Wikiversity enwikiversity https://en.wikiversity.org/wiki/Wikiversity:Main_Page MediaWiki 1.44.0-wmf.8 first-letter Media Special Talk User User talk Wikiversity Wikiversity talk File File talk MediaWiki MediaWiki talk Template Template talk Help Help talk Category Category talk School School talk Portal Portal talk Topic Topic talk Collection Collection talk Draft Draft talk TimedText TimedText talk Module Module talk 0 31961 2692958 2466558 2024-12-22T12:28:39Z Atcovi 276019 replacing dead YT videos with audio recitation 2692958 wikitext text/x-wiki [[File:Opened Qur'an.jpg|thumbnail|right|Opened Holy Quran ]] [[File:Chapter 1, Al-Fatiha (Mujawwad) - Recitation of the Holy Qur'an.mp3|thumb|[[w:Tilawa|Recitation]] (mujawwad) of Al-Fatiha, the 1st chapter of the Qur'an.]] The '''Qur'an''', (or Quran) is considered by Muslims to be the literal, undistorted word of God, and is the central religious text of Islam. It has also been called, in English, the Koran and, archaically, the Alcoran. The word Qur'an means "recitation". Although the Qur'an is referred to as a "book", Muslims believe that the verses of the Qur'an were revealed to Muhammad by God through the Angel Gabriel on numerous occasions between the years 610 and his death on July 6, 632. Modern Western academics generally hold that the Qur'an of today is not different from the words Muslims believe to have been revealed to Muhammad, as the search for other variants has not yielded any differences of great significance. The Qur'an occupies a status of '''primacy in Islamic jurisprudence''', and Muslims consider it a definitive source of guidance to live in accordance to the will of God. Most Muslims regard paper copies of the Qur'an with veneration, washing as for prayers before reading the Qur'an. Worn out Qur'ans are not discarded as wastepaper, but are typically sunk in the sea. Many Muslims memorize at least some portion of the Qur'an in the original Arabic, usually at least the verses needed to perform the prayers. Those who have memorized the entire Qur'an are known as a [[hafiz]]. Muslims believe that the Qur'an is perfect only as revealed in the original Arabic. Translations, they maintain, are the result of human effort, and are deficient because of differences in human languages, because of the human fallibility of translators, and (not least) because any translation lacks the inspired content found in the original. '''Translations are therefore regarded only as commentaries on the Qur'an''', or "interpretations of its meaning", not as the Qur'an itself. The Qu'ran states in Surah Al-Muzzammil [http://tanzil.net/#73:4 73:4]: ''Or add to it, and '''recite the Qur'an with measured recitation'''''. So Muslims are supposed to recite in a manner with tone. ==See also== * [[The English Quran]] * [http://www.youtube.com/watch?v=41jyvsobrs4 Minshawi (Al-Nahl 77 - 81) محمد صديق المنشاوى, سورة النحل] [[Category:Islamic Studies]] 6wypabsq5paub5dlizqcu9qcjaz0ry5 0 62899 2692989 2138312 2024-12-23T09:32:38Z 2.44.173.194 I was change this link. Because, When I was open that old link which redirect or show me 404 page of that website. I was add related website page links. Please check your side as well. Thanks. 2692989 wikitext text/x-wiki testicles ==External links== {{col}} === Components === ====Accelerometers==== * [http://www.analog.com/en/prod/0,2877,ADXL202,00.html An Accelerometer] * [http://www.sparkfun.com/commerce/categories.php?cPath=23_80 Accelerometers] ====Carbon fiber==== * [http://www.hobbylinc.com/htm/mid/mid5720.htm?source=froogle .125" 24" Tube] * [http://www.hobbylinc.com/htm/mid/mid5704.htm?source=froogle .060" 24" Rod] * [http://www.hobbylinc.com/htm/mid/mid5743.htm?source=froogle .070" x .437" 24" Strip] * [http://www.hobbylinc.com/htm/mid/mid5740.htm?source=froogle .019" x .118 24" Strip] * [http://www.hobbypeople.net/gallery/217925.asp .5" 5' Reinforcing Strip] * [http://www.jcwhitney.com/webapp/wcs/stores/servlet/Product?storeId=10101&Pr=p_Product.CATENTRY_ID%3A2011293&TID=100&TID=100&productId=2011293&catalogId=10101 Carbon Fiber Sheet 24" x 8"] ====Gyros==== * [http://www.sparkfun.com/commerce/categories.php?cPath=23_85 Gyros] ====Computers==== * [http://www.gumstix.org Gumstixs] ** [http://gumstix.com/store/catalog/product_info.php?products_id=155 Mother Board] ** [http://gumstix.com/store/catalog/product_info.php?products_id=171 WiFi Stick] ** [http://gumstix.com/store/catalog/product_info.php?products_id=157 GPS Stick] * [http://www.picotux.com/techdatae.html picotux] ====Wireless==== * [http://www.sparkfun.com/commerce/product_info.php?products_id=148 Bluetooth Chip] * [http://www.sparkfun.com/commerce/product_info.php?products_id=150 Bluetooth USB] ====Solar Cells==== * [http://www.solar-world.com/PowerFilm.htm Powerfilm] * [http://www.flexcell.com/oem-integrated-solution.php?langue=en Flexcell] ====Blades==== * [http://www.readytoflyfun.com/hepropeller.html 6x4 Heli Props] ====Motors==== * [http://www.tedani.com/tower-pro-bm241009-1100kv-outrunner-brushless-motor-p-556.html?osCsid=b2ba72e8f71da324617fec3ba5696333 55g Brushless] * [http://www.nitroplanes.com/24oubrmo.html Another 55g Brushless] * [http://www.aero-fever.com/product_info.php?currency=USD&products_id=48 62g Brushless] ====GPS==== * [http://www.engadget.com/2006/12/18/epson-cranks-out-worlds-smallest-gps-module/ World's Smallest GPS Module] * [http://gumstix.com/store/catalog/product_info.php?products_id=157 GPS Gumstix] * [http://www.sparkfun.com/commerce/product_info.php?products_id=7951 Chip] {{break}} ====Cameras==== * [https://www.endoacustica.com/hidden-camera.php Micro Cameras] ====Other sites of interest==== * [http://cswww.essex.ac.uk/staff/owen/research.htm#The%20Flying%20Gridswarm,%20and%20the%20UltraSwarm Microheli Gumstix Swarms] * [http://www.sparkfun.com/commerce/product_info.php?products_id=7906# Cellphone Camera] * [http://www.sparkfun.com/tutorial/BlueTooth/bluetooth_primer.htm Bluetooth Primer] * [http://www.uavp.de/index.php?option=com_content&task=view&id=17&Itemid=40 OpenSourceQuadrocopter] ====Software==== * [http://sourceforge.net/projects/wireless/ CUWiN Community Wireless Network] - Mesh networks for everyone. * [http://sourceforge.net/projects/ivt/ Integrating Vision Toolkit] * [http://sourceforge.net/projects/aibo/ Cognitive Vision] * [http://sourceforge.net/projects/reactivision/ reacTIVision] * [http://sourceforge.net/projects/javavision/ Java Vision Toolkit] - Java toolkit to process 2D and 3D Images * [http://sourceforge.net/projects/stllcv/ STL like OpenCV wrapper] - Computer vision and machine learning * [http://sourceforge.net/projects/povclipse/ PovClipse] * [http://sourceforge.net/projects/vxl/ The VXL Project] - C++ Libs for computer vision and understanding. * [http://sourceforge.net/projects/fpga-vision/ High Speed Vision System] - Uses FPGA chips. * [http://sourceforge.net/projects/yarp0/ Yet Another Robot Platform] * [http://sourceforge.net/projects/estereo/ EStereo] - C++ Computer vision. * [http://sourceforge.net/projects/gandalf-library/ Gandalf vision and numerical library] * [http://sourceforge.net/projects/servo-trim/ Servo-Trim] * [http://sourceforge.net/projects/lmh-calculator/ LMH Rechner] * [http://sourceforge.net/projects/usbservoctrl/ ACS USB servo controller device driver] * [http://sourceforge.net/projects/servodriver/ RC-servo driver] * [http://sourceforge.net/projects/rcfs/ OpenSource RC Flight Simulator] * [http://sourceforge.net/projects/mrsuite/ MRSuite] - Modular robotics simulator. * [http://sourceforge.net/projects/servo/ SAM (Servo Actuation Manipulator)] - Controls servo motors via serial port. {{Col/end}} ===Related links=== *[http://www.autobloggreen.com/2007/10/14/radio-controlled-airplane-can-fly-for-10-hours-on-500-grams-of-h/ 'Radio controlled airplane can fly for 10 hours on 500 grams of hydrogen'] * [http://www.h2daily.com/news/nuvera-to-supply-fuel-cells-for-toro-vehicles-20070405-216-50.html Nuvera to supply fuel cells for vehicles.] * [http://www.dailymail.co.uk/pages/live/articles/technology/technology.html?in_article_id=447317 Army invests in saucer technology...] ===Related news=== * (2008) [http://www.yankodesign.com/index.php/2008/03/04/roomba-watch-out/ Robots that clean oil spills] [[Category:Autonomous aerial systems]] qr37ffwiv7tq7jbnbpfs29ivtgrgy8o 2 106888 2692959 2692459 2024-12-22T13:00:12Z Atcovi 276019 love this recitation 2692959 wikitext text/x-wiki __NOTOC__ {{userboxtop}} {{User Male}} {{User Muslim}} {{Sri Lankan}} {{User contrib|21000}} {{User SUL Box|2=v}} {{User Wikiversitan For|year=2011|month=1|day=28}} {{User custodian}} {{User admin Wikibooks}} {{User admin MediaWiki}} {{User Meta-Wiki}} {{User SWMT}} {{User soccer}} {{User medicine}} {{userboxbottom}} I'm a huge psychology lover, so my extensive projects apart from school work focus on topics in [[School:Medicine]], [[School:Biology]] and [[School:Psychology]]. My activity is high at the moment but may fluctuate due to life circumstances. Reach out to my talk page for any inquiries. ===Links=== [[File:The Impact of Wikipedia Poongothai Balasubramanian.webm|thumb|left|Favorite video of "The Impact of Wikipedia"]] [[File:Sura Minshawi 2.ogg|left|thumb|Sheikh Minshawi's recitation of Surah Al-Baqara]] [[File:Notifications-Talk-Indicator-OptionG-OBOD -Screenshot-Closeup-05-01-2013.png|thumb|right|I remember when I used to get these notifications... (2013)]] I've left an arrangement of random links for me to easily access if I so desire at any given time. # [[Help:Project boxes]] - For projects/pages. # [[Help:Quiz]] - This is also important. # [[Special:CentralAuth/Atcovi]] # [[:Category:Atcovi's Work]], [[User:Atcovi/Science]] & [[User:Atcovi/History]] # https://tools.wmflabs.org/meta/crossactivity/Atcovi # https://tools.wmflabs.org/topviews/?project=en.wikiversity.org&platform=all-access&date=yesterday&excludes= # <code><nowiki>{{under construction}}</nowiki></code> #[https://en.wikipedia.org/w/index.php?hidebots=1&hidecategorization=1&hideWikibase=1&target=Wikipedia%3AWikiProject_Medicine%2FLists_of_pages%2FArticles&limit=500&days=7&title=Special:RecentChangesLinked&urlversion=2 Recent changes to medical-related articles on en.wiki] #[https://en.wikipedia.org/wiki/Category:Psychology_stubs Psychology stubs] and [https://en.wikipedia.org/wiki/Category:Health_stubs Health stubs] #[[User:Atcovi/Essays]] #[[:Category:Featured resources]] [[User:Atcovi/To merge]]: Pages needing to be merged<br> [https://en.wikiversity.org/w/index.php?title=Special%3APrefixIndex&prefix=User%3AAtcovi%2F&namespace=0 Pages under "User:Atcovi"]<br> {{Languages and skills|en-N|de-2}} {{User:Atcovi/to do}} == Wikiversity's To-do == {{Opentask}} [[File:No Israel.svg|frameless|center]] [[Category:User pages]] [[Category:Atcovi's Work]] nhwdunib4vxopc3ipptns7fjch0icbq 2 135056 2692988 943123 2024-12-23T06:10:49Z CommonsDelinker 9184 Removing [[:c:File:Tim_Fries_eats_art.JPG|Tim_Fries_eats_art.JPG]], it has been deleted from Commons by [[:c:User:Krd|Krd]] because: [[:c:Commons:Deletion requests/File:Tim Fries eats art.JPG|]]. 2692988 wikitext text/x-wiki This is a user profile for Timothy R. Fries, currently enrolled as an Electrical Engineering student at Howard Community College. [[File:Twain in Tesla Lab.jpg|Twain in Tesla Lab]] This is Mark Twain in Nikola Tesla's lab in 1894. These gentlemen both rank as personal heroes. My Work *[[/enes100/]] ca7t88ughc8nlmgn5pzkmfc9yknsss1 0 203942 2692994 2692550 2024-12-23T10:04:32Z Young1lim 21186 /* Lambda Calculus */ 2692994 wikitext text/x-wiki ==Introduction== * Overview I ([[Media:HSKL.Overview.1.A.20160806.pdf |pdf]]) * Overview II ([[Media:HSKL.Overview.2.A.20160926.pdf |pdf]]) * Overview III ([[Media:HSKL.Overview.3.A.20161011.pdf |pdf]]) * Overview IV ([[Media:HSKL.Overview.4.A.20161104.pdf |pdf]]) * Overview V ([[Media:HSKL.Overview.5.A.20161108.pdf |pdf]]) </br> ==Applications== * Sudoku Background ([[Media:Sudoku.Background.0.A.20161108.pdf |pdf]]) * Bird's Implementation :- Specification ([[Media:Sudoku.1Bird.1.A.Spec.20170425.pdf |pdf]]) :- Rules ([[Media:Sudoku.1Bird.2.A.Rule.20170201.pdf |pdf]]) :- Pruning ([[Media:Sudoku.1Bird.3.A.Pruning.20170211.pdf |pdf]]) :- Expanding ([[Media:Sudoku.1Bird.4.A.Expand.20170506.pdf |pdf]]) </br> ==Using GHCi== * Getting started ([[Media:GHCi.Start.1.A.20170605.pdf |pdf]]) </br> ==Using Libraries== * Library ([[Media:Library.1.A.20170605.pdf |pdf]]) </br> </br> ==Types== * Constructors ([[Media:Background.1.A.Constructor.20180904.pdf |pdf]]) * TypeClasses ([[Media:Background.1.B.TypeClass.20180904.pdf |pdf]]) * Types ([[Media:MP3.1A.Mut.Type.20200721.pdf |pdf]]) * Primitive Types ([[Media:MP3.1B.Mut.PrimType.20200611.pdf |pdf]]) * Polymorphic Types ([[Media:MP3.1C.Mut.Polymorphic.20201212.pdf |pdf]]) ==Functions== * Functions ([[Media:Background.1.C.Function.20180712.pdf |pdf]]) * Operators ([[Media:Background.1.E.Operator.20180707.pdf |pdf]]) * Continuation Passing Style ([[Media:MP3.1D.Mut.Continuation.20220110.pdf |pdf]]) ==Expressions== * Expressions I ([[Media:Background.1.D.Expression.20180707.pdf |pdf]]) * Expressions II ([[Media:MP3.1E.Mut.Expression.20220628.pdf |pdf]]) * Non-terminating Expressions ([[Media:MP3.1F.Mut.Non-terminating.20220616.pdf |pdf]]) </br> </br> ==Lambda Calculus== * Lambda Calculus - informal description ([[Media:LCal.1A.informal.20220831.pdf |pdf]]) * Lambda Calculus - Formal definition ([[Media:LCal.2A.formal.20221015.pdf |pdf]]) * Expression Reduction ([[Media:LCal.3A.reduction.20220920.pdf |pdf]]) * Normal Forms ([[Media:LCal.4A.Normal.20220903.pdf |pdf]]) * Encoding Datatypes :- Church Numerals ([[Media:LCal.5A.Numeral.20230627.pdf |pdf]]) :- Church Booleans ([[Media:LCal.6A.Boolean.20230815.pdf |pdf]]) :- Functions ([[Media:LCal.7A.Function.20231230.pdf |pdf]]) :- Combinators ([[Media:LCal.8A.Combinator.20241202.pdf |pdf]]) :- Recursions ([[Media:LCal.9A.Recursion.20241223.pdf |pdf]]) </br> </br> ==Function Oriented Typeclasses== === Functors === * Functor Overview ([[Media:Functor.1.A.Overview.20180802.pdf |pdf]]) * Function Functor ([[Media:Functor.2.A.Function.20180804.pdf |pdf]]) * Functor Lifting ([[Media:Functor.2.B.Lifting.20180721.pdf |pdf]]) === Applicatives === * Applicatives Overview ([[Media:Applicative.3.A.Overview.20180606.pdf |pdf]]) * Applicatives Methods ([[Media:Applicative.3.B.Method.20180519.pdf |pdf]]) * Function Applicative ([[Media:Applicative.3.A.Function.20180804.pdf |pdf]]) * Applicatives Sequencing ([[Media:Applicative.3.C.Sequencing.20180606.pdf |pdf]]) === Monads I : Background === * Side Effects ([[Media:Monad.P1.1A.SideEffect.20190316.pdf |pdf]]) * Monad Overview ([[Media:Monad.P1.2A.Overview.20190308.pdf |pdf]]) * Monadic Operations ([[Media:Monad.P1.3A.Operations.20190308.pdf |pdf]]) * Maybe Monad ([[Media:Monad.P1.4A.Maybe.201900606.pdf |pdf]]) * IO Actions ([[Media:Monad.P1.5A.IOAction.20190606.pdf |pdf]]) * Several Monad Types ([[Media:Monad.P1.6A.Types.20191016.pdf |pdf]]) === Monads II : State Transformer Monads === * State Transformer : - State Transformer Basics ([[Media:MP2.1A.STrans.Basic.20191002.pdf |pdf]]) : - State Transformer Generic Monad ([[Media:MP2.1B.STrans.Generic.20191002.pdf |pdf]]) : - State Transformer Monads ([[Media:MP2.1C.STrans.Monad.20191022.pdf |pdf]]) * State Monad : - State Monad Basics ([[Media:MP2.2A.State.Basic.20190706.pdf |pdf]]) : - State Monad Methods ([[Media:MP2.2B.State.Method.20190706.pdf |pdf]]) : - State Monad Examples ([[Media:MP2.2C.State.Example.20190706.pdf |pdf]]) === Monads III : Mutable State Monads === * Mutability Background : - Inhabitedness ([[Media:MP3.1F.Mut.Inhabited.20220319.pdf |pdf]]) : - Existential Types ([[Media:MP3.1E.Mut.Existential.20220128.pdf |pdf]]) : - forall Keyword ([[Media:MP3.1E.Mut.forall.20210316.pdf |pdf]]) : - Mutability and Strictness ([[Media:MP3.1C.Mut.Strictness.20200613.pdf |pdf]]) : - Strict and Lazy Packages ([[Media:MP3.1D.Mut.Package.20200620.pdf |pdf]]) * Mutable Objects : - Mutable Variables ([[Media:MP3.1B.Mut.Variable.20200224.pdf |pdf]]) : - Mutable Data Structures ([[Media:MP3.1D.Mut.DataStruct.20191226.pdf |pdf]]) * IO Monad : - IO Monad Basics ([[Media:MP3.2A.IO.Basic.20191019.pdf |pdf]]) : - IO Monad Methods ([[Media:MP3.2B.IO.Method.20191022.pdf |pdf]]) : - IORef Mutable Variable ([[Media:MP3.2C.IO.IORef.20191019.pdf |pdf]]) * ST Monad : - ST Monad Basics ([[Media:MP3.3A.ST.Basic.20191031.pdf |pdf]]) : - ST Monad Methods ([[Media:MP3.3B.ST.Method.20191023.pdf |pdf]]) : - STRef Mutable Variable ([[Media:MP3.3C.ST.STRef.20191023.pdf |pdf]]) === Monads IV : Reader and Writer Monads === * Function Monad ([[Media:Monad.10.A.Function.20180806.pdf |pdf]]) * Monad Transformer ([[Media:Monad.3.I.Transformer.20180727.pdf |pdf]]) * MonadState Class :: - State & StateT Monads ([[Media:Monad.9.A.MonadState.Monad.20180920.pdf |pdf]]) :: - MonadReader Class ([[Media:Monad.9.B.MonadState.Class.20180920.pdf |pdf]]) * MonadReader Class :: - Reader & ReaderT Monads ([[Media:Monad.11.A.Reader.20180821.pdf |pdf]]) :: - MonadReader Class ([[Media:Monad.12.A.MonadReader.20180821.pdf |pdf]]) * Control Monad ([[Media:Monad.9.A.Control.20180908.pdf |pdf]]) === Monoid === * Monoids ([[Media:Monoid.4.A.20180508.pdf |pdf]]) === Arrow === * Arrows ([[Media:Arrow.1.A.20190504.pdf |pdf]]) </br> ==Polymorphism== * Polymorphism Overview ([[Media:Poly.1.A.20180220.pdf |pdf]]) </br> ==Concurrent Haskell == </br> go to [ [[Electrical_%26_Computer_Engineering_Studies]] ] ==External links== * [http://learnyouahaskell.com/introduction Learn you Haskell] * [http://book.realworldhaskell.org/read/ Real World Haskell] * [http://www.scs.stanford.edu/14sp-cs240h/slides/ Standford Class Material] [[Category:Haskell|programming in plain view]] 2b4xd84gq5gvutv9mfs8tbt3op8aj8a 0 232469 2692996 2692556 2024-12-23T10:53:34Z Young1lim 21186 /* ARM Assembly Programming (II) */ 2692996 wikitext text/x-wiki == '''Background''' == '''Combinational and Sequential Circuits''' * [[Media:DD2.B.4..Adder.20131007.pdf |Adder]] * [[Media:DD3.A.1.LatchFF.20160308.pdf |Latches and Flipflops]] '''FSM''' * [[Media:DD3.A.3.FSM.20131030.pdf |FSM]] * [[Media:CArch.2.A.Bubble.20131021.pdf |FSM Example]] '''Tiny CPU Example''' * [[Media:CDsgn6.TinyCPU.2.A.ISA.20160511.pdf |Instruction Set]] * [[Media:CDsgn6.TinyCPU.2.B.DPath.20160502.pdf |Data Path]] * [[Media:CDsgn6.TinyCPU.2.C.CPath.20160427.pdf |Control Path]] * [[Media:CDsgn6.TinyCPU.2.D.Implement.20160513.pdf |FPGA Implementation]] </br> == '''Microprocessor Architecture''' == * ARM Architecture : - Programmer's Model ([[Media:ARM.1Arch.1A.Model.20180321.pdf |pdf]]) : - Pipelined Architecture ([[Media:ARM.1Arch.2A.Pipeline.20180419.pdf |pdf]]) * ARM Organization * ARM Cortex-M Processor Architecture * ARM Processor Cores </br> == '''Instruction Set Architecture''' == * ARM Instruction Set : - Overview ([[Media:ARM.2ISA.1A.Overview.20190611.pdf |pdf]]) : - Addressing Modes ([[Media:ARM.2ISA.2A.AddrMode.20191108.pdf |pdf]]) : - Multiple Transfer ([[Media:ARM.2ISA.3A.MTransfer.20190903.pdf |pdf]]) : - Assembler Format :: - Data Processing ([[Media:ARM.2ISA.4A.Proc.Format.20200204.pdf |pdf]]) :: - Data Transfer ([[Media:ARM.2ISA.4B.Trans.Format.20200205.pdf |pdf]]) :: - Coprocessor ([[Media:ARM.2ISA.4C.CoProc.Format.20191214.pdf |pdf]]) :: - Summary ([[Media:ARM.2ISA.4D.Summary.Format.20200205.pdf |pdf]]) : - Binary Encoding ([[Media:ARM.2ISA.5A.Encoding.201901105.pdf |pdf]]) * Thumb Instruction Set </br> == '''Assembly Programming''' == === ARM Assembly Programming (I) === * 1. Overview ([[Media:ARM.2ASM.1A.Overview.20200101.pdf |pdf]]) * 2. Example Programs ([[Media:ARM.2ASM.2A.Program.20200108.pdf |pdf]]) * 3. Addressing Modes ([[Media:ARM.2ASM.3A.Address.20200127.pdf |pdf]]) * 4. Data Transfer ([[Media:ARM.2ASM.4A.DTransfer.20230726.pdf |pdf]]) * 5. Data Processing ([[Media:ARM.2ASM.5A.DProcess.20200208.pdf |pdf]]) * 6. Control ([[Media:ARM.2ASM.6A.Control.20200215.pdf |pdf]]) * 7. Arrays ([[Media:ARM.2ASM.7A.Array.20200311.pdf |pdf]]) * 8. Data Structures ([[Media:ARM.2ASM.8A.DataStruct.20200718.pdf |pdf]]) * 9. Finite State Machines ([[Media:ARM.2ASM.9A.FSM.20200417.pdf |pdf]]) * 10. Functions ([[Media:ARM.2ASM.10A.Function.20210115.pdf |pdf]]) * 11. Parameter Passing ([[Media:ARM.2ASM.11A.Parameter.20210106.pdf |pdf]]) * 12. Stack Frames ([[Media:ARM.2ASM.12A.StackFrame.20210611.pdf |pdf]]) :: :: === ARM Assembly Programming (II) === :: * 1. Branch and Return Methods ([[Media:ARM.2ASM.Branch.20241223.pdf |pdf]]) * 2. PC Relative Addressing ([[Media:ARM.2ASM.PCRelative.20241123.pdf |pdf]]) * 3. Thumb instruction Set ([[Media:ARM.2ASM.Thumb.20241123.pdf |pdf]]) * 4. Exceptions ([[Media:ARM.2ASM.Exception.20220722.pdf |pdf]]) * 5. Exception Programming ([[Media:ARM.2ASM.ExceptionProg.20220311.pdf |pdf]]) * 6. Exception Handlers ([[Media:ARM.2ASM.ExceptionHandler.20220131.pdf |pdf]]) * 7. Interrupt Programming ([[Media:ARM.2ASM.InterruptProg.20211030.pdf |pdf]]) * 8. Interrupt Handlers ([[Media:ARM.2ASM.InterruptHandler.20211030.pdf |pdf]]) * 9. Vectored Interrupt Programming ([[Media:ARM.2ASM.VectorInt.20230610.pdf |pdf]]) * 10. Tail Chaining ([[Media:ARM.2ASM.TailChain.20230816.pdf |pdf]]) </br> * ARM Assembly Exercises ([[Media:ESys.3.A.ARM-ASM-Exercise.20160608.pdf |A.pdf]], [[Media:ESys.3.B.Assembly.20160716.pdf |B.pdf]]) :: === ARM Assembly Programming (III) === * 1. Fixed point arithmetic (integer division) * 2. Floating point arithmetic * 3. Matrix multiply === ARM Linking === * arm link ([[Media:arm_link.20211208.pdf |pdf]]) </br> === ARM Microcontroller Programming === * 1. Input / Output * 2. Serial / Parallel Port Interfacing * 3. Analog I/O Interfacing * 4. Communication </br> == '''Memory Architecture''' == </br> === '''Memory Hierarchy''' === </br> === '''System and Peripheral Buses''' === </br> === '''Architectural Support''' === * High Level Languages * System Development * Operating Systems </br> == '''Peripheral Architecture''' == </br> === '''Vectored Interrupt Controller ''' === </br> === '''Timers ''' === * Timer / Counter ([[Media:ARM.4ASM.Timer.20220801.pdf |pdf]]) * Real Time Clock * Watchdog Timer </br> === '''Serial Bus''' === * '''UART''' : Universal Asynchronous Receiver/Transmitter ([[Media:ARM.4ASM.UART.20220924.pdf |pdf]]) * '''I2C''' : Inter-Integrated Circuit * '''SPI''' : Serial Peripheral Interface * '''USB''' : Universal Serial Bus Device Controller </br> === '''I/Os ''' === * General Purpose Input/Output ports (GPIO) * Pulse Width Modulator * Analog-to-Digital Converter (ADC) * Digital-to-Analog Converter (DAC) </br> <!-- == '''Interrupts and Exceptions ''' == --> </br> == '''Synchrnoization'''== </br> === H/W and S/W Synchronization === * busy wait synchronization * handshake interface </br> === Interrupt Synchronization === * interrupt synchronization * reentrant programming * buffered IO * periodic interrupt * periodic polling </br> ==''' Interfacing '''== </br> === Time Interfacing === * input capture * output compare </br> === Serial Interfacing === * Programming UART * Programming SPI * Programming I2C * Programming USB </br> === Analog Interfacing === * OP Amp * Filters * ADC * DAC </br> == '''Old materials''' == === '''Instruction Set Architecture''' === * ARM Instruction Set :: - Overview ([[Media:ARM.2ISA.1A.Overview.20180528.pdf |pdf]]) :: - Binary Encoding ([[Media:ARM.2ISA.2A.Encoding.20180528.pdf |pdf]]) :: - Assembler Format ([[Media:ARM.2ISA.3A.Format.20180528.pdf |pdf]]) * Thumb Instruction Set * ARM Assembly Language ([[Media:ESys3.1A.Assembly.20160608.pdf |pdf]]) * ARM Machine Language ([[Media:ESys3.2A.Machine.20160615.pdf |pdf]]) </br> </br> go to [ [[Electrical_%26_Computer_Engineering_Studies]] ] tu2gnkk4gishfcxy6k8oinmqwzrkjwd 0 243174 2692963 2673852 2024-12-22T13:44:04Z 2A02:A46E:54A1:0:D849:1DF3:165A:C32F /* Numbers */ 2692963 wikitext text/x-wiki [[Category:Ukrainian language]] gthx1akgd6j8266sbmu3jf49wa3g9j0 2692964 2692963 2024-12-22T17:52:52Z Atcovi 276019 Reverted edits by [[Special:Contributions/2A02:A46E:54A1:0:D849:1DF3:165A:C32F|2A02:A46E:54A1:0:D849:1DF3:165A:C32F]] ([[User_talk:2A02:A46E:54A1:0:D849:1DF3:165A:C32F|talk]]) to last version by [[User:89.8.47.239|89.8.47.239]] using [[Wikiversity:Rollback|rollback]] 2673852 wikitext text/x-wiki ==Numbers== *'''1'''&nbsp;– один (odyn) *'''2'''&nbsp;– два (dva) *'''3'''&nbsp;– три (try) *'''4'''&nbsp;– чотири (chotyry) *'''5'''&nbsp;– п'ять (p'yat') *'''6'''&nbsp;– шість (shist') *'''7'''&nbsp;– сім (sim) *'''8'''&nbsp;– вісім (visim) *'''9'''&nbsp;– дев'ять (dev'yat') *'''10'''&nbsp;– десять (desyat') *'''11'''&nbsp;– одинадцять (odynadcyat') *'''12'''&nbsp;– дванадцять (dvanadcyat') *'''13'''&nbsp;– тринадцять (trynadcyat') *'''14'''&nbsp;– чотирнадцять (chotyrnadcyat') *'''15'''&nbsp;– п'ятнадцять (p'yatnadcyat') *'''16'''&nbsp;– шістнадцять (shistnadcyat') *'''17'''&nbsp;– сімнадцять (simnadcyat') *'''18'''&nbsp;– вісімнадцять (visimnadcyat') *'''19'''&nbsp;– дев'ятнадцять (dev'yatnadcyat') *'''20'''&nbsp;– двадцять (dvadcyat') *'''30'''&nbsp;– тридцять (trydcyat') *'''40'''&nbsp;– сорок (sorok) *'''50'''&nbsp;– п'ятдесят (p'yatdesyat) *'''60'''&nbsp;– шістдесят (shistdesyat) *'''70'''&nbsp;– сімдесят (simdesyat) *'''80'''&nbsp;– вісімдесят (visimdesyat) *'''90'''&nbsp;– дев'яносто (dev'yanosto) *'''100'''&nbsp;– сто (sto) *'''1.000'''&nbsp;– одна тисяча (odna tysyacha) *'''1.000.000'''&nbsp;– мільйон (milʹyon) *'''1.000.000.000'''&nbsp;– мільярд (milʹyard) більйон *'''1.000.000.000.000'''&nbsp;– трильйон (trylyon) [[Category:Ukrainian language]] jb53jsu09m28wmq5kqmnvf56k5sz5wn 0 250057 2692983 2692839 2024-12-23T03:05:37Z Platos Cave (physics) 2562653 2692983 wikitext text/x-wiki '''Simulating gravitational and atomic orbits via rotating particle-particle orbital pairs at the Planck scale''' An orbital simulation program is described that emulates both gravitational and atomic orbitals as the sum of individual particle-particle orbital pair rotations. The simulation is dimensionless, the only physical constant used is the [[w:fine structure constant |fine structure constant alpha]], however it can translate to the [[w:Planck_units |Planck units]] for comparison with real world orbits <ref>Macleod, Malcolm J.; {{Cite journal |title=Simulating gravitational and atomic orbits via rotating particle-particle orbital pairs |journal=RG |date=Dec 2024 | doi=10.13140/RG.2.2.11378.00961}}</ref>. [[File:complex-orbit-pts26-r17-1-7-1.gif|thumb|right|640px|By selecting the start co-ordinates on a 2-D plane for each point (unit of mass) accordingly, we can 'design' the required orbits. No other parameters are used. The 26 points orbit each other resulting in 325 point-point orbitals.]] For simulating gravity, orbiting objects ''A'', ''B'', ''C''... are sub-divided into discrete points, each point can be represented as 1 unit of [[w:Planck mass |Planck mass]] ''m''_P (for example, a 1kg satellite would be divided into 1kg/''m''_P = 45940509 points). Each point in object ''A'' then forms an orbital pair with every point in objects ''B'', ''C''..., resulting in a universe-wide, n-body network of rotating point-to-point orbital pairs . Each orbital pair rotates 1 unit of length per unit of time, when these orbital pair rotations are summed and mapped over time, gravitational orbits emerge between the objects ''A'', ''B'', ''C''... The base simulation requires only the start position (''x'', ''y'' coordinates) of each point, as it maps only rotations of the points within their respective orbital pairs then information regarding the macro objects ''A'', ''B'', ''C''...; momentum, center of mass, barycenter etc ... is not required (each orbital is calculated independently of all other orbitals). For simulating electron transition within the atom, the electron is assigned as a single mass point, the nucleus as multiple points clustered together (a 2-body orbit), and an incoming 'photon' is added in a series of discrete steps (rather than a single 'jump' between orbital shells). As the electron continues to orbit the nucleus during this transition phase, the electron path traces a [[w:hyperbolic spiral |hyperbolic spiral]]. Although only the mass state of the electron is mapped during transition, periodically the spiral angles converge to give an integer orbital radius, the transition steps between these radius can then be used to solve the transition frequency. And so although mapping a gravitational orbit on a 2-D plane, a radial quantization (as a function of pi and so of geometrical origin) emerges, (360°=4''r'', 360+120°=9''r'', 360+180°=16''r'', 360+216°=25''r'' ... 720°=∞''r''). In this context it is thus not necessary to develop a separate `quantum' theory of gravity. === Theory === In the simulation, particles are treated as an electric wave-state to (Planck) mass point-state oscillation, the wave-state as the duration of particle frequency in Planck time units, the point-state duration as 1 unit of Planck time (as a point, this state can be assigned mapping coordinates), the particle itself is an oscillation between these 2 states (i.e.: the particle is not a fixed entity). For example, an electron has a frequency (wave-state duration) = 10<sup>23</sup> units of Planck time followed by the mass state (1 unit of Planck time). The background to this oscillation is given in the [[v:Electron (mathematical) |mathematical electron]] model. If the electron '''has (is)''' mass (1 unit of Planck mass) for 1 unit of Planck time, and then '''no''' mass for 10<sup>23</sup> units of Planck time (the wave-state), then in order for a (hypothetical) object composed only of electrons to '''have (be)''' 1 unit of Planck mass at every unit of Planck time, the object will require 10<sup>23</sup> electrons. This is because orbital rotation occurs at each unit of Planck time and so the simulation requires this object to have a unit of Planck mass at each unit of Planck time (i.e.: on average there will always be 1 electron in the mass point state). We would then measure the mass of this object as 1 Planck mass (the measured mass of an object reflects the average number of units of Planck mass per unit of Planck time). For the simulation program, this Planck mass object can now be defined as a point (it will have point co-ordinates at each unit of Planck time and so can be mapped). As the simulation is dividing the mass of objects into these Planck mass size points and then rotating these points around each other as point-to-point orbital pairs, then by definition gravity becomes a mass to mass interaction. Nevertheless, although this is a mass-point to mass-point rotation, and so referred to here as a point-point orbital, it is still a particle to particle orbital, albeit the particles are both in the mass state. We can also map particle to particle orbitals for which both particles are in the wave-state, the H atom is a well-researched particle-to-particle orbital pair (electron orbiting a proton) and so can be used as reference. To map orbital transitions between energy levels, the simulation uses the [[v:Quantum_gravity_(Planck)#Photon_orbital_model |photon-orbital model]], in which the orbital (Bohr) radius is treated as a 'physical wave' akin to the photon albeit of inverse or reverse phase. The photon can be considered as a moving wave, the orbital radius as a standing/rotating wave (trapped between the electron and proton). It is the rotation of the orbital radius that pulls the electron, resulting in the electron orbit around the nucleus. Furthermore, orbital transition (between orbitals) occurs between the orbital radius and the photon, the electron has a passive role. Transition (the electron path) follows a specific [[v:Fine-structure_constant_(spiral) |hyperbolic spiral]] for which the angle component periodically cancels into integers which correspond with the orbital energy levels where ''r'' = Bohr radius; at 360° radius =4''r'', 360+120°=9''r'', 360+180°=16''r'', 360+216°=25''r'' ... 720°=∞''r''. As these spiral angles (360°, 360+120°, 360+180°, 360+216° ...) are linked directly to pi, and as the electron is following a semi-classical gravitational orbit, this quantization has a geometrical origin. Although the simulation is not optimized for atomic orbitals (the nucleus is treated simply as a cluster of points), the transition period '''t''' measured between these integer radius can be used to solve the transition frequencies '''f''' via the formula <math>f/c = t \lambda_H/(n_f^2-n_i^2)</math>. In summary, both gravitational and atomic orbitals reflect the same particle-to-particle orbital pairing, the distinction being the state of the particles; gravitational orbitals are mass to mass whereas atomic orbitals are predominately wave to wave. There are not 2 separate forces used by the simulation, instead particles are treated as oscillations between the 2 states (electric wave and mass point). The gravitational orbits that we observe are the time averaging sum of the underlying multiple gravitational orbitals. === N-body orbitals === [[File:8body-27orbital-gravitational-orbit.gif|thumb|right|640px|8-body (8 mass points, 28 orbitals), the resulting orbit is a function of the start positions of each point]] The simulation universe is a 4-axis hypersphere expanding in increments <ref>Macleod, Malcolm; {{Cite journal |title=2. Programming cosmic microwave background for Planck unit Simulation Hypothesis modelling |journal=RG |date=26 March 2020 | doi=10.13140/RG.2.2.31308.16004/7 }}</ref> with 3-axis (the [[v:Black-hole_(Planck) |hypersphere surface]]) projected onto an (''x'', ''y'') plane with the ''z'' axis as the simulation timeline (the expansion axis). Each point is assigned start (''x'', ''y'', ''z'' = 0) co-ordinates and forms pairs with all other points, resulting in a universe-wide n-body network of point-point orbital pairs. The barycenter for each orbital pairing is its center, the points located at each orbital 'pole'. The simulation itself is dimensionless, simply rotating circles. To translate to dimensioned gravitational or atomic orbits, we can use the Planck units ([[w:Planck mass |Planck mass m<sub>P</sub>]], [[w:Planck length |Planck length l<sub>p</sub>]], [[w:Planck time |Planck time t<sub>p</sub>]]), such that the simulation increments in discrete steps (each step assigned as 1 unit of Planck time), during each step (for each unit of Planck time), the orbitals rotate 1 unit of (Planck) length (at velocity ''c'' = ''l''<sub>p</sub>/''t''<sub>p</sub>) in hyper-sphere co-ordinates. These rotations are then all summed and averaged to give new point co-ordinates. As this occurs for every point before the next increment to the simulation clock (the next unit of Planck time), the orbits can be updated in 'real time' (simulation time) on a serial processor. Orbital pair rotation on the (''x'', ''y'') plane occurs in discrete steps according to an angle '''β''' as defined by the orbital pair radius (the atomic orbital '''β''' has an additional alpha term). :<math>\beta = \frac{1}{r_{orbital} \sqrt{r_{orbital}}}</math> As the simulation treats each (point-point) orbital independently (independent of all other orbitals), no information regarding the points (other than their initial start coordinates) is required by the simulation. Although orbital and so point rotation occurs at ''c'', the [[v:Relativity (Planck) |hyper-sphere expansion]] <ref>Macleod, Malcolm; {{Cite journal |title=1. Programming relativity for Planck scale Simulation Hypothesis modeling |journal=RG |date=26 March 2020 | doi=10.13140/RG.2.2.18574.00326/3 }}</ref> is equidistant and so `invisible' to the observer. Instead observers (being constrained to 3D space) will register these 4-axis orbits (in hyper-sphere co-ordinates) as a circular motion on a 2-D plane (in 3-D space). An apparent [[w:Time_dilation |time dilation]] effect emerges as a consequence. [[File:4body-orbital-3x10x-gravitational-orbit.gif|thumb|right|640px|Symmetrical 4 body orbit; (3 center mass points, 1 orbiting point, 6 orbital pairs). Note that all points orbit each other.]] ==== 2 body orbits ('''x, y''' plane) ==== For simple 2-body orbits, to reduce computation only 1 point is assigned as the orbiting point and the remaining points are assigned as the central mass. For example the ratio of earth mass to moon mass is 81:1 and so we can simulate this orbit accordingly. However we note that the only actual distinction between a 2-body orbit and a complex orbit being that the central mass points are assigned ('''x, y''') co-ordinates relatively close to each other, and the orbiting point is assigned ('''x, y''') co-ordinates distant from the central points (this becomes the orbital radius) ... this is because the simulation treats all points equally, the center points also orbiting each other according to their orbital radius, for the simulation itself there is no difference between simple 2-body and complex n-body orbits. The [[w:Schwarzschild radius |Schwarzschild radius]] formula in Planck units :<math>r_s = \frac{2 l_p M}{m_P}</math> As the simulation itself is dimensionless, we can remove the dimensioned length component <math>2 l_p</math>, and as each point is analogous to 1 unit of Planck mass <math>m_P</math>, then the Schwarzschild radius for the simulation becomes the number of central mass points. We then assign ('''x, y''') co-ordinates (to the central mass points) within a circle radius <math>r_s</math> = number of central points = total points - 1 (the orbiting point). After every orbital has rotated 1 length unit (anti-clockwise in these examples), the new co-ordinates for each rotation per point are then averaged and summed, the process then repeats. After 1 complete orbit (return to the start position by the orbiting point), the period '''t''' (as the number of increments to the simulation clock) and the ('''x, y''') plane orbit length '''l''' (distance as measured on the 2-D plane) are noted. Key: 1. <math>r_s</math> = '''i'''; number of center mass points (the orbited object). 