289<sub>7</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 9, 11, 11, 7) ||class="c"| (42560, 37055, 42669, 52214, 64543, 41613) ||class="entry q2 g1"| 42560<sub>5</sub> ||class="entry q3 g1"| 37055<sub>9</sub> ||class="entry q3 g1"| 42669<sub>9</sub> ||class="entry q2 g1"| 52214<sub>11</sub> ||class="entry q3 g1"| 64543<sub>11</sub> ||class="entry q3 g1"| 41613<sub>7</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 9, 11, 11, 7) ||class="c"| (46088, 33527, 46309, 55742, 61015, 45253) ||class="entry q2 g1"| 46088<sub>5</sub> ||class="entry q3 g1"| 33527<sub>9</sub> ||class="entry q3 g1"| 46309<sub>9</sub> ||class="entry q2 g1"| 55742<sub>11</sub> ||class="entry q3 g1"| 61015<sub>11</sub> ||class="entry q3 g1"| 45253<sub>7</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 9, 11, 11, 7) ||class="c"| (49760, 40207, 52331, 45014, 61871, 51275) ||class="entry q2 g1"| 49760<sub>5</sub> ||class="entry q3 g1"| 40207<sub>9</sub> ||class="entry q3 g1"| 52331<sub>9</sub> ||class="entry q2 g1"| 45014<sub>11</sub> ||class="entry q3 g1"| 61871<sub>11</sub> ||class="entry q3 g1"| 51275<sub>7</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 9, 11, 11, 7) ||class="c"| (53288, 36679, 56867, 48542, 58343, 55811) ||class="entry q2 g1"| 53288<sub>5</sub> ||class="entry q3 g1"| 36679<sub>9</sub> ||class="entry q3 g1"| 56867<sub>9</sub> ||class="entry q2 g1"| 48542<sub>11</sub> ||class="entry q3 g1"| 58343<sub>11</sub> ||class="entry q3 g1"| 55811<sub>7</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 11, 7, 7, 13) ||class="c"| (32946, 60691, 59867, 60676, 33203, 60923) ||class="entry q2 g1"| 32946<sub>5</sub> ||class="entry q3 g1"| 60691<sub>9</sub> ||class="entry q3 g1"| 59867<sub>11</sub> ||class="entry q2 g1"| 60676<sub>7</sub> ||class="entry q3 g1"| 33203<sub>7</sub> ||class="entry q3 g1"| 60923<sub>13</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 11, 7, 7, 13) ||class="c"| (36100, 57781, 60317, 57522, 36117, 61373) ||class="entry q2 g1"| 36100<sub>5</sub> ||class="entry q3 g1"| 57781<sub>9</sub> ||class="entry q3 g1"| 60317<sub>11</sub> ||class="entry q2 g1"| 57522<sub>7</sub> ||class="entry q3 g1"| 36117<sub>7</sub> ||class="entry q3 g1"| 61373<sub>13</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 11, 7, 7, 13) ||class="c"| (41010, 52375, 64151, 52612, 41015, 65207) ||class="entry q2 g1"| 41010<sub>5</sub> ||class="entry q3 g1"| 52375<sub>9</sub> ||class="entry q3 g1"| 64151<sub>11</sub> ||class="entry q2 g1"| 52612<sub>7</sub> ||class="entry q3 g1"| 41015<sub>7</sub> ||class="entry q3 g1"| 65207<sub>13</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 11, 7, 7, 13) ||class="c"| (50436, 43431, 63943, 43186, 50439, 64999) ||class="entry q2 g1"| 50436<sub>5</sub> ||class="entry q3 g1"| 43431<sub>9</sub> ||class="entry q3 g1"| 63943<sub>11</sub> ||class="entry q2 g1"| 43186<sub>7</sub> ||class="entry q3 g1"| 50439<sub>7</sub> ||class="entry q3 g1"| 64999<sub>13</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 11, 7, 11, 9) ||class="c"| (32934, 53563, 54759, 60688, 48539, 53703) ||class="entry q2 g1"| 32934<sub>5</sub> ||class="entry q3 g1"| 53563<sub>9</sub> ||class="entry q3 g1"| 54759<sub>11</sub> ||class="entry q2 g1"| 60688<sub>7</sub> ||class="entry q3 g1"| 48539<sub>11</sub> ||class="entry q3 g1"| 53703<sub>9</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 11, 7, 11, 9) ||class="c"| (32948, 35699, 36797, 60674, 59347, 35741) ||class="entry q2 g1"| 32948<sub>5</sub> ||class="entry q3 g1"| 35699<sub>9</sub> ||class="entry q3 g1"| 36797<sub>11</sub> ||class="entry q2 g1"| 60674<sub>7</sub> ||class="entry q3 g1"| 59347<sub>11</sub> ||class="entry q3 g1"| 35741<sub>9</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 11, 7, 11, 9) ||class="c"| (35846, 46365, 48695, 57776, 55741, 47639) ||class="entry q2 g1"| 35846<sub>5</sub> ||class="entry q3 g1"| 46365<sub>9</sub> ||class="entry q3 g1"| 48695<sub>11</sub> ||class="entry q2 g1"| 57776<sub>7</sub> ||class="entry q3 g1"| 55741<sub>11</sub> ||class="entry q3 g1"| 47639<sub>9</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 11, 7, 11, 9) ||class="c"| (36098, 34773, 36347, 57524, 60277, 35291) ||class="entry q2 g1"| 36098<sub>5</sub> ||class="entry q3 g1"| 34773<sub>9</sub> ||class="entry q3 g1"| 36347<sub>11</sub> ||class="entry q2 g1"| 57524<sub>7</sub> ||class="entry q3 g1"| 60277<sub>11</sub> ||class="entry q3 g1"| 35291<sub>9</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 11, 7, 11, 9) ||class="c"| (41250, 49783, 62823, 52372, 44759, 61767) ||class="entry q2 g1"| 41250<sub>5</sub> ||class="entry q3 g1"| 49783<sub>9</sub> ||class="entry q3 g1"| 62823<sub>11</sub> ||class="entry q2 g1"| 52372<sub>7</sub> ||class="entry q3 g1"| 44759<sub>11</sub> ||class="entry q3 g1"| 61767<sub>9</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 11, 7, 11, 9) ||class="c"| (41264, 38975, 44861, 52358, 62623, 43805) ||class="entry q2 g1"| 41264<sub>5</sub> ||class="entry q3 g1"| 38975<sub>9</sub> ||class="entry q3 g1"| 44861<sub>11</sub> ||class="entry q2 g1"| 52358<sub>7</sub> ||class="entry q3 g1"| 62623<sub>11</sub> ||class="entry q3 g1"| 43805<sub>9</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 11, 7, 11, 9) ||class="c"| (50196, 42823, 63031, 43426, 52199, 61975) ||class="entry q2 g1"| 50196<sub>5</sub> ||class="entry q3 g1"| 42823<sub>9</sub> ||class="entry q3 g1"| 63031<sub>11</sub> ||class="entry q2 g1"| 43426<sub>7</sub> ||class="entry q3 g1"| 52199<sub>11</sub> ||class="entry q3 g1"| 61975<sub>9</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 11, 7, 11, 9) ||class="c"| (50448, 38287, 50683, 43174, 63791, 49627) ||class="entry q2 g1"| 50448<sub>5</sub> ||class="entry q3 g1"| 38287<sub>9</sub> ||class="entry q3 g1"| 50683<sub>11</sub> ||class="entry q2 g1"| 43174<sub>7</sub> ||class="entry q3 g1"| 63791<sub>11</sub> ||class="entry q3 g1"| 49627<sub>9</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 11, 9, 7, 9, 7) ||class="c"| (33186, 58355, 58923, 60436, 36691, 57867) ||class="entry q2 g1"| 33186<sub>5</sub> ||class="entry q3 g1"| 58355<sub>11</sub> ||class="entry q3 g1"| 58923<sub>9</sub> ||class="entry q2 g1"| 60436<sub>7</sub> ||class="entry q3 g1"| 36691<sub>9</sub> ||class="entry q3 g1"| 57867<sub>7</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 11, 9, 7, 9, 7) ||class="c"| (33200, 47547, 48241, 60422, 54555, 47185) ||class="entry q2 g1"| 33200<sub>5</sub> ||class="entry q3 g1"| 47547<sub>11</sub> ||class="entry q3 g1"| 48241<sub>9</sub> ||class="entry q2 g1"| 60422<sub>7</sub> ||class="entry q3 g1"| 54555<sub>9</sub> ||class="entry q3 g1"| 47185<sub>7</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 11, 9, 7, 9, 7) ||class="c"| (35860, 61269, 58477, 57762, 33781, 57421) ||class="entry q2 g1"| 35860<sub>5</sub> ||class="entry q3 g1"| 61269<sub>11</sub> ||class="entry q3 g1"| 58477<sub>9</sub> ||class="entry q2 g1"| 57762<sub>7</sub> ||class="entry q3 g1"| 33781<sub>9</sub> ||class="entry q3 g1"| 57421<sub>7</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 11, 9, 7, 9, 7) ||class="c"| (36112, 56733, 55201, 57510, 45373, 54145) ||class="entry q2 g1"| 36112<sub>5</sub> ||class="entry q3 g1"| 56733<sub>11</sub> ||class="entry q3 g1"| 55201<sub>9</sub> ||class="entry q2 g1"| 57510<sub>7</sub> ||class="entry q3 g1"| 45373<sub>9</sub> ||class="entry q3 g1"| 54145<sub>7</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 11, 9, 7, 9, 7) ||class="c"| (40998, 61631, 50859, 52624, 39967, 49803) ||class="entry q2 g1"| 40998<sub>5</sub> ||class="entry q3 g1"| 61631<sub>11</sub> ||class="entry q3 g1"| 50859<sub>9</sub> ||class="entry q2 g1"| 52624<sub>7</sub> ||class="entry q3 g1"| 39967<sub>9</sub> ||class="entry q3 g1"| 49803<sub>7</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 11, 9, 7, 9, 7) ||class="c"| (41012, 43767, 40177, 52610, 50775, 39121) ||class="entry q2 g1"| 41012<sub>5</sub> ||class="entry q3 g1"| 43767<sub>11</sub> ||class="entry q3 g1"| 40177<sub>9</sub> ||class="entry q2 g1"| 52610<sub>7</sub> ||class="entry q3 g1"| 50775<sub>9</sub> ||class="entry q3 g1"| 39121<sub>7</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 11, 9, 7, 9, 7) ||class="c"| (50182, 64783, 44141, 43440, 37295, 43085) ||class="entry q2 g1"| 50182<sub>5</sub> ||class="entry q3 g1"| 64783<sub>11</sub> ||class="entry q3 g1"| 44141<sub>9</sub> ||class="entry q2 g1"| 43440<sub>7</sub> ||class="entry q3 g1"| 37295<sub>9</sub> ||class="entry q3 g1"| 43085<sub>7</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 11, 9, 7, 9, 7) ||class="c"| (50434, 53191, 40865, 43188, 41831, 39809) ||class="entry q2 g1"| 50434<sub>5</sub> ||class="entry q3 g1"| 53191<sub>11</sub> ||class="entry q3 g1"| 40865<sub>9</sub> ||class="entry q2 g1"| 43188<sub>7</sub> ||class="entry q3 g1"| 41831<sub>9</sub> ||class="entry q3 g1"| 39809<sub>7</sub>
|-
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|-
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|-
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|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 11, 9, 7, 9, 11) ||class="c"| (59424, 57037, 45719, 34198, 45677, 46775) ||class="entry q2 g1"| 59424<sub>5</sub> ||class="entry q3 g1"| 57037<sub>11</sub> ||class="entry q3 g1"| 45719<sub>9</sub> ||class="entry q2 g1"| 34198<sub>7</sub> ||class="entry q3 g1"| 45677<sub>9</sub> ||class="entry q3 g1"| 46775<sub>11</sub>
|-
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|-
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|-
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|-
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|-
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|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 5, 9, 9, 7, 7) ||class="c"| (36436, 33923, 36525, 58338, 59427, 35469) ||class="entry q2 g1"| 36436<sub>7</sub> ||class="entry q3 g1"| 33923<sub>5</sub> ||class="entry q3 g1"| 36525<sub>9</sub> ||class="entry q2 g1"| 58338<sub>9</sub> ||class="entry q3 g1"| 59427<sub>7</sub> ||class="entry q3 g1"| 35469<sub>7</sub>
|-
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|-
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|-
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|-
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|-
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|-
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|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 7, 9, 9, 9) ||class="c"| (33522, 34501, 33563, 61252, 60005, 34619) ||class="entry q2 g1"| 33522<sub>7</sub> ||class="entry q3 g1"| 34501<sub>7</sub> ||class="entry q3 g1"| 33563<sub>7</sub> ||class="entry q2 g1"| 61252<sub>9</sub> ||class="entry q3 g1"| 60005<sub>9</sub> ||class="entry q3 g1"| 34619<sub>9</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 7, 9, 9, 9) ||class="c"| (36676, 35427, 33117, 58098, 59075, 34173) ||class="entry q2 g1"| 36676<sub>7</sub> ||class="entry q3 g1"| 35427<sub>7</sub> ||class="entry q3 g1"| 33117<sub>7</sub> ||class="entry q2 g1"| 58098<sub>9</sub> ||class="entry q3 g1"| 59075<sub>9</sub> ||class="entry q3 g1"| 34173<sub>9</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 7, 9, 9, 9) ||class="c"| (37050, 38029, 37203, 64780, 63533, 38259) ||class="entry q2 g1"| 37050<sub>7</sub> ||class="entry q3 g1"| 38029<sub>7</sub> ||class="entry q3 g1"| 37203<sub>7</sub> ||class="entry q2 g1"| 64780<sub>9</sub> ||class="entry q3 g1"| 63533<sub>9</sub> ||class="entry q3 g1"| 38259<sub>9</sub>
|-
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|-
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|-
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|-
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|-
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|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 7, 13, 9, 9) ||class="c"| (37608, 42259, 41417, 65374, 51635, 42473) ||class="entry q2 g1"| 37608<sub>7</sub> ||class="entry q3 g1"| 42259<sub>7</sub> ||class="entry q3 g1"| 41417<sub>7</sub> ||class="entry q2 g1"| 65374<sub>13</sub> ||class="entry q3 g1"| 51635<sub>9</sub> ||class="entry q3 g1"| 42473<sub>9</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 7, 13, 9, 9) ||class="c"| (40520, 49461, 51737, 62462, 44437, 52793) ||class="entry q2 g1"| 40520<sub>7</sub> ||class="entry q3 g1"| 49461<sub>7</sub> ||class="entry q3 g1"| 51737<sub>7</sub> ||class="entry q2 g1"| 62462<sub>13</sub> ||class="entry q3 g1"| 44437<sub>9</sub> ||class="entry q3 g1"| 52793<sub>9</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 7, 13, 9, 9) ||class="c"| (45672, 33943, 45701, 57310, 59447, 46757) ||class="entry q2 g1"| 45672<sub>7</sub> ||class="entry q3 g1"| 33943<sub>7</sub> ||class="entry q3 g1"| 45701<sub>7</sub> ||class="entry q2 g1"| 57310<sub>13</sub> ||class="entry q3 g1"| 59447<sub>9</sub> ||class="entry q3 g1"| 46757<sub>9</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 7, 13, 9, 9) ||class="c"| (54856, 35111, 55363, 48126, 58759, 56419) ||class="entry q2 g1"| 54856<sub>7</sub> ||class="entry q3 g1"| 35111<sub>7</sub> ||class="entry q3 g1"| 55363<sub>7</sub> ||class="entry q2 g1"| 48126<sub>13</sub> ||class="entry q3 g1"| 58759<sub>9</sub> ||class="entry q3 g1"| 56419<sub>9</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 9, 9, 9, 11) ||class="c"| (34516, 36115, 35293, 60258, 57779, 36349) ||class="entry q2 g1"| 34516<sub>7</sub> ||class="entry q3 g1"| 36115<sub>7</sub> ||class="entry q3 g1"| 35293<sub>9</sub> ||class="entry q2 g1"| 60258<sub>9</sub> ||class="entry q3 g1"| 57779<sub>9</sub> ||class="entry q3 g1"| 36349<sub>11</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 9, 9, 9, 11) ||class="c"| (35682, 33205, 35739, 59092, 60693, 36795) ||class="entry q2 g1"| 35682<sub>7</sub> ||class="entry q3 g1"| 33205<sub>7</sub> ||class="entry q3 g1"| 35739<sub>9</sub> ||class="entry q2 g1"| 59092<sub>9</sub> ||class="entry q3 g1"| 60693<sub>9</sub> ||class="entry q3 g1"| 36795<sub>11</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 9, 9, 9, 11) ||class="c"| (38030, 50451, 49615, 63800, 43443, 50671) ||class="entry q2 g1"| 38030<sub>7</sub> ||class="entry q3 g1"| 50451<sub>7</sub> ||class="entry q3 g1"| 49615<sub>9</sub> ||class="entry q2 g1"| 63800<sub>9</sub> ||class="entry q3 g1"| 43443<sub>9</sub> ||class="entry q3 g1"| 50671<sub>11</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 9, 9, 9, 11) ||class="c"| (38958, 41269, 43551, 62872, 52629, 44607) ||class="entry q2 g1"| 38958<sub>7</sub> ||class="entry q3 g1"| 41269<sub>7</sub> ||class="entry q3 g1"| 43551<sub>9</sub> ||class="entry q2 g1"| 62872<sub>9</sub> ||class="entry q3 g1"| 52629<sub>9</sub> ||class="entry q3 g1"| 44607<sub>11</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 9, 9, 9, 11) ||class="c"| (42818, 50199, 62215, 51956, 43191, 63271) ||class="entry q2 g1"| 42818<sub>7</sub> ||class="entry q3 g1"| 50199<sub>7</sub> ||class="entry q3 g1"| 62215<sub>9</sub> ||class="entry q2 g1"| 51956<sub>9</sub> ||class="entry q3 g1"| 43191<sub>9</sub> ||class="entry q3 g1"| 63271<sub>11</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 9, 9, 9, 11) ||class="c"| (46360, 35863, 47893, 55470, 57527, 48949) ||class="entry q2 g1"| 46360<sub>7</sub> ||class="entry q3 g1"| 35863<sub>7</sub> ||class="entry q3 g1"| 47893<sub>9</sub> ||class="entry q2 g1"| 55470<sub>9</sub> ||class="entry q3 g1"| 57527<sub>9</sub> ||class="entry q3 g1"| 48949<sub>11</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 9, 9, 9, 11) ||class="c"| (49780, 41255, 61527, 44994, 52615, 62583) ||class="entry q2 g1"| 49780<sub>7</sub> ||class="entry q3 g1"| 41255<sub>7</sub> ||class="entry q3 g1"| 61527<sub>9</sub> ||class="entry q2 g1"| 44994<sub>9</sub> ||class="entry q3 g1"| 52615<sub>9</sub> ||class="entry q3 g1"| 62583<sub>11</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 9, 9, 9, 11) ||class="c"| (53560, 33191, 53715, 48270, 60679, 54771) ||class="entry q2 g1"| 53560<sub>7</sub> ||class="entry q3 g1"| 33191<sub>7</sub> ||class="entry q3 g1"| 53715<sub>9</sub> ||class="entry q2 g1"| 48270<sub>9</sub> ||class="entry q3 g1"| 60679<sub>9</sub> ||class="entry q3 g1"| 54771<sub>11</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 9, 13, 5, 7) ||class="c"| (38600, 51365, 52585, 64382, 41989, 51529) ||class="entry q2 g1"| 38600<sub>7</sub> ||class="entry q3 g1"| 51365<sub>7</sub> ||class="entry q3 g1"| 52585<sub>9</sub> ||class="entry q2 g1"| 64382<sub>13</sub> ||class="entry q3 g1"| 41989<sub>5</sub> ||class="entry q3 g1"| 51529<sub>7</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 9, 13, 5, 7) ||class="c"| (39528, 44163, 42681, 63454, 49187, 41625) ||class="entry q2 g1"| 39528<sub>7</sub> ||class="entry q3 g1"| 44163<sub>7</sub> ||class="entry q3 g1"| 42681<sub>9</sub> ||class="entry q2 g1"| 63454<sub>13</sub> ||class="entry q3 g1"| 49187<sub>5</sub> ||class="entry q3 g1"| 41625<sub>7</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 9, 13, 5, 7) ||class="c"| (46664, 59681, 56869, 56318, 34177, 55813) ||class="entry q2 g1"| 46664<sub>7</sub> ||class="entry q3 g1"| 59681<sub>7</sub> ||class="entry q3 g1"| 56869<sub>9</sub> ||class="entry q2 g1"| 56318<sub>13</sub> ||class="entry q3 g1"| 34177<sub>5</sub> ||class="entry q3 g1"| 55813<sub>7</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 9, 13, 5, 7) ||class="c"| (53864, 58513, 46307, 49118, 34865, 45251) ||class="entry q2 g1"| 53864<sub>7</sub> ||class="entry q3 g1"| 58513<sub>7</sub> ||class="entry q3 g1"| 46307<sub>9</sub> ||class="entry q2 g1"| 49118<sub>13</sub> ||class="entry q3 g1"| 34865<sub>5</sub> ||class="entry q3 g1"| 45251<sub>7</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 11, 9, 5, 9) ||class="c"| (33524, 57509, 58749, 61250, 35845, 57693) ||class="entry q2 g1"| 33524<sub>7</sub> ||class="entry q3 g1"| 57509<sub>7</sub> ||class="entry q3 g1"| 58749<sub>11</sub> ||class="entry q2 g1"| 61250<sub>9</sub> ||class="entry q3 g1"| 35845<sub>5</sub> ||class="entry q3 g1"| 57693<sub>9</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 11, 9, 5, 9) ||class="c"| (36674, 60419, 59195, 58100, 32931, 58139) ||class="entry q2 g1"| 36674<sub>7</sub> ||class="entry q3 g1"| 60419<sub>7</sub> ||class="entry q3 g1"| 59195<sub>11</sub> ||class="entry q2 g1"| 58100<sub>9</sub> ||class="entry q3 g1"| 32931<sub>5</sub> ||class="entry q3 g1"| 58139<sub>9</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 11, 9, 5, 9) ||class="c"| (37038, 43173, 44399, 64792, 50181, 43343) ||class="entry q2 g1"| 37038<sub>7</sub> ||class="entry q3 g1"| 43173<sub>7</sub> ||class="entry q3 g1"| 44399<sub>11</sub> ||class="entry q2 g1"| 64792<sub>9</sub> ||class="entry q3 g1"| 50181<sub>5</sub> ||class="entry q3 g1"| 43343<sub>9</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 11, 9, 5, 9) ||class="c"| (39950, 52355, 50879, 61880, 40995, 49823) ||class="entry q2 g1"| 39950<sub>7</sub> ||class="entry q3 g1"| 52355<sub>7</sub> ||class="entry q3 g1"| 50879<sub>11</sub> ||class="entry q2 g1"| 61880<sub>9</sub> ||class="entry q3 g1"| 40995<sub>5</sub> ||class="entry q3 g1"| 49823<sub>9</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 11, 9, 5, 9) ||class="c"| (41826, 43425, 40871, 52948, 50433, 39815) ||class="entry q2 g1"| 41826<sub>7</sub> ||class="entry q3 g1"| 43425<sub>7</sub> ||class="entry q3 g1"| 40871<sub>11</sub> ||class="entry q2 g1"| 52948<sub>9</sub> ||class="entry q3 g1"| 50433<sub>5</sub> ||class="entry q3 g1"| 39815<sub>9</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 11, 9, 5, 9) ||class="c"| (45368, 57761, 55221, 56462, 36097, 54165) ||class="entry q2 g1"| 45368<sub>7</sub> ||class="entry q3 g1"| 57761<sub>7</sub> ||class="entry q3 g1"| 55221<sub>11</sub> ||class="entry q2 g1"| 56462<sub>9</sub> ||class="entry q3 g1"| 36097<sub>5</sub> ||class="entry q3 g1"| 54165<sub>9</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 11, 9, 5, 9) ||class="c"| (50772, 52369, 40183, 44002, 41009, 39127) ||class="entry q2 g1"| 50772<sub>7</sub> ||class="entry q3 g1"| 52369<sub>7</sub> ||class="entry q3 g1"| 40183<sub>11</sub> ||class="entry q2 g1"| 44002<sub>9</sub> ||class="entry q3 g1"| 41009<sub>5</sub> ||class="entry q3 g1"| 39127<sub>9</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 11, 9, 5, 9) ||class="c"| (54552, 60433, 48499, 47278, 32945, 47443) ||class="entry q2 g1"| 54552<sub>7</sub> ||class="entry q3 g1"| 60433<sub>7</sub> ||class="entry q3 g1"| 48499<sub>11</sub> ||class="entry q2 g1"| 47278<sub>9</sub> ||class="entry q3 g1"| 32945<sub>5</sub> ||class="entry q3 g1"| 47443<sub>9</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 7, 9) ||class="c"| (34754, 58771, 57419, 60020, 35123, 58475) ||class="entry q2 g1"| 34754<sub>7</sub> ||class="entry q3 g1"| 58771<sub>9</sub> ||class="entry q3 g1"| 57419<sub>7</sub> ||class="entry q2 g1"| 60020<sub>9</sub> ||class="entry q3 g1"| 35123<sub>7</sub> ||class="entry q3 g1"| 58475<sub>9</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 7, 9) ||class="c"| (35444, 59701, 57869, 59330, 34197, 58925) ||class="entry q2 g1"| 35444<sub>7</sub> ||class="entry q3 g1"| 59701<sub>9</sub> ||class="entry q3 g1"| 57869<sub>7</sub> ||class="entry q2 g1"| 59330<sub>9</sub> ||class="entry q3 g1"| 34197<sub>7</sub> ||class="entry q3 g1"| 58925<sub>9</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 7, 9) ||class="c"| (38296, 44435, 43097, 63534, 49459, 44153) ||class="entry q2 g1"| 38296<sub>7</sub> ||class="entry q3 g1"| 44435<sub>9</sub> ||class="entry q3 g1"| 43097<sub>7</sub> ||class="entry q2 g1"| 63534<sub>9</sub> ||class="entry q3 g1"| 49459<sub>7</sub> ||class="entry q3 g1"| 44153<sub>9</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 7, 9) ||class="c"| (39224, 51637, 50057, 62606, 42261, 51113) ||class="entry q2 g1"| 39224<sub>7</sub> ||class="entry q3 g1"| 51637<sub>9</sub> ||class="entry q3 g1"| 50057<sub>7</sub> ||class="entry q2 g1"| 62606<sub>9</sub> ||class="entry q3 g1"| 42261<sub>7</sub> ||class="entry q3 g1"| 51113<sub>9</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 7, 9) ||class="c"| (42580, 44183, 39569, 52194, 49207, 40625) ||class="entry q2 g1"| 42580<sub>7</sub> ||class="entry q3 g1"| 44183<sub>9</sub> ||class="entry q3 g1"| 39569<sub>7</sub> ||class="entry q2 g1"| 52194<sub>9</sub> ||class="entry q3 g1"| 49207<sub>7</sub> ||class="entry q3 g1"| 40625<sub>9</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 7, 9) ||class="c"| (46094, 58519, 53891, 55736, 34871, 54947) ||class="entry q2 g1"| 46094<sub>7</sub> ||class="entry q3 g1"| 58519<sub>9</sub> ||class="entry q3 g1"| 53891<sub>7</sub> ||class="entry q2 g1"| 55736<sub>9</sub> ||class="entry q3 g1"| 34871<sub>7</sub> ||class="entry q3 g1"| 54947<sub>9</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 7, 9) ||class="c"| (50018, 51623, 39361, 44756, 42247, 40417) ||class="entry q2 g1"| 50018<sub>7</sub> ||class="entry q3 g1"| 51623<sub>9</sub> ||class="entry q3 g1"| 39361<sub>7</sub> ||class="entry q2 g1"| 44756<sub>9</sub> ||class="entry q3 g1"| 42247<sub>7</sub> ||class="entry q3 g1"| 40417<sub>9</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 7, 9) ||class="c"| (53294, 59687, 47173, 48536, 34183, 48229) ||class="entry q2 g1"| 53294<sub>7</sub> ||class="entry q3 g1"| 59687<sub>9</sub> ||class="entry q3 g1"| 47173<sub>7</sub> ||class="entry q2 g1"| 48536<sub>9</sub> ||class="entry q3 g1"| 34183<sub>7</sub> ||class="entry q3 g1"| 48229<sub>9</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 11, 5) ||class="c"| (34756, 33779, 34349, 60018, 61267, 33293) ||class="entry q2 g1"| 34756<sub>7</sub> ||class="entry q3 g1"| 33779<sub>9</sub> ||class="entry q3 g1"| 34349<sub>7</sub> ||class="entry q2 g1"| 60018<sub>9</sub> ||class="entry q3 g1"| 61267<sub>11</sub> ||class="entry q3 g1"| 33293<sub>5</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 11, 5) ||class="c"| (35442, 36693, 33899, 59332, 58357, 32843) ||class="entry q2 g1"| 35442<sub>7</sub> ||class="entry q3 g1"| 36693<sub>9</sub> ||class="entry q3 g1"| 33899<sub>7</sub> ||class="entry q2 g1"| 59332<sub>9</sub> ||class="entry q3 g1"| 58357<sub>11</sub> ||class="entry q3 g1"| 32843<sub>5</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 11, 5) ||class="c"| (38284, 37307, 37989, 63546, 64795, 36933) ||class="entry q2 g1"| 38284<sub>7</sub> ||class="entry q3 g1"| 37307<sub>9</sub> ||class="entry q3 g1"| 37989<sub>7</sub> ||class="entry q2 g1"| 63546<sub>9</sub> ||class="entry q3 g1"| 64795<sub>11</sub> ||class="entry q3 g1"| 36933<sub>5</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 11, 5) ||class="c"| (38970, 40221, 38435, 62860, 61885, 37379) ||class="entry q2 g1"| 38970<sub>7</sub> ||class="entry q3 g1"| 40221<sub>9</sub> ||class="entry q3 g1"| 38435<sub>7</sub> ||class="entry q2 g1"| 62860<sub>9</sub> ||class="entry q3 g1"| 61885<sub>11</sub> ||class="entry q3 g1"| 37379<sub>5</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 11, 5) ||class="c"| (42820, 41591, 38241, 51954, 52951, 37185) ||class="entry q2 g1"| 42820<sub>7</sub> ||class="entry q3 g1"| 41591<sub>9</sub> ||class="entry q3 g1"| 38241<sub>7</sub> ||class="entry q2 g1"| 51954<sub>9</sub> ||class="entry q3 g1"| 52951<sub>11</sub> ||class="entry q3 g1"| 37185<sub>5</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 11, 5) ||class="c"| (46348, 45119, 34601, 55482, 56479, 33545) ||class="entry q2 g1"| 46348<sub>7</sub> ||class="entry q3 g1"| 45119<sub>9</sub> ||class="entry q3 g1"| 34601<sub>7</sub> ||class="entry q2 g1"| 55482<sub>9</sub> ||class="entry q3 g1"| 56479<sub>11</sub> ||class="entry q3 g1"| 33545<sub>5</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 11, 5) ||class="c"| (49778, 51015, 38449, 44996, 44007, 37393) ||class="entry q2 g1"| 49778<sub>7</sub> ||class="entry q3 g1"| 51015<sub>9</sub> ||class="entry q3 g1"| 38449<sub>7</sub> ||class="entry q2 g1"| 44996<sub>9</sub> ||class="entry q3 g1"| 44007<sub>11</sub> ||class="entry q3 g1"| 37393<sub>5</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 11, 5) ||class="c"| (53306, 54543, 33913, 48524, 47535, 32857) ||class="entry q2 g1"| 53306<sub>7</sub> ||class="entry q3 g1"| 54543<sub>9</sub> ||class="entry q3 g1"| 33913<sub>7</sub> ||class="entry q2 g1"| 48524<sub>9</sub> ||class="entry q3 g1"| 47535<sub>11</sub> ||class="entry q3 g1"| 32857<sub>5</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 7, 11) ||class="c"| (33764, 60997, 60045, 61010, 33509, 61101) ||class="entry q2 g1"| 33764<sub>7</sub> ||class="entry q3 g1"| 60997<sub>9</sub> ||class="entry q3 g1"| 60045<sub>9</sub> ||class="entry q2 g1"| 61010<sub>9</sub> ||class="entry q3 g1"| 33509<sub>7</sub> ||class="entry q3 g1"| 61101<sub>11</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 7, 11) ||class="c"| (36434, 58083, 59595, 58340, 36419, 60651) ||class="entry q2 g1"| 36434<sub>7</sub> ||class="entry q3 g1"| 58083<sub>9</sub> ||class="entry q3 g1"| 59595<sub>9</sub> ||class="entry q2 g1"| 58340<sub>9</sub> ||class="entry q3 g1"| 36419<sub>7</sub> ||class="entry q3 g1"| 60651<sub>11</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 7, 11) ||class="c"| (37292, 64525, 63685, 64538, 37037, 64741) ||class="entry q2 g1"| 37292<sub>7</sub> ||class="entry q3 g1"| 64525<sub>9</sub> ||class="entry q3 g1"| 63685<sub>9</sub> ||class="entry q2 g1"| 64538<sub>9</sub> ||class="entry q3 g1"| 37037<sub>7</sub> ||class="entry q3 g1"| 64741<sub>11</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 7, 11) ||class="c"| (39962, 61611, 64131, 61868, 39947, 65187) ||class="entry q2 g1"| 39962<sub>7</sub> ||class="entry q3 g1"| 61611<sub>9</sub> ||class="entry q3 g1"| 64131<sub>9</sub> ||class="entry q2 g1"| 61868<sub>9</sub> ||class="entry q3 g1"| 39947<sub>7</sub> ||class="entry q3 g1"| 65187<sub>11</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 7, 11) ||class="c"| (41828, 53185, 63937, 52946, 41825, 64993) ||class="entry q2 g1"| 41828<sub>7</sub> ||class="entry q3 g1"| 53185<sub>9</sub> ||class="entry q3 g1"| 63937<sub>9</sub> ||class="entry q2 g1"| 52946<sub>9</sub> ||class="entry q3 g1"| 41825<sub>7</sub> ||class="entry q3 g1"| 64993<sub>11</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 7, 11) ||class="c"| (45356, 56713, 60297, 56474, 45353, 61353) ||class="entry q2 g1"| 45356<sub>7</sub> ||class="entry q3 g1"| 56713<sub>9</sub> ||class="entry q3 g1"| 60297<sub>9</sub> ||class="entry q2 g1"| 56474<sub>9</sub> ||class="entry q3 g1"| 45353<sub>7</sub> ||class="entry q3 g1"| 61353<sub>11</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 7, 11) ||class="c"| (50770, 43761, 64145, 44004, 50769, 65201) ||class="entry q2 g1"| 50770<sub>7</sub> ||class="entry q3 g1"| 43761<sub>9</sub> ||class="entry q3 g1"| 64145<sub>9</sub> ||class="entry q2 g1"| 44004<sub>9</sub> ||class="entry q3 g1"| 50769<sub>7</sub> ||class="entry q3 g1"| 65201<sub>11</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 7, 11) ||class="c"| (54298, 47289, 59609, 47532, 54297, 60665) ||class="entry q2 g1"| 54298<sub>7</sub> ||class="entry q3 g1"| 47289<sub>9</sub> ||class="entry q3 g1"| 59609<sub>9</sub> ||class="entry q2 g1"| 47532<sub>9</sub> ||class="entry q3 g1"| 54297<sub>7</sub> ||class="entry q3 g1"| 60665<sub>11</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 11, 7) ||class="c"| (33776, 53869, 54961, 60998, 48845, 53905) ||class="entry q2 g1"| 33776<sub>7</sub> ||class="entry q3 g1"| 53869<sub>9</sub> ||class="entry q3 g1"| 54961<sub>9</sub> ||class="entry q2 g1"| 60998<sub>9</sub> ||class="entry q3 g1"| 48845<sub>11</sub> ||class="entry q3 g1"| 53905<sub>7</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 11, 7) ||class="c"| (36688, 46667, 48481, 58086, 56043, 47425) ||class="entry q2 g1"| 36688<sub>7</sub> ||class="entry q3 g1"| 46667<sub>9</sub> ||class="entry q3 g1"| 48481<sub>9</sub> ||class="entry q2 g1"| 58086<sub>9</sub> ||class="entry q3 g1"| 56043<sub>11</sub> ||class="entry q3 g1"| 47425<sub>7</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 11, 7) ||class="c"| (37290, 39533, 40611, 64540, 63181, 39555) ||class="entry q2 g1"| 37290<sub>7</sub> ||class="entry q3 g1"| 39533<sub>9</sub> ||class="entry q3 g1"| 40611<sub>9</sub> ||class="entry q2 g1"| 64540<sub>9</sub> ||class="entry q3 g1"| 63181<sub>11</sub> ||class="entry q3 g1"| 39555<sub>7</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 11, 7) ||class="c"| (39964, 38603, 40165, 61866, 64107, 39109) ||class="entry q2 g1"| 39964<sub>7</sub> ||class="entry q3 g1"| 38603<sub>9</sub> ||class="entry q3 g1"| 40165<sub>9</sub> ||class="entry q2 g1"| 61866<sub>9</sub> ||class="entry q3 g1"| 64107<sub>11</sub> ||class="entry q3 g1"| 39109<sub>7</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 11, 7) ||class="c"| (41574, 39785, 44139, 53200, 63433, 43083) ||class="entry q2 g1"| 41574<sub>7</sub> ||class="entry q3 g1"| 39785<sub>9</sub> ||class="entry q3 g1"| 44139<sub>9</sub> ||class="entry q2 g1"| 53200<sub>9</sub> ||class="entry q3 g1"| 63433<sub>11</sub> ||class="entry q3 g1"| 43083<sub>7</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 11, 7) ||class="c"| (45116, 54121, 58489, 56714, 49097, 57433) ||class="entry q2 g1"| 45116<sub>7</sub> ||class="entry q3 g1"| 54121<sub>9</sub> ||class="entry q3 g1"| 58489<sub>9</sub> ||class="entry q2 g1"| 56714<sub>9</sub> ||class="entry q3 g1"| 49097<sub>11</sub> ||class="entry q3 g1"| 57433<sub>7</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 11, 7) ||class="c"| (50758, 38617, 50861, 44016, 64121, 49805) ||class="entry q2 g1"| 50758<sub>7</sub> ||class="entry q3 g1"| 38617<sub>9</sub> ||class="entry q3 g1"| 50861<sub>9</sub> ||class="entry q2 g1"| 44016<sub>9</sub> ||class="entry q3 g1"| 64121<sub>11</sub> ||class="entry q3 g1"| 49805<sub>7</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 11, 7) ||class="c"| (54538, 46681, 59177, 47292, 56057, 58121) ||class="entry q2 g1"| 54538<sub>7</sub> ||class="entry q3 g1"| 46681<sub>9</sub> ||class="entry q3 g1"| 59177<sub>9</sub> ||class="entry q2 g1"| 47292<sub>9</sub> ||class="entry q3 g1"| 56057<sub>11</sub> ||class="entry q3 g1"| 58121<sub>7</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 11, 11) ||class="c"| (44736, 39215, 41437, 50038, 62863, 42493) ||class="entry q2 g1"| 44736<sub>7</sub> ||class="entry q3 g1"| 39215<sub>9</sub> ||class="entry q3 g1"| 41437<sub>9</sub> ||class="entry q2 g1"| 50038<sub>9</sub> ||class="entry q3 g1"| 62863<sub>11</sub> ||class="entry q3 g1"| 42493<sub>11</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 11, 11) ||class="c"| (48264, 35687, 45973, 53566, 59335, 47029) ||class="entry q2 g1"| 48264<sub>7</sub> ||class="entry q3 g1"| 35687<sub>9</sub> ||class="entry q3 g1"| 45973<sub>9</sub> ||class="entry q2 g1"| 53566<sub>9</sub> ||class="entry q3 g1"| 59335<sub>11</sub> ||class="entry q3 g1"| 47029<sub>11</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 11, 11) ||class="c"| (51936, 38047, 51995, 42838, 63551, 53051) ||class="entry q2 g1"| 51936<sub>7</sub> ||class="entry q3 g1"| 38047<sub>9</sub> ||class="entry q3 g1"| 51995<sub>9</sub> ||class="entry q2 g1"| 42838<sub>9</sub> ||class="entry q3 g1"| 63551<sub>11</sub> ||class="entry q3 g1"| 53051<sub>11</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 11, 11) ||class="c"| (55464, 34519, 55635, 46366, 60023, 56691) ||class="entry q2 g1"| 55464<sub>7</sub> ||class="entry q3 g1"| 34519<sub>9</sub> ||class="entry q3 g1"| 55635<sub>9</sub> ||class="entry q2 g1"| 46366<sub>9</sub> ||class="entry q3 g1"| 60023<sub>11</sub> ||class="entry q3 g1"| 56691<sub>11</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 11, 11) ||class="c"| (59072, 53565, 45959, 35702, 48541, 47015) ||class="entry q2 g1"| 59072<sub>7</sub> ||class="entry q3 g1"| 53565<sub>9</sub> ||class="entry q3 g1"| 45959<sub>9</sub> ||class="entry q2 g1"| 35702<sub>9</sub> ||class="entry q3 g1"| 48541<sub>11</sub> ||class="entry q3 g1"| 47015<sub>11</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 11, 11) ||class="c"| (60000, 46363, 55383, 34774, 55739, 56439) ||class="entry q2 g1"| 60000<sub>7</sub> ||class="entry q3 g1"| 46363<sub>9</sub> ||class="entry q3 g1"| 55383<sub>9</sub> ||class="entry q2 g1"| 34774<sub>9</sub> ||class="entry q3 g1"| 55739<sub>11</sub> ||class="entry q3 g1"| 56439<sub>11</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 11, 11) ||class="c"| (62600, 50037, 41423, 39230, 45013, 42479) ||class="entry q2 g1"| 62600<sub>7</sub> ||class="entry q3 g1"| 50037<sub>9</sub> ||class="entry q3 g1"| 41423<sub>9</sub> ||class="entry q2 g1"| 39230<sub>9</sub> ||class="entry q3 g1"| 45013<sub>11</sub> ||class="entry q3 g1"| 42479<sub>11</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 11, 11) ||class="c"| (63528, 42835, 51743, 38302, 52211, 52799) ||class="entry q2 g1"| 63528<sub>7</sub> ||class="entry q3 g1"| 42835<sub>9</sub> ||class="entry q3 g1"| 51743<sub>9</sub> ||class="entry q2 g1"| 38302<sub>9</sub> ||class="entry q3 g1"| 52211<sub>11</sub> ||class="entry q3 g1"| 52799<sub>11</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 11, 9, 7, 9) ||class="c"| (43744, 62617, 52605, 51030, 38969, 51549) ||class="entry q2 g1"| 43744<sub>7</sub> ||class="entry q3 g1"| 62617<sub>9</sub> ||class="entry q3 g1"| 52605<sub>11</sub> ||class="entry q2 g1"| 51030<sub>9</sub> ||class="entry q3 g1"| 38969<sub>7</sub> ||class="entry q3 g1"| 51549<sub>9</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 11, 9, 7, 9) ||class="c"| (47272, 59089, 57141, 54558, 35441, 56085) ||class="entry q2 g1"| 47272<sub>7</sub> ||class="entry q3 g1"| 59089<sub>9</sub> ||class="entry q3 g1"| 57141<sub>11</sub> ||class="entry q2 g1"| 54558<sub>9</sub> ||class="entry q3 g1"| 35441<sub>7</sub> ||class="entry q3 g1"| 56085<sub>9</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 11, 9, 7, 9) ||class="c"| (52928, 63785, 42939, 41846, 38281, 41883) ||class="entry q2 g1"| 52928<sub>7</sub> ||class="entry q3 g1"| 63785<sub>9</sub> ||class="entry q3 g1"| 42939<sub>11</sub> ||class="entry q2 g1"| 41846<sub>9</sub> ||class="entry q3 g1"| 38281<sub>7</sub> ||class="entry q3 g1"| 41883<sub>9</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 11, 9, 7, 9) ||class="c"| (56456, 60257, 46579, 45374, 34753, 45523) ||class="entry q2 g1"| 56456<sub>7</sub> ||class="entry q3 g1"| 60257<sub>9</sub> ||class="entry q3 g1"| 46579<sub>11</sub> ||class="entry q2 g1"| 45374<sub>9</sub> ||class="entry q3 g1"| 34753<sub>7</sub> ||class="entry q3 g1"| 45523<sub>9</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 11, 9, 7, 9) ||class="c"| (58080, 48267, 57127, 36694, 53291, 56071) ||class="entry q2 g1"| 58080<sub>7</sub> ||class="entry q3 g1"| 48267<sub>9</sub> ||class="entry q3 g1"| 57127<sub>11</sub> ||class="entry q2 g1"| 36694<sub>9</sub> ||class="entry q3 g1"| 53291<sub>7</sub> ||class="entry q3 g1"| 56071<sub>9</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 11, 9, 7, 9) ||class="c"| (60992, 55469, 46327, 33782, 46093, 45271) ||class="entry q2 g1"| 60992<sub>7</sub> ||class="entry q3 g1"| 55469<sub>9</sub> ||class="entry q3 g1"| 46327<sub>11</sub> ||class="entry q2 g1"| 33782<sub>9</sub> ||class="entry q3 g1"| 46093<sub>7</sub> ||class="entry q3 g1"| 45271<sub>9</sub>
|-
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|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 11, 9, 7, 9) ||class="c"| (64520, 51941, 42687, 37310, 42565, 41631) ||class="entry q2 g1"| 64520<sub>7</sub> ||class="entry q3 g1"| 51941<sub>9</sub> ||class="entry q3 g1"| 42687<sub>11</sub> ||class="entry q2 g1"| 37310<sub>9</sub> ||class="entry q3 g1"| 42565<sub>7</sub> ||class="entry q3 g1"| 41631<sub>9</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 9, 9, 13, 11) ||class="c"| (34502, 55131, 54151, 60272, 48123, 55207) ||class="entry q2 g1"| 34502<sub>7</sub> ||class="entry q3 g1"| 55131<sub>11</sub> ||class="entry q3 g1"| 54151<sub>9</sub> ||class="entry q2 g1"| 60272<sub>9</sub> ||class="entry q3 g1"| 48123<sub>13</sub> ||class="entry q3 g1"| 55207<sub>11</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 9, 9, 13, 11) ||class="c"| (35430, 45949, 47191, 59344, 57309, 48247) ||class="entry q2 g1"| 35430<sub>7</sub> ||class="entry q3 g1"| 45949<sub>11</sub> ||class="entry q3 g1"| 47191<sub>9</sub> ||class="entry q2 g1"| 59344<sub>9</sub> ||class="entry q3 g1"| 57309<sub>13</sub> ||class="entry q3 g1"| 48247<sub>11</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 9, 9, 13, 11) ||class="c"| (38044, 40795, 39829, 63786, 62459, 40885) ||class="entry q2 g1"| 38044<sub>7</sub> ||class="entry q3 g1"| 40795<sub>11</sub> ||class="entry q3 g1"| 39829<sub>9</sub> ||class="entry q2 g1"| 63786<sub>9</sub> ||class="entry q3 g1"| 62459<sub>13</sub> ||class="entry q3 g1"| 40885<sub>11</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 9, 9, 13, 11) ||class="c"| (39210, 37885, 39379, 62620, 65373, 40435) ||class="entry q2 g1"| 39210<sub>7</sub> ||class="entry q3 g1"| 37885<sub>11</sub> ||class="entry q3 g1"| 39379<sub>9</sub> ||class="entry q2 g1"| 62620<sub>9</sub> ||class="entry q3 g1"| 65373<sub>13</sub> ||class="entry q3 g1"| 40435<sub>11</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 9, 9, 13, 11) ||class="c"| (42832, 40543, 43357, 51942, 62207, 44413) ||class="entry q2 g1"| 42832<sub>7</sub> ||class="entry q3 g1"| 40543<sub>11</sub> ||class="entry q3 g1"| 43357<sub>9</sub> ||class="entry q2 g1"| 51942<sub>9</sub> ||class="entry q3 g1"| 62207<sub>13</sub> ||class="entry q3 g1"| 44413<sub>11</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 9, 9, 13, 11) ||class="c"| (46346, 54879, 57679, 55484, 47871, 58735) ||class="entry q2 g1"| 46346<sub>7</sub> ||class="entry q3 g1"| 54879<sub>11</sub> ||class="entry q3 g1"| 57679<sub>9</sub> ||class="entry q2 g1"| 55484<sub>9</sub> ||class="entry q3 g1"| 47871<sub>13</sub> ||class="entry q3 g1"| 58735<sub>11</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 9, 9, 13, 11) ||class="c"| (50032, 37871, 50075, 44742, 65359, 51131) ||class="entry q2 g1"| 50032<sub>7</sub> ||class="entry q3 g1"| 37871<sub>11</sub> ||class="entry q3 g1"| 50075<sub>9</sub> ||class="entry q2 g1"| 44742<sub>9</sub> ||class="entry q3 g1"| 65359<sub>13</sub> ||class="entry q3 g1"| 51131<sub>11</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 9, 9, 13, 11) ||class="c"| (53308, 45935, 57887, 48522, 57295, 58943) ||class="entry q2 g1"| 53308<sub>7</sub> ||class="entry q3 g1"| 45935<sub>11</sub> ||class="entry q3 g1"| 57887<sub>9</sub> ||class="entry q2 g1"| 48522<sub>9</sub> ||class="entry q3 g1"| 57295<sub>13</sub> ||class="entry q3 g1"| 58943<sub>11</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 11, 9, 9, 9) ||class="c"| (33510, 47853, 48935, 61264, 54861, 47879) ||class="entry q2 g1"| 33510<sub>7</sub> ||class="entry q3 g1"| 47853<sub>11</sub> ||class="entry q3 g1"| 48935<sub>11</sub> ||class="entry q2 g1"| 61264<sub>9</sub> ||class="entry q3 g1"| 54861<sub>9</sub> ||class="entry q3 g1"| 47879<sub>9</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 11, 9, 9, 9) ||class="c"| (36422, 57035, 54519, 58352, 45675, 53463) ||class="entry q2 g1"| 36422<sub>7</sub> ||class="entry q3 g1"| 57035<sub>11</sub> ||class="entry q3 g1"| 54519<sub>11</sub> ||class="entry q2 g1"| 58352<sub>9</sub> ||class="entry q3 g1"| 45675<sub>9</sub> ||class="entry q3 g1"| 53463<sub>9</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 11, 9, 9, 9) ||class="c"| (37052, 62189, 63285, 64778, 40525, 62229) ||class="entry q2 g1"| 37052<sub>7</sub> ||class="entry q3 g1"| 62189<sub>11</sub> ||class="entry q3 g1"| 63285<sub>11</sub> ||class="entry q2 g1"| 64778<sub>9</sub> ||class="entry q3 g1"| 40525<sub>9</sub> ||class="entry q3 g1"| 62229<sub>9</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 11, 9, 9, 9) ||class="c"| (40202, 65099, 62835, 61628, 37611, 61779) ||class="entry q2 g1"| 40202<sub>7</sub> ||class="entry q3 g1"| 65099<sub>11</sub> ||class="entry q3 g1"| 62835<sub>11</sub> ||class="entry q2 g1"| 61628<sub>9</sub> ||class="entry q3 g1"| 37611<sub>9</sub> ||class="entry q3 g1"| 61779<sub>9</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 11, 9, 9, 9) ||class="c"| (41840, 62441, 50685, 52934, 40777, 49629) ||class="entry q2 g1"| 41840<sub>7</sub> ||class="entry q3 g1"| 62441<sub>11</sub> ||class="entry q3 g1"| 50685<sub>11</sub> ||class="entry q2 g1"| 52934<sub>9</sub> ||class="entry q3 g1"| 40777<sub>9</sub> ||class="entry q3 g1"| 49629<sub>9</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 11, 9, 9, 9) ||class="c"| (45354, 48105, 36335, 56476, 55113, 35279) ||class="entry q2 g1"| 45354<sub>7</sub> ||class="entry q3 g1"| 48105<sub>11</sub> ||class="entry q3 g1"| 36335<sub>11</sub> ||class="entry q2 g1"| 56476<sub>9</sub> ||class="entry q3 g1"| 55113<sub>9</sub> ||class="entry q3 g1"| 35279<sub>9</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 11, 9, 9, 9) ||class="c"| (51024, 65113, 44859, 43750, 37625, 43803) ||class="entry q2 g1"| 51024<sub>7</sub> ||class="entry q3 g1"| 65113<sub>11</sub> ||class="entry q3 g1"| 44859<sub>11</sub> ||class="entry q2 g1"| 43750<sub>9</sub> ||class="entry q3 g1"| 37625<sub>9</sub> ||class="entry q3 g1"| 43803<sub>9</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 11, 9, 9, 9) ||class="c"| (54300, 57049, 36543, 47530, 45689, 35487) ||class="entry q2 g1"| 54300<sub>7</sub> ||class="entry q3 g1"| 57049<sub>11</sub> ||class="entry q3 g1"| 36543<sub>11</sub> ||class="entry q2 g1"| 47530<sub>9</sub> ||class="entry q3 g1"| 45689<sub>9</sub> ||class="entry q3 g1"| 35487<sub>9</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 13, 9, 9, 11) ||class="c"| (34514, 60275, 61371, 60260, 34771, 60315) ||class="entry q2 g1"| 34514<sub>7</sub> ||class="entry q3 g1"| 60275<sub>11</sub> ||class="entry q3 g1"| 61371<sub>13</sub> ||class="entry q2 g1"| 60260<sub>9</sub> ||class="entry q3 g1"| 34771<sub>9</sub> ||class="entry q3 g1"| 60315<sub>11</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 13, 9, 9, 11) ||class="c"| (35684, 59349, 60925, 59090, 35701, 59869) ||class="entry q2 g1"| 35684<sub>7</sub> ||class="entry q3 g1"| 59349<sub>11</sub> ||class="entry q3 g1"| 60925<sub>13</sub> ||class="entry q2 g1"| 59090<sub>9</sub> ||class="entry q3 g1"| 35701<sub>9</sub> ||class="entry q3 g1"| 59869<sub>11</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 13, 9, 9, 11) ||class="c"| (38042, 63803, 65011, 63788, 38299, 63955) ||class="entry q2 g1"| 38042<sub>7</sub> ||class="entry q3 g1"| 63803<sub>11</sub> ||class="entry q3 g1"| 65011<sub>13</sub> ||class="entry q2 g1"| 63788<sub>9</sub> ||class="entry q3 g1"| 38299<sub>9</sub> ||class="entry q3 g1"| 63955<sub>11</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 13, 9, 9, 11) ||class="c"| (39212, 62877, 65461, 62618, 39229, 64405) ||class="entry q2 g1"| 39212<sub>7</sub> ||class="entry q3 g1"| 62877<sub>11</sub> ||class="entry q3 g1"| 65461<sub>13</sub> ||class="entry q2 g1"| 62618<sub>9</sub> ||class="entry q3 g1"| 39229<sub>9</sub> ||class="entry q3 g1"| 64405<sub>11</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 13, 9, 9, 11) ||class="c"| (42578, 51959, 64759, 52196, 42583, 63703) ||class="entry q2 g1"| 42578<sub>7</sub> ||class="entry q3 g1"| 51959<sub>11</sub> ||class="entry q3 g1"| 64759<sub>13</sub> ||class="entry q2 g1"| 52196<sub>9</sub> ||class="entry q3 g1"| 42583<sub>9</sub> ||class="entry q3 g1"| 63703<sub>11</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 13, 9, 9, 11) ||class="c"| (46106, 55487, 61119, 55724, 46111, 60063) ||class="entry q2 g1"| 46106<sub>7</sub> ||class="entry q3 g1"| 55487<sub>11</sub> ||class="entry q3 g1"| 61119<sub>13</sub> ||class="entry q2 g1"| 55724<sub>9</sub> ||class="entry q3 g1"| 46111<sub>9</sub> ||class="entry q3 g1"| 60063<sub>11</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 13, 9, 9, 11) ||class="c"| (50020, 44999, 65447, 44754, 50023, 64391) ||class="entry q2 g1"| 50020<sub>7</sub> ||class="entry q3 g1"| 44999<sub>11</sub> ||class="entry q3 g1"| 65447<sub>13</sub> ||class="entry q2 g1"| 44754<sub>9</sub> ||class="entry q3 g1"| 50023<sub>9</sub> ||class="entry q3 g1"| 64391<sub>11</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 13, 9, 9, 11) ||class="c"| (53548, 48527, 60911, 48282, 53551, 59855) ||class="entry q2 g1"| 53548<sub>7</sub> ||class="entry q3 g1"| 48527<sub>11</sub> ||class="entry q3 g1"| 60911<sub>13</sub> ||class="entry q2 g1"| 48282<sub>9</sub> ||class="entry q3 g1"| 53551<sub>9</sub> ||class="entry q3 g1"| 59855<sub>11</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 13, 7, 9, 11, 9) ||class="c"| (34768, 49115, 47633, 60006, 54139, 48689) ||class="entry q2 g1"| 34768<sub>7</sub> ||class="entry q3 g1"| 49115<sub>13</sub> ||class="entry q3 g1"| 47633<sub>7</sub> ||class="entry q2 g1"| 60006<sub>9</sub> ||class="entry q3 g1"| 54139<sub>11</sub> ||class="entry q3 g1"| 48689<sub>9</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 13, 7, 9, 11, 9) ||class="c"| (35696, 56317, 53697, 59078, 46941, 54753) ||class="entry q2 g1"| 35696<sub>7</sub> ||class="entry q3 g1"| 56317<sub>13</sub> ||class="entry q3 g1"| 53697<sub>7</sub> ||class="entry q2 g1"| 59078<sub>9</sub> ||class="entry q3 g1"| 46941<sub>11</sub> ||class="entry q3 g1"| 54753<sub>9</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 13, 7, 9, 11, 9) ||class="c"| (38282, 63451, 61955, 63548, 39803, 63011) ||class="entry q2 g1"| 38282<sub>7</sub> ||class="entry q3 g1"| 63451<sub>13</sub> ||class="entry q3 g1"| 61955<sub>7</sub> ||class="entry q2 g1"| 63548<sub>9</sub> ||class="entry q3 g1"| 39803<sub>11</sub> ||class="entry q3 g1"| 63011<sub>9</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 13, 7, 9, 11, 9) ||class="c"| (38972, 64381, 61509, 62858, 38877, 62565) ||class="entry q2 g1"| 38972<sub>7</sub> ||class="entry q3 g1"| 64381<sub>13</sub> ||class="entry q3 g1"| 61509<sub>7</sub> ||class="entry q2 g1"| 62858<sub>9</sub> ||class="entry q3 g1"| 38877<sub>11</sub> ||class="entry q3 g1"| 62565<sub>9</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 13, 7, 9, 11, 9) ||class="c"| (42566, 63199, 49355, 52208, 39551, 50411) ||class="entry q2 g1"| 42566<sub>7</sub> ||class="entry q3 g1"| 63199<sub>13</sub> ||class="entry q3 g1"| 49355<sub>7</sub> ||class="entry q2 g1"| 52208<sub>9</sub> ||class="entry q3 g1"| 39551<sub>11</sub> ||class="entry q3 g1"| 50411<sub>9</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 13, 7, 9, 11, 9) ||class="c"| (46108, 48863, 35033, 55722, 53887, 36089) ||class="entry q2 g1"| 46108<sub>7</sub> ||class="entry q3 g1"| 48863<sub>13</sub> ||class="entry q3 g1"| 35033<sub>7</sub> ||class="entry q2 g1"| 55722<sub>9</sub> ||class="entry q3 g1"| 53887<sub>11</sub> ||class="entry q3 g1"| 36089<sub>9</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 13, 7, 9, 11, 9) ||class="c"| (49766, 64367, 43533, 45008, 38863, 44589) ||class="entry q2 g1"| 49766<sub>7</sub> ||class="entry q3 g1"| 64367<sub>13</sub> ||class="entry q3 g1"| 43533<sub>7</sub> ||class="entry q2 g1"| 45008<sub>9</sub> ||class="entry q3 g1"| 38863<sub>11</sub> ||class="entry q3 g1"| 44589<sub>9</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 13, 7, 9, 11, 9) ||class="c"| (53546, 56303, 35721, 48284, 46927, 36777) ||class="entry q2 g1"| 53546<sub>7</sub> ||class="entry q3 g1"| 56303<sub>13</sub> ||class="entry q3 g1"| 35721<sub>7</sub> ||class="entry q2 g1"| 48284<sub>9</sub> ||class="entry q3 g1"| 46927<sub>11</sub> ||class="entry q3 g1"| 36777<sub>9</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 7, 7, 11, 9, 9) ||class="c"| (38858, 39949, 39107, 64124, 61613, 40163) ||class="entry q2 g1"| 38858<sub>9</sub> ||class="entry q3 g1"| 39949<sub>7</sub> ||class="entry q3 g1"| 39107<sub>7</sub> ||class="entry q2 g1"| 64124<sub>11</sub> ||class="entry q3 g1"| 61613<sub>9</sub> ||class="entry q3 g1"| 40163<sub>9</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 7, 7, 11, 9, 9) ||class="c"| (38872, 50757, 49817, 64110, 43749, 50873) ||class="entry q2 g1"| 38872<sub>9</sub> ||class="entry q3 g1"| 50757<sub>7</sub> ||class="entry q3 g1"| 49817<sub>7</sub> ||class="entry q2 g1"| 64110<sub>11</sub> ||class="entry q3 g1"| 43749<sub>9</sub> ||class="entry q3 g1"| 50873<sub>9</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 7, 7, 11, 9, 9) ||class="c"| (39548, 37035, 39557, 63434, 64523, 40613) ||class="entry q2 g1"| 39548<sub>9</sub> ||class="entry q3 g1"| 37035<sub>7</sub> ||class="entry q3 g1"| 39557<sub>7</sub> ||class="entry q2 g1"| 63434<sub>11</sub> ||class="entry q3 g1"| 64523<sub>9</sub> ||class="entry q3 g1"| 40613<sub>9</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 7, 7, 11, 9, 9) ||class="c"| (39800, 41571, 43337, 63182, 52931, 44393) ||class="entry q2 g1"| 39800<sub>9</sub> ||class="entry q3 g1"| 41571<sub>7</sub> ||class="entry q3 g1"| 43337<sub>7</sub> ||class="entry q2 g1"| 63182<sub>11</sub> ||class="entry q3 g1"| 52931<sub>9</sub> ||class="entry q3 g1"| 44393<sub>9</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 7, 7, 11, 9, 9) ||class="c"| (46670, 36673, 47171, 56312, 58337, 48227) ||class="entry q2 g1"| 46670<sub>9</sub> ||class="entry q3 g1"| 36673<sub>7</sub> ||class="entry q3 g1"| 47171<sub>7</sub> ||class="entry q2 g1"| 56312<sub>11</sub> ||class="entry q3 g1"| 58337<sub>9</sub> ||class="entry q3 g1"| 48227<sub>9</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 7, 7, 11, 9, 9) ||class="c"| (46684, 54537, 57881, 56298, 47529, 58937) ||class="entry q2 g1"| 46684<sub>9</sub> ||class="entry q3 g1"| 54537<sub>7</sub> ||class="entry q3 g1"| 57881<sub>7</sub> ||class="entry q2 g1"| 56298<sub>11</sub> ||class="entry q3 g1"| 47529<sub>9</sub> ||class="entry q3 g1"| 58937<sub>9</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 7, 7, 11, 9, 9) ||class="c"| (53870, 33521, 53893, 49112, 61009, 54949) ||class="entry q2 g1"| 53870<sub>9</sub> ||class="entry q3 g1"| 33521<sub>7</sub> ||class="entry q3 g1"| 53893<sub>7</sub> ||class="entry q2 g1"| 49112<sub>11</sub> ||class="entry q3 g1"| 61009<sub>9</sub> ||class="entry q3 g1"| 54949<sub>9</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 7, 7, 11, 9, 9) ||class="c"| (54122, 45113, 57673, 48860, 56473, 58729) ||class="entry q2 g1"| 54122<sub>9</sub> ||class="entry q3 g1"| 45113<sub>7</sub> ||class="entry q3 g1"| 57673<sub>7</sub> ||class="entry q2 g1"| 48860<sub>11</sub> ||class="entry q3 g1"| 56473<sub>9</sub> ||class="entry q3 g1"| 58729<sub>9</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 7, 9, 11, 5, 11) ||class="c"| (48840, 57521, 55637, 54142, 35857, 56693) ||class="entry q2 g1"| 48840<sub>9</sub> ||class="entry q3 g1"| 57521<sub>7</sub> ||class="entry q3 g1"| 55637<sub>9</sub> ||class="entry q2 g1"| 54142<sub>11</sub> ||class="entry q3 g1"| 35857<sub>5</sub> ||class="entry q3 g1"| 56693<sub>11</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 7, 9, 11, 5, 11) ||class="c"| (56040, 60673, 45971, 46942, 33185, 47027) ||class="entry q2 g1"| 56040<sub>9</sub> ||class="entry q3 g1"| 60673<sub>7</sub> ||class="entry q3 g1"| 45971<sub>9</sub> ||class="entry q2 g1"| 46942<sub>11</sub> ||class="entry q3 g1"| 33185<sub>5</sub> ||class="entry q3 g1"| 47027<sub>11</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 7, 9, 11, 5, 11) ||class="c"| (63176, 43171, 51983, 39806, 50179, 53039) ||class="entry q2 g1"| 63176<sub>9</sub> ||class="entry q3 g1"| 43171<sub>7</sub> ||class="entry q3 g1"| 51983<sub>9</sub> ||class="entry q2 g1"| 39806<sub>11</sub> ||class="entry q3 g1"| 50179<sub>5</sub> ||class="entry q3 g1"| 53039<sub>11</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 7, 9, 11, 5, 11) ||class="c"| (64104, 52357, 41183, 38878, 40997, 42239) ||class="entry q2 g1"| 64104<sub>9</sub> ||class="entry q3 g1"| 52357<sub>7</sub> ||class="entry q3 g1"| 41183<sub>9</sub> ||class="entry q2 g1"| 38878<sub>11</sub> ||class="entry q3 g1"| 40997<sub>5</sub> ||class="entry q3 g1"| 42239<sub>11</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 7, 11, 11, 9, 9) ||class="c"| (47848, 36103, 46581, 55134, 57767, 45525) ||class="entry q2 g1"| 47848<sub>9</sub> ||class="entry q3 g1"| 36103<sub>7</sub> ||class="entry q3 g1"| 46581<sub>11</sub> ||class="entry q2 g1"| 55134<sub>11</sub> ||class="entry q3 g1"| 57767<sub>9</sub> ||class="entry q3 g1"| 45525<sub>9</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 7, 11, 11, 9, 9) ||class="c"| (57032, 32951, 57139, 45950, 60439, 56083) ||class="entry q2 g1"| 57032<sub>9</sub> ||class="entry q3 g1"| 32951<sub>7</sub> ||class="entry q3 g1"| 57139<sub>11</sub> ||class="entry q2 g1"| 45950<sub>11</sub> ||class="entry q3 g1"| 60439<sub>9</sub> ||class="entry q3 g1"| 56083<sub>9</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 7, 11, 11, 9, 9) ||class="c"| (62184, 50453, 42927, 40798, 43445, 41871) ||class="entry q2 g1"| 62184<sub>9</sub> ||class="entry q3 g1"| 50453<sub>7</sub> ||class="entry q3 g1"| 42927<sub>11</sub> ||class="entry q2 g1"| 40798<sub>11</sub> ||class="entry q3 g1"| 43445<sub>9</sub> ||class="entry q3 g1"| 41871<sub>9</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 7, 11, 11, 9, 9) ||class="c"| (65096, 41267, 52351, 37886, 52627, 51295) ||class="entry q2 g1"| 65096<sub>9</sub> ||class="entry q3 g1"| 41267<sub>7</sub> ||class="entry q3 g1"| 52351<sub>11</sub> ||class="entry q2 g1"| 37886<sub>11</sub> ||class="entry q3 g1"| 52627<sub>9</sub> ||class="entry q3 g1"| 51295<sub>9</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 9, 9, 11, 7, 11) ||class="c"| (38606, 44741, 43791, 64376, 49765, 44847) ||class="entry q2 g1"| 38606<sub>9</sub> ||class="entry q3 g1"| 44741<sub>9</sub> ||class="entry q3 g1"| 43791<sub>9</sub> ||class="entry q2 g1"| 64376<sub>11</sub> ||class="entry q3 g1"| 49765<sub>7</sub> ||class="entry q3 g1"| 44847<sub>11</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 9, 9, 11, 7, 11) ||class="c"| (38620, 62605, 61781, 64362, 38957, 62837) ||class="entry q2 g1"| 38620<sub>9</sub> ||class="entry q3 g1"| 62605<sub>9</sub> ||class="entry q3 g1"| 61781<sub>9</sub> ||class="entry q2 g1"| 64362<sub>11</sub> ||class="entry q3 g1"| 38957<sub>7</sub> ||class="entry q3 g1"| 62837<sub>11</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 9, 9, 11, 7, 11) ||class="c"| (39534, 51939, 49375, 63448, 42563, 50431) ||class="entry q2 g1"| 39534<sub>9</sub> ||class="entry q3 g1"| 51939<sub>9</sub> ||class="entry q3 g1"| 49375<sub>9</sub> ||class="entry q2 g1"| 63448<sub>11</sub> ||class="entry q3 g1"| 42563<sub>7</sub> ||class="entry q3 g1"| 50431<sub>11</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 9, 9, 11, 7, 11) ||class="c"| (39786, 63531, 62227, 63196, 38027, 63283) ||class="entry q2 g1"| 39786<sub>9</sub> ||class="entry q3 g1"| 63531<sub>9</sub> ||class="entry q3 g1"| 62227<sub>9</sub> ||class="entry q2 g1"| 63196<sub>11</sub> ||class="entry q3 g1"| 38027<sub>7</sub> ||class="entry q3 g1"| 63283<sub>11</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 9, 9, 11, 7, 11) ||class="c"| (46922, 48521, 35727, 56060, 53545, 36783) ||class="entry q2 g1"| 46922<sub>9</sub> ||class="entry q3 g1"| 48521<sub>9</sub> ||class="entry q3 g1"| 35727<sub>9</sub> ||class="entry q2 g1"| 56060<sub>11</sub> ||class="entry q3 g1"| 53545<sub>7</sub> ||class="entry q3 g1"| 36783<sub>11</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 9, 9, 11, 7, 11) ||class="c"| (46936, 59329, 53717, 56046, 35681, 54773) ||class="entry q2 g1"| 46936<sub>9</sub> ||class="entry q3 g1"| 59329<sub>9</sub> ||class="entry q3 g1"| 53717<sub>9</sub> ||class="entry q2 g1"| 56046<sub>11</sub> ||class="entry q3 g1"| 35681<sub>7</sub> ||class="entry q3 g1"| 54773<sub>11</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 9, 9, 11, 7, 11) ||class="c"| (53884, 55481, 35039, 49098, 46105, 36095) ||class="entry q2 g1"| 53884<sub>9</sub> ||class="entry q3 g1"| 55481<sub>9</sub> ||class="entry q3 g1"| 35039<sub>9</sub> ||class="entry q2 g1"| 49098<sub>11</sub> ||class="entry q3 g1"| 46105<sub>7</sub> ||class="entry q3 g1"| 36095<sub>11</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 9, 9, 11, 7, 11) ||class="c"| (54136, 60017, 47891, 48846, 34513, 48947) ||class="entry q2 g1"| 54136<sub>9</sub> ||class="entry q3 g1"| 60017<sub>9</sub> ||class="entry q3 g1"| 47891<sub>9</sub> ||class="entry q2 g1"| 48846<sub>11</sub> ||class="entry q3 g1"| 34513<sub>7</sub> ||class="entry q3 g1"| 48947<sub>11</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 9, 9, 11, 11, 7) ||class="c"| (38618, 37613, 38707, 64364, 65101, 37651) ||class="entry q2 g1"| 38618<sub>9</sub> ||class="entry q3 g1"| 37613<sub>9</sub> ||class="entry q3 g1"| 38707<sub>9</sub> ||class="entry q2 g1"| 64364<sub>11</sub> ||class="entry q3 g1"| 65101<sub>11</sub> ||class="entry q3 g1"| 37651<sub>7</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 9, 9, 11, 11, 7) ||class="c"| (39788, 40523, 38261, 63194, 62187, 37205) ||class="entry q2 g1"| 39788<sub>9</sub> ||class="entry q3 g1"| 40523<sub>9</sub> ||class="entry q3 g1"| 38261<sub>9</sub> ||class="entry q2 g1"| 63194<sub>11</sub> ||class="entry q3 g1"| 62187<sub>11</sub> ||class="entry q3 g1"| 37205<sub>7</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 9, 9, 11, 11, 7) ||class="c"| (46682, 45929, 33919, 56300, 57289, 32863) ||class="entry q2 g1"| 46682<sub>9</sub> ||class="entry q3 g1"| 45929<sub>9</sub> ||class="entry q3 g1"| 33919<sub>9</sub> ||class="entry q2 g1"| 56300<sub>11</sub> ||class="entry q3 g1"| 57289<sub>11</sub> ||class="entry q3 g1"| 32863<sub>7</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 9, 9, 11, 11, 7) ||class="c"| (54124, 54873, 34607, 48858, 47865, 33551) ||class="entry q2 g1"| 54124<sub>9</sub> ||class="entry q3 g1"| 54873<sub>9</sub> ||class="entry q3 g1"| 34607<sub>9</sub> ||class="entry q2 g1"| 48858<sub>11</sub> ||class="entry q3 g1"| 47865<sub>11</sub> ||class="entry q3 g1"| 33551<sub>7</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 9, 11, 11, 11, 9) ||class="c"| (37614, 50035, 51119, 65368, 45011, 50063) ||class="entry q2 g1"| 37614<sub>9</sub> ||class="entry q3 g1"| 50035<sub>9</sub> ||class="entry q3 g1"| 51119<sub>11</sub> ||class="entry q2 g1"| 65368<sub>11</sub> ||class="entry q3 g1"| 45011<sub>11</sub> ||class="entry q3 g1"| 50063<sub>9</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 9, 11, 11, 11, 9) ||class="c"| (37628, 39227, 40437, 65354, 62875, 39381) ||class="entry q2 g1"| 37628<sub>9</sub> ||class="entry q3 g1"| 39227<sub>9</sub> ||class="entry q3 g1"| 40437<sub>11</sub> ||class="entry q2 g1"| 65354<sub>11</sub> ||class="entry q3 g1"| 62875<sub>11</sub> ||class="entry q3 g1"| 39381<sub>9</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 9, 11, 11, 11, 9) ||class="c"| (40526, 42837, 44159, 62456, 52213, 43103) ||class="entry q2 g1"| 40526<sub>9</sub> ||class="entry q3 g1"| 42837<sub>9</sub> ||class="entry q3 g1"| 44159<sub>11</sub> ||class="entry q2 g1"| 62456<sub>11</sub> ||class="entry q3 g1"| 52213<sub>11</sub> ||class="entry q3 g1"| 43103<sub>9</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 9, 11, 11, 11, 9) ||class="c"| (40778, 38301, 40883, 62204, 63805, 39827) ||class="entry q2 g1"| 40778<sub>9</sub> ||class="entry q3 g1"| 38301<sub>9</sub> ||class="entry q3 g1"| 40883<sub>11</sub> ||class="entry q2 g1"| 62204<sub>11</sub> ||class="entry q3 g1"| 63805<sub>11</sub> ||class="entry q3 g1"| 39827<sub>9</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 9, 11, 11, 11, 9) ||class="c"| (45930, 53311, 59183, 57052, 48287, 58127) ||class="entry q2 g1"| 45930<sub>9</sub> ||class="entry q3 g1"| 53311<sub>9</sub> ||class="entry q3 g1"| 59183<sub>11</sub> ||class="entry q2 g1"| 57052<sub>11</sub> ||class="entry q3 g1"| 48287<sub>11</sub> ||class="entry q3 g1"| 58127<sub>9</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 9, 11, 11, 11, 9) ||class="c"| (45944, 35447, 48501, 57038, 59095, 47445) ||class="entry q2 g1"| 45944<sub>9</sub> ||class="entry q3 g1"| 35447<sub>9</sub> ||class="entry q3 g1"| 48501<sub>11</sub> ||class="entry q2 g1"| 57038<sub>11</sub> ||class="entry q3 g1"| 59095<sub>11</sub> ||class="entry q3 g1"| 47445<sub>9</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 9, 11, 11, 11, 9) ||class="c"| (54876, 46351, 58495, 48106, 55727, 57439) ||class="entry q2 g1"| 54876<sub>9</sub> ||class="entry q3 g1"| 46351<sub>9</sub> ||class="entry q3 g1"| 58495<sub>11</sub> ||class="entry q2 g1"| 48106<sub>11</sub> ||class="entry q3 g1"| 55727<sub>11</sub> ||class="entry q3 g1"| 57439<sub>9</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 9, 11, 11, 11, 9) ||class="c"| (55128, 34759, 55219, 47854, 60263, 54163) ||class="entry q2 g1"| 55128<sub>9</sub> ||class="entry q3 g1"| 34759<sub>9</sub> ||class="entry q3 g1"| 55219<sub>11</sub> ||class="entry q2 g1"| 47854<sub>11</sub> ||class="entry q3 g1"| 60263<sub>11</sub> ||class="entry q3 g1"| 54163<sub>9</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 11, 5, 11, 13, 7) ||class="c"| (37868, 38875, 37381, 65114, 64379, 38437) ||class="entry q2 g1"| 37868<sub>9</sub> ||class="entry q3 g1"| 38875<sub>11</sub> ||class="entry q3 g1"| 37381<sub>5</sub> ||class="entry q2 g1"| 65114<sub>11</sub> ||class="entry q3 g1"| 64379<sub>13</sub> ||class="entry q3 g1"| 38437<sub>7</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 11, 5, 11, 13, 7) ||class="c"| (40538, 39805, 36931, 62444, 63453, 37987) ||class="entry q2 g1"| 40538<sub>9</sub> ||class="entry q3 g1"| 39805<sub>11</sub> ||class="entry q3 g1"| 36931<sub>5</sub> ||class="entry q2 g1"| 62444<sub>11</sub> ||class="entry q3 g1"| 63453<sub>13</sub> ||class="entry q3 g1"| 37987<sub>7</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 11, 5, 11, 13, 7) ||class="c"| (45932, 46687, 33097, 57050, 56063, 34153) ||class="entry q2 g1"| 45932<sub>9</sub> ||class="entry q3 g1"| 46687<sub>11</sub> ||class="entry q3 g1"| 33097<sub>5</sub> ||class="entry q2 g1"| 57050<sub>11</sub> ||class="entry q3 g1"| 56063<sub>13</sub> ||class="entry q3 g1"| 34153<sub>7</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 11, 5, 11, 13, 7) ||class="c"| (54874, 54127, 33305, 48108, 49103, 34361) ||class="entry q2 g1"| 54874<sub>9</sub> ||class="entry q3 g1"| 54127<sub>11</sub> ||class="entry q3 g1"| 33305<sub>5</sub> ||class="entry q2 g1"| 48108<sub>11</sub> ||class="entry q3 g1"| 49103<sub>13</sub> ||class="entry q3 g1"| 34361<sub>7</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 11, 9, 11, 9, 7) ||class="c"| (37866, 61883, 62563, 65116, 40219, 61507) ||class="entry q2 g1"| 37866<sub>9</sub> ||class="entry q3 g1"| 61883<sub>11</sub> ||class="entry q3 g1"| 62563<sub>9</sub> ||class="entry q2 g1"| 65116<sub>11</sub> ||class="entry q3 g1"| 40219<sub>9</sub> ||class="entry q3 g1"| 61507<sub>7</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 11, 9, 11, 9, 7) ||class="c"| (37880, 44019, 44601, 65102, 51027, 43545) ||class="entry q2 g1"| 37880<sub>9</sub> ||class="entry q3 g1"| 44019<sub>11</sub> ||class="entry q3 g1"| 44601<sub>9</sub> ||class="entry q2 g1"| 65102<sub>11</sub> ||class="entry q3 g1"| 51027<sub>9</sub> ||class="entry q3 g1"| 43545<sub>7</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 11, 9, 11, 9, 7) ||class="c"| (40540, 64797, 63013, 62442, 37309, 61957) ||class="entry q2 g1"| 40540<sub>9</sub> ||class="entry q3 g1"| 64797<sub>11</sub> ||class="entry q3 g1"| 63013<sub>9</sub> ||class="entry q2 g1"| 62442<sub>11</sub> ||class="entry q3 g1"| 37309<sub>9</sub> ||class="entry q3 g1"| 61957<sub>7</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 11, 9, 11, 9, 7) ||class="c"| (40792, 53205, 50665, 62190, 41845, 49609) ||class="entry q2 g1"| 40792<sub>9</sub> ||class="entry q3 g1"| 53205<sub>11</sub> ||class="entry q3 g1"| 50665<sub>9</sub> ||class="entry q2 g1"| 62190<sub>11</sub> ||class="entry q3 g1"| 41845<sub>9</sub> ||class="entry q3 g1"| 49609<sub>7</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 11, 9, 11, 9, 7) ||class="c"| (45678, 58103, 54499, 57304, 36439, 53443) ||class="entry q2 g1"| 45678<sub>9</sub> ||class="entry q3 g1"| 58103<sub>11</sub> ||class="entry q3 g1"| 54499<sub>9</sub> ||class="entry q2 g1"| 57304<sub>11</sub> ||class="entry q3 g1"| 36439<sub>9</sub> ||class="entry q3 g1"| 53443<sub>7</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 11, 9, 11, 9, 7) ||class="c"| (45692, 47295, 36537, 57290, 54303, 35481) ||class="entry q2 g1"| 45692<sub>9</sub> ||class="entry q3 g1"| 47295<sub>11</sub> ||class="entry q3 g1"| 36537<sub>9</sub> ||class="entry q2 g1"| 57290<sub>11</sub> ||class="entry q3 g1"| 54303<sub>9</sub> ||class="entry q3 g1"| 35481<sub>7</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 11, 9, 11, 9, 7) ||class="c"| (54862, 61255, 48677, 48120, 33767, 47621) ||class="entry q2 g1"| 54862<sub>9</sub> ||class="entry q3 g1"| 61255<sub>11</sub> ||class="entry q3 g1"| 48677<sub>9</sub> ||class="entry q2 g1"| 48120<sub>11</sub> ||class="entry q3 g1"| 33767<sub>9</sub> ||class="entry q3 g1"| 47621<sub>7</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 11, 9, 11, 9, 7) ||class="c"| (55114, 56719, 36329, 47868, 45359, 35273) ||class="entry q2 g1"| 55114<sub>9</sub> ||class="entry q3 g1"| 56719<sub>11</sub> ||class="entry q3 g1"| 36329<sub>9</sub> ||class="entry q2 g1"| 47868<sub>11</sub> ||class="entry q3 g1"| 45359<sub>9</sub> ||class="entry q3 g1"| 35273<sub>7</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 11, 11, 11, 9, 9) ||class="c"| (38860, 64109, 65189, 64122, 38605, 64133) ||class="entry q2 g1"| 38860<sub>9</sub> ||class="entry q3 g1"| 64109<sub>11</sub> ||class="entry q3 g1"| 65189<sub>11</sub> ||class="entry q2 g1"| 64122<sub>11</sub> ||class="entry q3 g1"| 38605<sub>9</sub> ||class="entry q3 g1"| 64133<sub>9</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 11, 11, 11, 9, 9) ||class="c"| (39546, 63179, 64739, 63436, 39531, 63683) ||class="entry q2 g1"| 39546<sub>9</sub> ||class="entry q3 g1"| 63179<sub>11</sub> ||class="entry q3 g1"| 64739<sub>11</sub> ||class="entry q2 g1"| 63436<sub>11</sub> ||class="entry q3 g1"| 39531<sub>9</sub> ||class="entry q3 g1"| 63683<sub>9</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 11, 11, 11, 9, 9) ||class="c"| (46924, 56297, 60905, 56058, 46921, 59849) ||class="entry q2 g1"| 46924<sub>9</sub> ||class="entry q3 g1"| 56297<sub>11</sub> ||class="entry q3 g1"| 60905<sub>11</sub> ||class="entry q2 g1"| 56058<sub>11</sub> ||class="entry q3 g1"| 46921<sub>9</sub> ||class="entry q3 g1"| 59849<sub>9</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 11, 11, 11, 9, 9) ||class="c"| (53882, 48857, 61113, 49100, 53881, 60057) ||class="entry q2 g1"| 53882<sub>9</sub> ||class="entry q3 g1"| 48857<sub>11</sub> ||class="entry q3 g1"| 61113<sub>11</sub> ||class="entry q2 g1"| 49100<sub>11</sub> ||class="entry q3 g1"| 53881<sub>9</sub> ||class="entry q3 g1"| 60057<sub>9</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 13, 11, 11, 11, 13) ||class="c"| (37626, 65371, 64403, 65356, 37883, 65459) ||class="entry q2 g1"| 37626<sub>9</sub> ||class="entry q3 g1"| 65371<sub>13</sub> ||class="entry q3 g1"| 64403<sub>11</sub> ||class="entry q2 g1"| 65356<sub>11</sub> ||class="entry q3 g1"| 37883<sub>11</sub> ||class="entry q3 g1"| 65459<sub>13</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 13, 11, 11, 11, 13) ||class="c"| (40780, 62461, 63957, 62202, 40797, 65013) ||class="entry q2 g1"| 40780<sub>9</sub> ||class="entry q3 g1"| 62461<sub>13</sub> ||class="entry q3 g1"| 63957<sub>11</sub> ||class="entry q2 g1"| 62202<sub>11</sub> ||class="entry q3 g1"| 40797<sub>11</sub> ||class="entry q3 g1"| 65013<sub>13</sub>
|-
|class="f"| 1334 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 13, 11, 11, 11, 13) ||class="c"| (45690, 57055, 59615, 57292, 45695, 60671) ||class="entry q2 g1"| 45690<sub>9</sub> ||class="entry q3 g1"| 57055<sub>13</sub> ||class="entry q3 g1"| 59615<sub>11</sub> ||class="entry q2 g1"| 57292<sub>11</sub> ||class="entry q3 g1"| 45695<sub>11</sub> ||class="entry q3 g1"| 60671<sub>13</sub>
|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
|class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 1, 0, 0) ||class="w"| (10, 8, 10, 4, 8, 8) ||class="c"| (24306, 25771, 20735, 12306, 2909, 22409) ||class="entry q0 g0"| 24306<sub>10</sub> ||class="entry q1 g0"| 25771<sub>8</sub> ||class="entry q1 g0"| 20735<sub>10</sub> ||class="entry q0 g1"| 12306<sub>4</sub> ||class="entry q1 g0"| 2909<sub>8</sub> ||class="entry q1 g0"| 22409<sub>8</sub>
|-
|class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 1, 0, 0) ||class="w"| (10, 8, 10, 4, 8, 8) ||class="c"| (24546, 27211, 24335, 12546, 1469, 22649) ||class="entry q0 g0"| 24546<sub>10</sub> ||class="entry q1 g0"| 27211<sub>8</sub> ||class="entry q1 g0"| 24335<sub>10</sub> ||class="entry q0 g1"| 12546<sub>4</sub> ||class="entry q1 g0"| 1469<sub>8</sub> ||class="entry q1 g0"| 22649<sub>8</sub>
|-
|class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 1, 0, 0) ||class="w"| (10, 8, 10, 4, 8, 8) ||class="c"| (26602, 15297, 13243, 2314, 21559, 13517) ||class="entry q0 g0"| 26602<sub>10</sub> ||class="entry q1 g0"| 15297<sub>8</sub> ||class="entry q1 g0"| 13243<sub>10</sub> ||class="entry q0 g1"| 2314<sub>4</sub> ||class="entry q1 g0"| 21559<sub>8</sub> ||class="entry q1 g0"| 13517<sub>8</sub>
|-
|class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 1, 0, 0) ||class="w"| (10, 8, 10, 4, 8, 8) ||class="c"| (26604, 23969, 21981, 2316, 12887, 21163) ||class="entry q0 g0"| 26604<sub>10</sub> ||class="entry q1 g0"| 23969<sub>8</sub> ||class="entry q1 g0"| 21981<sub>10</sub> ||class="entry q0 g1"| 2316<sub>4</sub> ||class="entry q1 g0"| 12887<sub>8</sub> ||class="entry q1 g0"| 21163<sub>8</sub>
|-
|class="f"| 1632 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 1, 0, 0) ||class="w"| (10, 8, 10, 4, 8, 8) ||class="c"| (28282, 15537, 15163, 154, 21319, 15437) ||class="entry q0 g0"| 28282<sub>10</sub> ||class="entry q1 g0"| 15537<sub>8</sub> ||class="entry q1 g0"| 15163<sub>10</sub> ||class="entry q0 g1"| 154<sub>4</sub> ||class="entry q1 g0"| 21319<sub>8</sub> ||class="entry q1 g0"| 15437<sub>8</sub>
|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
|class="f"| 4382 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 9, 7, 5, 11) ||class="c"| (34128, 38232, 60614, 37966, 33862, 64984) ||class="entry q2 g1"| 34128<sub>5</sub> ||class="entry q2 g1"| 38232<sub>7</sub> ||class="entry q2 g1"| 60614<sub>9</sub> ||class="entry q2 g1"| 37966<sub>7</sub> ||class="entry q2 g1"| 33862<sub>5</sub> ||class="entry q2 g1"| 64984<sub>11</sub>
|-
|class="f"| 4382 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 9, 7, 5, 11) ||class="c"| (34310, 38414, 61328, 38680, 34576, 65166) ||class="entry q2 g1"| 34310<sub>5</sub> ||class="entry q2 g1"| 38414<sub>7</sub> ||class="entry q2 g1"| 61328<sub>9</sub> ||class="entry q2 g1"| 38680<sub>7</sub> ||class="entry q2 g1"| 34576<sub>5</sub> ||class="entry q2 g1"| 65166<sub>11</sub>
|-
|class="f"| 4382 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 9, 7, 9, 7) ||class="c"| (32882, 44102, 54732, 37228, 48472, 50386) ||class="entry q2 g1"| 32882<sub>5</sub> ||class="entry q2 g1"| 44102<sub>7</sub> ||class="entry q2 g1"| 54732<sub>9</sub> ||class="entry q2 g1"| 37228<sub>7</sub> ||class="entry q2 g1"| 48472<sub>9</sub> ||class="entry q2 g1"| 50386<sub>7</sub>
|-
|class="f"| 4382 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 9, 7, 9, 7) ||class="c"| (32884, 51750, 45994, 37226, 56120, 41652) ||class="entry q2 g1"| 32884<sub>5</sub> ||class="entry q2 g1"| 51750<sub>7</sub> ||class="entry q2 g1"| 45994<sub>9</sub> ||class="entry q2 g1"| 37226<sub>7</sub> ||class="entry q2 g1"| 56120<sub>9</sub> ||class="entry q2 g1"| 41652<sub>7</sub>
|-
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|-
|class="f"| 4382 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 9, 7, 9, 7) ||class="c"| (33124, 50374, 48218, 36986, 54744, 44356) ||class="entry q2 g1"| 33124<sub>5</sub> ||class="entry q2 g1"| 50374<sub>7</sub> ||class="entry q2 g1"| 48218<sub>9</sub> ||class="entry q2 g1"| 36986<sub>7</sub> ||class="entry q2 g1"| 54744<sub>9</sub> ||class="entry q2 g1"| 44356<sub>7</sub>
|-
|class="f"| 4382 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 9, 7, 9, 7) ||class="c"| (33330, 51088, 48908, 37676, 54926, 44562) ||class="entry q2 g1"| 33330<sub>5</sub> ||class="entry q2 g1"| 51088<sub>7</sub> ||class="entry q2 g1"| 48908<sub>9</sub> ||class="entry q2 g1"| 37676<sub>7</sub> ||class="entry q2 g1"| 54926<sub>9</sub> ||class="entry q2 g1"| 44562<sub>7</sub>
|-
|class="f"| 4382 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 9, 7, 9, 7) ||class="c"| (33332, 41456, 55658, 37674, 45294, 51316) ||class="entry q2 g1"| 33332<sub>5</sub> ||class="entry q2 g1"| 41456<sub>7</sub> ||class="entry q2 g1"| 55658<sub>9</sub> ||class="entry q2 g1"| 37674<sub>7</sub> ||class="entry q2 g1"| 45294<sub>9</sub> ||class="entry q2 g1"| 51316<sub>7</sub>
|-
|class="f"| 4382 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 9, 7, 9, 7) ||class="c"| (33570, 51568, 45308, 37436, 55406, 41442) ||class="entry q2 g1"| 33570<sub>5</sub> ||class="entry q2 g1"| 51568<sub>7</sub> ||class="entry q2 g1"| 45308<sub>9</sub> ||class="entry q2 g1"| 37436<sub>7</sub> ||class="entry q2 g1"| 55406<sub>9</sub> ||class="entry q2 g1"| 41442<sub>7</sub>
|-
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|-
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|-
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|-
|class="f"| 4382 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 9, 7, 9, 7) ||class="c"| (34114, 53008, 46748, 37980, 56846, 42882) ||class="entry q2 g1"| 34114<sub>5</sub> ||class="entry q2 g1"| 53008<sub>7</sub> ||class="entry q2 g1"| 46748<sub>9</sub> ||class="entry q2 g1"| 37980<sub>7</sub> ||class="entry q2 g1"| 56846<sub>9</sub> ||class="entry q2 g1"| 42882<sub>7</sub>
|-
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|-
|class="f"| 4382 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 9, 7, 9, 7) ||class="c"| (34322, 43558, 54188, 38668, 47928, 49842) ||class="entry q2 g1"| 34322<sub>5</sub> ||class="entry q2 g1"| 43558<sub>7</sub> ||class="entry q2 g1"| 54188<sub>9</sub> ||class="entry q2 g1"| 38668<sub>7</sub> ||class="entry q2 g1"| 47928<sub>9</sub> ||class="entry q2 g1"| 49842<sub>7</sub>
|-
|class="f"| 4382 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 9, 7, 9, 7) ||class="c"| (34324, 52294, 46538, 38666, 56664, 42196) ||class="entry q2 g1"| 34324<sub>5</sub> ||class="entry q2 g1"| 52294<sub>7</sub> ||class="entry q2 g1"| 46538<sub>9</sub> ||class="entry q2 g1"| 38666<sub>7</sub> ||class="entry q2 g1"| 56664<sub>9</sub> ||class="entry q2 g1"| 42196<sub>7</sub>
|-
|class="f"| 4382 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 9, 7, 9, 7) ||class="c"| (34562, 42182, 56412, 38428, 46552, 52546) ||class="entry q2 g1"| 34562<sub>5</sub> ||class="entry q2 g1"| 42182<sub>7</sub> ||class="entry q2 g1"| 56412<sub>9</sub> ||class="entry q2 g1"| 38428<sub>7</sub> ||class="entry q2 g1"| 46552<sub>9</sub> ||class="entry q2 g1"| 52546<sub>7</sub>
|-
|class="f"| 4382 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 9, 7, 9, 7) ||class="c"| (34564, 49830, 47674, 38426, 54200, 43812) ||class="entry q2 g1"| 34564<sub>5</sub> ||class="entry q2 g1"| 49830<sub>7</sub> ||class="entry q2 g1"| 47674<sub>9</sub> ||class="entry q2 g1"| 38426<sub>7</sub> ||class="entry q2 g1"| 54200<sub>9</sub> ||class="entry q2 g1"| 43812<sub>7</sub>
|-
|class="f"| 4382 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 9, 11, 7, 7) ||class="c"| (41184, 55178, 40154, 45566, 50836, 36292) ||class="entry q2 g1"| 41184<sub>5</sub> ||class="entry q2 g1"| 55178<sub>9</sub> ||class="entry q2 g1"| 40154<sub>9</sub> ||class="entry q2 g1"| 45566<sub>11</sub> ||class="entry q2 g1"| 50836<sub>7</sub> ||class="entry q2 g1"| 36292<sub>7</sub>
|-
|class="f"| 4382 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 9, 11, 7, 7) ||class="c"| (41632, 48220, 63002, 46014, 44354, 59140) ||class="entry q2 g1"| 41632<sub>5</sub> ||class="entry q2 g1"| 48220<sub>9</sub> ||class="entry q2 g1"| 63002<sub>9</sub> ||class="entry q2 g1"| 46014<sub>11</sub> ||class="entry q2 g1"| 44354<sub>7</sub> ||class="entry q2 g1"| 59140<sub>7</sub>
|-
|class="f"| 4382 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 9, 11, 7, 7) ||class="c"| (42176, 47676, 61562, 46558, 43810, 57700) ||class="entry q2 g1"| 42176<sub>5</sub> ||class="entry q2 g1"| 47676<sub>9</sub> ||class="entry q2 g1"| 61562<sub>9</sub> ||class="entry q2 g1"| 46558<sub>11</sub> ||class="entry q2 g1"| 43810<sub>7</sub> ||class="entry q2 g1"| 57700<sub>7</sub>
|-
|class="f"| 4382 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 9, 11, 7, 7) ||class="c"| (42624, 53738, 39610, 47006, 49396, 35748) ||class="entry q2 g1"| 42624<sub>5</sub> ||class="entry q2 g1"| 53738<sub>9</sub> ||class="entry q2 g1"| 39610<sub>9</sub> ||class="entry q2 g1"| 47006<sub>11</sub> ||class="entry q2 g1"| 49396<sub>7</sub> ||class="entry q2 g1"| 35748<sub>7</sub>
|-
|class="f"| 4382 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 9, 11, 7, 7) ||class="c"| (43104, 56858, 39850, 47486, 52996, 35508) ||class="entry q2 g1"| 43104<sub>5</sub> ||class="entry q2 g1"| 56858<sub>9</sub> ||class="entry q2 g1"| 39850<sub>9</sub> ||class="entry q2 g1"| 47486<sub>11</sub> ||class="entry q2 g1"| 52996<sub>7</sub> ||class="entry q2 g1"| 35508<sub>7</sub>
|-
|class="f"| 4382 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 9, 11, 7, 7) ||class="c"| (43552, 46540, 61802, 47934, 42194, 57460) ||class="entry q2 g1"| 43552<sub>5</sub> ||class="entry q2 g1"| 46540<sub>9</sub> ||class="entry q2 g1"| 61802<sub>9</sub> ||class="entry q2 g1"| 47934<sub>11</sub> ||class="entry q2 g1"| 42194<sub>7</sub> ||class="entry q2 g1"| 57460<sub>7</sub>
|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
|class="f"| 4382 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 7, 9, 9, 9) ||class="c"| (55400, 51074, 58692, 51574, 54940, 62554) ||class="entry q2 g1"| 55400<sub>7</sub> ||class="entry q2 g1"| 51074<sub>7</sub> ||class="entry q2 g1"| 58692<sub>7</sub> ||class="entry q2 g1"| 51574<sub>9</sub> ||class="entry q2 g1"| 54940<sub>9</sub> ||class="entry q2 g1"| 62554<sub>9</sub>
|-
|class="f"| 4382 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 7, 9, 9, 9) ||class="c"| (55848, 44116, 36740, 52022, 48458, 40602) ||class="entry q2 g1"| 55848<sub>7</sub> ||class="entry q2 g1"| 44116<sub>7</sub> ||class="entry q2 g1"| 36740<sub>7</sub> ||class="entry q2 g1"| 52022<sub>9</sub> ||class="entry q2 g1"| 48458<sub>9</sub> ||class="entry q2 g1"| 40602<sub>9</sub>
|-
|class="f"| 4382 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 7, 9, 9, 9) ||class="c"| (56392, 43572, 35300, 52566, 47914, 39162) ||class="entry q2 g1"| 56392<sub>7</sub> ||class="entry q2 g1"| 43572<sub>7</sub> ||class="entry q2 g1"| 35300<sub>7</sub> ||class="entry q2 g1"| 52566<sub>9</sub> ||class="entry q2 g1"| 47914<sub>9</sub> ||class="entry q2 g1"| 39162<sub>9</sub>
|-
|class="f"| 4382 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 7, 9, 9, 9) ||class="c"| (56840, 49634, 58148, 53014, 53500, 62010) ||class="entry q2 g1"| 56840<sub>7</sub> ||class="entry q2 g1"| 49634<sub>7</sub> ||class="entry q2 g1"| 58148<sub>7</sub> ||class="entry q2 g1"| 53014<sub>9</sub> ||class="entry q2 g1"| 53500<sub>9</sub> ||class="entry q2 g1"| 62010<sub>9</sub>
|-
|class="f"| 4382 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 11, 9, 9, 9) ||class="c"| (35760, 39592, 60886, 39598, 35766, 64712) ||class="entry q2 g1"| 35760<sub>7</sub> ||class="entry q2 g1"| 39592<sub>7</sub> ||class="entry q2 g1"| 60886<sub>11</sub> ||class="entry q2 g1"| 39598<sub>9</sub> ||class="entry q2 g1"| 35766<sub>9</sub> ||class="entry q2 g1"| 64712<sub>9</sub>
|-
|class="f"| 4382 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 11, 9, 9, 9) ||class="c"| (36304, 40136, 60342, 40142, 36310, 64168) ||class="entry q2 g1"| 36304<sub>7</sub> ||class="entry q2 g1"| 40136<sub>7</sub> ||class="entry q2 g1"| 60342<sub>11</sub> ||class="entry q2 g1"| 40142<sub>9</sub> ||class="entry q2 g1"| 36310<sub>9</sub> ||class="entry q2 g1"| 64168<sub>9</sub>
|-
|class="f"| 4382 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 11, 9, 9, 9) ||class="c"| (57446, 61544, 61334, 61816, 57718, 65160) ||class="entry q2 g1"| 57446<sub>7</sub> ||class="entry q2 g1"| 61544<sub>7</sub> ||class="entry q2 g1"| 61334<sub>11</sub> ||class="entry q2 g1"| 61816<sub>9</sub> ||class="entry q2 g1"| 57718<sub>9</sub> ||class="entry q2 g1"| 65160<sub>9</sub>
|-
|class="f"| 4382 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 11, 9, 9, 9) ||class="c"| (58886, 62984, 59894, 63256, 59158, 63720) ||class="entry q2 g1"| 58886<sub>7</sub> ||class="entry q2 g1"| 62984<sub>7</sub> ||class="entry q2 g1"| 59894<sub>11</sub> ||class="entry q2 g1"| 63256<sub>9</sub> ||class="entry q2 g1"| 59158<sub>9</sub> ||class="entry q2 g1"| 63720<sub>9</sub>
|-
|class="f"| 4382 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 11, 9) ||class="c"| (41204, 60322, 41190, 45546, 64188, 45560) ||class="entry q2 g1"| 41204<sub>7</sub> ||class="entry q2 g1"| 60322<sub>9</sub> ||class="entry q2 g1"| 41190<sub>7</sub> ||class="entry q2 g1"| 45546<sub>9</sub> ||class="entry q2 g1"| 64188<sub>11</sub> ||class="entry q2 g1"| 45560<sub>9</sub>
|-
|class="f"| 4382 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 11, 9) ||class="c"| (41890, 59636, 41904, 45756, 63978, 45742) ||class="entry q2 g1"| 41890<sub>7</sub> ||class="entry q2 g1"| 59636<sub>9</sub> ||class="entry q2 g1"| 41904<sub>7</sub> ||class="entry q2 g1"| 45756<sub>9</sub> ||class="entry q2 g1"| 63978<sub>11</sub> ||class="entry q2 g1"| 45742<sub>9</sub>
|-
|class="f"| 4382 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 11, 9) ||class="c"| (42434, 61076, 42448, 46300, 65418, 46286) ||class="entry q2 g1"| 42434<sub>7</sub> ||class="entry q2 g1"| 61076<sub>9</sub> ||class="entry q2 g1"| 42448<sub>7</sub> ||class="entry q2 g1"| 46300<sub>9</sub> ||class="entry q2 g1"| 65418<sub>11</sub> ||class="entry q2 g1"| 46286<sub>9</sub>
|-
|class="f"| 4382 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 11, 9) ||class="c"| (42644, 60866, 42630, 46986, 64732, 47000) ||class="entry q2 g1"| 42644<sub>7</sub> ||class="entry q2 g1"| 60866<sub>9</sub> ||class="entry q2 g1"| 42630<sub>7</sub> ||class="entry q2 g1"| 46986<sub>9</sub> ||class="entry q2 g1"| 64732<sub>11</sub> ||class="entry q2 g1"| 47000<sub>9</sub>
|-
|class="f"| 4382 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 11, 9) ||class="c"| (43364, 60626, 43110, 47226, 64972, 47480) ||class="entry q2 g1"| 43364<sub>7</sub> ||class="entry q2 g1"| 60626<sub>9</sub> ||class="entry q2 g1"| 43110<sub>7</sub> ||class="entry q2 g1"| 47226<sub>9</sub> ||class="entry q2 g1"| 64972<sub>11</sub> ||class="entry q2 g1"| 47480<sub>9</sub>
|-
|class="f"| 4382 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 11, 9) ||class="c"| (43570, 61316, 43824, 47916, 65178, 47662) ||class="entry q2 g1"| 43570<sub>7</sub> ||class="entry q2 g1"| 61316<sub>9</sub> ||class="entry q2 g1"| 43824<sub>7</sub> ||class="entry q2 g1"| 47916<sub>9</sub> ||class="entry q2 g1"| 65178<sub>11</sub> ||class="entry q2 g1"| 47662<sub>9</sub>
|-
|class="f"| 4382 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 11, 9) ||class="c"| (44114, 59876, 44368, 48460, 63738, 48206) ||class="entry q2 g1"| 44114<sub>7</sub> ||class="entry q2 g1"| 59876<sub>9</sub> ||class="entry q2 g1"| 44368<sub>7</sub> ||class="entry q2 g1"| 48460<sub>9</sub> ||class="entry q2 g1"| 63738<sub>11</sub> ||class="entry q2 g1"| 48206<sub>9</sub>
|-
|class="f"| 4382 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 11, 9) ||class="c"| (44804, 60082, 44550, 48666, 64428, 48920) ||class="entry q2 g1"| 44804<sub>7</sub> ||class="entry q2 g1"| 60082<sub>9</sub> ||class="entry q2 g1"| 44550<sub>7</sub> ||class="entry q2 g1"| 48666<sub>9</sub> ||class="entry q2 g1"| 64428<sub>11</sub> ||class="entry q2 g1"| 48920<sub>9</sub>
