/*
 *  /MathJax/extensions/TeX/AMSmath.js
 *
 *  Copyright (c) 2009-2018 The MathJax Consortium
 *
 *  Licensed under the Apache License, Version 2.0 (the "License");
 *  you may not use this file except in compliance with the License.
 *  You may obtain a copy of the License at
 *
 *      http://www.apache.org/licenses/LICENSE-2.0
 *
 *  Unless required by applicable law or agreed to in writing, software
 *  distributed under the License is distributed on an "AS IS" BASIS,
 *  WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
 *  See the License for the specific language governing permissions and
 *  limitations under the License.
 */

MathJax.Extension["TeX/AMSmath"]={version:"2.7.6",number:0,startNumber:0,IDs:{},eqIDs:{},labels:{},eqlabels:{},refs:[]};MathJax.Hub.Register.StartupHook("TeX Jax Ready",function(){var b=MathJax.ElementJax.mml,h=MathJax.InputJax.TeX,g=MathJax.Extension["TeX/AMSmath"];var d=h.Definitions,f=h.Stack.Item,a=h.config.equationNumbers;var c=function(k){var n=[];for(var l=0,j=k.length;l<j;l++){n[l]=h.Parse.prototype.Em(k[l])}return n.join(" ")};var e=(document.getElementsByTagName("base").length===0)?"":String(document.location).replace(/#.*$/,"");d.Add({mathchar0mo:{iiiint:["2A0C",{texClass:b.TEXCLASS.OP}]},macros:{mathring:["Accent","2DA"],nobreakspace:"Tilde",negmedspace:["Spacer",b.LENGTH.NEGATIVEMEDIUMMATHSPACE],negthickspace:["Spacer",b.LENGTH.NEGATIVETHICKMATHSPACE],idotsint:["MultiIntegral","\\int\\cdots\\int"],dddot:["Accent","20DB"],ddddot:["Accent","20DC"],sideset:["Macro","\\mathop{\\mathop{\\rlap{\\phantom{#3}}}\\nolimits#1\\!\\mathop{#3}\\nolimits#2}",3],boxed:["Macro","\\fbox{$\\displaystyle{#1}$}",1],tag:"HandleTag",notag:"HandleNoTag",label:"HandleLabel",ref:"HandleRef",eqref:["HandleRef",true],substack:["Macro","\\begin{subarray}{c}#1\\end{subarray}",1],injlim:["NamedOp","inj&thinsp;lim"],projlim:["NamedOp","proj&thinsp;lim"],varliminf:["Macro","\\mathop{\\underline{\\mmlToken{mi}{lim}}}"],varlimsup:["Macro","\\mathop{\\overline{\\mmlToken{mi}{lim}}}"],varinjlim:["Macro","\\mathop{\\underrightarrow{\\mmlToken{mi}{lim}}}"],varprojlim:["Macro","\\mathop{\\underleftarrow{\\mmlToken{mi}{lim}}}"],DeclareMathOperator:"HandleDeclareOp",operatorname:"HandleOperatorName",SkipLimits:"SkipLimits",genfrac:"Genfrac",frac:["Genfrac","","","",""],tfrac:["Genfrac","","","",1],dfrac:["Genfrac","","","",0],binom:["Genfrac","(",")","0",""],tbinom:["Genfrac","(",")","0",1],dbinom:["Genfrac","(",")","0",0],cfrac:"CFrac",shoveleft:["HandleShove",b.ALIGN.LEFT],shoveright:["HandleShove",b.ALIGN.RIGHT],xrightarrow:["xArrow",8594,5,6],xleftarrow:["xArrow",8592,7,3]},environment:{align:["AMSarray",null,true,true,"rlrlrlrlrlrl",c([0,2,0,2,0,2,0,2,0,2,0])],"align*":["AMSarray",null,false,true,"rlrlrlrlrlrl",c([0,2,0,2,0,2,0,2,0,2,0])],multline:["Multline",null,true],"multline*":["Multline",null,false],split:["AMSarray",null,false,false,"rl",c([0])],gather:["AMSarray",null,true,true,"c"],"gather*":["AMSarray",null,false,true,"c"],alignat:["AlignAt",null,true,true],"alignat*":["AlignAt",null,false,true],alignedat:["AlignAt",null,false,false],aligned:["AlignedAMSArray",null,null,null,"rlrlrlrlrlrl",c([0,2,0,2,0,2,0,2,0,2,0]),".5em","D"],gathered:["AlignedAMSArray",null,null,null,"c",null,".5em","D"],subarray:["Array",null,null,null,null,c([0]),"0.1em","S",1],smallmatrix:["Array",null,null,null,"c",c([1/3]),".2em","S",1],equation:["EquationBegin","Equation",true],"equation*":["EquationBegin","EquationStar",false],eqnarray:["AMSarray",null,true,true,"rcl","0 "+b.LENGTH.THICKMATHSPACE,".5em"],"eqnarray*":["AMSarray",null,false,true,"rcl","0 "+b.LENGTH.THICKMATHSPACE,".5em"]},delimiter:{"\\lvert":["007C",{texClass:b.TEXCLASS.OPEN}],"\\rvert":["007C",{texClass:b.TEXCLASS.CLOSE}],"\\lVert":["2016",{texClass:b.TEXCLASS.OPEN}],"\\rVert":["2016",{texClass:b.TEXCLASS.CLOSE}]}},null,true);h.Parse.Augment({HandleTag:function(k){var m=this.GetStar();var j=this.trimSpaces(this.GetArgument(k)),i=j;if(!m){j=a.formatTag(j)}var l=this.stack.global;l.tagID=i;if(l.notags){h.Error(["CommandNotAllowedInEnv","%1 not allowed in %2 environment",k,l.notags])}if(l.tag){h.Error(["MultipleCommand","Multiple %1",k])}l.tag=b.mtd.apply(b,this.InternalMath(j)).With({id:a.formatID(i)})},HandleNoTag:function(i){if(this.stack.global.tag){delete this.stack.global.tag}this.stack.global.notag=true},HandleLabel:function(j){var k=this.stack.global,i=this.GetArgument(j);if(i===""){return}if(!g.refUpdate){if(k.label){h.Error(["MultipleCommand","Multiple %1",j])}k.label=i;if(g.labels[i]||g.eqlabels[i]){h.Error(["MultipleLabel","Label '%1' multiply defined",i])}g.eqlabels[i]={tag:"???",id:""}}},HandleRef:function(k,m){var j=this.GetArgument(k);var l=g.labels[j]||g.eqlabels[j];if(!l){l={tag:"???",id:""};g.badref=!g.refUpdate}var i=l.tag;if(m){i=a.formatTag(i)}this.Push(b.mrow.apply(b,this.InternalMath(i)).With({href:a.formatURL(l.id,e),"class":"MathJax_ref"}))},HandleDeclareOp:function(j){var i=(this.GetStar()?"":"\\nolimits\\SkipLimits");var k=this.trimSpaces(this.GetArgument(j));if(k.charAt(0)=="\\"){k=k.substr(1)}var l=this.GetArgument(j);l=l.replace(/\*/g,"\\text{*}").replace(/-/g,"\\text{-}");this.setDef(k,["Macro","\\mathop{\\rm "+l+"}"+i])},HandleOperatorName:function(j){var i=(this.GetStar()?"":"\\nolimits\\SkipLimits");var k=this.trimSpaces(this.GetArgument(j));k=k.replace(/\*/g,"\\text{*}").replace(/-/g,"\\text{-}");this.string="\\mathop{\\rm "+k+"}"+i+" "+this.string.slice(this.i);this.i=0},SkipLimits:function(j){var l=this.GetNext(),k=this.i;if(l==="\\"&&++this.i&&this.GetCS()!=="limits"){this.i=k}},HandleShove:function(j,i){var k=this.stack.Top();if(k.type!=="multline"){h.Error(["CommandInMultline","%1 can only appear within the multline environment",j])}if(k.data.length){h.Error(["CommandAtTheBeginingOfLine","%1 must come at the beginning of the line",j])}k.data.shove=i},CFrac:function(l){var i=this.trimSpaces(this.GetBrackets(l,"")),k=this.GetArgument(l),m=this.GetArgument(l);var j=b.mfrac(h.Parse("\\strut\\textstyle{"+k+"}",this.stack.env).mml(),h.Parse("\\strut\\textstyle{"+m+"}",this.stack.env).mml());i=({l:b.ALIGN.LEFT,r:b.ALIGN.RIGHT,"":""})[i];if(i==null){h.Error(["IllegalAlign","Illegal alignment specified in %1",l])}if(i){j.numalign=j.denomalign=i}this.Push(j)},Genfrac:function(j,l,q,n,i){if(l==null){l=this.GetDelimiterArg(j)}if(q==null){q=this.GetDelimiterArg(j)}if(n==null){n=this.GetArgument(j)}if(i==null){i=this.trimSpaces(this.GetArgument(j))}var m=this.ParseArg(j);var p=this.ParseArg(j);var k=b.mfrac(m,p);if(n!==""){k.linethickness=n}if(l||q){k=h.fixedFence(l,k.With({texWithDelims:true}),q)}if(i!