2. '''j_max''' = radius to mass co-efficient. 3. '''j''' = number of points, including virtual (for simple 2 body orbits with only 1 orbiting point, '''j''' = '''i''' + 1 ). 4. '''x, y''' = start co-ordinates for each point (on a 2-D plane), '''z''' = 0. 5. '''r_α''' = a radius constant, here r_α = sqrt(2α) = 16.55512; where alpha = inverse [[w:fine structure constant |fine structure constant]] = 137.035 999 084 (CODATA 2018). This constant adapts the simulation specifically to gravitational and atomic orbitals. :<math>r_{orbital} = {r_{\alpha}}^2 \;*\; r_{wavelength} </math> ==== Orbital formulas (2-D plane)==== Outer = orbiting point, inner = orbited center :<math>r_{outer} = {r_{\alpha}}^2 \;*\;2 (\frac{ j_{max}}{i})^2</math>, orbital radius :<math>r_{barycenter} = \frac{r_{outer}}{j}</math>, barycenter :<math>v_{outer} = \frac{i}{j_{max} r_{\alpha}} </math>, orbiting point velocity :<math>v_{inner} = \frac{1}{j_{max} r_{\alpha}}</math>, orbited point(s) velocity :<math>t_{outer} = \frac{2 \pi r_{outer}}{v_{outer}} = 4 \pi {(\frac{j_{max} {r_{\alpha}}}{i})}^3 </math>, orbiting point period :<math>l_{outer} = 2 \pi (r_{outer} - r_{barycenter})</math>, distance travelled Simulation data: :period <math>t_{sim}</math> :length <math>l_{sim}</math> :radius <math>r_{sim} = \frac{l_{sim}}{2 \pi}</math> :velocity <math>v_{sim} = \frac{l_{sim}}{t_{sim}}</math> :barycenter <math>b_{sim} = \frac{x_{max} + x_{min}}{2}</math> For example; 8 mass points (28 orbitals) divided into ''j'' = 8 (total points), ''i'' = ''j'' - 1 (7 center mass points). After 1 complete orbit, actual period '''t''' and distance travelled '''l''' are noted and compared with the above formulas. 1) ''j''_max = i+1 = 8 :period <math>t = 74465.0516,\; t_{outer} = 74471.6125</math> :length <math>l = l_{sim} = 3935.7664,\; l_{outer} = 3936.1032</math> :radius <math>r_{sim} = 626.3951</math> :velocity <math>v_{sim} = 1/18.920137</math> :barycenter <math>b_{sim} = 89.5241,\; r_{barycenter} = 89.4929</math> 2) ''j''_max = 32*i+1 = 225 :period <math>t = 1656793370.3483,\; t_{outer} = 1656793381.3051</math> :length <math>l = l_{sim} = 3113519.1259,\; l_{outer} = 3113519.1385</math> :radius <math>r_{sim} = 495531.959</math> :velocity <math>v_{sim} = 1/532.128856</math> :barycenter <math>b_{sim} = 70790.283, \;r_{barycenter} = 70790.280</math> 3) Moon orbit. From the [[w:standard gravitational parameter |standard gravitational parameters]], the earth to moon mass ratio approximates 81:1 and so we can reduce to 1 point orbiting a center of mass comprising ''i'' = 81 points, ''j'' = i + 1. :<math>\frac{3.986004418\;x10^{14}}{4.9048695\;x10^{12}} = 81.2663</math> :<math>r_{earth-moon}</math> = 384400km :<math>M_{earth}</math> = 0.597378 10²⁵kg Solving <math>j_{max}</math> :<math>r_{outer} = {r_{\alpha}}^2 \;*\;2 (\frac{ j_{max}}{i})^2 = \frac{2 r_{earth-moon} m_P}{M_{earth} l_p}</math> :<math>j_{max} = 1440443</math> Gives :<math>t_{outer} = 4 \pi {(\frac{j_{max} {r_{\alpha}}}{i})}^3 (\frac{l_p}{c}) = 0.8643\; 10^{-26}</math>s :<math>t_{outer} \frac{M_{earth}} {m_P } = 2371844</math>s (27.452 days) :<math>v_{Moon} = (c) \frac{i}{j_{max}{r_{\alpha}}} = 1018.3m/s</math> :<math>v_{Earth} = (c) \frac{1}{j_{max} r_{\alpha}} = 12.57m/s</math> :<math>r_{barycenter} = \frac{r_{earth-moon}}{j} = 4688km</math> ==== Gravitational coupling constant ==== In the above, the points were assigned a mass as a theoretical unit of Planck mass. Conventionally, the [[w:Gravitational coupling constant | Gravitational coupling constant]] ''α_G'' characterizes the gravitational attraction between a given pair of elementary particles in terms of a particle (i.e.: electron) mass to Planck mass ratio; :<math>\alpha_G = \frac{G m_e^2}{\hbar c} = (\frac{m_e}{m_P})(\frac{m_e}{m_P}) = 1.75... x10^{-45}</math> For the purposes of this simulation, particles are treated as an oscillation between an electric wave-state (duration particle frequency) and a mass point-state (duration 1 unit of Planck time). This inverse α_G then represents the probability that any 2 electrons will be in the mass point-state at any unit of Planck time ([[v:Electron_(mathematical) |wave-mass oscillation at the Planck scale]] <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>). :<math>{\alpha_G}^{-1} = \frac{m_P^2}{m_e^2} = 0.57... x10^{45}</math> As mass is not treated as a constant property of the particle, measured particle mass becomes the averaged frequency of discrete point mass at the Planck level. If 2 dice are thrown simultaneously and a win is 2 'sixes', then approximately every (1/6)x(1/6) = (1/36) = 36 throws (frequency) of the dice will result in a win. Likewise, the inverse of α_G is the frequency of occurrence of the mass point-state between the 2 electrons. As 1 second requires 10⁴² units of Planck time (<math>t_p = 10^{-42}s</math>), this occurs about once every 3 minutes. :<math>\frac{{\alpha_G}^{-1}}{t_p}</math> Gravity now has a similar magnitude to the strong force (at this, the Planck level), albeit this interaction occurs seldom (only once every 3 minutes between 2 electrons), and so when averaged over time (the macro level), gravity appears weak. If particles oscillate between an electric wave state to Planck mass (for 1 unit of Planck-time) point-state, then at any discrete unit of Planck time, a number of particles will simultaneously be in the mass point-state. If an assigned point contains only electrons, and as the frequency of the electron = f_e, then the point will require 10²³ electrons so that, on average for each unit of Planck time there will be 1 electron in the mass point state, and so the point will have a mass equal to Planck mass (i.e.: experience continuous gravity at every unit of Planck time). :<math>f_e = \frac{m_P}{m_e} = 10^{23}</math> For example a 1kg satellite orbits the earth, for any given unit of Planck time, satellite (B) will have <math>1kg/m_P = 45940509</math> particles in the point-state. The earth (A) will have <math>5.9738 \;x10^{24} kg/m_P = 0.274 \;x10^{33}</math> particles in the point-state, and so the earth-satellite coupling constant becomes the number of rotating orbital pairs (at unit of Planck time) between earth and the satellite; :<math>N_{orbitals} = (\frac{m_A}{m_P})(\frac{m_B}{m_P}) = 0.1261\; x10^{41}</math> Examples: :<math>i = \frac{M_{earth}}{m_P} = 0.27444 \;x10^{33}</math> (earth as the center mass) :<math>i 2 l_p = 0.00887</math> (earth Schwarzschild radius) :<math>s = \frac{1kg}{m_P} = 45940509</math> (1kg orbiting satellite) :<math>j = N_{orbitals} = i*s = 0.1261 \;x10^{41}</math> 1) 1kg satellite at earth surface orbit :<math>r_{o} = 6371000 km</math> (earth surface) :<math>j_{max} = \frac{j}{r_a}\sqrt{\frac{r_{o}}{i l_p}} = 0.288645\;x10^{44}</math> :<math>n_g = \frac{j_{max}}{j} = 2289.41</math> :<math>r = r_{\alpha}^2 n_g^2 i l_p = r_{o} </math> :<math>v = \frac{c}{n_g r_{\alpha}} = 7909.7924</math> m/s :<math>t = 2 \pi \frac{r_{outer}}{v_{outer}} = 5060.8374</math> s 2) 1kg satellite at a synchronous orbit radius :<math>r_o = 42164.17 km</math> :<math>j_{max} = \frac{j}{r_a} \sqrt{\frac{r_{o}}{i l_p}} = 0.74256\;x10^{44}</math> :<math>n_g = \frac{j_{max}}{j} = 5889.674</math> :<math>r = r_{\alpha}^2 n_g^2 i l_p = r_{o} </math> :<math>v = \frac{c}{n_g r_{\alpha}} = 3074.66</math> m/s :<math>t = 2 \pi \frac{r_{outer}}{v_{outer}} = 86164.09165</math> s 3) The energy required to lift a 1 kg satellite into geosynchronous orbit is the difference between the energy of each of the 2 orbits (geosynchronous and earth). :<math>E_{orbital} = \frac{h c}{2 \pi r_{6371}} - \frac{h c}{2 \pi r_{42164}} = 0.412 x10^{-32}J</math> (energy per orbital) :<math>N_{orbitals} = \frac{M_{earth}m_{satellite}}{m_P^2} = 0.126 x10^{41}</math> (number of orbitals) :<math>E_{total} = E_{orbital} N_{orbitals} = 53 MJ/kg</math> 4) The orbital angular momentum of the planets derived from the angular momentum of the respective orbital pairs. :<math>N_{sun} = \frac{M_{sun}}{m_P} </math> :<math>N_{planet} = \frac{M_{planet}}{m_P} </math> :<math>N_{orbitals} = N_{sun}N_{planet} </math> :<math>n_g = \sqrt{\frac{R_{radius} m_P}{2 \alpha l_p M_{sun}}} </math> :<math>L_{oam} = 2\pi \frac{M r^2}{T} = N_{orbitals} n_g\frac{h}{2\pi} \sqrt{2 \alpha},\;\frac{kg m^2}{s} </math> The orbital angular momentum of the planets; mercury = .9153 x10³⁹ venus = .1844 x10⁴¹ earth = .2662 x10⁴¹ mars = .3530 x10⁴⁰ jupiter = .1929 x10⁴⁴ pluto = .365 x10³⁹ Orbital angular momentum combined with orbit velocity cancels ''n_g'' giving an orbit constant. Adding momentum to an orbit will therefore result in a greater distance of separation and a corresponding reduction in orbit velocity accordingly. :<math>L_{oam}v_g = N_{orbitals} \frac{h c}{2\pi},\;\frac{kg m^3}{s^2} </math> [[File:orbit-points32-orbitals496-clumping-over-time.gif|thumb|right|640px|32 mass points (496 orbitals) begin with random co-ordinates, after 2³² steps they have clumped to form 1 large mass and 2 orbiting masses.]] ==== Freely moving points ==== The simulation calculates each point as if freely moving in space, and so is useful with 'dust' clouds where the freedom of movement is not restricted. In this animation, 32 mass points begin with random co-ordinates (the only input parameter here are the start (''x'', ''y'') coordinates of each point). We then fast-forward 2³² steps to see that the points have now clumped to form 1 larger mass and 2 orbiting masses. The larger center mass is then zoomed in on to show the component points are still orbiting each other, there are still 32 freely orbiting points, only the proximity between them has changed, they have formed ''planets''. [[File:Gravitational-potential-energy-8body-1-2.gif|thumb|right|640px|8-body circular orbit plus 1-body with opposing orbitals 1:2]] ==== Orbital trajectory (circular vs. straight) ==== Orbital trajectory is a measure of alignment of the orbitals. In the above examples, all orbitals rotate in the same direction = aligned. If all orbitals are unaligned the object will appear to 'fall' = straight line orbit. In this example, for comparison, onto an 8-body orbit (blue circle orbiting the center mass green circle), is imposed a single point (yellow dot) with a ratio of 1 orbital (anti-clockwise around the center mass) to 2 orbitals (clockwise around the center mass) giving an elliptical orbit. The change in orbit velocity (acceleration towards the center and deceleration from the center) derives automatically from the change in the orbital radius (there is no barycenter). The orbital drift (as determined where the blue and yellow meet) is due to these orbiting points rotating around each other. ==== Precession ==== Can the orbital plane also rotate? semi-minor axis: <math>b = \alpha l^2 \lambda_A</math> semi-major axis: <math>a = \alpha n^2 \lambda_A</math> radius of curvature :<math>L = \frac{b^2}{a} = \frac{a l^4 \lambda_A}{n^2}</math> :<math>\frac{3 \lambda_A}{2 L} = \frac{3 n^2}{2 \alpha l^4}</math> arc secs per 100 years (drift): :<math>T_{earth}</math> = 365.25 days drift = <math>\frac{3 n^2}{2 \alpha l^4} 1296000 \frac{100 T_{earth}}{T_{planet}}</math> Mercury (eccentricity = 0.205630) T = 87.9691 days a = 57909050 km (''n'' = 378.2734) b = 56671523 km (''l'' = 374.2096) drift = 42.98 Venus (eccentricity = 0.006772) T = 224.701 days a = 108208000 km (''n'' = 517.085) b = 108205519 km (''l'' = 517.079) drift = 8.6247 Earth (eccentricity = 0.0167) T = 365.25 days a = 149598000 km (''n'' = 607.989) b = 149577138 km (''l'' = 607.946) drift = 3.8388 Mars (eccentricity = 0.0934) T = 686.980 days a = 227939366 km (''n'' = 750.485) b = 226942967 km (''l'' = 748.843) drift = 1.351 [[File:relativistic-quantum-gravity-orbitals-codingthecosmos.png|thumb|right|480px|Illustration of B's cylindrical orbit relative to A's time-line axis]] ==== Hyper-sphere orbit ==== {{main|Relativity (Planck)}} Each point moves 1 unit of (Planck) length per 1 unit of (Planck) time in '''x, y, z''' (hyper-sphere) co-ordinates, the simulation 4-axis hyper-sphere universe expanding in uniform (Planck) steps (the simulation clock-rate) as the origin of the speed of light, and so (hyper-sphere) time and velocity are constants. Particles are pulled along by this expansion, the expansion as the origin of motion, and so all objects, including orbiting objects, travel at, and only at, the speed of light in these hyper-sphere co-ordinates <ref>Macleod, Malcolm; {{Cite journal |title=1. Programming relativity for Planck unit Simulation Hypothesis modelling |journal=RG |date=26 March 2020 | doi=10.13140/RG.2.2.18574.00326/3 }}</ref>. Time becomes [[v:God_(programmer)#Universe_time-line |time-line]]. While ''B'' (satellite) has a circular orbit period on a 2-axis plane (the horizontal axis representing 3-D space) around ''A'' (planet), it also follows a cylindrical orbit (from B¹ to B¹¹) around the ''A'' time-line (vertical expansion) axis ('''t_d''') in hyper-sphere co-ordinates. ''A'' is moving with the universe expansion (along the time-line axis) at (''v = c''), but is stationary in 3-D space (''v'' = 0). ''B'' is orbiting ''A'' at (''v = c''), but the time-line axis motion is equivalent (and so `invisible') to both ''A'' and ''B'', as a result the orbital period and velocity measures will be defined in terms of 3-D space co-ordinates by observers on ''A'' and ''B''. For object '''B''' :<math>t_d = t \sqrt{1 - v_{outer}^2}</math> For object '''A''' :<math>t_d = t \sqrt{1 - v_{inner}^2}</math> === Atomic orbitals === [[File:H-orbit-transitions-n1-n2-n3-n1.gif|thumb|right|640px|fig 5. H atom orbital transitions from n1-n2, n2-n3, n3-n1 via 2 photon capture, photons expand/contract the orbital radius. The spiral pattern emerges because the electron is continuously pulled in an anti-clockwise direction by the rotating orbital.]] In the atom we find individual particle to particle orbitals, and as such the atomic orbital is principally a wave-state orbital (during the orbit the electron is predominately in the electric wave-state). The wave-state is defined by a wave-function, we can however map (assign co-ordinates to) the mass point-states and so follow the electron orbit, for example, in 1 orbit at the lowest energy level in the H atom, the electron will oscillate between wave-state to point-state approximately 471960 times. This means that we can treat the atomic orbital as a simple 2-body orbit with the electron as the orbiting point. Although this approach can only map the electron point-state (and so offers no direct information regarding the electron as a wave), during electron transition between ''n''-shell orbitals, we find the electron follows a [[v:Fine-structure_constant_(spiral) |hyperbolic spiral]], this is significant because periodically the spiral angle components converge reducing to integer radius values (360°=4''r'', 360+120°=9''r'', 360+180°=16''r'', 360+216°=25''r'' ... 720°=∞''r''). As these spiral angles (360°, 360+120°, 360+180°, 360+216° ...) are linked directly to pi via this spiral geometry, we may ask if quantization of the atom has a geometrical origin. <ref>Macleod, Malcolm J.; {{Cite journal |title=Simulating gravitational and atomic orbits via rotating particle-particle orbital pairs |journal=RG |date=Dec 2024 | doi=10.13140/RG.2.2.11378.00961}}</ref>. ==== Theory ==== {{see|Fine-structure_constant_(spiral)}} =====Hyperbolic spiral===== [[File:Hyperbol-spiral-1.svg|thumb|right|320px|Hyperbolic spiral]] A [[w:hyperbolic spiral |hyperbolic spiral]] is a type of [[w:spiral|spiral]] with a pitch angle that increases with distance from its center. As this curve widens (radius '''r''' increases), it approaches an [[w:asymptotic line|asymptotic line]] (the '''y'''-axis) with the limit set by a scaling factor '''a''' (as '''r''' approaches infinity, the '''y''' axis approaches '''a'''). The spiral shape that the electron maps can be represented in Cartesian coordinates. Periodically the angles converge to give integer radius, the general form (beginning at the outer limit ranging inwards) gives; :<math>x = a^2 \frac{cos(\varphi)}{\varphi^2},\; y = a^2 \frac{sin(\varphi)}{\varphi^2},\;0 < \varphi < 4\pi</math> :radius = <math>\sqrt(x^2 + y^2) r</math> :<math>\varphi = (2)\pi, \; 4r</math> (360°) :<math>\varphi = (4/3)\pi,\; 9r</math> (240°) :<math>\varphi = (1)\pi, \; 16r</math> (180°) :<math>\varphi = (4/5)\pi, \; 25r</math> (144°) :<math>\varphi = (2/3)\pi, \; 36r</math> (120°) [[File:Bohr_atom_model_English.svg|thumb|right|320px|Electron at different ''n'' level orbitals]] =====Principal quantum number '''n'''===== The H atom has 1 proton and 1 electron orbiting the proton, in the [[w:Bohr model |Bohr model]] (which approximates a gravitational orbit), the electron can be found at select radius ([[w:Bohr radius |the Bohr radius]]) from the proton (nucleus), these radius represent the permitted energy levels (orbital regions) at which the electron may orbit the proton. Electron transition (to a higher energy level) occurs when an incoming photon provides the required energy (momentum). Conversely emission of a photon will result in electron transition to a lower energy level. The [[w:principal quantum number |principal quantum number ''n'']] denotes the energy level for each orbital. As ''n'' increases, the electron is at a higher energy level and is therefore less tightly bound to the nucleus (as ''n'' increases, the electron orbit is further from the nucleus). Each shell can accommodate up to ''n''<sup>2</sup> (1, 4, 9, 16 ... ) electrons. Accounting for two states of spin this becomes 2''n''<sup>2</sup> electrons. As these energy levels are fixed according to this integer ''n'', the orbitals may be said to be quantized. =====(Bohr) orbital===== The basic orbital radius has 2 components, dimensionless (the [[w:fine structure constant|fine structure constant alpha]]) and dimensioned (electron + proton wavelength); wavelength = <math>\lambda_H = \lambda_p + \lambda_e</math> radius = <math>r_{orbital} = 2\alpha n^2 (\lambda_H)</math> As a mass point, the electron orbits the proton at a fixed radius (the Bohr radius) in a series of steps (the duration of each step corresponds to the wavelength component). The distance travelled per step (per wave-point oscillation) equates to the distance between mass point states and is the inverse of the radius [[File:atomic-orbital-rotation-step.png|thumb|right|208px|electron (blue dot) moving 1 step anti-clockwise along the alpha orbital circumference]] length = <math>l_{orbital} = \frac{1}{r_{orbital}}</math> Duration = 1 step per wavelength and so velocity velocity = <math>v_{orbital} = \frac{1}{2\alpha n}</math> Giving period of orbit period = <math>t_{orbital} = \frac{2\pi r_{orbital}} {v_{orbital}} = 2\pi 2\alpha 2\alpha n^3 \lambda_H</math> As we are not mapping the wavelength component, a base (reference) orbital (''n''=1) :<math>t_{ref} = 2\pi 4\alpha^2</math> = 471964.356... The angle of rotation depends on the orbital radius :<math>\beta = \frac{1}{r_{orbital} \sqrt{r_{orbital}}\sqrt{2\alpha}}</math> ===== Photon orbital model ===== The electron can jump between ''n'' energy levels via the absorption or emission of a photon. In the Photon-orbital model<ref>Macleod, Malcolm J.; {{Cite journal |title=Simulating gravitational and atomic orbits via rotating particle-particle orbital pairs |journal=RG |date=Dec 2024 | doi=10.13140/RG.2.2.11378.00961}}</ref>, the orbital (Bohr) radius is treated as a 'physical wave' akin to the photon albeit of inverse or reverse phase such that <math>orbital \;radius + photon = zero</math> (cancel). The photon can be considered as a moving wave, the orbital radius as a standing/rotating wave (trapped between the electron and proton), as such it is the orbital radius that absorbs or emits the photon during transition, in the process the orbital radius is extended or reduced (until the photon is completely absorbed/emitted). The electron itself has a `passive' role in the transition phase. It is the rotation of the orbital radius that pulls the electron, resulting in the electron orbit around the nucleus (orbital momentum comes from the orbital radius), and this rotation continues during the transition phase resulting in the electron following a spiral path. The photon is actually 2 photons as per the Rydberg formula (denoted initial and final). :<math>\lambda_{photon} = R.(\frac{1}{n_i^2}-\frac{1}{n_f^2}) = \frac{R}{n_i^2}-\frac{R}{n_f^2}</math> :<math>\lambda_{photon} = (+\lambda_i) - (+\lambda_f)</math> The wavelength of the (<math>\lambda_i</math>) photon corresponds to the wavelength of the orbital radius. The (+<math>\lambda_i</math>) will then delete the orbital radius as described above (''orbital'' + ''photon'' = ''zero''), however the (-<math>\lambda_f</math>), because of the Rydberg minus term, will have the same phase as the orbital radius and so conversely will increase the orbital radius. And so for the duration of the (+<math>\lambda_i</math>) photon wavelength, the orbital radius does not change as the 2 photons cancel each other; :<math>r_{orbital} = r_{orbital} + (\lambda_i - \lambda_f)</math> However, the (<math>\lambda_f</math>) has the longer wavelength, and so after the (<math>\lambda_i</math>) photon has been absorbed, and for the remaining duration of this (<math>\lambda_f</math>) photon wavelength, the orbital radius will be extended until the (<math>\lambda_f</math>) is also absorbed. For example, the electron is at the ''n'' = 1 orbital. To jump from an initial <math>n_i = 1</math> orbital to a final <math>n_f = 2</math> orbital, first the (<math>\lambda_i</math>) photon is absorbed (<math>\lambda_i + \lambda_{orbital} = zero</math> which corresponds to 1 complete ''n'' = 1 orbit by the electron, the '''orbital phase'''), then the remaining (<math>\lambda_f</math>) photon continues until it too is absorbed (the '''transition phase'''). :<math>t_{ref} \sim 2\pi 4\alpha^2 </math> :<math>\lambda_i = 1t_{ref}</math> :<math>\lambda_f = 4t_{ref}</math> (''n'' = 2) After the (<math>\lambda_i</math>) photon is absorbed, the (<math>\lambda_f</math>) photon still has <math>\lambda_f = (n_f^2 - n_i^2)t_{ref} = 3 t_{ref}</math> steps remaining until it too is absorbed. [[File:atomic-orbital-transition-alpha-steps.png|thumb|right|277px|orbital transition during orbital rotation]] This process does not occur as a single `jump' between energy levels by the electron, but rather absorption/emission of the photon takes place in discrete steps, each step corresponds to a unit of <math>r_{incr}</math> (both photon and orbital radius may be considered as constructs from multiple units of this geometry); :<math>r_{incr} = -\frac{1}{2 \pi 2\alpha r_{wavelength}}</math> In summary; the (<math>\lambda_i</math>) photon, which has the same wavelength as the orbital radius, deletes the orbital radius in step <math>r = r_{orbital}</math> WHILE (<math>\lambda_i > 0</math>) :<math>r = r + r_{incr}</math> ://<math>\lambda_i</math> photon Conversely, because of its minus term, the (<math>\lambda_i</math>) photon will simultaneously extend the orbital radius accordingly; WHILE (<math>r < 4 r_{orbital}</math>) :<math>r = r - r_{incr}</math> ://<math>\lambda_f</math> photon The model assumes orbits also follow along a [[Quantum_gravity_(Planck)#Hyper-sphere_orbit|timeline ''z''-axis]] :<math>t_{orbital} = t_{ref} \sqrt{1 - \frac{1}{(v_{orbital})^2}}</math> The orbital phase has a fixed radius, however at the transition phase this needs to be calculated for each discrete step as the orbital velocity depends on the radius; :<math>t_{transition} = t_{ref} \sqrt{1 - \frac{1}{(v_{transition})^2}}</math> ==== Simulation ==== The simulation treats the atomic orbital as a 2-body gravitational orbit with the electron (single point) orbiting a central mass - the nucleus. The nucleus is a set of individual points (also orbiting each other) and not a static mass (static entity). The difference between gravitational and atomic orbits is only in the angle of rotation <math>\beta</math>' which has an additional <math>r_{\alpha}</math> term included as the atomic orbital wavelength component is dominated by the particle wave-state (the mass-state is treated as a point), and so velocity along the 2-D (gravitational) plane (we are only mapping the radial component of the orbital) will decrease proportionately. :<math>\beta = \frac{1}{r_{orbital} \sqrt{r_{orbital}} \sqrt{2\alpha}}</math> [[File:Alpha-hyperbolic-spiral.gif|thumb|right|640px|Bohr radius during ionization, as the H atom electron reaches each ''n'' level, it completes 1 orbit (for illustration) then continues outward (actual velocity will become slower as radius increases according to angle β)]] =====Spiral angle===== For an idealized Rydberg atom (a nucleus of point size, infinite mass and disregarding wavelength). In this example the electron transition starts at the initial (''n''<sub>i</sub> = 1) orbital :<math>\varphi = 0, \;r_{orbital} = 2\alpha</math> For each step during transition; :<math>\beta = \frac{1}{r_{orbital} \sqrt{r_{orbital}}\sqrt{2\alpha}}</math> :<math>\varphi = \varphi + \beta</math> Setting t = step number (FOR t = 1 TO ...), we can calculate the radius ''r'' and <math>n_f^2</math> at each step. :<math>r = r_{orbital} + \frac{t}{2\pi 2\alpha}</math> (number of increments ''t'' of <math>r_{incr}</math>) :<math>n_f^2 = 1 + \frac{t}{2\pi 4\alpha^2}</math> (<math>n_f^2</math> as a function of ''t'') :<math>\varphi =4 \pi \frac{(n_f^2 - n_f)}{n_f^2}</math> (<math>\varphi</math> at any <math>n_f^2</math>) We can then re-write (<math>n_f</math> is only an integer at prescribed spiral angles); :<math>\beta = \frac{1}{{r_{orbital}}^2 n_f^3}</math> Giving integer values at these spiral angles :<math>\varphi = (2)\pi, \; r = 4 r_{orbital}</math> (360°) :<math>\varphi = (8/3)\pi,\; r = 9 r_{orbital}</math> (360+120°) :<math>\varphi = (3)\pi, \; r = 16 r_{orbital}</math> (360+180°) :<math>\varphi = (16/5)\pi, \; r = 25 r_{orbital}</math> (360+216°) :<math>\varphi = (10/3)\pi, \; r = 36 r_{orbital}</math> (360+240°) :<math>\varphi = (7/4)\pi, \; r = 49 r_{orbital}</math> :<math>\varphi = (7/2)\pi, \; r = 64 r_{orbital}</math> (360+270°) ===== Rydberg atom ===== At the ''n'' = 1 orbital, 1 complete rotation becomes (dimensionless terms measured on a 2-D plane); :<math>t_{ref} = \frac{2\pi r_{orbital}}{v_{orbital}} = 2\pi 2\alpha 2\alpha</math> :<math>1t_{ref}</math> = 471964.3563... :<math>4t_{ref}</math> = 1887857.4255... :<math>9t_{ref}</math> = 4247679.2074... :<math>16t_{ref}</math> = 7551429.7021... ===== H atom ===== Experimental values for H(1s-ns) transitions (''n'' the [[w:principal quantum number |principal quantum number]]). H(1s-2s) = 2466 061 413 187.035 kHz <ref>http://www2.mpq.mpg.de/~haensch/pdf/Improved%20Measurement%20of%20the%20Hydrogen%201S-2S%20Transition%20Frequency.pdf</ref> H(1s-3s) = 2922 743 278 665.79 kHz <ref>https://pubmed.ncbi.nlm.nih.gov/33243883/</ref> H(1s-4s) = 3082 581 563 822.63 kHz <ref>https://codata.org/</ref> H(1s-∞s) = 3288 086 857 127.60 kHz <ref>https://codata.org/ (109678.77174307cm-1)</ref> (''n'' = ∞) R = 10973731.568157 <ref>https://codata.org/ (mean)</ref> ([[w:Rydberg constant |Rydberg constant]]) α =137.035999177 (inverse fine structure constant <ref>https://codata.org/ (mean)</ref> The wavelength of the H atom, for simplification the respective particle wavelengths are presumed constant irrespective of the vicinity of the electron to the proton. <math>r_{wavelength} = \lambda_H = \frac{2c}{\lambda_e + \lambda_p}</math> Dividing (dimensioned) wavelength (<math>r_{wavelength}</math>) by the (dimensioned) transition frequency returns a dimensionless number (the alpha component of the photon). The <math>(n^2 - 1)</math> term gives the number of orbital wavelengths in the transition phase; :<math>h_{(1s-ns)} = (n^2 - 1) \frac{\lambda_H }{H(1s-ns)}</math> <math>h_{(1s-2s)}</math> = 1887839.82626... <math>h_{(1s-3s)}</math> = 4247634.04874... <math>h_{(1s-4s)}</math> = 7551347.55306... ===== Simulation atom ===== The following example simulates an electron transition, the electron begins at radius <math>r = r_{orbital}</math> and makes a 360° rotation at orbital radius (the orbital phase) and then moves in incremental steps to higher orbitals (the transition phase) mapping a hyperbolic spiral path (red line) in the process (photon orbital model). The period <math>t_{sim}</math> and length <math>l_{sim}</math> are measured at integer <math>n^2 r</math> (''n'' = 1, 2, 3...) radius. For a Rydberg atom, these radius correspond precisely to the electron path at the [[v:Fine-structure_constant_(spiral) |(hyperbolic) spiral]] angles; (360°(''1r''), 360°(''4r''), 360+120°(''9r''); 360+180°(''16r''), 360+216°(''25r''), 360+240°(''36r'') ...) (the angles converge to give integer values at these radius), and so we find that as the simulation nucleus mass increases, the integer radius values approach these angles (table 2.). The period <math>t_{sim}</math> can then be used to calculate the transition frequencies. In this example, the nucleus = 249 mass points (start ''x'', ''y'' co-ordinates close to 0, 0) and the electron = 1 mass point (at radius ''x'' = ''r'', ''y'' = 0), ''t''<sub>sim</sub> = period and ''l''<sub>sim</sub> = distance travelled by the electron (<math>l_{orbital} = l_{sim}</math> at ''n''=1), the radius coefficient ''r''<sub>n</sub> = radius divided by <math>r_{orbital}</math>. As this is a gravitational orbit, although the nucleus comprises 249 points clumped close together, these points are independent of each other (they also rotate around each other), and so the `nucleus' size and shape is not static (the simulation is not optimised for a nucleus). Table 1. gives the relative values and the ''x'', ''y'' co-ordinates for the electron, nucleus center and barycenter. [[File:H-atom-electron-transition-nucleus-plot.gif|thumb|right|640px|H atom electron transition spiral plotting the nucleus and barycenter as the electron transitions from n=1 to n=8]] :<math>j_{atom} = 250</math> (atomic mass) :<math>i_{nucleus} = j_{atom} -1 = 249</math> (relative nucleus mass) :<math>r_{wavelength} = 2 (\frac{j_{atom}}{i_{nucleus}})^2</math> = 2.0160965 :<math>r_{orbital} = 2 \alpha \;*\; r_{wavelength} </math> (radius) = 552.5556 :<math>t_n = \frac{t_{sim}}{r_{wavelength}}</math> :<math>l_n = \frac{l_{sim}}{l_{orbital}} - l_{orbital}</math> :<math>r_b = r_{sim} - \frac{r_{sim}}{j_{atom}}</math> :<math>r_n = \frac{r_b}{r_{orbital}}</math> {| class="wikitable" |+table 1. Electron transition (mass = 250; ''r''_n= 1 to 5) ! ''r''_n ! ''t''_sim ! ''l''_n ! angle ! ''x'', ''y'' (electron) ! ''x'', ''y'' (nucleus) ! ''x'', ''y'' (barycenter) |-1 | 1 | 471957.072 | 0.9999897 | 360° | 550.334, 0.0036 | -2.2102, -0.00002 | -0.00004, -0.00001 |- | 4 | 1887867.293 | 2.000012 | 359.952489° | 2202.8558, 0.0001 | -7.9565, -1.9475 | 0.8868, -1.9397 |- | 9 | 4247689.502 | 4.000014 | 119.92712° | -2473.180, 4296.283 | 13.558, -10.325 | 3.611, 6.901 |- | 16 | 7551439.538 | 6.000014 | 179.91669° | -8815.254, 12.818 | 25.636, 13.303 | -9.728, 13.301 |- | 25 | 11799118.905 | 8.000014 | 215.9122° | -11158.64, -8081.13 | 16.580, 39.083 | -28.118, 6.602 |} Comparison of the spiral angle at ''r''_n = 4, 9, 16 (360, 360+120, 360+180) with different mass (''m'' = 64, 128, 250, 500, Rydberg). For the proton:electron mass ratio; ''m'' = 1836.15267... {| class="wikitable" |+ table 2. Spiral angle at <math>r_n</math> = 4, 9, 16 ! mass ! ''r''_n = 4 ! ''r''_n = 9 ! ''r''_n = 16 |- | ''m'' = 64 | 359.80318° | 119.70323° | 179.66239° |- | ''m'' = 128 | 359.90394° | 119.85415° | 179.83377° |- | ''m'' = 250 | 359.95249° | 119.92711° | 179.91669° |- | ''m'' = 500 | 359.97706° | 119.96501° | |- | Rydberg | 360° | 360+120° | 360+180° |} == External links == * [[v:Fine-structure_constant_(spiral) | Fine structure constant hyperbolic spiral]] * [[v:Physical_constant_(anomaly) | Physical constant anomalies]] * [[v:Planck_units_(geometrical) | Planck units as geometrical objects]] * [[v:electron_(mathematical) | The mathematical electron]] * [[v:Relativity_(Planck) | Programming relativity at the Planck scale]] * [[v:Black-hole_(Planck) | Programming the cosmic microwave background at the Planck level]] * [[v:Sqrt_Planck_momentum | The sqrt of Planck momentum]] * [[v:God_(programmer) | The Programmer God]] * [https://codingthecosmos.com/ Simulation hypothesis modelling at the Planck scale using geometrical objects] * [https://theprogrammergod.com/ The Programmer God, are we in a computer simulation? - eBook] ==References== {{Reflist}} [[Category:Physics| ]] [[Category:Philosophy of science| ]] 6wi0wyfsgmkab9gveuugwmzhghn15o2 0 263813 2692974 2692834 2024-12-22T22:50:10Z Scogdill 1331941 2692974 wikitext text/x-wiki ==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. 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, usually with a quick description of what she wore. The Hon. Algernon Bourke and Mr. Algernon, 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. == Acquaintances, Friends and Enemies == === Mr. 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> == Organizations == === The Hon. 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 * 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> * Stock Exchange * 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> *One of the directors, the Franco-English Tunisian Esparto Fibre Supply Company, Ltd.<ref>''Money Market Review'', 20 Jan 1883 (Vol 46): 124.</ref> *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> *[[Social Victorians/Timeline/1896#25 March 1896, Wednesday|The Sala Memorial Fund]], member of the committee (from 25 March 1896) === Mr. Algernon Bourke === * Head, Messrs. Bourke and Sandys, "that well-known firm of stockbrokers"<ref name=":8" /> == 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> '''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> '''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 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> '''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.