<!DOCTYPE html> <html> <head> <title>Dynamic Preview of Textarea with MathJax Content</title> <!– Copyright © 2012-2018 The MathJax Consortium –> <meta http-equiv=“Content-Type” content=“text/html; charset=UTF-8” /> <meta http-equiv=“X-UA-Compatible” content=“IE=edge” />
<style> .changed { color: red } </style>
<script type=“text/x-mathjax-config”>
MathJax.Hub.Config({ TeX: { equationNumbers: {autoNumber: "AMS"}, extensions: ["begingroup.js"], noErrors: {disabled: true} }, showProcessingMessages: false, tex2jax: { inlineMath: [['$','$'],['\\(','\\)']] } });
//MathJax.Hub.signal.Interest(function (message) {console.log(message)}); </script> <script type=“text/javascript” src=“../MathJax.js?config=TeX-AMS-MML_HTMLorMML”></script>
<script> var Preview = {
typeset: null, // the typeset preview area (filled in by Init below) preview: null, // the untypeset preview (filled in by Init below) buffer: null, // the new preview to be typeset (filled in by Init below) par: [], // paragraph-specific data refs: [], // undefined references needing to be reprocessed updateNeeded: 0, // number of paragraphs needing update oldtext: '', // used to see if an update is needed pending: false, // true when a restart is in the MathJax queue classDelay: 400, // how long to leave changed paragraphs colored ctimeout: null, // timeout for changed style remover labelDelay: 1250, // how long to wait before reprocessing for label changes ltimeout: null, // timeout for changed labels keytimes: [], // tracks the times between keypresses keyrate: 100, // the average of the keytimes (default value) keyn: 0, // key index to replace next keysize: 10, // use this many keypresses // // Get the preview and buffer DIV's // Init: function () { this.typeset = document.getElementById("MathPreview"); this.buffer = document.createElement("div"); this.preview = document.createElement("div"); for (var i = 0; i < this.keysize; i++) {this.keytimes[i] = this.keyrate} }, // // This gets called when a key is pressed in the textarea. // Update: function (up) { if (up) { // // Determine the typing speed as a rolling average of the last few keystrokes // var time = new Date().getTime(); if (this.lasttime) { var delta = time - this.lasttime; if (delta < 4*this.keyrate) { this.keyrate = (this.keysize*this.keyrate+delta-this.keytimes[this.keyn])/this.keysize; this.keytimes[this.keyn++] = delta; if (this.keyn === this.keysize) {this.keyn = 0} } } this.lasttime = time; } var text = document.getElementById("MathInput").value; text = text.replace(/^\s+/,'').replace(/\s+$/,'').replace(/\r\n?/g,"\n"); if (text !== this.oldtext) { this.oldtext = text; if (!this.pending) { this.pending = true; if (this.ctimeout) {clearTimeout(this.ctimeout); this.ctimeout = null} if (this.ltimeout) {clearTimeout(this.ltimeout); this.ltimeout = null} MathJax.Hub.Queue( // allow a little time for additional typing ["Delay",MathJax.Callback,Math.min(200,Math.floor(this.keyrate/2)+1)], ["Restart",this] ); } } }, Restart: function () { this.pending = false; var text = this.oldtext.replace(/&/g,'&').replace(/</g,'<').replace(/>/g,'>'); var text = text.replace(/\n\n+/g,"<p>"); this.buffer.innerHTML = text; var update = this.CompareBuffers(); if (update.needed) { MathJax.Hub.Queue( ["CopyChanges",this,update], ["PreTypeset",this,update], ["Typeset",this,update], ["PostTypeset",this,update] ); } }, CompareBuffers: function () { var b1 = this.buffer.childNodes, b2 = this.preview.childNodes, i, m1 = b1.length, m2 = b2.length, m = Math.min(m1,m2); // // Make sure all top-level elements are containers // for (i = 0; i < m1; i++) { var node = b1[i]; if (typeof(node.innerHTML) === "undefined") { this.buffer.replaceChild(document.createElement("span"),node); b1[i].appendChild(node); } } // // Find first non-matching element, if any, // and the last non-matching element // for (i = 0; i < m; i++) {if (b1[i].innerHTML !== b2[i].innerHTML) break} if (i === m && m1 === m2) {return {needed: false}} while (m1 > i && m2 > i) {if (b1[--m1].innerHTML !