MathML formulae – Schlage-Puchta, Jan-Christoph: Finiteness of a class of Rabinowitsch polynomials. (English). Archivum Mathematicum, vol. 40 (2004), issue 3, pp. 259-261

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Formula 1

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Original

m m m m

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="m1" class="ltx_Math" alttext="m" display="inline">
  <semantics>
    <mi>m</mi>
    <annotation-xml encoding="MathML-Content">
      <ci>m</ci>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMTok role="UNKNOWN" font="italic">m</XMTok>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">m</annotation>
  </semantics>
</math>

Canonicalized

m m m m

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics encoding="">
    <mi>m</mi>
    <annotation-xml encoding="MathML-Content">
      <ci encoding="">m</ci>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMTok>m</XMTok>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">m</annotation>
  </semantics>
</math>

Diff

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@@ -1,15 +1,15 @@
 <?xml version="1.0" encoding="UTF-8"?>
-<math xmlns="http://www.w3.org/1998/Math/MathML" id="m1" class="ltx_Math" alttext="m" display="inline">
-  <semantics>
+<math xmlns="http://www.w3.org/1998/Math/MathML">
+  <semantics encoding="">
     <mi>m</mi>
     <annotation-xml encoding="MathML-Content">
-      <ci>m</ci>
+      <ci encoding="">m</ci>
     </annotation-xml>
     <annotation-xml encoding="application/x-latexml">
       <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
-        <XMTok role="UNKNOWN" font="italic">m</XMTok>
+        <XMTok>m</XMTok>
       </XMath>
     </annotation-xml>
     <annotation encoding="application/x-tex">m</annotation>
   </semantics>
 </math>

Formula 2

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Original

t t t t

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="m2" class="ltx_Math" alttext="t" display="inline">
  <semantics>
    <mi>t</mi>
    <annotation-xml encoding="MathML-Content">
      <ci>t</ci>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMTok role="UNKNOWN" font="italic">t</XMTok>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">t</annotation>
  </semantics>
</math>

Canonicalized

t t t t

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics encoding="">
    <mi>t</mi>
    <annotation-xml encoding="MathML-Content">
      <ci encoding="">t</ci>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMTok>t</XMTok>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">t</annotation>
  </semantics>
</math>

Diff

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@@ -1,15 +1,15 @@
 <?xml version="1.0" encoding="UTF-8"?>
-<math xmlns="http://www.w3.org/1998/Math/MathML" id="m2" class="ltx_Math" alttext="t" display="inline">
-  <semantics>
+<math xmlns="http://www.w3.org/1998/Math/MathML">
+  <semantics encoding="">
     <mi>t</mi>
     <annotation-xml encoding="MathML-Content">
-      <ci>t</ci>
+      <ci encoding="">t</ci>
     </annotation-xml>
     <annotation-xml encoding="application/x-latexml">
       <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
-        <XMTok role="UNKNOWN" font="italic">t</XMTok>
+        <XMTok>t</XMTok>
       </XMath>
     </annotation-xml>
     <annotation encoding="application/x-tex">t</annotation>
   </semantics>
 </math>

Formula 3

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Original

| n 2 + n - m | superscript n 2 n m - + n 2 n m |n^{2}+n-m|

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="m3" class="ltx_Math" alttext="|n^{2}+n-m|" display="inline">
  <semantics>
    <mrow>
      <mo fence="true">|</mo>
      <mrow>
        <msup>
          <mi>n</mi>
          <mn>2</mn>
        </msup>
        <mo>+</mo>
        <mi>n</mi>
        <mo>-</mo>
        <mi>m</mi>
      </mrow>
      <mo fence="true">|</mo>
    </mrow>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <abs/>
        <apply>
          <minus/>
          <apply>
            <plus/>
            <apply>
              <csymbol cd="ambiguous">superscript</csymbol>
              <ci>n</ci>
              <cn type="integer">2</cn>
            </apply>
            <ci>n</ci>
          </apply>
          <ci>m</ci>
        </apply>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok argclose="|" argopen="|" meaning="absolute-value" role="FENCED"/>
          <XMApp>
            <XMTok meaning="minus" role="ADDOP">-</XMTok>
            <XMApp>
              <XMTok meaning="plus" role="ADDOP">+</XMTok>
              <XMApp>
                <XMTok role="SUPERSCRIPTOP" scriptpos="post3"/>
                <XMTok role="UNKNOWN" font="italic">n</XMTok>
                <XMTok meaning="2" role="NUMBER">2</XMTok>
              </XMApp>
              <XMTok role="UNKNOWN" font="italic">n</XMTok>
            </XMApp>
            <XMTok role="UNKNOWN" font="italic">m</XMTok>
          </XMApp>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">|n^{2}+n-m|</annotation>
  </semantics>
</math>

Canonicalized

| n 2 + n - m | superscript n 2 n m - + n 2 n m |n^{2}+n-m|

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics encoding="">
    <mrow>
      <mo>|</mo>
      <mrow>
        <msup>
          <mi>n</mi>
          <mn>2</mn>
        </msup>
        <mo>+</mo>
        <mi>n</mi>
        <mo>-</mo>
        <mi>m</mi>
      </mrow>
      <mo>|</mo>
    </mrow>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <abs/>
        <apply>
          <minus/>
          <apply>
            <plus/>
            <apply>
              <csymbol encoding="">superscript</csymbol>
              <ci encoding="">n</ci>
              <cn encoding="" base="10" type="integer">2</cn>
            </apply>
            <ci encoding="">n</ci>
          </apply>
          <ci encoding="">m</ci>
        </apply>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok/>
          <XMApp>
            <XMTok>-</XMTok>
            <XMApp>
              <XMTok>+</XMTok>
              <XMApp>
                <XMTok/>
                <XMTok>n</XMTok>
                <XMTok>2</XMTok>
              </XMApp>
              <XMTok>n</XMTok>
            </XMApp>
            <XMTok>m</XMTok>
          </XMApp>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">|n^{2}+n-m|</annotation>
  </semantics>
</math>

Diff

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@@ -1,58 +1,58 @@
 <?xml version="1.0" encoding="UTF-8"?>
-<math xmlns="http://www.w3.org/1998/Math/MathML" id="m3" class="ltx_Math" alttext="|n^{2}+n-m|" display="inline">
-  <semantics>
+<math xmlns="http://www.w3.org/1998/Math/MathML">
+  <semantics encoding="">
     <mrow>
-      <mo fence="true">|</mo>
+      <mo>|</mo>
       <mrow>
         <msup>
           <mi>n</mi>
           <mn>2</mn>
         </msup>
         <mo>+</mo>
         <mi>n</mi>
         <mo>-</mo>
         <mi>m</mi>
       </mrow>
-      <mo fence="true">|</mo>
+      <mo>|</mo>
     </mrow>
     <annotation-xml encoding="MathML-Content">
       <apply>
         <abs/>
         <apply>
           <minus/>
           <apply>
             <plus/>
             <apply>
-              <csymbol cd="ambiguous">superscript</csymbol>
-              <ci>n</ci>
-              <cn type="integer">2</cn>
+              <csymbol encoding="">superscript</csymbol>
+              <ci encoding="">n</ci>
+              <cn encoding="" base="10" type="integer">2</cn>
             </apply>
-            <ci>n</ci>
+            <ci encoding="">n</ci>
           </apply>
-          <ci>m</ci>
+          <ci encoding="">m</ci>
         </apply>
       </apply>
     </annotation-xml>
     <annotation-xml encoding="application/x-latexml">
       <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
         <XMApp>
-          <XMTok argclose="|" argopen="|" meaning="absolute-value" role="FENCED"/>
+          <XMTok/>
           <XMApp>
-            <XMTok meaning="minus" role="ADDOP">-</XMTok>
+            <XMTok>-</XMTok>
             <XMApp>
-              <XMTok meaning="plus" role="ADDOP">+</XMTok>
+              <XMTok>+</XMTok>
               <XMApp>
-                <XMTok role="SUPERSCRIPTOP" scriptpos="post3"/>
-                <XMTok role="UNKNOWN" font="italic">n</XMTok>
-                <XMTok meaning="2" role="NUMBER">2</XMTok>
+                <XMTok/>
+                <XMTok>n</XMTok>
+                <XMTok>2</XMTok>
               </XMApp>
-              <XMTok role="UNKNOWN" font="italic">n</XMTok>
+              <XMTok>n</XMTok>
             </XMApp>
-            <XMTok role="UNKNOWN" font="italic">m</XMTok>
+            <XMTok>m</XMTok>
           </XMApp>
         </XMApp>
       </XMath>
     </annotation-xml>
     <annotation encoding="application/x-tex">|n^{2}+n-m|</annotation>
   </semantics>
 </math>

Formula 4

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Original

n ∈ [ t + 1 , t + m ] n t 1 t m ∈ n + t 1 + t m n\in[t+1,t+\sqrt{m}]

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="m4" class="ltx_Math" alttext="n\in[t+1,t+\sqrt{m}]" display="inline">
  <semantics>
    <mrow>
      <mi>n</mi>
      <mo>∈</mo>
      <mrow>
        <mo>[</mo>
        <mrow>
          <mrow>
            <mi>t</mi>
            <mo>+</mo>
            <mn>1</mn>
          </mrow>
          <mo>,</mo>
          <mrow>
            <mi>t</mi>
            <mo>+</mo>
            <msqrt>
              <mi>m</mi>
            </msqrt>
          </mrow>
        </mrow>
        <mo>]</mo>
      </mrow>
    </mrow>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <in/>
        <ci>n</ci>
        <apply>
          <interval closure="closed"/>
          <apply>
            <plus/>
            <ci>t</ci>
            <cn type="integer">1</cn>
          </apply>
          <apply>
            <plus/>
            <ci>t</ci>
            <apply>
              <root/>
              <ci>m</ci>
            </apply>
          </apply>
        </apply>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok meaning="element-of" name="in" role="RELOP">∈</XMTok>
          <XMTok role="UNKNOWN" font="italic">n</XMTok>
          <XMApp>
            <XMTok argclose="]" argopen="[" meaning="closed-interval" role="FENCED" separators=","/>
            <XMApp>
              <XMTok meaning="plus" role="ADDOP">+</XMTok>
              <XMTok role="UNKNOWN" font="italic">t</XMTok>
              <XMTok meaning="1" role="NUMBER">1</XMTok>
            </XMApp>
            <XMApp>
              <XMTok meaning="plus" role="ADDOP">+</XMTok>
              <XMTok role="UNKNOWN" font="italic">t</XMTok>
              <XMApp>
                <XMTok meaning="square-root"/>
                <XMTok role="UNKNOWN" font="italic">m</XMTok>
              </XMApp>
            </XMApp>
          </XMApp>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">n\in[t+1,t+\sqrt{m}]</annotation>
  </semantics>
</math>

Canonicalized

n ∈ [ t + 1 , t + m ] n t 1 t m ∈ n + t 1 + t m n\in[t+1,t+\sqrt{m}]

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics encoding="">
    <mrow>
      <mi>n</mi>
      <mo>∈</mo>
      <mrow>
        <mo>[</mo>
        <mrow>
          <mrow>
            <mi>t</mi>
            <mo>+</mo>
            <mn>1</mn>
          </mrow>
          <mo>,</mo>
          <mrow>
            <mi>t</mi>
            <mo>+</mo>
            <msqrt>
              <mi>m</mi>
            </msqrt>
          </mrow>
        </mrow>
        <mo>]</mo>
      </mrow>
    </mrow>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <in/>
        <ci encoding="">n</ci>
        <apply>
          <interval closure="closed"/>
          <apply>
            <plus/>
            <ci encoding="">t</ci>
            <cn encoding="" base="10" type="integer">1</cn>
          </apply>
          <apply>
            <plus/>
            <ci encoding="">t</ci>
            <apply>
              <root/>
              <ci encoding="">m</ci>
            </apply>
          </apply>
        </apply>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok>∈</XMTok>
          <XMTok>n</XMTok>
          <XMApp>
            <XMTok/>
            <XMApp>
              <XMTok>+</XMTok>
              <XMTok>t</XMTok>
              <XMTok>1</XMTok>
            </XMApp>
            <XMApp>
              <XMTok>+</XMTok>
              <XMTok>t</XMTok>
              <XMApp>
                <XMTok/>
                <XMTok>m</XMTok>
              </XMApp>
            </XMApp>
          </XMApp>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">n\in[t+1,t+\sqrt{m}]</annotation>
  </semantics>
</math>

Diff

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@@ -1,75 +1,75 @@
 <?xml version="1.0" encoding="UTF-8"?>
-<math xmlns="http://www.w3.org/1998/Math/MathML" id="m4" class="ltx_Math" alttext="n\in[t+1,t+\sqrt{m}]" display="inline">
-  <semantics>
+<math xmlns="http://www.w3.org/1998/Math/MathML">
+  <semantics encoding="">
     <mrow>
       <mi>n</mi>
       <mo>∈</mo>
       <mrow>
         <mo>[</mo>
         <mrow>
           <mrow>
             <mi>t</mi>
             <mo>+</mo>
             <mn>1</mn>
           </mrow>
           <mo>,</mo>
           <mrow>
             <mi>t</mi>
             <mo>+</mo>
             <msqrt>
               <mi>m</mi>
             </msqrt>
           </mrow>
         </mrow>
         <mo>]</mo>
       </mrow>
     </mrow>
     <annotation-xml encoding="MathML-Content">
       <apply>
         <in/>
-        <ci>n</ci>
+        <ci encoding="">n</ci>
         <apply>
           <interval closure="closed"/>
           <apply>
             <plus/>
-            <ci>t</ci>
-            <cn type="integer">1</cn>
+            <ci encoding="">t</ci>
+            <cn encoding="" base="10" type="integer">1</cn>
           </apply>
           <apply>
             <plus/>
-            <ci>t</ci>
+            <ci encoding="">t</ci>
             <apply>
               <root/>
-              <ci>m</ci>
+              <ci encoding="">m</ci>
             </apply>
           </apply>
         </apply>
       </apply>
     </annotation-xml>
     <annotation-xml encoding="application/x-latexml">
       <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
         <XMApp>
-          <XMTok meaning="element-of" name="in" role="RELOP">∈</XMTok>
-          <XMTok role="UNKNOWN" font="italic">n</XMTok>
+          <XMTok>∈</XMTok>
+          <XMTok>n</XMTok>
           <XMApp>
-            <XMTok argclose="]" argopen="[" meaning="closed-interval" role="FENCED" separators=","/>
+            <XMTok/>
             <XMApp>
-              <XMTok meaning="plus" role="ADDOP">+</XMTok>
-              <XMTok role="UNKNOWN" font="italic">t</XMTok>
-              <XMTok meaning="1" role="NUMBER">1</XMTok>
+              <XMTok>+</XMTok>
+              <XMTok>t</XMTok>
+              <XMTok>1</XMTok>
             </XMApp>
             <XMApp>
-              <XMTok meaning="plus" role="ADDOP">+</XMTok>
-              <XMTok role="UNKNOWN" font="italic">t</XMTok>
+              <XMTok>+</XMTok>
+              <XMTok>t</XMTok>
               <XMApp>
-                <XMTok meaning="square-root"/>
-                <XMTok role="UNKNOWN" font="italic">m</XMTok>
+                <XMTok/>
+                <XMTok>m</XMTok>
               </XMApp>
             </XMApp>
           </XMApp>
         </XMApp>
       </XMath>
     </annotation-xml>
     <annotation encoding="application/x-tex">n\in[t+1,t+\sqrt{m}]</annotation>
   </semantics>
 </math>

Formula 5

( show / hide formula code)

Original

m m m m

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="p2.m1" class="ltx_Math" alttext="m" display="inline">
  <semantics>
    <mi>m</mi>
    <annotation-xml encoding="MathML-Content">
      <ci>m</ci>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMTok role="UNKNOWN" font="italic">m</XMTok>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">m</annotation>
  </semantics>
</math>

Canonicalized

m m m m

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics encoding="">
    <mi>m</mi>
    <annotation-xml encoding="MathML-Content">
      <ci encoding="">m</ci>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMTok>m</XMTok>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">m</annotation>
  </semantics>
</math>

Diff

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@@ -1,15 +1,15 @@
 <?xml version="1.0" encoding="UTF-8"?>
-<math xmlns="http://www.w3.org/1998/Math/MathML" id="p2.m1" class="ltx_Math" alttext="m" display="inline">
-  <semantics>
+<math xmlns="http://www.w3.org/1998/Math/MathML">
+  <semantics encoding="">
     <mi>m</mi>
     <annotation-xml encoding="MathML-Content">
-      <ci>m</ci>
+      <ci encoding="">m</ci>
     </annotation-xml>
     <annotation-xml encoding="application/x-latexml">
       <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
-        <XMTok role="UNKNOWN" font="italic">m</XMTok>
+        <XMTok>m</XMTok>
       </XMath>
     </annotation-xml>
     <annotation encoding="application/x-tex">m</annotation>
   </semantics>
 </math>

Formula 6

( show / hide formula code)

Original

4 ⁢ m - 1 4 m 1 - ⁢ 4 m 1 4m-1

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="p2.m2" class="ltx_Math" alttext="4m-1" display="inline">
  <semantics>
    <mrow>
      <mrow>
        <mn>4</mn>
        <mo>⁢</mo>
        <mi>m</mi>
      </mrow>
      <mo>-</mo>
      <mn>1</mn>
    </mrow>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <minus/>
        <apply>
          <times/>
          <cn type="integer">4</cn>
          <ci>m</ci>
        </apply>
        <cn type="integer">1</cn>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok meaning="minus" role="ADDOP">-</XMTok>
          <XMApp>
            <XMTok meaning="times" role="MULOP">⁢</XMTok>
            <XMTok meaning="4" role="NUMBER">4</XMTok>
            <XMTok role="UNKNOWN" font="italic">m</XMTok>
          </XMApp>
          <XMTok meaning="1" role="NUMBER">1</XMTok>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">4m-1</annotation>
  </semantics>
</math>

Canonicalized

4 m - 1 4 m 1 - ⁢ 4 m 1 4m-1

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics encoding="">
    <mrow>
      <mrow>
        <mn>4</mn>
        <mi>m</mi>
      </mrow>
      <mo>-</mo>
      <mn>1</mn>
    </mrow>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <minus/>
        <apply>
          <times/>
          <cn encoding="" base="10" type="integer">4</cn>
          <ci encoding="">m</ci>
        </apply>
        <cn encoding="" base="10" type="integer">1</cn>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok>-</XMTok>
          <XMApp>
            <XMTok>⁢</XMTok>
            <XMTok>4</XMTok>
            <XMTok>m</XMTok>
          </XMApp>
          <XMTok>1</XMTok>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">4m-1</annotation>
  </semantics>
</math>

Diff

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@@ -1,39 +1,38 @@
 <?xml version="1.0" encoding="UTF-8"?>
-<math xmlns="http://www.w3.org/1998/Math/MathML" id="p2.m2" class="ltx_Math" alttext="4m-1" display="inline">
-  <semantics>
+<math xmlns="http://www.w3.org/1998/Math/MathML">
+  <semantics encoding="">
     <mrow>
       <mrow>
         <mn>4</mn>
-        <mo>⁢</mo>
         <mi>m</mi>
       </mrow>
       <mo>-</mo>
       <mn>1</mn>
     </mrow>
     <annotation-xml encoding="MathML-Content">
       <apply>
         <minus/>
         <apply>
           <times/>
-          <cn type="integer">4</cn>
-          <ci>m</ci>
+          <cn encoding="" base="10" type="integer">4</cn>
+          <ci encoding="">m</ci>
         </apply>
-        <cn type="integer">1</cn>
+        <cn encoding="" base="10" type="integer">1</cn>
       </apply>
     </annotation-xml>
     <annotation-xml encoding="application/x-latexml">
       <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
         <XMApp>
-          <XMTok meaning="minus" role="ADDOP">-</XMTok>
+          <XMTok>-</XMTok>
           <XMApp>
-            <XMTok meaning="times" role="MULOP">⁢</XMTok>
-            <XMTok meaning="4" role="NUMBER">4</XMTok>
-            <XMTok role="UNKNOWN" font="italic">m</XMTok>
+            <XMTok>⁢</XMTok>
+            <XMTok>4</XMTok>
+            <XMTok>m</XMTok>
           </XMApp>
-          <XMTok meaning="1" role="NUMBER">1</XMTok>
+          <XMTok>1</XMTok>
         </XMApp>
       </XMath>
     </annotation-xml>
     <annotation encoding="application/x-tex">4m-1</annotation>
   </semantics>
 </math>

Formula 7

( show / hide formula code)

Original

ℚ ⁢ ( 1 - 4 ⁢ m ) ℚ 1 4 m ⁢ Q - 1 ⁢ 4 m \mathbb{Q}(\sqrt{1-4m})

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="p2.m3" class="ltx_Math" alttext="\mathbb{Q}(\sqrt{1-4m})" display="inline">
  <semantics>
    <mrow>
      <mi>ℚ</mi>
      <mo>⁢</mo>
      <mrow>
        <mo>(</mo>
        <msqrt>
          <mrow>
            <mn>1</mn>
            <mo>-</mo>
            <mrow>
              <mn>4</mn>
              <mo>⁢</mo>
              <mi>m</mi>
            </mrow>
          </mrow>
        </msqrt>
        <mo>)</mo>
      </mrow>
    </mrow>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <times/>
        <ci>ℚ</ci>
        <apply>
          <root/>
          <apply>
            <minus/>
            <cn type="integer">1</cn>
            <apply>
              <times/>
              <cn type="integer">4</cn>
              <ci>m</ci>
            </apply>
          </apply>
        </apply>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok meaning="times" role="MULOP">⁢</XMTok>
          <XMTok possibleFunction="yes" role="UNKNOWN" font="blackboard">Q</XMTok>
          <XMApp close=")" open="(">
            <XMTok meaning="square-root"/>
            <XMApp>
              <XMTok meaning="minus" role="ADDOP">-</XMTok>
              <XMTok meaning="1" role="NUMBER">1</XMTok>
              <XMApp>
                <XMTok meaning="times" role="MULOP">⁢</XMTok>
                <XMTok meaning="4" role="NUMBER">4</XMTok>
                <XMTok role="UNKNOWN" font="italic">m</XMTok>
              </XMApp>
            </XMApp>
          </XMApp>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">\mathbb{Q}(\sqrt{1-4m})</annotation>
  </semantics>
</math>

Canonicalized

Q ( 1 - 4 m ) Q 1 4 m ⁢ Q - 1 ⁢ 4 m \mathbb{Q}(\sqrt{1-4m})

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics encoding="">
    <mrow>
      <mi>Q</mi>
      <mrow>
        <mo>(</mo>
        <msqrt>
          <mn>1</mn>
          <mo>-</mo>
          <mrow>
            <mn>4</mn>
            <mi>m</mi>
          </mrow>
        </msqrt>
        <mo>)</mo>
      </mrow>
    </mrow>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <times/>
        <ci encoding="">Q</ci>
        <apply>
          <root/>
          <apply>
            <minus/>
            <cn encoding="" base="10" type="integer">1</cn>
            <apply>
              <times/>
              <cn encoding="" base="10" type="integer">4</cn>
              <ci encoding="">m</ci>
            </apply>
          </apply>
        </apply>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok>⁢</XMTok>
          <XMTok>Q</XMTok>
          <XMApp>
            <XMTok/>
            <XMApp>
              <XMTok>-</XMTok>
              <XMTok>1</XMTok>
              <XMApp>
                <XMTok>⁢</XMTok>
                <XMTok>4</XMTok>
                <XMTok>m</XMTok>
              </XMApp>
            </XMApp>
          </XMApp>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">\mathbb{Q}(\sqrt{1-4m})</annotation>
  </semantics>
</math>