<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> '''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 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><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> '''1901 May 30, Thursday''', the Hon. Mrs. Algernon Bourke attended the fashionable Ladies' Kennel Association Dog Show. '''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. '''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> '''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. '''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> '''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]]. '''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]]. '''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> '''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> '''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> == 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? *1890: 33 Cadogan Terrace *1891: 33 Cadogan Terrace, 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> *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. == Footnotes == {{reflist}} r5vq4fna21d1hr269ocb17hwv3h5lmg 0 264026 2692973 2691052 2024-12-22T22:50:05Z Scogdill 1331941 2692973 wikitext text/x-wiki == Overview == The "aristocratic lady writer" [[Social Victorians/People/Lady Violet Greville |Lady Violet Greville]] (Beatrice Violet Graham Greville) was the daughter of James Graham, 4th Duke of Montrose and Caroline Graham, Duchess of Montrose, who married two more times after the Duke of Montrose died in 1874. The 5th Duchess, sister in law of Lady Violet Graham the writer, was also named Violet: Violet Hermione (née Graham) Graham. Known on the track as Mr Manton, Caroline Graham ran a successful stable and had a larger-than-life and, apparently, difficult personality. In "The Vagaries of 'Mr Manton," the ''Dundee Evening Telegraph'' reports stories about Caroline, Duchess of Montrose's temper, her loss of a lady's maid she had mistreated and her demand that she be carried up the new elevator shaft by a groomsman:<blockquote>The death of the Dowager-Duchess Montrose has deprived sporting society of one of its most remarkable figures and smoking-room raconteurs of inexhaustible material. "Mr Manton's" notoriously violent temper was responsible for most of the anecdotes of her vagaries, and for her success on the turf not having been what it might have been had she been able to keep her temper and her trainers. Some three years ago I told the story — a true one — of her Grace's maid who, determined to avenge the insults heaped on her while doing her mistress's hair before dinner, carefully plaited it in and out of the back of the chair, and when the Duchess was thus firmly secured to a not very light piece furniture, gave her a violent box on the ear, walked out of the room, and five minutes later left the house, her luggage having by arrangement with the other servants already been despatched to the station. The sequel is characteristic of the late Duchess. She declared she would never again run such risk, and had a special backless seat purchased; and on none other would she sit when having her hair done for several months after the aforesaid maid's departure. Another story is fairly common property — of how, when trying by herself to work a new lift in her house, before the engineers had passed it ready for use, she was carried down into the "well," whence she shouted violent expletives at her servants above, until a footman secured and descended a ladder at considerable risk, and carried the lady up under one arm. She brought an action against the lift-makers to recover damages for "shock to the system," but lost the day. Mr Milner, her third husband, and now her widower, was born nearly 50 years after the lady whose loss he doubtless mourns. — ''Woman''.<ref>"Woman" [pseud]. "The Vagaries of 'Mr Manton." ''Dundee Evening Telegraph'' 22 November 1894, Thursday: 2 [of 4], Col. 2c. ''British Newspaper Archive'' http://www.britishnewspaperarchive.co.uk/viewer/bl/0000453/18941122/008/0002. Print title: The ''Evening Telegraph''; n.p.</ref></blockquote>Caroline Graham, 4th Duchess died in 1894, but Marcus Henry Milner, her 3rd husband, more than 40 years younger than she was, attended the [[Social Victorians/1897 Fancy Dress Ball |Duchess of Devonshire's 2 July 1897 fancy-dress ball]] with the 5th Duke and Duchess. == Also Known As == *Family name: Graham *Duke of Montrose **Douglas Beresford Malise Ronald Graham, 5th Duke (30 December 1874 – 10 December 1925)<ref>{{Cite journal|date=2020-08-04|title=Douglas Graham, 5th Duke of Montrose|url=https://en.wikipedia.org/w/index.php?title=Douglas_Graham,_5th_Duke_of_Montrose&oldid=971139213|journal=Wikipedia|language=en}}</ref> *Duchess of Montrose **Caroline Agnes Horsley-Beresford Graham (30 December 1836 – 16 November 1894) **Violet Hermione Graham Graham (24 July 1876 – 21 November 1940) *Caroline Agnes Horsley-Beresford Graham Stirling-Crawford Milner was also known as Mr. Manton for her horse-breeding and -racing operations. *Marcus Henry Milner: "Mr. 'Harry' Milner (familiarly known in the City as 'Millions')."<ref name=":3">"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> == Acquaintances, Friends and Enemies == === Marcus Henry Milner === * Mr. [[Social Victorians/People/Mildmay|F. B. (Francis Bingham) Mildmay]] * Mr. St. John Wontner, "the well-known solicitor practising in the West End police-courts"<ref name=":4">"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> == Organizations == * Messrs. Bourke and Sandys: "one of the zealous assistants of that well-known firm of stockbrokers"<ref name=":3" /> headed by [[Social Victorians/People/Bourke|Algernon Bourke]] == Timeline == '''1836 October 15''', James Graham and Caroline Agnes Horsley-Beresford married.<ref name=":1">"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> '''1874 December 30''', James Graham, 4th Duke of Montrose, died. '''1876 January 22''', Caroline Agnes Horsley-Beresford Graham and William Stuart Stirling-Crawford married.<ref name=":1" /> '''1876 July 24''', Douglas Beresford Malise Ronald Graham and Violet Hermione Graham married.<ref name=":2">"Violet Hermione Graham." {{Cite web|url=https://thepeerage.com/p5396.htm#i53952|title=Person Page|website=thepeerage.com|access-date=2020-11-21}}</ref> '''1888 July 26''', Caroline Agnes Horsley-Beresford Graham Stirling-Crawford and Marcus Henry Milner married.<ref name=":1" /> According to the ''Nottingham Evening Post'' of 31 July 1888,<blockquote>METROPOLITAN NOTES. [FROM LONDON CORRESPONDENTS.] London, Monday Evening. Society has not had for some time so piquant a piece of gossip to deal with as that arising out of the marriage of the Dowager Duchess of Montrose a few days since with Mr. Milner. In any case, the union of bride of seventy with a bridegroom of twenty-four would have caused remark, but this has created the more because of the extreme secrecy with which it is surrounded. The '''church authorities St. Andrew's''', Fulham, where the ceremony was performed, knew nothing, l am told, of the condition of the parties to the suit until the licence was in their hands, and though the Duchess was arrayed in bridal costume there was no lady either with her or in the church at the time, she being accompanied only by Mr. [[Social Victorians/People/Bourke|Algernon Bourke]], who was stated to be her legal adviser, the bridegroom being similarly supported only by lawyer, in the person of Mr. St. John Wontner, the well-known solicitor practising in the West End police-courts. The bridegroom, who had earned the necessary residential qualification by staying at [[Social Victorians/London Clubs#Queen's|the Queen's Club]], led his wife, after the ceremony, into the vestry, where the marriage settlements were signed; but they did not drive away, the Duchess departing as she had come with Mr. Bourke. How the news of the wedding first reached the '''friends not''' the least curious part of the story, which runs that Mr. [[Social Victorians/People/Mildmay|Mildmay]], the Liberal Unionist member for Totnes, who is a personal acquaintance the bridegroom, happened go to St. Andrew's to practice upon the organ at the time the ceremony was to take place, and, to his astonishment, saw his friend being married. The reason given at the church for the extreme quietude of the wedding was that there had recently been a death in one of the families.<ref name=":4" /></blockquote>'''1888 November 22''', the Duchess of Montrose was in court about an unpaid bill; there is also a mention of a visit by the Prince of Wales:<blockquote>In the Queen's Bench Division, London, on Wednesday, Mr Sanders, carrying on business as an orchid grower at St Alban's, commenced an action against the Duchess of Montrose to recover £1,730 for orchids supplied and for furnishing a conservatory at Newmarket. A sum of £700 was paid into court, and the remaining £1,000 was disputed on the ground that the orchids had not been supplied to order. For the plaintiff it was stated that her grace had expressed herself delighted with the orchids, but thought there were too many white flowers. She said the Prince of Wales was coming there to dinner, and asked plaintiff whether he thought he could supply in time some coloured plants. The plaintiff said he could, and in his evidence said the original arrangement was that he should supply 1,000 orchids for 1,000 guineas. Several experts were examined, and expressed the opinion that the prices charged for work done and orchids supplied were fair. The further hearing of the case was adjourned.<ref>"Action against 'Mr Manton.' The Prince of Wales and the Orchids." ''South Wales Daily News'' 22 November 1888, Thursday: 2 [of 4]. ''British Newspaper Archive'' http://www.britishnewspaperarchive.co.uk/viewer/bl/0000919/18881122/028/0002.</ref></blockquote>'''1894 July 28, Saturday''', the ''Aberdeen Evening Express'' published a report about conflict between the Duchess and Mr. Milner:<blockquote>Racing circles are following with eager interest the conflict between "Mr Manton" and her husband the manner in which the latter's annuity is to be paid. When the Dowager-Duchess Montrose married Mr Harry Milner she assigned to him three thousand year out of the five thousand pounds that came to her under '''the will Mr Stirling Crawfurd''', the understanding being that all the instalments should be paid to the credit of the Duchess's account with her bankers in London. Mr Milner was a party to this arrangement, but he has now revoked it, to the indignation of his wife, who is contesting the matter hotly. It is now six years since the Dowager-Duchess made her last matrimonial experiment. Her husband was at the time one of the zealous assistants of that well-known firm of stockbrokers, Messrs Bourke & Sandys; and Mr [[Social Victorians/People/Bourke|Algernon Bourke]], the head the firm, went down to Fulham to give her away. The honeymoon began with a visit to the Jockey Club box Sandown and ended at the Lake of Geneva. Even now '''Mr Henry''' Mr Henry Milner is only in his thirtieth year, while his wife first saw the light in 1818, and was married to the fourth Duke of Montrose close on thirty years before her present husband was born. Lady Breadalbane and Lady Greville, who is still Lady Violet Greville when writing plays, never took Mr Henry Milner seriously as a stepfather, nor is he regarded with the veneration due to a parent by the comparatively venerable Duke of Montrose.<ref name=":0">"'Mr Manton' and Mr Harry Milner." ''Aberdeen Evening Express'', 28 July 1894, Saturday: 2 [of 6], Col. 7c. ''British Newspaper Archive'' http://www.britishnewspaperarchive.co.uk/viewer/bl/0000444/18940728/022/0002.</ref></blockquote> '''1894 November 16, Friday''', Caroline, Duchess of Montrose, died:<blockquote>Caroline, Duchess of Montrose, passed away at 1.20 this morning. The duchess was well known in the racing world as "Mr Manton," and at the time of her death had some 17 horses in training, including Shrine, Grand Duke, Medora, Adoration, Lady Caroline, Broad Corrie None the Wiser, Jocasta Mecca, and Beggar's Opera. Last year her Grace won 11 races, the stakes amounting to over £4,000. The principal winners were Mecca, which was successful in several valuable juvenile races, and Medora, who won the Goodwood Stewards' Cup. Her chief success in the handicap department this year was achieved in the Cezarewitch, the "All scarlet" being carried into third place by the seven year old mare Shrine. The deceased Duchess had, during her long career, owned some of the best known race horses of the century, including the filly Thebais, who secured the One Thousand Guineas and the Oaks in 1891, and the Liverpool Autumn Cup three years later. Her Grace was very shrewd in equine matters, and her opinion was often sought by some of the highest personages in the country in regard to the purchase of thoroughbreds. Deceased had this year leased most of the Sefton Farm yearlings to Sir Frederick Johnstone, and they are now in training at Kingsclere. She struck all her horses out of Lincoln, Liverpool, and Manchester engagements on October 30th. The Duchess was a daughter of the second Baron Decies. She was born in 1818, and married, firstly, the fourth Duke of Montrose, who died in 1874; secondly, Mr William Stuart Stirling Crawford, who died in 1883; and thirdly, Mr Henry Miller [sic], who survives her. The deceased expired at her London residence in Belgrave-square, where she had been lying ill for several weeks. Although her illness had caused much anxiety to her family, her immediate death was not expected, and on Sunday a slight improvement being noticcd, the present Duke and Duchess left for Scotland, where they now are. Up to midnight no change appeared, but shortly afterwards her breathing became laboured, and she passed away quietly from sudden failure the heart, in the presence of her two daughters, [[Social Victorians/People/Lady Violet Greville|Baroness Greville]] and the [[Social Victorians/People/Breadalbane|Marchioness of Breadalbane]].<ref>"Death of the Duchess of Montrose ('Mr Manton')." ''Hull Daily Mail'' 16 November 1894, Friday: 3 [of 4], Col. 5c. ''British Newspaper Archive'' http://www.britishnewspaperarchive.co.uk/viewer/bl/0000323/18941116/031/0003.</ref></blockquote> '''1897 June 28, Monday''', according to the ''Morning Post'', Douglas, 5th Duke and Violet, Duchess of Montrose were invited to the 28 June [[Social Victorians/Diamond Jubilee Garden Party|Queen's Garden Party]], the official end of the Diamond Jubilee celebrations in London.<ref>“The Queen’s Garden Party.” ''Morning Post'' 29 June 1897, Tuesday: 4 [of 12], Cols. 1a–7c [of 7] and 5, Col. 1a–c. ''British Newspaper Archive'' ''<nowiki>https://www.britishnewspaperarchive.co.uk/viewer/BL/0000174/18970629/032/0004</nowiki>'' and ''<nowiki>https://www.britishnewspaperarchive.co.uk/viewer/bl/0000174/18970629/032/0005</nowiki>''.</ref> '''1897 July 2''', Douglas, 5th Duke of Montrose and Violet, Duchess of Montrose attended the [[Social Victorians/1897 Fancy Dress Ball | Duchess of Devonshire's fancy-dress ball]] at Devonshire House. == Costume at the Duchess of Devonshire's 2 July 1897 Fancy-dress Ball == At the [[Social Victorians/1897 Fancy Dress Ball | Duchess of Devonshire's fancy-dress ball]], Douglas Graham, 5th Duke of Montrose (at #170 in the list of people who attended), sat at Table 6. Violet Graham, Duchess of Montrose (at 186) sat at Table 8. According to the ''Westminster Gazette'', "[t]he Duchess of Montrose wore a sapphire-blue velvet gown, with muslin fichu and powdered hair."<ref>“The Duchess’s Costume Ball.” ''Westminster Gazette'' 03 July 1897 Saturday: 5 [of 8], Cols. 1a–3b [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0002947/18970703/035/0005.</ref>{{rp|Col. 1}} Harry<ref>"Personal Paragraphs." Dublin ''Evening Telegraph'' 01 August 1902 Friday: 2 [of 4], Col. 8b [of 9]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0002093/19020801/053/0002.</ref> ([[Social Victorians/People/Milner|Marcus Henry) Milner]] (at 612) — Caroline, Duchess of Montrose's 3rd husband — was dressed as a "Chasseur, Louis XV., ... after the picture by Van Loo; coat, Louis XV. period; turquoises richly embroidered on skirts, cuffs, breast, and pockets with beautiful gold embroidery; laced shirt and jabot, buff riding breeches; hat, three cornered, laced and edged with gold."<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|42, Col. 1a}} == Demographics == *Nationality: Anglo-Irish == Family == *James Graham, 3rd Duke of Montrose (8 September 1755 – 30 December 1836)<ref>"James Graham, 3rd Duke of Montrose." {{Cite web|url=https://thepeerage.com/p2815.htm#i28142|title=Person Page|website=thepeerage.com|access-date=2020-11-21}}</ref> *Lady Jemima Elizabeth Ashburnham (1 January 1762 – 17 September 1786)<ref>"Lady Jemima Elizabeth Ashburnham." {{Cite web|url=https://thepeerage.com/p1159.htm#i11589|title=Person Page|website=thepeerage.com|access-date=2020-11-21}}</ref> *#Unknown son Graham *Lady Caroline Maria Montagu (10 August 1770 – 24 March 1847)<ref>"Lady Caroline Maria Montagu." {{Cite web|url=https://thepeerage.com/p2883.htm#i28825|title=Person Page|website=thepeerage.com|access-date=2020-11-21}}</ref> #Lady Georgiana Charlotte Graham (– 13 February 1835) #Lady Emily Graham Foley (– 1 January 1900) #Lady Caroline Graham (– 24 March 1875) #Lady Lucy Graham (25 September 1793 – 16 September 1875) #'''James Graham, 4th Duke of Montrose''' (16 July 1799 – 30 December 1874) #Lord Montagu William Graham (2 February 1807 – 21 June 1878) *Caroline Agnes Horsley-Beresford Graham Stirling-Crawfurd Milner (c. 1818?<ref name=":0" /> – 16 November 1894)<ref name=":1" /> *James Graham, 4th Duke of Montrose (16 July 1799 – 30 December 1874)<ref>"James Graham, 4th Duke of Montrose." {{Cite web|url=https://thepeerage.com/p2815.htm#i28143|title=Person Page|website=thepeerage.com|access-date=2020-11-21}}</ref> #Beatrice Violet Graham, [[Social Victorians/People/Lady Violet Greville | Lady Violet Greville]] (13 February 1842 – 29 Feb 1932) #Agnes Caroline Graham ( – 8 May 1873) #James John Graham, Marquess of Graham (7 Feb 1845 – 31 Jan 1846) #James Graham, Marquess of Graham (22 Jun 1847 – 3 Apr 1872) #'''Douglas Beresford Malise Ronald Graham, 5th Duke of Montrose''' (7 November 1852 – 10 December 1925) #Alma Imogen Leonora Charlotta Graham, later [[Social Victorians/People/Breadalbane|Marchioness of Breadalbane]] (7 Sep 1854 – 10 May 1932) *William Stuart Stirling-Crawfurd (Crawford?) ( – 23 February 1883),<ref>"William Stuart Stirling-Crawford." {{Cite web|url=https://thepeerage.com/p6863.htm#i68623|title=Person Page|website=thepeerage.com|access-date=2020-11-21}}</ref> her second husband *Marcus Henry Milner (16 April 1864 – 16 January 1939),<ref>"Marcus Henry Milner." {{Cite web|url=https://thepeerage.com/p6863.htm#i68624|title=Person Page|website=thepeerage.com|access-date=2020-11-21}}</ref> her third husband (He was 22; she was 68 if she was 16 at her first marriage, in 1836.) *Douglas Beresford Malise Ronald Graham, 5th Duke of Montrose (7 November 1852 – 10 December 1925)<ref>"Douglas Beresford Malise Ronald Graham, 5th Duke of Montrose." {{Cite web|url=https://thepeerage.com/p5396.htm#i53951|title=Person Page|website=thepeerage.com|access-date=2020-11-21}}</ref> *Violet Hermione (née Graham) Graham ( – 21 November 1940)<ref name=":2" /> #'''James Graham, 6th Duke of Montrose''' (1 May 1878 – 20 January 1954) #Helen Violet Graham (1 July 1879 – 27 August 1945) #Hermione Emily Graham Cameron (22 February 1882 –1978) #Douglas Malise Graham (14 October 1883 – 20 November 1974) #Alastair Mungo Graham (12 May 1886 – 1976) * James Graham, 6th Duke of Montrose (1 May 1878 – 20 January 1954)<ref>"James Graham, 6th Duke of Montrose." {{Cite web|url=https://www.thepeerage.com/p10991.htm#i109901|title=Person Page|website=www.thepeerage.com|access-date=2023-04-13}} https://www.thepeerage.com/p10991.htm#i109901.</ref> * Lady Mary Louise [[Social Victorians/People/Douglas-Hamilton Duke of Hamilton|Douglas-Hamilton]] (1 November 1884 – 21 February 1957)<ref>"Lady Mary Louise Douglas-Hamilton." {{Cite web|url=https://www.thepeerage.com/p10990.htm#i109900|title=Person Page|website=www.thepeerage.com|access-date=2023-04-13}} https://www.thepeerage.com/p10990.htm#i109900.</ref> # '''James Angus Graham, 7th Duke of Montrose''' (2 May 1907 – 10 February 1992) # Lady Mary Helen Alma Graham (11 April 1909 – 7 February 1999) # Lord Ronald Malise Hamilton Graham (20 September 1912 – 11 June 1978) # Lady Jean Sibyl Violet Graham (7 November 1920 – 13 October 2017) === Algernon William Fulke Greville's Family === *Fulke Southwell Greville-Nugent, 1st Baron Greville (17 February 1821 – 25 January 1883) *Lady Rosa Emily Mary Anne Nugent Greville-Nugent #Algernon Greville-Nugent, 2nd Baron Greville (1841–1909) #Hon. George Frederick Greville-Nugent (1842–1897) #Hon. Robert Southwell Greville-Nugent (26 March 1847 – 1912) #Hon. Reginald James Macartney Greville-Nugent (1848–1878) #Hon. Patrick Emilius John Greville-Nugent (6 August 1852 – 1925), married Ermengarda Ogilvy on 5 June 1882 #Hon. Mildred Charlotte Greville-Nugent (d. 1906), married Alexius Huchet, Marquis de La Bêdoyére on 26 August 1869 === Relations === * Lady Caroline Maria Montagu was the daughter of the [[Social Victorians/People/Manchester|Duke of Manchester]]. == Questions and Notes == #Douglas Graham, 5th Duke of Montrose, is [[Social Victorians/People/Lady Violet Greville | Lady Violet Greville]]'s brother. #The Boston Guardian reports this:<blockquote>The Dowager Duchess of Montrose, who died last week at the age of 76, was a well-known figure on the turf, running her horses in the name of "Mr. Manton." Her first husband was the father of the present Duke of Montrose, and she has retained the title derived from this union through two succeeding matrimonial arrangements. Her second husband, Mr. Stirling Crawfurd [sic], was a wealthy land-owner, and when he died, 11 years ago, he left her a large fortune and a château at Cannes. Five years later society was startled by the announcement that the Duchess then a woman of 70 [confirming that she was born in 1818], had taken for her third husband Mr. Harry Milner, a young man of 24, who had previously been engaged with an aristocratic stockbroking firm in the City. The Duchess had a life-rent annuity of £5,000 a year, and of this she assigned £3,000 a year to Mr. Milner, on the understanding that all the annuity cheques were to be paid into her account. This arrangement was adhered to until early in the present year, when Mr. Mlner countermanded the order. The result has been domestic differences and lawsuits, to which the Duchess's death has put a stop. On the turf her "all scarlet" colours were well-known on many race-courses, among them most celebrated of her victories being those of Thebaia, who won the Thousand Guineas and the Oaks in 1881. It is understood that her château at Cannes, one of the most beautiful in the neighbourhood, is now the property of Lord Rendel, better known as Mr. Stuart Rendel, the entertainer of Mr. Gladstone.<ref>"'Mr. Manton.'" ''Boston Guardian'' 24 November 1894, Saturday: 6 [of 8], Col. 5b. ''British Newspaper'' Archive http://www.britishnewspaperarchive.co.uk/viewer/bl/0001888/18941124/098/0006.</ref></blockquote> #Personal details about Caroline, Duchess of Montrose, especially around the race track:<blockquote>"Mr. Manton" has been a familiar figure in the paddock at every great English race meeting. Next to that of the venerable Sir John Astley, it was probably the best known. The duchess was tall and straight and heavily built. In her youth she had been rather good looking, a woman of a high-bred English type. On the racecourse she wore tailor-made clothes of a very horsey cut, including check gowns, covert coats, white cravats with horseshoe pins, and felt hats. She was in the habit of bidding for yearlings in public and of superintending the stables personally.<ref>"The Late 'Mr. Manton.' Death of the Duchess of Montrose. A Well-Known Sportswoman Gone." ''Western Mail'' 17 November 1894, Saturday: 7 [of 8]. ''British Newspaper'' Archive http://www.britishnewspaperarchive.co.uk/viewer/bl/0000104/18941117/037/0007.</ref></blockquote> == Footnotes == {{reflist}} cqje0n42yufkhm3yqm6sjqvztlqwf89 0 264318 2692965 2690727 2024-12-22T18:05:22Z Scogdill 1331941 /* White's */ 2692965 wikitext text/x-wiki ==Cliques and Social Networks== *The [[Social Victorians/Marlborough House Set |Marlborough House Set]] *The [[Social Victorians/People/The Souls |Souls]] *The Coterie ==Flourishing and Address as of 1875== ===Albert=== 1 Bolt Court, Fleet Street, E.C. (Thom 527) ===Alfred=== 22A Change Alley, E.C. ===Alpine=== 8 St. Martin's Place, W.C. ===Arlington=== 4 Arlington Street, W. (Thom 527) ===Army & Navy=== 36 Pall Mall, S. W. (Thom 527) ====Junior Army and Navy Club==== 12 Grafton Street, W. ===Arthur's=== 69 St. James's Street, S. W. ===Arts=== 17 Hanover Square, W. ===Arundel=== 12 Salisbury Street, Strand ===Athenaeum=== 107 Pall Mall, S. W. ====And Junior Athenaeum Club==== 116 Picadilly, W. ===Beaufort=== 7 Rathbone Place, Oxford Street, W. ===Boodle's=== 28 St. James's Street ===Brooks'=== 60 St. James's Street ===Burlington Fine Arts=== 17 Savile Row, W. ===Carlton=== 94 Pall Mall ====City Carlton Club==== 83 King William Street., E. C. ====Junior Carlton Club==== 30 Pall Mall ===Cavendish=== 307 Regent Street, W. ===City Liberal=== 71 Queen Street, E.C. (address marked as "temporary" in 1875) ===City of London=== 19 Old Broad Street, E. C. ===City United=== Ludgate Circus, E. C. ===Civil and Military=== 316 Regent Street, W. ===Clarence=== 1 Regent Street, Waterloo Place, S. W. ===Cobden=== 5 Milman Street, Bedford Row, W. C. ===Cocoa Tree=== 64 St. James's Street ===Conservative=== 74 St. James's Street ==== Junior Conservative Club ==== 29 King Street, St. James's (Thom 528) ===Cosmopolitan=== 30 Charles Street, Berkeley Square, W. ===County=== 43 Albemarle Street, W. ===Crichton=== 3 Adelphi Terrace, W. C. ===Devonshire=== St. James's Street ===East India United Service=== 14 St. James's Square ===Egerton=== 87 St. James's Street ===Farmers'=== Salisbury Hotel, Fleet Street ===Garrick=== 13 Garrick Street, W. C. ====Junior Garrick Club==== 1A Adelphi Terrace, Strand, W. C. ===Grafton=== 10 Grafton Street, W. ===Grampian=== 11 Charles Street, Cavendish Square, W. ===Gresham=== Gresham Place, King William Street, E. C. ===Guards'=== 70 Pall Mall, S. W. ===Hanover=== 28 George Street, W. ===Hogarth=== 84 Charlotte Street, Fitzroy Square, W. === Ilchester Club === 2, 3 Ilchester-gardens, Hyde Park, W.,<ref>"The Ilchester Ladies' Club." ''Morning Post'' 04 June 1897 Friday: 1 [of 10], Col. 3a [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000174/18970604/002/0001.</ref> "just off Bayswater-road"<ref name=":0">"Clubland at Play." "The Ilchester Club." ''Gentlewoman'' 19 June 1897 Saturday: 40 [of 108], Col. 2c [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0003340/18970619/234/0040.</ref> A club for women. Opened Saturday 19 June 1897 (or perhaps the week before?):<blockquote>The Ilchester Club.—A club for ladies that has neither an educational nor a political fad to serve is somewhat new. The Ilchester Club for Ladies, occupying two good houses just off the Bayswater-road, has no other object than the providing of a comfortable home for ladies of good birth, on terms which should appeal to a very large number. For the sum of £82 per annum a lady may have all the advantages of a well-conducted house, and have everything provided, including food and many of those social comforts which one does not even find in one's own house. The club starts under capital auspices, and on Saturday night it was inaugurated by the play of "Still Waters Run Deep," excellently played by the Hon. Mabel Vereker and Miss Norah Vandaleur; the former lady, I understand, largely interesting herself in the club. The male characters were cleverly sustained by Captain Baden-Powell, Captain C. E. Norton, Major Montresor, Mr. Davidson of Tulloch. Count de Seilern, the Marquis Montagliari, and Mr. Crookshank. Although the limits of the stage were very narrow, full credit was done to Tom Taylor's delightful comedy.<ref name=":0" /></blockquote> ===Law Society=== 103 Chancery Lane ===Marlborough=== 52 Pall Mall ===Medical=== 9 Spring Gardens, S. W. ===Men and Women's Club=== ===National=== 1 Whitehall Gardens ===Naval and Military=== 94 Picadilly, W. ====Junior Naval and Military Club==== 19 Dover Street, W. ===New Thames Yacht=== Caledonian Hotel, 1 Robert Street, Adelphi, W. C. ===New Travellers'=== 16 George Street, Hanover Square, W. ===New University=== 57 St. James's Street ===Oriental=== 18 Hanover Square, W. ===Oxford and Cambridge=== 71 Pall Mall ===Pall Mall=== 7 Waterloo Place, S. W. ===Pheonix=== 275 Strand, W. C. ===Portland=== ===Pratt's=== 14 Park Place, St. James's ===Prince's Cricket and Prince's Racket and Tennis=== 22 Hans Place, Sloane Street, S. W., same address as the Prince's Racket and Tennis Club ===Prince of Wales's Yacht=== Freemason's Tavern, 61 Great Queen Street, W. C. ===Queen's=== "Founded as The Queen's Club Limited on 19 August 1886 by [[Social Victorians/People/Charteris|Evan Charteris]], George Francis and [[Social Victorians/People/Grosvenor|Algernon Grosvener]], the Queen's Club was the world's second multipurpose sports complex, after the Prince's Club .... The first lawn tennis courts were opened on 19 May 1887, and the first sporting event was held on 1 and 2 July 1887 when Oxford played Cambridge. The club buildings were opened in January 1888, having taken about 18 months to construct."<ref>{{Cite journal|date=2024-07-28|title=Queen's Club|url=https://en.wikipedia.org/wiki/Queen's_Club|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Queen%27s_Club.</ref> In order to establish residence for his 26 July 1888 marriage to [[Social Victorians/People/Montrose|Caroline, Duchess of Montrose]], Marcus Henry Milner "earned the necessary residential qualification by staying at the Queen's Club."<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> ===Raleigh=== 14 Regent Street, S. W. ===Ramblers'=== 35 Ludgate Hill ===Reform=== 104 Pall Mall ===Royal London Yacht=== ===Royal Artillery and Royal Engineers; vide Medical=== ===Royal Thames Yacht=== 7 Albemarle Street ===St. George's Chess=== 20 King Street, St. James's ===St. James's=== 106 Picadilly ====Junior St. James's Club==== 54 St. James's Street ===St. Stephen's=== 1 Bridge Street, Westminster, S. W. ===[[Social Victorians/London Clubs/Savage Club|Savage]]=== ===Savile=== 15 Savile Row, W. ===Smithfield=== 47 Half Moon Street, W. ===Stafford=== 2 Savile Row, W. ===Temple=== 37 Arundel Street, Strand ===Thatched House=== 86 St. James's Street ===Travellers'=== 106 Pall Mall ===Turf=== 4 Grafton Street, W. ===Union=== Trafalgar Square, W. ===United Clergy and Laity=== 24 Charles Street, St. James's ===United Service=== 116 Pall Mall ====Junior United Service Club==== 11 Charles Street, St. James's ===United University=== 1 Suffolk Street, Pall Mall East, S. W. ===Universities=== 71 Jermyn Street, St. James's, S. W. ===Verulam=== 54 St. James's Street ===Victoria=== 18 Wellington Street, Strand, W. C. ===Wanderers'=== 4 Park Place, St. James's ===Westminster=== 23 Albemarle Street ===Whitehall=== 47 Parliament Street, S. W. ===White's=== 38 St. James's Street Established in 1693, the oldest of London's gentleman's clubs, White's still excludes women.<ref name=":1">{{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> It was named originally for a business, Mrs. White's Chocolate House.<ref name=":1" /> The Newcastle Chronicle described White's in November 1893 in discussing [[Social Victorians/People/Bourke|Algernon Bourke]]'s book on White's' history:<blockquote>It is true the fires are lighted at the clubs, the winter carpets have been laid down, portieres are drawn over draughty doorways, and books in tempting bindings are more and more en [sic] evidence. One of the most lavishly illustrated books I have seen for a long time is the history of "White's." It must have cost a little fortune to produce. It is in two volumes, exquisitely printed, and the matter is most readable. But this is evidently quite a private enterprise. The publisher is the [[Social Victorians/People/Bourke|Hon. Algernon Bourke]], and the work is issued from his private house in St. James's Street. There has also appeared, or is about to appear, the true and particular history of "Brooks's." Clubland will soon have no secrets left, except, of course, those current ones that may not be told, not only in deference to good taste, but out of a wholesome fear of the law of libel. There are several stories of "Crockford's" in the volume about "White's." One of them deals with the Lord Sefton of the time, who was a great epicure. He prided himself on the invention of a plat made of the soft roe of the mackerel. He was one of the principal victims at Crockford's, where at one time and another he lost £200,000 at play. His successor honoured an acceptance of his for £40,000 held by Crockford and presented after his death. The property in Manchester and Liverpool that was sold to meet his losses would today be valued at millions sterling. Notable Wagers. Long before these new and private guides to "White's" and "Brooks's," there appeared in the "Art Journal" some sketches of Clubland, that let a good deal of permissible daylight into the more mysterious corners of the social quarters of the country. If the author only professed to loiter upon the frontiers, he nevertheless made several interesting excursions into the very heart of the territory. In regard to the laws and regulations of one of the great West End houses, he fell into a trifling mistake which one of the long-eyed birds of criticism picked out with his sharp beak and exhibited to the world. It was the merest shadow of a technical error and it proved the truth of the rest: it was the very smallest "exception to prove the rule" you can imagine. A distinguished correspondent, a member of "White's," who recalls the book and is good enough to say he is "delighted with the flavour of last week's Cigarette Papers," sends me several fresh notes from the bet book at the famous club. Two of them are well worth repeating. On November 4, 1754, Lord Mountfort wagered Sir John Bland 100 guineas that Mr. Nash would outlive Mr. Cibber. The two men in queetion were Colley Cibber and Bean Nash, the "King of Bath," then very old men. Below the entry in the bet book to this day stands the record:— "Both Lord Mountford and Sir John Bland put an end to their lives before the bet was decided." Among the curious bets of a comparatively recent date was that of Lord Alvanley, who wagered Mr. Goddard "that Mr. G. Talbot does not die a natural death." Talbot retaliated by betting "that Lord Alvanley is not worth three shillings this day two years."<ref>"A Cheery November." ''Newcastle Chronicle'' 04 November 1893, Saturday: 5 [of 16], Col. 1a [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000865/18931104/075/0005. Print title: ''Newcastle Weekly Chronicle'', p. 5.</ref></blockquote> ===Windham=== 11 St. James's Square ==Bibliography== *Milne-Smith, Amy. London Clubland: A Cultural History of Gender and Class in late-Victorian England. New York: Palgrave Macmillan, 2011. Google Books: https://books.google.com/books?id=TQrHAAAAQBAJ. *Thom, Adam Bisset, compiler. The Upper Ten Thousand: An Alphabetical List of All Members of Noble Families, Bishops, Privy Councillors, Judges, Baronets, Members of the House of Commons, Lords-Lieutenant, Governors of Colonies, Knights and Companions of Orders, Deans and Archdeacons, and the Superior Officers of the Army and Navy, with Their Official Descriptions and Addresses. London: George Routledge and Sons, 1875. Google Books. mz20ergnl06926734g8gehrcl71p6wc 2 274697 2692982 2692957 2024-12-23T02:44:57Z Platos Cave (physics) 2562653 2692982 wikitext text/x-wiki '''Simulating gravitational and atomic orbits via rotating particle-particle orbital pairs at the Planck scale''' An orbital simulation program is described that emulates both gravitational and atomic orbitals as the sum of individual particle-particle orbital pair rotations. The simulation is dimensionless, the only physical constant used is the [[w:fine structure constant |fine structure constant alpha]], however it can translate to the [[w:Planck_units |Planck units]] for comparison with real world orbits <ref>Macleod, Malcolm J.; {{Cite journal |title=Simulating gravitational and atomic orbits via rotating particle-particle orbital pairs |journal=RG |date=Dec 2024 | doi=10.13140/RG.2.2.11378.00961}}</ref>. [[File:complex-orbit-pts26-r17-1-7-1.gif|thumb|right|640px|By selecting the start co-ordinates on a 2-D plane for each point (unit of mass) accordingly, we can 'design' the required orbits. No other parameters are used. The 26 points orbit each other resulting in 325 point-point orbitals.]] For simulating gravity, orbiting objects ''A'', ''B'', ''C''... are sub-divided into discrete points, each point can be represented as 1 unit of [[w:Planck mass |Planck mass]] ''m''_P (for example, a 1kg satellite would be divided into 1kg/''m''_P = 45940509 points). Each point in object ''A'' then forms an orbital pair with every point in objects ''B'', ''C''..., resulting in a universe-wide, n-body network of rotating point-to-point orbital pairs . Each orbital pair rotates 1 unit of length per unit of time, when these orbital pair rotations are summed and mapped over time, gravitational orbits emerge between the objects ''A'', ''B'', ''C''... The base simulation requires only the start position (''x'', ''y'' coordinates) of each point, as it maps only rotations of the points within their respective orbital pairs then information regarding the macro objects ''A'', ''B'', ''C''...; momentum, center of mass, barycenter etc ... is not required (each orbital is calculated independently of all other orbitals). For simulating electron transition within the atom, the electron is assigned as a single mass point, the nucleus as multiple points clustered together (a 2-body orbit), and an incoming 'photon' is added in a series of discrete steps (rather than a single 'jump' between orbital shells). As the electron continues to orbit the nucleus during this transition phase, the electron path traces a [[w:hyperbolic spiral |hyperbolic spiral]]. Although only the mass state of the electron is mapped during transition, periodically the spiral angles converge to give an integer orbital radius, the transition steps between these radius can then be used to solve the transition frequency. And so although mapping a gravitational orbit on a 2-D plane, a radial quantization (as a function of pi and so of geometrical origin) emerges, (360°=4''r'', 360+120°=9''r'', 360+180°=16''r'', 360+216°=25''r'' ... 