|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 7, 7, 9, 9) ||class="c"| (37250, 45649, 41635, 59420, 51929, 45739) ||class="entry q2 g1"| 37250<sub>5</sub> ||class="entry q3 g1"| 45649<sub>7</sub> ||class="entry q3 g1"| 41635<sub>7</sub> ||class="entry q2 g1"| 59420<sub>7</sub> ||class="entry q3 g1"| 51929<sub>9</sub> ||class="entry q3 g1"| 45739<sub>9</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 7, 7, 9, 9) ||class="c"| (37252, 54321, 50373, 59418, 44217, 54477) ||class="entry q2 g1"| 37252<sub>5</sub> ||class="entry q3 g1"| 54321<sub>7</sub> ||class="entry q3 g1"| 50373<sub>7</sub> ||class="entry q2 g1"| 59418<sub>7</sub> ||class="entry q3 g1"| 44217<sub>9</sub> ||class="entry q3 g1"| 54477<sub>9</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 7, 7, 9, 9) ||class="c"| (38930, 46369, 43555, 57740, 52649, 47659) ||class="entry q2 g1"| 38930<sub>5</sub> ||class="entry q3 g1"| 46369<sub>7</sub> ||class="entry q3 g1"| 43555<sub>7</sub> ||class="entry q2 g1"| 57740<sub>7</sub> ||class="entry q3 g1"| 52649<sub>9</sub> ||class="entry q3 g1"| 47659<sub>9</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 7, 7, 9, 9) ||class="c"| (38932, 54081, 52293, 57738, 43977, 56397) ||class="entry q2 g1"| 38932<sub>5</sub> ||class="entry q3 g1"| 54081<sub>7</sub> ||class="entry q3 g1"| 52293<sub>7</sub> ||class="entry q2 g1"| 57738<sub>7</sub> ||class="entry q3 g1"| 43977<sub>9</sub> ||class="entry q3 g1"| 56397<sub>9</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 7, 7, 9, 9) ||class="c"| (40988, 33483, 41201, 55682, 64067, 45305) ||class="entry q2 g1"| 40988<sub>5</sub> ||class="entry q3 g1"| 33483<sub>7</sub> ||class="entry q3 g1"| 41201<sub>7</sub> ||class="entry q2 g1"| 55682<sub>7</sub> ||class="entry q3 g1"| 64067<sub>9</sub> ||class="entry q3 g1"| 45305<sub>9</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 7, 7, 9, 9) ||class="c"| (41228, 35883, 44801, 55442, 62627, 48905) ||class="entry q2 g1"| 41228<sub>5</sub> ||class="entry q3 g1"| 35883<sub>7</sub> ||class="entry q3 g1"| 44801<sub>7</sub> ||class="entry q2 g1"| 55442<sub>7</sub> ||class="entry q3 g1"| 62627<sub>9</sub> ||class="entry q3 g1"| 48905<sub>9</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 7, 7, 9, 9) ||class="c"| (49178, 33965, 49393, 47492, 64549, 53497) ||class="entry q2 g1"| 49178<sub>5</sub> ||class="entry q3 g1"| 33965<sub>7</sub> ||class="entry q3 g1"| 49393<sub>7</sub> ||class="entry q2 g1"| 47492<sub>7</sub> ||class="entry q3 g1"| 64549<sub>9</sub> ||class="entry q3 g1"| 53497<sub>9</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 7, 7, 9, 9) ||class="c"| (49418, 35405, 52993, 47252, 62149, 57097) ||class="entry q2 g1"| 49418<sub>5</sub> ||class="entry q3 g1"| 35405<sub>7</sub> ||class="entry q3 g1"| 52993<sub>7</sub> ||class="entry q2 g1"| 47252<sub>7</sub> ||class="entry q3 g1"| 62149<sub>9</sub> ||class="entry q3 g1"| 57097<sub>9</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 11, 7, 5, 9) ||class="c"| (37264, 59417, 63737, 59406, 37009, 59633) ||class="entry q2 g1"| 37264<sub>5</sub> ||class="entry q3 g1"| 59417<sub>7</sub> ||class="entry q3 g1"| 63737<sub>11</sub> ||class="entry q2 g1"| 59406<sub>7</sub> ||class="entry q3 g1"| 37009<sub>5</sub> ||class="entry q3 g1"| 59633<sub>9</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 11, 7, 5, 9) ||class="c"| (39184, 57737, 65417, 57486, 39169, 61313) ||class="entry q2 g1"| 39184<sub>5</sub> ||class="entry q3 g1"| 57737<sub>7</sub> ||class="entry q3 g1"| 65417<sub>11</sub> ||class="entry q2 g1"| 57486<sub>7</sub> ||class="entry q3 g1"| 39169<sub>5</sub> ||class="entry q3 g1"| 61313<sub>9</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 11, 7, 5, 9) ||class="c"| (40974, 55427, 64171, 55696, 40971, 60067) ||class="entry q2 g1"| 40974<sub>5</sub> ||class="entry q3 g1"| 55427<sub>7</sub> ||class="entry q3 g1"| 64171<sub>11</sub> ||class="entry q2 g1"| 55696<sub>7</sub> ||class="entry q3 g1"| 40971<sub>5</sub> ||class="entry q3 g1"| 60067<sub>9</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 7, 11, 7, 5, 9) ||class="c"| (49166, 47237, 64717, 47504, 49165, 60613) ||class="entry q2 g1"| 49166<sub>5</sub> ||class="entry q3 g1"| 47237<sub>7</sub> ||class="entry q3 g1"| 64717<sub>11</sub> ||class="entry q2 g1"| 47504<sub>7</sub> ||class="entry q3 g1"| 49165<sub>5</sub> ||class="entry q3 g1"| 60613<sub>9</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 7, 11, 7, 9) ||class="c"| (33480, 62641, 58689, 64342, 35897, 62793) ||class="entry q2 g1"| 33480<sub>5</sub> ||class="entry q3 g1"| 62641<sub>9</sub> ||class="entry q3 g1"| 58689<sub>7</sub> ||class="entry q2 g1"| 64342<sub>11</sub> ||class="entry q3 g1"| 35897<sub>7</sub> ||class="entry q3 g1"| 62793<sub>9</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 7, 11, 7, 9) ||class="c"| (33960, 62161, 58145, 64822, 35417, 62249) ||class="entry q2 g1"| 33960<sub>5</sub> ||class="entry q3 g1"| 62161<sub>9</sub> ||class="entry q3 g1"| 58145<sub>7</sub> ||class="entry q2 g1"| 64822<sub>11</sub> ||class="entry q3 g1"| 35417<sub>7</sub> ||class="entry q3 g1"| 62249<sub>9</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 7, 11, 7, 9) ||class="c"| (35400, 64801, 57905, 62422, 34217, 62009) ||class="entry q2 g1"| 35400<sub>5</sub> ||class="entry q3 g1"| 64801<sub>9</sub> ||class="entry q3 g1"| 57905<sub>7</sub> ||class="entry q2 g1"| 62422<sub>11</sub> ||class="entry q3 g1"| 34217<sub>7</sub> ||class="entry q3 g1"| 62009<sub>9</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 7, 11, 7, 9) ||class="c"| (35880, 64321, 58449, 62902, 33737, 62553) ||class="entry q2 g1"| 35880<sub>5</sub> ||class="entry q3 g1"| 64321<sub>9</sub> ||class="entry q3 g1"| 58449<sub>7</sub> ||class="entry q2 g1"| 62902<sub>11</sub> ||class="entry q3 g1"| 33737<sub>7</sub> ||class="entry q3 g1"| 62553<sub>9</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 7, 11, 7, 9) ||class="c"| (45632, 44203, 36485, 52190, 54307, 40589) ||class="entry q2 g1"| 45632<sub>5</sub> ||class="entry q3 g1"| 44203<sub>9</sub> ||class="entry q3 g1"| 36485<sub>7</sub> ||class="entry q2 g1"| 52190<sub>11</sub> ||class="entry q3 g1"| 54307<sub>7</sub> ||class="entry q3 g1"| 40589<sub>9</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 7, 11, 7, 9) ||class="c"| (46112, 43723, 35045, 52670, 53827, 39149) ||class="entry q2 g1"| 46112<sub>5</sub> ||class="entry q3 g1"| 43723<sub>9</sub> ||class="entry q3 g1"| 35045<sub>7</sub> ||class="entry q2 g1"| 52670<sub>11</sub> ||class="entry q3 g1"| 53827<sub>7</sub> ||class="entry q3 g1"| 39149<sub>9</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 7, 11, 7, 9) ||class="c"| (53824, 52397, 35043, 43998, 46117, 39147) ||class="entry q2 g1"| 53824<sub>5</sub> ||class="entry q3 g1"| 52397<sub>9</sub> ||class="entry q3 g1"| 35043<sub>7</sub> ||class="entry q2 g1"| 43998<sub>11</sub> ||class="entry q3 g1"| 46117<sub>7</sub> ||class="entry q3 g1"| 39147<sub>9</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 7, 11, 7, 9) ||class="c"| (54304, 51917, 36483, 44478, 45637, 40587) ||class="entry q2 g1"| 54304<sub>5</sub> ||class="entry q3 g1"| 51917<sub>9</sub> ||class="entry q3 g1"| 36483<sub>7</sub> ||class="entry q2 g1"| 44478<sub>11</sub> ||class="entry q3 g1"| 45637<sub>7</sub> ||class="entry q3 g1"| 40587<sub>9</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 9, 7, 7, 11) ||class="c"| (37010, 48305, 44371, 59660, 50233, 48475) ||class="entry q2 g1"| 37010<sub>5</sub> ||class="entry q3 g1"| 48305<sub>9</sub> ||class="entry q3 g1"| 44371<sub>9</sub> ||class="entry q2 g1"| 59660<sub>7</sub> ||class="entry q3 g1"| 50233<sub>7</sub> ||class="entry q3 g1"| 48475<sub>11</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 9, 7, 7, 11) ||class="c"| (37012, 56017, 52021, 59658, 41561, 56125) ||class="entry q2 g1"| 37012<sub>5</sub> ||class="entry q3 g1"| 56017<sub>9</sub> ||class="entry q3 g1"| 52021<sub>9</sub> ||class="entry q2 g1"| 59658<sub>7</sub> ||class="entry q3 g1"| 41561<sub>7</sub> ||class="entry q3 g1"| 56125<sub>11</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 9, 7, 7, 11) ||class="c"| (39170, 48065, 42451, 57500, 49993, 46555) ||class="entry q2 g1"| 39170<sub>5</sub> ||class="entry q3 g1"| 48065<sub>9</sub> ||class="entry q3 g1"| 42451<sub>9</sub> ||class="entry q2 g1"| 57500<sub>7</sub> ||class="entry q3 g1"| 49993<sub>7</sub> ||class="entry q3 g1"| 46555<sub>11</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 9, 7, 7, 11) ||class="c"| (39172, 56737, 50101, 57498, 42281, 54205) ||class="entry q2 g1"| 39172<sub>5</sub> ||class="entry q3 g1"| 56737<sub>9</sub> ||class="entry q3 g1"| 50101<sub>9</sub> ||class="entry q2 g1"| 57498<sub>7</sub> ||class="entry q3 g1"| 42281<sub>7</sub> ||class="entry q3 g1"| 54205<sub>11</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 9, 7, 7, 11) ||class="c"| (40986, 58539, 50839, 55684, 39971, 54943) ||class="entry q2 g1"| 40986<sub>5</sub> ||class="entry q3 g1"| 58539<sub>9</sub> ||class="entry q3 g1"| 50839<sub>9</sub> ||class="entry q2 g1"| 55684<sub>7</sub> ||class="entry q3 g1"| 39971<sub>7</sub> ||class="entry q3 g1"| 54943<sub>11</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 9, 7, 7, 11) ||class="c"| (41226, 59979, 51559, 55444, 37571, 55663) ||class="entry q2 g1"| 41226<sub>5</sub> ||class="entry q3 g1"| 59979<sub>9</sub> ||class="entry q3 g1"| 51559<sub>9</sub> ||class="entry q2 g1"| 55444<sub>7</sub> ||class="entry q3 g1"| 37571<sub>7</sub> ||class="entry q3 g1"| 55663<sub>11</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 9, 7, 7, 11) ||class="c"| (49180, 58061, 42647, 47490, 39493, 46751) ||class="entry q2 g1"| 49180<sub>5</sub> ||class="entry q3 g1"| 58061<sub>9</sub> ||class="entry q3 g1"| 42647<sub>9</sub> ||class="entry q2 g1"| 47490<sub>7</sub> ||class="entry q3 g1"| 39493<sub>7</sub> ||class="entry q3 g1"| 46751<sub>11</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 9, 7, 7, 11) ||class="c"| (49420, 60461, 43367, 47250, 38053, 47471) ||class="entry q2 g1"| 49420<sub>5</sub> ||class="entry q3 g1"| 60461<sub>9</sub> ||class="entry q3 g1"| 43367<sub>9</sub> ||class="entry q2 g1"| 47250<sub>7</sub> ||class="entry q3 g1"| 38053<sub>7</sub> ||class="entry q3 g1"| 47471<sub>11</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 9, 11, 11, 7) ||class="c"| (37568, 36143, 40393, 60254, 62887, 36289) ||class="entry q2 g1"| 37568<sub>5</sub> ||class="entry q3 g1"| 36143<sub>9</sub> ||class="entry q3 g1"| 40393<sub>9</sub> ||class="entry q2 g1"| 60254<sub>11</sub> ||class="entry q3 g1"| 62887<sub>11</sub> ||class="entry q3 g1"| 36289<sub>7</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 9, 11, 11, 7) ||class="c"| (38048, 35663, 39849, 60734, 62407, 35745) ||class="entry q2 g1"| 38048<sub>5</sub> ||class="entry q3 g1"| 35663<sub>9</sub> ||class="entry q3 g1"| 39849<sub>9</sub> ||class="entry q2 g1"| 60734<sub>11</sub> ||class="entry q3 g1"| 62407<sub>11</sub> ||class="entry q3 g1"| 35745<sub>7</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 9, 11, 11, 7) ||class="c"| (39488, 33983, 39609, 58334, 64567, 35505) ||class="entry q2 g1"| 39488<sub>5</sub> ||class="entry q3 g1"| 33983<sub>9</sub> ||class="entry q3 g1"| 39609<sub>9</sub> ||class="entry q2 g1"| 58334<sub>11</sub> ||class="entry q3 g1"| 64567<sub>11</sub> ||class="entry q3 g1"| 35505<sub>7</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 9, 11, 11, 7) ||class="c"| (39968, 33503, 40153, 58814, 64087, 36049) ||class="entry q2 g1"| 39968<sub>5</sub> ||class="entry q3 g1"| 33503<sub>9</sub> ||class="entry q3 g1"| 40153<sub>9</sub> ||class="entry q2 g1"| 58814<sub>11</sub> ||class="entry q3 g1"| 64087<sub>11</sub> ||class="entry q3 g1"| 36049<sub>7</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 9, 11, 11, 7) ||class="c"| (41544, 54581, 62989, 56278, 44477, 58885) ||class="entry q2 g1"| 41544<sub>5</sub> ||class="entry q3 g1"| 54581<sub>9</sub> ||class="entry q3 g1"| 62989<sub>9</sub> ||class="entry q2 g1"| 56278<sub>11</sub> ||class="entry q3 g1"| 44477<sub>11</sub> ||class="entry q3 g1"| 58885<sub>7</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 9, 11, 11, 7) ||class="c"| (42024, 54101, 61549, 56758, 43997, 57445) ||class="entry q2 g1"| 42024<sub>5</sub> ||class="entry q3 g1"| 54101<sub>9</sub> ||class="entry q3 g1"| 61549<sub>9</sub> ||class="entry q2 g1"| 56758<sub>11</sub> ||class="entry q3 g1"| 43997<sub>11</sub> ||class="entry q3 g1"| 57445<sub>7</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 9, 11, 11, 7) ||class="c"| (49736, 46387, 61547, 48086, 52667, 57443) ||class="entry q2 g1"| 49736<sub>5</sub> ||class="entry q3 g1"| 46387<sub>9</sub> ||class="entry q3 g1"| 61547<sub>9</sub> ||class="entry q2 g1"| 48086<sub>11</sub> ||class="entry q3 g1"| 52667<sub>11</sub> ||class="entry q3 g1"| 57443<sub>7</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 9, 11, 11, 7) ||class="c"| (50216, 45907, 62987, 48566, 52187, 58883) ||class="entry q2 g1"| 50216<sub>5</sub> ||class="entry q3 g1"| 45907<sub>9</sub> ||class="entry q3 g1"| 62987<sub>9</sub> ||class="entry q2 g1"| 48566<sub>11</sub> ||class="entry q3 g1"| 52187<sub>11</sub> ||class="entry q3 g1"| 58883<sub>7</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 11, 7, 7, 13) ||class="c"| (32910, 63751, 59879, 63760, 33167, 63983) ||class="entry q2 g1"| 32910<sub>5</sub> ||class="entry q3 g1"| 63751<sub>9</sub> ||class="entry q3 g1"| 59879<sub>11</sub> ||class="entry q2 g1"| 63760<sub>7</sub> ||class="entry q3 g1"| 33167<sub>7</sub> ||class="entry q3 g1"| 63983<sub>13</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 11, 7, 7, 13) ||class="c"| (34830, 61591, 61079, 61840, 34847, 65183) ||class="entry q2 g1"| 34830<sub>5</sub> ||class="entry q3 g1"| 61591<sub>9</sub> ||class="entry q3 g1"| 61079<sub>11</sub> ||class="entry q2 g1"| 61840<sub>7</sub> ||class="entry q3 g1"| 34847<sub>7</sub> ||class="entry q3 g1"| 65183<sub>13</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 11, 7, 7, 13) ||class="c"| (45328, 51613, 60341, 51342, 45333, 64445) ||class="entry q2 g1"| 45328<sub>5</sub> ||class="entry q3 g1"| 51613<sub>9</sub> ||class="entry q3 g1"| 60341<sub>11</sub> ||class="entry q2 g1"| 51342<sub>7</sub> ||class="entry q3 g1"| 45333<sub>7</sub> ||class="entry q3 g1"| 64445<sub>13</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 11, 7, 7, 13) ||class="c"| (53520, 43419, 60883, 43150, 53523, 64987) ||class="entry q2 g1"| 53520<sub>5</sub> ||class="entry q3 g1"| 43419<sub>9</sub> ||class="entry q3 g1"| 60883<sub>11</sub> ||class="entry q2 g1"| 43150<sub>7</sub> ||class="entry q3 g1"| 53523<sub>7</sub> ||class="entry q3 g1"| 64987<sub>13</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 11, 7, 11, 9) ||class="c"| (32922, 50479, 54747, 63748, 48551, 50643) ||class="entry q2 g1"| 32922<sub>5</sub> ||class="entry q3 g1"| 50479<sub>9</sub> ||class="entry q3 g1"| 54747<sub>11</sub> ||class="entry q2 g1"| 63748<sub>7</sub> ||class="entry q3 g1"| 48551<sub>11</sub> ||class="entry q3 g1"| 50643<sub>9</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 11, 7, 11, 9) ||class="c"| (32924, 41807, 46013, 63746, 56263, 41909) ||class="entry q2 g1"| 32924<sub>5</sub> ||class="entry q3 g1"| 41807<sub>9</sub> ||class="entry q3 g1"| 46013<sub>11</sub> ||class="entry q2 g1"| 63746<sub>7</sub> ||class="entry q3 g1"| 56263<sub>11</sub> ||class="entry q3 g1"| 41909<sub>9</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 11, 7, 11, 9) ||class="c"| (35082, 49759, 56667, 61588, 47831, 52563) ||class="entry q2 g1"| 35082<sub>5</sub> ||class="entry q3 g1"| 49759<sub>9</sub> ||class="entry q3 g1"| 56667<sub>11</sub> ||class="entry q2 g1"| 61588<sub>7</sub> ||class="entry q3 g1"| 47831<sub>11</sub> ||class="entry q3 g1"| 52563<sub>9</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 11, 7, 11, 9) ||class="c"| (35084, 42047, 47933, 61586, 56503, 43829) ||class="entry q2 g1"| 35084<sub>5</sub> ||class="entry q3 g1"| 42047<sub>9</sub> ||class="entry q3 g1"| 47933<sub>11</sub> ||class="entry q2 g1"| 61586<sub>7</sub> ||class="entry q3 g1"| 56503<sub>11</sub> ||class="entry q3 g1"| 43829<sub>9</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 11, 7, 11, 9) ||class="c"| (45074, 40245, 48671, 51596, 58813, 44567) ||class="entry q2 g1"| 45074<sub>5</sub> ||class="entry q3 g1"| 40245<sub>9</sub> ||class="entry q3 g1"| 48671<sub>11</sub> ||class="entry q2 g1"| 51596<sub>7</sub> ||class="entry q3 g1"| 58813<sub>11</sub> ||class="entry q3 g1"| 44567<sub>9</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 11, 7, 11, 9) ||class="c"| (45314, 37845, 45551, 51356, 60253, 41447) ||class="entry q2 g1"| 45314<sub>5</sub> ||class="entry q3 g1"| 37845<sub>9</sub> ||class="entry q3 g1"| 45551<sub>11</sub> ||class="entry q2 g1"| 51356<sub>7</sub> ||class="entry q3 g1"| 60253<sub>11</sub> ||class="entry q3 g1"| 41447<sub>9</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 11, 7, 11, 9) ||class="c"| (53268, 39763, 56863, 43402, 58331, 52759) ||class="entry q2 g1"| 53268<sub>5</sub> ||class="entry q3 g1"| 39763<sub>9</sub> ||class="entry q3 g1"| 56863<sub>11</sub> ||class="entry q2 g1"| 43402<sub>7</sub> ||class="entry q3 g1"| 58331<sub>11</sub> ||class="entry q3 g1"| 52759<sub>9</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 11, 7, 11, 9) ||class="c"| (53508, 38323, 53743, 43162, 60731, 49639) ||class="entry q2 g1"| 53508<sub>5</sub> ||class="entry q3 g1"| 38323<sub>9</sub> ||class="entry q3 g1"| 53743<sub>11</sub> ||class="entry q2 g1"| 43162<sub>7</sub> ||class="entry q3 g1"| 60731<sub>11</sub> ||class="entry q3 g1"| 49639<sub>9</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 11, 9, 7, 9, 7) ||class="c"| (33162, 52175, 55851, 63508, 45895, 51747) ||class="entry q2 g1"| 33162<sub>5</sub> ||class="entry q3 g1"| 52175<sub>11</sub> ||class="entry q3 g1"| 55851<sub>9</sub> ||class="entry q2 g1"| 63508<sub>7</sub> ||class="entry q3 g1"| 45895<sub>9</sub> ||class="entry q3 g1"| 51747<sub>7</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 11, 9, 7, 9, 7) ||class="c"| (33164, 44463, 48205, 63506, 54567, 44101) ||class="entry q2 g1"| 33164<sub>5</sub> ||class="entry q3 g1"| 44463<sub>11</sub> ||class="entry q3 g1"| 48205<sub>9</sub> ||class="entry q2 g1"| 63506<sub>7</sub> ||class="entry q3 g1"| 54567<sub>9</sub> ||class="entry q3 g1"| 44101<sub>7</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 11, 9, 7, 9, 7) ||class="c"| (34842, 52415, 53931, 61828, 46135, 49827) ||class="entry q2 g1"| 34842<sub>5</sub> ||class="entry q3 g1"| 52415<sub>11</sub> ||class="entry q3 g1"| 53931<sub>9</sub> ||class="entry q2 g1"| 61828<sub>7</sub> ||class="entry q3 g1"| 46135<sub>9</sub> ||class="entry q3 g1"| 49827<sub>7</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 11, 9, 7, 9, 7) ||class="c"| (34844, 43743, 46285, 61826, 53847, 42181) ||class="entry q2 g1"| 34844<sub>5</sub> ||class="entry q3 g1"| 43743<sub>11</sub> ||class="entry q3 g1"| 46285<sub>9</sub> ||class="entry q2 g1"| 61826<sub>7</sub> ||class="entry q3 g1"| 53847<sub>9</sub> ||class="entry q3 g1"| 42181<sub>7</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 11, 9, 7, 9, 7) ||class="c"| (45076, 64341, 55417, 51594, 33757, 51313) ||class="entry q2 g1"| 45076<sub>5</sub> ||class="entry q3 g1"| 64341<sub>11</sub> ||class="entry q3 g1"| 55417<sub>9</sub> ||class="entry q2 g1"| 51594<sub>7</sub> ||class="entry q3 g1"| 33757<sub>9</sub> ||class="entry q3 g1"| 51313<sub>7</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 11, 9, 7, 9, 7) ||class="c"| (45316, 62901, 55177, 51354, 36157, 51073) ||class="entry q2 g1"| 45316<sub>5</sub> ||class="entry q3 g1"| 62901<sub>11</sub> ||class="entry q3 g1"| 55177<sub>9</sub> ||class="entry q2 g1"| 51354<sub>7</sub> ||class="entry q3 g1"| 36157<sub>9</sub> ||class="entry q3 g1"| 51073<sub>7</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 11, 9, 7, 9, 7) ||class="c"| (53266, 64819, 47225, 43404, 34235, 43121) ||class="entry q2 g1"| 53266<sub>5</sub> ||class="entry q3 g1"| 64819<sub>11</sub> ||class="entry q3 g1"| 47225<sub>9</sub> ||class="entry q2 g1"| 43404<sub>7</sub> ||class="entry q3 g1"| 34235<sub>9</sub> ||class="entry q3 g1"| 43121<sub>7</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 11, 9, 7, 9, 7) ||class="c"| (53506, 62419, 46985, 43164, 35675, 42881) ||class="entry q2 g1"| 53506<sub>5</sub> ||class="entry q3 g1"| 62419<sub>11</sub> ||class="entry q3 g1"| 46985<sub>9</sub> ||class="entry q2 g1"| 43164<sub>7</sub> ||class="entry q3 g1"| 35675<sub>9</sub> ||class="entry q3 g1"| 42881<sub>7</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 11, 9, 7, 9, 11) ||class="c"| (47232, 52973, 58165, 49438, 46693, 62269) ||class="entry q2 g1"| 47232<sub>5</sub> ||class="entry q3 g1"| 52973<sub>11</sub> ||class="entry q3 g1"| 58165<sub>9</sub> ||class="entry q2 g1"| 49438<sub>7</sub> ||class="entry q3 g1"| 46693<sub>9</sub> ||class="entry q3 g1"| 62269<sub>11</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 11, 9, 7, 9, 11) ||class="c"| (55424, 44779, 58707, 41246, 54883, 62811) ||class="entry q2 g1"| 55424<sub>5</sub> ||class="entry q3 g1"| 44779<sub>11</sub> ||class="entry q3 g1"| 58707<sub>9</sub> ||class="entry q2 g1"| 41246<sub>7</sub> ||class="entry q3 g1"| 54883<sub>9</sub> ||class="entry q3 g1"| 62811<sub>11</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 11, 9, 7, 9, 11) ||class="c"| (57480, 65377, 35303, 39190, 34793, 39407) ||class="entry q2 g1"| 57480<sub>5</sub> ||class="entry q3 g1"| 65377<sub>11</sub> ||class="entry q3 g1"| 35303<sub>9</sub> ||class="entry q2 g1"| 39190<sub>7</sub> ||class="entry q3 g1"| 34793<sub>9</sub> ||class="entry q3 g1"| 39407<sub>11</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 11, 9, 7, 9, 11) ||class="c"| (59400, 63217, 36503, 37270, 36473, 40607) ||class="entry q2 g1"| 59400<sub>5</sub> ||class="entry q3 g1"| 63217<sub>11</sub> ||class="entry q3 g1"| 36503<sub>9</sub> ||class="entry q2 g1"| 37270<sub>7</sub> ||class="entry q3 g1"| 36473<sub>9</sub> ||class="entry q3 g1"| 40607<sub>11</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 11, 11, 7, 13, 9) ||class="c"| (43144, 46963, 39869, 53526, 53243, 35765) ||class="entry q2 g1"| 43144<sub>5</sub> ||class="entry q3 g1"| 46963<sub>11</sub> ||class="entry q3 g1"| 39869<sub>11</sub> ||class="entry q2 g1"| 53526<sub>7</sub> ||class="entry q3 g1"| 53243<sub>13</sub> ||class="entry q3 g1"| 35765<sub>9</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 11, 11, 7, 13, 9) ||class="c"| (51336, 55157, 40411, 45334, 45053, 36307) ||class="entry q2 g1"| 51336<sub>5</sub> ||class="entry q3 g1"| 55157<sub>11</sub> ||class="entry q3 g1"| 40411<sub>11</sub> ||class="entry q2 g1"| 45334<sub>7</sub> ||class="entry q3 g1"| 45053<sub>13</sub> ||class="entry q3 g1"| 36307<sub>9</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 11, 11, 7, 13, 9) ||class="c"| (61568, 34559, 61807, 35102, 65143, 57703) ||class="entry q2 g1"| 61568<sub>5</sub> ||class="entry q3 g1"| 34559<sub>11</sub> ||class="entry q3 g1"| 61807<sub>11</sub> ||class="entry q2 g1"| 35102<sub>7</sub> ||class="entry q3 g1"| 65143<sub>13</sub> ||class="entry q3 g1"| 57703<sub>9</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 11, 11, 7, 13, 9) ||class="c"| (63488, 36719, 63007, 33182, 63463, 58903) ||class="entry q2 g1"| 63488<sub>5</sub> ||class="entry q3 g1"| 36719<sub>11</sub> ||class="entry q3 g1"| 63007<sub>11</sub> ||class="entry q2 g1"| 33182<sub>7</sub> ||class="entry q3 g1"| 63463<sub>13</sub> ||class="entry q3 g1"| 58903<sub>9</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 5, 9, 9, 7, 7) ||class="c"| (33738, 40985, 45291, 64084, 55441, 41187) ||class="entry q2 g1"| 33738<sub>7</sub> ||class="entry q3 g1"| 40985<sub>5</sub> ||class="entry q3 g1"| 45291<sub>9</sub> ||class="entry q2 g1"| 64084<sub>9</sub> ||class="entry q3 g1"| 55441<sub>7</sub> ||class="entry q3 g1"| 41187<sub>7</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 5, 9, 9, 7, 7) ||class="c"| (34220, 49177, 53485, 64562, 47249, 49381) ||class="entry q2 g1"| 34220<sub>7</sub> ||class="entry q3 g1"| 49177<sub>5</sub> ||class="entry q3 g1"| 53485<sub>9</sub> ||class="entry q2 g1"| 64562<sub>9</sub> ||class="entry q3 g1"| 47249<sub>7</sub> ||class="entry q3 g1"| 49381<sub>7</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 5, 9, 9, 7, 7) ||class="c"| (35420, 49417, 56845, 62402, 47489, 52741) ||class="entry q2 g1"| 35420<sub>7</sub> ||class="entry q3 g1"| 49417<sub>5</sub> ||class="entry q3 g1"| 56845<sub>9</sub> ||class="entry q2 g1"| 62402<sub>9</sub> ||class="entry q3 g1"| 47489<sub>7</sub> ||class="entry q3 g1"| 52741<sub>7</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 5, 9, 9, 7, 7) ||class="c"| (35898, 41225, 48651, 62884, 55681, 44547) ||class="entry q2 g1"| 35898<sub>7</sub> ||class="entry q3 g1"| 41225<sub>5</sub> ||class="entry q3 g1"| 48651<sub>9</sub> ||class="entry q2 g1"| 62884<sub>9</sub> ||class="entry q3 g1"| 55681<sub>7</sub> ||class="entry q3 g1"| 44547<sub>7</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 5, 9, 9, 7, 7) ||class="c"| (45652, 36995, 45753, 52170, 59403, 41649) ||class="entry q2 g1"| 45652<sub>7</sub> ||class="entry q3 g1"| 36995<sub>5</sub> ||class="entry q3 g1"| 45753<sub>9</sub> ||class="entry q2 g1"| 52170<sub>9</sub> ||class="entry q3 g1"| 59403<sub>7</sub> ||class="entry q3 g1"| 41649<sub>7</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 5, 9, 9, 7, 7) ||class="c"| (46372, 38915, 47913, 52410, 57483, 43809) ||class="entry q2 g1"| 46372<sub>7</sub> ||class="entry q3 g1"| 38915<sub>5</sub> ||class="entry q3 g1"| 47913<sub>9</sub> ||class="entry q2 g1"| 52410<sub>9</sub> ||class="entry q3 g1"| 57483<sub>7</sub> ||class="entry q3 g1"| 43809<sub>7</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 5, 9, 9, 7, 7) ||class="c"| (54082, 38917, 56649, 43740, 57485, 52545) ||class="entry q2 g1"| 54082<sub>7</sub> ||class="entry q3 g1"| 38917<sub>5</sub> ||class="entry q3 g1"| 56649<sub>9</sub> ||class="entry q2 g1"| 43740<sub>9</sub> ||class="entry q3 g1"| 57485<sub>7</sub> ||class="entry q3 g1"| 52545<sub>7</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 5, 9, 9, 7, 7) ||class="c"| (54322, 36997, 54489, 44460, 59405, 50385) ||class="entry q2 g1"| 54322<sub>7</sub> ||class="entry q3 g1"| 36997<sub>5</sub> ||class="entry q3 g1"| 54489<sub>9</sub> ||class="entry q2 g1"| 44460<sub>9</sub> ||class="entry q3 g1"| 59405<sub>7</sub> ||class="entry q3 g1"| 50385<sub>7</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 7, 9, 9, 9) ||class="c"| (33486, 37585, 33575, 64336, 59993, 37679) ||class="entry q2 g1"| 33486<sub>7</sub> ||class="entry q3 g1"| 37585<sub>7</sub> ||class="entry q3 g1"| 33575<sub>7</sub> ||class="entry q2 g1"| 64336<sub>9</sub> ||class="entry q3 g1"| 59993<sub>9</sub> ||class="entry q3 g1"| 37679<sub>9</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 7, 9, 9, 9) ||class="c"| (33966, 38065, 34119, 64816, 60473, 38223) ||class="entry q2 g1"| 33966<sub>7</sub> ||class="entry q3 g1"| 38065<sub>7</sub> ||class="entry q3 g1"| 34119<sub>7</sub> ||class="entry q2 g1"| 64816<sub>9</sub> ||class="entry q3 g1"| 60473<sub>9</sub> ||class="entry q3 g1"| 38223<sub>9</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 7, 9, 9, 9) ||class="c"| (35406, 39745, 33879, 62416, 58313, 37983) ||class="entry q2 g1"| 35406<sub>7</sub> ||class="entry q3 g1"| 39745<sub>7</sub> ||class="entry q3 g1"| 33879<sub>7</sub> ||class="entry q2 g1"| 62416<sub>9</sub> ||class="entry q3 g1"| 58313<sub>9</sub> ||class="entry q3 g1"| 37983<sub>9</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 7, 9, 9, 9) ||class="c"| (35886, 40225, 33335, 62896, 58793, 37439) ||class="entry q2 g1"| 35886<sub>7</sub> ||class="entry q3 g1"| 40225<sub>7</sub> ||class="entry q3 g1"| 33335<sub>7</sub> ||class="entry q2 g1"| 62896<sub>9</sub> ||class="entry q3 g1"| 58793<sub>9</sub> ||class="entry q3 g1"| 37439<sub>9</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 7, 9, 9, 9) ||class="c"| (45904, 41547, 33141, 51918, 56003, 37245) ||class="entry q2 g1"| 45904<sub>7</sub> ||class="entry q3 g1"| 41547<sub>7</sub> ||class="entry q3 g1"| 33141<sub>7</sub> ||class="entry q2 g1"| 51918<sub>9</sub> ||class="entry q3 g1"| 56003<sub>9</sub> ||class="entry q3 g1"| 37245<sub>9</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 7, 9, 9, 9) ||class="c"| (46384, 42027, 34581, 52398, 56483, 38685) ||class="entry q2 g1"| 46384<sub>7</sub> ||class="entry q3 g1"| 42027<sub>7</sub> ||class="entry q3 g1"| 34581<sub>7</sub> ||class="entry q2 g1"| 52398<sub>9</sub> ||class="entry q3 g1"| 56483<sub>9</sub> ||class="entry q3 g1"| 38685<sub>9</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 7, 9, 9, 9) ||class="c"| (54096, 49741, 34579, 43726, 47813, 38683) ||class="entry q2 g1"| 54096<sub>7</sub> ||class="entry q3 g1"| 49741<sub>7</sub> ||class="entry q3 g1"| 34579<sub>7</sub> ||class="entry q2 g1"| 43726<sub>9</sub> ||class="entry q3 g1"| 47813<sub>9</sub> ||class="entry q3 g1"| 38683<sub>9</sub>
|-
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|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 7, 13, 9, 9) ||class="c"| (34536, 39175, 35297, 65398, 57743, 39401) ||class="entry q2 g1"| 34536<sub>7</sub> ||class="entry q3 g1"| 39175<sub>7</sub> ||class="entry q3 g1"| 35297<sub>7</sub> ||class="entry q2 g1"| 65398<sub>13</sub> ||class="entry q3 g1"| 57743<sub>9</sub> ||class="entry q3 g1"| 39401<sub>9</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 7, 13, 9, 9) ||class="c"| (36456, 37015, 36497, 63478, 59423, 40601) ||class="entry q2 g1"| 36456<sub>7</sub> ||class="entry q3 g1"| 37015<sub>7</sub> ||class="entry q3 g1"| 36497<sub>7</sub> ||class="entry q2 g1"| 63478<sub>13</sub> ||class="entry q3 g1"| 59423<sub>9</sub> ||class="entry q3 g1"| 40601<sub>9</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 7, 13, 9, 9) ||class="c"| (46688, 49437, 57893, 53246, 47509, 61997) ||class="entry q2 g1"| 46688<sub>7</sub> ||class="entry q3 g1"| 49437<sub>7</sub> ||class="entry q3 g1"| 57893<sub>7</sub> ||class="entry q2 g1"| 53246<sub>13</sub> ||class="entry q3 g1"| 47509<sub>9</sub> ||class="entry q3 g1"| 61997<sub>9</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 7, 13, 9, 9) ||class="c"| (54880, 41243, 58435, 45054, 55699, 62539) ||class="entry q2 g1"| 54880<sub>7</sub> ||class="entry q3 g1"| 41243<sub>7</sub> ||class="entry q3 g1"| 58435<sub>7</sub> ||class="entry q2 g1"| 45054<sub>13</sub> ||class="entry q3 g1"| 55699<sub>9</sub> ||class="entry q3 g1"| 62539<sub>9</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 9, 9, 9, 11) ||class="c"| (37588, 45319, 41461, 60234, 51599, 45565) ||class="entry q2 g1"| 37588<sub>7</sub> ||class="entry q3 g1"| 45319<sub>7</sub> ||class="entry q3 g1"| 41461<sub>9</sub> ||class="entry q2 g1"| 60234<sub>9</sub> ||class="entry q3 g1"| 51599<sub>9</sub> ||class="entry q3 g1"| 45565<sub>11</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 9, 9, 9, 11) ||class="c"| (38066, 53511, 49651, 60716, 43407, 53755) ||class="entry q2 g1"| 38066<sub>7</sub> ||class="entry q3 g1"| 53511<sub>7</sub> ||class="entry q3 g1"| 49651<sub>9</sub> ||class="entry q2 g1"| 60716<sub>9</sub> ||class="entry q3 g1"| 43407<sub>9</sub> ||class="entry q3 g1"| 53755<sub>11</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 9, 9, 9, 11) ||class="c"| (39746, 53271, 53011, 58076, 43167, 57115) ||class="entry q2 g1"| 39746<sub>7</sub> ||class="entry q3 g1"| 53271<sub>7</sub> ||class="entry q3 g1"| 53011<sub>9</sub> ||class="entry q2 g1"| 58076<sub>9</sub> ||class="entry q3 g1"| 43167<sub>9</sub> ||class="entry q3 g1"| 57115<sub>11</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 9, 9, 9, 11) ||class="c"| (40228, 45079, 44821, 58554, 51359, 48925) ||class="entry q2 g1"| 40228<sub>7</sub> ||class="entry q3 g1"| 45079<sub>7</sub> ||class="entry q3 g1"| 44821<sub>9</sub> ||class="entry q2 g1"| 58554<sub>9</sub> ||class="entry q3 g1"| 51359<sub>9</sub> ||class="entry q3 g1"| 48925<sub>11</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 9, 9, 9, 11) ||class="c"| (41802, 33181, 41895, 56020, 63765, 45999) ||class="entry q2 g1"| 41802<sub>7</sub> ||class="entry q3 g1"| 33181<sub>7</sub> ||class="entry q3 g1"| 41895<sub>9</sub> ||class="entry q2 g1"| 56020<sub>9</sub> ||class="entry q3 g1"| 63765<sub>9</sub> ||class="entry q3 g1"| 45999<sub>11</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 9, 9, 9, 11) ||class="c"| (42042, 35101, 43575, 56740, 61845, 47679) ||class="entry q2 g1"| 42042<sub>7</sub> ||class="entry q3 g1"| 35101<sub>7</sub> ||class="entry q3 g1"| 43575<sub>9</sub> ||class="entry q2 g1"| 56740<sub>9</sub> ||class="entry q3 g1"| 61845<sub>9</sub> ||class="entry q3 g1"| 47679<sub>11</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 9, 9, 9, 11) ||class="c"| (49756, 35099, 52311, 48066, 61843, 56415) ||class="entry q2 g1"| 49756<sub>7</sub> ||class="entry q3 g1"| 35099<sub>7</sub> ||class="entry q3 g1"| 52311<sub>9</sub> ||class="entry q2 g1"| 48066<sub>9</sub> ||class="entry q3 g1"| 61843<sub>9</sub> ||class="entry q3 g1"| 56415<sub>11</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 9, 9, 9, 11) ||class="c"| (50476, 33179, 50631, 48306, 63763, 54735) ||class="entry q2 g1"| 50476<sub>7</sub> ||class="entry q3 g1"| 33179<sub>7</sub> ||class="entry q3 g1"| 50631<sub>9</sub> ||class="entry q2 g1"| 48306<sub>9</sub> ||class="entry q3 g1"| 63763<sub>9</sub> ||class="entry q3 g1"| 54735<sub>11</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 9, 13, 5, 7) ||class="c"| (38624, 57497, 61801, 61310, 38929, 57697) ||class="entry q2 g1"| 38624<sub>7</sub> ||class="entry q3 g1"| 57497<sub>7</sub> ||class="entry q3 g1"| 61801<sub>9</sub> ||class="entry q2 g1"| 61310<sub>13</sub> ||class="entry q3 g1"| 38929<sub>5</sub> ||class="entry q3 g1"| 57697<sub>7</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 9, 13, 5, 7) ||class="c"| (40544, 59657, 63001, 59390, 37249, 58897) ||class="entry q2 g1"| 40544<sub>7</sub> ||class="entry q3 g1"| 59657<sub>7</sub> ||class="entry q3 g1"| 63001<sub>9</sub> ||class="entry q2 g1"| 59390<sub>13</sub> ||class="entry q3 g1"| 37249<sub>5</sub> ||class="entry q3 g1"| 58897<sub>7</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 9, 13, 5, 7) ||class="c"| (42600, 47235, 39597, 57334, 49163, 35493) ||class="entry q2 g1"| 42600<sub>7</sub> ||class="entry q3 g1"| 47235<sub>7</sub> ||class="entry q3 g1"| 39597<sub>9</sub> ||class="entry q2 g1"| 57334<sub>13</sub> ||class="entry q3 g1"| 49163<sub>5</sub> ||class="entry q3 g1"| 35493<sub>7</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 9, 13, 5, 7) ||class="c"| (50792, 55429, 40139, 49142, 40973, 36035) ||class="entry q2 g1"| 50792<sub>7</sub> ||class="entry q3 g1"| 55429<sub>7</sub> ||class="entry q3 g1"| 40139<sub>9</sub> ||class="entry q2 g1"| 49142<sub>13</sub> ||class="entry q3 g1"| 40973<sub>5</sub> ||class="entry q3 g1"| 36035<sub>7</sub>
|-
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|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 11, 9, 5, 9) ||class="c"| (33978, 43161, 47483, 64804, 53265, 43379) ||class="entry q2 g1"| 33978<sub>7</sub> ||class="entry q3 g1"| 43161<sub>7</sub> ||class="entry q3 g1"| 47483<sub>11</sub> ||class="entry q2 g1"| 64804<sub>9</sub> ||class="entry q3 g1"| 53265<sub>5</sub> ||class="entry q3 g1"| 43379<sub>9</sub>
|-
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|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 11, 9, 5, 9) ||class="c"| (36140, 51593, 55197, 62642, 45313, 51093) ||class="entry q2 g1"| 36140<sub>7</sub> ||class="entry q3 g1"| 51593<sub>7</sub> ||class="entry q3 g1"| 55197<sub>11</sub> ||class="entry q2 g1"| 62642<sub>9</sub> ||class="entry q3 g1"| 45313<sub>5</sub> ||class="entry q3 g1"| 51093<sub>9</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 11, 9, 5, 9) ||class="c"| (45890, 63491, 56111, 51932, 32907, 52007) ||class="entry q2 g1"| 45890<sub>7</sub> ||class="entry q3 g1"| 63491<sub>7</sub> ||class="entry q3 g1"| 56111<sub>11</sub> ||class="entry q2 g1"| 51932<sub>9</sub> ||class="entry q3 g1"| 32907<sub>5</sub> ||class="entry q3 g1"| 52007<sub>9</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 11, 9, 5, 9) ||class="c"| (46130, 61571, 53951, 52652, 34827, 49847) ||class="entry q2 g1"| 46130<sub>7</sub> ||class="entry q3 g1"| 61571<sub>7</sub> ||class="entry q3 g1"| 53951<sub>11</sub> ||class="entry q2 g1"| 52652<sub>9</sub> ||class="entry q3 g1"| 34827<sub>5</sub> ||class="entry q3 g1"| 49847<sub>9</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 11, 9, 5, 9) ||class="c"| (53844, 61573, 46303, 43978, 34829, 42199) ||class="entry q2 g1"| 53844<sub>7</sub> ||class="entry q3 g1"| 61573<sub>7</sub> ||class="entry q3 g1"| 46303<sub>11</sub> ||class="entry q2 g1"| 43978<sub>9</sub> ||class="entry q3 g1"| 34829<sub>5</sub> ||class="entry q3 g1"| 42199<sub>9</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 7, 11, 9, 5, 9) ||class="c"| (54564, 63493, 48463, 44218, 32909, 44359) ||class="entry q2 g1"| 54564<sub>7</sub> ||class="entry q3 g1"| 63493<sub>7</sub> ||class="entry q3 g1"| 48463<sub>11</sub> ||class="entry q2 g1"| 44218<sub>9</sub> ||class="entry q3 g1"| 32909<sub>5</sub> ||class="entry q3 g1"| 44359<sub>9</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 7, 9) ||class="c"| (37826, 55687, 51299, 59996, 41231, 55403) ||class="entry q2 g1"| 37826<sub>7</sub> ||class="entry q3 g1"| 55687<sub>9</sub> ||class="entry q3 g1"| 51299<sub>7</sub> ||class="entry q2 g1"| 59996<sub>9</sub> ||class="entry q3 g1"| 41231<sub>7</sub> ||class="entry q3 g1"| 55403<sub>9</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 7, 9) ||class="c"| (38308, 47495, 43109, 60474, 49423, 47213) ||class="entry q2 g1"| 38308<sub>7</sub> ||class="entry q3 g1"| 47495<sub>9</sub> ||class="entry q3 g1"| 43109<sub>7</sub> ||class="entry q2 g1"| 60474<sub>9</sub> ||class="entry q3 g1"| 49423<sub>7</sub> ||class="entry q3 g1"| 47213<sub>9</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 7, 9) ||class="c"| (39508, 47255, 42629, 58314, 49183, 46733) ||class="entry q2 g1"| 39508<sub>7</sub> ||class="entry q3 g1"| 47255<sub>9</sub> ||class="entry q3 g1"| 42629<sub>7</sub> ||class="entry q2 g1"| 58314<sub>9</sub> ||class="entry q3 g1"| 49183<sub>7</sub> ||class="entry q3 g1"| 46733<sub>9</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 7, 9) ||class="c"| (39986, 55447, 50819, 58796, 40991, 54923) ||class="entry q2 g1"| 39986<sub>7</sub> ||class="entry q3 g1"| 55447<sub>9</sub> ||class="entry q3 g1"| 50819<sub>7</sub> ||class="entry q2 g1"| 58796<sub>9</sub> ||class="entry q3 g1"| 40991<sub>7</sub> ||class="entry q3 g1"| 54923<sub>9</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 7, 9) ||class="c"| (41564, 59677, 51761, 56258, 37269, 55865) ||class="entry q2 g1"| 41564<sub>7</sub> ||class="entry q3 g1"| 59677<sub>9</sub> ||class="entry q3 g1"| 51761<sub>7</sub> ||class="entry q2 g1"| 56258<sub>9</sub> ||class="entry q3 g1"| 37269<sub>7</sub> ||class="entry q3 g1"| 55865<sub>9</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 7, 9) ||class="c"| (42284, 57757, 50081, 56498, 39189, 54185) ||class="entry q2 g1"| 42284<sub>7</sub> ||class="entry q3 g1"| 57757<sub>9</sub> ||class="entry q3 g1"| 50081<sub>7</sub> ||class="entry q2 g1"| 56498<sub>9</sub> ||class="entry q3 g1"| 39189<sub>7</sub> ||class="entry q3 g1"| 54185<sub>9</sub>
|-
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|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 7, 9) ||class="c"| (50234, 59675, 44113, 48548, 37267, 48217) ||class="entry q2 g1"| 50234<sub>7</sub> ||class="entry q3 g1"| 59675<sub>9</sub> ||class="entry q3 g1"| 44113<sub>7</sub> ||class="entry q2 g1"| 48548<sub>9</sub> ||class="entry q3 g1"| 37267<sub>7</sub> ||class="entry q3 g1"| 48217<sub>9</sub>
|-
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|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 11, 5) ||class="c"| (38320, 34223, 37977, 60462, 64807, 33873) ||class="entry q2 g1"| 38320<sub>7</sub> ||class="entry q3 g1"| 34223<sub>9</sub> ||class="entry q3 g1"| 37977<sub>7</sub> ||class="entry q2 g1"| 60462<sub>9</sub> ||class="entry q3 g1"| 64807<sub>11</sub> ||class="entry q3 g1"| 33873<sub>5</sub>
|-
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|-
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|-
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|-
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|-
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|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 11, 5) ||class="c"| (50222, 54579, 36973, 48560, 44475, 32869) ||class="entry q2 g1"| 50222<sub>7</sub> ||class="entry q3 g1"| 54579<sub>9</sub> ||class="entry q3 g1"| 36973<sub>7</sub> ||class="entry q2 g1"| 48560<sub>9</sub> ||class="entry q3 g1"| 44475<sub>11</sub> ||class="entry q3 g1"| 32869<sub>5</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 7, 11) ||class="c"| (33752, 64081, 60081, 64070, 33497, 64185) ||class="entry q2 g1"| 33752<sub>7</sub> ||class="entry q3 g1"| 64081<sub>9</sub> ||class="entry q3 g1"| 60081<sub>9</sub> ||class="entry q2 g1"| 64070<sub>9</sub> ||class="entry q3 g1"| 33497<sub>7</sub> ||class="entry q3 g1"| 64185<sub>11</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 7, 11) ||class="c"| (34232, 64561, 60625, 64550, 33977, 64729) ||class="entry q2 g1"| 34232<sub>7</sub> ||class="entry q3 g1"| 64561<sub>9</sub> ||class="entry q3 g1"| 60625<sub>9</sub> ||class="entry q2 g1"| 64550<sub>9</sub> ||class="entry q3 g1"| 33977<sub>7</sub> ||class="entry q3 g1"| 64729<sub>11</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 7, 11) ||class="c"| (35672, 62401, 60865, 62150, 35657, 64969) ||class="entry q2 g1"| 35672<sub>7</sub> ||class="entry q3 g1"| 62401<sub>9</sub> ||class="entry q3 g1"| 60865<sub>9</sub> ||class="entry q2 g1"| 62150<sub>9</sub> ||class="entry q3 g1"| 35657<sub>7</sub> ||class="entry q3 g1"| 64969<sub>11</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 7, 11) ||class="c"| (36152, 62881, 60321, 62630, 36137, 64425) ||class="entry q2 g1"| 36152<sub>7</sub> ||class="entry q3 g1"| 62881<sub>9</sub> ||class="entry q3 g1"| 60321<sub>9</sub> ||class="entry q2 g1"| 62630<sub>9</sub> ||class="entry q3 g1"| 36137<sub>7</sub> ||class="entry q3 g1"| 64425<sub>11</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 7, 11) ||class="c"| (45638, 51915, 59619, 52184, 45635, 63723) ||class="entry q2 g1"| 45638<sub>7</sub> ||class="entry q3 g1"| 51915<sub>9</sub> ||class="entry q3 g1"| 59619<sub>9</sub> ||class="entry q2 g1"| 52184<sub>9</sub> ||class="entry q3 g1"| 45635<sub>7</sub> ||class="entry q3 g1"| 63723<sub>11</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 7, 11) ||class="c"| (46118, 52395, 61059, 52664, 46115, 65163) ||class="entry q2 g1"| 46118<sub>7</sub> ||class="entry q3 g1"| 52395<sub>9</sub> ||class="entry q3 g1"| 61059<sub>9</sub> ||class="entry q2 g1"| 52664<sub>9</sub> ||class="entry q3 g1"| 46115<sub>7</sub> ||class="entry q3 g1"| 65163<sub>11</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 7, 11) ||class="c"| (53830, 43725, 61061, 43992, 53829, 65165) ||class="entry q2 g1"| 53830<sub>7</sub> ||class="entry q3 g1"| 43725<sub>9</sub> ||class="entry q3 g1"| 61061<sub>9</sub> ||class="entry q2 g1"| 43992<sub>9</sub> ||class="entry q3 g1"| 53829<sub>7</sub> ||class="entry q3 g1"| 65165<sub>11</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 7, 11) ||class="c"| (54310, 44205, 59621, 44472, 54309, 63725) ||class="entry q2 g1"| 54310<sub>7</sub> ||class="entry q3 g1"| 44205<sub>9</sub> ||class="entry q3 g1"| 59621<sub>9</sub> ||class="entry q2 g1"| 44472<sub>9</sub> ||class="entry q3 g1"| 54309<sub>7</sub> ||class="entry q3 g1"| 63725<sub>11</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 11, 7) ||class="c"| (33740, 50809, 54925, 64082, 48881, 50821) ||class="entry q2 g1"| 33740<sub>7</sub> ||class="entry q3 g1"| 50809<sub>9</sub> ||class="entry q3 g1"| 54925<sub>9</sub> ||class="entry q2 g1"| 64082<sub>9</sub> ||class="entry q3 g1"| 48881<sub>11</sub> ||class="entry q3 g1"| 50821<sub>7</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 11, 7) ||class="c"| (34218, 42617, 46731, 64564, 57073, 42627) ||class="entry q2 g1"| 34218<sub>7</sub> ||class="entry q3 g1"| 42617<sub>9</sub> ||class="entry q3 g1"| 46731<sub>9</sub> ||class="entry q2 g1"| 64564<sub>9</sub> ||class="entry q3 g1"| 57073<sub>11</sub> ||class="entry q3 g1"| 42627<sub>7</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 11, 7) ||class="c"| (35418, 42857, 47211, 62404, 57313, 43107) ||class="entry q2 g1"| 35418<sub>7</sub> ||class="entry q3 g1"| 42857<sub>9</sub> ||class="entry q3 g1"| 47211<sub>9</sub> ||class="entry q2 g1"| 62404<sub>9</sub> ||class="entry q3 g1"| 57313<sub>11</sub> ||class="entry q3 g1"| 43107<sub>7</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 11, 7) ||class="c"| (35900, 51049, 55405, 62882, 49121, 51301) ||class="entry q2 g1"| 35900<sub>7</sub> ||class="entry q3 g1"| 51049<sub>9</sub> ||class="entry q3 g1"| 55405<sub>9</sub> ||class="entry q2 g1"| 62882<sub>9</sub> ||class="entry q3 g1"| 49121<sub>11</sub> ||class="entry q3 g1"| 51301<sub>7</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 11, 7) ||class="c"| (45892, 40547, 48457, 51930, 59115, 44353) ||class="entry q2 g1"| 45892<sub>7</sub> ||class="entry q3 g1"| 40547<sub>9</sub> ||class="entry q3 g1"| 48457<sub>9</sub> ||class="entry q2 g1"| 51930<sub>9</sub> ||class="entry q3 g1"| 59115<sub>11</sub> ||class="entry q3 g1"| 44353<sub>7</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 11, 7) ||class="c"| (46132, 38627, 46297, 52650, 61035, 42193) ||class="entry q2 g1"| 46132<sub>7</sub> ||class="entry q3 g1"| 38627<sub>9</sub> ||class="entry q3 g1"| 46297<sub>9</sub> ||class="entry q2 g1"| 52650<sub>9</sub> ||class="entry q3 g1"| 61035<sub>11</sub> ||class="entry q3 g1"| 42193<sub>7</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 11, 7) ||class="c"| (53842, 38629, 53945, 43980, 61037, 49841) ||class="entry q2 g1"| 53842<sub>7</sub> ||class="entry q3 g1"| 38629<sub>9</sub> ||class="entry q3 g1"| 53945<sub>9</sub> ||class="entry q2 g1"| 43980<sub>9</sub> ||class="entry q3 g1"| 61037<sub>11</sub> ||class="entry q3 g1"| 49841<sub>7</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 11, 7) ||class="c"| (54562, 40549, 56105, 44220, 59117, 52001) ||class="entry q2 g1"| 54562<sub>7</sub> ||class="entry q3 g1"| 40549<sub>9</sub> ||class="entry q3 g1"| 56105<sub>9</sub> ||class="entry q2 g1"| 44220<sub>9</sub> ||class="entry q3 g1"| 59117<sub>11</sub> ||class="entry q3 g1"| 52001<sub>7</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 11, 11) ||class="c"| (47808, 42299, 35317, 50014, 56755, 39421) ||class="entry q2 g1"| 47808<sub>7</sub> ||class="entry q3 g1"| 42299<sub>9</sub> ||class="entry q3 g1"| 35317<sub>9</sub> ||class="entry q2 g1"| 50014<sub>9</sub> ||class="entry q3 g1"| 56755<sub>11</sub> ||class="entry q3 g1"| 39421<sub>11</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 11, 11) ||class="c"| (48288, 41819, 36757, 50494, 56275, 40861) ||class="entry q2 g1"| 48288<sub>7</sub> ||class="entry q3 g1"| 41819<sub>9</sub> ||class="entry q3 g1"| 36757<sub>9</sub> ||class="entry q2 g1"| 50494<sub>9</sub> ||class="entry q3 g1"| 56275<sub>11</sub> ||class="entry q3 g1"| 40861<sub>11</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 11, 11) ||class="c"| (56000, 50493, 36755, 41822, 48565, 40859) ||class="entry q2 g1"| 56000<sub>7</sub> ||class="entry q3 g1"| 50493<sub>9</sub> ||class="entry q3 g1"| 36755<sub>9</sub> ||class="entry q2 g1"| 41822<sub>9</sub> ||class="entry q3 g1"| 48565<sub>11</sub> ||class="entry q3 g1"| 40859<sub>11</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 11, 11) ||class="c"| (56480, 50013, 35315, 42302, 48085, 39419) ||class="entry q2 g1"| 56480<sub>7</sub> ||class="entry q3 g1"| 50013<sub>9</sub> ||class="entry q3 g1"| 35315<sub>9</sub> ||class="entry q2 g1"| 42302<sub>9</sub> ||class="entry q3 g1"| 48085<sub>11</sub> ||class="entry q3 g1"| 39419<sub>11</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 11, 11) ||class="c"| (58056, 38071, 58151, 39766, 60479, 62255) ||class="entry q2 g1"| 58056<sub>7</sub> ||class="entry q3 g1"| 38071<sub>9</sub> ||class="entry q3 g1"| 58151<sub>9</sub> ||class="entry q2 g1"| 39766<sub>9</sub> ||class="entry q3 g1"| 60479<sub>11</sub> ||class="entry q3 g1"| 62255<sub>11</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 11, 11) ||class="c"| (58536, 37591, 58695, 40246, 59999, 62799) ||class="entry q2 g1"| 58536<sub>7</sub> ||class="entry q3 g1"| 37591<sub>9</sub> ||class="entry q3 g1"| 58695<sub>9</sub> ||class="entry q2 g1"| 40246<sub>9</sub> ||class="entry q3 g1"| 59999<sub>11</sub> ||class="entry q3 g1"| 62799<sub>11</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 11, 11) ||class="c"| (59976, 40231, 58455, 37846, 58799, 62559) ||class="entry q2 g1"| 59976<sub>7</sub> ||class="entry q3 g1"| 40231<sub>9</sub> ||class="entry q3 g1"| 58455<sub>9</sub> ||class="entry q2 g1"| 37846<sub>9</sub> ||class="entry q3 g1"| 58799<sub>11</sub> ||class="entry q3 g1"| 62559<sub>11</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 9, 9, 11, 11) ||class="c"| (60456, 39751, 57911, 38326, 58319, 62015) ||class="entry q2 g1"| 60456<sub>7</sub> ||class="entry q3 g1"| 39751<sub>9</sub> ||class="entry q3 g1"| 57911<sub>9</sub> ||class="entry q2 g1"| 38326<sub>9</sub> ||class="entry q3 g1"| 58319<sub>11</sub> ||class="entry q3 g1"| 62015<sub>11</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 11, 9, 7, 9) ||class="c"| (43720, 56485, 61821, 54102, 42029, 57717) ||class="entry q2 g1"| 43720<sub>7</sub> ||class="entry q3 g1"| 56485<sub>9</sub> ||class="entry q3 g1"| 61821<sub>11</sub> ||class="entry q2 g1"| 54102<sub>9</sub> ||class="entry q3 g1"| 42029<sub>7</sub> ||class="entry q3 g1"| 57717<sub>9</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 11, 9, 7, 9) ||class="c"| (44200, 56005, 63261, 54582, 41549, 59157) ||class="entry q2 g1"| 44200<sub>7</sub> ||class="entry q3 g1"| 56005<sub>9</sub> ||class="entry q3 g1"| 63261<sub>11</sub> ||class="entry q2 g1"| 54582<sub>9</sub> ||class="entry q3 g1"| 41549<sub>7</sub> ||class="entry q3 g1"| 59157<sub>9</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 11, 9, 7, 9) ||class="c"| (51912, 48291, 63259, 45910, 50219, 59155) ||class="entry q2 g1"| 51912<sub>7</sub> ||class="entry q3 g1"| 48291<sub>9</sub> ||class="entry q3 g1"| 63259<sub>11</sub> ||class="entry q2 g1"| 45910<sub>9</sub> ||class="entry q3 g1"| 50219<sub>7</sub> ||class="entry q3 g1"| 59155<sub>9</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 11, 9, 7, 9) ||class="c"| (52392, 47811, 61819, 46390, 49739, 57715) ||class="entry q2 g1"| 52392<sub>7</sub> ||class="entry q3 g1"| 47811<sub>9</sub> ||class="entry q3 g1"| 61819<sub>11</sub> ||class="entry q2 g1"| 46390<sub>9</sub> ||class="entry q3 g1"| 49739<sub>7</sub> ||class="entry q3 g1"| 57715<sub>9</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 11, 9, 7, 9) ||class="c"| (62144, 60713, 39855, 35678, 38305, 35751) ||class="entry q2 g1"| 62144<sub>7</sub> ||class="entry q3 g1"| 60713<sub>9</sub> ||class="entry q3 g1"| 39855<sub>11</sub> ||class="entry q2 g1"| 35678<sub>9</sub> ||class="entry q3 g1"| 38305<sub>7</sub> ||class="entry q3 g1"| 35751<sub>9</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 11, 9, 7, 9) ||class="c"| (62624, 60233, 40399, 36158, 37825, 36295) ||class="entry q2 g1"| 62624<sub>7</sub> ||class="entry q3 g1"| 60233<sub>9</sub> ||class="entry q3 g1"| 40399<sub>11</sub> ||class="entry q2 g1"| 36158<sub>9</sub> ||class="entry q3 g1"| 37825<sub>7</sub> ||class="entry q3 g1"| 36295<sub>9</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 11, 9, 7, 9) ||class="c"| (64064, 58553, 40159, 33758, 39985, 36055) ||class="entry q2 g1"| 64064<sub>7</sub> ||class="entry q3 g1"| 58553<sub>9</sub> ||class="entry q3 g1"| 40159<sub>11</sub> ||class="entry q2 g1"| 33758<sub>9</sub> ||class="entry q3 g1"| 39985<sub>7</sub> ||class="entry q3 g1"| 36055<sub>9</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 11, 9, 7, 9) ||class="c"| (64544, 58073, 39615, 34238, 39505, 35511) ||class="entry q2 g1"| 64544<sub>7</sub> ||class="entry q3 g1"| 58073<sub>9</sub> ||class="entry q3 g1"| 39615<sub>11</sub> ||class="entry q2 g1"| 34238<sub>9</sub> ||class="entry q3 g1"| 39505<sub>7</sub> ||class="entry q3 g1"| 35511<sub>9</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 9, 9, 13, 11) ||class="c"| (37586, 55143, 51091, 60236, 45039, 55195) ||class="entry q2 g1"| 37586<sub>7</sub> ||class="entry q3 g1"| 55143<sub>11</sub> ||class="entry q3 g1"| 51091<sub>9</sub> ||class="entry q2 g1"| 60236<sub>9</sub> ||class="entry q3 g1"| 45039<sub>13</sub> ||class="entry q3 g1"| 55195<sub>11</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 9, 9, 13, 11) ||class="c"| (38068, 46951, 42901, 60714, 53231, 47005) ||class="entry q2 g1"| 38068<sub>7</sub> ||class="entry q3 g1"| 46951<sub>11</sub> ||class="entry q3 g1"| 42901<sub>9</sub> ||class="entry q2 g1"| 60714<sub>9</sub> ||class="entry q3 g1"| 53231<sub>13</sub> ||class="entry q3 g1"| 47005<sub>11</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 9, 9, 13, 11) ||class="c"| (39748, 46711, 43381, 58074, 52991, 47485) ||class="entry q2 g1"| 39748<sub>7</sub> ||class="entry q3 g1"| 46711<sub>11</sub> ||class="entry q3 g1"| 43381<sub>9</sub> ||class="entry q2 g1"| 58074<sub>9</sub> ||class="entry q3 g1"| 52991<sub>13</sub> ||class="entry q3 g1"| 47485<sub>11</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 9, 9, 13, 11) ||class="c"| (40226, 54903, 51571, 58556, 44799, 55675) ||class="entry q2 g1"| 40226<sub>7</sub> ||class="entry q3 g1"| 54903<sub>11</sub> ||class="entry q3 g1"| 51571<sub>9</sub> ||class="entry q2 g1"| 58556<sub>9</sub> ||class="entry q3 g1"| 44799<sub>13</sub> ||class="entry q3 g1"| 55675<sub>11</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 9, 9, 13, 11) ||class="c"| (41562, 36733, 44119, 56260, 63477, 48223) ||class="entry q2 g1"| 41562<sub>7</sub> ||class="entry q3 g1"| 36733<sub>11</sub> ||class="entry q3 g1"| 44119<sub>9</sub> ||class="entry q2 g1"| 56260<sub>9</sub> ||class="entry q3 g1"| 63477<sub>13</sub> ||class="entry q3 g1"| 48223<sub>11</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 9, 9, 13, 11) ||class="c"| (42282, 34813, 42439, 56500, 65397, 46543) ||class="entry q2 g1"| 42282<sub>7</sub> ||class="entry q3 g1"| 34813<sub>11</sub> ||class="entry q3 g1"| 42439<sub>9</sub> ||class="entry q2 g1"| 56500<sub>9</sub> ||class="entry q3 g1"| 65397<sub>13</sub> ||class="entry q3 g1"| 46543<sub>11</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 9, 9, 13, 11) ||class="c"| (49996, 34811, 50087, 47826, 65395, 54191) ||class="entry q2 g1"| 49996<sub>7</sub> ||class="entry q3 g1"| 34811<sub>11</sub> ||class="entry q3 g1"| 50087<sub>9</sub> ||class="entry q2 g1"| 47826<sub>9</sub> ||class="entry q3 g1"| 65395<sub>13</sub> ||class="entry q3 g1"| 54191<sub>11</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 9, 9, 13, 11) ||class="c"| (50236, 36731, 51767, 48546, 63475, 55871) ||class="entry q2 g1"| 50236<sub>7</sub> ||class="entry q3 g1"| 36731<sub>11</sub> ||class="entry q3 g1"| 51767<sub>9</sub> ||class="entry q2 g1"| 48546<sub>9</sub> ||class="entry q3 g1"| 63475<sub>13</sub> ||class="entry q3 g1"| 55871<sub>11</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 11, 9, 9, 9) ||class="c"| (33498, 44793, 48923, 64324, 54897, 44819) ||class="entry q2 g1"| 33498<sub>7</sub> ||class="entry q3 g1"| 44793<sub>11</sub> ||class="entry q3 g1"| 48923<sub>11</sub> ||class="entry q2 g1"| 64324<sub>9</sub> ||class="entry q3 g1"| 54897<sub>9</sub> ||class="entry q3 g1"| 44819<sub>9</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 11, 9, 9, 9) ||class="c"| (33980, 52985, 57117, 64802, 46705, 53013) ||class="entry q2 g1"| 33980<sub>7</sub> ||class="entry q3 g1"| 52985<sub>11</sub> ||class="entry q3 g1"| 57117<sub>11</sub> ||class="entry q2 g1"| 64802<sub>9</sub> ||class="entry q3 g1"| 46705<sub>9</sub> ||class="entry q3 g1"| 53013<sub>9</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 11, 9, 9, 9) ||class="c"| (35660, 53225, 53757, 62162, 46945, 49653) ||class="entry q2 g1"| 35660<sub>7</sub> ||class="entry q3 g1"| 53225<sub>11</sub> ||class="entry q3 g1"| 53757<sub>11</sub> ||class="entry q2 g1"| 62162<sub>9</sub> ||class="entry q3 g1"| 46945<sub>9</sub> ||class="entry q3 g1"| 49653<sub>9</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 11, 9, 9, 9) ||class="c"| (36138, 45033, 45563, 62644, 55137, 41459) ||class="entry q2 g1"| 36138<sub>7</sub> ||class="entry q3 g1"| 45033<sub>11</sub> ||class="entry q3 g1"| 45563<sub>11</sub> ||class="entry q2 g1"| 62644<sub>9</sub> ||class="entry q3 g1"| 55137<sub>9</sub> ||class="entry q3 g1"| 41459<sub>9</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 11, 9, 9, 9) ||class="c"| (45650, 63203, 54495, 52172, 36459, 50391) ||class="entry q2 g1"| 45650<sub>7</sub> ||class="entry q3 g1"| 63203<sub>11</sub> ||class="entry q3 g1"| 54495<sub>11</sub> ||class="entry q2 g1"| 52172<sub>9</sub> ||class="entry q3 g1"| 36459<sub>9</sub> ||class="entry q3 g1"| 50391<sub>9</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 11, 9, 9, 9) ||class="c"| (46370, 65123, 56655, 52412, 34539, 52551) ||class="entry q2 g1"| 46370<sub>7</sub> ||class="entry q3 g1"| 65123<sub>11</sub> ||class="entry q3 g1"| 56655<sub>11</sub> ||class="entry q2 g1"| 52412<sub>9</sub> ||class="entry q3 g1"| 34539<sub>9</sub> ||class="entry q3 g1"| 52551<sub>9</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 11, 9, 9, 9) ||class="c"| (54084, 65125, 47919, 43738, 34541, 43815) ||class="entry q2 g1"| 54084<sub>7</sub> ||class="entry q3 g1"| 65125<sub>11</sub> ||class="entry q3 g1"| 47919<sub>11</sub> ||class="entry q2 g1"| 43738<sub>9</sub> ||class="entry q3 g1"| 34541<sub>9</sub> ||class="entry q3 g1"| 43815<sub>9</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 11, 9, 9, 9) ||class="c"| (54324, 63205, 45759, 44458, 36461, 41655) ||class="entry q2 g1"| 54324<sub>7</sub> ||class="entry q3 g1"| 63205<sub>11</sub> ||class="entry q3 g1"| 45759<sub>11</sub> ||class="entry q2 g1"| 44458<sub>9</sub> ||class="entry q3 g1"| 36461<sub>9</sub> ||class="entry q3 g1"| 41655<sub>9</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 13, 9, 9, 11) ||class="c"| (37574, 60239, 64431, 60248, 37831, 60327) ||class="entry q2 g1"| 37574<sub>7</sub> ||class="entry q3 g1"| 60239<sub>11</sub> ||class="entry q3 g1"| 64431<sub>13</sub> ||class="entry q2 g1"| 60248<sub>9</sub> ||class="entry q3 g1"| 37831<sub>9</sub> ||class="entry q3 g1"| 60327<sub>11</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 13, 9, 9, 11) ||class="c"| (38054, 60719, 64975, 60728, 38311, 60871) ||class="entry q2 g1"| 38054<sub>7</sub> ||class="entry q3 g1"| 60719<sub>11</sub> ||class="entry q3 g1"| 64975<sub>13</sub> ||class="entry q2 g1"| 60728<sub>9</sub> ||class="entry q3 g1"| 38311<sub>9</sub> ||class="entry q3 g1"| 60871<sub>11</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 13, 9, 9, 11) ||class="c"| (39494, 58079, 64735, 58328, 39511, 60631) ||class="entry q2 g1"| 39494<sub>7</sub> ||class="entry q3 g1"| 58079<sub>11</sub> ||class="entry q3 g1"| 64735<sub>13</sub> ||class="entry q2 g1"| 58328<sub>9</sub> ||class="entry q3 g1"| 39511<sub>9</sub> ||class="entry q3 g1"| 60631<sub>11</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 13, 9, 9, 11) ||class="c"| (39974, 58559, 64191, 58808, 39991, 60087) ||class="entry q2 g1"| 39974<sub>7</sub> ||class="entry q3 g1"| 58559<sub>11</sub> ||class="entry q3 g1"| 64191<sub>13</sub> ||class="entry q2 g1"| 58808<sub>9</sub> ||class="entry q3 g1"| 39991<sub>9</sub> ||class="entry q3 g1"| 60087<sub>11</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 13, 9, 9, 11) ||class="c"| (41816, 56277, 63997, 56006, 41821, 59893) ||class="entry q2 g1"| 41816<sub>7</sub> ||class="entry q3 g1"| 56277<sub>11</sub> ||class="entry q3 g1"| 63997<sub>13</sub> ||class="entry q2 g1"| 56006<sub>9</sub> ||class="entry q3 g1"| 41821<sub>9</sub> ||class="entry q3 g1"| 59893<sub>11</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 13, 9, 9, 11) ||class="c"| (42296, 56757, 65437, 56486, 42301, 61333) ||class="entry q2 g1"| 42296<sub>7</sub> ||class="entry q3 g1"| 56757<sub>11</sub> ||class="entry q3 g1"| 65437<sub>13</sub> ||class="entry q2 g1"| 56486<sub>9</sub> ||class="entry q3 g1"| 42301<sub>9</sub> ||class="entry q3 g1"| 61333<sub>11</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 13, 9, 9, 11) ||class="c"| (50008, 48083, 65435, 47814, 50011, 61331) ||class="entry q2 g1"| 50008<sub>7</sub> ||class="entry q3 g1"| 48083<sub>11</sub> ||class="entry q3 g1"| 65435<sub>13</sub> ||class="entry q2 g1"| 47814<sub>9</sub> ||class="entry q3 g1"| 50011<sub>9</sub> ||class="entry q3 g1"| 61331<sub>11</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 13, 9, 9, 11) ||class="c"| (50488, 48563, 63995, 48294, 50491, 59891) ||class="entry q2 g1"| 50488<sub>7</sub> ||class="entry q3 g1"| 48563<sub>11</sub> ||class="entry q3 g1"| 63995<sub>13</sub> ||class="entry q2 g1"| 48294<sub>9</sub> ||class="entry q3 g1"| 50491<sub>9</sub> ||class="entry q3 g1"| 59891<sub>11</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 13, 7, 9, 11, 9) ||class="c"| (37828, 49127, 44549, 59994, 51055, 48653) ||class="entry q2 g1"| 37828<sub>7</sub> ||class="entry q3 g1"| 49127<sub>13</sub> ||class="entry q3 g1"| 44549<sub>7</sub> ||class="entry q2 g1"| 59994<sub>9</sub> ||class="entry q3 g1"| 51055<sub>11</sub> ||class="entry q3 g1"| 48653<sub>9</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 13, 7, 9, 11, 9) ||class="c"| (38306, 57319, 52739, 60476, 42863, 56843) ||class="entry q2 g1"| 38306<sub>7</sub> ||class="entry q3 g1"| 57319<sub>13</sub> ||class="entry q3 g1"| 52739<sub>7</sub> ||class="entry q2 g1"| 60476<sub>9</sub> ||class="entry q3 g1"| 42863<sub>11</sub> ||class="entry q3 g1"| 56843<sub>9</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 13, 7, 9, 11, 9) ||class="c"| (39506, 57079, 49379, 58316, 42623, 53483) ||class="entry q2 g1"| 39506<sub>7</sub> ||class="entry q3 g1"| 57079<sub>13</sub> ||class="entry q3 g1"| 49379<sub>7</sub> ||class="entry q2 g1"| 58316<sub>9</sub> ||class="entry q3 g1"| 42623<sub>11</sub> ||class="entry q3 g1"| 53483<sub>9</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 13, 7, 9, 11, 9) ||class="c"| (39988, 48887, 41189, 58794, 50815, 45293) ||class="entry q2 g1"| 39988<sub>7</sub> ||class="entry q3 g1"| 48887<sub>13</sub> ||class="entry q3 g1"| 41189<sub>7</sub> ||class="entry q2 g1"| 58794<sub>9</sub> ||class="entry q3 g1"| 50815<sub>11</sub> ||class="entry q3 g1"| 45293<sub>9</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 13, 7, 9, 11, 9) ||class="c"| (41804, 59389, 50625, 56018, 40821, 54729) ||class="entry q2 g1"| 41804<sub>7</sub> ||class="entry q3 g1"| 59389<sub>13</sub> ||class="entry q3 g1"| 50625<sub>7</sub> ||class="entry q2 g1"| 56018<sub>9</sub> ||class="entry q3 g1"| 40821<sub>11</sub> ||class="entry q3 g1"| 54729<sub>9</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 13, 7, 9, 11, 9) ||class="c"| (42044, 61309, 52305, 56738, 38901, 56409) ||class="entry q2 g1"| 42044<sub>7</sub> ||class="entry q3 g1"| 61309<sub>13</sub> ||class="entry q3 g1"| 52305<sub>7</sub> ||class="entry q2 g1"| 56738<sub>9</sub> ||class="entry q3 g1"| 38901<sub>11</sub> ||class="entry q3 g1"| 56409<sub>9</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 13, 7, 9, 11, 9) ||class="c"| (49754, 61307, 43569, 48068, 38899, 47673) ||class="entry q2 g1"| 49754<sub>7</sub> ||class="entry q3 g1"| 61307<sub>13</sub> ||class="entry q3 g1"| 43569<sub>7</sub> ||class="entry q2 g1"| 48068<sub>9</sub> ||class="entry q3 g1"| 38899<sub>11</sub> ||class="entry q3 g1"| 47673<sub>9</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 13, 7, 9, 11, 9) ||class="c"| (50474, 59387, 41889, 48308, 40819, 45993) ||class="entry q2 g1"| 50474<sub>7</sub> ||class="entry q3 g1"| 59387<sub>13</sub> ||class="entry q3 g1"| 41889<sub>7</sub> ||class="entry q2 g1"| 48308<sub>9</sub> ||class="entry q3 g1"| 40819<sub>11</sub> ||class="entry q3 g1"| 45993<sub>9</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 7, 7, 11, 9, 9) ||class="c"| (38882, 46129, 42179, 61052, 52409, 46283) ||class="entry q2 g1"| 38882<sub>9</sub> ||class="entry q3 g1"| 46129<sub>7</sub> ||class="entry q3 g1"| 42179<sub>7</sub> ||class="entry q2 g1"| 61052<sub>11</sub> ||class="entry q3 g1"| 52409<sub>9</sub> ||class="entry q3 g1"| 46283<sub>9</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 7, 7, 11, 9, 9) ||class="c"| (38884, 53841, 49829, 61050, 43737, 53933) ||class="entry q2 g1"| 38884<sub>9</sub> ||class="entry q3 g1"| 53841<sub>7</sub> ||class="entry q3 g1"| 49829<sub>7</sub> ||class="entry q2 g1"| 61050<sub>11</sub> ||class="entry q3 g1"| 43737<sub>9</sub> ||class="entry q3 g1"| 53933<sub>9</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 7, 7, 11, 9, 9) ||class="c"| (40562, 45889, 44099, 59372, 52169, 48203) ||class="entry q2 g1"| 40562<sub>9</sub> ||class="entry q3 g1"| 45889<sub>7</sub> ||class="entry q3 g1"| 44099<sub>7</sub> ||class="entry q2 g1"| 59372<sub>11</sub> ||class="entry q3 g1"| 52169<sub>9</sub> ||class="entry q3 g1"| 48203<sub>9</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 7, 7, 11, 9, 9) ||class="c"| (40564, 54561, 51749, 59370, 44457, 55853) ||class="entry q2 g1"| 40564<sub>9</sub> ||class="entry q3 g1"| 54561<sub>7</sub> ||class="entry q3 g1"| 51749<sub>7</sub> ||class="entry q2 g1"| 59370<sub>11</sub> ||class="entry q3 g1"| 44457<sub>9</sub> ||class="entry q3 g1"| 55853<sub>9</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 7, 7, 11, 9, 9) ||class="c"| (42620, 33963, 42641, 57314, 64547, 46745) ||class="entry q2 g1"| 42620<sub>9</sub> ||class="entry q3 g1"| 33963<sub>7</sub> ||class="entry q3 g1"| 42641<sub>7</sub> ||class="entry q2 g1"| 57314<sub>11</sub> ||class="entry q3 g1"| 64547<sub>9</sub> ||class="entry q3 g1"| 46745<sub>9</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 7, 7, 11, 9, 9) ||class="c"| (42860, 35403, 43361, 57074, 62147, 47465) ||class="entry q2 g1"| 42860<sub>9</sub> ||class="entry q3 g1"| 35403<sub>7</sub> ||class="entry q3 g1"| 43361<sub>7</sub> ||class="entry q2 g1"| 57074<sub>11</sub> ||class="entry q3 g1"| 62147<sub>9</sub> ||class="entry q3 g1"| 47465<sub>9</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 7, 7, 11, 9, 9) ||class="c"| (50810, 33485, 50833, 49124, 64069, 54937) ||class="entry q2 g1"| 50810<sub>9</sub> ||class="entry q3 g1"| 33485<sub>7</sub> ||class="entry q3 g1"| 50833<sub>7</sub> ||class="entry q2 g1"| 49124<sub>11</sub> ||class="entry q3 g1"| 64069<sub>9</sub> ||class="entry q3 g1"| 54937<sub>9</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 7, 7, 11, 9, 9) ||class="c"| (51050, 35885, 51553, 48884, 62629, 55657) ||class="entry q2 g1"| 51050<sub>9</sub> ||class="entry q3 g1"| 35885<sub>7</sub> ||class="entry q3 g1"| 51553<sub>7</sub> ||class="entry q2 g1"| 48884<sub>11</sub> ||class="entry q3 g1"| 62629<sub>9</sub> ||class="entry q3 g1"| 55657<sub>9</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 7, 9, 11, 5, 11) ||class="c"| (48864, 51341, 58709, 51070, 45061, 62813) ||class="entry q2 g1"| 48864<sub>9</sub> ||class="entry q3 g1"| 51341<sub>7</sub> ||class="entry q3 g1"| 58709<sub>9</sub> ||class="entry q2 g1"| 51070<sub>11</sub> ||class="entry q3 g1"| 45061<sub>5</sub> ||class="entry q3 g1"| 62813<sub>11</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 7, 9, 11, 5, 11) ||class="c"| (57056, 43147, 58163, 42878, 53251, 62267) ||class="entry q2 g1"| 57056<sub>9</sub> ||class="entry q3 g1"| 43147<sub>7</sub> ||class="entry q3 g1"| 58163<sub>9</sub> ||class="entry q2 g1"| 42878<sub>11</sub> ||class="entry q3 g1"| 53251<sub>5</sub> ||class="entry q3 g1"| 62267<sub>11</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 7, 9, 11, 5, 11) ||class="c"| (59112, 63745, 36743, 40822, 33161, 40847) ||class="entry q2 g1"| 59112<sub>9</sub> ||class="entry q3 g1"| 63745<sub>7</sub> ||class="entry q3 g1"| 36743<sub>9</sub> ||class="entry q2 g1"| 40822<sub>11</sub> ||class="entry q3 g1"| 33161<sub>5</sub> ||class="entry q3 g1"| 40847<sub>11</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 7, 9, 11, 5, 11) ||class="c"| (61032, 61585, 35063, 38902, 34841, 39167) ||class="entry q2 g1"| 61032<sub>9</sub> ||class="entry q3 g1"| 61585<sub>7</sub> ||class="entry q3 g1"| 35063<sub>9</sub> ||class="entry q2 g1"| 38902<sub>11</sub> ||class="entry q3 g1"| 34841<sub>5</sub> ||class="entry q3 g1"| 39167<sub>11</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 7, 11, 11, 9, 9) ||class="c"| (44776, 45331, 40413, 55158, 51611, 36309) ||class="entry q2 g1"| 44776<sub>9</sub> ||class="entry q3 g1"| 45331<sub>7</sub> ||class="entry q3 g1"| 40413<sub>11</sub> ||class="entry q2 g1"| 55158<sub>11</sub> ||class="entry q3 g1"| 51611<sub>9</sub> ||class="entry q3 g1"| 36309<sub>9</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 7, 11, 11, 9, 9) ||class="c"| (52968, 53525, 39867, 46966, 43421, 35763) ||class="entry q2 g1"| 52968<sub>9</sub> ||class="entry q3 g1"| 53525<sub>7</sub> ||class="entry q3 g1"| 39867<sub>11</sub> ||class="entry q2 g1"| 46966<sub>11</sub> ||class="entry q3 g1"| 43421<sub>9</sub> ||class="entry q3 g1"| 35763<sub>9</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 7, 11, 11, 9, 9) ||class="c"| (63200, 32927, 63247, 36734, 63511, 59143) ||class="entry q2 g1"| 63200<sub>9</sub> ||class="entry q3 g1"| 32927<sub>7</sub> ||class="entry q3 g1"| 63247<sub>11</sub> ||class="entry q2 g1"| 36734<sub>11</sub> ||class="entry q3 g1"| 63511<sub>9</sub> ||class="entry q3 g1"| 59143<sub>9</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 7, 11, 11, 9, 9) ||class="c"| (65120, 35087, 61567, 34814, 61831, 57463) ||class="entry q2 g1"| 65120<sub>9</sub> ||class="entry q3 g1"| 35087<sub>7</sub> ||class="entry q3 g1"| 61567<sub>11</sub> ||class="entry q2 g1"| 34814<sub>11</sub> ||class="entry q3 g1"| 61831<sub>9</sub> ||class="entry q3 g1"| 57463<sub>9</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 9, 9, 11, 7, 11) ||class="c"| (38642, 47825, 43827, 61292, 49753, 47931) ||class="entry q2 g1"| 38642<sub>9</sub> ||class="entry q3 g1"| 47825<sub>9</sub> ||class="entry q3 g1"| 43827<sub>9</sub> ||class="entry q2 g1"| 61292<sub>11</sub> ||class="entry q3 g1"| 49753<sub>7</sub> ||class="entry q3 g1"| 47931<sub>11</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 9, 9, 11, 7, 11) ||class="c"| (38644, 56497, 52565, 61290, 42041, 56669) ||class="entry q2 g1"| 38644<sub>9</sub> ||class="entry q3 g1"| 56497<sub>9</sub> ||class="entry q3 g1"| 52565<sub>9</sub> ||class="entry q2 g1"| 61290<sub>11</sub> ||class="entry q3 g1"| 42041<sub>7</sub> ||class="entry q3 g1"| 56669<sub>11</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 9, 9, 11, 7, 11) ||class="c"| (40802, 48545, 41907, 59132, 50473, 46011) ||class="entry q2 g1"| 40802<sub>9</sub> ||class="entry q3 g1"| 48545<sub>9</sub> ||class="entry q3 g1"| 41907<sub>9</sub> ||class="entry q2 g1"| 59132<sub>11</sub> ||class="entry q3 g1"| 50473<sub>7</sub> ||class="entry q3 g1"| 46011<sub>11</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 9, 9, 11, 7, 11) ||class="c"| (40804, 56257, 50645, 59130, 41801, 54749) ||class="entry q2 g1"| 40804<sub>9</sub> ||class="entry q3 g1"| 56257<sub>9</sub> ||class="entry q3 g1"| 50645<sub>9</sub> ||class="entry q2 g1"| 59130<sub>11</sub> ||class="entry q3 g1"| 41801<sub>7</sub> ||class="entry q3 g1"| 54749<sub>11</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 9, 9, 11, 7, 11) ||class="c"| (42618, 58059, 49399, 57316, 39491, 53503) ||class="entry q2 g1"| 42618<sub>9</sub> ||class="entry q3 g1"| 58059<sub>9</sub> ||class="entry q3 g1"| 49399<sub>9</sub> ||class="entry q2 g1"| 57316<sub>11</sub> ||class="entry q3 g1"| 39491<sub>7</sub> ||class="entry q3 g1"| 53503<sub>11</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 9, 9, 11, 7, 11) ||class="c"| (42858, 60459, 52999, 57076, 38051, 57103) ||class="entry q2 g1"| 42858<sub>9</sub> ||class="entry q3 g1"| 60459<sub>9</sub> ||class="entry q3 g1"| 52999<sub>9</sub> ||class="entry q2 g1"| 57076<sub>11</sub> ||class="entry q3 g1"| 38051<sub>7</sub> ||class="entry q3 g1"| 57103<sub>11</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 9, 9, 11, 7, 11) ||class="c"| (50812, 58541, 41207, 49122, 39973, 45311) ||class="entry q2 g1"| 50812<sub>9</sub> ||class="entry q3 g1"| 58541<sub>9</sub> ||class="entry q3 g1"| 41207<sub>9</sub> ||class="entry q2 g1"| 49122<sub>11</sub> ||class="entry q3 g1"| 39973<sub>7</sub> ||class="entry q3 g1"| 45311<sub>11</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 9, 9, 11, 7, 11) ||class="c"| (51052, 59981, 44807, 48882, 37573, 48911) ||class="entry q2 g1"| 51052<sub>9</sub> ||class="entry q3 g1"| 59981<sub>9</sub> ||class="entry q3 g1"| 44807<sub>9</sub> ||class="entry q2 g1"| 48882<sub>11</sub> ||class="entry q3 g1"| 37573<sub>7</sub> ||class="entry q3 g1"| 48911<sub>11</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 9, 9, 11, 11, 7) ||class="c"| (38630, 34553, 38671, 61304, 65137, 34567) ||class="entry q2 g1"| 38630<sub>9</sub> ||class="entry q3 g1"| 34553<sub>9</sub> ||class="entry q3 g1"| 38671<sub>9</sub> ||class="entry q2 g1"| 61304<sub>11</sub> ||class="entry q3 g1"| 65137<sub>11</sub> ||class="entry q3 g1"| 34567<sub>7</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 9, 9, 11, 11, 7) ||class="c"| (40550, 36713, 36991, 59384, 63457, 32887) ||class="entry q2 g1"| 40550<sub>9</sub> ||class="entry q3 g1"| 36713<sub>9</sub> ||class="entry q3 g1"| 36991<sub>9</sub> ||class="entry q2 g1"| 59384<sub>11</sub> ||class="entry q3 g1"| 63457<sub>11</sub> ||class="entry q3 g1"| 32887<sub>7</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 9, 9, 11, 11, 7) ||class="c"| (42872, 46691, 38237, 57062, 52971, 34133) ||class="entry q2 g1"| 42872<sub>9</sub> ||class="entry q3 g1"| 46691<sub>9</sub> ||class="entry q3 g1"| 38237<sub>9</sub> ||class="entry q2 g1"| 57062<sub>11</sub> ||class="entry q3 g1"| 52971<sub>11</sub> ||class="entry q3 g1"| 34133<sub>7</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 9, 9, 11, 11, 7) ||class="c"| (51064, 54885, 37691, 48870, 44781, 33587) ||class="entry q2 g1"| 51064<sub>9</sub> ||class="entry q3 g1"| 54885<sub>9</sub> ||class="entry q3 g1"| 37691<sub>9</sub> ||class="entry q2 g1"| 48870<sub>11</sub> ||class="entry q3 g1"| 44781<sub>11</sub> ||class="entry q3 g1"| 33587<sub>7</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 9, 11, 11, 11, 9) ||class="c"| (34554, 49999, 54203, 65380, 48071, 50099) ||class="entry q2 g1"| 34554<sub>9</sub> ||class="entry q3 g1"| 49999<sub>9</sub> ||class="entry q3 g1"| 54203<sub>11</sub> ||class="entry q2 g1"| 65380<sub>11</sub> ||class="entry q3 g1"| 48071<sub>11</sub> ||class="entry q3 g1"| 50099<sub>9</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 9, 11, 11, 11, 9) ||class="c"| (34556, 42287, 46557, 65378, 56743, 42453) ||class="entry q2 g1"| 34556<sub>9</sub> ||class="entry q3 g1"| 42287<sub>9</sub> ||class="entry q3 g1"| 46557<sub>11</sub> ||class="entry q2 g1"| 65378<sub>11</sub> ||class="entry q3 g1"| 56743<sub>11</sub> ||class="entry q3 g1"| 42453<sub>9</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 9, 11, 11, 11, 9) ||class="c"| (36714, 50239, 56123, 63220, 48311, 52019) ||class="entry q2 g1"| 36714<sub>9</sub> ||class="entry q3 g1"| 50239<sub>9</sub> ||class="entry q3 g1"| 56123<sub>11</sub> ||class="entry q2 g1"| 63220<sub>11</sub> ||class="entry q3 g1"| 48311<sub>11</sub> ||class="entry q3 g1"| 52019<sub>9</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 9, 11, 11, 11, 9) ||class="c"| (36716, 41567, 48477, 63218, 56023, 44373) ||class="entry q2 g1"| 36716<sub>9</sub> ||class="entry q3 g1"| 41567<sub>9</sub> ||class="entry q3 g1"| 48477<sub>11</sub> ||class="entry q2 g1"| 63218<sub>11</sub> ||class="entry q3 g1"| 56023<sub>11</sub> ||class="entry q3 g1"| 44373<sub>9</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 9, 11, 11, 11, 9) ||class="c"| (46706, 39765, 47231, 53228, 58333, 43127) ||class="entry q2 g1"| 46706<sub>9</sub> ||class="entry q3 g1"| 39765<sub>9</sub> ||class="entry q3 g1"| 47231<sub>11</sub> ||class="entry q2 g1"| 53228<sub>11</sub> ||class="entry q3 g1"| 58333<sub>11</sub> ||class="entry q3 g1"| 43127<sub>9</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 9, 11, 11, 11, 9) ||class="c"| (46946, 38325, 46991, 52988, 60733, 42887) ||class="entry q2 g1"| 46946<sub>9</sub> ||class="entry q3 g1"| 38325<sub>9</sub> ||class="entry q3 g1"| 46991<sub>11</sub> ||class="entry q2 g1"| 52988<sub>11</sub> ||class="entry q3 g1"| 60733<sub>11</sub> ||class="entry q3 g1"| 42887<sub>9</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 9, 11, 11, 11, 9) ||class="c"| (54900, 40243, 55423, 45034, 58811, 51319) ||class="entry q2 g1"| 54900<sub>9</sub> ||class="entry q3 g1"| 40243<sub>9</sub> ||class="entry q3 g1"| 55423<sub>11</sub> ||class="entry q2 g1"| 45034<sub>11</sub> ||class="entry q3 g1"| 58811<sub>11</sub> ||class="entry q3 g1"| 51319<sub>9</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 9, 11, 11, 11, 9) ||class="c"| (55140, 37843, 55183, 44794, 60251, 51079) ||class="entry q2 g1"| 55140<sub>9</sub> ||class="entry q3 g1"| 37843<sub>9</sub> ||class="entry q3 g1"| 55183<sub>11</sub> ||class="entry q2 g1"| 44794<sub>11</sub> ||class="entry q3 g1"| 60251<sub>11</sub> ||class="entry q3 g1"| 51079<sub>9</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 11, 5, 11, 13, 7) ||class="c"| (34808, 38887, 34321, 65126, 61295, 38425) ||class="entry q2 g1"| 34808<sub>9</sub> ||class="entry q3 g1"| 38887<sub>11</sub> ||class="entry q3 g1"| 34321<sub>5</sub> ||class="entry q2 g1"| 65126<sub>11</sub> ||class="entry q3 g1"| 61295<sub>13</sub> ||class="entry q3 g1"| 38425<sub>7</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 11, 5, 11, 13, 7) ||class="c"| (36728, 40567, 33121, 63206, 59135, 37225) ||class="entry q2 g1"| 36728<sub>9</sub> ||class="entry q3 g1"| 40567<sub>11</sub> ||class="entry q3 g1"| 33121<sub>5</sub> ||class="entry q2 g1"| 63206<sub>11</sub> ||class="entry q3 g1"| 59135<sub>13</sub> ||class="entry q3 g1"| 37225<sub>7</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 11, 5, 11, 13, 7) ||class="c"| (46694, 42877, 33859, 53240, 57333, 37963) ||class="entry q2 g1"| 46694<sub>9</sub> ||class="entry q3 g1"| 42877<sub>11</sub> ||class="entry q3 g1"| 33859<sub>5</sub> ||class="entry q2 g1"| 53240<sub>11</sub> ||class="entry q3 g1"| 57333<sub>13</sub> ||class="entry q3 g1"| 37963<sub>7</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 11, 5, 11, 13, 7) ||class="c"| (54886, 51067, 33317, 45048, 49139, 37421) ||class="entry q2 g1"| 54886<sub>9</sub> ||class="entry q3 g1"| 51067<sub>11</sub> ||class="entry q3 g1"| 33317<sub>5</sub> ||class="entry q2 g1"| 45048<sub>11</sub> ||class="entry q3 g1"| 49139<sub>13</sub> ||class="entry q3 g1"| 37421<sub>7</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 11, 9, 11, 9, 7) ||class="c"| (34794, 52655, 56395, 65140, 46375, 52291) ||class="entry q2 g1"| 34794<sub>9</sub> ||class="entry q3 g1"| 52655<sub>11</sub> ||class="entry q3 g1"| 56395<sub>9</sub> ||class="entry q2 g1"| 65140<sub>11</sub> ||class="entry q3 g1"| 46375<sub>9</sub> ||class="entry q3 g1"| 52291<sub>7</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 11, 9, 11, 9, 7) ||class="c"| (34796, 43983, 47661, 65138, 54087, 43557) ||class="entry q2 g1"| 34796<sub>9</sub> ||class="entry q3 g1"| 43983<sub>11</sub> ||class="entry q3 g1"| 47661<sub>9</sub> ||class="entry q2 g1"| 65138<sub>11</sub> ||class="entry q3 g1"| 54087<sub>9</sub> ||class="entry q3 g1"| 43557<sub>7</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 11, 9, 11, 9, 7) ||class="c"| (36474, 51935, 54475, 63460, 45655, 50371) ||class="entry q2 g1"| 36474<sub>9</sub> ||class="entry q3 g1"| 51935<sub>11</sub> ||class="entry q3 g1"| 54475<sub>9</sub> ||class="entry q2 g1"| 63460<sub>11</sub> ||class="entry q3 g1"| 45655<sub>9</sub> ||class="entry q3 g1"| 50371<sub>7</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 11, 9, 11, 9, 7) ||class="c"| (36476, 44223, 45741, 63458, 54327, 41637) ||class="entry q2 g1"| 36476<sub>9</sub> ||class="entry q3 g1"| 44223<sub>11</sub> ||class="entry q3 g1"| 45741<sub>9</sub> ||class="entry q2 g1"| 63458<sub>11</sub> ||class="entry q3 g1"| 54327<sub>9</sub> ||class="entry q3 g1"| 41637<sub>7</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 11, 9, 11, 9, 7) ||class="c"| (46708, 64821, 56857, 53226, 34237, 52753) ||class="entry q2 g1"| 46708<sub>9</sub> ||class="entry q3 g1"| 64821<sub>11</sub> ||class="entry q3 g1"| 56857<sub>9</sub> ||class="entry q2 g1"| 53226<sub>11</sub> ||class="entry q3 g1"| 34237<sub>9</sub> ||class="entry q3 g1"| 52753<sub>7</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 11, 9, 11, 9, 7) ||class="c"| (46948, 62421, 53737, 52986, 35677, 49633) ||class="entry q2 g1"| 46948<sub>9</sub> ||class="entry q3 g1"| 62421<sub>11</sub> ||class="entry q3 g1"| 53737<sub>9</sub> ||class="entry q2 g1"| 52986<sub>11</sub> ||class="entry q3 g1"| 35677<sub>9</sub> ||class="entry q3 g1"| 49633<sub>7</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 11, 9, 11, 9, 7) ||class="c"| (54898, 64339, 48665, 45036, 33755, 44561) ||class="entry q2 g1"| 54898<sub>9</sub> ||class="entry q3 g1"| 64339<sub>11</sub> ||class="entry q3 g1"| 48665<sub>9</sub> ||class="entry q2 g1"| 45036<sub>11</sub> ||class="entry q3 g1"| 33755<sub>9</sub> ||class="entry q3 g1"| 44561<sub>7</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 11, 9, 11, 9, 7) ||class="c"| (55138, 62899, 45545, 44796, 36155, 41441) ||class="entry q2 g1"| 55138<sub>9</sub> ||class="entry q3 g1"| 62899<sub>11</sub> ||class="entry q3 g1"| 45545<sub>9</sub> ||class="entry q2 g1"| 44796<sub>11</sub> ||class="entry q3 g1"| 36155<sub>9</sub> ||class="entry q3 g1"| 41441<sub>7</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 11, 11, 11, 9, 9) ||class="c"| (38896, 61049, 65177, 61038, 38641, 61073) ||class="entry q2 g1"| 38896<sub>9</sub> ||class="entry q3 g1"| 61049<sub>11</sub> ||class="entry q3 g1"| 65177<sub>11</sub> ||class="entry q2 g1"| 61038<sub>11</sub> ||class="entry q3 g1"| 38641<sub>9</sub> ||class="entry q3 g1"| 61073<sub>9</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 11, 11, 11, 9, 9) ||class="c"| (40816, 59369, 63977, 59118, 40801, 59873) ||class="entry q2 g1"| 40816<sub>9</sub> ||class="entry q3 g1"| 59369<sub>11</sub> ||class="entry q3 g1"| 63977<sub>11</sub> ||class="entry q2 g1"| 59118<sub>11</sub> ||class="entry q3 g1"| 40801<sub>9</sub> ||class="entry q3 g1"| 59873<sub>9</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 11, 11, 11, 9, 9) ||class="c"| (42606, 57059, 64715, 57328, 42603, 60611) ||class="entry q2 g1"| 42606<sub>9</sub> ||class="entry q3 g1"| 57059<sub>11</sub> ||class="entry q3 g1"| 64715<sub>11</sub> ||class="entry q2 g1"| 57328<sub>11</sub> ||class="entry q3 g1"| 42603<sub>9</sub> ||class="entry q3 g1"| 60611<sub>9</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 11, 11, 11, 9, 9) ||class="c"| (50798, 48869, 64173, 49136, 50797, 60069) ||class="entry q2 g1"| 50798<sub>9</sub> ||class="entry q3 g1"| 48869<sub>11</sub> ||class="entry q3 g1"| 64173<sub>11</sub> ||class="entry q2 g1"| 49136<sub>11</sub> ||class="entry q3 g1"| 50797<sub>9</sub> ||class="entry q3 g1"| 60069<sub>9</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 13, 11, 11, 11, 13) ||class="c"| (34542, 65383, 61319, 65392, 34799, 65423) ||class="entry q2 g1"| 34542<sub>9</sub> ||class="entry q3 g1"| 65383<sub>13</sub> ||class="entry q3 g1"| 61319<sub>11</sub> ||class="entry q2 g1"| 65392<sub>11</sub> ||class="entry q3 g1"| 34799<sub>11</sub> ||class="entry q3 g1"| 65423<sub>13</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 13, 11, 11, 11, 13) ||class="c"| (36462, 63223, 59639, 63472, 36479, 63743) ||class="entry q2 g1"| 36462<sub>9</sub> ||class="entry q3 g1"| 63223<sub>13</sub> ||class="entry q3 g1"| 59639<sub>11</sub> ||class="entry q2 g1"| 63472<sub>11</sub> ||class="entry q3 g1"| 36479<sub>11</sub> ||class="entry q3 g1"| 63743<sub>13</sub>
|-
|class="f"| 4382 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (9, 13, 11, 11, 11, 13) ||class="c"| (46960, 53245, 60885, 52974, 46965, 64989) ||class="entry q2 g1"| 46960<sub>9</sub> ||class="entry q3 g1"| 53245<sub>13</sub> ||class="entry q3 g1"| 60885<sub>11</sub> ||class="entry q2 g1"| 52974<sub>11</sub> ||class="entry q3 g1"| 46965<sub>11</sub> ||class="entry q3 g1"| 64989<sub>13</sub>
|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
|class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (742, 11397, 22193, 30766, 22363, 17903) ||class="entry q0 g0"| 742<sub>6</sub> ||class="entry q1 g0"| 11397<sub>6</sub> ||class="entry q1 g0"| 22193<sub>8</sub> ||class="entry q0 g0"| 30766<sub>8</sub> ||class="entry q1 g0"| 22363<sub>10</sub> ||class="entry q1 g0"| 17903<sub>10</sub>
|-
|class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (754, 4269, 27277, 30778, 27507, 31187) ||class="entry q0 g0"| 754<sub>6</sub> ||class="entry q1 g0"| 4269<sub>6</sub> ||class="entry q1 g0"| 27277<sub>8</sub> ||class="entry q0 g0"| 30778<sub>8</sub> ||class="entry q1 g0"| 27507<sub>10</sub> ||class="entry q1 g0"| 31187<sub>10</sub>
|-
|class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (3654, 18595, 15713, 29838, 13181, 11839) ||class="entry q0 g0"| 3654<sub>6</sub> ||class="entry q1 g0"| 18595<sub>6</sub> ||class="entry q1 g0"| 15713<sub>8</sub> ||class="entry q0 g0"| 29838<sub>8</sub> ||class="entry q1 g0"| 13181<sub>10</sub> ||class="entry q1 g0"| 11839<sub>10</sub>
|-
|class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (3908, 7179, 26827, 30092, 26581, 31637) ||class="entry q0 g0"| 3908<sub>6</sub> ||class="entry q1 g0"| 7179<sub>6</sub> ||class="entry q1 g0"| 26827<sub>8</sub> ||class="entry q0 g0"| 30092<sub>8</sub> ||class="entry q1 g0"| 26581<sub>10</sub> ||class="entry q1 g0"| 31637<sub>10</sub>
|-
|class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (4282, 741, 30917, 27250, 31035, 27547) ||class="entry q0 g0"| 4282<sub>6</sub> ||class="entry q1 g0"| 741<sub>6</sub> ||class="entry q1 g0"| 30917<sub>8</sub> ||class="entry q0 g0"| 27250<sub>8</sub> ||class="entry q1 g0"| 31035<sub>10</sub> ||class="entry q1 g0"| 27547<sub>10</sub>
|-
|class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (4284, 25733, 7843, 27252, 8027, 3581) ||class="entry q0 g0"| 4284<sub>6</sub> ||class="entry q1 g0"| 25733<sub>6</sub> ||class="entry q1 g0"| 7843<sub>8</sub> ||class="entry q0 g0"| 27252<sub>8</sub> ||class="entry q1 g0"| 8027<sub>10</sub> ||class="entry q1 g0"| 3581<sub>10</sub>
|-
|class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (7434, 26659, 7397, 26562, 5117, 4027) ||class="entry q0 g0"| 7434<sub>6</sub> ||class="entry q1 g0"| 26659<sub>6</sub> ||class="entry q1 g0"| 7397<sub>8</sub> ||class="entry q0 g0"| 26562<sub>8</sub> ||class="entry q1 g0"| 5117<sub>10</sub> ||class="entry q1 g0"| 4027<sub>10</sub>
|-
|class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (7436, 3651, 31363, 26564, 30109, 27101) ||class="entry q0 g0"| 7436<sub>6</sub> ||class="entry q1 g0"| 3651<sub>6</sub> ||class="entry q1 g0"| 31363<sub>8</sub> ||class="entry q0 g0"| 26564<sub>8</sub> ||class="entry q1 g0"| 30109<sub>10</sub> ||class="entry q1 g0"| 27101<sub>10</sub>
|-
|class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (8818, 12585, 31169, 22714, 19191, 27295) ||class="entry q0 g0"| 8818<sub>6</sub> ||class="entry q1 g0"| 12585<sub>6</sub> ||class="entry q1 g0"| 31169<sub>8</sub> ||class="entry q0 g0"| 22714<sub>8</sub> ||class="entry q1 g0"| 19191<sub>10</sub> ||class="entry q1 g0"| 27295<sub>10</sub>
|-
|class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (9072, 25985, 11371, 22968, 7775, 16181) ||class="entry q0 g0"| 9072<sub>6</sub> ||class="entry q1 g0"| 25985<sub>6</sub> ||class="entry q1 g0"| 11371<sub>8</sub> ||class="entry q0 g0"| 22968<sub>8</sub> ||class="entry q1 g0"| 7775<sub>10</sub> ||class="entry q1 g0"| 16181<sub>10</sub>
|-
|class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (11398, 753, 17645, 22094, 31023, 22451) ||class="entry q0 g0"| 11398<sub>6</sub> ||class="entry q1 g0"| 753<sub>6</sub> ||class="entry q1 g0"| 17645<sub>8</sub> ||class="entry q0 g0"| 22094<sub>8</sub> ||class="entry q1 g0"| 31023<sub>10</sub> ||class="entry q1 g0"| 22451<sub>10</sub>
|-
|class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (11650, 12345, 30497, 22346, 19431, 25727) ||class="entry q0 g0"| 11650<sub>6</sub> ||class="entry q1 g0"| 12345<sub>6</sub> ||class="entry q1 g0"| 30497<sub>8</sub> ||class="entry q0 g0"| 22346<sub>8</sub> ||class="entry q1 g0"| 19431<sub>10</sub> ||class="entry q1 g0"| 25727<sub>10</sub>
|-
|class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (12346, 9057, 27529, 19186, 22719, 30935) ||class="entry q0 g0"| 12346<sub>6</sub> ||class="entry q1 g0"| 9057<sub>6</sub> ||class="entry q1 g0"| 27529<sub>8</sub> ||class="entry q0 g0"| 19186<sub>8</sub> ||class="entry q1 g0"| 22719<sub>10</sub> ||class="entry q1 g0"| 30935<sub>10</sub>
|-
|class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (12586, 11649, 25721, 19426, 22111, 30503) ||class="entry q0 g0"| 12586<sub>6</sub> ||class="entry q1 g0"| 11649<sub>6</sub> ||class="entry q1 g0"| 25721<sub>8</sub> ||class="entry q0 g0"| 19426<sub>8</sub> ||class="entry q1 g0"| 22111<sub>10</sub> ||class="entry q1 g0"| 30503<sub>10</sub>
|-
|class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (18244, 21529, 31377, 15756, 12231, 27087) ||class="entry q0 g0"| 18244<sub>6</sub> ||class="entry q1 g0"| 21529<sub>6</sub> ||class="entry q1 g0"| 31377<sub>8</sub> ||class="entry q0 g0"| 15756<sub>8</sub> ||class="entry q1 g0"| 12231<sub>10</sub> ||class="entry q1 g0"| 27087<sub>10</sub>
|-
|class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (18256, 26673, 18093, 15768, 5103, 22003) ||class="entry q0 g0"| 18256<sub>6</sub> ||class="entry q1 g0"| 26673<sub>6</sub> ||class="entry q1 g0"| 18093<sub>8</sub> ||class="entry q0 g0"| 15768<sub>8</sub> ||class="entry q1 g0"| 5103<sub>10</sub> ||class="entry q1 g0"| 22003<sub>10</sub>
|-
|class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (18598, 3905, 11819, 12910, 29855, 15733) ||class="entry q0 g0"| 18598<sub>6</sub> ||class="entry q1 g0"| 3905<sub>6</sub> ||class="entry q1 g0"| 11819<sub>8</sub> ||class="entry q0 g0"| 12910<sub>8</sub> ||class="entry q1 g0"| 29855<sub>10</sub> ||class="entry q1 g0"| 15733<sub>10</sub>
|-
|class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (18612, 21769, 29809, 12924, 11991, 26415) ||class="entry q0 g0"| 18612<sub>6</sub> ||class="entry q1 g0"| 21769<sub>6</sub> ||class="entry q1 g0"| 29809<sub>8</sub> ||class="entry q0 g0"| 12924<sub>8</sub> ||class="entry q1 g0"| 11991<sub>10</sub> ||class="entry q1 g0"| 26415<sub>10</sub>
|-
|class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (21532, 18609, 26409, 11988, 13167, 29815) ||class="entry q0 g0"| 21532<sub>6</sub> ||class="entry q1 g0"| 18609<sub>6</sub> ||class="entry q1 g0"| 26409<sub>8</sub> ||class="entry q0 g0"| 11988<sub>8</sub> ||class="entry q1 g0"| 13167<sub>10</sub> ||class="entry q1 g0"| 29815<sub>10</sub>
|-
|class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (21772, 18001, 26841, 12228, 15759, 31623) ||class="entry q0 g0"| 21772<sub>6</sub> ||class="entry q1 g0"| 18001<sub>6</sub> ||class="entry q1 g0"| 26841<sub>8</sub> ||class="entry q0 g0"| 12228<sub>8</sub> ||class="entry q1 g0"| 15759<sub>10</sub> ||class="entry q1 g0"| 31623<sub>10</sub>
|-
|class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (25748, 4267, 3309, 7772, 27509, 8115) ||class="entry q0 g0"| 25748<sub>6</sub> ||class="entry q1 g0"| 4267<sub>6</sub> ||class="entry q1 g0"| 3309<sub>8</sub> ||class="entry q0 g0"| 7772<sub>8</sub> ||class="entry q1 g0"| 27509<sub>10</sub> ||class="entry q1 g0"| 8115<sub>10</sub>
|-
|class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (26000, 8803, 16161, 8024, 22973, 11391) ||class="entry q0 g0"| 26000<sub>6</sub> ||class="entry q1 g0"| 8803<sub>6</sub> ||class="entry q1 g0"| 16161<sub>8</sub> ||class="entry q0 g0"| 8024<sub>8</sub> ||class="entry q1 g0"| 22973<sub>10</sub> ||class="entry q1 g0"| 11391<sub>10</sub>
|-
|class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (26914, 7181, 3755, 5098, 26579, 7669) ||class="entry q0 g0"| 26914<sub>6</sub> ||class="entry q1 g0"| 7181<sub>6</sub> ||class="entry q1 g0"| 3755<sub>8</sub> ||class="entry q0 g0"| 5098<sub>8</sub> ||class="entry q1 g0"| 26579<sub>10</sub> ||class="entry q1 g0"| 7669<sub>10</sub>
|-
|class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (26928, 17989, 21745, 5112, 15771, 18351) ||class="entry q0 g0"| 26928<sub>6</sub> ||class="entry q1 g0"| 17989<sub>6</sub> ||class="entry q1 g0"| 21745<sub>8</sub> ||class="entry q0 g0"| 5112<sub>8</sub> ||class="entry q1 g0"| 15771<sub>10</sub> ||class="entry q1 g0"| 18351<sub>10</sub>
|-
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|-
|class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 6, 8, 8, 4) ||class="c"| (3666, 29835, 349, 29850, 3925, 4611) ||class="entry q0 g0"| 3666<sub>6</sub> ||class="entry q1 g0"| 29835<sub>8</sub> ||class="entry q1 g0"| 349<sub>6</sub> ||class="entry q0 g0"| 29850<sub>8</sub> ||class="entry q1 g0"| 3925<sub>8</sub> ||class="entry q1 g0"| 4611<sub>4</sub>
|-
|class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 6, 8, 8, 4) ||class="c"| (4524, 27237, 4435, 27492, 4539, 525) ||class="entry q0 g0"| 4524<sub>6</sub> ||class="entry q1 g0"| 27237<sub>8</sub> ||class="entry q1 g0"| 4435<sub>6</sub> ||class="entry q0 g0"| 27492<sub>8</sub> ||class="entry q1 g0"| 4539<sub>8</sub> ||class="entry q1 g0"| 525<sub>4</sub>
|-
|class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 6, 8, 8, 4) ||class="c"| (7194, 26307, 4885, 26322, 7453, 75) ||class="entry q0 g0"| 7194<sub>6</sub> ||class="entry q1 g0"| 26307<sub>8</sub> ||class="entry q1 g0"| 4885<sub>6</sub> ||class="entry q0 g0"| 26322<sub>8</sub> ||class="entry q1 g0"| 7453<sub>8</sub> ||class="entry q1 g0"| 75<sub>4</sub>
|-
|class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 6, 8, 8, 4) ||class="c"| (9060, 22953, 4183, 22956, 8823, 777) ||class="entry q0 g0"| 9060<sub>6</sub> ||class="entry q1 g0"| 22953<sub>8</sub> ||class="entry q1 g0"| 4183<sub>6</sub> ||class="entry q0 g0"| 22956<sub>8</sub> ||class="entry q1 g0"| 8823<sub>8</sub> ||class="entry q1 g0"| 777<sub>4</sub>
|-
|class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 6, 8, 8, 4) ||class="c"| (11652, 22105, 4423, 22348, 11655, 537) ||class="entry q0 g0"| 11652<sub>6</sub> ||class="entry q1 g0"| 22105<sub>8</sub> ||class="entry q1 g0"| 4423<sub>6</sub> ||class="entry q0 g0"| 22348<sub>8</sub> ||class="entry q1 g0"| 11655<sub>8</sub> ||class="entry q1 g0"| 537<sub>4</sub>
|-
|class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 6, 8, 8, 4) ||class="c"| (12588, 19425, 543, 19428, 12351, 4417) ||class="entry q0 g0"| 12588<sub>6</sub> ||class="entry q1 g0"| 19425<sub>8</sub> ||class="entry q1 g0"| 543<sub>6</sub> ||class="entry q0 g0"| 19428<sub>8</sub> ||class="entry q1 g0"| 12351<sub>8</sub> ||class="entry q1 g0"| 4417<sub>4</sub>
|-
|class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 6, 8, 8, 4) ||class="c"| (18002, 15513, 4871, 15514, 18247, 89) ||class="entry q0 g0"| 18002<sub>6</sub> ||class="entry q1 g0"| 15513<sub>8</sub> ||class="entry q1 g0"| 4871<sub>6</sub> ||class="entry q0 g0"| 15514<sub>8</sub> ||class="entry q1 g0"| 18247<sub>8</sub> ||class="entry q1 g0"| 89<sub>4</sub>
|-
|class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 6, 8, 8, 4) ||class="c"| (18610, 13161, 4631, 12922, 18615, 329) ||class="entry q0 g0"| 18610<sub>6</sub> ||class="entry q1 g0"| 13161<sub>8</sub> ||class="entry q1 g0"| 4631<sub>6</sub> ||class="entry q0 g0"| 12922<sub>8</sub> ||class="entry q1 g0"| 18615<sub>8</sub> ||class="entry q1 g0"| 329<sub>4</sub>
|-
|class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 6, 8, 8, 4) ||class="c"| (21530, 11985, 335, 11986, 21775, 4625) ||class="entry q0 g0"| 21530<sub>6</sub> ||class="entry q1 g0"| 11985<sub>8</sub> ||class="entry q1 g0"| 335<sub>6</sub> ||class="entry q0 g0"| 11986<sub>8</sub> ||class="entry q1 g0"| 21775<sub>8</sub> ||class="entry q1 g0"| 4625<sub>4</sub>
|-
|class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 6, 8, 8, 4) ||class="c"| (25988, 7755, 797, 8012, 26005, 4163) ||class="entry q0 g0"| 25988<sub>6</sub> ||class="entry q1 g0"| 7755<sub>8</sub> ||class="entry q1 g0"| 797<sub>6</sub> ||class="entry q0 g0"| 8012<sub>8</sub> ||class="entry q1 g0"| 26005<sub>8</sub> ||class="entry q1 g0"| 4163<sub>4</sub>
