ASCIIMathML.js is a JavaScript that translates simple ASCII mathematics expressions to MathML while the webpage is loading. It works in InternetExplorer+MathPlayer and Firefox. See http://www1.chapman.edu/~jipsen/asciimath.html for more details.\n

ASCIIsvg.js is a JavaScript that translates simple descriptions of diagrams into [[SVG]] while the webpage is loading. It works in InternetExplorer+SVGview and [[Firefox 1.5]]. See http://www1.chapman.edu/~jipsen/asciisvg.html for more details.\n

HelloThere MathExamples Download MainFeatures

Let `N` be a subgroup of a group `G`. We say that `N` is //normal// if the representative operation on right cosets of `N`, defined by `(Nx)(Ny)=N(xy)` is well-defined, i.e. if `Nx=Nx'` and `Ny=Ny'` then `N(xy)=N(x'y')`.

You can download your own copy of this math notebook by right-clicking on http://math.chapman.edu/~jipsen/tiddly/tiddlymath.html. This will allow you to add your own content.\n\nTo enable the math features you also should download the following ''three'' files into the same directory:\n\nhttp://math.chapman.edu/~jipsen/tiddly/ASCIIMathML.js\nhttp://math.chapman.edu/~jipsen/tiddly/ASCIIsvg.js\nhttp://math.chapman.edu/~jipsen/tiddly/d.svg\n\nOptionally you can also download [[jsMath]] from\n\nhttp://www.math.union.edu/~dpvc/jsMath/download/jsMath.html\n\nand unzip jsMath-2.3f.zip (or later version) in the same directory. This script parses a bigger subset of LaTeX and does not require MathML support.\n

Firefox 1.5 is a version of the Firefox browser that supports both MathML and [[SVG]] right out of the box. It's free, its easy to install, its standards compliant, and it works on many platforms. Don't hesitate to get your own copy at http://www.mozilla.org/products/firefox/

This is an adaptation of Jeremy Ruston's TiddlyWiki (version 1.2.37) that is suitable for writing mathematics notes (e.g. lecture notes, homework, projects, research notes, ...). See MainFeatures for a brief description of the three scripts that enable the math formulas, diagrams and graphs.\n\nThese features require InternetExplorer+MathPlayer+SVGview or [[Firefox 1.5]]. Take a look at the MathExamples below to see if your browser is setup okay.

JavaScript is a computer language that is implemented in many webbrowsers and is used to create dynamic webpages. In InternetExplorer it is called ~JScript, and part of the language has been standardized as ~ECMAscript. It is a weakly typed language with a syntax based on c and a prototype mechnism for handling objects. JavaScript programs are downloaded with the webpage, cached and interpreted on the client machine.

LaTeX is a standard mathematical typesetting language used all over the world for publishing scientific texts. An introduction can be found at http://www.latex-project.org/.

''Lemma'' Let `H` be a subgroup of a group `G`. Then for all `x,y in G`\n`x in Hy` if and only if `Hx=Hy` and\n`x in yH` if and only if `xH=yH`.\n\n//Proof// (hint). Recall that the left and right cosets of `H` are defined by `xH={xh|h in H}` and `Hx={hx|h in H}`. Now the "if" part is obvious since the identity `e` is in `H`. For the forward implication we have to make use of some very basic properties of groups (try it).

The MathML and SVG features work in Firefox 1.5 (RC2) and Internet Explorer (but the latter requires some plugins).\n\nASCIIMathML is used to translate notation like {{{\s`x^2\s`}}} to MathML. InternetExplorer requires the MathPlayer plugin to display MathML. It can be downloaded free from http://dessci.com. For details about the syntax of ASCIIMathML see http://www.chapman.edu/~jipsen/mathml/asciimath.html. \nIf the [[jsMath]] scripts are installed, LaTeX formulas (enclosed in dollar-signs or double-dollar-signs) are translated to math notation by these scripts (and otherwise ASCIIMathML tries to convert them, but it handles only a small subset of LaTeX).\n\nASCIIsvg is used to display diagrams described by an <embed> tag. InternetExplorer requires the Adobe SVGview plugin to display SVG. It can be downloaded free from http://adobe.com/svg. For details about the syntax of ASCIIsvg see http://www.chapman.edu/~jipsen/svg/asciisvg.html.