Diff

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@@ -1,63 +1,59 @@
 <?xml version="1.0" encoding="UTF-8"?>
-<math xmlns="http://www.w3.org/1998/Math/MathML" id="p2.m3" class="ltx_Math" alttext="\mathbb{Q}(\sqrt{1-4m})" display="inline">
-  <semantics>
+<math xmlns="http://www.w3.org/1998/Math/MathML">
+  <semantics encoding="">
     <mrow>
-      <mi>ℚ</mi>
-      <mo>⁢</mo>
+      <mi>Q</mi>
       <mrow>
         <mo>(</mo>
         <msqrt>
+          <mn>1</mn>
+          <mo>-</mo>
           <mrow>
-            <mn>1</mn>
-            <mo>-</mo>
-            <mrow>
-              <mn>4</mn>
-              <mo>⁢</mo>
-              <mi>m</mi>
-            </mrow>
+            <mn>4</mn>
+            <mi>m</mi>
           </mrow>
         </msqrt>
         <mo>)</mo>
       </mrow>
     </mrow>
     <annotation-xml encoding="MathML-Content">
       <apply>
         <times/>
-        <ci>ℚ</ci>
+        <ci encoding="">Q</ci>
         <apply>
           <root/>
           <apply>
             <minus/>
-            <cn type="integer">1</cn>
+            <cn encoding="" base="10" type="integer">1</cn>
             <apply>
               <times/>
-              <cn type="integer">4</cn>
-              <ci>m</ci>
+              <cn encoding="" base="10" type="integer">4</cn>
+              <ci encoding="">m</ci>
             </apply>
           </apply>
         </apply>
       </apply>
     </annotation-xml>
     <annotation-xml encoding="application/x-latexml">
       <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
         <XMApp>
-          <XMTok meaning="times" role="MULOP">⁢</XMTok>
-          <XMTok possibleFunction="yes" role="UNKNOWN" font="blackboard">Q</XMTok>
-          <XMApp close=")" open="(">
-            <XMTok meaning="square-root"/>
+          <XMTok>⁢</XMTok>
+          <XMTok>Q</XMTok>
+          <XMApp>
+            <XMTok/>
             <XMApp>
-              <XMTok meaning="minus" role="ADDOP">-</XMTok>
-              <XMTok meaning="1" role="NUMBER">1</XMTok>
+              <XMTok>-</XMTok>
+              <XMTok>1</XMTok>
               <XMApp>
-                <XMTok meaning="times" role="MULOP">⁢</XMTok>
-                <XMTok meaning="4" role="NUMBER">4</XMTok>
-                <XMTok role="UNKNOWN" font="italic">m</XMTok>
+                <XMTok>⁢</XMTok>
+                <XMTok>4</XMTok>
+                <XMTok>m</XMTok>
               </XMApp>
             </XMApp>
           </XMApp>
         </XMApp>
       </XMath>
     </annotation-xml>
     <annotation encoding="application/x-tex">\mathbb{Q}(\sqrt{1-4m})</annotation>
   </semantics>
 </math>

Formula 8

( show / hide formula code)

Original

n 2 + n + m superscript n 2 n m + n 2 n m n^{2}+n+m

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="p2.m4" class="ltx_Math" alttext="n^{2}+n+m" display="inline">
  <semantics>
    <mrow>
      <msup>
        <mi>n</mi>
        <mn>2</mn>
      </msup>
      <mo>+</mo>
      <mi>n</mi>
      <mo>+</mo>
      <mi>m</mi>
    </mrow>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <plus/>
        <apply>
          <csymbol cd="ambiguous">superscript</csymbol>
          <ci>n</ci>
          <cn type="integer">2</cn>
        </apply>
        <ci>n</ci>
        <ci>m</ci>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok meaning="plus" role="ADDOP">+</XMTok>
          <XMApp>
            <XMTok role="SUPERSCRIPTOP" scriptpos="post2"/>
            <XMTok role="UNKNOWN" font="italic">n</XMTok>
            <XMTok meaning="2" role="NUMBER">2</XMTok>
          </XMApp>
          <XMTok role="UNKNOWN" font="italic">n</XMTok>
          <XMTok role="UNKNOWN" font="italic">m</XMTok>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">n^{2}+n+m</annotation>
  </semantics>
</math>

Canonicalized

n 2 + n + m superscript n 2 n m + n 2 n m n^{2}+n+m

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics encoding="">
    <mrow>
      <msup>
        <mi>n</mi>
        <mn>2</mn>
      </msup>
      <mo>+</mo>
      <mi>n</mi>
      <mo>+</mo>
      <mi>m</mi>
    </mrow>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <plus/>
        <apply>
          <csymbol encoding="">superscript</csymbol>
          <ci encoding="">n</ci>
          <cn encoding="" base="10" type="integer">2</cn>
        </apply>
        <ci encoding="">n</ci>
        <ci encoding="">m</ci>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok>+</XMTok>
          <XMApp>
            <XMTok/>
            <XMTok>n</XMTok>
            <XMTok>2</XMTok>
          </XMApp>
          <XMTok>n</XMTok>
          <XMTok>m</XMTok>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">n^{2}+n+m</annotation>
  </semantics>
</math>

Diff

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@@ -1,42 +1,42 @@
 <?xml version="1.0" encoding="UTF-8"?>
-<math xmlns="http://www.w3.org/1998/Math/MathML" id="p2.m4" class="ltx_Math" alttext="n^{2}+n+m" display="inline">
-  <semantics>
+<math xmlns="http://www.w3.org/1998/Math/MathML">
+  <semantics encoding="">
     <mrow>
       <msup>
         <mi>n</mi>
         <mn>2</mn>
       </msup>
       <mo>+</mo>
       <mi>n</mi>
       <mo>+</mo>
       <mi>m</mi>
     </mrow>
     <annotation-xml encoding="MathML-Content">
       <apply>
         <plus/>
         <apply>
-          <csymbol cd="ambiguous">superscript</csymbol>
-          <ci>n</ci>
-          <cn type="integer">2</cn>
+          <csymbol encoding="">superscript</csymbol>
+          <ci encoding="">n</ci>
+          <cn encoding="" base="10" type="integer">2</cn>
         </apply>
-        <ci>n</ci>
-        <ci>m</ci>
+        <ci encoding="">n</ci>
+        <ci encoding="">m</ci>
       </apply>
     </annotation-xml>
     <annotation-xml encoding="application/x-latexml">
       <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
         <XMApp>
-          <XMTok meaning="plus" role="ADDOP">+</XMTok>
+          <XMTok>+</XMTok>
           <XMApp>
-            <XMTok role="SUPERSCRIPTOP" scriptpos="post2"/>
-            <XMTok role="UNKNOWN" font="italic">n</XMTok>
-            <XMTok meaning="2" role="NUMBER">2</XMTok>
+            <XMTok/>
+            <XMTok>n</XMTok>
+            <XMTok>2</XMTok>
           </XMApp>
-          <XMTok role="UNKNOWN" font="italic">n</XMTok>
-          <XMTok role="UNKNOWN" font="italic">m</XMTok>
+          <XMTok>n</XMTok>
+          <XMTok>m</XMTok>
         </XMApp>
       </XMath>
     </annotation-xml>
     <annotation encoding="application/x-tex">n^{2}+n+m</annotation>
   </semantics>
 </math>

Formula 9

( show / hide formula code)

Original

0 ≤ n ≤ m - 3 0 n m 3 0 ≤ n ≤ - m 3 0\leq n\leq m-3

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="p2.m5" class="ltx_Math" alttext="0\leq n\leq m-3" display="inline">
  <semantics>
    <mrow>
      <mn>0</mn>
      <mo>≤</mo>
      <mi>n</mi>
      <mo>≤</mo>
      <mrow>
        <mi>m</mi>
        <mo>-</mo>
        <mn>3</mn>
      </mrow>
    </mrow>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <and/>
        <apply>
          <leq/>
          <cn type="integer">0</cn>
          <ci xml:id="p2.m5.sh1.cmml">n</ci>
        </apply>
        <apply>
          <leq/>
          <share href="#p2.m5.sh1.cmml"/>
          <apply xml:id="p2.m5.sh2c.cmml">
            <minus xml:id="p2.m5.sh2.cmml"/>
            <ci xml:id="p2.m5.sh2a.cmml">m</ci>
            <cn type="integer" xml:id="p2.m5.sh2b.cmml">3</cn>
          </apply>
        </apply>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok meaning="multirelation"/>
          <XMTok meaning="0" role="NUMBER">0</XMTok>
          <XMTok meaning="less-than-or-equals" name="leq" role="RELOP">≤</XMTok>
          <XMTok role="UNKNOWN" font="italic" xml:id="p2.m5.sh1.xm" fragid="p2.m5.sh1.xm">n</XMTok>
          <XMTok meaning="less-than-or-equals" name="leq" role="RELOP">≤</XMTok>
          <XMApp xml:id="p2.m5.sh2.xm" fragid="p2.m5.sh2.xm">
            <XMTok meaning="minus" role="ADDOP">-</XMTok>
            <XMTok role="UNKNOWN" font="italic">m</XMTok>
            <XMTok meaning="3" role="NUMBER">3</XMTok>
          </XMApp>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">0\leq n\leq m-3</annotation>
  </semantics>
</math>

Canonicalized

0 ≤ n ≤ m - 3 0 n m 3 0 ≤ n ≤ - m 3 0\leq n\leq m-3

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics encoding="">
    <mrow>
      <mn>0</mn>
      <mo>≤</mo>
      <mi>n</mi>
      <mo>≤</mo>
      <mrow>
        <mi>m</mi>
        <mo>-</mo>
        <mn>3</mn>
      </mrow>
    </mrow>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <and/>
        <apply>
          <leq/>
          <cn encoding="" base="10" type="integer">0</cn>
          <ci encoding="">n</ci>
        </apply>
        <apply>
          <leq/>
          <share/>
          <apply>
            <minus encoding=""/>
            <ci encoding="">m</ci>
            <cn encoding="" base="10" type="integer">3</cn>
          </apply>
        </apply>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok/>
          <XMTok>0</XMTok>
          <XMTok>≤</XMTok>
          <XMTok>n</XMTok>
          <XMTok>≤</XMTok>
          <XMApp>
            <XMTok>-</XMTok>
            <XMTok>m</XMTok>
            <XMTok>3</XMTok>
          </XMApp>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">0\leq n\leq m-3</annotation>
  </semantics>
</math>

Diff

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@@ -1,52 +1,52 @@
 <?xml version="1.0" encoding="UTF-8"?>
-<math xmlns="http://www.w3.org/1998/Math/MathML" id="p2.m5" class="ltx_Math" alttext="0\leq n\leq m-3" display="inline">
-  <semantics>
+<math xmlns="http://www.w3.org/1998/Math/MathML">
+  <semantics encoding="">
     <mrow>
       <mn>0</mn>
       <mo>≤</mo>
       <mi>n</mi>
       <mo>≤</mo>
       <mrow>
         <mi>m</mi>
         <mo>-</mo>
         <mn>3</mn>
       </mrow>
     </mrow>
     <annotation-xml encoding="MathML-Content">
       <apply>
         <and/>
         <apply>
           <leq/>
-          <cn type="integer">0</cn>
-          <ci xml:id="p2.m5.sh1.cmml">n</ci>
+          <cn encoding="" base="10" type="integer">0</cn>
+          <ci encoding="">n</ci>
         </apply>
         <apply>
           <leq/>
-          <share href="#p2.m5.sh1.cmml"/>
-          <apply xml:id="p2.m5.sh2c.cmml">
-            <minus xml:id="p2.m5.sh2.cmml"/>
-            <ci xml:id="p2.m5.sh2a.cmml">m</ci>
-            <cn type="integer" xml:id="p2.m5.sh2b.cmml">3</cn>
+          <share/>
+          <apply>
+            <minus encoding=""/>
+            <ci encoding="">m</ci>
+            <cn encoding="" base="10" type="integer">3</cn>
           </apply>
         </apply>
       </apply>
     </annotation-xml>
     <annotation-xml encoding="application/x-latexml">
       <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
         <XMApp>
-          <XMTok meaning="multirelation"/>
-          <XMTok meaning="0" role="NUMBER">0</XMTok>
-          <XMTok meaning="less-than-or-equals" name="leq" role="RELOP">≤</XMTok>
-          <XMTok role="UNKNOWN" font="italic" xml:id="p2.m5.sh1.xm" fragid="p2.m5.sh1.xm">n</XMTok>
-          <XMTok meaning="less-than-or-equals" name="leq" role="RELOP">≤</XMTok>
-          <XMApp xml:id="p2.m5.sh2.xm" fragid="p2.m5.sh2.xm">
-            <XMTok meaning="minus" role="ADDOP">-</XMTok>
-            <XMTok role="UNKNOWN" font="italic">m</XMTok>
-            <XMTok meaning="3" role="NUMBER">3</XMTok>
+          <XMTok/>
+          <XMTok>0</XMTok>
+          <XMTok>≤</XMTok>
+          <XMTok>n</XMTok>
+          <XMTok>≤</XMTok>
+          <XMApp>
+            <XMTok>-</XMTok>
+            <XMTok>m</XMTok>
+            <XMTok>3</XMTok>
           </XMApp>
         </XMApp>
       </XMath>
     </annotation-xml>
     <annotation encoding="application/x-tex">0\leq n\leq m-3</annotation>
   </semantics>
 </math>

Formula 10

( show / hide formula code)

Original

f m ⁢ ( x ) = x 2 + x - m subscript f m x superscript x 2 x m = ⁢ f m x - + x 2 x m f_{m}(x)=x^{2}+x-m

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="p2.m6" class="ltx_Math" alttext="f_{m}(x)=x^{2}+x-m" display="inline">
  <semantics>
    <mrow>
      <mrow>
        <msub>
          <mi>f</mi>
          <mi>m</mi>
        </msub>
        <mo>⁢</mo>
        <mrow>
          <mo>(</mo>
          <mi>x</mi>
          <mo>)</mo>
        </mrow>
      </mrow>
      <mo>=</mo>
      <mrow>
        <msup>
          <mi>x</mi>
          <mn>2</mn>
        </msup>
        <mo>+</mo>
        <mi>x</mi>
        <mo>-</mo>
        <mi>m</mi>
      </mrow>
    </mrow>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <eq/>
        <apply>
          <times/>
          <apply>
            <csymbol cd="ambiguous">subscript</csymbol>
            <ci>f</ci>
            <ci>m</ci>
          </apply>
          <ci>x</ci>
        </apply>
        <apply>
          <minus/>
          <apply>
            <plus/>
            <apply>
              <csymbol cd="ambiguous">superscript</csymbol>
              <ci>x</ci>
              <cn type="integer">2</cn>
            </apply>
            <ci>x</ci>
          </apply>
          <ci>m</ci>
        </apply>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok meaning="equals" role="RELOP">=</XMTok>
          <XMApp>
            <XMTok meaning="times" role="MULOP">⁢</XMTok>
            <XMApp>
              <XMTok role="SUBSCRIPTOP" scriptpos="post2"/>
              <XMTok possibleFunction="yes" role="UNKNOWN" font="italic">f</XMTok>
              <XMTok role="UNKNOWN" font="italic">m</XMTok>
            </XMApp>
            <XMTok close=")" open="(" role="UNKNOWN" font="italic">x</XMTok>
          </XMApp>
          <XMApp>
            <XMTok meaning="minus" role="ADDOP">-</XMTok>
            <XMApp>
              <XMTok meaning="plus" role="ADDOP">+</XMTok>
              <XMApp>
                <XMTok role="SUPERSCRIPTOP" scriptpos="post2"/>
                <XMTok role="UNKNOWN" font="italic">x</XMTok>
                <XMTok meaning="2" role="NUMBER">2</XMTok>
              </XMApp>
              <XMTok role="UNKNOWN" font="italic">x</XMTok>
            </XMApp>
            <XMTok role="UNKNOWN" font="italic">m</XMTok>
          </XMApp>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">f_{m}(x)=x^{2}+x-m</annotation>
  </semantics>
</math>

Canonicalized

f m ( x ) = x 2 + x - m subscript f m x superscript x 2 x m = ⁢ f m x - + x 2 x m f_{m}(x)=x^{2}+x-m

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics encoding="">
    <mrow>
      <mrow>
        <msub>
          <mi>f</mi>
          <mi>m</mi>
        </msub>
        <mrow>
          <mo>(</mo>
          <mi>x</mi>
          <mo>)</mo>
        </mrow>
      </mrow>
      <mo>=</mo>
      <mrow>
        <msup>
          <mi>x</mi>
          <mn>2</mn>
        </msup>
        <mo>+</mo>
        <mi>x</mi>
        <mo>-</mo>
        <mi>m</mi>
      </mrow>
    </mrow>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <eq/>
        <apply>
          <times/>
          <apply>
            <csymbol encoding="">subscript</csymbol>
            <ci encoding="">f</ci>
            <ci encoding="">m</ci>
          </apply>
          <ci encoding="">x</ci>
        </apply>
        <apply>
          <minus/>
          <apply>
            <plus/>
            <apply>
              <csymbol encoding="">superscript</csymbol>
              <ci encoding="">x</ci>
              <cn encoding="" base="10" type="integer">2</cn>
            </apply>
            <ci encoding="">x</ci>
          </apply>
          <ci encoding="">m</ci>
        </apply>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok>=</XMTok>
          <XMApp>
            <XMTok>⁢</XMTok>
            <XMApp>
              <XMTok/>
              <XMTok>f</XMTok>
              <XMTok>m</XMTok>
            </XMApp>
            <XMTok>x</XMTok>
          </XMApp>
          <XMApp>
            <XMTok>-</XMTok>
            <XMApp>
              <XMTok>+</XMTok>
              <XMApp>
                <XMTok/>
                <XMTok>x</XMTok>
                <XMTok>2</XMTok>
              </XMApp>
              <XMTok>x</XMTok>
            </XMApp>
            <XMTok>m</XMTok>
          </XMApp>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">f_{m}(x)=x^{2}+x-m</annotation>
  </semantics>
</math>

Diff

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@@ -1,87 +1,86 @@
 <?xml version="1.0" encoding="UTF-8"?>
-<math xmlns="http://www.w3.org/1998/Math/MathML" id="p2.m6" class="ltx_Math" alttext="f_{m}(x)=x^{2}+x-m" display="inline">
-  <semantics>
+<math xmlns="http://www.w3.org/1998/Math/MathML">
+  <semantics encoding="">
     <mrow>
       <mrow>
         <msub>
           <mi>f</mi>
           <mi>m</mi>
         </msub>
-        <mo>⁢</mo>
         <mrow>
           <mo>(</mo>
           <mi>x</mi>
           <mo>)</mo>
         </mrow>
       </mrow>
       <mo>=</mo>
       <mrow>
         <msup>
           <mi>x</mi>
           <mn>2</mn>
         </msup>
         <mo>+</mo>
         <mi>x</mi>
         <mo>-</mo>
         <mi>m</mi>
       </mrow>
     </mrow>
     <annotation-xml encoding="MathML-Content">
       <apply>
         <eq/>
         <apply>
           <times/>
           <apply>
-            <csymbol cd="ambiguous">subscript</csymbol>
-            <ci>f</ci>
-            <ci>m</ci>
+            <csymbol encoding="">subscript</csymbol>
+            <ci encoding="">f</ci>
+            <ci encoding="">m</ci>
           </apply>
-          <ci>x</ci>
+          <ci encoding="">x</ci>
         </apply>
         <apply>
           <minus/>
           <apply>
             <plus/>
             <apply>
-              <csymbol cd="ambiguous">superscript</csymbol>
-              <ci>x</ci>
-              <cn type="integer">2</cn>
+              <csymbol encoding="">superscript</csymbol>
+              <ci encoding="">x</ci>
+              <cn encoding="" base="10" type="integer">2</cn>
             </apply>
-            <ci>x</ci>
+            <ci encoding="">x</ci>
           </apply>
-          <ci>m</ci>
+          <ci encoding="">m</ci>
         </apply>
       </apply>
     </annotation-xml>
     <annotation-xml encoding="application/x-latexml">
       <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
         <XMApp>
-          <XMTok meaning="equals" role="RELOP">=</XMTok>
+          <XMTok>=</XMTok>
           <XMApp>
-            <XMTok meaning="times" role="MULOP">⁢</XMTok>
+            <XMTok>⁢</XMTok>
             <XMApp>
-              <XMTok role="SUBSCRIPTOP" scriptpos="post2"/>
-              <XMTok possibleFunction="yes" role="UNKNOWN" font="italic">f</XMTok>
-              <XMTok role="UNKNOWN" font="italic">m</XMTok>
+              <XMTok/>
+              <XMTok>f</XMTok>
+              <XMTok>m</XMTok>
             </XMApp>
-            <XMTok close=")" open="(" role="UNKNOWN" font="italic">x</XMTok>
+            <XMTok>x</XMTok>
           </XMApp>
           <XMApp>
-            <XMTok meaning="minus" role="ADDOP">-</XMTok>
+            <XMTok>-</XMTok>
             <XMApp>
-              <XMTok meaning="plus" role="ADDOP">+</XMTok>
+              <XMTok>+</XMTok>
               <XMApp>
-                <XMTok role="SUPERSCRIPTOP" scriptpos="post2"/>
-                <XMTok role="UNKNOWN" font="italic">x</XMTok>
-                <XMTok meaning="2" role="NUMBER">2</XMTok>
+                <XMTok/>
+                <XMTok>x</XMTok>
+                <XMTok>2</XMTok>
               </XMApp>
-              <XMTok role="UNKNOWN" font="italic">x</XMTok>
+              <XMTok>x</XMTok>
             </XMApp>
-            <XMTok role="UNKNOWN" font="italic">m</XMTok>
+            <XMTok>m</XMTok>
           </XMApp>
         </XMApp>
       </XMath>
     </annotation-xml>
     <annotation encoding="application/x-tex">f_{m}(x)=x^{2}+x-m</annotation>
   </semantics>
 </math>

Formula 11

( show / hide formula code)

Original

t t t t

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="p2.m7" class="ltx_Math" alttext="t" display="inline">
  <semantics>
    <mi>t</mi>
    <annotation-xml encoding="MathML-Content">
      <ci>t</ci>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMTok role="UNKNOWN" font="italic">t</XMTok>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">t</annotation>
  </semantics>
</math>

Canonicalized

t t t t

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics encoding="">
    <mi>t</mi>
    <annotation-xml encoding="MathML-Content">
      <ci encoding="">t</ci>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMTok>t</XMTok>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">t</annotation>
  </semantics>
</math>

Diff

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@@ -1,15 +1,15 @@
 <?xml version="1.0" encoding="UTF-8"?>
-<math xmlns="http://www.w3.org/1998/Math/MathML" id="p2.m7" class="ltx_Math" alttext="t" display="inline">
-  <semantics>
+<math xmlns="http://www.w3.org/1998/Math/MathML">
+  <semantics encoding="">
     <mi>t</mi>
     <annotation-xml encoding="MathML-Content">
-      <ci>t</ci>
+      <ci encoding="">t</ci>
     </annotation-xml>
     <annotation-xml encoding="application/x-latexml">
       <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
-        <XMTok role="UNKNOWN" font="italic">t</XMTok>
+        <XMTok>t</XMTok>
       </XMath>
     </annotation-xml>
     <annotation encoding="application/x-tex">t</annotation>
   </semantics>
 </math>

Formula 12

( show / hide formula code)

Original

| f m ⁢ ( n ) | subscript f m n ⁢ f m n |f_{m}(n)|

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="p2.m8" class="ltx_Math" alttext="|f_{m}(n)|" display="inline">
  <semantics>
    <mrow>
      <mo fence="true">|</mo>
      <mrow>
        <msub>
          <mi>f</mi>
          <mi>m</mi>
        </msub>
        <mo>⁢</mo>
        <mrow>
          <mo>(</mo>
          <mi>n</mi>
          <mo>)</mo>
        </mrow>
      </mrow>
      <mo fence="true">|</mo>
    </mrow>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <abs/>
        <apply>
          <times/>
          <apply>
            <csymbol cd="ambiguous">subscript</csymbol>
            <ci>f</ci>
            <ci>m</ci>
          </apply>
          <ci>n</ci>
        </apply>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok argclose="|" argopen="|" meaning="absolute-value" role="FENCED"/>
          <XMApp>
            <XMTok meaning="times" role="MULOP">⁢</XMTok>
            <XMApp>
              <XMTok role="SUBSCRIPTOP" scriptpos="post2"/>
              <XMTok possibleFunction="yes" role="UNKNOWN" font="italic">f</XMTok>
              <XMTok role="UNKNOWN" font="italic">m</XMTok>
            </XMApp>
            <XMTok close=")" open="(" role="UNKNOWN" font="italic">n</XMTok>
          </XMApp>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">|f_{m}(n)|</annotation>
  </semantics>
</math>