== b2[--m2].innerHTML) break} return {needed:true, start:i, end1:m1, end2:m2}; }, CopyChanges: function (update) { var i = update.start, m1 = update.end1, m2 = update.end2; var b1 = this.buffer.childNodes, b2 = this.typeset.childNodes; update.indices = []; update.nodes = []; update.replace = true; // // Remove differing elements from typeset copy // and add in the new (untypeset) elements. // this.recordOldData(this.par.splice(i,m2+1-i)); var tail = b2[m2+1]; while (m2 >= i && b2[i]) {this.typeset.removeChild(b2[i]); m2--} while (i <= m1 && b1[i]) { this.par.splice(i,0,{number:0, labels:[], defs:[], refs:[], replaced:true, update:true}); var node = b1[i].cloneNode(true); update.nodes.push(node); update.indices.push(i++); this.updateNeeded++; if (tail) {this.typeset.insertBefore(node,tail)} else {this.typeset.appendChild(node)} this.addChanged(node); } // // Swap buffers and set up the new buffer for the next change // this.preview = this.buffer; this.buffer = document.createElement("div"); }, PreTypeset: function (update) { var TEX = MathJax.InputJax.TeX; this.incremental = true; this.i = this.j = 0; this.eqNum = 0; this.update = update.indices; this.replace = update.replace; // // Pop any left over \begingroups and push a new one // Reset the equation numbers (but not labels) // while (TEX.rootStack.top > 1) {TEX.rootStack.stack.pop(); TEX.rootStack.top--} TEX.rootStack.Push(TEX.nsStack.nsFrame()); }, recordOldData: function (par) { var AMS = MathJax.Extension["TeX/AMSmath"]; var labels = [], defs = []; this.oldnumber = 0; for (var i = 0, m = par.length; i < m; i++) { this.oldnumber += par[i].number; defs.push(par[i].defs.all); for (var j = 0, n = par[i].labels.length; j < n; j++) { delete AMS.labels[par[i].labels[j].split(/=/)[0]]; labels.push(par[i].labels[j]); } } this.oldlabels = labels.join(''); this.olddefs = defs.join(''); }, Typeset: function (update) { return MathJax.Hub.Typeset(update.nodes,{}); }, BeginMath: function () { // // Save the start time for this paragraph // this.time = new Date().getTime(); }, BeginInput: function () { // // Skip any paragraphs that aren't being updated, and // update the equation numbers and macro definitions // accordingly // var TEX = MathJax.InputJax.TeX, par; while (this.i < this.update[this.j]) { par = this.par[this.i++]; this.eqNum += par.number; for (i = 0, m = par.defs.length; i < m; i++) { TEX.rootStack.Def.apply(TEX.rootStack,par.defs[i]); } } TEX.resetEquationNumbers(this.eqNum,true); // // Store new macro and label definitions here // par = this.par[this.i]; if (par) { if (!par.replaced) {par.olddefs = par.defs.all; par.oldlabels = par.labels.join('')} par.defs = []; par.defs.all = []; par.labels = []; } }, TeXFilter: function (data) { // // Get any new labels for this paragraph // var AMS = MathJax.Extension["TeX/AMSmath"]; var labels = this.par[this.i].labels; for (var id in AMS.eqlabels) {if (AMS.eqlabels.hasOwnProperty(id)) { labels.push(id+"="+AMS.eqlabels[id]) }} }, TeXDef: function (def) { var defs = this.par[this.i].defs; defs.push(def); defs.all.push(def[0]+"{"+def[1]+"}"); }, EndInput: function () { // // Record the undefined references, // the new definitions, and the equation number // for this paragraph // var AMS = MathJax.Extension["TeX/AMSmath"]; var par = this.par[this.i]; if (par) { par.refs = AMS.refs; AMS.refs = []; par.defs.all = par.defs.all.join(""); par.number = AMS.startNumber - this.eqNum; this.eqNum = AMS.startNumber; if (!par.replaced) { delete par.update; if (par.defs.all !== par.olddefs) {this.refreshRest = true} if (par.labels.join('') !