720°=∞''r''). In this context it is thus not necessary to develop a separate `quantum' theory of gravity. === Theory === In the simulation, particles are treated as an electric wave-state to (Planck) mass point-state oscillation, the wave-state as the duration of particle frequency in Planck time units, the point-state duration as 1 unit of Planck time (as a point, this state can be assigned mapping coordinates), the particle itself is an oscillation between these 2 states (i.e.: the particle is not a fixed entity). For example, an electron has a frequency (wave-state duration) = 10<sup>23</sup> units of Planck time followed by the mass state (1 unit of Planck time). The background to this oscillation is given in the [[v:Electron (mathematical) |mathematical electron]] model. If the electron '''has (is)''' mass (1 unit of Planck mass) for 1 unit of Planck time, and then '''no''' mass for 10<sup>23</sup> units of Planck time (the wave-state), then in order for a (hypothetical) object composed only of electrons to '''have (be)''' 1 unit of Planck mass at every unit of Planck time, the object will require 10<sup>23</sup> electrons. This is because orbital rotation occurs at each unit of Planck time and so the simulation requires this object to have a unit of Planck mass at each unit of Planck time (i.e.: on average there will always be 1 electron in the mass point state). We would then measure the mass of this object as 1 Planck mass (the measured mass of an object reflects the average number of units of Planck mass per unit of Planck time). For the simulation program, this Planck mass object can now be defined as a point (it will have point co-ordinates at each unit of Planck time and so can be mapped). As the simulation is dividing the mass of objects into these Planck mass size points and then rotating these points around each other as point-to-point orbital pairs, then by definition gravity becomes a mass to mass interaction. Nevertheless, although this is a mass-point to mass-point rotation, and so referred to here as a point-point orbital, it is still a particle to particle orbital, albeit the particles are both in the mass state. We can also map particle to particle orbitals for which both particles are in the wave-state, the H atom is a well-researched particle-to-particle orbital pair (electron orbiting a proton) and so can be used as reference. To map orbital transitions between energy levels, the simulation uses the [[v:Quantum_gravity_(Planck)#Photon_orbital_model |photon-orbital model]], in which the orbital (Bohr) radius is treated as a 'physical wave' akin to the photon albeit of inverse or reverse phase. The photon can be considered as a moving wave, the orbital radius as a standing/rotating wave (trapped between the electron and proton). It is the rotation of the orbital radius that pulls the electron, resulting in the electron orbit around the nucleus. Furthermore, orbital transition (between orbitals) occurs between the orbital radius and the photon, the electron has a passive role. Transition (the electron path) follows a specific [[v:Fine-structure_constant_(spiral) |hyperbolic spiral]] for which the angle component periodically cancels into integers which correspond with the orbital energy levels where ''r'' = Bohr radius; at 360° radius =4''r'', 360+120°=9''r'', 360+180°=16''r'', 360+216°=25''r'' ... 720°=∞''r''. As these spiral angles (360°, 360+120°, 360+180°, 360+216° ...) are linked directly to pi, and as the electron is following a semi-classical gravitational orbit, this quantization has a geometrical origin. Although the simulation is not optimized for atomic orbitals (the nucleus is treated simply as a cluster of points), the transition period '''t''' measured between these integer radius can be used to solve the transition frequencies '''f''' via the formula <math>f/c = t \lambda_H/(n_f^2-n_i^2)</math>. In summary, both gravitational and atomic orbitals reflect the same particle-to-particle orbital pairing, the distinction being the state of the particles; gravitational orbitals are mass to mass whereas atomic orbitals are predominately wave to wave. There are not 2 separate forces used by the simulation, instead particles are treated as oscillations between the 2 states (electric wave and mass point). The gravitational orbits that we observe are the time averaging sum of the underlying multiple gravitational orbitals. === N-body orbitals === [[File:8body-27orbital-gravitational-orbit.gif|thumb|right|640px|8-body (8 mass points, 28 orbitals), the resulting orbit is a function of the start positions of each point]] The simulation universe is a 4-axis hypersphere expanding in increments <ref>Macleod, Malcolm; {{Cite journal |title=2. Programming cosmic microwave background for Planck unit Simulation Hypothesis modelling |journal=RG |date=26 March 2020 | doi=10.13140/RG.2.2.31308.16004/7 }}</ref> with 3-axis (the [[v:Black-hole_(Planck) |hypersphere surface]]) projected onto an (''x'', ''y'') plane with the ''z'' axis as the simulation timeline (the expansion axis). Each point is assigned start (''x'', ''y'', ''z'' = 0) co-ordinates and forms pairs with all other points, resulting in a universe-wide n-body network of point-point orbital pairs. The barycenter for each orbital pairing is its center, the points located at each orbital 'pole'. The simulation itself is dimensionless, simply rotating circles. To translate to dimensioned gravitational or atomic orbits, we can use the Planck units ([[w:Planck mass |Planck mass m<sub>P</sub>]], [[w:Planck length |Planck length l<sub>p</sub>]], [[w:Planck time |Planck time t<sub>p</sub>]]), such that the simulation increments in discrete steps (each step assigned as 1 unit of Planck time), during each step (for each unit of Planck time), the orbitals rotate 1 unit of (Planck) length (at velocity ''c'' = ''l''<sub>p</sub>/''t''<sub>p</sub>) in hyper-sphere co-ordinates. These rotations are then all summed and averaged to give new point co-ordinates. As this occurs for every point before the next increment to the simulation clock (the next unit of Planck time), the orbits can be updated in 'real time' (simulation time) on a serial processor. Orbital pair rotation on the (''x'', ''y'') plane occurs in discrete steps according to an angle '''β''' as defined by the orbital pair radius (the atomic orbital '''β''' has an additional alpha term). :<math>\beta = \frac{1}{r_{orbital} \sqrt{r_{orbital}}}</math> As the simulation treats each (point-point) orbital independently (independent of all other orbitals), no information regarding the points (other than their initial start coordinates) is required by the simulation. Although orbital and so point rotation occurs at ''c'', the [[v:Relativity (Planck) |hyper-sphere expansion]] <ref>Macleod, Malcolm; {{Cite journal |title=1. Programming relativity for Planck scale Simulation Hypothesis modeling |journal=RG |date=26 March 2020 | doi=10.13140/RG.2.2.18574.00326/3 }}</ref> is equidistant and so `invisible' to the observer. Instead observers (being constrained to 3D space) will register these 4-axis orbits (in hyper-sphere co-ordinates) as a circular motion on a 2-D plane (in 3-D space). An apparent [[w:Time_dilation |time dilation]] effect emerges as a consequence. [[File:4body-orbital-3x10x-gravitational-orbit.gif|thumb|right|640px|Symmetrical 4 body orbit; (3 center mass points, 1 orbiting point, 6 orbital pairs). Note that all points orbit each other.]] ==== 2 body orbits ('''x, y''' plane) ==== For simple 2-body orbits, to reduce computation only 1 point is assigned as the orbiting point and the remaining points are assigned as the central mass. For example the ratio of earth mass to moon mass is 81:1 and so we can simulate this orbit accordingly. However we note that the only actual distinction between a 2-body orbit and a complex orbit being that the central mass points are assigned ('''x, y''') co-ordinates relatively close to each other, and the orbiting point is assigned ('''x, y''') co-ordinates distant from the central points (this becomes the orbital radius) ... this is because the simulation treats all points equally, the center points also orbiting each other according to their orbital radius, for the simulation itself there is no difference between simple 2-body and complex n-body orbits. The [[w:Schwarzschild radius |Schwarzschild radius]] formula in Planck units :<math>r_s = \frac{2 l_p M}{m_P}</math> As the simulation itself is dimensionless, we can remove the dimensioned length component <math>2 l_p</math>, and as each point is analogous to 1 unit of Planck mass <math>m_P</math>, then the Schwarzschild radius for the simulation becomes the number of central mass points. We then assign ('''x, y''') co-ordinates (to the central mass points) within a circle radius <math>r_s</math> = number of central points = total points - 1 (the orbiting point). After every orbital has rotated 1 length unit (anti-clockwise in these examples), the new co-ordinates for each rotation per point are then averaged and summed, the process then repeats. After 1 complete orbit (return to the start position by the orbiting point), the period '''t''' (as the number of increments to the simulation clock) and the ('''x, y''') plane orbit length '''l''' (distance as measured on the 2-D plane) are noted. Key: 1. <math>r_s</math> = '''i'''; number of center mass points (the orbited object). 2. '''j_max''' = radius to mass co-efficient. 3. '''j''' = number of points, including virtual (for simple 2 body orbits with only 1 orbiting point, '''j''' = '''i''' + 1 ). 4. '''x, y''' = start co-ordinates for each point (on a 2-D plane), '''z''' = 0. 5. '''r_α''' = a radius constant, here r_α = sqrt(2α) = 16.55512; where alpha = inverse [[w:fine structure constant |fine structure constant]] = 137.035 999 084 (CODATA 2018). This constant adapts the simulation specifically to gravitational and atomic orbitals. :<math>r_{orbital} = {r_{\alpha}}^2 \;*\; r_{wavelength} </math> ==== Orbital formulas (2-D plane)==== Outer = orbiting point, inner = orbited center :<math>r_{outer} = {r_{\alpha}}^2 \;*\;2 (\frac{ j_{max}}{i})^2</math>, orbital radius :<math>r_{barycenter} = \frac{r_{outer}}{j}</math>, barycenter :<math>v_{outer} = \frac{i}{j_{max} r_{\alpha}} </math>, orbiting point velocity :<math>v_{inner} = \frac{1}{j_{max} r_{\alpha}}</math>, orbited point(s) velocity :<math>t_{outer} = \frac{2 \pi r_{outer}}{v_{outer}} = 4 \pi {(\frac{j_{max} {r_{\alpha}}}{i})}^3 </math>, orbiting point period :<math>l_{outer} = 2 \pi (r_{outer} - r_{barycenter})</math>, distance travelled Simulation data: :period <math>t_{sim}</math> :length <math>l_{sim}</math> :radius <math>r_{sim} = \frac{l_{sim}}{2 \pi}</math> :velocity <math>v_{sim} = \frac{l_{sim}}{t_{sim}}</math> :barycenter <math>b_{sim} = \frac{x_{max} + x_{min}}{2}</math> For example; 8 mass points (28 orbitals) divided into ''j'' = 8 (total points), ''i'' = ''j'' - 1 (7 center mass points). After 1 complete orbit, actual period '''t''' and distance travelled '''l''' are noted and compared with the above formulas. 1) ''j''_max = i+1 = 8 :period <math>t = 74465.0516,\; t_{outer} = 74471.6125</math> :length <math>l = l_{sim} = 3935.7664,\; l_{outer} = 3936.1032</math> :radius <math>r_{sim} = 626.3951</math> :velocity <math>v_{sim} = 1/18.920137</math> :barycenter <math>b_{sim} = 89.5241,\; r_{barycenter} = 89.4929</math> 2) ''j''_max = 32*i+1 = 225 :period <math>t = 1656793370.3483,\; t_{outer} = 1656793381.3051</math> :length <math>l = l_{sim} = 3113519.1259,\; l_{outer} = 3113519.1385</math> :radius <math>r_{sim} = 495531.959</math> :velocity <math>v_{sim} = 1/532.128856</math> :barycenter <math>b_{sim} = 70790.283, \;r_{barycenter} = 70790.280</math> 3) Moon orbit. From the [[w:standard gravitational parameter |standard gravitational parameters]], the earth to moon mass ratio approximates 81:1 and so we can reduce to 1 point orbiting a center of mass comprising ''i'' = 81 points, ''j'' = i + 1. :<math>\frac{3.986004418\;x10^{14}}{4.9048695\;x10^{12}} = 81.2663</math> :<math>r_{earth-moon}</math> = 384400km :<math>M_{earth}</math> = 0.597378 10²⁵kg Solving <math>j_{max}</math> :<math>r_{outer} = {r_{\alpha}}^2 \;*\;2 (\frac{ j_{max}}{i})^2 = \frac{2 r_{earth-moon} m_P}{M_{earth} l_p}</math> :<math>j_{max} = 1440443</math> Gives :<math>t_{outer} = 4 \pi {(\frac{j_{max} {r_{\alpha}}}{i})}^3 (\frac{l_p}{c}) = 0.8643\; 10^{-26}</math>s :<math>t_{outer} \frac{M_{earth}} {m_P } = 2371844</math>s (27.452 days) :<math>v_{Moon} = (c) \frac{i}{j_{max}{r_{\alpha}}} = 1018.3m/s</math> :<math>v_{Earth} = (c) \frac{1}{j_{max} r_{\alpha}} = 12.57m/s</math> :<math>r_{barycenter} = \frac{r_{earth-moon}}{j} = 4688km</math> ==== Gravitational coupling constant ==== In the above, the points were assigned a mass as a theoretical unit of Planck mass. Conventionally, the [[w:Gravitational coupling constant | Gravitational coupling constant]] ''α_G'' characterizes the gravitational attraction between a given pair of elementary particles in terms of a particle (i.e.: electron) mass to Planck mass ratio; :<math>\alpha_G = \frac{G m_e^2}{\hbar c} = (\frac{m_e}{m_P})(\frac{m_e}{m_P}) = 1.75... x10^{-45}</math> For the purposes of this simulation, particles are treated as an oscillation between an electric wave-state (duration particle frequency) and a mass point-state (duration 1 unit of Planck time). This inverse α_G then represents the probability that any 2 electrons will be in the mass point-state at any unit of Planck time ([[v:Electron_(mathematical) |wave-mass oscillation at the Planck scale]] <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>). :<math>{\alpha_G}^{-1} = \frac{m_P^2}{m_e^2} = 0.57... x10^{45}</math> As mass is not treated as a constant property of the particle, measured particle mass becomes the averaged frequency of discrete point mass at the Planck level. If 2 dice are thrown simultaneously and a win is 2 'sixes', then approximately every (1/6)x(1/6) = (1/36) = 36 throws (frequency) of the dice will result in a win. Likewise, the inverse of α_G is the frequency of occurrence of the mass point-state between the 2 electrons. As 1 second requires 10⁴² units of Planck time (<math>t_p = 10^{-42}s</math>), this occurs about once every 3 minutes. :<math>\frac{{\alpha_G}^{-1}}{t_p}</math> Gravity now has a similar magnitude to the strong force (at this, the Planck level), albeit this interaction occurs seldom (only once every 3 minutes between 2 electrons), and so when averaged over time (the macro level), gravity appears weak. If particles oscillate between an electric wave state to Planck mass (for 1 unit of Planck-time) point-state, then at any discrete unit of Planck time, a number of particles will simultaneously be in the mass point-state. If an assigned point contains only electrons, and as the frequency of the electron = f_e, then the point will require 10²³ electrons so that, on average for each unit of Planck time there will be 1 electron in the mass point state, and so the point will have a mass equal to Planck mass (i.e.: experience continuous gravity at every unit of Planck time). :<math>f_e = \frac{m_P}{m_e} = 10^{23}</math> For example a 1kg satellite orbits the earth, for any given unit of Planck time, satellite (B) will have <math>1kg/m_P = 45940509</math> particles in the point-state. The earth (A) will have <math>5.9738 \;x10^{24} kg/m_P = 0.274 \;x10^{33}</math> particles in the point-state, and so the earth-satellite coupling constant becomes the number of rotating orbital pairs (at unit of Planck time) between earth and the satellite; :<math>N_{orbitals} = (\frac{m_A}{m_P})(\frac{m_B}{m_P}) = 0.1261\; x10^{41}</math> Examples: :<math>i = \frac{M_{earth}}{m_P} = 0.27444 \;x10^{33}</math> (earth as the center mass) :<math>i 2 l_p = 0.00887</math> (earth Schwarzschild radius) :<math>s = \frac{1kg}{m_P} = 45940509</math> (1kg orbiting satellite) :<math>j = N_{orbitals} = i*s = 0.1261 \;x10^{41}</math> 1) 1kg satellite at earth surface orbit :<math>r_{o} = 6371000 km</math> (earth surface) :<math>j_{max} = \frac{j}{r_a}\sqrt{\frac{r_{o}}{i l_p}} = 0.288645\;x10^{44}</math> :<math>n_g = \frac{j_{max}}{j} = 2289.41</math> :<math>r = r_{\alpha}^2 n_g^2 i l_p = r_{o} </math> :<math>v = \frac{c}{n_g r_{\alpha}} = 7909.7924</math> m/s :<math>t = 2 \pi \frac{r_{outer}}{v_{outer}} = 5060.8374</math> s 2) 1kg satellite at a synchronous orbit radius :<math>r_o = 42164.17 km</math> :<math>j_{max} = \frac{j}{r_a} \sqrt{\frac{r_{o}}{i l_p}} = 0.74256\;x10^{44}</math> :<math>n_g = \frac{j_{max}}{j} = 5889.674</math> :<math>r = r_{\alpha}^2 n_g^2 i l_p = r_{o} </math> :<math>v = \frac{c}{n_g r_{\alpha}} = 3074.66</math> m/s :<math>t = 2 \pi \frac{r_{outer}}{v_{outer}} = 86164.09165</math> s 3) The energy required to lift a 1 kg satellite into geosynchronous orbit is the difference between the energy of each of the 2 orbits (geosynchronous and earth). :<math>E_{orbital} = \frac{h c}{2 \pi r_{6371}} - \frac{h c}{2 \pi r_{42164}} = 0.412 x10^{-32}J</math> (energy per orbital) :<math>N_{orbitals} = \frac{M_{earth}m_{satellite}}{m_P^2} = 0.126 x10^{41}</math> (number of orbitals) :<math>E_{total} = E_{orbital} N_{orbitals} = 53 MJ/kg</math> 4) The orbital angular momentum of the planets derived from the angular momentum of the respective orbital pairs. :<math>N_{sun} = \frac{M_{sun}}{m_P} </math> :<math>N_{planet} = \frac{M_{planet}}{m_P} </math> :<math>N_{orbitals} = N_{sun}N_{planet} </math> :<math>n_g = \sqrt{\frac{R_{radius} m_P}{2 \alpha l_p M_{sun}}} </math> :<math>L_{oam} = 2\pi \frac{M r^2}{T} = N_{orbitals} n_g\frac{h}{2\pi} \sqrt{2 \alpha},\;\frac{kg m^2}{s} </math> The orbital angular momentum of the planets; mercury = .9153 x10³⁹ venus = .1844 x10⁴¹ earth = .2662 x10⁴¹ mars = .3530 x10⁴⁰ jupiter = .1929 x10⁴⁴ pluto = .365 x10³⁹ Orbital angular momentum combined with orbit velocity cancels ''n_g'' giving an orbit constant. Adding momentum to an orbit will therefore result in a greater distance of separation and a corresponding reduction in orbit velocity accordingly. :<math>L_{oam}v_g = N_{orbitals} \frac{h c}{2\pi},\;\frac{kg m^3}{s^2} </math> [[File:orbit-points32-orbitals496-clumping-over-time.gif|thumb|right|640px|32 mass points (496 orbitals) begin with random co-ordinates, after 2³² steps they have clumped to form 1 large mass and 2 orbiting masses.]] ==== Freely moving points ==== The simulation calculates each point as if freely moving in space, and so is useful with 'dust' clouds where the freedom of movement is not restricted. In this animation, 32 mass points begin with random co-ordinates (the only input parameter here are the start (''x'', ''y'') coordinates of each point). We then fast-forward 2³² steps to see that the points have now clumped to form 1 larger mass and 2 orbiting masses. The larger center mass is then zoomed in on to show the component points are still orbiting each other, there are still 32 freely orbiting points, only the proximity between them has changed, they have formed ''planets''. [[File:Gravitational-potential-energy-8body-1-2.gif|thumb|right|640px|8-body circular orbit plus 1-body with opposing orbitals 1:2]] ==== Orbital trajectory (circular vs. straight) ==== Orbital trajectory is a measure of alignment of the orbitals. In the above examples, all orbitals rotate in the same direction = aligned. If all orbitals are unaligned the object will appear to 'fall' = straight line orbit. In this example, for comparison, onto an 8-body orbit (blue circle orbiting the center mass green circle), is imposed a single point (yellow dot) with a ratio of 1 orbital (anti-clockwise around the center mass) to 2 orbitals (clockwise around the center mass) giving an elliptical orbit. The change in orbit velocity (acceleration towards the center and deceleration from the center) derives automatically from the change in the orbital radius (there is no barycenter). The orbital drift (as determined where the blue and yellow meet) is due to these orbiting points rotating around each other. ==== Precession ==== Can the orbital plane also rotate? semi-minor axis: <math>b = \alpha l^2 \lambda_A</math> semi-major axis: <math>a = \alpha n^2 \lambda_A</math> radius of curvature :<math>L = \frac{b^2}{a} = \frac{a l^4 \lambda_A}{n^2}</math> :<math>\frac{3 \lambda_A}{2 L} = \frac{3 n^2}{2 \alpha l^4}</math> arc secs per 100 years (drift): :<math>T_{earth}</math> = 365.25 days drift = <math>\frac{3 n^2}{2 \alpha l^4} 1296000 \frac{100 T_{earth}}{T_{planet}}</math> Mercury (eccentricity = 0.205630) T = 87.9691 days a = 57909050 km (''n'' = 378.2734) b = 56671523 km (''l'' = 374.2096) drift = 42.98 Venus (eccentricity = 0.006772) T = 224.701 days a = 108208000 km (''n'' = 517.085) b = 108205519 km (''l'' = 517.079) drift = 8.6247 Earth (eccentricity = 0.0167) T = 365.25 days a = 149598000 km (''n'' = 607.989) b = 149577138 km (''l'' = 607.946) drift = 3.8388 Mars (eccentricity = 0.0934) T = 686.980 days a = 227939366 km (''n'' = 750.485) b = 226942967 km (''l'' = 748.843) drift = 1.351 [[File:relativistic-quantum-gravity-orbitals-codingthecosmos.png|thumb|right|480px|Illustration of B's cylindrical orbit relative to A's time-line axis]] ==== Hyper-sphere orbit ==== {{main|Relativity (Planck)}} Each point moves 1 unit of (Planck) length per 1 unit of (Planck) time in '''x, y, z''' (hyper-sphere) co-ordinates, the simulation 4-axis hyper-sphere universe expanding in uniform (Planck) steps (the simulation clock-rate) as the origin of the speed of light, and so (hyper-sphere) time and velocity are constants. Particles are pulled along by this expansion, the expansion as the origin of motion, and so all objects, including orbiting objects, travel at, and only at, the speed of light in these hyper-sphere co-ordinates <ref>Macleod, Malcolm; {{Cite journal |title=1. Programming relativity for Planck unit Simulation Hypothesis modelling |journal=RG |date=26 March 2020 | doi=10.13140/RG.2.2.18574.00326/3 }}</ref>. Time becomes [[v:God_(programmer)#Universe_time-line |time-line]]. While ''B'' (satellite) has a circular orbit period on a 2-axis plane (the horizontal axis representing 3-D space) around ''A'' (planet), it also follows a cylindrical orbit (from B¹ to B¹¹) around the ''A'' time-line (vertical expansion) axis ('''t_d''') in hyper-sphere co-ordinates. ''A'' is moving with the universe expansion (along the time-line axis) at (''v = c''), but is stationary in 3-D space (''v'' = 0). ''B'' is orbiting ''A'' at (''v = c''), but the time-line axis motion is equivalent (and so `invisible') to both ''A'' and ''B'', as a result the orbital period and velocity measures will be defined in terms of 3-D space co-ordinates by observers on ''A'' and ''B''. For object '''B''' :<math>t_d = t \sqrt{1 - v_{outer}^2}</math> For object '''A''' :<math>t_d = t \sqrt{1 - v_{inner}^2}</math> === Atomic orbitals === [[File:H-orbit-transitions-n1-n2-n3-n1.gif|thumb|right|640px|fig 5. H atom orbital transitions from n1-n2, n2-n3, n3-n1 via 2 photon capture, photons expand/contract the orbital radius. The spiral pattern emerges because the electron is continuously pulled in an anti-clockwise direction by the rotating orbital.]] In the atom we find individual particle to particle orbitals, and as such the atomic orbital is principally a wave-state orbital (during the orbit the electron is predominately in the electric wave-state). The wave-state is defined by a wave-function, we can however map (assign co-ordinates to) the mass point-states and so follow the electron orbit, for example, in 1 orbit at the lowest energy level in the H atom, the electron will oscillate between wave-state to point-state approximately 471960 times. This means that we can treat the atomic orbital as a simple 2-body orbit with the electron as the orbiting point. Although this approach can only map the electron point-state (and so offers no direct information regarding the electron as a wave), during electron transition between ''n''-shell orbitals, we find the electron follows a [[v:Fine-structure_constant_(spiral) |hyperbolic spiral]], this is significant because periodically the spiral angle components converge reducing to integer radius values (360°=4''r'', 360+120°=9''r'', 360+180°=16''r'', 360+216°=25''r'' ... 720°=∞''r''). As these spiral angles (360°, 360+120°, 360+180°, 360+216° ...) are linked directly to pi via this spiral geometry, we may ask if quantization of the atom has a geometrical origin. <ref>Macleod, Malcolm J.; {{Cite journal |title=Simulating gravitational and atomic orbits via rotating particle-particle orbital pairs |journal=RG |date=Dec 2024 | doi=10.13140/RG.2.2.11378.00961}}</ref>. ==== Theory ==== {{see|Fine-structure_constant_(spiral)}} =====Hyperbolic spiral===== [[File:Hyperbol-spiral-1.svg|thumb|right|320px|Hyperbolic spiral]] A [[w:hyperbolic spiral |hyperbolic spiral]] is a type of [[w:spiral|spiral]] with a pitch angle that increases with distance from its center. As this curve widens (radius '''r''' increases), it approaches an [[w:asymptotic line|asymptotic line]] (the '''y'''-axis) with the limit set by a scaling factor '''a''' (as '''r''' approaches infinity, the '''y''' axis approaches '''a'''). The spiral shape that the electron maps can be represented in Cartesian coordinates. Periodically the angles converge to give integer radius, the general form (beginning at the outer limit ranging inwards) gives; :<math>x = a^2 \frac{cos(\varphi)}{\varphi^2},\; y = a^2 \frac{sin(\varphi)}{\varphi^2},\;0 < \varphi < 4\pi</math> :radius = <math>\sqrt(x^2 + y^2) r</math> :<math>\varphi = (2)\pi, \; 4r</math> (360°) :<math>\varphi = (4/3)\pi,\; 9r</math> (240°) :<math>\varphi = (1)\pi, \; 16r</math> (180°) :<math>\varphi = (4/5)\pi, \; 25r</math> (144°) :<math>\varphi = (2/3)\pi, \; 36r</math> (120°) [[File:Bohr_atom_model_English.svg|thumb|right|320px|Electron at different ''n'' level orbitals]] =====Principal quantum number '''n'''===== The H atom has 1 proton and 1 electron orbiting the proton, in the [[w:Bohr model |Bohr model]] (which approximates a gravitational orbit), the electron can be found at select radius ([[w:Bohr radius |the Bohr radius]]) from the proton (nucleus), these radius represent the permitted energy levels (orbital regions) at which the electron may orbit the proton. Electron transition (to a higher energy level) occurs when an incoming photon provides the required energy (momentum). Conversely emission of a photon will result in electron transition to a lower energy level. The [[w:principal quantum number |principal quantum number ''n'']] denotes the energy level for each orbital. As ''n'' increases, the electron is at a higher energy level and is therefore less tightly bound to the nucleus (as ''n'' increases, the electron orbit is further from the nucleus). Each shell can accommodate up to ''n''<sup>2</sup> (1, 4, 9, 16 ... ) electrons. Accounting for two states of spin this becomes 2''n''<sup>2</sup> electrons. As these energy levels are fixed according to this integer ''n'', the orbitals may be said to be quantized. =====(Bohr) orbital===== The basic orbital radius has 2 components, dimensionless (the [[w:fine structure constant|fine structure constant alpha]]) and dimensioned (electron + proton wavelength); wavelength = <math>\lambda_H = \lambda_p + \lambda_e</math> radius = <math>r_{orbital} = 2\alpha n^2 (\lambda_H)</math> As a mass point, the electron orbits the proton at a fixed radius (the Bohr radius) in a series of steps (the duration of each step corresponds to the wavelength component). The distance travelled per step (per wave-point oscillation) equates to the distance between mass point states and is the inverse of the radius [[File:atomic-orbital-rotation-step.png|thumb|right|208px|electron (blue dot) moving 1 step anti-clockwise along the alpha orbital circumference]] length = <math>l_{orbital} = \frac{1}{r_{orbital}}</math> Duration = 1 step per wavelength and so velocity velocity = <math>v_{orbital} = \frac{1}{2\alpha n}</math> Giving period of orbit period = <math>t_{orbital} = \frac{2\pi r_{orbital}} {v_{orbital}} = 2\pi 2\alpha 2\alpha n^3 \lambda_H</math> As we are not mapping the wavelength component, a base (reference) orbital (''n''=1) :<math>t_{ref} = 2\pi 4\alpha^2</math> = 471964.356... The angle of rotation depends on the orbital radius :<math>\beta = \frac{1}{r_{orbital} \sqrt{r_{orbital}}\sqrt{2\alpha}}</math> ===== Photon orbital model ===== The electron can jump between ''n'' energy levels via the absorption or emission of a photon. In the Photon-orbital model<ref>Macleod, Malcolm J.; {{Cite journal |title=Simulating gravitational and atomic orbits via rotating particle-particle orbital pairs |journal=RG |date=Dec 2024 | doi=10.13140/RG.2.2.11378.00961}}</ref>, the orbital (Bohr) radius is treated as a 'physical wave' akin to the photon albeit of inverse or reverse phase such that <math>orbital \;radius + photon = zero</math> (cancel). The photon can be considered as a moving wave, the orbital radius as a standing/rotating wave (trapped between the electron and proton), as such it is the orbital radius that absorbs or emits the photon during transition, in the process the orbital radius is extended or reduced (until the photon is completely absorbed/emitted). The electron itself has a `passive' role in the transition phase. It is the rotation of the orbital radius that pulls the electron, resulting in the electron orbit around the nucleus (orbital momentum comes from the orbital radius), and this rotation continues during the transition phase resulting in the electron following a spiral path. The photon is actually 2 photons as per the Rydberg formula (denoted initial and final). :<math>\lambda_{photon} = R.(\frac{1}{n_i^2}-\frac{1}{n_f^2}) = \frac{R}{n_i^2}-\frac{R}{n_f^2}</math> :<math>\lambda_{photon} = (+\lambda_i) - (+\lambda_f)</math> The wavelength of the (<math>\lambda_i</math>) photon corresponds to the wavelength of the orbital radius. The (+<math>\lambda_i</math>) will then delete the orbital radius as described above (''orbital'' + ''photon'' = ''zero''), however the (-<math>\lambda_f</math>), because of the Rydberg minus term, will have the same phase as the orbital radius and so conversely will increase the orbital radius. And so for the duration of the (+<math>\lambda_i</math>) photon wavelength, the orbital radius does not change as the 2 photons cancel each other; :<math>r_{orbital} = r_{orbital} + (\lambda_i - \lambda_f)</math> However, the (<math>\lambda_f</math>) has the longer wavelength, and so after the (<math>\lambda_i</math>) photon has been absorbed, and for the remaining duration of this (<math>\lambda_f</math>) photon wavelength, the orbital radius will be extended until the (<math>\lambda_f</math>) is also absorbed. For example, the electron is at the ''n'' = 1 orbital. To jump from an initial <math>n_i = 1</math> orbital to a final <math>n_f = 2</math> orbital, first the (<math>\lambda_i</math>) photon is absorbed (<math>\lambda_i + \lambda_{orbital} = zero</math> which corresponds to 1 complete ''n'' = 1 orbit by the electron, the '''orbital phase'''), then the remaining (<math>\lambda_f</math>) photon continues until it too is absorbed (the '''transition phase'''). :<math>t_{ref} \sim 2\pi 4\alpha^2 </math> :<math>\lambda_i = 1t_{ref}</math> :<math>\lambda_f = 4t_{ref}</math> (''n'' = 2) After the (<math>\lambda_i</math>) photon is absorbed, the (<math>\lambda_f</math>) photon still has <math>\lambda_f = (n_f^2 - n_i^2)t_{ref} = 3 t_{ref}</math> steps remaining until it too is absorbed. [[File:atomic-orbital-transition-alpha-steps.png|thumb|right|277px|orbital transition during orbital rotation]] This process does not occur as a single `jump' between energy levels by the electron, but rather absorption/emission of the photon takes place in discrete steps, each step corresponds to a unit of <math>r_{incr}</math> (both photon and orbital radius may be considered as constructs from multiple units of this geometry); :<math>r_{incr} = -\frac{1}{2 \pi 2\alpha r_{wavelength}}</math> In summary; the (<math>\lambda_i</math>) photon, which has the same wavelength as the orbital radius, deletes the orbital radius in step <math>r = r_{orbital}</math> WHILE (<math>\lambda_i > 0</math>) :<math>r = r + r_{incr}</math> ://<math>\lambda_i</math> photon Conversely, because of its minus term, the (<math>\lambda_i</math>) photon will simultaneously extend the orbital radius accordingly; WHILE (<math>r < 4 r_{orbital}</math>) :<math>r = r - r_{incr}</math> ://<math>\lambda_f</math> photon The model assumes orbits also follow along a [[Quantum_gravity_(Planck)#Hyper-sphere_orbit|timeline ''z''-axis]] :<math>t_{orbital} = t_{ref} \sqrt{1 - \frac{1}{(v_{orbital})^2}}</math> The orbital phase has a fixed radius, however at the transition phase this needs to be calculated for each discrete step as the orbital velocity depends on the radius; :<math>t_{transition} = t_{ref} \sqrt{1 - \frac{1}{(v_{transition})^2}}</math> ==== Simulation ==== The simulation treats the atomic orbital as a 2-body gravitational orbit with the electron (single point) orbiting a central mass - the nucleus. The nucleus is a set of individual points (also orbiting each other) and not a static mass (static entity). The difference between gravitational and atomic orbits is only in the angle of rotation <math>\beta</math>' which has an additional <math>r_{\alpha}</math> term included as the atomic orbital wavelength component is dominated by the particle wave-state (the mass-state is treated as a point), and so velocity along the 2-D (gravitational) plane (we are only mapping the radial component of the orbital) will decrease proportionately. :<math>\beta = \frac{1}{r_{orbital} \sqrt{r_{orbital}} \sqrt{2\alpha}}</math> [[File:Alpha-hyperbolic-spiral.gif|thumb|right|640px|Bohr radius during ionization, as the H atom electron reaches each ''n'' level, it completes 1 orbit (for illustration) then continues outward (actual velocity will become slower as radius increases according to angle β)]] =====Spiral angle===== For an idealized Rydberg atom (a nucleus of point size, infinite mass and disregarding wavelength). In this example the electron transition starts at the initial (''n''<sub>i</sub> = 1) orbital :<math>\varphi = 0, \;r_{orbital} = 2\alpha</math> For each step during transition; :<math>\beta = \frac{1}{r_{orbital} \sqrt{r_{orbital}}\sqrt{2\alpha}}</math> :<math>\varphi = \varphi + \beta</math> Setting t = step number (FOR t = 1 TO ...), we can calculate the radius ''r'' and <math>n_f^2</math> at each step. :<math>r = r_{orbital} + \frac{t}{2\pi 2\alpha}</math> (number of increments ''t'' of <math>r_{incr}</math>) :<math>n_f^2 = 1 + \frac{t}{2\pi 4\alpha^2}</math> (<math>n_f^2</math> as a function of ''t'') :<math>\varphi =4 \pi \frac{(n_f^2 - n_f)}{n_f^2}</math> (<math>\varphi</math> at any <math>n_f^2</math>) We can then re-write (<math>n_f</math> is only an integer at prescribed spiral angles); :<math>\beta = \frac{1}{{r_{orbital}}^2 n_f^3}</math> Giving integer values at these spiral angles :<math>\varphi = (2)\pi, \; r = 4 r_{orbital}</math> (360°) :<math>\varphi = (8/3)\pi,\; r = 9 r_{orbital}</math> (360+120°) :<math>\varphi = (3)\pi, \; r = 16 r_{orbital}</math> (360+180°) :<math>\varphi = (16/5)\pi, \; r = 25 r_{orbital}</math> (360+216°) :<math>\varphi = (10/3)\pi, \; r = 36 r_{orbital}</math> (360+240°) :<math>\varphi = (7/4)\pi, \; r = 49 r_{orbital}</math> :<math>\varphi = (7/2)\pi, \; r = 64 