|-
|class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 6, 8, 8, 4) ||class="c"| (26674, 4845, 347, 4858, 26931, 4613) ||class="entry q0 g0"| 26674<sub>6</sub> ||class="entry q1 g0"| 4845<sub>8</sub> ||class="entry q1 g0"| 347<sub>6</sub> ||class="entry q0 g0"| 4858<sub>8</sub> ||class="entry q1 g0"| 26931<sub>8</sub> ||class="entry q1 g0"| 4613<sub>4</sub>
|-
|class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 6, 12, 8, 8) ||class="c"| (438, 18867, 13185, 31614, 12909, 8415) ||class="entry q0 g0"| 438<sub>6</sub> ||class="entry q1 g0"| 18867<sub>8</sub> ||class="entry q1 g0"| 13185<sub>6</sub> ||class="entry q0 g0"| 31614<sub>12</sub> ||class="entry q1 g0"| 12909<sub>8</sub> ||class="entry q1 g0"| 8415<sub>8</sub>
|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
|class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 6, 8, 6, 8) ||class="c"| (1746, 32027, 1581, 31770, 1733, 5491) ||class="entry q0 g0"| 1746<sub>6</sub> ||class="entry q1 g0"| 32027<sub>10</sub> ||class="entry q1 g0"| 1581<sub>6</sub> ||class="entry q0 g0"| 31770<sub>8</sub> ||class="entry q1 g0"| 1733<sub>6</sub> ||class="entry q1 g0"| 5491<sub>8</sub>
|-
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|-
|class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 6, 8, 6, 8) ||class="c"| (2914, 6109, 25101, 29098, 27651, 29011) ||class="entry q0 g0"| 2914<sub>6</sub> ||class="entry q1 g0"| 6109<sub>10</sub> ||class="entry q1 g0"| 25101<sub>6</sub> ||class="entry q0 g0"| 29098<sub>8</sub> ||class="entry q1 g0"| 27651<sub>6</sub> ||class="entry q1 g0"| 29011<sub>8</sub>
|-
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|-
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|-
|class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 6, 8, 6, 8) ||class="c"| (5274, 28499, 5221, 28242, 5261, 1851) ||class="entry q0 g0"| 5274<sub>6</sub> ||class="entry q1 g0"| 28499<sub>10</sub> ||class="entry q1 g0"| 5221<sub>6</sub> ||class="entry q0 g0"| 28242<sub>8</sub> ||class="entry q1 g0"| 5261<sub>6</sub> ||class="entry q1 g0"| 1851<sub>8</sub>
|-
|class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 6, 8, 6, 8) ||class="c"| (6190, 14173, 17289, 25318, 19587, 20695) ||class="entry q0 g0"| 6190<sub>6</sub> ||class="entry q1 g0"| 14173<sub>10</sub> ||class="entry q1 g0"| 17289<sub>6</sub> ||class="entry q0 g0"| 25318<sub>8</sub> ||class="entry q1 g0"| 19587<sub>6</sub> ||class="entry q1 g0"| 20695<sub>8</sub>
|-
|class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 6, 8, 6, 8) ||class="c"| (6444, 25589, 5667, 25572, 6187, 1405) ||class="entry q0 g0"| 6444<sub>6</sub> ||class="entry q1 g0"| 25589<sub>10</sub> ||class="entry q1 g0"| 5667<sub>6</sub> ||class="entry q0 g0"| 25572<sub>8</sub> ||class="entry q1 g0"| 6187<sub>6</sub> ||class="entry q1 g0"| 1405<sub>8</sub>
|-
|class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 6, 8, 6, 8) ||class="c"| (9810, 23711, 5473, 23706, 10049, 1599) ||class="entry q0 g0"| 9810<sub>6</sub> ||class="entry q1 g0"| 23711<sub>10</sub> ||class="entry q1 g0"| 5473<sub>6</sub> ||class="entry q0 g0"| 23706<sub>8</sub> ||class="entry q1 g0"| 10049<sub>6</sub> ||class="entry q1 g0"| 1599<sub>8</sub>
|-
|class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 6, 8, 6, 8) ||class="c"| (10050, 21119, 6801, 23946, 10657, 2511) ||class="entry q0 g0"| 10050<sub>6</sub> ||class="entry q1 g0"| 21119<sub>10</sub> ||class="entry q1 g0"| 6801<sub>6</sub> ||class="entry q0 g0"| 23946<sub>8</sub> ||class="entry q1 g0"| 10657<sub>6</sub> ||class="entry q1 g0"| 2511<sub>8</sub>
|-
|class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 6, 8, 6, 8) ||class="c"| (10406, 28487, 10317, 21102, 5273, 15123) ||class="entry q0 g0"| 10406<sub>6</sub> ||class="entry q1 g0"| 28487<sub>10</sub> ||class="entry q1 g0"| 10317<sub>6</sub> ||class="entry q0 g0"| 21102<sub>8</sub> ||class="entry q1 g0"| 5273<sub>6</sub> ||class="entry q1 g0"| 15123<sub>8</sub>
|-
|class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 6, 8, 6, 8) ||class="c"| (10418, 21359, 5233, 21114, 10417, 1839) ||class="entry q0 g0"| 10418<sub>6</sub> ||class="entry q1 g0"| 21359<sub>10</sub> ||class="entry q1 g0"| 5233<sub>6</sub> ||class="entry q0 g0"| 21114<sub>8</sub> ||class="entry q1 g0"| 10417<sub>6</sub> ||class="entry q1 g0"| 1839<sub>8</sub>
|-
|class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 6, 8, 6, 8) ||class="c"| (10658, 23951, 7041, 21354, 9809, 2271) ||class="entry q0 g0"| 10658<sub>6</sub> ||class="entry q1 g0"| 23951<sub>10</sub> ||class="entry q1 g0"| 7041<sub>6</sub> ||class="entry q0 g0"| 21354<sub>8</sub> ||class="entry q1 g0"| 9809<sub>6</sub> ||class="entry q1 g0"| 2271<sub>8</sub>
|-
|class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 6, 8, 6, 8) ||class="c"| (13338, 20183, 1833, 20178, 13577, 5239) ||class="entry q0 g0"| 13338<sub>6</sub> ||class="entry q1 g0"| 20183<sub>10</sub> ||class="entry q1 g0"| 1833<sub>6</sub> ||class="entry q0 g0"| 20178<sub>8</sub> ||class="entry q1 g0"| 13577<sub>6</sub> ||class="entry q1 g0"| 5239<sub>8</sub>
|-
|class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 10, 6, 8, 6, 8) ||class="c"| (13592, 6783, 21123, 20432, 24993, 16861) ||class="entry q0 g0"| 13592<sub>6</sub> ||class="entry q1 g0"| 6783<sub>10</sub> ||class="entry q1 g0"| 21123<sub>6</sub> ||class="entry q0 g0"| 20432<sub>8</sub> ||class="entry q1 g0"| 24993<sub>6</sub> ||class="entry q1 g0"| 16861<sub>8</sub>
|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
|class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 12, 8, 8, 4, 6) ||class="c"| (17250, 24527, 28759, 14762, 9233, 25353) ||class="entry q0 g0"| 17250<sub>6</sub> ||class="entry q1 g0"| 24527<sub>12</sub> ||class="entry q1 g0"| 28759<sub>8</sub> ||class="entry q0 g0"| 14762<sub>8</sub> ||class="entry q1 g0"| 9233<sub>4</sub> ||class="entry q1 g0"| 25353<sub>6</sub>
|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
|class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 1, 0) ||class="w"| (8, 12, 8, 6, 4, 10) ||class="c"| (26324, 31613, 26157, 7196, 163, 30067) ||class="entry q0 g0"| 26324<sub>8</sub> ||class="entry q1 g0"| 31613<sub>12</sub> ||class="entry q1 g0"| 26157<sub>8</sub> ||class="entry q0 g0"| 7196<sub>6</sub> ||class="entry q1 g1"| 163<sub>4</sub> ||class="entry q1 g0"| 30067<sub>10</sub>
|-
|class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 1, 0) ||class="w"| (8, 12, 8, 6, 4, 10) ||class="c"| (27490, 30683, 25707, 4522, 3077, 30517) ||class="entry q0 g0"| 27490<sub>8</sub> ||class="entry q1 g0"| 30683<sub>12</sub> ||class="entry q1 g0"| 25707<sub>8</sub> ||class="entry q0 g0"| 4522<sub>6</sub> ||class="entry q1 g1"| 3077<sub>4</sub> ||class="entry q1 g0"| 30517<sub>10</sub>
|-
|class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 1, 0) ||class="w"| (8, 12, 8, 6, 4, 10) ||class="c"| (30104, 23549, 18345, 3920, 8227, 21751) ||class="entry q0 g0"| 30104<sub>8</sub> ||class="entry q1 g0"| 23549<sub>12</sub> ||class="entry q1 g0"| 18345<sub>8</sub> ||class="entry q0 g0"| 3920<sub>6</sub> ||class="entry q1 g1"| 8227<sub>4</sub> ||class="entry q1 g0"| 21751<sub>10</sub>
|-
|class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 1, 0) ||class="w"| (8, 12, 8, 6, 4, 10) ||class="c"| (31032, 16347, 11385, 1008, 17413, 16167) ||class="entry q0 g0"| 31032<sub>8</sub> ||class="entry q1 g0"| 16347<sub>12</sub> ||class="entry q1 g0"| 11385<sub>8</sub> ||class="entry q0 g0"| 1008<sub>6</sub> ||class="entry q1 g1"| 17413<sub>4</sub> ||class="entry q1 g0"| 16167<sub>10</sub>
|-
|class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 1, 0, 0) ||class="w"| (10, 8, 10, 4, 8, 8) ||class="c"| (16092, 19185, 3327, 17428, 12591, 8097) ||class="entry q0 g0"| 16092<sub>10</sub> ||class="entry q1 g0"| 19185<sub>8</sub> ||class="entry q1 g0"| 3327<sub>10</sub> ||class="entry q0 g1"| 17428<sub>4</sub> ||class="entry q1 g0"| 12591<sub>8</sub> ||class="entry q1 g0"| 8097<sub>8</sub>
|-
|class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 1, 0, 0) ||class="w"| (10, 8, 10, 4, 8, 8) ||class="c"| (16344, 30777, 16179, 17680, 999, 11373) ||class="entry q0 g0"| 16344<sub>10</sub> ||class="entry q1 g0"| 30777<sub>8</sub> ||class="entry q1 g0"| 16179<sub>10</sub> ||class="entry q0 g1"| 17680<sub>4</sub> ||class="entry q1 g0"| 999<sub>8</sub> ||class="entry q1 g0"| 11373<sub>8</sub>
|-
|class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 1, 0, 0) ||class="w"| (10, 8, 10, 4, 8, 8) ||class="c"| (23530, 12225, 4015, 8482, 21535, 7409) ||class="entry q0 g0"| 23530<sub>10</sub> ||class="entry q1 g0"| 12225<sub>8</sub> ||class="entry q1 g0"| 4015<sub>10</sub> ||class="entry q0 g1"| 8482<sub>4</sub> ||class="entry q1 g0"| 21535<sub>8</sub> ||class="entry q1 g0"| 7409<sub>8</sub>
|-
|class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 1, 0, 0) ||class="w"| (10, 8, 10, 4, 8, 8) ||class="c"| (23544, 30089, 22005, 8496, 3671, 18091) ||class="entry q0 g0"| 23544<sub>10</sub> ||class="entry q1 g0"| 30089<sub>8</sub> ||class="entry q1 g0"| 22005<sub>10</sub> ||class="entry q0 g1"| 8496<sub>4</sub> ||class="entry q1 g0"| 3671<sub>8</sub> ||class="entry q1 g0"| 18091<sub>8</sub>
|-
|class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 1, 0, 0) ||class="w"| (10, 8, 10, 4, 8, 8) ||class="c"| (30414, 22699, 17663, 3078, 9077, 22433) ||class="entry q0 g0"| 30414<sub>10</sub> ||class="entry q1 g0"| 22699<sub>8</sub> ||class="entry q1 g0"| 17663<sub>10</sub> ||class="entry q0 g1"| 3078<sub>4</sub> ||class="entry q1 g0"| 9077<sub>8</sub> ||class="entry q1 g0"| 22433<sub>8</sub>
|-
|class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 1, 0, 0) ||class="w"| (10, 8, 10, 4, 8, 8) ||class="c"| (30666, 27235, 30515, 3330, 4541, 25709) ||class="entry q0 g0"| 30666<sub>10</sub> ||class="entry q1 g0"| 27235<sub>8</sub> ||class="entry q1 g0"| 30515<sub>10</sub> ||class="entry q0 g1"| 3330<sub>4</sub> ||class="entry q1 g0"| 4541<sub>8</sub> ||class="entry q1 g0"| 25709<sub>8</sub>
|-
|class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 1, 0, 0) ||class="w"| (10, 8, 10, 4, 8, 8) ||class="c"| (31342, 15501, 12079, 166, 18259, 15473) ||class="entry q0 g0"| 31342<sub>10</sub> ||class="entry q1 g0"| 15501<sub>8</sub> ||class="entry q1 g0"| 12079<sub>10</sub> ||class="entry q0 g1"| 166<sub>4</sub> ||class="entry q1 g0"| 18259<sub>8</sub> ||class="entry q1 g0"| 15473<sub>8</sub>
|-
|class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 1, 0, 0) ||class="w"| (10, 8, 10, 4, 8, 8) ||class="c"| (31356, 26309, 30069, 180, 7451, 26155) ||class="entry q0 g0"| 31356<sub>10</sub> ||class="entry q1 g0"| 26309<sub>8</sub> ||class="entry q1 g0"| 30069<sub>10</sub> ||class="entry q0 g1"| 180<sub>4</sub> ||class="entry q1 g0"| 7451<sub>8</sub> ||class="entry q1 g0"| 26155<sub>8</sub>
|-
|class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 1, 1, 0) ||class="w"| (8, 10, 6, 2, 6, 8) ||class="c"| (16072, 30425, 12483, 17408, 3335, 9117) ||class="entry q0 g0"| 16072<sub>8</sub> ||class="entry q1 g0"| 30425<sub>10</sub> ||class="entry q1 g0"| 12483<sub>6</sub> ||class="entry q0 g1"| 17408<sub>2</sub> ||class="entry q1 g1"| 3335<sub>6</sub> ||class="entry q1 g0"| 9117<sub>8</sub>
|-
|class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 1, 1, 0) ||class="w"| (8, 10, 6, 2, 6, 8) ||class="c"| (23272, 31593, 23045, 8224, 183, 18779) ||class="entry q0 g0"| 23272<sub>8</sub> ||class="entry q1 g0"| 31593<sub>10</sub> ||class="entry q1 g0"| 23045<sub>6</sub> ||class="entry q0 g1"| 8224<sub>2</sub> ||class="entry q1 g1"| 183<sub>6</sub> ||class="entry q1 g0"| 18779<sub>8</sub>
|-
|class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 1, 1, 0) ||class="w"| (8, 10, 6, 2, 6, 8) ||class="c"| (30408, 16075, 8857, 3072, 17685, 12743) ||class="entry q0 g0"| 30408<sub>8</sub> ||class="entry q1 g0"| 16075<sub>10</sub> ||class="entry q1 g0"| 8857<sub>6</sub> ||class="entry q0 g1"| 3072<sub>2</sub> ||class="entry q1 g1"| 17685<sub>6</sub> ||class="entry q1 g0"| 12743<sub>8</sub>
|-
|class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 1, 1, 0) ||class="w"| (8, 10, 6, 2, 6, 8) ||class="c"| (31336, 23277, 18761, 160, 8499, 23063) ||class="entry q0 g0"| 31336<sub>8</sub> ||class="entry q1 g0"| 23277<sub>10</sub> ||class="entry q1 g0"| 18761<sub>6</sub> ||class="entry q0 g1"| 160<sub>2</sub> ||class="entry q1 g1"| 8499<sub>6</sub> ||class="entry q1 g0"| 23063<sub>8</sub>
|-
|class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 1, 0, 1, 0, 0) ||class="w"| (10, 4, 6, 4, 12, 4) ||class="c"| (16332, 17425, 783, 17668, 16335, 4177) ||class="entry q0 g0"| 16332<sub>10</sub> ||class="entry q1 g1"| 17425<sub>4</sub> ||class="entry q1 g0"| 783<sub>6</sub> ||class="entry q0 g1"| 17668<sub>4</sub> ||class="entry q1 g0"| 16335<sub>12</sub> ||class="entry q1 g0"| 4177<sub>4</sub>
|-
|class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 1, 0, 1, 0, 0) ||class="w"| (10, 4, 6, 4, 12, 4) ||class="c"| (23290, 8481, 95, 8242, 23295, 4865) ||class="entry q0 g0"| 23290<sub>10</sub> ||class="entry q1 g1"| 8481<sub>4</sub> ||class="entry q1 g0"| 95<sub>6</sub> ||class="entry q0 g1"| 8242<sub>4</sub> ||class="entry q1 g0"| 23295<sub>12</sub> ||class="entry q1 g0"| 4865<sub>4</sub>
|-
|class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 1, 0, 1, 0, 0) ||class="w"| (10, 4, 6, 4, 12, 4) ||class="c"| (30668, 3075, 4437, 3332, 30685, 523) ||class="entry q0 g0"| 30668<sub>10</sub> ||class="entry q1 g1"| 3075<sub>4</sub> ||class="entry q1 g0"| 4437<sub>6</sub> ||class="entry q0 g1"| 3332<sub>4</sub> ||class="entry q1 g0"| 30685<sub>12</sub> ||class="entry q1 g0"| 523<sub>4</sub>
|-
|class="f"| 4680 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 1, 0, 1, 0, 0) ||class="w"| (10, 4, 6, 4, 12, 4) ||class="c"| (31354, 165, 4883, 178, 31611, 77) ||class="entry q0 g0"| 31354<sub>10</sub> ||class="entry q1 g1"| 165<sub>4</sub> ||class="entry q1 g0"| 4883<sub>6</sub> ||class="entry q0 g1"| 178<sub>4</sub> ||class="entry q1 g0"| 31611<sub>12</sub> ||class="entry q1 g0"| 77<sub>4</sub>
|-
|class="f"| 5160 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (2, 6, 4, 2, 6, 4) ||class="c"| (40, 10300, 15360, 5120, 15380, 10280) ||class="entry q0 g0"| 40<sub>2</sub> ||class="entry q0 g0"| 10300<sub>6</sub> ||class="entry q0 g0"| 15360<sub>4</sub> ||class="entry q0 g0"| 5120<sub>2</sub> ||class="entry q0 g0"| 15380<sub>6</sub> ||class="entry q0 g0"| 10280<sub>4</sub>
|-
|class="f"| 5160 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (2, 6, 4, 2, 6, 4) ||class="c"| (1032, 17802, 20640, 4128, 20898, 17544) ||class="entry q0 g0"| 1032<sub>2</sub> ||class="entry q0 g0"| 17802<sub>6</sub> ||class="entry q0 g0"| 20640<sub>4</sub> ||class="entry q0 g0"| 4128<sub>2</sub> ||class="entry q0 g0"| 20898<sub>6</sub> ||class="entry q0 g0"| 17544<sub>4</sub>
|-
|class="f"| 5160 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 6, 8, 4, 6, 8) ||class="c"| (2216, 8620, 15216, 7296, 13700, 12120) ||class="entry q0 g0"| 2216<sub>4</sub> ||class="entry q0 g0"| 8620<sub>6</sub> ||class="entry q0 g0"| 15216<sub>8</sub> ||class="entry q0 g0"| 7296<sub>4</sub> ||class="entry q0 g0"| 13700<sub>6</sub> ||class="entry q0 g0"| 12120<sub>8</sub>
|-
|class="f"| 5160 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 6, 8, 4, 6, 8) ||class="c"| (3208, 19482, 22480, 6304, 22578, 17400) ||class="entry q0 g0"| 3208<sub>4</sub> ||class="entry q0 g0"| 19482<sub>6</sub> ||class="entry q0 g0"| 22480<sub>8</sub> ||class="entry q0 g0"| 6304<sub>4</sub> ||class="entry q0 g0"| 22578<sub>6</sub> ||class="entry q0 g0"| 17400<sub>8</sub>
|-
|class="f"| 5160 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 6, 8, 4, 6, 8) ||class="c"| (8360, 2488, 12108, 13440, 7568, 15204) ||class="entry q0 g0"| 8360<sub>4</sub> ||class="entry q0 g0"| 2488<sub>6</sub> ||class="entry q0 g0"| 12108<sub>8</sub> ||class="entry q0 g0"| 13440<sub>4</sub> ||class="entry q0 g0"| 7568<sub>6</sub> ||class="entry q0 g0"| 15204<sub>8</sub>
|-
|class="f"| 5160 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 6, 8, 4, 6, 8) ||class="c"| (9352, 25614, 17388, 12448, 28710, 22468) ||class="entry q0 g0"| 9352<sub>4</sub> ||class="entry q0 g0"| 25614<sub>6</sub> ||class="entry q0 g0"| 17388<sub>8</sub> ||class="entry q0 g0"| 12448<sub>4</sub> ||class="entry q0 g0"| 28710<sub>6</sub> ||class="entry q0 g0"| 22468<sub>8</sub>
|-
|class="f"| 5160 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 6, 8, 4, 6, 8) ||class="c"| (18472, 24622, 11866, 23552, 29702, 14962) ||class="entry q0 g0"| 18472<sub>4</sub> ||class="entry q0 g0"| 24622<sub>6</sub> ||class="entry q0 g0"| 11866<sub>8</sub> ||class="entry q0 g0"| 23552<sub>4</sub> ||class="entry q0 g0"| 29702<sub>6</sub> ||class="entry q0 g0"| 14962<sub>8</sub>
|-
|class="f"| 5160 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 6, 8, 4, 6, 8) ||class="c"| (19464, 3480, 17146, 22560, 6576, 22226) ||class="entry q0 g0"| 19464<sub>4</sub> ||class="entry q0 g0"| 3480<sub>6</sub> ||class="entry q0 g0"| 17146<sub>8</sub> ||class="entry q0 g0"| 22560<sub>4</sub> ||class="entry q0 g0"| 6576<sub>6</sub> ||class="entry q0 g0"| 22226<sub>8</sub>
|-
|class="f"| 5160 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 6, 8, 4, 6, 8) ||class="c"| (24616, 18490, 14950, 29696, 23570, 11854) ||class="entry q0 g0"| 24616<sub>4</sub> ||class="entry q0 g0"| 18490<sub>6</sub> ||class="entry q0 g0"| 14950<sub>8</sub> ||class="entry q0 g0"| 29696<sub>4</sub> ||class="entry q0 g0"| 23570<sub>6</sub> ||class="entry q0 g0"| 11854<sub>8</sub>
|-
|class="f"| 5160 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 6, 8, 4, 6, 8) ||class="c"| (25608, 9612, 22214, 28704, 12708, 17134) ||class="entry q0 g0"| 25608<sub>4</sub> ||class="entry q0 g0"| 9612<sub>6</sub> ||class="entry q0 g0"| 22214<sub>8</sub> ||class="entry q0 g0"| 28704<sub>4</sub> ||class="entry q0 g0"| 12708<sub>6</sub> ||class="entry q0 g0"| 17134<sub>8</sub>
|-
|class="f"| 5160 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 6, 4, 8, 6) ||class="c"| (616, 17386, 22208, 5696, 22466, 17128) ||class="entry q0 g0"| 616<sub>4</sub> ||class="entry q0 g0"| 17386<sub>8</sub> ||class="entry q0 g0"| 22208<sub>6</sub> ||class="entry q0 g0"| 5696<sub>4</sub> ||class="entry q0 g0"| 22466<sub>8</sub> ||class="entry q0 g0"| 17128<sub>6</sub>
|-
|class="f"| 5160 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 6, 4, 8, 6) ||class="c"| (1608, 11868, 14944, 4704, 14964, 11848) ||class="entry q0 g0"| 1608<sub>4</sub> ||class="entry q0 g0"| 11868<sub>8</sub> ||class="entry q0 g0"| 14944<sub>6</sub> ||class="entry q0 g0"| 4704<sub>4</sub> ||class="entry q0 g0"| 14964<sub>8</sub> ||class="entry q0 g0"| 11848<sub>6</sub>
|-
|class="f"| 5160 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 8, 4, 8, 8) ||class="c"| (46, 20060, 23142, 5126, 23156, 20046) ||class="entry q0 g0"| 46<sub>4</sub> ||class="entry q0 g0"| 20060<sub>8</sub> ||class="entry q0 g0"| 23142<sub>8</sub> ||class="entry q0 g0"| 5126<sub>4</sub> ||class="entry q0 g0"| 23156<sub>8</sub> ||class="entry q0 g0"| 20046<sub>8</sub>
|-
|class="f"| 5160 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 8, 4, 8, 8) ||class="c"| (58, 29300, 26202, 5138, 26204, 29298) ||class="entry q0 g0"| 58<sub>4</sub> ||class="entry q0 g0"| 29300<sub>8</sub> ||class="entry q0 g0"| 26202<sub>8</sub> ||class="entry q0 g0"| 5138<sub>4</sub> ||class="entry q0 g0"| 26204<sub>8</sub> ||class="entry q0 g0"| 29298<sub>8</sub>
|-
|class="f"| 5160 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 8, 4, 8, 8) ||class="c"| (298, 31892, 27050, 5378, 26812, 32130) ||class="entry q0 g0"| 298<sub>4</sub> ||class="entry q0 g0"| 31892<sub>8</sub> ||class="entry q0 g0"| 27050<sub>8</sub> ||class="entry q0 g0"| 5378<sub>4</sub> ||class="entry q0 g0"| 26812<sub>8</sub> ||class="entry q0 g0"| 32130<sub>8</sub>
|-
|class="f"| 5160 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 8, 4, 8, 8) ||class="c"| (300, 6900, 4044, 5380, 3804, 7140) ||class="entry q0 g0"| 300<sub>4</sub> ||class="entry q0 g0"| 6900<sub>8</sub> ||class="entry q0 g0"| 4044<sub>8</sub> ||class="entry q0 g0"| 5380<sub>4</sub> ||class="entry q0 g0"| 3804<sub>8</sub> ||class="entry q0 g0"| 7140<sub>8</sub>
|-
|class="f"| 5160 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 8, 4, 8, 8) ||class="c"| (312, 9948, 13296, 5392, 13044, 10200) ||class="entry q0 g0"| 312<sub>4</sub> ||class="entry q0 g0"| 9948<sub>8</sub> ||class="entry q0 g0"| 13296<sub>8</sub> ||class="entry q0 g0"| 5392<sub>4</sub> ||class="entry q0 g0"| 13044<sub>8</sub> ||class="entry q0 g0"| 10200<sub>8</sub>
|-
|class="f"| 5160 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 8, 4, 8, 8) ||class="c"| (1038, 9194, 14022, 4134, 14274, 8942) ||class="entry q0 g0"| 1038<sub>4</sub> ||class="entry q0 g0"| 9194<sub>8</sub> ||class="entry q0 g0"| 14022<sub>8</sub> ||class="entry q0 g0"| 4134<sub>4</sub> ||class="entry q0 g0"| 14274<sub>8</sub> ||class="entry q0 g0"| 8942<sub>8</sub>
|-
|class="f"| 5160 ||class="q"| (0, 0, 0, 0, 0, 0) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 8, 8, 4, 8, 8) ||class="c"| (1050, 8130, 2810, 4146, 3050, 7890) ||class="entry q0 g0"| 1050<sub>4</sub> ||class="entry q0 g0"| 8130<sub>8</sub> ||class="entry q0 g0"| 2810<sub>8</sub> ||class="entry q0 g0"| 4146<sub>4</sub> ||class="entry q0 g0"| 3050<sub>8</sub> ||class="entry q0 g0"| 7890<sub>8</sub>
|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
|class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 10, 10, 10, 6, 8) ||class="c"| (2580, 7995, 28059, 30396, 25221, 30885) ||class="entry q0 g0"| 2580<sub>4</sub> ||class="entry q1 g0"| 7995<sub>10</sub> ||class="entry q1 g0"| 28059<sub>10</sub> ||class="entry q0 g0"| 30396<sub>10</sub> ||class="entry q1 g0"| 25221<sub>6</sub> ||class="entry q1 g0"| 30885<sub>8</sub>
|-
|class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 10, 10, 10, 6, 8) ||class="c"| (2832, 11763, 24151, 30648, 20557, 19305) ||class="entry q0 g0"| 2832<sub>4</sub> ||class="entry q1 g0"| 11763<sub>10</sub> ||class="entry q1 g0"| 24151<sub>10</sub> ||class="entry q0 g0"| 30648<sub>10</sub> ||class="entry q1 g0"| 20557<sub>6</sub> ||class="entry q1 g0"| 19305<sub>8</sub>
|-
|class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 10, 10, 10, 6, 8) ||class="c"| (8710, 28007, 9213, 24238, 4313, 14019) ||class="entry q0 g0"| 8710<sub>4</sub> ||class="entry q1 g0"| 28007<sub>10</sub> ||class="entry q1 g0"| 9213<sub>10</sub> ||class="entry q0 g0"| 24238<sub>10</sub> ||class="entry q1 g0"| 4313<sub>6</sub> ||class="entry q1 g0"| 14019<sub>8</sub>
|-
|class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 10, 10, 10, 6, 8) ||class="c"| (8724, 14127, 31143, 24252, 19089, 27801) ||class="entry q0 g0"| 8724<sub>4</sub> ||class="entry q1 g0"| 14127<sub>10</sub> ||class="entry q1 g0"| 31143<sub>10</sub> ||class="entry q0 g0"| 24252<sub>10</sub> ||class="entry q1 g0"| 19089<sub>6</sub> ||class="entry q1 g0"| 27801<sub>8</sub>
|-
|class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 10, 10, 10, 6, 8) ||class="c"| (8964, 14799, 30295, 24492, 17521, 25449) ||class="entry q0 g0"| 8964<sub>4</sub> ||class="entry q1 g0"| 14799<sub>10</sub> ||class="entry q1 g0"| 30295<sub>10</sub> ||class="entry q0 g0"| 24492<sub>10</sub> ||class="entry q1 g0"| 17521<sub>6</sub> ||class="entry q1 g0"| 25449<sub>8</sub>
|-
|class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 10, 10, 10, 6, 8) ||class="c"| (16454, 26295, 20315, 15598, 6921, 23141) ||class="entry q0 g0"| 16454<sub>4</sub> ||class="entry q1 g0"| 26295<sub>10</sub> ||class="entry q1 g0"| 20315<sub>10</sub> ||class="entry q0 g0"| 15598<sub>10</sub> ||class="entry q1 g0"| 6921<sub>6</sub> ||class="entry q1 g0"| 23141<sub>8</sub>
|-
|class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 10, 10, 10, 6, 8) ||class="c"| (16466, 23199, 29543, 15610, 10017, 26201) ||class="entry q0 g0"| 16466<sub>4</sub> ||class="entry q1 g0"| 23199<sub>10</sub> ||class="entry q1 g0"| 29543<sub>10</sub> ||class="entry q0 g0"| 15610<sub>10</sub> ||class="entry q1 g0"| 10017<sub>6</sub> ||class="entry q1 g0"| 26201<sub>8</sub>
|-
|class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (4, 10, 10, 10, 6, 8) ||class="c"| (16706, 21631, 31895, 15850, 10689, 27049) ||class="entry q0 g0"| 16706<sub>4</sub> ||class="entry q1 g0"| 21631<sub>10</sub> ||class="entry q1 g0"| 31895<sub>10</sub> ||class="entry q0 g0"| 15850<sub>10</sub> ||class="entry q1 g0"| 10689<sub>6</sub> ||class="entry q1 g0"| 27049<sub>8</sub>
|-
|class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 4, 8, 12, 12, 10) ||class="c"| (918, 16899, 16193, 32574, 16317, 10879) ||class="entry q0 g0"| 918<sub>6</sub> ||class="entry q1 g0"| 16899<sub>4</sub> ||class="entry q1 g0"| 16193<sub>8</sub> ||class="entry q0 g0"| 32574<sub>12</sub> ||class="entry q1 g0"| 16317<sub>12</sub> ||class="entry q1 g0"| 10879<sub>10</sub>
|-
|class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 4, 8, 12, 12, 10) ||class="c"| (2390, 8261, 21233, 30206, 24059, 18383) ||class="entry q0 g0"| 2390<sub>6</sub> ||class="entry q1 g0"| 8261<sub>4</sub> ||class="entry q1 g0"| 21233<sub>8</sub> ||class="entry q0 g0"| 30206<sub>12</sub> ||class="entry q1 g0"| 24059<sub>12</sub> ||class="entry q1 g0"| 18383<sub>10</sub>
|-
|class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 4, 8, 12, 12, 10) ||class="c"| (8534, 2129, 18125, 24062, 30191, 21491) ||class="entry q0 g0"| 8534<sub>6</sub> ||class="entry q1 g0"| 2129<sub>4</sub> ||class="entry q1 g0"| 18125<sub>8</sub> ||class="entry q0 g0"| 24062<sub>12</sub> ||class="entry q1 g0"| 30191<sub>12</sub> ||class="entry q1 g0"| 21491<sub>10</sub>
|-
|class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 4, 8, 12, 12, 10) ||class="c"| (17174, 897, 10859, 16318, 32319, 16213) ||class="entry q0 g0"| 17174<sub>6</sub> ||class="entry q1 g0"| 897<sub>4</sub> ||class="entry q1 g0"| 10859<sub>8</sub> ||class="entry q0 g0"| 16318<sub>12</sub> ||class="entry q1 g0"| 32319<sub>12</sub> ||class="entry q1 g0"| 16213<sub>10</sub>
|-
|class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 6, 8, 10, 8) ||class="c"| (1702, 8533, 23569, 31246, 23787, 18735) ||class="entry q0 g0"| 1702<sub>6</sub> ||class="entry q1 g0"| 8533<sub>6</sub> ||class="entry q1 g0"| 23569<sub>6</sub> ||class="entry q0 g0"| 31246<sub>8</sub> ||class="entry q1 g0"| 23787<sub>10</sub> ||class="entry q1 g0"| 18735<sub>8</sub>
|-
|class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 6, 8, 10, 8) ||class="c"| (3174, 17171, 12705, 28878, 16045, 9375) ||class="entry q0 g0"| 3174<sub>6</sub> ||class="entry q1 g0"| 17171<sub>6</sub> ||class="entry q1 g0"| 12705<sub>6</sub> ||class="entry q0 g0"| 28878<sub>8</sub> ||class="entry q1 g0"| 16045<sub>10</sub> ||class="entry q1 g0"| 9375<sub>8</sub>
|-
|class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 6, 8, 10, 8) ||class="c"| (4762, 2389, 29701, 28210, 29931, 24891) ||class="entry q0 g0"| 4762<sub>6</sub> ||class="entry q1 g0"| 2389<sub>6</sub> ||class="entry q1 g0"| 29701<sub>6</sub> ||class="entry q0 g0"| 28210<sub>8</sub> ||class="entry q1 g0"| 29931<sub>10</sub> ||class="entry q1 g0"| 24891<sub>8</sub>
|-
|class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 6, 8, 10, 8) ||class="c"| (6476, 915, 28707, 26084, 32301, 25885) ||class="entry q0 g0"| 6476<sub>6</sub> ||class="entry q1 g0"| 915<sub>6</sub> ||class="entry q1 g0"| 28707<sub>6</sub> ||class="entry q0 g0"| 26084<sub>8</sub> ||class="entry q1 g0"| 32301<sub>10</sub> ||class="entry q1 g0"| 25885<sub>8</sub>
|-
|class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 6, 8, 10, 8) ||class="c"| (9584, 903, 19467, 23000, 32313, 22837) ||class="entry q0 g0"| 9584<sub>6</sub> ||class="entry q1 g0"| 903<sub>6</sub> ||class="entry q1 g0"| 19467<sub>6</sub> ||class="entry q0 g0"| 23000<sub>8</sub> ||class="entry q1 g0"| 32313<sub>10</sub> ||class="entry q1 g0"| 22837<sub>8</sub>
|-
|class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 6, 8, 10, 8) ||class="c"| (12378, 17159, 3465, 19698, 16057, 6327) ||class="entry q0 g0"| 12378<sub>6</sub> ||class="entry q1 g0"| 17159<sub>6</sub> ||class="entry q1 g0"| 3465<sub>6</sub> ||class="entry q0 g0"| 19698<sub>8</sub> ||class="entry q1 g0"| 16057<sub>10</sub> ||class="entry q1 g0"| 6327<sub>8</sub>
|-
|class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 6, 8, 10, 8) ||class="c"| (18224, 2135, 8365, 15256, 30185, 13715) ||class="entry q0 g0"| 18224<sub>6</sub> ||class="entry q1 g0"| 2135<sub>6</sub> ||class="entry q1 g0"| 8365<sub>6</sub> ||class="entry q0 g0"| 15256<sub>8</sub> ||class="entry q1 g0"| 30185<sub>10</sub> ||class="entry q1 g0"| 13715<sub>8</sub>
|-
|class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 6, 8, 10, 8) ||class="c"| (21260, 8279, 2233, 12196, 24041, 7559) ||class="entry q0 g0"| 21260<sub>6</sub> ||class="entry q1 g0"| 8279<sub>6</sub> ||class="entry q1 g0"| 2233<sub>6</sub> ||class="entry q0 g0"| 12196<sub>8</sub> ||class="entry q1 g0"| 24041<sub>10</sub> ||class="entry q1 g0"| 7559<sub>8</sub>
|-
|class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (1254, 19075, 14033, 30798, 14141, 9199) ||class="entry q0 g0"| 1254<sub>6</sub> ||class="entry q1 g0"| 19075<sub>6</sub> ||class="entry q1 g0"| 14033<sub>8</sub> ||class="entry q0 g0"| 30798<sub>8</sub> ||class="entry q1 g0"| 14141<sub>10</sub> ||class="entry q1 g0"| 9199<sub>10</sub>
|-
|class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (1268, 4299, 27787, 30812, 28021, 31157) ||class="entry q0 g0"| 1268<sub>6</sub> ||class="entry q1 g0"| 4299<sub>6</sub> ||class="entry q1 g0"| 27787<sub>8</sub> ||class="entry q0 g0"| 30812<sub>8</sub> ||class="entry q1 g0"| 28021<sub>10</sub> ||class="entry q1 g0"| 31157<sub>10</sub>
|-
|class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (3622, 10437, 23393, 29326, 21883, 20063) ||class="entry q0 g0"| 3622<sub>6</sub> ||class="entry q1 g0"| 10437<sub>6</sub> ||class="entry q1 g0"| 23393<sub>8</sub> ||class="entry q0 g0"| 29326<sub>8</sub> ||class="entry q1 g0"| 21883<sub>10</sub> ||class="entry q1 g0"| 20063<sub>10</sub>
|-
|class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (3874, 6669, 26797, 29578, 26547, 32147) ||class="entry q0 g0"| 3874<sub>6</sub> ||class="entry q1 g0"| 6669<sub>6</sub> ||class="entry q1 g0"| 26797<sub>8</sub> ||class="entry q0 g0"| 29578<sub>8</sub> ||class="entry q1 g0"| 26547<sub>10</sub> ||class="entry q1 g0"| 32147<sub>10</sub>
|-
|class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (4314, 25219, 7877, 27762, 7997, 3067) ||class="entry q0 g0"| 4314<sub>6</sub> ||class="entry q1 g0"| 25219<sub>6</sub> ||class="entry q1 g0"| 7877<sub>8</sub> ||class="entry q0 g0"| 27762<sub>8</sub> ||class="entry q1 g0"| 7997<sub>10</sub> ||class="entry q1 g0"| 3067<sub>10</sub>
|-
|class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (4316, 1251, 30883, 27764, 31069, 28061) ||class="entry q0 g0"| 4316<sub>6</sub> ||class="entry q1 g0"| 1251<sub>6</sub> ||class="entry q1 g0"| 30883<sub>8</sub> ||class="entry q0 g0"| 27764<sub>8</sub> ||class="entry q1 g0"| 31069<sub>10</sub> ||class="entry q1 g0"| 28061<sub>10</sub>
|-
|class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (6922, 3621, 31877, 26530, 29595, 27067) ||class="entry q0 g0"| 6922<sub>6</sub> ||class="entry q1 g0"| 3621<sub>6</sub> ||class="entry q1 g0"| 31877<sub>8</sub> ||class="entry q0 g0"| 26530<sub>8</sub> ||class="entry q1 g0"| 29595<sub>10</sub> ||class="entry q1 g0"| 27067<sub>10</sub>
|-
|class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (6924, 26693, 6883, 26532, 5627, 4061) ||class="entry q0 g0"| 6924<sub>6</sub> ||class="entry q1 g0"| 26693<sub>6</sub> ||class="entry q1 g0"| 6883<sub>8</sub> ||class="entry q0 g0"| 26532<sub>8</sub> ||class="entry q1 g0"| 5627<sub>10</sub> ||class="entry q1 g0"| 4061<sub>10</sub>
|-
|class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (10018, 12825, 31889, 23434, 20391, 27055) ||class="entry q0 g0"| 10018<sub>6</sub> ||class="entry q1 g0"| 12825<sub>6</sub> ||class="entry q1 g0"| 31889<sub>8</sub> ||class="entry q0 g0"| 23434<sub>8</sub> ||class="entry q1 g0"| 20391<sub>10</sub> ||class="entry q1 g0"| 27055<sub>10</sub>
|-
|class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (10032, 26705, 9931, 23448, 5615, 13301) ||class="entry q0 g0"| 10032<sub>6</sub> ||class="entry q1 g0"| 26705<sub>6</sub> ||class="entry q1 g0"| 9931<sub>8</sub> ||class="entry q0 g0"| 23448<sub>8</sub> ||class="entry q1 g0"| 5615<sub>10</sub> ||class="entry q1 g0"| 13301<sub>10</sub>
|-
|class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (10438, 3873, 20045, 21614, 29343, 23411) ||class="entry q0 g0"| 10438<sub>6</sub> ||class="entry q1 g0"| 3873<sub>6</sub> ||class="entry q1 g0"| 20045<sub>8</sub> ||class="entry q0 g0"| 21614<sub>8</sub> ||class="entry q1 g0"| 29343<sub>10</sub> ||class="entry q1 g0"| 23411<sub>10</sub>
|-
|class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (10450, 13065, 29297, 21626, 20151, 26447) ||class="entry q0 g0"| 10450<sub>6</sub> ||class="entry q1 g0"| 13065<sub>6</sub> ||class="entry q1 g0"| 29297<sub>8</sub> ||class="entry q0 g0"| 21626<sub>8</sub> ||class="entry q1 g0"| 20151<sub>10</sub> ||class="entry q1 g0"| 26447<sub>10</sub>
|-
|class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (12826, 10449, 26441, 20146, 21871, 29303) ||class="entry q0 g0"| 12826<sub>6</sub> ||class="entry q1 g0"| 10449<sub>6</sub> ||class="entry q1 g0"| 26441<sub>8</sub> ||class="entry q0 g0"| 20146<sub>8</sub> ||class="entry q1 g0"| 21871<sub>10</sub> ||class="entry q1 g0"| 29303<sub>10</sub>
|-
|class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (13066, 9777, 26809, 20386, 23439, 32135) ||class="entry q0 g0"| 13066<sub>6</sub> ||class="entry q1 g0"| 9777<sub>6</sub> ||class="entry q1 g0"| 26809<sub>8</sub> ||class="entry q0 g0"| 20386<sub>8</sub> ||class="entry q1 g0"| 23439<sub>10</sub> ||class="entry q1 g0"| 32135<sub>10</sub>
|-
|class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (17524, 20809, 31137, 14556, 11511, 27807) ||class="entry q0 g0"| 17524<sub>6</sub> ||class="entry q1 g0"| 20809<sub>6</sub> ||class="entry q1 g0"| 31137<sub>8</sub> ||class="entry q0 g0"| 14556<sub>8</sub> ||class="entry q1 g0"| 11511<sub>10</sub> ||class="entry q1 g0"| 27807<sub>10</sub>
|-
|class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (17776, 25473, 19053, 14808, 7743, 24403) ||class="entry q0 g0"| 17776<sub>6</sub> ||class="entry q1 g0"| 25473<sub>6</sub> ||class="entry q1 g0"| 19053<sub>8</sub> ||class="entry q0 g0"| 14808<sub>8</sub> ||class="entry q1 g0"| 7743<sub>10</sub> ||class="entry q1 g0"| 24403<sub>10</sub>
|-
|class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (19078, 1265, 8939, 13870, 31055, 14293) ||class="entry q0 g0"| 19078<sub>6</sub> ||class="entry q1 g0"| 1265<sub>6</sub> ||class="entry q1 g0"| 8939<sub>8</sub> ||class="entry q0 g0"| 13870<sub>8</sub> ||class="entry q1 g0"| 31055<sub>10</sub> ||class="entry q1 g0"| 14293<sub>10</sub>
|-
|class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (19332, 20569, 30529, 14124, 11751, 25215) ||class="entry q0 g0"| 19332<sub>6</sub> ||class="entry q1 g0"| 20569<sub>6</sub> ||class="entry q1 g0"| 30529<sub>8</sub> ||class="entry q0 g0"| 14124<sub>8</sub> ||class="entry q1 g0"| 11751<sub>10</sub> ||class="entry q1 g0"| 25215<sub>10</sub>
|-
|class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (20572, 17761, 28041, 11508, 14559, 30903) ||class="entry q0 g0"| 20572<sub>6</sub> ||class="entry q1 g0"| 17761<sub>6</sub> ||class="entry q1 g0"| 28041<sub>8</sub> ||class="entry q0 g0"| 11508<sub>8</sub> ||class="entry q1 g0"| 14559<sub>10</sub> ||class="entry q1 g0"| 30903<sub>10</sub>
|-
|class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (20812, 19329, 25209, 11748, 13887, 30535) ||class="entry q0 g0"| 20812<sub>6</sub> ||class="entry q1 g0"| 19329<sub>6</sub> ||class="entry q1 g0"| 25209<sub>8</sub> ||class="entry q0 g0"| 11748<sub>8</sub> ||class="entry q1 g0"| 13887<sub>10</sub> ||class="entry q1 g0"| 30535<sub>10</sub>
|-
|class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (25234, 4301, 2795, 7738, 28019, 8149) ||class="entry q0 g0"| 25234<sub>6</sub> ||class="entry q1 g0"| 4301<sub>6</sub> ||class="entry q1 g0"| 2795<sub>8</sub> ||class="entry q0 g0"| 7738<sub>8</sub> ||class="entry q1 g0"| 28019<sub>10</sub> ||class="entry q1 g0"| 8149<sub>10</sub>
|-
|class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (25488, 17509, 24385, 7992, 14811, 19071) ||class="entry q0 g0"| 25488<sub>6</sub> ||class="entry q1 g0"| 17509<sub>6</sub> ||class="entry q1 g0"| 24385<sub>8</sub> ||class="entry q0 g0"| 7992<sub>8</sub> ||class="entry q1 g0"| 14811<sub>10</sub> ||class="entry q1 g0"| 19071<sub>10</sub>
|-
|class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (26948, 6667, 3789, 5612, 26549, 7155) ||class="entry q0 g0"| 26948<sub>6</sub> ||class="entry q1 g0"| 6667<sub>6</sub> ||class="entry q1 g0"| 3789<sub>8</sub> ||class="entry q0 g0"| 5612<sub>8</sub> ||class="entry q1 g0"| 26549<sub>10</sub> ||class="entry q1 g0"| 7155<sub>10</sub>
|-
|class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 6, 8, 8, 10, 10) ||class="c"| (26960, 9763, 13041, 5624, 23453, 10191) ||class="entry q0 g0"| 26960<sub>6</sub> ||class="entry q1 g0"| 9763<sub>6</sub> ||class="entry q1 g0"| 13041<sub>8</sub> ||class="entry q0 g0"| 5624<sub>8</sub> ||class="entry q1 g0"| 23453<sub>10</sub> ||class="entry q1 g0"| 10191<sub>10</sub>
|-
|class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 6, 8, 8, 4) ||class="c"| (1506, 30795, 1309, 31050, 1525, 4131) ||class="entry q0 g0"| 1506<sub>6</sub> ||class="entry q1 g0"| 30795<sub>8</sub> ||class="entry q1 g0"| 1309<sub>6</sub> ||class="entry q0 g0"| 31050<sub>8</sub> ||class="entry q1 g0"| 1525<sub>8</sub> ||class="entry q1 g0"| 4131<sub>4</sub>
|-
|class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 6, 8, 8, 4) ||class="c"| (3636, 29325, 315, 29340, 3891, 5125) ||class="entry q0 g0"| 3636<sub>6</sub> ||class="entry q1 g0"| 29325<sub>8</sub> ||class="entry q1 g0"| 315<sub>6</sub> ||class="entry q0 g0"| 29340<sub>8</sub> ||class="entry q1 g0"| 3891<sub>8</sub> ||class="entry q1 g0"| 5125<sub>4</sub>
|-
|class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 6, 8, 8, 4) ||class="c"| (4554, 27747, 4405, 28002, 4573, 1035) ||class="entry q0 g0"| 4554<sub>6</sub> ||class="entry q1 g0"| 27747<sub>8</sub> ||class="entry q1 g0"| 4405<sub>6</sub> ||class="entry q0 g0"| 28002<sub>8</sub> ||class="entry q1 g0"| 4573<sub>8</sub> ||class="entry q1 g0"| 1035<sub>4</sub>
|-
|class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 6, 8, 8, 4) ||class="c"| (6684, 26277, 5395, 26292, 6939, 45) ||class="entry q0 g0"| 6684<sub>6</sub> ||class="entry q1 g0"| 26277<sub>8</sub> ||class="entry q1 g0"| 5395<sub>6</sub> ||class="entry q0 g0"| 26292<sub>8</sub> ||class="entry q1 g0"| 6939<sub>8</sub> ||class="entry q1 g0"| 45<sub>4</sub>
|-
|class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 6, 8, 8, 4) ||class="c"| (9780, 23193, 5383, 23196, 10023, 57) ||class="entry q0 g0"| 9780<sub>6</sub> ||class="entry q1 g0"| 23193<sub>8</sub> ||class="entry q1 g0"| 5383<sub>6</sub> ||class="entry q0 g0"| 23196<sub>8</sub> ||class="entry q1 g0"| 10023<sub>8</sub> ||class="entry q1 g0"| 57<sub>4</sub>
|-
|class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 6, 8, 8, 4) ||class="c"| (10452, 21865, 5143, 21628, 10455, 297) ||class="entry q0 g0"| 10452<sub>6</sub> ||class="entry q1 g0"| 21865<sub>8</sub> ||class="entry q1 g0"| 5143<sub>6</sub> ||class="entry q0 g0"| 21628<sub>8</sub> ||class="entry q1 g0"| 10455<sub>8</sub> ||class="entry q1 g0"| 297<sub>4</sub>
|-
|class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 6, 8, 8, 4) ||class="c"| (12828, 20145, 303, 20148, 13071, 5137) ||class="entry q0 g0"| 12828<sub>6</sub> ||class="entry q1 g0"| 20145<sub>8</sub> ||class="entry q1 g0"| 303<sub>6</sub> ||class="entry q0 g0"| 20148<sub>8</sub> ||class="entry q1 g0"| 13071<sub>8</sub> ||class="entry q1 g0"| 5137<sub>4</sub>
|-
|class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 6, 8, 8, 4) ||class="c"| (17762, 14793, 4151, 14794, 17527, 1289) ||class="entry q0 g0"| 17762<sub>6</sub> ||class="entry q1 g0"| 14793<sub>8</sub> ||class="entry q1 g0"| 4151<sub>6</sub> ||class="entry q0 g0"| 14794<sub>8</sub> ||class="entry q1 g0"| 17527<sub>8</sub> ||class="entry q1 g0"| 1289<sub>4</sub>
|-
|class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 6, 8, 8, 4) ||class="c"| (19330, 13881, 4391, 14122, 19335, 1049) ||class="entry q0 g0"| 19330<sub>6</sub> ||class="entry q1 g0"| 13881<sub>8</sub> ||class="entry q1 g0"| 4391<sub>6</sub> ||class="entry q0 g0"| 14122<sub>8</sub> ||class="entry q1 g0"| 19335<sub>8</sub> ||class="entry q1 g0"| 1049<sub>4</sub>
|-
|class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 6, 8, 8, 4) ||class="c"| (20810, 11745, 1055, 11746, 20575, 4385) ||class="entry q0 g0"| 20810<sub>6</sub> ||class="entry q1 g0"| 11745<sub>8</sub> ||class="entry q1 g0"| 1055<sub>6</sub> ||class="entry q0 g0"| 11746<sub>8</sub> ||class="entry q1 g0"| 20575<sub>8</sub> ||class="entry q1 g0"| 4385<sub>4</sub>
|-
|class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 6, 8, 8, 4) ||class="c"| (25474, 7725, 1307, 7978, 25491, 4133) ||class="entry q0 g0"| 25474<sub>6</sub> ||class="entry q1 g0"| 7725<sub>8</sub> ||class="entry q1 g0"| 1307<sub>6</sub> ||class="entry q0 g0"| 7978<sub>8</sub> ||class="entry q1 g0"| 25491<sub>8</sub> ||class="entry q1 g0"| 4133<sub>4</sub>
|-
|class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 6, 8, 8, 4) ||class="c"| (26708, 5355, 317, 5372, 26965, 5123) ||class="entry q0 g0"| 26708<sub>6</sub> ||class="entry q1 g0"| 5355<sub>8</sub> ||class="entry q1 g0"| 317<sub>6</sub> ||class="entry q0 g0"| 5372<sub>8</sub> ||class="entry q1 g0"| 26965<sub>8</sub> ||class="entry q1 g0"| 5123<sub>4</sub>
|-
|class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 6, 12, 8, 8) ||class="c"| (470, 10709, 21889, 32126, 21611, 16575) ||class="entry q0 g0"| 470<sub>6</sub> ||class="entry q1 g0"| 10709<sub>8</sub> ||class="entry q1 g0"| 21889<sub>6</sub> ||class="entry q0 g0"| 32126<sub>12</sub> ||class="entry q1 g0"| 21611<sub>8</sub> ||class="entry q1 g0"| 16575<sub>8</sub>
|-
|class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 6, 12, 8, 8) ||class="c"| (2838, 19347, 14385, 30654, 13869, 11535) ||class="entry q0 g0"| 2838<sub>6</sub> ||class="entry q1 g0"| 19347<sub>8</sub> ||class="entry q1 g0"| 14385<sub>6</sub> ||class="entry q0 g0"| 30654<sub>12</sub> ||class="entry q1 g0"| 13869<sub>8</sub> ||class="entry q1 g0"| 11535<sub>8</sub>
|-
|class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 6, 12, 8, 8) ||class="c"| (8982, 25479, 11277, 24510, 7737, 14643) ||class="entry q0 g0"| 8982<sub>6</sub> ||class="entry q1 g0"| 25479<sub>8</sub> ||class="entry q1 g0"| 11277<sub>6</sub> ||class="entry q0 g0"| 24510<sub>12</sub> ||class="entry q1 g0"| 7737<sub>8</sub> ||class="entry q1 g0"| 14643<sub>8</sub>
|-
|class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 6, 12, 8, 8) ||class="c"| (16726, 26711, 16555, 15870, 5609, 21909) ||class="entry q0 g0"| 16726<sub>6</sub> ||class="entry q1 g0"| 26711<sub>8</sub> ||class="entry q1 g0"| 16555<sub>6</sub> ||class="entry q0 g0"| 15870<sub>12</sub> ||class="entry q1 g0"| 5609<sub>8</sub> ||class="entry q1 g0"| 21909<sub>8</sub>
|-
|class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 8, 8, 8, 6) ||class="c"| (1968, 18901, 13703, 31512, 13419, 8377) ||class="entry q0 g0"| 1968<sub>6</sub> ||class="entry q1 g0"| 18901<sub>8</sub> ||class="entry q1 g0"| 13703<sub>8</sub> ||class="entry q0 g0"| 31512<sub>8</sub> ||class="entry q1 g0"| 13419<sub>8</sub> ||class="entry q1 g0"| 8377<sub>6</sub>
|-
|class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 8, 8, 8, 6) ||class="c"| (3440, 11155, 22583, 29144, 22061, 19721) ||class="entry q0 g0"| 3440<sub>6</sub> ||class="entry q1 g0"| 11155<sub>8</sub> ||class="entry q1 g0"| 22583<sub>8</sub> ||class="entry q0 g0"| 29144<sub>8</sub> ||class="entry q1 g0"| 22061<sub>8</sub> ||class="entry q1 g0"| 19721<sub>6</sub>
|-
|class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 8, 8, 8, 6) ||class="c"| (5004, 25045, 7571, 28452, 7275, 2221) ||class="entry q0 g0"| 5004<sub>6</sub> ||class="entry q1 g0"| 25045<sub>8</sub> ||class="entry q1 g0"| 7571<sub>8</sub> ||class="entry q0 g0"| 28452<sub>8</sub> ||class="entry q1 g0"| 7275<sub>8</sub> ||class="entry q1 g0"| 2221<sub>6</sub>
|-
|class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 8, 8, 8, 6) ||class="c"| (6234, 27411, 6581, 25842, 5805, 3211) ||class="entry q0 g0"| 6234<sub>6</sub> ||class="entry q1 g0"| 27411<sub>8</sub> ||class="entry q1 g0"| 6581<sub>8</sub> ||class="entry q0 g0"| 25842<sub>8</sub> ||class="entry q1 g0"| 5805<sub>8</sub> ||class="entry q1 g0"| 3211<sub>6</sub>
|-
|class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 8, 8, 8, 6) ||class="c"| (9318, 27399, 9629, 22734, 5817, 12451) ||class="entry q0 g0"| 9318<sub>6</sub> ||class="entry q1 g0"| 27399<sub>8</sub> ||class="entry q1 g0"| 9629<sub>8</sub> ||class="entry q0 g0"| 22734<sub>8</sub> ||class="entry q1 g0"| 5817<sub>8</sub> ||class="entry q1 g0"| 12451<sub>6</sub>
|-
|class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 8, 8, 8, 6) ||class="c"| (11140, 12383, 28967, 22316, 19937, 25625) ||class="entry q0 g0"| 11140<sub>6</sub> ||class="entry q1 g0"| 12383<sub>8</sub> ||class="entry q1 g0"| 28967<sub>8</sub> ||class="entry q0 g0"| 22316<sub>8</sub> ||class="entry q1 g0"| 19937<sub>8</sub> ||class="entry q1 g0"| 25625<sub>6</sub>
|-
|class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 8, 8, 8, 6) ||class="c"| (11152, 3191, 19739, 22328, 29129, 22565) ||class="entry q0 g0"| 11152<sub>6</sub> ||class="entry q1 g0"| 3191<sub>8</sub> ||class="entry q1 g0"| 19739<sub>8</sub> ||class="entry q0 g0"| 22328<sub>8</sub> ||class="entry q1 g0"| 29129<sub>8</sub> ||class="entry q1 g0"| 22565<sub>6</sub>
|-
|class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 8, 8, 8, 6) ||class="c"| (12620, 11143, 25631, 19940, 22073, 28961) ||class="entry q0 g0"| 12620<sub>6</sub> ||class="entry q1 g0"| 11143<sub>8</sub> ||class="entry q1 g0"| 25631<sub>8</sub> ||class="entry q0 g0"| 19940<sub>8</sub> ||class="entry q1 g0"| 22073<sub>8</sub> ||class="entry q1 g0"| 28961<sub>6</sub>
|-
|class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 8, 8, 8, 6) ||class="c"| (17958, 24791, 18747, 14990, 7529, 23557) ||class="entry q0 g0"| 17958<sub>6</sub> ||class="entry q1 g0"| 24791<sub>8</sub> ||class="entry q1 g0"| 18747<sub>8</sub> ||class="entry q0 g0"| 14990<sub>8</sub> ||class="entry q1 g0"| 7529<sub>8</sub> ||class="entry q1 g0"| 23557<sub>6</sub>
|-
|class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 8, 8, 8, 6) ||class="c"| (18642, 21263, 29719, 13434, 11953, 24873) ||class="entry q0 g0"| 18642<sub>6</sub> ||class="entry q1 g0"| 21263<sub>8</sub> ||class="entry q1 g0"| 29719<sub>8</sub> ||class="entry q0 g0"| 13434<sub>8</sub> ||class="entry q1 g0"| 11953<sub>8</sub> ||class="entry q1 g0"| 24873<sub>6</sub>
|-
|class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 8, 8, 8, 6) ||class="c"| (18896, 1959, 8637, 13688, 31257, 13443) ||class="entry q0 g0"| 18896<sub>6</sub> ||class="entry q1 g0"| 1959<sub>8</sub> ||class="entry q1 g0"| 8637<sub>8</sub> ||class="entry q0 g0"| 13688<sub>8</sub> ||class="entry q1 g0"| 31257<sub>8</sub> ||class="entry q1 g0"| 13443<sub>6</sub>
|-
|class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 8, 8, 8, 6) ||class="c"| (21018, 18647, 24879, 11954, 13673, 29713) ||class="entry q0 g0"| 21018<sub>6</sub> ||class="entry q1 g0"| 18647<sub>8</sub> ||class="entry q1 g0"| 24879<sub>8</sub> ||class="entry q0 g0"| 11954<sub>8</sub> ||class="entry q1 g0"| 13673<sub>8</sub> ||class="entry q1 g0"| 29713<sub>6</sub>
|-
|class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 8, 8, 8, 6) ||class="c"| (24774, 18227, 23575, 7278, 14989, 18729) ||class="entry q0 g0"| 24774<sub>6</sub> ||class="entry q1 g0"| 18227<sub>8</sub> ||class="entry q1 g0"| 23575<sub>8</sub> ||class="entry q0 g0"| 7278<sub>8</sub> ||class="entry q1 g0"| 14989<sub>8</sub> ||class="entry q1 g0"| 18729<sub>6</sub>
|-
|class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 8, 8, 8, 6) ||class="c"| (25028, 5019, 2493, 7532, 28197, 7299) ||class="entry q0 g0"| 25028<sub>6</sub> ||class="entry q1 g0"| 5019<sub>8</sub> ||class="entry q1 g0"| 2493<sub>8</sub> ||class="entry q0 g0"| 7532<sub>8</sub> ||class="entry q1 g0"| 28197<sub>8</sub> ||class="entry q1 g0"| 7299<sub>6</sub>
|-
|class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 8, 8, 8, 6) ||class="c"| (27142, 9589, 12711, 5806, 22731, 9369) ||class="entry q0 g0"| 27142<sub>6</sub> ||class="entry q1 g0"| 9589<sub>8</sub> ||class="entry q1 g0"| 12711<sub>8</sub> ||class="entry q0 g0"| 5806<sub>8</sub> ||class="entry q1 g0"| 22731<sub>8</sub> ||class="entry q1 g0"| 9369<sub>6</sub>
|-
|class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 8, 8, 8, 6) ||class="c"| (27154, 6493, 3483, 5818, 25827, 6309) ||class="entry q0 g0"| 27154<sub>6</sub> ||class="entry q1 g0"| 6493<sub>8</sub> ||class="entry q1 g0"| 3483<sub>8</sub> ||class="entry q0 g0"| 5818<sub>8</sub> ||class="entry q1 g0"| 25827<sub>8</sub> ||class="entry q1 g0"| 6309<sub>6</sub>
|-
|class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 10, 8, 8, 8) ||class="c"| (1508, 7723, 25467, 31052, 25493, 30277) ||class="entry q0 g0"| 1508<sub>6</sub> ||class="entry q1 g0"| 7723<sub>8</sub> ||class="entry q1 g0"| 25467<sub>10</sub> ||class="entry q0 g0"| 31052<sub>8</sub> ||class="entry q1 g0"| 25493<sub>8</sub> ||class="entry q1 g0"| 30277<sub>8</sub>
|-
|class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 10, 8, 8, 8) ||class="c"| (3634, 5357, 26461, 29338, 26963, 29283) ||class="entry q0 g0"| 3634<sub>6</sub> ||class="entry q1 g0"| 5357<sub>8</sub> ||class="entry q1 g0"| 26461<sub>10</sub> ||class="entry q0 g0"| 29338<sub>8</sub> ||class="entry q1 g0"| 26963<sub>8</sub> ||class="entry q1 g0"| 29283<sub>8</sub>
|-
|class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 10, 8, 8, 8) ||class="c"| (4568, 13867, 19311, 28016, 19349, 24145) ||class="entry q0 g0"| 4568<sub>6</sub> ||class="entry q1 g0"| 13867<sub>8</sub> ||class="entry q1 g0"| 19311<sub>10</sub> ||class="entry q0 g0"| 28016<sub>8</sub> ||class="entry q1 g0"| 19349<sub>8</sub> ||class="entry q1 g0"| 24145<sub>8</sub>
|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
|class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (6, 8, 10, 8, 8, 8) ||class="c"| (26706, 29323, 26459, 5370, 3893, 29285) ||class="entry q0 g0"| 26706<sub>6</sub> ||class="entry q1 g0"| 29323<sub>8</sub> ||class="entry q1 g0"| 26459<sub>10</sub> ||class="entry q0 g0"| 5370<sub>8</sub> ||class="entry q1 g0"| 3893<sub>8</sub> ||class="entry q1 g0"| 29285<sub>8</sub>
|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
|class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 6, 6, 10, 10, 8) ||class="c"| (10038, 3633, 16557, 23454, 29583, 21907) ||class="entry q0 g0"| 10038<sub>8</sub> ||class="entry q1 g0"| 3633<sub>6</sub> ||class="entry q1 g0"| 16557<sub>6</sub> ||class="entry q0 g0"| 23454<sub>10</sub> ||class="entry q1 g0"| 29583<sub>10</sub> ||class="entry q1 g0"| 21907<sub>8</sub>
|-
|class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 6, 6, 10, 10, 8) ||class="c"| (13086, 6681, 21637, 20406, 26535, 16827) ||class="entry q0 g0"| 13086<sub>8</sub> ||class="entry q1 g0"| 6681<sub>6</sub> ||class="entry q1 g0"| 21637<sub>6</sub> ||class="entry q0 g0"| 20406<sub>10</sub> ||class="entry q1 g0"| 26535<sub>10</sub> ||class="entry q1 g0"| 16827<sub>8</sub>
|-
|class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 6, 6, 10, 10, 8) ||class="c"| (17782, 1505, 11275, 14814, 30815, 14645) ||class="entry q0 g0"| 17782<sub>8</sub> ||class="entry q1 g0"| 1505<sub>6</sub> ||class="entry q1 g0"| 11275<sub>6</sub> ||class="entry q0 g0"| 14814<sub>10</sub> ||class="entry q1 g0"| 30815<sub>10</sub> ||class="entry q1 g0"| 14645<sub>8</sub>
|-
|class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 6, 6, 10, 10, 8) ||class="c"| (20830, 4553, 14371, 11766, 27767, 11549) ||class="entry q0 g0"| 20830<sub>8</sub> ||class="entry q1 g0"| 4553<sub>6</sub> ||class="entry q1 g0"| 14371<sub>6</sub> ||class="entry q0 g0"| 11766<sub>10</sub> ||class="entry q1 g0"| 27767<sub>10</sub> ||class="entry q1 g0"| 11549<sub>8</sub>
|-
|class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 8, 10, 10, 8, 8) ||class="c"| (11158, 27159, 11133, 22334, 6057, 15939) ||class="entry q0 g0"| 11158<sub>8</sub> ||class="entry q1 g0"| 27159<sub>8</sub> ||class="entry q1 g0"| 11133<sub>10</sub> ||class="entry q0 g0"| 22334<sub>10</sub> ||class="entry q1 g0"| 6057<sub>8</sub> ||class="entry q1 g0"| 15939<sub>8</sub>
|-
|class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 8, 10, 10, 8, 8) ||class="c"| (18902, 25031, 18395, 13694, 7289, 21221) ||class="entry q0 g0"| 18902<sub>8</sub> ||class="entry q1 g0"| 25031<sub>8</sub> ||class="entry q1 g0"| 18395<sub>10</sub> ||class="entry q0 g0"| 13694<sub>10</sub> ||class="entry q1 g0"| 7289<sub>8</sub> ||class="entry q1 g0"| 21221<sub>8</sub>
|-
|class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 8, 10, 10, 8, 8) ||class="c"| (25046, 18899, 21479, 7550, 13421, 18137) ||class="entry q0 g0"| 25046<sub>8</sub> ||class="entry q1 g0"| 18899<sub>8</sub> ||class="entry q1 g0"| 21479<sub>10</sub> ||class="entry q0 g0"| 7550<sub>10</sub> ||class="entry q1 g0"| 13421<sub>8</sub> ||class="entry q1 g0"| 18137<sub>8</sub>
|-
|class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 8, 10, 10, 8, 8) ||class="c"| (27414, 11157, 15959, 6078, 22059, 11113) ||class="entry q0 g0"| 27414<sub>8</sub> ||class="entry q1 g0"| 11157<sub>8</sub> ||class="entry q1 g0"| 15959<sub>10</sub> ||class="entry q0 g0"| 6078<sub>10</sub> ||class="entry q1 g0"| 22059<sub>8</sub> ||class="entry q1 g0"| 11113<sub>8</sub>
|-
|class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 10, 8, 10, 6, 10) ||class="c"| (1974, 12213, 21473, 31518, 21003, 18143) ||class="entry q0 g0"| 1974<sub>8</sub> ||class="entry q1 g0"| 12213<sub>10</sub> ||class="entry q1 g0"| 21473<sub>8</sub> ||class="entry q0 g0"| 31518<sub>10</sub> ||class="entry q1 g0"| 21003<sub>6</sub> ||class="entry q1 g0"| 18143<sub>10</sub>
|-
|class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 10, 8, 10, 6, 10) ||class="c"| (3446, 19955, 15953, 29150, 12365, 11119) ||class="entry q0 g0"| 3446<sub>8</sub> ||class="entry q1 g0"| 19955<sub>10</sub> ||class="entry q1 g0"| 15953<sub>8</sub> ||class="entry q0 g0"| 29150<sub>10</sub> ||class="entry q1 g0"| 12365<sub>6</sub> ||class="entry q1 g0"| 11119<sub>10</sub>
|-
|class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 10, 8, 10, 6, 10) ||class="c"| (5022, 15261, 18377, 28470, 17955, 21239) ||class="entry q0 g0"| 5022<sub>8</sub> ||class="entry q1 g0"| 15261<sub>10</sub> ||class="entry q1 g0"| 18377<sub>8</sub> ||class="entry q0 g0"| 28470<sub>10</sub> ||class="entry q1 g0"| 17955<sub>6</sub> ||class="entry q1 g0"| 21239<sub>10</sub>
|-
|class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 10, 8, 10, 6, 10) ||class="c"| (6494, 23003, 10873, 26102, 9317, 16199) ||class="entry q0 g0"| 6494<sub>8</sub> ||class="entry q1 g0"| 23003<sub>10</sub> ||class="entry q1 g0"| 10873<sub>8</sub> ||class="entry q0 g0"| 26102<sub>10</sub> ||class="entry q1 g0"| 9317<sub>6</sub> ||class="entry q1 g0"| 16199<sub>10</sub>
|-
|class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 10, 8, 10, 6, 10) ||class="c"| (9590, 26087, 10861, 23006, 6233, 16211) ||class="entry q0 g0"| 9590<sub>8</sub> ||class="entry q1 g0"| 26087<sub>10</sub> ||class="entry q1 g0"| 10861<sub>8</sub> ||class="entry q0 g0"| 23006<sub>10</sub> ||class="entry q1 g0"| 6233<sub>6</sub> ||class="entry q1 g0"| 16211<sub>10</sub>
|-
|class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 10, 8, 10, 6, 10) ||class="c"| (12638, 29135, 15941, 19958, 3185, 11131) ||class="entry q0 g0"| 12638<sub>8</sub> ||class="entry q1 g0"| 29135<sub>10</sub> ||class="entry q1 g0"| 15941<sub>8</sub> ||class="entry q0 g0"| 19958<sub>10</sub> ||class="entry q1 g0"| 3185<sub>6</sub> ||class="entry q1 g0"| 11131<sub>10</sub>
|-
|class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 10, 8, 10, 6, 10) ||class="c"| (18230, 28215, 18123, 15262, 5001, 21493) ||class="entry q0 g0"| 18230<sub>8</sub> ||class="entry q1 g0"| 28215<sub>10</sub> ||class="entry q1 g0"| 18123<sub>8</sub> ||class="entry q0 g0"| 15262<sub>10</sub> ||class="entry q1 g0"| 5001<sub>6</sub> ||class="entry q1 g0"| 21493<sub>10</sub>
|-
|class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 0, 0) ||class="w"| (8, 10, 8, 10, 6, 10) ||class="c"| (21278, 31263, 21219, 12214, 1953, 18397) ||class="entry q0 g0"| 21278<sub>8</sub> ||class="entry q1 g0"| 31263<sub>10</sub> ||class="entry q1 g0"| 21219<sub>8</sub> ||class="entry q0 g0"| 12214<sub>10</sub> ||class="entry q1 g0"| 1953<sub>6</sub> ||class="entry q1 g0"| 18397<sub>10</sub>
|-
|class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 1, 0) ||class="w"| (8, 12, 8, 6, 4, 10) ||class="c"| (11506, 24255, 7889, 20570, 8961, 3055) ||class="entry q0 g0"| 11506<sub>8</sub> ||class="entry q1 g0"| 24255<sub>12</sub> ||class="entry q1 g0"| 7889<sub>8</sub> ||class="entry q0 g0"| 20570<sub>6</sub> ||class="entry q1 g1"| 8961<sub>4</sub> ||class="entry q1 g0"| 3055<sub>10</sub>
|-
|class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 1, 0) ||class="w"| (8, 12, 8, 6, 4, 10) ||class="c"| (14542, 30399, 14021, 17510, 2817, 9211) ||class="entry q0 g0"| 14542<sub>8</sub> ||class="entry q1 g0"| 30399<sub>12</sub> ||class="entry q1 g0"| 14021<sub>8</sub> ||class="entry q0 g0"| 17510<sub>6</sub> ||class="entry q1 g1"| 2817<sub>4</sub> ||class="entry q1 g0"| 9211<sub>10</sub>
|-
|class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 1, 0) ||class="w"| (8, 12, 8, 6, 4, 10) ||class="c"| (20388, 15855, 7137, 13068, 16465, 3807) ||class="entry q0 g0"| 20388<sub>8</sub> ||class="entry q1 g0"| 15855<sub>12</sub> ||class="entry q1 g0"| 7137<sub>8</sub> ||class="entry q0 g0"| 13068<sub>6</sub> ||class="entry q1 g1"| 16465<sub>4</sub> ||class="entry q1 g0"| 3807<sub>10</sub>
|-
|class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 1, 0) ||class="w"| (8, 12, 8, 6, 4, 10) ||class="c"| (23182, 32111, 23139, 9766, 209, 20317) ||class="entry q0 g0"| 23182<sub>8</sub> ||class="entry q1 g0"| 32111<sub>12</sub> ||class="entry q1 g0"| 23139<sub>8</sub> ||class="entry q0 g0"| 9766<sub>6</sub> ||class="entry q1 g1"| 209<sub>4</sub> ||class="entry q1 g0"| 20317<sub>10</sub>
|-
|class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 1, 0) ||class="w"| (8, 12, 8, 6, 4, 10) ||class="c"| (26290, 32123, 26187, 6682, 197, 29557) ||class="entry q0 g0"| 26290<sub>8</sub> ||class="entry q1 g0"| 32123<sub>12</sub> ||class="entry q1 g0"| 26187<sub>8</sub> ||class="entry q0 g0"| 6682<sub>6</sub> ||class="entry q1 g1"| 197<sub>4</sub> ||class="entry q1 g0"| 29557<sub>10</sub>
|-
|class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 1, 0) ||class="w"| (8, 12, 8, 6, 4, 10) ||class="c"| (28004, 30653, 25197, 4556, 2563, 30547) ||class="entry q0 g0"| 28004<sub>8</sub> ||class="entry q1 g0"| 30653<sub>12</sub> ||class="entry q1 g0"| 25197<sub>8</sub> ||class="entry q0 g0"| 4556<sub>6</sub> ||class="entry q1 g1"| 2563<sub>4</sub> ||class="entry q1 g0"| 30547<sub>10</sub>
|-
|class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 1, 0) ||class="w"| (8, 12, 8, 6, 4, 10) ||class="c"| (29592, 15867, 10185, 3888, 16453, 13047) ||class="entry q0 g0"| 29592<sub>8</sub> ||class="entry q1 g0"| 15867<sub>12</sub> ||class="entry q1 g0"| 10185<sub>8</sub> ||class="entry q0 g0"| 3888<sub>6</sub> ||class="entry q1 g1"| 16453<sub>4</sub> ||class="entry q1 g0"| 13047<sub>10</sub>
|-
|class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 0, 1, 0) ||class="w"| (8, 12, 8, 6, 4, 10) ||class="c"| (31064, 24509, 19065, 1520, 8707, 24391) ||class="entry q0 g0"| 31064<sub>8</sub> ||class="entry q1 g0"| 24509<sub>12</sub> ||class="entry q1 g0"| 19065<sub>8</sub> ||class="entry q0 g0"| 1520<sub>6</sub> ||class="entry q1 g1"| 8707<sub>4</sub> ||class="entry q1 g0"| 24391<sub>10</sub>
|-
|class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 1, 0, 0) ||class="w"| (10, 8, 10, 4, 8, 8) ||class="c"| (15852, 20385, 4047, 16708, 12831, 6897) ||class="entry q0 g0"| 15852<sub>10</sub> ||class="entry q1 g0"| 20385<sub>8</sub> ||class="entry q1 g0"| 4047<sub>10</sub> ||class="entry q0 g1"| 16708<sub>4</sub> ||class="entry q1 g0"| 12831<sub>8</sub> ||class="entry q1 g0"| 6897<sub>8</sub>
|-
|class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 1, 0, 0) ||class="w"| (10, 8, 10, 4, 8, 8) ||class="c"| (15864, 29577, 13299, 16720, 3639, 9933) ||class="entry q0 g0"| 15864<sub>10</sub> ||class="entry q1 g0"| 29577<sub>8</sub> ||class="entry q1 g0"| 13299<sub>10</sub> ||class="entry q0 g1"| 16720<sub>4</sub> ||class="entry q1 g0"| 3639<sub>8</sub> ||class="entry q1 g0"| 9933<sub>8</sub>
|-
|class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 1, 0, 0) ||class="w"| (10, 8, 10, 4, 8, 8) ||class="c"| (24250, 11505, 2815, 8722, 20815, 8129) ||class="entry q0 g0"| 24250<sub>10</sub> ||class="entry q1 g0"| 11505<sub>8</sub> ||class="entry q1 g0"| 2815<sub>10</sub> ||class="entry q0 g1"| 8722<sub>4</sub> ||class="entry q1 g0"| 20815<sub>8</sub> ||class="entry q1 g0"| 8129<sub>8</sub>
|-
|class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 1, 0, 0) ||class="w"| (10, 8, 10, 4, 8, 8) ||class="c"| (24504, 30809, 24405, 8976, 1511, 19051) ||class="entry q0 g0"| 24504<sub>10</sub> ||class="entry q1 g0"| 30809<sub>8</sub> ||class="entry q1 g0"| 24405<sub>10</sub> ||class="entry q0 g1"| 8976<sub>4</sub> ||class="entry q1 g0"| 1511<sub>8</sub> ||class="entry q1 g0"| 19051<sub>8</sub>
|-
|class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 1, 0, 0) ||class="w"| (10, 8, 10, 4, 8, 8) ||class="c"| (30382, 14541, 8959, 2566, 17779, 14273) ||class="entry q0 g0"| 30382<sub>10</sub> ||class="entry q1 g0"| 14541<sub>8</sub> ||class="entry q1 g0"| 8959<sub>10</sub> ||class="entry q0 g1"| 2566<sub>4</sub> ||class="entry q1 g0"| 17779<sub>8</sub> ||class="entry q1 g0"| 14273<sub>8</sub>
|-
|class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 1, 0, 0) ||class="w"| (10, 8, 10, 4, 8, 8) ||class="c"| (30636, 27749, 30549, 2820, 4571, 25195) ||class="entry q0 g0"| 30636<sub>10</sub> ||class="entry q1 g0"| 27749<sub>8</sub> ||class="entry q1 g0"| 30549<sub>10</sub> ||class="entry q0 g1"| 2820<sub>4</sub> ||class="entry q1 g0"| 4571<sub>8</sub> ||class="entry q1 g0"| 25195<sub>8</sub>
|-
|class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 1, 0, 0) ||class="w"| (10, 8, 10, 4, 8, 8) ||class="c"| (31854, 23179, 20303, 198, 10037, 23153) ||class="entry q0 g0"| 31854<sub>10</sub> ||class="entry q1 g0"| 23179<sub>8</sub> ||class="entry q1 g0"| 20303<sub>10</sub> ||class="entry q0 g1"| 198<sub>4</sub> ||class="entry q1 g0"| 10037<sub>8</sub> ||class="entry q1 g0"| 23153<sub>8</sub>
|-
|class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 1, 0, 0) ||class="w"| (10, 8, 10, 4, 8, 8) ||class="c"| (31866, 26275, 29555, 210, 6941, 26189) ||class="entry q0 g0"| 31866<sub>10</sub> ||class="entry q1 g0"| 26275<sub>8</sub> ||class="entry q1 g0"| 29555<sub>10</sub> ||class="entry q0 g1"| 210<sub>4</sub> ||class="entry q1 g0"| 6941<sub>8</sub> ||class="entry q1 g0"| 26189<sub>8</sub>
|-
|class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 1, 1, 0) ||class="w"| (8, 10, 6, 2, 6, 8) ||class="c"| (15592, 32105, 15363, 16448, 215, 10557) ||class="entry q0 g0"| 15592<sub>8</sub> ||class="entry q1 g0"| 32105<sub>10</sub> ||class="entry q1 g0"| 15363<sub>6</sub> ||class="entry q0 g1"| 16448<sub>2</sub> ||class="entry q1 g1"| 215<sub>6</sub> ||class="entry q1 g0"| 10557<sub>8</sub>
|-
|class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 1, 1, 0) ||class="w"| (8, 10, 6, 2, 6, 8) ||class="c"| (24232, 30393, 20645, 8704, 2823, 17819) ||class="entry q0 g0"| 24232<sub>8</sub> ||class="entry q1 g0"| 30393<sub>10</sub> ||class="entry q1 g0"| 20645<sub>6</sub> ||class="entry q0 g1"| 8704<sub>2</sub> ||class="entry q1 g1"| 2823<sub>6</sub> ||class="entry q1 g0"| 17819<sub>8</sub>
|-
|class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 1, 1, 0) ||class="w"| (8, 10, 6, 2, 6, 8) ||class="c"| (30376, 24237, 17561, 2560, 8979, 20903) ||class="entry q0 g0"| 30376<sub>8</sub> ||class="entry q1 g0"| 24237<sub>10</sub> ||class="entry q1 g0"| 17561<sub>6</sub> ||class="entry q0 g1"| 2560<sub>2</sub> ||class="entry q1 g1"| 8979<sub>6</sub> ||class="entry q1 g0"| 20903<sub>8</sub>
|-
|class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 0, 0, 1, 1, 0) ||class="w"| (8, 10, 6, 2, 6, 8) ||class="c"| (31848, 15595, 10537, 192, 16725, 15383) ||class="entry q0 g0"| 31848<sub>8</sub> ||class="entry q1 g0"| 15595<sub>10</sub> ||class="entry q1 g0"| 10537<sub>6</sub> ||class="entry q0 g1"| 192<sub>2</sub> ||class="entry q1 g1"| 16725<sub>6</sub> ||class="entry q1 g0"| 15383<sub>8</sub>
|-
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|-
|class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 1, 0, 1, 0, 0) ||class="w"| (10, 4, 6, 4, 12, 4) ||class="c"| (24490, 8721, 1295, 8962, 24495, 4145) ||class="entry q0 g0"| 24490<sub>10</sub> ||class="entry q1 g1"| 8721<sub>4</sub> ||class="entry q1 g0"| 1295<sub>6</sub> ||class="entry q0 g1"| 8962<sub>4</sub> ||class="entry q1 g0"| 24495<sub>12</sub> ||class="entry q1 g0"| 4145<sub>4</sub>
|-
|class="f"| 5160 ||class="q"| (0, 1, 1, 0, 1, 1) ||class="g"| (0, 1, 0, 1, 0, 0) ||class="w"| (10, 4, 6, 4, 12, 4) ||class="c"| (30634, 2565, 4403, 2818, 30651, 1037) ||class="entry q0 g0"| 30634<sub>10</sub> ||class="entry q1 g1"| 2565<sub>4</sub> ||class="entry q1 g0"| 4403<sub>6</sub> ||class="entry q0 g1"| 2818<sub>4</sub> ||class="entry q1 g0"| 30651<sub>12</sub> ||class="entry q1 g0"| 1037<sub>4</sub>
|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
|class="f"| 6014 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (3, 11, 11, 13, 5, 9) ||class="c"| (43008, 48764, 64938, 49022, 43266, 60116) ||class="entry q2 g1"| 43008<sub>3</sub> ||class="entry q2 g1"| 48764<sub>11</sub> ||class="entry q2 g1"| 64938<sub>11</sub> ||class="entry q2 g1"| 49022<sub>13</sub> ||class="entry q2 g1"| 43266<sub>5</sub> ||class="entry q2 g1"| 60116<sub>9</sub>
|-
|class="f"| 6014 ||class="q"| (2, 2, 2, 2, 2, 2) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (3, 11, 11, 13, 5, 9) ||class="c"| (49280, 55274, 64700, 55294, 49300, 60354) ||class="entry q2 g1"| 49280<sub>3</sub> ||class="entry q2 g1"| 55274<sub>11</sub> ||class="entry q2 g1"| 64700<sub>11</sub> ||class="entry q2 g1"| 55294<sub>13</sub> ||class="entry q2 g1"| 49300<sub>5</sub> ||class="entry q2 g1"| 60354<sub>9</sub>
|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
|class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 5, 11, 11, 7) ||class="c"| (33992, 37559, 34081, 64310, 60511, 37705) ||class="entry q2 g1"| 33992<sub>5</sub> ||class="entry q3 g1"| 37559<sub>9</sub> ||class="entry q3 g1"| 34081<sub>5</sub> ||class="entry q2 g1"| 64310<sub>11</sub> ||class="entry q3 g1"| 60511<sub>11</sub> ||class="entry q3 g1"| 37705<sub>7</sub>
|-
|class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 5, 11, 11, 7) ||class="c"| (35368, 40263, 33841, 62934, 58287, 37465) ||class="entry q2 g1"| 35368<sub>5</sub> ||class="entry q3 g1"| 40263<sub>9</sub> ||class="entry q3 g1"| 33841<sub>5</sub> ||class="entry q2 g1"| 62934<sub>11</sub> ||class="entry q3 g1"| 58287<sub>11</sub> ||class="entry q3 g1"| 37465<sub>7</sub>
|-
|class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 5, 11, 11, 7) ||class="c"| (35912, 39719, 33361, 62390, 58831, 37945) ||class="entry q2 g1"| 35912<sub>5</sub> ||class="entry q3 g1"| 39719<sub>9</sub> ||class="entry q3 g1"| 33361<sub>5</sub> ||class="entry q2 g1"| 62390<sub>11</sub> ||class="entry q3 g1"| 58831<sub>11</sub> ||class="entry q3 g1"| 37945<sub>7</sub>
|-
|class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 5, 11, 11, 7) ||class="c"| (37088, 34463, 37129, 61214, 63607, 34657) ||class="entry q2 g1"| 37088<sub>5</sub> ||class="entry q3 g1"| 34463<sub>9</sub> ||class="entry q3 g1"| 37129<sub>5</sub> ||class="entry q2 g1"| 61214<sub>11</sub> ||class="entry q3 g1"| 63607<sub>11</sub> ||class="entry q3 g1"| 34657<sub>7</sub>
|-
|class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 5, 11, 11, 7) ||class="c"| (40448, 35183, 36889, 57854, 63367, 34417) ||class="entry q2 g1"| 40448<sub>5</sub> ||class="entry q3 g1"| 35183<sub>9</sub> ||class="entry q3 g1"| 36889<sub>5</sub> ||class="entry q2 g1"| 57854<sub>11</sub> ||class="entry q3 g1"| 63367<sub>11</sub> ||class="entry q3 g1"| 34417<sub>7</sub>
|-
|class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 5, 11, 11, 7) ||class="c"| (41512, 46419, 36877, 56790, 52155, 34405) ||class="entry q2 g1"| 41512<sub>5</sub> ||class="entry q3 g1"| 46419<sub>9</sub> ||class="entry q3 g1"| 36877<sub>5</sub> ||class="entry q2 g1"| 56790<sub>11</sub> ||class="entry q3 g1"| 52155<sub>11</sub> ||class="entry q3 g1"| 34405<sub>7</sub>
|-
|class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 5, 11, 11, 7) ||class="c"| (45152, 42779, 33349, 53150, 55795, 37933) ||class="entry q2 g1"| 45152<sub>5</sub> ||class="entry q3 g1"| 42779<sub>9</sub> ||class="entry q3 g1"| 33349<sub>5</sub> ||class="entry q2 g1"| 53150<sub>11</sub> ||class="entry q3 g1"| 55795<sub>11</sub> ||class="entry q3 g1"| 37933<sub>7</sub>
|-
|class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 5, 11, 11, 7) ||class="c"| (46592, 41339, 33829, 51710, 57235, 37453) ||class="entry q2 g1"| 46592<sub>5</sub> ||class="entry q3 g1"| 41339<sub>9</sub> ||class="entry q3 g1"| 33829<sub>5</sub> ||class="entry q2 g1"| 51710<sub>11</sub> ||class="entry q3 g1"| 57235<sub>11</sub> ||class="entry q3 g1"| 37453<sub>7</sub>
|-
|class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 5, 11, 11, 7) ||class="c"| (50248, 54069, 36875, 48054, 44509, 34403) ||class="entry q2 g1"| 50248<sub>5</sub> ||class="entry q3 g1"| 54069<sub>9</sub> ||class="entry q3 g1"| 36875<sub>5</sub> ||class="entry q2 g1"| 48054<sub>11</sub> ||class="entry q3 g1"| 44509<sub>11</sub> ||class="entry q3 g1"| 34403<sub>7</sub>
|-
|class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 5, 11, 11, 7) ||class="c"| (53344, 50973, 33827, 44958, 47605, 37451) ||class="entry q2 g1"| 53344<sub>5</sub> ||class="entry q3 g1"| 50973<sub>9</sub> ||class="entry q3 g1"| 33827<sub>5</sub> ||class="entry q2 g1"| 44958<sub>11</sub> ||class="entry q3 g1"| 47605<sub>11</sub> ||class="entry q3 g1"| 37451<sub>7</sub>
|-
|class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (5, 9, 5, 11, 11, 7) ||class="c"| (54784, 49533, 33347, 43518, 49045, 37931) ||class="entry q2 g1"| 54784<sub>5</sub> ||class="entry q3 g1"| 49533<sub>9</sub> ||class="entry q3 g1"| 33347<sub>5</sub> ||class="entry q2 g1"| 43518<sub>11</sub> ||class="entry q3 g1"| 49045<sub>11</sub> ||class="entry q3 g1"| 37931<sub>7</sub>
|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
|class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 5, 9, 7, 11) ||class="c"| (37796, 57217, 51205, 60506, 41321, 56941) ||class="entry q2 g1"| 37796<sub>7</sub> ||class="entry q3 g1"| 57217<sub>9</sub> ||class="entry q3 g1"| 51205<sub>5</sub> ||class="entry q2 g1"| 60506<sub>9</sub> ||class="entry q3 g1"| 41321<sub>7</sub> ||class="entry q3 g1"| 56941<sub>11</sub>
|-
|class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 5, 9, 7, 11) ||class="c"| (38338, 49025, 43011, 59964, 49513, 48747) ||class="entry q2 g1"| 38338<sub>7</sub> ||class="entry q3 g1"| 49025<sub>9</sub> ||class="entry q3 g1"| 43011<sub>5</sub> ||class="entry q2 g1"| 59964<sub>9</sub> ||class="entry q3 g1"| 49513<sub>7</sub> ||class="entry q3 g1"| 48747<sub>11</sub>
|-
|class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 5, 9, 7, 11) ||class="c"| (39476, 55537, 49285, 58826, 42521, 55021) ||class="entry q2 g1"| 39476<sub>7</sub> ||class="entry q3 g1"| 55537<sub>9</sub> ||class="entry q3 g1"| 49285<sub>5</sub> ||class="entry q2 g1"| 58826<sub>9</sub> ||class="entry q3 g1"| 42521<sub>7</sub> ||class="entry q3 g1"| 55021<sub>11</sub>
|-
|class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 5, 9, 7, 11) ||class="c"| (40018, 47345, 41091, 58284, 50713, 46827) ||class="entry q2 g1"| 40018<sub>7</sub> ||class="entry q3 g1"| 47345<sub>9</sub> ||class="entry q3 g1"| 41091<sub>5</sub> ||class="entry q2 g1"| 58284<sub>9</sub> ||class="entry q3 g1"| 50713<sub>7</sub> ||class="entry q3 g1"| 46827<sub>11</sub>
|-
|class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 5, 9, 7, 11) ||class="c"| (41324, 60493, 51457, 56978, 37541, 57193) ||class="entry q2 g1"| 41324<sub>7</sub> ||class="entry q3 g1"| 60493<sub>9</sub> ||class="entry q3 g1"| 51457<sub>5</sub> ||class="entry q2 g1"| 56978<sub>9</sub> ||class="entry q3 g1"| 37541<sub>7</sub> ||class="entry q3 g1"| 57193<sub>11</sub>
|-
|class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 5, 9, 7, 11) ||class="c"| (42524, 58573, 49297, 55778, 39461, 55033) ||class="entry q2 g1"| 42524<sub>7</sub> ||class="entry q3 g1"| 58573<sub>9</sub> ||class="entry q3 g1"| 49297<sub>5</sub> ||class="entry q2 g1"| 55778<sub>9</sub> ||class="entry q3 g1"| 39461<sub>7</sub> ||class="entry q3 g1"| 55033<sub>11</sub>
|-
|class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 5, 9, 7, 11) ||class="c"| (46150, 44237, 34947, 52152, 53797, 40683) ||class="entry q2 g1"| 46150<sub>7</sub> ||class="entry q3 g1"| 44237<sub>9</sub> ||class="entry q3 g1"| 34947<sub>5</sub> ||class="entry q2 g1"| 52152<sub>9</sub> ||class="entry q3 g1"| 53797<sub>7</sub> ||class="entry q3 g1"| 40683<sub>11</sub>
|-
|class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 5, 9, 7, 11) ||class="c"| (49514, 59947, 43265, 48788, 38083, 49001) ||class="entry q2 g1"| 49514<sub>7</sub> ||class="entry q3 g1"| 59947<sub>9</sub> ||class="entry q3 g1"| 43265<sub>5</sub> ||class="entry q2 g1"| 48788<sub>9</sub> ||class="entry q3 g1"| 38083<sub>7</sub> ||class="entry q3 g1"| 49001<sub>11</sub>
|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
|class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 7, 9, 11, 9) ||class="c"| (42316, 33275, 42401, 55986, 65299, 46025) ||class="entry q2 g1"| 42316<sub>7</sub> ||class="entry q3 g1"| 33275<sub>9</sub> ||class="entry q3 g1"| 42401<sub>7</sub> ||class="entry q2 g1"| 55986<sub>9</sub> ||class="entry q3 g1"| 65299<sub>11</sub> ||class="entry q3 g1"| 46025<sub>9</sub>
|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
|class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 13, 9, 11, 11) ||class="c"| (43240, 55061, 64957, 55062, 43517, 60373) ||class="entry q2 g1"| 43240<sub>7</sub> ||class="entry q3 g1"| 55061<sub>9</sub> ||class="entry q3 g1"| 64957<sub>13</sub> ||class="entry q2 g1"| 55062<sub>9</sub> ||class="entry q3 g1"| 43517<sub>11</sub> ||class="entry q3 g1"| 60373<sub>11</sub>
|-
|class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 13, 9, 11, 11) ||class="c"| (44680, 53621, 64477, 53622, 44957, 60853) ||class="entry q2 g1"| 44680<sub>7</sub> ||class="entry q3 g1"| 53621<sub>9</sub> ||class="entry q3 g1"| 64477<sub>13</sub> ||class="entry q2 g1"| 53622<sub>9</sub> ||class="entry q3 g1"| 44957<sub>11</sub> ||class="entry q3 g1"| 60853<sub>11</sub>
|-
|class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 13, 9, 11, 11) ||class="c"| (47776, 50525, 61429, 50526, 48053, 63901) ||class="entry q2 g1"| 47776<sub>7</sub> ||class="entry q3 g1"| 50525<sub>9</sub> ||class="entry q3 g1"| 61429<sub>13</sub> ||class="entry q2 g1"| 50526<sub>9</sub> ||class="entry q3 g1"| 48053<sub>11</sub> ||class="entry q3 g1"| 63901<sub>11</sub>
|-
|class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 13, 9, 11, 11) ||class="c"| (51432, 46867, 64475, 46870, 51707, 60851) ||class="entry q2 g1"| 51432<sub>7</sub> ||class="entry q3 g1"| 46867<sub>9</sub> ||class="entry q3 g1"| 64475<sub>13</sub> ||class="entry q2 g1"| 46870<sub>9</sub> ||class="entry q3 g1"| 51707<sub>11</sub> ||class="entry q3 g1"| 60851<sub>11</sub>
|-
|class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 13, 9, 11, 11) ||class="c"| (52872, 45427, 64955, 45430, 53147, 60371) ||class="entry q2 g1"| 52872<sub>7</sub> ||class="entry q3 g1"| 45427<sub>9</sub> ||class="entry q3 g1"| 64955<sub>13</sub> ||class="entry q2 g1"| 45430<sub>9</sub> ||class="entry q3 g1"| 53147<sub>11</sub> ||class="entry q3 g1"| 60371<sub>11</sub>
|-
|class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 13, 9, 11, 11) ||class="c"| (56512, 41787, 61427, 41790, 56787, 63899) ||class="entry q2 g1"| 56512<sub>7</sub> ||class="entry q3 g1"| 41787<sub>9</sub> ||class="entry q3 g1"| 61427<sub>13</sub> ||class="entry q2 g1"| 41790<sub>9</sub> ||class="entry q3 g1"| 56787<sub>11</sub> ||class="entry q3 g1"| 63899<sub>11</sub>
|-
|class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 13, 9, 11, 11) ||class="c"| (57576, 40711, 61415, 40726, 57839, 63887) ||class="entry q2 g1"| 57576<sub>7</sub> ||class="entry q3 g1"| 40711<sub>9</sub> ||class="entry q3 g1"| 61415<sub>13</sub> ||class="entry q2 g1"| 40726<sub>9</sub> ||class="entry q3 g1"| 57839<sub>11</sub> ||class="entry q3 g1"| 63887<sub>11</sub>
|-
|class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 13, 9, 11, 11) ||class="c"| (60936, 37111, 61175, 37366, 60959, 63647) ||class="entry q2 g1"| 60936<sub>7</sub> ||class="entry q3 g1"| 37111<sub>9</sub> ||class="entry q3 g1"| 61175<sub>13</sub> ||class="entry q2 g1"| 37366<sub>9</sub> ||class="entry q3 g1"| 60959<sub>11</sub> ||class="entry q3 g1"| 63647<sub>11</sub>
|-
|class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 13, 9, 11, 11) ||class="c"| (62112, 36175, 64943, 36190, 62375, 60359) ||class="entry q2 g1"| 62112<sub>7</sub> ||class="entry q3 g1"| 36175<sub>9</sub> ||class="entry q3 g1"| 64943<sub>13</sub> ||class="entry q2 g1"| 36190<sub>9</sub> ||class="entry q3 g1"| 62375<sub>11</sub> ||class="entry q3 g1"| 60359<sub>11</sub>
|-
|class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 13, 9, 11, 11) ||class="c"| (62656, 35631, 64463, 35646, 62919, 60839) ||class="entry q2 g1"| 62656<sub>7</sub> ||class="entry q3 g1"| 35631<sub>9</sub> ||class="entry q3 g1"| 64463<sub>13</sub> ||class="entry q2 g1"| 35646<sub>9</sub> ||class="entry q3 g1"| 62919<sub>11</sub> ||class="entry q3 g1"| 60839<sub>11</sub>
|-
|class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 13, 9, 11, 11) ||class="c"| (64032, 34015, 64223, 34270, 64055, 60599) ||class="entry q2 g1"| 64032<sub>7</sub> ||class="entry q3 g1"| 34015<sub>9</sub> ||class="entry q3 g1"| 64223<sub>13</sub> ||class="entry q2 g1"| 34270<sub>9</sub> ||class="entry q3 g1"| 64055<sub>11</sub> ||class="entry q3 g1"| 60599<sub>11</sub>
|-
|class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 9, 13, 9, 11, 11) ||class="c"| (64576, 33471, 64703, 33726, 64599, 60119) ||class="entry q2 g1"| 64576<sub>7</sub> ||class="entry q3 g1"| 33471<sub>9</sub> ||class="entry q3 g1"| 64703<sub>13</sub> ||class="entry q2 g1"| 33726<sub>9</sub> ||class="entry q3 g1"| 64599<sub>11</sub> ||class="entry q3 g1"| 60119<sub>11</sub>
|-
|class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 9, 9, 9, 11) ||class="c"| (33454, 62135, 58663, 64848, 35935, 62287) ||class="entry q2 g1"| 33454<sub>7</sub> ||class="entry q3 g1"| 62135<sub>11</sub> ||class="entry q3 g1"| 58663<sub>9</sub> ||class="entry q2 g1"| 64848<sub>9</sub> ||class="entry q3 g1"| 35935<sub>9</sub> ||class="entry q3 g1"| 62287<sub>11</sub>
|-
|class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 9, 9, 9, 11) ||class="c"| (33466, 52895, 55579, 64836, 45175, 53107) ||class="entry q2 g1"| 33466<sub>7</sub> ||class="entry q3 g1"| 52895<sub>11</sub> ||class="entry q3 g1"| 55579<sub>9</sub> ||class="entry q2 g1"| 64836<sub>9</sub> ||class="entry q3 g1"| 45175<sub>9</sub> ||class="entry q3 g1"| 53107<sub>11</sub>
|-
|class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 9, 9, 9, 11) ||class="c"| (33998, 62679, 58183, 64304, 35391, 62767) ||class="entry q2 g1"| 33998<sub>7</sub> ||class="entry q3 g1"| 62679<sub>11</sub> ||class="entry q3 g1"| 58183<sub>9</sub> ||class="entry q2 g1"| 64304<sub>9</sub> ||class="entry q3 g1"| 35391<sub>9</sub> ||class="entry q3 g1"| 62767<sub>11</sub>
|-
|class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 9, 9, 9, 11) ||class="c"| (34012, 44703, 47389, 64290, 53367, 44917) ||class="entry q2 g1"| 34012<sub>7</sub> ||class="entry q3 g1"| 44703<sub>11</sub> ||class="entry q3 g1"| 47389<sub>9</sub> ||class="entry q2 g1"| 64290<sub>9</sub> ||class="entry q3 g1"| 53367<sub>9</sub> ||class="entry q3 g1"| 44917<sub>11</sub>
|-
|class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 9, 9, 9, 11) ||class="c"| (35374, 64295, 57943, 62928, 34255, 62527) ||class="entry q2 g1"| 35374<sub>7</sub> ||class="entry q3 g1"| 64295<sub>11</sub> ||class="entry q3 g1"| 57943<sub>9</sub> ||class="entry q2 g1"| 62928<sub>9</sub> ||class="entry q3 g1"| 34255<sub>9</sub> ||class="entry q3 g1"| 62527<sub>11</sub>
|-
|class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 9, 9, 9, 11) ||class="c"| (35626, 51695, 53659, 62676, 46855, 51187) ||class="entry q2 g1"| 35626<sub>7</sub> ||class="entry q3 g1"| 51695<sub>11</sub> ||class="entry q3 g1"| 53659<sub>9</sub> ||class="entry q2 g1"| 62676<sub>9</sub> ||class="entry q3 g1"| 46855<sub>9</sub> ||class="entry q3 g1"| 51187<sub>11</sub>
|-
|class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 9, 9, 9, 11) ||class="c"| (35918, 64839, 58423, 62384, 33711, 62047) ||class="entry q2 g1"| 35918<sub>7</sub> ||class="entry q3 g1"| 64839<sub>11</sub> ||class="entry q3 g1"| 58423<sub>9</sub> ||class="entry q2 g1"| 62384<sub>9</sub> ||class="entry q3 g1"| 33711<sub>9</sub> ||class="entry q3 g1"| 62047<sub>11</sub>
|-
|class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 9, 9, 9, 11) ||class="c"| (36172, 43503, 45469, 62130, 55047, 42997) ||class="entry q2 g1"| 36172<sub>7</sub> ||class="entry q3 g1"| 43503<sub>11</sub> ||class="entry q3 g1"| 45469<sub>9</sub> ||class="entry q2 g1"| 62130<sub>9</sub> ||class="entry q3 g1"| 55047<sub>9</sub> ||class="entry q3 g1"| 42997<sub>11</sub>
|-
|class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 9, 9, 9, 11) ||class="c"| (37106, 56535, 52051, 61196, 41535, 56635) ||class="entry q2 g1"| 37106<sub>7</sub> ||class="entry q3 g1"| 56535<sub>11</sub> ||class="entry q3 g1"| 52051<sub>9</sub> ||class="entry q2 g1"| 61196<sub>9</sub> ||class="entry q3 g1"| 41535<sub>9</sub> ||class="entry q3 g1"| 56635<sub>11</sub>
|-
|class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 9, 9, 9, 11) ||class="c"| (37108, 47799, 44341, 61194, 50271, 47965) ||class="entry q2 g1"| 37108<sub>7</sub> ||class="entry q3 g1"| 47799<sub>11</sub> ||class="entry q3 g1"| 44341<sub>9</sub> ||class="entry q2 g1"| 61194<sub>9</sub> ||class="entry q3 g1"| 50271<sub>9</sub> ||class="entry q3 g1"| 47965<sub>11</sub>
|-
|class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 9, 9, 9, 11) ||class="c"| (38534, 59039, 61711, 59768, 39031, 59239) ||class="entry q2 g1"| 38534<sub>7</sub> ||class="entry q3 g1"| 59039<sub>11</sub> ||class="entry q3 g1"| 61711<sub>9</sub> ||class="entry q2 g1"| 59768<sub>9</sub> ||class="entry q3 g1"| 39031<sub>9</sub> ||class="entry q3 g1"| 59239<sub>11</sub>
|-
|class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 9, 9, 9, 11) ||class="c"| (38546, 55991, 52531, 59756, 42079, 56155) ||class="entry q2 g1"| 38546<sub>7</sub> ||class="entry q3 g1"| 55991<sub>11</sub> ||class="entry q3 g1"| 52531<sub>9</sub> ||class="entry q2 g1"| 59756<sub>9</sub> ||class="entry q3 g1"| 42079<sub>9</sub> ||class="entry q3 g1"| 56155<sub>11</sub>
|-
|class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 9, 9, 9, 11) ||class="c"| (38548, 48343, 43861, 59754, 49727, 48445) ||class="entry q2 g1"| 38548<sub>7</sub> ||class="entry q3 g1"| 48343<sub>11</sub> ||class="entry q3 g1"| 43861<sub>9</sub> ||class="entry q2 g1"| 59754<sub>9</sub> ||class="entry q3 g1"| 49727<sub>9</sub> ||class="entry q3 g1"| 48445<sub>11</sub>
|-
|class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 9, 9, 9, 11) ||class="c"| (39014, 59759, 61471, 59288, 38791, 58999) ||class="entry q2 g1"| 39014<sub>7</sub> ||class="entry q3 g1"| 59759<sub>11</sub> ||class="entry q3 g1"| 61471<sub>9</sub> ||class="entry q2 g1"| 59288<sub>9</sub> ||class="entry q3 g1"| 38791<sub>9</sub> ||class="entry q3 g1"| 58999<sub>11</sub>
|-
|class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 9, 9, 9, 11) ||class="c"| (39266, 56231, 50131, 59036, 42319, 54715) ||class="entry q2 g1"| 39266<sub>7</sub> ||class="entry q3 g1"| 56231<sub>11</sub> ||class="entry q3 g1"| 50131<sub>9</sub> ||class="entry q2 g1"| 59036<sub>9</sub> ||class="entry q3 g1"| 42319<sub>9</sub> ||class="entry q3 g1"| 54715<sub>11</sub>
|-
|class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 9, 9, 9, 11) ||class="c"| (39268, 48583, 42421, 59034, 49967, 46045) ||class="entry q2 g1"| 39268<sub>7</sub> ||class="entry q3 g1"| 48583<sub>11</sub> ||class="entry q3 g1"| 42421<sub>9</sub> ||class="entry q2 g1"| 59034<sub>9</sub> ||class="entry q3 g1"| 49967<sub>9</sub> ||class="entry q3 g1"| 46045<sub>11</sub>
|-
|class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 9, 9, 9, 11) ||class="c"| (40706, 56775, 50611, 57596, 41775, 54235) ||class="entry q2 g1"| 40706<sub>7</sub> ||class="entry q3 g1"| 56775<sub>11</sub> ||class="entry q3 g1"| 50611<sub>9</sub> ||class="entry q2 g1"| 57596<sub>9</sub> ||class="entry q3 g1"| 41775<sub>9</sub> ||class="entry q3 g1"| 54235<sub>11</sub>
|-
|class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 9, 9, 9, 11) ||class="c"| (40708, 48039, 41941, 57594, 50511, 46525) ||class="entry q2 g1"| 40708<sub>7</sub> ||class="entry q3 g1"| 48039<sub>11</sub> ||class="entry q3 g1"| 41941<sub>9</sub> ||class="entry q2 g1"| 57594<sub>9</sub> ||class="entry q3 g1"| 50511<sub>9</sub> ||class="entry q3 g1"| 46525<sub>11</sub>
|-
|class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 9, 9, 9, 11) ||class="c"| (41530, 61211, 51799, 56772, 37363, 56383) ||class="entry q2 g1"| 41530<sub>7</sub> ||class="entry q3 g1"| 61211<sub>11</sub> ||class="entry q3 g1"| 51799<sub>9</sub> ||class="entry q2 g1"| 56772<sub>9</sub> ||class="entry q3 g1"| 37363<sub>9</sub> ||class="entry q3 g1"| 56383<sub>11</sub>
|-
|class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 9, 9, 9, 11) ||class="c"| (41770, 57851, 50599, 56532, 40723, 54223) ||class="entry q2 g1"| 41770<sub>7</sub> ||class="entry q3 g1"| 57851<sub>11</sub> ||class="entry q3 g1"| 50599<sub>9</sub> ||class="entry q2 g1"| 56532<sub>9</sub> ||class="entry q3 g1"| 40723<sub>9</sub> ||class="entry q3 g1"| 54223<sub>11</sub>
|-
|class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 9, 9, 9, 11) ||class="c"| (42074, 59771, 52279, 56228, 38803, 55903) ||class="entry q2 g1"| 42074<sub>7</sub> ||class="entry q3 g1"| 59771<sub>11</sub> ||class="entry q3 g1"| 52279<sub>9</sub> ||class="entry q2 g1"| 56228<sub>9</sub> ||class="entry q3 g1"| 38803<sub>9</sub> ||class="entry q3 g1"| 55903<sub>11</sub>
|-
|class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 9, 9, 9, 11) ||class="c"| (42314, 59291, 50119, 55988, 39283, 54703) ||class="entry q2 g1"| 42314<sub>7</sub> ||class="entry q3 g1"| 59291<sub>11</sub> ||class="entry q3 g1"| 50119<sub>9</sub> ||class="entry q2 g1"| 55988<sub>9</sub> ||class="entry q3 g1"| 39283<sub>9</sub> ||class="entry q3 g1"| 54703<sub>11</sub>
|-
|class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 9, 9, 9, 11) ||class="c"| (42328, 48595, 39325, 55974, 49979, 36853) ||class="entry q2 g1"| 42328<sub>7</sub> ||class="entry q3 g1"| 48595<sub>11</sub> ||class="entry q3 g1"| 39325<sub>9</sub> ||class="entry q2 g1"| 55974<sub>9</sub> ||class="entry q3 g1"| 49979<sub>9</sub> ||class="entry q3 g1"| 36853<sub>11</sub>
|-
|class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 9, 9, 9, 11) ||class="c"| (45170, 64851, 55327, 53132, 33723, 52855) ||class="entry q2 g1"| 45170<sub>7</sub> ||class="entry q3 g1"| 64851<sub>11</sub> ||class="entry q3 g1"| 55327<sub>9</sub> ||class="entry q2 g1"| 53132<sub>9</sub> ||class="entry q3 g1"| 33723<sub>9</sub> ||class="entry q3 g1"| 52855<sub>11</sub>
|-
|class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 9, 9, 9, 11) ||class="c"| (45424, 43515, 36277, 52878, 55059, 39901) ||class="entry q2 g1"| 45424<sub>7</sub> ||class="entry q3 g1"| 43515<sub>11</sub> ||class="entry q3 g1"| 36277<sub>9</sub> ||class="entry q2 g1"| 52878<sub>9</sub> ||class="entry q3 g1"| 55059<sub>9</sub> ||class="entry q3 g1"| 39901<sub>11</sub>
|-
|class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 9, 9, 9, 11) ||class="c"| (46850, 62931, 53647, 51452, 35643, 51175) ||class="entry q2 g1"| 46850<sub>7</sub> ||class="entry q3 g1"| 62931<sub>11</sub> ||class="entry q3 g1"| 53647<sub>9</sub> ||class="entry q2 g1"| 51452<sub>9</sub> ||class="entry q3 g1"| 35643<sub>9</sub> ||class="entry q3 g1"| 51175<sub>11</sub>
|-
|class="f"| 6014 ||class="q"| (2, 3, 3, 2, 3, 3) ||class="g"| (1, 1, 1, 1, 1, 1) ||class="w"| (7, 11, 9, 9, 9, 11) ||class="c"| (46864, 44955, 35797, 51438, 53619, 40381) ||class="entry q2 g1"| 46864<sub>7</sub> ||class="entry q3 g1"| 44955<sub>11</sub> ||class="entry q3 g1"| 35797<sub>9</sub> ||class="entry q2 g1"| 51438<sub>9</sub> ||class="entry q3 g1"| 53619<sub>9</sub> ||class="entry q3 g1"| 40381<sub>11</sub>
|-
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|-
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[[Category:Mentors of Boolean functions; chains]]
</noinclude>
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{| class="wikitable"
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| [[File:4-ary noble with quadrant 1; zhe 4382.svg|200px]]
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|style="border-right: 3px solid #a2a9b1;"| [[File:4-ary noble with quadrant 1; zhe 6014.svg|200px]]
| [[File:4-ary noble with quadrant 0; zhe 5160.svg|200px]]
| [[File:4-ary noble with quadrant 0; zhe 4680.svg|200px]]
| [[File:4-ary noble with quadrant 0; zhe 1632.svg|200px]]
|}<noinclude>
[[Category:Mentors of Boolean functions; chains]]
</noinclude>
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User talk:TiwariA.23
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==Welcome==
{{Robelbox|theme=9|title='''[[Wikiversity:Welcome|Welcome]] to [[Wikiversity:What is Wikiversity|Wikiversity]], TiwariA.23!'''|width=100%}}
<div style="{{Robelbox/pad}}">
You can [[Wikiversity:Contact|contact us]] with [[Wikiversity:Questions|questions]] at the [[Wikiversity:Colloquium|colloquium]] or get in touch with [[User talk:Atcovi|me personally]] if you would like some [[Help:Contents|help]].
