mathmlben

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<link rel="stylesheet" href="/w/A.load.php,,qdebug==false,,alang==en,,amodules==ext.math.styles,,P7Cmediawiki.legacy.commonPrint,,P2Cshared,,P7Cmediawiki.sectionAnchor,,P7Cmediawiki.skinning.interface,,P7Cskins.vector.styles,,P7Cskins.vector.styles.responsive,,aonly==styles,,askin==vector+load.php,,qdebug==false,,alang==en,,amodules==ext.smw.style,,P7Cext.smw.tooltip.styles,,aonly==styles,,askin==vector+load.php,,qdebug==false,,alang==en,,amodules==site.styles,,aonly==styles,,askin==vector,Mcc.W0_lySKyC-.css.pagespeed.cf.pJBQ32dAd4.css"/> <script async="" src="https://en.formulasearchengine.com/w/load.php,qdebug=false,alang=en,amodules=startup,aonly=scripts,askin=vector.pagespeed.jm.0aTVPPbYss.js"></script> <meta name="ResourceLoaderDynamicStyles" content=""/> <meta name="generator" content="MediaWiki 1.30.0-alpha"/> <meta name="robots" content="noindex,nofollow"/> <meta name="viewport" content="width=device-width, initial-scale=1"/> <link rel="shortcut icon" 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href="#mw-head">navigation</a>, <a href="#p-search">search</a> </div> <div id="mw-content-text"><p>⧼math-wmc-Queries⧽ </p><table class="wikitable sortable"> <tr> <th> # </th> <th> fId </th> <th> #Var </th> <th> #matches </th> <th> query </th> <th> reference </th></tr> <tr> <td> <a href="https://en.formulasearchengine.com/wiki/Special:DisplayTopics/1" title="Special:DisplayTopics/1">1</a> </td> <td> <a href="https://en.formulasearchengine.com/wiki/Special:PermanentLink/2459#math.2459.3" title="Special:PermanentLink/2459">Van_der_Waerden's_theorem-3</a> </td> <td> 0 </td> <td> 1 </td> <td> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style=""><math xmlns="http://www.w3.org/1998/Math/MathML" id="p1.1.m1.1" class="ltx_Math" alttext="{\displaystyle W(2,k)>2^{{k}}/k^{{\varepsilon}}}"> <semantics id="p1.1.m1.1a"> <mrow id="p1.1.m1.1.13" xref="p1.1.m1.1.13.cmml"> <mrow id="p1.1.m1.1.13.1" xref="p1.1.m1.1.13.1.cmml"> <mi id="p1.1.m1.1.1" xref="p1.1.m1.1.1.cmml">W</mi> <mo id="p1.1.m1.1.13.1.1" xref="p1.1.m1.1.13.1.1.cmml">⁢</mo> <mrow id="p1.1.m1.1.13.1.2c" xref="p1.1.m1.1.13.1.2.cmml"> <mo stretchy="false" id="p1.1.m1.1.13.1.2" xref="p1.1.m1.1.13.1.2.cmml">(</mo> <mn id="p1.1.m1.1.3" xref="p1.1.m1.1.3.cmml">2</mn> <mo id="p1.1.m1.1.13.1.2a" xref="p1.1.m1.1.13.1.2.cmml">,</mo> <mi id="p1.1.m1.1.5" xref="p1.1.m1.1.5.cmml">k</mi> <mo stretchy="false" id="p1.1.m1.1.13.1.2b" xref="p1.1.m1.1.13.1.2.cmml">)</mo> </mrow> </mrow> <mo id="p1.1.m1.1.7" xref="p1.1.m1.1.7.cmml">></mo> <mrow id="p1.1.m1.1.13.2" xref="p1.1.m1.1.13.2.cmml"> <msup id="p1.1.m1.1.13.2.1" xref="p1.1.m1.1.13.2.1.cmml"> <mn id="p1.1.m1.1.8" xref="p1.1.m1.1.8.cmml">2</mn> <mi id="p1.1.m1.1.9.1" xref="p1.1.m1.1.9.1.cmml">k</mi> </msup> <mo id="p1.1.m1.1.10" xref="p1.1.m1.1.10.cmml">/</mo> <msup id="p1.1.m1.1.13.2.2" xref="p1.1.m1.1.13.2.2.cmml"> <mi id="p1.1.m1.1.11" xref="p1.1.m1.1.11.cmml">k</mi> <mi