Canonicalized

| f m ( n ) | subscript f m n ⁢ f m n |f_{m}(n)|

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics encoding="">
    <mrow>
      <mo>|</mo>
      <mrow>
        <msub>
          <mi>f</mi>
          <mi>m</mi>
        </msub>
        <mrow>
          <mo>(</mo>
          <mi>n</mi>
          <mo>)</mo>
        </mrow>
      </mrow>
      <mo>|</mo>
    </mrow>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <abs/>
        <apply>
          <times/>
          <apply>
            <csymbol encoding="">subscript</csymbol>
            <ci encoding="">f</ci>
            <ci encoding="">m</ci>
          </apply>
          <ci encoding="">n</ci>
        </apply>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok/>
          <XMApp>
            <XMTok>⁢</XMTok>
            <XMApp>
              <XMTok/>
              <XMTok>f</XMTok>
              <XMTok>m</XMTok>
            </XMApp>
            <XMTok>n</XMTok>
          </XMApp>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">|f_{m}(n)|</annotation>
  </semantics>
</math>

Diff

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@@ -1,52 +1,51 @@
 <?xml version="1.0" encoding="UTF-8"?>
-<math xmlns="http://www.w3.org/1998/Math/MathML" id="p2.m8" class="ltx_Math" alttext="|f_{m}(n)|" display="inline">
-  <semantics>
+<math xmlns="http://www.w3.org/1998/Math/MathML">
+  <semantics encoding="">
     <mrow>
-      <mo fence="true">|</mo>
+      <mo>|</mo>
       <mrow>
         <msub>
           <mi>f</mi>
           <mi>m</mi>
         </msub>
-        <mo>⁢</mo>
         <mrow>
           <mo>(</mo>
           <mi>n</mi>
           <mo>)</mo>
         </mrow>
       </mrow>
-      <mo fence="true">|</mo>
+      <mo>|</mo>
     </mrow>
     <annotation-xml encoding="MathML-Content">
       <apply>
         <abs/>
         <apply>
           <times/>
           <apply>
-            <csymbol cd="ambiguous">subscript</csymbol>
-            <ci>f</ci>
-            <ci>m</ci>
+            <csymbol encoding="">subscript</csymbol>
+            <ci encoding="">f</ci>
+            <ci encoding="">m</ci>
           </apply>
-          <ci>n</ci>
+          <ci encoding="">n</ci>
         </apply>
       </apply>
     </annotation-xml>
     <annotation-xml encoding="application/x-latexml">
       <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
         <XMApp>
-          <XMTok argclose="|" argopen="|" meaning="absolute-value" role="FENCED"/>
+          <XMTok/>
           <XMApp>
-            <XMTok meaning="times" role="MULOP">⁢</XMTok>
+            <XMTok>⁢</XMTok>
             <XMApp>
-              <XMTok role="SUBSCRIPTOP" scriptpos="post2"/>
-              <XMTok possibleFunction="yes" role="UNKNOWN" font="italic">f</XMTok>
-              <XMTok role="UNKNOWN" font="italic">m</XMTok>
+              <XMTok/>
+              <XMTok>f</XMTok>
+              <XMTok>m</XMTok>
             </XMApp>
-            <XMTok close=")" open="(" role="UNKNOWN" font="italic">n</XMTok>
+            <XMTok>n</XMTok>
           </XMApp>
         </XMApp>
       </XMath>
     </annotation-xml>
     <annotation encoding="application/x-tex">|f_{m}(n)|</annotation>
   </semantics>
 </math>

Formula 13

( show / hide formula code)

Original

n ∈ [ t + 1 , t + m ] n t 1 t m ∈ n + t 1 + t m n\in[t+1,t+\sqrt{m}]

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="p2.m9" class="ltx_Math" alttext="n\in[t+1,t+\sqrt{m}]" display="inline">
  <semantics>
    <mrow>
      <mi>n</mi>
      <mo>∈</mo>
      <mrow>
        <mo>[</mo>
        <mrow>
          <mrow>
            <mi>t</mi>
            <mo>+</mo>
            <mn>1</mn>
          </mrow>
          <mo>,</mo>
          <mrow>
            <mi>t</mi>
            <mo>+</mo>
            <msqrt>
              <mi>m</mi>
            </msqrt>
          </mrow>
        </mrow>
        <mo>]</mo>
      </mrow>
    </mrow>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <in/>
        <ci>n</ci>
        <apply>
          <interval closure="closed"/>
          <apply>
            <plus/>
            <ci>t</ci>
            <cn type="integer">1</cn>
          </apply>
          <apply>
            <plus/>
            <ci>t</ci>
            <apply>
              <root/>
              <ci>m</ci>
            </apply>
          </apply>
        </apply>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok meaning="element-of" name="in" role="RELOP">∈</XMTok>
          <XMTok role="UNKNOWN" font="italic">n</XMTok>
          <XMApp>
            <XMTok argclose="]" argopen="[" meaning="closed-interval" role="FENCED" separators=","/>
            <XMApp>
              <XMTok meaning="plus" role="ADDOP">+</XMTok>
              <XMTok role="UNKNOWN" font="italic">t</XMTok>
              <XMTok meaning="1" role="NUMBER">1</XMTok>
            </XMApp>
            <XMApp>
              <XMTok meaning="plus" role="ADDOP">+</XMTok>
              <XMTok role="UNKNOWN" font="italic">t</XMTok>
              <XMApp>
                <XMTok meaning="square-root"/>
                <XMTok role="UNKNOWN" font="italic">m</XMTok>
              </XMApp>
            </XMApp>
          </XMApp>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">n\in[t+1,t+\sqrt{m}]</annotation>
  </semantics>
</math>

Canonicalized

n ∈ [ t + 1 , t + m ] n t 1 t m ∈ n + t 1 + t m n\in[t+1,t+\sqrt{m}]

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics encoding="">
    <mrow>
      <mi>n</mi>
      <mo>∈</mo>
      <mrow>
        <mo>[</mo>
        <mrow>
          <mrow>
            <mi>t</mi>
            <mo>+</mo>
            <mn>1</mn>
          </mrow>
          <mo>,</mo>
          <mrow>
            <mi>t</mi>
            <mo>+</mo>
            <msqrt>
              <mi>m</mi>
            </msqrt>
          </mrow>
        </mrow>
        <mo>]</mo>
      </mrow>
    </mrow>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <in/>
        <ci encoding="">n</ci>
        <apply>
          <interval closure="closed"/>
          <apply>
            <plus/>
            <ci encoding="">t</ci>
            <cn encoding="" base="10" type="integer">1</cn>
          </apply>
          <apply>
            <plus/>
            <ci encoding="">t</ci>
            <apply>
              <root/>
              <ci encoding="">m</ci>
            </apply>
          </apply>
        </apply>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok>∈</XMTok>
          <XMTok>n</XMTok>
          <XMApp>
            <XMTok/>
            <XMApp>
              <XMTok>+</XMTok>
              <XMTok>t</XMTok>
              <XMTok>1</XMTok>
            </XMApp>
            <XMApp>
              <XMTok>+</XMTok>
              <XMTok>t</XMTok>
              <XMApp>
                <XMTok/>
                <XMTok>m</XMTok>
              </XMApp>
            </XMApp>
          </XMApp>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">n\in[t+1,t+\sqrt{m}]</annotation>
  </semantics>
</math>

Diff

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@@ -1,75 +1,75 @@
 <?xml version="1.0" encoding="UTF-8"?>
-<math xmlns="http://www.w3.org/1998/Math/MathML" id="p2.m9" class="ltx_Math" alttext="n\in[t+1,t+\sqrt{m}]" display="inline">
-  <semantics>
+<math xmlns="http://www.w3.org/1998/Math/MathML">
+  <semantics encoding="">
     <mrow>
       <mi>n</mi>
       <mo>∈</mo>
       <mrow>
         <mo>[</mo>
         <mrow>
           <mrow>
             <mi>t</mi>
             <mo>+</mo>
             <mn>1</mn>
           </mrow>
           <mo>,</mo>
           <mrow>
             <mi>t</mi>
             <mo>+</mo>
             <msqrt>
               <mi>m</mi>
             </msqrt>
           </mrow>
         </mrow>
         <mo>]</mo>
       </mrow>
     </mrow>
     <annotation-xml encoding="MathML-Content">
       <apply>
         <in/>
-        <ci>n</ci>
+        <ci encoding="">n</ci>
         <apply>
           <interval closure="closed"/>
           <apply>
             <plus/>
-            <ci>t</ci>
-            <cn type="integer">1</cn>
+            <ci encoding="">t</ci>
+            <cn encoding="" base="10" type="integer">1</cn>
           </apply>
           <apply>
             <plus/>
-            <ci>t</ci>
+            <ci encoding="">t</ci>
             <apply>
               <root/>
-              <ci>m</ci>
+              <ci encoding="">m</ci>
             </apply>
           </apply>
         </apply>
       </apply>
     </annotation-xml>
     <annotation-xml encoding="application/x-latexml">
       <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
         <XMApp>
-          <XMTok meaning="element-of" name="in" role="RELOP">∈</XMTok>
-          <XMTok role="UNKNOWN" font="italic">n</XMTok>
+          <XMTok>∈</XMTok>
+          <XMTok>n</XMTok>
           <XMApp>
-            <XMTok argclose="]" argopen="[" meaning="closed-interval" role="FENCED" separators=","/>
+            <XMTok/>
             <XMApp>
-              <XMTok meaning="plus" role="ADDOP">+</XMTok>
-              <XMTok role="UNKNOWN" font="italic">t</XMTok>
-              <XMTok meaning="1" role="NUMBER">1</XMTok>
+              <XMTok>+</XMTok>
+              <XMTok>t</XMTok>
+              <XMTok>1</XMTok>
             </XMApp>
             <XMApp>
-              <XMTok meaning="plus" role="ADDOP">+</XMTok>
-              <XMTok role="UNKNOWN" font="italic">t</XMTok>
+              <XMTok>+</XMTok>
+              <XMTok>t</XMTok>
               <XMApp>
-                <XMTok meaning="square-root"/>
-                <XMTok role="UNKNOWN" font="italic">m</XMTok>
+                <XMTok/>
+                <XMTok>m</XMTok>
               </XMApp>
             </XMApp>
           </XMApp>
         </XMApp>
       </XMath>
     </annotation-xml>
     <annotation encoding="application/x-tex">n\in[t+1,t+\sqrt{m}]</annotation>
   </semantics>
 </math>

Formula 14

( show / hide formula code)

Original

f m subscript f m f m f_{m}

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="I1.i1.p1.m1" class="ltx_Math" alttext="f_{m}" display="inline">
  <semantics>
    <msub>
      <mi>f</mi>
      <mi>m</mi>
    </msub>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <csymbol cd="ambiguous">subscript</csymbol>
        <ci>f</ci>
        <ci>m</ci>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok role="SUBSCRIPTOP" scriptpos="post4"/>
          <XMTok role="UNKNOWN" font="italic">f</XMTok>
          <XMTok role="UNKNOWN" font="italic">m</XMTok>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">f_{m}</annotation>
  </semantics>
</math>

Canonicalized

f m subscript f m f m f_{m}

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics encoding="">
    <msub>
      <mi>f</mi>
      <mi>m</mi>
    </msub>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <csymbol encoding="">subscript</csymbol>
        <ci encoding="">f</ci>
        <ci encoding="">m</ci>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok/>
          <XMTok>f</XMTok>
          <XMTok>m</XMTok>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">f_{m}</annotation>
  </semantics>
</math>

Diff

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@@ -1,26 +1,26 @@
 <?xml version="1.0" encoding="UTF-8"?>
-<math xmlns="http://www.w3.org/1998/Math/MathML" id="I1.i1.p1.m1" class="ltx_Math" alttext="f_{m}" display="inline">
-  <semantics>
+<math xmlns="http://www.w3.org/1998/Math/MathML">
+  <semantics encoding="">
     <msub>
       <mi>f</mi>
       <mi>m</mi>
     </msub>
     <annotation-xml encoding="MathML-Content">
       <apply>
-        <csymbol cd="ambiguous">subscript</csymbol>
-        <ci>f</ci>
-        <ci>m</ci>
+        <csymbol encoding="">subscript</csymbol>
+        <ci encoding="">f</ci>
+        <ci encoding="">m</ci>
       </apply>
     </annotation-xml>
     <annotation-xml encoding="application/x-latexml">
       <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
         <XMApp>
-          <XMTok role="SUBSCRIPTOP" scriptpos="post4"/>
-          <XMTok role="UNKNOWN" font="italic">f</XMTok>
-          <XMTok role="UNKNOWN" font="italic">m</XMTok>
+          <XMTok/>
+          <XMTok>f</XMTok>
+          <XMTok>m</XMTok>
         </XMApp>
       </XMath>
     </annotation-xml>
     <annotation encoding="application/x-tex">f_{m}</annotation>
   </semantics>
 </math>

Formula 15

( show / hide formula code)

Original

m = 1 m 1 = m 1 m=1

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="I1.i1.p1.m2" class="ltx_Math" alttext="m=1" display="inline">
  <semantics>
    <mrow>
      <mi>m</mi>
      <mo>=</mo>
      <mn>1</mn>
    </mrow>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <eq/>
        <ci>m</ci>
        <cn type="integer">1</cn>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok meaning="equals" role="RELOP">=</XMTok>
          <XMTok role="UNKNOWN" font="italic">m</XMTok>
          <XMTok meaning="1" role="NUMBER">1</XMTok>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">m=1</annotation>
  </semantics>
</math>

Canonicalized

m = 1 m 1 = m 1 m=1

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics encoding="">
    <mrow>
      <mi>m</mi>
      <mo>=</mo>
      <mn>1</mn>
    </mrow>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <eq/>
        <ci encoding="">m</ci>
        <cn encoding="" base="10" type="integer">1</cn>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok>=</XMTok>
          <XMTok>m</XMTok>
          <XMTok>1</XMTok>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">m=1</annotation>
  </semantics>
</math>

Diff

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@@ -1,27 +1,27 @@
 <?xml version="1.0" encoding="UTF-8"?>
-<math xmlns="http://www.w3.org/1998/Math/MathML" id="I1.i1.p1.m2" class="ltx_Math" alttext="m=1" display="inline">
-  <semantics>
+<math xmlns="http://www.w3.org/1998/Math/MathML">
+  <semantics encoding="">
     <mrow>
       <mi>m</mi>
       <mo>=</mo>
       <mn>1</mn>
     </mrow>
     <annotation-xml encoding="MathML-Content">
       <apply>
         <eq/>
-        <ci>m</ci>
-        <cn type="integer">1</cn>
+        <ci encoding="">m</ci>
+        <cn encoding="" base="10" type="integer">1</cn>
       </apply>
     </annotation-xml>
     <annotation-xml encoding="application/x-latexml">
       <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
         <XMApp>
-          <XMTok meaning="equals" role="RELOP">=</XMTok>
-          <XMTok role="UNKNOWN" font="italic">m</XMTok>
-          <XMTok meaning="1" role="NUMBER">1</XMTok>
+          <XMTok>=</XMTok>
+          <XMTok>m</XMTok>
+          <XMTok>1</XMTok>
         </XMApp>
       </XMath>
     </annotation-xml>
     <annotation encoding="application/x-tex">m=1</annotation>
   </semantics>
 </math>

Formula 16

( show / hide formula code)

Original

m = 2 m 2 = m 2 m=2

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="I1.i1.p1.m3" class="ltx_Math" alttext="m=2" display="inline">
  <semantics>
    <mrow>
      <mi>m</mi>
      <mo>=</mo>
      <mn>2</mn>
    </mrow>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <eq/>
        <ci>m</ci>
        <cn type="integer">2</cn>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok meaning="equals" role="RELOP">=</XMTok>
          <XMTok role="UNKNOWN" font="italic">m</XMTok>
          <XMTok meaning="2" role="NUMBER">2</XMTok>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">m=2</annotation>
  </semantics>
</math>

Canonicalized

m = 2 m 2 = m 2 m=2

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics encoding="">
    <mrow>
      <mi>m</mi>
      <mo>=</mo>
      <mn>2</mn>
    </mrow>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <eq/>
        <ci encoding="">m</ci>
        <cn encoding="" base="10" type="integer">2</cn>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok>=</XMTok>
          <XMTok>m</XMTok>
          <XMTok>2</XMTok>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">m=2</annotation>
  </semantics>
</math>

Diff

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@@ -1,27 +1,27 @@
 <?xml version="1.0" encoding="UTF-8"?>
-<math xmlns="http://www.w3.org/1998/Math/MathML" id="I1.i1.p1.m3" class="ltx_Math" alttext="m=2" display="inline">
-  <semantics>
+<math xmlns="http://www.w3.org/1998/Math/MathML">
+  <semantics encoding="">
     <mrow>
       <mi>m</mi>
       <mo>=</mo>
       <mn>2</mn>
     </mrow>
     <annotation-xml encoding="MathML-Content">
       <apply>
         <eq/>
-        <ci>m</ci>
-        <cn type="integer">2</cn>
+        <ci encoding="">m</ci>
+        <cn encoding="" base="10" type="integer">2</cn>
       </apply>
     </annotation-xml>
     <annotation-xml encoding="application/x-latexml">
       <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
         <XMApp>
-          <XMTok meaning="equals" role="RELOP">=</XMTok>
-          <XMTok role="UNKNOWN" font="italic">m</XMTok>
-          <XMTok meaning="2" role="NUMBER">2</XMTok>
+          <XMTok>=</XMTok>
+          <XMTok>m</XMTok>
+          <XMTok>2</XMTok>
         </XMApp>
       </XMath>
     </annotation-xml>
     <annotation encoding="application/x-tex">m=2</annotation>
   </semantics>
 </math>

Formula 17

( show / hide formula code)

Original

m = p 2 m superscript p 2 = m p 2 m=p^{2}

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="I1.i1.p1.m4" class="ltx_Math" alttext="m=p^{2}" display="inline">
  <semantics>
    <mrow>
      <mi>m</mi>
      <mo>=</mo>
      <msup>
        <mi>p</mi>
        <mn>2</mn>
      </msup>
    </mrow>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <eq/>
        <ci>m</ci>
        <apply>
          <csymbol cd="ambiguous">superscript</csymbol>
          <ci>p</ci>
          <cn type="integer">2</cn>
        </apply>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok meaning="equals" role="RELOP">=</XMTok>
          <XMTok role="UNKNOWN" font="italic">m</XMTok>
          <XMApp>
            <XMTok role="SUPERSCRIPTOP" scriptpos="post4"/>
            <XMTok role="UNKNOWN" font="italic">p</XMTok>
            <XMTok meaning="2" role="NUMBER">2</XMTok>
          </XMApp>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">m=p^{2}</annotation>
  </semantics>
</math>

Canonicalized

m = p 2 m superscript p 2 = m p 2 m=p^{2}

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics encoding="">
    <mrow>
      <mi>m</mi>
      <mo>=</mo>
      <msup>
        <mi>p</mi>
        <mn>2</mn>
      </msup>
    </mrow>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <eq/>
        <ci encoding="">m</ci>
        <apply>
          <csymbol encoding="">superscript</csymbol>
          <ci encoding="">p</ci>
          <cn encoding="" base="10" type="integer">2</cn>
        </apply>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok>=</XMTok>
          <XMTok>m</XMTok>
          <XMApp>
            <XMTok/>
            <XMTok>p</XMTok>
            <XMTok>2</XMTok>
          </XMApp>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">m=p^{2}</annotation>
  </semantics>
</math>

Diff

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@@ -1,38 +1,38 @@
 <?xml version="1.0" encoding="UTF-8"?>
-<math xmlns="http://www.w3.org/1998/Math/MathML" id="I1.i1.p1.m4" class="ltx_Math" alttext="m=p^{2}" display="inline">
-  <semantics>
+<math xmlns="http://www.w3.org/1998/Math/MathML">
+  <semantics encoding="">
     <mrow>
       <mi>m</mi>
       <mo>=</mo>
       <msup>
         <mi>p</mi>
         <mn>2</mn>
       </msup>
     </mrow>
     <annotation-xml encoding="MathML-Content">
       <apply>
         <eq/>
-        <ci>m</ci>
+        <ci encoding="">m</ci>
         <apply>
-          <csymbol cd="ambiguous">superscript</csymbol>
-          <ci>p</ci>
-          <cn type="integer">2</cn>
+          <csymbol encoding="">superscript</csymbol>
+          <ci encoding="">p</ci>
+          <cn encoding="" base="10" type="integer">2</cn>
         </apply>
       </apply>
     </annotation-xml>
     <annotation-xml encoding="application/x-latexml">
       <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
         <XMApp>
-          <XMTok meaning="equals" role="RELOP">=</XMTok>
-          <XMTok role="UNKNOWN" font="italic">m</XMTok>
+          <XMTok>=</XMTok>
+          <XMTok>m</XMTok>
           <XMApp>
-            <XMTok role="SUPERSCRIPTOP" scriptpos="post4"/>
-            <XMTok role="UNKNOWN" font="italic">p</XMTok>
-            <XMTok meaning="2" role="NUMBER">2</XMTok>
+            <XMTok/>
+            <XMTok>p</XMTok>
+            <XMTok>2</XMTok>
           </XMApp>
         </XMApp>
       </XMath>
     </annotation-xml>
     <annotation encoding="application/x-tex">m=p^{2}</annotation>
   </semantics>
 </math>

Formula 18

( show / hide formula code)

Original

p p p p

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="I1.i1.p1.m5" class="ltx_Math" alttext="p" display="inline">
  <semantics>
    <mi>p</mi>
    <annotation-xml encoding="MathML-Content">
      <ci>p</ci>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMTok role="UNKNOWN" font="italic">p</XMTok>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">p</annotation>
  </semantics>
</math>

Canonicalized

p p p p

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics encoding="">
    <mi>p</mi>
    <annotation-xml encoding="MathML-Content">
      <ci encoding="">p</ci>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMTok>p</XMTok>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">p</annotation>
  </semantics>
</math>

Diff

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@@ -1,15 +1,15 @@
 <?xml version="1.0" encoding="UTF-8"?>
-<math xmlns="http://www.w3.org/1998/Math/MathML" id="I1.i1.p1.m5" class="ltx_Math" alttext="p" display="inline">
-  <semantics>
+<math xmlns="http://www.w3.org/1998/Math/MathML">
+  <semantics encoding="">
     <mi>p</mi>
     <annotation-xml encoding="MathML-Content">
-      <ci>p</ci>
+      <ci encoding="">p</ci>
     </annotation-xml>
     <annotation-xml encoding="application/x-latexml">
       <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
-        <XMTok role="UNKNOWN" font="italic">p</XMTok>
+        <XMTok>p</XMTok>
       </XMath>
     </annotation-xml>
     <annotation encoding="application/x-tex">p</annotation>
   </semantics>
 </math>

Formula 19

( show / hide formula code)

Original

m = t 2 + t ± 1 m plus-or-minus superscript t 2 t 1 = m ± + t 2 t 1 m=t^{2}+t\pm 1

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="I1.i1.p1.m6" class="ltx_Math" alttext="m=t^{2}+t\pm 1" display="inline">
  <semantics>
    <mrow>
      <mi>m</mi>
      <mo>=</mo>
      <mrow>
        <msup>
          <mi>t</mi>
          <mn>2</mn>
        </msup>
        <mo>+</mo>
        <mi>t</mi>
        <mo>±</mo>
        <mn>1</mn>
      </mrow>
    </mrow>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <eq/>
        <ci>m</ci>
        <apply>
          <csymbol cd="latexml">plus-or-minus</csymbol>
          <apply>
            <plus/>
            <apply>
              <csymbol cd="ambiguous">superscript</csymbol>
              <ci>t</ci>
              <cn type="integer">2</cn>
            </apply>
            <ci>t</ci>
          </apply>
          <cn type="integer">1</cn>
        </apply>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok meaning="equals" role="RELOP">=</XMTok>
          <XMTok role="UNKNOWN" font="italic">m</XMTok>
          <XMApp>
            <XMTok meaning="plus-or-minus" name="pm" role="ADDOP">±</XMTok>
            <XMApp>
              <XMTok meaning="plus" role="ADDOP">+</XMTok>
              <XMApp>
                <XMTok role="SUPERSCRIPTOP" scriptpos="post4"/>
                <XMTok role="UNKNOWN" font="italic">t</XMTok>
                <XMTok meaning="2" role="NUMBER">2</XMTok>
              </XMApp>
              <XMTok role="UNKNOWN" font="italic">t</XMTok>
            </XMApp>
            <XMTok meaning="1" role="NUMBER">1</XMTok>
          </XMApp>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">m=t^{2}+t\pm 1</annotation>
  </semantics>
</math>

Canonicalized

m = t 2 + t ± 1 m plus-or-minus superscript t 2 t 1 = m ± + t 2 t 1 m=t^{2}+t\pm 1