Remember to [[Wikiversity:Signature#How to add your signature|sign]] your comments when [[Wikiversity:Who are Wikiversity participants?|participating]] in [[Wikiversity:Talk page|discussions]]. Using the signature icon [[File:OOjs UI icon signature-ltr.svg]] makes it simple.
We invite you to [[Wikiversity:Be bold|be bold]] and [[Wikiversity|assume good faith]]. Please abide by our [[Wikiversity:Civility|civility]], [[Wikiversity:Privacy policy|privacy]], and [[Foundation:Terms of Use|terms of use]] policies.
To find your way around, check out:
<!-- The Left column -->
<div style="width:50.0%; float:left">
* [[Wikiversity:Introduction|Introduction to Wikiversity]]
* [[Help:Guides|Take a guided tour]] and learn [[Help:Editing|how to edit]]
* [[Wikiversity:Browse|Browse]] or visit an educational level portal:<br>[[Portal:Pre-school Education|pre-school]] | [[Portal:Primary Education|primary]] | [[Portal:Secondary Education|secondary]] | [[Portal:Tertiary Education|tertiary]] | [[Portal:Non-formal Education|non-formal]]
* [[Wikiversity:Introduction explore|Explore]] links in left-hand navigation menu
</div>
<!-- The Right column -->
<div style="width:50.0%; float:left">
* Read an [[Wikiversity:Wikiversity teachers|introduction for teachers]]
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</div>
<br clear="both"/>
To get started, experiment in the [[wikiversity:sandbox|sandbox]] or on [[special:mypage|your userpage]].
See you around Wikiversity! --—[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 18:17, 8 December 2024 (UTC)</div>
<!-- Template:Welcome -->
{{Robelbox/close}}
== [[Subject Summary for Aaryan Tiwari (8CJB)]] ==
Hello. I'm not sure why you decided to revert my edit where I moved this page into your userspace, but I've gone ahead and deleted the page as it seemed to be a test page and had no [[WV:What is Wikiversity?|learning value]]. Could you please clarify your revert and what you intend to make out of this page? Thanks, —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 19:29, 8 December 2024 (UTC)
== [[Partridge, Mrs N]] ==
Hi. In the future for test edits like this, please use the [[Wikiversity:Sandbox|sandbox]]. Thanks, —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 00:17, 18 December 2024 (UTC)
== Warning ==
Hi TiwariA.23. This is the 3rd occasion (creating unhelpful, redundant redirects) where you've conducted unhelpful edits. Please take heed of this warning or you may be blocked from editing Wikiversity. Thanks, —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 19:19, 1 January 2025 (UTC)
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==Welcome==
{{Robelbox|theme=9|title='''[[Wikiversity:Welcome|Welcome]] to [[Wikiversity:What is Wikiversity|Wikiversity]], TiwariA.23!'''|width=100%}}
<div style="{{Robelbox/pad}}">
You can [[Wikiversity:Contact|contact us]] with [[Wikiversity:Questions|questions]] at the [[Wikiversity:Colloquium|colloquium]] or get in touch with [[User talk:Atcovi|me personally]] if you would like some [[Help:Contents|help]].
Remember to [[Wikiversity:Signature#How to add your signature|sign]] your comments when [[Wikiversity:Who are Wikiversity participants?|participating]] in [[Wikiversity:Talk page|discussions]]. Using the signature icon [[File:OOjs UI icon signature-ltr.svg]] makes it simple.
We invite you to [[Wikiversity:Be bold|be bold]] and [[Wikiversity|assume good faith]]. Please abide by our [[Wikiversity:Civility|civility]], [[Wikiversity:Privacy policy|privacy]], and [[Foundation:Terms of Use|terms of use]] policies.
To find your way around, check out:
<!-- The Left column -->
<div style="width:50.0%; float:left">
* [[Wikiversity:Introduction|Introduction to Wikiversity]]
* [[Help:Guides|Take a guided tour]] and learn [[Help:Editing|how to edit]]
* [[Wikiversity:Browse|Browse]] or visit an educational level portal:<br>[[Portal:Pre-school Education|pre-school]] | [[Portal:Primary Education|primary]] | [[Portal:Secondary Education|secondary]] | [[Portal:Tertiary Education|tertiary]] | [[Portal:Non-formal Education|non-formal]]
* [[Wikiversity:Introduction explore|Explore]] links in left-hand navigation menu
</div>
<!-- The Right column -->
<div style="width:50.0%; float:left">
* Read an [[Wikiversity:Wikiversity teachers|introduction for teachers]]
* Learn [[Help:How to write an educational resource|how to write an educational resource]]
* Find out about [[Wikiversity:Research|research]] activities
* Give [[Wikiversity:Feedback|feedback]] about your observations
* Discuss issues or ask questions at the [[Wikiversity:Colloquium|colloquium]]
</div>
<br clear="both"/>
To get started, experiment in the [[wikiversity:sandbox|sandbox]] or on [[special:mypage|your userpage]].
See you around Wikiversity! --—[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 18:17, 8 December 2024 (UTC)</div>
<!-- Template:Welcome -->
{{Robelbox/close}}
== [[Subject Summary for Aaryan Tiwari (8CJB)]] ==
Hello. I'm not sure why you decided to revert my edit where I moved this page into your userspace, but I've gone ahead and deleted the page as it seemed to be a test page and had no [[WV:What is Wikiversity?|learning value]]. Could you please clarify your revert and what you intend to make out of this page? Thanks, —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 19:29, 8 December 2024 (UTC)
== [[Partridge, Mrs N]] ==
Hi. In the future for test edits like this, please use the [[Wikiversity:Sandbox|sandbox]]. Thanks, —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 00:17, 18 December 2024 (UTC)
== Warning ==
Hi TiwariA.23. This is the 3rd occasion (creating unhelpful, redundant redirects) where you've conducted unhelpful edits. Please take heed of this warning or you may be blocked from editing Wikiversity. Thanks, —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 19:19, 1 January 2025 (UTC)
== Blocked ==
<div class="user-block" style="background:#ffe0e0; border:1px solid pink; padding:0.5em; margin:0.5em auto; min-height: 40px">
[[File:Modern clock chris kemps 01 with Octagon-warning.svg|40px|left|alt=Stop icon with clock]] You have been '''[[Wikiversity:Blocking policy|blocked]]''' temporarily from editing for abuse of editing privileges. Once the block has expired, you are welcome to make constructive contributions to Wikiversity. If you think you were blocked in error and/or want to contest this block, consider adding the following tag below this notice: <code><nowiki>{{unblock|your reason here ~~~~}}</nowiki></code>
<hr>
If using the above tag does not help, either an administrator may have declined the request after the unblock request was reviewed by an administrator or you may have been blocked from editing your talk page.</div>
—[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 21:04, 1 January 2025 (UTC)
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User:Atcovi/Health Psychology/Chapter 10: Diverse Approaches to Pain
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== 10.1 - The Experience of Pain and Its Cultural Variations ==
* '''Pain''' - Defined as "an unpleasant sensory and emotional experience associated with actual or potential tissue damage, or described in terms of such damage". Highlights study where armor in the virtual world reduced subjective pain, despite an actual sting in the arm.
**'''Nociception''' - Activation of specialized nerve fibers and receptors in response to harmful stimuli (most basic level of pain). Signals occurence of tissue damage, but does not ''automatically'' lead to pain as subjective interpretation takes place. Accompanied by cognitive, behavioral, and affective states.
**Important to take a '''biopsychosocial''' approach when dealing with pain, and is necessary for survival.
'''Cultural Variations in the Experience of Pain'''
* Boys are taught to hide the pain, while girls express pain.
* Majority of studies show pain intensity and unpleasantness are not different between the two genders. Women experience more social support for their pain.
* Pain experiences and treatments differ significantly across ethnic groups due to cultural, socioeconomic, and psychological factors. Research shows that Black Americans, Asian Americans, and Latinx/e individuals generally exhibit higher pain sensitivity and lower tolerance than non-Hispanic European Americans. Black Americans, in particular, face consistent disparities in pain treatment, regardless of pain type or setting, while Latinx/e people experience disparities in specific contexts. Cultural barriers, such as language differences and stress from socioeconomic challenges, further influence pain management and perception. Studies also highlight unique concerns in Asian populations regarding cancer pain and suggest that ethnic identity mediates some differences in pain experiences among groups. While individual variations exist within cultures, limitations in research methods, such as small or non-random samples, complicate the understanding of these phenomena. Future research should aim for larger, more diverse samples and explore traditional healing practices to enhance the understanding of cultural differences in pain.
'''Typologies and Biology of Pain'''
* Pain is a complex experience described through various terms and classifications, including '''acute pain''' (short-term) and '''chronic pain''' (long-term). Chronic pain can further be categorized as '''malignant''' (associated with diseases like cancer) or '''noncancer pain''' (e.g., lower back pain). Acute pain typically resolves within three months, while chronic pain may persist for years and is often considered a disease. Pain is also classified by its origin, such as psychogenic (psychological), neuropathic (pure nociception), or somatic (physiological without tissue damage). Pain involves four physiological processes: '''transduction''', where stimuli (chemical, mechanical, or thermal) are converted into nerve signals; '''transmission''', the relay of these signals to the central nervous system; '''modulation''', which controls the pain signals between brain regions; and '''perception''', where these signals result in the subjective experience of pain. Key structures in these processes include sensory receptors, afferent fibers, the spinal cord, and various brain regions such as the thalamus and cortex. This intricate system highlights the multidimensional nature of pain, emphasizing the need for precise classification and measurement methods.
== 10.2 - Measuring Pain ==
* Pain is a multifaceted and subjective experience, making it challenging to measure objectively. Physical damage, like a broken limb, may cause varying pain levels in different individuals, and language barriers can further complicate pain assessment. The Initiative on Methods, Measurement, and Pain Assessment in Clinical Trials (IMMPACT) highlights four key areas for assessing pain: intensity, physical functioning, emotional functioning, and overall well-being. Pain is now widely recognized as a vital sign, alongside temperature, pulse, blood pressure, and respiration. Efforts like the National Pain Strategy aim to improve pain measurement through large-scale surveys and healthcare data. However, immediate measurement remains difficult in cases where language barriers exist or when assessing non-verbal individuals, such as young children, underscoring the need for innovative and inclusive assessment methods.
'''Basic Pain Measures'''
Hospitals employ various tools and techniques to assess pain across diverse cultural and demographic groups. Visual tools, such as pain scales with illustrative faces ranging from smiling (no pain) to frowning (extreme pain), are widely used. These scales often include multilingual instructions for accessibility. Numeric (NRS), verbal (VRS), and visual analog (VAS) scales are also common, though each has unique challenges. For instance, older adults or individuals from certain cultural backgrounds may find numerical pain ratings abstract.
Advanced tools, like the '''McGill Pain Questionnaire (MPQ)''' and the '''Multidimensional Pain Inventory (MPI)''', delve into sensory, emotional, and evaluative aspects of pain. Innovative approaches such as '''Ecological Momentary Assessment (EMA)''' allow patients to report pain in real-time using digital devices, reducing reliance on memory. Behavioral observation also provides critical insights, especially for non-verbal patients, using tools like the '''UAB Pain Behavior Scale'''. Psychological factors, such as catastrophizing, are evaluated with specialized questionnaires, linking mental outlook to pain perception.
While physiological measures like EEG and EMG exist, their efficacy in pain assessment remains limited, highlighting the importance of a biopsychosocial approach to understanding and managing pain.
== 10.3 - Theories of Pain ==
'''Early Physiological and Psychological Approaches'''
The history of pain theories reveals humanity's evolving attempts to understand and address this universal experience. Ancient beliefs attributed pain to supernatural causes, such as evil spirits or divine will. By 500 BCE, Greek philosophers viewed pain as a consequence of ''irrational thinking'', integrating it with their rational approach to human experiences.
In 1664, '''Descartes''' introduced one of the earliest mechanistic explanations, proposing that pain resulted from specific stimuli transmitted through nerves to the brain, prompting a coordinated response. This unidimensional "specificity" theory was later formalized by '''Von Frey''' in 1894, suggesting dedicated pain receptors and pathways. Around the same time, '''Goldschneider’s "pattern theory"''' argued that pain arises from the integration of nerve impulses rather than specific receptors.
Neither theory fully explained individual differences in pain perception or variability in treatment outcomes. '''Engel’s 1959''' "pain-prone personality" model introduced psychological influences, though it lacked empirical support. Later advancements included the '''gate control theory''' (Melzack & Wall, 1965), emphasizing the interaction between psychological and physiological factors, and '''Melzack's neuromatrix theory''' (1999), which proposed a brain-centered model incorporating genetics, emotions, and experience.
Modern perspectives recognize pain as a complex interplay of biological, psychological, and social factors, reflecting the biopsychosocial approach to health.
'''Biopsychological Theories of Pain'''
'''Theories and Models of Pain'''
# '''Early Theories'''
#* Early models were primarily physiological or psychological.
#* Cognitive-behavioral model: Suggests learned expectations condition pain experiences (e.g., fear of dental visits).
#* Diathesis-stress model: Links physiological predispositions (e.g., low pain thresholds) with psychological factors.
# '''Gate Control Theory of Pain (GCT)'''
#* Proposed by Melzack & Wall (1965), it integrates biopsychosocial aspects.
#* Pain signals from receptors (A-beta, A-delta, C fibers) pass through a "gate" in the dorsal horn of the spinal cord.
#** '''A-beta fibers''': Large, myelinated, and inhibit pain by closing the gate.
#** '''A-delta & C fibers''': Small, unmyelinated, and open the gate, intensifying pain.
#* The balance of fiber activity influences pain intensity and duration.
#* Chronic pain may occur when large fibers (A-beta) are deactivated.
# '''Pain Management Insights'''
#* Counterirritation (e.g., scratching or electrical stimulation) activates A-beta fibers, temporarily closing the gate.
#* Descending pathways from the brain modulate pain:
#** Positive emotions reduce pain by closing the gate.
#** Negative emotions increase pain by keeping the gate open.
# '''Psychological Factors in Pain'''
#* Depression, anxiety, and personality disorders are linked to increased pain.
#* Internal locus of control correlates with less severe pain perception.
#* Cognitive appraisals and beliefs about pain significantly affect tolerance and severity.
# '''Learning and Behavior in Pain'''
#* '''Social learning''': Observing others’ reactions influences pain expectations (e.g., Bandura’s findings).
#* '''Operant conditioning''': Reinforced pain behaviors (e.g., attention from others) can amplify pain perception.
#* '''Classical conditioning''': Associating pain with specific stimuli (e.g., dentist visits) increases pain anticipation.
# '''Innovative Pain Treatments'''
#* Virtual Reality (VR): Emerging tools like VR exercises help alleviate pain by redirecting focus.
#* Psychosocial strategies: Changing attitudes and beliefs about pain can significantly reduce its impact.
== 10.4 - Pain Management Techniques ==
There are three main categories of pain management: physiological, psychological, and self-regulation techniques. These methods vary based on individual pain thresholds (the level at which pain is perceived) and tolerance (the level beyond which pain becomes unbearable). Cultural and gender differences primarily affect pain tolerance.
'''Key Techniques:'''
* '''Physiological:''' Medications like aspirin and morphine, and interventions like acupuncture or electrical nerve stimulation.
* '''Psychological:''' Techniques such as relaxation, guided imagery, hypnosis, and distraction.
* '''Self-Regulation:''' Long-term approaches like biofeedback, meditation, and self-management programs for chronic pain.
These methods aim to achieve analgesia (pain relief) and are applied differently depending on whether the pain is acute or chronic.
'''Physiological Treatments'''
# '''Chemical Methods:'''
#* '''Over-the-Counter Medications:''' Aspirin, acetaminophen, and ibuprofen are common for minor pains.
#* '''Prescription Drugs:''' Includes opioids (e.g., morphine, oxycodone) for severe or chronic pain. Opioids act on brain receptors but risk addiction and tolerance.
#* '''Cultural and Usage Trends:''' Disparities exist in opioid prescriptions across ethnic groups.
# '''Acupuncture:'''
#* Based on stimulating specific points to restore energy flow or close pain gates.
#* Shown to release endorphins and provide analgesia, though effectiveness varies and may involve placebo effects.
# '''Surgery:'''
#* Nerve severing or brain lesioning can temporarily reduce pain but often fails long-term due to nerve regrowth.
# '''Other Physiological Methods:'''
#* '''Non-Medication Treatments:''' Use of heat, cold, vibrations, and decompression systems for localized pain relief.
#* '''Stress-Induced Analgesia (SIA):''' Exercise can trigger natural pain relief through endogenous opioid release, like the “runner’s high.”
These methods cater to both acute and chronic pain management but vary in effectiveness and risk profiles.
'''Psychological Treatments'''
# '''Expectations and Placebo Effect:'''
#* Beliefs about pain or treatment can influence pain perception. Positive expectations often result in pain relief.
# '''Psychological States and Cognitive Styles:'''
#* Negative emotions like anxiety and depression intensify pain.
#* Cognitive biases (e.g., catastrophizing or learned helplessness) can worsen pain but are modifiable through therapy.
# '''Distraction:'''
#* Activities like reading, watching TV, or listening to music can redirect focus from pain.
#* Techniques like guided imagery and cognitive distraction are effective, especially in acute pain scenarios.
# '''Hypnosis:'''
#* Combines relaxation and suggestion to reduce pain perception. Self-hypnosis shows promise for chronic pain relief.
# '''Cognitive and Relaxation Methods:'''
#* Biofeedback, meditation, mindfulness, and relaxation reduce muscle tension and anxiety, aiding pain management.
#* Massages and guided imagery are also effective.
# '''Environment and Visual Influence:'''
#* Natural views or simulations (e.g., greenery, art, or nature sounds) can lower pain and enhance recovery.
# '''Virtual Reality (VR):'''
#* Immersive VR environments distract patients from pain, mimicking the analgesic effects of nature or interactive distractions like video games.
These techniques show how psychology and environment significantly influence pain perception and management.
'''Self-Management of Chronic Pain'''
* '''Self-Management Programs:'''
** Shift responsibility for change to the patient rather than solely relying on doctors or medical staff.
** Emphasize psychological and behavioral changes over medication or physical procedures, reducing side effects.
* '''Focus Areas:'''
** Address emotional, cognitive, and sensory experiences of pain.
** Consider pain-related behaviors and social consequences, such as daily activities and relationships.
** Modify cognitive processes like attention focus, pain-related memories, coping strategies, expectations, and self-perceptions.
* '''Goals of Self-Management Programs:'''
** Teach skills to redirect attention away from pain.
** Enhance physical fitness and increase daily physical activity.
** Provide coping mechanisms for severe pain episodes without relying on medication.
** Equip patients to manage emotions like depression, anger, and anxiety.
** Reduce stress, interpersonal conflicts, and unhealthy behaviors.
* '''Program Structure:'''
** Begins with an intensive interview and evaluation of medical history, pain, and functional status.
** Patients collaborate with staff to set program goals and sign a contract to work toward them.
** Includes education, skills training, relaxation techniques, and cognitive-behavioral strategies to change maladaptive thoughts and actions.
** Covers relapse prevention and follow-up to ensure long-term progress.
* '''Broader Perspective on Pain Management:'''
** Pain is a universal experience, and coping methods vary across cultures.
** A comprehensive understanding of diverse pain management techniques enhances adaptability and resilience.
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User:Atcovi/Health Psychology/Chapter 11: Disability, Terminal Illness, and Death
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== 11.1 - What is Disability and Chronic Illness? ==
* Disability affects 15% of the world's population.
'''Disability Models'''
This passage explores disability through three main models: moral, medical, and social. The '''moral model''' associates disability with divine intervention or punishment, emphasizing cultural and religious beliefs, and remains prevalent globally, though less so in Western cultures. The '''medical model''', dominant in the West, views disability as a biological dysfunction to be treated or cured, focusing responsibility on medical professionals and individuals. In contrast, the '''social model''' attributes disability to societal structures and attitudes, urging systemic changes to dismantle ableism and emphasizing a possible "cure" and "return to normalcy" for disabled people. The '''biopsychosocial model''', adopted by the World Health Organization (WHO) in the International Classification of Functioning (ICF), integrates aspects of all three models, highlighting how physical, personal, and societal factors interact to shape disability experiences. Disability dimensions, including visibility, functionality, and progression, influence individual adaptation and societal responses. Chronic illnesses like cardiovascular disease and diabetes are prevalent today due to increased life expectancy and medical advancements. The discussion also underscores the impact of cultural norms, health equity, and chronic conditions on life expectancy and societal roles in health and disability.
== 11.2 - Coping With Disability and Chronic Illness ==
'''Goals of Treatment'''
* Adaptation/coping is key. Can be defined as the affective, mental, and behavior chances that allow a disabled person to embrace their life despite their disability. The five major areas are success in performing daily tasks, reducing psych. disorders, reduce negative affect and improve positive affect, maintain a satisfactory functional status, and experience happiness in other areas of life.
'''Quality of Life'''
* '''QOL''' measures how well someone copes with chronic illness, considering physical, psychological, and social factors.
* Initially assessed by physicians, QOL is now better evaluated by patients themselves based on their experiences with pain, emotional well-being, and functional status.
* Tools like PROMIS, PROQOLID, OLGA, and Optum provide assessments for QOL.
'''Biopsychosocial components of adaptation:'''
# '''Biological Issues''':
#* Chronic illnesses like cancer, coronary heart disease (CHD), and diabetes impact physical functioning and often involve pain, necessitating physical rehabilitation.
#* Such conditions also influence psychological outlooks.
# '''Psychological Issues''':
#* Adaptation involves cognitive, emotional, and interpersonal adjustments, with key factors including:
#** '''Appraisals''': Seeing illness as a challenge and perceiving social support improves QOL.
#** '''Personality''': Traits like optimism and positive affect aid in coping, while depression and anxiety hinder it.
#** '''Comparison''': Social comparisons (upward or downward) can impact self-esteem.
#** '''Meaning''': Finding meaning in illness can enhance or detract from well-being, depending on context.
'''Research Highlights'''
* '''Optimism''': Strong predictor of better adaptation, linked to active coping and less distress across illnesses.
* '''Gratitude''': Reduces depression and promotes better outcomes.
* '''Social Support''': Critical for managing emotional and practical challenges.
* '''Cultural Variations''': Different coping mechanisms observed across cultures (e.g., Chinese women with breast cancer).
'''Challenges and Strategies:'''
* Psychological reactions vary by illness and individual differences (e.g., premorbid mental health, cultural beliefs).
* Interventions that foster optimism, gratitude, and supportive relationships are effective in improving QOL.
== 11.3 - Culture, Community, Chronic Illness, and Disability ==
The text explores how chronic illness and disability intersect with various social and cultural factors, influencing how individuals cope and adapt. Marginalized groups, such as those who are older, low-income, or from ethnic minorities, face unique challenges that are compounded by social discrimination. The environment, including family dynamics, neighborhood safety, and social support, significantly affects the management of chronic illnesses. Cultural beliefs and practices also play a role, with different groups relying on religion, community support, or traditional medicine for coping. Social support, both formal and informal, is crucial but must be tailored to individual needs, as excessive or mismatched support can sometimes be counterproductive. The importance of close relationships, such as family and community, is highlighted as a sustainable approach to managing chronic illnesses. Additionally, interventions like support groups and positive psychology strategies offer varying degrees of effectiveness, underscoring the need for personalized approaches to care.
== 11.4 - Coping With Terminal Illness and Death ==
'''Introduction'''
'''Chronic and Terminal Illnesses:'''
* Chronic illnesses are treatable but often not curable, with some being fatal (e.g., certain cancers, also called terminal illnesses).
* Facing terminal illnesses presents significant psychological challenges, requiring patients to confront their mortality and adjust their outlook on life.
'''End-of-Life Care:'''
* Effective end-of-life care addresses emotional, psychological, and logistical challenges faced by patients and their families.
** '''Hospital Challenges:''' Issues like fragmented care, strict visiting hours, and excessive administrative processes can be distressing.
** '''Support Measures:''' Clear communication, informed consent, psychological counseling, and preparation for death are critical.
** '''Family Support:''' Families need help with grief, caregiving stress, and emotional communication, such as saying goodbye.
'''Role of Religion and Spirituality:'''
* Religion, an integral aspect of culture, often becomes more important at the end of life and serves as a coping mechanism.
** Religious practices and beliefs can aid in pain management, emotional resilience, and acceptance of death.
** Studies show correlations between religious coping (e.g., prayer, rituals) and psychological well-being in patients, though causation is not established.
** Cultural and ethnic diversity in religious beliefs highlights the need for tailored spiritual care.
'''Perspectives Across Religions:'''
* '''Christianity (Catholicism):''' Suffering is tied to original sin; death is seen as liberation of the soul. Rituals like last rites provide solace.
* '''Islam:''' Death is the soul’s detachment from the body, viewed as a blessing. Preparing includes settling debts, penance, and cleanliness, with a preference for dying at home.
* '''Hinduism:''' Death and suffering are part of the karmic cycle; liberation comes from transcending this cycle. Karma’s role reduces anxiety about death.
* '''Buddhism:''' Death is inevitable; fear of death reflects unfulfilled life. Contemplating mortality helps reduce fear and fosters healthier attitudes toward life and death.
* '''Other Traditions:''' Irish wakes celebrate life with joy, while Sikhs see death as an opportunity for soul cleansing and reunification with the creator.
'''Broader Implications:'''
Religious and cultural practices help frame death positively, emphasizing liberation, spiritual growth, and joy. This understanding should guide health psychologists and caregivers to use holistic and culturally sensitive approaches to end-of-life care.
'''Death Across the Life Span'''
==== Physiological and Psychological Changes in Dying Patients ====
* Near the end of life, patients often experience physiological changes, such as incontinence, reduced digestive function, increased pain, memory problems, and difficulty concentrating.
* High doses of morphine, often self-administered, are used to manage pain.
* Emotional and social difficulties arise as family and friends may struggle to interact with the patient in this vulnerable state, compounded by societal taboos around discussing death.
* Death education, particularly for younger populations, is critical in helping people cope with loss.
----
==== Ethical Dilemmas in End-of-Life Care ====
* '''Euthanasia:''' Active termination of life via a lethal drug.
* '''Assisted Suicide:''' Physician provides a lethal drug but does not administer it.
* '''Passive Euthanasia:''' Withdrawing life-sustaining treatment and allowing the disease to progress naturally.
These practices are controversial, with ethical, moral, religious, and societal implications.
----
==== Notable Cases and Debates ====
# '''Terri Schiavo Case (2005):'''
#* Persistent vegetative state for 15 years.
#* Conflict between her husband (advocating for life support removal) and her parents (wanting to continue life support).
#* Highlights challenges in determining consciousness, psychological pain, and ethical considerations.
# '''Jack Kevorkian:'''
#* Assisted in 130 suicides, sparking national debates and legal reforms.
#* His actions influenced legislation, such as Oregon's 1994 legalization of physician-assisted suicide.
# '''Medical Advances:'''
#* A vagus nerve implant in 2017 partially restored consciousness to a patient in a persistent vegetative state, complicating future decision-making.
----
==== Perspectives and Advocacy ====
* '''Religious and Disability Groups:'''
** Many religious groups oppose euthanasia on moral grounds.
** Organizations like ''Not Dead Yet'' argue that euthanasia and assisted suicide discriminate against disabled and elderly individuals, emphasizing the need for better palliative care and support systems.
* '''Patient Autonomy:'''
** Respect for the patient’s wishes, often expressed through advance directives or living wills, is a cornerstone of ethical medical care.
** Family members play a significant role but may misinterpret or override patient preferences.
----
==== Palliative Care and Ethical Considerations ====
* Palliative care focuses on alleviating suffering without directly treating the cause.
* Decisions to withdraw life support are easier when advance directives are in place, yet families and healthcare providers may still face challenges aligning their actions with the patient’s wishes.
----
==== Key Questions for Decision-Making ====
* Is the patient’s condition reversible?
* Does the patient have an advance directive or living will?
* What is the role of family in interpreting the patient’s wishes?
* What societal, religious, or cultural factors influence the decision?
* Can palliative care adequately address pain and suffering?
The interplay of these considerations highlights the complexity and sensitivity of end-of-life decisions.
=== Simplified Summary of the Stages of Death and Cultural Perspectives on Death and Dying ===
'''Stages of Death by Kübler-Ross:'''
* Elisabeth Kübler-Ross proposed five stages of dying: '''denial''', '''anger''', '''bargaining''', '''depression''', and '''acceptance'''.
* These stages are not universally sequential or empirically supported; people may experience emotions dynamically and not in this order.
* Factors such as culture, social support, and disease progression influence individual experiences of death.
'''Cultural Perspectives on Death:'''
* Cultural practices surrounding death differ significantly:
** '''American Indians''': Use sage-burning rituals for spiritual preparation, often conflicting with hospital policies.
** '''Muslims''': Follow strict protocols like facing the body toward Mecca and reciting the Qur'an.
** '''Ghana''': Unique coffin shapes as part of burial traditions.
* '''Health Practitioners''': Need to respect cultural practices for effective support, despite challenges due to ethnocentric biases or institutional limitations.
'''Gender Differences in Grief:'''
* '''Women''': Tend to openly express emotions and seek social support.
* '''Men''': Less likely to express emotions, leading to greater health consequences during grief, including higher rates of depression and illness.
* Widowers face more challenges than widows, both mentally and physically, during acute grief periods.
'''Health Practitioner Guidelines:'''
* Be culturally sensitive and informed about the unique needs of different groups.
* Understand that cultural mistrust may affect relationships, especially with groups like African Americans and American Indians.
* Tailor support approaches to include socioeconomic status, religion, and spirituality.
Understanding death as a universal yet deeply individual and cultural experience is crucial for providing compassionate care.
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User:Atcovi/Health Psychology/Chapter 13: Cancer
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In 2019, more than 1.7 million new cases of cancer were diagnosed and nearly 600,000 people died of cancer in the United States (U.S. Cancer Statistics Working Group, 2022). Cancer is the second leading cause of death in the United States (after heart disease), with nearly 1 in 4 deaths due to cancer (U.S. Cancer Statistics Working Group, 2022).
==13.1 - Cancer: Definitions and Prevalence==
'''Origins and Terminology''':
* Cancer originates from the Greek terms "carcinos" and "carcinoma," referring to tumor types. Its name is linked to the crab-like appearance of spreading cancer cells.
* Cancer encompasses over 100 diseases, differing in incidence, risk factors, and treatment.
'''Biological Basis''':
* Cancer arises from malignant tumors characterized by uncontrolled cell growth and tissue destruction.
* Mutations in DNA disrupt normal cell processes, often caused by genetic factors or exposure to carcinogens (e.g., cigarette smoke).
* Normal cells divide and die systematically, while cancer cells grow indefinitely, forming abnormal cells.
'''Types of Cancer''':
* '''Carcinomas''': The most common type, originating in epithelial cells.
* '''Sarcomas''': Affect muscles, bones, and cartilage.
* '''Lymphomas''': Involve the lymphatic system (e.g., Hodgkin’s and non-Hodgkin’s lymphoma).
* '''Leukemias''': Found in blood and bone marrow, leading to anemia and immune issues.
'''Staging and Severity''':
* The TNM system assesses tumor size (T), lymph node involvement (N), and metastasis (M).
* Stages range from 0 (localized) to IV (widespread and severe).
'''Epidemiology and Risk''':
* About 1 in 2 men and 1 in 3 women are diagnosed with cancer during their lifetimes, with risks increasing with age.
* Prevalence varies by sex, ethnicity, and geography, with significant disparities in survival rates among ethnic groups.
'''Cultural and Psychosocial Factors''':
* Understanding cancer requires considering cultural influences, including ethnicity, gender, and socioeconomic factors.
* Minority groups face higher risks of death from cancer and lower survival rates, highlighting healthcare inequities.
'''Advancements and Trends''':
* Overall cancer incidence and mortality have decreased over decades, especially for males, though disparities persist.
* Early diagnosis and treatment advances have contributed to better outcomes in many populations.
'''Notable Trends''':
* Death rates from cancer decreased by 32% between 1991 and 2019.
* Minority populations experience lower survival rates, with Black Americans and Native Americans facing the highest relative risk of cancer death compared to White Americans.
By addressing cultural, biological, and environmental factors, we can better understand and combat cancer effectively
== 13.2 - Cultural Variations and Developmental Issues ==
'''Cancer Across Cultures and Groups'''
Cancer affects people differently based on their ethnicity, race, sex, and socioeconomic status. Black Americans face higher cancer death rates than other groups in the U.S., despite White Americans having higher cancer diagnosis rates. For example, Black women are 41% more likely to die from breast cancer than White women, even though they are diagnosed less often. Similarly, Black men have double the cancer death rate compared to Asian/Pacific Islander men.
Cultural factors also influence cancer risks. Asian and Latino Americans show higher stomach and liver cancer rates, while Native Americans often see cancer as a punishment or a spiritual issue. Low socioeconomic status worsens outcomes due to limited knowledge about cancer and later diagnoses.
'''Sex Differences'''
Men are generally diagnosed with cancer more often than women, with prostate, lung, and colorectal cancers being the most common. For women, breast, lung, and colorectal cancers are most prevalent. Research on male-specific cancers, like testicular cancer, lags behind studies on female cancers, but awareness campaigns like No-Shave November are helping.
'''Beliefs and Knowledge About Cancer'''
Cultural beliefs affect understanding and attitudes toward cancer. Some groups attribute cancer to punishment or fate, leading to misconceptions. For instance, some Native Americans believe cancer can result from curses or violations of traditions. Fatalism—the idea that cancer always leads to death—is common in many communities, which can discourage seeking treatment.
'''Cancer as a Developmental Disease'''
Cancer risk increases with age, but many risk factors start early in life. For instance, smoking—a leading cause of cancer—often begins in teenage years. Children and teens diagnosed with cancer experience unique challenges, such as different coping mechanisms and impacts on their quality of life. Understanding how cancer affects individuals at different life stages is essential for providing appropriate support.
== 13.3 - Correlates of Cancer ==
=== Physiological Correlates of Cancer ===
The '''physiological symptoms''' of cancer depend on its '''size, location, and stage''' (Weinberg, 2013). If cancer has reached an advanced stage and '''metastasizes''' (spreads to other parts of the body), symptoms may occur at different locations. As '''mutant cells''' divide, they exert '''pressure''' on surrounding organs, blood vessels, and nerves, causing '''pain''' and discomfort (Pecorino, 2021).
Certain cancers, like '''pancreatic cancer''', often remain '''asymptomatic''' until they reach an advanced stage. '''General symptoms''' include fever, fatigue, pain, skin changes, and weight loss, though these symptoms are not exclusive to cancer. Sometimes, '''cancer cells''' release '''substances''' into the bloodstream that trigger unusual symptoms. For instance:
* '''Pancreatic cancer''' may release substances causing '''blood clots''' in leg veins.
* '''Lung cancers''' can produce '''hormone-like substances''' that alter '''blood calcium levels''', leading to '''weakness and dizziness'''.
Specific symptoms may also indicate particular cancers:
* '''Excretory function changes''': Could signal '''colon cancer''' (e.g., chronic diarrhea or altered stool consistency) or '''bladder/prostate cancer''' (e.g., painful urination, blood in urine).
* '''Sores that do not heal''': Signs of '''skin cancer''', particularly in the mouth or sexual organs.
* '''Unusual bleeding''': Blood in saliva, urine, stool, or other fluids should be reported immediately.
* '''Lumps''': In the breast, testicles, or lymph nodes could suggest cancer.
Being familiar with your body and recognizing changes is key to identifying potential issues early.
----
=== Psychological Correlates of Cancer ===
==== Psychological Factors in Cancer Incidence ====
Although there is little evidence that psychological factors '''cause cancer''', '''stress''' has been recognized as a contributing factor (Stanton, 2019). There are three main pathways through which '''psychological processes''' may influence cancer:
# '''Direct effects''' on bodily systems.
# '''Health behaviors''' (e.g., smoking or diet).
# Responses to '''illness or treatment''', such as screening behaviors.
Key psychological influences include:
* '''Personality''': Early studies linked '''anger suppression''' to a '''Type C personality'''—characterized by being unassertive and suppressing emotions—but this connection has since been discredited (Johansen, 2018).
* '''Social Support''': While it plays a major role in '''coping''', its impact on '''cancer incidence''' is inconsistent. For example, '''social isolation''' is linked to higher cancer mortality among White Americans but not Black Americans (Alcaraz et al., 2019).
* '''Depression''': Stronger evidence links '''clinical depression''' to higher cancer incidence and poorer survival rates (Wang, Li et al., 2020).
==== Psychological Responses to Diagnosis ====
The experience of a '''cancer diagnosis''' can trigger:
* '''Existential plight''': Heightened thoughts about life and death (Weisman & Worden, 1972).
* '''Anxiety and depression''', particularly after diagnosis and during treatment (Civilotti et al., 2021).
* '''Stress''', reducing '''quality of life (QOL)''' (Dornelas, 2018).
Depression varies with '''age, sex''', and '''cancer type''':
* '''Younger patients''' often experience higher rates of depression.
* '''Females''' report almost twice the depression rates as males.
* Certain cancers, like '''lung''' or '''gynecological cancers''', are associated with higher depression levels (Linden et al., 2012).
----
=== Psychological Factors in Cancer Progression and Coping ===
Key factors influencing '''cancer progression''' include:
# '''Personality''': '''Optimism''' is strongly linked to better coping and recovery (Scheier & Carver, 2018).
# '''Social Support''': A larger and stronger '''support network''' correlates with better survival outcomes (Nausheen et al., 2009).
# '''Depression''': Associated with faster disease progression and lower QOL (Bortolato et al., 2017).
----
=== Cancer, Stress, and Immunity ===
'''Stress''' has been shown to:
* Increase cancer risk and worsen progression (Chida et al., 2008).
* Reduce '''natural killer (NK) cell activity''', critical for fighting cancer (Capellino et al., 2020).
* Trigger '''unhealthy behaviors''' (e.g., smoking, poor diet), which accelerate progression.
Mechanisms include activation of the '''sympathetic nervous system (SNS)''' and '''hypothalamic-pituitary-adrenal (HPA) axis''', leading to immune dysregulation (Dai et al., 2020).
== 13.4 - Health Behaviors and Cancer ==
'''Health Behaviors and Cancer:'''
Your daily habits can significantly impact your chances of developing or surviving cancer. Some health risks, like smoking and unhealthy diets, are widely recognized, but others might surprise you.
'''Tobacco Use:'''
* Smoking is the leading preventable cause of cancer, contributing to 19% of all cases in the U.S.
* Smokers face 7.5 to 9 times the risk of lung cancer compared to nonsmokers.
* Tobacco use also increases risks for cancers of the throat, mouth, esophagus, and more.
* Cultural and regional differences affect tobacco usage rates and cancer risks.
'''Diet:'''
* Diet is the second biggest factor after smoking for cancer risk.
* Diets high in fat and low in fiber raise cancer risks, while fruits, vegetables, and certain fats (like omega-3s) help lower them.
* Obesity, often linked to poor diets, increases the risk of at least 13 cancers.
* Some cultural diets (e.g., Mediterranean) are associated with lower cancer risks.
'''Physical Activity:'''
* Regular physical activity reduces the risk of many cancers, including breast, colon, and bladder cancer.
* Adults should aim for 150–300 minutes of moderate exercise per week.
'''Sun Exposure:'''
* Excessive sun exposure leads to skin cancer, including melanoma.
* Regular sunscreen use and avoiding tanning beds significantly lower risks.
* Moderate sun exposure is beneficial for vitamin D production and mood.
'''Key Takeaway:'''
Prevent cancer by avoiding smoking, eating a healthy diet, staying physically active, protecting your skin from UV rays, and being mindful of your overall lifestyle
Cancer is no longer an automatic death sentence, as advances in detection and treatment have improved survival rates. While early detection increases the chances of successful treatment, some advanced cancers can still be treated effectively. Cancer treatments aim to remove or control abnormal cells, with three main approaches: surgery, chemotherapy, and radiation therapy.
'''1. Surgery'''
Surgery is the oldest cancer treatment, primarily used to remove tumors. It is most effective when cancer is localized.
* '''Preventive Surgery:''' Removes tissue likely to become cancerous, such as in individuals with BRCA gene mutations.
* '''Diagnostic Surgery:''' Extracts small tissue samples to confirm or stage cancer.
* '''Curative Surgery:''' Targets localized cancer, often combined with other treatments.
* '''Other Types:'''
** '''Debulking Surgery:''' Reduces tumor size when complete removal isn’t possible.
** '''Palliative Surgery:''' Eases complications in advanced cancer.
** '''Restorative Surgery:''' Reconstructs appearance post-surgery (e.g., breast reconstruction).
'''2. Chemotherapy'''
Chemotherapy involves administering drugs to halt cancer growth and can be used alone or with other treatments.
* '''Forms:''' Pills, injections, or intravenous delivery.
* '''Purposes:'''
** '''Neuroadjuvant Therapy:''' Shrinks tumors before surgery/radiation.
** '''Adjuvant Therapy:''' Destroys residual cancer cells after primary treatment.
* '''Effects:'''
** '''Biological:''' Reduces white blood cells, causing fatigue and vulnerability to infections.
** '''Psychological and Social:''' Side effects like hair loss, nausea, and fatigue impact self-esteem and relationships.
'''3. Radiation Therapy'''
Radiation therapy uses ionizing radiation to damage cancer DNA, leading to cell death.
* '''Techniques:'''
** '''External Radiation:''' Targets tumors with precise beams over weeks.
** '''Internal Radiation (Brachytherapy):''' Places radioactive sources near the tumor.
** '''Systemic Radiation:''' Uses radioactive substances administered orally or intravenously.
* '''Side Effects:''' Fatigue, localized irritation, difficulty swallowing (if applied to the throat), and potential fertility issues.
'''Other Treatments'''
* '''Immunotherapy:''' Activates the immune system against cancer.
* '''Targeted Therapy:''' Uses drugs that specifically target cancer cells.
* '''Hormonal Therapy:''' Alters hormone activity to slow cancer growth.
* '''Complementary and Alternative Medicine (CAM):''' Includes yoga, acupuncture, and special diets for symptom management and psychological benefits. However, these methods require caution due to varying effectiveness and potential risks when delaying standard treatments.
'''Survivorship and Challenges'''
Cancer survivors face risks like secondary cancers, PTSD, anxiety, and depression. Many turn to CAM for holistic healing, but more research is needed to validate its efficacy. Health professionals advise patients to discuss CAM use to avoid harmful interactions or delays in effective treatment.
Advances in treatment and care have significantly improved outcomes for cancer patients, but ongoing research and psychological support remain crucial for long-term recovery and well-being.
=== Raw Textbook Page ===
Cancer need no longer be a death sentence, although many people still believe it is. Although earlier cancer detection increases its chances of being completely treated, even some cancers in their later stages can be successfully treated. Because cancer is essentially cells that are out of control, the main goal of treatment is to remove the cells from the body. There are three major ways to treat cancer.
Surgery
The oldest and most straightforward way to treat cancer is surgery, during which the surgeon can remove the tumor. Surgery is most successful when the cancer has not spread because this provides the best chance of removing all the mutant cells. Surgery also has other uses in cancer (ACS, 2019a). Preventive surgery is performed to remove tissue that is not malignant as yet but has a high chance of turning malignant. This happens in the case of females with a family history of breast cancer who also have a mutant breast cancer gene (BRCA1 or BRCA2). Preventive surgery may also be used to remove parts of the colon if polyps, small stalk-shaped growths (not the little marine animals), are found.
Diagnostic surgery is the process of removing a small amount of tissue to either identify a cancer or to make a diagnosis (ACS, 2019a). If the patient receives a positive diagnosis of cancer, sometimes staging surgery is needed to ascertain what stage of development the cancer is in. This form of surgery helps determine how far the cancer has spread, provides a clinical stage for the growth and can guide treatment decisions. If the cancer has been localized to a small area, curative surgery can be used to remove the growth. This is the primary form of treatment for cancer and is often used in conjunction with other treatments. If it is not possible to remove the entire tumor without damaging the surrounding tissues, debulking surgery reduces the tumor mass. Finally, palliative surgery is used to treat complications of advanced disease (not as a cure), and restorative surgery is used to modify a person’s appearance after curative surgery (e.g., breast reconstructive surgery after a mastectomy or breast removal) (ACS, 2019a).
Chemotherapy
The second major form of treatment for cancer involves taking medications with the aim of disabling the cancer growth, a process referred to as chemotherapy (ACS, 2019b). The medications are either given in pill form, in the form of an injection, or in the form of an intravenous injection (medication delivered through a catheter right into a vein). Chemotherapy can be used alone or in combination with other treatments. Neuroadjuvant chemotherapy involves using chemotherapy to reduce the size of a tumor before surgery or radiation therapy. At other times, chemotherapy is adjuvant, that is, used after surgery or radiation therapy to destroy any remaining cancer cells. Chemotherapy can also be used in conjunction with targeted therapy, hormone therapy or immunotherapy.
The type of cancer and its severity determine the frequency of chemotherapy, and it can range from daily to monthly medication. This form of treatment can have very strong side effects but often results in successful outcomes. Patients who undergo chemotherapy (or chemo for short) may lose all their hair (not just the hair on their head as is commonly believed), experience severe nausea, and have a dry mouth and skin.
Chemotherapy also has biopsychosocial effects (Lorusso et al., 2017). Biologically, the treatment lowers both red and white blood cell counts. Fewer red blood cells make a person anemic and feel weak and tired. A reduced number of white blood cells makes a person more prone to infection, and a person undergoing chemotherapy needs to take special care to not be exposed to germs or sources of contamination. Psychologically, the fatigue can lead to low moods and also a loss of sexual desire and low sociability. Social interactions become strained as well. The patient often feels embarrassed by not having any hair and may not want to be seen by other people or may be too tired to interact. Simultaneously, many visitors and friends feel uncomfortable at the sight of the hairless, fatigued patient. As with most chronic illnesses it is important for the patient’s support networks to be prepared for and compensate for the effects of treatment.
Radiation Therapy
The third major form of treatment for cancer involves the use of radioactive particles aimed at the DNA of the cancer cells, a process referred to as radiation therapy (ACS, 2019c). Radiation used for cancer treatment is called ionizing radiation because it forms ions as it passes through tissues and dislodges electrons from atoms. The ionization causes cell death or a genetic change in the cancerous cells (Sia et al., 2020). There are many different types of external radiation treatments (e.g., electron beams, high-energy photons, protons, and neutrons), each sounding like something out of a science fiction movie and varying in intensity and energy. The process for getting radiation therapy is a little more complex than that for chemotherapy and surgery, although the preliminary stages are the same. Medical personnel first need to identify the location and size of the tumor and then pick the correct level of radiation. The key is to be able to do the most damage to the cancerous cells without damaging the normal cells. This is hard to do because the radiation stream cannot differentiate between types of cells, and normal cells often end up being affected as part of the process, resulting in treatment side effects. For external radiation, radiation is delivered from outside the body into the tumor. The total dose of radiation, a rad, is often broken down into fractions, and delivered as an outpatient over several weeks. Radiation therapy is perhaps the most involved type of therapy, with treatments usually being given daily, 5 days a week, for 5 to 7 weeks (ACS, 2019c). Radiation therapy can also be delivered internally, known as brachytherapy. In this treatment, a radioactive source is put inside the body or located close to the tumor; it may be left there or removed after a specific period of time. Lastly, radiotherapy may involve systemic radiation, where radioactive substances given by mouth or through a vein (ACS, 2019c). The type of radiotherapy will vary depending on the type and location of the cancer; sometimes multiple types will be used.
The main side effects of radiotherapy are fatigue and irritation of the body areas close to the radiation site (for external radiation), often accompanied by some disruption of functioning. For example, radiation to the mouth and throat area can cause loss of salivary function, difficulty in swallowing, and a redness of the neck and surrounding areas. Depending on the location (i.e., ovaries or testicles), radiation may also affect future fertility, with females advised not to conceive while undergoing radiation therapy (ACS, 2019c).
Other Treatments
In addition to these three main forms of treatment, there are also a variety of other possible ways to treat cancer. For example, immunotherapy involves the activation of the body’s own immune system to fight the cancer (ACS, 2019d), while targeted therapy uses medications that are designed to target cancer cells without damaging normal cells (ACS, 2021b) and hormonal therapy involves altering or blocking hormones to slow or stop cancer growth (ACS, 2020). There are also a number of alternative and complementary therapies that help people cope with cancer (some of which are believed to keep cancer at bay as well). These include aromatherapy, music therapy, yoga, massage therapy, meditation to reduce stress, special diets like taking peppermint tea for nausea, and acupuncture. There is growing public interest, especially among those living with cancer or the relatives of people with cancer, in obtaining information about complementary and alternative medicine (CAM) as discussed in Chapter 3 and methods of treatment. This interest is even more prevalent among individuals from different cultural groups who may have approaches and beliefs about cancer and its treatment that vary greatly from the view held by Western biomedicine. Very often cancer patients do not tell their doctors that they are also trying other treatments. Although there may be many treatments for cancer used by other cultures that are actually beneficial, very few methods have been tested by Western science and correspondingly North American health practitioners recommend very few alternative methods.
The stance of Western biomedicine is reflected particularly well in how the ACS refers to complementary and alternative medicine. The American Cancer Society defines alternative methods as “unproven or disproven methods used instead of standard medical treatments to prevent, diagnose, or treat cancer” (ACS, 2021c) and complementary methods as “supportive methods used along with standard medical treatment . . . used to help relieve symptoms of cancer and side effects of treatment” (ACS, 2021d). They differentiate between the two, stating that complementary methods are used with standard treatments, while alternative methods are used instead of standard treatments (ACS, 2021d). The American Cancer Society acknowledges that more research is needed to determine the safety and effectiveness of many of these methods and advocates for peer-reviewed scientific evidence of the safety and efficacy of these methods. Health-care practitioners recognize the need to balance access to complementary and alternative methods while protecting patients against methods that might be harmful to them. For example, the American Cancer Society supports patients having access to complementary methods but cautions them that the evidence for such methods is of lower quality than for standard treatments, and that their use should be discussed with treating health professionals. Part of the problem arises from the fact that harmful drug interactions may occur and must be recognized. In addition, sometimes use of the other treatments causes delays in starting standard therapies and is detrimental to the success of cancer treatment. The American Cancer Society also give greater warning with regards to alternative methods, indicating that the level of harm from such methods can vary (from few harms to life-threatening), these methods have not been sufficiently studied to demonstrate effectiveness and that people who use alternative methods instead of standard treatment for the most common cancers have a greater risk of cancer-related death.
A patient is lying on a stretcher, wearing an immobilization thermoplastic mask that covers his head and face. Two individuals are standing on both sides of the patient.
Radiation Therapy. A cancer patient has his head immobilized before targeted laser-guided radiation treatment.
When treatments succeed in keeping cancer at bay, the health psychologist’s job is not done. There is evidence that cancer survivors may develop secondary cancers. This risk is particularly evident for survivors of childhood cancer (Turcotte et al., 2018) as well as adolescent and young adult survivors due to greater sensitivity to treatments and longer life-expectancy (Demoor-Goldschmidt & de Vathaire, 2019). Similarly problematic, a significant number of adolescent cancer survivors report posttraumatic stress disorder (PTSD), anxiety and depression (Kosir et al., 2019). Many cancer survivors begin to (or continue to) use complementary and alternative treatments. For example, Keene et al. (2019) conducted a systematic review of 61 studies exploring complementary and alternative treatment use among cancer patients, finding that on average, 51% of cancer patients used complementary and alternative treatments, with the main motivation reported to be to treat or cure cancer (73.8%). People also used complementary and alternative treatments to manage treatment complications (62.3%), meet holistic needs (57.4%), influence their general health (55.7%), and take control of their treatment (45.9%). Complementary and alternative treatments may provide the survivors with psychological benefits and prevent relapse, though this possibility has not been tested.
== 13.6 ==
'''The Primary Strategies for Reducing Cancer Mortality'''
The most effective approach to reducing cancer mortality rates involves promoting healthy behaviors while minimizing risk-enhancing habits. Key strategies include decreasing tobacco use, improving nutrition, and encouraging physical activity. Despite challenges in changing long-established habits like dietary practices, recent interventions have shown success in these areas. Additionally, health psychologists have developed various psychosocial interventions to support cancer patients.
'''Screening as a Preventative Measure'''
Routine cancer screening is crucial for early detection, significantly improving treatment outcomes. Screenings can be clinical (e.g., mammograms) or self-conducted (e.g., breast or testicular self-examinations). The Health Belief Model (HBM) frequently guides interventions to enhance screening adherence by addressing perceptions of susceptibility, severity, barriers, and benefits. Evidence suggests that increasing awareness about cancer risks and benefits of screening boosts screening rates, while perceived barriers hinder participation.
'''Cultural and Socioeconomic Influences on Screening'''
Screening disparities exist due to socioeconomic and cultural factors. Low-income groups and non-White populations generally exhibit lower screening rates, often facing barriers like cost, access, and cultural norms. For instance, modesty concerns and mistrust toward male healthcare providers deter screenings among many Asian, Black, and Native American women. Fear of cancer or the screening process itself further complicates efforts to increase adherence.
'''Psychosocial Interventions for Cancer Patients'''
Beyond prevention, interventions aim to extend life expectancy and ease anxiety for diagnosed patients. Social support programs, like group therapy, have demonstrated significant benefits in reducing emotional distress and improving immune function. Mindfulness-based practices, relaxation training, and hypnosis also hold promise for managing cancer-related pain, fatigue, and treatment side effects. However, empirical support for some techniques remains limited.
'''Addressing Cultural Sensitivity in Cancer Care'''
Culturally adapted interventions have proven effective in changing cancer-related health behaviors, such as smoking cessation. Tailored approaches that incorporate cultural values and community-specific messaging significantly enhance the efficacy of these programs. Integrating cultural awareness into broader cancer care initiatives can improve access and outcomes for underserved populations.
'''Summary'''
Efforts to combat cancer mortality require a multifaceted approach encompassing behavior modification, routine screenings, and culturally sensitive interventions. Combining medical advances with psychosocial support ensures comprehensive care, improving both prevention and quality of life for individuals affected by cancer.
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User:Atcovi/Health Psychology/Chapter 14: Cardiovascular Disease
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== 14.1 - Cardiovascular Disease: Definitions and Prevalence ==
* '''Definition:''' Diseases affecting the heart and circulatory system, including:
** '''Common Types:''' Coronary heart disease (CHD), heart failure, strokes, and hypertension.
** '''Other Types:''' Abnormal heart rhythms, congenital heart disease, cardiomyopathy, rheumatic heart disease, pulmonary heart disease, peripheral artery disease, cerebrovascular disease, and vascular diseases.
* '''Risk Factors:''' High blood pressure, diabetes, kidney disease, and obesity.
----
==== Coronary Heart Disease (CHD): ====
* '''Definition:''' Narrowing of coronary arteries due to fat/scar tissue buildup, also called Coronary Artery Disease (CAD).
* '''Symptoms:''' Chest pain, shortness of breath, nausea, cold sweats, and discomfort in the back, neck, jaw, or arms.
* '''Key Statistics:'''
** Leading cause of death in the U.S. (1 in 5 deaths in 2020).
** Affects 20.5 million Americans aged 20+.
** Higher prevalence in Native Americans (8.6%) and males (8.7% vs. 5.8% in females).
** An American has a heart attack every 40 seconds.
----
==== Hypertension (High Blood Pressure): ====
* '''Definition:''' Chronic elevation of blood pressure.
** Normal: <120/80 mm Hg.
** Elevated: 120–129/<80 mm Hg.
** Stage 1: 130–139/80–89 mm Hg.
** Stage 2: >140/90 mm Hg.
** Hypertensive Crisis: >180/120 mm Hg.
* '''Prevalence:'''
** Affects 116.4 million Americans (58.7M males, 57.7M females).
** Higher rates in Black Americans (57.6% males, 53.2% females).
* '''Risk Factors:''' Obesity, poor diet, lack of physical activity.
* '''Awareness Issue:''' Over 35% of cases go undiagnosed.
----
==== Stroke: ====
* '''Definition:''' Blockage or rupture of blood vessels in the brain, leading to oxygen deprivation.
* '''Types:'''
** '''Ischemic Stroke:''' Blocked blood vessel (87% of cases).
** '''Hemorrhagic Stroke:''' Ruptured vessel causing brain bleeding (13%).
** '''Others:''' Transient ischemic attacks (mini-strokes), brainstem strokes, cryptogenic strokes.
* '''Impact:'''
** 7 million Americans have had a stroke.
** Leading cause of disability and 4th (females) and 5th (males) leading cause of death.
** Higher mortality in Southeastern U.S. ("Stroke Belt").
* '''Risk Disparities:'''
** Males have higher incidence; females experience more strokes due to longer life expectancy.
----
==== Global and Cultural Trends: ====
* '''CVD Mortality Rates:'''
** Highest in Eastern Europe (e.g., Lithuania, Russia).
** Declining rates in the U.S., U.K., and Brazil.
** High disparities in low-income countries.
* '''Framingham Heart Study:'''
** Longitudinal study initiated in 1948 in Framingham, Massachusetts.
** Key findings: Poor diet, sedentary living, smoking, and weight gain are major CVD risk factors.
** Continues to provide insights with third-generation participants.
----
==== Key Terms and Concepts: ====
* '''Blood Pressure Categories:''' Normal, Elevated, Stages of Hypertension.
* '''CVD Risk Factors:''' Diet, physical activity, smoking, obesity.
* '''CHD Warning Signs:''' Jaw/neck/back discomfort, light-headedness, chest pain, arm/shoulder discomfort, and shortness of breath.
* '''CVD Prevention:''' Early detection (e.g., monitoring blood pressure), lifestyle changes, and increased awareness.
----
==== Trends and Recommendations: ====
* Efforts to reduce mortality through public health education and lifestyle interventions.
* Emphasis on addressing disparities in diagnosis and treatment across different demographic groups.
== 14.2 - Cultural Variations and Developmental Issues ==
Cardiovascular diseases (CVDs) are a significant global health issue, with prevalence and mortality rates doubling between 1990 and 2019 (Roth et al., 2020). Regions like Northern Africa, Central Asia, and Latin America are particularly affected due to aging populations. Countries like Uzbekistan and Tajikistan have the highest CVD mortality, while France, Peru, and Japan have the lowest. Ethnic and cultural differences influence CVD prevalence, with variations in health literacy, behaviors, and risk factors among groups. For example, South Asians have higher CVD risks due to diabetes prevalence, and Black Americans exhibit higher rates of hypertension (Hamner & Wilder, 2008; Tsao et al., 2023).
Cultural dimensions also play a role. For instance, individualistic cultures may foster Type A personality traits (e.g., competitiveness), increasing CVD risks, while collectivist cultures offer stronger social support, mitigating stress. Time orientation differences (fixed vs. fluid) affect stress and blood pressure dynamics across cultures. Access to healthcare and patient–doctor interactions also vary; South Asians often delay seeking care, and gender concordance between patient and doctor significantly impacts outcomes (Greenwood et al., 2018).
Biological markers like C-reactive protein (CRP) differ among ethnic groups, influenced by psychosocial factors such as discrimination and educational disadvantage. Ethnic differences in calcifications tied to atherosclerosis have been identified, though their link to race remains inconclusive (Nasir et al., 2008; Budoff et al., 2018).
Developmentally, life events (e.g., marriage, divorce, natural disasters) and transitions increase CVD risks. Social support changes over time, with loneliness being a critical risk factor. Surprisingly, extensive social networks in older women may increase mental burdens, correlating with cardiac issues (Valtorta et al., 2016). Life span approaches are essential to understand the psychosocial aspects of CVD.
'''Key Points:'''
# '''Global Impact:''' Doubling of CVD prevalence and mortality from 1990–2019; regional disparities.
# '''Ethnic Variations:''' Risk factors like diabetes and hypertension differ by ethnicity.
# '''Cultural Influences:''' Social support, time orientation, and Type A traits affect CVD risks.
# '''Healthcare Disparities:''' Delayed care and gender concordance affect outcomes.
# '''Biological Markers:''' Ethnic disparities in CRP levels and atherosclerosis calcifications.
# '''Developmental Factors:''' Life transitions, social isolation, and loneliness influence risks
== 14.3 - Correlates of Cardiovascular Disease ==
=== Physiological Correlates ===
# '''Atherosclerosis and Arteriosclerosis:'''
#* '''Atherosclerosis:''' Fat accumulates in arteries, leading to plaques that restrict or block blood flow, causing heart attacks.
#* '''Arteriosclerosis:''' Hardening of arteries reduces elasticity, increasing the risk of blockages.
# '''Risk Factors:'''
#* '''Non-modifiable:''' Age (risk increases with age), sex (males under 50 and postmenopausal females are at higher risk), and family history.
#* '''Modifiable:''' High blood pressure, diabetes, high cholesterol, obesity, inactivity, and smoking.
#* '''Ethnic and genetic predispositions:''' Certain groups show increased susceptibility to conditions like hypertension and obesity.
# '''Diabetes and CVD:'''
#* Both type 1 and type 2 diabetes significantly increase CVD risk.
#* Diabetic dyslipidemia (imbalanced fat metabolism) contributes to atherosclerosis.
#* Gestational diabetes elevates CVD risk even without progression to type 2 diabetes.
# '''Discrimination and CVD:'''
#* Studies show discrimination influences cardiovascular reactivity (e.g., blood pressure and stress responses), with variations by race/ethnicity and sexual orientation.
----
=== Psychological Correlates ===
# '''Emotions and Personality Traits:'''
#* Hostility and anger are strongly linked to CVD, triggering events like heart attacks and increasing long-term risk.
#* Depression and hopelessness can independently predict CVD and worsen outcomes after heart attacks.
# '''Social Support:'''
#* Social networks and emotional support play crucial roles in preventing and managing CVD.
#* Low social support is linked to higher risks of mortality and poor health outcomes post-heart attack.
#* Marital status, social participation, and perceived support significantly affect outcomes.
# '''Socioeconomic Status (SES):'''
#* In high-income countries, low SES correlates with increased CVD risk.
#* In low- and middle-income countries, limited access to healthcare exacerbates risks despite better baseline risk profiles.
----
=== Key Findings ===
* Modifiable lifestyle changes (diet, exercise, stress management) are vital in mitigating CVD risks.
* Psychological and social factors are as critical as physiological factors in both prevention and recovery.
* Comprehensive interventions addressing behavioral, emotional, and socioeconomic aspects are essential for effective CVD management.
=== Stress and Cardiovascular Health ===
City living can significantly impact your health. Factors like traffic, pollution, noise, limited green spaces, and access to unhealthy food options increase stress, which can lead to cardiovascular diseases (CVD). Noise, for example, can raise stress levels and affect blood pressure, heart rate, and blood flow.
Stress affects the body by releasing hormones like cortisol, which can increase blood pressure and heart rate. While short-term stress prepares us for emergencies, chronic stress can harm the heart and circulatory system. Personal and work-related stress can raise the risk of heart conditions, strokes, and other health problems.
Even enjoyable activities, like watching sports, can be stressful. Research shows that emotional stress during games, especially when a favorite team loses, may increase the risk of heart attacks or strokes. Major events like natural disasters or the loss of a loved one can also trigger heart issues, particularly in people already at risk.
Work stress is a major focus of research. Long hours, unclear roles, or lack of support at work can increase the likelihood of CVD. People in lower-status jobs generally face a higher risk of heart disease in Western countries, though this pattern differs in places like Japan and Korea. For example, Japan recognizes "karoshi," or death from overwork, highlighting the toll high-status jobs can take.
Stress also leads to harmful behaviors like smoking or exhaustion, which further strain the heart. People with heart disease who experience "vital exhaustion" — marked by fatigue, irritability, and feeling low — are more likely to have recurring heart attacks.
In summary, stress from various sources—city life, work, or personal challenges—can take a significant toll on heart health. Managing stress is essential for reducing the risk of cardiovascular problems
== 14.4 - Health Behaviors and Cardiovascular Disease ==
==== Impact of Smoking on Global Health and CVD ====
* '''Prevalence:''' Over 1 billion active tobacco users globally (Roth et al., 2020).
* '''Mortality:''' Tobacco use caused 8.71 million deaths in 2019; 36.7% linked to cardiovascular diseases (CVDs).
* '''Gender Disparity:''' Smoking rates are higher in males (33.5%) compared to females (6.8%), with 75.4% of smoking-related deaths among men.
* '''Passive Smoking:''' Increased CVD risk by 28% at home/work (case-control studies) and by 12% (cohort studies).
==== Physiological Effects of Smoking ====
* Smoking impacts physiological systems, contributing to high blood pressure, arterial stiffening, inflammation, insulin resistance, and cholesterol changes.
* Historical data from Doll et al. (2004) revealed a 10-year reduction in life expectancy for lifelong smokers born between 1900–1930, with life expectancy improvements for those who quit at any age.
==== Benefits of Smoking Reduction and Cessation ====
* Reducing daily cigarette consumption significantly lowers risks of coronary heart disease (CHD) and stroke (Chang et al., 2021).
* Smoking cessation shows rapid risk reduction within 5 years, with near-normalized CVD risk after 10–15 years (Duncan et al., 2019).
==== Policy Interventions and Effectiveness ====
* Massachusetts Tobacco Control Program (MTCP, 1993):
** Smoking prevalence decreased by 29%.
** CVD-related deaths reduced by 31% over a decade.
* Smoke-free laws lower smoking initiation rates and reduce exposure to secondhand smoke, including among children.
----
=== Diet and CVD: Role and Patterns ===
==== Cholesterol and Risk ====
* High LDL cholesterol is a primary risk factor, particularly in conjunction with diabetes (20–50% reduced CV risk when cholesterol is controlled).
==== Dietary Patterns and Recommendations ====
* '''Saturated Fat Debate:'''
** Earlier guidelines on reducing saturated fat intake have been questioned; focus is now on overall dietary patterns and quality (Astrup et al., 2020).
* '''Beneficial Diets:'''
** DASH Diet: Emphasizes fruits, vegetables, whole grains, and low sodium; reduces risks for CHD, stroke, and high CRP levels.
** Mediterranean Diet: Similar to DASH, includes olive oil, nuts, and moderate wine consumption; linked to lower CHD/stroke incidence and mortality.
** Indo-Mediterranean Diet: Enhances DASH/Mediterranean diets with anti-inflammatory ingredients like turmeric, brown rice, and spices, offering greater diversity and benefits.
==== Alcohol's Role ====
* Moderate wine consumption is associated with lower CVD risk (French Paradox). However, binge drinking or excess consumption negates these benefits.
----
=== Physical Activity: Prevention and Recovery ===
==== Global Trends and Impact ====
* Inactivity increases CVD risks (INTERHEART study). Ownership of cars/TVs correlates with sedentary behavior and CVD prevalence.
==== Rehabilitation and Outcomes ====
* Exercise-based cardiac rehabilitation (6–12 months): Fewer deaths and hospitalizations for CVD patients (Dibben et al., 2021).
* Physical activity significantly reduces all-cause and CVD-specific mortality, even in patients with existing CVD (Jeong et al., 2019).
==== Prescriptive Exercise Plans ====
* Designed for intensity and progression: Moderate aerobic activity transitioning to resistance training for tailored recovery (Tucker et al., 2022).
This structured summary highlights the multifaceted approach to CVD prevention and management, focusing on lifestyle changes, policy interventions, and cultural considerations.
== 14.5 - Treatment Options ==
=== Treatment for Cardiovascular Disease (CVD) ===
Treatment of CVD varies based on symptom severity, ischemic areas, left ventricle function, and other medical factors. Unhealthy lifestyles are key contributors to CVD development, making behavioral changes a critical aspect of prevention and treatment.
==== Cardiac Rehabilitation Programs ====
* '''Purpose:''' Improve functioning, quality of life, and reduce mortality.
* '''Components:''' Physical activity, nutrition counseling, psychological and social support.
* '''Types:'''
** '''Center-based programs''' (traditional, supervised facilities).
** '''Home-based and technology-assisted programs''' (remote or hybrid models).
** '''Yoga-based rehabilitation''' (emerging option, especially in low-income areas).
* '''Effectiveness:''' Linked to reduced mortality and increased quality of life, though barriers like low participation rates persist.
==== Aspirin Therapy ====
* '''Benefits:''' Reduces risk of CVD events, especially in patients with prior cardiac events (Bartolucci et al., 2011).
* '''Risks:''' Bleeding complications; not recommended for migraines or healthy older adults (Weisman & Brunton, 2022; McNeil et al., 2018).
* '''Usage:''' Debated for primary prevention but necessary in severe ischemias or worsening disease.
==== Surgical Interventions ====
# '''Angioplasty:''' Opens blocked arteries using a balloon or alternative tools (e.g., lasers or shavers). Often followed by stent placement to maintain artery openness.
# '''Cardiac Bypass Surgery:'''
#* Creates alternative blood routes using grafts.
#* Variants include minimally invasive options (e.g., robotic surgery, port-access procedures).
#* Recovery involves 4–7 hospital days.
==== Behavioral and Psychological Interventions ====
# '''Stress Management:'''
#* Training combined with cardiac rehabilitation reduces stress and cardiac event rates (Blumenthal et al., 2016).
#* Cognitive-behavioral therapy (CBT) and relaxation techniques improve psychological well-being and reduce hostility (Friedman et al., 1986; Hamieh et al., 2020).
# '''Dietary Interventions:'''
#* Healthy diets reduce inflammation and CVD risks (Kovell et al., 2020).
#* Emphasis should be on healthy food choices rather than specific macronutrient patterns
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Complex Analysis/development in Laurent series
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== Laurent Expansion around a Point ==
Let <math>G \subseteq \mathbb{C}</math> be a domain, <math>z_0 \in G</math>, and <math>f \colon G \setminus {z_0} \to \mathbb{C}</math> a [[w:en:Holomorphic function|holomorphic]] function. A Laurent expansion of <math>f</math> around <math>z_0</math> is a representation of <math>f</math> as a [[Laurent Series|Laurent Series]]:
<center><math>f(z) = \sum_{n=-\infty}^\infty a_n (z-z_0)^n</math></center>
with <math>a_n \in \mathbb{C}</math>, which converges on a punctured disk (i.e., excluding the center <math>z_0</math>) around <math>z_0</math>.
== Laurent Expansion on an Annulus ==
A more general case than the above is the following: Let <math>0 \leq r_1 < r_2</math> be two radii (the expansion around a point corresponds to <math>r_1 = 0</math>), and let
<math>A_{r_1,r_2} := {z \in \mathbb{C} : r_1 < |z-z_0| < r_2}</math>
be an annulus around <math>z_0</math>. If <math>f \colon A_{r_1,r_2} \to \mathbb{C}</math> is a holomorphic function, then the [[Laurent Series|Laurent Series]]
<center><math>f(z) = \sum_{n=-\infty}^\infty a_n (z-z_0)^n</math></center>
with <math>a_n \in \mathbb{C}</math> is a Laurent expansion of <math>f</math> on <math>A_{r_1,r_2}</math>, provided the series converges for all <math>z \in A_{r_1,r_2}</math>.
=== Existence ===
Every holomorphic function on <math>A_{r_1,r_2}</math> possesses a Laurent expansion around <math>z_0</math>. The coefficients <math>a_n</math> in the above representation are given by:
<center><math>a_n = \frac{1}{2\pi i} \int_{|z-z_0| = r} \frac{f(z)}{(z-z_0)^{n+1}} \, dz</math></center>
for a radius <math>r</math> with <math>r_1 < r < r_2</math>.
=== Uniqueness ===
The coefficients are uniquely determined by:
<center><math>a_n = \frac{1}{2\pi i} \int_{|z-z_0| = r} \frac{f(z)}{(z-z_0)^{n+1}} \, dz</math></center>
=== Proof of Existence and Uniqueness of a Laurent Expansion ===
The uniqueness follows from the identity theorem for [[Laurent Series|Laurent Series]]. For existence, choose <math>r</math> with <math>r_1 < r < r_2</math> and <math>R_1, R_2</math> such that <math>r_1 < R_1 < r < R_2 < r_2</math>. Let <math>z \in A_{R_1,R_2}</math> be arbitrary. "Cut" the annulus <math>A_{R_1,R_2}</math> at two points using radii <math>D_1</math> and <math>D_2</math> such that the cycle <math>\partial K_{R_2} - \partial K_{R_1}</math> can be expressed as the sum of two closed, null-homotopic curves <math>C_1</math> and <math>C_2</math>. Choose <math>D_1</math> and <math>D_2</math> such that <math>z</math> is enclosed by <math>C_1</math>. By the [[Cauchy Integral Theorem|Cauchy Integral Theorem]], we have:
<center><math>f(z) = \frac{1}{2\pi i} \int_{C_1} \frac{f(w)}{w-z} \, dw</math></center>
and
<center><math>0 = \frac{1}{2\pi i} \int_{C_2} \frac{f(w)}{w-z} \, dw</math></center>
since <math>C_2</math> does not enclose <math>z</math>. Hence, due to <math>C_1 + C_2 = \partial K_{R_2} - \partial K_{R_1}</math>, we obtain:
<center><math>f(z) = \frac{1}{2\pi i} \int_{|w-z_0| = R_2} \frac{f(w)}{w-z} \, dw - \frac{1}{2\pi i} \int_{|w-z_0| = R_1} \frac{f(w)}{w-z} \, dw</math></center> We now expand <math>\frac{1}{w-z}</math> for <math>|w-z_0| = R_2</math> using:
<center><math>
\begin{array}{rl}
\frac{1}{w-z} &= \frac{1}{(w-z_0) - (z-z_0)} \\
&= \frac{1}{w-z_0} \cdot \frac{1}{1 - \frac{z-z_0}{w-z_0}} \\
&= \frac{1}{w-z_0} \sum_{n=0}^\infty \frac{(z-z_0)^n}{(w-z_0)^n}
\end{array}
</math></center>
This series converges absolutely for <math>|z-z_0| < |w-z_0|</math>, yielding:
<center><math>
\frac{1}{2\pi i} \int_{|w-z_0| = R_2} \frac{f(w)}{w-z} \, dw = \sum_{n=0}^\infty \left(\frac{1}{2\pi i} \int_{|w-z_0| = R_2} \frac{f(w)}{(w-z_0)^{n+1}} \, dw\right) (z-z_0)^n.
</math></center>
Similarly, for the inner circle <math>|w-z_0| = R_1</math>, we expand and calculate analogously. The final result shows that for <math>z \in A_{R_1,R_2}</math>, the Laurent series converges, proving the existence of the Laurent expansion.
== See Also ==
*[[Complex Analysis/Example Computation with Laurent Series|Example Computation with Laurent Series]]
*[[Laurent Series|Laurent Series]]
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WikiJournal Preprints/Mobility-aware Scheduling in Fog Computing: Analysis and Challenges
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{{Article info
| journal = WikiJournal of Science
| last1 = Islam
| orcid1 = 0000-0001-5750-1058
| first1 = Mir Salim Ul
| last2 = Kumar
| orcid2 = 0000-0003-3279-5111
| first2 = Ashok
| affiliation2 = Department of Computer Application, Chandigarh School of Business, Chandigarh Group of Colleges, Jhanjeri, Punjab, India
| et_al =
| affiliation1 = Department of Computer Science and Engineering, National Institute of Technology Srinagar, Jammu and Kashmir, India
| correspondence1 = mir.salim27@gmail.com
| affiliations =
| keywords = Fog computing, Mobility-aware, Scheduling, Resource Management, Resource Allocation
| license = CC BY 4.0
| abstract = The increasing focus on Fog-IoT results in billions of Internet-connected devices that demand substantial computational power and network bandwidth. These devices are geographically distributed, heterogeneous, computational capacity constrained, inconsistent in behaviour, and generally mobile. Therefore, providing seamless service, irrespective of location and movement of the devices as well as the end-users, makes resource scheduling a significant challenge in the Fog computing paradigm. Several mobility-aware scheduling strategies have been proposed in the literature to efficiently manage the resources for mobile users and devices in the Fog environment. This paper gives a survey of mobility-aware scheduling in the Fog computing environment. It describes the many strategies presented and their benefits and drawbacks. It also includes a complete study and taxonomy of the mobility-aware scheduling field. Further, it delineates open issues and challenges. This work will provide researchers with future research directions and aid them in recognizing the gaps before planning for further research in mobility-aware scheduling.
}}
== Introduction ==
With the advancement in technology and the exponential growth of mobile devices, network traffic has increased manifold in cloud computing. Due to this reason, Latency reduction and faster processing of data for mobile users have become critical challenges in providing seamless connectivity and minimal disruption while the user is moving.<ref>{{Cite journal|last=Abdullah|first=Fatima|last2=Kimovski|first2=Dragi|last3=Prodan|first3=Radu|last4=Munir|first4=Kashif|date=2021-06-11|title=Handover authentication latency reduction using mobile edge computing and mobility patterns|url=https://doi.org/10.1007/s00607-021-00969-z|journal=Computing|volume=103|issue=11|pages=2667–2686|doi=10.1007/s00607-021-00969-z|issn=0010-485X}}</ref> The data movement brings the additional issue of integrity and confidentiality because data is moving via a wireless connection to a far distant cloud. Additionally, due to the cloud's location is far from mobile users, so data movement is also affected by variable network strength and phone bandwidth. The solution proposed by [2] is to extend cloud capabilities through fog computing architecture. The Fog architecture allows substantial computation, storage, and processing using the Fog devices installed close to the user’s access point. Fog computing, therefore, reduces Latency and bandwidth consumption, improves security, provides context awareness, and renders more efficient services to mobile users [3, 4, 5].
[[File:Fog Computing Figure 1.jpg|left|thumb|Figure 1: Mobility Scenario in Fog Computing]]
However, mobility also imposes severe challenges for Fog computing due to its distributed and diverse environment. Mobility is recognized by either user-level or device-level contextual information [6]. As the user moves from one location to another, the geographical location of the smart devices also changes. The change in location of the devices raises the issue of searching and rescheduling with mobility management. Efficient re-scheduling requires a well-planned handoff mechanism that is accountable for smoothly de-registering a sensor node from a source access point where the application was initially hosted and registering it to a new access point. Figure 1 depicts these mentioned problems of change in access points while the user moves from one area to another. The services may also get interrupted when there is more distance between the Fog nodes and users [7]. Further, any disruption in communication may lead to an increase in Latency for mobile users [8].
The significant challenges in mobility management of Fog computing are: (i) ''Fog node discovery''- due to the heterogeneous and mobile nature of the smart mobile devices, the Fog task scheduler faces the issue of finding an optimal Fog node for scheduling the task, ''(ii) Handover mechanism between users and Fog nodes''- imagine a Fog user is moving from location to location and accessing information about his surroundings with the help of a smart device. Due to the frequent change in the user's location, the Fog scheduler may need to repeatedly migrate the user’s task to a different Fog node available in his vicinity. The frequent migration of tasks increases the overhead of scheduling and, further, the restricted signal strength in certain places may lead to a breakdown of task or result delivery, and (iii) ''the Handover mechanism between Cloud and Fog''- Fog nodes have limited capacities and need to continuously communicate with Cloud computing for passing information about the tasks [9]. Due to strict requirements for security, Latency, network coverage, and reliability, It becomes challenging to implement an efficient handover mechanism for full mobility support in critical domains such as healthcare [10, 11, 12] and vehicular systems [13, 14]. Thus, mobility significantly impacts the overhead of scheduling policies and the applications' performance, eventually affecting the Quality of Experience (QoE). Therefore, mobility-aware scheduling in Fog computing has observed strong attention from researchers. Our main goal is to provide a detailed review of mobility-aware scheduling in fog computing. The review provides a detailed analysis of existing scheduling strategies that concentrate specifically on mobility awareness in the Fog environment. The following are the main contributions of this paper:
* This paper presents a detailed survey of mobility-aware scheduling in the Fog computing environment.
* It provides the details of the different techniques proposed, as well as their advantages and limitations.
* It provides a detailed analysis and taxonomy of the mobility-aware scheduling field.
* It identifies several open challenges for future research directions.
=== Related Surveys ===
Numerous survey studies on Fog computing focus on resource management, job scheduling, and context-aware scheduling. For example, Ghobaei-Arani et al. [15] presented a survey on resource management techniques in fog computing in the form of taxonomy to highlight cutting-edge methods while also addressing unresolved challenges. The various authors in the cloud-fog area provided the task scheduling review. Their benefits and drawbacks, as well as numerous tools and challenges concerning the scheduling techniques and their limitations, were analyzed by Alizadeh et al. [16]. Islam et al. [6] thoroughly analyze relevant literature on context-aware scheduling in fog computing. Further, Mouradian et al. [17] review fog computing's significant issues and challenges. But, the critical area of mobility research is still in its initial stage, and most of the review papers contain very few documents on mobility-aware task scheduling in the area of fog computing. Therefore, an extensive and comparative study is required in mobility-constrained fog computing. A deep insight into various techniques which can impact the user QoS is necessary to understand mobility-aware fog computing. This motivates us to carry out a comprehensive survey; to the best of our knowledge, this is the first detailed survey. This survey paper thoroughly examines existing scheduling solutions that focus on user mobility. Moreover, various mobility-aware scheduling techniques are discussed, along with their pros and cons. Further, the impact of mobility parameters on various QoS parameters and context-awareness is also analyzed thoroughly.
=== Paper Organization ===
The remainder of this review article is divided into 5 sections: Section 2 discusses mobility-aware scheduling in Fog computing. Section 3 presents the review methodology. Section 4 analyzes and summarizes the considered research papers and compares existing mobility-aware scheduling strategies. Section 5 provides the results drawn after critically examining the existing literature on mobility-aware scheduling policies. Finally, Section 6 presents the conclusion.
== Mobility-aware Scheduling in Fog Computing ==
Mobile device management compromises the fundamental features of Fog computing because whenever a user moves, the distance between them increases, impacting the QoS. Therefore, to keep the computing fog node close to the associated mobile device, the services or tasks need to be migrated from one fog node to another appropriate fog device. The selection of such appropriate fog nodes in a mobile environment deals with two main processes: Estimation of user mobility patterns: User mobility estimation techniques can be probabilistic and deterministic [18]. In a deterministic method, the source and destination are known beforehand, whereas in a non-deterministic technique, periodic estimation has to be done regarding the user's route. Many authors estimate the route of mobile users by leveraging external services like Open Street Maps (<nowiki>http://www.openstreetmap.org</nowiki>), Google Maps APIs (<nowiki>https://cloud.google.com/</nowiki> maps-platform/) [18], logistics maps [19], GPRS Here APIs (<nowiki>https://developer</nowiki>. here.com/) [20, 21], Lyapunov estimation technique [22]. Specific QoS requirements: The selection of a fog node also depends upon specific quality requirements such as Latency, which many authors are using to select an efficient fog node [18, 10, 9], workload [23, 24], cost [16], etc.
== Review Methodology ==
The review process was conducted considering a review methodology consisting of four phases. The first phase was searching using traditional online database sources based on outlined search keywords. Table 1 lists the
Keywords used to find the relevant research articles to conduct the review in the Fog computing mobility-aware scheduling area. Second Phase: Limit the search of research articles beginning in 2015, and inclusion and exclusion principles are also used to refine research articles that specifically deal with mobility issues in task scheduling. Finally, in the Third Phase, A total of 20 papers are shortlisted for the review process. Further, Table 2 presents the research questions drafted for this study in mobility-aware scheduling in Fog computing.
{| class="wikitable"
|+Table 1: List of keywords used in the review process
!Sno
!Keyword
!Description
!Years
|-
|1
|Mobility
|Mobility-aware Fog task scheduling
| rowspan="5" |2015- 2021
|-
|2
|Mobility environment
|Mobility environment in Fog task scheduling
|-
|3
|Mobility factors
|Mobility factors in Fog task scheduling
|-
|4
|Mobility Awareness
|Mobility awareness in Fog task scheduling
|-
|5
|Mobility management
|Mobility management in Fog task scheduling
|}
{| class="wikitable"
|+Table 2: List of research questions used to complete the review process
!Q.No
!Research questions
|-
|1
|What scheduling approaches are used in the Fog computing environment to manage mobility awareness?
|-
|2
|What are the main limitations considered for mobility-aware scheduling techniques?
|-
|3
|Which case studies are applied to mobility-aware scheduling techniques?
|-
|4
|What evaluation tools are used to assess mobility-aware scheduling techniques?
|-
|5
|What performance indicators are utilized to evaluate mobility-aware scheduling techniques?
|-
|6
|What are the major open issues concerns in the field of mobility-aware scheduling for future research directions?
|}
=== Source of Information ===
To conduct this review, various online sources are listed below, and the research articles were searched using the different keywords mentioned in Table 1.
* Google Scholar (<nowiki>http://www.scholar.google.co.in</nowiki>)
* John Wiley & Sons Inc. (<nowiki>https://onlinelibrary.wiley.com/</nowiki>)
* Elsevier (<nowiki>https://www.elsevier.com/en-in</nowiki>)
* ACM Digital Library (<nowiki>https://www.acm.org/</nowiki>)
* Springer (<nowiki>https://www.springer.com/in</nowiki>)
* IEEE Xplore Digital Library (<nowiki>https://www.ieee.org/</nowiki>)
=== Quality Assessment ===
Research papers used quality assessment to filter out the most suitable mobility-based scheduling research articles in fog computing utilising the principle of inclusion and exclusion. Furthermore, in order to obtain high-quality research publications, the Center for Reviews and Dissemination (CRD) recommendations were followed, and each study item was examined for internal and external validation of results.
== Literature Analysis ==
Verma et al. [25] proposed a Server Cloudlet(SC) migration technique to handle users' mobility. The strategy is to select the target SC to migrate the services based on its highest rank. The rank of the SC further depends upon its available RAM, MIPS, and bandwidth.
Maleki et al. [26] designed two mobility-aware computation offloading approaches, sampling-based Online mobile applications to cloudlets(S-OAMC) and Greedy online mobile applications to cloudlets(G-OAMC). The developed system works in three major steps: ''(i)'' the future specifications of mobile applications are predicted using the Machine Learning (ML) method, named matrix completion. ''(ii)'' The predicted specifications of mobile applications, including future location prediction, also help estimate the offloading cost. ''(iii)'' Apply S-OAMC or G-OAMC offloading technique based on minimum cost value.
Abdelmoneem et al. [10] work on a fog-based healthcare architecture system that supports patient movement without altering their Quality of Service (QoS). The QoS in a mobile environment is maintained with the help of their proposed handoff mechanism. The authors modified the traditional Horizontal Handoff (HHO) mechanism [27] into two different phases: ''(i) handoff decision policy'': Received signal Strength (RSS) method is used to detect the network change information of the mobile patients and (ii) ''handoff strategy'': a handoff decision is made based on a comparison between RSS received from the old fog gateways, current threshold value and new fog gateways that are in close proximity to the mobile patient.
Javanmardi et al. [28] consider an imaginative city-based mobility scenario where the user is placing delay-sensitive service while moving. The authors proposed a task scheduling technique that jointly employs fuzzy logic and particle swarm optimization (PSO) algorithm to improve the QoS for mobile users within the city. The proposed algorithm is deployed at the Fog gateway, which further distributes the tasks to Fog devices available in an entire region in order to provide seamless service to mobile users. The main motive of the work is to improve resource utilization in a mobility-aware environment.
Puliafito et al. [29] develop an extension of ifogsim [30], which supports mobile environment. The authors implement different migration techniques inspired by Bittencourt et al. [31]. Different phases are devised to perform migration in a fog mobile environment, which: ''(i) Before migration phase'': migration decision is taken in this phase, based on specific parameters, like user location, speed, the direction of movement, zone, and migration point in order to select the appropriate cloudlet for offloading the services, ''(ii) During the migration phase'': this phase manages, monitors and synchronizes the whole selected migration process and ''(iii) After migration phase'': this phase involves closing the older cloudlet connections with the user and using the new cloudlet for services.
A Blockchain-based Mobility-aware Offloading (BMO) mechanism is designed by Dou et al. [32], where user mobility prediction is implemented using the Individual-Mobility (IM) model [33]. The idea behind the offloading mechanism is to shift the computational workload to different available Fog Servers (FSs) in the geo-location predicted by the IM model. Further, blockchain technology is deployed to check the authenticity of the forthcoming Fog servers. Finally, accounting is being managed by ''Fogcoin'', similar to Bitcoin, which stores the entire transaction history between the online Fog server and mobile users.
Martin et al. [7] proposed a framework that supports the migration of containers while satisfying the QoS requirements of mobile users. The migration of containers is done in an autonomic manner, by adopting the Monitor-Analyze-Plan-Execute (MAPE) autonomic control loop. The MAPE control loops discuss various steps of migration, like ''(i) Monitor'': used to constantly monitor the environment context, such as the mobility of users that is subsequently used to determine the need to migrate an application module to some other Fog node called target node; ''(ii) Analyze'': applies forecasting techniques to predict the user possible location in the next time step. If the distance between the user and the device is not under certain acceptable limits, a migration decision is made. ''(iii) Plan'': a Genetic Algorithm (GA) is used to identify a suitable Fog node closest to the forecasted location, where migration of the container running user application can be done. ''(iv) Execute'': this step ensures the whole migration process should take place smoothly.
Mass et al. propose a mobility and delay-aware fog server selection scheme. [34] called Edge-Process management (EPM) system. The EPM system depends upon the trajectory of a user’s movement, Fog server workload, and user location to select the appropriate Fog server for executing user applications. The system selects or re-selects a Fog Server (FS) based on a score value calculated through available bandwidth, power, distance from the user, and finally, duration of availability in a region.
Mobi-IoST (Mobility-aware Internet of Spatial Things), a real-time mobility-aware framework is presented by Ghosh et al. [35]. The authors considered the mobile nature of both IoT devices and Fog nodes, collaboratively called mobile agents. The proposed mobility-aware framework collects a vast amount of Global Positioning System (GPS) data of these mobile agents to predict their movement patterns using various machine learning algorithms. The major components of the framework are, ''(i) Movement pattern model''ling, collecting and modelling GPS log, stay-point, and other contextual location information; ''(ii) Predicting the following location'': human movement semantics is analyzed using all modelled information; ''(iii) Delivery of result:'' after the user movement prediction in the previous phase, the system intelligently discovers a capable fog device for data processing in a timely manner.
A middleware solution, URMILA, for managing resources and scheduling tasks in the Fog environment is presented by Shekhar et al. [18]. Ubiquitous Resource Management for Interference and Latency-Aware services (URMILA), ensures minimum Service Level Objectives (SLO) violation for latency-sensitive mobile applications across the cloud-Fog environment. The major modules of the proposed system are ''(i) Route calculation'', which calculates the user's possible routes using Google Maps or GPS data; ''(ii) Latency calculation'', the system deploys a data-driven model to estimate the Latency on predicted user routes; ''(iii) Fog node selection:'' the system selects a fog server for execution of task on the basis of its instantaneous utilization of the available resources. Further, it selects the Fog server for the entire period of execution, during which mobile users can still access their application through various Wireless Access Points (WAP).
Gia et al. [8] proposed a Handover mechanism for mobility management between fog nodes with the overall objective of consuming minimum energy and delay during handovers. Handover methods frequently rely on one or more measures, such as the Received Signal Strength Indicator (RSSI), the Link Quality Indicator (LQI), and the velocity of objects, to make handover decisions. This proposed system provides emergency services to health monitoring systems and basically works in two different mobility scenarios: ''(i) Node mobility between indoor or outdoor locations'': nodes belonging to indoor location or outdoor location only are considered to be similar, and they're calculated metrics value like RSSI, LQI, velocity, etc.; can be directly used for the handover of services to appropriate gateway, ''(ii) Node mobility between indoor and outdoor locations'': nodes are considered dissimilar, if they belong to indoor and outdoor location both, So, the metrics are re-calculated which introduce some additional parameters like temperature and interference signals in order to make a decision over handover gateway.
Babu and Biswash [36] proposed a mobility management technique that supports node-to-node communication and Fog computing-based architecture for 5G networks. It addresses the technical problems between 5G networks and Fog servers. The mobility-based approach assists mobile nodes in establishing communication while they are in motion. The mobility management technique may also be used to begin N2N communication in dynamic environments. N2N communication schemes for fog networks, on the other hand, provide an effective communication environment for mobile users with highly minimal network usage.
Wang et al. [37] proposed a mobility-aware offloading scheme, that considers an adequate quality and a computation allocation system that deals with the user equipment affairs to maximize the total revenue. The quality of user equipment is delineated by the sojourn time that follows the exponential distribution to reduce the chance of migration and maximize the entire income of user equipment. MILP (mixed-integer non-linear programming) NP-hard problem is modelled and consists of resource allocation and task offloading schemes. So, to solve this problem, a Gini coefficient and genetic algorithm are used to estimate the allocation of resources. The proposed approach can easily handle the mobility of users by minimizing the chances of migration.
Waqas et al. [38] provided a forward-thinking analysis of quality about-mobility in Fog computing by identifying quality challenges, requirements, and options for numerous ideas. The authors also identified outstanding concerns from previous research and summarized the advantages of quality for readers. It allows researchers and developers to avoid common misunderstandings and capture real-world scenarios such as businesses, governments, and educational institutions. Furthermore, it revolutionizes follow-up analysis and differentiates and foregrounds futurity orientations in real-life events involving humans and vehicles in a highly dynamic Fog setting.
Bi et al. [9] introduced software-defined networking-based fog computing architecture by decoupling mobility control and data forwarding. When mobile consumers travel between several access networks, the authors suggested an Optimal Path Selection (OPS) algorithm to preserve service continuity. Mobile customers received seamless and transparent mobility support thanks to efficient signalling operations. In mobile fog computing, the suggested algorithm ensured service continuity, increased handover performance, and achieved high data transfer efficiency.
Niu et al. [21] established a system called mobility-aware and multihop-D2D relaying-based scheduling scheme (MHRC) at Edge nodes near hotspots. The authors exploited concurrent transmissions to improve the performance of the system. The mmWave (millimetre-wave) band of Fog computing was cached, and extensive performance evaluation confirms that MHRC delivers more than the higher expected cached data amount. Name et al. [19] proposed an efficient algorithm to address the problem of resource allocation and user mobility from the Edge of the network to cloud data centres. This algorithm operates on a seamless handover scheme for mobile IPV6 to ease the user mobility challenge and reduce the application response time. The study showed that the task of service delay and packet loss was decreased due to the effect of change in the mobile node position.
Bittencourt et al. [23] examined the subject of resource allocation in the Fog/Cloud environment, taking into account the hierarchical structure. In the context of the Fog paradigm, the authors developed three scheduling algorithms (First come, First serve, delay-priority, and concurrent) that address user mobility and edge computing capabilities. The authors demonstrated that scheduling techniques may be designed to cope with different application classes based on demand from mobile users by leveraging both Fog to the end-user and cloud characteristics in this study.
Velasquez et al. [39] proposed a hybrid strategy for the Fog environment to manage resources for mobility scenarios. The authors applied the orchestrator technique to offer mobility support in a Smart City situation. In this technique, three components, the status monitor, the Planner, and the VM/Container, are employed to monitor, plan and execute the applications. The main aim of this study was to guarantee the QoS and QoE requirements of mobility-based applications and services.
Bittencourt et al. [31] presented a Fog computing architecture focusing on Virtual Machine (VM) migration where each user has a VM running in a cloudlet. In this architecture, the user's location is identified by using GPS, and then the VM is moved to a nearby Fog Cloud. The main aim of this study was to migrate users' data according to their mobility in order to maintain QoS for applications demanding lower Latency and allow smooth handoff mechanisms for mobile users.
From the extensive analysis of the literature, the various mobility-aware scheduling techniques have been classified as shown in Table 3. Further, it presents the advantages and limitations of each technique.
{| class="wikitable"
|+Table 3: Classification of Mobility-aware scheduling techniques
!Ref.
!Technique
!Advantage(s)
!Limitation(s)
|-
|[25]
|Ranking of VM
|
* Decrease in delay time, migration time, tuple lost value and downtime
|
* Case study not discussed
|-
|[26]
|S-OAMC, G-OAMC, Machine learning matrix completion
|
* Migration rate decreased
* Better Scalability
|
* Energy utilization of devices not investigated
|-
|[32]
|IM model
|
* Provides better mobility support and security
|
* Did not investigate synchronization overhead
|-
|[10]
|RSS
|
* Reliable and Heterogeneous execution
|
* Low scalability
* No distributed scheduling to minimize response time
|-
|[7]
|MAPE control loop
|
* Improved QoS
* Reduced service downtime
|
* No real-time evaluation
* High energy consumption
* Low robustness and security
|-
|[28]
|Copy of task to over a region
|
* Network Utilization developed
* Low-Loop delay
|
* Fault tolerance reliability is based on Fog gateways only
|-
|[18]
|URMILA
|
* Service availability is maintained by delivering the desired QoS
* Deployment cost minimized
* Battery longevity ensured
|
* No empirical validations
* No user probabilistic routes
* Low scalability in terms of distance and speed
|-
|[8]
|RSSI, LQI
|
* Promises to keep the connection active with a low latency rate between the system and sensor nodes
|
* Consumes more energy
* Overhead is large for network transmission
* Coverage and overhead area are undefined between gateways
|-
|[29]
|User trajectories pre- diction using GPS log
|
* Provides better mobility support
* Reduces migration time
|
* Low scalability
|-
|[36]
|N2N communication, Data Analytics
|
* Fast data access
* High reliability and scalability- city
* Minimum overhead and cost
* High throughput and less delay
|
* No real-time cellular network evaluation
* Low network efficiency
|-
|[38]
|Mobility facets analysis
|
* Improved QoS and QoE
* Latency rate reduced
|
* No real-life implementation
* No reliability and low Latency between dynamic users and fog servers
|-
|[37]
|M-ILP, Sojourn time
|
* Cost-effective
* Migration time reduced
|
* Migration cost not considered
* No real-time implementation
|-
|[34]
|User trajectories prediction using GPS log
|
* Conventional delay tolerance
* High QoS
* Avoided local task processing cost
* Efficient in saving battery
* Handles subtle scenarios with high Latency
|
* Smart city not directed through the use of accurate city maps with aid from stimulation setting
|-
|[35]
|Prediction of user location
|
* Power consumption and de- lay handled proficiently
* Power and delay are reduced
|
* No acquiring of mobile data usage where location sense and time-series data can be projected to achieve the bandwidth
|-
|[21]
|Relay path planning algorithm
|
* Power efficient
* High spectral efficiency
* Data is relayed on cached edge nodes and relay nodes
|
* Blockage problem due to weak diffraction
|-
|[9]
|OPS
|
* Handover performance improved
* Efficiency of high data communication achieved
* Guarantees continuity of services
|
* It does not guarantee privacy and security
* Virtual Machine migration not determined
* The handover process during the optimal path for more logical routing could have been more efficient
|-
|[23]
|Assignment of FS
|
* Low Latency
* Supports dynamic computing and user behaviour
|
* There is no prediction of mobility failure
* Bandwidth and processing not considered in scheduling
|-
|[39]
|Orchestrator
|
* Maintains trustworthiness, resilience, and low Latency in a dynamic environment
|
* No real implementation has been carried out
|-
|[19]
|Pattern modelling, dictating the following location
|
* Application Response time reduced
* Latency time reduced
|
* Services become temporarily inaccessible for some mobile nodes
|-
|[31]
|Forecasting technique
|
* Computing capacity provided for storage and processing of data
|
* Security concerns associated with both user data and applications not considered
|}
[[File:Fog Computing Figure 2.jpg|center|thumb|Figure 2: Year-wise count of research articles]]
== Analytical Discussion ==
The existing research on Fog computing Mobility-aware scheduling has been analyzed thoroughly. The analysis was performed using the answers given in Table 2. The results drawn through the thorough analysis of the literature are presented in various figures as follows:
===Subheading===
==Second Heading==
==Third Heading, etc==
==Additional information==
===Acknowledgements===
Any people, organisations, or funding sources that you would like to thank.