id="p1.1.m1.1.12.1" xref="p1.1.m1.1.12.1.cmml">ε</mi> </msup> </mrow> </mrow> <annotation-xml encoding="MathML-Content" id="p1.1.m1.1b"> <apply id="p1.1.m1.1.13.cmml" xref="p1.1.m1.1.13"> <gt id="p1.1.m1.1.7.cmml" xref="p1.1.m1.1.7"/> <apply id="p1.1.m1.1.13.1.cmml" xref="p1.1.m1.1.13.1"> <times id="p1.1.m1.1.13.1.1.cmml" xref="p1.1.m1.1.13.1.1"/> <ci id="p1.1.m1.1.1.cmml" xref="p1.1.m1.1.1">W</ci> <interval closure="open" id="p1.1.m1.1.13.1.2.cmml" xref="p1.1.m1.1.13.1.2"> <cn type="integer" id="p1.1.m1.1.3.cmml" xref="p1.1.m1.1.3">2</cn> <ci id="p1.1.m1.1.5.cmml" xref="p1.1.m1.1.5">k</ci> </interval> </apply> <apply id="p1.1.m1.1.13.2.cmml" xref="p1.1.m1.1.13.2"> <divide id="p1.1.m1.1.10.cmml" xref="p1.1.m1.1.10"/> <apply id="p1.1.m1.1.13.2.1.cmml" xref="p1.1.m1.1.13.2.1"> <csymbol cd="ambiguous" id="p1.1.m1.1.13.2.1.1.cmml">superscript</csymbol> <cn type="integer" id="p1.1.m1.1.8.cmml" xref="p1.1.m1.1.8">2</cn> <ci id="p1.1.m1.1.9.1.cmml" xref="p1.1.m1.1.9.1">k</ci> </apply> <apply id="p1.1.m1.1.13.2.2.cmml" xref="p1.1.m1.1.13.2.2"> <csymbol cd="ambiguous" id="p1.1.m1.1.13.2.2.1.cmml">superscript</csymbol> <ci id="p1.1.m1.1.11.cmml" xref="p1.1.m1.1.11">k</ci> <ci id="p1.1.m1.1.12.1.cmml" xref="p1.1.m1.1.12.1">ε</ci> </apply> </apply> </apply> </annotation-xml> <annotation encoding="application/x-tex" id="p1.1.m1.1c">{\displaystyle W(2,k)>2^{{k}}/k^{{\varepsilon}}}</annotation> </semantics> </math></span><img src="https://en.formulasearchengine.com/w/index.php?title=Special:MathShowImage&hash=00cecdd7d2afa32b326e78ff2717120e&mode=latexml" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.667ex; height: 2.833ex; width: 16.833ex;" alt="W(2,k)>2^{{k}}/k^{{\varepsilon }}" data-pagespeed-url-hash="3727682694" onload="pagespeed.CriticalImages.checkImageForCriticality(this);"/></span> </td> <td> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style=""><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W(2,k)>2^{k}/k^{\varepsilon }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mo>,</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>></mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ε</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W(2,k)>2^{k}/k^{\varepsilon }}</annotation> </semantics> </math></span><img src="https://api.formulasearchengine.com/v1/media/math/render/svg/045045535ed2655006f05a36e645307bd33a4e92" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:16.374ex; height:3.176ex;" alt="{\displaystyle W(2,k)>2^{k}/k^{\varepsilon }}" data-pagespeed-url-hash="3022000311" onload="pagespeed.CriticalImages.checkImageForCriticality(this);"/></span> </td></tr> <tr> <td> <a href="https://en.formulasearchengine.com/wiki/Special:DisplayTopics/2" title="Special:DisplayTopics/2">2</a> </td> <td> <a href="https://en.formulasearchengine.com/wiki/Special:PermanentLink/4050#math.4050.189" title="Special:PermanentLink/4050">Bounded_variation-189</a> </td> <td> 0 </td> <td> 9 </td> <td> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style=""><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (X,\Sigma )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi mathvariant="normal">Σ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (X,\Sigma )}</annotation> </semantics> </math></span><img src="https://en.formulasearchengine.com/w/index.php?title=Special:MathShowImage&hash=000138f6a2210ff1f9bb5eb7bc25ab6c&mode=latexml" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; height: 2.843ex; width: 6.501ex;" alt="{\displaystyle (X,\Sigma)}" data-pagespeed-url-hash="3556134968" onload="pagespeed.CriticalImages.checkImageForCriticality(this);"/></span> </td> <td> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style=""><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (X,\Sigma )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi mathvariant="normal">Σ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (X,\Sigma )}</annotation> </semantics> </math></span><img src="https://api.formulasearchengine.com/v1/media/math/render/svg/63c4fcf15808c3dbf52cc4b3fd473e97e231f53f" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:6.501ex; height:2.843ex;" alt="{\displaystyle (X,\Sigma )}" data-pagespeed-url-hash="1104805632" onload="pagespeed.CriticalImages.checkImageForCriticality(this);"/></span> </td></tr> <tr> <td> <a href="https://en.formulasearchengine.com/wiki/Special:DisplayTopics/3" title="Special:DisplayTopics/3">3</a> </td> <td> <a href="https://en.formulasearchengine.com/wiki/Special:PermanentLink/4189#math.4189.50" title="Special:PermanentLink/4189">Lindemann–Weierstrass_theorem-50</a> </td> <td> 0 </td> <td> 1 </td> <td> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style=""><math xmlns="http://www.w3.org/1998/Math/MathML" id="p1.1.m1.1" class="ltx_Math" alttext="{\displaystyle(p-1)!^{{n}}}"> <semantics id="p1.1.m1.1a"> <msup id="p1.1.m1.1.8" xref="p1.1.m1.1.8.cmml"> <mrow id="p1.1.m1.1.8.2" xref="p1.1.m1.1.8.2.cmml"> <mrow id="p1.1.m1.1.8.2.1b"> <mo stretchy="false" id="p1.1.m1.1.8.2.1">(</mo> <mrow id="XM1" xref="XM1.cmml"> <mi id="p1.1.m1.1.2" xref="p1.1.m1.1.2.cmml">p</mi> <mo 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(p-1)!^{n}}</annotation> </semantics> </math></span><img src="https://api.formulasearchengine.com/v1/media/math/render/svg/caaaaaab187a02e4be6f46e48ce9b80cd9e5d6d1" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:8.847ex; height:2.843ex;" alt="{\displaystyle (p-1)!^{n}}" data-pagespeed-url-hash="2835232102" onload="pagespeed.CriticalImages.checkImageForCriticality(this);"/></span> </td></tr> <tr> <td> <a href="https://en.formulasearchengine.com/wiki/Special:DisplayTopics/4" title="Special:DisplayTopics/4">4</a> </td> <td> <a href="https://en.formulasearchengine.com/wiki/Special:PermanentLink/15332#math.15332.68" title="Special:PermanentLink/15332">Orbit_portrait-68</a> </td> <td> 1 </td> <td> 2 </td> <td> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style=""><math