HelloThere\nTiddlyWiki\n[[Download]]\nMainFeatures\n[[Thanks]]\n\n<<newTiddler>>\n<<newJournal "DD MMM YYYY">>

A famous result due to [[Euler|http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Euler.html]] in ASCIIMathML: `sum_(n=1)^oo 1/n^2=pi^2/6`\n\nand again in LaTeX: $\ssum_{n=1}^\sinfty \sfrac1{n^2}=\sfrac{\spi^2}6$\n\nHere is a graph of `x-x^3/6` and `sin(x)` (double click on this tiddler to see the <embed> tag that produced it).\n\n<html>\n<center>\n<embed src="d.svg" script='\nwidth="300"; height="150"; nobutton()\nxmin=-2pi; xmax=2pi; xscl=1; axes(); stroke="red"\nplot(sin(x))\nstroke="blue"; strokedasharray="10,10"\nplot(x-x^3/6,-4,4)\n'>\n</center>\n</html>\n\nand a diagram illustrating why Pythagoras' Theorem holds:\n<html>\n<center>\n<embed src="d.svg" script='\nwidth="400"; height="200"; nobutton()\nxmin=-4; xmax=4; xscl=1; ymin=1; noaxes()\nfill="yellow"\nrect([1,1],[4,4])\nrect([-4,1],[-1,4])\nfill="red"\nrect([-2,1],[-1,2])\nfill="blue"\nrect([-4,2],[-2,4])\nfill="green"\npath([[1,2],[2,4],[4,3],[3,1],[1,2]])\nfill="none"\npath([[-4,1],[-2,2],[-1,4]])'>\n</center>\n</html>\n\n

MathML is the W3C standard for mathematics notation on webpages. See http://w3.org/Math for more details.

MathPlayer is a free plugin for InternetExplorer that allows this browser to display MathML. The plugin is freely available from http://dessci.com.

/***\nASCIIMathML plugin: see http://www1.chapman.edu/~jipsen/mathml/asciimathsyntax.html for syntax help.\n\n!Code:\n***/\n//{{{\nwindow.createTiddlerViewer_original_MathPlugin = window.createTiddlerViewer;\nwindow.createTiddlerViewer=function(title,highlightText,highlightCaseSensitive) {\n window.createTiddlerViewer_original_MathPlugin(title,highlightText,highlightCaseSensitive);\n var theViewer = document.getElementById("viewer"+title);\n if (theViewer) {\n if (jsMath.ConvertTeX) {\n jsMath.ConvertTeX(theViewer);\n jsMath.ProcessBeforeShowing(theViewer);\n }\n if (AMprocessNode) AMprocessNode(theViewer,false);\n if (drawPictures) setTimeout('drawPictures()',100);\n }\n}\n//}}}\n\n

SVG is the W3C standard for scalable vector graphics on webpages. See http://www.w3.org/Graphics/SVG/ for more details.

SVGview.exe is the filename of the Adobe SVG Viewer plugin for InternetExplorer. It is freely available from http://www.adobe.com/svg.

a web notebook for mathematics

TiddlyMath

http://www.math.chapman.edu/~jipsen/tiddly

Many thanks to all who have contributed to making this math notebook possible.\n\nTiddlyWiki is due to Jeremy Ruston (with contributions from other volunteers).\nThe MathPlugin is adapted from [[Franco Bagnoli's version of TiddlyWiki| http://didattica.dma.unifi.it/WebWrite/pub/Fisica/MeccanicaETermodinamicaIAT05-06/MTIAT0506.html]]\n[[jsMath]] is due to Davide Cervione.\nASCIIMathML and ASCIIsvg are scripts I wrote in early 2004.\nThanks also to the many individuals who have worked on creating MathML, LaTeX, [[SVG]], JavaScript, [[Firefox 1.5]], InternetExplorer, MathPlayer and Adobe SVGview.