===Competing interests===
Any conflicts of interest that you would like to declare. Otherwise, a statement that the authors have no competing interest.
===Ethics statement===
An ethics statement, if appropriate, on any animal or human research performed should be included here or in the methods section.
==References==
{{reflist|35em}}
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{{Article info
| journal = WikiJournal of Science
| last1 = Islam
| orcid1 = 0000-0001-5750-1058
| first1 = Mir Salim Ul
| last2 = Kumar
| orcid2 = 0000-0003-3279-5111
| first2 = Ashok
| affiliation2 = Department of Computer Application, Chandigarh School of Business, Chandigarh Group of Colleges, Jhanjeri, Punjab, India
| et_al =
| affiliation1 = Department of Computer Science and Engineering, National Institute of Technology Srinagar, Jammu and Kashmir, India
| correspondence1 = mir.salim27@gmail.com
| affiliations =
| keywords = Fog computing, Mobility-aware, Scheduling, Resource Management, Resource Allocation
| license = CC BY 4.0
| abstract = The increasing focus on Fog-IoT results in billions of Internet-connected devices that demand substantial computational power and network bandwidth. These devices are geographically distributed, heterogeneous, computational capacity constrained, inconsistent in behaviour, and generally mobile. Therefore, providing seamless service, irrespective of location and movement of the devices as well as the end-users, makes resource scheduling a significant challenge in the Fog computing paradigm. Several mobility-aware scheduling strategies have been proposed in the literature to efficiently manage the resources for mobile users and devices in the Fog environment. This paper gives a survey of mobility-aware scheduling in the Fog computing environment. It describes the many strategies presented and their benefits and drawbacks. It also includes a complete study and taxonomy of the mobility-aware scheduling field. Further, it delineates open issues and challenges. This work will provide researchers with future research directions and aid them in recognizing the gaps before planning for further research in mobility-aware scheduling.
}}
== Introduction ==
With the advancement in technology and the exponential growth of mobile devices, network traffic has increased manifold in cloud computing. Due to this reason, Latency reduction and faster processing of data for mobile users have become critical challenges in providing seamless connectivity and minimal disruption while the user is moving.<ref>{{Cite journal|last=Abdullah|first=Fatima|last2=Kimovski|first2=Dragi|last3=Prodan|first3=Radu|last4=Munir|first4=Kashif|date=2021-06-11|title=Handover authentication latency reduction using mobile edge computing and mobility patterns|url=https://doi.org/10.1007/s00607-021-00969-z|journal=Computing|volume=103|issue=11|pages=2667–2686|doi=10.1007/s00607-021-00969-z|issn=0010-485X}}</ref> The data movement brings the additional issue of integrity and confidentiality because data is moving via a wireless connection to a far distant cloud. Additionally, due to the cloud's location is far from mobile users, so data movement is also affected by variable network strength and phone bandwidth. The solution proposed by [2] is to extend cloud capabilities through fog computing architecture. The Fog architecture allows substantial computation, storage, and processing using the Fog devices installed close to the user’s access point. Fog computing, therefore, reduces Latency and bandwidth consumption, improves security, provides context awareness, and renders more efficient services to mobile users [3, 4, 5].
[[File:Fog Computing Figure 1.jpg|left|thumb|Figure 1: Mobility Scenario in Fog Computing]]
However, mobility also imposes severe challenges for Fog computing due to its distributed and diverse environment. Mobility is recognized by either user-level or device-level contextual information [6]. As the user moves from one location to another, the geographical location of the smart devices also changes. The change in location of the devices raises the issue of searching and rescheduling with mobility management. Efficient re-scheduling requires a well-planned handoff mechanism that is accountable for smoothly de-registering a sensor node from a source access point where the application was initially hosted and registering it to a new access point. Figure 1 depicts these mentioned problems of change in access points while the user moves from one area to another. The services may also get interrupted when there is more distance between the Fog nodes and users [7]. Further, any disruption in communication may lead to an increase in Latency for mobile users [8].
The significant challenges in mobility management of Fog computing are: (i) ''Fog node discovery''- due to the heterogeneous and mobile nature of the smart mobile devices, the Fog task scheduler faces the issue of finding an optimal Fog node for scheduling the task, ''(ii) Handover mechanism between users and Fog nodes''- imagine a Fog user is moving from location to location and accessing information about his surroundings with the help of a smart device. Due to the frequent change in the user's location, the Fog scheduler may need to repeatedly migrate the user’s task to a different Fog node available in his vicinity. The frequent migration of tasks increases the overhead of scheduling and, further, the restricted signal strength in certain places may lead to a breakdown of task or result delivery, and (iii) ''the Handover mechanism between Cloud and Fog''- Fog nodes have limited capacities and need to continuously communicate with Cloud computing for passing information about the tasks [9]. Due to strict requirements for security, Latency, network coverage, and reliability, It becomes challenging to implement an efficient handover mechanism for full mobility support in critical domains such as healthcare [10, 11, 12] and vehicular systems [13, 14]. Thus, mobility significantly impacts the overhead of scheduling policies and the applications' performance, eventually affecting the Quality of Experience (QoE). Therefore, mobility-aware scheduling in Fog computing has observed strong attention from researchers. Our main goal is to provide a detailed review of mobility-aware scheduling in fog computing. The review provides a detailed analysis of existing scheduling strategies that concentrate specifically on mobility awareness in the Fog environment. The following are the main contributions of this paper:
* This paper presents a detailed survey of mobility-aware scheduling in the Fog computing environment.
* It provides the details of the different techniques proposed, as well as their advantages and limitations.
* It provides a detailed analysis and taxonomy of the mobility-aware scheduling field.
* It identifies several open challenges for future research directions.
=== Related Surveys ===
Numerous survey studies on Fog computing focus on resource management, job scheduling, and context-aware scheduling. For example, Ghobaei-Arani et al. [15] presented a survey on resource management techniques in fog computing in the form of taxonomy to highlight cutting-edge methods while also addressing unresolved challenges. The various authors in the cloud-fog area provided the task scheduling review. Their benefits and drawbacks, as well as numerous tools and challenges concerning the scheduling techniques and their limitations, were analyzed by Alizadeh et al. [16]. Islam et al. [6] thoroughly analyze relevant literature on context-aware scheduling in fog computing. Further, Mouradian et al. [17] review fog computing's significant issues and challenges. But, the critical area of mobility research is still in its initial stage, and most of the review papers contain very few documents on mobility-aware task scheduling in the area of fog computing. Therefore, an extensive and comparative study is required in mobility-constrained fog computing. A deep insight into various techniques which can impact the user QoS is necessary to understand mobility-aware fog computing. This motivates us to carry out a comprehensive survey; to the best of our knowledge, this is the first detailed survey. This survey paper thoroughly examines existing scheduling solutions that focus on user mobility. Moreover, various mobility-aware scheduling techniques are discussed, along with their pros and cons. Further, the impact of mobility parameters on various QoS parameters and context-awareness is also analyzed thoroughly.
=== Paper Organization ===
The remainder of this review article is divided into 5 sections: Section 2 discusses mobility-aware scheduling in Fog computing. Section 3 presents the review methodology. Section 4 analyzes and summarizes the considered research papers and compares existing mobility-aware scheduling strategies. Section 5 provides the results drawn after critically examining the existing literature on mobility-aware scheduling policies. Finally, Section 6 presents the conclusion.
== Mobility-aware Scheduling in Fog Computing ==
Mobile device management compromises the fundamental features of Fog computing because whenever a user moves, the distance between them increases, impacting the QoS. Therefore, to keep the computing fog node close to the associated mobile device, the services or tasks need to be migrated from one fog node to another appropriate fog device. The selection of such appropriate fog nodes in a mobile environment deals with two main processes: Estimation of user mobility patterns: User mobility estimation techniques can be probabilistic and deterministic [18]. In a deterministic method, the source and destination are known beforehand, whereas in a non-deterministic technique, periodic estimation has to be done regarding the user's route. Many authors estimate the route of mobile users by leveraging external services like Open Street Maps (<nowiki>http://www.openstreetmap.org</nowiki>), Google Maps APIs (<nowiki>https://cloud.google.com/</nowiki> maps-platform/) [18], logistics maps [19], GPRS Here APIs (<nowiki>https://developer</nowiki>. here.com/) [20, 21], Lyapunov estimation technique [22]. Specific QoS requirements: The selection of a fog node also depends upon specific quality requirements such as Latency, which many authors are using to select an efficient fog node [18, 10, 9], workload [23, 24], cost [16], etc.
== Review Methodology ==
The review process was conducted considering a review methodology consisting of four phases. The first phase was searching using traditional online database sources based on outlined search keywords. Table 1 lists the
Keywords used to find the relevant research articles to conduct the review in the Fog computing mobility-aware scheduling area. Second Phase: Limit the search of research articles beginning in 2015, and inclusion and exclusion principles are also used to refine research articles that specifically deal with mobility issues in task scheduling. Finally, in the Third Phase, A total of 20 papers are shortlisted for the review process. Further, Table 2 presents the research questions drafted for this study in mobility-aware scheduling in Fog computing.
{| class="wikitable"
|+Table 1: List of keywords used in the review process
!Sno
!Keyword
!Description
!Years
|-
|1
|Mobility
|Mobility-aware Fog task scheduling
| rowspan="5" |2015- 2021
|-
|2
|Mobility environment
|Mobility environment in Fog task scheduling
|-
|3
|Mobility factors
|Mobility factors in Fog task scheduling
|-
|4
|Mobility Awareness
|Mobility awareness in Fog task scheduling
|-
|5
|Mobility management
|Mobility management in Fog task scheduling
|}
{| class="wikitable"
|+Table 2: List of research questions used to complete the review process
!Q.No
!Research questions
|-
|1
|What scheduling approaches are used in the Fog computing environment to manage mobility awareness?
|-
|2
|What are the main limitations considered for mobility-aware scheduling techniques?
|-
|3
|Which case studies are applied to mobility-aware scheduling techniques?
|-
|4
|What evaluation tools are used to assess mobility-aware scheduling techniques?
|-
|5
|What performance indicators are utilized to evaluate mobility-aware scheduling techniques?
|-
|6
|What are the major open issues concerns in the field of mobility-aware scheduling for future research directions?
|}
=== Source of Information ===
To conduct this review, various online sources are listed below, and the research articles were searched using the different keywords mentioned in Table 1.
* Google Scholar (<nowiki>http://www.scholar.google.co.in</nowiki>)
* John Wiley & Sons Inc. (<nowiki>https://onlinelibrary.wiley.com/</nowiki>)
* Elsevier (<nowiki>https://www.elsevier.com/en-in</nowiki>)
* ACM Digital Library (<nowiki>https://www.acm.org/</nowiki>)
* Springer (<nowiki>https://www.springer.com/in</nowiki>)
* IEEE Xplore Digital Library (<nowiki>https://www.ieee.org/</nowiki>)
=== Quality Assessment ===
Research papers used quality assessment to filter out the most suitable mobility-based scheduling research articles in fog computing utilising the principle of inclusion and exclusion. Furthermore, in order to obtain high-quality research publications, the Center for Reviews and Dissemination (CRD) recommendations were followed, and each study item was examined for internal and external validation of results.
== Literature Analysis ==
Verma et al. [25] proposed a Server Cloudlet(SC) migration technique to handle users' mobility. The strategy is to select the target SC to migrate the services based on its highest rank. The rank of the SC further depends upon its available RAM, MIPS, and bandwidth.
Maleki et al. [26] designed two mobility-aware computation offloading approaches, sampling-based Online mobile applications to cloudlets(S-OAMC) and Greedy online mobile applications to cloudlets(G-OAMC). The developed system works in three major steps: ''(i)'' the future specifications of mobile applications are predicted using the Machine Learning (ML) method, named matrix completion. ''(ii)'' The predicted specifications of mobile applications, including future location prediction, also help estimate the offloading cost. ''(iii)'' Apply S-OAMC or G-OAMC offloading technique based on minimum cost value.
Abdelmoneem et al. [10] work on a fog-based healthcare architecture system that supports patient movement without altering their Quality of Service (QoS). The QoS in a mobile environment is maintained with the help of their proposed handoff mechanism. The authors modified the traditional Horizontal Handoff (HHO) mechanism [27] into two different phases: ''(i) handoff decision policy'': Received signal Strength (RSS) method is used to detect the network change information of the mobile patients and (ii) ''handoff strategy'': a handoff decision is made based on a comparison between RSS received from the old fog gateways, current threshold value and new fog gateways that are in close proximity to the mobile patient.
Javanmardi et al. [28] consider an imaginative city-based mobility scenario where the user is placing delay-sensitive service while moving. The authors proposed a task scheduling technique that jointly employs fuzzy logic and particle swarm optimization (PSO) algorithm to improve the QoS for mobile users within the city. The proposed algorithm is deployed at the Fog gateway, which further distributes the tasks to Fog devices available in an entire region in order to provide seamless service to mobile users. The main motive of the work is to improve resource utilization in a mobility-aware environment.
Puliafito et al. [29] develop an extension of ifogsim [30], which supports mobile environment. The authors implement different migration techniques inspired by Bittencourt et al. [31]. Different phases are devised to perform migration in a fog mobile environment, which: ''(i) Before migration phase'': migration decision is taken in this phase, based on specific parameters, like user location, speed, the direction of movement, zone, and migration point in order to select the appropriate cloudlet for offloading the services, ''(ii) During the migration phase'': this phase manages, monitors and synchronizes the whole selected migration process and ''(iii) After migration phase'': this phase involves closing the older cloudlet connections with the user and using the new cloudlet for services.
A Blockchain-based Mobility-aware Offloading (BMO) mechanism is designed by Dou et al. [32], where user mobility prediction is implemented using the Individual-Mobility (IM) model [33]. The idea behind the offloading mechanism is to shift the computational workload to different available Fog Servers (FSs) in the geo-location predicted by the IM model. Further, blockchain technology is deployed to check the authenticity of the forthcoming Fog servers. Finally, accounting is being managed by ''Fogcoin'', similar to Bitcoin, which stores the entire transaction history between the online Fog server and mobile users.
Martin et al. [7] proposed a framework that supports the migration of containers while satisfying the QoS requirements of mobile users. The migration of containers is done in an autonomic manner, by adopting the Monitor-Analyze-Plan-Execute (MAPE) autonomic control loop. The MAPE control loops discuss various steps of migration, like ''(i) Monitor'': used to constantly monitor the environment context, such as the mobility of users that is subsequently used to determine the need to migrate an application module to some other Fog node called target node; ''(ii) Analyze'': applies forecasting techniques to predict the user possible location in the next time step. If the distance between the user and the device is not under certain acceptable limits, a migration decision is made. ''(iii) Plan'': a Genetic Algorithm (GA) is used to identify a suitable Fog node closest to the forecasted location, where migration of the container running user application can be done. ''(iv) Execute'': this step ensures the whole migration process should take place smoothly.
Mass et al. propose a mobility and delay-aware fog server selection scheme. [34] called Edge-Process management (EPM) system. The EPM system depends upon the trajectory of a user’s movement, Fog server workload, and user location to select the appropriate Fog server for executing user applications. The system selects or re-selects a Fog Server (FS) based on a score value calculated through available bandwidth, power, distance from the user, and finally, duration of availability in a region.
Mobi-IoST (Mobility-aware Internet of Spatial Things), a real-time mobility-aware framework is presented by Ghosh et al. [35]. The authors considered the mobile nature of both IoT devices and Fog nodes, collaboratively called mobile agents. The proposed mobility-aware framework collects a vast amount of Global Positioning System (GPS) data of these mobile agents to predict their movement patterns using various machine learning algorithms. The major components of the framework are, ''(i) Movement pattern model''ling, collecting and modelling GPS log, stay-point, and other contextual location information; ''(ii) Predicting the following location'': human movement semantics is analyzed using all modelled information; ''(iii) Delivery of result:'' after the user movement prediction in the previous phase, the system intelligently discovers a capable fog device for data processing in a timely manner.
A middleware solution, URMILA, for managing resources and scheduling tasks in the Fog environment is presented by Shekhar et al. [18]. Ubiquitous Resource Management for Interference and Latency-Aware services (URMILA), ensures minimum Service Level Objectives (SLO) violation for latency-sensitive mobile applications across the cloud-Fog environment. The major modules of the proposed system are ''(i) Route calculation'', which calculates the user's possible routes using Google Maps or GPS data; ''(ii) Latency calculation'', the system deploys a data-driven model to estimate the Latency on predicted user routes; ''(iii) Fog node selection:'' the system selects a fog server for execution of task on the basis of its instantaneous utilization of the available resources. Further, it selects the Fog server for the entire period of execution, during which mobile users can still access their application through various Wireless Access Points (WAP).
Gia et al. [8] proposed a Handover mechanism for mobility management between fog nodes with the overall objective of consuming minimum energy and delay during handovers. Handover methods frequently rely on one or more measures, such as the Received Signal Strength Indicator (RSSI), the Link Quality Indicator (LQI), and the velocity of objects, to make handover decisions. This proposed system provides emergency services to health monitoring systems and basically works in two different mobility scenarios: ''(i) Node mobility between indoor or outdoor locations'': nodes belonging to indoor location or outdoor location only are considered to be similar, and they're calculated metrics value like RSSI, LQI, velocity, etc.; can be directly used for the handover of services to appropriate gateway, ''(ii) Node mobility between indoor and outdoor locations'': nodes are considered dissimilar, if they belong to indoor and outdoor location both, So, the metrics are re-calculated which introduce some additional parameters like temperature and interference signals in order to make a decision over handover gateway.
Babu and Biswash [36] proposed a mobility management technique that supports node-to-node communication and Fog computing-based architecture for 5G networks. It addresses the technical problems between 5G networks and Fog servers. The mobility-based approach assists mobile nodes in establishing communication while they are in motion. The mobility management technique may also be used to begin N2N communication in dynamic environments. N2N communication schemes for fog networks, on the other hand, provide an effective communication environment for mobile users with highly minimal network usage.
Wang et al. [37] proposed a mobility-aware offloading scheme, that considers an adequate quality and a computation allocation system that deals with the user equipment affairs to maximize the total revenue. The quality of user equipment is delineated by the sojourn time that follows the exponential distribution to reduce the chance of migration and maximize the entire income of user equipment. MILP (mixed-integer non-linear programming) NP-hard problem is modelled and consists of resource allocation and task offloading schemes. So, to solve this problem, a Gini coefficient and genetic algorithm are used to estimate the allocation of resources. The proposed approach can easily handle the mobility of users by minimizing the chances of migration.
Waqas et al. [38] provided a forward-thinking analysis of quality about-mobility in Fog computing by identifying quality challenges, requirements, and options for numerous ideas. The authors also identified outstanding concerns from previous research and summarized the advantages of quality for readers. It allows researchers and developers to avoid common misunderstandings and capture real-world scenarios such as businesses, governments, and educational institutions. Furthermore, it revolutionizes follow-up analysis and differentiates and foregrounds futurity orientations in real-life events involving humans and vehicles in a highly dynamic Fog setting.
Bi et al. [9] introduced software-defined networking-based fog computing architecture by decoupling mobility control and data forwarding. When mobile consumers travel between several access networks, the authors suggested an Optimal Path Selection (OPS) algorithm to preserve service continuity. Mobile customers received seamless and transparent mobility support thanks to efficient signalling operations. In mobile fog computing, the suggested algorithm ensured service continuity, increased handover performance, and achieved high data transfer efficiency.
Niu et al. [21] established a system called mobility-aware and multihop-D2D relaying-based scheduling scheme (MHRC) at Edge nodes near hotspots. The authors exploited concurrent transmissions to improve the performance of the system. The mmWave (millimetre-wave) band of Fog computing was cached, and extensive performance evaluation confirms that MHRC delivers more than the higher expected cached data amount. Name et al. [19] proposed an efficient algorithm to address the problem of resource allocation and user mobility from the Edge of the network to cloud data centres. This algorithm operates on a seamless handover scheme for mobile IPV6 to ease the user mobility challenge and reduce the application response time. The study showed that the task of service delay and packet loss was decreased due to the effect of change in the mobile node position.
Bittencourt et al. [23] examined the subject of resource allocation in the Fog/Cloud environment, taking into account the hierarchical structure. In the context of the Fog paradigm, the authors developed three scheduling algorithms (First come, First serve, delay-priority, and concurrent) that address user mobility and edge computing capabilities. The authors demonstrated that scheduling techniques may be designed to cope with different application classes based on demand from mobile users by leveraging both Fog to the end-user and cloud characteristics in this study.
Velasquez et al. [39] proposed a hybrid strategy for the Fog environment to manage resources for mobility scenarios. The authors applied the orchestrator technique to offer mobility support in a Smart City situation. In this technique, three components, the status monitor, the Planner, and the VM/Container, are employed to monitor, plan and execute the applications. The main aim of this study was to guarantee the QoS and QoE requirements of mobility-based applications and services.
Bittencourt et al. [31] presented a Fog computing architecture focusing on Virtual Machine (VM) migration where each user has a VM running in a cloudlet. In this architecture, the user's location is identified by using GPS, and then the VM is moved to a nearby Fog Cloud. The main aim of this study was to migrate users' data according to their mobility in order to maintain QoS for applications demanding lower Latency and allow smooth handoff mechanisms for mobile users.
From the extensive analysis of the literature, the various mobility-aware scheduling techniques have been classified as shown in Table 3. Further, it presents the advantages and limitations of each technique.
{| class="wikitable"
|+Table 3: Classification of Mobility-aware scheduling techniques
!Ref.
!Technique
!Advantage(s)
!Limitation(s)
|-
|[25]
|Ranking of VM
|
* Decrease in delay time, migration time, tuple lost value and downtime
|
* Case study not discussed
|-
|[26]
|S-OAMC, G-OAMC, Machine learning matrix completion
|
* Migration rate decreased
* Better Scalability
|
* Energy utilization of devices not investigated
|-
|[32]
|IM model
|
* Provides better mobility support and security
|
* Did not investigate synchronization overhead
|-
|[10]
|RSS
|
* Reliable and Heterogeneous execution
|
* Low scalability
* No distributed scheduling to minimize response time
|-
|[7]
|MAPE control loop
|
* Improved QoS
* Reduced service downtime
|
* No real-time evaluation
* High energy consumption
* Low robustness and security
|-
|[28]
|Copy of task to over a region
|
* Network Utilization developed
* Low-Loop delay
|
* Fault tolerance reliability is based on Fog gateways only
|-
|[18]
|URMILA
|
* Service availability is maintained by delivering the desired QoS
* Deployment cost minimized
* Battery longevity ensured
|
* No empirical validations
* No user probabilistic routes
* Low scalability in terms of distance and speed
|-
|[8]
|RSSI, LQI
|
* Promises to keep the connection active with a low latency rate between the system and sensor nodes
|
* Consumes more energy
* Overhead is large for network transmission
* Coverage and overhead area are undefined between gateways
|-
|[29]
|User trajectories pre- diction using GPS log
|
* Provides better mobility support
* Reduces migration time
|
* Low scalability
|-
|[36]
|N2N communication, Data Analytics
|
* Fast data access
* High reliability and scalability- city
* Minimum overhead and cost
* High throughput and less delay
|
* No real-time cellular network evaluation
* Low network efficiency
|-
|[38]
|Mobility facets analysis
|
* Improved QoS and QoE
* Latency rate reduced
|
* No real-life implementation
* No reliability and low Latency between dynamic users and fog servers
|-
|[37]
|M-ILP, Sojourn time
|
* Cost-effective
* Migration time reduced
|
* Migration cost not considered
* No real-time implementation
|-
|[34]
|User trajectories prediction using GPS log
|
* Conventional delay tolerance
* High QoS
* Avoided local task processing cost
* Efficient in saving battery
* Handles subtle scenarios with high Latency
|
* Smart city not directed through the use of accurate city maps with aid from stimulation setting
|-
|[35]
|Prediction of user location
|
* Power consumption and de- lay handled proficiently
* Power and delay are reduced
|
* No acquiring of mobile data usage where location sense and time-series data can be projected to achieve the bandwidth
|-
|[21]
|Relay path planning algorithm
|
* Power efficient
* High spectral efficiency
* Data is relayed on cached edge nodes and relay nodes
|
* Blockage problem due to weak diffraction
|-
|[9]
|OPS
|
* Handover performance improved
* Efficiency of high data communication achieved
* Guarantees continuity of services
|
* It does not guarantee privacy and security
* Virtual Machine migration not determined
* The handover process during the optimal path for more logical routing could have been more efficient
|-
|[23]
|Assignment of FS
|
* Low Latency
* Supports dynamic computing and user behaviour
|
* There is no prediction of mobility failure
* Bandwidth and processing not considered in scheduling
|-
|[39]
|Orchestrator
|
* Maintains trustworthiness, resilience, and low Latency in a dynamic environment
|
* No real implementation has been carried out
|-
|[19]
|Pattern modelling, dictating the following location
|
* Application Response time reduced
* Latency time reduced
|
* Services become temporarily inaccessible for some mobile nodes
|-
|[31]
|Forecasting technique
|
* Computing capacity provided for storage and processing of data
|
* Security concerns associated with both user data and applications not considered
|}
[[File:Fog Computing Figure 2.jpg|center|thumb|Figure 2: Year-wise count of research articles]]
== Analytical Discussion ==
The existing research on Fog computing Mobility-aware scheduling has been analyzed thoroughly. The analysis was performed using the answers given in Table 2. The results drawn through the thorough analysis of the literature are presented in various figures as follows:
Figure 2 lists the year-wise count of research papers that are considered for this survey. The bar graph represents the total number of research papers from journals and Conferences from the year 2015 - 2021. The research articles from the journal are 16, and the conference papers are 4. It is observed that more research needs to be conducted on mobility-aware scheduling in Fog computing.
Figure 3 displays an analytical comparison of mobility-aware scheduling approaches in Fog computing based on the content of the represented taxonomy in Figure 7. From the thorough analysis of the literature, four methods have been considered: migration, task offloading, handoff/handover mechanism, and task scheduling. The handoff/handover mechanism has the highest percentage of usage in mobility-aware scheduling, at 30%. The task scheduling and offloading have 25% of us- age in mobility-aware scheduling each. Finally, migration is only 20% of the usage in mobility-aware scheduling. Therefore, these approaches, specifically migration, are still open challenges to address for further research.
Figure 4 depicts various tools that were used for evaluating the mobility-aware scheduling approaches. 18% and 9% of the research articles used iFogSim and Mob-FogSim simulation tools for implementation, respectively. Besides, other simulation tools such as ONE (9%), NS2(5%), MATLAB(4%), Mininet(5%), and Docker(9%) have been utilized for implementing the proposed techniques in the research articles. Further, pro-Programming languages such as C++ (9%) and Python (9%) and hardware deployments such as Raspberry Pi (5%) and Ardunio (4%) were used for implementing existing case studies based on mobility-aware scheduling.
[[File:Fog Computing Figure 3.jpg|center|thumb|Figure 3: Percentage of the presented classified approaches in mobility-aware scheduling]]
The applied case studies are shown in Figure 5, which shows a maximum of 20% of research articles have implemented IoT-based applications. After that, 15% of each research article used Health care and Mobile-based applications. Besides, Smart City and 5G-based applications have been applied in 10% of research articles. Moreover, Surveillance and gaming, Mobile IPV6, and Wireless computing applications are the case studies on which only 5% of research articles exist.
After reviewing numerous research articles based on mobility-aware scheduling, it has been observed that researchers employed various parameters for evaluating the performance of the Mobility-scheduling approaches, as represented in Figure 6. It shows that Time completion (18%) followed by Delay (12%), Network usage (12%), Latency (12%), Energy consumption (10%), and cost (10%) are generally utilized. However, Downtime (4%), Migration time (4%), Makespan (2%), Success ratio (2%), Signal level (2%), Deadline (2%), Makespan (2%), Migration rate (2%), Mobility patterns (2%), Tuple lost (2%), and power consumption (2%) are less exploited parameters.
A taxonomy was compiled after going through the detailed review process, and various techniques have been categorized in Fog computing-based mobility-aware scheduling. Figure 7 presents these categories broadly in Migration, Offloading, Handoff/Handover mechanism, and Scheduling.
[[File:Fog Computing Figure 4.jpg|center|thumb|Figure 4: Percentage of tools utilized in the literature]]
===Subheading===
==Second Heading==
==Third Heading, etc==
==Additional information==
===Acknowledgements===
Any people, organisations, or funding sources that you would like to thank.
===Competing interests===
Any conflicts of interest that you would like to declare. Otherwise, a statement that the authors have no competing interest.
===Ethics statement===
An ethics statement, if appropriate, on any animal or human research performed should be included here or in the methods section.
==References==
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{{Article info
| journal = WikiJournal of Science
| last1 = Islam
| orcid1 = 0000-0001-5750-1058
| first1 = Mir Salim Ul
| last2 = Kumar
| orcid2 = 0000-0003-3279-5111
| first2 = Ashok
| affiliation2 = Department of Computer Application, Chandigarh School of Business, Chandigarh Group of Colleges, Jhanjeri, Punjab, India
| et_al =
| affiliation1 = Department of Computer Science and Engineering, National Institute of Technology Srinagar, Jammu and Kashmir, India
| correspondence1 = mir.salim27@gmail.com
| affiliations =
| keywords = Fog computing, Mobility-aware, Scheduling, Resource Management, Resource Allocation
| license = CC BY 4.0
| abstract = The increasing focus on Fog-IoT results in billions of Internet-connected devices that demand substantial computational power and network bandwidth. These devices are geographically distributed, heterogeneous, computational capacity constrained, inconsistent in behaviour, and generally mobile. Therefore, providing seamless service, irrespective of location and movement of the devices as well as the end-users, makes resource scheduling a significant challenge in the Fog computing paradigm. Several mobility-aware scheduling strategies have been proposed in the literature to efficiently manage the resources for mobile users and devices in the Fog environment. This paper gives a survey of mobility-aware scheduling in the Fog computing environment. It describes the many strategies presented and their benefits and drawbacks. It also includes a complete study and taxonomy of the mobility-aware scheduling field. Further, it delineates open issues and challenges. This work will provide researchers with future research directions and aid them in recognizing the gaps before planning for further research in mobility-aware scheduling.
}}
== Introduction ==
With the advancement in technology and the exponential growth of mobile devices, network traffic has increased manifold in cloud computing. Due to this reason, Latency reduction and faster processing of data for mobile users have become critical challenges in providing seamless connectivity and minimal disruption while the user is moving.<ref>{{Cite journal|last=Abdullah|first=Fatima|last2=Kimovski|first2=Dragi|last3=Prodan|first3=Radu|last4=Munir|first4=Kashif|date=2021-06-11|title=Handover authentication latency reduction using mobile edge computing and mobility patterns|url=https://doi.org/10.1007/s00607-021-00969-z|journal=Computing|volume=103|issue=11|pages=2667–2686|doi=10.1007/s00607-021-00969-z|issn=0010-485X}}</ref> The data movement brings the additional issue of integrity and confidentiality because data is moving via a wireless connection to a far distant cloud. Additionally, due to the cloud's location is far from mobile users, so data movement is also affected by variable network strength and phone bandwidth. The solution proposed by [2] is to extend cloud capabilities through fog computing architecture. The Fog architecture allows substantial computation, storage, and processing using the Fog devices installed close to the user’s access point. Fog computing, therefore, reduces Latency and bandwidth consumption, improves security, provides context awareness, and renders more efficient services to mobile users [3, 4, 5].
[[File:Fog Computing Figure 1.jpg|left|thumb|Figure 1: Mobility Scenario in Fog Computing]]
However, mobility also imposes severe challenges for Fog computing due to its distributed and diverse environment. Mobility is recognized by either user-level or device-level contextual information [6]. As the user moves from one location to another, the geographical location of the smart devices also changes. The change in location of the devices raises the issue of searching and rescheduling with mobility management. Efficient re-scheduling requires a well-planned handoff mechanism that is accountable for smoothly de-registering a sensor node from a source access point where the application was initially hosted and registering it to a new access point. Figure 1 depicts these mentioned problems of change in access points while the user moves from one area to another. The services may also get interrupted when there is more distance between the Fog nodes and users [7]. Further, any disruption in communication may lead to an increase in Latency for mobile users [8].
The significant challenges in mobility management of Fog computing are: (i) ''Fog node discovery''- due to the heterogeneous and mobile nature of the smart mobile devices, the Fog task scheduler faces the issue of finding an optimal Fog node for scheduling the task, ''(ii) Handover mechanism between users and Fog nodes''- imagine a Fog user is moving from location to location and accessing information about his surroundings with the help of a smart device. Due to the frequent change in the user's location, the Fog scheduler may need to repeatedly migrate the user’s task to a different Fog node available in his vicinity. The frequent migration of tasks increases the overhead of scheduling and, further, the restricted signal strength in certain places may lead to a breakdown of task or result delivery, and (iii) ''the Handover mechanism between Cloud and Fog''- Fog nodes have limited capacities and need to continuously communicate with Cloud computing for passing information about the tasks [9]. Due to strict requirements for security, Latency, network coverage, and reliability, It becomes challenging to implement an efficient handover mechanism for full mobility support in critical domains such as healthcare [10, 11, 12] and vehicular systems [13, 14]. Thus, mobility significantly impacts the overhead of scheduling policies and the applications' performance, eventually affecting the Quality of Experience (QoE). Therefore, mobility-aware scheduling in Fog computing has observed strong attention from researchers. Our main goal is to provide a detailed review of mobility-aware scheduling in fog computing. The review provides a detailed analysis of existing scheduling strategies that concentrate specifically on mobility awareness in the Fog environment. The following are the main contributions of this paper:
* This paper presents a detailed survey of mobility-aware scheduling in the Fog computing environment.
* It provides the details of the different techniques proposed, as well as their advantages and limitations.
* It provides a detailed analysis and taxonomy of the mobility-aware scheduling field.
* It identifies several open challenges for future research directions.
=== Related Surveys ===
Numerous survey studies on Fog computing focus on resource management, job scheduling, and context-aware scheduling. For example, Ghobaei-Arani et al. [15] presented a survey on resource management techniques in fog computing in the form of taxonomy to highlight cutting-edge methods while also addressing unresolved challenges. The various authors in the cloud-fog area provided the task scheduling review. Their benefits and drawbacks, as well as numerous tools and challenges concerning the scheduling techniques and their limitations, were analyzed by Alizadeh et al. [16]. Islam et al. [6] thoroughly analyze relevant literature on context-aware scheduling in fog computing. Further, Mouradian et al. [17] review fog computing's significant issues and challenges. But, the critical area of mobility research is still in its initial stage, and most of the review papers contain very few documents on mobility-aware task scheduling in the area of fog computing. Therefore, an extensive and comparative study is required in mobility-constrained fog computing. A deep insight into various techniques which can impact the user QoS is necessary to understand mobility-aware fog computing. This motivates us to carry out a comprehensive survey; to the best of our knowledge, this is the first detailed survey. This survey paper thoroughly examines existing scheduling solutions that focus on user mobility. Moreover, various mobility-aware scheduling techniques are discussed, along with their pros and cons. Further, the impact of mobility parameters on various QoS parameters and context-awareness is also analyzed thoroughly.
=== Paper Organization ===
The remainder of this review article is divided into 5 sections: Section 2 discusses mobility-aware scheduling in Fog computing. Section 3 presents the review methodology. Section 4 analyzes and summarizes the considered research papers and compares existing mobility-aware scheduling strategies. Section 5 provides the results drawn after critically examining the existing literature on mobility-aware scheduling policies. Finally, Section 6 presents the conclusion.
== Mobility-aware Scheduling in Fog Computing ==
Mobile device management compromises the fundamental features of Fog computing because whenever a user moves, the distance between them increases, impacting the QoS. Therefore, to keep the computing fog node close to the associated mobile device, the services or tasks need to be migrated from one fog node to another appropriate fog device. The selection of such appropriate fog nodes in a mobile environment deals with two main processes: Estimation of user mobility patterns: User mobility estimation techniques can be probabilistic and deterministic [18]. In a deterministic method, the source and destination are known beforehand, whereas in a non-deterministic technique, periodic estimation has to be done regarding the user's route. Many authors estimate the route of mobile users by leveraging external services like Open Street Maps (<nowiki>http://www.openstreetmap.org</nowiki>), Google Maps APIs (<nowiki>https://cloud.google.com/</nowiki> maps-platform/) [18], logistics maps [19], GPRS Here APIs (<nowiki>https://developer</nowiki>. here.com/) [20, 21], Lyapunov estimation technique [22]. Specific QoS requirements: The selection of a fog node also depends upon specific quality requirements such as Latency, which many authors are using to select an efficient fog node [18, 10, 9], workload [23, 24], cost [16], etc.
== Review Methodology ==
The review process was conducted considering a review methodology consisting of four phases. The first phase was searching using traditional online database sources based on outlined search keywords. Table 1 lists the
Keywords used to find the relevant research articles to conduct the review in the Fog computing mobility-aware scheduling area. Second Phase: Limit the search of research articles beginning in 2015, and inclusion and exclusion principles are also used to refine research articles that specifically deal with mobility issues in task scheduling. Finally, in the Third Phase, A total of 20 papers are shortlisted for the review process. Further, Table 2 presents the research questions drafted for this study in mobility-aware scheduling in Fog computing.
{| class="wikitable"
|+Table 1: List of keywords used in the review process
!Sno
!Keyword
!Description
!Years
|-
|1
|Mobility
|Mobility-aware Fog task scheduling
| rowspan="5" |2015- 2021
|-
|2
|Mobility environment
|Mobility environment in Fog task scheduling
|-
|3
|Mobility factors
|Mobility factors in Fog task scheduling
|-
|4
|Mobility Awareness
|Mobility awareness in Fog task scheduling
|-
|5
|Mobility management
|Mobility management in Fog task scheduling
|}
{| class="wikitable"
|+Table 2: List of research questions used to complete the review process
!Q.No
!Research questions
|-
|1
|What scheduling approaches are used in the Fog computing environment to manage mobility awareness?
|-
|2
|What are the main limitations considered for mobility-aware scheduling techniques?
|-
|3
|Which case studies are applied to mobility-aware scheduling techniques?
|-
|4
|What evaluation tools are used to assess mobility-aware scheduling techniques?
|-
|5
|What performance indicators are utilized to evaluate mobility-aware scheduling techniques?
|-
|6
|What are the major open issues concerns in the field of mobility-aware scheduling for future research directions?
|}
=== Source of Information ===
To conduct this review, various online sources are listed below, and the research articles were searched using the different keywords mentioned in Table 1.
* Google Scholar (<nowiki>http://www.scholar.google.co.in</nowiki>)
* John Wiley & Sons Inc. (<nowiki>https://onlinelibrary.wiley.com/</nowiki>)
* Elsevier (<nowiki>https://www.elsevier.com/en-in</nowiki>)
* ACM Digital Library (<nowiki>https://www.acm.org/</nowiki>)
* Springer (<nowiki>https://www.springer.com/in</nowiki>)
* IEEE Xplore Digital Library (<nowiki>https://www.ieee.org/</nowiki>)
=== Quality Assessment ===
Research papers used quality assessment to filter out the most suitable mobility-based scheduling research articles in fog computing utilising the principle of inclusion and exclusion. Furthermore, in order to obtain high-quality research publications, the Center for Reviews and Dissemination (CRD) recommendations were followed, and each study item was examined for internal and external validation of results.
== Literature Analysis ==
Verma et al. [25] proposed a Server Cloudlet(SC) migration technique to handle users' mobility. The strategy is to select the target SC to migrate the services based on its highest rank. The rank of the SC further depends upon its available RAM, MIPS, and bandwidth.
Maleki et al. [26] designed two mobility-aware computation offloading approaches, sampling-based Online mobile applications to cloudlets(S-OAMC) and Greedy online mobile applications to cloudlets(G-OAMC). The developed system works in three major steps: ''(i)'' the future specifications of mobile applications are predicted using the Machine Learning (ML) method, named matrix completion. ''(ii)'' The predicted specifications of mobile applications, including future location prediction, also help estimate the offloading cost. ''(iii)'' Apply S-OAMC or G-OAMC offloading technique based on minimum cost value.
Abdelmoneem et al. [10] work on a fog-based healthcare architecture system that supports patient movement without altering their Quality of Service (QoS). The QoS in a mobile environment is maintained with the help of their proposed handoff mechanism. The authors modified the traditional Horizontal Handoff (HHO) mechanism [27] into two different phases: ''(i) handoff decision policy'': Received signal Strength (RSS) method is used to detect the network change information of the mobile patients and (ii) ''handoff strategy'': a handoff decision is made based on a comparison between RSS received from the old fog gateways, current threshold value and new fog gateways that are in close proximity to the mobile patient.
Javanmardi et al. [28] consider an imaginative city-based mobility scenario where the user is placing delay-sensitive service while moving. The authors proposed a task scheduling technique that jointly employs fuzzy logic and particle swarm optimization (PSO) algorithm to improve the QoS for mobile users within the city. The proposed algorithm is deployed at the Fog gateway, which further distributes the tasks to Fog devices available in an entire region in order to provide seamless service to mobile users. The main motive of the work is to improve resource utilization in a mobility-aware environment.
Puliafito et al. [29] develop an extension of ifogsim [30], which supports mobile environment. The authors implement different migration techniques inspired by Bittencourt et al. [31]. Different phases are devised to perform migration in a fog mobile environment, which: ''(i) Before migration phase'': migration decision is taken in this phase, based on specific parameters, like user location, speed, the direction of movement, zone, and migration point in order to select the appropriate cloudlet for offloading the services, ''(ii) During the migration phase'': this phase manages, monitors and synchronizes the whole selected migration process and ''(iii) After migration phase'': this phase involves closing the older cloudlet connections with the user and using the new cloudlet for services.
A Blockchain-based Mobility-aware Offloading (BMO) mechanism is designed by Dou et al. [32], where user mobility prediction is implemented using the Individual-Mobility (IM) model [33]. The idea behind the offloading mechanism is to shift the computational workload to different available Fog Servers (FSs) in the geo-location predicted by the IM model. Further, blockchain technology is deployed to check the authenticity of the forthcoming Fog servers. Finally, accounting is being managed by ''Fogcoin'', similar to Bitcoin, which stores the entire transaction history between the online Fog server and mobile users.
Martin et al. [7] proposed a framework that supports the migration of containers while satisfying the QoS requirements of mobile users. The migration of containers is done in an autonomic manner, by adopting the Monitor-Analyze-Plan-Execute (MAPE) autonomic control loop. The MAPE control loops discuss various steps of migration, like ''(i) Monitor'': used to constantly monitor the environment context, such as the mobility of users that is subsequently used to determine the need to migrate an application module to some other Fog node called target node; ''(ii) Analyze'': applies forecasting techniques to predict the user possible location in the next time step. If the distance between the user and the device is not under certain acceptable limits, a migration decision is made. ''(iii) Plan'': a Genetic Algorithm (GA) is used to identify a suitable Fog node closest to the forecasted location, where migration of the container running user application can be done. ''(iv) Execute'': this step ensures the whole migration process should take place smoothly.
Mass et al. propose a mobility and delay-aware fog server selection scheme. [34] called Edge-Process management (EPM) system. The EPM system depends upon the trajectory of a user’s movement, Fog server workload, and user location to select the appropriate Fog server for executing user applications. The system selects or re-selects a Fog Server (FS) based on a score value calculated through available bandwidth, power, distance from the user, and finally, duration of availability in a region.
Mobi-IoST (Mobility-aware Internet of Spatial Things), a real-time mobility-aware framework is presented by Ghosh et al. [35]. The authors considered the mobile nature of both IoT devices and Fog nodes, collaboratively called mobile agents. The proposed mobility-aware framework collects a vast amount of Global Positioning System (GPS) data of these mobile agents to predict their movement patterns using various machine learning algorithms. The major components of the framework are, ''(i) Movement pattern model''ling, collecting and modelling GPS log, stay-point, and other contextual location information; ''(ii) Predicting the following location'': human movement semantics is analyzed using all modelled information; ''(iii) Delivery of result:'' after the user movement prediction in the previous phase, the system intelligently discovers a capable fog device for data processing in a timely manner.
A middleware solution, URMILA, for managing resources and scheduling tasks in the Fog environment is presented by Shekhar et al. [18]. Ubiquitous Resource Management for Interference and Latency-Aware services (URMILA), ensures minimum Service Level Objectives (SLO) violation for latency-sensitive mobile applications across the cloud-Fog environment. The major modules of the proposed system are ''(i) Route calculation'', which calculates the user's possible routes using Google Maps or GPS data; ''(ii) Latency calculation'', the system deploys a data-driven model to estimate the Latency on predicted user routes; ''(iii) Fog node selection:'' the system selects a fog server for execution of task on the basis of its instantaneous utilization of the available resources. Further, it selects the Fog server for the entire period of execution, during which mobile users can still access their application through various Wireless Access Points (WAP).
Gia et al. [8] proposed a Handover mechanism for mobility management between fog nodes with the overall objective of consuming minimum energy and delay during handovers. Handover methods frequently rely on one or more measures, such as the Received Signal Strength Indicator (RSSI), the Link Quality Indicator (LQI), and the velocity of objects, to make handover decisions. This proposed system provides emergency services to health monitoring systems and basically works in two different mobility scenarios: ''(i) Node mobility between indoor or outdoor locations'': nodes belonging to indoor location or outdoor location only are considered to be similar, and they're calculated metrics value like RSSI, LQI, velocity, etc.; can be directly used for the handover of services to appropriate gateway, ''(ii) Node mobility between indoor and outdoor locations'': nodes are considered dissimilar, if they belong to indoor and outdoor location both, So, the metrics are re-calculated which introduce some additional parameters like temperature and interference signals in order to make a decision over handover gateway.
Babu and Biswash [36] proposed a mobility management technique that supports node-to-node communication and Fog computing-based architecture for 5G networks. It addresses the technical problems between 5G networks and Fog servers. The mobility-based approach assists mobile nodes in establishing communication while they are in motion. The mobility management technique may also be used to begin N2N communication in dynamic environments. N2N communication schemes for fog networks, on the other hand, provide an effective communication environment for mobile users with highly minimal network usage.
Wang et al. [37] proposed a mobility-aware offloading scheme, that considers an adequate quality and a computation allocation system that deals with the user equipment affairs to maximize the total revenue. The quality of user equipment is delineated by the sojourn time that follows the exponential distribution to reduce the chance of migration and maximize the entire income of user equipment. MILP (mixed-integer non-linear programming) NP-hard problem is modelled and consists of resource allocation and task offloading schemes. So, to solve this problem, a Gini coefficient and genetic algorithm are used to estimate the allocation of resources. The proposed approach can easily handle the mobility of users by minimizing the chances of migration.
Waqas et al. [38] provided a forward-thinking analysis of quality about-mobility in Fog computing by identifying quality challenges, requirements, and options for numerous ideas. The authors also identified outstanding concerns from previous research and summarized the advantages of quality for readers. It allows researchers and developers to avoid common misunderstandings and capture real-world scenarios such as businesses, governments, and educational institutions. Furthermore, it revolutionizes follow-up analysis and differentiates and foregrounds futurity orientations in real-life events involving humans and vehicles in a highly dynamic Fog setting.
Bi et al. [9] introduced software-defined networking-based fog computing architecture by decoupling mobility control and data forwarding. When mobile consumers travel between several access networks, the authors suggested an Optimal Path Selection (OPS) algorithm to preserve service continuity. Mobile customers received seamless and transparent mobility support thanks to efficient signalling operations. In mobile fog computing, the suggested algorithm ensured service continuity, increased handover performance, and achieved high data transfer efficiency.
Niu et al. [21] established a system called mobility-aware and multihop-D2D relaying-based scheduling scheme (MHRC) at Edge nodes near hotspots. The authors exploited concurrent transmissions to improve the performance of the system. The mmWave (millimetre-wave) band of Fog computing was cached, and extensive performance evaluation confirms that MHRC delivers more than the higher expected cached data amount. Name et al. [19] proposed an efficient algorithm to address the problem of resource allocation and user mobility from the Edge of the network to cloud data centres. This algorithm operates on a seamless handover scheme for mobile IPV6 to ease the user mobility challenge and reduce the application response time. The study showed that the task of service delay and packet loss was decreased due to the effect of change in the mobile node position.
Bittencourt et al. [23] examined the subject of resource allocation in the Fog/Cloud environment, taking into account the hierarchical structure. In the context of the Fog paradigm, the authors developed three scheduling algorithms (First come, First serve, delay-priority, and concurrent) that address user mobility and edge computing capabilities. The authors demonstrated that scheduling techniques may be designed to cope with different application classes based on demand from mobile users by leveraging both Fog to the end-user and cloud characteristics in this study.
Velasquez et al. [39] proposed a hybrid strategy for the Fog environment to manage resources for mobility scenarios. The authors applied the orchestrator technique to offer mobility support in a Smart City situation. In this technique, three components, the status monitor, the Planner, and the VM/Container, are employed to monitor, plan and execute the applications. The main aim of this study was to guarantee the QoS and QoE requirements of mobility-based applications and services.
Bittencourt et al. [31] presented a Fog computing architecture focusing on Virtual Machine (VM) migration where each user has a VM running in a cloudlet. In this architecture, the user's location is identified by using GPS, and then the VM is moved to a nearby Fog Cloud. The main aim of this study was to migrate users' data according to their mobility in order to maintain QoS for applications demanding lower Latency and allow smooth handoff mechanisms for mobile users.
From the extensive analysis of the literature, the various mobility-aware scheduling techniques have been classified as shown in Table 3. Further, it presents the advantages and limitations of each technique.
{| class="wikitable"
|+Table 3: Classification of Mobility-aware scheduling techniques
!Ref.
!Technique
!Advantage(s)
!Limitation(s)
|-
|[25]
|Ranking of VM
|
* Decrease in delay time, migration time, tuple lost value and downtime
|
* Case study not discussed
|-
|[26]
|S-OAMC, G-OAMC, Machine learning matrix completion
|
* Migration rate decreased
* Better Scalability
|
* Energy utilization of devices not investigated
|-
|[32]
|IM model
|
* Provides better mobility support and security
|
* Did not investigate synchronization overhead
|-
|[10]
|RSS
|
* Reliable and Heterogeneous execution
|
* Low scalability
* No distributed scheduling to minimize response time
|-
|[7]
|MAPE control loop
|
* Improved QoS
* Reduced service downtime
|
* No real-time evaluation
* High energy consumption
* Low robustness and security
|-
|[28]
|Copy of task to over a region
|
* Network Utilization developed
* Low-Loop delay
|
* Fault tolerance reliability is based on Fog gateways only
|-
|[18]
|URMILA
|
* Service availability is maintained by delivering the desired QoS
* Deployment cost minimized
* Battery longevity ensured
|
* No empirical validations
* No user probabilistic routes
* Low scalability in terms of distance and speed
|-
|[8]
|RSSI, LQI
|
* Promises to keep the connection active with a low latency rate between the system and sensor nodes
|
* Consumes more energy
* Overhead is large for network transmission
* Coverage and overhead area are undefined between gateways
|-
|[29]
|User trajectories pre- diction using GPS log
|
* Provides better mobility support
* Reduces migration time
|
* Low scalability
|-
|[36]
|N2N communication, Data Analytics
|
* Fast data access
* High reliability and scalability- city
* Minimum overhead and cost
* High throughput and less delay
|
* No real-time cellular network evaluation
* Low network efficiency
|-
|[38]
|Mobility facets analysis
|
* Improved QoS and QoE
* Latency rate reduced
|
* No real-life implementation
* No reliability and low Latency between dynamic users and fog servers
|-
|[37]
|M-ILP, Sojourn time
|
* Cost-effective
* Migration time reduced
|
* Migration cost not considered
* No real-time implementation
|-
|[34]
|User trajectories prediction using GPS log
|
* Conventional delay tolerance
* High QoS
* Avoided local task processing cost
* Efficient in saving battery
* Handles subtle scenarios with high Latency
|
* Smart city not directed through the use of accurate city maps with aid from stimulation setting
|-
|[35]
|Prediction of user location
|
* Power consumption and de- lay handled proficiently
* Power and delay are reduced
|
* No acquiring of mobile data usage where location sense and time-series data can be projected to achieve the bandwidth
|-
|[21]
|Relay path planning algorithm
|
* Power efficient
* High spectral efficiency
* Data is relayed on cached edge nodes and relay nodes
|
* Blockage problem due to weak diffraction
|-
|[9]
|OPS
|
* Handover performance improved
* Efficiency of high data communication achieved
* Guarantees continuity of services
|
* It does not guarantee privacy and security
* Virtual Machine migration not determined
* The handover process during the optimal path for more logical routing could have been more efficient
|-
|[23]
|Assignment of FS
|
* Low Latency
* Supports dynamic computing and user behaviour
|
* There is no prediction of mobility failure
* Bandwidth and processing not considered in scheduling
|-
|[39]
|Orchestrator
|
* Maintains trustworthiness, resilience, and low Latency in a dynamic environment
|
* No real implementation has been carried out
|-
|[19]
|Pattern modelling, dictating the following location
|
* Application Response time reduced
* Latency time reduced
|
* Services become temporarily inaccessible for some mobile nodes
|-
|[31]
|Forecasting technique
|
* Computing capacity provided for storage and processing of data
|
* Security concerns associated with both user data and applications not considered
|}
[[File:Fog Computing Figure 2.jpg|center|thumb|Figure 2: Year-wise count of research articles]]
== Analytical Discussion ==
The existing research on Fog computing Mobility-aware scheduling has been analyzed thoroughly. The analysis was performed using the answers given in Table 2. The results drawn through the thorough analysis of the literature are presented in various figures as follows:
Figure 2 lists the year-wise count of research papers that are considered for this survey. The bar graph represents the total number of research papers from journals and Conferences from the year 2015 - 2021. The research articles from the journal are 16, and the conference papers are 4. It is observed that more research needs to be conducted on mobility-aware scheduling in Fog computing.
Figure 3 displays an analytical comparison of mobility-aware scheduling approaches in Fog computing based on the content of the represented taxonomy in Figure 7. From the thorough analysis of the literature, four methods have been considered: migration, task offloading, handoff/handover mechanism, and task scheduling. The handoff/handover mechanism has the highest percentage of usage in mobility-aware scheduling, at 30%. The task scheduling and offloading have 25% of us- age in mobility-aware scheduling each. Finally, migration is only 20% of the usage in mobility-aware scheduling. Therefore, these approaches, specifically migration, are still open challenges to address for further research.
Figure 4 depicts various tools that were used for evaluating the mobility-aware scheduling approaches. 18% and 9% of the research articles used iFogSim and Mob-FogSim simulation tools for implementation, respectively. Besides, other simulation tools such as ONE (9%), NS2(5%), MATLAB(4%), Mininet(5%), and Docker(9%) have been utilized for implementing the proposed techniques in the research articles. Further, pro-Programming languages such as C++ (9%) and Python (9%) and hardware deployments such as Raspberry Pi (5%) and Ardunio (4%) were used for implementing existing case studies based on mobility-aware scheduling.
[[File:Fog Computing Figure 3.jpg|center|thumb|Figure 3: Percentage of the presented classified approaches in mobility-aware scheduling]]
The applied case studies are shown in Figure 5, which shows a maximum of 20% of research articles have implemented IoT-based applications. After that, 15% of each research article used Health care and Mobile-based applications. Besides, Smart City and 5G-based applications have been applied in 10% of research articles. Moreover, Surveillance and gaming, Mobile IPV6, and Wireless computing applications are the case studies on which only 5% of research articles exist.
After reviewing numerous research articles based on mobility-aware scheduling, it has been observed that researchers employed various parameters for evaluating the performance of the Mobility-scheduling approaches, as represented in Figure 6. It shows that Time completion (18%) followed by Delay (12%), Network usage (12%), Latency (12%), Energy consumption (10%), and cost (10%) are generally utilized. However, Downtime (4%), Migration time (4%), Makespan (2%), Success ratio (2%), Signal level (2%), Deadline (2%), Makespan (2%), Migration rate (2%), Mobility patterns (2%), Tuple lost (2%), and power consumption (2%) are less exploited parameters.
A taxonomy was compiled after going through the detailed review process, and various techniques have been categorized in Fog computing-based mobility-aware scheduling. Figure 7 presents these categories broadly in Migration, Offloading, Handoff/Handover mechanism, and Scheduling.
[[File:Fog Computing Figure 4.jpg|center|thumb|Figure 4: Percentage of tools utilized in the literature]]
[[File:Fog Computing Figure 5.jpg|center|thumb|Figure 5: Percentage of case studies employed in the literature]]
===Subheading===
==Second Heading==
==Third Heading, etc==
==Additional information==
===Acknowledgements===
Any people, organisations, or funding sources that you would like to thank.
===Competing interests===
Any conflicts of interest that you would like to declare. Otherwise, a statement that the authors have no competing interest.
===Ethics statement===
An ethics statement, if appropriate, on any animal or human research performed should be included here or in the methods section.
==References==
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{{Article info
| journal = WikiJournal of Science
| last1 = Islam
| orcid1 = 0000-0001-5750-1058
| first1 = Mir Salim Ul
| last2 = Kumar
| orcid2 = 0000-0003-3279-5111
| first2 = Ashok
| affiliation2 = Department of Computer Application, Chandigarh School of Business, Chandigarh Group of Colleges, Jhanjeri, Punjab, India
| et_al =
| affiliation1 = Department of Computer Science and Engineering, National Institute of Technology Srinagar, Jammu and Kashmir, India
| correspondence1 = mir.salim27@gmail.com
| affiliations =
| keywords = Fog computing, Mobility-aware, Scheduling, Resource Management, Resource Allocation
| license = CC BY 4.0
| abstract = The increasing focus on Fog-IoT results in billions of Internet-connected devices that demand substantial computational power and network bandwidth. These devices are geographically distributed, heterogeneous, computational capacity constrained, inconsistent in behaviour, and generally mobile. Therefore, providing seamless service, irrespective of location and movement of the devices as well as the end-users, makes resource scheduling a significant challenge in the Fog computing paradigm. Several mobility-aware scheduling strategies have been proposed in the literature to efficiently manage the resources for mobile users and devices in the Fog environment. This paper gives a survey of mobility-aware scheduling in the Fog computing environment. It describes the many strategies presented and their benefits and drawbacks. It also includes a complete study and taxonomy of the mobility-aware scheduling field. Further, it delineates open issues and challenges. This work will provide researchers with future research directions and aid them in recognizing the gaps before planning for further research in mobility-aware scheduling.
}}
== Introduction ==
With the advancement in technology and the exponential growth of mobile devices, network traffic has increased manifold in cloud computing. Due to this reason, Latency reduction and faster processing of data for mobile users have become critical challenges in providing seamless connectivity and minimal disruption while the user is moving.<ref>{{Cite journal|last=Abdullah|first=Fatima|last2=Kimovski|first2=Dragi|last3=Prodan|first3=Radu|last4=Munir|first4=Kashif|date=2021-06-11|title=Handover authentication latency reduction using mobile edge computing and mobility patterns|url=https://doi.org/10.1007/s00607-021-00969-z|journal=Computing|volume=103|issue=11|pages=2667–2686|doi=10.1007/s00607-021-00969-z|issn=0010-485X}}</ref> The data movement brings the additional issue of integrity and confidentiality because data is moving via a wireless connection to a far distant cloud. Additionally, due to the cloud's location is far from mobile users, so data movement is also affected by variable network strength and phone bandwidth. The solution proposed by [2] is to extend cloud capabilities through fog computing architecture. The Fog architecture allows substantial computation, storage, and processing using the Fog devices installed close to the user’s access point. Fog computing, therefore, reduces Latency and bandwidth consumption, improves security, provides context awareness, and renders more efficient services to mobile users [3, 4, 5].
[[File:Fog Computing Figure 1.jpg|left|thumb|Figure 1: Mobility Scenario in Fog Computing]]
However, mobility also imposes severe challenges for Fog computing due to its distributed and diverse environment. Mobility is recognized by either user-level or device-level contextual information [6]. As the user moves from one location to another, the geographical location of the smart devices also changes. The change in location of the devices raises the issue of searching and rescheduling with mobility management. Efficient re-scheduling requires a well-planned handoff mechanism that is accountable for smoothly de-registering a sensor node from a source access point where the application was initially hosted and registering it to a new access point. Figure 1 depicts these mentioned problems of change in access points while the user moves from one area to another. The services may also get interrupted when there is more distance between the Fog nodes and users [7]. Further, any disruption in communication may lead to an increase in Latency for mobile users [8].
The significant challenges in mobility management of Fog computing are: (i) ''Fog node discovery''- due to the heterogeneous and mobile nature of the smart mobile devices, the Fog task scheduler faces the issue of finding an optimal Fog node for scheduling the task, ''(ii) Handover mechanism between users and Fog nodes''- imagine a Fog user is moving from location to location and accessing information about his surroundings with the help of a smart device. Due to the frequent change in the user's location, the Fog scheduler may need to repeatedly migrate the user’s task to a different Fog node available in his vicinity. The frequent migration of tasks increases the overhead of scheduling and, further, the restricted signal strength in certain places may lead to a breakdown of task or result delivery, and (iii) ''the Handover mechanism between Cloud and Fog''- Fog nodes have limited capacities and need to continuously communicate with Cloud computing for passing information about the tasks [9]. Due to strict requirements for security, Latency, network coverage, and reliability, It becomes challenging to implement an efficient handover mechanism for full mobility support in critical domains such as healthcare [10, 11, 12] and vehicular systems [13, 14]. Thus, mobility significantly impacts the overhead of scheduling policies and the applications' performance, eventually affecting the Quality of Experience (QoE). Therefore, mobility-aware scheduling in Fog computing has observed strong attention from researchers. Our main goal is to provide a detailed review of mobility-aware scheduling in fog computing. The review provides a detailed analysis of existing scheduling strategies that concentrate specifically on mobility awareness in the Fog environment. The following are the main contributions of this paper:
* This paper presents a detailed survey of mobility-aware scheduling in the Fog computing environment.
* It provides the details of the different techniques proposed, as well as their advantages and limitations.
* It provides a detailed analysis and taxonomy of the mobility-aware scheduling field.
* It identifies several open challenges for future research directions.
=== Related Surveys ===
Numerous survey studies on Fog computing focus on resource management, job scheduling, and context-aware scheduling. For example, Ghobaei-Arani et al. [15] presented a survey on resource management techniques in fog computing in the form of taxonomy to highlight cutting-edge methods while also addressing unresolved challenges. The various authors in the cloud-fog area provided the task scheduling review. Their benefits and drawbacks, as well as numerous tools and challenges concerning the scheduling techniques and their limitations, were analyzed by Alizadeh et al. [16]. Islam et al. [6] thoroughly analyze relevant literature on context-aware scheduling in fog computing. Further, Mouradian et al. [17] review fog computing's significant issues and challenges. But, the critical area of mobility research is still in its initial stage, and most of the review papers contain very few documents on mobility-aware task scheduling in the area of fog computing. Therefore, an extensive and comparative study is required in mobility-constrained fog computing. A deep insight into various techniques which can impact the user QoS is necessary to understand mobility-aware fog computing. This motivates us to carry out a comprehensive survey; to the best of our knowledge, this is the first detailed survey. This survey paper thoroughly examines existing scheduling solutions that focus on user mobility. Moreover, various mobility-aware scheduling techniques are discussed, along with their pros and cons. Further, the impact of mobility parameters on various QoS parameters and context-awareness is also analyzed thoroughly.
=== Paper Organization ===
The remainder of this review article is divided into 5 sections: Section 2 discusses mobility-aware scheduling in Fog computing. Section 3 presents the review methodology. Section 4 analyzes and summarizes the considered research papers and compares existing mobility-aware scheduling strategies. Section 5 provides the results drawn after critically examining the existing literature on mobility-aware scheduling policies. Finally, Section 6 presents the conclusion.
== Mobility-aware Scheduling in Fog Computing ==
Mobile device management compromises the fundamental features of Fog computing because whenever a user moves, the distance between them increases, impacting the QoS. Therefore, to keep the computing fog node close to the associated mobile device, the services or tasks need to be migrated from one fog node to another appropriate fog device. The selection of such appropriate fog nodes in a mobile environment deals with two main processes: Estimation of user mobility patterns: User mobility estimation techniques can be probabilistic and deterministic [18]. In a deterministic method, the source and destination are known beforehand, whereas in a non-deterministic technique, periodic estimation has to be done regarding the user's route. Many authors estimate the route of mobile users by leveraging external services like Open Street Maps (<nowiki>http://www.openstreetmap.org</nowiki>), Google Maps APIs (<nowiki>https://cloud.google.com/</nowiki> maps-platform/) [18], logistics maps [19], GPRS Here APIs (<nowiki>https://developer</nowiki>. here.com/) [20, 21], Lyapunov estimation technique [22]. Specific QoS requirements: The selection of a fog node also depends upon specific quality requirements such as Latency, which many authors are using to select an efficient fog node [18, 10, 9], workload [23, 24], cost [16], etc.
== Review Methodology ==
The review process was conducted considering a review methodology consisting of four phases. The first phase was searching using traditional online database sources based on outlined search keywords. Table 1 lists the
Keywords used to find the relevant research articles to conduct the review in the Fog computing mobility-aware scheduling area. Second Phase: Limit the search of research articles beginning in 2015, and inclusion and exclusion principles are also used to refine research articles that specifically deal with mobility issues in task scheduling. Finally, in the Third Phase, A total of 20 papers are shortlisted for the review process. Further, Table 2 presents the research questions drafted for this study in mobility-aware scheduling in Fog computing.
{| class="wikitable"
|+Table 1: List of keywords used in the review process
!Sno
!Keyword
!Description
!Years
|-
|1
|Mobility
|Mobility-aware Fog task scheduling
| rowspan="5" |2015- 2021
|-
|2
|Mobility environment
|Mobility environment in Fog task scheduling
|-
|3
|Mobility factors
|Mobility factors in Fog task scheduling
|-
|4
|Mobility Awareness
|Mobility awareness in Fog task scheduling
|-
|5
|Mobility management
|Mobility management in Fog task scheduling
|}
{| class="wikitable"
|+Table 2: List of research questions used to complete the review process
!Q.No
!Research questions
|-
|1
|What scheduling approaches are used in the Fog computing environment to manage mobility awareness?
|-
|2
|What are the main limitations considered for mobility-aware scheduling techniques?
|-
|3
|Which case studies are applied to mobility-aware scheduling techniques?
|-
|4
|What evaluation tools are used to assess mobility-aware scheduling techniques?
|-
|5
|What performance indicators are utilized to evaluate mobility-aware scheduling techniques?
|-
|6
|What are the major open issues concerns in the field of mobility-aware scheduling for future research directions?
|}
=== Source of Information ===
To conduct this review, various online sources are listed below, and the research articles were searched using the different keywords mentioned in Table 1.
* Google Scholar (<nowiki>http://www.scholar.google.co.in</nowiki>)
* John Wiley & Sons Inc. (<nowiki>https://onlinelibrary.wiley.com/</nowiki>)
* Elsevier (<nowiki>https://www.elsevier.com/en-in</nowiki>)
* ACM Digital Library (<nowiki>https://www.acm.org/</nowiki>)
* Springer (<nowiki>https://www.springer.com/in</nowiki>)
* IEEE Xplore Digital Library (<nowiki>https://www.ieee.org/</nowiki>)
=== Quality Assessment ===
Research papers used quality assessment to filter out the most suitable mobility-based scheduling research articles in fog computing utilising the principle of inclusion and exclusion. Furthermore, in order to obtain high-quality research publications, the Center for Reviews and Dissemination (CRD) recommendations were followed, and each study item was examined for internal and external validation of results.
== Literature Analysis ==
Verma et al. [25] proposed a Server Cloudlet(SC) migration technique to handle users' mobility. The strategy is to select the target SC to migrate the services based on its highest rank. The rank of the SC further depends upon its available RAM, MIPS, and bandwidth.
Maleki et al. [26] designed two mobility-aware computation offloading approaches, sampling-based Online mobile applications to cloudlets(S-OAMC) and Greedy online mobile applications to cloudlets(G-OAMC). The developed system works in three major steps: ''(i)'' the future specifications of mobile applications are predicted using the Machine Learning (ML) method, named matrix completion. ''(ii)'' The predicted specifications of mobile applications, including future location prediction, also help estimate the offloading cost. ''(iii)'' Apply S-OAMC or G-OAMC offloading technique based on minimum cost value.
Abdelmoneem et al. [10] work on a fog-based healthcare architecture system that supports patient movement without altering their Quality of Service (QoS). The QoS in a mobile environment is maintained with the help of their proposed handoff mechanism. The authors modified the traditional Horizontal Handoff (HHO) mechanism [27] into two different phases: ''(i) handoff decision policy'': Received signal Strength (RSS) method is used to detect the network change information of the mobile patients and (ii) ''handoff strategy'': a handoff decision is made based on a comparison between RSS received from the old fog gateways, current threshold value and new fog gateways that are in close proximity to the mobile patient.
Javanmardi et al. [28] consider an imaginative city-based mobility scenario where the user is placing delay-sensitive service while moving. The authors proposed a task scheduling technique that jointly employs fuzzy logic and particle swarm optimization (PSO) algorithm to improve the QoS for mobile users within the city. The proposed algorithm is deployed at the Fog gateway, which further distributes the tasks to Fog devices available in an entire region in order to provide seamless service to mobile users. The main motive of the work is to improve resource utilization in a mobility-aware environment.
Puliafito et al. [29] develop an extension of ifogsim [30], which supports mobile environment. The authors implement different migration techniques inspired by Bittencourt et al. [31]. Different phases are devised to perform migration in a fog mobile environment, which: ''(i) Before migration phase'': migration decision is taken in this phase, based on specific parameters, like user location, speed, the direction of movement, zone, and migration point in order to select the appropriate cloudlet for offloading the services, ''(ii) During the migration phase'': this phase manages, monitors and synchronizes the whole selected migration process and ''(iii) After migration phase'': this phase involves closing the older cloudlet connections with the user and using the new cloudlet for services.
A Blockchain-based Mobility-aware Offloading (BMO) mechanism is designed by Dou et al. [32], where user mobility prediction is implemented using the Individual-Mobility (IM) model [33]. The idea behind the offloading mechanism is to shift the computational workload to different available Fog Servers (FSs) in the geo-location predicted by the IM model. Further, blockchain technology is deployed to check the authenticity of the forthcoming Fog servers. Finally, accounting is being managed by ''Fogcoin'', similar to Bitcoin, which stores the entire transaction history between the online Fog server and mobile users.
Martin et al. [7] proposed a framework that supports the migration of containers while satisfying the QoS requirements of mobile users. The migration of containers is done in an autonomic manner, by adopting the Monitor-Analyze-Plan-Execute (MAPE) autonomic control loop. The MAPE control loops discuss various steps of migration, like ''(i) Monitor'': used to constantly monitor the environment context, such as the mobility of users that is subsequently used to determine the need to migrate an application module to some other Fog node called target node; ''(ii) Analyze'': applies forecasting techniques to predict the user possible location in the next time step. If the distance between the user and the device is not under certain acceptable limits, a migration decision is made. ''(iii) Plan'': a Genetic Algorithm (GA) is used to identify a suitable Fog node closest to the forecasted location, where migration of the container running user application can be done. ''(iv) Execute'': this step ensures the whole migration process should take place smoothly.
Mass et al. propose a mobility and delay-aware fog server selection scheme. [34] called Edge-Process management (EPM) system. The EPM system depends upon the trajectory of a user’s movement, Fog server workload, and user location to select the appropriate Fog server for executing user applications. The system selects or re-selects a Fog Server (FS) based on a score value calculated through available bandwidth, power, distance from the user, and finally, duration of availability in a region.
Mobi-IoST (Mobility-aware Internet of Spatial Things), a real-time mobility-aware framework is presented by Ghosh et al. [35]. The authors considered the mobile nature of both IoT devices and Fog nodes, collaboratively called mobile agents. The proposed mobility-aware framework collects a vast amount of Global Positioning System (GPS) data of these mobile agents to predict their movement patterns using various machine learning algorithms. The major components of the framework are, ''(i) Movement pattern model''ling, collecting and modelling GPS log, stay-point, and other contextual location information; ''(ii) Predicting the following location'': human movement semantics is analyzed using all modelled information; ''(iii) Delivery of result:'' after the user movement prediction in the previous phase, the system intelligently discovers a capable fog device for data processing in a timely manner.
A middleware solution, URMILA, for managing resources and scheduling tasks in the Fog environment is presented by Shekhar et al. [18]. Ubiquitous Resource Management for Interference and Latency-Aware services (URMILA), ensures minimum Service Level Objectives (SLO) violation for latency-sensitive mobile applications across the cloud-Fog environment. The major modules of the proposed system are ''(i) Route calculation'', which calculates the user's possible routes using Google Maps or GPS data; ''(ii) Latency calculation'', the system deploys a data-driven model to estimate the Latency on predicted user routes; ''(iii) Fog node selection:'' the system selects a fog server for execution of task on the basis of its instantaneous utilization of the available resources. Further, it selects the Fog server for the entire period of execution, during which mobile users can still access their application through various Wireless Access Points (WAP).
Gia et al. [8] proposed a Handover mechanism for mobility management between fog nodes with the overall objective of consuming minimum energy and delay during handovers. Handover methods frequently rely on one or more measures, such as the Received Signal Strength Indicator (RSSI), the Link Quality Indicator (LQI), and the velocity of objects, to make handover decisions. This proposed system provides emergency services to health monitoring systems and basically works in two different mobility scenarios: ''(i) Node mobility between indoor or outdoor locations'': nodes belonging to indoor location or outdoor location only are considered to be similar, and they're calculated metrics value like RSSI, LQI, velocity, etc.; can be directly used for the handover of services to appropriate gateway, ''(ii) Node mobility between indoor and outdoor locations'': nodes are considered dissimilar, if they belong to indoor and outdoor location both, So, the metrics are re-calculated which introduce some additional parameters like temperature and interference signals in order to make a decision over handover gateway.
Babu and Biswash [36] proposed a mobility management technique that supports node-to-node communication and Fog computing-based architecture for 5G networks. It addresses the technical problems between 5G networks and Fog servers. The mobility-based approach assists mobile nodes in establishing communication while they are in motion. The mobility management technique may also be used to begin N2N communication in dynamic environments. N2N communication schemes for fog networks, on the other hand, provide an effective communication environment for mobile users with highly minimal network usage.
Wang et al. [37] proposed a mobility-aware offloading scheme, that considers an adequate quality and a computation allocation system that deals with the user equipment affairs to maximize the total revenue. The quality of user equipment is delineated by the sojourn time that follows the exponential distribution to reduce the chance of migration and maximize the entire income of user equipment. MILP (mixed-integer non-linear programming) NP-hard problem is modelled and consists of resource allocation and task offloading schemes. So, to solve this problem, a Gini coefficient and genetic algorithm are used to estimate the allocation of resources. The proposed approach can easily handle the mobility of users by minimizing the chances of migration.
Waqas et al. [38] provided a forward-thinking analysis of quality about-mobility in Fog computing by identifying quality challenges, requirements, and options for numerous ideas. The authors also identified outstanding concerns from previous research and summarized the advantages of quality for readers. It allows researchers and developers to avoid common misunderstandings and capture real-world scenarios such as businesses, governments, and educational institutions. Furthermore, it revolutionizes follow-up analysis and differentiates and foregrounds futurity orientations in real-life events involving humans and vehicles in a highly dynamic Fog setting.
Bi et al. [9] introduced software-defined networking-based fog computing architecture by decoupling mobility control and data forwarding. When mobile consumers travel between several access networks, the authors suggested an Optimal Path Selection (OPS) algorithm to preserve service continuity. Mobile customers received seamless and transparent mobility support thanks to efficient signalling operations. In mobile fog computing, the suggested algorithm ensured service continuity, increased handover performance, and achieved high data transfer efficiency.
Niu et al. [21] established a system called mobility-aware and multihop-D2D relaying-based scheduling scheme (MHRC) at Edge nodes near hotspots. The authors exploited concurrent transmissions to improve the performance of the system. The mmWave (millimetre-wave) band of Fog computing was cached, and extensive performance evaluation confirms that MHRC delivers more than the higher expected cached data amount. Name et al. [19] proposed an efficient algorithm to address the problem of resource allocation and user mobility from the Edge of the network to cloud data centres. This algorithm operates on a seamless handover scheme for mobile IPV6 to ease the user mobility challenge and reduce the application response time. The study showed that the task of service delay and packet loss was decreased due to the effect of change in the mobile node position.
Bittencourt et al. [23] examined the subject of resource allocation in the Fog/Cloud environment, taking into account the hierarchical structure. In the context of the Fog paradigm, the authors developed three scheduling algorithms (First come, First serve, delay-priority, and concurrent) that address user mobility and edge computing capabilities. The authors demonstrated that scheduling techniques may be designed to cope with different application classes based on demand from mobile users by leveraging both Fog to the end-user and cloud characteristics in this study.
Velasquez et al. [39] proposed a hybrid strategy for the Fog environment to manage resources for mobility scenarios. The authors applied the orchestrator technique to offer mobility support in a Smart City situation. In this technique, three components, the status monitor, the Planner, and the VM/Container, are employed to monitor, plan and execute the applications. The main aim of this study was to guarantee the QoS and QoE requirements of mobility-based applications and services.
Bittencourt et al. [31] presented a Fog computing architecture focusing on Virtual Machine (VM) migration where each user has a VM running in a cloudlet. In this architecture, the user's location is identified by using GPS, and then the VM is moved to a nearby Fog Cloud. The main aim of this study was to migrate users' data according to their mobility in order to maintain QoS for applications demanding lower Latency and allow smooth handoff mechanisms for mobile users.
From the extensive analysis of the literature, the various mobility-aware scheduling techniques have been classified as shown in Table 3. Further, it presents the advantages and limitations of each technique.
{| class="wikitable"
|+Table 3: Classification of Mobility-aware scheduling techniques
!Ref.
!Technique
!Advantage(s)
!Limitation(s)
|-
|[25]
|Ranking of VM
|
* Decrease in delay time, migration time, tuple lost value and downtime
|
* Case study not discussed
|-
|[26]
|S-OAMC, G-OAMC, Machine learning matrix completion
|
* Migration rate decreased
* Better Scalability
|
* Energy utilization of devices not investigated
|-
|[32]
|IM model
|
* Provides better mobility support and security
|
* Did not investigate synchronization overhead
|-
|[10]
|RSS
|
* Reliable and Heterogeneous execution
|
* Low scalability
* No distributed scheduling to minimize response time
|-
|[7]
|MAPE control loop
|
* Improved QoS
* Reduced service downtime
|
* No real-time evaluation
* High energy consumption
* Low robustness and security
|-
|[28]
|Copy of task to over a region
|
* Network Utilization developed
* Low-Loop delay
|
* Fault tolerance reliability is based on Fog gateways only
|-
|[18]
|URMILA
|
* Service availability is maintained by delivering the desired QoS
* Deployment cost minimized
* Battery longevity ensured
|
* No empirical validations
* No user probabilistic routes
* Low scalability in terms of distance and speed
|-
|[8]
|RSSI, LQI
|
* Promises to keep the connection active with a low latency rate between the system and sensor nodes
|
* Consumes more energy
* Overhead is large for network transmission
* Coverage and overhead area are undefined between gateways
|-
|[29]
|User trajectories pre- diction using GPS log
|
* Provides better mobility support
* Reduces migration time
|
* Low scalability
|-
|[36]
|N2N communication, Data Analytics
|
* Fast data access
* High reliability and scalability- city
* Minimum overhead and cost
* High throughput and less delay
|
* No real-time cellular network evaluation
* Low network efficiency
|-
|[38]
|Mobility facets analysis
|
* Improved QoS and QoE
* Latency rate reduced
|
* No real-life implementation
* No reliability and low Latency between dynamic users and fog servers
|-
|[37]
|M-ILP, Sojourn time
|
* Cost-effective
* Migration time reduced
|
* Migration cost not considered
* No real-time implementation
|-
|[34]
|User trajectories prediction using GPS log
|
* Conventional delay tolerance
* High QoS
* Avoided local task processing cost
* Efficient in saving battery
* Handles subtle scenarios with high Latency
|
* Smart city not directed through the use of accurate city maps with aid from stimulation setting
|-
|[35]
|Prediction of user location
|
* Power consumption and de- lay handled proficiently
* Power and delay are reduced
|
* No acquiring of mobile data usage where location sense and time-series data can be projected to achieve the bandwidth
|-
|[21]
|Relay path planning algorithm
|
* Power efficient
* High spectral efficiency
* Data is relayed on cached edge nodes and relay nodes
|
* Blockage problem due to weak diffraction
|-
|[9]
|OPS
|
* Handover performance improved
* Efficiency of high data communication achieved
* Guarantees continuity of services
|
* It does not guarantee privacy and security
* Virtual Machine migration not determined
* The handover process during the optimal path for more logical routing could have been more efficient
|-
|[23]
|Assignment of FS
|
* Low Latency
* Supports dynamic computing and user behaviour
|
* There is no prediction of mobility failure
* Bandwidth and processing not considered in scheduling
|-
|[39]
|Orchestrator
|
* Maintains trustworthiness, resilience, and low Latency in a dynamic environment
|
* No real implementation has been carried out
|-
|[19]
|Pattern modelling, dictating the following location
|
* Application Response time reduced
* Latency time reduced
|
* Services become temporarily inaccessible for some mobile nodes
|-
|[31]
|Forecasting technique
|
* Computing capacity provided for storage and processing of data
|
* Security concerns associated with both user data and applications not considered
|}
[[File:Fog Computing Figure 2.jpg|center|thumb|Figure 2: Year-wise count of research articles]]
== Analytical Discussion ==
The existing research on Fog computing Mobility-aware scheduling has been analyzed thoroughly. The analysis was performed using the answers given in Table 2. The results drawn through the thorough analysis of the literature are presented in various figures as follows:
Figure 2 lists the year-wise count of research papers that are considered for this survey. The bar graph represents the total number of research papers from journals and Conferences from the year 2015 - 2021. The research articles from the journal are 16, and the conference papers are 4. It is observed that more research needs to be conducted on mobility-aware scheduling in Fog computing.
Figure 3 displays an analytical comparison of mobility-aware scheduling approaches in Fog computing based on the content of the represented taxonomy in Figure 7. From the thorough analysis of the literature, four methods have been considered: migration, task offloading, handoff/handover mechanism, and task scheduling. The handoff/handover mechanism has the highest percentage of usage in mobility-aware scheduling, at 30%. The task scheduling and offloading have 25% of us- age in mobility-aware scheduling each. Finally, migration is only 20% of the usage in mobility-aware scheduling. Therefore, these approaches, specifically migration, are still open challenges to address for further research.
Figure 4 depicts various tools that were used for evaluating the mobility-aware scheduling approaches. 18% and 9% of the research articles used iFogSim and Mob-FogSim simulation tools for implementation, respectively. Besides, other simulation tools such as ONE (9%), NS2(5%), MATLAB(4%), Mininet(5%), and Docker(9%) have been utilized for implementing the proposed techniques in the research articles. Further, pro-Programming languages such as C++ (9%) and Python (9%) and hardware deployments such as Raspberry Pi (5%) and Ardunio (4%) were used for implementing existing case studies based on mobility-aware scheduling.
[[File:Fog Computing Figure 3.jpg|center|thumb|Figure 3: Percentage of the presented classified approaches in mobility-aware scheduling]]
The applied case studies are shown in Figure 5, which shows a maximum of 20% of research articles have implemented IoT-based applications. After that, 15% of each research article used Health care and Mobile-based applications. Besides, Smart City and 5G-based applications have been applied in 10% of research articles. Moreover, Surveillance and gaming, Mobile IPV6, and Wireless computing applications are the case studies on which only 5% of research articles exist.
After reviewing numerous research articles based on mobility-aware scheduling, it has been observed that researchers employed various parameters for evaluating the performance of the Mobility-scheduling approaches, as represented in Figure 6. It shows that Time completion (18%) followed by Delay (12%), Network usage (12%), Latency (12%), Energy consumption (10%), and cost (10%) are generally utilized. However, Downtime (4%), Migration time (4%), Makespan (2%), Success ratio (2%), Signal level (2%), Deadline (2%), Makespan (2%), Migration rate (2%), Mobility patterns (2%), Tuple lost (2%), and power consumption (2%) are less exploited parameters.
A taxonomy was compiled after going through the detailed review process, and various techniques have been categorized in Fog computing-based mobility-aware scheduling. Figure 7 presents these categories broadly in Migration, Offloading, Handoff/Handover mechanism, and Scheduling.
[[File:Fog Computing Figure 4.jpg|center|thumb|Figure 4: Percentage of tools utilized in the literature]]
[[File:Fog Computing Figure 5.jpg|center|thumb|Figure 5: Percentage of case studies employed in the literature]]
[[File:Fog Computing Figure 6.jpg|center|thumb|Figure 6: Percentage of parameters for evaluating Mobility-aware scheduling in the literature]]
=== Open Issues and Challenges ===
From the thorough analysis of the literature, several open issues and challenges pertaining to the area of mobility-aware scheduling in Fog computing have been identified in order to provide directions for future research exploration. The identified open problems and challenges, depicted in Figure 8, are discussed below.
=== Task Scheduling ===
Fog computing consists of several Fog nodes, each of which is a mini Cloud in the vicinity of mobile devices near the Edge of the network. When a mobile device submits a task, the Fog scheduler assigns it to a nearby Fog node(s) for execution. However, as the device moves from one network to another, the task needs to be rescheduled when the device enters a different network. Additionally, Fog nodes have limited capacity and availability; if the mobile user enters into a network where there is no nearby Fog service available, then this leads to a significant delay in service and raises a significant issue of task scheduling for mobile users [10, 40].
=== Resource Provisioning ===
Fog computing reduces the workload of Cloud computing by processing the tasks locally near the Edge of the network. However, due to the mobility of the user, the Fog node primarily assigned to a task might not be optimal over time. Therefore, the migration of the task to another Fog node near the user's mobile device is perceived as a necessary solution to resolve this concern [41]. However, such frequent migration over a short time poses the challenge of providing an efficient resource for the task that is capable of performing computation on time and delivering results to users while adhering to QoE.
===Subheading===
==Second Heading==
==Third Heading, etc==
==Additional information==
===Acknowledgements===
Any people, organisations, or funding sources that you would like to thank.
===Competing interests===
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===Ethics statement===
An ethics statement, if appropriate, on any animal or human research performed should be included here or in the methods section.
==References==
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{{Article info
| journal = WikiJournal of Science
| last1 = Islam
| orcid1 = 0000-0001-5750-1058
| first1 = Mir Salim Ul
| last2 = Kumar
| orcid2 = 0000-0003-3279-5111
| first2 = Ashok
| affiliation2 = Department of Computer Application, Chandigarh School of Business, Chandigarh Group of Colleges, Jhanjeri, Punjab, India
| et_al =
| affiliation1 = Department of Computer Science and Engineering, National Institute of Technology Srinagar, Jammu and Kashmir, India
| correspondence1 = mir.salim27@gmail.com
| affiliations =
| keywords = Fog computing, Mobility-aware, Scheduling, Resource Management, Resource Allocation
| license = CC BY 4.0
| abstract = The increasing focus on Fog-IoT results in billions of Internet-connected devices that demand substantial computational power and network bandwidth. These devices are geographically distributed, heterogeneous, computational capacity constrained, inconsistent in behaviour, and generally mobile. Therefore, providing seamless service, irrespective of location and movement of the devices as well as the end-users, makes resource scheduling a significant challenge in the Fog computing paradigm. Several mobility-aware scheduling strategies have been proposed in the literature to efficiently manage the resources for mobile users and devices in the Fog environment. This paper gives a survey of mobility-aware scheduling in the Fog computing environment. It describes the many strategies presented and their benefits and drawbacks. It also includes a complete study and taxonomy of the mobility-aware scheduling field. Further, it delineates open issues and challenges. This work will provide researchers with future research directions and aid them in recognizing the gaps before planning for further research in mobility-aware scheduling.
}}
== Introduction ==
With the advancement in technology and the exponential growth of mobile devices, network traffic has increased manifold in cloud computing. Due to this reason, Latency reduction and faster processing of data for mobile users have become critical challenges in providing seamless connectivity and minimal disruption while the user is moving.<ref>{{Cite journal|last=Abdullah|first=Fatima|last2=Kimovski|first2=Dragi|last3=Prodan|first3=Radu|last4=Munir|first4=Kashif|date=2021-06-11|title=Handover authentication latency reduction using mobile edge computing and mobility patterns|url=https://doi.org/10.1007/s00607-021-00969-z|journal=Computing|volume=103|issue=11|pages=2667–2686|doi=10.1007/s00607-021-00969-z|issn=0010-485X}}</ref> The data movement brings the additional issue of integrity and confidentiality because data is moving via a wireless connection to a far distant cloud. Additionally, due to the cloud's location is far from mobile users, so data movement is also affected by variable network strength and phone bandwidth. The solution proposed by [2] is to extend cloud capabilities through fog computing architecture. The Fog architecture allows substantial computation, storage, and processing using the Fog devices installed close to the user’s access point. Fog computing, therefore, reduces Latency and bandwidth consumption, improves security, provides context awareness, and renders more efficient services to mobile users [3, 4, 5].
[[File:Fog Computing Figure 1.jpg|left|thumb|Figure 1: Mobility Scenario in Fog Computing]]
However, mobility also imposes severe challenges for Fog computing due to its distributed and diverse environment. Mobility is recognized by either user-level or device-level contextual information [6]. As the user moves from one location to another, the geographical location of the smart devices also changes. The change in location of the devices raises the issue of searching and rescheduling with mobility management. Efficient re-scheduling requires a well-planned handoff mechanism that is accountable for smoothly de-registering a sensor node from a source access point where the application was initially hosted and registering it to a new access point. Figure 1 depicts these mentioned problems of change in access points while the user moves from one area to another. The services may also get interrupted when there is more distance between the Fog nodes and users [7]. Further, any disruption in communication may lead to an increase in Latency for mobile users [8].
The significant challenges in mobility management of Fog computing are: (i) ''Fog node discovery''- due to the heterogeneous and mobile nature of the smart mobile devices, the Fog task scheduler faces the issue of finding an optimal Fog node for scheduling the task, ''(ii) Handover mechanism between users and Fog nodes''- imagine a Fog user is moving from location to location and accessing information about his surroundings with the help of a smart device. Due to the frequent change in the user's location, the Fog scheduler may need to repeatedly migrate the user’s task to a different Fog node available in his vicinity. The frequent migration of tasks increases the overhead of scheduling and, further, the restricted signal strength in certain places may lead to a breakdown of task or result delivery, and (iii) ''the Handover mechanism between Cloud and Fog''- Fog nodes have limited capacities and need to continuously communicate with Cloud computing for passing information about the tasks [9]. Due to strict requirements for security, Latency, network coverage, and reliability, It becomes challenging to implement an efficient handover mechanism for full mobility support in critical domains such as healthcare [10, 11, 12] and vehicular systems [13, 14]. Thus, mobility significantly impacts the overhead of scheduling policies and the applications' performance, eventually affecting the Quality of Experience (QoE). Therefore, mobility-aware scheduling in Fog computing has observed strong attention from researchers. Our main goal is to provide a detailed review of mobility-aware scheduling in fog computing. The review provides a detailed analysis of existing scheduling strategies that concentrate specifically on mobility awareness in the Fog environment. The following are the main contributions of this paper:
* This paper presents a detailed survey of mobility-aware scheduling in the Fog computing environment.
* It provides the details of the different techniques proposed, as well as their advantages and limitations.
* It provides a detailed analysis and taxonomy of the mobility-aware scheduling field.
* It identifies several open challenges for future research directions.
=== Related Surveys ===
Numerous survey studies on Fog computing focus on resource management, job scheduling, and context-aware scheduling. For example, Ghobaei-Arani et al. [15] presented a survey on resource management techniques in fog computing in the form of taxonomy to highlight cutting-edge methods while also addressing unresolved challenges. The various authors in the cloud-fog area provided the task scheduling review. Their benefits and drawbacks, as well as numerous tools and challenges concerning the scheduling techniques and their limitations, were analyzed by Alizadeh et al. [16]. Islam et al. [6] thoroughly analyze relevant literature on context-aware scheduling in fog computing. Further, Mouradian et al. [17] review fog computing's significant issues and challenges. But, the critical area of mobility research is still in its initial stage, and most of the review papers contain very few documents on mobility-aware task scheduling in the area of fog computing. Therefore, an extensive and comparative study is required in mobility-constrained fog computing. A deep insight into various techniques which can impact the user QoS is necessary to understand mobility-aware fog computing. This motivates us to carry out a comprehensive survey; to the best of our knowledge, this is the first detailed survey. This survey paper thoroughly examines existing scheduling solutions that focus on user mobility. Moreover, various mobility-aware scheduling techniques are discussed, along with their pros and cons. Further, the impact of mobility parameters on various QoS parameters and context-awareness is also analyzed thoroughly.
=== Paper Organization ===
The remainder of this review article is divided into 5 sections: Section 2 discusses mobility-aware scheduling in Fog computing. Section 3 presents the review methodology. Section 4 analyzes and summarizes the considered research papers and compares existing mobility-aware scheduling strategies. Section 5 provides the results drawn after critically examining the existing literature on mobility-aware scheduling policies. Finally, Section 6 presents the conclusion.
== Mobility-aware Scheduling in Fog Computing ==
Mobile device management compromises the fundamental features of Fog computing because whenever a user moves, the distance between them increases, impacting the QoS. Therefore, to keep the computing fog node close to the associated mobile device, the services or tasks need to be migrated from one fog node to another appropriate fog device. The selection of such appropriate fog nodes in a mobile environment deals with two main processes: Estimation of user mobility patterns: User mobility estimation techniques can be probabilistic and deterministic [18]. In a deterministic method, the source and destination are known beforehand, whereas in a non-deterministic technique, periodic estimation has to be done regarding the user's route. Many authors estimate the route of mobile users by leveraging external services like Open Street Maps (<nowiki>http://www.openstreetmap.org</nowiki>), Google Maps APIs (<nowiki>https://cloud.google.com/</nowiki> maps-platform/) [18], logistics maps [19], GPRS Here APIs (<nowiki>https://developer</nowiki>. here.com/) [20, 21], Lyapunov estimation technique [22]. Specific QoS requirements: The selection of a fog node also depends upon specific quality requirements such as Latency, which many authors are using to select an efficient fog node [18, 10, 9], workload [23, 24], cost [16], etc.
== Review Methodology ==
The review process was conducted considering a review methodology consisting of four phases. The first phase was searching using traditional online database sources based on outlined search keywords. Table 1 lists the
Keywords used to find the relevant research articles to conduct the review in the Fog computing mobility-aware scheduling area. Second Phase: Limit the search of research articles beginning in 2015, and inclusion and exclusion principles are also used to refine research articles that specifically deal with mobility issues in task scheduling. Finally, in the Third Phase, A total of 20 papers are shortlisted for the review process. Further, Table 2 presents the research questions drafted for this study in mobility-aware scheduling in Fog computing.
{| class="wikitable"
|+Table 1: List of keywords used in the review process
!Sno
!Keyword
!Description
!Years
|-
|1
|Mobility
|Mobility-aware Fog task scheduling
| rowspan="5" |2015- 2021
|-
|2
|Mobility environment
|Mobility environment in Fog task scheduling
|-
|3
|Mobility factors
|Mobility factors in Fog task scheduling
|-
|4
|Mobility Awareness
|Mobility awareness in Fog task scheduling
|-
|5
|Mobility management
|Mobility management in Fog task scheduling
|}
{| class="wikitable"
|+Table 2: List of research questions used to complete the review process
!Q.No
!Research questions
|-
|1
|What scheduling approaches are used in the Fog computing environment to manage mobility awareness?
|-
|2
|What are the main limitations considered for mobility-aware scheduling techniques?
|-
|3
|Which case studies are applied to mobility-aware scheduling techniques?
|-
|4
|What evaluation tools are used to assess mobility-aware scheduling techniques?
|-
|5
|What performance indicators are utilized to evaluate mobility-aware scheduling techniques?
|-
|6
|What are the major open issues concerns in the field of mobility-aware scheduling for future research directions?
|}
=== Source of Information ===
To conduct this review, various online sources are listed below, and the research articles were searched using the different keywords mentioned in Table 1.
* Google Scholar (<nowiki>http://www.scholar.google.co.in</nowiki>)
* John Wiley & Sons Inc. (<nowiki>https://onlinelibrary.wiley.com/</nowiki>)
* Elsevier (<nowiki>https://www.elsevier.com/en-in</nowiki>)
* ACM Digital Library (<nowiki>https://www.acm.org/</nowiki>)
* Springer (<nowiki>https://www.springer.com/in</nowiki>)
* IEEE Xplore Digital Library (<nowiki>https://www.ieee.org/</nowiki>)
=== Quality Assessment ===
Research papers used quality assessment to filter out the most suitable mobility-based scheduling research articles in fog computing utilising the principle of inclusion and exclusion. Furthermore, in order to obtain high-quality research publications, the Center for Reviews and Dissemination (CRD) recommendations were followed, and each study item was examined for internal and external validation of results.
== Literature Analysis ==
Verma et al. [25] proposed a Server Cloudlet(SC) migration technique to handle users' mobility. The strategy is to select the target SC to migrate the services based on its highest rank. The rank of the SC further depends upon its available RAM, MIPS, and bandwidth.
Maleki et al. [26] designed two mobility-aware computation offloading approaches, sampling-based Online mobile applications to cloudlets(S-OAMC) and Greedy online mobile applications to cloudlets(G-OAMC). The developed system works in three major steps: ''(i)'' the future specifications of mobile applications are predicted using the Machine Learning (ML) method, named matrix completion. ''(ii)'' The predicted specifications of mobile applications, including future location prediction, also help estimate the offloading cost. ''(iii)'' Apply S-OAMC or G-OAMC offloading technique based on minimum cost value.
Abdelmoneem et al. [10] work on a fog-based healthcare architecture system that supports patient movement without altering their Quality of Service (QoS). The QoS in a mobile environment is maintained with the help of their proposed handoff mechanism. The authors modified the traditional Horizontal Handoff (HHO) mechanism [27] into two different phases: ''(i) handoff decision policy'': Received signal Strength (RSS) method is used to detect the network change information of the mobile patients and (ii) ''handoff strategy'': a handoff decision is made based on a comparison between RSS received from the old fog gateways, current threshold value and new fog gateways that are in close proximity to the mobile patient.
Javanmardi et al. [28] consider an imaginative city-based mobility scenario where the user is placing delay-sensitive service while moving. The authors proposed a task scheduling technique that jointly employs fuzzy logic and particle swarm optimization (PSO) algorithm to improve the QoS for mobile users within the city. The proposed algorithm is deployed at the Fog gateway, which further distributes the tasks to Fog devices available in an entire region in order to provide seamless service to mobile users. The main motive of the work is to improve resource utilization in a mobility-aware environment.
Puliafito et al. [29] develop an extension of ifogsim [30], which supports mobile environment. The authors implement different migration techniques inspired by Bittencourt et al. [31]. Different phases are devised to perform migration in a fog mobile environment, which: ''(i) Before migration phase'': migration decision is taken in this phase, based on specific parameters, like user location, speed, the direction of movement, zone, and migration point in order to select the appropriate cloudlet for offloading the services, ''(ii) During the migration phase'': this phase manages, monitors and synchronizes the whole selected migration process and ''(iii) After migration phase'': this phase involves closing the older cloudlet connections with the user and using the new cloudlet for services.
A Blockchain-based Mobility-aware Offloading (BMO) mechanism is designed by Dou et al. [32], where user mobility prediction is implemented using the Individual-Mobility (IM) model [33]. The idea behind the offloading mechanism is to shift the computational workload to different available Fog Servers (FSs) in the geo-location predicted by the IM model. Further, blockchain technology is deployed to check the authenticity of the forthcoming Fog servers. Finally, accounting is being managed by ''Fogcoin'', similar to Bitcoin, which stores the entire transaction history between the online Fog server and mobile users.
Martin et al. [7] proposed a framework that supports the migration of containers while satisfying the QoS requirements of mobile users. The migration of containers is done in an autonomic manner, by adopting the Monitor-Analyze-Plan-Execute (MAPE) autonomic control loop. The MAPE control loops discuss various steps of migration, like ''(i) Monitor'': used to constantly monitor the environment context, such as the mobility of users that is subsequently used to determine the need to migrate an application module to some other Fog node called target node; ''(ii) Analyze'': applies forecasting techniques to predict the user possible location in the next time step. If the distance between the user and the device is not under certain acceptable limits, a migration decision is made. ''(iii) Plan'': a Genetic Algorithm (GA) is used to identify a suitable Fog node closest to the forecasted location, where migration of the container running user application can be done. ''(iv) Execute'': this step ensures the whole migration process should take place smoothly.
Mass et al. propose a mobility and delay-aware fog server selection scheme. [34] called Edge-Process management (EPM) system. The EPM system depends upon the trajectory of a user’s movement, Fog server workload, and user location to select the appropriate Fog server for executing user applications. The system selects or re-selects a Fog Server (FS) based on a score value calculated through available bandwidth, power, distance from the user, and finally, duration of availability in a region.
Mobi-IoST (Mobility-aware Internet of Spatial Things), a real-time mobility-aware framework is presented by Ghosh et al. [35]. The authors considered the mobile nature of both IoT devices and Fog nodes, collaboratively called mobile agents. The proposed mobility-aware framework collects a vast amount of Global Positioning System (GPS) data of these mobile agents to predict their movement patterns using various machine learning algorithms. The major components of the framework are, ''(i) Movement pattern model''ling, collecting and modelling GPS log, stay-point, and other contextual location information; ''(ii) Predicting the following location'': human movement semantics is analyzed using all modelled information; ''(iii) Delivery of result:'' after the user movement prediction in the previous phase, the system intelligently discovers a capable fog device for data processing in a timely manner.
A middleware solution, URMILA, for managing resources and scheduling tasks in the Fog environment is presented by Shekhar et al. [18]. Ubiquitous Resource Management for Interference and Latency-Aware services (URMILA), ensures minimum Service Level Objectives (SLO) violation for latency-sensitive mobile applications across the cloud-Fog environment. The major modules of the proposed system are ''(i) Route calculation'', which calculates the user's possible routes using Google Maps or GPS data; ''(ii) Latency calculation'', the system deploys a data-driven model to estimate the Latency on predicted user routes; ''(iii) Fog node selection:'' the system selects a fog server for execution of task on the basis of its instantaneous utilization of the available resources. Further, it selects the Fog server for the entire period of execution, during which mobile users can still access their application through various Wireless Access Points (WAP).
Gia et al. [8] proposed a Handover mechanism for mobility management between fog nodes with the overall objective of consuming minimum energy and delay during handovers. Handover methods frequently rely on one or more measures, such as the Received Signal Strength Indicator (RSSI), the Link Quality Indicator (LQI), and the velocity of objects, to make handover decisions. This proposed system provides emergency services to health monitoring systems and basically works in two different mobility scenarios: ''(i) Node mobility between indoor or outdoor locations'': nodes belonging to indoor location or outdoor location only are considered to be similar, and they're calculated metrics value like RSSI, LQI, velocity, etc.; can be directly used for the handover of services to appropriate gateway, ''(ii) Node mobility between indoor and outdoor locations'': nodes are considered dissimilar, if they belong to indoor and outdoor location both, So, the metrics are re-calculated which introduce some additional parameters like temperature and interference signals in order to make a decision over handover gateway.
Babu and Biswash [36] proposed a mobility management technique that supports node-to-node communication and Fog computing-based architecture for 5G networks. It addresses the technical problems between 5G networks and Fog servers. The mobility-based approach assists mobile nodes in establishing communication while they are in motion. The mobility management technique may also be used to begin N2N communication in dynamic environments. N2N communication schemes for fog networks, on the other hand, provide an effective communication environment for mobile users with highly minimal network usage.
Wang et al. [37] proposed a mobility-aware offloading scheme, that considers an adequate quality and a computation allocation system that deals with the user equipment affairs to maximize the total revenue. The quality of user equipment is delineated by the sojourn time that follows the exponential distribution to reduce the chance of migration and maximize the entire income of user equipment. MILP (mixed-integer non-linear programming) NP-hard problem is modelled and consists of resource allocation and task offloading schemes. So, to solve this problem, a Gini coefficient and genetic algorithm are used to estimate the allocation of resources. The proposed approach can easily handle the mobility of users by minimizing the chances of migration.
Waqas et al. [38] provided a forward-thinking analysis of quality about-mobility in Fog computing by identifying quality challenges, requirements, and options for numerous ideas. The authors also identified outstanding concerns from previous research and summarized the advantages of quality for readers. It allows researchers and developers to avoid common misunderstandings and capture real-world scenarios such as businesses, governments, and educational institutions. Furthermore, it revolutionizes follow-up analysis and differentiates and foregrounds futurity orientations in real-life events involving humans and vehicles in a highly dynamic Fog setting.
Bi et al. [9] introduced software-defined networking-based fog computing architecture by decoupling mobility control and data forwarding. When mobile consumers travel between several access networks, the authors suggested an Optimal Path Selection (OPS) algorithm to preserve service continuity. Mobile customers received seamless and transparent mobility support thanks to efficient signalling operations. In mobile fog computing, the suggested algorithm ensured service continuity, increased handover performance, and achieved high data transfer efficiency.
Niu et al. [21] established a system called mobility-aware and multihop-D2D relaying-based scheduling scheme (MHRC) at Edge nodes near hotspots. The authors exploited concurrent transmissions to improve the performance of the system. The mmWave (millimetre-wave) band of Fog computing was cached, and extensive performance evaluation confirms that MHRC delivers more than the higher expected cached data amount. Name et al. [19] proposed an efficient algorithm to address the problem of resource allocation and user mobility from the Edge of the network to cloud data centres. This algorithm operates on a seamless handover scheme for mobile IPV6 to ease the user mobility challenge and reduce the application response time. The study showed that the task of service delay and packet loss was decreased due to the effect of change in the mobile node position.
Bittencourt et al. [23] examined the subject of resource allocation in the Fog/Cloud environment, taking into account the hierarchical structure. In the context of the Fog paradigm, the authors developed three scheduling algorithms (First come, First serve, delay-priority, and concurrent) that address user mobility and edge computing capabilities. The authors demonstrated that scheduling techniques may be designed to cope with different application classes based on demand from mobile users by leveraging both Fog to the end-user and cloud characteristics in this study.
Velasquez et al. [39] proposed a hybrid strategy for the Fog environment to manage resources for mobility scenarios. The authors applied the orchestrator technique to offer mobility support in a Smart City situation. In this technique, three components, the status monitor, the Planner, and the VM/Container, are employed to monitor, plan and execute the applications. The main aim of this study was to guarantee the QoS and QoE requirements of mobility-based applications and services.
Bittencourt et al. [31] presented a Fog computing architecture focusing on Virtual Machine (VM) migration where each user has a VM running in a cloudlet. In this architecture, the user's location is identified by using GPS, and then the VM is moved to a nearby Fog Cloud. The main aim of this study was to migrate users' data according to their mobility in order to maintain QoS for applications demanding lower Latency and allow smooth handoff mechanisms for mobile users.
From the extensive analysis of the literature, the various mobility-aware scheduling techniques have been classified as shown in Table 3. Further, it presents the advantages and limitations of each technique.
{| class="wikitable"
|+Table 3: Classification of Mobility-aware scheduling techniques
!Ref.
!Technique
!Advantage(s)
!Limitation(s)
|-
|[25]
|Ranking of VM
|
* Decrease in delay time, migration time, tuple lost value and downtime
|
* Case study not discussed
|-
|[26]
|S-OAMC, G-OAMC, Machine learning matrix completion
|
* Migration rate decreased
* Better Scalability
|
* Energy utilization of devices not investigated
|-
|[32]
|IM model
|
* Provides better mobility support and security
|
* Did not investigate synchronization overhead
|-
|[10]
|RSS
|
* Reliable and Heterogeneous execution
|
* Low scalability
* No distributed scheduling to minimize response time
|-
|[7]
|MAPE control loop
|
* Improved QoS
* Reduced service downtime
|
* No real-time evaluation
* High energy consumption
* Low robustness and security
|-
|[28]
|Copy of task to over a region
|
* Network Utilization developed
* Low-Loop delay
|
* Fault tolerance reliability is based on Fog gateways only
|-
|[18]
|URMILA
|
* Service availability is maintained by delivering the desired QoS
* Deployment cost minimized
* Battery longevity ensured
|
* No empirical validations
* No user probabilistic routes
* Low scalability in terms of distance and speed
|-
|[8]
|RSSI, LQI
|
* Promises to keep the connection active with a low latency rate between the system and sensor nodes
|
* Consumes more energy
* Overhead is large for network transmission
* Coverage and overhead area are undefined between gateways
|-
|[29]
|User trajectories pre- diction using GPS log
|
* Provides better mobility support
* Reduces migration time
|
* Low scalability
|-
|[36]
|N2N communication, Data Analytics
|
* Fast data access
* High reliability and scalability- city
* Minimum overhead and cost
* High throughput and less delay
|
* No real-time cellular network evaluation
* Low network efficiency
|-
|[38]
|Mobility facets analysis
|
* Improved QoS and QoE
* Latency rate reduced
|
* No real-life implementation
* No reliability and low Latency between dynamic users and fog servers
|-
|[37]
|M-ILP, Sojourn time
|
* Cost-effective
* Migration time reduced
|
* Migration cost not considered
* No real-time implementation
|-
|[34]
|User trajectories prediction using GPS log
|
* Conventional delay tolerance
* High QoS
* Avoided local task processing cost
* Efficient in saving battery
* Handles subtle scenarios with high Latency
|
* Smart city not directed through the use of accurate city maps with aid from stimulation setting
|-
|[35]
|Prediction of user location
|
* Power consumption and de- lay handled proficiently
* Power and delay are reduced
|
* No acquiring of mobile data usage where location sense and time-series data can be projected to achieve the bandwidth
|-
|[21]
|Relay path planning algorithm
|
* Power efficient
* High spectral efficiency
* Data is relayed on cached edge nodes and relay nodes
|
* Blockage problem due to weak diffraction
|-
|[9]
|OPS
|
* Handover performance improved
* Efficiency of high data communication achieved
* Guarantees continuity of services
|
* It does not guarantee privacy and security
* Virtual Machine migration not determined
* The handover process during the optimal path for more logical routing could have been more efficient
|-
|[23]
|Assignment of FS
|
* Low Latency
* Supports dynamic computing and user behaviour
|
* There is no prediction of mobility failure
* Bandwidth and processing not considered in scheduling
|-
|[39]
|Orchestrator
|
* Maintains trustworthiness, resilience, and low Latency in a dynamic environment
|
* No real implementation has been carried out
|-
|[19]
|Pattern modelling, dictating the following location
|
* Application Response time reduced
* Latency time reduced
|
* Services become temporarily inaccessible for some mobile nodes
|-
|[31]
|Forecasting technique
|
* Computing capacity provided for storage and processing of data
|
* Security concerns associated with both user data and applications not considered
|}
[[File:Fog Computing Figure 2.jpg|center|thumb|Figure 2: Year-wise count of research articles]]
== Analytical Discussion ==
The existing research on Fog computing Mobility-aware scheduling has been analyzed thoroughly. The analysis was performed using the answers given in Table 2. The results drawn through the thorough analysis of the literature are presented in various figures as follows:
Figure 2 lists the year-wise count of research papers that are considered for this survey. The bar graph represents the total number of research papers from journals and Conferences from the year 2015 - 2021. The research articles from the journal are 16, and the conference papers are 4. It is observed that more research needs to be conducted on mobility-aware scheduling in Fog computing.