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics encoding="">
    <mrow>
      <mi>m</mi>
      <mo>=</mo>
      <mrow>
        <msup>
          <mi>t</mi>
          <mn>2</mn>
        </msup>
        <mo>+</mo>
        <mi>t</mi>
        <mo>±</mo>
        <mn>1</mn>
      </mrow>
    </mrow>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <eq/>
        <ci encoding="">m</ci>
        <apply>
          <csymbol encoding="">plus-or-minus</csymbol>
          <apply>
            <plus/>
            <apply>
              <csymbol encoding="">superscript</csymbol>
              <ci encoding="">t</ci>
              <cn encoding="" base="10" type="integer">2</cn>
            </apply>
            <ci encoding="">t</ci>
          </apply>
          <cn encoding="" base="10" type="integer">1</cn>
        </apply>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok>=</XMTok>
          <XMTok>m</XMTok>
          <XMApp>
            <XMTok>±</XMTok>
            <XMApp>
              <XMTok>+</XMTok>
              <XMApp>
                <XMTok/>
                <XMTok>t</XMTok>
                <XMTok>2</XMTok>
              </XMApp>
              <XMTok>t</XMTok>
            </XMApp>
            <XMTok>1</XMTok>
          </XMApp>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">m=t^{2}+t\pm 1</annotation>
  </semantics>
</math>

Diff

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@@ -1,60 +1,60 @@
 <?xml version="1.0" encoding="UTF-8"?>
-<math xmlns="http://www.w3.org/1998/Math/MathML" id="I1.i1.p1.m6" class="ltx_Math" alttext="m=t^{2}+t\pm 1" display="inline">
-  <semantics>
+<math xmlns="http://www.w3.org/1998/Math/MathML">
+  <semantics encoding="">
     <mrow>
       <mi>m</mi>
       <mo>=</mo>
       <mrow>
         <msup>
           <mi>t</mi>
           <mn>2</mn>
         </msup>
         <mo>+</mo>
         <mi>t</mi>
         <mo>±</mo>
         <mn>1</mn>
       </mrow>
     </mrow>
     <annotation-xml encoding="MathML-Content">
       <apply>
         <eq/>
-        <ci>m</ci>
+        <ci encoding="">m</ci>
         <apply>
-          <csymbol cd="latexml">plus-or-minus</csymbol>
+          <csymbol encoding="">plus-or-minus</csymbol>
           <apply>
             <plus/>
             <apply>
-              <csymbol cd="ambiguous">superscript</csymbol>
-              <ci>t</ci>
-              <cn type="integer">2</cn>
+              <csymbol encoding="">superscript</csymbol>
+              <ci encoding="">t</ci>
+              <cn encoding="" base="10" type="integer">2</cn>
             </apply>
-            <ci>t</ci>
+            <ci encoding="">t</ci>
           </apply>
-          <cn type="integer">1</cn>
+          <cn encoding="" base="10" type="integer">1</cn>
         </apply>
       </apply>
     </annotation-xml>
     <annotation-xml encoding="application/x-latexml">
       <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
         <XMApp>
-          <XMTok meaning="equals" role="RELOP">=</XMTok>
-          <XMTok role="UNKNOWN" font="italic">m</XMTok>
+          <XMTok>=</XMTok>
+          <XMTok>m</XMTok>
           <XMApp>
-            <XMTok meaning="plus-or-minus" name="pm" role="ADDOP">±</XMTok>
+            <XMTok>±</XMTok>
             <XMApp>
-              <XMTok meaning="plus" role="ADDOP">+</XMTok>
+              <XMTok>+</XMTok>
               <XMApp>
-                <XMTok role="SUPERSCRIPTOP" scriptpos="post4"/>
-                <XMTok role="UNKNOWN" font="italic">t</XMTok>
-                <XMTok meaning="2" role="NUMBER">2</XMTok>
+                <XMTok/>
+                <XMTok>t</XMTok>
+                <XMTok>2</XMTok>
               </XMApp>
-              <XMTok role="UNKNOWN" font="italic">t</XMTok>
+              <XMTok>t</XMTok>
             </XMApp>
-            <XMTok meaning="1" role="NUMBER">1</XMTok>
+            <XMTok>1</XMTok>
           </XMApp>
         </XMApp>
       </XMath>
     </annotation-xml>
     <annotation encoding="application/x-tex">m=t^{2}+t\pm 1</annotation>
   </semantics>
 </math>

Formula 20

( show / hide formula code)

Original

m = t 2 + t ± 2 ⁢ t + 1 3 m plus-or-minus superscript t 2 t 2 t 1 3 = m ± + t 2 t + ⁢ 2 t 1 3 m=t^{2}+t\pm\frac{2t+1}{3}

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="I1.i1.p1.m7" class="ltx_Math" alttext="m=t^{2}+t\pm\frac{2t+1}{3}" display="inline">
  <semantics>
    <mrow>
      <mi>m</mi>
      <mo>=</mo>
      <mrow>
        <msup>
          <mi>t</mi>
          <mn>2</mn>
        </msup>
        <mo>+</mo>
        <mi>t</mi>
        <mo>±</mo>
        <mfrac>
          <mrow>
            <mrow>
              <mn>2</mn>
              <mo>⁢</mo>
              <mi>t</mi>
            </mrow>
            <mo>+</mo>
            <mn>1</mn>
          </mrow>
          <mn>3</mn>
        </mfrac>
      </mrow>
    </mrow>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <eq/>
        <ci>m</ci>
        <apply>
          <csymbol cd="latexml">plus-or-minus</csymbol>
          <apply>
            <plus/>
            <apply>
              <csymbol cd="ambiguous">superscript</csymbol>
              <ci>t</ci>
              <cn type="integer">2</cn>
            </apply>
            <ci>t</ci>
          </apply>
          <apply>
            <divide/>
            <apply>
              <plus/>
              <apply>
                <times/>
                <cn type="integer">2</cn>
                <ci>t</ci>
              </apply>
              <cn type="integer">1</cn>
            </apply>
            <cn type="integer">3</cn>
          </apply>
        </apply>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok meaning="equals" role="RELOP">=</XMTok>
          <XMTok role="UNKNOWN" font="italic">m</XMTok>
          <XMApp>
            <XMTok meaning="plus-or-minus" name="pm" role="ADDOP">±</XMTok>
            <XMApp>
              <XMTok meaning="plus" role="ADDOP">+</XMTok>
              <XMApp>
                <XMTok role="SUPERSCRIPTOP" scriptpos="post4"/>
                <XMTok role="UNKNOWN" font="italic">t</XMTok>
                <XMTok meaning="2" role="NUMBER">2</XMTok>
              </XMApp>
              <XMTok role="UNKNOWN" font="italic">t</XMTok>
            </XMApp>
            <XMApp>
              <XMTok mathstyle="text" meaning="divide" role="MULOP"/>
              <XMApp>
                <XMTok meaning="plus" role="ADDOP">+</XMTok>
                <XMApp>
                  <XMTok meaning="times" role="MULOP">⁢</XMTok>
                  <XMTok meaning="2" role="NUMBER">2</XMTok>
                  <XMTok role="UNKNOWN" font="italic">t</XMTok>
                </XMApp>
                <XMTok meaning="1" role="NUMBER">1</XMTok>
              </XMApp>
              <XMTok meaning="3" role="NUMBER">3</XMTok>
            </XMApp>
          </XMApp>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">m=t^{2}+t\pm\frac{2t+1}{3}</annotation>
  </semantics>
</math>

Canonicalized

m = t 2 + t ± 2 t + 1 3 m plus-or-minus superscript t 2 t 2 t 1 3 = m ± + t 2 t + ⁢ 2 t 1 3 m=t^{2}+t\pm\frac{2t+1}{3}

(show/hide code)
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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics encoding="">
    <mrow>
      <mi>m</mi>
      <mo>=</mo>
      <mrow>
        <msup>
          <mi>t</mi>
          <mn>2</mn>
        </msup>
        <mo>+</mo>
        <mi>t</mi>
        <mo>±</mo>
        <mfrac>
          <mrow>
            <mrow>
              <mn>2</mn>
              <mi>t</mi>
            </mrow>
            <mo>+</mo>
            <mn>1</mn>
          </mrow>
          <mn>3</mn>
        </mfrac>
      </mrow>
    </mrow>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <eq/>
        <ci encoding="">m</ci>
        <apply>
          <csymbol encoding="">plus-or-minus</csymbol>
          <apply>
            <plus/>
            <apply>
              <csymbol encoding="">superscript</csymbol>
              <ci encoding="">t</ci>
              <cn encoding="" base="10" type="integer">2</cn>
            </apply>
            <ci encoding="">t</ci>
          </apply>
          <apply>
            <divide/>
            <apply>
              <plus/>
              <apply>
                <times/>
                <cn encoding="" base="10" type="integer">2</cn>
                <ci encoding="">t</ci>
              </apply>
              <cn encoding="" base="10" type="integer">1</cn>
            </apply>
            <cn encoding="" base="10" type="integer">3</cn>
          </apply>
        </apply>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok>=</XMTok>
          <XMTok>m</XMTok>
          <XMApp>
            <XMTok>±</XMTok>
            <XMApp>
              <XMTok>+</XMTok>
              <XMApp>
                <XMTok/>
                <XMTok>t</XMTok>
                <XMTok>2</XMTok>
              </XMApp>
              <XMTok>t</XMTok>
            </XMApp>
            <XMApp>
              <XMTok/>
              <XMApp>
                <XMTok>+</XMTok>
                <XMApp>
                  <XMTok>⁢</XMTok>
                  <XMTok>2</XMTok>
                  <XMTok>t</XMTok>
                </XMApp>
                <XMTok>1</XMTok>
              </XMApp>
              <XMTok>3</XMTok>
            </XMApp>
          </XMApp>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">m=t^{2}+t\pm\frac{2t+1}{3}</annotation>
  </semantics>
</math>

Diff

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@@ -1,95 +1,94 @@
 <?xml version="1.0" encoding="UTF-8"?>
-<math xmlns="http://www.w3.org/1998/Math/MathML" id="I1.i1.p1.m7" class="ltx_Math" alttext="m=t^{2}+t\pm\frac{2t+1}{3}" display="inline">
-  <semantics>
+<math xmlns="http://www.w3.org/1998/Math/MathML">
+  <semantics encoding="">
     <mrow>
       <mi>m</mi>
       <mo>=</mo>
       <mrow>
         <msup>
           <mi>t</mi>
           <mn>2</mn>
         </msup>
         <mo>+</mo>
         <mi>t</mi>
         <mo>±</mo>
         <mfrac>
           <mrow>
             <mrow>
               <mn>2</mn>
-              <mo>⁢</mo>
               <mi>t</mi>
             </mrow>
             <mo>+</mo>
             <mn>1</mn>
           </mrow>
           <mn>3</mn>
         </mfrac>
       </mrow>
     </mrow>
     <annotation-xml encoding="MathML-Content">
       <apply>
         <eq/>
-        <ci>m</ci>
+        <ci encoding="">m</ci>
         <apply>
-          <csymbol cd="latexml">plus-or-minus</csymbol>
+          <csymbol encoding="">plus-or-minus</csymbol>
           <apply>
             <plus/>
             <apply>
-              <csymbol cd="ambiguous">superscript</csymbol>
-              <ci>t</ci>
-              <cn type="integer">2</cn>
+              <csymbol encoding="">superscript</csymbol>
+              <ci encoding="">t</ci>
+              <cn encoding="" base="10" type="integer">2</cn>
             </apply>
-            <ci>t</ci>
+            <ci encoding="">t</ci>
           </apply>
           <apply>
             <divide/>
             <apply>
               <plus/>
               <apply>
                 <times/>
-                <cn type="integer">2</cn>
-                <ci>t</ci>
+                <cn encoding="" base="10" type="integer">2</cn>
+                <ci encoding="">t</ci>
               </apply>
-              <cn type="integer">1</cn>
+              <cn encoding="" base="10" type="integer">1</cn>
             </apply>
-            <cn type="integer">3</cn>
+            <cn encoding="" base="10" type="integer">3</cn>
           </apply>
         </apply>
       </apply>
     </annotation-xml>
     <annotation-xml encoding="application/x-latexml">
       <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
         <XMApp>
-          <XMTok meaning="equals" role="RELOP">=</XMTok>
-          <XMTok role="UNKNOWN" font="italic">m</XMTok>
+          <XMTok>=</XMTok>
+          <XMTok>m</XMTok>
           <XMApp>
-            <XMTok meaning="plus-or-minus" name="pm" role="ADDOP">±</XMTok>
+            <XMTok>±</XMTok>
             <XMApp>
-              <XMTok meaning="plus" role="ADDOP">+</XMTok>
+              <XMTok>+</XMTok>
               <XMApp>
-                <XMTok role="SUPERSCRIPTOP" scriptpos="post4"/>
-                <XMTok role="UNKNOWN" font="italic">t</XMTok>
-                <XMTok meaning="2" role="NUMBER">2</XMTok>
+                <XMTok/>
+                <XMTok>t</XMTok>
+                <XMTok>2</XMTok>
               </XMApp>
-              <XMTok role="UNKNOWN" font="italic">t</XMTok>
+              <XMTok>t</XMTok>
             </XMApp>
             <XMApp>
-              <XMTok mathstyle="text" meaning="divide" role="MULOP"/>
+              <XMTok/>
               <XMApp>
-                <XMTok meaning="plus" role="ADDOP">+</XMTok>
+                <XMTok>+</XMTok>
                 <XMApp>
-                  <XMTok meaning="times" role="MULOP">⁢</XMTok>
-                  <XMTok meaning="2" role="NUMBER">2</XMTok>
-                  <XMTok role="UNKNOWN" font="italic">t</XMTok>
+                  <XMTok>⁢</XMTok>
+                  <XMTok>2</XMTok>
+                  <XMTok>t</XMTok>
                 </XMApp>
-                <XMTok meaning="1" role="NUMBER">1</XMTok>
+                <XMTok>1</XMTok>
               </XMApp>
-              <XMTok meaning="3" role="NUMBER">3</XMTok>
+              <XMTok>3</XMTok>
             </XMApp>
           </XMApp>
         </XMApp>
       </XMath>
     </annotation-xml>
     <annotation encoding="application/x-tex">m=t^{2}+t\pm\frac{2t+1}{3}</annotation>
   </semantics>
 </math>

Formula 21

( show / hide formula code)

Original

2 ⁢ t + 1 3 2 t 1 3 + ⁢ 2 t 1 3 \frac{2t+1}{3}

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="I1.i1.p1.m8" class="ltx_Math" alttext="\frac{2t+1}{3}" display="inline">
  <semantics>
    <mfrac>
      <mrow>
        <mrow>
          <mn>2</mn>
          <mo>⁢</mo>
          <mi>t</mi>
        </mrow>
        <mo>+</mo>
        <mn>1</mn>
      </mrow>
      <mn>3</mn>
    </mfrac>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <divide/>
        <apply>
          <plus/>
          <apply>
            <times/>
            <cn type="integer">2</cn>
            <ci>t</ci>
          </apply>
          <cn type="integer">1</cn>
        </apply>
        <cn type="integer">3</cn>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok mathstyle="text" meaning="divide" role="MULOP"/>
          <XMApp>
            <XMTok meaning="plus" role="ADDOP">+</XMTok>
            <XMApp>
              <XMTok meaning="times" role="MULOP">⁢</XMTok>
              <XMTok meaning="2" role="NUMBER">2</XMTok>
              <XMTok role="UNKNOWN" font="italic">t</XMTok>
            </XMApp>
            <XMTok meaning="1" role="NUMBER">1</XMTok>
          </XMApp>
          <XMTok meaning="3" role="NUMBER">3</XMTok>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">\frac{2t+1}{3}</annotation>
  </semantics>
</math>

Canonicalized

2 t + 1 3 2 t 1 3 + ⁢ 2 t 1 3 \frac{2t+1}{3}

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics encoding="">
    <mfrac>
      <mrow>
        <mrow>
          <mn>2</mn>
          <mi>t</mi>
        </mrow>
        <mo>+</mo>
        <mn>1</mn>
      </mrow>
      <mn>3</mn>
    </mfrac>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <divide/>
        <apply>
          <plus/>
          <apply>
            <times/>
            <cn encoding="" base="10" type="integer">2</cn>
            <ci encoding="">t</ci>
          </apply>
          <cn encoding="" base="10" type="integer">1</cn>
        </apply>
        <cn encoding="" base="10" type="integer">3</cn>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok/>
          <XMApp>
            <XMTok>+</XMTok>
            <XMApp>
              <XMTok>⁢</XMTok>
              <XMTok>2</XMTok>
              <XMTok>t</XMTok>
            </XMApp>
            <XMTok>1</XMTok>
          </XMApp>
          <XMTok>3</XMTok>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">\frac{2t+1}{3}</annotation>
  </semantics>
</math>

Diff

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@@ -1,50 +1,49 @@
 <?xml version="1.0" encoding="UTF-8"?>
-<math xmlns="http://www.w3.org/1998/Math/MathML" id="I1.i1.p1.m8" class="ltx_Math" alttext="\frac{2t+1}{3}" display="inline">
-  <semantics>
+<math xmlns="http://www.w3.org/1998/Math/MathML">
+  <semantics encoding="">
     <mfrac>
       <mrow>
         <mrow>
           <mn>2</mn>
-          <mo>⁢</mo>
           <mi>t</mi>
         </mrow>
         <mo>+</mo>
         <mn>1</mn>
       </mrow>
       <mn>3</mn>
     </mfrac>
     <annotation-xml encoding="MathML-Content">
       <apply>
         <divide/>
         <apply>
           <plus/>
           <apply>
             <times/>
-            <cn type="integer">2</cn>
-            <ci>t</ci>
+            <cn encoding="" base="10" type="integer">2</cn>
+            <ci encoding="">t</ci>
           </apply>
-          <cn type="integer">1</cn>
+          <cn encoding="" base="10" type="integer">1</cn>
         </apply>
-        <cn type="integer">3</cn>
+        <cn encoding="" base="10" type="integer">3</cn>
       </apply>
     </annotation-xml>
     <annotation-xml encoding="application/x-latexml">
       <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
         <XMApp>
-          <XMTok mathstyle="text" meaning="divide" role="MULOP"/>
+          <XMTok/>
           <XMApp>
-            <XMTok meaning="plus" role="ADDOP">+</XMTok>
+            <XMTok>+</XMTok>
             <XMApp>
-              <XMTok meaning="times" role="MULOP">⁢</XMTok>
-              <XMTok meaning="2" role="NUMBER">2</XMTok>
-              <XMTok role="UNKNOWN" font="italic">t</XMTok>
+              <XMTok>⁢</XMTok>
+              <XMTok>2</XMTok>
+              <XMTok>t</XMTok>
             </XMApp>
-            <XMTok meaning="1" role="NUMBER">1</XMTok>
+            <XMTok>1</XMTok>
           </XMApp>
-          <XMTok meaning="3" role="NUMBER">3</XMTok>
+          <XMTok>3</XMTok>
         </XMApp>
       </XMath>
     </annotation-xml>
     <annotation encoding="application/x-tex">\frac{2t+1}{3}</annotation>
   </semantics>
 </math>

Formula 22

( show / hide formula code)

Original

f m subscript f m f m f_{m}

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="I1.i2.p1.m1" class="ltx_Math" alttext="f_{m}" display="inline">
  <semantics>
    <msub>
      <mi>f</mi>
      <mi>m</mi>
    </msub>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <csymbol cd="ambiguous">subscript</csymbol>
        <ci>f</ci>
        <ci>m</ci>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok role="SUBSCRIPTOP" scriptpos="post4"/>
          <XMTok role="UNKNOWN" font="italic">f</XMTok>
          <XMTok role="UNKNOWN" font="italic">m</XMTok>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">f_{m}</annotation>
  </semantics>
</math>

Canonicalized

f m subscript f m f m f_{m}

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics encoding="">
    <msub>
      <mi>f</mi>
      <mi>m</mi>
    </msub>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <csymbol encoding="">subscript</csymbol>
        <ci encoding="">f</ci>
        <ci encoding="">m</ci>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok/>
          <XMTok>f</XMTok>
          <XMTok>m</XMTok>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">f_{m}</annotation>
  </semantics>
</math>

Diff

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@@ -1,26 +1,26 @@
 <?xml version="1.0" encoding="UTF-8"?>
-<math xmlns="http://www.w3.org/1998/Math/MathML" id="I1.i2.p1.m1" class="ltx_Math" alttext="f_{m}" display="inline">
-  <semantics>
+<math xmlns="http://www.w3.org/1998/Math/MathML">
+  <semantics encoding="">
     <msub>
       <mi>f</mi>
       <mi>m</mi>
     </msub>
     <annotation-xml encoding="MathML-Content">
       <apply>
-        <csymbol cd="ambiguous">subscript</csymbol>
-        <ci>f</ci>
-        <ci>m</ci>
+        <csymbol encoding="">subscript</csymbol>
+        <ci encoding="">f</ci>
+        <ci encoding="">m</ci>
       </apply>
     </annotation-xml>
     <annotation-xml encoding="application/x-latexml">
       <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
         <XMApp>
-          <XMTok role="SUBSCRIPTOP" scriptpos="post4"/>
-          <XMTok role="UNKNOWN" font="italic">f</XMTok>
-          <XMTok role="UNKNOWN" font="italic">m</XMTok>
+          <XMTok/>
+          <XMTok>f</XMTok>
+          <XMTok>m</XMTok>
         </XMApp>
       </XMath>
     </annotation-xml>
     <annotation encoding="application/x-tex">f_{m}</annotation>
   </semantics>
 </math>

Formula 23

( show / hide formula code)

Original

ℚ ⁢ ( 4 ⁢ m + 1 ) ℚ 4 m 1 ⁢ Q + ⁢ 4 m 1 \mathbb{Q}(\sqrt{4m+1})

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="I1.i2.p1.m2" class="ltx_Math" alttext="\mathbb{Q}(\sqrt{4m+1})" display="inline">
  <semantics>
    <mrow>
      <mi>ℚ</mi>
      <mo>⁢</mo>
      <mrow>
        <mo>(</mo>
        <msqrt>
          <mrow>
            <mrow>
              <mn>4</mn>
              <mo>⁢</mo>
              <mi>m</mi>
            </mrow>
            <mo>+</mo>
            <mn>1</mn>
          </mrow>
        </msqrt>
        <mo>)</mo>
      </mrow>
    </mrow>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <times/>
        <ci>ℚ</ci>
        <apply>
          <root/>
          <apply>
            <plus/>
            <apply>
              <times/>
              <cn type="integer">4</cn>
              <ci>m</ci>
            </apply>
            <cn type="integer">1</cn>
          </apply>
        </apply>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok meaning="times" role="MULOP">⁢</XMTok>
          <XMTok possibleFunction="yes" role="UNKNOWN" font="blackboard">Q</XMTok>
          <XMApp close=")" open="(">
            <XMTok meaning="square-root"/>
            <XMApp>
              <XMTok meaning="plus" role="ADDOP">+</XMTok>
              <XMApp>
                <XMTok meaning="times" role="MULOP">⁢</XMTok>
                <XMTok meaning="4" role="NUMBER">4</XMTok>
                <XMTok role="UNKNOWN" font="italic">m</XMTok>
              </XMApp>
              <XMTok meaning="1" role="NUMBER">1</XMTok>
            </XMApp>
          </XMApp>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">\mathbb{Q}(\sqrt{4m+1})</annotation>
  </semantics>
</math>

Canonicalized

Q ( 4 m + 1 ) Q 4 m 1 ⁢ Q + ⁢ 4 m 1 \mathbb{Q}(\sqrt{4m+1})

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics encoding="">
    <mrow>
      <mi>Q</mi>
      <mrow>
        <mo>(</mo>
        <msqrt>
          <mrow>
            <mn>4</mn>
            <mi>m</mi>
          </mrow>
          <mo>+</mo>
          <mn>1</mn>
        </msqrt>
        <mo>)</mo>
      </mrow>
    </mrow>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <times/>
        <ci encoding="">Q</ci>
        <apply>
          <root/>
          <apply>
            <plus/>
            <apply>
              <times/>
              <cn encoding="" base="10" type="integer">4</cn>
              <ci encoding="">m</ci>
            </apply>
            <cn encoding="" base="10" type="integer">1</cn>
          </apply>
        </apply>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok>⁢</XMTok>
          <XMTok>Q</XMTok>
          <XMApp>
            <XMTok/>
            <XMApp>
              <XMTok>+</XMTok>
              <XMApp>
                <XMTok>⁢</XMTok>
                <XMTok>4</XMTok>
                <XMTok>m</XMTok>
              </XMApp>
              <XMTok>1</XMTok>
            </XMApp>
          </XMApp>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">\mathbb{Q}(\sqrt{4m+1})</annotation>
  </semantics>
</math>