xmlns="http://www.w3.org/1998/Math/MathML" id="p1.1.m1.1" class="ltx_Math" alttext="{\displaystyle f_{\qvar@construct{x0}}(z)=z^{2}+c}"> <semantics id="p1.1.m1.1a"> <mrow id="p1.1.m1.1.11" xref="p1.1.m1.1.11.cmml"> <mrow id="p1.1.m1.1.11.1" xref="p1.1.m1.1.11.1.cmml"> <msub id="p1.1.m1.1.11.1.2" xref="p1.1.m1.1.11.1.2.cmml"> <mi id="p1.1.m1.1.1" xref="p1.1.m1.1.1.cmml">f</mi> <mi mathcolor="red" mathvariant="italic" id="p1.1.m1.1.2.1" xref="p1.1.m1.1.2.1.cmml">x0</mi> </msub> <mo id="p1.1.m1.1.11.1.1" xref="p1.1.m1.1.11.1.1.cmml">⁢</mo> <mrow id="p1.1.m1.1.11.1.3b"> <mo stretchy="false" id="p1.1.m1.1.11.1.3">(</mo> <mi id="p1.1.m1.1.4" xref="p1.1.m1.1.4.cmml">z</mi> <mo stretchy="false" id="p1.1.m1.1.11.1.3a">)</mo> </mrow> </mrow> <mo id="p1.1.m1.1.6" xref="p1.1.m1.1.6.cmml">=</mo> <mrow id="p1.1.m1.1.11.2" xref="p1.1.m1.1.11.2.cmml"> <msup id="p1.1.m1.1.11.2.1" xref="p1.1.m1.1.11.2.1.cmml"> <mi id="p1.1.m1.1.7" xref="p1.1.m1.1.7.cmml">z</mi> <mn id="p1.1.m1.1.8.1" xref="p1.1.m1.1.8.1.cmml">2</mn> </msup> <mo id="p1.1.m1.1.9" xref="p1.1.m1.1.9.cmml">+</mo> <mi id="p1.1.m1.1.10" 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xref="p1.1.m1.1.8.1">2</cn> </apply> <ci id="p1.1.m1.1.10.cmml" xref="p1.1.m1.1.10">c</ci> </apply> </apply> </annotation-xml> <annotation encoding="application/x-tex" id="p1.1.m1.1c">{\displaystyle f_{\qvar@construct{x0}}(z)=z^{2}+c}</annotation> </semantics> </math></span><img src="https://en.formulasearchengine.com/w/index.php?title=Special:MathShowImage&hash=96dbbd2846e7c7ecb56194a3e6a53732&mode=latexml" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.667ex; height: 2.667ex; width: 15.667ex;" alt="f_{\qvar{x0}}(z)=z^{2}+c" data-pagespeed-url-hash="3236431601" onload="pagespeed.CriticalImages.checkImageForCriticality(this);"/></span> </td> <td> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style=""><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{c}(z)=z^{2}+c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{c}(z)=z^{2}+c}</annotation> </semantics> </math></span><img src="https://api.formulasearchengine.com/v1/media/math/render/svg/191627a3eebdd6608c9b226786defc468b747502" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:14.071ex; height:3.176ex;" alt="{\displaystyle f_{c}(z)=z^{2}+c}" data-pagespeed-url-hash="1019378453" onload="pagespeed.CriticalImages.checkImageForCriticality(this);"/></span> </td></tr> <tr> <td> <a href="https://en.formulasearchengine.com/wiki/Special:DisplayTopics/5" title="Special:DisplayTopics/5">5</a> </td> <td> <a href="https://en.formulasearchengine.com/wiki/Special:PermanentLink/391#math.391.114" title="Special:PermanentLink/391">First-order_logic-114</a> </td> <td> 0 </td> <td> 1 </td> <td> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style=""><math xmlns="http://www.w3.org/1998/Math/MathML" id="p1.1.m1.1" class="ltx_Math" alttext="{\displaystyle\forall x\,\forall y\,P(x,y)\Leftrightarrow\forall y\,\forall x%