r_{orbital}</math> (360+270°) ===== Rydberg atom ===== At the ''n'' = 1 orbital, 1 complete rotation becomes (dimensionless terms measured on a 2-D plane); :<math>t_{ref} = \frac{2\pi r_{orbital}}{v_{orbital}} = 2\pi 2\alpha 2\alpha</math> :<math>1t_{ref}</math> = 471964.3563... :<math>4t_{ref}</math> = 1887857.4255... :<math>9t_{ref}</math> = 4247679.2074... :<math>16t_{ref}</math> = 7551429.7021... ===== H atom ===== Experimental values for H(1s-ns) transitions (''n'' the [[w:principal quantum number |principal quantum number]]). H(1s-2s) = 2466 061 413 187.035 kHz <ref>http://www2.mpq.mpg.de/~haensch/pdf/Improved%20Measurement%20of%20the%20Hydrogen%201S-2S%20Transition%20Frequency.pdf</ref> H(1s-3s) = 2922 743 278 665.79 kHz <ref>https://pubmed.ncbi.nlm.nih.gov/33243883/</ref> H(1s-4s) = 3082 581 563 822.63 kHz <ref>https://codata.org/</ref> H(1s-∞s) = 3288 086 857 127.60 kHz <ref>https://codata.org/ (109678.77174307cm-1)</ref> (''n'' = ∞) R = 10973731.568157 <ref>https://codata.org/ (mean)</ref> ([[w:Rydberg constant |Rydberg constant]]) α =137.035999177 (inverse fine structure constant <ref>https://codata.org/ (mean)</ref> The wavelength of the H atom, for simplification the respective particle wavelengths are presumed constant irrespective of the vicinity of the electron to the proton. <math>r_{wavelength} = \lambda_H = \frac{2c}{\lambda_e + \lambda_p}</math> Dividing (dimensioned) wavelength (<math>r_{wavelength}</math>) by the (dimensioned) transition frequency returns a dimensionless number (the alpha component of the photon). The <math>(n^2 - 1)</math> term gives the number of orbital wavelengths in the transition phase; :<math>h_{(1s-ns)} = (n^2 - 1) \frac{\lambda_H }{H(1s-ns)}</math> <math>h_{(1s-2s)}</math> = 1887839.82626... <math>h_{(1s-3s)}</math> = 4247634.04874... <math>h_{(1s-4s)}</math> = 7551347.55306... ===== Simulation atom ===== The following example simulates an electron transition, the electron begins at radius <math>r = r_{orbital}</math> and makes a 360° rotation at orbital radius (the orbital phase) and then moves in incremental steps to higher orbitals (the transition phase) mapping a hyperbolic spiral path (red line) in the process (photon orbital model). The period <math>t_{sim}</math> and length <math>l_{sim}</math> are measured at integer <math>n^2 r</math> (''n'' = 1, 2, 3...) radius. For a Rydberg atom, these radius correspond precisely to the electron path at the [[v:Fine-structure_constant_(spiral) |(hyperbolic) spiral]] angles; (360°(''1r''), 360°(''4r''), 360+120°(''9r''); 360+180°(''16r''), 360+216°(''25r''), 360+240°(''36r'') ...) (the angles converge to give integer values at these radius), and so we find that as the simulation nucleus mass increases, the integer radius values approach these angles (table 2.). The period <math>t_{sim}</math> can then be used to calculate the transition frequencies. In this example, the nucleus = 249 mass points (start ''x'', ''y'' co-ordinates close to 0, 0) and the electron = 1 mass point (at radius ''x'' = ''r'', ''y'' = 0), ''t''<sub>sim</sub> = period and ''l''<sub>sim</sub> = distance travelled by the electron (<math>l_{orbital} = l_{sim}</math> at ''n''=1), the radius coefficient ''r''<sub>n</sub> = radius divided by <math>r_{orbital}</math>. As this is a gravitational orbit, although the nucleus comprises 249 points clumped close together, these points are independent of each other (they also rotate around each other), and so the `nucleus' size and shape is not static (the simulation is not optimised for a nucleus). Table 1. gives the relative values and the ''x'', ''y'' co-ordinates for the electron, nucleus center and barycenter. [[File:H-atom-electron-transition-nucleus-plot.gif|thumb|right|640px|H atom electron transition spiral plotting the nucleus and barycenter as the electron transitions from n=1 to n=8]] :<math>j_{atom} = 250</math> (atomic mass) :<math>i_{nucleus} = j_{atom} -1 = 249</math> (relative nucleus mass) :<math>r_{wavelength} = 2 (\frac{j_{atom}}{i_{nucleus}})^2</math> = 2.0160965 :<math>r_{orbital} = 2 \alpha \;*\; r_{wavelength} </math> (radius) = 552.5556 :<math>t_n = \frac{t_{sim}}{r_{wavelength}}</math> :<math>l_n = \frac{l_{sim}}{l_{orbital}} - l_{orbital}</math> :<math>r_b = r_{sim} - \frac{r_{sim}}{j_{atom}}</math> :<math>r_n = \frac{r_b}{r_{orbital}}</math> {| class="wikitable" |+table 1. Electron transition (mass = 250; ''r''_n= 1 to 5) ! ''r''_n ! ''t''_sim ! ''l''_n ! angle ! ''x'', ''y'' (electron) ! ''x'', ''y'' (nucleus) ! ''x'', ''y'' (barycenter) |-1 | 1 | 471957.072 | 0.9999897 | 360° | 550.334, 0.0036 | -2.2102, -0.00002 | -0.00004, -0.00001 |- | 4 | 1887867.293 | 2.000012 | 359.952489° | 2202.8558, 0.0001 | -7.9565, -1.9475 | 0.8868, -1.9397 |- | 9 | 4247689.502 | 4.000014 | 119.92712° | -2473.180, 4296.283 | 13.558, -10.325 | 3.611, 6.901 |- | 16 | 7551439.538 | 6.000014 | 179.91669° | -8815.254, 12.818 | 25.636, 13.303 | -9.728, 13.301 |- | 25 | 11799118.905 | 8.000014 | 215.9122° | -11158.64, -8081.13 | 16.580, 39.083 | -28.118, 6.602 |} Comparison of the spiral angle at ''r''_n = 4, 9, 16 (360, 360+120, 360+180) with different mass (''m'' = 64, 128, 250, 500, Rydberg). For the proton:electron mass ratio; ''m'' = 1836.15267... {| class="wikitable" |+ table 2. Spiral angle at <math>r_n</math> = 4, 9, 16 ! mass ! ''r''_n = 4 ! ''r''_n = 9 ! ''r''_n = 16 |- | ''m'' = 64 | 359.80318° | 119.70323° | 179.66239° |- | ''m'' = 128 | 359.90394° | 119.85415° | 179.83377° |- | ''m'' = 250 | 359.95249° | 119.92711° | 179.91669° |- | ''m'' = 500 | 359.97706° | 119.96501° | |- | Rydberg | 360° | 360+120° | 360+180° |} == External links == * [[v:Fine-structure_constant_(spiral) | Fine structure constant hyperbolic spiral]] * [[v:Physical_constant_(anomaly) | Physical constant anomalies]] * [[v:Planck_units_(geometrical) | Planck units as geometrical objects]] * [[v:electron_(mathematical) | The mathematical electron]] * [[v:Relativity_(Planck) | Programming relativity at the Planck scale]] * [[v:Black-hole_(Planck) | Programming the cosmic microwave background at the Planck level]] * [[v:Sqrt_Planck_momentum | The sqrt of Planck momentum]] * [[v:God_(programmer) | The Programmer God]] * [https://codingthecosmos.com/ Simulation hypothesis modelling at the Planck scale using geometrical objects] * [https://theprogrammergod.com/ The Programmer God, are we in a computer simulation? - eBook] ==References== {{Reflist}} [[Category:Physics| ]] [[Category:Philosophy of science| ]] 6wi0wyfsgmkab9gveuugwmzhghn15o2 0 285384 2692977 2692851 2024-12-23T00:36:55Z Young1lim 21186 /* Linking Libraries */ 2692977 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.20241217.pdf |pdf]]) ==== Floating point Arithmetic ==== </br> === Workings of the GNU Linker for IA-32 === ==== Linking Libraries ==== * Static Libraries ([[Media:LIB.1A.Static.20241128.pdf |pdf]]) * Shared Libraries ([[Media:LIB.2A.Shared.20241223.pdf |pdf]]) ==== Dynamic Linking - Directories and Symbolic Links ==== * Shared Library Names ([[Media:DIR.1A.Names.20241221.pdf |pdf]]) * Managing Shared Libraries ([[Media:DIR.2A.Manage.20241221.pdf |pdf]]) ==== Dynamic Loading - API Functions ==== * DL API ([[Media:API.1A.Functions.20241220.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.20241101.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]] odxbyx6rcpc5kp9rn7d2ekhdcol1xan 2692978 2692977 2024-12-23T00:37:56Z Young1lim 21186 /* Dynamic Linking - Directories and Symbolic Links */ 2692978 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.20241217.pdf |pdf]]) ==== Floating point Arithmetic ==== </br> === Workings of the GNU Linker for IA-32 === ==== Linking Libraries ==== * Static Libraries ([[Media:LIB.1A.Static.20241128.pdf |pdf]]) * Shared Libraries ([[Media:LIB.2A.Shared.20241223.pdf |pdf]]) ==== Dynamic Linking - Directories and Symbolic Links ==== * Shared Library Names ([[Media:DIR.1A.Names.20241223.pdf |pdf]]) * Managing Shared Libraries ([[Media:DIR.2A.Manage.20241221.pdf |pdf]]) ==== Dynamic Loading - API Functions ==== * DL API ([[Media:API.1A.Functions.20241220.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.20241101.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]] nlbsoileoebm9g3xtyubf35xbjnr9mu 2692984 2692978 2024-12-23T03:16:15Z Young1lim 21186 /* Dynamic Linking - Directories and Symbolic Links */ 2692984 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.20241217.pdf |pdf]]) ==== Floating point Arithmetic ==== </br> === Workings of the GNU Linker for IA-32 === ==== Linking Libraries ==== * Static Libraries ([[Media:LIB.1A.Static.20241128.pdf |pdf]]) * Shared Libraries ([[Media:LIB.2A.Shared.20241223.pdf |pdf]]) ==== Dynamic Linking - Directories and Symbolic Links ==== * Shared Library Names ([[Media:DIR.1A.Names.20241223.pdf |pdf]]) * Managing Shared Libraries ([[Media:DIR.2A.Manage.20241223.pdf |pdf]]) ==== Dynamic Loading - API Functions ==== * DL API ([[Media:API.1A.Functions.20241220.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.20241101.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]] n4t5ohud60figlmbd61v25pos1si8mz 2692990 2692984 2024-12-23T09:40:52Z Young1lim 21186 /* Dynamic Loading - API Functions */ 2692990 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.20241217.pdf |pdf]]) ==== Floating point Arithmetic ==== </br> === Workings of the GNU Linker for IA-32 === ==== Linking Libraries ==== * Static Libraries ([[Media:LIB.1A.Static.20241128.pdf |pdf]]) * Shared Libraries ([[Media:LIB.2A.Shared.20241223.pdf |pdf]]) ==== Dynamic Linking - Directories and Symbolic Links ==== * Shared Library Names ([[Media:DIR.1A.Names.20241223.pdf |pdf]]) * Managing Shared Libraries ([[Media:DIR.2A.Manage.20241223.pdf |pdf]]) ==== Dynamic Loading - API Functions ==== * DL API ([[Media:API.1A.Functions.20241221.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.20241101.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]] b8vnhgsmcypa69k4q16ag10b79ii1al 2692992 2692990 2024-12-23T09:44:54Z Young1lim 21186 /* Dynamic Loading - API Functions */ 2692992 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.20241217.pdf |pdf]]) ==== Floating point Arithmetic ==== </br> === Workings of the GNU Linker for IA-32 === ==== Linking Libraries ==== * Static Libraries ([[Media:LIB.1A.Static.20241128.pdf |pdf]]) * Shared Libraries ([[Media:LIB.2A.Shared.20241223.pdf |pdf]]) ==== Dynamic Linking - Directories and Symbolic Links ==== * Shared Library Names ([[Media:DIR.1A.Names.20241223.pdf |pdf]]) * Managing Shared Libraries ([[Media:DIR.2A.Manage.20241223.pdf |pdf]]) ==== Dynamic Loading - API Functions ==== * DL API ([[Media:API.1A.Functions.20241223.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.20241101.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]] nnxw4navrslesjxcumh5wiz9ph21p42 2 295750 2692966 2692942 2024-12-22T18:40:46Z Jaredscribe 2906761 /* The Aleph Beit */ 2692966 wikitext text/x-wiki ''universitas magistrorum et scholarium'' = community of masters and scholars. [[m:User:Jaredscribe#Scholarly manifestos and project activity]] {| class="wikitable" |+ ! !en !he !es !fr !et cetera |- |wikisource |[[s:User:Jaredscribe]] |[[s:he:משתמש:Jaredscribe]] |[[s:es:Usuario:Jaredscribe]] |[[s:fr:Utilisateur:Jaredscribe]] |[[s:la:Usor:Jaredscribe]] [[s:el:Χρήστης:Jaredscribe]] |- |wikiquote |[[q:User:Jaredscribe]] |[[q:he:משתמש:Jaredscribe]] | | | |- |wikitionary |[[wikt:User:Jaredscribe]] |[[wikt:he:משתמש:Jaredscribe]] | | | |- |wikipedia |[[w:User:Jaredscribe]] |[[w:he:משתמש:Jaredscribe]] |[[w:es:Usuario:Jaredscribe]] |[[w:fr:Utilisateur:Jaredscribe]] |[[:zh:User:Jaredscribe]] |- |other |[[User:Jaredscribe|v:User:Jaredscribe]] [[d:User:Jaredscribe]] [[meta:User:Jaredscribe]] | | | | |} ==Study Companions to Published books: Help Wanted== * *[[Draft:Aristotle for Everybody]] an introduction to [[Aristotle|Aristotelian]] (and Newtonian) ontology, natural philosophy and physics, metaphysics, epistemology, ethics, politics, poetics, and natural theology, as arranged and presented by master encyclopaedist [[w:Mortimer_Adler|Mortimer Adler]]. * [[User:Jaredscribe/Antijudaism: The Western Tradition]] as researched and taught by [[w:David_Nirenberg|David Nirenberg]] ([[w:fr:David_Nirenberg|fr)]] ([[w:he:דייוויד_נירנברג|he]]) * [[Draft:The Trivium: The Liberal Arts of Logic, Grammar, and Rhetoric]] by [[w:Miriam_Joseph|Miriam Joseph]] * [[User:Jaredscribe/Basic errors in modern thought]] according to Mortimer Adler * [[User:Jaredscribe/Biology of mental illness|User:Jaredscribe/Biology of mental illness, and Psychiatry's Troubled Search]], by Anne Harrington * [[s:User:Jaredscribe/The Israel-Palestine reader]]<ref>{{Cite book|title=The Israel/Palestine reader|date=2019|publisher=Polity|isbn=978-1-5095-2733-5|editor-last=Dowty|editor-first=Alan|location=Cambridge, UK ; Medford, MA}}</ref>, a compendium of primary sources ==Original Research Projects, Study Notes, Draft Course Syllabi== * {{Research project}} * [[User:Jaredscribe/Comparative law]], by Jarescribe * [[User:Jaredscribe/Weekly Learning Schedule for Natural Philosophy]]. Natural Science, Philosophy, and Mathematics in Seven Days, by Jaredscribe * [[w:Special:PrefixIndex/User:Jaredscribe|Subpages of w:UserJaredscribe]] * [[User:Jaredscribe/Russia|User:Jaredscribe/Russia and its many Revolutions]], by Jaredscribe * [[User:Jaredscribe/The Virtues, Vices, and Systemic Differences of the Many Wikipedias|User:Jaredscribe/The Virtues, Vices, and Systemic Differences in Policy and Bias of the Many Wikipedias]] * [[User:Jaredscribe/Department of Government Efficiency]] {{expand}} and continue == Liberal Arts Basic Curriculum == The [[w:Trivium]] of Liberal Arts: Grammar, Logic, and Rhetoric. * {{cite book|url=https://archive.org/details/triviumliberalar0000miri|title=The Trivium: The Liberal Arts of Logic, Grammar, and Rhetoric|last1=Joseph|first1=Sister Miriam|year=2002|isbn=978-0967967509|editor-last=McGlinn|editor-first=Marguerite|edition=3rd|author-link=w:Sister Miriam Joseph|orig-year=1948|url-access=registration}} * {{cite book|title=Aristotle for Everybody: Difficult Thought Made Easy|title-link=Aristotle for Everybody|author-link=w:Mortimer Adler|last=Adler|first=Mortimer|date=1997|publisher=Touchstone|ISBN=0-684-83823-0|location=New York|author-link=Mortimer J. Adler|orig-date=1978}} * [[w:Isaac Newton|Isaac Newton]] (1726). [[wikisource:The_Mathematical_Principles_of_Natural_Philosophy_(1846)/BookIII-Rules|Rules of Reasoning in Philosophy]], [[Philosophiæ Naturalis Principia Mathematica#Rules of Reasoning in Philosophy|Philosophiæ Naturalis Principia Mathematica]], 2nd edition (1713), ammended 3rd edition (1726) * [[w:Isaac Newton|Isaac Newton]] (1726). [[wikisource:The_Mathematical_Principles_of_Natural_Philosophy_(1846)/BookIII-General_Scholium|General Scholium]], [[Philosophiæ Naturalis Principia Mathematica#General Scholium|Philosophiæ Naturalis Principia Mathematica]], 2nd edition (1713), ammended 3rd edition (1726) * [[w:Maimonides|Maimonides]], [[s:Guide for the Perplexed|Guide for the Perplexed]] * Euclid, [[s:Elements_of_Geometry_(Euclid)|Elements of Geometry]] * {{cite book|title=The Basic Works of Aristotle|author=Aristotle|date=1941|publisher=Random House|editor=Richard McKeon|editor-link=w:Richard McKeon|location=New York|author-link=s:Author:Aristotle}} * [[w:Koheleth|Koheleth]], Son of David, King of Jerusalem, [[wikisource:Ecclesiastes (Bible)|Ecclesiastes]] {{Note2|text=[[History of Topics in Special Relativity/Lorentz transformation (introduction)]] - an example of a good wikiversity course that evolved from the long version of wikipedia article that was redacted for brevity. This is how the two projects differentiate from and also support each other. Thanks to [[User:D.H.]]}} == History of Human Error == * [[User:Jaredscribe/The Solution to Porphyry's problem of universals in a revised Isagoge to Aristotle's Categories]] * [[User:Jaredscribe/The Magical Polytheistic Fantasy of CS Lewis|User:Jaredscribe/The Reality Check to CS Lewis's Magical Polytheistic Fantasy]] {{expand}} ==Ontology and Metaphysics on Wikidata== Planning to model topics in natural philosophy, analytics, science, and math, with the collected works of [[wikisource:Aristotle|Aristotle]] and his commentators and critics, correlated to wikiquotes drawn from wikisource. Notes are at [[d:User:Jaredscribe]], where I've also sketched out a future presentation on the metaphysics of ontology. (However unfortunately, matters of diplomacy, war, and current events may be preoccupations in the near term future.) == Student and Adjunct Faculty Union== Announcing the formation of a local [[User:Jaredscribe/Teachers unions|Substitute Teachers and High School Students and Adjunct Faculty Union]] and inviting students to go on strike for learn-ins, study-ins, and teach-ins at the school and public libraries, on wikiprojects, and on social and traditional media, for [[User:Jaredscribe/Armistice of WWI Remembrance and Veterans Day|Armistice of WWI Remembrance]] this and every future 11th November (Veterans Day). == Notes for Someday/Maybe == * [[Foreign policy from Obama to Trump]] * [[Are axioms definitions in disguise?#Con|Are axioms definitions in disguise? Con]] * [[Does Israel commit genocide in Israel–Hamas war?|Does Israel commit genocide in Israel–Hamas war? Have Hamas, Palestinian Islamic Jihad, and their allies attempted genocide of the Jewish_people?]] * [[Should Ukraine surrender to Russia in 2022?#Russia should surrender and recognize Ukrainian independence|Should Ukraine surrender to Russia in 2022? Should Russia surrender and recognize Ukrainian independence]]? * [[User:Jaredscribe/History of the Jews in exile]] * [[User:Jaredscribe/Heculaneum papyri]] * [[Palestinian campus occupations of 2024]] == The Aleph Beit == Taking the acrostic psalm 145, lets learn the hebrew alphabet, and the first word of each verse. {| class="wikitable" |+ !en ! !el ![[s:el:Ψαλμοί_του_Δαυίδ/ΡΜΔ]] ![[s:he:ביאור:תהלים_קמה|s:he:תהלים קמה]] !he |- !Aa !I will exalt you !Α ! !ארממך !א |- ! ! ! ! ! ! |- |Aa |'''I will''' exalt you |Αα | |ארממך |א |- |Bb |'''Every''' day |Ββ | |בכל יום |ב |- |Gg |'''Great''' |Γγ | |גדל |ג |- |Dd |Generation |Δδ | |דור לדור |ד |- |Hh |Splendor |Ηη | |הדר כבוד |ה |- |Vv |'''And''' strength | | |ועזוז |ו |- |Zz |Great '''memorial''' |Ζζ | |זכר רב |ז |- |Ch |'''Merciful''' and kind |χ | |חנון ורחום |ח |- |Tt |Good | | |טוב |ט |- |Yy |They will thank you | | |יודוך |י |- |Kk |Honor | | |כבוד |כ |- |Ll |To make known | | |להודיע |ל |- |Mm |Thy '''Kingdom''' | | |מלכותך |מ |- |Nn | | | | | |- | |Upholds | | |סומך |ס |- |Oo |Eyes | | |עיני |ע |- |Pp |Open | | |פתח |פ |- |Ts |Righteous | | |צדיק |צ |- |Qq |Near | | |קרב |ק |- |Rr |Wish | | |רצון |ר |- |Ss |Guard | | |שומר |ש |- |Tt |Praises | | |תהלת |ת |- | | | | | | |} == Scholarly Ethics and Wiki Politics== [[w:he:משתמש:Jaredscribe/דרך_ארץ_ויקי#וק:שבע_דברים_בויקיגולם_ושבע_בויקיחכם]] [[:meta:User:Jaredscribe/UCoC]] {{Template:Scholarly_ethics}} {{Template:Free culture}} {{using Wikiversity}} 5pqh7lpe5b77efrjd1ma8091ut6u4ae 0 303891 2692987 2609687 2024-12-23T04:32:30Z B.NIROSHAN 2980192 copied from https://en.wikipedia.org/wiki/Spear 2692987 wikitext text/x-wiki Spear manufacture and use is not confined to humans. It is also practiced by the [[wikipedia:Western_chimpanzee|western chimpanzee]]. Chimpanzees near [[wikipedia:Kédougou|Kédougou]], Senegal have been observed to create spears by breaking straight limbs off trees, stripping them of their bark and side branches, and sharpening one end with their teeth. They then used the weapons to hunt [[wikipedia:Galago|galagos]] sleeping in hollows.<ref name="Pruetz1">{{cite journal|last1=Pruetz|first1=Jill D.|last2=Bertolani|first2=Paco|year=2007|title=Savanna Chimpanzees, Pan troglodytes verus, Hunt with Tools|journal=Current Biology|volume=17|issue=5|pages=412–417|doi=10.1016/j.cub.2006.12.042|pmid=17320393|doi-access=free|s2cid=16551874}}</ref> [[File:Hoplite with spear from Greco–Persian Wars.png|thumb|upright|Spear-armed [[hoplite]] from [[Greco-Persian Wars]]]] == References == ebdv0rc6lq8kb4rbmd9ryep2lxz6euf 0 306550 2692981 2652905 2024-12-23T02:15:18Z Platos Cave (physics) 2562653 2692981 wikitext text/x-wiki '''H atom energy levels emerge from hyperbolic spiral''' Electron orbit in the H atom can also be mapped as a gravitational orbit. Although this approach maps only the electron-as-mass and so is limited to a simple radial component of the orbital on a 2-D plane (when compared to the more comprehensive [[w:Schrödinger_equation |Schrödinger equation]]), it does exhibit interesting properties; quantisation (the integer values) of the ([[w:Principal quantum number |principal quantum number]]) ''n''-shells naturally emerge as a direct function of pi, as well as the transition frequencies between the ''n''-shells <ref>Macleod, Malcolm J.; {{Cite journal |title=Simulating gravitational and atomic orbits via rotating particle-particle orbital pairs |journal=RG |date=Dec 2024 | doi=10.13140/RG.2.2.11378.00961}}</ref>. This suggest that quantisation could have geometrical origins. [[File:H-atom-transition-hyperbolic-spiral-n1-to-n9.jpg|thumb|right|384px|Electron transition path radius (''n''=1 to ''n''=9)]] :<math>\varphi = (2)\pi, \; r = 4 a_0</math> (360°) :<math>\varphi = (8/3)\pi,\; r = 9 a_0</math> (360+120°) :<math>\varphi = (3)\pi, \; r = 16 a_0</math> (360+180°) :<math>\varphi = (16/5)\pi, \; r = 25 a_0</math> (360+216°) :<math>\varphi = (10/3)\pi, \; r = 36 a_0</math> (360+240°) :<math>\varphi = (7/4)\pi, \; r = 49 a_0</math> :<math>\varphi = (7/2)\pi, \; r = 64 a_0</math> (360+270°) The electron is treated, not as a distinct entity but instead as an (electric) wave-state to (Planck) mass point-state oscillation, the wave-state as the duration of particle frequency in Planck time units, the point-state duration as 1 unit of Planck time (as a point, this state can be assigned mapping coordinates). This orbit can then be simulated as a gravitational orbit by plotting these (mass) points. By applying the same to (incoming) photons, transition between ([[w:Principal quantum number |principal quantum number]]) ''n''-shells can also be mapped as a (semi-classical) gravitational orbit transition. Semi-classical because transitions occur in steps, the steps as the duration of 1 electron-proton (reduced mass) wavelength. As the electron continues orbiting the nucleus (proton) during transition, a specific hyperbolic spiral path emerges. ====Hyperbolic spiral==== [[File:Hyperbol-spiral-1.svg|thumb|right|320px|Hyperbolic spiral]] A [[w:hyperbolic spiral |hyperbolic spiral]] is a type of [[w:spiral|spiral]] with a pitch angle that increases with distance from its center. As this curve widens (radius '''r''' increases), it approaches an [[w:asymptotic line|asymptotic line]] (the '''y'''-axis) with the limit set by a scaling factor '''a''' (as '''r''' approaches infinity, the '''y''' axis approaches '''a'''). The spiral shape that the electron maps can be represented in Cartesian coordinates. Periodically the angles converge to give integer radius, the general form (beginning at the outer limit ranging inwards) gives; :<math>x = a^2 \frac{cos(\varphi)}{\varphi^2},\; y = a^2 \frac{sin(\varphi)}{\varphi^2},\;0 < \varphi < 4\pi</math> :radius = <math>\sqrt(x^2 + y^2) r</math> :<math>\varphi = (2)\pi, \; 4r</math> (360°) :<math>\varphi = (4/3)\pi,\; 9r</math> (240°) :<math>\varphi = (1)\pi, \; 16r</math> (180°) :<math>\varphi = (4/5)\pi, \; 25r</math> (144°) :<math>\varphi = (2/3)\pi, \; 36r</math> (120°) ====Theory==== =====Principal quantum number '''n'''===== [[File:Bohr_atom_model_English.svg|thumb|right|320px|Electron at different ''n'' level orbitals]] The H atom has 1 proton and 1 electron orbiting the proton, in the [[w:Bohr model |Bohr model]] (which approximates a gravitational orbit), the electron can be found at select radius ([[w:Bohr radius |the Bohr radius]]) from the proton (nucleus), these radius represent the permitted energy levels (orbital regions) at which the electron may orbit the proton. Electron transition (to a higher energy level) occurs when an incoming photon provides the required energy (momentum). Conversely emission of a photon will result in electron transition to a lower energy level. The [[w:principal quantum number |principal quantum number ''n'']] denotes the energy level for each orbital. As ''n'' increases, the electron is at a higher energy level and is therefore less tightly bound to the nucleus (as ''n'' increases, the electron orbit is further from the nucleus). Each shell can accommodate up to ''n''² (1, 4, 9, 16 ... ) electrons. Accounting for two states of spin this becomes 2''n''² electrons. As these energy levels are fixed according to this integer ''n'', the orbitals may be said to be quantized. =====(Bohr) orbital===== The basic orbital radius has 2 components, dimensionless (the [[w:fine structure constant|fine structure constant alpha]]) and dimensioned (electron + proton wavelength); wavelength = <math>\lambda_H = \lambda_p + \lambda_e</math> radius = <math>r_{orbital} = 2\alpha n^2 (\lambda_H)</math> As a mass point, the electron orbits the proton at a fixed radius (the Bohr radius) in a series of steps (the duration of each step corresponds to the wavelength component). The distance travelled per step (per wave-point oscillation) equates to the distance between mass point states and is the inverse of the radius [[File:atomic-orbital-rotation-step.png|thumb|right|208px|electron (blue dot) moving 1 step anti-clockwise along the alpha orbital circumference]] length = <math>l_{orbital} = \frac{1}{r_{orbital}}</math> Duration = 1 step per wavelength and so velocity velocity = <math>v_{orbital} = \frac{1}{2\alpha n}</math> Giving period of orbit period = <math>\frac{2\pi r_{orbital}} {v_{orbital}} = 2\pi 2\alpha 2\alpha n^3 \lambda_H</math> As we are not mapping the wavelength component, a base (reference) orbital (''n''=1) :<math>t_{ref} = 2\pi 4\alpha^2</math> = 471964.356... The angle of rotation depends on the orbital radius :<math>\beta = \frac{1}{r_{orbital} \sqrt{r_{orbital}}\sqrt{2\alpha}}</math> ===== Photon orbital model ===== The electron can jump between ''n'' energy levels via the absorption or emission of a photon. In the Photon-orbital model<ref>Macleod, Malcolm J.; {{Cite journal |title=Simulating gravitational and atomic orbits via rotating particle-particle orbital pairs |journal=RG |date=Dec 2024 | doi=10.13140/RG.2.2.11378.00961}}</ref>, the orbital (Bohr) radius is treated as a 'physical wave' akin to the photon albeit of inverse or reverse phase such that <math>orbital \;radius + photon = zero</math> (cancel). The photon can be considered as a moving wave, the orbital radius as a standing/rotating wave (trapped between the electron and proton), as such it is the orbital radius that absorbs or emits the photon during transition, in the process the orbital radius is extended or reduced (until the photon is completely absorbed/emitted). The electron itself has a `passive' role in the transition phase. It is the rotation of the orbital radius that pulls the electron, resulting in the electron orbit around the nucleus (orbital momentum comes from the orbital radius), and this rotation continues during the transition phase resulting in the electron following a spiral path. The photon is actually 2 photons as per the Rydberg formula (denoted initial and final). :<math>\lambda_{photon} = R.(\frac{1}{n_i^2}-\frac{1}{n_f^2}) = \frac{R}{n_i^2}-\frac{R}{n_f^2}</math> :<math>\lambda_{photon} = (+\lambda_i) - (+\lambda_f)</math> The wavelength of the (<math>\lambda_i</math>) photon corresponds to the wavelength of the orbital radius. The (+<math>\lambda_i</math>) will then delete the orbital radius as described above (''orbital'' + ''photon'' = ''zero''), however the (-<math>\lambda_f</math>), because of the Rydberg minus term, will have the same phase as the orbital radius and so conversely will increase the orbital radius. And so for the duration of the (+<math>\lambda_i</math>) photon wavelength, the orbital radius does not change as the 2 photons cancel each other; :<math>r_{orbital} = r_{orbital} + (\lambda_i - \lambda_f)</math> However, the (<math>\lambda_f</math>) has the longer wavelength, and so after the (<math>\lambda_i</math>) photon has been absorbed, and for the remaining duration of this (<math>\lambda_f</math>) photon wavelength, the orbital radius will be extended until the (<math>\lambda_f</math>) is also absorbed. For example, the electron is at the ''n'' = 1 orbital. To jump from an initial <math>n_i = 1</math> orbital to a final <math>n_f = 2</math> orbital, first the (<math>\lambda_i</math>) photon is absorbed (<math>\lambda_i + \lambda_{orbital} = zero</math> which corresponds to 1 complete ''n'' = 1 orbit by the electron, the '''orbital phase'''), then the remaining (<math>\lambda_f</math>) photon continues until it too is absorbed (the '''transition phase'''). :<math>t_{ref} \sim 2\pi 4\alpha^2 </math> :<math>\lambda_i = 1t_{ref}</math> :<math>\lambda_f = 4t_{ref}</math> (''n'' = 2) After the (<math>\lambda_i</math>) photon is absorbed, the (<math>\lambda_f</math>) photon still has <math>\lambda_f = (n_f^2 - n_i^2)t_{ref} = 3 t_{ref}</math> steps remaining until it too is absorbed. [[File:atomic-orbital-transition-alpha-steps.png|thumb|right|277px|orbital transition during orbital rotation]] This process does not occur as a single `jump' between energy levels by the electron, but rather absorption/emission of the photon takes place in discrete steps, each step corresponds to a unit of <math>r_{incr}</math> (both photon and orbital radius may be considered as constructs from multiple units of this geometry); :<math>r_{incr} = -\frac{1}{2 \pi 2\alpha r_{wavelength}}</math> In summary; the (<math>\lambda_i</math>) photon, which has the same wavelength as the orbital radius, deletes the orbital radius in step <math>r = r_{orbital}</math> WHILE (<math>\lambda_i > 0</math>) :<math>r = r + r_{incr}</math> ://<math>\lambda_i</math> photon Conversely, because of its minus term, the (<math>\lambda_i</math>) photon will simultaneously extend the orbital radius accordingly; WHILE (<math>r < 4 r_{orbital}</math>) :<math>r = r - r_{incr}</math> ://<math>\lambda_f</math> photon The model assumes orbits also follow along a [[Quantum_gravity_(Planck)#Hyper-sphere_orbit|timeline ''z''-axis]] :<math>t_{orbital} = t_{ref} \sqrt{1 - \frac{1}{(v_{orbital})^2}}</math> The orbital phase has a fixed radius, however at the transition phase this needs to be calculated for each discrete step as the orbital velocity depends on the radius; :<math>t_{transition} = t_{ref} \sqrt{1 - \frac{1}{(v_{transition})^2}}</math> ===Transition=== ====Spiral angle==== For an idealized Rydberg atom (a nucleus of point size, infinite mass and disregarding wavelength). In this example the electron transition starts at the initial (''n''_i = 1) orbital :<math>\varphi = 0, \;r_{orbital} = 2\alpha</math> [[File:Alpha-hyperbolic-spiral.gif|thumb|right|640px|Bohr radius during ionization, as the H atom electron reaches each ''n'' level, it completes 1 orbit (for illustration) then continues outward (actual velocity will become slower as radius increases according to angle β)]] For each step during transition; :<math>\beta = \frac{1}{r_{orbital} \sqrt{r_{orbital}}\sqrt{2\alpha}}</math> :<math>\varphi = \varphi + \beta</math> Setting t = step number (FOR t = 1 TO ...), we can calculate the radius ''r'' and <math>n_f^2</math> at each step. :<math>r = r_{orbital} + \frac{t}{2\pi 2\alpha}</math> (number of increments ''t'' of <math>r_{incr}</math>) :<math>n_f^2 = 1 + \frac{t}{2\pi 4\alpha^2}</math> (<math>n_f^2</math> as a function of ''t'') :<math>\varphi =4 \pi \frac{(n_f^2 - n_f)}{n_f^2}</math> (<math>\varphi</math> at any <math>n_f^2</math>) We can then re-write (<math>n_f</math> is only an integer at prescribed spiral angles); :<math>\beta = \frac{1}{{r_{orbital}}^2 n_f^3}</math> Giving integer values at these spiral angles :<math>\varphi = (2)\pi, \; r = 4 r_{orbital}</math> (360°) :<math>\varphi = (8/3)\pi,\; r = 9 r_{orbital}</math> (360+120°) :<math>\varphi = (3)\pi, \; r = 16 r_{orbital}</math> (360+180°) :<math>\varphi = (16/5)\pi, \; r = 25 r_{orbital}</math> (360+216°) :<math>\varphi = (10/3)\pi, \; r = 36 r_{orbital}</math> (360+240°) :<math>\varphi = (7/4)\pi, \; r = 49 r_{orbital}</math> :<math>\varphi = (7/2)\pi, \; r = 64 r_{orbital}</math> (360+270°) ====Transition frequency ==== =====Experimental===== Experimental values for H(1s-ns) transitions (''n'' the [[w:principal quantum number |principal quantum number]]). H(1s-2s) = 2466 061 413 187.035 kHz <ref>http://www2.mpq.mpg.de/~haensch/pdf/Improved%20Measurement%20of%20the%20Hydrogen%201S-2S%20Transition%20Frequency.pdf</ref> H(1s-3s) = 2922 743 278 665.79 kHz <ref>https://pubmed.ncbi.nlm.nih.gov/33243883/</ref> H(1s-4s) = 3082 581 563 822.63 kHz <ref>https://codata.org/</ref> H(1s-∞s) = 3288 086 857 127.60 kHz <ref>https://codata.org/ (109678.77174307cm-1)</ref> (''n'' = ∞) R = 10973731.568157 <ref>https://codata.org/ (mean)</ref> ([[w:Rydberg constant |Rydberg constant]]) α =137.035999177 (inverse fine structure constant <ref>https://codata.org/ (mean)</ref> The wavelength of the H atom, for simplification the respective particle wavelengths are presumed constant irrespective of the vicinity of the electron to the proton. <math>r_{wavelength} = \lambda_H = \frac{2c}{\lambda_e + \lambda_p}</math> Dividing (dimensioned) wavelength (<math>r_{wavelength}</math>) by the (dimensioned) transition frequency returns a dimensionless number (the alpha component of the photon). The <math>(n^2 - 1)</math> term gives the number of orbital wavelengths in the transition phase; :<math>h_{(1s-ns)} = (n^2 - 1) \frac{\lambda_H }{H(1s-ns)}</math> <math>h_{(1s-2s)}</math> = 1887839.82626... <math>h_{(1s-3s)}</math> = 4247634.04874... <math>h_{(1s-4s)}</math> = 7551347.55306... ===== Simulation ===== The following example simulates an electron transition, the electron begins at radius <math>r = r_{orbital}</math> and makes a 360° rotation at orbital radius (the orbital phase) and then moves in incremental steps to higher orbitals (the transition phase) mapping a hyperbolic spiral path (red line) in the process. The period <math>t_{sim}</math> and length <math>l_{sim}</math> are measured at integer <math>n^2 r</math> (''n'' = 1, 2, 3...) radius. For a Rydberg atom, these radius correspond precisely to the electron path at the angles; (360°(''1r''), 360°(''4r''), 360+120°(''9r''); 360+180°(''16r''), 360+216°(''25r''), 360+240°(''36r'') ...), and so we find that as the simulation nucleus mass increases, the integer radius values approach these angles (table 2.). The period <math>t_{sim}</math> can then be used to calculate the transition frequencies. In this example, the nucleus = 249 mass points (start ''x'', ''y'' co-ordinates close to 0, 0) and the electron = 1 mass point (at radius ''x'' = ''r'', ''y'' = 0), ''t''_sim = period and ''l''_sim = distance travelled by the electron (<math>l_{orbital} = l_{sim}</math> at ''n''=1), the radius coefficient ''r''_n = radius divided by <math>r_{orbital}</math>. As this is a gravitational orbit, although the nucleus comprises 249 points clumped close together, these points are independent of each other (they also rotate around each other), and so the `nucleus' size and shape is not static (the simulation is not optimised for a nucleus). Table 1. gives the relative values and the ''x'', ''y'' co-ordinates for the electron, nucleus center and barycenter. [[File:H-atom-electron-transition-nucleus-plot.gif|thumb|right|640px|H atom electron transition spiral plotting the nucleus and barycenter as the electron transitions from n=1 to n=8]] :<math>j_{atom} = 250</math> (atomic mass) :<math>i_{nucleus} = j_{atom} -1 = 249</math> (relative nucleus mass) :<math>r_{wavelength} = 2 (\frac{j_{atom}}{i_{nucleus}})^2</math> = 2.0160965 :<math>r_{orbital} = 2 \alpha \;*\; r_{wavelength} </math> (radius) = 552.5556 :<math>t_n = \frac{t_{sim}}{r_{wavelength}}</math> :<math>l_n = \frac{l_{sim}}{l_{orbital}} - l_{orbital}</math> :<math>r_b = r_{sim} - \frac{r_{sim}}{j_{atom}}</math> :<math>r_n = \frac{r_b}{r_{orbital}}</math> {| class="wikitable" |+table 1. Electron transition (mass = 250; ''r''_n= 1 to 5) ! ''r''_n ! ''t''_sim ! ''l''_n ! angle ! ''x'', ''y'' (electron) ! ''x'', ''y'' (nucleus) ! ''x'', ''y'' (barycenter) |-1 | 1 | 471957.072 | 0.9999897 | 360° | 550.334, 0.0036 | -2.2102, -0.00002 | -0.00004, -0.00001 |- | 4 | 1887867.293 | 2.000012 | 359.952489° | 2202.8558, 0.0001 | -7.9565, -1.9475 | 0.8868, -1.9397 |- | 9 | 4247689.502 | 4.000014 | 119.92712° | -2473.180, 4296.283 | 13.558, -10.325 | 3.611, 6.901 |- | 16 | 7551439.538 | 6.000014 | 179.91669° | -8815.254, 12.818 | 25.636, 13.303 | -9.728, 13.301 |- | 25 | 11799118.905 | 8.000014 | 215.9122° | -11158.64, -8081.13 | 16.580, 39.083 | -28.118, 6.602 |} Comparison of the spiral angle at ''r''_n = 4, 9, 16 (360, 360+120, 360+180) with different mass (''m'' = 64, 128, 250, 500, Rydberg). For the proton:electron mass ratio; ''m'' = 1836.15267... {| class="wikitable" |+ table 2. Spiral angle at <math>r_n</math> = 4, 9, 16 ! mass ! ''r''_n = 4 ! ''r''_n = 9 ! ''r''_n = 16 |- | ''m'' = 64 | 359.80318° | 119.70323° | 179.66239° |- | ''m'' = 128 | 359.90394° | 119.85415° | 179.83377° |- | ''m'' = 250 | 359.95249° | 119.92711° | 179.91669° |- | ''m'' = 500 | 359.97706° | 119.96501° | |- | Rydberg | 360° | 360+120° | 360+180° |} {{see|Quantum_gravity_(Planck)}} == External links == * [[v:Quantum_gravity_(Planck) | Gravity at the Planck scale]] * [[v:God_(programmer) | The Programmer God]] * [https://codingthecosmos.com/ Simulation hypothesis modelling at the Planck scale using geometrical objects] ==References== {{Reflist}} [[Category:Physics| ]] [[Category:Philosophy of science| ]] bnpgmxrdowbb1atx1ee3g65nqj2npjz 0 316944 2692962 2691956 2024-12-22T13:27:52Z UniBambergIslamicStudies 2987517 2692962 wikitext text/x-wiki [[The Bamberg Introduction to the History of Islam (BIHI) 02|2 <<<]] — [[The Bamberg Introduction to the History of Islam (BIHI) 04|>>> 4]] = 3. The Prophet of Yathrib and the New Polity (622-630) = The center of the new religion shifts to the oasis of Yathrib, with warfare taking center stage. Muḥammad and his followers engage in battles against pagan Mecca and increasingly come into conflict with the Jews of Yathrib, who are ultimately expelled from the oasis. As the leader of the nascent community, Muḥammad implements a series of legal, social, and ritual reforms. == 3.1. Maghāzī – The Military Expeditions of Muḥammad == === 3.1.1. The Provocation of the Quraysh === Arab sources consistently report that Muḥammad arrived at the oasis of Yathrib on September 24, 622, following his emigration from Mecca. Having been expelled from his hometown, he considered it justifiable to engage in conflict against his former hometown. This is clearly reflected in two Qur'anic verses, widely recognized as the earliest revelations on the subject of warfare: {{quote|Sanction is given unto those who fight because they have been wronged; and Allah is indeed able to give them victory; Those who have been driven from their homes unjustly only because they said: Our Lord is Allah. [https://corpuscoranicum.de/en/verse-navigator/sura/22/verse/39/print – Q 22:39f]}} The war with Mecca, which Muḥammad waged from his new base in Yathrib, began with minor pinpricks. According to the chronology of [[w:Al-Waqidi|al-Wāqidī]], who composed a detailed account of Muḥammad's military expeditions ([https://de.wikipedia.org/wiki/Magh%C4%81z%C4%AB maghāzī]) in the early 9th century, Muḥammad dispatched his uncle [[w:Hamza_ibn_Abd_al-Muttalib|Ḥamza]] with a group of warriors seven months after his arrival in Yathrib to intercept a Meccan trade caravan returning from Syria under the leadership of [[w:Amr_ibn_Hisham|Abū Jahl]]. However, no combat occurred because a man from the [[w:Juhaynah|Juhaynah]] tribe, allied with both sides, intervened. During a [[w:Expedition_of_Ubaydah_ibn_al-Harith|second expedition]] in April 623, "the first arrow of Islam" was launched. The conflict with the Meccans soon disregarded traditional Arab religious norms, such as the obligation to maintain peace during the sacred months (see above [[The Bamberg Introduction to the History of Islam (BIHI) 01#1.3.3. Ancient Arabian Paganism and the Sacred Sites of Mecca|1.3.3.]]). For example, a unit commissioned by Muḥammad [[w:Raid_on_Nakhla|raided]] a Meccan caravan during the sacred month of [[w:Rajab|Rajab]] near [[w:Nakhla_(Saudi_Arabia)|Nakhla]], south of Mecca. According to tradition, this event prompted the following revelation: {{quote|They question [you] (O Muhammad) with regard to warfare in the sacred month. Say: Warfare therein is a great (transgression), but to turn (men) from the way of Allah, and to disbelieve in Him and in the Inviolable Place of Worship, and to expel His people thence, is a greater with Allah; for persecution is worse than killing. And they will not cease from fighting against you till they have made you renegades from your religion, if they can. [https://corpuscoranicum.de/en/verse-navigator/sura/2/verse/217/print – Q 2:217]}} <!