Figure 3 displays an analytical comparison of mobility-aware scheduling approaches in Fog computing based on the content of the represented taxonomy in Figure 7. From the thorough analysis of the literature, four methods have been considered: migration, task offloading, handoff/handover mechanism, and task scheduling. The handoff/handover mechanism has the highest percentage of usage in mobility-aware scheduling, at 30%. The task scheduling and offloading have 25% of us- age in mobility-aware scheduling each. Finally, migration is only 20% of the usage in mobility-aware scheduling. Therefore, these approaches, specifically migration, are still open challenges to address for further research.
Figure 4 depicts various tools that were used for evaluating the mobility-aware scheduling approaches. 18% and 9% of the research articles used iFogSim and Mob-FogSim simulation tools for implementation, respectively. Besides, other simulation tools such as ONE (9%), NS2(5%), MATLAB(4%), Mininet(5%), and Docker(9%) have been utilized for implementing the proposed techniques in the research articles. Further, pro-Programming languages such as C++ (9%) and Python (9%) and hardware deployments such as Raspberry Pi (5%) and Ardunio (4%) were used for implementing existing case studies based on mobility-aware scheduling.
[[File:Fog Computing Figure 3.jpg|center|thumb|Figure 3: Percentage of the presented classified approaches in mobility-aware scheduling]]
The applied case studies are shown in Figure 5, which shows a maximum of 20% of research articles have implemented IoT-based applications. After that, 15% of each research article used Health care and Mobile-based applications. Besides, Smart City and 5G-based applications have been applied in 10% of research articles. Moreover, Surveillance and gaming, Mobile IPV6, and Wireless computing applications are the case studies on which only 5% of research articles exist.
After reviewing numerous research articles based on mobility-aware scheduling, it has been observed that researchers employed various parameters for evaluating the performance of the Mobility-scheduling approaches, as represented in Figure 6. It shows that Time completion (18%) followed by Delay (12%), Network usage (12%), Latency (12%), Energy consumption (10%), and cost (10%) are generally utilized. However, Downtime (4%), Migration time (4%), Makespan (2%), Success ratio (2%), Signal level (2%), Deadline (2%), Makespan (2%), Migration rate (2%), Mobility patterns (2%), Tuple lost (2%), and power consumption (2%) are less exploited parameters.
A taxonomy was compiled after going through the detailed review process, and various techniques have been categorized in Fog computing-based mobility-aware scheduling. Figure 7 presents these categories broadly in Migration, Offloading, Handoff/Handover mechanism, and Scheduling.
[[File:Fog Computing Figure 4.jpg|center|thumb|Figure 4: Percentage of tools utilized in the literature]]
[[File:Fog Computing Figure 5.jpg|center|thumb|Figure 5: Percentage of case studies employed in the literature]]
[[File:Fog Computing Figure 6.jpg|center|thumb|Figure 6: Percentage of parameters for evaluating Mobility-aware scheduling in the literature]]
=== Open Issues and Challenges ===
From the thorough analysis of the literature, several open issues and challenges pertaining to the area of mobility-aware scheduling in Fog computing have been identified in order to provide directions for future research exploration. The identified open problems and challenges, depicted in Figure 8, are discussed below.
=== Task Scheduling ===
Fog computing consists of several Fog nodes, each of which is a mini Cloud in the vicinity of mobile devices near the Edge of the network. When a mobile device submits a task, the Fog scheduler assigns it to a nearby Fog node(s) for execution. However, as the device moves from one network to another, the task needs to be rescheduled when the device enters a different network. Additionally, Fog nodes have limited capacity and availability; if the mobile user enters into a network where there is no nearby Fog service available, then this leads to a significant delay in service and raises a significant issue of task scheduling for mobile users [10, 40].
=== Resource Provisioning ===
Fog computing reduces the workload of Cloud computing by processing the tasks locally near the Edge of the network. However, due to the mobility of the user, the Fog node primarily assigned to a task might not be optimal over time. Therefore, the migration of the task to another Fog node near the user's mobile device is perceived as a necessary solution to resolve this concern [41]. However, such frequent migration over a short time poses the challenge of providing an efficient resource for the task that is capable of performing computation on time and delivering results to users while adhering to QoE.
[[File:Fog Computing Figure 7.jpg|center|thumb|Figure 7: Mobility-aware Fog Scheduling Taxonomy]]
=== Energy Consumption ===
The placement of fog services at the Edge of the network can provide better QoS to mobile users, resulting in a shorter response time. However, it is practically impossible due to the high deployment cost of new Fog devices, which further raises the significant issue of energy consumption. If too many deployments are done, there will be lots of communication traffic from the Cloud to Fog nodes and servers in order to create copies of the task from one network to another in case of mobility [42]. This results in considerable energy wastage in the form of high bandwidth consumption. This means that where and when to reschedule the task to an efficient Fog node must be carefully determined to minimize energy, response time, and deployment cost.
=== Quality of Experience (QoE) ===
Several mobility-based scheduling algorithms exist, but they need to focus on maximizing the user QoE [29, 8, 10, 18]. Further, they do not analyze the user performance; hence, the QoE of using a service or product is not determined. Therefore, to understand the user gain and loss, the scheduling algorithm needs to focus on enhancing the user QoE.
===Subheading===
==Second Heading==
==Third Heading, etc==
==Additional information==
===Acknowledgements===
Any people, organisations, or funding sources that you would like to thank.
===Competing interests===
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===Ethics statement===
An ethics statement, if appropriate, on any animal or human research performed should be included here or in the methods section.
==References==
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{{Article info
| journal = WikiJournal of Science
| last1 = Islam
| orcid1 = 0000-0001-5750-1058
| first1 = Mir Salim Ul
| last2 = Kumar
| orcid2 = 0000-0003-3279-5111
| first2 = Ashok
| affiliation2 = Department of Computer Application, Chandigarh School of Business, Chandigarh Group of Colleges, Jhanjeri, Punjab, India
| et_al =
| affiliation1 = Department of Computer Science and Engineering, National Institute of Technology Srinagar, Jammu and Kashmir, India
| correspondence1 = mir.salim27@gmail.com
| affiliations =
| keywords = Fog computing, Mobility-aware, Scheduling, Resource Management, Resource Allocation
| license = CC BY 4.0
| abstract = The increasing focus on Fog-IoT results in billions of Internet-connected devices that demand substantial computational power and network bandwidth. These devices are geographically distributed, heterogeneous, computational capacity constrained, inconsistent in behaviour, and generally mobile. Therefore, providing seamless service, irrespective of location and movement of the devices as well as the end-users, makes resource scheduling a significant challenge in the Fog computing paradigm. Several mobility-aware scheduling strategies have been proposed in the literature to efficiently manage the resources for mobile users and devices in the Fog environment. This paper gives a survey of mobility-aware scheduling in the Fog computing environment. It describes the many strategies presented and their benefits and drawbacks. It also includes a complete study and taxonomy of the mobility-aware scheduling field. Further, it delineates open issues and challenges. This work will provide researchers with future research directions and aid them in recognizing the gaps before planning for further research in mobility-aware scheduling.
}}
== Introduction ==
With the advancement in technology and the exponential growth of mobile devices, network traffic has increased manifold in cloud computing. Due to this reason, Latency reduction and faster processing of data for mobile users have become critical challenges in providing seamless connectivity and minimal disruption while the user is moving.<ref>{{Cite journal|last=Abdullah|first=Fatima|last2=Kimovski|first2=Dragi|last3=Prodan|first3=Radu|last4=Munir|first4=Kashif|date=2021-06-11|title=Handover authentication latency reduction using mobile edge computing and mobility patterns|url=https://doi.org/10.1007/s00607-021-00969-z|journal=Computing|volume=103|issue=11|pages=2667–2686|doi=10.1007/s00607-021-00969-z|issn=0010-485X}}</ref> The data movement brings the additional issue of integrity and confidentiality because data is moving via a wireless connection to a far distant cloud. Additionally, due to the cloud's location is far from mobile users, so data movement is also affected by variable network strength and phone bandwidth. The solution proposed by [2] is to extend cloud capabilities through fog computing architecture. The Fog architecture allows substantial computation, storage, and processing using the Fog devices installed close to the user’s access point. Fog computing, therefore, reduces Latency and bandwidth consumption, improves security, provides context awareness, and renders more efficient services to mobile users [3, 4, 5].
[[File:Fog Computing Figure 1.jpg|left|thumb|Figure 1: Mobility Scenario in Fog Computing]]
However, mobility also imposes severe challenges for Fog computing due to its distributed and diverse environment. Mobility is recognized by either user-level or device-level contextual information [6]. As the user moves from one location to another, the geographical location of the smart devices also changes. The change in location of the devices raises the issue of searching and rescheduling with mobility management. Efficient re-scheduling requires a well-planned handoff mechanism that is accountable for smoothly de-registering a sensor node from a source access point where the application was initially hosted and registering it to a new access point. Figure 1 depicts these mentioned problems of change in access points while the user moves from one area to another. The services may also get interrupted when there is more distance between the Fog nodes and users [7]. Further, any disruption in communication may lead to an increase in Latency for mobile users [8].
The significant challenges in mobility management of Fog computing are: (i) ''Fog node discovery''- due to the heterogeneous and mobile nature of the smart mobile devices, the Fog task scheduler faces the issue of finding an optimal Fog node for scheduling the task, ''(ii) Handover mechanism between users and Fog nodes''- imagine a Fog user is moving from location to location and accessing information about his surroundings with the help of a smart device. Due to the frequent change in the user's location, the Fog scheduler may need to repeatedly migrate the user’s task to a different Fog node available in his vicinity. The frequent migration of tasks increases the overhead of scheduling and, further, the restricted signal strength in certain places may lead to a breakdown of task or result delivery, and (iii) ''the Handover mechanism between Cloud and Fog''- Fog nodes have limited capacities and need to continuously communicate with Cloud computing for passing information about the tasks [9]. Due to strict requirements for security, Latency, network coverage, and reliability, It becomes challenging to implement an efficient handover mechanism for full mobility support in critical domains such as healthcare [10, 11, 12] and vehicular systems [13, 14]. Thus, mobility significantly impacts the overhead of scheduling policies and the applications' performance, eventually affecting the Quality of Experience (QoE). Therefore, mobility-aware scheduling in Fog computing has observed strong attention from researchers. Our main goal is to provide a detailed review of mobility-aware scheduling in fog computing. The review provides a detailed analysis of existing scheduling strategies that concentrate specifically on mobility awareness in the Fog environment. The following are the main contributions of this paper:
* This paper presents a detailed survey of mobility-aware scheduling in the Fog computing environment.
* It provides the details of the different techniques proposed, as well as their advantages and limitations.
* It provides a detailed analysis and taxonomy of the mobility-aware scheduling field.
* It identifies several open challenges for future research directions.
=== Related Surveys ===
Numerous survey studies on Fog computing focus on resource management, job scheduling, and context-aware scheduling. For example, Ghobaei-Arani et al. [15] presented a survey on resource management techniques in fog computing in the form of taxonomy to highlight cutting-edge methods while also addressing unresolved challenges. The various authors in the cloud-fog area provided the task scheduling review. Their benefits and drawbacks, as well as numerous tools and challenges concerning the scheduling techniques and their limitations, were analyzed by Alizadeh et al. [16]. Islam et al. [6] thoroughly analyze relevant literature on context-aware scheduling in fog computing. Further, Mouradian et al. [17] review fog computing's significant issues and challenges. But, the critical area of mobility research is still in its initial stage, and most of the review papers contain very few documents on mobility-aware task scheduling in the area of fog computing. Therefore, an extensive and comparative study is required in mobility-constrained fog computing. A deep insight into various techniques which can impact the user QoS is necessary to understand mobility-aware fog computing. This motivates us to carry out a comprehensive survey; to the best of our knowledge, this is the first detailed survey. This survey paper thoroughly examines existing scheduling solutions that focus on user mobility. Moreover, various mobility-aware scheduling techniques are discussed, along with their pros and cons. Further, the impact of mobility parameters on various QoS parameters and context-awareness is also analyzed thoroughly.
=== Paper Organization ===
The remainder of this review article is divided into 5 sections: Section 2 discusses mobility-aware scheduling in Fog computing. Section 3 presents the review methodology. Section 4 analyzes and summarizes the considered research papers and compares existing mobility-aware scheduling strategies. Section 5 provides the results drawn after critically examining the existing literature on mobility-aware scheduling policies. Finally, Section 6 presents the conclusion.
== Mobility-aware Scheduling in Fog Computing ==
Mobile device management compromises the fundamental features of Fog computing because whenever a user moves, the distance between them increases, impacting the QoS. Therefore, to keep the computing fog node close to the associated mobile device, the services or tasks need to be migrated from one fog node to another appropriate fog device. The selection of such appropriate fog nodes in a mobile environment deals with two main processes: Estimation of user mobility patterns: User mobility estimation techniques can be probabilistic and deterministic [18]. In a deterministic method, the source and destination are known beforehand, whereas in a non-deterministic technique, periodic estimation has to be done regarding the user's route. Many authors estimate the route of mobile users by leveraging external services like Open Street Maps (<nowiki>http://www.openstreetmap.org</nowiki>), Google Maps APIs (<nowiki>https://cloud.google.com/</nowiki> maps-platform/) [18], logistics maps [19], GPRS Here APIs (<nowiki>https://developer</nowiki>. here.com/) [20, 21], Lyapunov estimation technique [22]. Specific QoS requirements: The selection of a fog node also depends upon specific quality requirements such as Latency, which many authors are using to select an efficient fog node [18, 10, 9], workload [23, 24], cost [16], etc.
== Review Methodology ==
The review process was conducted considering a review methodology consisting of four phases. The first phase was searching using traditional online database sources based on outlined search keywords. Table 1 lists the
Keywords used to find the relevant research articles to conduct the review in the Fog computing mobility-aware scheduling area. Second Phase: Limit the search of research articles beginning in 2015, and inclusion and exclusion principles are also used to refine research articles that specifically deal with mobility issues in task scheduling. Finally, in the Third Phase, A total of 20 papers are shortlisted for the review process. Further, Table 2 presents the research questions drafted for this study in mobility-aware scheduling in Fog computing.
{| class="wikitable"
|+Table 1: List of keywords used in the review process
!Sno
!Keyword
!Description
!Years
|-
|1
|Mobility
|Mobility-aware Fog task scheduling
| rowspan="5" |2015- 2021
|-
|2
|Mobility environment
|Mobility environment in Fog task scheduling
|-
|3
|Mobility factors
|Mobility factors in Fog task scheduling
|-
|4
|Mobility Awareness
|Mobility awareness in Fog task scheduling
|-
|5
|Mobility management
|Mobility management in Fog task scheduling
|}
{| class="wikitable"
|+Table 2: List of research questions used to complete the review process
!Q.No
!Research questions
|-
|1
|What scheduling approaches are used in the Fog computing environment to manage mobility awareness?
|-
|2
|What are the main limitations considered for mobility-aware scheduling techniques?
|-
|3
|Which case studies are applied to mobility-aware scheduling techniques?
|-
|4
|What evaluation tools are used to assess mobility-aware scheduling techniques?
|-
|5
|What performance indicators are utilized to evaluate mobility-aware scheduling techniques?
|-
|6
|What are the major open issues concerns in the field of mobility-aware scheduling for future research directions?
|}
=== Source of Information ===
To conduct this review, various online sources are listed below, and the research articles were searched using the different keywords mentioned in Table 1.
* Google Scholar (<nowiki>http://www.scholar.google.co.in</nowiki>)
* John Wiley & Sons Inc. (<nowiki>https://onlinelibrary.wiley.com/</nowiki>)
* Elsevier (<nowiki>https://www.elsevier.com/en-in</nowiki>)
* ACM Digital Library (<nowiki>https://www.acm.org/</nowiki>)
* Springer (<nowiki>https://www.springer.com/in</nowiki>)
* IEEE Xplore Digital Library (<nowiki>https://www.ieee.org/</nowiki>)
=== Quality Assessment ===
Research papers used quality assessment to filter out the most suitable mobility-based scheduling research articles in fog computing utilising the principle of inclusion and exclusion. Furthermore, in order to obtain high-quality research publications, the Center for Reviews and Dissemination (CRD) recommendations were followed, and each study item was examined for internal and external validation of results.
== Literature Analysis ==
Verma et al. [25] proposed a Server Cloudlet(SC) migration technique to handle users' mobility. The strategy is to select the target SC to migrate the services based on its highest rank. The rank of the SC further depends upon its available RAM, MIPS, and bandwidth.
Maleki et al. [26] designed two mobility-aware computation offloading approaches, sampling-based Online mobile applications to cloudlets(S-OAMC) and Greedy online mobile applications to cloudlets(G-OAMC). The developed system works in three major steps: ''(i)'' the future specifications of mobile applications are predicted using the Machine Learning (ML) method, named matrix completion. ''(ii)'' The predicted specifications of mobile applications, including future location prediction, also help estimate the offloading cost. ''(iii)'' Apply S-OAMC or G-OAMC offloading technique based on minimum cost value.
Abdelmoneem et al. [10] work on a fog-based healthcare architecture system that supports patient movement without altering their Quality of Service (QoS). The QoS in a mobile environment is maintained with the help of their proposed handoff mechanism. The authors modified the traditional Horizontal Handoff (HHO) mechanism [27] into two different phases: ''(i) handoff decision policy'': Received signal Strength (RSS) method is used to detect the network change information of the mobile patients and (ii) ''handoff strategy'': a handoff decision is made based on a comparison between RSS received from the old fog gateways, current threshold value and new fog gateways that are in close proximity to the mobile patient.
Javanmardi et al. [28] consider an imaginative city-based mobility scenario where the user is placing delay-sensitive service while moving. The authors proposed a task scheduling technique that jointly employs fuzzy logic and particle swarm optimization (PSO) algorithm to improve the QoS for mobile users within the city. The proposed algorithm is deployed at the Fog gateway, which further distributes the tasks to Fog devices available in an entire region in order to provide seamless service to mobile users. The main motive of the work is to improve resource utilization in a mobility-aware environment.
Puliafito et al. [29] develop an extension of ifogsim [30], which supports mobile environment. The authors implement different migration techniques inspired by Bittencourt et al. [31]. Different phases are devised to perform migration in a fog mobile environment, which: ''(i) Before migration phase'': migration decision is taken in this phase, based on specific parameters, like user location, speed, the direction of movement, zone, and migration point in order to select the appropriate cloudlet for offloading the services, ''(ii) During the migration phase'': this phase manages, monitors and synchronizes the whole selected migration process and ''(iii) After migration phase'': this phase involves closing the older cloudlet connections with the user and using the new cloudlet for services.
A Blockchain-based Mobility-aware Offloading (BMO) mechanism is designed by Dou et al. [32], where user mobility prediction is implemented using the Individual-Mobility (IM) model [33]. The idea behind the offloading mechanism is to shift the computational workload to different available Fog Servers (FSs) in the geo-location predicted by the IM model. Further, blockchain technology is deployed to check the authenticity of the forthcoming Fog servers. Finally, accounting is being managed by ''Fogcoin'', similar to Bitcoin, which stores the entire transaction history between the online Fog server and mobile users.
Martin et al. [7] proposed a framework that supports the migration of containers while satisfying the QoS requirements of mobile users. The migration of containers is done in an autonomic manner, by adopting the Monitor-Analyze-Plan-Execute (MAPE) autonomic control loop. The MAPE control loops discuss various steps of migration, like ''(i) Monitor'': used to constantly monitor the environment context, such as the mobility of users that is subsequently used to determine the need to migrate an application module to some other Fog node called target node; ''(ii) Analyze'': applies forecasting techniques to predict the user possible location in the next time step. If the distance between the user and the device is not under certain acceptable limits, a migration decision is made. ''(iii) Plan'': a Genetic Algorithm (GA) is used to identify a suitable Fog node closest to the forecasted location, where migration of the container running user application can be done. ''(iv) Execute'': this step ensures the whole migration process should take place smoothly.
Mass et al. propose a mobility and delay-aware fog server selection scheme. [34] called Edge-Process management (EPM) system. The EPM system depends upon the trajectory of a user’s movement, Fog server workload, and user location to select the appropriate Fog server for executing user applications. The system selects or re-selects a Fog Server (FS) based on a score value calculated through available bandwidth, power, distance from the user, and finally, duration of availability in a region.
Mobi-IoST (Mobility-aware Internet of Spatial Things), a real-time mobility-aware framework is presented by Ghosh et al. [35]. The authors considered the mobile nature of both IoT devices and Fog nodes, collaboratively called mobile agents. The proposed mobility-aware framework collects a vast amount of Global Positioning System (GPS) data of these mobile agents to predict their movement patterns using various machine learning algorithms. The major components of the framework are, ''(i) Movement pattern model''ling, collecting and modelling GPS log, stay-point, and other contextual location information; ''(ii) Predicting the following location'': human movement semantics is analyzed using all modelled information; ''(iii) Delivery of result:'' after the user movement prediction in the previous phase, the system intelligently discovers a capable fog device for data processing in a timely manner.
A middleware solution, URMILA, for managing resources and scheduling tasks in the Fog environment is presented by Shekhar et al. [18]. Ubiquitous Resource Management for Interference and Latency-Aware services (URMILA), ensures minimum Service Level Objectives (SLO) violation for latency-sensitive mobile applications across the cloud-Fog environment. The major modules of the proposed system are ''(i) Route calculation'', which calculates the user's possible routes using Google Maps or GPS data; ''(ii) Latency calculation'', the system deploys a data-driven model to estimate the Latency on predicted user routes; ''(iii) Fog node selection:'' the system selects a fog server for execution of task on the basis of its instantaneous utilization of the available resources. Further, it selects the Fog server for the entire period of execution, during which mobile users can still access their application through various Wireless Access Points (WAP).
Gia et al. [8] proposed a Handover mechanism for mobility management between fog nodes with the overall objective of consuming minimum energy and delay during handovers. Handover methods frequently rely on one or more measures, such as the Received Signal Strength Indicator (RSSI), the Link Quality Indicator (LQI), and the velocity of objects, to make handover decisions. This proposed system provides emergency services to health monitoring systems and basically works in two different mobility scenarios: ''(i) Node mobility between indoor or outdoor locations'': nodes belonging to indoor location or outdoor location only are considered to be similar, and they're calculated metrics value like RSSI, LQI, velocity, etc.; can be directly used for the handover of services to appropriate gateway, ''(ii) Node mobility between indoor and outdoor locations'': nodes are considered dissimilar, if they belong to indoor and outdoor location both, So, the metrics are re-calculated which introduce some additional parameters like temperature and interference signals in order to make a decision over handover gateway.
Babu and Biswash [36] proposed a mobility management technique that supports node-to-node communication and Fog computing-based architecture for 5G networks. It addresses the technical problems between 5G networks and Fog servers. The mobility-based approach assists mobile nodes in establishing communication while they are in motion. The mobility management technique may also be used to begin N2N communication in dynamic environments. N2N communication schemes for fog networks, on the other hand, provide an effective communication environment for mobile users with highly minimal network usage.
Wang et al. [37] proposed a mobility-aware offloading scheme, that considers an adequate quality and a computation allocation system that deals with the user equipment affairs to maximize the total revenue. The quality of user equipment is delineated by the sojourn time that follows the exponential distribution to reduce the chance of migration and maximize the entire income of user equipment. MILP (mixed-integer non-linear programming) NP-hard problem is modelled and consists of resource allocation and task offloading schemes. So, to solve this problem, a Gini coefficient and genetic algorithm are used to estimate the allocation of resources. The proposed approach can easily handle the mobility of users by minimizing the chances of migration.
Waqas et al. [38] provided a forward-thinking analysis of quality about-mobility in Fog computing by identifying quality challenges, requirements, and options for numerous ideas. The authors also identified outstanding concerns from previous research and summarized the advantages of quality for readers. It allows researchers and developers to avoid common misunderstandings and capture real-world scenarios such as businesses, governments, and educational institutions. Furthermore, it revolutionizes follow-up analysis and differentiates and foregrounds futurity orientations in real-life events involving humans and vehicles in a highly dynamic Fog setting.
Bi et al. [9] introduced software-defined networking-based fog computing architecture by decoupling mobility control and data forwarding. When mobile consumers travel between several access networks, the authors suggested an Optimal Path Selection (OPS) algorithm to preserve service continuity. Mobile customers received seamless and transparent mobility support thanks to efficient signalling operations. In mobile fog computing, the suggested algorithm ensured service continuity, increased handover performance, and achieved high data transfer efficiency.
Niu et al. [21] established a system called mobility-aware and multihop-D2D relaying-based scheduling scheme (MHRC) at Edge nodes near hotspots. The authors exploited concurrent transmissions to improve the performance of the system. The mmWave (millimetre-wave) band of Fog computing was cached, and extensive performance evaluation confirms that MHRC delivers more than the higher expected cached data amount. Name et al. [19] proposed an efficient algorithm to address the problem of resource allocation and user mobility from the Edge of the network to cloud data centres. This algorithm operates on a seamless handover scheme for mobile IPV6 to ease the user mobility challenge and reduce the application response time. The study showed that the task of service delay and packet loss was decreased due to the effect of change in the mobile node position.
Bittencourt et al. [23] examined the subject of resource allocation in the Fog/Cloud environment, taking into account the hierarchical structure. In the context of the Fog paradigm, the authors developed three scheduling algorithms (First come, First serve, delay-priority, and concurrent) that address user mobility and edge computing capabilities. The authors demonstrated that scheduling techniques may be designed to cope with different application classes based on demand from mobile users by leveraging both Fog to the end-user and cloud characteristics in this study.
Velasquez et al. [39] proposed a hybrid strategy for the Fog environment to manage resources for mobility scenarios. The authors applied the orchestrator technique to offer mobility support in a Smart City situation. In this technique, three components, the status monitor, the Planner, and the VM/Container, are employed to monitor, plan and execute the applications. The main aim of this study was to guarantee the QoS and QoE requirements of mobility-based applications and services.
Bittencourt et al. [31] presented a Fog computing architecture focusing on Virtual Machine (VM) migration where each user has a VM running in a cloudlet. In this architecture, the user's location is identified by using GPS, and then the VM is moved to a nearby Fog Cloud. The main aim of this study was to migrate users' data according to their mobility in order to maintain QoS for applications demanding lower Latency and allow smooth handoff mechanisms for mobile users.
From the extensive analysis of the literature, the various mobility-aware scheduling techniques have been classified as shown in Table 3. Further, it presents the advantages and limitations of each technique.
{| class="wikitable"
|+Table 3: Classification of Mobility-aware scheduling techniques
!Ref.
!Technique
!Advantage(s)
!Limitation(s)
|-
|[25]
|Ranking of VM
|
* Decrease in delay time, migration time, tuple lost value and downtime
|
* Case study not discussed
|-
|[26]
|S-OAMC, G-OAMC, Machine learning matrix completion
|
* Migration rate decreased
* Better Scalability
|
* Energy utilization of devices not investigated
|-
|[32]
|IM model
|
* Provides better mobility support and security
|
* Did not investigate synchronization overhead
|-
|[10]
|RSS
|
* Reliable and Heterogeneous execution
|
* Low scalability
* No distributed scheduling to minimize response time
|-
|[7]
|MAPE control loop
|
* Improved QoS
* Reduced service downtime
|
* No real-time evaluation
* High energy consumption
* Low robustness and security
|-
|[28]
|Copy of task to over a region
|
* Network Utilization developed
* Low-Loop delay
|
* Fault tolerance reliability is based on Fog gateways only
|-
|[18]
|URMILA
|
* Service availability is maintained by delivering the desired QoS
* Deployment cost minimized
* Battery longevity ensured
|
* No empirical validations
* No user probabilistic routes
* Low scalability in terms of distance and speed
|-
|[8]
|RSSI, LQI
|
* Promises to keep the connection active with a low latency rate between the system and sensor nodes
|
* Consumes more energy
* Overhead is large for network transmission
* Coverage and overhead area are undefined between gateways
|-
|[29]
|User trajectories pre- diction using GPS log
|
* Provides better mobility support
* Reduces migration time
|
* Low scalability
|-
|[36]
|N2N communication, Data Analytics
|
* Fast data access
* High reliability and scalability- city
* Minimum overhead and cost
* High throughput and less delay
|
* No real-time cellular network evaluation
* Low network efficiency
|-
|[38]
|Mobility facets analysis
|
* Improved QoS and QoE
* Latency rate reduced
|
* No real-life implementation
* No reliability and low Latency between dynamic users and fog servers
|-
|[37]
|M-ILP, Sojourn time
|
* Cost-effective
* Migration time reduced
|
* Migration cost not considered
* No real-time implementation
|-
|[34]
|User trajectories prediction using GPS log
|
* Conventional delay tolerance
* High QoS
* Avoided local task processing cost
* Efficient in saving battery
* Handles subtle scenarios with high Latency
|
* Smart city not directed through the use of accurate city maps with aid from stimulation setting
|-
|[35]
|Prediction of user location
|
* Power consumption and de- lay handled proficiently
* Power and delay are reduced
|
* No acquiring of mobile data usage where location sense and time-series data can be projected to achieve the bandwidth
|-
|[21]
|Relay path planning algorithm
|
* Power efficient
* High spectral efficiency
* Data is relayed on cached edge nodes and relay nodes
|
* Blockage problem due to weak diffraction
|-
|[9]
|OPS
|
* Handover performance improved
* Efficiency of high data communication achieved
* Guarantees continuity of services
|
* It does not guarantee privacy and security
* Virtual Machine migration not determined
* The handover process during the optimal path for more logical routing could have been more efficient
|-
|[23]
|Assignment of FS
|
* Low Latency
* Supports dynamic computing and user behaviour
|
* There is no prediction of mobility failure
* Bandwidth and processing not considered in scheduling
|-
|[39]
|Orchestrator
|
* Maintains trustworthiness, resilience, and low Latency in a dynamic environment
|
* No real implementation has been carried out
|-
|[19]
|Pattern modelling, dictating the following location
|
* Application Response time reduced
* Latency time reduced
|
* Services become temporarily inaccessible for some mobile nodes
|-
|[31]
|Forecasting technique
|
* Computing capacity provided for storage and processing of data
|
* Security concerns associated with both user data and applications not considered
|}
[[File:Fog Computing Figure 2.jpg|center|thumb|Figure 2: Year-wise count of research articles]]
== Analytical Discussion ==
The existing research on Fog computing Mobility-aware scheduling has been analyzed thoroughly. The analysis was performed using the answers given in Table 2. The results drawn through the thorough analysis of the literature are presented in various figures as follows:
Figure 2 lists the year-wise count of research papers that are considered for this survey. The bar graph represents the total number of research papers from journals and Conferences from the year 2015 - 2021. The research articles from the journal are 16, and the conference papers are 4. It is observed that more research needs to be conducted on mobility-aware scheduling in Fog computing.
Figure 3 displays an analytical comparison of mobility-aware scheduling approaches in Fog computing based on the content of the represented taxonomy in Figure 7. From the thorough analysis of the literature, four methods have been considered: migration, task offloading, handoff/handover mechanism, and task scheduling. The handoff/handover mechanism has the highest percentage of usage in mobility-aware scheduling, at 30%. The task scheduling and offloading have 25% of us- age in mobility-aware scheduling each. Finally, migration is only 20% of the usage in mobility-aware scheduling. Therefore, these approaches, specifically migration, are still open challenges to address for further research.
Figure 4 depicts various tools that were used for evaluating the mobility-aware scheduling approaches. 18% and 9% of the research articles used iFogSim and Mob-FogSim simulation tools for implementation, respectively. Besides, other simulation tools such as ONE (9%), NS2(5%), MATLAB(4%), Mininet(5%), and Docker(9%) have been utilized for implementing the proposed techniques in the research articles. Further, pro-Programming languages such as C++ (9%) and Python (9%) and hardware deployments such as Raspberry Pi (5%) and Ardunio (4%) were used for implementing existing case studies based on mobility-aware scheduling.
[[File:Fog Computing Figure 3.jpg|center|thumb|Figure 3: Percentage of the presented classified approaches in mobility-aware scheduling]]
The applied case studies are shown in Figure 5, which shows a maximum of 20% of research articles have implemented IoT-based applications. After that, 15% of each research article used Health care and Mobile-based applications. Besides, Smart City and 5G-based applications have been applied in 10% of research articles. Moreover, Surveillance and gaming, Mobile IPV6, and Wireless computing applications are the case studies on which only 5% of research articles exist.
After reviewing numerous research articles based on mobility-aware scheduling, it has been observed that researchers employed various parameters for evaluating the performance of the Mobility-scheduling approaches, as represented in Figure 6. It shows that Time completion (18%) followed by Delay (12%), Network usage (12%), Latency (12%), Energy consumption (10%), and cost (10%) are generally utilized. However, Downtime (4%), Migration time (4%), Makespan (2%), Success ratio (2%), Signal level (2%), Deadline (2%), Makespan (2%), Migration rate (2%), Mobility patterns (2%), Tuple lost (2%), and power consumption (2%) are less exploited parameters.
A taxonomy was compiled after going through the detailed review process, and various techniques have been categorized in Fog computing-based mobility-aware scheduling. Figure 7 presents these categories broadly in Migration, Offloading, Handoff/Handover mechanism, and Scheduling.
[[File:Fog Computing Figure 4.jpg|center|thumb|Figure 4: Percentage of tools utilized in the literature]]
[[File:Fog Computing Figure 5.jpg|center|thumb|Figure 5: Percentage of case studies employed in the literature]]
[[File:Fog Computing Figure 6.jpg|center|thumb|Figure 6: Percentage of parameters for evaluating Mobility-aware scheduling in the literature]]
=== Open Issues and Challenges ===
From the thorough analysis of the literature, several open issues and challenges pertaining to the area of mobility-aware scheduling in Fog computing have been identified in order to provide directions for future research exploration. The identified open problems and challenges, depicted in Figure 8, are discussed below.
=== Task Scheduling ===
Fog computing consists of several Fog nodes, each of which is a mini Cloud in the vicinity of mobile devices near the Edge of the network. When a mobile device submits a task, the Fog scheduler assigns it to a nearby Fog node(s) for execution. However, as the device moves from one network to another, the task needs to be rescheduled when the device enters a different network. Additionally, Fog nodes have limited capacity and availability; if the mobile user enters into a network where there is no nearby Fog service available, then this leads to a significant delay in service and raises a significant issue of task scheduling for mobile users [10, 40].
=== Resource Provisioning ===
Fog computing reduces the workload of Cloud computing by processing the tasks locally near the Edge of the network. However, due to the mobility of the user, the Fog node primarily assigned to a task might not be optimal over time. Therefore, the migration of the task to another Fog node near the user's mobile device is perceived as a necessary solution to resolve this concern [41]. However, such frequent migration over a short time poses the challenge of providing an efficient resource for the task that is capable of performing computation on time and delivering results to users while adhering to QoE.
[[File:Fog Computing Figure 7.jpg|center|thumb|Figure 7: Mobility-aware Fog Scheduling Taxonomy]]
=== Energy Consumption ===
The placement of fog services at the Edge of the network can provide better QoS to mobile users, resulting in a shorter response time. However, it is practically impossible due to the high deployment cost of new Fog devices, which further raises the significant issue of energy consumption. If too many deployments are done, there will be lots of communication traffic from the Cloud to Fog nodes and servers in order to create copies of the task from one network to another in case of mobility [42]. This results in considerable energy wastage in the form of high bandwidth consumption. This means that where and when to reschedule the task to an efficient Fog node must be carefully determined to minimize energy, response time, and deployment cost.
=== Quality of Experience (QoE) ===
[[File:Fog Computing Figure 8.jpg|left|thumb|Figure 8: Mobility-aware scheduling open issues and challenges]]
Several mobility-based scheduling algorithms exist, but they need to focus on maximizing the user QoE [29, 8, 10, 18]. Further, they do not analyze the user performance; hence, the QoE of using a service or product is not determined. Therefore, to understand the user gain and loss, the scheduling algorithm needs to focus on enhancing the user QoE.
=== Resource Management ===
The mobility of Fog nodes/users demands efficient resource discovery and sharing, resource availability, and task offloading [43]. Few techniques that were proposed to manage the resources effectively did not consider more constraints such as density, latency sensitivity, and mobility of Edge and Fog nodes, and as the number of nodes increases, issues such as scalability and distributing the algorithms arise [44, 45, 46]. Therefore, more attention needs to be paid towards the mobile Fog computing environment to manage the resources effectively.
=== Privacy and Security ===
In [47], a scheduling policy is proposed for the mobile device system to minimize the cost. However, the privacy issues of location and usage patterns were ignored. Additionally, data privacy, access control, and intrusion detection in scheduling policies have been overlooked [7, 48, 28]. Besides, Fog node devices are normally deployed near the end-user; hence, protection and surveillance are comparatively weak, which can result in a malicious attack [49, 50].
== Data availability statement ==
Not applicable.
== Conclusions ==
Fog computing infrastructure provides services at the Edge of the network. So, to provide support for scheduling and management of mobility awareness, efficient techniques and mechanisms have been proposed. In this survey, research articles on the mobility-aware-scheduling strategies in Fog computing have been thoroughly analyzed. It provides a comparative study among existing mobility-aware scheduling strategies based on vital factors such as techniques proposed, parameters considered, tools utilized for implementation, and case studies considered, along with the advantages and limitations. Further, several open issues and challenges have been identified for future research direction.
===Subheading===
==Second Heading==
==Third Heading, etc==
==Additional information==
===Acknowledgements===
Any people, organisations, or funding sources that you would like to thank.
===Competing interests===
Any conflicts of interest that you would like to declare. Otherwise, a statement that the authors have no competing interest.
===Ethics statement===
An ethics statement, if appropriate, on any animal or human research performed should be included here or in the methods section.
==References==
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{{Article info
| journal = WikiJournal of Science
| last1 = Islam
| orcid1 = 0000-0001-5750-1058
| first1 = Mir Salim Ul
| last2 = Kumar
| orcid2 = 0000-0003-3279-5111
| first2 = Ashok
| affiliation2 = Department of Computer Application, Chandigarh School of Business, Chandigarh Group of Colleges, Jhanjeri, Punjab, India
| et_al =
| affiliation1 = Department of Computer Science and Engineering, National Institute of Technology Srinagar, Jammu and Kashmir, India
| correspondence1 = mir.salim27@gmail.com
| affiliations =
| keywords = Fog computing, Mobility-aware, Scheduling, Resource Management, Resource Allocation
| license = CC BY 4.0
| abstract = The increasing focus on Fog-IoT results in billions of Internet-connected devices that demand substantial computational power and network bandwidth. These devices are geographically distributed, heterogeneous, computational capacity constrained, inconsistent in behaviour, and generally mobile. Therefore, providing seamless service, irrespective of location and movement of the devices as well as the end-users, makes resource scheduling a significant challenge in the Fog computing paradigm. Several mobility-aware scheduling strategies have been proposed in the literature to efficiently manage the resources for mobile users and devices in the Fog environment. This paper gives a survey of mobility-aware scheduling in the Fog computing environment. It describes the many strategies presented and their benefits and drawbacks. It also includes a complete study and taxonomy of the mobility-aware scheduling field. Further, it delineates open issues and challenges. This work will provide researchers with future research directions and aid them in recognizing the gaps before planning for further research in mobility-aware scheduling.
}}
== Introduction ==
With the advancement in technology and the exponential growth of mobile devices, network traffic has increased manifold in cloud computing. Due to this reason, Latency reduction and faster processing of data for mobile users have become critical challenges in providing seamless connectivity and minimal disruption while the user is moving.<ref>{{Cite journal|last=Abdullah|first=Fatima|last2=Kimovski|first2=Dragi|last3=Prodan|first3=Radu|last4=Munir|first4=Kashif|date=2021-06-11|title=Handover authentication latency reduction using mobile edge computing and mobility patterns|url=https://doi.org/10.1007/s00607-021-00969-z|journal=Computing|volume=103|issue=11|pages=2667–2686|doi=10.1007/s00607-021-00969-z|issn=0010-485X}}</ref> The data movement brings the additional issue of integrity and confidentiality because data is moving via a wireless connection to a far distant cloud. Additionally, due to the cloud's location is far from mobile users, so data movement is also affected by variable network strength and phone bandwidth. The solution proposed by [2] is to extend cloud capabilities through fog computing architecture. The Fog architecture allows substantial computation, storage, and processing using the Fog devices installed close to the user’s access point. Fog computing, therefore, reduces Latency and bandwidth consumption, improves security, provides context awareness, and renders more efficient services to mobile users [3, 4, 5]. <ref>{{Cite journal|last=Dastjerdi|first=Amir Vahid|last2=Buyya|first2=Rajkumar|date=2016-08|title=Fog Computing: Helping the Internet of Things Realize Its Potential|url=https://doi.org/10.1109/mc.2016.245|journal=Computer|volume=49|issue=8|pages=112–116|doi=10.1109/mc.2016.245|issn=0018-9162}}</ref>
[[File:Fog Computing Figure 1.jpg|left|thumb|Figure 1: Mobility Scenario in Fog Computing]]
However, mobility also imposes severe challenges for Fog computing due to its distributed and diverse environment. Mobility is recognized by either user-level or device-level contextual information [6]. <ref name=":0">{{Cite journal|last=Islam|first=Mir Salim Ul|last2=Kumar|first2=Ashok|last3=Hu|first3=Yu-Chen|date=2021-04|title=Context-aware scheduling in Fog computing: A survey, taxonomy, challenges and future directions|url=https://doi.org/10.1016/j.jnca.2021.103008|journal=Journal of Network and Computer Applications|volume=180|pages=103008|doi=10.1016/j.jnca.2021.103008|issn=1084-8045}}</ref> As the user moves from one location to another, the geographical location of the smart devices also changes. The change in location of the devices raises the issue of searching and rescheduling with mobility management. Efficient re-scheduling requires a well-planned handoff mechanism that is accountable for smoothly de-registering a sensor node from a source access point where the application was initially hosted and registering it to a new access point. Figure 1 depicts these mentioned problems of change in access points while the user moves from one area to another. The services may also get interrupted when there is more distance between the Fog nodes and users [7]. <ref name=":1">{{Cite journal|last=Martin|first=John Paul|last2=Kandasamy|first2=A|last3=Chandrasekaran|first3=K|date=2020-03-09|title=Mobility aware autonomic approach for the migration of application modules in fog computing environment|url=https://doi.org/10.1007/s12652-020-01854-x|journal=Journal of Ambient Intelligence and Humanized Computing|volume=11|issue=11|pages=5259–5278|doi=10.1007/s12652-020-01854-x|issn=1868-5137}}</ref> Further, any disruption in communication may lead to an increase in Latency for mobile users [8].
The significant challenges in mobility management of Fog computing are: (i) ''Fog node discovery''- due to the heterogeneous and mobile nature of the smart mobile devices, the Fog task scheduler faces the issue of finding an optimal Fog node for scheduling the task, ''(ii) Handover mechanism between users and Fog nodes''- imagine a Fog user is moving from location to location and accessing information about his surroundings with the help of a smart device. Due to the frequent change in the user's location, the Fog scheduler may need to repeatedly migrate the user’s task to a different Fog node available in his vicinity. The frequent migration of tasks increases the overhead of scheduling and, further, the restricted signal strength in certain places may lead to a breakdown of task or result delivery, and (iii) ''the Handover mechanism between Cloud and Fog''- Fog nodes have limited capacities and need to continuously communicate with Cloud computing for passing information about the tasks [9]. <ref name=":2">{{Cite journal|last=Bi|first=Yuanguo|last2=Han|first2=Guangjie|last3=Lin|first3=Chuan|last4=Deng|first4=Qingxu|last5=Guo|first5=Lei|last6=Li|first6=Fuliang|date=2018-05|title=Mobility Support for Fog Computing: An SDN Approach|url=https://doi.org/10.1109/mcom.2018.1700908|journal=IEEE Communications Magazine|volume=56|issue=5|pages=53–59|doi=10.1109/mcom.2018.1700908|issn=0163-6804}}</ref> Due to strict requirements for security, Latency, network coverage, and reliability, It becomes challenging to implement an efficient handover mechanism for full mobility support in critical domains such as healthcare [10, 11, 12] <ref name=":3">{{Cite journal|last=Abdelmoneem|first=Randa M.|last2=Benslimane|first2=Abderrahim|last3=Shaaban|first3=Eman|date=2020-10|title=Mobility-aware task scheduling in cloud-Fog IoT-based healthcare architectures|url=https://doi.org/10.1016/j.comnet.2020.107348|journal=Computer Networks|volume=179|pages=107348|doi=10.1016/j.comnet.2020.107348|issn=1389-1286}}</ref> <ref>{{Cite journal|last=Rahmani|first=Amir M.|last2=Gia|first2=Tuan Nguyen|last3=Negash|first3=Behailu|last4=Anzanpour|first4=Arman|last5=Azimi|first5=Iman|last6=Jiang|first6=Mingzhe|last7=Liljeberg|first7=Pasi|date=2018-01|title=Exploiting smart e-Health gateways at the edge of healthcare Internet-of-Things: A fog computing approach|url=https://doi.org/10.1016/j.future.2017.02.014|journal=Future Generation Computer Systems|volume=78|pages=641–658|doi=10.1016/j.future.2017.02.014|issn=0167-739X}}</ref> and vehicular systems [13, 14]. <ref>{{Cite journal|last=Zhu|first=Chao|last2=Pastor|first2=Giancarlo|last3=Xiao|first3=Yu|last4=Li|first4=Yong|last5=Ylae-Jaeaeski|first5=Antti|date=2018-06|title=Fog Following Me: Latency and Quality Balanced Task Allocation in Vehicular Fog Computing|url=https://doi.org/10.1109/sahcn.2018.8397129|journal=2018 15th Annual IEEE International Conference on Sensing, Communication, and Networking (SECON)|publisher=IEEE|pages=1–9|doi=10.1109/sahcn.2018.8397129}}</ref> <ref>{{Cite journal|last=Aljeri|first=Noura|last2=Boukerche|first2=Azzedine|date=2020-01-17|title=Fog‐enabled vehicular networks: A new challenge for mobility management|url=https://doi.org/10.1002/itl2.141|journal=Internet Technology Letters|volume=3|issue=6|doi=10.1002/itl2.141|issn=2476-1508}}</ref> Thus, mobility significantly impacts the overhead of scheduling policies and the applications' performance, eventually affecting the Quality of Experience (QoE). Therefore, mobility-aware scheduling in Fog computing has observed strong attention from researchers. Our main goal is to provide a detailed review of mobility-aware scheduling in fog computing. The review provides a detailed analysis of existing scheduling strategies that concentrate specifically on mobility awareness in the Fog environment. The following are the main contributions of this paper:
* This paper presents a detailed survey of mobility-aware scheduling in the Fog computing environment.
* It provides the details of the different techniques proposed, as well as their advantages and limitations.
* It provides a detailed analysis and taxonomy of the mobility-aware scheduling field.
* It identifies several open challenges for future research directions.
=== Related Surveys ===
Numerous survey studies on Fog computing focus on resource management, job scheduling, and context-aware scheduling. For example, Ghobaei-Arani et al. [15] <ref>{{Cite journal|last=Ghobaei-Arani|first=Mostafa|last2=Souri|first2=Alireza|last3=Rahmanian|first3=Ali A.|date=2019-09-06|title=Resource Management Approaches in Fog Computing: a Comprehensive Review|url=https://doi.org/10.1007/s10723-019-09491-1|journal=Journal of Grid Computing|volume=18|issue=1|pages=1–42|doi=10.1007/s10723-019-09491-1|issn=1570-7873}}</ref> presented a survey on resource management techniques in fog computing in the form of taxonomy to highlight cutting-edge methods while also addressing unresolved challenges. The various authors in the cloud-fog area provided the task scheduling review. Their benefits and drawbacks, as well as numerous tools and challenges concerning the scheduling techniques and their limitations, were analyzed by Alizadeh et al. [16]. Islam et al. [6] <ref name=":0" /> thoroughly analyze relevant literature on context-aware scheduling in fog computing. Further, Mouradian et al. [17] review fog computing's significant issues and challenges. But, the critical area of mobility research is still in its initial stage, and most of the review papers contain very few documents on mobility-aware task scheduling in the area of fog computing. Therefore, an extensive and comparative study is required in mobility-constrained fog computing. A deep insight into various techniques which can impact the user QoS is necessary to understand mobility-aware fog computing. This motivates us to carry out a comprehensive survey; to the best of our knowledge, this is the first detailed survey. This survey paper thoroughly examines existing scheduling solutions that focus on user mobility. Moreover, various mobility-aware scheduling techniques are discussed, along with their pros and cons. Further, the impact of mobility parameters on various QoS parameters and context-awareness is also analyzed thoroughly.
=== Paper Organization ===
The remainder of this review article is divided into 5 sections: Section 2 discusses mobility-aware scheduling in Fog computing. Section 3 presents the review methodology. Section 4 analyzes and summarizes the considered research papers and compares existing mobility-aware scheduling strategies. Section 5 provides the results drawn after critically examining the existing literature on mobility-aware scheduling policies. Finally, Section 6 presents the conclusion.
== Mobility-aware Scheduling in Fog Computing ==
Mobile device management compromises the fundamental features of Fog computing because whenever a user moves, the distance between them increases, impacting the QoS. Therefore, to keep the computing fog node close to the associated mobile device, the services or tasks need to be migrated from one fog node to another appropriate fog device. The selection of such appropriate fog nodes in a mobile environment deals with two main processes: Estimation of user mobility patterns: User mobility estimation techniques can be probabilistic and deterministic [18]. <ref name=":4">{{Cite journal|last=Shekhar|first=Shashank|last2=Chhokra|first2=Ajay|last3=Sun|first3=Hongyang|last4=Gokhale|first4=Aniruddha|last5=Dubey|first5=Abhishek|last6=Koutsoukos|first6=Xenofon|date=2019-05|title=URMILA: A Performance and Mobility-Aware Fog/Edge Resource Management Middleware|url=https://doi.org/10.1109/isorc.2019.00033|journal=2019 IEEE 22nd International Symposium on Real-Time Distributed Computing (ISORC)|publisher=IEEE|pages=118–125|doi=10.1109/isorc.2019.00033}}</ref> In a deterministic method, the source and destination are known beforehand, whereas in a non-deterministic technique, periodic estimation has to be done regarding the user's route. Many authors estimate the route of mobile users by leveraging external services like Open Street Maps (<nowiki>http://www.openstreetmap.org</nowiki>), Google Maps APIs (<nowiki>https://cloud.google.com/</nowiki> maps-platform/) [18] <ref name=":4" />, logistics maps [19], GPRS Here APIs (<nowiki>https://developer</nowiki>. here.com/) [20, 21], Lyapunov estimation technique [22]. Specific QoS requirements: The selection of a fog node also depends upon specific quality requirements such as Latency, which many authors are using to select an efficient fog node [18, 10, 9], <ref name=":2" /> <ref name=":3" /> <ref name=":4" /> workload [23, 24], cost [16], etc.
== Review Methodology ==
The review process was conducted considering a review methodology consisting of four phases. The first phase was searching using traditional online database sources based on outlined search keywords. Table 1 lists the keywords used to find the relevant research articles to conduct the review in the Fog computing mobility-aware scheduling area. Second Phase: Limit the search of research articles beginning in 2015, and inclusion and exclusion principles are also used to refine research articles that specifically deal with mobility issues in task scheduling. Finally, in the Third Phase, A total of 20 papers are shortlisted for the review process. Further, Table 2 presents the research questions drafted for this study in mobility-aware scheduling in Fog computing.
{| class="wikitable"
|+Table 1: List of keywords used in the review process
!Sno
!Keyword
!Description
!Years
|-
|1
|Mobility
|Mobility-aware Fog task scheduling
| rowspan="5" |2015- 2021
|-
|2
|Mobility environment
|Mobility environment in Fog task scheduling
|-
|3
|Mobility factors
|Mobility factors in Fog task scheduling
|-
|4
|Mobility Awareness
|Mobility awareness in Fog task scheduling
|-
|5
|Mobility management
|Mobility management in Fog task scheduling
|}
{| class="wikitable"
|+Table 2: List of research questions used to complete the review process
!Q.No
!Research questions
|-
|1
|What scheduling approaches are used in the Fog computing environment to manage mobility awareness?
|-
|2
|What are the main limitations considered for mobility-aware scheduling techniques?
|-
|3
|Which case studies are applied to mobility-aware scheduling techniques?
|-
|4
|What evaluation tools are used to assess mobility-aware scheduling techniques?
|-
|5
|What performance indicators are utilized to evaluate mobility-aware scheduling techniques?
|-
|6
|What are the major open issues concerns in the field of mobility-aware scheduling for future research directions?
|}
=== Source of Information ===
To conduct this review, various online sources are listed below, and the research articles were searched using the different keywords mentioned in Table 1.
* Google Scholar (<nowiki>http://www.scholar.google.co.in</nowiki>)
* John Wiley & Sons Inc. (<nowiki>https://onlinelibrary.wiley.com/</nowiki>)
* Elsevier (<nowiki>https://www.elsevier.com/en-in</nowiki>)
* ACM Digital Library (<nowiki>https://www.acm.org/</nowiki>)
* Springer (<nowiki>https://www.springer.com/in</nowiki>)
* IEEE Xplore Digital Library (<nowiki>https://www.ieee.org/</nowiki>)
=== Quality Assessment ===
Research papers used quality assessment to filter out the most suitable mobility-based scheduling research articles in fog computing utilising the principle of inclusion and exclusion. Furthermore, in order to obtain high-quality research publications, the Center for Reviews and Dissemination (CRD) recommendations were followed, and each study item was examined for internal and external validation of results.
== Literature Analysis ==
Verma et al. [25] proposed a Server Cloudlet(SC) migration technique to handle users' mobility. The strategy is to select the target SC to migrate the services based on its highest rank. The rank of the SC further depends upon its available RAM, MIPS, and bandwidth.
Maleki et al. [26] designed two mobility-aware computation offloading approaches, sampling-based Online mobile applications to cloudlets(S-OAMC) and Greedy online mobile applications to cloudlets(G-OAMC). The developed system works in three major steps: ''(i)'' the future specifications of mobile applications are predicted using the Machine Learning (ML) method, named matrix completion. ''(ii)'' The predicted specifications of mobile applications, including future location prediction, also help estimate the offloading cost. ''(iii)'' Apply S-OAMC or G-OAMC offloading technique based on minimum cost value.
Abdelmoneem et al. [10] <ref name=":3" /> work on a fog-based healthcare architecture system that supports patient movement without altering their Quality of Service (QoS). The QoS in a mobile environment is maintained with the help of their proposed handoff mechanism. The authors modified the traditional Horizontal Handoff (HHO) mechanism [27] into two different phases: ''(i) handoff decision policy'': Received signal Strength (RSS) method is used to detect the network change information of the mobile patients and (ii) ''handoff strategy'': a handoff decision is made based on a comparison between RSS received from the old fog gateways, current threshold value and new fog gateways that are in close proximity to the mobile patient.
Javanmardi et al. [28] consider an imaginative city-based mobility scenario where the user is placing delay-sensitive service while moving. The authors proposed a task scheduling technique that jointly employs fuzzy logic and particle swarm optimization (PSO) algorithm to improve the QoS for mobile users within the city. The proposed algorithm is deployed at the Fog gateway, which further distributes the tasks to Fog devices available in an entire region in order to provide seamless service to mobile users. The main motive of the work is to improve resource utilization in a mobility-aware environment.
Puliafito et al. [29] develop an extension of ifogsim [30], which supports mobile environment. The authors implement different migration techniques inspired by Bittencourt et al. [31]. Different phases are devised to perform migration in a fog mobile environment, which: ''(i) Before migration phase'': migration decision is taken in this phase, based on specific parameters, like user location, speed, the direction of movement, zone, and migration point in order to select the appropriate cloudlet for offloading the services, ''(ii) During the migration phase'': this phase manages, monitors and synchronizes the whole selected migration process and ''(iii) After migration phase'': this phase involves closing the older cloudlet connections with the user and using the new cloudlet for services.
A Blockchain-based Mobility-aware Offloading (BMO) mechanism is designed by Dou et al. [32], where user mobility prediction is implemented using the Individual-Mobility (IM) model [33]. The idea behind the offloading mechanism is to shift the computational workload to different available Fog Servers (FSs) in the geo-location predicted by the IM model. Further, blockchain technology is deployed to check the authenticity of the forthcoming Fog servers. Finally, accounting is being managed by ''Fogcoin'', similar to Bitcoin, which stores the entire transaction history between the online Fog server and mobile users.
Martin et al. [7] <ref name=":1" /> proposed a framework that supports the migration of containers while satisfying the QoS requirements of mobile users. The migration of containers is done in an autonomic manner, by adopting the Monitor-Analyze-Plan-Execute (MAPE) autonomic control loop. The MAPE control loops discuss various steps of migration, like ''(i) Monitor'': used to constantly monitor the environment context, such as the mobility of users that is subsequently used to determine the need to migrate an application module to some other Fog node called target node; ''(ii) Analyze'': applies forecasting techniques to predict the user possible location in the next time step. If the distance between the user and the device is not under certain acceptable limits, a migration decision is made. ''(iii) Plan'': a Genetic Algorithm (GA) is used to identify a suitable Fog node closest to the forecasted location, where migration of the container running user application can be done. ''(iv) Execute'': this step ensures the whole migration process should take place smoothly.
Mass et al. propose a mobility and delay-aware fog server selection scheme. [34] called Edge-Process management (EPM) system. The EPM system depends upon the trajectory of a user’s movement, Fog server workload, and user location to select the appropriate Fog server for executing user applications. The system selects or re-selects a Fog Server (FS) based on a score value calculated through available bandwidth, power, distance from the user, and finally, duration of availability in a region.
Mobi-IoST (Mobility-aware Internet of Spatial Things), a real-time mobility-aware framework is presented by Ghosh et al. [35]. The authors considered the mobile nature of both IoT devices and Fog nodes, collaboratively called mobile agents. The proposed mobility-aware framework collects a vast amount of Global Positioning System (GPS) data of these mobile agents to predict their movement patterns using various machine learning algorithms. The major components of the framework are, ''(i) Movement pattern model''ling, collecting and modelling GPS log, stay-point, and other contextual location information; ''(ii) Predicting the following location'': human movement semantics is analyzed using all modelled information; ''(iii) Delivery of result:'' after the user movement prediction in the previous phase, the system intelligently discovers a capable fog device for data processing in a timely manner.
A middleware solution, URMILA, for managing resources and scheduling tasks in the Fog environment is presented by Shekhar et al. [18]. <ref name=":4" /> Ubiquitous Resource Management for Interference and Latency-Aware services (URMILA), ensures minimum Service Level Objectives (SLO) violation for latency-sensitive mobile applications across the cloud-Fog environment. The major modules of the proposed system are ''(i) Route calculation'', which calculates the user's possible routes using Google Maps or GPS data; ''(ii) Latency calculation'', the system deploys a data-driven model to estimate the Latency on predicted user routes; ''(iii) Fog node selection:'' the system selects a fog server for execution of task on the basis of its instantaneous utilization of the available resources. Further, it selects the Fog server for the entire period of execution, during which mobile users can still access their application through various Wireless Access Points (WAP).
Gia et al. [8] proposed a Handover mechanism for mobility management between fog nodes with the overall objective of consuming minimum energy and delay during handovers. Handover methods frequently rely on one or more measures, such as the Received Signal Strength Indicator (RSSI), the Link Quality Indicator (LQI), and the velocity of objects, to make handover decisions. This proposed system provides emergency services to health monitoring systems and basically works in two different mobility scenarios: ''(i) Node mobility between indoor or outdoor locations'': nodes belonging to indoor location or outdoor location only are considered to be similar, and they're calculated metrics value like RSSI, LQI, velocity, etc.; can be directly used for the handover of services to appropriate gateway, ''(ii) Node mobility between indoor and outdoor locations'': nodes are considered dissimilar, if they belong to indoor and outdoor location both, So, the metrics are re-calculated which introduce some additional parameters like temperature and interference signals in order to make a decision over handover gateway.
Babu and Biswash [36] proposed a mobility management technique that supports node-to-node communication and Fog computing-based architecture for 5G networks. It addresses the technical problems between 5G networks and Fog servers. The mobility-based approach assists mobile nodes in establishing communication while they are in motion. The mobility management technique may also be used to begin N2N communication in dynamic environments. N2N communication schemes for fog networks, on the other hand, provide an effective communication environment for mobile users with highly minimal network usage.
Wang et al. [37] proposed a mobility-aware offloading scheme, that considers an adequate quality and a computation allocation system that deals with the user equipment affairs to maximize the total revenue. The quality of user equipment is delineated by the sojourn time that follows the exponential distribution to reduce the chance of migration and maximize the entire income of user equipment. MILP (mixed-integer non-linear programming) NP-hard problem is modelled and consists of resource allocation and task offloading schemes. So, to solve this problem, a Gini coefficient and genetic algorithm are used to estimate the allocation of resources. The proposed approach can easily handle the mobility of users by minimizing the chances of migration.
Waqas et al. [38] provided a forward-thinking analysis of quality about-mobility in Fog computing by identifying quality challenges, requirements, and options for numerous ideas. The authors also identified outstanding concerns from previous research and summarized the advantages of quality for readers. It allows researchers and developers to avoid common misunderstandings and capture real-world scenarios such as businesses, governments, and educational institutions. Furthermore, it revolutionizes follow-up analysis and differentiates and foregrounds futurity orientations in real-life events involving humans and vehicles in a highly dynamic Fog setting.
Bi et al. [9] <ref name=":2" /> introduced software-defined networking-based fog computing architecture by decoupling mobility control and data forwarding. When mobile consumers travel between several access networks, the authors suggested an Optimal Path Selection (OPS) algorithm to preserve service continuity. Mobile customers received seamless and transparent mobility support thanks to efficient signalling operations. In mobile fog computing, the suggested algorithm ensured service continuity, increased handover performance, and achieved high data transfer efficiency.
Niu et al. [21] established a system called mobility-aware and multihop-D2D relaying-based scheduling scheme (MHRC) at Edge nodes near hotspots. The authors exploited concurrent transmissions to improve the performance of the system. The mmWave (millimetre-wave) band of Fog computing was cached, and extensive performance evaluation confirms that MHRC delivers more than the higher expected cached data amount. Name et al. [19] proposed an efficient algorithm to address the problem of resource allocation and user mobility from the Edge of the network to cloud data centres. This algorithm operates on a seamless handover scheme for mobile IPV6 to ease the user mobility challenge and reduce the application response time. The study showed that the task of service delay and packet loss was decreased due to the effect of change in the mobile node position.
Bittencourt et al. [23] examined the subject of resource allocation in the Fog/Cloud environment, taking into account the hierarchical structure. In the context of the Fog paradigm, the authors developed three scheduling algorithms (First come, First serve, delay-priority, and concurrent) that address user mobility and edge computing capabilities. The authors demonstrated that scheduling techniques may be designed to cope with different application classes based on demand from mobile users by leveraging both Fog to the end-user and cloud characteristics in this study.
Velasquez et al. [39] proposed a hybrid strategy for the Fog environment to manage resources for mobility scenarios. The authors applied the orchestrator technique to offer mobility support in a Smart City situation. In this technique, three components, the status monitor, the Planner, and the VM/Container, are employed to monitor, plan and execute the applications. The main aim of this study was to guarantee the QoS and QoE requirements of mobility-based applications and services.
Bittencourt et al. [31] presented a Fog computing architecture focusing on Virtual Machine (VM) migration where each user has a VM running in a cloudlet. In this architecture, the user's location is identified by using GPS, and then the VM is moved to a nearby Fog Cloud. The main aim of this study was to migrate users' data according to their mobility in order to maintain QoS for applications demanding lower Latency and allow smooth handoff mechanisms for mobile users.
From the extensive analysis of the literature, the various mobility-aware scheduling techniques have been classified as shown in Table 3. Further, it presents the advantages and limitations of each technique.
{| class="wikitable"
|+Table 3: Classification of Mobility-aware scheduling techniques
!Ref.
!Technique
!Advantage(s)
!Limitation(s)
|-
|[25]
|Ranking of VM
|
* Decrease in delay time, migration time, tuple lost value and downtime
|
* Case study not discussed
|-
|[26]
|S-OAMC, G-OAMC, Machine learning matrix completion
|
* Migration rate decreased
* Better Scalability
|
* Energy utilization of devices not investigated
|-
|[32]
|IM model
|
* Provides better mobility support and security
|
* Did not investigate synchronization overhead
|-
|[10] <ref name=":3" />
|RSS
|
* Reliable and Heterogeneous execution
|
* Low scalability
* No distributed scheduling to minimize response time
|-
|[7] <ref name=":1" />
|MAPE control loop
|
* Improved QoS
* Reduced service downtime
|
* No real-time evaluation
* High energy consumption
* Low robustness and security
|-
|[28]
|Copy of task to over a region
|
* Network Utilization developed
* Low-Loop delay
|
* Fault tolerance reliability is based on Fog gateways only
|-
|[18] <ref name=":4" />
|URMILA
|
* Service availability is maintained by delivering the desired QoS
* Deployment cost minimized
* Battery longevity ensured
|
* No empirical validations
* No user probabilistic routes
* Low scalability in terms of distance and speed
|-
|[8]
|RSSI, LQI
|
* Promises to keep the connection active with a low latency rate between the system and sensor nodes
|
* Consumes more energy
* Overhead is large for network transmission
* Coverage and overhead area are undefined between gateways
|-
|[29]
|User trajectories pre- diction using GPS log
|
* Provides better mobility support
* Reduces migration time
|
* Low scalability
|-
|[36]
|N2N communication, Data Analytics
|
* Fast data access
* High reliability and scalability- city
* Minimum overhead and cost
* High throughput and less delay
|
* No real-time cellular network evaluation
* Low network efficiency
|-
|[38]
|Mobility facets analysis
|
* Improved QoS and QoE
* Latency rate reduced
|
* No real-life implementation
* No reliability and low Latency between dynamic users and fog servers
|-
|[37]
|M-ILP, Sojourn time
|
* Cost-effective
* Migration time reduced
|
* Migration cost not considered
* No real-time implementation
|-
|[34]
|User trajectories prediction using GPS log
|
* Conventional delay tolerance
* High QoS
* Avoided local task processing cost
* Efficient in saving battery
* Handles subtle scenarios with high Latency
|
* Smart city not directed through the use of accurate city maps with aid from stimulation setting
|-
|[35]
|Prediction of user location
|
* Power consumption and de- lay handled proficiently
* Power and delay are reduced
|
* No acquiring of mobile data usage where location sense and time-series data can be projected to achieve the bandwidth
|-
|[21]
|Relay path planning algorithm
|
* Power efficient
* High spectral efficiency
* Data is relayed on cached edge nodes and relay nodes
|
* Blockage problem due to weak diffraction
|-
|[9] <ref name=":2" />
|OPS
|
* Handover performance improved
* Efficiency of high data communication achieved
* Guarantees continuity of services
|
* It does not guarantee privacy and security
* Virtual Machine migration not determined
* The handover process during the optimal path for more logical routing could have been more efficient
|-
|[23]
|Assignment of FS
|
* Low Latency
* Supports dynamic computing and user behaviour
|
* There is no prediction of mobility failure
* Bandwidth and processing not considered in scheduling
|-
|[39]
|Orchestrator
|
* Maintains trustworthiness, resilience, and low Latency in a dynamic environment
|
* No real implementation has been carried out
|-
|[19]
|Pattern modelling, dictating the following location
|
* Application Response time reduced
* Latency time reduced
|
* Services become temporarily inaccessible for some mobile nodes
|-
|[31]
|Forecasting technique
|
* Computing capacity provided for storage and processing of data
|
* Security concerns associated with both user data and applications not considered
|}
[[File:Fog Computing Figure 2.jpg|center|thumb|Figure 2: Year-wise count of research articles]]
== Analytical Discussion ==
The existing research on Fog computing Mobility-aware scheduling has been analyzed thoroughly. The analysis was performed using the answers given in Table 2. The results drawn through the thorough analysis of the literature are presented in various figures as follows:
Figure 2 lists the year-wise count of research papers that are considered for this survey. The bar graph represents the total number of research papers from journals and Conferences from the year 2015 - 2021. The research articles from the journal are 16, and the conference papers are 4. It is observed that more research needs to be conducted on mobility-aware scheduling in Fog computing.
Figure 3 displays an analytical comparison of mobility-aware scheduling approaches in Fog computing based on the content of the represented taxonomy in Figure 7. From the thorough analysis of the literature, four methods have been considered: migration, task offloading, handoff/handover mechanism, and task scheduling. The handoff/handover mechanism has the highest percentage of usage in mobility-aware scheduling, at 30%. The task scheduling and offloading have 25% of us- age in mobility-aware scheduling each. Finally, migration is only 20% of the usage in mobility-aware scheduling. Therefore, these approaches, specifically migration, are still open challenges to address for further research.
Figure 4 depicts various tools that were used for evaluating the mobility-aware scheduling approaches. 18% and 9% of the research articles used iFogSim and Mob-FogSim simulation tools for implementation, respectively. Besides, other simulation tools such as ONE (9%), NS2(5%), MATLAB (4%), Mininet (5%), and Docker (9%) have been utilized for implementing the proposed techniques in the research articles. Further, pro-Programming languages such as C++ (9%) and Python (9%) and hardware deployments such as Raspberry Pi (5%) and Ardunio (4%) were used for implementing existing case studies based on mobility-aware scheduling.
[[File:Fog Computing Figure 3.jpg|center|thumb|Figure 3: Percentage of the presented classified approaches in mobility-aware scheduling]]
The applied case studies are shown in Figure 5, which shows a maximum of 20% of research articles have implemented IoT-based applications. After that, 15% of each research article used Health care and Mobile-based applications. Besides, Smart City and 5G-based applications have been applied in 10% of research articles. Moreover, Surveillance and gaming, Mobile IPV6, and Wireless computing applications are the case studies on which only 5% of research articles exist.
After reviewing numerous research articles based on mobility-aware scheduling, it has been observed that researchers employed various parameters for evaluating the performance of the Mobility-scheduling approaches, as represented in Figure 6. It shows that Time completion (18%) followed by Delay (12%), Network usage (12%), Latency (12%), Energy consumption (10%), and cost (10%) are generally utilized. However, Downtime (4%), Migration time (4%), Makespan (2%), Success ratio (2%), Signal level (2%), Deadline (2%), Makespan (2%), Migration rate (2%), Mobility patterns (2%), Tuple lost (2%), and power consumption (2%) are less exploited parameters.
A taxonomy was compiled after going through the detailed review process, and various techniques have been categorized in Fog computing-based mobility-aware scheduling. Figure 7 presents these categories broadly in Migration, Offloading, Handoff/Handover mechanism, and Scheduling.
[[File:Fog Computing Figure 4.jpg|center|thumb|Figure 4: Percentage of tools utilized in the literature]]
[[File:Fog Computing Figure 5.jpg|center|thumb|Figure 5: Percentage of case studies employed in the literature]]
[[File:Fog Computing Figure 6.jpg|center|thumb|Figure 6: Percentage of parameters for evaluating Mobility-aware scheduling in the literature]]
=== Open Issues and Challenges ===
From the thorough analysis of the literature, several open issues and challenges pertaining to the area of mobility-aware scheduling in Fog computing have been identified in order to provide directions for future research exploration. The identified open problems and challenges, depicted in Figure 8, are discussed below.
=== Task Scheduling ===
Fog computing consists of several Fog nodes, each of which is a mini Cloud in the vicinity of mobile devices near the Edge of the network. When a mobile device submits a task, the Fog scheduler assigns it to a nearby Fog node(s) for execution. However, as the device moves from one network to another, the task needs to be rescheduled when the device enters a different network. Additionally, Fog nodes have limited capacity and availability; if the mobile user enters into a network where there is no nearby Fog service available, then this leads to a significant delay in service and raises a significant issue of task scheduling for mobile users [10, 40]. <ref name=":3" />
=== Resource Provisioning ===
Fog computing reduces the workload of Cloud computing by processing the tasks locally near the Edge of the network. However, due to the mobility of the user, the Fog node primarily assigned to a task might not be optimal over time. Therefore, the migration of the task to another Fog node near the user's mobile device is perceived as a necessary solution to resolve this concern [41]. However, such frequent migration over a short time poses the challenge of providing an efficient resource for the task that is capable of performing computation on time and delivering results to users while adhering to QoE.
[[File:Fog Computing Figure 7.jpg|center|thumb|Figure 7: Mobility-aware Fog Scheduling Taxonomy]]
=== Energy Consumption ===
The placement of fog services at the Edge of the network can provide better QoS to mobile users, resulting in a shorter response time. However, it is practically impossible due to the high deployment cost of new Fog devices, which further raises the significant issue of energy consumption. If too many deployments are done, there will be lots of communication traffic from the Cloud to Fog nodes and servers in order to create copies of the task from one network to another in case of mobility [42]. This results in considerable energy wastage in the form of high bandwidth consumption. This means that where and when to reschedule the task to an efficient Fog node must be carefully determined to minimize energy, response time, and deployment cost.
=== Quality of Experience (QoE) ===
[[File:Fog Computing Figure 8.jpg|left|thumb|Figure 8: Mobility-aware scheduling open issues and challenges]]
Several mobility-based scheduling algorithms exist, but they need to focus on maximizing the user QoE [29, 8, 10, 18]. <ref name=":3" /> <ref name=":4" /> Further, they do not analyze the user performance; hence, the QoE of using a service or product is not determined. Therefore, to understand the user gain and loss, the scheduling algorithm needs to focus on enhancing the user QoE.
=== Resource Management ===
The mobility of Fog nodes/users demands efficient resource discovery and sharing, resource availability, and task offloading [43]. Few techniques that were proposed to manage the resources effectively did not consider more constraints such as density, latency sensitivity, and mobility of Edge and Fog nodes, and as the number of nodes increases, issues such as scalability and distributing the algorithms arise [44, 45, 46]. Therefore, more attention needs to be paid towards the mobile Fog computing environment to manage the resources effectively.
=== Privacy and Security ===
In [47], a scheduling policy is proposed for the mobile device system to minimize the cost. However, the privacy issues of location and usage patterns were ignored. Additionally, data privacy, access control, and intrusion detection in scheduling policies have been overlooked [7, 48, 28]. <ref name=":1" /> Besides, Fog node devices are normally deployed near the end-user; hence, protection and surveillance are comparatively weak, which can result in a malicious attack [49, 50].
== Data availability statement ==
Not applicable.
== Conclusions ==
Fog computing infrastructure provides services at the Edge of the network. So, to provide support for scheduling and management of mobility awareness, efficient techniques and mechanisms have been proposed. In this survey, research articles on the mobility-aware-scheduling strategies in Fog computing have been thoroughly analyzed. It provides a comparative study among existing mobility-aware scheduling strategies based on vital factors such as techniques proposed, parameters considered, tools utilized for implementation, and case studies considered, along with the advantages and limitations. Further, several open issues and challenges have been identified for future research direction.
===Subheading===
==Second Heading==
==Third Heading, etc==
==Additional information==
===Acknowledgements===
Any people, organisations, or funding sources that you would like to thank.
===Competing interests===
Any conflicts of interest that you would like to declare. Otherwise, a statement that the authors have no competing interest.
===Ethics statement===
An ethics statement, if appropriate, on any animal or human research performed should be included here or in the methods section.
==References==
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{{Article info
| journal = WikiJournal of Science
| last1 = Islam
| orcid1 = 0000-0001-5750-1058
| first1 = Mir Salim Ul
| last2 = Kumar
| orcid2 = 0000-0003-3279-5111
| first2 = Ashok
| affiliation2 = Department of Computer Application, Chandigarh School of Business, Chandigarh Group of Colleges, Jhanjeri, Punjab, India
| et_al =
| affiliation1 = Department of Computer Science and Engineering, National Institute of Technology Srinagar, Jammu and Kashmir, India
| correspondence1 = mir.salim27@gmail.com
| affiliations =
| keywords = Fog computing, Mobility-aware, Scheduling, Resource Management, Resource Allocation
| license = CC BY 4.0
| submitted = 2024-08-22
| abstract = The increasing focus on Fog-IoT results in billions of Internet-connected devices that demand substantial computational power and network bandwidth. These devices are geographically distributed, heterogeneous, computational capacity constrained, inconsistent in behaviour, and generally mobile. Therefore, providing seamless service, irrespective of location and movement of the devices as well as the end-users, makes resource scheduling a significant challenge in the Fog computing paradigm. Several mobility-aware scheduling strategies have been proposed in the literature to efficiently manage the resources for mobile users and devices in the Fog environment. This paper gives a survey of mobility-aware scheduling in the Fog computing environment. It describes the many strategies presented and their benefits and drawbacks. It also includes a complete study and taxonomy of the mobility-aware scheduling field. Further, it delineates open issues and challenges. This work will provide researchers with future research directions and aid them in recognizing the gaps before planning for further research in mobility-aware scheduling.
}}
== Introduction ==
With the advancement in technology and the exponential growth of mobile devices, network traffic has increased manifold in cloud computing. Due to this reason, Latency reduction and faster processing of data for mobile users have become critical challenges in providing seamless connectivity and minimal disruption while the user is moving.<ref>{{Cite journal|last=Abdullah|first=Fatima|last2=Kimovski|first2=Dragi|last3=Prodan|first3=Radu|last4=Munir|first4=Kashif|date=2021-06-11|title=Handover authentication latency reduction using mobile edge computing and mobility patterns|url=https://doi.org/10.1007/s00607-021-00969-z|journal=Computing|volume=103|issue=11|pages=2667–2686|doi=10.1007/s00607-021-00969-z|issn=0010-485X}}</ref> The data movement brings the additional issue of integrity and confidentiality because data is moving via a wireless connection to a far distant cloud. Additionally, due to the cloud's location is far from mobile users, so data movement is also affected by variable network strength and phone bandwidth. The solution proposed by [2] is to extend cloud capabilities through fog computing architecture. The Fog architecture allows substantial computation, storage, and processing using the Fog devices installed close to the user’s access point. Fog computing, therefore, reduces Latency and bandwidth consumption, improves security, provides context awareness, and renders more efficient services to mobile users [3, 4, 5]. <ref>{{Cite journal|last=Dastjerdi|first=Amir Vahid|last2=Buyya|first2=Rajkumar|date=2016-08|title=Fog Computing: Helping the Internet of Things Realize Its Potential|url=https://doi.org/10.1109/mc.2016.245|journal=Computer|volume=49|issue=8|pages=112–116|doi=10.1109/mc.2016.245|issn=0018-9162}}</ref>
[[File:Fog Computing Figure 1.jpg|left|thumb|Figure 1: Mobility Scenario in Fog Computing]]
However, mobility also imposes severe challenges for Fog computing due to its distributed and diverse environment. Mobility is recognized by either user-level or device-level contextual information [6]. <ref name=":0">{{Cite journal|last=Islam|first=Mir Salim Ul|last2=Kumar|first2=Ashok|last3=Hu|first3=Yu-Chen|date=2021-04|title=Context-aware scheduling in Fog computing: A survey, taxonomy, challenges and future directions|url=https://doi.org/10.1016/j.jnca.2021.103008|journal=Journal of Network and Computer Applications|volume=180|pages=103008|doi=10.1016/j.jnca.2021.103008|issn=1084-8045}}</ref> As the user moves from one location to another, the geographical location of the smart devices also changes. The change in location of the devices raises the issue of searching and rescheduling with mobility management. Efficient re-scheduling requires a well-planned handoff mechanism that is accountable for smoothly de-registering a sensor node from a source access point where the application was initially hosted and registering it to a new access point. Figure 1 depicts these mentioned problems of change in access points while the user moves from one area to another. The services may also get interrupted when there is more distance between the Fog nodes and users [7]. <ref name=":1">{{Cite journal|last=Martin|first=John Paul|last2=Kandasamy|first2=A|last3=Chandrasekaran|first3=K|date=2020-03-09|title=Mobility aware autonomic approach for the migration of application modules in fog computing environment|url=https://doi.org/10.1007/s12652-020-01854-x|journal=Journal of Ambient Intelligence and Humanized Computing|volume=11|issue=11|pages=5259–5278|doi=10.1007/s12652-020-01854-x|issn=1868-5137}}</ref> Further, any disruption in communication may lead to an increase in Latency for mobile users [8].
The significant challenges in mobility management of Fog computing are: (i) ''Fog node discovery''- due to the heterogeneous and mobile nature of the smart mobile devices, the Fog task scheduler faces the issue of finding an optimal Fog node for scheduling the task, ''(ii) Handover mechanism between users and Fog nodes''- imagine a Fog user is moving from location to location and accessing information about his surroundings with the help of a smart device. Due to the frequent change in the user's location, the Fog scheduler may need to repeatedly migrate the user’s task to a different Fog node available in his vicinity. The frequent migration of tasks increases the overhead of scheduling and, further, the restricted signal strength in certain places may lead to a breakdown of task or result delivery, and (iii) ''the Handover mechanism between Cloud and Fog''- Fog nodes have limited capacities and need to continuously communicate with Cloud computing for passing information about the tasks [9]. <ref name=":2">{{Cite journal|last=Bi|first=Yuanguo|last2=Han|first2=Guangjie|last3=Lin|first3=Chuan|last4=Deng|first4=Qingxu|last5=Guo|first5=Lei|last6=Li|first6=Fuliang|date=2018-05|title=Mobility Support for Fog Computing: An SDN Approach|url=https://doi.org/10.1109/mcom.2018.1700908|journal=IEEE Communications Magazine|volume=56|issue=5|pages=53–59|doi=10.1109/mcom.2018.1700908|issn=0163-6804}}</ref> Due to strict requirements for security, Latency, network coverage, and reliability, It becomes challenging to implement an efficient handover mechanism for full mobility support in critical domains such as healthcare [10, 11, 12] <ref name=":3">{{Cite journal|last=Abdelmoneem|first=Randa M.|last2=Benslimane|first2=Abderrahim|last3=Shaaban|first3=Eman|date=2020-10|title=Mobility-aware task scheduling in cloud-Fog IoT-based healthcare architectures|url=https://doi.org/10.1016/j.comnet.2020.107348|journal=Computer Networks|volume=179|pages=107348|doi=10.1016/j.comnet.2020.107348|issn=1389-1286}}</ref> <ref>{{Cite journal|last=Rahmani|first=Amir M.|last2=Gia|first2=Tuan Nguyen|last3=Negash|first3=Behailu|last4=Anzanpour|first4=Arman|last5=Azimi|first5=Iman|last6=Jiang|first6=Mingzhe|last7=Liljeberg|first7=Pasi|date=2018-01|title=Exploiting smart e-Health gateways at the edge of healthcare Internet-of-Things: A fog computing approach|url=https://doi.org/10.1016/j.future.2017.02.014|journal=Future Generation Computer Systems|volume=78|pages=641–658|doi=10.1016/j.future.2017.02.014|issn=0167-739X}}</ref> and vehicular systems [13, 14]. <ref>{{Cite journal|last=Zhu|first=Chao|last2=Pastor|first2=Giancarlo|last3=Xiao|first3=Yu|last4=Li|first4=Yong|last5=Ylae-Jaeaeski|first5=Antti|date=2018-06|title=Fog Following Me: Latency and Quality Balanced Task Allocation in Vehicular Fog Computing|url=https://doi.org/10.1109/sahcn.2018.8397129|journal=2018 15th Annual IEEE International Conference on Sensing, Communication, and Networking (SECON)|publisher=IEEE|pages=1–9|doi=10.1109/sahcn.2018.8397129}}</ref> <ref>{{Cite journal|last=Aljeri|first=Noura|last2=Boukerche|first2=Azzedine|date=2020-01-17|title=Fog‐enabled vehicular networks: A new challenge for mobility management|url=https://doi.org/10.1002/itl2.141|journal=Internet Technology Letters|volume=3|issue=6|doi=10.1002/itl2.141|issn=2476-1508}}</ref> Thus, mobility significantly impacts the overhead of scheduling policies and the applications' performance, eventually affecting the Quality of Experience (QoE). Therefore, mobility-aware scheduling in Fog computing has observed strong attention from researchers. Our main goal is to provide a detailed review of mobility-aware scheduling in fog computing. The review provides a detailed analysis of existing scheduling strategies that concentrate specifically on mobility awareness in the Fog environment. The following are the main contributions of this paper:
* This paper presents a detailed survey of mobility-aware scheduling in the Fog computing environment.
* It provides the details of the different techniques proposed, as well as their advantages and limitations.
* It provides a detailed analysis and taxonomy of the mobility-aware scheduling field.
* It identifies several open challenges for future research directions.
=== Related Surveys ===
Numerous survey studies on Fog computing focus on resource management, job scheduling, and context-aware scheduling. For example, Ghobaei-Arani et al. [15] <ref>{{Cite journal|last=Ghobaei-Arani|first=Mostafa|last2=Souri|first2=Alireza|last3=Rahmanian|first3=Ali A.|date=2019-09-06|title=Resource Management Approaches in Fog Computing: a Comprehensive Review|url=https://doi.org/10.1007/s10723-019-09491-1|journal=Journal of Grid Computing|volume=18|issue=1|pages=1–42|doi=10.1007/s10723-019-09491-1|issn=1570-7873}}</ref> presented a survey on resource management techniques in fog computing in the form of taxonomy to highlight cutting-edge methods while also addressing unresolved challenges. The various authors in the cloud-fog area provided the task scheduling review. Their benefits and drawbacks, as well as numerous tools and challenges concerning the scheduling techniques and their limitations, were analyzed by Alizadeh et al. [16]. Islam et al. [6] <ref name=":0" /> thoroughly analyze relevant literature on context-aware scheduling in fog computing. Further, Mouradian et al. [17] review fog computing's significant issues and challenges. But, the critical area of mobility research is still in its initial stage, and most of the review papers contain very few documents on mobility-aware task scheduling in the area of fog computing. Therefore, an extensive and comparative study is required in mobility-constrained fog computing. A deep insight into various techniques which can impact the user QoS is necessary to understand mobility-aware fog computing. This motivates us to carry out a comprehensive survey; to the best of our knowledge, this is the first detailed survey. This survey paper thoroughly examines existing scheduling solutions that focus on user mobility. Moreover, various mobility-aware scheduling techniques are discussed, along with their pros and cons. Further, the impact of mobility parameters on various QoS parameters and context-awareness is also analyzed thoroughly.
=== Paper Organization ===
The remainder of this review article is divided into 5 sections: Section 2 discusses mobility-aware scheduling in Fog computing. Section 3 presents the review methodology. Section 4 analyzes and summarizes the considered research papers and compares existing mobility-aware scheduling strategies. Section 5 provides the results drawn after critically examining the existing literature on mobility-aware scheduling policies. Finally, Section 6 presents the conclusion.
== Mobility-aware Scheduling in Fog Computing ==
Mobile device management compromises the fundamental features of Fog computing because whenever a user moves, the distance between them increases, impacting the QoS. Therefore, to keep the computing fog node close to the associated mobile device, the services or tasks need to be migrated from one fog node to another appropriate fog device. The selection of such appropriate fog nodes in a mobile environment deals with two main processes: Estimation of user mobility patterns: User mobility estimation techniques can be probabilistic and deterministic [18]. <ref name=":4">{{Cite journal|last=Shekhar|first=Shashank|last2=Chhokra|first2=Ajay|last3=Sun|first3=Hongyang|last4=Gokhale|first4=Aniruddha|last5=Dubey|first5=Abhishek|last6=Koutsoukos|first6=Xenofon|date=2019-05|title=URMILA: A Performance and Mobility-Aware Fog/Edge Resource Management Middleware|url=https://doi.org/10.1109/isorc.2019.00033|journal=2019 IEEE 22nd International Symposium on Real-Time Distributed Computing (ISORC)|publisher=IEEE|pages=118–125|doi=10.1109/isorc.2019.00033}}</ref> In a deterministic method, the source and destination are known beforehand, whereas in a non-deterministic technique, periodic estimation has to be done regarding the user's route. Many authors estimate the route of mobile users by leveraging external services like Open Street Maps (<nowiki>http://www.openstreetmap.org</nowiki>), Google Maps APIs (<nowiki>https://cloud.google.com/</nowiki> maps-platform/) [18] <ref name=":4" />, logistics maps [19], GPRS Here APIs (<nowiki>https://developer</nowiki>. here.com/) [20, 21], Lyapunov estimation technique [22]. Specific QoS requirements: The selection of a fog node also depends upon specific quality requirements such as Latency, which many authors are using to select an efficient fog node [18, 10, 9], <ref name=":2" /> <ref name=":3" /> <ref name=":4" /> workload [23, 24], cost [16], etc.
== Review Methodology ==
The review process was conducted considering a review methodology consisting of four phases. The first phase was searching using traditional online database sources based on outlined search keywords. Table 1 lists the keywords used to find the relevant research articles to conduct the review in the Fog computing mobility-aware scheduling area. Second Phase: Limit the search of research articles beginning in 2015, and inclusion and exclusion principles are also used to refine research articles that specifically deal with mobility issues in task scheduling. Finally, in the Third Phase, A total of 20 papers are shortlisted for the review process. Further, Table 2 presents the research questions drafted for this study in mobility-aware scheduling in Fog computing.
{| class="wikitable"
|+Table 1: List of keywords used in the review process
!Sno
!Keyword
!Description
!Years
|-
|1
|Mobility
|Mobility-aware Fog task scheduling
| rowspan="5" |2015- 2021
|-
|2
|Mobility environment
|Mobility environment in Fog task scheduling
|-
|3
|Mobility factors
|Mobility factors in Fog task scheduling
|-
|4
|Mobility Awareness
|Mobility awareness in Fog task scheduling
|-
|5
|Mobility management
|Mobility management in Fog task scheduling
|}
{| class="wikitable"
|+Table 2: List of research questions used to complete the review process
!Q.No
!Research questions
|-
|1
|What scheduling approaches are used in the Fog computing environment to manage mobility awareness?
|-
|2
|What are the main limitations considered for mobility-aware scheduling techniques?
|-
|3
|Which case studies are applied to mobility-aware scheduling techniques?
|-
|4
|What evaluation tools are used to assess mobility-aware scheduling techniques?
|-
|5
|What performance indicators are utilized to evaluate mobility-aware scheduling techniques?
|-
|6
|What are the major open issues concerns in the field of mobility-aware scheduling for future research directions?
|}
=== Source of Information ===
To conduct this review, various online sources are listed below, and the research articles were searched using the different keywords mentioned in Table 1.
* Google Scholar (<nowiki>http://www.scholar.google.co.in</nowiki>)
* John Wiley & Sons Inc. (<nowiki>https://onlinelibrary.wiley.com/</nowiki>)
* Elsevier (<nowiki>https://www.elsevier.com/en-in</nowiki>)
* ACM Digital Library (<nowiki>https://www.acm.org/</nowiki>)
* Springer (<nowiki>https://www.springer.com/in</nowiki>)
* IEEE Xplore Digital Library (<nowiki>https://www.ieee.org/</nowiki>)
=== Quality Assessment ===
Research papers used quality assessment to filter out the most suitable mobility-based scheduling research articles in fog computing utilising the principle of inclusion and exclusion. Furthermore, in order to obtain high-quality research publications, the Center for Reviews and Dissemination (CRD) recommendations were followed, and each study item was examined for internal and external validation of results.
== Literature Analysis ==
Verma et al. [25] proposed a Server Cloudlet(SC) migration technique to handle users' mobility. The strategy is to select the target SC to migrate the services based on its highest rank. The rank of the SC further depends upon its available RAM, MIPS, and bandwidth.
Maleki et al. [26] designed two mobility-aware computation offloading approaches, sampling-based Online mobile applications to cloudlets(S-OAMC) and Greedy online mobile applications to cloudlets(G-OAMC). The developed system works in three major steps: ''(i)'' the future specifications of mobile applications are predicted using the Machine Learning (ML) method, named matrix completion. ''(ii)'' The predicted specifications of mobile applications, including future location prediction, also help estimate the offloading cost. ''(iii)'' Apply S-OAMC or G-OAMC offloading technique based on minimum cost value.
Abdelmoneem et al. [10] <ref name=":3" /> work on a fog-based healthcare architecture system that supports patient movement without altering their Quality of Service (QoS). The QoS in a mobile environment is maintained with the help of their proposed handoff mechanism. The authors modified the traditional Horizontal Handoff (HHO) mechanism [27] into two different phases: ''(i) handoff decision policy'': Received signal Strength (RSS) method is used to detect the network change information of the mobile patients and (ii) ''handoff strategy'': a handoff decision is made based on a comparison between RSS received from the old fog gateways, current threshold value and new fog gateways that are in close proximity to the mobile patient.
Javanmardi et al. [28] consider an imaginative city-based mobility scenario where the user is placing delay-sensitive service while moving. The authors proposed a task scheduling technique that jointly employs fuzzy logic and particle swarm optimization (PSO) algorithm to improve the QoS for mobile users within the city. The proposed algorithm is deployed at the Fog gateway, which further distributes the tasks to Fog devices available in an entire region in order to provide seamless service to mobile users. The main motive of the work is to improve resource utilization in a mobility-aware environment.
Puliafito et al. [29] develop an extension of ifogsim [30], which supports mobile environment. The authors implement different migration techniques inspired by Bittencourt et al. [31]. Different phases are devised to perform migration in a fog mobile environment, which: ''(i) Before migration phase'': migration decision is taken in this phase, based on specific parameters, like user location, speed, the direction of movement, zone, and migration point in order to select the appropriate cloudlet for offloading the services, ''(ii) During the migration phase'': this phase manages, monitors and synchronizes the whole selected migration process and ''(iii) After migration phase'': this phase involves closing the older cloudlet connections with the user and using the new cloudlet for services.
A Blockchain-based Mobility-aware Offloading (BMO) mechanism is designed by Dou et al. [32], where user mobility prediction is implemented using the Individual-Mobility (IM) model [33]. The idea behind the offloading mechanism is to shift the computational workload to different available Fog Servers (FSs) in the geo-location predicted by the IM model. Further, blockchain technology is deployed to check the authenticity of the forthcoming Fog servers. Finally, accounting is being managed by ''Fogcoin'', similar to Bitcoin, which stores the entire transaction history between the online Fog server and mobile users.
Martin et al. [7] <ref name=":1" /> proposed a framework that supports the migration of containers while satisfying the QoS requirements of mobile users. The migration of containers is done in an autonomic manner, by adopting the Monitor-Analyze-Plan-Execute (MAPE) autonomic control loop. The MAPE control loops discuss various steps of migration, like ''(i) Monitor'': used to constantly monitor the environment context, such as the mobility of users that is subsequently used to determine the need to migrate an application module to some other Fog node called target node; ''(ii) Analyze'': applies forecasting techniques to predict the user possible location in the next time step. If the distance between the user and the device is not under certain acceptable limits, a migration decision is made. ''(iii) Plan'': a Genetic Algorithm (GA) is used to identify a suitable Fog node closest to the forecasted location, where migration of the container running user application can be done. ''(iv) Execute'': this step ensures the whole migration process should take place smoothly.
Mass et al. propose a mobility and delay-aware fog server selection scheme. [34] called Edge-Process management (EPM) system. The EPM system depends upon the trajectory of a user’s movement, Fog server workload, and user location to select the appropriate Fog server for executing user applications. The system selects or re-selects a Fog Server (FS) based on a score value calculated through available bandwidth, power, distance from the user, and finally, duration of availability in a region.
Mobi-IoST (Mobility-aware Internet of Spatial Things), a real-time mobility-aware framework is presented by Ghosh et al. [35]. The authors considered the mobile nature of both IoT devices and Fog nodes, collaboratively called mobile agents. The proposed mobility-aware framework collects a vast amount of Global Positioning System (GPS) data of these mobile agents to predict their movement patterns using various machine learning algorithms. The major components of the framework are, ''(i) Movement pattern model''ling, collecting and modelling GPS log, stay-point, and other contextual location information; ''(ii) Predicting the following location'': human movement semantics is analyzed using all modelled information; ''(iii) Delivery of result:'' after the user movement prediction in the previous phase, the system intelligently discovers a capable fog device for data processing in a timely manner.
A middleware solution, URMILA, for managing resources and scheduling tasks in the Fog environment is presented by Shekhar et al. [18]. <ref name=":4" /> Ubiquitous Resource Management for Interference and Latency-Aware services (URMILA), ensures minimum Service Level Objectives (SLO) violation for latency-sensitive mobile applications across the cloud-Fog environment. The major modules of the proposed system are ''(i) Route calculation'', which calculates the user's possible routes using Google Maps or GPS data; ''(ii) Latency calculation'', the system deploys a data-driven model to estimate the Latency on predicted user routes; ''(iii) Fog node selection:'' the system selects a fog server for execution of task on the basis of its instantaneous utilization of the available resources. Further, it selects the Fog server for the entire period of execution, during which mobile users can still access their application through various Wireless Access Points (WAP).
Gia et al. [8] proposed a Handover mechanism for mobility management between fog nodes with the overall objective of consuming minimum energy and delay during handovers. Handover methods frequently rely on one or more measures, such as the Received Signal Strength Indicator (RSSI), the Link Quality Indicator (LQI), and the velocity of objects, to make handover decisions. This proposed system provides emergency services to health monitoring systems and basically works in two different mobility scenarios: ''(i) Node mobility between indoor or outdoor locations'': nodes belonging to indoor location or outdoor location only are considered to be similar, and they're calculated metrics value like RSSI, LQI, velocity, etc.; can be directly used for the handover of services to appropriate gateway, ''(ii) Node mobility between indoor and outdoor locations'': nodes are considered dissimilar, if they belong to indoor and outdoor location both, So, the metrics are re-calculated which introduce some additional parameters like temperature and interference signals in order to make a decision over handover gateway.
Babu and Biswash [36] proposed a mobility management technique that supports node-to-node communication and Fog computing-based architecture for 5G networks. It addresses the technical problems between 5G networks and Fog servers. The mobility-based approach assists mobile nodes in establishing communication while they are in motion. The mobility management technique may also be used to begin N2N communication in dynamic environments. N2N communication schemes for fog networks, on the other hand, provide an effective communication environment for mobile users with highly minimal network usage.
Wang et al. [37] proposed a mobility-aware offloading scheme, that considers an adequate quality and a computation allocation system that deals with the user equipment affairs to maximize the total revenue. The quality of user equipment is delineated by the sojourn time that follows the exponential distribution to reduce the chance of migration and maximize the entire income of user equipment. MILP (mixed-integer non-linear programming) NP-hard problem is modelled and consists of resource allocation and task offloading schemes. So, to solve this problem, a Gini coefficient and genetic algorithm are used to estimate the allocation of resources. The proposed approach can easily handle the mobility of users by minimizing the chances of migration.
Waqas et al. [38] provided a forward-thinking analysis of quality about-mobility in Fog computing by identifying quality challenges, requirements, and options for numerous ideas. The authors also identified outstanding concerns from previous research and summarized the advantages of quality for readers. It allows researchers and developers to avoid common misunderstandings and capture real-world scenarios such as businesses, governments, and educational institutions. Furthermore, it revolutionizes follow-up analysis and differentiates and foregrounds futurity orientations in real-life events involving humans and vehicles in a highly dynamic Fog setting.
Bi et al. [9] <ref name=":2" /> introduced software-defined networking-based fog computing architecture by decoupling mobility control and data forwarding. When mobile consumers travel between several access networks, the authors suggested an Optimal Path Selection (OPS) algorithm to preserve service continuity. Mobile customers received seamless and transparent mobility support thanks to efficient signalling operations. In mobile fog computing, the suggested algorithm ensured service continuity, increased handover performance, and achieved high data transfer efficiency.
Niu et al. [21] established a system called mobility-aware and multihop-D2D relaying-based scheduling scheme (MHRC) at Edge nodes near hotspots. The authors exploited concurrent transmissions to improve the performance of the system. The mmWave (millimetre-wave) band of Fog computing was cached, and extensive performance evaluation confirms that MHRC delivers more than the higher expected cached data amount. Name et al. [19] proposed an efficient algorithm to address the problem of resource allocation and user mobility from the Edge of the network to cloud data centres. This algorithm operates on a seamless handover scheme for mobile IPV6 to ease the user mobility challenge and reduce the application response time. The study showed that the task of service delay and packet loss was decreased due to the effect of change in the mobile node position.
Bittencourt et al. [23] examined the subject of resource allocation in the Fog/Cloud environment, taking into account the hierarchical structure. In the context of the Fog paradigm, the authors developed three scheduling algorithms (First come, First serve, delay-priority, and concurrent) that address user mobility and edge computing capabilities. The authors demonstrated that scheduling techniques may be designed to cope with different application classes based on demand from mobile users by leveraging both Fog to the end-user and cloud characteristics in this study.
Velasquez et al. [39] proposed a hybrid strategy for the Fog environment to manage resources for mobility scenarios. The authors applied the orchestrator technique to offer mobility support in a Smart City situation. In this technique, three components, the status monitor, the Planner, and the VM/Container, are employed to monitor, plan and execute the applications. The main aim of this study was to guarantee the QoS and QoE requirements of mobility-based applications and services.
Bittencourt et al. [31] presented a Fog computing architecture focusing on Virtual Machine (VM) migration where each user has a VM running in a cloudlet. In this architecture, the user's location is identified by using GPS, and then the VM is moved to a nearby Fog Cloud. The main aim of this study was to migrate users' data according to their mobility in order to maintain QoS for applications demanding lower Latency and allow smooth handoff mechanisms for mobile users.
From the extensive analysis of the literature, the various mobility-aware scheduling techniques have been classified as shown in Table 3. Further, it presents the advantages and limitations of each technique.
{| class="wikitable"
|+Table 3: Classification of Mobility-aware scheduling techniques
!Ref.
!Technique
!Advantage(s)
!Limitation(s)
|-
|[25]
|Ranking of VM
|
* Decrease in delay time, migration time, tuple lost value and downtime
|
* Case study not discussed
|-
|[26]
|S-OAMC, G-OAMC, Machine learning matrix completion
|
* Migration rate decreased
* Better Scalability
|
* Energy utilization of devices not investigated
|-
|[32]
|IM model
|
* Provides better mobility support and security
|
* Did not investigate synchronization overhead
|-
|[10] <ref name=":3" />
|RSS
|
* Reliable and Heterogeneous execution
|
* Low scalability
* No distributed scheduling to minimize response time
|-
|[7] <ref name=":1" />
|MAPE control loop
|
* Improved QoS
* Reduced service downtime
|
* No real-time evaluation
* High energy consumption
* Low robustness and security
|-
|[28]
|Copy of task to over a region
|
* Network Utilization developed
* Low-Loop delay
|
* Fault tolerance reliability is based on Fog gateways only
|-
|[18] <ref name=":4" />
|URMILA
|
* Service availability is maintained by delivering the desired QoS
* Deployment cost minimized
* Battery longevity ensured
|
* No empirical validations
* No user probabilistic routes
* Low scalability in terms of distance and speed
|-
|[8]
|RSSI, LQI
|
* Promises to keep the connection active with a low latency rate between the system and sensor nodes
|
* Consumes more energy
* Overhead is large for network transmission
* Coverage and overhead area are undefined between gateways
|-
|[29]
|User trajectories pre- diction using GPS log
|
* Provides better mobility support
* Reduces migration time
|
* Low scalability
|-
|[36]
|N2N communication, Data Analytics
|
* Fast data access
* High reliability and scalability- city
* Minimum overhead and cost
* High throughput and less delay
|
* No real-time cellular network evaluation
* Low network efficiency
|-
|[38]
|Mobility facets analysis
|
* Improved QoS and QoE
* Latency rate reduced
|
* No real-life implementation
* No reliability and low Latency between dynamic users and fog servers
|-
|[37]
|M-ILP, Sojourn time
|
* Cost-effective
* Migration time reduced
|
* Migration cost not considered
* No real-time implementation
|-
|[34]
|User trajectories prediction using GPS log
|
* Conventional delay tolerance
* High QoS
* Avoided local task processing cost
* Efficient in saving battery
* Handles subtle scenarios with high Latency
|
* Smart city not directed through the use of accurate city maps with aid from stimulation setting
|-
|[35]
|Prediction of user location
|
* Power consumption and de- lay handled proficiently
* Power and delay are reduced
|
* No acquiring of mobile data usage where location sense and time-series data can be projected to achieve the bandwidth
|-
|[21]
|Relay path planning algorithm
|
* Power efficient
* High spectral efficiency
* Data is relayed on cached edge nodes and relay nodes
|
* Blockage problem due to weak diffraction
|-
|[9] <ref name=":2" />
|OPS
|
* Handover performance improved
* Efficiency of high data communication achieved
* Guarantees continuity of services
|
* It does not guarantee privacy and security
* Virtual Machine migration not determined
* The handover process during the optimal path for more logical routing could have been more efficient
|-
|[23]
|Assignment of FS
|
* Low Latency
* Supports dynamic computing and user behaviour
|
* There is no prediction of mobility failure
* Bandwidth and processing not considered in scheduling
|-
|[39]
|Orchestrator
|
* Maintains trustworthiness, resilience, and low Latency in a dynamic environment
|
* No real implementation has been carried out
|-
|[19]
|Pattern modelling, dictating the following location
|
* Application Response time reduced
* Latency time reduced
|
* Services become temporarily inaccessible for some mobile nodes
|-
|[31]
|Forecasting technique
|
* Computing capacity provided for storage and processing of data
|
* Security concerns associated with both user data and applications not considered
|}
[[File:Fog Computing Figure 2.jpg|center|thumb|Figure 2: Year-wise count of research articles]]
== Analytical Discussion ==
The existing research on Fog computing Mobility-aware scheduling has been analyzed thoroughly. The analysis was performed using the answers given in Table 2. The results drawn through the thorough analysis of the literature are presented in various figures as follows:
Figure 2 lists the year-wise count of research papers that are considered for this survey. The bar graph represents the total number of research papers from journals and Conferences from the year 2015 - 2021. The research articles from the journal are 16, and the conference papers are 4. It is observed that more research needs to be conducted on mobility-aware scheduling in Fog computing.
Figure 3 displays an analytical comparison of mobility-aware scheduling approaches in Fog computing based on the content of the represented taxonomy in Figure 7. From the thorough analysis of the literature, four methods have been considered: migration, task offloading, handoff/handover mechanism, and task scheduling. The handoff/handover mechanism has the highest percentage of usage in mobility-aware scheduling, at 30%. The task scheduling and offloading have 25% of us- age in mobility-aware scheduling each. Finally, migration is only 20% of the usage in mobility-aware scheduling. Therefore, these approaches, specifically migration, are still open challenges to address for further research.
Figure 4 depicts various tools that were used for evaluating the mobility-aware scheduling approaches. 18% and 9% of the research articles used iFogSim and Mob-FogSim simulation tools for implementation, respectively. Besides, other simulation tools such as ONE (9%), NS2(5%), MATLAB (4%), Mininet (5%), and Docker (9%) have been utilized for implementing the proposed techniques in the research articles. Further, pro-Programming languages such as C++ (9%) and Python (9%) and hardware deployments such as Raspberry Pi (5%) and Ardunio (4%) were used for implementing existing case studies based on mobility-aware scheduling.
[[File:Fog Computing Figure 3.jpg|center|thumb|Figure 3: Percentage of the presented classified approaches in mobility-aware scheduling]]
The applied case studies are shown in Figure 5, which shows a maximum of 20% of research articles have implemented IoT-based applications. After that, 15% of each research article used Health care and Mobile-based applications. Besides, Smart City and 5G-based applications have been applied in 10% of research articles. Moreover, Surveillance and gaming, Mobile IPV6, and Wireless computing applications are the case studies on which only 5% of research articles exist.
After reviewing numerous research articles based on mobility-aware scheduling, it has been observed that researchers employed various parameters for evaluating the performance of the Mobility-scheduling approaches, as represented in Figure 6. It shows that Time completion (18%) followed by Delay (12%), Network usage (12%), Latency (12%), Energy consumption (10%), and cost (10%) are generally utilized. However, Downtime (4%), Migration time (4%), Makespan (2%), Success ratio (2%), Signal level (2%), Deadline (2%), Makespan (2%), Migration rate (2%), Mobility patterns (2%), Tuple lost (2%), and power consumption (2%) are less exploited parameters.
A taxonomy was compiled after going through the detailed review process, and various techniques have been categorized in Fog computing-based mobility-aware scheduling. Figure 7 presents these categories broadly in Migration, Offloading, Handoff/Handover mechanism, and Scheduling.
[[File:Fog Computing Figure 4.jpg|center|thumb|Figure 4: Percentage of tools utilized in the literature]]
[[File:Fog Computing Figure 5.jpg|center|thumb|Figure 5: Percentage of case studies employed in the literature]]
[[File:Fog Computing Figure 6.jpg|center|thumb|Figure 6: Percentage of parameters for evaluating Mobility-aware scheduling in the literature]]
=== Open Issues and Challenges ===
From the thorough analysis of the literature, several open issues and challenges pertaining to the area of mobility-aware scheduling in Fog computing have been identified in order to provide directions for future research exploration. The identified open problems and challenges, depicted in Figure 8, are discussed below.