\,P(x,y)}"> <semantics id="p1.1.m1.1a"> <mrow id="p1.1.m1.1.26" xref="p1.1.m1.1.26.cmml"> <mrow id="p1.1.m1.1.26.1" xref="p1.1.m1.1.26.1.cmml"> <mo id="p1.1.m1.1.1" xref="p1.1.m1.1.1.cmml">∀</mo> <mrow id="p1.1.m1.1.26.1.1" xref="p1.1.m1.1.26.1.1.cmml"> <mpadded width="+1.7pt" id="p1.1.m1.1.2" xref="p1.1.m1.1.2.cmml"> <mi id="p1.1.m1.1.2a" xref="p1.1.m1.1.2.cmml">x</mi> </mpadded> <mo id="p1.1.m1.1.26.1.1.1" xref="p1.1.m1.1.26.1.1.1.cmml">⁢</mo> <mrow id="p1.1.m1.1.26.1.1.2" xref="p1.1.m1.1.26.1.1.2.cmml"> <mo id="p1.1.m1.1.4" xref="p1.1.m1.1.4.cmml">∀</mo> <mrow id="p1.1.m1.1.26.1.1.2.1" xref="p1.1.m1.1.26.1.1.2.1.cmml"> <mpadded width="+1.7pt" id="p1.1.m1.1.5" xref="p1.1.m1.1.5.cmml"> <mi id="p1.1.m1.1.5a" xref="p1.1.m1.1.5.cmml">y</mi> </mpadded> <mo id="p1.1.m1.1.26.1.1.2.1.1" xref="p1.1.m1.1.26.1.1.2.1.1.cmml">⁢</mo> <mi id="p1.1.m1.1.7" xref="p1.1.m1.1.7.cmml">P</mi> <mo id="p1.1.m1.1.26.1.1.2.1.1a" xref="p1.1.m1.1.26.1.1.2.1.1.cmml">⁢</mo> <mrow id="p1.1.m1.1.26.1.1.2.1.2c" xref="p1.1.m1.1.26.1.1.2.1.2.cmml"> <mo stretchy="false" id="p1.1.m1.1.26.1.1.2.1.2" xref="p1.1.m1.1.26.1.1.2.1.2.cmml">(</mo> <mi id="p1.1.m1.1.9" xref="p1.1.m1.1.9.cmml">x</mi> <mo id="p1.1.m1.1.26.1.1.2.1.2a" xref="p1.1.m1.1.26.1.1.2.1.2.cmml">,</mo> <mi id="p1.1.m1.1.11" xref="p1.1.m1.1.11.cmml">y</mi> <mo stretchy="false" id="p1.1.m1.1.26.1.1.2.1.2b" xref="p1.1.m1.1.26.1.1.2.1.2.cmml">)</mo> </mrow> </mrow> </mrow> </mrow> </mrow> <mo id="p1.1.m1.1.13" xref="p1.1.m1.1.13.cmml">⇔</mo> <mrow id="p1.1.m1.1.26.2" xref="p1.1.m1.1.26.2.cmml"> <mo id="p1.1.m1.1.14" xref="p1.1.m1.1.14.cmml">∀</mo> <mrow id="p1.1.m1.1.26.2.1" xref="p1.1.m1.1.26.2.1.cmml"> <mpadded width="+1.7pt" id="p1.1.m1.1.15" xref="p1.1.m1.1.15.cmml"> <mi id="p1.1.m1.1.15a" xref="p1.1.m1.1.15.cmml">y</mi> </mpadded> <mo id="p1.1.m1.1.26.2.1.1" xref="p1.1.m1.1.26.2.1.1.cmml">⁢</mo> <mrow id="p1.1.m1.1.26.2.1.2" xref="p1.1.m1.1.26.2.1.2.cmml"> <mo id="p1.1.m1.1.17" xref="p1.1.m1.1.17.cmml">∀</mo> <mrow id="p1.1.m1.1.26.2.1.2.1" xref="p1.1.m1.1.26.2.1.2.1.cmml"> <mpadded width="+1.7pt" id="p1.1.m1.1.18" xref="p1.1.m1.1.18.cmml"> <mi id="p1.1.m1.1.18a" xref="p1.1.m1.1.18.cmml">x</mi> </mpadded> <mo id="p1.1.m1.1.26.2.1.2.1.1" xref="p1.1.m1.1.26.2.1.2.1.1.cmml">⁢</mo> <mi id="p1.1.m1.1.20" xref="p1.1.m1.1.20.cmml">P</mi> <mo id="p1.1.m1.1.26.2.1.2.1.1a" xref="p1.1.m1.1.26.2.1.2.1.1.cmml">⁢</mo> <mrow id="p1.1.m1.1.26.2.1.2.1.2c" xref="p1.1.m1.1.26.2.1.2.1.2.cmml"> <mo stretchy="false" id="p1.1.m1.1.26.2.1.2.1.2" xref="p1.1.m1.1.26.2.1.2.1.2.cmml">(</mo> <mi id="p1.1.m1.1.22" xref="p1.1.m1.1.22.cmml">x</mi> <mo id="p1.1.m1.1.26.2.1.2.1.2a" xref="p1.1.m1.1.26.2.1.2.1.2.cmml">,</mo> <mi id="p1.1.m1.1.24" xref="p1.1.m1.1.24.cmml">y</mi> <mo stretchy="false" id="p1.1.m1.1.26.2.1.2.1.2b" xref="p1.1.m1.1.26.2.1.2.1.2.cmml">)</mo> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding="MathML-Content" id="p1.1.m1.1b"> <apply id="p1.1.m1.1.26.cmml" xref="p1.1.m1.1.26"> <ci