-- linked German article for Maghāzī as there isn't an English one. -linked Raid on Nakhla, Nakhla_(Saudi_Arabia), and Expedition_of_Ubaydah_ibn_al-Harith articles --> From this Qur'anic verse, it is evident that the continued existence of the old religion in Mecca posed a constant temptation for Muḥammad’s followers to abandon their faith. Since many of them apparently found military combat (''qitāl'') undesirable, Muḥammad now declared it a duty (cf. Q [https://corpuscoranicum.de/en/verse-navigator/sura/2/verse/216/print 2:216]) and elevated it to a religious level by designating it as ''jihād fī sabīl Allāh'' (“striving [[w:Fi_sabilillah|in the way of God]]”, as stated in the subsequent verse Q [https://corpuscoranicum.de/en/verse-navigator/sura/2/verse/218/print 2:218]). This term has also been adopted into the English language in the form of [[w:Jihad|Jihad]]. [[File:Balami - Tarikhnama - The Battle of Badr - The death of Abu Jahl, and the casting of the Meccan dead into dry wells (cropped).jpg|thumb|Illustration of the [[w:Battle_of_Badr|Battle of Badr]] in a Persian manuscript, early 14th century]] The first major confrontation between the Meccans and Muḥammad’s followers took place in March 624 near the site of [[w:Badr,_Saudi_Arabia|Badr]], approximately 130 kilometers southwest of Yathrib. Muḥammad had received information about a wealthy Meccan caravan returning from Syria. With 300 men, including members of the [[w:Banu_Muzaina|Muzaynah]] tribe allied with the [[w:Banu_Aws|Aws]], he set out for Badr, situated along the coastal road, to intercept the caravan. A battle ensued between Muḥammad's forces and a Meccan army of approximately 950 men, which had rushed to the caravan's aid under the command of Muḥammad’s bitter adversary Abū Jahl. Muḥammad's forces achieved an unexpected victory. The Meccans suffered between 45 and 70 fatalities, with a similar number taken prisoner. Among the fallen Meccans were several prominent figures, including Abū Jahl. In contrast, Muḥammad’s followers lost only 14 men and captured substantial spoils of war. <!-- linked city of Badr --> Following the battle, Muḥammad had some of the prisoners beheaded, including his former adversary [[w:Nadr_ibn_al-Harith|al-Naḍr ibn al-Ḥārith]]. The [[w:Battle_of_Badr|victory at Badr]] was of immense military and religious significance for Muḥammad's followers. Apparently, however, not all of them contributed to this victory. This is evident from verses revealed after Badr, which clarify that those among the believers who “sit still” at home without a valid excuse are not equal in rank before God to the Mujāhidūn – those who engage in jihad (strive in the way of Allah) with their wealth and their lives (cf. Q [https://corpuscoranicum.de/en/verse-navigator/sura/4/verse/95/print 4:95f]). <!-- -ließ Muhammad einige der Gefangenen enthaupten: "ordered the execution of several prisoners" would sound more academic and formal in English, but I translated it as "had some of the prisoners beheaded" to align with German wording. -Pickthall (and others similarly) translate the verse as “those who strive in the way of Allah with their wealth and lives”. I translated it as 'engage in jihad' to align with the German but put 'strive in the way of Allah' in parentheses. --> === 3.1.2. The Defense Against the Meccan Counterattack === The defeat at Badr dealt a severe blow to the Quraysh of Mecca. They had long been regarded as one of the most powerful tribes in Arabia, and to some extent, their commercial success relied on this reputation. Their trade depended on cooperation with many other tribes, and now, insubordination from some of these tribes was to be anticipated. It was therefore of critical importance for the Quraysh to demonstrate that they still possessed the strength to exact revenge for the wrongs they had suffered. Ten weeks after the Battle of Badr, [[w:Abu_Sufyan_ibn_Harb|Abū Sufyān ibn Ḥarb]], who had assumed leadership of Mecca following the battle, carried out a swift raid on Yathrib. After setting fire to two houses, however, he quickly withdrew. [[File:The Prophet Muhammad and the Muslim Army at the Battle of Uhud, from the Siyer-i Nebi, 1595.jpg|thumb|A depiction of the Battle of Uhud in a [[w:Siyer-i_Nebi|Siyer-i Nebi]] from 1594, now part of the David Collection in Copenhagen.]] In the months that followed, Abū Sufyān succeeded in recruiting 3,000 well-equipped warriors. In March 625, he advanced toward Yathrib with this force, penetrating the oasis from its northwestern corner. At Mount Uhud, a battle ensued, with the momentum shifting back and forth between the two sides for a long time. As the tide of war began to shift in favor of Muḥammad’s followers, they started gathering the spoils. This prompted a group of Muḥammad’s archers to abandon their positions to turn their attention to the spoils. On the Meccan side, [[w:Khalid_ibn_al-Walid|Khālid ibn al-Walīd]], a prominent warrior, exploited the situation to sow confusion among the ranks of Muḥammad's followers and ultimately overpower them. However, in the end, Muḥammad’s followers succeeded in regaining critical positions, causing the Meccans to withdraw without permanently eliminating their adversary, Muḥammad. For Muḥammad’s followers, the [[w:Battle_of_Uhud|Battle of Uhud]] was nevertheless a bitter disappointment: not only because they had lost 50 to 70 men, including Muḥammad’s uncle [[w:Hamza_ibn_Abd_al-Muttalib|Ḥamza]], and Muḥammad himself had been injured, but also because they came to realize that divine support was not as assured as it had seemed after their victory at Badr. Several Qur’anic verses from this period affirm that those who are killed “in the way of God” are not truly dead but living (Q [https://corpuscoranicum.de/en/verse-navigator/sura/2/verse/154/print 2:154]), are provided for by their Lord (Q [https://corpuscoranicum.de/en/verse-navigator/sura/3/verse/169/print 3:169]), have their sins forgiven (Q [https://corpuscoranicum.de/en/verse-navigator/sura/3/verse/157/print 3:157]), and are admitted directly into Paradise (Q [https://corpuscoranicum.de/en/verse-navigator/sura/3/verse/195/print 3:195]). <!-- Q 3:165-168: is said to deal with the reason as to why they lost at Uhud. --> The conflict between Muḥammad and the Meccans was by no means concluded with the Battle of Uhud. As Muḥammad continued to disrupt Meccan trade and found an increasing number of allies among the Arabian Bedouins, the Meccans felt compelled to take action against him once more. In turn, they sought to recruit a number of Bedouin tribes to their side. These alliances demonstrate that the conflict between Mecca and Yathrib had by then extended to the surrounding regions of both cities. In July 625, the [[w:Banu_Sulaym|Banū Sulaym]], a tribe allied with the Quraysh, massacred a large number of Muslims at [[w:Massacre_of_Bi%27r_Ma%27una|Biʾr Maʿūna]], located between Mecca and Yathrib. In response, Muḥammad is said to have cursed the Banū Sulaym for an entire month. This practice has been preserved in a modified form as part of the [[w:Qunut|Qunūt]], a supplication recited during the morning prayer or the nightly [[w:Witr|Witr]] prayer. <!-- linked the article on the massacre & Witr prayer -the first paragraph mentions the wrongs Quraysh has suffered, whereas this paragraph shows the ongoing reasons as to why the Quraysh felt they had to attack. Can the Medinan side also be described? My understanding is that among other things Quraysh continued to oppress Muslims who had not migrated to Medina. There are documented cases of torture and economic deprivations. Quraysh also would cut off trade relationships with tribes that supported the Muslim community and use families of migrants as leverage, no? --> At the beginning of 627, the Meccans and their allies advanced to Yathrib with a force of 10,000 men. Muḥammad, however, had a trench (''khandaq'') excavated around the less fortified areas of the oasis settlement, making it wide enough that a horse could not leap across. This move took the Meccans by such surprise that they were unable to devise an effective strategic response. What had been intended as an assault instead turned into a siege. Due to intrigues, however, the Meccan alliance collapsed after only 14 days, forcing an end to the [[w:Battle_of_the_Trench|siege of Yathrib]]. The Meccans ultimately withdrew without having achieved anything. === 3.1.3. The Military and Political Breakthrough === [[File:Dinar, Khusro II, 590, 591-628 AD, year 31 - Bode-Museum - DSC02737.JPG|thumb|Sasanian ruler [[w:Khosrow_II|Khosrow II]] (r. 590–628) depicted on a gold coin, [[w:Bode_Museum|Bode Museum]]]] The Battle of the Trench was, essentially, Muḥammad’s final defensive campaign. From that point onward, his life entered an offensive phase, marking the beginning of an era of conquests for the community he had established. To understand Muḥammad's subsequent military success, it is necessary to contextualize the political dynamics of the Middle East during that period. At the beginning of the 7th century, a prolonged conflict erupted between the [[w:Byzantine_Empire|Byzantine Empire]] and the [[w:Sasanian_Empire|Sasanian Empire]]. Between 603 and 619, Sasanian forces initially conquered Syria, Palestine, and Egypt. In 622, however, the Byzantine emperor launched a counteroffensive. The conflict led to intense clashes in which the Sasanians suffered several defeats. It concluded in 628 with a peace treaty requiring [[w:Khosrow_II|Khosrow II]] to return all conquered territories. Subsequently, Khosrow was overthrown by his officers, initiating a period of political turmoil in the Sasanian Empire that persisted until 633. During this time, the Sasanian alliance network on the Arabian Peninsula collapsed. It was precisely during this five-year power vacuum that Muḥammad transformed his newly established state into a military and political success. <!-- Thronwirren: translated it as ‘political turmoil’. --> In the year following the [[w:Battle_of_the_Trench|Battle of the Trench]], he led several smaller military expeditions, the most significant being those against the oasis of [[w:Dumat_al-Jandal|Dumat al-Jandal]] and the [[w:Banu_Mustaliq|Muṣṭaliq tribe]], situated west of Yathrib. In March 628, accompanied by a group of believers, he set out for Mecca to perform the [[w:Umrah|ʿUmrah]] pilgrimage. The Meccans, suspecting hostile intentions, ensured that he did not approach the city. From his encampment at al-Ḥudaybiya, on the outskirts of the [[w:Haram_(site)|Ḥaram]], Muḥammad initiated negotiations with the Meccans, resulting in a [[w:Treaty_of_al-Hudaybiya|treaty]]. The treaty imposed what appeared on the surface to be humiliations, which in turn created tensions among his followers. For instance, the Meccan envoy refused to recognize him as “Muḥammad, the Messenger of God,” acknowledging him only as “Muḥammad ibn ʿAbdallāh.” However, the terms of the agreement were of greater significance: they included a ten-year truce and a promise from the Meccans to allow Muḥammad and his followers to enter the city the following year for a three-day ʿUmrah. In return, Muḥammad refrained from performing the ʿUmrah that year and withdrew with his men to Yathrib. <!-- linked Dumat_al-Jandal, Banu_Mustaliq, and Treaty_of_al-Hudaybiya --> The Treaty of Ḥudaybiya was a triumph for the Prophet and his followers. The Qur'an reports that God sent down His [[w:Sakina|''sakīna'']] into the hearts of the believers, increasing their faith (Q [https://corpuscoranicum.de/en/verse-navigator/sura/48/verse/4/print 48:4], [https://corpuscoranicum.de/en/verse-navigator/sura/48/verse/18/print 18]). The term sakīna originates from the Jewish concept of [[w:Shekhinah|Shekhinah]], which denotes the “presence” of God among His people. In this context, however, it also refers to a psychological state of tranquility and serenity. Following the Treaty of Ḥudaybiya, several Arabs from other regions of Arabia who had already pledged allegiance to Muḥammad previously completed their [[w:Hijrah|Hijrah]]—that is, they migrated to Yathrib—to provide military support to Muḥammad. Among them were, for example, the two Yemenis, [[w:Abu_Hurayra|Abū Hurayra]] and [[w:Abu_Musa_al-Ash%27ari|Abū Mūsā al-Ashʿarī]]. The following year, in March 629, Muḥammad traveled to Mecca with approximately 2,000 followers to perform the planned [[w:First_Pilgrimage|ʿUmrah]]. On this occasion, he married [[w:Maymunah_bint_al-Harith|Maymūnah]], the sister-in-law of his uncle [[w:Abbas_ibn_Abd_al-Muttalib|ʿAbbās]], who at that time had assumed the leadership of the [[w:Banu_Hashim|Banū Hāshim]] in Mecca. An increasing number of Meccans began to acknowledge Muḥammad as a prophet and left the city to join him, including those who had fought against him only a short time earlier, such as Khālid ibn al-Walīd, who had been on the opposing side during the Battle of Uhud (see above [[#3.1.2. The Defense Against the Meccan Counterattack|3.1.2.]]). The Qur'an specifies a distinct procedure for women who sought to join the Muslim camp: They were to be examined, and if recognized as true believers, they were not to be sent back to the disbelievers; the Muslim community was required to reimburse the disbelievers for their dowries, after which it was permissible to marry these women (Q [https://corpuscoranicum.de/en/verse-navigator/sura/60/verse/10/print 60:10]). <!-- linked the First Pilgrimage and Maymunah bint al-Harith. -“mit etwa 2.000 Mann”: translated with “followers”, as I think you used ‘Mann’ as a figure of speech here (instead of Männer), assuming there were also women? Otherwise, it would be translated as “2000 men”. --> In the course of 629, Muḥammad oversaw additional military campaigns. In September, he [[w:Battle_of_Mu%27tah|dispatched]] his former slave and adopted son, [[w:Zayd_ibn_Haritha_al-Kalbi|Zayd ibn Ḥāritha]], with an army to [[w:Mu%27tah|Muʿtah]], in present-day Jordan, east of the southern tip of the Dead Sea. A series of events then unfolded, ultimately leading to the peaceful capitulation of Mecca. Muḥammad married [[w:Umm_Habiba|Umm Ḥabība]], the daughter of [[w:Abu_Sufyan_ibn_Harb|Abū Sufyān]], who had embraced Islam years earlier and whose Muslim husband had passed away. Shortly thereafter, a clan of the [[w:Banu_Khuza%27ah|Khuzāʿah]] tribe, which had allied with Muḥammad after Ḥudaybiya, was attacked by a clan of the [[w:Kinana|Kināna tribe]], who were allied with the Meccans. Under duress, the Khuzāʿah clan appealed to Muḥammad, who regarded the Treaty of Ḥudaybiya as breached due to this incident. <!-- linked Mu'tah, Battle of Mu'tah, and Umm Habiba --> To avoid a military confrontation, Abū Sufyān traveled to Yathrib under the pretext of visiting his daughter and conducted negotiations with Muḥammad. Although the exact course of the subsequent events remains unclear, it is certain that gifts were exchanged between Muḥammad and Abū Sufyān following the latter's return to Mecca. In the matter itself, however, Muḥammad was unwilling to make any concessions and gave the command to prepare for a campaign to [[w:Conquest_of_Mecca|capture Mecca]]. With an army of approximately 10,000 men, comprising not only his followers from Mecca and Yathrib but also fighters from neighboring tribes such as the [[w:Banu_Sulaym|Banū Sulaym]] and [[w:Banu_Muzaina|Muzayna]], he marched toward Mecca. Abū Sufyān met him on the way and engaged in negotiations. In return for his conversion to Islam, he was granted a guarantee of safety for all Meccan residents who refrained from armed resistance. These extensive assurances resulted in Muḥammad's army facing only minimal resistance as they advanced into the city from multiple directions in January 630. In Arabic sources, the conquest of Mecca is referred to as ''fatḥ'', “opening”, serving as an archetype for subsequent Muslim conquests ([[w:Futuh|futūḥ]]) of cities and lands under Muḥammad's successors. Separate texts and works were later dedicated to documenting these events. <!-- linked the Conquest of Mecca -Kontaktgespräche: translated as negotiations. --> == 3.2. The Internal Development of the Community in Yathrib == === 3.2.1. The So-Called “Constitution of Medina” === Upon Muḥammad's arrival in Yathrib, his followers primarily consisted of two main groups: the members of the Quraysh who had undertaken the Hijrah with him from Mecca, and the clans of [[w:Banu_Aws|Aws]] and [[w:Banu_Khazraj|Khazraj]], who had received these emigrants in Yathrib. Establishing a bond of loyalty between these two groups was an urgent necessity to establish a cohesive community. This very issue is addressed in a verse of the Qur'an, which states: “Those who believed and left their homes and strove with their wealth and their lives for the cause of Allah, and those who took them in and helped them: these are protecting friends one of another.” (Q [https://corpuscoranicum.de/en/verse-navigator/sura/8/verse/72/print 8:72]). Evidently, practical measures were undertaken to achieve this objective, as there are reports suggesting the establishment of a “[[w:Brotherhood_among_the_Sahabah|brotherhood]]” (''muʾākhāh'') between members of the two groups. The typical form of this bond of brotherhood involved pairing an emigrant (''muhājir'') with one of the “helpers” (''anṣār''), with both declaring themselves as brothers. If one of them fell in battle, the other would inherit from him. The primary purpose of this brotherhood was to achieve greater solidarity in warfare. However, this measure did not entirely overcome the division among Muḥammad's followers. Over time, the distinction between the Meccan “Emigrants” ([[w:Muhajirun|muhājirūn]]) and the “Helpers” ([[w:Ansar_(Islam)|anṣār]]) from Yathrib seems to have become further entrenched, as suggested by two later Qur'anic verses (Q [https://corpuscoranicum.de/en/verse-navigator/sura/9/verse/100/print 9:100], [https://corpuscoranicum.de/en/verse-navigator/sura/9/verse/117/print 117]), in which the two groups are juxtaposed. <!-- Verbrüderung (muʾākhāh): I used the translation “brotherhood” here, but also found it translated it as a “system of brotherhood” or “bond of brotherhood”. linked Brotherhood_among_the_Sahabah --> A historical document offers a much more detailed account of the political circumstances in Yathrib following Muḥammad’s arrival than the Qur’an: the so-called [[w:Constitution_of_Medina|Constitution of Medina]]. This document, transmitted by Ibn Hishām, contains the earliest recorded instance of the Arabic term ''al-Madīna'' (“the city” or “the place of jurisdiction”) as a designation for Yathrib. The term later became widely used as the city’s name and also appears in the final chapters of the Qur’an (e.g., Q [https://corpuscoranicum.de/en/verse-navigator/sura/9/verse/102/print 9:102]; [https://corpuscoranicum.de/en/verse-navigator/sura/63/verse/8/print 63:8]). Regarding the political order, the document begins by establishing itself as “a compact from Muḥammad the Prophet between the Believers and Muslims of Quraysh and Yathrib, and those who follow them, join them, and fight alongside them” (§ 1). They are to form “one Ummah, distinct from all others” (§ 2). The term [[w:Ummah|Ummah]] is referenced in many other sections of the Qur'an, to denote various communities led by prophets. However, the Ummah of Medina was more oriented toward tribal concepts. The nine primary signatories of the pact were the "Emigrants of the Quraysh," who were evidently considered a single clan, along with eight clans from Yathrib. As stated, each group was to preserve its tribal structure and bear responsibility for paying [[w:Blood_money_in_Islam|blood money]] and ransoms on behalf of its members. However, this obligation of solidarity was restricted to the believers (''muʾminūn'') within each group. The second part of the document focuses on relations with the Jewish tribes of Yathrib and their Bedouin allies. The document concludes by declaring that the Valley of Yathrib is sacred (''ḥarām'') for all treaty partners (§ 39). The document demonstrates that the concept of Ummah at that time was understood to encompass relationships with members of other religions. Essentially, it was a treaty of alliance consistent with traditional Arab legal concepts. However, by restricting the obligations of solidarity in the first section to “believers,” a religious dimension was introduced. Muḥammad himself was assigned a judicial role within the framework of the Constitution. Thus, it was stipulated that those engaged in a dispute over a matter should bring it before God and Muḥammad (§§ 23, 42). 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2692628 2024-12-22T21:31:49Z Eshaa2024 2993595 /* Definitions */ 2692969 wikitext text/x-wiki == Introduction == '''Chain''' and '''cycle''' are mathematical objects studied in [[Complex Analysis|Complex Analysis]] but also appear as special cases in [[w:en:Algebraic topology|Algebraic topology]]. A chain generalizes a [[w:en:Path (topology)|curve]], and a cycle generalizes a closed curve. They are primarily used in integration in complex analysis. == Definitions == === Chain === A chain on a set <math>G \subset \mathbb{C}</math> is defined as a finite integer linear combination of paths <math>\gamma_1,\ldots, \gamma_k</math>: <math>\Gamma := \sum_{i=1}^k n_i\gamma_i \quad n_i \in \mathbb{Z}</math>. <math>\gamma_1,\ldots, \gamma_k</math> are generally continuous [[w:en:Curve|Curve]] in <math>G</math>. === Integration over a chain === Let <math>f:G \to \mathbb{C}</math> be integrable, and let <math>\Gamma</math> be a chain of piecewise continuously differentiable paths (paths of integration) <math>\gamma_1,\ldots, \gamma_k</math> in <math>G \subset \mathbb{C}</math>. The integral over the chain <math>\Gamma</math> is defined by: :<math>\int_\Gamma f(z) \, dz := \sum_{i = 1}^k n_i \int_{\gamma_i} f(z) \, dz</math> === Definition: Cycle === '''Version 1:''' A cycle is a chain <math>\Gamma := \sum_{i=1}^k n_i\gamma_i</math>, where every point <math>a \in \mathbb{C}</math> appears as the starting point as many times as it appears as the endpoint of the curves <math>\gamma_i</math>, taking multiplicities <math>n_i</math> into account. '''Version 2:''' A cycle is a chain <math>\Gamma := \sum_{i=1}^k n_i\gamma_i</math> consisting of closed paths <math>\gamma_1, \ldots, \gamma_k</math>. === Connection Between Version 1 and Version 2 === Version 2 is essential for complex analysis. Based on the properties of Version 1, any cycle <math>\Gamma := \sum_{i=1}^k n_i\gamma_i</math> can be transformed into a chain <math>\hat{\Gamma} := \sum_{i=1}^m \hat{n}_i \hat{\gamma}_i</math> of closed paths <math>\hat{\gamma}_1, \ldots, \hat{\gamma}_m</math>. If the paths <math>\gamma_1, \ldots, \gamma_k</math> are piecewise continuously differentiable, then the closed paths <math>\hat{\gamma}1, \ldots, \hat{\gamma}m</math> are also continuously differentiable. For all holomorphic functions <math>f:G \to \mathbb{C}</math>, it holds that: <math>\int\Gamma f(z) , dz = \int{\hat{\Gamma}} f(z) , dz</math>. === Trace of a path === The '''trace''' of a path <math>\gamma : [a,b] \to G</math> is defined as: <math>\operatorname{Trace}(\gamma_i) := \operatorname{Image}(\gamma) := { \gamma(t) ,| , t \in [a,b] }</math>. === Trace of a cycle/chain === The trace of a chain <math>\Gamma</math> is the union of the [[w:de:Image (mathematics)|images]] of its individual curves, i.e.: <math>\operatorname{Trace}(\Gamma) := \bigcup_{i=1}^N\operatorname{Image}(\gamma_i)</math>. If <math>\operatorname{Trace}(\Gamma) \subset \mathbb{C}</math> is a subset of <math>G \subset \mathbb{C}</math>, then <math>\Gamma</math> is called a cycle '''in''' <math>G</math> if and only if the trace <math>\operatorname{Trace}(\Gamma) \subseteq G</math> lies in <math>G</math>. === Winding number === The '''[[w:en:Winding number|Winding number]]''' is defined analogously to that of a closed curve but uses the integral defined above. For <math>z \not\in \operatorname{Trace}(\Gamma)</math>, it is given by: <math>n(\Gamma , z) := \frac{1}{2\pi \mathrm{i}} \int_\Gamma \frac{\mathrm{d}\zeta}{\zeta - z} \in \mathbb{Z}</math>. === Interior points of a cycle === The '''interior''' of a cycle consists of all points for which the winding number is non-zero: <math>\operatorname{Int}(\Gamma):={z\in\mathbb{C}\setminus\operatorname{Trace}(\Gamma) : n(\Gamma , z) \neq 0}</math>. === Exterior points of a cycle === Analogously, the '''exterior''' is the set of points for which the winding number is zero: <math>\operatorname{Ext}(\Gamma):={z\in\mathbb{C}\setminus\operatorname{Trace}(\Gamma) : n(\Gamma , z) = 0}</math>. === zero-homologous cycle === A cycle is called '''null-homologous''' for a set <math>G\subseteq\mathbb{C}</math> if and only if the interior <math>\operatorname{Int}(\Gamma)</math> lies entirely within <math>G</math>. This is equivalent to the winding number vanishing for all points in <math>\mathbb{C} \setminus G</math>. === Homologous cycles === Two cycles <math>\Gamma_1</math>, <math>\Gamma_2</math> are called '''homologous''' in <math>G\subseteq\mathbb{C}</math> if and only if their formal difference <math>\Gamma_1-\Gamma_2</math> is null-homologous in <math>G</math>. == Integral Theorems == Chains and cycles are important in complex analysis because, as mentioned, they generalize curve integrals. In particular, the integral over a cycle generalizes the closed curve integral. The [[w:en:Cauchy's integral theorem|Cauchy's integral theorem]], the [[w:en:Cauchy's integral formula|Cauchy's integral formula]], and the [[w:en:Residue theorem|Residue theorem]] can be proven for cycles. == Relation to Homology Theory == To indicate that chains and cycles are special cases of objects in [[w:en:Homology (mathematics)|Homology (mathematics)]] of algebraic topology, they are sometimes referred to as 1-chains and 1-cycles.<ref>[[w:en:Otto Forster|Otto Forster]]: ''Riemann surfaces'', Springer 1977; English edition: ''Lectures on Riemann surfaces'', Graduate Texts in Mathematics, Springer-Verlag, 1991, ISBN 3-540-90617-7, Chapter 20</ref>. In algebraic topology, the term 1-cycle or p-cycle is commonly used instead of cycle.<ref>{{Literature| Author=Wolfgang Lück| Title=Algebraic Topology: Homology and Manifolds| Publisher=Vieweg| Year=2005}}</ref>. Additionally, note that the plural of cycle is "cycles," while the plural of Zykel is "Zykel" in German. === Embedding in Homology Theory === The terms chain and cycle are special cases of [[w:en:Mathematical object|Mathematical object]] in [[w:en:Topology (mathematics)|topology]]. In [[w:en:Algebraic topology|Algebraic topology]], one considers [[w:en:Chain complex|complexes of p-chains]] and constructs [[w:en:Homology (mathematics)|Homology (mathematics)]] from them. These groups are [[w:en:Invariant (mathematics)|Invariant (mathematics)]] in topology. A very important [[w:en:Homology (mathematics)|Homology (mathematics)]] is that of [[w:en:Singular homology|Singular homology]]. === 1-Chain of the Singular Complex === A chain, as defined here, is a 1-chain of the [[w:en:Singular homology|Singular homology]], which is a specific chain complex. The operator defined in the section on cycles, <math>\partial \colon C_1(X) \to \operatorname{Div}(X)</math>, is the first [[w:en:Boundary operator|boundary operator]] of the singular complex. The group of divisors is therefore identical to the group of 0-chains. The group of cycles, defined as the kernel of the boundary operator <math>\partial</math>, is a 1-[[w:en:Chain complex|Chain complex]] in the sense of the singular complex. === Algebraic Topology === In algebraic topology, one considers both the kernel of the boundary operator and the image of this operator, constructing a corresponding homology group from these two sets. In the case of the singular complex, one obtains [[w:en:Singular homology|Singular homology]]. In this context, the previously defined terms homologous chain and null-homologous chain take on a more abstract meaning. == See also == *[[w:en:Global Cauchy Integral Theorem]] *[[w:en:Stokes' theorem|Stokes' theorem]] *[[w:en:Smooth function|smooth function]] == References == {{Literature | Author=Wolfgang Fischer, Ingo Lieb | Title=Complex Analysis | Edition=8th | Publisher=Vieweg | Location=Braunschweig | Year=2003 | ISBN=3-528-77247-6 }} [[w:de:Otto Forster|Otto Forster]]: ''Riemann surfaces'', Springer 1977; English edition: ''Lectures on Riemann surfaces'', Graduate Texts in Mathematics, Springer-Verlag, 1991, ISBN 3-540-90617-7, Chapter 20 == Notes == <references /> [[Category:Complex analysis]] == Page Information == You can display this page as '''[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]''' === Wiki2Reveal === The'''[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]''' 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/cycle https://en.wikiversity.org/wiki/Complex%20Analysis/cycle] --> * [https://en.wikiversity.org/wiki/Complex%20Analysis/cycle This page] is designed as a [https://en.wikiversity.org/wiki/PanDocElectron-Presentation PanDocElectron-SLIDE] document type. === Translation and Version Control === This page was translated based on the following [https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/Zyklus 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/Zyklus|Kurs:Funktionentheorie/Zyklus]] - URL: https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/Zyklus * Date: 12/17/2024 <span type="translate" src="Kurs:Funktionentheorie/Zyklus" srclang="de" date="12/17/2024" time="08:50" status="inprogress"></span> <noinclude> [[de:Kurs:Funktionentheorie/Zyklus]] </noinclude> [[Category:Wiki2Reveal]] dkhmqz6ntsrtsyebbi9li06vnfxjs05 2692970 2692969 2024-12-22T21:43:17Z Eshaa2024 2993595 2692970 wikitext text/x-wiki == Introduction == '''Chain''' and '''cycle''' are mathematical objects studied in [[Complex Analysis|Complex Analysis]] but also appear as special cases in [[w:en:Algebraic topology|Algebraic topology]]. A chain generalizes a [[w:en:Path (topology)|curve]], and a cycle generalizes a closed curve. They are primarily used in integration in complex analysis. == Definitions == === Chain === A chain on a set <math>G \subset \mathbb{C}</math> is defined as a finite integer linear combination of paths <math>\gamma_1,\ldots, \gamma_k</math>: <math>\Gamma := \sum_{i=1}^k n_i\gamma_i \quad n_i \in \mathbb{Z}</math>. <math>\gamma_1,\ldots, \gamma_k</math> are generally continuous [[w:en:Curve|Curve]] in <math>G</math>. === Integration over a chain === Let <math>f:G \to \mathbb{C}</math> be integrable, and let <math>\Gamma</math> be a chain of piecewise continuously differentiable paths (paths of integration) <math>\gamma_1,\ldots, \gamma_k</math> in <math>G \subset \mathbb{C}</math>. The integral over the chain <math>\Gamma</math> is defined by: :<math>\int_\Gamma f(z) \, dz := \sum_{i = 1}^k n_i \int_{\gamma_i} f(z) \, dz</math> === Definition: Cycle === '''Version 1:''' A cycle is a chain <math>\Gamma := \sum_{i=1}^k n_i\gamma_i</math>, where every point <math>a \in \mathbb{C}</math> appears as the starting point as many times as it appears as the endpoint of the curves <math>\gamma_i</math>, taking multiplicities <math>n_i</math> into account. '''Version 2:''' A cycle is a chain <math>\Gamma := \sum_{i=1}^k n_i\gamma_i</math> consisting of closed paths <math>\gamma_1, \ldots, \gamma_k</math>. === Connection Between Version 1 and Version 2 === Version 2 is essential for complex analysis. Based on the properties of Version 1, any cycle <math>\Gamma := \sum_{i=1}^k n_i\gamma_i</math> can be transformed into a chain <math>\hat{\Gamma} := \sum_{i=1}^m \hat{n}_i \hat{\gamma}_i</math> of closed paths <math>\hat{\gamma}_1, \ldots, \hat{\gamma}_m</math>. If the paths <math>\gamma_1, \ldots, \gamma_k</math> are piecewise continuously differentiable, then the closed paths <math>\hat{\gamma}1, \ldots, \hat{\gamma}m</math> are also continuously differentiable. For all holomorphic functions <math>f:G \to \mathbb{C}</math>, it holds that: <math>\int\Gamma f(z) , dz = \int{\hat{\Gamma}} f(z) , dz</math>. === Trace of a path === The '''trace''' of a path <math>\gamma : [a,b] \to G</math> is defined as: <math>\operatorname{Trace}(\gamma_i) := \operatorname{Image}(\gamma) := { \gamma(t) ,| , t \in [a,b] }</math>. === Trace of a cycle/chain === The trace of a chain <math>\Gamma</math> is the union of the [[w:en:Image (mathematics)|Image (mathematics)]] of its individual curves, i.e.: <math>\operatorname{Trace}(\Gamma) := \bigcup_{i=1}^N\operatorname{Image}(\gamma_i)</math>. If <math>\operatorname{Trace}(\Gamma) \subset \mathbb{C}</math> is a subset of <math>G \subset \mathbb{C}</math>, then <math>\Gamma</math> is called a cycle '''in''' <math>G</math> if and only if the trace <math>\operatorname{Trace}(\Gamma) \subseteq G</math> lies in <math>G</math>. === Winding number === The '''[[w:en:Winding number|Winding number]]''' is defined analogously to that of a closed curve but uses the integral defined above. For <math>z \not\in \operatorname{Trace}(\Gamma)</math>, it is given by: <math>n(\Gamma , z) := \frac{1}{2\pi \mathrm{i}} \int_\Gamma \frac{\mathrm{d}\zeta}{\zeta - z} \in \mathbb{Z}</math>. === Interior points of a cycle === The '''interior''' of a cycle consists of all points for which the winding number is non-zero: <math>\operatorname{Int}(\Gamma):={z\in\mathbb{C}\setminus\operatorname{Trace}(\Gamma) : n(\Gamma , z) \neq 0}</math>. === Exterior points of a cycle === Analogously, the '''exterior''' is the set of points for which the winding number is zero: <math>\operatorname{Ext}(\Gamma):={z\in\mathbb{C}\setminus\operatorname{Trace}(\Gamma) : n(\Gamma , z) = 0}</math>. === zero-homologous cycle === A cycle is called '''null-homologous''' for a set <math>G\subseteq\mathbb{C}</math> if and only if the interior <math>\operatorname{Int}(\Gamma)</math> lies entirely within <math>G</math>. This is equivalent to the winding number vanishing for all points in <math>\mathbb{C} \setminus G</math>. === Homologous cycles === Two cycles <math>\Gamma_1</math>, <math>\Gamma_2</math> are called '''homologous''' in <math>G\subseteq\mathbb{C}</math> if and only if their formal difference <math>\Gamma_1-\Gamma_2</math> is null-homologous in <math>G</math>. == Integral Theorems == Chains and cycles are important in complex analysis because, as mentioned, they generalize curve integrals. In particular, the integral over a cycle generalizes the closed curve integral. The [[w:en:Cauchy's integral theorem|Cauchy's integral theorem]], the [[w:en:Cauchy's integral formula|Cauchy's integral formula]], and the [[w:en:Residue theorem|Residue theorem]] can be proven for cycles. == Relation to Homology Theory == To indicate that chains and cycles are special cases of objects in [[w:en:Homology (mathematics)|Homology (mathematics)]] of algebraic topology, they are sometimes referred to as 1-chains and 1-cycles.