=== Task Scheduling ===
Fog computing consists of several Fog nodes, each of which is a mini Cloud in the vicinity of mobile devices near the Edge of the network. When a mobile device submits a task, the Fog scheduler assigns it to a nearby Fog node(s) for execution. However, as the device moves from one network to another, the task needs to be rescheduled when the device enters a different network. Additionally, Fog nodes have limited capacity and availability; if the mobile user enters into a network where there is no nearby Fog service available, then this leads to a significant delay in service and raises a significant issue of task scheduling for mobile users [10, 40]. <ref name=":3" />
=== Resource Provisioning ===
Fog computing reduces the workload of Cloud computing by processing the tasks locally near the Edge of the network. However, due to the mobility of the user, the Fog node primarily assigned to a task might not be optimal over time. Therefore, the migration of the task to another Fog node near the user's mobile device is perceived as a necessary solution to resolve this concern [41]. However, such frequent migration over a short time poses the challenge of providing an efficient resource for the task that is capable of performing computation on time and delivering results to users while adhering to QoE.
[[File:Fog Computing Figure 7.jpg|center|thumb|Figure 7: Mobility-aware Fog Scheduling Taxonomy]]
=== Energy Consumption ===
The placement of fog services at the Edge of the network can provide better QoS to mobile users, resulting in a shorter response time. However, it is practically impossible due to the high deployment cost of new Fog devices, which further raises the significant issue of energy consumption. If too many deployments are done, there will be lots of communication traffic from the Cloud to Fog nodes and servers in order to create copies of the task from one network to another in case of mobility [42]. This results in considerable energy wastage in the form of high bandwidth consumption. This means that where and when to reschedule the task to an efficient Fog node must be carefully determined to minimize energy, response time, and deployment cost.
=== Quality of Experience (QoE) ===
[[File:Fog Computing Figure 8.jpg|left|thumb|Figure 8: Mobility-aware scheduling open issues and challenges]]
Several mobility-based scheduling algorithms exist, but they need to focus on maximizing the user QoE [29, 8, 10, 18]. <ref name=":3" /> <ref name=":4" /> Further, they do not analyze the user performance; hence, the QoE of using a service or product is not determined. Therefore, to understand the user gain and loss, the scheduling algorithm needs to focus on enhancing the user QoE.
=== Resource Management ===
The mobility of Fog nodes/users demands efficient resource discovery and sharing, resource availability, and task offloading [43]. Few techniques that were proposed to manage the resources effectively did not consider more constraints such as density, latency sensitivity, and mobility of Edge and Fog nodes, and as the number of nodes increases, issues such as scalability and distributing the algorithms arise [44, 45, 46]. Therefore, more attention needs to be paid towards the mobile Fog computing environment to manage the resources effectively.
=== Privacy and Security ===
In [47], a scheduling policy is proposed for the mobile device system to minimize the cost. However, the privacy issues of location and usage patterns were ignored. Additionally, data privacy, access control, and intrusion detection in scheduling policies have been overlooked [7, 48, 28]. <ref name=":1" /> Besides, Fog node devices are normally deployed near the end-user; hence, protection and surveillance are comparatively weak, which can result in a malicious attack [49, 50].
== Data availability statement ==
Not applicable.
== Conclusions ==
Fog computing infrastructure provides services at the Edge of the network. So, to provide support for scheduling and management of mobility awareness, efficient techniques and mechanisms have been proposed. In this survey, research articles on the mobility-aware-scheduling strategies in Fog computing have been thoroughly analyzed. It provides a comparative study among existing mobility-aware scheduling strategies based on vital factors such as techniques proposed, parameters considered, tools utilized for implementation, and case studies considered, along with the advantages and limitations. Further, several open issues and challenges have been identified for future research direction.
===Subheading===
==Second Heading==
==Third Heading, etc==
==Additional information==
===Acknowledgements===
Any people, organisations, or funding sources that you would like to thank.
===Competing interests===
Any conflicts of interest that you would like to declare. Otherwise, a statement that the authors have no competing interest.
===Ethics statement===
An ethics statement, if appropriate, on any animal or human research performed should be included here or in the methods section.
==References==
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{{Article info
| journal = WikiJournal of Science
| last1 = Islam
| orcid1 = 0000-0001-5750-1058
| first1 = Mir Salim Ul
| last2 = Kumar
| orcid2 = 0000-0003-3279-5111
| first2 = Ashok
| affiliation2 = Department of Computer Application, Chandigarh School of Business, Chandigarh Group of Colleges, Jhanjeri, Punjab, India
| et_al =
| affiliation1 = Department of Computer Science and Engineering, National Institute of Technology Srinagar, Jammu and Kashmir, India
| correspondence1 = mir.salim27@gmail.com
| affiliations =
| keywords = Fog computing, Mobility-aware, Scheduling, Resource Management, Resource Allocation
| license = CC BY 4.0
| submitted = 2024-08-22
| abstract = The increasing focus on Fog-IoT results in billions of Internet-connected devices that demand substantial computational power and network bandwidth. These devices are geographically distributed, heterogeneous, computational capacity constrained, inconsistent in behaviour, and generally mobile. Therefore, providing seamless service, irrespective of location and movement of the devices as well as the end-users, makes resource scheduling a significant challenge in the Fog computing paradigm. Several mobility-aware scheduling strategies have been proposed in the literature to efficiently manage the resources for mobile users and devices in the Fog environment. This paper gives a survey of mobility-aware scheduling in the Fog computing environment. It describes the many strategies presented and their benefits and drawbacks. It also includes a complete study and taxonomy of the mobility-aware scheduling field. Further, it delineates open issues and challenges. This work will provide researchers with future research directions and aid them in recognizing the gaps before planning for further research in mobility-aware scheduling.
}}
== Introduction ==
With the advancement in technology and the exponential growth of mobile devices, network traffic has increased manifold in cloud computing. Due to this reason, Latency reduction and faster processing of data for mobile users have become critical challenges in providing seamless connectivity and minimal disruption while the user is moving.<ref>{{Cite journal|last=Abdullah|first=Fatima|last2=Kimovski|first2=Dragi|last3=Prodan|first3=Radu|last4=Munir|first4=Kashif|date=2021-06-11|title=Handover authentication latency reduction using mobile edge computing and mobility patterns|url=https://doi.org/10.1007/s00607-021-00969-z|journal=Computing|volume=103|issue=11|pages=2667–2686|doi=10.1007/s00607-021-00969-z|issn=0010-485X}}</ref> The data movement brings the additional issue of integrity and confidentiality because data is moving via a wireless connection to a far distant cloud. Additionally, due to the cloud's location is far from mobile users, so data movement is also affected by variable network strength and phone bandwidth. The solution proposed by [2] is to extend cloud capabilities through fog computing architecture. The Fog architecture allows substantial computation, storage, and processing using the Fog devices installed close to the user’s access point. Fog computing, therefore, reduces Latency and bandwidth consumption, improves security, provides context awareness, and renders more efficient services to mobile users [3, 4, 5]. <ref>{{Cite journal|last=Dastjerdi|first=Amir Vahid|last2=Buyya|first2=Rajkumar|date=2016-08|title=Fog Computing: Helping the Internet of Things Realize Its Potential|url=https://doi.org/10.1109/mc.2016.245|journal=Computer|volume=49|issue=8|pages=112–116|doi=10.1109/mc.2016.245|issn=0018-9162}}</ref>
[[File:Fog Computing Figure 1.jpg|left|thumb|Figure 1: Mobility Scenario in Fog Computing]]
However, mobility also imposes severe challenges for Fog computing due to its distributed and diverse environment. Mobility is recognized by either user-level or device-level contextual information [6]. <ref name=":0">{{Cite journal|last=Islam|first=Mir Salim Ul|last2=Kumar|first2=Ashok|last3=Hu|first3=Yu-Chen|date=2021-04|title=Context-aware scheduling in Fog computing: A survey, taxonomy, challenges and future directions|url=https://doi.org/10.1016/j.jnca.2021.103008|journal=Journal of Network and Computer Applications|volume=180|pages=103008|doi=10.1016/j.jnca.2021.103008|issn=1084-8045}}</ref> As the user moves from one location to another, the geographical location of the smart devices also changes. The change in location of the devices raises the issue of searching and rescheduling with mobility management. Efficient re-scheduling requires a well-planned handoff mechanism that is accountable for smoothly de-registering a sensor node from a source access point where the application was initially hosted and registering it to a new access point. Figure 1 depicts these mentioned problems of change in access points while the user moves from one area to another. The services may also get interrupted when there is more distance between the Fog nodes and users [7]. <ref name=":1">{{Cite journal|last=Martin|first=John Paul|last2=Kandasamy|first2=A|last3=Chandrasekaran|first3=K|date=2020-03-09|title=Mobility aware autonomic approach for the migration of application modules in fog computing environment|url=https://doi.org/10.1007/s12652-020-01854-x|journal=Journal of Ambient Intelligence and Humanized Computing|volume=11|issue=11|pages=5259–5278|doi=10.1007/s12652-020-01854-x|issn=1868-5137}}</ref> Further, any disruption in communication may lead to an increase in Latency for mobile users [8].
The significant challenges in mobility management of Fog computing are: (i) ''Fog node discovery''- due to the heterogeneous and mobile nature of the smart mobile devices, the Fog task scheduler faces the issue of finding an optimal Fog node for scheduling the task, ''(ii) Handover mechanism between users and Fog nodes''- imagine a Fog user is moving from location to location and accessing information about his surroundings with the help of a smart device. Due to the frequent change in the user's location, the Fog scheduler may need to repeatedly migrate the user’s task to a different Fog node available in his vicinity. The frequent migration of tasks increases the overhead of scheduling and, further, the restricted signal strength in certain places may lead to a breakdown of task or result delivery, and (iii) ''the Handover mechanism between Cloud and Fog''- Fog nodes have limited capacities and need to continuously communicate with Cloud computing for passing information about the tasks [9]. <ref name=":2">{{Cite journal|last=Bi|first=Yuanguo|last2=Han|first2=Guangjie|last3=Lin|first3=Chuan|last4=Deng|first4=Qingxu|last5=Guo|first5=Lei|last6=Li|first6=Fuliang|date=2018-05|title=Mobility Support for Fog Computing: An SDN Approach|url=https://doi.org/10.1109/mcom.2018.1700908|journal=IEEE Communications Magazine|volume=56|issue=5|pages=53–59|doi=10.1109/mcom.2018.1700908|issn=0163-6804}}</ref> Due to strict requirements for security, Latency, network coverage, and reliability, It becomes challenging to implement an efficient handover mechanism for full mobility support in critical domains such as healthcare [10, 11, 12] <ref name=":3">{{Cite journal|last=Abdelmoneem|first=Randa M.|last2=Benslimane|first2=Abderrahim|last3=Shaaban|first3=Eman|date=2020-10|title=Mobility-aware task scheduling in cloud-Fog IoT-based healthcare architectures|url=https://doi.org/10.1016/j.comnet.2020.107348|journal=Computer Networks|volume=179|pages=107348|doi=10.1016/j.comnet.2020.107348|issn=1389-1286}}</ref> <ref>{{Cite journal|last=Rahmani|first=Amir M.|last2=Gia|first2=Tuan Nguyen|last3=Negash|first3=Behailu|last4=Anzanpour|first4=Arman|last5=Azimi|first5=Iman|last6=Jiang|first6=Mingzhe|last7=Liljeberg|first7=Pasi|date=2018-01|title=Exploiting smart e-Health gateways at the edge of healthcare Internet-of-Things: A fog computing approach|url=https://doi.org/10.1016/j.future.2017.02.014|journal=Future Generation Computer Systems|volume=78|pages=641–658|doi=10.1016/j.future.2017.02.014|issn=0167-739X}}</ref> and vehicular systems [13, 14]. <ref>{{Cite journal|last=Zhu|first=Chao|last2=Pastor|first2=Giancarlo|last3=Xiao|first3=Yu|last4=Li|first4=Yong|last5=Ylae-Jaeaeski|first5=Antti|date=2018-06|title=Fog Following Me: Latency and Quality Balanced Task Allocation in Vehicular Fog Computing|url=https://doi.org/10.1109/sahcn.2018.8397129|journal=2018 15th Annual IEEE International Conference on Sensing, Communication, and Networking (SECON)|publisher=IEEE|pages=1–9|doi=10.1109/sahcn.2018.8397129}}</ref> <ref>{{Cite journal|last=Aljeri|first=Noura|last2=Boukerche|first2=Azzedine|date=2020-01-17|title=Fog‐enabled vehicular networks: A new challenge for mobility management|url=https://doi.org/10.1002/itl2.141|journal=Internet Technology Letters|volume=3|issue=6|doi=10.1002/itl2.141|issn=2476-1508}}</ref> Thus, mobility significantly impacts the overhead of scheduling policies and the applications' performance, eventually affecting the Quality of Experience (QoE). Therefore, mobility-aware scheduling in Fog computing has observed strong attention from researchers. Our main goal is to provide a detailed review of mobility-aware scheduling in fog computing. The review provides a detailed analysis of existing scheduling strategies that concentrate specifically on mobility awareness in the Fog environment. The following are the main contributions of this paper:
* This paper presents a detailed survey of mobility-aware scheduling in the Fog computing environment.
* It provides the details of the different techniques proposed, as well as their advantages and limitations.
* It provides a detailed analysis and taxonomy of the mobility-aware scheduling field.
* It identifies several open challenges for future research directions.
=== Related Surveys ===
Numerous survey studies on Fog computing focus on resource management, job scheduling, and context-aware scheduling. For example, Ghobaei-Arani et al. [15] <ref>{{Cite journal|last=Ghobaei-Arani|first=Mostafa|last2=Souri|first2=Alireza|last3=Rahmanian|first3=Ali A.|date=2019-09-06|title=Resource Management Approaches in Fog Computing: a Comprehensive Review|url=https://doi.org/10.1007/s10723-019-09491-1|journal=Journal of Grid Computing|volume=18|issue=1|pages=1–42|doi=10.1007/s10723-019-09491-1|issn=1570-7873}}</ref> presented a survey on resource management techniques in fog computing in the form of taxonomy to highlight cutting-edge methods while also addressing unresolved challenges. The various authors in the cloud-fog area provided the task scheduling review. Their benefits and drawbacks, as well as numerous tools and challenges concerning the scheduling techniques and their limitations, were analyzed by Alizadeh et al. [16]. Islam et al. [6] <ref name=":0" /> thoroughly analyze relevant literature on context-aware scheduling in fog computing. Further, Mouradian et al. [17] review fog computing's significant issues and challenges. But, the critical area of mobility research is still in its initial stage, and most of the review papers contain very few documents on mobility-aware task scheduling in the area of fog computing. Therefore, an extensive and comparative study is required in mobility-constrained fog computing. A deep insight into various techniques which can impact the user QoS is necessary to understand mobility-aware fog computing. This motivates us to carry out a comprehensive survey; to the best of our knowledge, this is the first detailed survey. This survey paper thoroughly examines existing scheduling solutions that focus on user mobility. Moreover, various mobility-aware scheduling techniques are discussed, along with their pros and cons. Further, the impact of mobility parameters on various QoS parameters and context-awareness is also analyzed thoroughly.
=== Paper Organization ===
The remainder of this review article is divided into 5 sections: Section 2 discusses mobility-aware scheduling in Fog computing. Section 3 presents the review methodology. Section 4 analyzes and summarizes the considered research papers and compares existing mobility-aware scheduling strategies. Section 5 provides the results drawn after critically examining the existing literature on mobility-aware scheduling policies. Finally, Section 6 presents the conclusion.
== Mobility-aware Scheduling in Fog Computing ==
Mobile device management compromises the fundamental features of Fog computing because whenever a user moves, the distance between them increases, impacting the QoS. Therefore, to keep the computing fog node close to the associated mobile device, the services or tasks need to be migrated from one fog node to another appropriate fog device. The selection of such appropriate fog nodes in a mobile environment deals with two main processes: Estimation of user mobility patterns: User mobility estimation techniques can be probabilistic and deterministic [18]. <ref name=":4">{{Cite journal|last=Shekhar|first=Shashank|last2=Chhokra|first2=Ajay|last3=Sun|first3=Hongyang|last4=Gokhale|first4=Aniruddha|last5=Dubey|first5=Abhishek|last6=Koutsoukos|first6=Xenofon|date=2019-05|title=URMILA: A Performance and Mobility-Aware Fog/Edge Resource Management Middleware|url=https://doi.org/10.1109/isorc.2019.00033|journal=2019 IEEE 22nd International Symposium on Real-Time Distributed Computing (ISORC)|publisher=IEEE|pages=118–125|doi=10.1109/isorc.2019.00033}}</ref> In a deterministic method, the source and destination are known beforehand, whereas in a non-deterministic technique, periodic estimation has to be done regarding the user's route. Many authors estimate the route of mobile users by leveraging external services like Open Street Maps (<nowiki>http://www.openstreetmap.org</nowiki>), Google Maps APIs (<nowiki>https://cloud.google.com/</nowiki> maps-platform/) [18] <ref name=":4" />, logistics maps [19], <ref name=":5">{{Cite journal|last=Name|first=Haruna Abdu Manis|last2=Oladipo|first2=Francisca O.|last3=Ariwa|first3=Ezendu|date=2017-08|title=User mobility and resource scheduling and management in fog computing to support IoT devices|url=https://doi.org/10.1109/intech.2017.8102447|journal=2017 Seventh International Conference on Innovative Computing Technology (INTECH)|publisher=IEEE|pages=191–196|doi=10.1109/intech.2017.8102447}}</ref> GPRS Here APIs (<nowiki>https://developer</nowiki>. here.com/) [20, 21], <ref name=":6">{{Cite journal|last=Niu|first=Yong|last2=Liu|first2=Yu|last3=Li|first3=Yong|last4=Zhong|first4=Zhangdui|last5=Ai|first5=Bo|last6=Hui|first6=Pan|date=2018|title=Mobility-Aware Caching Scheduling for Fog Computing in mmWave Band|url=https://doi.org/10.1109/access.2018.2880031|journal=IEEE Access|volume=6|pages=69358–69370|doi=10.1109/access.2018.2880031|issn=2169-3536}}</ref> Lyapunov estimation technique [22]. <ref>{{Cite journal|last=Li|first=Yun|last2=Xia|first2=Shichao|last3=Zheng|first3=Mengyan|last4=Cao|first4=Bin|last5=Liu|first5=Qilie|date=2022-01-01|title=Lyapunov Optimization-Based Trade-Off Policy for Mobile Cloud Offloading in Heterogeneous Wireless Networks|url=https://doi.org/10.1109/tcc.2019.2938504|journal=IEEE Transactions on Cloud Computing|volume=10|issue=1|pages=491–505|doi=10.1109/tcc.2019.2938504|issn=2168-7161}}</ref> Specific QoS requirements: The selection of a fog node also depends upon specific quality requirements such as Latency, which many authors are using to select an efficient fog node [18, 10, 9], <ref name=":2" /> <ref name=":3" /> <ref name=":4" /> workload [23, 24], <ref name=":7">{{Cite journal|last=Bittencourt|first=Luiz F.|last2=Diaz-Montes|first2=Javier|last3=Buyya|first3=Rajkumar|last4=Rana|first4=Omer F.|last5=Parashar|first5=Manish|date=2017-03|title=Mobility-Aware Application Scheduling in Fog Computing|url=https://doi.org/10.1109/mcc.2017.27|journal=IEEE Cloud Computing|volume=4|issue=2|pages=26–35|doi=10.1109/mcc.2017.27|issn=2325-6095}}</ref><ref>{{Cite journal|last=Ahanger|first=Tariq Ahamed|last2=Tariq|first2=Usman|last3=Nusir|first3=Muneer|date=2022-04-17|title=Mobility of Internet of Things and Fog Computing: Serious Concerns and Future Directions|url=https://doi.org/10.17762/ijcnis.v10i3.3706|journal=International Journal of Communication Networks and Information Security (IJCNIS)|volume=10|issue=3|doi=10.17762/ijcnis.v10i3.3706|issn=2073-607X}}</ref> cost [16], etc.
== Review Methodology ==
The review process was conducted considering a review methodology consisting of four phases. The first phase was searching using traditional online database sources based on outlined search keywords. Table 1 lists the keywords used to find the relevant research articles to conduct the review in the Fog computing mobility-aware scheduling area. Second Phase: Limit the search of research articles beginning in 2015, and inclusion and exclusion principles are also used to refine research articles that specifically deal with mobility issues in task scheduling. Finally, in the Third Phase, A total of 20 papers are shortlisted for the review process. Further, Table 2 presents the research questions drafted for this study in mobility-aware scheduling in Fog computing.
{| class="wikitable"
|+Table 1: List of keywords used in the review process
!Sno
!Keyword
!Description
!Years
|-
|1
|Mobility
|Mobility-aware Fog task scheduling
| rowspan="5" |2015- 2021
|-
|2
|Mobility environment
|Mobility environment in Fog task scheduling
|-
|3
|Mobility factors
|Mobility factors in Fog task scheduling
|-
|4
|Mobility Awareness
|Mobility awareness in Fog task scheduling
|-
|5
|Mobility management
|Mobility management in Fog task scheduling
|}
{| class="wikitable"
|+Table 2: List of research questions used to complete the review process
!Q.No
!Research questions
|-
|1
|What scheduling approaches are used in the Fog computing environment to manage mobility awareness?
|-
|2
|What are the main limitations considered for mobility-aware scheduling techniques?
|-
|3
|Which case studies are applied to mobility-aware scheduling techniques?
|-
|4
|What evaluation tools are used to assess mobility-aware scheduling techniques?
|-
|5
|What performance indicators are utilized to evaluate mobility-aware scheduling techniques?
|-
|6
|What are the major open issues concerns in the field of mobility-aware scheduling for future research directions?
|}
=== Source of Information ===
To conduct this review, various online sources are listed below, and the research articles were searched using the different keywords mentioned in Table 1.
* Google Scholar (<nowiki>http://www.scholar.google.co.in</nowiki>)
* John Wiley & Sons Inc. (<nowiki>https://onlinelibrary.wiley.com/</nowiki>)
* Elsevier (<nowiki>https://www.elsevier.com/en-in</nowiki>)
* ACM Digital Library (<nowiki>https://www.acm.org/</nowiki>)
* Springer (<nowiki>https://www.springer.com/in</nowiki>)
* IEEE Xplore Digital Library (<nowiki>https://www.ieee.org/</nowiki>)
=== Quality Assessment ===
Research papers used quality assessment to filter out the most suitable mobility-based scheduling research articles in fog computing utilising the principle of inclusion and exclusion. Furthermore, in order to obtain high-quality research publications, the Center for Reviews and Dissemination (CRD) recommendations were followed, and each study item was examined for internal and external validation of results.
== Literature Analysis ==
Verma et al. [25] <ref name=":8">{{Cite journal|last=Verma|first=Kanupriya|last2=Kumar|first2=Ashok|last3=Ul Islam|first3=Mir Salim|last4=Kanwar|first4=Tulika|last5=Bhushan|first5=Megha|date=2021|title=Rank based mobility-aware scheduling in Fog computing|url=https://doi.org/10.1016/j.imu.2021.100619|journal=Informatics in Medicine Unlocked|volume=24|pages=100619|doi=10.1016/j.imu.2021.100619|issn=2352-9148}}</ref> proposed a Server Cloudlet(SC) migration technique to handle users' mobility. The strategy is to select the target SC to migrate the services based on its highest rank. The rank of the SC further depends upon its available RAM, MIPS, and bandwidth.
Maleki et al. [26] designed two mobility-aware computation offloading approaches, sampling-based Online mobile applications to cloudlets(S-OAMC) and Greedy online mobile applications to cloudlets(G-OAMC). The developed system works in three major steps: ''(i)'' the future specifications of mobile applications are predicted using the Machine Learning (ML) method, named matrix completion. ''(ii)'' The predicted specifications of mobile applications, including future location prediction, also help estimate the offloading cost. ''(iii)'' Apply S-OAMC or G-OAMC offloading technique based on minimum cost value.
Abdelmoneem et al. [10] <ref name=":3" /> work on a fog-based healthcare architecture system that supports patient movement without altering their Quality of Service (QoS). The QoS in a mobile environment is maintained with the help of their proposed handoff mechanism. The authors modified the traditional Horizontal Handoff (HHO) mechanism [27] <ref>{{Cite journal|last=Kassar|first=Meriem|last2=Kervella|first2=Brigitte|last3=Pujolle|first3=Guy|date=2008-06|title=An overview of vertical handover decision strategies in heterogeneous wireless networks|url=https://doi.org/10.1016/j.comcom.2008.01.044|journal=Computer Communications|volume=31|issue=10|pages=2607–2620|doi=10.1016/j.comcom.2008.01.044|issn=0140-3664}}</ref> into two different phases: ''(i) handoff decision policy'': Received signal Strength (RSS) method is used to detect the network change information of the mobile patients and (ii) ''handoff strategy'': a handoff decision is made based on a comparison between RSS received from the old fog gateways, current threshold value and new fog gateways that are in close proximity to the mobile patient.
Javanmardi et al. [28] <ref name=":9">{{Cite journal|last=Javanmardi|first=Saeed|last2=Shojafar|first2=Mohammad|last3=Persico|first3=Valerio|last4=Pescapè|first4=Antonio|date=2020-08-18|title=FPFTS: A joint fuzzy particle swarm optimization mobility‐aware approach to fog task scheduling algorithm for Internet of Things devices|url=https://doi.org/10.1002/spe.2867|journal=Software: Practice and Experience|volume=51|issue=12|pages=2519–2539|doi=10.1002/spe.2867|issn=0038-0644}}</ref> consider an imaginative city-based mobility scenario where the user is placing delay-sensitive service while moving. The authors proposed a task scheduling technique that jointly employs fuzzy logic and particle swarm optimization (PSO) algorithm to improve the QoS for mobile users within the city. The proposed algorithm is deployed at the Fog gateway, which further distributes the tasks to Fog devices available in an entire region in order to provide seamless service to mobile users. The main motive of the work is to improve resource utilization in a mobility-aware environment.
Puliafito et al. [29] develop an extension of ifogsim [30], <ref>{{Cite journal|last=Gupta|first=Harshit|last2=Vahid Dastjerdi|first2=Amir|last3=Ghosh|first3=Soumya K.|last4=Buyya|first4=Rajkumar|date=2017-06-21|title=iFogSim: A toolkit for modeling and simulation of resource management techniques in the Internet of Things, Edge and Fog computing environments|url=https://doi.org/10.1002/spe.2509|journal=Software: Practice and Experience|volume=47|issue=9|pages=1275–1296|doi=10.1002/spe.2509|issn=0038-0644}}</ref> which supports mobile environment. The authors implement different migration techniques inspired by Bittencourt et al. [31]. <ref name=":10">{{Cite journal|last=Bittencourt|first=Luiz Fernando|last2=Lopes|first2=Marcio Moraes|last3=Petri|first3=Ioan|last4=Rana|first4=Omer F.|date=2015-11|title=Towards Virtual Machine Migration in Fog Computing|url=https://doi.org/10.1109/3pgcic.2015.85|journal=2015 10th International Conference on P2P, Parallel, Grid, Cloud and Internet Computing (3PGCIC)|publisher=IEEE|doi=10.1109/3pgcic.2015.85}}</ref> Different phases are devised to perform migration in a fog mobile environment, which: ''(i) Before migration phase'': migration decision is taken in this phase, based on specific parameters, like user location, speed, the direction of movement, zone, and migration point in order to select the appropriate cloudlet for offloading the services, ''(ii) During the migration phase'': this phase manages, monitors and synchronizes the whole selected migration process and ''(iii) After migration phase'': this phase involves closing the older cloudlet connections with the user and using the new cloudlet for services.
A Blockchain-based Mobility-aware Offloading (BMO) mechanism is designed by Dou et al. [32], where user mobility prediction is implemented using the Individual-Mobility (IM) model [33]. <ref>{{Cite journal|last=Song|first=Chaoming|last2=Koren|first2=Tal|last3=Wang|first3=Pu|last4=Barabási|first4=Albert-László|date=2010-09-12|title=Modelling the scaling properties of human mobility|url=https://doi.org/10.1038/nphys1760|journal=Nature Physics|volume=6|issue=10|pages=818–823|doi=10.1038/nphys1760|issn=1745-2473}}</ref> The idea behind the offloading mechanism is to shift the computational workload to different available Fog Servers (FSs) in the geo-location predicted by the IM model. Further, blockchain technology is deployed to check the authenticity of the forthcoming Fog servers. Finally, accounting is being managed by ''Fogcoin'', similar to Bitcoin, which stores the entire transaction history between the online Fog server and mobile users.
Martin et al. [7] <ref name=":1" /> proposed a framework that supports the migration of containers while satisfying the QoS requirements of mobile users. The migration of containers is done in an autonomic manner, by adopting the Monitor-Analyze-Plan-Execute (MAPE) autonomic control loop. The MAPE control loops discuss various steps of migration, like ''(i) Monitor'': used to constantly monitor the environment context, such as the mobility of users that is subsequently used to determine the need to migrate an application module to some other Fog node called target node; ''(ii) Analyze'': applies forecasting techniques to predict the user possible location in the next time step. If the distance between the user and the device is not under certain acceptable limits, a migration decision is made. ''(iii) Plan'': a Genetic Algorithm (GA) is used to identify a suitable Fog node closest to the forecasted location, where migration of the container running user application can be done. ''(iv) Execute'': this step ensures the whole migration process should take place smoothly.
Mass et al. propose a mobility and delay-aware fog server selection scheme. [34] <ref name=":11">{{Cite journal|last=Mass|first=Jakob|last2=Chang|first2=Chii|last3=Srirama|first3=Satish Narayana|date=2019-06|title=Edge Process Management: A case study on adaptive task scheduling in mobile IoT|url=https://doi.org/10.1016/j.iot.2019.100051|journal=Internet of Things|volume=6|pages=100051|doi=10.1016/j.iot.2019.100051|issn=2542-6605}}</ref> called Edge-Process management (EPM) system. The EPM system depends upon the trajectory of a user’s movement, Fog server workload, and user location to select the appropriate Fog server for executing user applications. The system selects or re-selects a Fog Server (FS) based on a score value calculated through available bandwidth, power, distance from the user, and finally, duration of availability in a region.
Mobi-IoST (Mobility-aware Internet of Spatial Things), a real-time mobility-aware framework is presented by Ghosh et al. [35]. <ref name=":12">{{Cite journal|last=Ghosh|first=Shreya|last2=Mukherjee|first2=Anwesha|last3=Ghosh|first3=Soumya K.|last4=Buyya|first4=Rajkumar|date=2020-10-01|title=Mobi-IoST: Mobility-Aware Cloud-Fog-Edge-IoT Collaborative Framework for Time-Critical Applications|url=https://doi.org/10.1109/tnse.2019.2941754|journal=IEEE Transactions on Network Science and Engineering|volume=7|issue=4|pages=2271–2285|doi=10.1109/tnse.2019.2941754|issn=2327-4697}}</ref> The authors considered the mobile nature of both IoT devices and Fog nodes, collaboratively called mobile agents. The proposed mobility-aware framework collects a vast amount of Global Positioning System (GPS) data of these mobile agents to predict their movement patterns using various machine learning algorithms. The major components of the framework are, ''(i) Movement pattern model''ling, collecting and modelling GPS log, stay-point, and other contextual location information; ''(ii) Predicting the following location'': human movement semantics is analyzed using all modelled information; ''(iii) Delivery of result:'' after the user movement prediction in the previous phase, the system intelligently discovers a capable fog device for data processing in a timely manner.
A middleware solution, URMILA, for managing resources and scheduling tasks in the Fog environment is presented by Shekhar et al. [18]. <ref name=":4" /> Ubiquitous Resource Management for Interference and Latency-Aware services (URMILA), ensures minimum Service Level Objectives (SLO) violation for latency-sensitive mobile applications across the cloud-Fog environment. The major modules of the proposed system are ''(i) Route calculation'', which calculates the user's possible routes using Google Maps or GPS data; ''(ii) Latency calculation'', the system deploys a data-driven model to estimate the Latency on predicted user routes; ''(iii) Fog node selection:'' the system selects a fog server for execution of task on the basis of its instantaneous utilization of the available resources. Further, it selects the Fog server for the entire period of execution, during which mobile users can still access their application through various Wireless Access Points (WAP).
Gia et al. [8] proposed a Handover mechanism for mobility management between fog nodes with the overall objective of consuming minimum energy and delay during handovers. Handover methods frequently rely on one or more measures, such as the Received Signal Strength Indicator (RSSI), the Link Quality Indicator (LQI), and the velocity of objects, to make handover decisions. This proposed system provides emergency services to health monitoring systems and basically works in two different mobility scenarios: ''(i) Node mobility between indoor or outdoor locations'': nodes belonging to indoor location or outdoor location only are considered to be similar, and they're calculated metrics value like RSSI, LQI, velocity, etc.; can be directly used for the handover of services to appropriate gateway, ''(ii) Node mobility between indoor and outdoor locations'': nodes are considered dissimilar, if they belong to indoor and outdoor location both, So, the metrics are re-calculated which introduce some additional parameters like temperature and interference signals in order to make a decision over handover gateway.
Babu and Biswash [36] <ref name=":13">{{Cite journal|last=Babu|first=S.|last2=Biswash|first2=Sanjay Kumar|date=2019-09-03|title=Fog computing–based node‐to‐node communication and mobility management technique for 5G networks|url=https://doi.org/10.1002/ett.3738|journal=Transactions on Emerging Telecommunications Technologies|volume=30|issue=10|doi=10.1002/ett.3738|issn=2161-3915}}</ref> proposed a mobility management technique that supports node-to-node communication and Fog computing-based architecture for 5G networks. It addresses the technical problems between 5G networks and Fog servers. The mobility-based approach assists mobile nodes in establishing communication while they are in motion. The mobility management technique may also be used to begin N2N communication in dynamic environments. N2N communication schemes for fog networks, on the other hand, provide an effective communication environment for mobile users with highly minimal network usage.
Wang et al. [37] <ref name=":14">{{Cite journal|last=Wang|first=Dongyu|last2=Liu|first2=Zhaolin|last3=Wang|first3=Xiaoxiang|last4=Lan|first4=Yanwen|date=2019|title=Mobility-Aware Task Offloading and Migration Schemes in Fog Computing Networks|url=https://doi.org/10.1109/access.2019.2908263|journal=IEEE Access|volume=7|pages=43356–43368|doi=10.1109/access.2019.2908263|issn=2169-3536}}</ref> proposed a mobility-aware offloading scheme, that considers an adequate quality and a computation allocation system that deals with the user equipment affairs to maximize the total revenue. The quality of user equipment is delineated by the sojourn time that follows the exponential distribution to reduce the chance of migration and maximize the entire income of user equipment. MILP (mixed-integer non-linear programming) NP-hard problem is modelled and consists of resource allocation and task offloading schemes. So, to solve this problem, a Gini coefficient and genetic algorithm are used to estimate the allocation of resources. The proposed approach can easily handle the mobility of users by minimizing the chances of migration.
Waqas et al. [38] <ref name=":15">{{Cite journal|last=Waqas|first=Muhammad|last2=Niu|first2=Yong|last3=Ahmed|first3=Manzoor|last4=Li|first4=Yong|last5=Jin|first5=Depeng|last6=Han|first6=Zhu|date=2019|title=Mobility-Aware Fog Computing in Dynamic Environments: Understandings and Implementation|url=https://doi.org/10.1109/access.2018.2883662|journal=IEEE Access|volume=7|pages=38867–38879|doi=10.1109/access.2018.2883662|issn=2169-3536}}</ref> provided a forward-thinking analysis of quality about-mobility in Fog computing by identifying quality challenges, requirements, and options for numerous ideas. The authors also identified outstanding concerns from previous research and summarized the advantages of quality for readers. It allows researchers and developers to avoid common misunderstandings and capture real-world scenarios such as businesses, governments, and educational institutions. Furthermore, it revolutionizes follow-up analysis and differentiates and foregrounds futurity orientations in real-life events involving humans and vehicles in a highly dynamic Fog setting.
Bi et al. [9] <ref name=":2" /> introduced software-defined networking-based fog computing architecture by decoupling mobility control and data forwarding. When mobile consumers travel between several access networks, the authors suggested an Optimal Path Selection (OPS) algorithm to preserve service continuity. Mobile customers received seamless and transparent mobility support thanks to efficient signalling operations. In mobile fog computing, the suggested algorithm ensured service continuity, increased handover performance, and achieved high data transfer efficiency.
Niu et al. [21] <ref name=":6" /> established a system called mobility-aware and multihop-D2D relaying-based scheduling scheme (MHRC) at Edge nodes near hotspots. The authors exploited concurrent transmissions to improve the performance of the system. The mmWave (millimetre-wave) band of Fog computing was cached, and extensive performance evaluation confirms that MHRC delivers more than the higher expected cached data amount. Name et al. [19] <ref name=":5" /> proposed an efficient algorithm to address the problem of resource allocation and user mobility from the Edge of the network to cloud data centres. This algorithm operates on a seamless handover scheme for mobile IPV6 to ease the user mobility challenge and reduce the application response time. The study showed that the task of service delay and packet loss was decreased due to the effect of change in the mobile node position.
Bittencourt et al. [23] <ref name=":7" /> examined the subject of resource allocation in the Fog/Cloud environment, taking into account the hierarchical structure. In the context of the Fog paradigm, the authors developed three scheduling algorithms (First come, First serve, delay-priority, and concurrent) that address user mobility and edge computing capabilities. The authors demonstrated that scheduling techniques may be designed to cope with different application classes based on demand from mobile users by leveraging both Fog to the end-user and cloud characteristics in this study.
Velasquez et al. [39] proposed a hybrid strategy for the Fog environment to manage resources for mobility scenarios. The authors applied the orchestrator technique to offer mobility support in a Smart City situation. In this technique, three components, the status monitor, the Planner, and the VM/Container, are employed to monitor, plan and execute the applications. The main aim of this study was to guarantee the QoS and QoE requirements of mobility-based applications and services.
Bittencourt et al. [31] <ref name=":10" /> presented a Fog computing architecture focusing on Virtual Machine (VM) migration where each user has a VM running in a cloudlet. In this architecture, the user's location is identified by using GPS, and then the VM is moved to a nearby Fog Cloud. The main aim of this study was to migrate users' data according to their mobility in order to maintain QoS for applications demanding lower Latency and allow smooth handoff mechanisms for mobile users.
From the extensive analysis of the literature, the various mobility-aware scheduling techniques have been classified as shown in Table 3. Further, it presents the advantages and limitations of each technique.
{| class="wikitable"
|+Table 3: Classification of Mobility-aware scheduling techniques
!Ref.
!Technique
!Advantage(s)
!Limitation(s)
|-
|[25] <ref name=":8" />
|Ranking of VM
|
* Decrease in delay time, migration time, tuple lost value and downtime
|
* Case study not discussed
|-
|[26]
|S-OAMC, G-OAMC, Machine learning matrix completion
|
* Migration rate decreased
* Better Scalability
|
* Energy utilization of devices not investigated
|-
|[32]
|IM model
|
* Provides better mobility support and security
|
* Did not investigate synchronization overhead
|-
|[10] <ref name=":3" />
|RSS
|
* Reliable and Heterogeneous execution
|
* Low scalability
* No distributed scheduling to minimize response time
|-
|[7] <ref name=":1" />
|MAPE control loop
|
* Improved QoS
* Reduced service downtime
|
* No real-time evaluation
* High energy consumption
* Low robustness and security
|-
|[28] <ref name=":9" />
|Copy of task to over a region
|
* Network Utilization developed
* Low-Loop delay
|
* Fault tolerance reliability is based on Fog gateways only
|-
|[18] <ref name=":4" />
|URMILA
|
* Service availability is maintained by delivering the desired QoS
* Deployment cost minimized
* Battery longevity ensured
|
* No empirical validations
* No user probabilistic routes
* Low scalability in terms of distance and speed
|-
|[8]
|RSSI, LQI
|
* Promises to keep the connection active with a low latency rate between the system and sensor nodes
|
* Consumes more energy
* Overhead is large for network transmission
* Coverage and overhead area are undefined between gateways
|-
|[29]
|User trajectories pre- diction using GPS log
|
* Provides better mobility support
* Reduces migration time
|
* Low scalability
|-
|[36] <ref name=":13" />
|N2N communication, Data Analytics
|
* Fast data access
* High reliability and scalability- city
* Minimum overhead and cost
* High throughput and less delay
|
* No real-time cellular network evaluation
* Low network efficiency
|-
|[38] <ref name=":15" />
|Mobility facets analysis
|
* Improved QoS and QoE
* Latency rate reduced
|
* No real-life implementation
* No reliability and low Latency between dynamic users and fog servers
|-
|[37] <ref name=":14" />
|M-ILP, Sojourn time
|
* Cost-effective
* Migration time reduced
|
* Migration cost not considered
* No real-time implementation
|-
|[34] <ref name=":11" />
|User trajectories prediction using GPS log
|
* Conventional delay tolerance
* High QoS
* Avoided local task processing cost
* Efficient in saving battery
* Handles subtle scenarios with high Latency
|
* Smart city not directed through the use of accurate city maps with aid from stimulation setting
|-
|[35] <ref name=":12" />
|Prediction of user location
|
* Power consumption and de- lay handled proficiently
* Power and delay are reduced
|
* No acquiring of mobile data usage where location sense and time-series data can be projected to achieve the bandwidth
|-
|[21] <ref name=":6" />
|Relay path planning algorithm
|
* Power efficient
* High spectral efficiency
* Data is relayed on cached edge nodes and relay nodes
|
* Blockage problem due to weak diffraction
|-
|[9] <ref name=":2" />
|OPS
|
* Handover performance improved
* Efficiency of high data communication achieved
* Guarantees continuity of services
|
* It does not guarantee privacy and security
* Virtual Machine migration not determined
* The handover process during the optimal path for more logical routing could have been more efficient
|-
|[23] <ref name=":7" />
|Assignment of FS
|
* Low Latency
* Supports dynamic computing and user behaviour
|
* There is no prediction of mobility failure
* Bandwidth and processing not considered in scheduling
|-
|[39]
|Orchestrator
|
* Maintains trustworthiness, resilience, and low Latency in a dynamic environment
|
* No real implementation has been carried out
|-
|[19] <ref name=":5" />
|Pattern modelling, dictating the following location
|
* Application Response time reduced
* Latency time reduced
|
* Services become temporarily inaccessible for some mobile nodes
|-
|[31] <ref name=":10" />
|Forecasting technique
|
* Computing capacity provided for storage and processing of data
|
* Security concerns associated with both user data and applications not considered
|}
[[File:Fog Computing Figure 2.jpg|center|thumb|Figure 2: Year-wise count of research articles]]
== Analytical Discussion ==
The existing research on Fog computing Mobility-aware scheduling has been analyzed thoroughly. The analysis was performed using the answers given in Table 2. The results drawn through the thorough analysis of the literature are presented in various figures as follows:
Figure 2 lists the year-wise count of research papers that are considered for this survey. The bar graph represents the total number of research papers from journals and Conferences from the year 2015 - 2021. The research articles from the journal are 16, and the conference papers are 4. It is observed that more research needs to be conducted on mobility-aware scheduling in Fog computing.
Figure 3 displays an analytical comparison of mobility-aware scheduling approaches in Fog computing based on the content of the represented taxonomy in Figure 7. From the thorough analysis of the literature, four methods have been considered: migration, task offloading, handoff/handover mechanism, and task scheduling. The handoff/handover mechanism has the highest percentage of usage in mobility-aware scheduling, at 30%. The task scheduling and offloading have 25% of us- age in mobility-aware scheduling each. Finally, migration is only 20% of the usage in mobility-aware scheduling. Therefore, these approaches, specifically migration, are still open challenges to address for further research.
Figure 4 depicts various tools that were used for evaluating the mobility-aware scheduling approaches. 18% and 9% of the research articles used iFogSim and Mob-FogSim simulation tools for implementation, respectively. Besides, other simulation tools such as ONE (9%), NS2(5%), MATLAB (4%), Mininet (5%), and Docker (9%) have been utilized for implementing the proposed techniques in the research articles. Further, pro-Programming languages such as C++ (9%) and Python (9%) and hardware deployments such as Raspberry Pi (5%) and Ardunio (4%) were used for implementing existing case studies based on mobility-aware scheduling.
[[File:Fog Computing Figure 3.jpg|center|thumb|Figure 3: Percentage of the presented classified approaches in mobility-aware scheduling]]
The applied case studies are shown in Figure 5, which shows a maximum of 20% of research articles have implemented IoT-based applications. After that, 15% of each research article used Health care and Mobile-based applications. Besides, Smart City and 5G-based applications have been applied in 10% of research articles. Moreover, Surveillance and gaming, Mobile IPV6, and Wireless computing applications are the case studies on which only 5% of research articles exist.
After reviewing numerous research articles based on mobility-aware scheduling, it has been observed that researchers employed various parameters for evaluating the performance of the Mobility-scheduling approaches, as represented in Figure 6. It shows that Time completion (18%) followed by Delay (12%), Network usage (12%), Latency (12%), Energy consumption (10%), and cost (10%) are generally utilized. However, Downtime (4%), Migration time (4%), Makespan (2%), Success ratio (2%), Signal level (2%), Deadline (2%), Makespan (2%), Migration rate (2%), Mobility patterns (2%), Tuple lost (2%), and power consumption (2%) are less exploited parameters.
A taxonomy was compiled after going through the detailed review process, and various techniques have been categorized in Fog computing-based mobility-aware scheduling. Figure 7 presents these categories broadly in Migration, Offloading, Handoff/Handover mechanism, and Scheduling.
[[File:Fog Computing Figure 4.jpg|center|thumb|Figure 4: Percentage of tools utilized in the literature]]
[[File:Fog Computing Figure 5.jpg|center|thumb|Figure 5: Percentage of case studies employed in the literature]]
[[File:Fog Computing Figure 6.jpg|center|thumb|Figure 6: Percentage of parameters for evaluating Mobility-aware scheduling in the literature]]
=== Open Issues and Challenges ===
From the thorough analysis of the literature, several open issues and challenges pertaining to the area of mobility-aware scheduling in Fog computing have been identified in order to provide directions for future research exploration. The identified open problems and challenges, depicted in Figure 8, are discussed below.
=== Task Scheduling ===
Fog computing consists of several Fog nodes, each of which is a mini Cloud in the vicinity of mobile devices near the Edge of the network. When a mobile device submits a task, the Fog scheduler assigns it to a nearby Fog node(s) for execution. However, as the device moves from one network to another, the task needs to be rescheduled when the device enters a different network. Additionally, Fog nodes have limited capacity and availability; if the mobile user enters into a network where there is no nearby Fog service available, then this leads to a significant delay in service and raises a significant issue of task scheduling for mobile users [10, 40]. <ref name=":3" /><ref>{{Cite journal|last=Kaur|first=Navjeet|last2=Kumar|first2=Ashok|last3=Kumar|first3=Rajesh|date=2021-06-04|title=A systematic review on task scheduling in Fog computing: Taxonomy, tools, challenges, and future directions|url=https://doi.org/10.1002/cpe.6432|journal=Concurrency and Computation: Practice and Experience|volume=33|issue=21|doi=10.1002/cpe.6432|issn=1532-0626}}</ref>
=== Resource Provisioning ===
Fog computing reduces the workload of Cloud computing by processing the tasks locally near the Edge of the network. However, due to the mobility of the user, the Fog node primarily assigned to a task might not be optimal over time. Therefore, the migration of the task to another Fog node near the user's mobile device is perceived as a necessary solution to resolve this concern [41]. <ref>{{Cite journal|last=Maleki|first=Erfan Farhangi|last2=Mashayekhy|first2=Lena|date=2020-05|title=Mobility-aware computation offloading in edge computing using prediction|url=https://doi.org/10.1109/icfec50348.2020.00015|journal=2020 IEEE 4th International Conference on Fog and Edge Computing (ICFEC)|publisher=IEEE|pages=69–74|doi=10.1109/icfec50348.2020.00015}}</ref> However, such frequent migration over a short time poses the challenge of providing an efficient resource for the task that is capable of performing computation on time and delivering results to users while adhering to QoE.
[[File:Fog Computing Figure 7.jpg|center|thumb|Figure 7: Mobility-aware Fog Scheduling Taxonomy]]
=== Energy Consumption ===
The placement of fog services at the Edge of the network can provide better QoS to mobile users, resulting in a shorter response time. However, it is practically impossible due to the high deployment cost of new Fog devices, which further raises the significant issue of energy consumption. If too many deployments are done, there will be lots of communication traffic from the Cloud to Fog nodes and servers in order to create copies of the task from one network to another in case of mobility [42]. <ref>{{Cite journal|last=Zhao|first=Xuhui|last2=Shi|first2=Yan|last3=Chen|first3=Shanzhi|date=2020-12|title=MAESP: Mobility aware edge service placement in mobile edge networks|url=https://doi.org/10.1016/j.comnet.2020.107435|journal=Computer Networks|volume=182|pages=107435|doi=10.1016/j.comnet.2020.107435|issn=1389-1286}}</ref> This results in considerable energy wastage in the form of high bandwidth consumption. This means that where and when to reschedule the task to an efficient Fog node must be carefully determined to minimize energy, response time, and deployment cost.
=== Quality of Experience (QoE) ===
[[File:Fog Computing Figure 8.jpg|left|thumb|Figure 8: Mobility-aware scheduling open issues and challenges]]
Several mobility-based scheduling algorithms exist, but they need to focus on maximizing the user QoE [29, 8, 10, 18]. <ref name=":3" /> <ref name=":4" /> Further, they do not analyze the user performance; hence, the QoE of using a service or product is not determined. Therefore, to understand the user gain and loss, the scheduling algorithm needs to focus on enhancing the user QoE.
=== Resource Management ===
The mobility of Fog nodes/users demands efficient resource discovery and sharing, resource availability, and task offloading [43]. <ref>{{Cite journal|last=Yousefpour|first=Ashkan|last2=Fung|first2=Caleb|last3=Nguyen|first3=Tam|last4=Kadiyala|first4=Krishna|last5=Jalali|first5=Fatemeh|last6=Niakanlahiji|first6=Amirreza|last7=Kong|first7=Jian|last8=Jue|first8=Jason P.|date=2019-09|title=All one needs to know about fog computing and related edge computing paradigms: A complete survey|url=https://doi.org/10.1016/j.sysarc.2019.02.009|journal=Journal of Systems Architecture|volume=98|pages=289–330|doi=10.1016/j.sysarc.2019.02.009|issn=1383-7621}}</ref> Few techniques that were proposed to manage the resources effectively did not consider more constraints such as density, latency sensitivity, and mobility of Edge and Fog nodes, and as the number of nodes increases, issues such as scalability and distributing the algorithms arise [44, 45, 46]. <ref>{{Cite journal|last=Liu|first=Wei|last2=Nishio|first2=Takayuki|last3=Shinkuma|first3=Ryoichi|last4=Takahashi|first4=Tatsuro|date=2014-09|title=Adaptive resource discovery in mobile cloud computing|url=https://doi.org/10.1016/j.comcom.2014.02.006|journal=Computer Communications|volume=50|pages=119–129|doi=10.1016/j.comcom.2014.02.006|issn=0140-3664}}</ref><ref>{{Cite journal|last=He|first=Jianhua|last2=Wei|first2=Jian|last3=Chen|first3=Kai|last4=Tang|first4=Zuoyin|last5=Zhou|first5=Yi|last6=Zhang|first6=Yan|date=2018-04|title=Multitier Fog Computing With Large-Scale IoT Data Analytics for Smart Cities|url=https://doi.org/10.1109/jiot.2017.2724845|journal=IEEE Internet of Things Journal|volume=5|issue=2|pages=677–686|doi=10.1109/jiot.2017.2724845|issn=2327-4662}}</ref><ref>{{Cite journal|last=Nishio|first=Takayuki|last2=Shinkuma|first2=Ryoichi|last3=Takahashi|first3=Tatsuro|last4=Mandayam|first4=Narayan B.|date=2013|title=Service-oriented heterogeneous resource sharing for optimizing service latency in mobile cloud|url=https://doi.org/10.1145/2492348.2492354|journal=Proceedings of the first international workshop on Mobile cloud computing & networking - MobileCloud '13|location=New York, New York, USA|publisher=ACM Press|pages=19|doi=10.1145/2492348.2492354}}</ref> Therefore, more attention needs to be paid towards the mobile Fog computing environment to manage the resources effectively.
=== Privacy and Security ===
In [47], a scheduling policy is proposed for the mobile device system to minimize the cost. However, the privacy issues of location and usage patterns were ignored. Additionally, data privacy, access control, and intrusion detection in scheduling policies have been overlooked [7, 48, 28]. <ref name=":1" /><ref name=":9" /> Besides, Fog node devices are normally deployed near the end-user; hence, protection and surveillance are comparatively weak, which can result in a malicious attack [49, 50]. <ref>{{Cite journal|last=Hu|first=Pengfei|last2=Ning|first2=Huansheng|last3=Qiu|first3=Tie|last4=Song|first4=Houbing|last5=Wang|first5=Yanna|last6=Yao|first6=Xuanxia|date=2017-10|title=Security and Privacy Preservation Scheme of Face Identification and Resolution Framework Using Fog Computing in Internet of Things|url=https://doi.org/10.1109/jiot.2017.2659783|journal=IEEE Internet of Things Journal|volume=4|issue=5|pages=1143–1155|doi=10.1109/jiot.2017.2659783|issn=2327-4662}}</ref><ref>{{Cite journal|last=Han|first=Guangjie|last2=Liu|first2=Li|last3=Chan|first3=Sammy|last4=Yu|first4=Ruiyun|last5=Yang|first5=Yu|date=2017-03|title=HySense: A Hybrid Mobile CrowdSensing Framework for Sensing Opportunities Compensation under Dynamic Coverage Constraint|url=https://doi.org/10.1109/mcom.2017.1600658cm|journal=IEEE Communications Magazine|volume=55|issue=3|pages=93–99|doi=10.1109/mcom.2017.1600658cm|issn=0163-6804}}</ref>
== Data availability statement ==
Not applicable.
== Conclusions ==
Fog computing infrastructure provides services at the Edge of the network. So, to provide support for scheduling and management of mobility awareness, efficient techniques and mechanisms have been proposed. In this survey, research articles on the mobility-aware-scheduling strategies in Fog computing have been thoroughly analyzed. It provides a comparative study among existing mobility-aware scheduling strategies based on vital factors such as techniques proposed, parameters considered, tools utilized for implementation, and case studies considered, along with the advantages and limitations. Further, several open issues and challenges have been identified for future research direction.
===Subheading===
==Second Heading==
==Third Heading, etc==
==Additional information==
===Acknowledgements===
Any people, organisations, or funding sources that you would like to thank.
===Competing interests===
Any conflicts of interest that you would like to declare. Otherwise, a statement that the authors have no competing interest.
===Ethics statement===
An ethics statement, if appropriate, on any animal or human research performed should be included here or in the methods section.
==References==
{{reflist|35em}}
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==Welcome==
{{Robelbox|theme=9|title='''[[Wikiversity:Welcome|Welcome]] to [[Wikiversity:What is Wikiversity|Wikiversity]], Raph Williams65!'''|width=100%}}
<div style="{{Robelbox/pad}}">
You can [[Wikiversity:Contact|contact us]] with [[Wikiversity:Questions|questions]] at the [[Wikiversity:Colloquium|colloquium]] or get in touch with [[User talk:Atcovi|me personally]] if you would like some [[Help:Contents|help]].
Remember to [[Wikiversity:Signature#How to add your signature|sign]] your comments when [[Wikiversity:Who are Wikiversity participants?|participating]] in [[Wikiversity:Talk page|discussions]]. Using the signature icon [[File:OOjs UI icon signature-ltr.svg]] makes it simple.
We invite you to [[Wikiversity:Be bold|be bold]] and [[Wikiversity|assume good faith]]. Please abide by our [[Wikiversity:Civility|civility]], [[Wikiversity:Privacy policy|privacy]], and [[Foundation:Terms of Use|terms of use]] policies.
To find your way around, check out:
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<div style="width:50.0%; float:left">
* [[Wikiversity:Introduction|Introduction to Wikiversity]]
* [[Help:Guides|Take a guided tour]] and learn [[Help:Editing|how to edit]]
* [[Wikiversity:Browse|Browse]] or visit an educational level portal:<br>[[Portal:Pre-school Education|pre-school]] | [[Portal:Primary Education|primary]] | [[Portal:Secondary Education|secondary]] | [[Portal:Tertiary Education|tertiary]] | [[Portal:Non-formal Education|non-formal]]
* [[Wikiversity:Introduction explore|Explore]] links in left-hand navigation menu
</div>
<!-- The Right column -->
<div style="width:50.0%; float:left">
* Read an [[Wikiversity:Wikiversity teachers|introduction for teachers]]
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* Discuss issues or ask questions at the [[Wikiversity:Colloquium|colloquium]]
</div>
<br clear="both"/>
To get started, experiment in the [[wikiversity:sandbox|sandbox]] or on [[special:mypage|your userpage]].
See you around Wikiversity! --—[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 16:44, 1 January 2025 (UTC)</div>
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<templatestyles src="Template:Boolf triangles/style.css" />
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{| class="wikitable boolf-triangle" style="margin: 0;"
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!class="sum"| {{#expr: {{{6}}} + {{{7}}} + {{{8}}} + {{{9}}} + {{{10}}} }}
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|class="dummy"|
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!class="sum"| {{#expr: {{{11}}} + {{{12}}} + {{{13}}} + {{{14}}} + {{{15}}} + {{{16}}} + {{{17}}} + {{{18}}} + {{{19}}} }}
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{{Collapsible END}}<noinclude>
----
see e.g. {{tl|Boolf triangle Magnolia}}
[[Category:Boolf triangles with weight columns]]
[[Category:Some templates created by Watchduck]]
</noinclude>
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<templatestyles src="Template:Boolf triangles/style.css" />
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| {{{10}}}
!class="sum"| {{#expr: {{{6}}} + {{{7}}} + {{{8}}} + {{{9}}} + {{{10}}} }}
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!class="sum"| {{#expr: {{{6}}} + {{{7}}} + {{{8}}} + {{{9}}} + {{{10}}} }}
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!class="sum"| {{#expr: {{{11}}} + {{{12}}} + {{{13}}} + {{{14}}} + {{{15}}} + {{{16}}} + {{{17}}} + {{{18}}} + {{{19}}} }}
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<!-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------->
<templatestyles src="Template:Boolf triangles/style.css" />
{{Collapsible START|<small>(merged weights)</small>|collapsed wide light}}
{| class="wikitable boolf-triangle" style="margin: 0;"
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!style="color: gray; font-size: 60%;"| 0
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| {{{8}}}
| {{#expr: {{{9}}} + {{{10}}} }}
|class="dummy"|
|class="dummy"|
!class="sum"| {{#expr: {{{6}}} + {{{7}}} + {{{8}}} + {{{9}}} + {{{10}}} }}
|-
! 3
|style="font-size: 60%;"| {{{11}}}
| {{{12}}}
| {{{13}}}
| {{#expr: {{{14}}} + {{{15}}} }}
| {{#expr: {{{16}}} + {{{17}}} + {{{18}}} + {{{19}}} }}
|class="dummy"|
!class="sum"| {{#expr: {{{11}}} + {{{12}}} + {{{13}}} + {{{14}}} + {{{15}}} + {{{16}}} + {{{17}}} + {{{18}}} + {{{19}}} }}
|-
! 4
|style="font-size: 60%;"| {{{20}}}
| {{{21}}}
| {{{22}}}
| {{#expr: {{{23}}} + {{{24}}} }}
| {{#expr: {{{25}}} + {{{26}}} + {{{27}}} + {{{28}}} }}
| {{#expr: {{{29}}} + {{{30}}} + {{{31}}} + {{{32}}} + {{{33}}} + {{{34}}} + {{{35}}} + {{{36}}} }}
!class="sum"| {{#expr: {{{20}}} + {{{21}}} + {{{22}}} + {{{23}}} + {{{24}}} + {{{25}}} + {{{26}}} + {{{27}}} + {{{28}}} + {{{29}}} + {{{30}}} + {{{31}}} + {{{32}}} + {{{33}}} + {{{34}}} + {{{35}}} + {{{36}}} }}
|}
{{Collapsible END}}<noinclude>
There is also {{tl|Boolf weight triangle 5}}.
----
[[Category:Boolf triangles with weight columns]]
[[Category:Some templates created by Watchduck]]
</noinclude>
c0k69ymfczvtap9ute37z54mo754ga5
2694022
2694017
2025-01-01T20:22:31Z
Watchduck
137431
2694022
wikitext
text/x-wiki
<templatestyles src="Template:Boolf triangles/style.css" />
{{Collapsible START|rational weight|open wide light followed}}
{| class="wikitable boolf-triangle" style="margin: 0;"
! {{diagonal split header|''a''|''w''}}
! <math>0</math>
! <math>\frac{1}{16}</math>
! <math>\frac{1}{8}</math>
! <math>\frac{3}{16}</math>
! <math>\frac{1}{4}</math>
! <math>\frac{5}{16}</math>
! <math>\frac{3}{8}</math>
! <math>\frac{7}{16}</math>
! <math>\frac{1}{2}</math>
! <math>\frac{9}{16}</math>
! <math>\frac{5}{8}</math>
! <math>\frac{11}{16}</math>
! <math>\frac{3}{4}</math>
! <math>\frac{13}{16}</math>
! <math>\frac{7}{8}</math>
! <math>\frac{15}{16}</math>
! <math>1</math>
!class="sum"| sums
|-
! 0
| {{{1}}}
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
| {{{2}}}
!class="sum"| {{#expr: {{{1}}} + {{{2}}} }}
|-
! 1
| {{{3}}}
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
| {{{4}}}
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
| {{{5}}}
!class="sum"| {{#expr: {{{3}}} + {{{4}}} + {{{5}}} }}
|-
! 2
| {{{6}}}
|class="dummy"|
|class="dummy"|
|class="dummy"|
| {{{7}}}
|class="dummy"|
|class="dummy"|
|class="dummy"|
| {{{8}}}
|class="dummy"|
|class="dummy"|
|class="dummy"|
| {{{9}}}
|class="dummy"|
|class="dummy"|
|class="dummy"|
| {{{10}}}
!class="sum"| {{#expr: {{{6}}} + {{{7}}} + {{{8}}} + {{{9}}} + {{{10}}} }}
|- style="font-size: {{{76|80}}}%;"
! 3
| {{{11}}}
|class="dummy"|
| {{{12}}}
|class="dummy"|
| {{{13}}}
|class="dummy"|
| {{{14}}}
|class="dummy"|
| {{{15}}}
|class="dummy"|
| {{{16}}}
|class="dummy"|
| {{{17}}}
|class="dummy"|
| {{{18}}}
|class="dummy"|
| {{{19}}}
!class="sum"| {{#expr: {{{11}}} + {{{12}}} + {{{13}}} + {{{14}}} + {{{15}}} + {{{16}}} + {{{17}}} + {{{18}}} + {{{19}}} }}
|- style="font-size: {{{77|70}}}%;"
! 4
| {{{20}}}
| {{{21}}}
| {{{22}}}
| {{{23}}}
| {{{24}}}
| {{{25}}}
| {{{26}}}
| {{{27}}}
| {{{28}}}
| {{{29}}}
| {{{30}}}
| {{{31}}}
| {{{32}}}
| {{{33}}}
| {{{34}}}
| {{{35}}}
| {{{36}}}
!class="sum"| {{#expr: {{{20}}} + {{{21}}} + {{{22}}} + {{{23}}} + {{{24}}} + {{{25}}} + {{{26}}} + {{{27}}} + {{{28}}} + {{{29}}} + {{{30}}} + {{{31}}} + {{{32}}} + {{{33}}} + {{{34}}} + {{{35}}} + {{{36}}} }}
|}
{{Collapsible END}}
<!-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------->
{{Collapsible START|integer weight|collapsed light wide followed}}
{| class="wikitable boolf-triangle" style="margin: 0;"
! {{diagonal split header|''a''|''w''}}
! 0
! 1
! 2
! 3
! 4
! 5
! 6
! 7
! 8
! 9
! 10
! 11
! 12
! 13
! 14
! 15
! 16
!class="sum"| sums
|-
! 0
| {{{1}}}
| {{{2}}}
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
!class="sum"| {{#expr: {{{1}}} + {{{2}}} }}
|-
! 1
| {{{3}}}
| {{{4}}}
| {{{5}}}
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
!class="sum"| {{#expr: {{{3}}} + {{{4}}} + {{{5}}} }}
|-
! 2
| {{{6}}}
| {{{7}}}
| {{{8}}}
| {{{9}}}
| {{{10}}}
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
!class="sum"| {{#expr: {{{6}}} + {{{7}}} + {{{8}}} + {{{9}}} + {{{10}}} }}
|- style="font-size: {{{76|80}}}%;"
! 3
| {{{11}}}
| {{{12}}}
| {{{13}}}
| {{{14}}}
| {{{15}}}
| {{{16}}}
| {{{17}}}
| {{{18}}}
| {{{19}}}
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
!class="sum"| {{#expr: {{{11}}} + {{{12}}} + {{{13}}} + {{{14}}} + {{{15}}} + {{{16}}} + {{{17}}} + {{{18}}} + {{{19}}} }}
|- style="font-size: {{{77|70}}}%;"
! 4
| {{{20}}}
| {{{21}}}
| {{{22}}}
| {{{23}}}
| {{{24}}}
| {{{25}}}
| {{{26}}}
| {{{27}}}
| {{{28}}}
| {{{29}}}
| {{{30}}}
| {{{31}}}
| {{{32}}}
| {{{33}}}
| {{{34}}}
| {{{35}}}
| {{{36}}}
!class="sum"| {{#expr: {{{20}}} + {{{21}}} + {{{22}}} + {{{23}}} + {{{24}}} + {{{25}}} + {{{26}}} + {{{27}}} + {{{28}}} + {{{29}}} + {{{30}}} + {{{31}}} + {{{32}}} + {{{33}}} + {{{34}}} + {{{35}}} + {{{36}}} }}
|}
{{Collapsible END}}
<!-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------->
<templatestyles src="Template:Boolf triangles/style.css" />
{{Collapsible START|<small>(merged weights)</small>|collapsed wide light}}
{| class="wikitable boolf-triangle" style="margin: 0;"
|-
!style="color: gray;"| ''w''
!style="color: gray; font-size: 60%;"| 0
!style="color: gray;"| 1
!style="color: gray;"| 2
!style="color: gray;"| 3...4
!style="color: gray;"| 5...8
!style="color: gray;"| 9...16
!rowspan="2" class="sum"| sums
|-
! {{diagonal split header|''a''|''k''}}
!style="font-size: 60%;"| -1
! 0
! 1
! 2
! 3
! 4
|-
! 0
|style="font-size: 60%;"| {{{1}}}
| {{{2}}}
|class="dummy"|
|class="dummy"|
|class="dummy"|
|class="dummy"|
!class="sum"| {{#expr: {{{1}}} + {{{2}}} }}
|-
! 1
|style="font-size: 60%;"| {{{3}}}
| {{{4}}}
| {{{5}}}
|class="dummy"|
|class="dummy"|
|class="dummy"|
!class="sum"| {{#expr: {{{3}}} + {{{4}}} + {{{5}}} }}
|-
! 2
|style="font-size: 60%;"| {{{6}}}
| {{{7}}}
| {{{8}}}
| {{#expr: {{{9}}} + {{{10}}} }}
|class="dummy"|
|class="dummy"|
!class="sum"| {{#expr: {{{6}}} + {{{7}}} + {{{8}}} + {{{9}}} + {{{10}}} }}
|-
! 3
|style="font-size: 60%;"| {{{11}}}
| {{{12}}}
| {{{13}}}
| {{#expr: {{{14}}} + {{{15}}} }}
| {{#expr: {{{16}}} + {{{17}}} + {{{18}}} + {{{19}}} }}
|class="dummy"|
!class="sum"| {{#expr: {{{11}}} + {{{12}}} + {{{13}}} + {{{14}}} + {{{15}}} + {{{16}}} + {{{17}}} + {{{18}}} + {{{19}}} }}
|-
! 4
|style="font-size: 60%;"| {{{20}}}
| {{{21}}}
| {{{22}}}
| {{#expr: {{{23}}} + {{{24}}} }}
| {{#expr: {{{25}}} + {{{26}}} + {{{27}}} + {{{28}}} }}
| {{#expr: {{{29}}} + {{{30}}} + {{{31}}} + {{{32}}} + {{{33}}} + {{{34}}} + {{{35}}} + {{{36}}} }}
!class="sum"| {{#expr: {{{20}}} + {{{21}}} + {{{22}}} + {{{23}}} + {{{24}}} + {{{25}}} + {{{26}}} + {{{27}}} + {{{28}}} + {{{29}}} + {{{30}}} + {{{31}}} + {{{32}}} + {{{33}}} + {{{34}}} + {{{35}}} + {{{36}}} }}
|}
{{Collapsible END}}<noinclude>
There is also {{tl|Boolf weight triangle 5}}.
----
[[Category:Boolf triangles with weight columns| ]]
[[Category:Some templates created by Watchduck]]
</noinclude>
8gp4gwev4nsiita3u2odwqzxbnlg394
Template:Boolf triangle Magnolia
10
317664
2694019
2025-01-01T20:18:38Z
Watchduck
137431
Watchduck moved page [[Template:Boolf triangle Magnolia]] to [[Template:Boolf weight triangle; dense]]
2694019
wikitext
text/x-wiki
#REDIRECT [[Template:Boolf weight triangle; dense]]
4t1qy0tt4kelct4evv6a0d166r2tfcf
Template:Boolf weight triangle; all
10
317665
2694026
2025-01-01T20:51:42Z
Watchduck
137431
New resource with "{{Boolf weight triangle 5| 1|1| 1|2|1| 1|4|6|4|1| 1|8|28|56|70|56|28|8|1| 1|16|120|560|1820|4368|8008|11440|12870|11440|8008|4368|1820|560|120|16|1| 1|32|496|4960|35960|201376|906192|3365856|10518300|28048800|64512240|129024480|225792840|347373600|471435600|565722720|601080390|565722720|471435600|347373600|225792840|129024480|64512240|28048800|10518300|3365856|906192|201376|35960|4960|496|32|1}} <noinclude> ---- All Boolean functions by weight. Row <math>arity</math> o..."
2694026
wikitext
text/x-wiki
{{Boolf weight triangle 5|
1|1|
1|2|1|
1|4|6|4|1|
1|8|28|56|70|56|28|8|1|
1|16|120|560|1820|4368|8008|11440|12870|11440|8008|4368|1820|560|120|16|1|
1|32|496|4960|35960|201376|906192|3365856|10518300|28048800|64512240|129024480|225792840|347373600|471435600|565722720|601080390|565722720|471435600|347373600|225792840|129024480|64512240|28048800|10518300|3365856|906192|201376|35960|4960|496|32|1}}
<noinclude>
----
All Boolean functions by weight. Row <math>arity</math> of this triangle is row <math>2^{arity}</math> of {{w|Pascal's triangle}}.
[[Category:Boolf triangles with weight columns|all]]
</noinclude>
l47hcbo3ytfav58ep501zy62lupscfi
Template:Mentors of Boolean functions/code WP vectors
10
317666
2694029
2025-01-01T21:16:37Z
Watchduck
137431
New resource with "{{dh-box|vector of Ю calculated in Python}} <source lang="python"> from discretehelpers.boolf import Boolf from discretehelpers.walsh_perm import WalshPerm from discretehelpers.sig_perm import SigPerm for arity in range(1, 5): length = 2 ** arity linear = ~Boolf(multi_xor=range(arity)) # negated XOR of all arguments (top row of Щ matrix) right_bit = Boolf(multi_and=range(arity)) # AND of all arguments boolf = (linear ^ right_bit).reverse #..."
2694029
wikitext
text/x-wiki
{{dh-box|vector of Ю calculated in Python}}
<source lang="python">
from discretehelpers.boolf import Boolf
from discretehelpers.walsh_perm import WalshPerm
from discretehelpers.sig_perm import SigPerm
for arity in range(1, 5):
length = 2 ** arity
linear = ~Boolf(multi_xor=range(arity)) # negated XOR of all arguments (top row of Щ matrix)
right_bit = Boolf(multi_and=range(arity)) # AND of all arguments
boolf = (linear ^ right_bit).reverse # their XOR (top row of Ш matrix), then reversed
boolf = boolf.twin(arity).reverse # its Zhegalkin twin, then reversed (top row of Ч matrix)
vector_yu = []
for i in range(length):
sigperm = SigPerm(valneg_index=i, perm_index=0)
family_boolf = boolf.apply_sigperm(sigperm)
twin = family_boolf.twin(arity)
vector_yu.append(twin.tt(arity).intval)
wp_yu = WalshPerm(vector_yu)
print(arity, wp_yu.vector())
</source>
<source>
1 (2, 3)
2 (15, 10, 12, 8)
3 (233, 189, 219, 158, 231, 182, 214, 151)
4 (59521, 48321, 55969, 40945, 59017, 47053, 55211, 38654, 59497, 48253, 55931, 40814, 58991, 46970, 55164, 38504)
</source>
|}<!-- end of dh-box --><noinclude>
[[Category:Mentors of Boolean functions]]
</noinclude>
ew68mm9gqsjswxq8b36pgz1n472s04b
2694030
2694029
2025-01-01T21:43:30Z
Watchduck
137431
Watchduck moved page [[Template:Mentors of Boolean functions/code Yu vector]] to [[Template:Mentors of Boolean functions/code WP vectors]]
2694029
wikitext
text/x-wiki
{{dh-box|vector of Ю calculated in Python}}
<source lang="python">
from discretehelpers.boolf import Boolf
from discretehelpers.walsh_perm import WalshPerm
from discretehelpers.sig_perm import SigPerm
for arity in range(1, 5):
length = 2 ** arity
linear = ~Boolf(multi_xor=range(arity)) # negated XOR of all arguments (top row of Щ matrix)
right_bit = Boolf(multi_and=range(arity)) # AND of all arguments
boolf = (linear ^ right_bit).reverse # their XOR (top row of Ш matrix), then reversed
boolf = boolf.twin(arity).reverse # its Zhegalkin twin, then reversed (top row of Ч matrix)
vector_yu = []
for i in range(length):
sigperm = SigPerm(valneg_index=i, perm_index=0)
family_boolf = boolf.apply_sigperm(sigperm)
twin = family_boolf.twin(arity)
vector_yu.append(twin.tt(arity).intval)
wp_yu = WalshPerm(vector_yu)
print(arity, wp_yu.vector())
</source>
<source>
1 (2, 3)
2 (15, 10, 12, 8)
3 (233, 189, 219, 158, 231, 182, 214, 151)
4 (59521, 48321, 55969, 40945, 59017, 47053, 55211, 38654, 59497, 48253, 55931, 40814, 58991, 46970, 55164, 38504)
</source>
|}<!-- end of dh-box --><noinclude>
[[Category:Mentors of Boolean functions]]
</noinclude>
ew68mm9gqsjswxq8b36pgz1n472s04b
2694032
2694030
2025-01-01T21:44:47Z
Watchduck
137431
2694032
wikitext
text/x-wiki
{{dh-box|vectors of Walsh permutations calculated in Python}}
<source lang="python">
import numpy as np
from discretehelpers.boolf import Boolf
from discretehelpers.sig_perm import SigPerm
for arity in range(1, 5):
length = 2 ** arity
linear = ~Boolf(multi_xor=range(arity)) # negated XOR of all arguments (top row of Щ matrix)
right_bit = Boolf(multi_and=range(arity)) # AND of all arguments
boolf = (linear ^ right_bit).reverse # their XOR (top row of Ш matrix), then reversed
boolf = boolf.twin(arity).reverse # its Zhegalkin twin, then reversed (top row of Ч matrix)
vector_che = []
vector_yu = []
matrix_yu = np.zeros([length, length], dtype=int)
for i in range(length):
sigperm = SigPerm(valneg_index=i, perm_index=0)
family_boolf = boolf.apply_sigperm(sigperm)
vector_che.append(family_boolf.tt(arity).intval)
twin = family_boolf.twin(arity)
vector_yu.append(twin.tt(arity).intval)
matrix_yu[:, i] = twin.tt(arity)
reverse_vector_sha = []
for i in range(length):
row = list(matrix_yu[i, :])
reverse_row = row[::-1]
boolf = Boolf(reverse_row)
twin = boolf.twin(arity)
reverse_vector_sha.append(twin.tt(arity).intval)
vector_sha = reverse_vector_sha[::-1]
print(f'\n# arity {arity}')
print(f'vector_che_{arity} = {vector_che}')
print(f'vector_yu_{arity} = {vector_yu}')
print(f'vector_sha_{arity} = {vector_sha}')
</source>
<source>
# arity 1
vector_che_1 = [2, 1]
vector_yu_1 = [2, 3]
vector_sha_1 = [1, 3]
# arity 2
vector_che_2 = [1, 2, 4, 8]
vector_yu_2 = [15, 10, 12, 8]
vector_sha_2 = [1, 2, 4, 8]
# arity 3
vector_che_3 = [151, 107, 109, 158, 121, 182, 214, 233]
vector_yu_3 = [233, 189, 219, 158, 231, 182, 214, 151]
vector_sha_3 = [1, 2, 4, 9, 16, 33, 65, 151]
# arity 4
vector_che_4 = [6015, 11199, 19935, 36591, 29175, 45819, 54525, 59646, 32535, 48939, 57165, 61326, 63345, 64434, 64980, 65256]
vector_yu_4 = [59521, 48321, 55969, 40945, 59017, 47053, 55211, 38654, 59497, 48253, 55931, 40814, 58991, 46970, 55164, 38504]
vector_sha_4 = [1, 2, 4, 9, 16, 33, 65, 150, 256, 513, 1025, 2310, 4097, 8466, 16660, 38504]
</source>
|}<!-- end of dh-box --><noinclude>
[[Category:Mentors of Boolean functions]]
</noinclude>
q7nf8nbi3vef4e3i94z0pmpl74o12d8
2694036
2694032
2025-01-01T22:34:03Z
Watchduck
137431
2694036
wikitext
text/x-wiki
{{dh-box|vectors of Walsh permutations calculated in Python}}
<source lang="python">
import numpy as np
from discretehelpers.boolf import Boolf
from discretehelpers.sig_perm import SigPerm
for arity in range(1, 6):
length = 2 ** arity
linear = ~Boolf(multi_xor=range(arity)) # negated XOR of all arguments (top row of Щ matrix)
right_bit = Boolf(multi_and=range(arity)) # AND of all arguments
boolf = (linear ^ right_bit).reverse # their XOR (top row of Ш matrix), then reversed
boolf = boolf.twin(arity).reverse # its Zhegalkin twin, then reversed (top row of Ч matrix)
vector_che = []
vector_yu = []
matrix_yu = np.zeros([length, length], dtype=int)
for i in range(length):
sigperm = SigPerm(valneg_index=i, perm_index=0)
family_boolf = boolf.apply_sigperm(sigperm)
vector_che.append(family_boolf.tt(arity).intval)
twin = family_boolf.twin(arity)
vector_yu.append(twin.tt(arity).intval)
matrix_yu[:, i] = twin.tt(arity)
reverse_vector_sha = []
for i in range(length):
row = list(matrix_yu[i, :])
reverse_row = row[::-1]
boolf = Boolf(reverse_row)
twin = boolf.twin(arity)
reverse_vector_sha.append(twin.tt(arity).intval)
vector_sha = reverse_vector_sha[::-1]
print(f'\n# arity {arity}')
print(f'vector_che_{arity} = {vector_che}')
print(f'vector_yu_{arity} = {vector_yu}')
print(f'vector_sha_{arity} = {vector_sha}')
</source>
<source>
# arity 1
vector_che_1 = [2, 1]
vector_yu_1 = [2, 3]
vector_sha_1 = [1, 3]
# arity 2
vector_che_2 = [1, 2, 4, 8]
vector_yu_2 = [15, 10, 12, 8]
vector_sha_2 = [1, 2, 4, 8]
# arity 3
vector_che_3 = [151, 107, 109, 158, 121, 182, 214, 233]
vector_yu_3 = [233, 189, 219, 158, 231, 182, 214, 151]
vector_sha_3 = [1, 2, 4, 9, 16, 33, 65, 151]
# arity 4
vector_che_4 = [6015, 11199, 19935, 36591, 29175, 45819, 54525, 59646, 32535, 48939, 57165, 61326, 63345, 64434, 64980, 65256]
vector_yu_4 = [59521, 48321, 55969, 40945, 59017, 47053, 55211, 38654, 59497, 48253, 55931, 40814, 58991, 46970, 55164, 38504]
vector_sha_4 = [1, 2, 4, 9, 16, 33, 65, 150, 256, 513, 1025, 2310, 4097, 8466, 16660, 38504]
# arity 5
vector_che_5 = [2541715455, 1807728639, 1843388415, 2666524671, 2046294015, 3069967359, 3606969855, 3925802751, 2140667775, 3211526079, 3748528095, 4020174831, 4151967735, 4223074299, 4258725885, 4276748286, 2147456895, 3221187519, 3758058975, 4026506991, 4160715255, 4227839739, 4261402365, 4278184446, 4286545815, 4290756459, 4292861805, 4293914526, 4294440825, 4294704054, 4294835670, 4294901481]
vector_yu_5 = [3900735489, 3166748673, 3667959809, 2683367425, 3867707393, 3083652097, 3618286081, 2533359361, 3899162753, 3162292417, 3665469601, 2674913521, 3866003593, 3078343885, 3615337131, 2523529214, 3900729473, 3166731457, 3667950241, 2683334641, 3867700873, 3083631565, 3618274731, 2533321214, 3899156713, 3162275005, 3665459931, 2674880414, 3865997031, 3078323126, 3615325654, 2523490711]
vector_sha_5 = [1, 2, 4, 9, 16, 33, 65, 150, 256, 513, 1025, 2310, 4097, 8466, 16660, 38505, 65536, 131073, 262145, 589830, 1048577, 2162706, 4259860, 9830505, 16777217, 33620226, 67174660, 151389705, 268501264, 554832417, 1091834945, 2523490711]
</source>
|}<!-- end of dh-box --><noinclude>
[[Category:Mentors of Boolean functions]]
</noinclude>
lvw0gqyv1wts4z0duj7c8w8j72zss8e
Template:Mentors of Boolean functions/code Yu vector
10
317667
2694031
2025-01-01T21:43:30Z
Watchduck
137431
Watchduck moved page [[Template:Mentors of Boolean functions/code Yu vector]] to [[Template:Mentors of Boolean functions/code WP vectors]]
2694031
wikitext
text/x-wiki
#REDIRECT [[Template:Mentors of Boolean functions/code WP vectors]]
bnm340qmpe7qx80di11u8cypdqyeb03
Template:Boolf weight triangle; fixed points of Sha
10
317668
2694034
2025-01-01T22:25:22Z
Watchduck
137431
New resource with "{{Boolf weight triangle 4| 1|1| 1|1|0| 1|4|6|4|1| 1|4|9|16|19|12|3|0|0| 1|5|25|85|170|226|226|170|85|25|5|1|0|0|0|0|0}} <noinclude> ---- [[Mentors of Boolean functions]] [[Category:Boolf triangles with weight columns|fixed points of Sha]] </noinclude>"
2694034
wikitext
text/x-wiki
{{Boolf weight triangle 4|
1|1|
1|1|0|
1|4|6|4|1|
1|4|9|16|19|12|3|0|0|
1|5|25|85|170|226|226|170|85|25|5|1|0|0|0|0|0}}
<noinclude>
----
[[Mentors of Boolean functions]]
[[Category:Boolf triangles with weight columns|fixed points of Sha]]
</noinclude>
kqcpar0iaewyre2dzc50serd0l4glc2
Template:Boolf weight triangle; fixed points of Che
10
317669
2694035
2025-01-01T22:25:25Z
Watchduck
137431
New resource with "{{Boolf weight triangle 4| 1|1| 1|0|1| 1|4|6|4|1| 1|0|12|0|38|0|12|0|1| 1|0|0|0|60|0|256|0|390|0|256|0|60|0|0|0|1}} <noinclude> ---- [[Mentors of Boolean functions]] [[Category:Boolf triangles with weight columns|fixed points of Che]] </noinclude>"
2694035
wikitext
text/x-wiki
{{Boolf weight triangle 4|
1|1|
1|0|1|
1|4|6|4|1|
1|0|12|0|38|0|12|0|1|
1|0|0|0|60|0|256|0|390|0|256|0|60|0|0|0|1}}
<noinclude>
----
[[Mentors of Boolean functions]]
[[Category:Boolf triangles with weight columns|fixed points of Che]]
</noinclude>
mww13rk8swjddoggq0cr4ncege1md5s
Template:Boolf weight triangle; families
10
317670
2694039
2025-01-01T23:05:57Z
Watchduck
137431
New resource with "{{Boolf weight triangle 5| 1|1| 1|1|1| 1|1|3|1|1| 1|1|7|7|14|7|7|1|1| 1|1|15|35|140|273|553|715|870|715|553|273|140|35|15|1|1| 1|1|31|155|1240|6293|28861|105183|330460|876525|2020239|4032015|7063784|10855425|14743445|17678835|18796230|17678835|14743445|10855425|7063784|4032015|2020239|876525|330460|105183|28861|6293|1240|155|31|1|1|1}} <noinclude> ---- {{oeis|A054724}} with row sums {{oeis|A000231}} [[Category:Boolf triangles with weight columns|families]] </noincl..."
2694039
wikitext
text/x-wiki
{{Boolf weight triangle 5|
1|1|
1|1|1|
1|1|3|1|1|
1|1|7|7|14|7|7|1|1|
1|1|15|35|140|273|553|715|870|715|553|273|140|35|15|1|1|
1|1|31|155|1240|6293|28861|105183|330460|876525|2020239|4032015|7063784|10855425|14743445|17678835|18796230|17678835|14743445|10855425|7063784|4032015|2020239|876525|330460|105183|28861|6293|1240|155|31|1|1|1}}
<noinclude>
----
{{oeis|A054724}} with row sums {{oeis|A000231}}
[[Category:Boolf triangles with weight columns|families]]
</noinclude>
3dw366wi346ifg83kaybsj268ffzj6h
2694044
2694039
2025-01-01T23:25:19Z
Watchduck
137431
2694044
wikitext
text/x-wiki
{{Boolf weight triangle 5|
1|1|
1|1|1|
1|1|3|1|1|
1|1|7|7|14|7|7|1|1|
1|1|15|35|140|273|553|715|870|715|553|273|140|35|15|1|1|
1|1|31|155|1240|6293|28861|105183|330460|876525|2020239|4032015|7063784|10855425|14743445|17678835|18796230|17678835|14743445|10855425|7063784|4032015|2020239|876525|330460|105183|28861|6293|1240|155|31|1|1|1}}
<noinclude>
----
{{oeis|A054724}} with row sums {{oeis|A000231}}
Compare {{tl|Boolf weight triangle; clans}}.
The central values (1, 3, 14, 870, 18796230) are similar to {{oeis|A340259}}, the central values of {{oeis|A340312}}. See {{tl|Boolf weight triangle; A340312}}.
[[Category:Boolf triangles with weight columns|families]]
</noinclude>
1ofoz5wa21p7f8l5g6gpl67yi03qlo4
Template:Boolf weight triangle; clans
10
317671
2694041
2025-01-01T23:12:00Z
Watchduck
137431
New resource with "{{Boolf weight triangle 5| 1|1| 1|1|1| 1|1|2|1|1| 1|1|3|3|6|3|3|1|1| 1|1|4|6|19|27|50|56|74|56|50|27|19|6|4|1|1| 1|1|5|10|47|131|472|1326|3779|9013|19963|38073|65664|98804|133576|158658|169112|158658|133576|98804|65664|38073|19963|9013|3779|1326|472|131|47|10|5|1|1| }} <noinclude> ---- {{oeis|A039754}} with row sums {{oeis|A000616}} Central column is {{oeis|A000721}} [[Category:Boolf triangles with weight columns|families]] </noinclude>"
2694041
wikitext
text/x-wiki
{{Boolf weight triangle 5|
1|1|
1|1|1|
1|1|2|1|1|
1|1|3|3|6|3|3|1|1|
1|1|4|6|19|27|50|56|74|56|50|27|19|6|4|1|1|
1|1|5|10|47|131|472|1326|3779|9013|19963|38073|65664|98804|133576|158658|169112|158658|133576|98804|65664|38073|19963|9013|3779|1326|472|131|47|10|5|1|1|
}}
<noinclude>
----
{{oeis|A039754}} with row sums {{oeis|A000616}}
Central column is {{oeis|A000721}}
[[Category:Boolf triangles with weight columns|families]]
</noinclude>
d0r8dcr8wqulwxmzsvqc1dfvwxhw0su
2694042
2694041
2025-01-01T23:12:47Z
Watchduck
137431
2694042
wikitext
text/x-wiki
{{Boolf weight triangle 5|
1|1|
1|1|1|
1|1|2|1|1|
1|1|3|3|6|3|3|1|1|
1|1|4|6|19|27|50|56|74|56|50|27|19|6|4|1|1|
1|1|5|10|47|131|472|1326|3779|9013|19963|38073|65664|98804|133576|158658|169112|158658|133576|98804|65664|38073|19963|9013|3779|1326|472|131|47|10|5|1|1|
}}
<noinclude>
----
{{oeis|A039754}} with row sums {{oeis|A000616}}
Central column is {{oeis|A000721}}.
Compare {{tl|Boolf weight triangle; families}}.
[[Category:Boolf triangles with weight columns|families]]
</noinclude>
n2qspze24kftuuu96g3alx2nr12lkh3
2694046
2694042
2025-01-01T23:26:54Z
Watchduck
137431
2694046
wikitext
text/x-wiki
{{Boolf weight triangle 5|
1|1|
1|1|1|
1|1|2|1|1|
1|1|3|3|6|3|3|1|1|
1|1|4|6|19|27|50|56|74|56|50|27|19|6|4|1|1|
1|1|5|10|47|131|472|1326|3779|9013|19963|38073|65664|98804|133576|158658|169112|158658|133576|98804|65664|38073|19963|9013|3779|1326|472|131|47|10|5|1|1|
}}
<noinclude>
----
{{oeis|A039754}} with row sums {{oeis|A000616}}
Central column is {{oeis|A000721}}.
Compare {{tl|Boolf weight triangle; families}}.
[[Category:Boolf triangles with weight columns|clans]]
</noinclude>
1fz2pfw7laacpti2cqlu81ana5lehzo
Template:Boolf weight triangle; A340312
10
317672
2694043
2025-01-01T23:25:10Z
Watchduck
137431
New resource with "{{Boolf weight triangle 5| 1|1| 1|1|0| 1|1|0|1|1| 1|1|0|7|14|7|0|1|1| 1|1|0|35|140|273|448|715|870|715|448|273|140|35|0|1|1| 1|1|0|155|1240|6293|27776|105183|330460|876525|2011776|4032015|7063784|10855425|14721280|17678835|18796230|17678835|14721280|10855425|7063784|4032015|2011776|876525|330460|105183|27776|6293|1240|155|0|1|1| }} <noinclude> ---- {{oeis|A340312}} with row sums {{oeis|A300361}} = <math>2^{2^n-n}</math> The central values are similar to those of {{o..."
2694043
wikitext
text/x-wiki
{{Boolf weight triangle 5|
1|1|
1|1|0|
1|1|0|1|1|
1|1|0|7|14|7|0|1|1|
1|1|0|35|140|273|448|715|870|715|448|273|140|35|0|1|1|
1|1|0|155|1240|6293|27776|105183|330460|876525|2011776|4032015|7063784|10855425|14721280|17678835|18796230|17678835|14721280|10855425|7063784|4032015|2011776|876525|330460|105183|27776|6293|1240|155|0|1|1|
}}
<noinclude>
----
{{oeis|A340312}} with row sums {{oeis|A300361}} = <math>2^{2^n-n}</math>
The central values are similar to those of {{oeis|A054724}}. See {{tl|Boolf weight triangle; families}}.
[[Category:Boolf triangles with weight columns|families]]
</noinclude>
tgz96tvtfqdnkvn7eejnhs6u27p5rvc
2694045
2694043
2025-01-01T23:26:43Z
Watchduck
137431
2694045
wikitext
text/x-wiki
{{Boolf weight triangle 5|
1|1|
1|1|0|
1|1|0|1|1|
1|1|0|7|14|7|0|1|1|
1|1|0|35|140|273|448|715|870|715|448|273|140|35|0|1|1|
1|1|0|155|1240|6293|27776|105183|330460|876525|2011776|4032015|7063784|10855425|14721280|17678835|18796230|17678835|14721280|10855425|7063784|4032015|2011776|876525|330460|105183|27776|6293|1240|155|0|1|1|
}}
<noinclude>
----
{{oeis|A340312}} with row sums {{oeis|A300361}} = <math>2^{2^n-n}</math>
The central values are similar to those of {{oeis|A054724}}. See {{tl|Boolf weight triangle; families}}.
[[Category:Boolf triangles with weight columns|A340312]]
</noinclude>
sk1x8h9xhcp62vkqarsril5xhtsb0lk
File:LIB.2B.Shared.20250102.pdf
6
317673
2694056
2025-01-02T00:23:51Z
Young1lim
21186
{{Information
|Description=LIB.2B: Shared Libraries (20250102 - 20250101-1)
|Source={{own|Young1lim}}
|Date=2025-01-02
|Author=Young W. Lim
|Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}}
}}
2694056
wikitext
text/x-wiki
== Summary ==
{{Information
|Description=LIB.2B: Shared Libraries (20250102 - 20250101-1)
|Source={{own|Young1lim}}
|Date=2025-01-02
|Author=Young W. Lim
|Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}}
}}
== Licensing ==
{{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}}
j4un4mdy9ef8bbnbslez2zkm2jbk0mi
Arithmetic
0
317674
2694059
2025-01-02T01:10:21Z
FlatPanda55
2995721
Started a banger.
2694059
wikitext
text/x-wiki
Arithmetic is the field of elementary mathematics concerning numbers, operations, and computation.
rj2gajp8hr4e8f385i851l5wqdwdg2d
2694060
2694059
2025-01-02T01:12:29Z
FlatPanda55
2995721
2694060
wikitext
text/x-wiki
Arithmetic is the field of elementary mathematics concerning numbers, operations, and computation.
Numbers
Operations
Addition
Subtraction
Multiplacation
Division
Fractions, Ratios, Rates, Percentages, and Proportions
Exponentation
Roots
Computation
ntafph87zhe869vnv3i0xbftf48e1go
2694067
2694060
2025-01-02T01:26:40Z
Atcovi
276019
housekeeping
2694067
wikitext
text/x-wiki
{{cleanup|organization and formatting needed}}
{{stub}}
{{mathematics}}
Arithmetic is the field of elementary mathematics concerning numbers, operations, and computation.
Numbers
Operations
Addition
Subtraction
Multiplacation
Division
Fractions, Ratios, Rates, Percentages, and Proportions
Exponentation
Roots
Computation
[[Category:Mathematics]]
sf1t0pwi7grqwkirqaqwzqos4jyjrpb
Geometry/Chapter 1/Lesson 2
0
317675
2694061
2025-01-02T01:19:24Z
FlatPanda55
2995721
New resource with "A line segment is a straight line with two endpoints. A line segment is said to be closed if the set of all its points contains its endpoints, open if not, and half-open if only one of the end points is contained. Line segments are also refered to as segments when talking about lines. Note: Segment can also refer to part of a circle. Congruence is the relationship between two shapes of being the same size and shape regardless of position, mirroring, or orientation. By t..."
2694061
wikitext
text/x-wiki
A line segment is a straight line with two endpoints. A line segment is said to be closed if the set of all its points contains its endpoints, open if not, and half-open if only one of the end points is contained. Line segments are also refered to as segments when talking about lines. Note: Segment can also refer to part of a circle.
Congruence is the relationship between two shapes of being the same size and shape regardless of position, mirroring, or orientation. By this definition, two line segments are said to be congruent if they are the same length as any two lines will always be the same shape.
78hmfba4c5us5vlv2sd4b6ieh8nckj6
2694063
2694061
2025-01-02T01:23:53Z
Atcovi
276019
housekeeping
2694063
wikitext
text/x-wiki
{{stub}}
A line segment is a straight line with two endpoints. A line segment is said to be closed if the set of all its points contains its endpoints, open if not, and half-open if only one of the end points is contained. Line segments are also refered to as segments when talking about lines. Note: Segment can also refer to part of a circle.
Congruence is the relationship between two shapes of being the same size and shape regardless of position, mirroring, or orientation. By this definition, two line segments are said to be congruent if they are the same length as any two lines will always be the same shape.
{{stub}}
[[Category:Geometry]]
rflxazmfa8fwgj9rzmz0hn8azckqsxy
2694064
2694063
2025-01-02T01:24:06Z
Atcovi
276019
didn't mean to add this twice
2694064
wikitext
text/x-wiki
{{stub}}
A line segment is a straight line with two endpoints. A line segment is said to be closed if the set of all its points contains its endpoints, open if not, and half-open if only one of the end points is contained. Line segments are also refered to as segments when talking about lines. Note: Segment can also refer to part of a circle.
Congruence is the relationship between two shapes of being the same size and shape regardless of position, mirroring, or orientation. By this definition, two line segments are said to be congruent if they are the same length as any two lines will always be the same shape.
[[Category:Geometry]]
bxbwt6ut04es2so51pv9drlze81ofv0
Geometry/Chapter 1/Lesson 4
0
317676
2694062
2025-01-02T01:21:26Z
FlatPanda55
2995721
New resource with "Angle is the amount of turn or rotation that a ray from a certain vertex would need to turn to reach one point starting at the other."
2694062
wikitext
text/x-wiki
Angle is the amount of turn or rotation that a ray from a certain vertex would need to turn to reach one point starting at the other.
edpy6nwdjf6f3khgvjchv2yv4nd2deu
2694065
2694062
2025-01-02T01:24:18Z
Atcovi
276019
{{stub}}
2694065
wikitext
text/x-wiki
{{stub}}
Angle is the amount of turn or rotation that a ray from a certain vertex would need to turn to reach one point starting at the other.
mh4w3ckbub648pmami5nyowtl7qrud7
User talk:FlatPanda55
3
317677
2694066
2025-01-02T01:24:52Z
Atcovi
276019
/* Welcome */ new section
2694066
wikitext
text/x-wiki
==Welcome==
{{Robelbox|theme=9|title='''[[Wikiversity:Welcome|Welcome]] to [[Wikiversity:What is Wikiversity|Wikiversity]], FlatPanda55!'''|width=100%}}
<div style="{{Robelbox/pad}}">
You can [[Wikiversity:Contact|contact us]] with [[Wikiversity:Questions|questions]] at the [[Wikiversity:Colloquium|colloquium]] or get in touch with [[User talk:Atcovi|me personally]] if you would like some [[Help:Contents|help]].
Remember to [[Wikiversity:Signature#How to add your signature|sign]] your comments when [[Wikiversity:Who are Wikiversity participants?|participating]] in [[Wikiversity:Talk page|discussions]]. Using the signature icon [[File:OOjs UI icon signature-ltr.svg]] makes it simple.
We invite you to [[Wikiversity:Be bold|be bold]] and [[Wikiversity|assume good faith]]. Please abide by our [[Wikiversity:Civility|civility]], [[Wikiversity:Privacy policy|privacy]], and [[Foundation:Terms of Use|terms of use]] policies.
To find your way around, check out:
<!-- The Left column -->
<div style="width:50.0%; float:left">
* [[Wikiversity:Introduction|Introduction to Wikiversity]]
* [[Help:Guides|Take a guided tour]] and learn [[Help:Editing|how to edit]]
* [[Wikiversity:Browse|Browse]] or visit an educational level portal:<br>[[Portal:Pre-school Education|pre-school]] | [[Portal:Primary Education|primary]] | [[Portal:Secondary Education|secondary]] | [[Portal:Tertiary Education|tertiary]] | [[Portal:Non-formal Education|non-formal]]
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</div>
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<div style="width:50.0%; float:left">
* Read an [[Wikiversity:Wikiversity teachers|introduction for teachers]]
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* Give [[Wikiversity:Feedback|feedback]] about your observations
* Discuss issues or ask questions at the [[Wikiversity:Colloquium|colloquium]]
</div>
<br clear="both"/>
To get started, experiment in the [[wikiversity:sandbox|sandbox]] or on [[special:mypage|your userpage]].
See you around Wikiversity! --—[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 01:24, 2 January 2025 (UTC)</div>
<!-- Template:Welcome -->
{{Robelbox/close}}
l0tkqoe7202oijb1jeia796939wu7sf
User:Dhairyadigital
2
317678
2694070
2025-01-02T04:31:04Z
Dhairyadigital
2995728
New resource with "= Digital Marketing Demystified: Strategies to Elevate Your Online Presence = '''In the digital age, an effective online presence is no longer optional—it’s essential. Digital marketing has transformed how businesses connect with their audiences, offering innovative tools and strategies to engage, influence, and convert customers. By understanding and leveraging these tools, businesses can thrive in the competitive online landscape.''' '''Digital marketing encompass..."
2694070
wikitext
text/x-wiki
= Digital Marketing Demystified: Strategies to Elevate Your Online Presence =
'''In the digital age, an effective online presence is no longer optional—it’s essential. Digital marketing has transformed how businesses connect with their audiences, offering innovative tools and strategies to engage, influence, and convert customers. By understanding and leveraging these tools, businesses can thrive in the competitive online landscape.'''
'''Digital marketing encompasses a wide range of tactics designed to attract and engage target audiences. Search Engine Optimization (SEO) focuses on enhancing website visibility, ensuring your brand appears at the top of search engine results. Content Marketing, on the other hand, emphasizes creating valuable, engaging content that educates and resonates with your audience. Platforms like blogs, videos, and infographics are key tools in this strategy.'''
'''Social media platforms, such as Instagram, Facebook, LinkedIn, and TikTok, are integral to Social Media Marketing, where businesses can build meaningful connections and foster brand loyalty. Simultaneously, Pay-Per-Click (PPC) advertising enables companies to place highly targeted ads in front of potential customers, driving immediate traffic and conversions.'''
'''Another vital component is Email Marketing, which offers a personalized approach to engage customers, nurture leads, and boost retention. Automated email campaigns tailored to user behavior can significantly enhance customer relationships and sales.'''
'''What makes digital marketing unique is its ability to provide measurable results. Analytics tools such as Google Analytics and platform-specific insights allow businesses to track campaign performance, understand customer behavior, and refine strategies in real-time. This data-driven approach ensures optimal results with cost-effective efforts.'''
'''To succeed, businesses need a cohesive digital marketing strategy that aligns with their goals and audience preferences. Consistent branding, regular engagement, and high-quality content are the foundation of a successful online presence.'''
'''Looking to elevate your business with proven digital marketing strategies explore tailored solutions that suit your unique needs. From optimizing your website for search engines to creating impactful ad campaigns, AarCodes empowers businesses to thrive in the digital space.'''
'''Stay ahead of the competition, engage your audience effectively, and achieve lasting online success with the right digital marketing tools and expertise. Start your journey today with AarCodes!'''
2v9jnndqygksqly2kfvf36kycxnfmkj
WikiJournal Preprints/A Chittagong trade coin probably in the name of Amar Manikeya.
0
317679
2694071
2025-01-02T05:16:14Z
Shawkat26
2995723
New resource with " {{Article info | journal = WikiJournal Preprints <!-- WikiJournal of Medicine, Science, or Humanities --> | last1 = Kamal | orcid1 = https://orcid.org/0000-0002-1585-7475 | first1 = Sheikh Shawkat | last2 = | first2 = | last3 = | first3 = | last4 = | first4 = <!-- up to 9 authors can be added in this above format --> | et_al = <!-- if there are >9 authors, hyperlink to the list here --> | affiliation1 = Senior consultant Surgeon, Department of Otolaryngology,..."
2694071
wikitext
text/x-wiki
{{Article info
| journal = WikiJournal Preprints <!-- WikiJournal of Medicine, Science, or Humanities -->
| last1 = Kamal
| orcid1 = https://orcid.org/0000-0002-1585-7475
| first1 = Sheikh Shawkat
| last2 =
| first2 =
| last3 =
| first3 =
| last4 =
| first4 = <!-- up to 9 authors can be added in this above format -->
| et_al = <!-- if there are >9 authors, hyperlink to the list here -->
| affiliation1 = Senior consultant Surgeon, Department of Otolaryngology, Surgiscope Hospital, Chittagong, Bangladesh.
| correspondence1 = drshawkatent@gmail.com
| affiliations = institutes / affiliations
| correspondence = email@address.com
| keywords = Chittagong, Trade, Coin, Amar Manikeya, Tripura.
| license = <!-- default is CC-BY -->
| abstract = Chittagong trade coins were invariably discovered to be struck in the name(s) of powerful individuals. In this regard, numismatists are perplexed by the person called Wamar or Vamar Shah on the trade coin. Based on numismatic and historical facts, this article assesses the probability that King Amar Manikeya of Tripura was the specific individual.
}}
== Introduction: ==
[[File:Chittagong trade coins in the name of Amar Manikeya.png|left|thumb|677x677px|The collection includes coins named 'Wamar Shah,' each with a unique mint date. All photographs are courtesy of Noman Nasir.]]
It has already been a long time since a coin in the name of Wamar Shah or Vamar Shah, possibly minted in Chittagong, was mentioned in articles by expert numismatists <ref>Mitchiner M. (2000). The Land of Water, Coinage and History of Bangladesh and later Arakan, circa 300BC to the present day. Hawkins Publications. Pp. 115-126.</ref><ref>Deyell, J. (1995). The Trade Coinage of Chittagong Region in the Mid-Sixteenth Century. Journal of the Asiatic Society of Bangladesh, Vol. 4O, No. 2 (December). Pp. 207-235. </ref> . All of them considered it as a ‘Chittagong trade coin.’ Besides, a few of them speculated that the person named 'Wamar' or 'Vamar' could have been a governor appointed by the Arakan king <ref>Hauret, P. (2022). The The Chittagonian Coinage of Arakan’s Royal Sons. ''Advances in Social Sciences Research Journal'', ''9''(9), 361–378.</ref> . History does not provide a straightforward identity of this person.
According to numismatic records, the individual named Wamar Shah/Vamar Shah flourished during the late period of Bengal sultanates (about the 1570-80s AD). As all the Chittagong trade coins were struck in the name of an influential person, the person of interest should have been a potential one. In this respect, it is vital to comprehend the geopolitical context in Chittagong at that particular time by reviewing historical data from numerous sources. It is also critical to ensure that the individual's history data matches the numismatic data. The present article has tried to justify King Amar Manikeya of Tripura as the person engraved on the coin in light of historical and numismatic data.
== Searching the correct pronunciation of the name inscribed on the coin: ==
Before digging deep, it is initially important to confirm the correct pronunciation of the engraved name mentioned on that coin. The Persian language was introduced in Bengal by the Afghan rulers. The Afghan Persian language is called ‘Dari.’ Phonetically, it is a little different from the classical Iranian Persian language. In Dari Persian, the Arabic/Persian consonant “ <big>'''و'''</big> ” is pronounced bilabial and sounds like ['''wa''']. On the contrary, in classical Iranian Persian, it is pronounced labiodental and sounds like ['''v'''] <ref>Dari. (2024, December 7). In ''Wikipedia''. <nowiki>https://en.wikipedia.org/wiki/Dari</nowiki></ref> . The court of the Bengal sultan was likely using Afghan Persian, so the person whose name was engraved on the coin should be Wamar, not Vamar.
== Re-visiting the history: ==
In 933 BE (1572 AD), Min Phalaung ascended to the throne of Arakan. ‘Min Phalaung’ (meaning 'Prince Portuguese/foreigner') was his honorific name since he was born in the year when his father, the great king of Arakan, Min Pa Gyi (Min Bin), defeated the foreign invader, the Portuguese. His birth name was 'Phwa Daw Htwe' (meaning ‘Royal Youngest Birth), as he was the youngest son of Min Pa Gyi <ref>Lankara S. (Second Edition. 1997). ရခိုင်ရာဇဝင်သစ်ကျမ်း [Rakhain Razawin Thit- New History of Arakan]. Vol-2. Yangun, Myanmar. U Ye Myint. Pp. 47.</ref> . His designation in the Muslim community was 'Sikander Shah-I'. When Min Phalaung desired to expand his territorial supremacy, he first sent his eldest son, the crown prince, to attack the Ramu, the southern area of Chittagong. It happened in 936 BE (1575 AD) when Ramu was under the control of the Tripura king <ref>Ibid. Pp. 48-49.</ref> . In the Arakanese chronicle, Tripura and Ramu were titled as ‘Thet’ and ‘Kamboja,’ respectively. The crown prince defeated the ruler of Ramu. This successful expedition of Min Phalaung sent the message of his strength to the Pathan rulers of Chittagong and its surrounding territories. As a consequence, the Pathan ruler of Chittagong, Arakan chronicle termed him as ‘U Sisla,’ and other Pathan rulers were subdued to Min Phalaung <ref>Ibid. Pp. 49.</ref> . Throughout the mediaeval period, the Bengal sultan, Tripura king, and Arakan king subjugated the Chittagong port city at various times. According to Arakan historian, Sandamala Lankara, ‘Sisla’ is the Burmese corrupt of the Muslim name ‘Jalal’ (Jalal Khan). Mediaeval Chittagonian poet Muhammad Khan (who was alive in 1646 AD), in his hand written book (puthi) 'Maktul Hossain,' described the list of his ancestors, who were the early Muslim rulers in Chittagong <ref>Hoque M E & Karim A. (2021). আরাকান-রাজসভায় বাঙ্গালা সাহিত্য [Bengali Literature in the Court of Arakan King]. Dhaka, Bangladesh. Batighar. Pp. 133-134. </ref> . According to that depiction, Jalal Khan became the ruler of Chittagong following the death of his father, Nusrat Khan. After Jalal Khan, his son, Ibrahim Khan, became the ruler of Chittagong. Nusrat Khan was the ruler of Chittagong during the reign of Bengal Sultan Sulaiman Karrani. The Arakan chronicle mentioned Nusrat Khan as ‘Nakhtha Tharu Khin’.