== par.oldlabels) { // ### cancel typesetting and do all paragraphs this.refreshAll = true; } delete par.olddefs; delete par.oldlabels; } } }, EndMath: function () { // // Record the tie it took for this paragraph // and go on to the next one. // var par = (this.par[this.i]||{}); var time = new Date().getTime(); par.time = time - this.time; this.time = time; delete par.update; this.updateNeeded--; this.j++; this.i++; }, PostTypeset: function (update) { var incremental = this.incremental; this.incremental = false; // ### if cancelled return? // // Check if there are undefined references that might have been // defined in this update, and reprocess if so. // for (var i = 0, m = this.update.length; i < m; i++) { var par = this.par[this.update[i]]; if (par.refs.length) {this.refs = this.refs.concat(par.refs); par.refs = []} } if (incremental && this.refs.length) { var queue = MathJax.Callback.Queue( ["Reprocess",MathJax.Hub,this.refs,{}], function () {/* if not cancelled */ this.refs = []} ); return queue.Push(["PostTypeset",this,update]); } // // Set the timer for the color removal // this.ctimeout = setTimeout(this.Unmark,this.classDelay); // // var labels = [], defs = [], number = 0; if (this.replace) { for (i = 0, m = this.update.length; i < m; i++) { var par = this.par[this.update[i]]; if (par.replaced) { labels = labels.concat(par.labels.join('')); defs = defs.concat(par.defs.all); number += par.number; delete par.replaced; } } this.loopCount = 0; // avoid any possibility of infinite loop // (shouldn't happen anyway, but I'm paranoid) } if (update.nodes.length !== this.preview.childNodes.length) { if (this.refreshAll || labels.join('') !== this.oldlabels) { this.MarkForUpdate(0); this.refreshAll = this.refreshRest = false; } else if (this.refreshRest || number !== this.oldnumber || defs.join('') !== this.olddefs) { this.MarkForUpdate(this.i); this.refreshRest = false; } if (this.updateNeeded && this.loopCount++ < 10) { var delay = Math.min(this.labelDelay,3*this.keyRate); if (this.getTime() < 2*this.keyrate) {this.Refresh()} else {this.ltimeout = setTimeout(this.Refresh,delay)} } } }, MarkForUpdate: function (i) { for (var m = this.par.length; i < m; i++) { if (!this.par[i].update) {this.par[i].update = true; this.updateNeeded++} } }, GetMarked: function () { var AMS = MathJax.Extension["TeX/AMSmath"]; var nodes = [], indices = [], par = this.par; for (var i = 0, m = par.length; i < m; i++) { if (par[i].update) { var node = this.typeset.childNodes[i]; nodes.push(node); indices.push(i); this.addChanged(node); for (var j = 0, n = par[i].labels.length; j < n; j++) { delete AMS.labels[par[i].labels[j].split(/=/)[0]]; } } } return {nodes:nodes, indices:indices}; }, Unmark: function () { Preview.ctimeout = null; var nodes = Preview.typeset.childNodes; for (var i = 0, m = nodes.length; i < m; i++) {Preview.removeChanged(nodes[i])} }, Refresh: function () { var update = Preview.GetMarked(); this.oldlabels = this.olddefs = ""; this.oldnumber = 0; if (update.nodes.length) { MathJax.Hub.Queue( ["PreTypeset",Preview,update], ["Reprocess",MathJax.Hub,update.nodes,{}], ["PostTypeset",Preview,update] ); } }, getTime: function () { var time = 0, i = 0, m = this.par.length; while (i < m) {if (this.par[i].update) {time += this.par[i].time}; i++} return time; }, // // Remove the "changed" class from an element (leaving all other classes) // removeChanged: function (node) { if (node.className) { node.className = node.className.toString() .replace(/(^|\s+)changed(\s|$)/,"$2") .replace(/^\s+/,""); } }, addChanged: function (node) { if (node.className && node.className != "") {node.className += " changed"} else {node.className = "changed"} }
};
// // Hook into the math signals // MathJax.Hub.Register.MessageHook(“Begin Math”,function () {
if (Preview.incremental) {Preview.BeginMath()}
}); MathJax.Hub.Register.MessageHook(“End Math”,function () {
if (Preview.incremental) {Preview.EndMath()}
}); MathJax.Hub.Register.StartupHook(“TeX Jax Ready”,function () {
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}); MathJax.Hub.Register.MessageHook(“Begin Math Input”,function () {
if (Preview.incremental) {Preview.BeginInput()}
}); MathJax.Hub.Register.MessageHook(“End Math Input”,function () {
if (Preview.incremental) {Preview.EndInput()}
},5); // priority = 5 to make sure it is before AMS.eqlabels are removed.