id="p1.1.m1.1.13.cmml" xref="p1.1.m1.1.13">normal-⇔</ci> <apply id="p1.1.m1.1.26.1.cmml" xref="p1.1.m1.1.26.1"> <csymbol cd="latexml" id="p1.1.m1.1.1.cmml" xref="p1.1.m1.1.1">for-all</csymbol> <apply id="p1.1.m1.1.26.1.1.cmml" xref="p1.1.m1.1.26.1.1"> <times id="p1.1.m1.1.26.1.1.1.cmml" xref="p1.1.m1.1.26.1.1.1"/> <ci id="p1.1.m1.1.2.cmml" xref="p1.1.m1.1.2">x</ci> <apply id="p1.1.m1.1.26.1.1.2.cmml" xref="p1.1.m1.1.26.1.1.2"> <csymbol cd="latexml" id="p1.1.m1.1.4.cmml" xref="p1.1.m1.1.4">for-all</csymbol> <apply id="p1.1.m1.1.26.1.1.2.1.cmml" xref="p1.1.m1.1.26.1.1.2.1"> <times id="p1.1.m1.1.26.1.1.2.1.1.cmml" xref="p1.1.m1.1.26.1.1.2.1.1"/> <ci id="p1.1.m1.1.5.cmml" xref="p1.1.m1.1.5">y</ci> <ci id="p1.1.m1.1.7.cmml" xref="p1.1.m1.1.7">P</ci> <interval closure="open" id="p1.1.m1.1.26.1.1.2.1.2.cmml" xref="p1.1.m1.1.26.1.1.2.1.2"> <ci id="p1.1.m1.1.9.cmml" xref="p1.1.m1.1.9">x</ci> <ci id="p1.1.m1.1.11.cmml" xref="p1.1.m1.1.11">y</ci> </interval> </apply> </apply> </apply> </apply> <apply id="p1.1.m1.1.26.2.cmml" xref="p1.1.m1.1.26.2"> <csymbol cd="latexml" id="p1.1.m1.1.14.cmml" xref="p1.1.m1.1.14">for-all</csymbol> <apply id="p1.1.m1.1.26.2.1.cmml" xref="p1.1.m1.1.26.2.1"> <times id="p1.1.m1.1.26.2.1.1.cmml" xref="p1.1.m1.1.26.2.1.1"/> <ci id="p1.1.m1.1.15.cmml" xref="p1.1.m1.1.15">y</ci> <apply id="p1.1.m1.1.26.2.1.2.cmml" xref="p1.1.m1.1.26.2.1.2"> <csymbol cd="latexml" id="p1.1.m1.1.17.cmml" xref="p1.1.m1.1.17">for-all</csymbol> <apply id="p1.1.m1.1.26.2.1.2.1.cmml" xref="p1.1.m1.1.26.2.1.2.1"> <times id="p1.1.m1.1.26.2.1.2.1.1.cmml" xref="p1.1.m1.1.26.2.1.2.1.1"/> <ci id="p1.1.m1.1.18.cmml" xref="p1.1.m1.1.18">x</ci> <ci id="p1.1.m1.1.20.cmml" xref="p1.1.m1.1.20">P</ci> <interval closure="open" id="p1.1.m1.1.26.2.1.2.1.2.cmml" xref="p1.1.m1.1.26.2.1.2.1.2"> <ci id="p1.1.m1.1.22.cmml" xref="p1.1.m1.1.22">x</ci> <ci id="p1.1.m1.1.24.cmml" xref="p1.1.m1.1.24">y</ci> </interval> </apply> </apply> </apply> </apply> </apply> </annotation-xml> <annotation encoding="application/x-tex" id="p1.1.m1.1c">{\displaystyle\forall x\,\forall y\,P(x,y)\Leftrightarrow\forall y\,\forall x% \,P(x,y)}</annotation> </semantics> </math></span><img src="https://en.formulasearchengine.com/w/index.php?title=Special:MathShowImage&hash=0853e868640db6db59c7bf82483ae01d&mode=latexml" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.667ex; height: 2.5ex; width: 30.5ex;" alt="\forall x\,\forall y\,P(x,y)\Leftrightarrow \forall y\,\forall x\,P(x,y)" data-pagespeed-url-hash="3793527728" onload="pagespeed.CriticalImages.checkImageForCriticality(this);"/></span> </td> <td> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style=""><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall x\,\forall y\,P(x,y)\Leftrightarrow \forall y\,\forall x\,P(x,y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mspace width="thinmathspace"/> <mi mathvariant="normal">∀</mi> <mi>y</mi> <mspace width="thinmathspace"/> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo stretchy="false">⇔</mo> <mi mathvariant="normal">∀</mi> <mi>y</mi> <mspace width="thinmathspace"/> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mspace width="thinmathspace"/> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall x\,\forall y\,P(x,y)\Leftrightarrow \forall y\,\forall x\,P(x,y)}</annotation> </semantics> </math></span><img src="https://api.formulasearchengine.com/v1/media/math/render/svg/b2bd10da835e07d3f2df8a14ea462278e5a23409" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:29.451ex; height:2.843ex;" alt="{\displaystyle \forall x\,\forall y\,P(x,y)\Leftrightarrow \forall y\,\forall x\,P(x,y)}" data-pagespeed-url-hash="4075345840" onload="pagespeed.CriticalImages.checkImageForCriticality(this);"/></span> </td></tr> <tr> <td> <a href="https://en.formulasearchengine.com/wiki/Special:DisplayTopics/6" title="Special:DisplayTopics/6">6</a> </td> <td> <a href="https://en.formulasearchengine.com/wiki/Special:PermanentLink/6937#math.6937.5" title="Special:PermanentLink/6937">Clenshaw_algorithm-5</a> </td> <td> 0 </td> <td> 8 </td> <td> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style=""><math xmlns="http://www.w3.org/1998/Math/MathML"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha (x)}</annotation> </semantics> </math></span><img src="https://en.formulasearchengine.com/w/index.php?title=Special:MathShowImage&hash=000ad1eb8a2c2182ff048350cc9eb0e8&mode=latexml" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; height: 2.843ex; width: 4.668ex;" alt="{\displaystyle \alpha(x)}" data-pagespeed-url-hash="2106747584" onload="pagespeed.CriticalImages.checkImageForCriticality(this);"/></span> </td> <td> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style=""><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha (x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha (x)}</annotation> </semantics> </math></span><img src="https://api.formulasearchengine.com/v1/media/math/render/svg/43941ef40b4eb751a8daea118d0012ea998f5610" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:4.627ex; height:2.843ex;" alt="{\displaystyle \alpha (x)}" data-pagespeed-url-hash="3005649952" onload="pagespeed.CriticalImages.checkImageForCriticality(this);"/></span> </td></tr> <tr> <td> <a href="https://en.formulasearchengine.com/wiki/Special:DisplayTopics/7" title="Special:DisplayTopics/7">7</a> </td> <td> <a href="https://en.formulasearchengine.com/wiki/Special:PermanentLink/25610#math.25610.24" title="Special:PermanentLink/25610">Isolation_lemma-24</a> </td> <td> 0 </td> <td> 8 </td> <td> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style=""><math xmlns="http://www.w3.org/1998/Math/MathML"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha (x)}</annotation> </semantics> </math></span><img src="https://en.formulasearchengine.com/w/index.php?title=Special:MathShowImage&hash=000ad1eb8a2c2182ff048350cc9eb0e8&mode=latexml" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; height: 2.843ex; width: 