==""){var o=(["D","T","S","SS"])[i];if(o==null){h.Error(["BadMathStyleFor","Bad math style for %1",j])}k=b.mstyle(k);if(o==="D"){k.displaystyle=true;k.scriptlevel=0}else{k.displaystyle=false;k.scriptlevel=i-1}}this.Push(k)},Multline:function(j,i){this.Push(j);this.checkEqnEnv();return f.multline(i,this.stack).With({arraydef:{displaystyle:true,rowspacing:".5em",width:h.config.MultLineWidth,columnwidth:"100%",side:h.config.TagSide,minlabelspacing:h.config.TagIndent}})},AMSarray:function(k,j,i,m,l){this.Push(k);if(i){this.checkEqnEnv()}m=m.replace(/[^clr]/g,"").split("").join(" ");m=m.replace(/l/g,"left").replace(/r/g,"right").replace(/c/g,"center");return f.AMSarray(k.name,j,i,this.stack).With({arraydef:{displaystyle:true,rowspacing:".5em",columnalign:m,columnspacing:(l||"1em"),rowspacing:"3pt",side:h.config.TagSide,minlabelspacing:h.config.TagIndent}})},AlignedAMSArray:function(i){var j=this.GetBrackets("\\begin{"+i.name+"}");return this.setArrayAlign(this.AMSarray.apply(this,arguments),j)},AlignAt:function(l,j,i){var q,k,p="",o=[];if(!i){k=this.GetBrackets("\\begin{"+l.name+"}")}q=this.GetArgument("\\begin{"+l.name+"}");if(q.match(/[^0-9]/)){h.Error(["PositiveIntegerArg","Argument to %1 must me a positive integer","\\begin{"+l.name+"}"])}while(q>0){p+="rl";o.push("0em 0em");q--}o=o.join(" ");if(i){return this.AMSarray(l,j,i,p,o)}var m=this.AMSarray(l,j,i,p,o);return this.setArrayAlign(m,k)},EquationBegin:function(i,j){this.checkEqnEnv();this.stack.global.forcetag=(j&&a.autoNumber!=="none");return i},EquationStar:function(i,j){this.stack.global.tagged=true;return j},checkEqnEnv:function(){if(this.stack.global.eqnenv){h.Error(["ErroneousNestingEq","Erroneous nesting of equation structures"])}this.stack.global.eqnenv=true},MultiIntegral:function(j,m){var l=this.GetNext();if(l==="\\"){var k=this.i;l=this.GetArgument(j);this.i=k;if(l==="\\limits"){if(j==="\\idotsint"){m="\\!\\!\\mathop{\\,\\,"+m+"}"}else{m="\\!\\!\\!\\mathop{\\,\\,\\,"+m+"}"}}}this.string=m+" "+this.string.slice(this.i);this.i=0},xArrow:function(k,o,n,i){var m={width:"+"+(n+i)+"mu",lspace:n+"mu"};var p=this.GetBrackets(k),q=this.ParseArg(k);var s=b.mo(b.chars(String.fromCharCode(o))).With({stretchy:true,texClass:b.TEXCLASS.REL});var j=b.munderover(s);j.SetData(j.over,b.mpadded(q).With(m).With({voffset:".15em"}));if(p){p=h.Parse(p,this.stack.env).mml();j.SetData(j.under,b.mpadded(p).With(m).With({voffset:"-.24em"}))}this.Push(j.With({subsupOK:true}))},GetDelimiterArg:function(i){var j=this.trimSpaces(this.GetArgument(i));if(j==""){return null}if(j in d.delimiter){return j}h.Error(["MissingOrUnrecognizedDelim","Missing or unrecognized delimiter for %1",i])},GetStar:function(){var i=(this.GetNext()==="*");if(i){this.i++}return i}});f.Augment({autoTag:function(){var j=this.global;if(!j.notag){g.number++;j.tagID=a.formatNumber(g.number.toString());var i=h.Parse("\\text{"+a.formatTag(j.tagID)+"}",{}).mml();j.tag=b.mtd(i).With({id:a.formatID(j.tagID)})}},getTag:function(){var m=this.global,k=m.tag;m.tagged=true;if(m.label){if(a.useLabelIds){k.id=a.formatID(m.label)}g.eqlabels[m.label]={tag:m.tagID,id:k.id}}if(document.getElementById(k.id)||g.IDs[k.id]||g.eqIDs[k.id]){var l=0,j;do{l++;j=k.id+"_"+l}while(document.getElementById(j)||g.IDs[j]||g.eqIDs[j]);k.id=j;if(m.label){g.eqlabels[m.label].id=j}}g.eqIDs[k.id]=1;this.clearTag();return k},clearTag:function(){var i=this.global;delete i.tag;delete i.tagID;delete i.label},fixInitialMO:function(l){for(var k=0,j=l.length;k<j;k++){if(l[k]&&(l[k].type!=="mspace"&&(l[k].type!=="texatom"||(l[k].data[0]&&l[k].data[0].data.length)))){if(l[k].isEmbellished()){l.unshift(b.mi())}break}}}});f.multline=f.array.Subclass({type:"multline",Init:function(j,i){this.SUPER(arguments).Init.apply(this);this.numbered=(j&&a.autoNumber!