TiddlyMath can be helpful for students to create and keep track of definitions, lemmas, theorems, corollaries and their proofs. The trick is to develop a system for naming results, and for writing their proofs.\n\nEvery definition and result should have a (preferably short) meaningful name. Wellknown results mostly have names already, so that is no problem (e.g. [[Fundamental Theorem of Calculus Part I]], [[Cayley's Theorem]], ...). If a good name cannot be found, one can of course resort to something like [[Defn 1.1]], [[Lemma 2]], [[Theorem 3]], [[Cor 4.4]].\n\nProofs can be part of the same tiddler that states the result, or they can be in a separate one, e.g. ProofOfThm3. The latter allows tiddlers like HintForThm3 or ProofOutlineThm3. Steps within proofs should be justified by referring to the definitions and results in other tiddlers. Of course the nonlinear structure of TiddlyWiki can make it harder to spot circular reasoning. On the other hand it makes it easier to provide lots of detail in small pieces that can be skipped by experienced readers.

''Theorem'' Let `N` be a subgroup of a group `G`. Then the following are equivalent:\n\n(i) `N` is [[normal|DefnNormalSubgroup]].\n(ii) `Nx=xN` for all `x in G`.\n(iii) `N` is closed under conjugation, i.e. `x^-1ax in N` for all `x in G` and `a in N`.\n\n//Proof//: We will prove (i)`=>`(ii)`=>`(iii)`=>`(i).\n(i)`=>`(ii): Assume `N` is normal. This means the representative operation on right cosets is well-defined. So for all `x,u,y,v in G` if `Nx=Nu` and `Ny=Nv` then `N(xy)=N(uv)`. By LemmaEqualCosets this means `x in Nu` and `y in Nv` imply `xy in N(uv)`, or put another way, for all `a,b in N` if `x=au` and `y=bv` then `xy=cuv` for some `c in N`.\n\nSo we now know that for all `u,v in G` and all `a,b in N` there exists `c in N` such that `aubv=cuv`.\n\nWe want to show that `Nx=xN` for all `x in G`. This is equivalent to `Nx sube xN` and `xN sube Nx`.\nSo let `x in G`, and let `y in xN`. This means `y=xb` for some `b in N`. \nWe want to show that `y in Nx`, i.e. that `xb=cx` for some `c in N`. But this follows directly from the assumption if we let `a=v=e` and `u=x`. Hence we have proved `xN sube Nx`.\nSince this holds for all `x in G` we obtain `x^-1N sube Nx^-1`. Thus for all `b in N` there exists `c in N` such that `x^-1b=cx^-1`, or equivalently `bx=xc`. Therefore `Nx sube xN` also holds.\n\n(ii)`=>`(iii): Assume `Nx=xN` for all `x in G`. We want to show that `N` is closed under conjugation. Let `x in G` and `a in N`. Then `ax in Nx`, so by assumption `ax in xN`. Hence `ax=xb` for some `b in N`, and it follows that `x^-1ax=b in N`. Therefore `N` is closed under conjugation.\n\n(iii)`=>`(i): Assume `N` is closed under conjugation, and let `Nx=Nu`, `Ny=Nv` for some `x,y,u,v in G`, which means `x=au` and `y=bv` for some `a,b in N`. We want to show that `Nxy=Nuv`, i.e. `xy in Nuv`.\nNow `xy=aubv=a(u^-1)^-1bu^-1uv=acuv` where `c=(u^-1)^-1bu^-1 in N` since `N` is closed under conjugation. We know that `N` is a subgroup, so `ac in N`, and hence we have shown that `xy in Nuv`.

TiddlyMath is the current name for a version of Jeremy Ruston's TiddlyWiki that is suitable for writing mathematical tiddlers. It is freely available from http://math.chapman.edu/~jipsen/tiddly/tiddlymath.html.

Take a look at http://www.tiddlywiki.com for a description of TiddlyWiki and the latest version of Jeremy Ruston's fantastic script.

The abbreviation for the World Wide Web Consortium, online at http://w3.org.\n

jsMath is a collection of ~JavaScripts, written by Davide Cervone that convert LaTeX formulas to typeset expressions within a webpage while the page is loading. The scripts can be freely downloaded from http://www.math.union.edu/~dpvc/jsMath/.