<ref>[[w:en:Otto Forster|Otto Forster]]: ''Riemann surfaces'', Springer 1977; English edition: ''Lectures on Riemann surfaces'', Graduate Texts in Mathematics, Springer-Verlag, 1991, ISBN 3-540-90617-7, Chapter 20</ref>. In algebraic topology, the term 1-cycle or p-cycle is commonly used instead of cycle.<ref>{{Literature| Author=Wolfgang Lück| Title=Algebraic Topology: Homology and Manifolds| Publisher=Vieweg| Year=2005}}</ref>. Additionally, note that the plural of cycle is "cycles," while the plural of Zykel is "Zykel" in German. === Embedding in Homology Theory === The terms chain and cycle are special cases of [[w:en:Mathematical object|Mathematical object]] in [[w:en:Topology (mathematics)|topology]]. In [[w:en:Algebraic topology|Algebraic topology]], one considers [[w:en:Chain complex|complexes of p-chains]] and constructs [[w:en:Homology (mathematics)|Homology (mathematics)]] from them. These groups are [[w:en:Invariant (mathematics)|Invariant (mathematics)]] in topology. A very important [[w:en:Homology (mathematics)|Homology (mathematics)]] is that of [[w:en:Singular homology|Singular homology]]. === 1-Chain of the Singular Complex === A chain, as defined here, is a 1-chain of the [[w:en:Singular homology|Singular homology]], which is a specific chain complex. The operator defined in the section on cycles, <math>\partial \colon C_1(X) \to \operatorname{Div}(X)</math>, is the first [[w:en:Boundary operator|boundary operator]] of the singular complex. The group of divisors is therefore identical to the group of 0-chains. The group of cycles, defined as the kernel of the boundary operator <math>\partial</math>, is a 1-[[w:en:Chain complex|Chain complex]] in the sense of the singular complex. === Algebraic Topology === In algebraic topology, one considers both the kernel of the boundary operator and the image of this operator, constructing a corresponding homology group from these two sets. In the case of the singular complex, one obtains [[w:en:Singular homology|Singular homology]]. In this context, the previously defined terms homologous chain and null-homologous chain take on a more abstract meaning. == See also == *[[w:en:Global Cauchy Integral Theorem]] *[[w:en:Stokes' theorem|Stokes' theorem]] *[[w:en:Smooth function|smooth function]] == References == {{Literature | Author=Wolfgang Fischer, Ingo Lieb | Title=Complex Analysis | Edition=8th | Publisher=Vieweg | Location=Braunschweig | Year=2003 | ISBN=3-528-77247-6 }} [[w:en:Otto Forster|Otto Forster]]: ''Riemann surfaces'', Springer 1977; English edition: ''Lectures on Riemann surfaces'', Graduate Texts in Mathematics, Springer-Verlag, 1991, ISBN 3-540-90617-7, Chapter 20 == Notes == <references /> [[Category:Complex analysis]] == Page Information == You can display this page as '''[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]''' === Wiki2Reveal === The'''[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]''' 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/cycle https://en.wikiversity.org/wiki/Complex%20Analysis/cycle] --> * [https://en.wikiversity.org/wiki/Complex%20Analysis/cycle This page] is designed as a [https://en.wikiversity.org/wiki/PanDocElectron-Presentation PanDocElectron-SLIDE] document type. === Translation and Version Control === This page was translated based on the following [https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/Zyklus 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/Zyklus|Kurs:Funktionentheorie/Zyklus]] - URL: https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/Zyklus * Date: 12/17/2024 <span type="translate" src="Kurs:Funktionentheorie/Zyklus" srclang="de" date="12/17/2024" time="08:50" status="inprogress"></span> <noinclude> [[de:Kurs:Funktionentheorie/Zyklus]] </noinclude> [[Category:Wiki2Reveal]] a8tw26eifulphrsqihu2phs4f0d92qt 0 317290 2692971 2692481 2024-12-22T21:56:29Z Eshaa2024 2993595 2692971 wikitext text/x-wiki == Introduction == The '''Laurent series''' (named after [[w:en:Pierre Alphonse Laurent|Pierre Alphonse Laurent]]) is an [[w:en:Infinite series|infinite series]] similar to a [[w:en:Power series|power series]] but additionally includes negative [[w:en:Exponent (mathematics)|exponents]]. In general, a Laurent series in <math display="inline">x</math> with development point <math display="inline">c</math> has the following form: :<math>f(x) = \sum_{n=-\infty}^\infty c_n (x-a)^n</math> * <math display="inline">c_n</math> coefficients * <math display="inline">a</math> development point of series == Main Part and Regular Part == The series of terms with negative exponents is called the main part of the Laurent series, and the series of terms with non-negative exponents is called the regular part or the residual part. == Connection to Power Series == A Laurent series with a vanishing main part is a [[w:en:Power series|power series]]; if it also has only finitely many terms, then it is a [[w:en:Polynomial|polynomial]]. If a Laurent series has only finitely many terms in total (with negative or positive exponents), it is called a Laurent polynomial. == History == The Laurent series was introduced in 1843 by the French mathematician [[w:en:Pierre Alphonse Laurent|Pierre Alphonse Laurent]]. However, notes in the legacy of the German mathematician [[w:en:Karl Weierstrass|Karl Weierstrass]] suggest that he discovered it as early as 1841. == Laurent Decomposition == The principle of developing a [[w:en:Holomorphic function|holomorphic function]] into a Laurent series is based on the Laurent decomposition. To do this, consider an annular region <math>\mathcal{R} = {z \in \mathbb{C} ; |; r < |z| < R} </math>. Now define two holomorphic functions <math>g</math> and <math>h</math>: :<math>g\colon U_R(0) \rightarrow \mathbb{C}</math> :<math>h\colon U_{\frac{1}{r}}(0) \rightarrow \mathbb{C}</math>. == Representation of Laurent Series by Two Holomorphic Functions == Let <math>g:G \rightarrow \mathbb{C}</math> and <math>h:G \rightarrow \mathbb{C}</math> be two holomorphic functions with a development point <math>z_0 \in G</math>, :<math>f(z):=g(z-z_0)+\hat{h}(z-z_0)</math> with <math>\hat{h}(z):= h(1/z)</math>. <math>g</math> and <math>h</math> are holomorphic functions on <math>G_0:= {z-z_0 \in \mathbb{C} \ | \ z \in G }</math>, which can be developed into a power series around 0 in <math>G_0</math>. == Convergence Set of Laurent Series == The functions <math>g</math> and <math>h</math> can be locally represented as a power series on a disk in <math>G_o := {z - z_o \in \mathbb{C} \ | \ z \in G }</math> (holomorphy criterion). Then <math>\hat{h}</math> with <math>\hat{h}(z) := h(1/z)</math> converges on the complement of a disk. == Intersection of Convergence Domains == If <math>f(z)</math>'s principal part <math>g(z)</math> and <math>\hat{h}(z)</math> are convergent, then <math>z</math> lies in the intersection of the convergence sets. If <math>r > R</math>, the convergence set is empty because <math>z</math> would simultaneously have to lie on a disk of radius <math>R</math> and on the complement of a disk with radius <math>r</math>. === Convergence Radii === Let <math>R_g > 0</math> and <math>R_h > 0</math> be the convergence radii for the functions <math>g</math> and <math>h</math>. Calculate the radius <math>R_{\hat{h}} > 0</math> of the convergence set of <math>\hat{h}(z) := h(1/z)</math> for all <math>z \in G_o</math> with <math>|z| > R_{\hat{h}} > 0</math>. == Geometry of the Convergence Set == <math>h</math> converges holomorphically around the center on the disk with radius <math>1/r</math>. Since the argument of the function <math>h</math> must lie within the defined circular region, it quickly becomes evident that the function <math>h(1/z)</math> is defined for values <math>|z| > r</math>. Thus, the sum of the two functions :<math>f(z) = g(z) + h\left(\frac{1}{z}\right)</math> is analytic on the annulus <math>\mathcal{R}</math>. == Uniqueness of Decomposition == It can be shown that any holomorphic function on an annular domain can be decomposed in this way. If one also assumes <math>h(0) = 0</math>, the decomposition is unique. By expanding this function in the form of power series, the following representation arises: :<math>f(z) = g(z) + h\left(\frac{1}{z}\right) = \sum_{n=0}^{\infty} a_n z^n + \sum_{n=1}^{\infty} b_n z^{-n} = \sum_{n=-\infty}^{\infty} a_n z^n</math>. Here, <math>b_{n} \equiv a_{-n}</math> is defined. Additionally, <math>b_0 = 0</math> follows from the condition <math>h(0) = 0</math>. == Decomposition with Expansion Point == If these considerations are extended to an expansion around a point <math>c</math>, rather than the origin, the initially stated definition of the Laurent series for a holomorphic function <math>f</math> around the expansion point <math>c</math> results: :<math>f(z) = \sum_{n=-\infty}^{\infty} a_n (z-c)^n</math> == Example == In the following, <math display="inline">\mathbb{K}</math> refers to either the [[w:en:Real number|real numbers]] or the [[w:en:Complex number|complex numbers]]. :<math>f\colon \mathbb{K}\to \mathbb{K} \colon x\mapsto\begin{cases} \exp\left(-\frac{1}{x^2}\right), & x\neq 0\ 0, & \text{otherwise}\end{cases}</math>. The function is infinitely often [[w:en:Differentiable function|differentiable]] in the real sense, but it is not [[w:en:Holomorphic function|holomorphic]] at <math>x = 0</math>, where it has an [[w:en:Essential singularity|essential singularity]]. == Substituting into the Taylor Series == By substituting <math display="inline">z = -\frac{1}{x^2}</math> into the power series expansion of the [[w:en:Exponential function|exponential function]], :<math>e^z = \sum_{n=0}^\infty \frac{z^{n}}{n!} = \sum_{n=0}^\infty \frac{\left(-\frac{1}{x^2}\right)^{n}}{n!} </math> the Laurent series of <math display="inline">f</math> with the expansion point <math display="inline">0</math> is obtained: :<math>f(x) = \sum_{n=0}^\infty (-1)^n\frac{x^{-2n}}{n!} = \underbrace{\sum_{n=-\infty}^{-1} \frac{(-1)^n}{(-n)!} x^{2n} }_{\text{Principal part}} + 1</math> == Convergence Domain of the Laurent Series == The secondary part <math>g(x) = 1</math> converges throughout <math>\mathbb{C}</math>, and the principal part (and therefore the entire Laurent series) converges for every complex number <math display="inline">x \neq 0</math>. == Approximation of the Function by Partial Sums == [[File:Laurentreihe_Exp_-X-2.png|thumb|Approximation of Laurent series by partial sums]] The image shows how the partial sum sequence :<math>f_n(x) = \sum_{j=0}^n (-1)^j\frac{x^{-2j}}{j!}</math> approaches the function. == Comparison of Graphs of Partial Sums with the Function == [[File:Laurentreihe_Exp_-X-2.png|450px|Approximation of Laurent series by partial sums]]. Since graphs in <math>\mathbb{C}</math> are subsets of 4-dimensional <math>\mathbb{R}</math>-vector spaces, the graph is plotted here for values <math>x \in \mathbb{R} \setminus {0}</math>. The Laurent expansion can be continuously extended at 0. == Convergence of Laurent Series == Laurent series are important tools in [[w:en:Complex analysis|complex analysis]], especially for studying functions with [[w:en:Isolated singularity|isolated singularities]]. == Annuli and Disks == Laurent series describe complex functions that are [[w:en:Holomorphic function|holomorphic]] on an [[w:en:Annulus (mathematics)|annulus]], just as power series describe functions holomorphic on a [[w:en:Disk (mathematics)|disk]]. Let :<math>\sum_{n=-\infty}^\infty a_n (z-c)^n</math> be a Laurent series in <math>z</math> with complex coefficients <math>a_n</math> and expansion point <math>c</math>. == Convergence Radii - Interior of the Annulus == There are two uniquely determined numbers <math>r</math> and <math>R</math> such that: The Laurent series converges [[w:en:Uniform convergence|uniformly]] and [[w:en:Absolute convergence|absolutely]] on the open annulus <math>A := { z : r < \vert z - c \vert < R }</math>. It converges normally, meaning the principal and secondary parts converge normally. This defines a holomorphic function <math>f</math> on <math>A</math>. == Outside the Annulus == Outside the annulus, the Laurent series diverges. For every point in :<math>\mathbb{C}\setminus \overline{A} := { z : r > \vert z - c \vert \vee \vert z - c \vert > R }</math>, either the terms with positive (secondary part) or negative exponents (principal part) diverge. == Convergence Radii and Cauchy-Hadamard == The two radii can be calculated using the [[w:en:Cauchy-Hadamard theorem|Cauchy-Hadamard formula]]: :<math>r = \limsup_{n\to\infty} \vert a_{-n} \vert ^{1/n}</math> :<math>R = \frac{1}{\limsup_{n\to\infty} \vert a_n \vert ^{1/n}}</math> We set <math>\frac{1}{0}=\infty</math> and <math>\frac{1}{\infty}=0</math> in the second formula. == Functions Defined on Annuli == Conversely, one can start with an annulus <math>A := { z : r < \vert z - c \vert < R }</math> and a function <math>f</math> that is holomorphic on <math>A</math>. Then, there always exists a uniquely determined Laurent series with expansion point <math>c</math> that converges (at least) on <math>A</math> and coincides with <math>f</math> there. The coefficients satisfy: :<math>a_n=\frac{1}{2\pi\mathrm{i}}\oint_{\partial U_\varrho(c)}\frac{f(\zeta)}{\left(\zeta-c\right)^{n+1}}\mathrm{d}\zeta</math> for all <math>n\in\mathbb{Z}</math> and a <math>\varrho\in(r,R)</math>. Due to the [[w:en:Cauchy integral theorem|Cauchy integral theorem]], the choice of <math>\varrho</math> does not matter. == Punctured Disk == The case <math>r = 0</math>, i.e., a holomorphic function <math>f</math> on a punctured disk around <math>c</math>, is particularly important. The coefficient <math>a_{-1}</math> in the Laurent series expansion of <math>f</math> is called the [[w:en:Residue (complex analysis)|residue]] of <math>f</math> at the isolated singularity <math>c</math>. It plays a significant role in the [[w:en:Residue theorem|residue theorem]]. == Formal Laurent Series == Formal Laurent series are Laurent series in the indeterminate <math>X</math>, used without consideration of convergence. == Laurent Series on Commutative Rings == The coefficients <math>a_k</math> can then belong to any [[w:en:Commutative ring|commutative]] [[w:en:ring (mathematics)|ring]]. In this context, it only makes sense to consider Laurent series with finitely many negative exponents, known as a "finite principal part," and to omit the expansion point by setting <math>c = 0</math>. == Equality of Formal Laurent Series == Two such formal Laurent series are defined as equal if and only if all their coefficients agree. Laurent series are added by summing their respective coefficients. Since there are only finitely many terms with negative exponents, they can be multiplied by [[w:en:Convolution|convolution]] of their coefficient sequences, similar to power series. With these operations, the set of all Laurent series over a commutative ring <math>R</math> forms a commutative ring, denoted by <math>R \left(!\left( X \right)!\right)</math>. == Laurent Series and Integral Domains == If <math>K</math> is a [[w:en:Field (mathematics)|field]], the [[w:en:Formal power series|formal power series]] in the indeterminate <math>X</math> over <math>K</math> form an [[w:en:Integral domain|integral domain]], denoted by <math>K\left[!\left[ X \right]!\right]</math>. Its [[w:en:Field of fractions|field of fractions]] is [[w:en:Isomorphism|isomorphic]] to the field <math>K \left(!\left( X \right)!\right)</math> of Laurent series over <math>K</math>. == Exercises == Let <math>K_{r_1,r_2}:={z \in\mathbb{C} , | , r_1 <|z| < r_2}</math>. Construct a Laurent series with this annulus as its domain of convergence, which does not converge on <math>\mathbb{C}\setminus \overline{K_{r_1,r_2}}</math>. Use geometric series as an idea with <math>\sum_{n=0}^{+\infty} q^n</math> converging for <math>q \in \mathbb{C}</math> when <math>|q| < 1</math>. == Exercises on Laurent Series and b-adic Number Systems == Analyze the relationship between Laurent series and the p-adic number system (e.g., binary system, hexadecimal system)! What are the similarities and differences? Represent the number <math>\frac{1}{7}</math> as a value of a Laurent series in the 4-based number system <math>x=4</math>, where <math>c_n \in {0,1,2,3}</math>. Calculate the coefficients <math>c_n</math>: :: <math>f(x):= \sum_{n=-\infty}^{+\infty} c_n \cdot z^n</math>z == Literature == [[w:en:Eberhard Freitag|Eberhard Freitag]] & Rolf Busam: ''Complex Analysis 1'', Springer-Verlag, Berlin, ISBN 3-540-67641-4 == See Also == *[[w:en:Power series|Power series]] *[[w:en:Residue (complex analysis)|Residue]] *[[w:en:Laurent series#Examples|Example calculations with Laurent series for rational functions]] [[Category:Complex Analysis]] [[Category:Sequences and Series]] == Page Information == You can display this page as '''[https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Laurent%20Series&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Laurent%20Series&coursetitle=Complex%20Analysis Wiki2Reveal slides]''' === Wiki2Reveal === The '''[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]''' 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/Laurent%20Series https://en.wikiversity.org/wiki/Laurent%20Series] --> * [https://en.wikiversity.org/wiki/Laurent%20Series This page] is designed as a [https://en.wikiversity.org/wiki/PanDocElectron-Presentation PanDocElectron-SLIDE] document type. * Source: Wikiversity https://en.wikiversity.org/wiki/Laurent%20Series * see [[v:en:Wiki2Reveal|Wiki2Reveal]] for the functionality of [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]. <!-- * Next contents of the course are [[]] -->; === Translation and Version Control === This page was translated based on the following [https://de.wikiversity.org/wiki/Laurent-Reihe Wikiversity source page] and uses the concept of [[Translation and Version Control]] for a transparent language fork in a Wikiversity: * Source: [[v:de:Laurent-Reihe|Laurent-Reihe]] - URL: https://de.wikiversity.org/wiki/Laurent-Reihe * Date: 12/17/2024 <span type="translate" src="Kurvenintegral" srclang="de" date="12/17/2024" time="17:04" status="inprogress"></span> <noinclude> [[de:Laurent-Reihe]] </noinclude> [[Category:Wiki2Reveal]] 7tnsmrho4lsimvl1l4pemvfi0pzrixf 2692972 2692971 2024-12-22T22:22:27Z Eshaa2024 2993595 2692972 wikitext text/x-wiki == Introduction == The '''Laurent series''' (named after [[w:en:Pierre Alphonse Laurent|Pierre Alphonse Laurent]]) is an [[w:en:Infinite series|infinite series]] similar to a [[w:en:Power series|power series]] but additionally includes negative [[w:en:Exponent (mathematics)|exponents]]. In general, a Laurent series in <math display="inline">x</math> with development point <math display="inline">c</math> has the following form: :<math>f(x) = \sum_{n=-\infty}^\infty c_n (x-a)^n</math> * <math display="inline">c_n</math> coefficients * <math display="inline">a</math> development point of series == Main Part and Regular Part == The series of terms with negative exponents is called the main part of the Laurent series, and the series of terms with non-negative exponents is called the regular part or the residual part. == Connection to Power Series == A Laurent series with a vanishing main part is a [[w:en:Power series|power series]]; if it also has only finitely many terms, then it is a [[w:en:Polynomial|polynomial]]. If a Laurent series has only finitely many terms in total (with negative or positive exponents), it is called a Laurent polynomial. == History == The Laurent series was introduced in 1843 by the French mathematician [[w:en:Pierre Alphonse Laurent|Pierre Alphonse Laurent]]. However, notes in the legacy of the German mathematician [[w:en:Karl Weierstrass|Karl Weierstrass]] suggest that he discovered it as early as 1841. == Laurent Decomposition == The principle of developing a [[w:en:Holomorphic function|holomorphic function]] into a Laurent series is based on the Laurent decomposition. To do this, consider an annular region <math>\mathcal{R} = {z \in \mathbb{C} ; |; r < |z| < R} </math>. Now define two holomorphic functions <math>g</math> and <math>h</math>: :<math>g\colon U_R(0) \rightarrow \mathbb{C}</math> :<math>h\colon U_{\frac{1}{r}}(0) \rightarrow \mathbb{C}</math>. == Representation of Laurent Series by Two Holomorphic Functions == Let <math>g:G \rightarrow \mathbb{C}</math> and <math>h:G \rightarrow \mathbb{C}</math> be two holomorphic functions with a development point <math>z_0 \in G</math>, :<math>f(z):=g(z-z_0)+\hat{h}(z-z_0)</math> with <math>\hat{h}(z):= h(1/z)</math>. <math>g</math> and <math>h</math> are holomorphic functions on <math>G_0:= {z-z_0 \in \mathbb{C} \ | \ z \in G }</math>, which can be developed into a power series around 0 in <math>G_0</math>. == Convergence Set of Laurent Series == The functions <math>g</math> and <math>h</math> can be locally represented as a power series on a disk in <math>G_o := {z - z_o \in \mathbb{C} \ | \ z \in G }</math> (holomorphy criterion). Then <math>\hat{h}</math> with <math>\hat{h}(z) := h(1/z)</math> converges on the complement of a disk. == Intersection of Convergence Domains == If <math>f(z)</math>'s principal part <math>g(z)</math> and <math>\hat{h}(z)</math> are convergent, then <math>z</math> lies in the intersection of the convergence sets. If <math>r > R</math>, the convergence set is empty because <math>z</math> would simultaneously have to lie on a disk of radius <math>R</math> and on the complement of a disk with radius <math>r</math>. === Convergence Radii === Let <math>R_g > 0</math> and <math>R_h > 0</math> be the convergence radii for the functions <math>g</math> and <math>h</math>. Calculate the radius <math>R_{\hat{h}} > 0</math> of the convergence set of <math>\hat{h}(z) := h(1/z)</math> for all <math>z \in G_o</math> with <math>|z| > R_{\hat{h}} > 0</math>. == Geometry of the Convergence Set == <math>h</math> converges holomorphically around the center on the disk with radius <math>1/r</math>. Since the argument of the function <math>h</math> must lie within the defined circular region, it quickly becomes evident that the function <math>h(1/z)</math> is defined for values <math>|z| > r</math>. Thus, the sum of the two functions :<math>f(z) = g(z) + h\left(\frac{1}{z}\right)</math> is analytic on the annulus <math>\mathcal{R}</math>. == Uniqueness of Decomposition == It can be shown that any holomorphic function on an annular domain can be decomposed in this way. If one also assumes <math>h(0) = 0</math>, the decomposition is unique. By expanding this function in the form of power series, the following representation arises: :<math>f(z) = g(z) + h\left(\frac{1}{z}\right) = \sum_{n=0}^{\infty} a_n z^n + \sum_{n=1}^{\infty} b_n z^{-n} = \sum_{n=-\infty}^{\infty} a_n z^n</math>. Here, <math>b_{n} \equiv a_{-n}</math> is defined. Additionally, <math>b_0 = 0</math> follows from the condition <math>h(0) = 0</math>. == Decomposition with Expansion Point == If these considerations are extended to an expansion around a point <math>c</math>, rather than the origin, the initially stated definition of the Laurent series for a holomorphic function <math>f</math> around the expansion point <math>c</math> results: :<math>f(z) = \sum_{n=-\infty}^{\infty} a_n (z-c)^n</math> == Example == In the following, <math display="inline">\mathbb{K}</math> refers to either the [[w:en:Real number|real numbers]] or the [[w:en:Complex number|complex numbers]]. :<math>f\colon \mathbb{K}\to \mathbb{K} \colon x\mapsto\begin{cases} \exp\left(-\frac{1}{x^2}\right), & x\neq 0\ 0, & \text{otherwise}\end{cases}</math>. The function is infinitely often [[w:en:Differentiable function|differentiable]] in the real sense, but it is not [[w:en:Holomorphic function|holomorphic]] at <math>x = 0</math>, where it has an [[w:en:Essential singularity|essential singularity]]. == Substituting into the Taylor Series == By substituting <math display="inline">z = -\frac{1}{x^2}</math> into the power series expansion of the [[w:en:Exponential function|exponential function]], :<math>e^z = \sum_{n=0}^\infty \frac{z^{n}}{n!} = \sum_{n=0}^\infty \frac{\left(-\frac{1}{x^2}\right)^{n}}{n!} </math> the Laurent series of <math display="inline">f</math> with the expansion point <math display="inline">0</math> is obtained: :<math>f(x) = \sum_{n=0}^\infty (-1)^n\frac{x^{-2n}}{n!} = \underbrace{\sum_{n=-\infty}^{-1} \frac{(-1)^n}{(-n)!} x^{2n} }_{\text{Principal part}} + 1</math> == Convergence Domain of the Laurent Series == The secondary part <math>g(x) = 1</math> converges throughout <math>\mathbb{C}</math>, and the principal part (and therefore the entire Laurent series) converges for every complex number <math display="inline">x \neq 0</math>. == Approximation of the Function by Partial Sums == [[File:Laurentreihe_Exp_-X-2.png|thumb|Approximation of Laurent series by partial sums]] The image shows how the partial sum sequence :<math>f_n(x) = \sum_{j=0}^n (-1)^j\frac{x^{-2j}}{j!}</math> approaches the function. == Comparison of Graphs of Partial Sums with the Function == [[File:Laurentreihe_Exp_-X-2.png|450px|Approximation of Laurent series by partial sums]]. Since graphs in <math>\mathbb{C}</math> are subsets of 4-dimensional <math>\mathbb{R}</math>-vector spaces, the graph is plotted here for values <math>x \in \mathbb{R} \setminus {0}</math>. The Laurent expansion can be continuously extended at 0. == Convergence of Laurent Series == Laurent series are important tools in [[w:en:Complex analysis|complex analysis]], especially for studying functions with [[w:en:Isolated singularity|isolated singularities]]. == Annuli and Disks == Laurent series describe complex functions that are [[w:en:Holomorphic function|holomorphic]] on an [[w:en:Annulus (mathematics)|annulus]], just as power series describe functions holomorphic on a [[w:en:Disk (mathematics)|disk]]. Let :<math>\sum_{n=-\infty}^\infty a_n (z-c)^n</math> be a Laurent series in <math>z</math> with complex coefficients <math>a_n</math> and expansion point <math>c</math>. == Convergence Radii - Interior of the Annulus == There are two uniquely determined numbers <math>r</math> and <math>R</math> such that: The Laurent series converges [[w:en:Normal convergence|Normal convergence]] and [[w:en:Absolute convergence|Absolute convergence]] on the open annulus <math>A := { z : r < \vert z - c \vert < R }</math>. It converges normally, meaning the principal and secondary parts converge normally. This defines a holomorphic function <math>f</math> on <math>A</math>. == Outside the Annulus == Outside the annulus, the Laurent series diverges. For every point in :<math>\mathbb{C}\setminus \overline{A} := { z : r > \vert z - c \vert \vee \vert z - c \vert > R }</math>, either the terms with positive (secondary part) or negative exponents (principal part) diverge. == Convergence Radii and Cauchy-Hadamard == The two radii can be calculated using the [[w:en:Cauchy-Hadamard theorem|Cauchy-Hadamard formula]]: :<math>r = \limsup_{n\to\infty} \vert a_{-n} \vert ^{1/n}</math> :<math>R = \frac{1}{\limsup_{n\to\infty} \vert a_n \vert ^{1/n}}</math> We set <math>\frac{1}{0}=\infty</math> and <math>\frac{1}{\infty}=0</math> in the second formula. == Functions Defined on Annuli == Conversely, one can start with an annulus <math>A := { z : r < \vert z - c \vert < R }</math> and a function <math>f</math> that is holomorphic on <math>A</math>. Then, there always exists a uniquely determined Laurent series with expansion point <math>c</math> that converges (at least) on <math>A</math> and coincides with <math>f</math> there. The coefficients satisfy: :<math>a_n=\frac{1}{2\pi\mathrm{i}}\oint_{\partial U_\varrho(c)}\frac{f(\zeta)}{\left(\zeta-c\right)^{n+1}}\mathrm{d}\zeta</math> for all <math>n\in\mathbb{Z}</math> and a <math>\varrho\in(r,R)</math>. Due to the [[w:en:Cauchy integral theorem|Cauchy integral theorem]], the choice of <math>\varrho</math> does not matter. == Punctured Disk == The case <math>r = 0</math>, i.e., a holomorphic function <math>f</math> on a punctured disk around <math>c</math>, is particularly important. The coefficient <math>a_{-1}</math> in the Laurent series expansion of <math>f</math> is called the [[w:en:Residue (complex analysis)|residue]] of <math>f</math> at the isolated singularity <math>c</math>. It plays a significant role in the [[w:en:Residue theorem|residue theorem]]. == Formal Laurent Series == Formal Laurent series are Laurent series in the indeterminate <math>X</math>, used without consideration of convergence. == Laurent Series on Commutative Rings == The coefficients <math>a_k</math> can then belong to any [[w:en:Commutative property|Commutative]] [[w:en:Ring (mathematics)|Ring]]. In this context, it only makes sense to consider Laurent series with finitely many negative exponents, known as a "finite principal part," and to omit the expansion point by setting <math>c = 0</math>. == Equality of Formal Laurent Series == Two such formal Laurent series are defined as equal if and only if all their coefficients agree. Laurent series are added by summing their respective coefficients. Since there are only finitely many terms with negative exponents, they can be multiplied by [[w:en:Convolution|Convolution]] of their coefficient sequences, similar to power series. With these operations, the set of all Laurent series over a commutative ring <math>R</math> forms a commutative ring, denoted by <math>R \left(!\left( X \right)!\right)</math>. == Laurent Series and Integral Domains == If <math>K</math> is a [[w:en:Field (mathematics)|field]], the [[w:en:Formal power series|formal power series]] in the indeterminate <math>X</math> over <math>K</math> form an [[w:en:Integral domain|integral domain]], denoted by <math>K\left[!\left[ X \right]!\right]</math>. Its [[w:en:Field of fractions|field of fractions]] is [[w:en:Isomorphism|isomorphic]] to the field <math>K \left(!\left( X \right)!\right)</math> of Laurent series over <math>K</math>. == Exercises == Let <math>K_{r_1,r_2}:={z \in\mathbb{C} , | , r_1 <|z| < r_2}</math>. Construct a Laurent series with this annulus as its domain of convergence, which does not converge on <math>\mathbb{C}\setminus \overline{K_{r_1,r_2}}</math>. Use geometric series as an idea with <math>\sum_{n=0}^{+\infty} q^n</math> converging for <math>q \in \mathbb{C}</math> when <math>|q| < 1</math>. == Exercises on Laurent Series and b-adic Number Systems == Analyze the relationship between Laurent series and the p-adic number system (e.g., binary system, hexadecimal system)! What are the similarities and differences? Represent the number <math>\frac{1}{7}</math> as a value of a Laurent series in the 4-based number system <math>x=4</math>, where <math>c_n \in {0,1,2,3}</math>. Calculate the coefficients <math>c_n</math>: :: <math>f(x):= \sum_{n=-\infty}^{+\infty} c_n \cdot z^n</math>z == Literature == [[w:en:Eberhard Freitag|Eberhard Freitag]] & Rolf Busam: ''Complex Analysis 1'', Springer-Verlag, Berlin, ISBN 3-540-67641-4 == See Also == *[[w:en:Power series|Power series]] *[[w:en:Residue (complex analysis)|Residue]] *[[w:en:Laurent series#Examples|Example calculations with Laurent series for rational functions]] [[Category:Complex Analysis]] [[Category:Sequences and Series]] == Page Information == You can display this page as '''[https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Laurent%20Series&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Laurent%20Series&coursetitle=Complex%20Analysis Wiki2Reveal slides]''' === Wiki2Reveal === The '''[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]''' 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/Laurent%20Series https://en.wikiversity.org/wiki/Laurent%20Series] --> * [https://en.wikiversity.org/wiki/Laurent%20Series This page] is designed as a [https://en.wikiversity.org/wiki/PanDocElectron-Presentation PanDocElectron-SLIDE] document type. * Source: Wikiversity https://en.wikiversity.org/wiki/Laurent%20Series * see [[v:en:Wiki2Reveal|Wiki2Reveal]] for the functionality of [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]. <!-- * Next contents of the course are [[]] -->; === Translation and Version Control === This page was translated based on the following [https://de.wikiversity.org/wiki/Laurent-Reihe Wikiversity source page] and uses the concept of [[Translation and Version Control]] for a transparent language fork in a Wikiversity: * Source: [[v:de:Laurent-Reihe|Laurent-Reihe]] - URL: https://de.wikiversity.org/wiki/Laurent-Reihe * Date: 12/17/2024 <span type="translate" src="Kurvenintegral" srclang="de" date="12/17/2024" time="17:04" status="inprogress"></span> <noinclude> [[de:Laurent-Reihe]] </noinclude> [[Category:Wiki2Reveal]] 1xd85me7fa589yy3dz9xbvu2kqiduy6 0 317327 2692975 2692476 2024-12-22T22:52:11Z Eshaa2024 2993595 2692975 wikitext text/x-wiki ==Introduction== The Cauchy integral theorem is one of the central results of [[w:en:Complex Analysis|Complex Analysis]]. It exists in various versions, and in this article, we aim to present a basic one for convex regions and a relatively general one for [[w:en:Homology (mathematics)|nullhomologous cycles]]. ==For Convex Regions== === Statement === Let <math>G \subseteq \mathbb{C}</math> be a convex region, and let <math>\gamma</math> be a closed [[Complex Analysis/rectifiable curve|rectifiable curve]] [[Complex Analysis/Trace|Trace of Curve]] in <math>G</math>. Then, for every holomorphic function <math>f \colon G \to \mathbb{C}</math>, the following holds: <center><math>\int_\gamma f(z)\, dz = 0</math></center> === Proof 1: Primitive of <math>f</math> === First, we observe that <math>f</math> has a primitive in <math>G</math>. Fix a point <math>z_0 \in G</math>. For any point <math>z \in G</math>, let <math>[z_0, z]</math> denote the straight-line segment connecting <math>z_0</math> and <math>z</math>. == Proof 2: Definition of the Primitive == Define <math>F \colon G \to \mathbb{C}</math> by: :<math>F(z) := \int_{[z_0, z]} f(\zeta), d\zeta</math>. Due to the convexity of <math>G</math>, the triangle <math>D</math> with vertices <math>z_0, z, w</math> lies entirely within <math>G</math> for <math>z, w \in G</math>. === Proof 3: Application of Goursat’s Lemma === By [[Complex Analysis/Goursat's Lemma]] for the boundary <math>\partial \Delta</math> of a triangle <math>\Delta</math> with vertices <math>z_0, z, w \in \mathbb{C}</math>, we have: :<math> \begin{array}{rl} 0 &= \int_{\partial \Delta} f(z)\, dz \\ &= \int_{[z_{0},z]} f(\zeta)\, d\zeta - \int_{[z_{0},w]} f(\zeta)\, d\zeta + \int_{[z,w]} f(\zeta)\, d\zeta \\ & = F(z) - F(w) + \int_{[z,w]} f(\zeta)\, d\zeta \end{array} </math> === Proof 4: Conclusion Using Goursat's Lemma === This leads to: :<math>\begin{array}{rl} F(z) - F(w) &= \int_{[w,z]} f(\zeta)\,d\zeta\\ &= \int_0^1 f\bigl(w + t(z-w)\bigr)\cdot (z-w)\, dt\\ &= \underbrace{\int_0^1 f\bigl(w+t(z-w)\bigr)\, dt}_{A(z):=} \cdot (z-w) \end{array}</math> Thus, we have: :<math>A(z)=\frac{F(z) - F(w)}{(z-w)}</math> === Proof 5: Limit Process === Since <math>A</math> is continuous in <math>w</math>, taking the limit as <math>z \to w</math> gives: :<math>A(w) = \lim_{z \to w} A(z) = \lim_{z \to w} \frac{F(z) - F(w)}{(z-w)} = F'(w).