Diff

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@@ -1,63 +1,59 @@
 <?xml version="1.0" encoding="UTF-8"?>
-<math xmlns="http://www.w3.org/1998/Math/MathML" id="I1.i2.p1.m2" class="ltx_Math" alttext="\mathbb{Q}(\sqrt{4m+1})" display="inline">
-  <semantics>
+<math xmlns="http://www.w3.org/1998/Math/MathML">
+  <semantics encoding="">
     <mrow>
-      <mi>ℚ</mi>
-      <mo>⁢</mo>
+      <mi>Q</mi>
       <mrow>
         <mo>(</mo>
         <msqrt>
           <mrow>
-            <mrow>
-              <mn>4</mn>
-              <mo>⁢</mo>
-              <mi>m</mi>
-            </mrow>
-            <mo>+</mo>
-            <mn>1</mn>
+            <mn>4</mn>
+            <mi>m</mi>
           </mrow>
+          <mo>+</mo>
+          <mn>1</mn>
         </msqrt>
         <mo>)</mo>
       </mrow>
     </mrow>
     <annotation-xml encoding="MathML-Content">
       <apply>
         <times/>
-        <ci>ℚ</ci>
+        <ci encoding="">Q</ci>
         <apply>
           <root/>
           <apply>
             <plus/>
             <apply>
               <times/>
-              <cn type="integer">4</cn>
-              <ci>m</ci>
+              <cn encoding="" base="10" type="integer">4</cn>
+              <ci encoding="">m</ci>
             </apply>
-            <cn type="integer">1</cn>
+            <cn encoding="" base="10" type="integer">1</cn>
           </apply>
         </apply>
       </apply>
     </annotation-xml>
     <annotation-xml encoding="application/x-latexml">
       <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
         <XMApp>
-          <XMTok meaning="times" role="MULOP">⁢</XMTok>
-          <XMTok possibleFunction="yes" role="UNKNOWN" font="blackboard">Q</XMTok>
-          <XMApp close=")" open="(">
-            <XMTok meaning="square-root"/>
+          <XMTok>⁢</XMTok>
+          <XMTok>Q</XMTok>
+          <XMApp>
+            <XMTok/>
             <XMApp>
-              <XMTok meaning="plus" role="ADDOP">+</XMTok>
+              <XMTok>+</XMTok>
               <XMApp>
-                <XMTok meaning="times" role="MULOP">⁢</XMTok>
-                <XMTok meaning="4" role="NUMBER">4</XMTok>
-                <XMTok role="UNKNOWN" font="italic">m</XMTok>
+                <XMTok>⁢</XMTok>
+                <XMTok>4</XMTok>
+                <XMTok>m</XMTok>
               </XMApp>
-              <XMTok meaning="1" role="NUMBER">1</XMTok>
+              <XMTok>1</XMTok>
             </XMApp>
           </XMApp>
         </XMApp>
       </XMath>
     </annotation-xml>
     <annotation encoding="application/x-tex">\mathbb{Q}(\sqrt{4m+1})</annotation>
   </semantics>
 </math>

Formula 24

( show / hide formula code)

Original

1 1 1 1

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="I1.i2.p1.m3" class="ltx_Math" alttext="1" display="inline">
  <semantics>
    <mn>1</mn>
    <annotation-xml encoding="MathML-Content">
      <cn type="integer">1</cn>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMTok meaning="1" role="NUMBER">1</XMTok>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">1</annotation>
  </semantics>
</math>

Canonicalized

1 1 1 1

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics encoding="">
    <mn>1</mn>
    <annotation-xml encoding="MathML-Content">
      <cn encoding="" base="10" type="integer">1</cn>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMTok>1</XMTok>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">1</annotation>
  </semantics>
</math>

Diff

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@@ -1,15 +1,15 @@
 <?xml version="1.0" encoding="UTF-8"?>
-<math xmlns="http://www.w3.org/1998/Math/MathML" id="I1.i2.p1.m3" class="ltx_Math" alttext="1" display="inline">
-  <semantics>
+<math xmlns="http://www.w3.org/1998/Math/MathML">
+  <semantics encoding="">
     <mn>1</mn>
     <annotation-xml encoding="MathML-Content">
-      <cn type="integer">1</cn>
+      <cn encoding="" base="10" type="integer">1</cn>
     </annotation-xml>
     <annotation-xml encoding="application/x-latexml">
       <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
-        <XMTok meaning="1" role="NUMBER">1</XMTok>
+        <XMTok>1</XMTok>
       </XMath>
     </annotation-xml>
     <annotation encoding="application/x-tex">1</annotation>
   </semantics>
 </math>

Formula 25

( show / hide formula code)

Original

m m m m

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="I1.i3.p1.m1" class="ltx_Math" alttext="m" display="inline">
  <semantics>
    <mi>m</mi>
    <annotation-xml encoding="MathML-Content">
      <ci>m</ci>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMTok role="UNKNOWN" font="italic">m</XMTok>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">m</annotation>
  </semantics>
</math>

Canonicalized

m m m m

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics encoding="">
    <mi>m</mi>
    <annotation-xml encoding="MathML-Content">
      <ci encoding="">m</ci>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMTok>m</XMTok>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">m</annotation>
  </semantics>
</math>

Diff

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@@ -1,15 +1,15 @@
 <?xml version="1.0" encoding="UTF-8"?>
-<math xmlns="http://www.w3.org/1998/Math/MathML" id="I1.i3.p1.m1" class="ltx_Math" alttext="m" display="inline">
-  <semantics>
+<math xmlns="http://www.w3.org/1998/Math/MathML">
+  <semantics encoding="">
     <mi>m</mi>
     <annotation-xml encoding="MathML-Content">
-      <ci>m</ci>
+      <ci encoding="">m</ci>
     </annotation-xml>
     <annotation-xml encoding="application/x-latexml">
       <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
-        <XMTok role="UNKNOWN" font="italic">m</XMTok>
+        <XMTok>m</XMTok>
       </XMath>
     </annotation-xml>
     <annotation encoding="application/x-tex">m</annotation>
   </semantics>
 </math>

Formula 26

( show / hide formula code)

Original

4 ⁢ m + 1 4 m 1 + ⁢ 4 m 1 4m+1

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="I1.i3.p1.m2" class="ltx_Math" alttext="4m+1" display="inline">
  <semantics>
    <mrow>
      <mrow>
        <mn>4</mn>
        <mo>⁢</mo>
        <mi>m</mi>
      </mrow>
      <mo>+</mo>
      <mn>1</mn>
    </mrow>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <plus/>
        <apply>
          <times/>
          <cn type="integer">4</cn>
          <ci>m</ci>
        </apply>
        <cn type="integer">1</cn>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok meaning="plus" role="ADDOP">+</XMTok>
          <XMApp>
            <XMTok meaning="times" role="MULOP">⁢</XMTok>
            <XMTok meaning="4" role="NUMBER">4</XMTok>
            <XMTok role="UNKNOWN" font="italic">m</XMTok>
          </XMApp>
          <XMTok meaning="1" role="NUMBER">1</XMTok>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">4m+1</annotation>
  </semantics>
</math>

Canonicalized

4 m + 1 4 m 1 + ⁢ 4 m 1 4m+1

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics encoding="">
    <mrow>
      <mrow>
        <mn>4</mn>
        <mi>m</mi>
      </mrow>
      <mo>+</mo>
      <mn>1</mn>
    </mrow>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <plus/>
        <apply>
          <times/>
          <cn encoding="" base="10" type="integer">4</cn>
          <ci encoding="">m</ci>
        </apply>
        <cn encoding="" base="10" type="integer">1</cn>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok>+</XMTok>
          <XMApp>
            <XMTok>⁢</XMTok>
            <XMTok>4</XMTok>
            <XMTok>m</XMTok>
          </XMApp>
          <XMTok>1</XMTok>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">4m+1</annotation>
  </semantics>
</math>

Diff

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@@ -1,39 +1,38 @@
 <?xml version="1.0" encoding="UTF-8"?>
-<math xmlns="http://www.w3.org/1998/Math/MathML" id="I1.i3.p1.m2" class="ltx_Math" alttext="4m+1" display="inline">
-  <semantics>
+<math xmlns="http://www.w3.org/1998/Math/MathML">
+  <semantics encoding="">
     <mrow>
       <mrow>
         <mn>4</mn>
-        <mo>⁢</mo>
         <mi>m</mi>
       </mrow>
       <mo>+</mo>
       <mn>1</mn>
     </mrow>
     <annotation-xml encoding="MathML-Content">
       <apply>
         <plus/>
         <apply>
           <times/>
-          <cn type="integer">4</cn>
-          <ci>m</ci>
+          <cn encoding="" base="10" type="integer">4</cn>
+          <ci encoding="">m</ci>
         </apply>
-        <cn type="integer">1</cn>
+        <cn encoding="" base="10" type="integer">1</cn>
       </apply>
     </annotation-xml>
     <annotation-xml encoding="application/x-latexml">
       <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
         <XMApp>
-          <XMTok meaning="plus" role="ADDOP">+</XMTok>
+          <XMTok>+</XMTok>
           <XMApp>
-            <XMTok meaning="times" role="MULOP">⁢</XMTok>
-            <XMTok meaning="4" role="NUMBER">4</XMTok>
-            <XMTok role="UNKNOWN" font="italic">m</XMTok>
+            <XMTok>⁢</XMTok>
+            <XMTok>4</XMTok>
+            <XMTok>m</XMTok>
           </XMApp>
-          <XMTok meaning="1" role="NUMBER">1</XMTok>
+          <XMTok>1</XMTok>
         </XMApp>
       </XMath>
     </annotation-xml>
     <annotation encoding="application/x-tex">4m+1</annotation>
   </semantics>
 </math>

Formula 27

( show / hide formula code)

Original

f m subscript f m f m f_{m}

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="I1.i3.p1.m3" class="ltx_Math" alttext="f_{m}" display="inline">
  <semantics>
    <msub>
      <mi>f</mi>
      <mi>m</mi>
    </msub>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <csymbol cd="ambiguous">subscript</csymbol>
        <ci>f</ci>
        <ci>m</ci>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok role="SUBSCRIPTOP" scriptpos="post4"/>
          <XMTok role="UNKNOWN" font="italic">f</XMTok>
          <XMTok role="UNKNOWN" font="italic">m</XMTok>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">f_{m}</annotation>
  </semantics>
</math>

Canonicalized

f m subscript f m f m f_{m}

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics encoding="">
    <msub>
      <mi>f</mi>
      <mi>m</mi>
    </msub>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <csymbol encoding="">subscript</csymbol>
        <ci encoding="">f</ci>
        <ci encoding="">m</ci>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok/>
          <XMTok>f</XMTok>
          <XMTok>m</XMTok>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">f_{m}</annotation>
  </semantics>
</math>

Diff

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@@ -1,26 +1,26 @@
 <?xml version="1.0" encoding="UTF-8"?>
-<math xmlns="http://www.w3.org/1998/Math/MathML" id="I1.i3.p1.m3" class="ltx_Math" alttext="f_{m}" display="inline">
-  <semantics>
+<math xmlns="http://www.w3.org/1998/Math/MathML">
+  <semantics encoding="">
     <msub>
       <mi>f</mi>
       <mi>m</mi>
     </msub>
     <annotation-xml encoding="MathML-Content">
       <apply>
-        <csymbol cd="ambiguous">subscript</csymbol>
-        <ci>f</ci>
-        <ci>m</ci>
+        <csymbol encoding="">subscript</csymbol>
+        <ci encoding="">f</ci>
+        <ci encoding="">m</ci>
       </apply>
     </annotation-xml>
     <annotation-xml encoding="application/x-latexml">
       <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
         <XMApp>
-          <XMTok role="SUBSCRIPTOP" scriptpos="post4"/>
-          <XMTok role="UNKNOWN" font="italic">f</XMTok>
-          <XMTok role="UNKNOWN" font="italic">m</XMTok>
+          <XMTok/>
+          <XMTok>f</XMTok>
+          <XMTok>m</XMTok>
         </XMApp>
       </XMath>
     </annotation-xml>
     <annotation encoding="application/x-tex">f_{m}</annotation>
   </semantics>
 </math>

Formula 28

( show / hide formula code)

Original

m m m m

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="p3.m1" class="ltx_Math" alttext="m" display="inline">
  <semantics>
    <mi>m</mi>
    <annotation-xml encoding="MathML-Content">
      <ci>m</ci>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMTok role="UNKNOWN" font="italic">m</XMTok>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">m</annotation>
  </semantics>
</math>

Canonicalized

m m m m

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics encoding="">
    <mi>m</mi>
    <annotation-xml encoding="MathML-Content">
      <ci encoding="">m</ci>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMTok>m</XMTok>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">m</annotation>
  </semantics>
</math>

Diff

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@@ -1,15 +1,15 @@
 <?xml version="1.0" encoding="UTF-8"?>
-<math xmlns="http://www.w3.org/1998/Math/MathML" id="p3.m1" class="ltx_Math" alttext="m" display="inline">
-  <semantics>
+<math xmlns="http://www.w3.org/1998/Math/MathML">
+  <semantics encoding="">
     <mi>m</mi>
     <annotation-xml encoding="MathML-Content">
-      <ci>m</ci>
+      <ci encoding="">m</ci>
     </annotation-xml>
     <annotation-xml encoding="application/x-latexml">
       <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
-        <XMTok role="UNKNOWN" font="italic">m</XMTok>
+        <XMTok>m</XMTok>
       </XMath>
     </annotation-xml>
     <annotation encoding="application/x-tex">m</annotation>
   </semantics>
 </math>

Formula 29

( show / hide formula code)

Original

4 ⁢ m + 1 4 m 1 + ⁢ 4 m 1 4m+1

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="p3.m2" class="ltx_Math" alttext="4m+1" display="inline">
  <semantics>
    <mrow>
      <mrow>
        <mn>4</mn>
        <mo>⁢</mo>
        <mi>m</mi>
      </mrow>
      <mo>+</mo>
      <mn>1</mn>
    </mrow>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <plus/>
        <apply>
          <times/>
          <cn type="integer">4</cn>
          <ci>m</ci>
        </apply>
        <cn type="integer">1</cn>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok meaning="plus" role="ADDOP">+</XMTok>
          <XMApp>
            <XMTok meaning="times" role="MULOP">⁢</XMTok>
            <XMTok meaning="4" role="NUMBER">4</XMTok>
            <XMTok role="UNKNOWN" font="italic">m</XMTok>
          </XMApp>
          <XMTok meaning="1" role="NUMBER">1</XMTok>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">4m+1</annotation>
  </semantics>
</math>

Canonicalized

4 m + 1 4 m 1 + ⁢ 4 m 1 4m+1

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics encoding="">
    <mrow>
      <mrow>
        <mn>4</mn>
        <mi>m</mi>
      </mrow>
      <mo>+</mo>
      <mn>1</mn>
    </mrow>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <plus/>
        <apply>
          <times/>
          <cn encoding="" base="10" type="integer">4</cn>
          <ci encoding="">m</ci>
        </apply>
        <cn encoding="" base="10" type="integer">1</cn>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok>+</XMTok>
          <XMApp>
            <XMTok>⁢</XMTok>
            <XMTok>4</XMTok>
            <XMTok>m</XMTok>
          </XMApp>
          <XMTok>1</XMTok>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">4m+1</annotation>
  </semantics>
</math>

Diff

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@@ -1,39 +1,38 @@
 <?xml version="1.0" encoding="UTF-8"?>
-<math xmlns="http://www.w3.org/1998/Math/MathML" id="p3.m2" class="ltx_Math" alttext="4m+1" display="inline">
-  <semantics>
+<math xmlns="http://www.w3.org/1998/Math/MathML">
+  <semantics encoding="">
     <mrow>
       <mrow>
         <mn>4</mn>
-        <mo>⁢</mo>
         <mi>m</mi>
       </mrow>
       <mo>+</mo>
       <mn>1</mn>
     </mrow>
     <annotation-xml encoding="MathML-Content">
       <apply>
         <plus/>
         <apply>
           <times/>
-          <cn type="integer">4</cn>
-          <ci>m</ci>
+          <cn encoding="" base="10" type="integer">4</cn>
+          <ci encoding="">m</ci>
         </apply>
-        <cn type="integer">1</cn>
+        <cn encoding="" base="10" type="integer">1</cn>
       </apply>
     </annotation-xml>
     <annotation-xml encoding="application/x-latexml">
       <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
         <XMApp>
-          <XMTok meaning="plus" role="ADDOP">+</XMTok>
+          <XMTok>+</XMTok>
           <XMApp>
-            <XMTok meaning="times" role="MULOP">⁢</XMTok>
-            <XMTok meaning="4" role="NUMBER">4</XMTok>
-            <XMTok role="UNKNOWN" font="italic">m</XMTok>
+            <XMTok>⁢</XMTok>
+            <XMTok>4</XMTok>
+            <XMTok>m</XMTok>
           </XMApp>
-          <XMTok meaning="1" role="NUMBER">1</XMTok>
+          <XMTok>1</XMTok>
         </XMApp>
       </XMath>
     </annotation-xml>
     <annotation encoding="application/x-tex">4m+1</annotation>
   </semantics>
 </math>

Formula 30

( show / hide formula code)

Original

m ≥ 0 m 0 ≥ m 0 m\geq 0

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="ThmTheo2.p1.m1" class="ltx_Math" alttext="m\geq 0" display="inline">
  <semantics>
    <mrow>
      <mi>m</mi>
      <mo mathvariant="normal">≥</mo>
      <mn mathvariant="normal">0</mn>
    </mrow>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <geq/>
        <ci>m</ci>
        <cn type="integer">0</cn>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok meaning="greater-than-or-equals" name="geq" role="RELOP" font="upright">≥</XMTok>
          <XMTok role="UNKNOWN">m</XMTok>
          <XMTok meaning="0" role="NUMBER" font="upright">0</XMTok>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">m\geq 0</annotation>
  </semantics>
</math>

Canonicalized

m ≥ 0 m 0 ≥ m 0 m\geq 0

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics encoding="">
    <mrow>
      <mi>m</mi>
      <mo mathvariant="normal">≥</mo>
      <mn mathvariant="normal">0</mn>
    </mrow>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <geq/>
        <ci encoding="">m</ci>
        <cn encoding="" base="10" type="integer">0</cn>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok>≥</XMTok>
          <XMTok>m</XMTok>
          <XMTok>0</XMTok>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">m\geq 0</annotation>
  </semantics>
</math>

Diff

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@@ -1,27 +1,27 @@
 <?xml version="1.0" encoding="UTF-8"?>
-<math xmlns="http://www.w3.org/1998/Math/MathML" id="ThmTheo2.p1.m1" class="ltx_Math" alttext="m\geq 0" display="inline">
-  <semantics>
+<math xmlns="http://www.w3.org/1998/Math/MathML">
+  <semantics encoding="">
     <mrow>
       <mi>m</mi>
       <mo mathvariant="normal">≥</mo>
       <mn mathvariant="normal">0</mn>
     </mrow>
     <annotation-xml encoding="MathML-Content">
       <apply>
         <geq/>
-        <ci>m</ci>
-        <cn type="integer">0</cn>
+        <ci encoding="">m</ci>
+        <cn encoding="" base="10" type="integer">0</cn>
       </apply>
     </annotation-xml>
     <annotation-xml encoding="application/x-latexml">
       <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
         <XMApp>
-          <XMTok meaning="greater-than-or-equals" name="geq" role="RELOP" font="upright">≥</XMTok>
-          <XMTok role="UNKNOWN">m</XMTok>
-          <XMTok meaning="0" role="NUMBER" font="upright">0</XMTok>
+          <XMTok>≥</XMTok>
+          <XMTok>m</XMTok>
+          <XMTok>0</XMTok>
         </XMApp>
       </XMath>
     </annotation-xml>
     <annotation encoding="application/x-tex">m\geq 0</annotation>
   </semantics>
 </math>

Formula 31

( show / hide formula code)

Original

f m subscript f m f m f_{m}

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="ThmTheo2.p1.m2" class="ltx_Math" alttext="f_{m}" display="inline">
  <semantics>
    <msub>
      <mi>f</mi>
      <mi>m</mi>
    </msub>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <csymbol cd="ambiguous">subscript</csymbol>
        <ci>f</ci>
        <ci>m</ci>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok role="SUBSCRIPTOP" scriptpos="post3"/>
          <XMTok role="UNKNOWN">f</XMTok>
          <XMTok role="UNKNOWN">m</XMTok>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">f_{m}</annotation>
  </semantics>
</math>

Canonicalized

f m subscript f m f m f_{m}

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics encoding="">
    <msub>
      <mi>f</mi>
      <mi>m</mi>
    </msub>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <csymbol encoding="">subscript</csymbol>
        <ci encoding="">f</ci>
        <ci encoding="">m</ci>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok/>
          <XMTok>f</XMTok>
          <XMTok>m</XMTok>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">f_{m}</annotation>
  </semantics>
</math>

Diff

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@@ -1,26 +1,26 @@
 <?xml version="1.0" encoding="UTF-8"?>
-<math xmlns="http://www.w3.org/1998/Math/MathML" id="ThmTheo2.p1.m2" class="ltx_Math" alttext="f_{m}" display="inline">
-  <semantics>
+<math xmlns="http://www.w3.org/1998/Math/MathML">
+  <semantics encoding="">
     <msub>
       <mi>f</mi>
       <mi>m</mi>
     </msub>
     <annotation-xml encoding="MathML-Content">
       <apply>
-        <csymbol cd="ambiguous">subscript</csymbol>
-        <ci>f</ci>
-        <ci>m</ci>
+        <csymbol encoding="">subscript</csymbol>
+        <ci encoding="">f</ci>
+        <ci encoding="">m</ci>
       </apply>
     </annotation-xml>
     <annotation-xml encoding="application/x-latexml">
       <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
         <XMApp>
-          <XMTok role="SUBSCRIPTOP" scriptpos="post3"/>
-          <XMTok role="UNKNOWN">f</XMTok>
-          <XMTok role="UNKNOWN">m</XMTok>
+          <XMTok/>
+          <XMTok>f</XMTok>
+          <XMTok>m</XMTok>
         </XMApp>
       </XMath>
     </annotation-xml>
     <annotation encoding="application/x-tex">f_{m}</annotation>
   </semantics>
 </math>

Formula 32

( show / hide formula code)

Original

4 ⁢ m + 1 = u 2 ⁢ D 4 m 1 superscript u 2 D = + ⁢ 4 m 1 ⁢ u 2 D 4m+1=u^{2}D

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="p4.m1" class="ltx_Math" alttext="4m+1=u^{2}D" display="inline">
  <semantics>
    <mrow>
      <mrow>
        <mrow>
          <mn>4</mn>
          <mo>⁢</mo>
          <mi>m</mi>
        </mrow>
        <mo>+</mo>
        <mn>1</mn>
      </mrow>
      <mo>=</mo>
      <mrow>
        <msup>
          <mi>u</mi>
          <mn>2</mn>
        </msup>
        <mo>⁢</mo>
        <mi>D</mi>
      </mrow>
    </mrow>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <eq/>
        <apply>
          <plus/>
          <apply>
            <times/>
            <cn type="integer">4</cn>
            <ci>m</ci>
          </apply>
          <cn type="integer">1</cn>
        </apply>
        <apply>
          <times/>
          <apply>
            <csymbol cd="ambiguous">superscript</csymbol>
            <ci>u</ci>
            <cn type="integer">2</cn>
          </apply>
          <ci>D</ci>
        </apply>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok meaning="equals" role="RELOP">=</XMTok>
          <XMApp>
            <XMTok meaning="plus" role="ADDOP">+</XMTok>
            <XMApp>
              <XMTok meaning="times" role="MULOP">⁢</XMTok>
              <XMTok meaning="4" role="NUMBER">4</XMTok>
              <XMTok role="UNKNOWN" font="italic">m</XMTok>
            </XMApp>
            <XMTok meaning="1" role="NUMBER">1</XMTok>
          </XMApp>
          <XMApp>
            <XMTok meaning="times" role="MULOP">⁢</XMTok>
            <XMApp>
              <XMTok role="SUPERSCRIPTOP" scriptpos="post2"/>
              <XMTok role="UNKNOWN" font="italic">u</XMTok>
              <XMTok meaning="2" role="NUMBER">2</XMTok>
            </XMApp>
            <XMTok role="UNKNOWN" font="italic">D</XMTok>
          </XMApp>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">4m+1=u^{2}D</annotation>
  </semantics>
</math>