4.668ex;" alt="{\displaystyle \alpha(x)}" data-pagespeed-url-hash="2106747584" onload="pagespeed.CriticalImages.checkImageForCriticality(this);"/></span> </td> <td> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style=""><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha (x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha (x)}</annotation> </semantics> </math></span><img src="https://api.formulasearchengine.com/v1/media/math/render/svg/43941ef40b4eb751a8daea118d0012ea998f5610" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:4.627ex; height:2.843ex;" alt="{\displaystyle \alpha (x)}" data-pagespeed-url-hash="3005649952" onload="pagespeed.CriticalImages.checkImageForCriticality(this);"/></span> </td></tr> <tr> <td> <a href="https://en.formulasearchengine.com/wiki/Special:DisplayTopics/8" title="Special:DisplayTopics/8">8</a> </td> <td> <a href="https://en.formulasearchengine.com/wiki/Special:PermanentLink/27419#math.27419.4" title="Special:PermanentLink/27419">Singularity_spectrum-4</a> </td> <td> 0 </td> <td> 8 </td> <td> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style=""><math xmlns="http://www.w3.org/1998/Math/MathML"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha (x)}</annotation> </semantics> </math></span><img src="https://en.formulasearchengine.com/w/index.php?title=Special:MathShowImage&hash=000ad1eb8a2c2182ff048350cc9eb0e8&mode=latexml" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; height: 2.843ex; width: 4.668ex;" alt="{\displaystyle \alpha(x)}" data-pagespeed-url-hash="2106747584" onload="pagespeed.CriticalImages.checkImageForCriticality(this);"/></span> </td> <td> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style=""><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha (x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha (x)}</annotation> </semantics> </math></span><img src="https://api.formulasearchengine.com/v1/media/math/render/svg/43941ef40b4eb751a8daea118d0012ea998f5610" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:4.627ex; height:2.843ex;" alt="{\displaystyle \alpha (x)}" data-pagespeed-url-hash="3005649952" onload="pagespeed.CriticalImages.checkImageForCriticality(this);"/></span> </td></tr> <tr> <td> <a href="https://en.formulasearchengine.com/wiki/Special:DisplayTopics/9" title="Special:DisplayTopics/9">9</a> </td> <td> <a href="https://en.formulasearchengine.com/wiki/Special:PermanentLink/14488#math.14488.36" title="Special:PermanentLink/14488">Time-evolving_block_decimation-36</a> </td> <td> 7 </td> <td> 1 </td> <td> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style=""><math xmlns="http://www.w3.org/1998/Math/MathML" id="p1.1.m1.1" class="ltx_Math" alttext="{\displaystyle|{\Psi}\rangle=\sum_{\qvar@construct{x0}}\Gamma_{{%
\qvar@construct{x1}}}^{{[1]i_{1}}}\lambda_{{\alpha_{1}}}^{{[1]}}\Gamma_{{%
\alpha_{\qvar@construct{x2}}\alpha_{2}}}^{{[2]i_{\qvar@construct{x3}}}}\lambda%
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