</math> === Proof 5: Differentiability of <math>F</math> === Therefore, <math>A \colon G \to \mathbb{C}</math> is continuous, and <math>F</math> is differentiable in <math>w \in G</math>, with: <center><math>F'(w) = A(w) = f(w).</math></center> Since <math>w \in G</math> was arbitrary, we conclude <math>F' = f</math>, proving that <math>f</math> has a primitive. === Proof 6: Path Integration === Now, let <math>\gamma \colon [a, b] \to G</math> be a piecewise continuously differentiable, closed curve. Then: <center><math> \begin{array}{rl} \int_\gamma f(z)\, dz &= \int_a^b f(\gamma(t)) \gamma'(t)\, dt \\ &= \int_a^b F'(\gamma(t)) \gamma'(t)\, dt \\ &= \int_a^b (F \circ \gamma)'(t)\, dt \\ &= F(\gamma(b)) - F(\gamma(a)) = 0. \end{array} </math></center> === Proof 7: === Let <math>\gamma \colon [a, b] \to G</math> be an arbitrary integration path in <math>G</math>, and let <math>\epsilon > 0</math>. As shown [[w:en:Path integral#Approximation by polygonal paths|here]], we choose a polygonal path <math>\hat{\gamma} \colon [a, b] \to \mathbb{C}</math> such that <math>\hat{\gamma}(a) = \gamma(a)</math>, <math>\hat{\gamma}(b) = \gamma(b)</math>, and :<math>\left|\int_{\hat{\gamma}} f(z), dz - \int_{\gamma} f(z), dz\right| < \epsilon.</math> Since polygonal paths are piecewise continuously differentiable, the above result implies <math>\int_{\hat{\gamma}} f(z), dz = 0</math>. Consequently, :<math>\left|\int_{\gamma} f(z), dz\right| < \epsilon.</math> As <math>\epsilon > 0</math> was arbitrary, the claim follows. == For Cycles in Arbitrary Open Sets == In arbitrary open sets, one must ensure that cycles do not enclose singularities or poles in the complement of the domain. Enclosing such singularities may contribute a non-zero value to the integral (e.g., the function <math>f(z) = \frac{1}{z}</math> and <math>\gamma(t) := e^{it}</math> in a domain <math>G = \mathbb{C}\setminus \{0\}</math> . Even though <math>f</math> is holomorphic in <math>G</math>, the integral is not zero but <math>2\pi i</math> (see [[w:en:Homology (mathematics)#Nullhomologous cycle|nullhomologous cycle]]). === Statement === Let <math>G \subseteq \mathbb{C}</math> be open, and let <math>\Gamma</math> be a [[w:en:Homology (mathematics)#Nullhomologous cycle|nullhomologous]] cycle in <math>G</math>. Then, for every holomorphic function <math>f \colon G \to \mathbb{C}</math>, the following holds: <center><math>\int_\Gamma f(z)\, dz = 0</math></center> === Proof === Let <math>w \in G \setminus \text{trace}(\Gamma)</math>, and define <math>g \colon G \to \mathbb{C}</math> by <center><math>g(z) := (z-w) \cdot f(z).</math></center> Then, <math>g</math> is holomorphic, and by the [[w:en:Cauchy integral formula#For cycles in arbitrary open sets|global integral formula]], we have: :<math> \int_\Gamma f(z)\, dz = \int_{\Gamma} \frac{g(z)}{z-w}\, dz = 2\pi i \cdot n(\Gamma, w) \cdot g(w) = 0. </math> == See Also == *[[w:en:Complex Analysis|Complex Analysis]] *[[w:en:Cauchy's integral theorem|Cauchy integral theorem for disks]] == Page Information == You can display this page as ''' ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis%20Cauchy%20Integral%20Theorem&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Complex%20Analysis%20Cauchy%20Integral%20Theorem&coursetitle=Complex%20Analysis Wiki2Reveal slides])''' === Wiki2Reveal === The '''[https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis%20Cauchy%20Integral%20Theorem&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Complex%20Analysis%20Cauchy%20Integral%20Theorem&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%20Cauchy%20Integral%20Theoremhttps://en.wikiversity.org/wiki/Complex%20Analysis%20Cauchy%20Integral%20Theorem] --> * [https://en.wikiversity.org/wiki/Laurent%20Series This page] is designed as a [https://en.wikiversity.org/wiki/PanDocElectron-Presentation PanDocElectron-SLIDE] document type. * Source: Wikiversity https://en.wikiversity.org/wiki/Laurent%20Series * see [[v:en:Wiki2Reveal|Wiki2Reveal]] for the functionality of [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]. <!-- * Next contents of the course are [[]] -->; === Translation and Version Control === This page was translated based on the following [https://de.wikiversity.org/wiki/Integralsatz von Cauchy Wikiversity source page] and uses the concept of [[Translation and Version Control]] for a transparent language fork in a Wikiversity: * Source: [[v:de:Integralsatz von Cauchy|Integralsatz von Cauchy]] - URL: https://de.wikiversity.org/wiki/Integralsatz von Cauchy * Date: 12/18/2024 <span type="translate" src="Integralsatz von Cauchy" srclang="de" date="12/18/2024" time="09:15" status="inprogress"></span> <noinclude> [[de:Integralsatz von Cauchy]] </noinclude> [[Category:Wiki2Reveal]] giuoe6hsmwqyowlh5nkqe047yvynvbo 2692976 2692975 2024-12-22T23:01:55Z Eshaa2024 2993595 /* Proof 7: */ 2692976 wikitext text/x-wiki ==Introduction== The Cauchy integral theorem is one of the central results of [[w:en:Complex Analysis|Complex Analysis]]. It exists in various versions, and in this article, we aim to present a basic one for convex regions and a relatively general one for [[w:en:Homology (mathematics)|nullhomologous cycles]]. ==For Convex Regions== === Statement === Let <math>G \subseteq \mathbb{C}</math> be a convex region, and let <math>\gamma</math> be a closed [[Complex Analysis/rectifiable curve|rectifiable curve]] [[Complex Analysis/Trace|Trace of Curve]] in <math>G</math>. Then, for every holomorphic function <math>f \colon G \to \mathbb{C}</math>, the following holds: <center><math>\int_\gamma f(z)\, dz = 0</math></center> === Proof 1: Primitive of <math>f</math> === First, we observe that <math>f</math> has a primitive in <math>G</math>. Fix a point <math>z_0 \in G</math>. For any point <math>z \in G</math>, let <math>[z_0, z]</math> denote the straight-line segment connecting <math>z_0</math> and <math>z</math>. == Proof 2: Definition of the Primitive == Define <math>F \colon G \to \mathbb{C}</math> by: :<math>F(z) := \int_{[z_0, z]} f(\zeta), d\zeta</math>. Due to the convexity of <math>G</math>, the triangle <math>D</math> with vertices <math>z_0, z, w</math> lies entirely within <math>G</math> for <math>z, w \in G</math>. === Proof 3: Application of Goursat’s Lemma === By [[Complex Analysis/Goursat's Lemma]] for the boundary <math>\partial \Delta</math> of a triangle <math>\Delta</math> with vertices <math>z_0, z, w \in \mathbb{C}</math>, we have: :<math> \begin{array}{rl} 0 &= \int_{\partial \Delta} f(z)\, dz \\ &= \int_{[z_{0},z]} f(\zeta)\, d\zeta - \int_{[z_{0},w]} f(\zeta)\, d\zeta + \int_{[z,w]} f(\zeta)\, d\zeta \\ & = F(z) - F(w) + \int_{[z,w]} f(\zeta)\, d\zeta \end{array} </math> === Proof 4: Conclusion Using Goursat's Lemma === This leads to: :<math>\begin{array}{rl} F(z) - F(w) &= \int_{[w,z]} f(\zeta)\,d\zeta\\ &= \int_0^1 f\bigl(w + t(z-w)\bigr)\cdot (z-w)\, dt\\ &= \underbrace{\int_0^1 f\bigl(w+t(z-w)\bigr)\, dt}_{A(z):=} \cdot (z-w) \end{array}</math> Thus, we have: :<math>A(z)=\frac{F(z) - F(w)}{(z-w)}</math> === Proof 5: Limit Process === Since <math>A</math> is continuous in <math>w</math>, taking the limit as <math>z \to w</math> gives: :<math>A(w) = \lim_{z \to w} A(z) = \lim_{z \to w} \frac{F(z) - F(w)}{(z-w)} = F'(w).</math> === Proof 5: Differentiability of <math>F</math> === Therefore, <math>A \colon G \to \mathbb{C}</math> is continuous, and <math>F</math> is differentiable in <math>w \in G</math>, with: <center><math>F'(w) = A(w) = f(w).</math></center> Since <math>w \in G</math> was arbitrary, we conclude <math>F' = f</math>, proving that <math>f</math> has a primitive. === Proof 6: Path Integration === Now, let <math>\gamma \colon [a, b] \to G</math> be a piecewise continuously differentiable, closed curve. Then: <center><math> \begin{array}{rl} \int_\gamma f(z)\, dz &= \int_a^b f(\gamma(t)) \gamma'(t)\, dt \\ &= \int_a^b F'(\gamma(t)) \gamma'(t)\, dt \\ &= \int_a^b (F \circ \gamma)'(t)\, dt \\ &= F(\gamma(b)) - F(\gamma(a)) = 0. \end{array} </math></center> === Proof 7: === Let <math>\gamma \colon [a, b] \to G</math> be an arbitrary integration path in <math>G</math>, and let <math>\epsilon > 0</math>. As shown [[Complex Analysis/Curve Integral #Approximation by polygonal paths|here]], we choose a polygonal path <math>\hat{\gamma} \colon [a, b] \to \mathbb{C}</math> such that <math>\hat{\gamma}(a) = \gamma(a)</math>, <math>\hat{\gamma}(b) = \gamma(b)</math>, and :<math>\left|\int_{\hat{\gamma}} f(z), dz - \int_{\gamma} f(z), dz\right| < \epsilon.</math> Since polygonal paths are piecewise continuously differentiable, the above result implies <math>\int_{\hat{\gamma}} f(z), dz = 0</math>. Consequently, :<math>\left|\int_{\gamma} f(z), dz\right| < \epsilon.</math> As <math>\epsilon > 0</math> was arbitrary, the claim follows. == For Cycles in Arbitrary Open Sets == In arbitrary open sets, one must ensure that cycles do not enclose singularities or poles in the complement of the domain. Enclosing such singularities may contribute a non-zero value to the integral (e.g., the function <math>f(z) = \frac{1}{z}</math> and <math>\gamma(t) := e^{it}</math> in a domain <math>G = \mathbb{C}\setminus \{0\}</math> . Even though <math>f</math> is holomorphic in <math>G</math>, the integral is not zero but <math>2\pi i</math> (see [[w:en:Homology (mathematics)#Nullhomologous cycle|nullhomologous cycle]]). === Statement === Let <math>G \subseteq \mathbb{C}</math> be open, and let <math>\Gamma</math> be a [[w:en:Homology (mathematics)#Nullhomologous cycle|nullhomologous]] cycle in <math>G</math>. Then, for every holomorphic function <math>f \colon G \to \mathbb{C}</math>, the following holds: <center><math>\int_\Gamma f(z)\, dz = 0</math></center> === Proof === Let <math>w \in G \setminus \text{trace}(\Gamma)</math>, and define <math>g \colon G \to \mathbb{C}</math> by <center><math>g(z) := (z-w) \cdot f(z).</math></center> Then, <math>g</math> is holomorphic, and by the [[w:en:Cauchy integral formula#For cycles in arbitrary open sets|global integral formula]], we have: :<math> \int_\Gamma f(z)\, dz = \int_{\Gamma} \frac{g(z)}{z-w}\, dz = 2\pi i \cdot n(\Gamma, w) \cdot g(w) = 0. </math> == See Also == *[[w:en:Complex Analysis|Complex Analysis]] *[[w:en:Cauchy's integral theorem|Cauchy integral theorem for disks]] == Page Information == You can display this page as ''' ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis%20Cauchy%20Integral%20Theorem&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Complex%20Analysis%20Cauchy%20Integral%20Theorem&coursetitle=Complex%20Analysis Wiki2Reveal slides])''' === Wiki2Reveal === The '''[https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis%20Cauchy%20Integral%20Theorem&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Complex%20Analysis%20Cauchy%20Integral%20Theorem&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%20Cauchy%20Integral%20Theoremhttps://en.wikiversity.org/wiki/Complex%20Analysis%20Cauchy%20Integral%20Theorem] --> * [https://en.wikiversity.org/wiki/Laurent%20Series This page] is designed as a [https://en.wikiversity.org/wiki/PanDocElectron-Presentation PanDocElectron-SLIDE] document type. * Source: Wikiversity https://en.wikiversity.org/wiki/Laurent%20Series * see [[v:en:Wiki2Reveal|Wiki2Reveal]] for the functionality of [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]. <!-- * Next contents of the course are [[]] -->; === Translation and Version Control === This page was translated based on the following [https://de.wikiversity.org/wiki/Integralsatz von Cauchy Wikiversity source page] and uses the concept of [[Translation and Version Control]] for a transparent language fork in a Wikiversity: * Source: [[v:de:Integralsatz von Cauchy|Integralsatz von Cauchy]] - URL: https://de.wikiversity.org/wiki/Integralsatz von Cauchy * Date: 12/18/2024 <span type="translate" src="Integralsatz von Cauchy" srclang="de" date="12/18/2024" time="09:15" status="inprogress"></span> <noinclude> [[de:Integralsatz von Cauchy]] </noinclude> [[Category:Wiki2Reveal]] 1rglfuco6l25xb0otipbybx5t2d80wz 0 317394 2692986 2692822 2024-12-23T04:02:52Z JackBot 238563 Bot: Fixing double redirect to [[Boolf prop/3-ary/reverse family]] 2692986 wikitext text/x-wiki #REDIRECT [[Boolf prop/3-ary/reverse family]] 3ewubqj7ax9td9pwgv02a3s53e0gepv 2 317421 2692967 2692930 2024-12-22T18:54:54Z Atcovi 276019 /* 8.3 - Tobacco Use */ 2692967 wikitext text/x-wiki == 8.1 - Nutrition and Weight == * There is no single, specific, right answer for the nutritional standards for everyone. Everyone has a different metabolic and hormonal level. <u>Brief History</u> ''A public policy approach to making sense of the science of nutrition got started in the early 1900s when the U.S. Department of Agriculture (USDA) began to make recommendations to consumers about how much protein, fats, and carbohydrates to consume. Their first food guide was published in 1916 and consisted of five major food groups (e.g., fats, sugars; Welsh et al., 1993). The economic problems of the Great Depression in the 1930s greatly influenced American families’ food purchasing and consumption habits because they were forced to balance price and nutrition because affordable foods were often low in nutritional value. The rising inflation rates in 2021 and 2022 following the COVID-19 lockdown periods forced families to make similar decisions nearly 100 years later as food insecurity rates increased (NPR link 2022).'' * '''Food Guide Pyramid''' - A nutritional tool by the US Departement of Agriculture developed in the 80's - 2005, showing foods to consume in the bottom and foods to eat sparringly at the top. MyPyramid was released in 2005 to provide a sense of comprehensibility, something which the pyramid lacked. * In 2011, another guide was released, which is found on www.choosemyplate.gov. Changes included concise amounts of food consumption per age and gender, urging of more whole grains consumption + fruits&veggies consumption, and that 1/2 of the plate is fruits and veggies. ''Four factors were considered in establishing the serving sizes: typical portion sizes from food consumption surveys, ease of use, nutrient content, and traditional uses of foods (accounting for culture).'' **The Chinese culture includes "herbs of immortality" (bodybuilder type foods) and 'hot' and 'cold' foods, including spicy foods and expensive cuts of meats to vegetables, diary products, and inexpensive cuts of meat, respectively. **In Latin cultures, a "hot condition", like pregnancy, shouldn't be associated with "hot foods", which may bring about "hot illnesses", like skin rash. The opposite principle applies to the Chinese. **The "China Study" advocates for a vegetarian diet, but questions have been raised about its authenticity. *'''[[w:5_A_Day|5-A-Day program]]''' raises awareness regarding eating fruits and vegetables 5 times a day. '''Development of Food Preferences''' * Liking sweet and salty to sour is innate. Beyond this, preferences come into play. Behaviorist principles and culture plays a role as well. '''Obesity''' * Increase in obesity has occured between the 80s and now due to addition of [[w:high_fructose_corn_syrup|high fructose corn syrup]], change in play patterns for young children, and a rise in screen time. * '''Obesity''' - Having a BMI of 30>. Chance of obesity increases with age and is highest during mid-life. Commonly accepted cause is a combination of eating habits and lack of physical activity, but environment and genetics ([[w:ob/ob_mouse|''ob'' gene]]) can play a role too. Big cause is the supersizing fast food industry. * '''Overweight''' - Having a BMI of 25>. * BMI isn't entirely to be relied on, as it does not account for health risks, muscles, and cultural variations (difference in body fat distribution amongst various racial backgrounds). <u>Racial accounts in the US regarding obesity</u> ''The bottom line is that obesity varies significantly by different demographic groups (Hill et al., 2017; Wong et al., 2017). In a large national study, Black men have greater odds of obesity than White men in the South, West, and Midwest. In the South and West, Hispanic men also have greater odds of obesity than White men. In all regions, Asian men have lower odds of obesity than White men (Kelley et al., 2016). Further, Black and African American women are more likely to meet the criteria for obesity than either White women or Black men (Ogden et al., 2014; Wang & Beydoun, 2007). Recent work has also shown that transgender individuals experience higher risk of both underweight and overweight status, compared to their cisgender peers, at least in the college years (VanKim et al., 2014), while the risk for obesity is higher for transmasculine compared to transfeminine adults (Kyinn et al., 2021).'' * '''Sensory specific satiety''' - 1 food available, moderate amount; 2 foods available, 2nd food is eaten more than if it were to be presented by itself. * Presenting the food as 'healthier' or even 'cheaper' can change perceptions of that food. * Effects of weight discrimination and bullying due to weight also have severe impacts, including emotional distress, loneliness, and an increase in adhering metabolic syndromes and diabetes. '''[[Eating Disorders]]''' * '''Anorexia''' - Intense fear of gaining weight, characterized by excessive excersising, skipping meals, and indulging laxatives. * '''Bulimia''' - Binging on food, then purging. * '''Binge eating disorder (BED)''' - Binging food, but NOT purging. ''When compared with White American women, Black American women tend to have lower rates of anorexia nervosa but similar rates of BED, and Latinas may have slightly higher rates of eating disorders than both of these ethnic groups (Markey et al., 2009). Asian American women have the lowest rates of eating disorders among the major ethnic groups in the United States, while Native American women have the highest rates.'' == 8.2 - Physical Activity == * '''Physical activity''' is movement that is produced by contraction of the skeletal muscles, exerting energy. * Guidelines for physical activity include... ** Children & adolescenets (6-17yrs) should do 60m+ of activity daily and muscle strength work at least 3 days a week. ** Adults should engage in 150 minutes of moderate/75 minutes of high intensity excersise every week. Strengthening muscles should be done twice a week. * '''Basal metabolic rate''' - Number of calories you burn as your body performs basic (basal) life-sustaining function (https://www.garnethealth.org/news/basal-metabolic-rate-calculator) * '''Thermic effect of food''' - Amount of energy it takes for your body to digest, absorb, and metabolise the food you eat. * '''Exercise''' is physical activity with the goal of improving fitness. '''Cultural Variations in Physical Activity''' * The '''NHIS''', '''NHANES III''', and the '''Surgeon General's Report''', physical activity is lowest among people with low incomes and education. * Children with a TV in their bedroom are more likely to be less active. '''Health Compromising Behaviors''' * '''Behavioral cueing''' - A smoker always smokes on his work break, the work break becomes a cue to smoke. * '''Addiction''' - The body relies on substances over regular functioning. == 8.3 - Tobacco Use == * '''Tobacco use''' is the leading cause of preventable morbidity and mortality in the US. * Highest in ages 25 - 65, lowest in early adulthood. Memory recall and exposure increases risk of smoking. '''Cultural Variations''' * Men smoke more than women. * Less education & income, more likely to smoke. * KT & VA have the most smokers. * Native Americans, Alaskans, Blacks = Whites, Hispanic, Asian Americans (highest rate of smoking to least rate of smoking). '''Why do people smoke?''' * ''Why do people start'' and ''why do they keep smoking?'' ** Nicotine is pleasing, '''biologically'''. ** '''Genetics''': certain genotypes, like SLC6A3, DRD2, and DRD2-A1, have an association with smoking. ''Genes also influence how nicotine is broken down. Tang et al. (2012) investigated how nicotine metabolism and genetic variation in CYP2A6, a gene that mediates nicotine breakdown, influence the neural response to smoking cues. Tang et al. used functional magnetic resonance imaging to scan smokers with variations in the gene and hence high and low in nicotine metabolism. Fast metabolizers, by phenotype or genotype, had significantly greater responses to visual cigarette cues than slow metabolizers in the amygdala, hippocampus, striatum, insula, and cingulate cortex. This finding helps explain why fast metabolizers who smoke have lower cessation rates. Similar brain mapping of smokers with different genes show how variations in the amygdala (responsible for emotion and pleasure) may influence cessation (Jasinska et al., 2012).'' * '''Psychological''': Parental/peer modeling. Overcoming inferiority and establishing an identity may lead to smoking (Erikson). * '''Social/cultural''': Movies/ads. * '''Physiological''': Chances of dying increases. Has a '''synergistic effect''' (cumulative effects of two active ingredients). '''ETS,''' or ''environmental tobacco smoke'', is the tobacco smoke inhaled by non-smokers who are near a person who smokes. Passive smoking is linked to lung cancer, cardiovascular disease, and smoking addiction. tnkex3xom5cjh87i0pbpw4gh7508edb 2692968 2692967 2024-12-22T19:18:31Z Atcovi 276019 2692968 wikitext text/x-wiki == 8.1 - Nutrition and Weight == * There is no single, specific, right answer for the nutritional standards for everyone. Everyone has a different metabolic and hormonal level. <u>Brief History</u> ''A public policy approach to making sense of the science of nutrition got started in the early 1900s when the U.S. Department of Agriculture (USDA) began to make recommendations to consumers about how much protein, fats, and carbohydrates to consume. Their first food guide was published in 1916 and consisted of five major food groups (e.g., fats, sugars; Welsh et al., 1993). The economic problems of the Great Depression in the 1930s greatly influenced American families’ food purchasing and consumption habits because they were forced to balance price and nutrition because affordable foods were often low in nutritional value. The rising inflation rates in 2021 and 2022 following the COVID-19 lockdown periods forced families to make similar decisions nearly 100 years later as food insecurity rates increased (NPR link 2022).'' * '''Food Guide Pyramid''' - A nutritional tool by the US Departement of Agriculture developed in the 80's - 2005, showing foods to consume in the bottom and foods to eat sparringly at the top. MyPyramid was released in 2005 to provide a sense of comprehensibility, something which the pyramid lacked. * In 2011, another guide was released, which is found on www.choosemyplate.gov. Changes included concise amounts of food consumption per age and gender, urging of more whole grains consumption + fruits&veggies consumption, and that 1/2 of the plate is fruits and veggies. ''Four factors were considered in establishing the serving sizes: typical portion sizes from food consumption surveys, ease of use, nutrient content, and traditional uses of foods (accounting for culture).'' **The Chinese culture includes "herbs of immortality" (bodybuilder type foods) and 'hot' and 'cold' foods, including spicy foods and expensive cuts of meats to vegetables, diary products, and inexpensive cuts of meat, respectively. **In Latin cultures, a "hot condition", like pregnancy, shouldn't be associated with "hot foods", which may bring about "hot illnesses", like skin rash. The opposite principle applies to the Chinese. **The "China Study" advocates for a vegetarian diet, but questions have been raised about its authenticity. *'''[[w:5_A_Day|5-A-Day program]]''' raises awareness regarding eating fruits and vegetables 5 times a day. '''Development of Food Preferences''' * Liking sweet and salty to sour is innate. Beyond this, preferences come into play. Behaviorist principles and culture plays a role as well. '''Obesity''' * Increase in obesity has occured between the 80s and now due to addition of [[w:high_fructose_corn_syrup|high fructose corn syrup]], change in play patterns for young children, and a rise in screen time. * '''Obesity''' - Having a BMI of 30>. Chance of obesity increases with age and is highest during mid-life. Commonly accepted cause is a combination of eating habits and lack of physical activity, but environment and genetics ([[w:ob/ob_mouse|''ob'' gene]]) can play a role too. Big cause is the supersizing fast food industry. * '''Overweight''' - Having a BMI of 25>. * BMI isn't entirely to be relied on, as it does not account for health risks, muscles, and cultural variations (difference in body fat distribution amongst various racial backgrounds). <u>Racial accounts in the US regarding obesity</u> ''The bottom line is that obesity varies significantly by different demographic groups (Hill et al., 2017; Wong et al., 2017). In a large national study, Black men have greater odds of obesity than White men in the South, West, and Midwest. In the South and West, Hispanic men also have greater odds of obesity than White men. In all regions, Asian men have lower odds of obesity than White men (Kelley et al., 2016). Further, Black and African American women are more likely to meet the criteria for obesity than either White women or Black men (Ogden et al., 2014; Wang & Beydoun, 2007). Recent work has also shown that transgender individuals experience higher risk of both underweight and overweight status, compared to their cisgender peers, at least in the college years (VanKim et al., 2014), while the risk for obesity is higher for transmasculine compared to transfeminine adults (Kyinn et al., 2021).'' * '''Sensory specific satiety''' - 1 food available, moderate amount; 2 foods available, 2nd food is eaten more than if it were to be presented by itself. * Presenting the food as 'healthier' or even 'cheaper' can change perceptions of that food. * Effects of weight discrimination and bullying due to weight also have severe impacts, including emotional distress, loneliness, and an increase in adhering metabolic syndromes and diabetes. '''[[Eating Disorders]]''' * '''Anorexia''' - Intense fear of gaining weight, characterized by excessive excersising, skipping meals, and indulging laxatives. * '''Bulimia''' - Binging on food, then purging. * '''Binge eating disorder (BED)''' - Binging food, but NOT purging. ''When compared with White American women, Black American women tend to have lower rates of anorexia nervosa but similar rates of BED, and Latinas may have slightly higher rates of eating disorders than both of these ethnic groups (Markey et al., 2009). Asian American women have the lowest rates of eating disorders among the major ethnic groups in the United States, while Native American women have the highest rates.'' == 8.2 - Physical Activity == * '''Physical activity''' is movement that is produced by contraction of the skeletal muscles, exerting energy. * Guidelines for physical activity include... ** Children & adolescenets (6-17yrs) should do 60m+ of activity daily and muscle strength work at least 3 days a week. ** Adults should engage in 150 minutes of moderate/75 minutes of high intensity excersise every week. Strengthening muscles should be done twice a week. * '''Basal metabolic rate''' - Number of calories you burn as your body performs basic (basal) life-sustaining function (https://www.garnethealth.org/news/basal-metabolic-rate-calculator) * '''Thermic effect of food''' - Amount of energy it takes for your body to digest, absorb, and metabolise the food you eat. * '''Exercise''' is physical activity with the goal of improving fitness. '''Cultural Variations in Physical Activity''' * The '''NHIS''', '''NHANES III''', and the '''Surgeon General's Report''', physical activity is lowest among people with low incomes and education. * Children with a TV in their bedroom are more likely to be less active. '''Health Compromising Behaviors''' * '''Behavioral cueing''' - A smoker always smokes on his work break, the work break becomes a cue to smoke. * '''Addiction''' - The body relies on substances over regular functioning. == 8.3 - Tobacco Use == * '''Tobacco use''' is the leading cause of preventable morbidity and mortality in the US. * Highest in ages 25 - 65, lowest in early adulthood. Memory recall and exposure increases risk of smoking. '''Cultural Variations''' * Men smoke more than women. * Less education & income, more likely to smoke. * KT & VA have the most smokers. * Native Americans, Alaskans, Blacks = Whites, Hispanic, Asian Americans (highest rate of smoking to least rate of smoking). '''Why do people smoke?''' * ''Why do people start'' and ''why do they keep smoking?'' ** Nicotine is pleasing, '''biologically'''. ** '''Genetics''': certain genotypes, like SLC6A3, DRD2, and DRD2-A1, have an association with smoking. ''Genes also influence how nicotine is broken down. Tang et al. (2012) investigated how nicotine metabolism and genetic variation in CYP2A6, a gene that mediates nicotine breakdown, influence the neural response to smoking cues. Tang et al. used functional magnetic resonance imaging to scan smokers with variations in the gene and hence high and low in nicotine metabolism. Fast metabolizers, by phenotype or genotype, had significantly greater responses to visual cigarette cues than slow metabolizers in the amygdala, hippocampus, striatum, insula, and cingulate cortex. This finding helps explain why fast metabolizers who smoke have lower cessation rates. Similar brain mapping of smokers with different genes show how variations in the amygdala (responsible for emotion and pleasure) may influence cessation (Jasinska et al., 2012).'' * '''Psychological''': Parental/peer modeling. Overcoming inferiority and establishing an identity may lead to smoking (Erikson). * '''Social/cultural''': Movies/ads. * '''Physiological''': Chances of dying increases. Has a '''synergistic effect''' (cumulative effects of two active ingredients). '''ETS,''' or ''environmental tobacco smoke'', is the tobacco smoke inhaled by non-smokers who are near a person who smokes. Passive smoking is linked to lung cancer, cardiovascular disease, and smoking addiction. == 8.4 - Alcohol == * 3rd leading cause of death. '''Who Drinks, How Much, and Why?''' * '''Alcohol abuse''' - Diagnosed either through 1) lack of fulfilling responsibilities, 2) using alcohol repeatedly whilst putting others in danger, 3) getting in trouble with the law due to alcohol on a regular basis, and 4) using alcohol despite obvious social problems. ** Men who consume ''five or more'' drinks in a row, women who have consumed ''four or more'' drinks in a row at least once in the last 2 weeks. * ''Why?'' ** '''Biologically:''' Genetic predictors of alcoholism. Reduces stress. ** '''Psychologically''': High in neuroticism, impulsive, and extroverted. If someone were to 'expect good things' from alcohol, then they are more likely to continue indulging in alcohol. Social peers influence alcohol consumption as well. '''Consequences of Alcohol Abuse''' * 7th leading cause of death worldwide, leading cause of early mortality between ages 15 - 45. * Liver disease is a common consequence for older drinkers. '''Benefits?''' * Recent studies have put out a '''standard drink''', like a 12-ounce serving of beer, as a standard serving for alcohol. Also, a citing of the '''[[w:French_paradox|French paradox]].''' Though recent evidence have shown that these studies showing the so-called "positive" effects of alcohol don't account for other healthy factors. 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(20241223 - 20241221) |Source={{own|Young1lim}} |Date=2024-12-23 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} 2692979 wikitext text/x-wiki == Summary == {{Information |Description=DIR.1A: Shared Library Names (20241223 - 20241221) |Source={{own|Young1lim}} |Date=2024-12-23 |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}} 1a2o81pgv7h13yv5yi2drqsqstbaxz3 6 317428 2692980 2024-12-23T00:39:08Z Young1lim 21186 {{Information |Description=LIB.2A: Shared Libraries (20241223 - 20241221) |Source={{own|Young1lim}} |Date=2024-12-23 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} 2692980 wikitext text/x-wiki == Summary == {{Information |Description=LIB.2A: Shared Libraries (20241223 - 20241221) |Source={{own|Young1lim}} |Date=2024-12-23 |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}} m1tihomdhxp6fh11vvyd087rjm4mqya 6 317429 2692985 2024-12-23T03:16:56Z Young1lim 21186 {{Information |Description=DIR.2A: Managing Shared Libraries (20241223 - 20241221) |Source={{own|Young1lim}} |Date=2024-12-23 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} 2692985 wikitext text/x-wiki == Summary == {{Information |Description=DIR.2A: Managing Shared Libraries (20241223 - 20241221) |Source={{own|Young1lim}} |Date=2024-12-23 |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}} rzrdz23vjrcfxso0gsfk5t7zr86xsnf 6 317430 2692991 2024-12-23T09:42:08Z Young1lim 21186 {{Information |Description=API.2A: Functions (20241221 - 20241220) |Source={{own|Young1lim}} |Date=2024-12-23 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} 2692991 wikitext text/x-wiki == Summary == {{Information |Description=API.2A: Functions (20241221 - 20241220) |Source={{own|Young1lim}} |Date=2024-12-23 |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}} t181o0pvrgnu43ixvk8r7f6g9ete0ov 6 317431 2692993 2024-12-23T09:45:33Z Young1lim 21186 {{Information |Description=API.2A: Functions (20241223 - 20241221) |Source={{own|Young1lim}} |Date=2024-12-23 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} 2692993 wikitext text/x-wiki == Summary == {{Information |Description=API.2A: Functions (20241223 - 20241221) |Source={{own|Young1lim}} |Date=2024-12-23 |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}} alm8oquu864sfl0ynq1d6c031e0t59v 6 317432 2692995 2024-12-23T10:05:38Z Young1lim 21186 {{Information |Description=LCal.9A: Recursion (20241223 - 20241219) |Source={{own|Young1lim}} |Date=2024-12-23 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} 2692995 wikitext text/x-wiki == Summary == {{Information |Description=LCal.9A: Recursion (20241223 - 20241219) |Source={{own|Young1lim}} |Date=2024-12-23 |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}} febf2cie2bzok4bj8mvy3clzbky9rul 6 317433 2692997 2024-12-23T10:54:13Z Young1lim 21186 {{Information |Description=ARM.2ASM: Branch and Return Methods (20241223 - 20241219) |Source={{own|Young1lim}} |Date=2024-12-23 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} 2692997 wikitext text/x-wiki == Summary == {{Information |Description=ARM.2ASM: Branch and Return Methods (20241223 - 20241219) |Source={{own|Young1lim}} |Date=2024-12-23 |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}} 8hw3ogvgi9y3yo7g110z95gpdi36px6 7 317434 2692998 2024-12-23T11:03:48Z 180.244.208.121 /* 4ty */ new section 2692998 wikitext text/x-wiki == 4ty == fh5trh [[Special:Contributions/180.244.208.121|180.244.208.121]] ([[User talk:180.244.208.121|discuss]]) 11:03, 23 December 2024 (UTC) c67kbdjsm191hk255ylpbarb3l34wvd