// // Hook into the definition routines to record // new definitions that occur. // MathJax.Hub.Register.StartupHook(“TeX begingroup Ready”,function () {
var STACK = MathJax.InputJax.TeX.eqnStack; var DEF = STACK.Def; STACK.Def = function () { if (Preview.incremental) {Preview.TeXDef([].slice.call(arguments,0))} DEF.apply(this,arguments); } // // Temporary hack to fix typo in begingroup.js // MathJax.InputJax.TeX.rootStack.stack[0].environments = MathJax.InputJax.TeX.Definitions.environment;
});
</script> </head> <body>
Type text with embedded TeX in the box below:<br/>
<textarea id=“MathInput” cols=“60” rows=“10” onkeyup=“Preview.Update(true)” onkeydown=“Preview.Update()” style=“margin-top:5px”> There must be some missing constraints. If $alpha_n$ is allowed to be negative, we get the following counterexample. $smash{rlap{phantom{Bigg(}}}$
Define $$ u_{n+1}=(1-alpha_n)u_n+beta_ntag{1} $$ and $$ A_n=prod_{k=1}^{n-1}(1-alpha_k)tag{2} $$ By induction, it can be verified that $$ u_n=A_nleft(u_1+sum_{k=1}^{n-1}frac{beta_k}{A_{k+1}}right)tag{3} $$ For $jge1$, define $$ n_j=left{begin{array}{} 2^{j(j-1)/2}&text{when }jtext{ is odd}\ 2^{j(j-1)/2+1}&text{when }jtext{ is even} end{array}right.tag{4} $$ and for $nge1$, $$ alpha_n=left{begin{array}{} frac{1}{n+1}&text{for }n_jle n< n_{j+1}text{ when }jtext{ is odd}\ -frac1n&text{for }n_jle n< n_{j+1}text{ when }jtext{ is even} end{array}right.tag{5} $$ Obviously, $displaystylelim_{ntoinfty}alpha_n=0$.
Using telescoping products, it is not difficult to show that $$ frac{A_{n_{j+1}}}{A_{n_j}}=left{begin{array}{} frac{n_j}{n_{j+1}}=2^{-j-1}&text{when }jtext{ is odd}\ frac{n_{j+1}}{n_j}=2^{j-1}&text{when }jtext{ is even} end{array}right.tag{6} $$ Equation $(6)$ yields $$ A_{n_j}=left{begin{array}{} 2^{-(j-1)/2}&text{when }jtext{ is odd}\ 2^{-(3j-2)/2}&text{when }jtext{ is even} end{array}right.tag{7} $$ Furthermore, using the standard formula for the partial harmonic series, when $j$ is odd, $$ begin{align} sum_{n=n_j}^{n_{j+1}-1}alpha_n &=logleft(frac{n_{j+1}}{n_j}right)+Oleft(frac{1}{n_j}right)\ &=(j+1)log(2)+Oleft(2^{-j(j-1)/2}right)tag{8} end{align} $$ and when $j$ is even, $$ begin{align} sum_{n=n_j}^{n_{j+1}-1}alpha_n &=-logleft(frac{n_{j+1}}{n_j}right)+Oleft(frac{1}{n_j}right)\ &=-(j-1)log(2)+Oleft(2^{-j(j-1)/2}right)tag{9} end{align} $$ Combining $(8)$ and $(9)$ yields $$ sum_{n=1}^{n_j-1}alpha_n=left{begin{array}{} frac{j-1}{2}log(2)+O(1)&text{when }jtext{ is odd}\ frac{3j-2}{2}log(2)+O(1)&text{when }jtext{ is even} end{array}right.tag{10} $$ Equation $(10)$ says that $displaystylesum_{n=1}^inftyalpha_n=infty$.
Define $$ beta_n=left{begin{array}{} 2^{-j}&text{when }n=n_j-1text{ for }jtext{ even}\ 0&text{otherwise} end{array}right.tag{11} $$ Summing the geometric series yields $displaystylesum_{n=1}^inftybeta_n=frac13$.