During the early time of the Tripura King, Udaya Manikeya-I (reign 1567-72), the Bengal sultan Sulaiman Khan Karrani (reign 1565-72 AD) sent his troops against Tripura to capture Chittagong <ref>Sen K P. (1927). শ্রীরাজমালা- দ্বিতীয় লহর [ Rajmala- Second Volume]. Agartala, Tripura, India. Rajmala Office. Pp. 69.</ref> . The army of Sulaiman Khan Karrani defeated Tripura and established the Sultan’s hold over Chittagong. But Tripura’s king did not give up his claim over Chittagong for long and continued to fight against the Bengal Sultan’s army to regain its control over Chittagong. According to the Tripura chronicle, this battle lasted for around 5 years, and most of the time the Sultan’s army was victorious. Now the important question is when Tripura regained control of Chittagong. In 1569 AD, a notable Venetian named ‘Caesar Frederick’ was visiting Chittagong and Sandwip <ref>Campos J J A. (1998). History of the Portuguese in Bengal with Maps and Illustrations. New Delhi & Madras, India. Asian Educational Services. Pp. 269.</ref> . He stated in his travel diary that both the Chittagong and Sandwip were under the Bengal Sultan during his visit. If the battle between armies of Bengal Sultan and Tripura continued to the reign of Daud Khan Karrani (reign 1572-76 AD), the son of Sultan Sulaiman Khan Karrani, the hold of Bengal Sultan over Chittagong might become weak from 1574 AD because of the successful military campaigns by Mughal Emperor Akbar against him <ref>Salim G H. (1903). Riyazu-s- Salatin (A Salam, Trans.). Idarah-I Adabiyat-I Delli, India. (Original work published in 1788). Pp. 155.</ref>. This eventually might help Tripura to regain its supremacy over Chittagong. So the control of Chittagong might hand over to Tripura in between 1569 AD to 1574 AD.
During 1573 AD, Joy Manikeya-I (reign 1573-77 AD) was the king of Tripura <ref>Sarma R M. (1986). A Political History of Tripura. Puthipatra, Calcutta, India. Pp.77.</ref> . He was a puppet king in the hands of his army chief, Ranagan Narayan. Both Joy Manikeya-I and his father, Udaya Manikeya-I, were not members of the royal Manikeya dynasty. Udaya Manikeya-I (his previous name was Gopi Prasad) treacherously killed his earlier king, Ananta Manikeya, and ascended the throne of Tripura. Prince Amardeva, son of King Deva Manikeya, was an eligible contender for the kingship of Tripura. He, as an army commander, had been fighting against the Bengal sultan since the time of Udaya Manikeya-I and gained popularity in that frontier <ref>Sen K P. (1927). শ্রীরাজমালা- দ্বিতীয় লহর [ Rajmala- Second Volume]. Agartala, Tripura, India. Rajmala Office. Pp. 72.</ref> . The army chief of Joy Manikeya-I, Ranagan Narayan, was envious of Amardeva’s popularity since he had a desire to be the next king. Out of his jealousy, he planned to kill Amardeva. But the plan failed. Subsequently, Amardeva sent his loyal troops against Ranagan Narayan and killed him. The murder of Ranagan Narayan displeased Joy Manikeya-I. Sensing the upcoming retaliation from Joy Manikeya-I, Amardeva marched against him in advance and assassinated him <ref>Sen K P. (1927). শ্রীরাজমালা- দ্বিতীয় লহর [ Rajmala- Second Volume]. Agartala, Tripura, India. Rajmala Office. Pp. 76-77.</ref> . It is assumed that this political turmoil in Tripura supported the Arakan king, Min Phalaung, in capturing the Ramu from Tripura. In 1477 Saka (1577 AD), Amardeva, as Amar Manikeya, took the throne of Tripura <ref>Sarma R M. (1986). A Political History of Tripura. Puthipatra, Calcutta, India. Pp.77.</ref> .
Amar Manikeya proved his extraordinary kingship by expanding the territory of the Tripura kingdom. He turned areas situated in the northern and western borders of his kingdom—Sylhet, Bikrampur, Bhawal, Sarail, Bakla, Bhulua, etc.—into his vassal states <ref>Sen K P. (1931). শ্রীরাজমালা- তৃতীয় লহর [ Rajmala- Third Volume]. Agartala, Tripura, India. Rajmala Office.Pp. 11,12,87,88,89,177-180.</ref> . Amar Manikeya had an intimate relationship with Isha Khan, the potential leader of the ‘Baro-Bhuiya’ of Bhati regions. Both parties supported each other on various occasions <ref>Ibid.Pp.3,7,15.16.</ref> . His close relationship with Isha Khan might have influenced other Pathan rulers of the Chittagong areas The Arakan chronicle gave the sense that the rulers of areas that had previously been subdued to the Tripura monarch rose occasionally in rebellion against Min Phalaung due to the influence of a certain individual <ref>Lankara S. (Second Edition. 1997). ရခိုင်ရာဇဝင်သစ်ကျမ်း [Rakhain Razawin Thit- New History of Arakan]. Vol-2. Yangun, Myanmar. U Ye Myint. Pp. 49.</ref> . However, the Arakan chronicle did not mention his name. This person could have been the Amar Manikeya.
From 1580 AD, Min Phalaung had to remain busy repulsing the attack of the Toungoo king. However, in 1581 AD, the sudden death of the Toungoo king gave great relief to Min Phalaung as the invading army returned to their own country <ref>Ibid. Pp. 50.</ref> . After this, Min Phalaung took the initiative to establish his firm control over all of Chittagong situated in his western territory <ref>Ibid. Pp. 50-51.</ref>. During that time, all of Chittagong was divided into small feudal territories. Chittagong port and areas north of the Karnafully River formed Chittagong proper. According to the Tripura chronicle ‘Rajmala’, areas south of the Karnafully River were divided into 6 territories. Some of them were Rambu (Ramu), Chokria, Diang, and Uria (Ukhia). Min Phalaung appointed his second son, ‘Thato Min Saw,’ as the King of the West to look after the whole of Chittagong <ref>Ibid. Pp. 51.</ref> . This appointment displeased the feudal rulers of Chittagong. They rose against ‘Thato Min Saw.’ When Min Phalaung heard about this conspiracy, he became furious and vowed to punish them. To get rid of his anger, the ruler of Ramu and Chokoria, Adam fled and took shelter with Tripura king Amar Manikeya <ref>Ibid. Pp. 51.</ref> <ref>Sen K P. (1931). শ্রীরাজমালা- তৃতীয় লহর [ Rajmala- Third Volume]. Agartala, Tripura, India. Rajmala Office. Pp.38.</ref>. The northern ruler of Chittagong- Jalal Khan, neighbouring other Pathan rulers, and the Portuguese Captain in Chittagong allied with Amar Manikeya <ref>Lankara S. (Second Edition. 1997). ရခိုင်ရာဇဝင်သစ်ကျမ်း [Rakhain Razawin Thit- New History of Arakan]. Vol-2. Yangun, Myanmar. U Ye Myint. Pp. 51.</ref> . This incident ultimately torched the battle between Arakan and Tripura. The Tripura army marched to Chittagong in advance, and they met the Arakan army near Ramu. Initially, the Tripura army was victorious over its counterpart. A ceasefire agreement was made between these two rival parties for one year. But the king of Arakan proceeded to Chittagong before the end of the ceasefire period and, in a letter, demanded the handover of Adam (former ruler of Ramu and Chokoria) to him. Amar Manikeya refused it. The Arakan chronicle mentioned the date as 948 BE (1586 AD) <ref>Ibid. Pp. 51.</ref>. As a consequence, these two rival parties again entered into a war. During that time the English traveler Ralph Fitch was in Chittagong and mentioned this war in his write-up <ref>Purchas S. (1905). Hakluytus Posthumus of Purchas His Pilgrimes, Vol-X. Glasgow, United Kingdom. James Mac Lehose and Sons. Pp. 183.</ref> . Finally, the Arakan Army defeated Tripura and eventually captured the Tripura capital, ‘Udaipur’ <ref>Sen K P. (1931). শ্রীরাজমালা- তৃতীয় লহর [ Rajmala- Third Volume]. Agartala, Tripura, India. Rajmala Office. Pp. 41.</ref>. This Arakan victory erased Tripura’s hold over Chittagong forever.
== Correlating history with numismatic data: ==
It is well recognized that all the Chittagong trade coins were minted in the name of person(s) having influence either on political or religious grounds. In these trade coins, except for Wamar Shah, other persons, even the local rulers like Jalal Khan and Adam Humayun, have been identified in the regional history. History revealed that at the early period of Udaya Manikeya-I, Bengal sultan Sulaiman Khan Karrani snatched control of Chittagong from Tripura. Thereafter, the Tripura army engaged in fighting against the sultan’s army for 5 years to regain control of Chittagong. When the Mughal Emperor Akbar attacked the Bengal sultan, the firm grip of the Bengal sultan on Chittagong became loose, and ultimately Tripura regained it. During the battle between Tripura and Bengal, Prince Amardeva (later, he became King Amar Manikeya) was the famous commandant of the Tripura army on the Chittagong front. His popularity made him a competent contender for Tripura’s throne. After his ascension to the throne, Amar Manikeya extended Tripura’s northern and western territories. He had a close relationship with Isha Khan, the well-known leader of the 'Baro Bhuiya' of the Bhati regions. Even the Chittagong’s Pathan ruler, who had been under Arakan control since 1575 AD, inclined towards King Amar Manikeya and supported him during the battle between Tripura and Arakan. Contemporary history cannot recognize any other prominent person named Wamar who had potential influence like King Amar Manikeya on Chittagong. It is also understandable that the name Amar can be pronounced as Wamar in Persian dialect.
Chittagong trade coins in the name of ‘Wamar Shah’ are extremely rare. It is considered that this variety of coins would have been minted in north Chittagong since its flan shape, style, calligraphy, and provenance resemble the other Chittagong-minted trade coins. The mint dates in so far collected coins of this ruler are 983 AH (1575), 984 AH (1576 AD), 985 AH (1577AD), 988 AH (1580 AD) and 989 AH (1581) '''[Figure 1]'''. Previously the coin dated 984 AH was read as 977 AH (retrograde); later numismatists confirmed the date as 984 AH. One of the important characteristics of Chittagong trade coins is that their mint dates are incompatible with the reign of the inscribed ruler. Many of Chittagong's trade coins have fictitious mint dates, either anticipatory or posthumous <ref>Mitchiner M. (2000). The Land of Water, Coinage and History of Bangladesh and later Arakan, circa 300BC to the present day. Hawkins Publications. Pp. 118 & 124.</ref> . For example, a trade coin of Jalal Shah, son of Muhammad Shah, has a mint date (951 AH) that is about 17 years anticipatory to his coronation (968 AH). In the case of Wamar Shah’s coins, mint dates 983 AH and 984 AH can be considered anticipatory dates concerning the reign of Amar Manikeya. On the other hand, coins with mint dates 985 AH, 988 AH and 989 AH can be accepted as correct since they were minted during the reign of Amar Manikeya.
Some numismatists suggested that the Arakan king might appointed Wamar Shah as the governor of north Chittagong after Jalal Khan. However, history depicts that after Jalal Khan, his son, Ibrahim Khan became the governor of Chittagong. Besides, all the trade coins previously found with the inscriptions of Jalal Khan and Adam Humayun, the two rulers of Chittagong under the Arakan king, have no mint date. However, very recently two coins with a mint date of 992 AH have been found in the name of Adam Humayun, probably the only known two pieces in the whole world <ref>These coins of Adam Humayun are in the collection of a private collector living in Bangladesh.</ref> . Moreover, the early mint dates of Wamar Shah’s coins make it unreasonable to place the Wamar as governor succeeding Jalal Khan.
Interestingly, mint dates of some of Wamar Shah’s coinage coincide with the mint dates of Sikander Shah’s coinage (the Muslim name of Arakan King, Min Phalaung). Even though Chittagong had been subdued to the Arakan king since 1575 AD, the supremacy of the Arakan king on Chittagong was weak till 1581 AD. According to the Arakan chronicle, some Pathan rulers of Chittagong and surrounding areas occasionally revolted during these periods. When the threat of invasion by the Toungoo king on Arakan was over in 1581 AD, the Arakan king took the initiative to bring back Chittagong under his firm control. The simultaneous minting of Wamar’s and Sikander’s coinages may have indicated the ‘swing political state’ of Chittagong between the influences of Min Phalaung and Amar Manikeya from 1575 to 1581 AD. Remarkably, till now, no coin minted in the name of Wamar Shah was found after 1581 AD. Such connections between historical and numismatic evidence suggest that Amar Manikeya may be the Wamar Shah.
== Conclusion: ==
Based on Afghan Persian, the court language of the Bengal Sultanate, it is possible to recognize correctly the name carved on the coin as ‘Wamar’ and rule out the option of ‘Vamar’. Historical evidence from many sources narrates that, even though the Arakan king subdued Chittagong in 1575 AD, King Amar Manikeya of Tripura significantly influenced Chittagong throughout the 1570s and the middle of the 80s. This influence might have prompted the Pathan ruler of Chittagong to mint the trade coin in the name of Amar Manikeya. Persian dialect styled Amar Manikeya as ‘Wamar Shah’ on the coin.
==Additional information==
===Acknowledgements===
Special thanks to Noman Nasir for supplying images of the coins.
===Competing interests===
The author has no competing interest.
==References and notes:==
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==Introduction==
[[File:Kreisring r R Definition.svg|mini|Definition domain: Annular region with radii r and R]]
Using the [[Cauchy's integral formula|Cauchy's integral formula]], two holomorphic functions, <math>f_2:\mathbb{C}\setminus D_r(z_o)\to \mathbb{C}</math> and <math>f_1:D_R(z_o)\to \mathbb{C}</math>, are defined in the decomposition theorem. These are then utilized for the expansion into a [[Laurent Series|Laurent Series]].
== Fundamental Definitions ==
The following fundamental definitions are used in the decomposition theorem:
: <math> K_r^R(z_o):= { z\in \mathbb{C} , : , 0 \leq r < |z-z_o| < R }</math>
: <math> K_r^\infty(z_o):= { z\in \mathbb{C} , : , 0 \leq r < |z-z_o| }</math>
: <math> D_r(z_o) := { z\in \mathbb{C} , : , |z-z_o| < r } </math>
: <math> \overline{D_r(z_o)} := { z\in \mathbb{C} , : , |z-z_o| \leq r } </math>
: <math> \gamma_{r,z_0}: [0,2\pi] \to \mathbb{C} </math> with <math> t \mapsto \gamma_{r,z_0}(t) = z_o +r \cdot e^{it} </math>
:<math> \int_{\partial D_r(z_o)} f(\xi), d\xi := \int_{\gamma_{r,z_0}} f(\xi), d\xi</math>
== Decomposition Theorem for Annular Regions ==
Let <math>G \subseteq \mathbb C </math> be an open set with a holomorphic function on an annular region <math>\overline{K_r^R(z_o)} = { z\in \mathbb{C} , : , r \leq |z-z_o| \leq R } \subset G</math>. Then, <math>f:K_r^R(z_o) \to \mathbb{C}</math> can be decomposed as <math>f=f_1 +f_2</math> into two holomorphic functions <math>f_1:D_R(z_o) \to \mathbb{C}</math> and <math>f_2:K_r^\infty(z_o) \to \mathbb{C}</math>. The decomposition <math>f=f_1 +f_2</math> is unique under the condition <math>\lim_{z \to \infty} f_2(z) = 0</math>.
== Proof ==
The proof considers functions on annular regions centered at <math>z_o = 0 \in \mathbb{C}</math>. By suitable composition with a shift, the decomposition theorem can be generalized for arbitrary <math>K_r^R(z_o) = { z\in \mathbb{C} , : , r \leq |z-z_o| \leq R }</math> with <math>\overline{K_r^R(z_o)} \subset G</math>. Initially, the proof idea for the decomposition theorem is discussed.
==Proof Idea==
*The center <math>z_o = 0 \in \mathbb{C}</math> of circular rings is a special case that can be used to generalize the statement for arbitrary circular rings <math>z_o \in \mathbb{C}</math>.
*Definition of a boundary cycle over a circular ring with two integration paths over an outer boundary and an inner boundary with reversed orientation.
*Application of the Cauchy Integral Theorem for cycles.
*Decomposition of the integral over a cycle into two partial integration paths along the inner and outer boundaries.
*One partial integral will provide the principal part of the Laurent expansion, while the other partial integral will yield the remainder of the integral.
==Uniqueness of the Decomposition==
The decomposition is generally not unique because the constant cannot be uniquely assigned to either the principal part or the remainder. The additional condition for the limit <math>z \to \infty</math> ensures uniqueness, as the constant is then assigned to the remainder.
==Proof 1: Circular Rings with Center 0==
We consider holomorphic functions on circular rings around the point <math>z_o \in \mathbb{C}</math>. The point <math>z_o \in \mathbb{C}</math> serves as the expansion point of the Laurent series with <math>(z-z_o)^n</math> where <math>n\in \mathbb{Z}</math>. Initially, we can restrict to circular rings around 0 (and thus around the expansion point 0), since any function <math>f:G \to \mathbb{C}</math> with <math>z_o \in \mathbb{C}</math> and a circular ring <math>\overline{K_r^R(z_o)} \subset G</math> can be transformed into a function <math>f_{z_o}:G_{z_o} \to \mathbb{C}</math> with:
: <math>G_{z_o} = { z-z_o \in \mathbb{C} , : , z \in G }</math> and <math>0 \in G_{z_o}</math>, as <math>z_o \in G</math>.
: <math>f_{z_o}:G_{z_o} \to \mathbb{C}</math> with <math> f_{z_o}(z) := f(z-z_o)</math>.
: <math>\overline{K_r^R(0)} \subset G_{z_o}</math> as <math>\overline{K_r^R(z_o)} \subset G</math>.
==Proof 2: Definition of the Boundary Cycle for the Circular Ring==
We define two integration paths along the inner and outer boundaries of the circular ring. These two paths have opposite orientations.
: <math> \gamma_1: [0,2\pi] \to \mathbb{C} </math> with <math> t \mapsto \gamma_{R,z_0}(t) = z_o + R \cdot e^{it} </math>.
: <math> \gamma_2: [0,2\pi] \to \mathbb{C} </math> with <math> t \mapsto \gamma_{r,z_0}(t) = z_o + r \cdot e^{-it} </math>.
: <math> \Gamma := \gamma_1 + \gamma_2</math>.
==Proof 3: Null-Homologous Cycle==
Since <math>\overline{K_r^R(z_o)} = { z\in \mathbb{C} , : , r \leq |z-z_o| \leq R } \subset G</math>, <math>\Gamma</math> is a null-homologous cycle, as only the inner points of the circular ring have a winding number of 1, and all inner points (including the circular ring’s boundary) belong to <math>G</math>.
==Proof 4: Application of the Cauchy Integral Formula for Cycles==
For <math>z \in K_r^R(z_o)</math>, the following holds:
:<math>n(\Gamma,z) \cdot f(z) = \frac 1{2\pi i}\int\limits_{\Gamma} \frac{f(\xi)}{\xi-z}, d\xi = \frac{1}{2\pi i}\left( \int\limits_{\gamma_1} \frac{f(\xi)}{\xi-z}, d\xi + \int\limits_{\gamma_2} \frac{f(\xi)}{\xi-z}, d\xi \right)</math>.
=== Proof 5: Substitution for Integration Paths ===
Since <math>n(\Gamma,z)=1</math> for <math>z \in K_r^R(z_o)</math>, we get the following by substituting the integration paths:
:<math>f(z) = \underbrace{\frac{1}{2\pi i} \int\limits_{\partial D_R(z_o)} \frac{f(\xi)}{\xi-z}, d\xi}{=f_1(z)} + \bigg(\underbrace{-\frac{1}{2\pi i} \int\limits{\partial D_r(z_o)} \frac{f(\xi)}{\xi-z}, d\xi}_{=f_2(z)} \bigg)</math>
The minus sign before the second integral arises due to the reversed direction of the path <math>\gamma_2</math>.
=== Proof 6: Standard Estimation ===
We consider the limit of the integral <math>|f_2(z)|</math> for <math>z\to \infty</math> with <math>C := \max\limits_{\xi \in \partial D_r(z_o)} |f(\xi)|</math>:
:<math>|f_2(z)| = \left| - \frac{1}{2\pi i} \int\limits_{\partial D_r(z_o)} \frac{f(\xi)}{\xi-z}, d\xi \right| \leq \int\limits_{\partial D_r(z_o)} \left| \frac{f(\xi)}{\xi-z} \right|, d\xi</math>
:: <math> \leq \int\limits_{\partial D_r(z_o)} \frac{C}{|\xi-z|} , d\xi \leq \underbrace{2\pi r}{L(\gamma_2)} \cdot \max\limits{\xi \in \partial D_r(z_o)} \frac{C}{|\xi-z|}</math>
Since this is an upper bound and the prefactors are all smaller than 1, they can simply be omitted.
=== Proof 7: Limit Process for Standard Estimation ===
Thus, we obtain:
:: <math>\lim_{z\to \infty} |f_2(z)| \leq \lim_{z\to \infty} \underbrace{2\pi r}{L(\gamma_2)} \cdot \max\limits{\xi\in\partial D_r(z_o)} \frac{C}{|\xi-z|} = 0</math>
=== Proof 8: Taylor Expansion with Cauchy Kernel ===
The series expansion occurs with the [[Cauchy Kernel]] in the red-marked convergence area:
:<math>\begin{align} f_1(z) & =\frac{1}{2\pi\mathrm{i}}\oint_{\partial D_{R}(z_o)}\frac{f(\zeta)}{\zeta-z}\mathrm{d}\zeta=\frac{1}{2\pi\mathrm{i}}\oint_{\partial D_{R}(z_o)}\frac{f(\zeta)}{\zeta-z_o-(z-z_o)}\mathrm{d}\zeta \ & {=} \frac{1}{2\pi\mathrm{i}}\oint_{\partial D_{R}(z_o)}\frac{f(\zeta)}{\zeta-z_o}\cdot \frac{1}{1-\frac{z-z_o}{\zeta-z_o}}\mathrm{d}\zeta, \ &\overset{|\frac{z-z_o}{\zeta-z_o}|<1}{=} \frac{1}{2\pi\mathrm{i}}\oint_{\partial D_{R}(z_o)}\frac{f(\zeta)}{\zeta-z_o}\sum_{n=0}^{\infty}\left(\frac{z-z_o}{\zeta-z_o}\right)^{n}\mathrm{d}\zeta\ & =\sum_{n=0}^{\infty}\underbrace{\left(\frac{1}{2\pi\mathrm{i}}\oint_{\partial D_{R}(z_o)}\frac{f(\zeta)}{(\zeta-z_o)^{n+1}}\mathrm{d}\zeta\right)}{a{n}}(z-z_o)^{n}\end{align}</math>
(see also [[Abel's Lemma]]).
=== Proof 9: Extension of the Domain to the Interior of the Annulus ===
The domains of the two functions are currently limited to the annular region.
[[File:Kreisring_r_R_Erweiterung2a.svg|350px|Holomorphic Extension 1 of Domain to Annuli by Taylor Expansion]]
We extend the function <math>f_1</math> step by step to the interior of the annulus.
=== Proof 10: Development Point Moves to the Inner Circle ===
The Taylor expansion moves along a circular path in <math>K_{r}^{R}(z_o)</math>. Using the Cauchy integral formula and the local developability in power series/Taylor series.
Thus, we can extend the function using the identity theorem to a region that is the union of the (green-marked) annular region and the (red-marked) open disk. At the same time, the holomorphic criterion is incorporated, which states that a function is holomorphic if it can be locally developed in power series within a region.
=== Proof 11: Moving Circular Regions as Convergence Region for Power Series ===
[[File:Kreisring_r_R_Erweiterung1a.svg|300px|Annulus]]
=== Proof 11: Holomorphic Extension - Disk, Annulus ===
<p>Using the [[Complex Analysis/Identity Theorem|Identity Theorem]], two holomorphic functions agree if they agree on a non-discrete set. In this case, the set is the annular region <math>G=K_{r_o}^{R_o}</math>.</p>
[[File:Kreisring r R Erweiterung3a.svg|350px|Holomorphic Extension 3]]
<p>The red annulus represents the extension with all disks and development points on the trace.</p>
=== Proof 12: Convergence Region of the Power Series ===
For the integrand <math>f_1</math>, we can again develop a Taylor series using the [[Cauchy Kernel]] and the commutability of limit processes. These developments have a disk as the region of convergence, where the series converges for all <math>z</math> inside the disk (see [[Abel's Lemma]]). The following figure shows the extension of the domain after applying the identity theorem to the disks as the region of convergence for the Taylor expansion and the intersection with the annular region <math>K_r^R(z_o)</math>. The intersection is always a non-discrete set, and the holomorphic extension to the union is uniquely determined by the identity theorem.
=== Proofs 13: Iteratively Extend the Convergence Region ===
Extend the domain to cover the entire interior of the annulus by continuing the process. Repeatedly apply the identity theorem for the holomorphic extension of <math>f_1</math>.
[[File:Kreisring r R Erweiterung4a.svg|450px|Extended Domain to Cover Interior]]
The development points of the Taylor expansions now lie along a circular integration path with a smaller radius.
=== Proof 14a: Transformation of Domain for the Main Part ===
For the main part, we replace the function <math>f_2</math> with <math>h</math> using the transformation <math>T(z):=z_o + \frac{1}{z}</math>:
:<math>h(z):=(f_2 \circ T)(z) = f_2\left(T(z)\right) = f_2\left(z_o + \frac{1}{z}\right)</math>
With this transformation, we have:
:<math>|z| < \frac{1}{r} , \Longrightarrow |T(z)-z_o| > r</math>
Thus, an analogous approach for <math>f_1</math> can also be applied to the extension of <math>h</math>. For <math>h</math>, the annular region <math>K_{R_1}^{r_1}(0)</math> with <math>0 < R_1 := \frac{1}{R} < \frac{1}{r} =: r_1</math> is considered. The function <math>h</math> can also be holomorphically extended for <math>0 \in \mathbb{C}</math>. However, <math>0 \in \mathbb{C}</math> is not defined under the transformation <math>T</math> in <math>\mathbb{C}</math>. Thus, we can extend <math>f_2</math> holomorphically to <math>K_r^{\infty}(z_o)</math>.
=== Proof 14b: Analogous Approach for the Main Part ===
Analogous to the extension of the annulus from the outer region to the interior, the annulus can also be extended to the exterior by performing the Taylor expansion for the integral <math>f_2</math> using the Cauchy Kernel. It should be noted that the development point of the Taylor expansion moves along the path of <math>\gamma_R</math>, and the region of convergence of the Taylor expansion does not cover the inner path <math>\gamma_r</math>, as the integral of <math>f_2</math> would otherwise not be defined.
== See also ==
*[[Abel's Lemma]]
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==Introduction==
[[File:Kreisring r R Definition.svg|mini|Definition domain: Annular region with radii r and R]]
Using the [[Cauchy's integral formula|Cauchy's integral formula]], two holomorphic functions, <math>f_2:\mathbb{C}\setminus D_r(z_o)\to \mathbb{C}</math> and <math>f_1:D_R(z_o)\to \mathbb{C}</math>, are defined in the decomposition theorem. These are then utilized for the expansion into a [[Laurent Series|Laurent Series]].
== Fundamental Definitions ==
The following fundamental definitions are used in the decomposition theorem:
: <math> K_r^R(z_o):= { z\in \mathbb{C} , : , 0 \leq r < |z-z_o| < R }</math>
: <math> K_r^\infty(z_o):= { z\in \mathbb{C} , : , 0 \leq r < |z-z_o| }</math>
: <math> D_r(z_o) := { z\in \mathbb{C} , : , |z-z_o| < r } </math>
: <math> \overline{D_r(z_o)} := { z\in \mathbb{C} , : , |z-z_o| \leq r } </math>
: <math> \gamma_{r,z_0}: [0,2\pi] \to \mathbb{C} </math> with <math> t \mapsto \gamma_{r,z_0}(t) = z_o +r \cdot e^{it} </math>
:<math> \int_{\partial D_r(z_o)} f(\xi), d\xi := \int_{\gamma_{r,z_0}} f(\xi), d\xi</math>
== Decomposition Theorem for Annular Regions ==
Let <math>G \subseteq \mathbb C </math> be an open set with a holomorphic function on an annular region <math>\overline{K_r^R(z_o)} = { z\in \mathbb{C} , : , r \leq |z-z_o| \leq R } \subset G</math>. Then, <math>f:K_r^R(z_o) \to \mathbb{C}</math> can be decomposed as <math>f=f_1 +f_2</math> into two holomorphic functions <math>f_1:D_R(z_o) \to \mathbb{C}</math> and <math>f_2:K_r^\infty(z_o) \to \mathbb{C}</math>. The decomposition <math>f=f_1 +f_2</math> is unique under the condition <math>\lim_{z \to \infty} f_2(z) = 0</math>.
== Proof ==
The proof considers functions on annular regions centered at <math>z_o = 0 \in \mathbb{C}</math>. By suitable composition with a shift, the decomposition theorem can be generalized for arbitrary <math>K_r^R(z_o) = { z\in \mathbb{C} , : , r \leq |z-z_o| \leq R }</math> with <math>\overline{K_r^R(z_o)} \subset G</math>. Initially, the proof idea for the decomposition theorem is discussed.
==Proof Idea==
*The center <math>z_o = 0 \in \mathbb{C}</math> of circular rings is a special case that can be used to generalize the statement for arbitrary circular rings <math>z_o \in \mathbb{C}</math>.
*Definition of a boundary cycle over a circular ring with two integration paths over an outer boundary and an inner boundary with reversed orientation.
*Application of the Cauchy Integral Theorem for cycles.
*Decomposition of the integral over a cycle into two partial integration paths along the inner and outer boundaries.
*One partial integral will provide the principal part of the Laurent expansion, while the other partial integral will yield the remainder of the integral.
==Uniqueness of the Decomposition==
The decomposition is generally not unique because the constant cannot be uniquely assigned to either the principal part or the remainder. The additional condition for the limit <math>z \to \infty</math> ensures uniqueness, as the constant is then assigned to the remainder.
==Proof 1: Circular Rings with Center 0==
We consider holomorphic functions on circular rings around the point <math>z_o \in \mathbb{C}</math>. The point <math>z_o \in \mathbb{C}</math> serves as the expansion point of the Laurent series with <math>(z-z_o)^n</math> where <math>n\in \mathbb{Z}</math>. Initially, we can restrict to circular rings around 0 (and thus around the expansion point 0), since any function <math>f:G \to \mathbb{C}</math> with <math>z_o \in \mathbb{C}</math> and a circular ring <math>\overline{K_r^R(z_o)} \subset G</math> can be transformed into a function <math>f_{z_o}:G_{z_o} \to \mathbb{C}</math> with:
: <math>G_{z_o} = { z-z_o \in \mathbb{C} , : , z \in G }</math> and <math>0 \in G_{z_o}</math>, as <math>z_o \in G</math>.
: <math>f_{z_o}:G_{z_o} \to \mathbb{C}</math> with <math> f_{z_o}(z) := f(z-z_o)</math>.
: <math>\overline{K_r^R(0)} \subset G_{z_o}</math> as <math>\overline{K_r^R(z_o)} \subset G</math>.
==Proof 2: Definition of the Boundary Cycle for the Circular Ring==
We define two integration paths along the inner and outer boundaries of the circular ring. These two paths have opposite orientations.
: <math> \gamma_1: [0,2\pi] \to \mathbb{C} </math> with <math> t \mapsto \gamma_{R,z_0}(t) = z_o + R \cdot e^{it} </math>.
: <math> \gamma_2: [0,2\pi] \to \mathbb{C} </math> with <math> t \mapsto \gamma_{r,z_0}(t) = z_o + r \cdot e^{-it} </math>.
: <math> \Gamma := \gamma_1 + \gamma_2</math>.
==Proof 3: Null-Homologous Cycle==
Since <math>\overline{K_r^R(z_o)} = { z\in \mathbb{C} , : , r \leq |z-z_o| \leq R } \subset G</math>, <math>\Gamma</math> is a null-homologous cycle, as only the inner points of the circular ring have a winding number of 1, and all inner points (including the circular ring’s boundary) belong to <math>G</math>.
==Proof 4: Application of the Cauchy Integral Formula for Cycles==
For <math>z \in K_r^R(z_o)</math>, the following holds:
:<math>n(\Gamma,z) \cdot f(z) = \frac 1{2\pi i}\int\limits_{\Gamma} \frac{f(\xi)}{\xi-z}, d\xi = \frac{1}{2\pi i}\left( \int\limits_{\gamma_1} \frac{f(\xi)}{\xi-z}, d\xi + \int\limits_{\gamma_2} \frac{f(\xi)}{\xi-z}, d\xi \right)</math>.
=== Proof 5: Substitution for Integration Paths ===
Since <math>n(\Gamma,z)=1</math> for <math>z \in K_r^R(z_o)</math>, we get the following by substituting the integration paths:
:<math>f(z) = \underbrace{\frac{1}{2\pi i} \int\limits_{\partial D_R(z_o)} \frac{f(\xi)}{\xi-z}, d\xi}{=f_1(z)} + \bigg(\underbrace{-\frac{1}{2\pi i} \int\limits{\partial D_r(z_o)} \frac{f(\xi)}{\xi-z}, d\xi}_{=f_2(z)} \bigg)</math>
The minus sign before the second integral arises due to the reversed direction of the path <math>\gamma_2</math>.
=== Proof 6: Standard Estimation ===
We consider the limit of the integral <math>|f_2(z)|</math> for <math>z\to \infty</math> with <math>C := \max\limits_{\xi \in \partial D_r(z_o)} |f(\xi)|</math>:
:<math>|f_2(z)| = \left| - \frac{1}{2\pi i} \int\limits_{\partial D_r(z_o)} \frac{f(\xi)}{\xi-z}, d\xi \right| \leq \int\limits_{\partial D_r(z_o)} \left| \frac{f(\xi)}{\xi-z} \right|, d\xi</math>
:: <math> \leq \int\limits_{\partial D_r(z_o)} \frac{C}{|\xi-z|} , d\xi \leq \underbrace{2\pi r}{L(\gamma_2)} \cdot \max\limits{\xi \in \partial D_r(z_o)} \frac{C}{|\xi-z|}</math>
Since this is an upper bound and the prefactors are all smaller than 1, they can simply be omitted.
=== Proof 7: Limit Process for Standard Estimation ===
Thus, we obtain:
:: <math>\lim_{z\to \infty} |f_2(z)| \leq \lim_{z\to \infty} \underbrace{2\pi r}{L(\gamma_2)} \cdot \max\limits{\xi\in\partial D_r(z_o)} \frac{C}{|\xi-z|} = 0</math>
=== Proof 8: Taylor Expansion with Cauchy Kernel ===
The series expansion occurs with the [[Cauchy Kernel]] in the red-marked convergence area:
:<math>\begin{align} f_1(z) & =\frac{1}{2\pi\mathrm{i}}\oint_{\partial D_{R}(z_o)}\frac{f(\zeta)}{\zeta-z}\mathrm{d}\zeta=\frac{1}{2\pi\mathrm{i}}\oint_{\partial D_{R}(z_o)}\frac{f(\zeta)}{\zeta-z_o-(z-z_o)}\mathrm{d}\zeta \\ & {=} \frac{1}{2\pi\mathrm{i}}\oint_{\partial D_{R}(z_o)}\frac{f(\zeta)}{\zeta-z_o}\cdot \frac{1}{1-\frac{z-z_o}{\zeta-z_o}}\mathrm{d}\zeta\, \\ &\overset{|\frac{z-z_o}{\zeta-z_o}|<1}{=} \frac{1}{2\pi\mathrm{i}}\oint_{\partial D_{R}(z_o)}\frac{f(\zeta)}{\zeta-z_o}\sum_{n=0}^{\infty}\left(\frac{z-z_o}{\zeta-z_o}\right)^{n}\mathrm{d}\zeta\\ & =\sum_{n=0}^{\infty}\underbrace{\left(\frac{1}{2\pi\mathrm{i}}\oint_{\partial D_{R}(z_o)}\frac{f(\zeta)}{(\zeta-z_o)^{n+1}}\mathrm{d}\zeta\right)}_{a_{n}}(z-z_o)^{n}\end{align}</math>
(see also [[Abel's Lemma]]).
=== Proof 9: Extension of the Domain to the Interior of the Annulus ===
The domains of the two functions are currently limited to the annular region.
[[File:Kreisring_r_R_Erweiterung2a.svg|350px|Holomorphic Extension 1 of Domain to Annuli by Taylor Expansion]]
We extend the function <math>f_1</math> step by step to the interior of the annulus.
=== Proof 10: Development Point Moves to the Inner Circle ===
The Taylor expansion moves along a circular path in <math>K_{r}^{R}(z_o)</math>. Using the Cauchy integral formula and the local developability in power series/Taylor series.
Thus, we can extend the function using the identity theorem to a region that is the union of the (green-marked) annular region and the (red-marked) open disk. At the same time, the holomorphic criterion is incorporated, which states that a function is holomorphic if it can be locally developed in power series within a region.
=== Proof 11: Moving Circular Regions as Convergence Region for Power Series ===
[[File:Kreisring_r_R_Erweiterung1a.svg|300px|Annulus]]
=== Proof 11: Holomorphic Extension - Disk, Annulus ===
<p>Using the [[Complex Analysis/Identity Theorem|Identity Theorem]], two holomorphic functions agree if they agree on a non-discrete set. In this case, the set is the annular region <math>G=K_{r_o}^{R_o}</math>.</p>
[[File:Kreisring r R Erweiterung3a.svg|350px|Holomorphic Extension 3]]
<p>The red annulus represents the extension with all disks and development points on the trace.</p>
=== Proof 12: Convergence Region of the Power Series ===
For the integrand <math>f_1</math>, we can again develop a Taylor series using the [[Cauchy Kernel]] and the commutability of limit processes. These developments have a disk as the region of convergence, where the series converges for all <math>z</math> inside the disk (see [[Abel's Lemma]]). The following figure shows the extension of the domain after applying the identity theorem to the disks as the region of convergence for the Taylor expansion and the intersection with the annular region <math>K_r^R(z_o)</math>. The intersection is always a non-discrete set, and the holomorphic extension to the union is uniquely determined by the identity theorem.
=== Proofs 13: Iteratively Extend the Convergence Region ===
Extend the domain to cover the entire interior of the annulus by continuing the process. Repeatedly apply the identity theorem for the holomorphic extension of <math>f_1</math>.
[[File:Kreisring r R Erweiterung4a.svg|450px|Extended Domain to Cover Interior]]
The development points of the Taylor expansions now lie along a circular integration path with a smaller radius.
=== Proof 14a: Transformation of Domain for the Main Part ===
For the main part, we replace the function <math>f_2</math> with <math>h</math> using the transformation <math>T(z):=z_o + \frac{1}{z}</math>:
:<math>h(z):=(f_2 \circ T)(z) = f_2\left(T(z)\right) = f_2\left(z_o + \frac{1}{z}\right)</math>
With this transformation, we have:
:<math>|z| < \frac{1}{r} , \Longrightarrow |T(z)-z_o| > r</math>
Thus, an analogous approach for <math>f_1</math> can also be applied to the extension of <math>h</math>. For <math>h</math>, the annular region <math>K_{R_1}^{r_1}(0)</math> with <math>0 < R_1 := \frac{1}{R} < \frac{1}{r} =: r_1</math> is considered. The function <math>h</math> can also be holomorphically extended for <math>0 \in \mathbb{C}</math>. However, <math>0 \in \mathbb{C}</math> is not defined under the transformation <math>T</math> in <math>\mathbb{C}</math>. Thus, we can extend <math>f_2</math> holomorphically to <math>K_r^{\infty}(z_o)</math>.
=== Proof 14b: Analogous Approach for the Main Part ===
Analogous to the extension of the annulus from the outer region to the interior, the annulus can also be extended to the exterior by performing the Taylor expansion for the integral <math>f_2</math> using the Cauchy Kernel. It should be noted that the development point of the Taylor expansion moves along the path of <math>\gamma_R</math>, and the region of convergence of the Taylor expansion does not cover the inner path <math>\gamma_r</math>, as the integral of <math>f_2</math> would otherwise not be defined.
== See also ==
*[[Abel's Lemma]]
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==Introduction==
[[File:Kreisring r R Definition.svg|mini|Definition domain: Annular region with radii r and R]]
Using the [[Cauchy's integral formula|Cauchy's integral formula]], two holomorphic functions, <math>f_2:\mathbb{C}\setminus D_r(z_o)\to \mathbb{C}</math> and <math>f_1:D_R(z_o)\to \mathbb{C}</math>, are defined in the decomposition theorem. These are then utilized for the expansion into a [[Laurent Series|Laurent Series]].
== Fundamental Definitions ==
The following fundamental definitions are used in the decomposition theorem:
: <math> K_r^R(z_o):= { z\in \mathbb{C} , : , 0 \leq r < |z-z_o| < R }</math>
: <math> K_r^\infty(z_o):= { z\in \mathbb{C} , : , 0 \leq r < |z-z_o| }</math>
: <math> D_r(z_o) := { z\in \mathbb{C} , : , |z-z_o| < r } </math>
: <math> \overline{D_r(z_o)} := { z\in \mathbb{C} , : , |z-z_o| \leq r } </math>
: <math> \gamma_{r,z_0}: [0,2\pi] \to \mathbb{C} </math> with <math> t \mapsto \gamma_{r,z_0}(t) = z_o +r \cdot e^{it} </math>
:<math> \int_{\partial D_r(z_o)} f(\xi), d\xi := \int_{\gamma_{r,z_0}} f(\xi), d\xi</math>
== Decomposition Theorem for Annular Regions ==
Let <math>G \subseteq \mathbb C </math> be an open set with a holomorphic function on an annular region <math>\overline{K_r^R(z_o)} = { z\in \mathbb{C} , : , r \leq |z-z_o| \leq R } \subset G</math>. Then, <math>f:K_r^R(z_o) \to \mathbb{C}</math> can be decomposed as <math>f=f_1 +f_2</math> into two holomorphic functions <math>f_1:D_R(z_o) \to \mathbb{C}</math> and <math>f_2:K_r^\infty(z_o) \to \mathbb{C}</math>. The decomposition <math>f=f_1 +f_2</math> is unique under the condition <math>\lim_{z \to \infty} f_2(z) = 0</math>.
== Proof ==
The proof considers functions on annular regions centered at <math>z_o = 0 \in \mathbb{C}</math>. By suitable composition with a shift, the decomposition theorem can be generalized for arbitrary <math>K_r^R(z_o) = { z\in \mathbb{C} , : , r \leq |z-z_o| \leq R }</math> with <math>\overline{K_r^R(z_o)} \subset G</math>. Initially, the proof idea for the decomposition theorem is discussed.
==Proof Idea==
*The center <math>z_o = 0 \in \mathbb{C}</math> of circular rings is a special case that can be used to generalize the statement for arbitrary circular rings <math>z_o \in \mathbb{C}</math>.
*Definition of a boundary cycle over a circular ring with two integration paths over an outer boundary and an inner boundary with reversed orientation.
*Application of the Cauchy Integral Theorem for cycles.
*Decomposition of the integral over a cycle into two partial integration paths along the inner and outer boundaries.
*One partial integral will provide the principal part of the Laurent expansion, while the other partial integral will yield the remainder of the integral.
==Uniqueness of the Decomposition==
The decomposition is generally not unique because the constant cannot be uniquely assigned to either the principal part or the remainder. The additional condition for the limit <math>z \to \infty</math> ensures uniqueness, as the constant is then assigned to the remainder.
==Proof 1: Circular Rings with Center 0==
We consider holomorphic functions on circular rings around the point <math>z_o \in \mathbb{C}</math>. The point <math>z_o \in \mathbb{C}</math> serves as the expansion point of the Laurent series with <math>(z-z_o)^n</math> where <math>n\in \mathbb{Z}</math>. Initially, we can restrict to circular rings around 0 (and thus around the expansion point 0), since any function <math>f:G \to \mathbb{C}</math> with <math>z_o \in \mathbb{C}</math> and a circular ring <math>\overline{K_r^R(z_o)} \subset G</math> can be transformed into a function <math>f_{z_o}:G_{z_o} \to \mathbb{C}</math> with:
: <math>G_{z_o} = { z-z_o \in \mathbb{C} , : , z \in G }</math> and <math>0 \in G_{z_o}</math>, as <math>z_o \in G</math>.
: <math>f_{z_o}:G_{z_o} \to \mathbb{C}</math> with <math> f_{z_o}(z) := f(z-z_o)</math>.
: <math>\overline{K_r^R(0)} \subset G_{z_o}</math> as <math>\overline{K_r^R(z_o)} \subset G</math>.
==Proof 2: Definition of the Boundary Cycle for the Circular Ring==
We define two integration paths along the inner and outer boundaries of the circular ring. These two paths have opposite orientations.
: <math> \gamma_1: [0,2\pi] \to \mathbb{C} </math> with <math> t \mapsto \gamma_{R,z_0}(t) = z_o + R \cdot e^{it} </math>.
: <math> \gamma_2: [0,2\pi] \to \mathbb{C} </math> with <math> t \mapsto \gamma_{r,z_0}(t) = z_o + r \cdot e^{-it} </math>.
: <math> \Gamma := \gamma_1 + \gamma_2</math>.
==Proof 3: Null-Homologous Cycle==
Since <math>\overline{K_r^R(z_o)} = { z\in \mathbb{C} , : , r \leq |z-z_o| \leq R } \subset G</math>, <math>\Gamma</math> is a null-homologous cycle, as only the inner points of the circular ring have a winding number of 1, and all inner points (including the circular ring’s boundary) belong to <math>G</math>.
==Proof 4: Application of the Cauchy Integral Formula for Cycles==
For <math>z \in K_r^R(z_o)</math>, the following holds:
:<math>n(\Gamma,z) \cdot f(z) = \frac 1{2\pi i}\int\limits_{\Gamma} \frac{f(\xi)}{\xi-z}, d\xi = \frac{1}{2\pi i}\left( \int\limits_{\gamma_1} \frac{f(\xi)}{\xi-z}, d\xi + \int\limits_{\gamma_2} \frac{f(\xi)}{\xi-z}, d\xi \right)</math>.
=== Proof 5: Substitution for Integration Paths ===
Since <math>n(\Gamma,z)=1</math> for <math>z \in K_r^R(z_o)</math>, we get the following by substituting the integration paths:
:<math>f(z) = \underbrace{\frac{1}{2\pi i} \int\limits_{\partial D_R(z_o)} \frac{f(\xi)}{\xi-z}, d\xi}{=f_1(z)} + \bigg(\underbrace{-\frac{1}{2\pi i} \int\limits{\partial D_r(z_o)} \frac{f(\xi)}{\xi-z}, d\xi}_{=f_2(z)} \bigg)</math>
The minus sign before the second integral arises due to the reversed direction of the path <math>\gamma_2</math>.
=== Proof 6: Standard Estimation ===
We consider the limit of the integral <math>|f_2(z)|</math> for <math>z\to \infty</math> with <math>C := \max\limits_{\xi \in \partial D_r(z_o)} |f(\xi)|</math>:
:<math>|f_2(z)| = \left| - \frac{1}{2\pi i} \int\limits_{\partial D_r(z_o)} \frac{f(\xi)}{\xi-z}, d\xi \right| \leq \int\limits_{\partial D_r(z_o)} \left| \frac{f(\xi)}{\xi-z} \right|, d\xi</math>
:: <math> \leq \int\limits_{\partial D_r(z_o)} \frac{C}{|\xi-z|} , d\xi \leq \underbrace{2\pi r}{L(\gamma_2)} \cdot \max\limits{\xi \in \partial D_r(z_o)} \frac{C}{|\xi-z|}</math>
Since this is an upper bound and the prefactors are all smaller than 1, they can simply be omitted.
=== Proof 7: Limit Process for Standard Estimation ===
Thus, we obtain:
:: <math>\lim_{z\to \infty} |f_2(z)| \leq \lim_{z\to \infty} \underbrace{2\pi r}{L(\gamma_2)} \cdot \max\limits{\xi\in\partial D_r(z_o)} \frac{C}{|\xi-z|} = 0</math>
=== Proof 8: Taylor Expansion with Cauchy Kernel ===
The series expansion occurs with the [[Complex Analysis/Cauchy's Integral Theorem for Disks|Cauchy's Integral Theorem for Disks]] in the red-marked convergence area:
:<math>\begin{align} f_1(z) & =\frac{1}{2\pi\mathrm{i}}\oint_{\partial D_{R}(z_o)}\frac{f(\zeta)}{\zeta-z}\mathrm{d}\zeta=\frac{1}{2\pi\mathrm{i}}\oint_{\partial D_{R}(z_o)}\frac{f(\zeta)}{\zeta-z_o-(z-z_o)}\mathrm{d}\zeta \\ & {=} \frac{1}{2\pi\mathrm{i}}\oint_{\partial D_{R}(z_o)}\frac{f(\zeta)}{\zeta-z_o}\cdot \frac{1}{1-\frac{z-z_o}{\zeta-z_o}}\mathrm{d}\zeta\, \\ &\overset{|\frac{z-z_o}{\zeta-z_o}|<1}{=} \frac{1}{2\pi\mathrm{i}}\oint_{\partial D_{R}(z_o)}\frac{f(\zeta)}{\zeta-z_o}\sum_{n=0}^{\infty}\left(\frac{z-z_o}{\zeta-z_o}\right)^{n}\mathrm{d}\zeta\\ & =\sum_{n=0}^{\infty}\underbrace{\left(\frac{1}{2\pi\mathrm{i}}\oint_{\partial D_{R}(z_o)}\frac{f(\zeta)}{(\zeta-z_o)^{n+1}}\mathrm{d}\zeta\right)}_{a_{n}}(z-z_o)^{n}\end{align}</math>
(see also [[Abel's Lemma]]).
=== Proof 9: Extension of the Domain to the Interior of the Annulus ===
The domains of the two functions are currently limited to the annular region.
[[File:Kreisring_r_R_Erweiterung2a.svg|350px|Holomorphic Extension 1 of Domain to Annuli by Taylor Expansion]]
We extend the function <math>f_1</math> step by step to the interior of the annulus.
=== Proof 10: Development Point Moves to the Inner Circle ===
The Taylor expansion moves along a circular path in <math>K_{r}^{R}(z_o)</math>. Using the Cauchy integral formula and the local developability in power series/Taylor series.
Thus, we can extend the function using the identity theorem to a region that is the union of the (green-marked) annular region and the (red-marked) open disk. At the same time, the holomorphic criterion is incorporated, which states that a function is holomorphic if it can be locally developed in power series within a region.
=== Proof 11: Moving Circular Regions as Convergence Region for Power Series ===
[[File:Kreisring_r_R_Erweiterung1a.svg|300px|Annulus]]
=== Proof 11: Holomorphic Extension - Disk, Annulus ===
<p>Using the [[Complex Analysis/Identity Theorem|Identity Theorem]], two holomorphic functions agree if they agree on a non-discrete set. In this case, the set is the annular region <math>G=K_{r_o}^{R_o}</math>.</p>
[[File:Kreisring r R Erweiterung3a.svg|350px|Holomorphic Extension 3]]
<p>The red annulus represents the extension with all disks and development points on the trace.</p>
=== Proof 12: Convergence Region of the Power Series ===
For the integrand <math>f_1</math>, we can again develop a Taylor series using the [[Complex Analysis/Cauchy's Integral Theorem for Disks|Cauchy's Integral Theorem for Disks]] and the commutability of limit processes. These developments have a disk as the region of convergence, where the series converges for all <math>z</math> inside the disk (see [[Abel's Lemma]]). The following figure shows the extension of the domain after applying the identity theorem to the disks as the region of convergence for the Taylor expansion and the intersection with the annular region <math>K_r^R(z_o)</math>. The intersection is always a non-discrete set, and the holomorphic extension to the union is uniquely determined by the identity theorem.
=== Proofs 13: Iteratively Extend the Convergence Region ===
Extend the domain to cover the entire interior of the annulus by continuing the process. Repeatedly apply the identity theorem for the holomorphic extension of <math>f_1</math>.
[[File:Kreisring r R Erweiterung4a.svg|450px|Extended Domain to Cover Interior]]
The development points of the Taylor expansions now lie along a circular integration path with a smaller radius.
=== Proof 14a: Transformation of Domain for the Main Part ===
For the main part, we replace the function <math>f_2</math> with <math>h</math> using the transformation <math>T(z):=z_o + \frac{1}{z}</math>:
:<math>h(z):=(f_2 \circ T)(z) = f_2\left(T(z)\right) = f_2\left(z_o + \frac{1}{z}\right)</math>
With this transformation, we have:
:<math>|z| < \frac{1}{r} , \Longrightarrow |T(z)-z_o| > r</math>
Thus, an analogous approach for <math>f_1</math> can also be applied to the extension of <math>h</math>. For <math>h</math>, the annular region <math>K_{R_1}^{r_1}(0)</math> with <math>0 < R_1 := \frac{1}{R} < \frac{1}{r} =: r_1</math> is considered. The function <math>h</math> can also be holomorphically extended for <math>0 \in \mathbb{C}</math>. However, <math>0 \in \mathbb{C}</math> is not defined under the transformation <math>T</math> in <math>\mathbb{C}</math>. Thus, we can extend <math>f_2</math> holomorphically to <math>K_r^{\infty}(z_o)</math>.
=== Proof 14b: Analogous Approach for the Main Part ===
Analogous to the extension of the annulus from the outer region to the interior, the annulus can also be extended to the exterior by performing the Taylor expansion for the integral <math>f_2</math> using the Cauchy Kernel. It should be noted that the development point of the Taylor expansion moves along the path of <math>\gamma_R</math>, and the region of convergence of the Taylor expansion does not cover the inner path <math>\gamma_r</math>, as the integral of <math>f_2</math> would otherwise not be defined.
== See also ==
*[[Abel's Lemma]]
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==Introduction==
[[File:Kreisring r R Definition.svg|mini|Definition domain: Annular region with radii r and R]]
Using the [[Cauchy's integral formula|Cauchy's integral formula]], two holomorphic functions, <math>f_2:\mathbb{C}\setminus D_r(z_o)\to \mathbb{C}</math> and <math>f_1:D_R(z_o)\to \mathbb{C}</math>, are defined in the decomposition theorem. These are then utilized for the expansion into a [[Laurent Series|Laurent Series]].
== Fundamental Definitions ==
The following fundamental definitions are used in the decomposition theorem:
: <math> K_r^R(z_o):= { z\in \mathbb{C} , : , 0 \leq r < |z-z_o| < R }</math>
: <math> K_r^\infty(z_o):= { z\in \mathbb{C} , : , 0 \leq r < |z-z_o| }</math>
: <math> D_r(z_o) := { z\in \mathbb{C} , : , |z-z_o| < r } </math>
: <math> \overline{D_r(z_o)} := { z\in \mathbb{C} , : , |z-z_o| \leq r } </math>
: <math> \gamma_{r,z_0}: [0,2\pi] \to \mathbb{C} </math> with <math> t \mapsto \gamma_{r,z_0}(t) = z_o +r \cdot e^{it} </math>
:<math> \int_{\partial D_r(z_o)} f(\xi), d\xi := \int_{\gamma_{r,z_0}} f(\xi), d\xi</math>
== Decomposition Theorem for Annular Regions ==
Let <math>G \subseteq \mathbb C </math> be an open set with a holomorphic function on an annular region <math>\overline{K_r^R(z_o)} = { z\in \mathbb{C} , : , r \leq |z-z_o| \leq R } \subset G</math>. Then, <math>f:K_r^R(z_o) \to \mathbb{C}</math> can be decomposed as <math>f=f_1 +f_2</math> into two holomorphic functions <math>f_1:D_R(z_o) \to \mathbb{C}</math> and <math>f_2:K_r^\infty(z_o) \to \mathbb{C}</math>. The decomposition <math>f=f_1 +f_2</math> is unique under the condition <math>\lim_{z \to \infty} f_2(z) = 0</math>.
== Proof ==
The proof considers functions on annular regions centered at <math>z_o = 0 \in \mathbb{C}</math>. By suitable composition with a shift, the decomposition theorem can be generalized for arbitrary <math>K_r^R(z_o) = { z\in \mathbb{C} , : , r \leq |z-z_o| \leq R }</math> with <math>\overline{K_r^R(z_o)} \subset G</math>. Initially, the proof idea for the decomposition theorem is discussed.
==Proof Idea==
*The center <math>z_o = 0 \in \mathbb{C}</math> of circular rings is a special case that can be used to generalize the statement for arbitrary circular rings <math>z_o \in \mathbb{C}</math>.
*Definition of a boundary cycle over a circular ring with two integration paths over an outer boundary and an inner boundary with reversed orientation.
*Application of the Cauchy Integral Theorem for cycles.
*Decomposition of the integral over a cycle into two partial integration paths along the inner and outer boundaries.
*One partial integral will provide the principal part of the Laurent expansion, while the other partial integral will yield the remainder of the integral.
==Uniqueness of the Decomposition==
The decomposition is generally not unique because the constant cannot be uniquely assigned to either the principal part or the remainder. The additional condition for the limit <math>z \to \infty</math> ensures uniqueness, as the constant is then assigned to the remainder.
==Proof 1: Circular Rings with Center 0==
We consider holomorphic functions on circular rings around the point <math>z_o \in \mathbb{C}</math>. The point <math>z_o \in \mathbb{C}</math> serves as the expansion point of the Laurent series with <math>(z-z_o)^n</math> where <math>n\in \mathbb{Z}</math>. Initially, we can restrict to circular rings around 0 (and thus around the expansion point 0), since any function <math>f:G \to \mathbb{C}</math> with <math>z_o \in \mathbb{C}</math> and a circular ring <math>\overline{K_r^R(z_o)} \subset G</math> can be transformed into a function <math>f_{z_o}:G_{z_o} \to \mathbb{C}</math> with:
: <math>G_{z_o} = { z-z_o \in \mathbb{C} , : , z \in G }</math> and <math>0 \in G_{z_o}</math>, as <math>z_o \in G</math>.
: <math>f_{z_o}:G_{z_o} \to \mathbb{C}</math> with <math> f_{z_o}(z) := f(z-z_o)</math>.
: <math>\overline{K_r^R(0)} \subset G_{z_o}</math> as <math>\overline{K_r^R(z_o)} \subset G</math>.
==Proof 2: Definition of the Boundary Cycle for the Circular Ring==
We define two integration paths along the inner and outer boundaries of the circular ring. These two paths have opposite orientations.
: <math> \gamma_1: [0,2\pi] \to \mathbb{C} </math> with <math> t \mapsto \gamma_{R,z_0}(t) = z_o + R \cdot e^{it} </math>.
: <math> \gamma_2: [0,2\pi] \to \mathbb{C} </math> with <math> t \mapsto \gamma_{r,z_0}(t) = z_o + r \cdot e^{-it} </math>.
: <math> \Gamma := \gamma_1 + \gamma_2</math>.
==Proof 3: Null-Homologous Cycle==
Since <math>\overline{K_r^R(z_o)} = { z\in \mathbb{C} , : , r \leq |z-z_o| \leq R } \subset G</math>, <math>\Gamma</math> is a null-homologous cycle, as only the inner points of the circular ring have a winding number of 1, and all inner points (including the circular ring’s boundary) belong to <math>G</math>.
==Proof 4: Application of the Cauchy Integral Formula for Cycles==
For <math>z \in K_r^R(z_o)</math>, the following holds:
:<math>n(\Gamma,z) \cdot f(z) = \frac 1{2\pi i}\int\limits_{\Gamma} \frac{f(\xi)}{\xi-z}, d\xi = \frac{1}{2\pi i}\left( \int\limits_{\gamma_1} \frac{f(\xi)}{\xi-z}, d\xi + \int\limits_{\gamma_2} \frac{f(\xi)}{\xi-z}, d\xi \right)</math>.
=== Proof 5: Substitution for Integration Paths ===
Since <math>n(\Gamma,z)=1</math> for <math>z \in K_r^R(z_o)</math>, we get the following by substituting the integration paths:
:<math>f(z) = \underbrace{\frac{1}{2\pi i} \int\limits_{\partial D_R(z_o)} \frac{f(\xi)}{\xi-z}, d\xi}{=f_1(z)} + \bigg(\underbrace{-\frac{1}{2\pi i} \int\limits{\partial D_r(z_o)} \frac{f(\xi)}{\xi-z}, d\xi}_{=f_2(z)} \bigg)</math>
The minus sign before the second integral arises due to the reversed direction of the path <math>\gamma_2</math>.
=== Proof 6: Standard Estimation ===
We consider the limit of the integral <math>|f_2(z)|</math> for <math>z\to \infty</math> with <math>C := \max\limits_{\xi \in \partial D_r(z_o)} |f(\xi)|</math>:
:<math>|f_2(z)| = \left| - \frac{1}{2\pi i} \int\limits_{\partial D_r(z_o)} \frac{f(\xi)}{\xi-z}, d\xi \right| \leq \int\limits_{\partial D_r(z_o)} \left| \frac{f(\xi)}{\xi-z} \right|, d\xi</math>
:: <math> \leq \int\limits_{\partial D_r(z_o)} \frac{C}{|\xi-z|} , d\xi \leq \underbrace{2\pi r}{L(\gamma_2)} \cdot \max\limits{\xi \in \partial D_r(z_o)} \frac{C}{|\xi-z|}</math>
Since this is an upper bound and the prefactors are all smaller than 1, they can simply be omitted.
=== Proof 7: Limit Process for Standard Estimation ===
Thus, we obtain:
:: <math>\lim_{z\to \infty} |f_2(z)| \leq \lim_{z\to \infty} \underbrace{2\pi r}{L(\gamma_2)} \cdot \max\limits{\xi\in\partial D_r(z_o)} \frac{C}{|\xi-z|} = 0</math>
=== Proof 8: Taylor Expansion with Cauchy Kernel ===
The series expansion occurs with the [[Complex Analysis/Cauchy's Integral Theorem for Disks|Cauchy's Integral Theorem for Disks]] in the red-marked convergence area:
:<math>\begin{align} f_1(z) & =\frac{1}{2\pi\mathrm{i}}\oint_{\partial D_{R}(z_o)}\frac{f(\zeta)}{\zeta-z}\mathrm{d}\zeta=\frac{1}{2\pi\mathrm{i}}\oint_{\partial D_{R}(z_o)}\frac{f(\zeta)}{\zeta-z_o-(z-z_o)}\mathrm{d}\zeta \\ & {=} \frac{1}{2\pi\mathrm{i}}\oint_{\partial D_{R}(z_o)}\frac{f(\zeta)}{\zeta-z_o}\cdot \frac{1}{1-\frac{z-z_o}{\zeta-z_o}}\mathrm{d}\zeta\, \\ &\overset{|\frac{z-z_o}{\zeta-z_o}|<1}{=} \frac{1}{2\pi\mathrm{i}}\oint_{\partial D_{R}(z_o)}\frac{f(\zeta)}{\zeta-z_o}\sum_{n=0}^{\infty}\left(\frac{z-z_o}{\zeta-z_o}\right)^{n}\mathrm{d}\zeta\\ & =\sum_{n=0}^{\infty}\underbrace{\left(\frac{1}{2\pi\mathrm{i}}\oint_{\partial D_{R}(z_o)}\frac{f(\zeta)}{(\zeta-z_o)^{n+1}}\mathrm{d}\zeta\right)}_{a_{n}}(z-z_o)^{n}\end{align}</math>
(see also [[Complex Analysis/Abel's Lemma|Abel's Lemma]]).
=== Proof 9: Extension of the Domain to the Interior of the Annulus ===
The domains of the two functions are currently limited to the annular region.
[[File:Kreisring_r_R_Erweiterung2a.svg|350px|Holomorphic Extension 1 of Domain to Annuli by Taylor Expansion]]
We extend the function <math>f_1</math> step by step to the interior of the annulus.
=== Proof 10: Development Point Moves to the Inner Circle ===
The Taylor expansion moves along a circular path in <math>K_{r}^{R}(z_o)</math>. Using the Cauchy integral formula and the local developability in power series/Taylor series.
Thus, we can extend the function using the identity theorem to a region that is the union of the (green-marked) annular region and the (red-marked) open disk. At the same time, the holomorphic criterion is incorporated, which states that a function is holomorphic if it can be locally developed in power series within a region.
=== Proof 11: Moving Circular Regions as Convergence Region for Power Series ===
[[File:Kreisring_r_R_Erweiterung1a.svg|300px|Annulus]]
=== Proof 11: Holomorphic Extension - Disk, Annulus ===
<p>Using the [[Complex Analysis/Identity Theorem|Identity Theorem]], two holomorphic functions agree if they agree on a non-discrete set. In this case, the set is the annular region <math>G=K_{r_o}^{R_o}</math>.</p>
[[File:Kreisring r R Erweiterung3a.svg|350px|Holomorphic Extension 3]]
<p>The red annulus represents the extension with all disks and development points on the trace.</p>
=== Proof 12: Convergence Region of the Power Series ===
For the integrand <math>f_1</math>, we can again develop a Taylor series using the [[Complex Analysis/Cauchy's Integral Theorem for Disks|Cauchy's Integral Theorem for Disks]] and the commutability of limit processes. These developments have a disk as the region of convergence, where the series converges for all <math>z</math> inside the disk (see [[Complex Analysis/Abel's Lemma|Abel's Lemma]]). The following figure shows the extension of the domain after applying the identity theorem to the disks as the region of convergence for the Taylor expansion and the intersection with the annular region <math>K_r^R(z_o)</math>. The intersection is always a non-discrete set, and the holomorphic extension to the union is uniquely determined by the identity theorem.
=== Proofs 13: Iteratively Extend the Convergence Region ===
Extend the domain to cover the entire interior of the annulus by continuing the process. Repeatedly apply the identity theorem for the holomorphic extension of <math>f_1</math>.
[[File:Kreisring r R Erweiterung4a.svg|450px|Extended Domain to Cover Interior]]
The development points of the Taylor expansions now lie along a circular integration path with a smaller radius.
=== Proof 14a: Transformation of Domain for the Main Part ===
For the main part, we replace the function <math>f_2</math> with <math>h</math> using the transformation <math>T(z):=z_o + \frac{1}{z}</math>:
:<math>h(z):=(f_2 \circ T)(z) = f_2\left(T(z)\right) = f_2\left(z_o + \frac{1}{z}\right)</math>
With this transformation, we have:
:<math>|z| < \frac{1}{r} , \Longrightarrow |T(z)-z_o| > r</math>
Thus, an analogous approach for <math>f_1</math> can also be applied to the extension of <math>h</math>. For <math>h</math>, the annular region <math>K_{R_1}^{r_1}(0)</math> with <math>0 < R_1 := \frac{1}{R} < \frac{1}{r} =: r_1</math> is considered. The function <math>h</math> can also be holomorphically extended for <math>0 \in \mathbb{C}</math>. However, <math>0 \in \mathbb{C}</math> is not defined under the transformation <math>T</math> in <math>\mathbb{C}</math>. Thus, we can extend <math>f_2</math> holomorphically to <math>K_r^{\infty}(z_o)</math>.
=== Proof 14b: Analogous Approach for the Main Part ===
Analogous to the extension of the annulus from the outer region to the interior, the annulus can also be extended to the exterior by performing the Taylor expansion for the integral <math>f_2</math> using the Cauchy Kernel. It should be noted that the development point of the Taylor expansion moves along the path of <math>\gamma_R</math>, and the region of convergence of the Taylor expansion does not cover the inner path <math>\gamma_r</math>, as the integral of <math>f_2</math> would otherwise not be defined.
== See also ==
*[[Abel's Lemma]]
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==Introduction==
[[File:Kreisring r R Definition.svg|mini|Definition domain: Annular region with radii r and R]]
Using the [[Cauchy's integral formula|Cauchy's integral formula]], two holomorphic functions, <math>f_2:\mathbb{C}\setminus D_r(z_o)\to \mathbb{C}</math> and <math>f_1:D_R(z_o)\to \mathbb{C}</math>, are defined in the decomposition theorem. These are then utilized for the expansion into a [[Laurent Series|Laurent Series]].
== Fundamental Definitions ==
The following fundamental definitions are used in the decomposition theorem:
: <math> K_r^R(z_o):= { z\in \mathbb{C} , : , 0 \leq r < |z-z_o| < R }</math>
: <math> K_r^\infty(z_o):= { z\in \mathbb{C} , : , 0 \leq r < |z-z_o| }</math>
: <math> D_r(z_o) := { z\in \mathbb{C} , : , |z-z_o| < r } </math>
: <math> \overline{D_r(z_o)} := { z\in \mathbb{C} , : , |z-z_o| \leq r } </math>
: <math> \gamma_{r,z_0}: [0,2\pi] \to \mathbb{C} </math> with <math> t \mapsto \gamma_{r,z_0}(t) = z_o +r \cdot e^{it} </math>
:<math> \int_{\partial D_r(z_o)} f(\xi), d\xi := \int_{\gamma_{r,z_0}} f(\xi), d\xi</math>
== Decomposition Theorem for Annular Regions ==
Let <math>G \subseteq \mathbb C </math> be an open set with a holomorphic function on an annular region <math>\overline{K_r^R(z_o)} = { z\in \mathbb{C} , : , r \leq |z-z_o| \leq R } \subset G</math>. Then, <math>f:K_r^R(z_o) \to \mathbb{C}</math> can be decomposed as <math>f=f_1 +f_2</math> into two holomorphic functions <math>f_1:D_R(z_o) \to \mathbb{C}</math> and <math>f_2:K_r^\infty(z_o) \to \mathbb{C}</math>. The decomposition <math>f=f_1 +f_2</math> is unique under the condition <math>\lim_{z \to \infty} f_2(z) = 0</math>.
== Proof ==
The proof considers functions on annular regions centered at <math>z_o = 0 \in \mathbb{C}</math>. By suitable composition with a shift, the decomposition theorem can be generalized for arbitrary <math>K_r^R(z_o) = { z\in \mathbb{C} , : , r \leq |z-z_o| \leq R }</math> with <math>\overline{K_r^R(z_o)} \subset G</math>. Initially, the proof idea for the decomposition theorem is discussed.
==Proof Idea==
*The center <math>z_o = 0 \in \mathbb{C}</math> of circular rings is a special case that can be used to generalize the statement for arbitrary circular rings <math>z_o \in \mathbb{C}</math>.
*Definition of a boundary cycle over a circular ring with two integration paths over an outer boundary and an inner boundary with reversed orientation.
*Application of the Cauchy Integral Theorem for cycles.
*Decomposition of the integral over a cycle into two partial integration paths along the inner and outer boundaries.
*One partial integral will provide the principal part of the Laurent expansion, while the other partial integral will yield the remainder of the integral.
==Uniqueness of the Decomposition==
The decomposition is generally not unique because the constant cannot be uniquely assigned to either the principal part or the remainder. The additional condition for the limit <math>z \to \infty</math> ensures uniqueness, as the constant is then assigned to the remainder.
==Proof 1: Circular Rings with Center 0==
We consider holomorphic functions on circular rings around the point <math>z_o \in \mathbb{C}</math>. The point <math>z_o \in \mathbb{C}</math> serves as the expansion point of the Laurent series with <math>(z-z_o)^n</math> where <math>n\in \mathbb{Z}</math>. Initially, we can restrict to circular rings around 0 (and thus around the expansion point 0), since any function <math>f:G \to \mathbb{C}</math> with <math>z_o \in \mathbb{C}</math> and a circular ring <math>\overline{K_r^R(z_o)} \subset G</math> can be transformed into a function <math>f_{z_o}:G_{z_o} \to \mathbb{C}</math> with:
: <math>G_{z_o} = { z-z_o \in \mathbb{C} , : , z \in G }</math> and <math>0 \in G_{z_o}</math>, as <math>z_o \in G</math>.
: <math>f_{z_o}:G_{z_o} \to \mathbb{C}</math> with <math> f_{z_o}(z) := f(z-z_o)</math>.
: <math>\overline{K_r^R(0)} \subset G_{z_o}</math> as <math>\overline{K_r^R(z_o)} \subset G</math>.
==Proof 2: Definition of the Boundary Cycle for the Circular Ring==
We define two integration paths along the inner and outer boundaries of the circular ring. These two paths have opposite orientations.
: <math> \gamma_1: [0,2\pi] \to \mathbb{C} </math> with <math> t \mapsto \gamma_{R,z_0}(t) = z_o + R \cdot e^{it} </math>.
: <math> \gamma_2: [0,2\pi] \to \mathbb{C} </math> with <math> t \mapsto \gamma_{r,z_0}(t) = z_o + r \cdot e^{-it} </math>.
: <math> \Gamma := \gamma_1 + \gamma_2</math>.
==Proof 3: Null-Homologous Cycle==
Since <math>\overline{K_r^R(z_o)} = { z\in \mathbb{C} , : , r \leq |z-z_o| \leq R } \subset G</math>, <math>\Gamma</math> is a null-homologous cycle, as only the inner points of the circular ring have a winding number of 1, and all inner points (including the circular ring’s boundary) belong to <math>G</math>.
==Proof 4: Application of the Cauchy Integral Formula for Cycles==
For <math>z \in K_r^R(z_o)</math>, the following holds:
:<math>n(\Gamma,z) \cdot f(z) = \frac 1{2\pi i}\int\limits_{\Gamma} \frac{f(\xi)}{\xi-z}, d\xi = \frac{1}{2\pi i}\left( \int\limits_{\gamma_1} \frac{f(\xi)}{\xi-z}, d\xi + \int\limits_{\gamma_2} \frac{f(\xi)}{\xi-z}, d\xi \right)</math>.
=== Proof 5: Substitution for Integration Paths ===
Since <math>n(\Gamma,z)=1</math> for <math>z \in K_r^R(z_o)</math>, we get the following by substituting the integration paths:
:<math>f(z) = \underbrace{\frac{1}{2\pi i} \int\limits_{\partial D_R(z_o)} \frac{f(\xi)}{\xi-z}, d\xi}{=f_1(z)} + \bigg(\underbrace{-\frac{1}{2\pi i} \int\limits{\partial D_r(z_o)} \frac{f(\xi)}{\xi-z}, d\xi}_{=f_2(z)} \bigg)</math>
The minus sign before the second integral arises due to the reversed direction of the path <math>\gamma_2</math>.
=== Proof 6: Standard Estimation ===
We consider the limit of the integral <math>|f_2(z)|</math> for <math>z\to \infty</math> with <math>C := \max\limits_{\xi \in \partial D_r(z_o)} |f(\xi)|</math>:
:<math>|f_2(z)| = \left| - \frac{1}{2\pi i} \int\limits_{\partial D_r(z_o)} \frac{f(\xi)}{\xi-z}, d\xi \right| \leq \int\limits_{\partial D_r(z_o)} \left| \frac{f(\xi)}{\xi-z} \right|, d\xi</math>
:: <math> \leq \int\limits_{\partial D_r(z_o)} \frac{C}{|\xi-z|} , d\xi \leq \underbrace{2\pi r}{L(\gamma_2)} \cdot \max\limits{\xi \in \partial D_r(z_o)} \frac{C}{|\xi-z|}</math>
Since this is an upper bound and the prefactors are all smaller than 1, they can simply be omitted.
=== Proof 7: Limit Process for Standard Estimation ===
Thus, we obtain:
:: <math>\lim_{z\to \infty} |f_2(z)| \leq \lim_{z\to \infty} \underbrace{2\pi r}{L(\gamma_2)} \cdot \max\limits{\xi\in\partial D_r(z_o)} \frac{C}{|\xi-z|} = 0</math>
=== Proof 8: Taylor Expansion with Cauchy Kernel ===
The series expansion occurs with the [[Complex Analysis/Cauchy's Integral Theorem for Disks|Cauchy's Integral Theorem for Disks]] in the red-marked convergence area:
:<math>\begin{align} f_1(z) & =\frac{1}{2\pi\mathrm{i}}\oint_{\partial D_{R}(z_o)}\frac{f(\zeta)}{\zeta-z}\mathrm{d}\zeta=\frac{1}{2\pi\mathrm{i}}\oint_{\partial D_{R}(z_o)}\frac{f(\zeta)}{\zeta-z_o-(z-z_o)}\mathrm{d}\zeta \\ & {=} \frac{1}{2\pi\mathrm{i}}\oint_{\partial D_{R}(z_o)}\frac{f(\zeta)}{\zeta-z_o}\cdot \frac{1}{1-\frac{z-z_o}{\zeta-z_o}}\mathrm{d}\zeta\, \\ &\overset{|\frac{z-z_o}{\zeta-z_o}|<1}{=} \frac{1}{2\pi\mathrm{i}}\oint_{\partial D_{R}(z_o)}\frac{f(\zeta)}{\zeta-z_o}\sum_{n=0}^{\infty}\left(\frac{z-z_o}{\zeta-z_o}\right)^{n}\mathrm{d}\zeta\\ & =\sum_{n=0}^{\infty}\underbrace{\left(\frac{1}{2\pi\mathrm{i}}\oint_{\partial D_{R}(z_o)}\frac{f(\zeta)}{(\zeta-z_o)^{n+1}}\mathrm{d}\zeta\right)}_{a_{n}}(z-z_o)^{n}\end{align}</math>
(see also [[Abel's Lemma]]).
=== Proof 9: Extension of the Domain to the Interior of the Annulus ===
The domains of the two functions are currently limited to the annular region.
[[File:Kreisring_r_R_Erweiterung2a.svg|350px|Holomorphic Extension 1 of Domain to Annuli by Taylor Expansion]]
We extend the function <math>f_1</math> step by step to the interior of the annulus.
=== Proof 10: Development Point Moves to the Inner Circle ===
The Taylor expansion moves along a circular path in <math>K_{r}^{R}(z_o)</math>. Using the Cauchy integral formula and the local developability in power series/Taylor series.
Thus, we can extend the function using the identity theorem to a region that is the union of the (green-marked) annular region and the (red-marked) open disk. At the same time, the holomorphic criterion is incorporated, which states that a function is holomorphic if it can be locally developed in power series within a region.
=== Proof 11: Moving Circular Regions as Convergence Region for Power Series ===
[[File:Kreisring_r_R_Erweiterung1a.svg|300px|Annulus]]
=== Proof 11: Holomorphic Extension - Disk, Annulus ===
<p>Using the [[Complex Analysis/Identity Theorem|Identity Theorem]], two holomorphic functions agree if they agree on a non-discrete set. In this case, the set is the annular region <math>G=K_{r_o}^{R_o}</math>.</p>
[[File:Kreisring r R Erweiterung3a.svg|350px|Holomorphic Extension 3]]
<p>The red annulus represents the extension with all disks and development points on the trace.</p>
=== Proof 12: Convergence Region of the Power Series ===
For the integrand <math>f_1</math>, we can again develop a Taylor series using the [[Complex Analysis/Cauchy's Integral Theorem for Disks|Cauchy's Integral Theorem for Disks]] and the commutability of limit processes. These developments have a disk as the region of convergence, where the series converges for all <math>z</math> inside the disk (see [[Abel's Lemma]]). The following figure shows the extension of the domain after applying the identity theorem to the disks as the region of convergence for the Taylor expansion and the intersection with the annular region <math>K_r^R(z_o)</math>. The intersection is always a non-discrete set, and the holomorphic extension to the union is uniquely determined by the identity theorem.
=== Proofs 13: Iteratively Extend the Convergence Region ===
Extend the domain to cover the entire interior of the annulus by continuing the process. Repeatedly apply the identity theorem for the holomorphic extension of <math>f_1</math>.
[[File:Kreisring r R Erweiterung4a.svg|450px|Extended Domain to Cover Interior]]
The development points of the Taylor expansions now lie along a circular integration path with a smaller radius.
=== Proof 14a: Transformation of Domain for the Main Part ===
For the main part, we replace the function <math>f_2</math> with <math>h</math> using the transformation <math>T(z):=z_o + \frac{1}{z}</math>:
:<math>h(z):=(f_2 \circ T)(z) = f_2\left(T(z)\right) = f_2\left(z_o + \frac{1}{z}\right)</math>
With this transformation, we have:
:<math>|z| < \frac{1}{r} , \Longrightarrow |T(z)-z_o| > r</math>
Thus, an analogous approach for <math>f_1</math> can also be applied to the extension of <math>h</math>. For <math>h</math>, the annular region <math>K_{R_1}^{r_1}(0)</math> with <math>0 < R_1 := \frac{1}{R} < \frac{1}{r} =: r_1</math> is considered. The function <math>h</math> can also be holomorphically extended for <math>0 \in \mathbb{C}</math>. However, <math>0 \in \mathbb{C}</math> is not defined under the transformation <math>T</math> in <math>\mathbb{C}</math>. Thus, we can extend <math>f_2</math> holomorphically to <math>K_r^{\infty}(z_o)</math>.
=== Proof 14b: Analogous Approach for the Main Part ===
Analogous to the extension of the annulus from the outer region to the interior, the annulus can also be extended to the exterior by performing the Taylor expansion for the integral <math>f_2</math> using the Cauchy Kernel. It should be noted that the development point of the Taylor expansion moves along the path of <math>\gamma_R</math>, and the region of convergence of the Taylor expansion does not cover the inner path <math>\gamma_r</math>, as the integral of <math>f_2</math> would otherwise not be defined.
== See also ==
*[[Abel's Lemma]]
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The '''Abelian Lemma '''is a [[w:en:Lemma (mathematics)|Lemma (mathematics)]] used to investigate the [[w:en:convergence|convergence]] region of [[w:en:power series|power series]]. It is named after [[w:en:Niels Henrik Abel|Niels Henrik Abel]].
==Abel's Lemma==
Let <math>K_P := { z \in \mathbb{C} : P(z) = \sum_{k=0}^\infty a_k (z - z_0)^k \text{ converges} } \subseteq \mathbb{C}</math> be the region of convergence of the power series <math>P</math> given by: <math>P(z) = \sum_{k=0}^\infty a_k (z - z_0)^k</math>, then the following statements hold:
*For a given element <math>z_1 \in K_P</math> from the convergence region of <math>P</math>, the series <math>P(z)</math> converges absolutely for all <math>z \in \mathbb{C}</math> such that <math>|z - z_0| < |z_1 - z_0|</math>.
*For a given element <math>z_2 \notin K_P</math> where <math>P</math> diverges, all <math>z \in \mathbb{C}</math> with <math>|z_2 - z_0| < |z - z_0|</math> also cause <math>P(z)</math> to diverge.
==Task for Learners==
*Prove the statement of Abel's Lemma by utilizing the fact that a convergent series ([[w:en:in absolute terms|in absolute terms]]) has bounded coefficients. Then, use the majorant criterion and a geometric series as a majorant to show that <math>P</math> converges [[w:en:absolutely|absolutely]].
*Justify why the convergence region <math>K_P</math> contains an open disk <math>D_r(z_0) \subseteq { z \in \mathbb{C} : |z_0 - z| < r }</math> (where <math>r > 0</math> is maximally chosen), and why <math>P</math> diverges for all <math>z \in \mathbb{C}</math> with <math>r < |z - z_0|</math> when <math>P</math> diverges.
*Determine the radius of convergence <math>r > 0</math> for the following power series, and on the boundary <math>\partial D_{r}(0)</math> of the convergence region, identify two points <math>z_1, z_2 \in \partial D_{r}(0)</math>, such that <math>P(z_1)</math> converges and <math>P(z_2)</math> diverges. <math>P(z) = \sum_{k=1}^\infty \frac{1}{k} \cdot z^k</math> Use your knowledge of the harmonic series to choose the points <math>z_1, z_2 \in \partial D_{r}(0)</math>.
([[Complex Analysis/decomposition theorem|decomposition theorem]]) Analyze the [[Complex Analysis/decomposition theorem|decomposition theorem]] and explain how Abel's Lemma contributes to the extension of the domain to a ring and the use of the Identity Theorem.
==Consequence==
Taking into account that the series must always diverge at points <math>z \in \mathbb{C}</math> where the sequence of its summands is unbounded (by the [[w:en:Cauchy Criterion for Series|Cauchy Criterion for Series]], it follows from the lemma that every power series has a well-defined [[w:en:Radius of convergence|radius of convergence]] and converges uniformly on any [[w:en:Compact space|Compact space]] within the convergence disk. Outside the convergence disk, it diverges. No statement is made about the convergence for points on the boundary of the convergence disk.
==See also==
*[[Complex Analysis]]
*[[Complex Analysis/decomposition theorem|decomposition theorem]]
==Source==
Eberhard Freitag & Rolf Busam: Function Theory 1, Springer-Verlag, Berlin, ISBN 3-540-67641-4, p. 98
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The '''Abelian Lemma '''is a [[w:en:Lemma (mathematics)|Lemma (mathematics)]] used to investigate the [[w:en:convergence|convergence]] region of [[w:en:power series|power series]]. It is named after [[w:en:Niels Henrik Abel|Niels Henrik Abel]].
==Abel's Lemma==
Let <math>K_P := { z \in \mathbb{C} : P(z) = \sum_{k=0}^\infty a_k (z - z_0)^k \text{ converges} } \subseteq \mathbb{C}</math> be the region of convergence of the power series <math>P</math> given by: <math>P(z) = \sum_{k=0}^\infty a_k (z - z_0)^k</math>, then the following statements hold:
*For a given element <math>z_1 \in K_P</math> from the convergence region of <math>P</math>, the series <math>P(z)</math> converges absolutely for all <math>z \in \mathbb{C}</math> such that <math>|z - z_0| < |z_1 - z_0|</math>.
*For a given element <math>z_2 \notin K_P</math> where <math>P</math> diverges, all <math>z \in \mathbb{C}</math> with <math>|z_2 - z_0| < |z - z_0|</math> also cause <math>P(z)</math> to diverge.
==Task for Learners==
*Prove the statement of Abel's Lemma by utilizing the fact that a convergent series ([[w:en:in absolute terms|in absolute terms]]) has bounded coefficients. Then, use the majorant criterion and a geometric series as a majorant to show that <math>P</math> converges [[w:en:absolutely|absolutely]].
*Justify why the convergence region <math>K_P</math> contains an open disk <math>D_r(z_0) \subseteq { z \in \mathbb{C} : |z_0 - z| < r }</math> (where <math>r > 0</math> is maximally chosen), and why <math>P</math> diverges for all <math>z \in \mathbb{C}</math> with <math>r < |z - z_0|</math> when <math>P</math> diverges.
*Determine the radius of convergence <math>r > 0</math> for the following power series, and on the boundary <math>\partial D_{r}(0)</math> of the convergence region, identify two points <math>z_1, z_2 \in \partial D_{r}(0)</math>, such that <math>P(z_1)</math> converges and <math>P(z_2)</math> diverges. <math>P(z) = \sum_{k=1}^\infty \frac{1}{k} \cdot z^k</math> Use your knowledge of the harmonic series to choose the points <math>z_1, z_2 \in \partial D_{r}(0)</math>.
([[Complex Analysis/decomposition theorem|decomposition theorem]]) Analyze the [[Complex Analysis/decomposition theorem|decomposition theorem]] and explain how Abel's Lemma contributes to the extension of the domain to a ring and the use of the Identity Theorem.
==Consequence==
Taking into account that the series must always diverge at points <math>z \in \mathbb{C}</math> where the sequence of its summands is unbounded (by the [[w:en:Cauchy Criterion for Series|Cauchy Criterion for Series]], it follows from the lemma that every power series has a well-defined [[w:en:Radius of convergence|radius of convergence]] and converges uniformly on any [[w:en:Compact space|Compact space]] within the convergence disk. Outside the convergence disk, it diverges. No statement is made about the convergence for points on the boundary of the convergence disk.
==See also==
*[[Complex Analysis]]
*[[Complex Analysis/decomposition theorem|decomposition theorem]]
==Source==
Eberhard Freitag & Rolf Busam: Function Theory 1, Springer-Verlag, Berlin, ISBN 3-540-67641-4, p. 98
== Page Information ==
You can display this page as '''[https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Complex%20Analysis/Abel's%20Lemma&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Abel's%20Lemma&coursetitle=Complex%20Analysis Wiki2Reveal slides]'''
=== Wiki2Reveal ===
The'''[https://niebert.github.io/Wiki2Reveal/wiki2reveal.htmldomain=wikiversity&title=Complex%20Analysis/Abel's%20Lemma&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Abel's%20Lemma&coursetitle=Complex%20Analysis Wiki2Reveal slides]''' were created for the '''[https://en.wikiversity.org/wiki/Complex%20Analysis Complex Analysis]'''' and the Link for the [[v:en:Wiki2Reveal|Wiki2Reveal Slides]] was created with the [https://niebert.github.io/Wiki2Reveal/ link generator].
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=== Translation and Version Control ===
This page was translated based on the following [https://de.wikiversity.org/wiki/Abelsches Lemma Wikiversity source page] and uses the concept of [[Translation and Version Control]] for a transparent language fork in a Wikiversity:
* Source: [[v:de:Abelsches Lemma|Abelsches Lemma]] - URL:
https://de.wikiversity.org/wiki/Abelsches Lemma
* Date: 1/2/2025
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[[de:Abelsches Lemma]]
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The '''Abelian Lemma '''is a [[w:en:Lemma (mathematics)|Lemma (mathematics)]] used to investigate the [[w:en:convergence series|convergence series]] region of [[w:en:power series|power series]]. It is named after [[w:en:Niels Henrik Abel|Niels Henrik Abel]].
==Abel's Lemma==
Let <math>K_P := { z \in \mathbb{C} : P(z) = \sum_{k=0}^\infty a_k (z - z_0)^k \text{ converges} } \subseteq \mathbb{C}</math> be the region of convergence of the power series <math>P</math> given by: <math>P(z) = \sum_{k=0}^\infty a_k (z - z_0)^k</math>, then the following statements hold:
*For a given element <math>z_1 \in K_P</math> from the convergence region of <math>P</math>, the series <math>P(z)</math> converges absolutely for all <math>z \in \mathbb{C}</math> such that <math>|z - z_0| < |z_1 - z_0|</math>.
*For a given element <math>z_2 \notin K_P</math> where <math>P</math> diverges, all <math>z \in \mathbb{C}</math> with <math>|z_2 - z_0| < |z - z_0|</math> also cause <math>P(z)</math> to diverge.
==Task for Learners==
*Prove the statement of Abel's Lemma by utilizing the fact that a convergent series ([[w:en:Absolute value|Absolute value]]) has bounded coefficients. Then, use the majorant criterion and a geometric series as a majorant to show that <math>P</math> converges [[w:en:Absolute convergence|Absolute convergence]].
*Justify why the convergence region <math>K_P</math> contains an open disk <math>D_r(z_0) \subseteq { z \in \mathbb{C} : |z_0 - z| < r }</math> (where <math>r > 0</math> is maximally chosen), and why <math>P</math> diverges for all <math>z \in \mathbb{C}</math> with <math>r < |z - z_0|</math> when <math>P</math> diverges.
*Determine the radius of convergence <math>r > 0</math> for the following power series, and on the boundary <math>\partial D_{r}(0)</math> of the convergence region, identify two points <math>z_1, z_2 \in \partial D_{r}(0)</math>, such that <math>P(z_1)</math> converges and <math>P(z_2)</math> diverges. <math>P(z) = \sum_{k=1}^\infty \frac{1}{k} \cdot z^k</math> Use your knowledge of the harmonic series to choose the points <math>z_1, z_2 \in \partial D_{r}(0)</math>.
([[Complex Analysis/decomposition theorem|decomposition theorem]]) Analyze the [[Complex Analysis/decomposition theorem|decomposition theorem]] and explain how Abel's Lemma contributes to the extension of the domain to a ring and the use of the Identity Theorem.
==Consequence==
Taking into account that the series must always diverge at points <math>z \in \mathbb{C}</math> where the sequence of its summands is unbounded (by the [[w:en:Cauchy's convergence test|Cauchy's convergence test]], it follows from the lemma that every power series has a well-defined [[w:en:Radius of convergence|radius of convergence]] and converges uniformly on any [[w:en:Compact space|Compact space]] within the convergence disk. Outside the convergence disk, it diverges. No statement is made about the convergence for points on the boundary of the convergence disk.
==See also==
*[[Complex Analysis]]
*[[Complex Analysis/decomposition theorem|decomposition theorem]]
==Source==
Eberhard Freitag & Rolf Busam: Function Theory 1, Springer-Verlag, Berlin, ISBN 3-540-67641-4, p. 98
== Page Information ==
You can display this page as '''[https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Complex%20Analysis/Abel's%20Lemma&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Abel's%20Lemma&coursetitle=Complex%20Analysis Wiki2Reveal slides]'''
=== Wiki2Reveal ===
The'''[https://niebert.github.io/Wiki2Reveal/wiki2reveal.htmldomain=wikiversity&title=Complex%20Analysis/Abel's%20Lemma&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Abel's%20Lemma&coursetitle=Complex%20Analysis Wiki2Reveal slides]''' were created for the '''[https://en.wikiversity.org/wiki/Complex%20Analysis Complex Analysis]'''' and the Link for the [[v:en:Wiki2Reveal|Wiki2Reveal Slides]] was created with the [https://niebert.github.io/Wiki2Reveal/ link generator].
<!--
* Contents of the page are based on:
** [https://en.wikipedia.org/wiki/Complex%20Analysis/Abel's%20Lemma https://en.wikiversity.org/wiki/Complex%20Analysis/Abel's%20Lemma]
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=== Translation and Version Control ===
This page was translated based on the following [https://de.wikiversity.org/wiki/Abelsches Lemma Wikiversity source page] and uses the concept of [[Translation and Version Control]] for a transparent language fork in a Wikiversity:
* Source: [[v:de:Abelsches Lemma|Abelsches Lemma]] - URL:
https://de.wikiversity.org/wiki/Abelsches Lemma
* Date: 1/2/2025
<span type="translate" src="Abelsches Lemma" srclang="de" date="1/2/2025" time="12:02" status="inprogress"></span>
<noinclude>
[[de:Abelsches Lemma]]
</noinclude>
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Complex Analysis/Abel's Lemma
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New resource with "The '''Abelian Lemma '''is a [[w:en:Lemma (mathematics)|Lemma (mathematics)]] used to investigate the [[w:en:convergence|convergence]] region of [[w:en:power series|power series]]. It is named after [[w:en:Niels Henrik Abel|Niels Henrik Abel]]. ==Abel's Lemma== Let <math>K_P := { z \in \mathbb{C} : P(z) = \sum_{k=0}^\infty a_k (z - z_0)^k \text{ converges} } \subseteq \mathbb{C}</math> be the region of convergence of the power series <math>P</math> given by: <math>P(z) =..."
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The '''Abelian Lemma '''is a [[w:en:Lemma (mathematics)|Lemma (mathematics)]] used to investigate the [[w:en:convergence|convergence]] region of [[w:en:power series|power series]]. It is named after [[w:en:Niels Henrik Abel|Niels Henrik Abel]].
==Abel's Lemma==
Let <math>K_P := { z \in \mathbb{C} : P(z) = \sum_{k=0}^\infty a_k (z - z_0)^k \text{ converges} } \subseteq \mathbb{C}</math> be the region of convergence of the power series <math>P</math> given by: <math>P(z) = \sum_{k=0}^\infty a_k (z - z_0)^k</math>, then the following statements hold:
*For a given element <math>z_1 \in K_P</math> from the convergence region of <math>P</math>, the series <math>P(z)</math> converges absolutely for all <math>z \in \mathbb{C}</math> such that <math>|z - z_0| < |z_1 - z_0|</math>.
*For a given element <math>z_2 \notin K_P</math> where <math>P</math> diverges, all <math>z \in \mathbb{C}</math> with <math>|z_2 - z_0| < |z - z_0|</math> also cause <math>P(z)</math> to diverge.
==Task for Learners==
*Prove the statement of Abel's Lemma by utilizing the fact that a convergent series ([[w:en:in absolute terms|in absolute terms]]) has bounded coefficients. Then, use the majorant criterion and a geometric series as a majorant to show that <math>P</math> converges [[w:en:absolutely|absolutely]].
*Justify why the convergence region <math>K_P</math> contains an open disk <math>D_r(z_0) \subseteq { z \in \mathbb{C} : |z_0 - z| < r }</math> (where <math>r > 0</math> is maximally chosen), and why <math>P</math> diverges for all <math>z \in \mathbb{C}</math> with <math>r < |z - z_0|</math> when <math>P</math> diverges.
*Determine the radius of convergence <math>r > 0</math> for the following power series, and on the boundary <math>\partial D_{r}(0)</math> of the convergence region, identify two points <math>z_1, z_2 \in \partial D_{r}(0)</math>, such that <math>P(z_1)</math> converges and <math>P(z_2)</math> diverges. <math>P(z) = \sum_{k=1}^\infty \frac{1}{k} \cdot z^k</math> Use your knowledge of the harmonic series to choose the points <math>z_1, z_2 \in \partial D_{r}(0)</math>.
([[Complex Analysis/decomposition theorem|decomposition theorem]]) Analyze the [[Complex Analysis/decomposition theorem|decomposition theorem]] and explain how Abel's Lemma contributes to the extension of the domain to a ring and the use of the Identity Theorem.
==Consequence==
Taking into account that the series must always diverge at points <math>z \in \mathbb{C}</math> where the sequence of its summands is unbounded (by the [[w:en:Cauchy Criterion for Series|Cauchy Criterion for Series]], it follows from the lemma that every power series has a well-defined [[w:en:Radius of convergence|radius of convergence]] and converges uniformly on any [[w:en:Compact space|Compact space]] within the convergence disk. Outside the convergence disk, it diverges. No statement is made about the convergence for points on the boundary of the convergence disk.
==See also==
*[[Complex Analysis]]
*[[Complex Analysis/decomposition theorem|decomposition theorem]]
==Source==
Eberhard Freitag & Rolf Busam: Function Theory 1, Springer-Verlag, Berlin, ISBN 3-540-67641-4, p. 98
== Page Information ==
You can display this page as '''[https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Complex%20Analysis/Abel's%20Lemma&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Abel's%20Lemma&coursetitle=Complex%20Analysis Wiki2Reveal slides]'''
=== Wiki2Reveal ===
The'''[https://niebert.github.io/Wiki2Reveal/wiki2reveal.htmldomain=wikiversity&title=Complex%20Analysis/Abel's%20Lemma&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Abel's%20Lemma&coursetitle=Complex%20Analysis Wiki2Reveal slides]''' were created for the '''[https://en.wikiversity.org/wiki/Complex%20Analysis Complex Analysis]'''' and the Link for the [[v:en:Wiki2Reveal|Wiki2Reveal Slides]] was created with the [https://niebert.github.io/Wiki2Reveal/ link generator].
<!--
* Contents of the page are based on:
** [https://en.wikipedia.org/wiki/Complex%20Analysis/Abel's%20Lemma https://en.wikiversity.org/wiki/Complex%20Analysis/Abel's%20Lemma]
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* [https://en.wikiversity.org/wiki/Complex%20Analysis/Abel's%20Lemma This page] is designed as a [https://en.wikiversity.org/wiki/PanDocElectron-Presentation PanDocElectron-SLIDE] document type.
* Source: Wikiversity https://en.wikiversity.org/wiki/Complex%20Analysis/Abel's%20Lemma
* see [[v:en:Wiki2Reveal|Wiki2Reveal]] for the functionality of [https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Abel's%20Lemma&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Abel's%20Lemma&coursetitle=Complex%20Analysis Wiki2Reveal].
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[[Category:Wiki2Reveal]]
=== Translation and Version Control ===
This page was translated based on the following [https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/Abelsches Lemma Wikiversity source page] and uses the concept of [[Translation and Version Control]] for a transparent language fork in a Wikiversity:
* Source: [[v:de:Kurs:Funktionentheorie/Abelsches Lemma|Kurs:Funktionentheorie/Abelsches Lemma]] - URL:
https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/Abelsches Lemma
* Date: 1/2/2025
<span type="translate" src="Kurs:Funktionentheorie/Abelsches Lemma" srclang="de" date="1/2/2025" time="12:02" status="inprogress"></span>
<noinclude>
[[de:Kurs:Funktionentheorie/Abelsches Lemma]]
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The Casorati-Weierstrass theorem is a statement about the behavior of [[Holomorphic function|Holomorphic function]] in the vicinity of [[Complex Analysis/Isolated singularity|Isolated singularity]]. It essentially states that in every neighborhood of an essential singularity, every complex number can be arbitrarily closely approximated by the values of the function. It is a significantly easier-to-prove weakening of the great Picard theorem, which states that in every neighborhood of an essential singularity, every complex number (except possibly one) occurs infinitely often as a value.
== Statement ==
Let <math>G \subseteq \mathbb{C}</math> be open, and <math>z_0 \in G</math>. Let <math>f \colon G \setminus {z_0} \to \mathbb{C}</math> be a [[Holomorphic function|Holomorphic function]]. Then, <math>f</math> has an [[Complex Analysis/Isolated singularity|Isolated singularity]] at <math>z_0</math> if and only if for every neighborhood <math>U \subseteq G</math> of <math>z_0</math>, it holds that <math>\overline{f(U \setminus {z_0})} = \mathbb{C}</math>.
=== Proof ===
First, assume that <math>z_0</math> is an essential singularity of <math>f</math>, and suppose there exists an <math>r > 0</math> such that <math>f(B_r(z_0) \setminus {z_0})</math> is not dense in <math>\mathbb{C}</math>. Then there exists an <math>\epsilon > 0</math> and a <math>w_0 \in \mathbb{C}</math> such that <math>B_\epsilon(w_0)</math> and <math>f(B_r(z_0) \setminus {z_0})</math> are disjoint. Consider the function <math>g(z) := \frac{1}{f(z) - w_0}</math> on <math>B_r(z_0) \setminus {z_0}</math>. Let <math>r</math> be chosen so that <math>z_0</math> is the only <math>w_0</math>-pole in <math>f(B_r(z_0))</math>. This is possible by the [[Complex Analysis/Identity Theorem|Identity Theorem]] for non-constant holomorphic functions. Since <math>f</math> is not constant (as it has an essential singularity), it is holomorphic and bounded by <math>\frac{1}{\epsilon}</math>. By the [[Riemann Removability Theorem]], <math>g</math> is therefore holomorphically extendable to all of <math>B_r(z_0)</math>. Since <math>g \neq 0</math>, there exists an <math>m \geq 0</math> and a holomorphic function <math>g_0 \colon B_r(z_0) \to \mathbb{C}</math> with <math>g_0(z_0) \neq 0</math>, such that
<center><math> g(z) = (z-z_0)^m g_0(z), \qquad |z-z_0| < r. </math></center>
It follows that
<center><math> f(z) = w_0 + \frac{1}{(z-z_0)^m g_0(z)} </math></center>
and thus
<center><math> f(z)(z-z_0)^m = w_0(z-z_0)^m + \frac{1}{g_0(z)} </math></center>
Since <math>g_0(z_0) \neq 0</math>, <math>\frac{1}{g_0(z)}</math> is holomorphic in a neighborhood of <math>z_0</math>. Therefore, <math>f \cdot (z-z_0)^m</math> is holomorphic in a neighborhood of <math>z_0</math>, meaning that <math>f</math> has at most a pole of order <math>m</math> at <math>z_0</math>, which leads to a contradiction.Conversely, let <math>z_0</math> be a removable singularity or a pole of <math>f</math>. If <math>z_0</math> is a removable singularity, there exists a neighborhood <math>U</math> of <math>z_0</math> where <math>f</math> is bounded, say <math>|f(z)| \leq M</math> for <math>z \in U \setminus {z_0}</math>. Then it follows that
<center><math> \overline{f(U \setminus \{z_0\})} \subseteq \bar{B}_M(0) \neq \mathbb{C}. </math></center>
If <math>z_0</math> is a pole of order <math>m</math> for <math>f</math>, there exists a neighborhood <math>U</math> of <math>z_0</math> and a holomorphic function <math>g \colon U \to \mathbb{C}</math> with <math>g(z_0) \neq 0</math> and <math>f(z) = g(z)(z-z_0)^{-m}</math>. Choose a neighborhood <math>\epsilon > 0</math> such that <math>|g(z)| \geq \frac{1}{2}|g(z_0)|</math> for <math>|z-z_0| < \epsilon</math>. Then it follows that
<center><math> |f(z)| = |g(z)| |z-z_0|^{-m} \geq \frac{1}{2}|g(z_0)| \cdot \epsilon^{-m}, \quad 0 < |z-z_0| < \epsilon </mathcenter>
Thus, <math>0 \notin \overline{f(B_\epsilon(z_0) \setminus \{z_0\})}</math>, and this proves the claim.
== See also ==
*[[w:en:Complex analysis|Complex Analysis]]
*[[Complex Analysis/Identity Theorem|Identity Theorem]]
=== Translation and Version Control ===
This page was translated based on the following [https://de.wikiversity.org/wiki/Satz von Casorati-Weierstraß Wikiversity source page] and uses the concept of [[Translation and Version Control]] for a transparent language fork in a Wikiversity:
* Source: [[v:de:Satz von Casorati-Weierstraß|Satz von Casorati-Weierstraß]] - URL:
https://de.wikiversity.org/wiki/Satz von Casorati-Weierstraß
* Date: 1/2/2025
<span type="translate" src="Satz von Casorati-Weierstraß" srclang="de" date="1/2/2025" time="11:12" status="inprogress"></span>
<noinclude>
[[de:Satz von Casorati-Weierstraß]]
</noinclude>
[[Category:Wiki2Reveal]]
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The Casorati-Weierstrass theorem is a statement about the behavior of [[Holomorphic function|Holomorphic function]] in the vicinity of [[Complex Analysis/Isolated singularity|Isolated singularity]]. It essentially states that in every neighborhood of an essential singularity, every complex number can be arbitrarily closely approximated by the values of the function. It is a significantly easier-to-prove weakening of the great Picard theorem, which states that in every neighborhood of an essential singularity, every complex number (except possibly one) occurs infinitely often as a value.
== Statement ==
Let <math>G \subseteq \mathbb{C}</math> be open, and <math>z_0 \in G</math>. Let <math>f \colon G \setminus {z_0} \to \mathbb{C}</math> be a [[Holomorphic function|Holomorphic function]]. Then, <math>f</math> has an [[Complex Analysis/Isolated singularity|Isolated singularity]] at <math>z_0</math> if and only if for every neighborhood <math>U \subseteq G</math> of <math>z_0</math>, it holds that <math>\overline{f(U \setminus {z_0})} = \mathbb{C}</math>.
=== Proof ===
First, assume that <math>z_0</math> is an essential singularity of <math>f</math>, and suppose there exists an <math>r > 0</math> such that <math>f(B_r(z_0) \setminus {z_0})</math> is not dense in <math>\mathbb{C}</math>. Then there exists an <math>\epsilon > 0</math> and a <math>w_0 \in \mathbb{C}</math> such that <math>B_\epsilon(w_0)</math> and <math>f(B_r(z_0) \setminus {z_0})</math> are disjoint. Consider the function <math>g(z) := \frac{1}{f(z) - w_0}</math> on <math>B_r(z_0) \setminus {z_0}</math>. Let <math>r</math> be chosen so that <math>z_0</math> is the only <math>w_0</math>-pole in <math>f(B_r(z_0))</math>. This is possible by the [[Complex Analysis/Identity Theorem|Identity Theorem]] for non-constant holomorphic functions. Since <math>f</math> is not constant (as it has an essential singularity), it is holomorphic and bounded by <math>\frac{1}{\epsilon}</math>. By the [[Riemann Removability Theorem]], <math>g</math> is therefore holomorphically extendable to all of <math>B_r(z_0)</math>. Since <math>g \neq 0</math>, there exists an <math>m \geq 0</math> and a holomorphic function <math>g_0 \colon B_r(z_0) \to \mathbb{C}</math> with <math>g_0(z_0) \neq 0</math>, such that
<center><math> g(z) = (z-z_0)^m g_0(z), \qquad |z-z_0| < r. </math></center>
It follows that
<center><math> f(z) = w_0 + \frac{1}{(z-z_0)^m g_0(z)} </math></center>
and thus
<center><math> f(z)(z-z_0)^m = w_0(z-z_0)^m + \frac{1}{g_0(z)} </math></center>
Since <math>g_0(z_0) \neq 0</math>, <math>\frac{1}{g_0(z)}</math> is holomorphic in a neighborhood of <math>z_0</math>. Therefore, <math>f \cdot (z-z_0)^m</math> is holomorphic in a neighborhood of <math>z_0</math>, meaning that <math>f</math> has at most a pole of order <math>m</math> at <math>z_0</math>, which leads to a contradiction.Conversely, let <math>z_0</math> be a removable singularity or a pole of <math>f</math>. If <math>z_0</math> is a removable singularity, there exists a neighborhood <math>U</math> of <math>z_0</math> where <math>f</math> is bounded, say <math>|f(z)| \leq M</math> for <math>z \in U \setminus {z_0}</math>. Then it follows that
<center><math> \overline{f(U \setminus \{z_0\})} \subseteq \bar{B}_M(0) \neq \mathbb{C}. </math></center>
If <math>z_0</math> is a pole of order <math>m</math> for <math>f</math>, there exists a neighborhood <math>U</math> of <math>z_0</math> and a holomorphic function <math>g \colon U \to \mathbb{C}</math> with <math>g(z_0) \neq 0</math> and <math>f(z) = g(z)(z-z_0)^{-m}</math>. Choose a neighborhood <math>\epsilon > 0</math> such that <math>|g(z)| \geq \frac{1}{2}|g(z_0)|</math> for <math>|z-z_0| < \epsilon</math>. Then it follows that
<center><math> |f(z)| = |g(z)| |z-z_0|^{-m} \geq \frac{1}{2}|g(z_0)| \cdot \epsilon^{-m}, \quad 0 < |z-z_0| < \epsilon </mathcenter>Thus, <math>0 \notin \overline{f(B_\epsilon(z_0) \setminus \{z_0\})}</math>,and this proves the claim.
== see also==
*[[w:en:Complex analysis|Complex Analysis]]
*[[Complex Analysis/Identity Theorem|Identity Theorem]]
==Translation and Version Control==
This page was translated based on the following [https://de.wikiversity.org/wiki/Satz von Casorati-Weierstraß Wikiversity source page] and uses the concept of [[Translation and Version Control]] for a transparent language fork in a Wikiversity:
* Source: [[v:de:Satz von Casorati-Weierstraß|Satz von Casorati-Weierstraß]] - URL:
https://de.wikiversity.org/wiki/Satz von Casorati-Weierstraß
* Date: 1/2/2025<span type="translate" src="Satz von Casorati-Weierstraß" srclang="de" date="1/2/2025" time="12:55" status="inprogress"></span>
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[[de:Satz von Casorati-Weierstraß]]
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