Canonicalized

4 m + 1 = u 2 D 4 m 1 superscript u 2 D = + ⁢ 4 m 1 ⁢ u 2 D 4m+1=u^{2}D

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics encoding="">
    <mrow>
      <mrow>
        <mrow>
          <mn>4</mn>
          <mi>m</mi>
        </mrow>
        <mo>+</mo>
        <mn>1</mn>
      </mrow>
      <mo>=</mo>
      <mrow>
        <msup>
          <mi>u</mi>
          <mn>2</mn>
        </msup>
        <mi>D</mi>
      </mrow>
    </mrow>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <eq/>
        <apply>
          <plus/>
          <apply>
            <times/>
            <cn encoding="" base="10" type="integer">4</cn>
            <ci encoding="">m</ci>
          </apply>
          <cn encoding="" base="10" type="integer">1</cn>
        </apply>
        <apply>
          <times/>
          <apply>
            <csymbol encoding="">superscript</csymbol>
            <ci encoding="">u</ci>
            <cn encoding="" base="10" type="integer">2</cn>
          </apply>
          <ci encoding="">D</ci>
        </apply>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok>=</XMTok>
          <XMApp>
            <XMTok>+</XMTok>
            <XMApp>
              <XMTok>⁢</XMTok>
              <XMTok>4</XMTok>
              <XMTok>m</XMTok>
            </XMApp>
            <XMTok>1</XMTok>
          </XMApp>
          <XMApp>
            <XMTok>⁢</XMTok>
            <XMApp>
              <XMTok/>
              <XMTok>u</XMTok>
              <XMTok>2</XMTok>
            </XMApp>
            <XMTok>D</XMTok>
          </XMApp>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">4m+1=u^{2}D</annotation>
  </semantics>
</math>

Diff

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@@ -1,74 +1,72 @@
 <?xml version="1.0" encoding="UTF-8"?>
-<math xmlns="http://www.w3.org/1998/Math/MathML" id="p4.m1" class="ltx_Math" alttext="4m+1=u^{2}D" display="inline">
-  <semantics>
+<math xmlns="http://www.w3.org/1998/Math/MathML">
+  <semantics encoding="">
     <mrow>
       <mrow>
         <mrow>
           <mn>4</mn>
-          <mo>⁢</mo>
           <mi>m</mi>
         </mrow>
         <mo>+</mo>
         <mn>1</mn>
       </mrow>
       <mo>=</mo>
       <mrow>
         <msup>
           <mi>u</mi>
           <mn>2</mn>
         </msup>
-        <mo>⁢</mo>
         <mi>D</mi>
       </mrow>
     </mrow>
     <annotation-xml encoding="MathML-Content">
       <apply>
         <eq/>
         <apply>
           <plus/>
           <apply>
             <times/>
-            <cn type="integer">4</cn>
-            <ci>m</ci>
+            <cn encoding="" base="10" type="integer">4</cn>
+            <ci encoding="">m</ci>
           </apply>
-          <cn type="integer">1</cn>
+          <cn encoding="" base="10" type="integer">1</cn>
         </apply>
         <apply>
           <times/>
           <apply>
-            <csymbol cd="ambiguous">superscript</csymbol>
-            <ci>u</ci>
-            <cn type="integer">2</cn>
+            <csymbol encoding="">superscript</csymbol>
+            <ci encoding="">u</ci>
+            <cn encoding="" base="10" type="integer">2</cn>
           </apply>
-          <ci>D</ci>
+          <ci encoding="">D</ci>
         </apply>
       </apply>
     </annotation-xml>
     <annotation-xml encoding="application/x-latexml">
       <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
         <XMApp>
-          <XMTok meaning="equals" role="RELOP">=</XMTok>
+          <XMTok>=</XMTok>
           <XMApp>
-            <XMTok meaning="plus" role="ADDOP">+</XMTok>
+            <XMTok>+</XMTok>
             <XMApp>
-              <XMTok meaning="times" role="MULOP">⁢</XMTok>
-              <XMTok meaning="4" role="NUMBER">4</XMTok>
-              <XMTok role="UNKNOWN" font="italic">m</XMTok>
+              <XMTok>⁢</XMTok>
+              <XMTok>4</XMTok>
+              <XMTok>m</XMTok>
             </XMApp>
-            <XMTok meaning="1" role="NUMBER">1</XMTok>
+            <XMTok>1</XMTok>
           </XMApp>
           <XMApp>
-            <XMTok meaning="times" role="MULOP">⁢</XMTok>
+            <XMTok>⁢</XMTok>
             <XMApp>
-              <XMTok role="SUPERSCRIPTOP" scriptpos="post2"/>
-              <XMTok role="UNKNOWN" font="italic">u</XMTok>
-              <XMTok meaning="2" role="NUMBER">2</XMTok>
+              <XMTok/>
+              <XMTok>u</XMTok>
+              <XMTok>2</XMTok>
             </XMApp>
-            <XMTok role="UNKNOWN" font="italic">D</XMTok>
+            <XMTok>D</XMTok>
           </XMApp>
         </XMApp>
       </XMath>
     </annotation-xml>
     <annotation encoding="application/x-tex">4m+1=u^{2}D</annotation>
   </semantics>
 </math>

Formula 33

( show / hide formula code)

Original

D D D D

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="p4.m2" class="ltx_Math" alttext="D" display="inline">
  <semantics>
    <mi>D</mi>
    <annotation-xml encoding="MathML-Content">
      <ci>D</ci>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMTok role="UNKNOWN" font="italic">D</XMTok>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">D</annotation>
  </semantics>
</math>

Canonicalized

D D D D

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics encoding="">
    <mi>D</mi>
    <annotation-xml encoding="MathML-Content">
      <ci encoding="">D</ci>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMTok>D</XMTok>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">D</annotation>
  </semantics>
</math>

Diff

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@@ -1,15 +1,15 @@
 <?xml version="1.0" encoding="UTF-8"?>
-<math xmlns="http://www.w3.org/1998/Math/MathML" id="p4.m2" class="ltx_Math" alttext="D" display="inline">
-  <semantics>
+<math xmlns="http://www.w3.org/1998/Math/MathML">
+  <semantics encoding="">
     <mi>D</mi>
     <annotation-xml encoding="MathML-Content">
-      <ci>D</ci>
+      <ci encoding="">D</ci>
     </annotation-xml>
     <annotation-xml encoding="application/x-latexml">
       <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
-        <XMTok role="UNKNOWN" font="italic">D</XMTok>
+        <XMTok>D</XMTok>
       </XMath>
     </annotation-xml>
     <annotation encoding="application/x-tex">D</annotation>
   </semantics>
 </math>

Formula 34

( show / hide formula code)

Original

u u u u

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="p4.m3" class="ltx_Math" alttext="u" display="inline">
  <semantics>
    <mi>u</mi>
    <annotation-xml encoding="MathML-Content">
      <ci>u</ci>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMTok role="UNKNOWN" font="italic">u</XMTok>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">u</annotation>
  </semantics>
</math>

Canonicalized

u u u u

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics encoding="">
    <mi>u</mi>
    <annotation-xml encoding="MathML-Content">
      <ci encoding="">u</ci>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMTok>u</XMTok>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">u</annotation>
  </semantics>
</math>

Diff

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@@ -1,15 +1,15 @@
 <?xml version="1.0" encoding="UTF-8"?>
-<math xmlns="http://www.w3.org/1998/Math/MathML" id="p4.m3" class="ltx_Math" alttext="u" display="inline">
-  <semantics>
+<math xmlns="http://www.w3.org/1998/Math/MathML">
+  <semantics encoding="">
     <mi>u</mi>
     <annotation-xml encoding="MathML-Content">
-      <ci>u</ci>
+      <ci encoding="">u</ci>
     </annotation-xml>
     <annotation-xml encoding="application/x-latexml">
       <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
-        <XMTok role="UNKNOWN" font="italic">u</XMTok>
+        <XMTok>u</XMTok>
       </XMath>
     </annotation-xml>
     <annotation encoding="application/x-tex">u</annotation>
   </semantics>
 </math>

Formula 35

( show / hide formula code)

Original

D = 1 D 1 = D 1 D=1

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="p4.m4" class="ltx_Math" alttext="D=1" display="inline">
  <semantics>
    <mrow>
      <mi>D</mi>
      <mo>=</mo>
      <mn>1</mn>
    </mrow>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <eq/>
        <ci>D</ci>
        <cn type="integer">1</cn>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok meaning="equals" role="RELOP">=</XMTok>
          <XMTok role="UNKNOWN" font="italic">D</XMTok>
          <XMTok meaning="1" role="NUMBER">1</XMTok>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">D=1</annotation>
  </semantics>
</math>

Canonicalized

D = 1 D 1 = D 1 D=1

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics encoding="">
    <mrow>
      <mi>D</mi>
      <mo>=</mo>
      <mn>1</mn>
    </mrow>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <eq/>
        <ci encoding="">D</ci>
        <cn encoding="" base="10" type="integer">1</cn>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok>=</XMTok>
          <XMTok>D</XMTok>
          <XMTok>1</XMTok>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">D=1</annotation>
  </semantics>
</math>

Diff

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@@ -1,27 +1,27 @@
 <?xml version="1.0" encoding="UTF-8"?>
-<math xmlns="http://www.w3.org/1998/Math/MathML" id="p4.m4" class="ltx_Math" alttext="D=1" display="inline">
-  <semantics>
+<math xmlns="http://www.w3.org/1998/Math/MathML">
+  <semantics encoding="">
     <mrow>
       <mi>D</mi>
       <mo>=</mo>
       <mn>1</mn>
     </mrow>
     <annotation-xml encoding="MathML-Content">
       <apply>
         <eq/>
-        <ci>D</ci>
-        <cn type="integer">1</cn>
+        <ci encoding="">D</ci>
+        <cn encoding="" base="10" type="integer">1</cn>
       </apply>
     </annotation-xml>
     <annotation-xml encoding="application/x-latexml">
       <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
         <XMApp>
-          <XMTok meaning="equals" role="RELOP">=</XMTok>
-          <XMTok role="UNKNOWN" font="italic">D</XMTok>
-          <XMTok meaning="1" role="NUMBER">1</XMTok>
+          <XMTok>=</XMTok>
+          <XMTok>D</XMTok>
+          <XMTok>1</XMTok>
         </XMApp>
       </XMath>
     </annotation-xml>
     <annotation encoding="application/x-tex">D=1</annotation>
   </semantics>
 </math>

Formula 36

( show / hide formula code)

Original

1 < D < m 1 / 12 1 D superscript m 1 12 1 < D < m / 1 12 1<D<m^{1/12}

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="p4.m5" class="ltx_Math" alttext="1&lt;D&lt;m^{1/12}" display="inline">
  <semantics>
    <mrow>
      <mn>1</mn>
      <mo>&lt;</mo>
      <mi>D</mi>
      <mo>&lt;</mo>
      <msup>
        <mi>m</mi>
        <mrow>
          <mn>1</mn>
          <mo>/</mo>
          <mn>12</mn>
        </mrow>
      </msup>
    </mrow>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <and/>
        <apply>
          <lt/>
          <cn type="integer">1</cn>
          <ci xml:id="p4.m5.sh1.cmml">D</ci>
        </apply>
        <apply>
          <lt/>
          <share href="#p4.m5.sh1.cmml"/>
          <apply xml:id="p4.m5.sh2f.cmml">
            <csymbol cd="ambiguous" xml:id="p4.m5.sh2.cmml">superscript</csymbol>
            <ci xml:id="p4.m5.sh2a.cmml">m</ci>
            <apply xml:id="p4.m5.sh2e.cmml">
              <divide xml:id="p4.m5.sh2b.cmml"/>
              <cn type="integer" xml:id="p4.m5.sh2c.cmml">1</cn>
              <cn type="integer" xml:id="p4.m5.sh2d.cmml">12</cn>
            </apply>
          </apply>
        </apply>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok meaning="multirelation"/>
          <XMTok meaning="1" role="NUMBER">1</XMTok>
          <XMTok meaning="less-than" role="RELOP">&lt;</XMTok>
          <XMTok role="UNKNOWN" font="italic" xml:id="p4.m5.sh1.xm" fragid="p4.m5.sh1.xm">D</XMTok>
          <XMTok meaning="less-than" role="RELOP">&lt;</XMTok>
          <XMApp xml:id="p4.m5.sh2.xm" fragid="p4.m5.sh2.xm">
            <XMTok role="SUPERSCRIPTOP" scriptpos="post2"/>
            <XMTok role="UNKNOWN" font="italic">m</XMTok>
            <XMApp>
              <XMTok mathstyle="inline" meaning="divide" role="MULOP">/</XMTok>
              <XMTok meaning="1" role="NUMBER">1</XMTok>
              <XMTok meaning="12" role="NUMBER">12</XMTok>
            </XMApp>
          </XMApp>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">1&lt;D&lt;m^{1/12}</annotation>
  </semantics>
</math>

Canonicalized

1 < D < m 1 / 12 1 D superscript m 1 12 1 < D < m / 1 12 1<D<m^{1/12}

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics encoding="">
    <mrow>
      <mn>1</mn>
      <mo>&lt;</mo>
      <mi>D</mi>
      <mo>&lt;</mo>
      <msup>
        <mi>m</mi>
        <mrow>
          <mn>1</mn>
          <mo>/</mo>
          <mn>12</mn>
        </mrow>
      </msup>
    </mrow>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <and/>
        <apply>
          <lt/>
          <cn encoding="" base="10" type="integer">1</cn>
          <ci encoding="">D</ci>
        </apply>
        <apply>
          <lt/>
          <share/>
          <apply>
            <csymbol encoding="">superscript</csymbol>
            <ci encoding="">m</ci>
            <apply>
              <divide encoding=""/>
              <cn encoding="" base="10" type="integer">1</cn>
              <cn encoding="" base="10" type="integer">12</cn>
            </apply>
          </apply>
        </apply>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok/>
          <XMTok>1</XMTok>
          <XMTok>&lt;</XMTok>
          <XMTok>D</XMTok>
          <XMTok>&lt;</XMTok>
          <XMApp>
            <XMTok/>
            <XMTok>m</XMTok>
            <XMApp>
              <XMTok>/</XMTok>
              <XMTok>1</XMTok>
              <XMTok>12</XMTok>
            </XMApp>
          </XMApp>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">1&lt;D&lt;m^{1/12}</annotation>
  </semantics>
</math>

Diff

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@@ -1,63 +1,63 @@
 <?xml version="1.0" encoding="UTF-8"?>
-<math xmlns="http://www.w3.org/1998/Math/MathML" id="p4.m5" class="ltx_Math" alttext="1&lt;D&lt;m^{1/12}" display="inline">
-  <semantics>
+<math xmlns="http://www.w3.org/1998/Math/MathML">
+  <semantics encoding="">
     <mrow>
       <mn>1</mn>
       <mo>&lt;</mo>
       <mi>D</mi>
       <mo>&lt;</mo>
       <msup>
         <mi>m</mi>
         <mrow>
           <mn>1</mn>
           <mo>/</mo>
           <mn>12</mn>
         </mrow>
       </msup>
     </mrow>
     <annotation-xml encoding="MathML-Content">
       <apply>
         <and/>
         <apply>
           <lt/>
-          <cn type="integer">1</cn>
-          <ci xml:id="p4.m5.sh1.cmml">D</ci>
+          <cn encoding="" base="10" type="integer">1</cn>
+          <ci encoding="">D</ci>
         </apply>
         <apply>
           <lt/>
-          <share href="#p4.m5.sh1.cmml"/>
-          <apply xml:id="p4.m5.sh2f.cmml">
-            <csymbol cd="ambiguous" xml:id="p4.m5.sh2.cmml">superscript</csymbol>
-            <ci xml:id="p4.m5.sh2a.cmml">m</ci>
-            <apply xml:id="p4.m5.sh2e.cmml">
-              <divide xml:id="p4.m5.sh2b.cmml"/>
-              <cn type="integer" xml:id="p4.m5.sh2c.cmml">1</cn>
-              <cn type="integer" xml:id="p4.m5.sh2d.cmml">12</cn>
+          <share/>
+          <apply>
+            <csymbol encoding="">superscript</csymbol>
+            <ci encoding="">m</ci>
+            <apply>
+              <divide encoding=""/>
+              <cn encoding="" base="10" type="integer">1</cn>
+              <cn encoding="" base="10" type="integer">12</cn>
             </apply>
           </apply>
         </apply>
       </apply>
     </annotation-xml>
     <annotation-xml encoding="application/x-latexml">
       <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
         <XMApp>
-          <XMTok meaning="multirelation"/>
-          <XMTok meaning="1" role="NUMBER">1</XMTok>
-          <XMTok meaning="less-than" role="RELOP">&lt;</XMTok>
-          <XMTok role="UNKNOWN" font="italic" xml:id="p4.m5.sh1.xm" fragid="p4.m5.sh1.xm">D</XMTok>
-          <XMTok meaning="less-than" role="RELOP">&lt;</XMTok>
-          <XMApp xml:id="p4.m5.sh2.xm" fragid="p4.m5.sh2.xm">
-            <XMTok role="SUPERSCRIPTOP" scriptpos="post2"/>
-            <XMTok role="UNKNOWN" font="italic">m</XMTok>
+          <XMTok/>
+          <XMTok>1</XMTok>
+          <XMTok>&lt;</XMTok>
+          <XMTok>D</XMTok>
+          <XMTok>&lt;</XMTok>
+          <XMApp>
+            <XMTok/>
+            <XMTok>m</XMTok>
             <XMApp>
-              <XMTok mathstyle="inline" meaning="divide" role="MULOP">/</XMTok>
-              <XMTok meaning="1" role="NUMBER">1</XMTok>
-              <XMTok meaning="12" role="NUMBER">12</XMTok>
+              <XMTok>/</XMTok>
+              <XMTok>1</XMTok>
+              <XMTok>12</XMTok>
             </XMApp>
           </XMApp>
         </XMApp>
       </XMath>
     </annotation-xml>
     <annotation encoding="application/x-tex">1&lt;D&lt;m^{1/12}</annotation>
   </semantics>
 </math>

Formula 37

( show / hide formula code)

Original

D ≥ m 1 / 12 D superscript m 1 12 ≥ D m / 1 12 D\geq m^{1/12}

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="p4.m6" class="ltx_Math" alttext="D\geq m^{1/12}" display="inline">
  <semantics>
    <mrow>
      <mi>D</mi>
      <mo>≥</mo>
      <msup>
        <mi>m</mi>
        <mrow>
          <mn>1</mn>
          <mo>/</mo>
          <mn>12</mn>
        </mrow>
      </msup>
    </mrow>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <geq/>
        <ci>D</ci>
        <apply>
          <csymbol cd="ambiguous">superscript</csymbol>
          <ci>m</ci>
          <apply>
            <divide/>
            <cn type="integer">1</cn>
            <cn type="integer">12</cn>
          </apply>
        </apply>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok meaning="greater-than-or-equals" name="geq" role="RELOP">≥</XMTok>
          <XMTok role="UNKNOWN" font="italic">D</XMTok>
          <XMApp>
            <XMTok role="SUPERSCRIPTOP" scriptpos="post2"/>
            <XMTok role="UNKNOWN" font="italic">m</XMTok>
            <XMApp>
              <XMTok mathstyle="inline" meaning="divide" role="MULOP">/</XMTok>
              <XMTok meaning="1" role="NUMBER">1</XMTok>
              <XMTok meaning="12" role="NUMBER">12</XMTok>
            </XMApp>
          </XMApp>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">D\geq m^{1/12}</annotation>
  </semantics>
</math>

Canonicalized

D ≥ m 1 / 12 D superscript m 1 12 ≥ D m / 1 12 D\geq m^{1/12}

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics encoding="">
    <mrow>
      <mi>D</mi>
      <mo>≥</mo>
      <msup>
        <mi>m</mi>
        <mrow>
          <mn>1</mn>
          <mo>/</mo>
          <mn>12</mn>
        </mrow>
      </msup>
    </mrow>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <geq/>
        <ci encoding="">D</ci>
        <apply>
          <csymbol encoding="">superscript</csymbol>
          <ci encoding="">m</ci>
          <apply>
            <divide/>
            <cn encoding="" base="10" type="integer">1</cn>
            <cn encoding="" base="10" type="integer">12</cn>
          </apply>
        </apply>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok>≥</XMTok>
          <XMTok>D</XMTok>
          <XMApp>
            <XMTok/>
            <XMTok>m</XMTok>
            <XMApp>
              <XMTok>/</XMTok>
              <XMTok>1</XMTok>
              <XMTok>12</XMTok>
            </XMApp>
          </XMApp>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">D\geq m^{1/12}</annotation>
  </semantics>
</math>

Diff

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@@ -1,50 +1,50 @@
 <?xml version="1.0" encoding="UTF-8"?>
-<math xmlns="http://www.w3.org/1998/Math/MathML" id="p4.m6" class="ltx_Math" alttext="D\geq m^{1/12}" display="inline">
-  <semantics>
+<math xmlns="http://www.w3.org/1998/Math/MathML">
+  <semantics encoding="">
     <mrow>
       <mi>D</mi>
       <mo>≥</mo>
       <msup>
         <mi>m</mi>
         <mrow>
           <mn>1</mn>
           <mo>/</mo>
           <mn>12</mn>
         </mrow>
       </msup>
     </mrow>
     <annotation-xml encoding="MathML-Content">
       <apply>
         <geq/>
-        <ci>D</ci>
+        <ci encoding="">D</ci>
         <apply>
-          <csymbol cd="ambiguous">superscript</csymbol>
-          <ci>m</ci>
+          <csymbol encoding="">superscript</csymbol>
+          <ci encoding="">m</ci>
           <apply>
             <divide/>
-            <cn type="integer">1</cn>
-            <cn type="integer">12</cn>
+            <cn encoding="" base="10" type="integer">1</cn>
+            <cn encoding="" base="10" type="integer">12</cn>
           </apply>
         </apply>
       </apply>
     </annotation-xml>
     <annotation-xml encoding="application/x-latexml">
       <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
         <XMApp>
-          <XMTok meaning="greater-than-or-equals" name="geq" role="RELOP">≥</XMTok>
-          <XMTok role="UNKNOWN" font="italic">D</XMTok>
+          <XMTok>≥</XMTok>
+          <XMTok>D</XMTok>
           <XMApp>
-            <XMTok role="SUPERSCRIPTOP" scriptpos="post2"/>
-            <XMTok role="UNKNOWN" font="italic">m</XMTok>
+            <XMTok/>
+            <XMTok>m</XMTok>
             <XMApp>
-              <XMTok mathstyle="inline" meaning="divide" role="MULOP">/</XMTok>
-              <XMTok meaning="1" role="NUMBER">1</XMTok>
-              <XMTok meaning="12" role="NUMBER">12</XMTok>
+              <XMTok>/</XMTok>
+              <XMTok>1</XMTok>
+              <XMTok>12</XMTok>
             </XMApp>
           </XMApp>
         </XMApp>
       </XMath>
     </annotation-xml>
     <annotation encoding="application/x-tex">D\geq m^{1/12}</annotation>
   </semantics>
 </math>

Formula 38

( show / hide formula code)

Original

4 ⁢ m + 1 4 m 1 + ⁢ 4 m 1 4m+1

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="p4.m7" class="ltx_Math" alttext="4m+1" display="inline">
  <semantics>
    <mrow>
      <mrow>
        <mn>4</mn>
        <mo>⁢</mo>
        <mi>m</mi>
      </mrow>
      <mo>+</mo>
      <mn>1</mn>
    </mrow>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <plus/>
        <apply>
          <times/>
          <cn type="integer">4</cn>
          <ci>m</ci>
        </apply>
        <cn type="integer">1</cn>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok meaning="plus" role="ADDOP">+</XMTok>
          <XMApp>
            <XMTok meaning="times" role="MULOP">⁢</XMTok>
            <XMTok meaning="4" role="NUMBER">4</XMTok>
            <XMTok role="UNKNOWN" font="italic">m</XMTok>
          </XMApp>
          <XMTok meaning="1" role="NUMBER">1</XMTok>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">4m+1</annotation>
  </semantics>
</math>