Using $(3)$, we get $$ begin{align} u_{n_{j+1}} &=A_{n_{j+1}}left(u_1+sum_{k=1}^{n_{j+1}-1}frac{beta_k}{A_{k+1}}right)\ &gefrac{A_{n_{j+1}}}{A_{n_j}}beta_{n_j-1}\ &=2^{j-1}cdot2^{-j}\ &=frac12tag{12} end{align} $$ when $j$ is even. $(12)$ says that $displaystylelim_{ntoinfty}u_nnot=0$. </textarea> <br/><br/> <div id=“MoreMath”></div> Preview is shown here: <div id=“MathPreview” style=“border:1px solid; padding: 3px; width:50%; margin-top:5px”></div> <div style=“display:none”>Force loading: $x$</div> <script> Preview.Init(); MathJax.Hub.Queue(); </script>
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<!–
| There must be some missing constraints. If $\alpha_n$ is allowed to be negative, we get the following counterexample. $\smash{\rlap{\phantom{\Bigg(}}}$ | | Define | $$ | u_{n+1}=(1-\alpha_n)u_n+\beta_n\tag{1} | $$ | and | $$ | A_n=\prod_{k=1}^{n-1}(1-\alpha_k)\tag{2} | $$ | By induction, it can be verified that | $$ | u_n=A_n\left(u_1+\sum_{k=1}^{n-1}\frac{\beta_k}{A_{k+1}}\right)\tag{3} | $$ | For $j\ge1$, define | $$ | n_j=\left\{\begin{array}{} | 2^{j(j-1)/2}&\text{when }j\text{ is odd}\\ | 2^{j(j-1)/2+1}&\text{when }j\text{ is even} | \end{array}\right.\tag{4} | $$ | and for $n\ge1$, | $$ | \alpha_n=\left\{\begin{array}{} | \frac{1}{n+1}&\text{for }n_j\le n< n_{j+1}\text{ when }j\text{ is odd}\\ | -\frac1n&\text{for }n_j\le n< n_{j+1}\text{ when }j\text{ is even} | \end{array}\right.\tag{5} | $$ | Obviously, $\displaystyle\lim_{n\to\infty}\alpha_n=0$. | | Using telescoping products, it is not difficult to show that | $$ | \frac{A_{n_{j+1}}}{A_{n_j}}=\left\{\begin{array}{} | \frac{n_j}{n_{j+1}}=2^{-j-1}&\text{when }j\text{ is odd}\\ | \frac{n_{j+1}}{n_j}=2^{j-1}&\text{when }j\text{ is even} | \end{array}\right.\tag{6} | $$ | Equation $(6)$ yields | $$ | A_{n_j}=\left\{\begin{array}{} | 2^{-(j-1)/2}&\text{when }j\text{ is odd}\\ | 2^{-(3j-2)/2}&\text{when }j\text{ is even} | \end{array}\right.\tag{7} | $$ | Furthermore, using the standard formula for the partial harmonic series, when $j$ is odd, | $$ | \begin{align} | \sum_{n=n_j}^{n_{j+1}-1}\alpha_n | &=\log\left(\frac{n_{j+1}}{n_j}\right)+O\left(\frac{1}{n_j}\right)\\ | &=(j+1)\log(2)+O\left(2^{-j(j-1)/2}\right)\tag{8} | \end{align} | $$ | and when $j$ is even, | $$ | \begin{align} | \sum_{n=n_j}^{n_{j+1}-1}\alpha_n | &=-\log\left(\frac{n_{j+1}}{n_j}\right)+O\left(\frac{1}{n_j}\right)\\ | &=-(j-1)\log(2)+O\left(2^{-j(j-1)/2}\right)\tag{9} | \end{align} | $$ | Combining $(8)$ and $(9)$ yields | $$ | \sum_{n=1}^{n_j-1}\alpha_n=\left\{\begin{array}{} | \frac{j-1}{2}\log(2)+O(1)&\text{when }j\text{ is odd}\\ | \frac{3j-2}{2}\log(2)+O(1)&\text{when }j\text{ is even} | \end{array}\right.\tag{10} | $$ | Equation $(10)$ says that $\displaystyle\sum_{n=1}^\infty\alpha_n=\infty$. | | Define | $$ | \beta_n=\left\{\begin{array}{} | 2^{-j}&\text{when }n=n_j-1\text{ for }j\text{ even}\\ | 0&\text{otherwise} | \end{array}\right.\tag{11} | $$ | Summing the geometric series yields $\displaystyle\sum_{n=1}^\infty\beta_n=\frac13$. | | Using $(3)$, we get | $$ | \begin{align} | u_{n_{j+1}} | &=A_{n_{j+1}}\left(u_1+\sum_{k=1}^{n_{j+1}-1}\frac{\beta_k}{A_{k+1}}\right)\\ | &\ge\frac{A_{n_{j+1}}}{A_{n_j}}\beta_{n_j-1}\\ | &=2^{j-1}\cdot2^{-j}\\ | &=\frac12\tag{12} | \end{align} | $$ | when $j$ is even. $(12)$ says that $\displaystyle\lim_{n\to\infty}u_n\not=0$.
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