=="none");this.save={notag:i.global.notag};i.global.tagged=!j&&!i.global.forcetag},EndEntry:function(){if(this.table.length){this.fixInitialMO(this.data)}var i=b.mtd.apply(b,this.data);if(this.data.shove){i.columnalign=this.data.shove}this.row.push(i);this.data=[]},EndRow:function(){if(this.row.length!=1){h.Error(["MultlineRowsOneCol","The rows within the %1 environment must have exactly one column","multline"])}this.table.push(this.row);this.row=[]},EndTable:function(){this.SUPER(arguments).EndTable.call(this);if(this.table.length){var k=this.table.length-1,n,l=-1;if(!this.table[0][0].columnalign){this.table[0][0].columnalign=b.ALIGN.LEFT}if(!this.table[k][0].columnalign){this.table[k][0].columnalign=b.ALIGN.RIGHT}if(!this.global.tag&&this.numbered){this.autoTag()}if(this.global.tag&&!this.global.notags){l=(this.arraydef.side==="left"?0:this.table.length-1);this.table[l]=[this.getTag()].concat(this.table[l])}for(n=0,k=this.table.length;n<k;n++){var j=(n===l?b.mlabeledtr:b.mtr);this.table[n]=j.apply(b,this.table[n])}}this.global.notag=this.save.notag}});f.AMSarray=f.array.Subclass({type:"AMSarray",Init:function(l,k,j,i){this.SUPER(arguments).Init.apply(this);this.numbered=(k&&a.autoNumber!=="none");this.save={notags:i.global.notags,notag:i.global.notag};i.global.notags=(j?null:l);i.global.tagged=!k&&!i.global.forcetag},EndEntry:function(){if(this.row.length%2===1){this.fixInitialMO(this.data)}this.row.push(b.mtd.apply(b,this.data));this.data=[]},EndRow:function(){var i=b.mtr;if(!this.global.tag&&this.numbered){this.autoTag()}if(this.global.tag&&!this.global.notags){this.row=[this.getTag()].concat(this.row);i=b.mlabeledtr}else{this.clearTag()}if(this.numbered){delete this.global.notag}this.table.push(i.apply(b,this.row));this.row=[]},EndTable:function(){this.SUPER(arguments).EndTable.call(this);this.global.notags=this.save.notags;this.global.notag=this.save.notag}});f.start.Augment({oldCheckItem:f.start.prototype.checkItem,checkItem:function(k){if(k.type==="stop"){var i=this.mmlData(),j=this.global;if(g.display&&!j.tag&&!j.tagged&&!j.isInner&&(a.autoNumber==="all"||j.forcetag)){this.autoTag()}if(j.tag){var m=[this.getTag(),b.mtd(i)];var l={side:h.config.TagSide,minlabelspacing:h.config.TagIndent,displaystyle:"inherit"};i=b.mtable(b.mlabeledtr.apply(b,m)).With(l)}return f.mml(i)}return this.oldCheckItem.call(this,k)}});h.prefilterHooks.Add(function(i){g.display=i.display;g.number=g.startNumber;g.eqlabels={};g.eqIDs={};g.badref=false;if(g.refUpdate){g.number=i.script.MathJax.startNumber}});h.postfilterHooks.Add(function(i){i.script.MathJax.startNumber=g.startNumber;g.startNumber=g.number;MathJax.Hub.Insert(g.IDs,g.eqIDs);MathJax.Hub.Insert(g.labels,g.eqlabels);if(g.badref&&!i.math.texError){g.refs.push(i.script)}},100);MathJax.Hub.Register.MessageHook("Begin Math Input",function(){g.refs=[];g.refUpdate=false});MathJax.Hub.Register.MessageHook("End Math Input",function(l){if(g.refs.length){g.refUpdate=true;for(var k=0,j=g.refs.length;k<j;k++){g.refs[k].MathJax.state=MathJax.ElementJax.STATE.UPDATE}return MathJax.Hub.processInput({scripts:g.refs,start:new Date().getTime(),i:0,j:0,jax:{},jaxIDs:[]})}return null});h.resetEquationNumbers=function(j,i){g.startNumber=(j||0);if(!i){g.labels={};g.IDs={}}};MathJax.Hub.Startup.signal.Post("TeX AMSmath Ready")});MathJax.Ajax.loadComplete("[MathJax]/extensions/TeX/AMSmath.js");