Canonicalized

4 m + 1 4 m 1 + ⁢ 4 m 1 4m+1

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics encoding="">
    <mrow>
      <mrow>
        <mn>4</mn>
        <mi>m</mi>
      </mrow>
      <mo>+</mo>
      <mn>1</mn>
    </mrow>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <plus/>
        <apply>
          <times/>
          <cn encoding="" base="10" type="integer">4</cn>
          <ci encoding="">m</ci>
        </apply>
        <cn encoding="" base="10" type="integer">1</cn>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok>+</XMTok>
          <XMApp>
            <XMTok>⁢</XMTok>
            <XMTok>4</XMTok>
            <XMTok>m</XMTok>
          </XMApp>
          <XMTok>1</XMTok>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">4m+1</annotation>
  </semantics>
</math>

Diff

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@@ -1,39 +1,38 @@
 <?xml version="1.0" encoding="UTF-8"?>
-<math xmlns="http://www.w3.org/1998/Math/MathML" id="p4.m7" class="ltx_Math" alttext="4m+1" display="inline">
-  <semantics>
+<math xmlns="http://www.w3.org/1998/Math/MathML">
+  <semantics encoding="">
     <mrow>
       <mrow>
         <mn>4</mn>
-        <mo>⁢</mo>
         <mi>m</mi>
       </mrow>
       <mo>+</mo>
       <mn>1</mn>
     </mrow>
     <annotation-xml encoding="MathML-Content">
       <apply>
         <plus/>
         <apply>
           <times/>
-          <cn type="integer">4</cn>
-          <ci>m</ci>
+          <cn encoding="" base="10" type="integer">4</cn>
+          <ci encoding="">m</ci>
         </apply>
-        <cn type="integer">1</cn>
+        <cn encoding="" base="10" type="integer">1</cn>
       </apply>
     </annotation-xml>
     <annotation-xml encoding="application/x-latexml">
       <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
         <XMApp>
-          <XMTok meaning="plus" role="ADDOP">+</XMTok>
+          <XMTok>+</XMTok>
           <XMApp>
-            <XMTok meaning="times" role="MULOP">⁢</XMTok>
-            <XMTok meaning="4" role="NUMBER">4</XMTok>
-            <XMTok role="UNKNOWN" font="italic">m</XMTok>
+            <XMTok>⁢</XMTok>
+            <XMTok>4</XMTok>
+            <XMTok>m</XMTok>
           </XMApp>
-          <XMTok meaning="1" role="NUMBER">1</XMTok>
+          <XMTok>1</XMTok>
         </XMApp>
       </XMath>
     </annotation-xml>
     <annotation encoding="application/x-tex">4m+1</annotation>
   </semantics>
 </math>

Formula 39

( show / hide formula code)

Original

f m subscript f m f m f_{m}

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="ThmLem1.p1.m1" class="ltx_Math" alttext="f_{m}" display="inline">
  <semantics>
    <msub>
      <mi>f</mi>
      <mi>m</mi>
    </msub>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <csymbol cd="ambiguous">subscript</csymbol>
        <ci>f</ci>
        <ci>m</ci>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok role="SUBSCRIPTOP" scriptpos="post3"/>
          <XMTok role="UNKNOWN">f</XMTok>
          <XMTok role="UNKNOWN">m</XMTok>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">f_{m}</annotation>
  </semantics>
</math>

Canonicalized

f m subscript f m f m f_{m}

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics encoding="">
    <msub>
      <mi>f</mi>
      <mi>m</mi>
    </msub>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <csymbol encoding="">subscript</csymbol>
        <ci encoding="">f</ci>
        <ci encoding="">m</ci>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok/>
          <XMTok>f</XMTok>
          <XMTok>m</XMTok>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">f_{m}</annotation>
  </semantics>
</math>

Diff

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@@ -1,26 +1,26 @@
 <?xml version="1.0" encoding="UTF-8"?>
-<math xmlns="http://www.w3.org/1998/Math/MathML" id="ThmLem1.p1.m1" class="ltx_Math" alttext="f_{m}" display="inline">
-  <semantics>
+<math xmlns="http://www.w3.org/1998/Math/MathML">
+  <semantics encoding="">
     <msub>
       <mi>f</mi>
       <mi>m</mi>
     </msub>
     <annotation-xml encoding="MathML-Content">
       <apply>
-        <csymbol cd="ambiguous">subscript</csymbol>
-        <ci>f</ci>
-        <ci>m</ci>
+        <csymbol encoding="">subscript</csymbol>
+        <ci encoding="">f</ci>
+        <ci encoding="">m</ci>
       </apply>
     </annotation-xml>
     <annotation-xml encoding="application/x-latexml">
       <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
         <XMApp>
-          <XMTok role="SUBSCRIPTOP" scriptpos="post3"/>
-          <XMTok role="UNKNOWN">f</XMTok>
-          <XMTok role="UNKNOWN">m</XMTok>
+          <XMTok/>
+          <XMTok>f</XMTok>
+          <XMTok>m</XMTok>
         </XMApp>
       </XMath>
     </annotation-xml>
     <annotation encoding="application/x-tex">f_{m}</annotation>
   </semantics>
 </math>

Formula 40

( show / hide formula code)

Original

D = 1 D 1 = D 1 D=1

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="ThmLem1.p1.m2" class="ltx_Math" alttext="D=1" display="inline">
  <semantics>
    <mrow>
      <mi>D</mi>
      <mo mathvariant="normal">=</mo>
      <mn mathvariant="normal">1</mn>
    </mrow>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <eq/>
        <ci>D</ci>
        <cn type="integer">1</cn>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok meaning="equals" role="RELOP" font="upright">=</XMTok>
          <XMTok role="UNKNOWN">D</XMTok>
          <XMTok meaning="1" role="NUMBER" font="upright">1</XMTok>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">D=1</annotation>
  </semantics>
</math>

Canonicalized

D = 1 D 1 = D 1 D=1

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics encoding="">
    <mrow>
      <mi>D</mi>
      <mo mathvariant="normal">=</mo>
      <mn mathvariant="normal">1</mn>
    </mrow>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <eq/>
        <ci encoding="">D</ci>
        <cn encoding="" base="10" type="integer">1</cn>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok>=</XMTok>
          <XMTok>D</XMTok>
          <XMTok>1</XMTok>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">D=1</annotation>
  </semantics>
</math>

Diff

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@@ -1,27 +1,27 @@
 <?xml version="1.0" encoding="UTF-8"?>
-<math xmlns="http://www.w3.org/1998/Math/MathML" id="ThmLem1.p1.m2" class="ltx_Math" alttext="D=1" display="inline">
-  <semantics>
+<math xmlns="http://www.w3.org/1998/Math/MathML">
+  <semantics encoding="">
     <mrow>
       <mi>D</mi>
       <mo mathvariant="normal">=</mo>
       <mn mathvariant="normal">1</mn>
     </mrow>
     <annotation-xml encoding="MathML-Content">
       <apply>
         <eq/>
-        <ci>D</ci>
-        <cn type="integer">1</cn>
+        <ci encoding="">D</ci>
+        <cn encoding="" base="10" type="integer">1</cn>
       </apply>
     </annotation-xml>
     <annotation-xml encoding="application/x-latexml">
       <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
         <XMApp>
-          <XMTok meaning="equals" role="RELOP" font="upright">=</XMTok>
-          <XMTok role="UNKNOWN">D</XMTok>
-          <XMTok meaning="1" role="NUMBER" font="upright">1</XMTok>
+          <XMTok>=</XMTok>
+          <XMTok>D</XMTok>
+          <XMTok>1</XMTok>
         </XMApp>
       </XMath>
     </annotation-xml>
     <annotation encoding="application/x-tex">D=1</annotation>
   </semantics>
 </math>

Formula 41

( show / hide formula code)

Original

m = 2 m 2 = m 2 m=2

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="ThmLem1.p1.m3" class="ltx_Math" alttext="m=2" display="inline">
  <semantics>
    <mrow>
      <mi>m</mi>
      <mo mathvariant="normal">=</mo>
      <mn mathvariant="normal">2</mn>
    </mrow>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <eq/>
        <ci>m</ci>
        <cn type="integer">2</cn>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok meaning="equals" role="RELOP" font="upright">=</XMTok>
          <XMTok role="UNKNOWN">m</XMTok>
          <XMTok meaning="2" role="NUMBER" font="upright">2</XMTok>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">m=2</annotation>
  </semantics>
</math>

Canonicalized

m = 2 m 2 = m 2 m=2

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics encoding="">
    <mrow>
      <mi>m</mi>
      <mo mathvariant="normal">=</mo>
      <mn mathvariant="normal">2</mn>
    </mrow>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <eq/>
        <ci encoding="">m</ci>
        <cn encoding="" base="10" type="integer">2</cn>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok>=</XMTok>
          <XMTok>m</XMTok>
          <XMTok>2</XMTok>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">m=2</annotation>
  </semantics>
</math>

Diff

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@@ -1,27 +1,27 @@
 <?xml version="1.0" encoding="UTF-8"?>
-<math xmlns="http://www.w3.org/1998/Math/MathML" id="ThmLem1.p1.m3" class="ltx_Math" alttext="m=2" display="inline">
-  <semantics>
+<math xmlns="http://www.w3.org/1998/Math/MathML">
+  <semantics encoding="">
     <mrow>
       <mi>m</mi>
       <mo mathvariant="normal">=</mo>
       <mn mathvariant="normal">2</mn>
     </mrow>
     <annotation-xml encoding="MathML-Content">
       <apply>
         <eq/>
-        <ci>m</ci>
-        <cn type="integer">2</cn>
+        <ci encoding="">m</ci>
+        <cn encoding="" base="10" type="integer">2</cn>
       </apply>
     </annotation-xml>
     <annotation-xml encoding="application/x-latexml">
       <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
         <XMApp>
-          <XMTok meaning="equals" role="RELOP" font="upright">=</XMTok>
-          <XMTok role="UNKNOWN">m</XMTok>
-          <XMTok meaning="2" role="NUMBER" font="upright">2</XMTok>
+          <XMTok>=</XMTok>
+          <XMTok>m</XMTok>
+          <XMTok>2</XMTok>
         </XMApp>
       </XMath>
     </annotation-xml>
     <annotation encoding="application/x-tex">m=2</annotation>
   </semantics>
 </math>

Formula 42

( show / hide formula code)

Original

m = t 2 + t + 2 ⁢ t + 1 3 m superscript t 2 t 2 t 1 3 = m + t 2 t + ⁢ 2 t 1 3 m=t^{2}+t+\frac{2t+1}{3}

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="p5.m1" class="ltx_Math" alttext="m=t^{2}+t+\frac{2t+1}{3}" display="inline">
  <semantics>
    <mrow>
      <mi>m</mi>
      <mo>=</mo>
      <mrow>
        <msup>
          <mi>t</mi>
          <mn>2</mn>
        </msup>
        <mo>+</mo>
        <mi>t</mi>
        <mo>+</mo>
        <mfrac>
          <mrow>
            <mrow>
              <mn>2</mn>
              <mo>⁢</mo>
              <mi>t</mi>
            </mrow>
            <mo>+</mo>
            <mn>1</mn>
          </mrow>
          <mn>3</mn>
        </mfrac>
      </mrow>
    </mrow>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <eq/>
        <ci>m</ci>
        <apply>
          <plus/>
          <apply>
            <csymbol cd="ambiguous">superscript</csymbol>
            <ci>t</ci>
            <cn type="integer">2</cn>
          </apply>
          <ci>t</ci>
          <apply>
            <divide/>
            <apply>
              <plus/>
              <apply>
                <times/>
                <cn type="integer">2</cn>
                <ci>t</ci>
              </apply>
              <cn type="integer">1</cn>
            </apply>
            <cn type="integer">3</cn>
          </apply>
        </apply>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok meaning="equals" role="RELOP">=</XMTok>
          <XMTok role="UNKNOWN" font="italic">m</XMTok>
          <XMApp>
            <XMTok meaning="plus" role="ADDOP">+</XMTok>
            <XMApp>
              <XMTok role="SUPERSCRIPTOP" scriptpos="post2"/>
              <XMTok role="UNKNOWN" font="italic">t</XMTok>
              <XMTok meaning="2" role="NUMBER">2</XMTok>
            </XMApp>
            <XMTok role="UNKNOWN" font="italic">t</XMTok>
            <XMApp>
              <XMTok mathstyle="text" meaning="divide" role="MULOP"/>
              <XMApp>
                <XMTok meaning="plus" role="ADDOP">+</XMTok>
                <XMApp>
                  <XMTok meaning="times" role="MULOP">⁢</XMTok>
                  <XMTok meaning="2" role="NUMBER">2</XMTok>
                  <XMTok role="UNKNOWN" font="italic">t</XMTok>
                </XMApp>
                <XMTok meaning="1" role="NUMBER">1</XMTok>
              </XMApp>
              <XMTok meaning="3" role="NUMBER">3</XMTok>
            </XMApp>
          </XMApp>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">m=t^{2}+t+\frac{2t+1}{3}</annotation>
  </semantics>
</math>

Canonicalized

m = t 2 + t + 2 t + 1 3 m superscript t 2 t 2 t 1 3 = m + t 2 t + ⁢ 2 t 1 3 m=t^{2}+t+\frac{2t+1}{3}

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics encoding="">
    <mrow>
      <mi>m</mi>
      <mo>=</mo>
      <mrow>
        <msup>
          <mi>t</mi>
          <mn>2</mn>
        </msup>
        <mo>+</mo>
        <mi>t</mi>
        <mo>+</mo>
        <mfrac>
          <mrow>
            <mrow>
              <mn>2</mn>
              <mi>t</mi>
            </mrow>
            <mo>+</mo>
            <mn>1</mn>
          </mrow>
          <mn>3</mn>
        </mfrac>
      </mrow>
    </mrow>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <eq/>
        <ci encoding="">m</ci>
        <apply>
          <plus/>
          <apply>
            <csymbol encoding="">superscript</csymbol>
            <ci encoding="">t</ci>
            <cn encoding="" base="10" type="integer">2</cn>
          </apply>
          <ci encoding="">t</ci>
          <apply>
            <divide/>
            <apply>
              <plus/>
              <apply>
                <times/>
                <cn encoding="" base="10" type="integer">2</cn>
                <ci encoding="">t</ci>
              </apply>
              <cn encoding="" base="10" type="integer">1</cn>
            </apply>
            <cn encoding="" base="10" type="integer">3</cn>
          </apply>
        </apply>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok>=</XMTok>
          <XMTok>m</XMTok>
          <XMApp>
            <XMTok>+</XMTok>
            <XMApp>
              <XMTok/>
              <XMTok>t</XMTok>
              <XMTok>2</XMTok>
            </XMApp>
            <XMTok>t</XMTok>
            <XMApp>
              <XMTok/>
              <XMApp>
                <XMTok>+</XMTok>
                <XMApp>
                  <XMTok>⁢</XMTok>
                  <XMTok>2</XMTok>
                  <XMTok>t</XMTok>
                </XMApp>
                <XMTok>1</XMTok>
              </XMApp>
              <XMTok>3</XMTok>
            </XMApp>
          </XMApp>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">m=t^{2}+t+\frac{2t+1}{3}</annotation>
  </semantics>
</math>

Diff

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@@ -1,89 +1,88 @@
 <?xml version="1.0" encoding="UTF-8"?>
-<math xmlns="http://www.w3.org/1998/Math/MathML" id="p5.m1" class="ltx_Math" alttext="m=t^{2}+t+\frac{2t+1}{3}" display="inline">
-  <semantics>
+<math xmlns="http://www.w3.org/1998/Math/MathML">
+  <semantics encoding="">
     <mrow>
       <mi>m</mi>
       <mo>=</mo>
       <mrow>
         <msup>
           <mi>t</mi>
           <mn>2</mn>
         </msup>
         <mo>+</mo>
         <mi>t</mi>
         <mo>+</mo>
         <mfrac>
           <mrow>
             <mrow>
               <mn>2</mn>
-              <mo>⁢</mo>
               <mi>t</mi>
             </mrow>
             <mo>+</mo>
             <mn>1</mn>
           </mrow>
           <mn>3</mn>
         </mfrac>
       </mrow>
     </mrow>
     <annotation-xml encoding="MathML-Content">
       <apply>
         <eq/>
-        <ci>m</ci>
+        <ci encoding="">m</ci>
         <apply>
           <plus/>
           <apply>
-            <csymbol cd="ambiguous">superscript</csymbol>
-            <ci>t</ci>
-            <cn type="integer">2</cn>
+            <csymbol encoding="">superscript</csymbol>
+            <ci encoding="">t</ci>
+            <cn encoding="" base="10" type="integer">2</cn>
           </apply>
-          <ci>t</ci>
+          <ci encoding="">t</ci>
           <apply>
             <divide/>
             <apply>
               <plus/>
               <apply>
                 <times/>
-                <cn type="integer">2</cn>
-                <ci>t</ci>
+                <cn encoding="" base="10" type="integer">2</cn>
+                <ci encoding="">t</ci>
               </apply>
-              <cn type="integer">1</cn>
+              <cn encoding="" base="10" type="integer">1</cn>
             </apply>
-            <cn type="integer">3</cn>
+            <cn encoding="" base="10" type="integer">3</cn>
           </apply>
         </apply>
       </apply>
     </annotation-xml>
     <annotation-xml encoding="application/x-latexml">
       <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
         <XMApp>
-          <XMTok meaning="equals" role="RELOP">=</XMTok>
-          <XMTok role="UNKNOWN" font="italic">m</XMTok>
+          <XMTok>=</XMTok>
+          <XMTok>m</XMTok>
           <XMApp>
-            <XMTok meaning="plus" role="ADDOP">+</XMTok>
+            <XMTok>+</XMTok>
             <XMApp>
-              <XMTok role="SUPERSCRIPTOP" scriptpos="post2"/>
-              <XMTok role="UNKNOWN" font="italic">t</XMTok>
-              <XMTok meaning="2" role="NUMBER">2</XMTok>
+              <XMTok/>
+              <XMTok>t</XMTok>
+              <XMTok>2</XMTok>
             </XMApp>
-            <XMTok role="UNKNOWN" font="italic">t</XMTok>
+            <XMTok>t</XMTok>
             <XMApp>
-              <XMTok mathstyle="text" meaning="divide" role="MULOP"/>
+              <XMTok/>
               <XMApp>
-                <XMTok meaning="plus" role="ADDOP">+</XMTok>
+                <XMTok>+</XMTok>
                 <XMApp>
-                  <XMTok meaning="times" role="MULOP">⁢</XMTok>
-                  <XMTok meaning="2" role="NUMBER">2</XMTok>
-                  <XMTok role="UNKNOWN" font="italic">t</XMTok>
+                  <XMTok>⁢</XMTok>
+                  <XMTok>2</XMTok>
+                  <XMTok>t</XMTok>
                 </XMApp>
-                <XMTok meaning="1" role="NUMBER">1</XMTok>
+                <XMTok>1</XMTok>
               </XMApp>
-              <XMTok meaning="3" role="NUMBER">3</XMTok>
+              <XMTok>3</XMTok>
             </XMApp>
           </XMApp>
         </XMApp>
       </XMath>
     </annotation-xml>
     <annotation encoding="application/x-tex">m=t^{2}+t+\frac{2t+1}{3}</annotation>
   </semantics>
 </math>

Formula 43

( show / hide formula code)

Original

D = 1 D 1 = D 1 D=1

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="p5.m2" class="ltx_Math" alttext="D=1" display="inline">
  <semantics>
    <mrow>
      <mi>D</mi>
      <mo>=</mo>
      <mn>1</mn>
    </mrow>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <eq/>
        <ci>D</ci>
        <cn type="integer">1</cn>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok meaning="equals" role="RELOP">=</XMTok>
          <XMTok role="UNKNOWN" font="italic">D</XMTok>
          <XMTok meaning="1" role="NUMBER">1</XMTok>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">D=1</annotation>
  </semantics>
</math>

Canonicalized

D = 1 D 1 = D 1 D=1

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics encoding="">
    <mrow>
      <mi>D</mi>
      <mo>=</mo>
      <mn>1</mn>
    </mrow>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <eq/>
        <ci encoding="">D</ci>
        <cn encoding="" base="10" type="integer">1</cn>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok>=</XMTok>
          <XMTok>D</XMTok>
          <XMTok>1</XMTok>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">D=1</annotation>
  </semantics>
</math>

Diff

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@@ -1,27 +1,27 @@
 <?xml version="1.0" encoding="UTF-8"?>
-<math xmlns="http://www.w3.org/1998/Math/MathML" id="p5.m2" class="ltx_Math" alttext="D=1" display="inline">
-  <semantics>
+<math xmlns="http://www.w3.org/1998/Math/MathML">
+  <semantics encoding="">
     <mrow>
       <mi>D</mi>
       <mo>=</mo>
       <mn>1</mn>
     </mrow>
     <annotation-xml encoding="MathML-Content">
       <apply>
         <eq/>
-        <ci>D</ci>
-        <cn type="integer">1</cn>
+        <ci encoding="">D</ci>
+        <cn encoding="" base="10" type="integer">1</cn>
       </apply>
     </annotation-xml>
     <annotation-xml encoding="application/x-latexml">
       <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
         <XMApp>
-          <XMTok meaning="equals" role="RELOP">=</XMTok>
-          <XMTok role="UNKNOWN" font="italic">D</XMTok>
-          <XMTok meaning="1" role="NUMBER">1</XMTok>
+          <XMTok>=</XMTok>
+          <XMTok>D</XMTok>
+          <XMTok>1</XMTok>
         </XMApp>
       </XMath>
     </annotation-xml>
     <annotation encoding="application/x-tex">D=1</annotation>
   </semantics>
 </math>

Formula 44

( show / hide formula code)

Original

4 ⁢ t 2 + 20 ⁢ t 3 + 7 3 = u 2 4 superscript t 2 20 t 3 7 3 superscript u 2 = + ⁢ 4 t 2 ⁢ 20 t 3 7 3 u 2 4t^{2}+\frac{20t}{3}+\frac{7}{3}=u^{2}

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="p5.m3" class="ltx_Math" alttext="4t^{2}+\frac{20t}{3}+\frac{7}{3}=u^{2}" display="inline">
  <semantics>
    <mrow>
      <mrow>
        <mrow>
          <mn>4</mn>
          <mo>⁢</mo>
          <msup>
            <mi>t</mi>
            <mn>2</mn>
          </msup>
        </mrow>
        <mo>+</mo>
        <mfrac>
          <mrow>
            <mn>20</mn>
            <mo>⁢</mo>
            <mi>t</mi>
          </mrow>
          <mn>3</mn>
        </mfrac>
        <mo>+</mo>
        <mfrac>
          <mn>7</mn>
          <mn>3</mn>
        </mfrac>
      </mrow>
      <mo>=</mo>
      <msup>
        <mi>u</mi>
        <mn>2</mn>
      </msup>
    </mrow>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <eq/>
        <apply>
          <plus/>
          <apply>
            <times/>
            <cn type="integer">4</cn>
            <apply>
              <csymbol cd="ambiguous">superscript</csymbol>
              <ci>t</ci>
              <cn type="integer">2</cn>
            </apply>
          </apply>
          <apply>
            <divide/>
            <apply>
              <times/>
              <cn type="integer">20</cn>
              <ci>t</ci>
            </apply>
            <cn type="integer">3</cn>
          </apply>
          <apply>
            <divide/>
            <cn type="integer">7</cn>
            <cn type="integer">3</cn>
          </apply>
        </apply>
        <apply>
          <csymbol cd="ambiguous">superscript</csymbol>
          <ci>u</ci>
          <cn type="integer">2</cn>
        </apply>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok meaning="equals" role="RELOP">=</XMTok>
          <XMApp>
            <XMTok meaning="plus" role="ADDOP">+</XMTok>
            <XMApp>
              <XMTok meaning="times" role="MULOP">⁢</XMTok>
              <XMTok meaning="4" role="NUMBER">4</XMTok>
              <XMApp>
                <XMTok role="SUPERSCRIPTOP" scriptpos="post2"/>
                <XMTok role="UNKNOWN" font="italic">t</XMTok>
                <XMTok meaning="2" role="NUMBER">2</XMTok>
              </XMApp>
            </XMApp>
            <XMApp>
              <XMTok mathstyle="text" meaning="divide" role="MULOP"/>
              <XMApp>
                <XMTok meaning="times" role="MULOP">⁢</XMTok>
                <XMTok meaning="20" role="NUMBER">20</XMTok>
                <XMTok role="UNKNOWN" font="italic">t</XMTok>
              </XMApp>
              <XMTok meaning="3" role="NUMBER">3</XMTok>
            </XMApp>
            <XMApp>
              <XMTok mathstyle="text" meaning="divide" role="MULOP"/>
              <XMTok meaning="7" role="NUMBER">7</XMTok>
              <XMTok meaning="3" role="NUMBER">3</XMTok>
            </XMApp>
          </XMApp>
          <XMApp>
            <XMTok role="SUPERSCRIPTOP" scriptpos="post2"/>
            <XMTok role="UNKNOWN" font="italic">u</XMTok>
            <XMTok meaning="2" role="NUMBER">2</XMTok>
          </XMApp>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">4t^{2}+\frac{20t}{3}+\frac{7}{3}=u^{2}</annotation>
  </semantics>
</math>

Canonicalized

4 t 2 + 20 t 3 + 7 3 = u 2 4 superscript t 2 20 t 3 7 3 superscript u 2 = + ⁢ 4 t 2 ⁢ 20 t 3 7 3 u 2 4t^{2}+\frac{20t}{3}+\frac{7}{3}=u^{2}

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics encoding="">
    <mrow>
      <mrow>
        <mrow>
          <mn>4</mn>
          <msup>
            <mi>t</mi>
            <mn>2</mn>
          </msup>
        </mrow>
        <mo>+</mo>
        <mfrac>
          <mrow>
            <mn>20</mn>
            <mi>t</mi>
          </mrow>
          <mn>3</mn>
        </mfrac>
        <mo>+</mo>
        <mfrac>
          <mn>7</mn>
          <mn>3</mn>
        </mfrac>
      </mrow>
      <mo>=</mo>
      <msup>
        <mi>u</mi>
        <mn>2</mn>
      </msup>
    </mrow>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <eq/>
        <apply>
          <plus/>
          <apply>
            <times/>
            <cn encoding="" base="10" type="integer">4</cn>
            <apply>
              <csymbol encoding="">superscript</csymbol>
              <ci encoding="">t</ci>
              <cn encoding="" base="10" type="integer">2</cn>
            </apply>
          </apply>
          <apply>
            <divide/>
            <apply>
              <times/>
              <cn encoding="" base="10" type="integer">20</cn>
              <ci encoding="">t</ci>
            </apply>
            <cn encoding="" base="10" type="integer">3</cn>
          </apply>
          <apply>
            <divide/>
            <cn encoding="" base="10" type="integer">7</cn>
            <cn encoding="" base="10" type="integer">3</cn>
          </apply>
        </apply>
        <apply>
          <csymbol encoding="">superscript</csymbol>
          <ci encoding="">u</ci>
          <cn encoding="" base="10" type="integer">2</cn>
        </apply>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok>=</XMTok>
          <XMApp>
            <XMTok>+</XMTok>
            <XMApp>
              <XMTok>⁢</XMTok>
              <XMTok>4</XMTok>
              <XMApp>
                <XMTok/>
                <XMTok>t</XMTok>
                <XMTok>2</XMTok>
              </XMApp>
            </XMApp>
            <XMApp>
              <XMTok/>
              <XMApp>
                <XMTok>⁢</XMTok>
                <XMTok>20</XMTok>
                <XMTok>t</XMTok>
              </XMApp>
              <XMTok>3</XMTok>
            </XMApp>
            <XMApp>
              <XMTok/>
              <XMTok>7</XMTok>
              <XMTok>3</XMTok>
            </XMApp>
          </XMApp>
          <XMApp>
            <XMTok/>
            <XMTok>u</XMTok>
            <XMTok>2</XMTok>
          </XMApp>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">4t^{2}+\frac{20t}{3}+\frac{7}{3}=u^{2}</annotation>
  </semantics>
</math>

Diff

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@@ -1,111 +1,109 @@
 <?xml version="1.0" encoding="UTF-8"?>
-<math xmlns="http://www.w3.org/1998/Math/MathML" id="p5.m3" class="ltx_Math" alttext="4t^{2}+\frac{20t}{3}+\frac{7}{3}=u^{2}" display="inline">
-  <semantics>
+<math xmlns="http://www.w3.org/1998/Math/MathML">
+  <semantics encoding="">
     <mrow>
       <mrow>
         <mrow>
           <mn>4</mn>
-          <mo>⁢</mo>
           <msup>
             <mi>t</mi>
             <mn>2</mn>
           </msup>
         </mrow>
         <mo>+</mo>
         <mfrac>
           <mrow>
             <mn>20</mn>
-            <mo>⁢</mo>
             <mi>t</mi>
           </mrow>
           <mn>3</mn>
         </mfrac>
         <mo>+</mo>
         <mfrac>
           <mn>7</mn>
           <mn>3</mn>
         </mfrac>
       </mrow>
       <mo>=</mo>
       <msup>
         <mi>u</mi>
         <mn>2</mn>
       </msup>
     </mrow>
     <annotation-xml encoding="MathML-Content">
       <apply>
         <eq/>
         <apply>
           <plus/>
           <apply>
             <times/>
-            <cn type="integer">4</cn>
+            <cn encoding="" base="10" type="integer">4</cn>
             <apply>
-              <csymbol cd="ambiguous">superscript</csymbol>
-              <ci>t</ci>
-              <cn type="integer">2</cn>
+              <csymbol encoding="">superscript</csymbol>
+              <ci encoding="">t</ci>
+              <cn encoding="" base="10" type="integer">2</cn>
             </apply>
           </apply>
           <apply>
             <divide/>
             <apply>
               <times/>
-              <cn type="integer">20</cn>
-              <ci>t</ci>
+              <cn encoding="" base="10" type="integer">20</cn>
+              <ci encoding="">t</ci>
             </apply>
-            <cn type="integer">3</cn>
+            <cn encoding="" base="10" type="integer">3</cn>
           </apply>
           <apply>
             <divide/>
-            <cn type="integer">7</cn>
-            <cn type="integer">3</cn>
+            <cn encoding="" base="10" type="integer">7</cn>
+            <cn encoding="" base="10" type="integer">3</cn>
           </apply>
         </apply>
         <apply>
-          <csymbol cd="ambiguous">superscript</csymbol>
-          <ci>u</ci>
-          <cn type="integer">2</cn>
+          <csymbol encoding="">superscript</csymbol>
+          <ci encoding="">u</ci>
+          <cn encoding="" base="10" type="integer">2</cn>
         </apply>
       </apply>
     </annotation-xml>
     <annotation-xml encoding="application/x-latexml">
       <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
         <XMApp>
-          <XMTok meaning="equals" role="RELOP">=</XMTok>
+          <XMTok>=</XMTok>
           <XMApp>
-            <XMTok meaning="plus" role="ADDOP">+</XMTok>
+            <XMTok>+</XMTok>
             <XMApp>
-              <XMTok meaning="times" role="MULOP">⁢</XMTok>
-              <XMTok meaning="4" role="NUMBER">4</XMTok>
+              <XMTok>⁢</XMTok>
+              <XMTok>4</XMTok>
               <XMApp>
-                <XMTok role="SUPERSCRIPTOP" scriptpos="post2"/>
-                <XMTok role="UNKNOWN" font="italic">t</XMTok>
-                <XMTok meaning="2" role="NUMBER">2</XMTok>
+                <XMTok/>
+                <XMTok>t</XMTok>
+                <XMTok>2</XMTok>
               </XMApp>
             </XMApp>
             <XMApp>
-              <XMTok mathstyle="text" meaning="divide" role="MULOP"/>
+              <XMTok/>
               <XMApp>
-                <XMTok meaning="times" role="MULOP">⁢</XMTok>
-                <XMTok meaning="20" role="NUMBER">20</XMTok>
-                <XMTok role="UNKNOWN" font="italic">t</XMTok>
+                <XMTok>⁢</XMTok>
+                <XMTok>20</XMTok>
+                <XMTok>t</XMTok>
               </XMApp>
-              <XMTok meaning="3" role="NUMBER">3</XMTok>
+              <XMTok>3</XMTok>
             </XMApp>
             <XMApp>
-              <XMTok mathstyle="text" meaning="divide" role="MULOP"/>
-              <XMTok meaning="7" role="NUMBER">7</XMTok>
-              <XMTok meaning="3" role="NUMBER">3</XMTok>
+              <XMTok/>
+              <XMTok>7</XMTok>
+              <XMTok>3</XMTok>
             </XMApp>
           </XMApp>
           <XMApp>
-            <XMTok role="SUPERSCRIPTOP" scriptpos="post2"/>
-            <XMTok role="UNKNOWN" font="italic">u</XMTok>
-            <XMTok meaning="2" role="NUMBER">2</XMTok>
+            <XMTok/>
+            <XMTok>u</XMTok>
+            <XMTok>2</XMTok>
           </XMApp>
         </XMApp>
       </XMath>
     </annotation-xml>
     <annotation encoding="application/x-tex">4t^{2}+\frac{20t}{3}+\frac{7}{3}=u^{2}</annotation>
   </semantics>
 </math>

Formula 45

( show / hide formula code)

Original

4 ⁢ t 2 + 4 ⁢ t + 1 < 4 ⁢ t 2 + 20 ⁢ t 3 + 7 3 < 4 ⁢ t 2 + 8 ⁢ t + 4 4 superscript t 2 4 t 1 4 superscript t 2 20 t 3 7 3 4 superscript t 2 8 t 4 + ⁢ 4 t 2 ⁢ 4 t 1 < + ⁢ 4 t 2 ⁢ 20 t 3 7 3 < + ⁢ 4 t 2 ⁢ 8 t 4 4t^{2}+4t+1<4t^{2}+\frac{20t}{3}+\frac{7}{3}<4t^{2}+8t+4

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="S0.Ex1.m1" class="ltx_Math" alttext="4t^{2}+4t+1&lt;4t^{2}+\frac{20t}{3}+\frac{7}{3}&lt;4t^{2}+8t+4" display="block">
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          <mn>4</mn>
          <mo>⁢</mo>
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            <mn>2</mn>
          </msup>
        </mrow>
        <mo>+</mo>
        <mrow>
          <mn>4</mn>
          <mo>⁢</mo>
          <mi>t</mi>
        </mrow>
        <mo>+</mo>
        <mn>1</mn>
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      <mo>&lt;</mo>
      <mrow>
        <mrow>
          <mn>4</mn>
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        <mo>+</mo>
        <mfrac>
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            <mn>20</mn>
            <mo>⁢</mo>
            <mi>t</mi>
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          <mn>3</mn>
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          <mn>7</mn>
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        </mfrac>
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      <mo>&lt;</mo>
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          <mn>8</mn>
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        <and/>
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                <csymbol cd="ambiguous">superscript</csymbol>
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          <XMTok meaning="less-than" role="RELOP">&lt;</XMTok>
          <XMApp xml:id="S0.Ex1.m1.sh1.xm" fragid="S0.Ex1.m1.sh1.xm">
            <XMTok meaning="plus" role="ADDOP">+</XMTok>
            <XMApp>
              <XMTok meaning="times" role="MULOP">⁢</XMTok>
              <XMTok meaning="4" role="NUMBER">4</XMTok>
              <XMApp>
                <XMTok role="SUPERSCRIPTOP" scriptpos="post2"/>
                <XMTok role="UNKNOWN" font="italic">t</XMTok>
                <XMTok meaning="2" role="NUMBER">2</XMTok>
              </XMApp>
            </XMApp>
            <XMApp>
              <XMTok mathstyle="display" meaning="divide" role="MULOP"/>
              <XMApp>
                <XMTok meaning="times" role="MULOP">⁢</XMTok>
                <XMTok meaning="20" role="NUMBER">20</XMTok>
                <XMTok role="UNKNOWN" font="italic">t</XMTok>
              </XMApp>
              <XMTok meaning="3" role="NUMBER">3</XMTok>
            </XMApp>
            <XMApp>
              <XMTok mathstyle="display" meaning="divide" role="MULOP"/>
              <XMTok meaning="7" role="NUMBER">7</XMTok>
              <XMTok meaning="3" role="NUMBER">3</XMTok>
            </XMApp>
          </XMApp>
          <XMTok meaning="less-than" role="RELOP">&lt;</XMTok>
          <XMApp xml:id="S0.Ex1.m1.sh2.xm" fragid="S0.Ex1.m1.sh2.xm">
            <XMTok meaning="plus" role="ADDOP">+</XMTok>
            <XMApp>
              <XMTok meaning="times" role="MULOP">⁢</XMTok>
              <XMTok meaning="4" role="NUMBER">4</XMTok>
              <XMApp>
                <XMTok role="SUPERSCRIPTOP" scriptpos="post2"/>
                <XMTok role="UNKNOWN" font="italic">t</XMTok>
                <XMTok meaning="2" role="NUMBER">2</XMTok>
              </XMApp>
            </XMApp>
            <XMApp>
              <XMTok meaning="times" role="MULOP">⁢</XMTok>
              <XMTok meaning="8" role="NUMBER">8</XMTok>
              <XMTok role="UNKNOWN" font="italic">t</XMTok>
            </XMApp>
            <XMTok meaning="4" role="NUMBER">4</XMTok>
          </XMApp>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">4t^{2}+4t+1&lt;4t^{2}+\frac{20t}{3}+\frac{7}{3}&lt;4t^{2}+8t+4</annotation>
  </semantics>
</math>

Canonicalized

4 t 2 + 4 t + 1 < 4 t 2 + 20 t 3 + 7 3 < 4 t 2 + 8 t + 4 4 superscript t 2 4 t 1 4 superscript t 2 20 t 3 7 3 4 superscript t 2 8 t 4 + ⁢ 4 t 2 ⁢ 4 t 1 < + ⁢ 4 t 2 ⁢ 20 t 3 7 3 < + ⁢ 4 t 2 ⁢ 8 t 4 4t^{2}+4t+1<4t^{2}+\frac{20t}{3}+\frac{7}{3}<4t^{2}+8t+4

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics encoding="">
    <mrow>
      <mrow>
        <mrow>
          <mn>4</mn>
          <msup>
            <mi>t</mi>
            <mn>2</mn>
          </msup>
        </mrow>
        <mo>+</mo>
        <mrow>
          <mn>4</mn>
          <mi>t</mi>
        </mrow>
        <mo>+</mo>
        <mn>1</mn>
      </mrow>
      <mo>&lt;</mo>
      <mrow>
        <mrow>
          <mn>4</mn>
          <msup>
            <mi>t</mi>
            <mn>2</mn>
          </msup>
        </mrow>
        <mo>+</mo>
        <mfrac>
          <mrow>
            <mn>20</mn>
            <mi>t</mi>
          </mrow>
          <mn>3</mn>
        </mfrac>
        <mo>+</mo>
        <mfrac>
          <mn>7</mn>
          <mn>3</mn>
        </mfrac>
      </mrow>
      <mo>&lt;</mo>
      <mrow>
        <mrow>
          <mn>4</mn>
          <msup>
            <mi>t</mi>
            <mn>2</mn>
          </msup>
        </mrow>
        <mo>+</mo>
        <mrow>
          <mn>8</mn>
          <mi>t</mi>
        </mrow>
        <mo>+</mo>
        <mn>4</mn>
      </mrow>
    </mrow>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <and/>
        <apply>
          <lt/>
          <apply>
            <plus/>
            <apply>
              <times/>
              <cn encoding="" base="10" type="integer">4</cn>
              <apply>
                <csymbol encoding="">superscript</csymbol>
                <ci encoding="">t</ci>
                <cn encoding="" base="10" type="integer">2</cn>
              </apply>
            </apply>
            <apply>
              <times/>
              <cn encoding="" base="10" type="integer">4</cn>
              <ci encoding="">t</ci>
            </apply>
            <cn encoding="" base="10" type="integer">1</cn>
          </apply>
          <apply>
            <plus encoding=""/>
            <apply>
              <times encoding=""/>
              <cn encoding="" base="10" type="integer">4</cn>
              <apply>
                <csymbol encoding="">superscript</csymbol>
                <ci encoding="">t</ci>
                <cn encoding="" base="10" type="integer">2</cn>
              </apply>
            </apply>
            <apply>
              <divide encoding=""/>
              <apply>
                <times encoding=""/>
                <cn encoding="" base="10" type="integer">20</cn>
                <ci encoding="">t</ci>
              </apply>
              <cn encoding="" base="10" type="integer">3</cn>
            </apply>
            <apply>
              <divide encoding=""/>
              <cn encoding="" base="10" type="integer">7</cn>
              <cn encoding="" base="10" type="integer">3</cn>
            </apply>
          </apply>
        </apply>
        <apply>
          <lt/>
          <share/>
          <apply>
            <plus encoding=""/>
            <apply>
              <times encoding=""/>
              <cn encoding="" base="10" type="integer">4</cn>
              <apply>
                <csymbol encoding="">superscript</csymbol>
                <ci encoding="">t</ci>
                <cn encoding="" base="10" type="integer">2</cn>
              </apply>
            </apply>
            <apply>
              <times encoding=""/>
              <cn encoding="" base="10" type="integer">8</cn>
              <ci encoding="">t</ci>
            </apply>
            <cn encoding="" base="10" type="integer">4</cn>
          </apply>
        </apply>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok/>
          <XMApp>
            <XMTok>+</XMTok>
            <XMApp>
              <XMTok>⁢</XMTok>
              <XMTok>4</XMTok>
              <XMApp>
                <XMTok/>
                <XMTok>t</XMTok>
                <XMTok>2</XMTok>
              </XMApp>
            </XMApp>
            <XMApp>
              <XMTok>⁢</XMTok>
              <XMTok>4</XMTok>
              <XMTok>t</XMTok>
            </XMApp>
            <XMTok>1</XMTok>
          </XMApp>
          <XMTok>&lt;</XMTok>
          <XMApp>
            <XMTok>+</XMTok>
            <XMApp>
              <XMTok>⁢</XMTok>
              <XMTok>4</XMTok>
              <XMApp>
                <XMTok/>
                <XMTok>t</XMTok>
                <XMTok>2</XMTok>
              </XMApp>
            </XMApp>
            <XMApp>
              <XMTok/>
              <XMApp>
                <XMTok>⁢</XMTok>
                <XMTok>20</XMTok>
                <XMTok>t</XMTok>
              </XMApp>
              <XMTok>3</XMTok>
            </XMApp>
            <XMApp>
              <XMTok/>
              <XMTok>7</XMTok>
              <XMTok>3</XMTok>
            </XMApp>
          </XMApp>
          <XMTok>&lt;</XMTok>
          <XMApp>
            <XMTok>+</XMTok>
            <XMApp>
              <XMTok>⁢</XMTok>
              <XMTok>4</XMTok>
              <XMApp>
                <XMTok/>
                <XMTok>t</XMTok>
                <XMTok>2</XMTok>
              </XMApp>
            </XMApp>
            <XMApp>
              <XMTok>⁢</XMTok>
              <XMTok>8</XMTok>
              <XMTok>t</XMTok>
            </XMApp>
            <XMTok>4</XMTok>
          </XMApp>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">4t^{2}+4t+1&lt;4t^{2}+\frac{20t}{3}+\frac{7}{3}&lt;4t^{2}+8t+4</annotation>
  </semantics>
</math>

Diff

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@@ -1,215 +1,209 @@
 <?xml version="1.0" encoding="UTF-8"?>
-<math xmlns="http://www.w3.org/1998/Math/MathML" id="S0.Ex1.m1" class="ltx_Math" alttext="4t^{2}+4t+1&lt;4t^{2}+\frac{20t}{3}+\frac{7}{3}&lt;4t^{2}+8t+4" display="block">
-  <semantics>
+<math xmlns="http://www.w3.org/1998/Math/MathML">
+  <semantics encoding="">
     <mrow>
       <mrow>
         <mrow>
           <mn>4</mn>
-          <mo>⁢</mo>
           <msup>
             <mi>t</mi>
             <mn>2</mn>
           </msup>
         </mrow>
         <mo>+</mo>
         <mrow>
           <mn>4</mn>
-          <mo>⁢</mo>
           <mi>t</mi>
         </mrow>
         <mo>+</mo>
         <mn>1</mn>
       </mrow>
       <mo>&lt;</mo>
       <mrow>
         <mrow>
           <mn>4</mn>
-          <mo>⁢</mo>
           <msup>
             <mi>t</mi>
             <mn>2</mn>
           </msup>
         </mrow>
         <mo>+</mo>
         <mfrac>
           <mrow>
             <mn>20</mn>
-            <mo>⁢</mo>
             <mi>t</mi>
           </mrow>
           <mn>3</mn>
         </mfrac>
         <mo>+</mo>
         <mfrac>
           <mn>7</mn>
           <mn>3</mn>
         </mfrac>
       </mrow>
       <mo>&lt;</mo>
       <mrow>
         <mrow>
           <mn>4</mn>
-          <mo>⁢</mo>
           <msup>
             <mi>t</mi>
             <mn>2</mn>
           </msup>
         </mrow>
         <mo>+</mo>
         <mrow>
           <mn>8</mn>
-          <mo>⁢</mo>
           <mi>t</mi>
         </mrow>
         <mo>+</mo>
         <mn>4</mn>
       </mrow>
     </mrow>
     <annotation-xml encoding="MathML-Content">
       <apply>
         <and/>
         <apply>
           <lt/>
           <apply>
             <plus/>
             <apply>
               <times/>
-              <cn type="integer">4</cn>
+              <cn encoding="" base="10" type="integer">4</cn>
               <apply>
-                <csymbol cd="ambiguous">superscript</csymbol>
-                <ci>t</ci>
-                <cn type="integer">2</cn>
+                <csymbol encoding="">superscript</csymbol>
+                <ci encoding="">t</ci>
+                <cn encoding="" base="10" type="integer">2</cn>
               </apply>
             </apply>
             <apply>
               <times/>
-              <cn type="integer">4</cn>
-              <ci>t</ci>
+              <cn encoding="" base="10" type="integer">4</cn>
+              <ci encoding="">t</ci>
             </apply>
-            <cn type="integer">1</cn>
+            <cn encoding="" base="10" type="integer">1</cn>
           </apply>
-          <apply xml:id="S0.Ex1.m1.sh1s.cmml">
-            <plus xml:id="S0.Ex1.m1.sh1.cmml"/>
-            <apply xml:id="S0.Ex1.m1.sh1g.cmml">
-              <times xml:id="S0.Ex1.m1.sh1a.cmml"/>
-              <cn type="integer" xml:id="S0.Ex1.m1.sh1b.cmml">4</cn>
-              <apply xml:id="S0.Ex1.m1.sh1f.cmml">
-                <csymbol cd="ambiguous" xml:id="S0.Ex1.m1.sh1c.cmml">superscript</csymbol>
-                <ci xml:id="S0.Ex1.m1.sh1d.cmml">t</ci>
-                <cn type="integer" xml:id="S0.Ex1.m1.sh1e.cmml">2</cn>
+          <apply>
+            <plus encoding=""/>
+            <apply>
+              <times encoding=""/>
+              <cn encoding="" base="10" type="integer">4</cn>
+              <apply>
+                <csymbol encoding="">superscript</csymbol>
+                <ci encoding="">t</ci>
+                <cn encoding="" base="10" type="integer">2</cn>
               </apply>
             </apply>
-            <apply xml:id="S0.Ex1.m1.sh1n.cmml">
-              <divide xml:id="S0.Ex1.m1.sh1h.cmml"/>
-              <apply xml:id="S0.Ex1.m1.sh1l.cmml">
-                <times xml:id="S0.Ex1.m1.sh1i.cmml"/>
-                <cn type="integer" xml:id="S0.Ex1.m1.sh1j.cmml">20</cn>
-                <ci xml:id="S0.Ex1.m1.sh1k.cmml">t</ci>
+            <apply>
+              <divide encoding=""/>
+              <apply>
+                <times encoding=""/>
+                <cn encoding="" base="10" type="integer">20</cn>
+                <ci encoding="">t</ci>
               </apply>
-              <cn type="integer" xml:id="S0.Ex1.m1.sh1m.cmml">3</cn>
+              <cn encoding="" base="10" type="integer">3</cn>
             </apply>
-            <apply xml:id="S0.Ex1.m1.sh1r.cmml">
-              <divide xml:id="S0.Ex1.m1.sh1o.cmml"/>
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-              <cn type="integer" xml:id="S0.Ex1.m1.sh1q.cmml">3</cn>
+            <apply>
+              <divide encoding=""/>
+              <cn encoding="" base="10" type="integer">7</cn>
+              <cn encoding="" base="10" type="integer">3</cn>
             </apply>
           </apply>
         </apply>
         <apply>
           <lt/>
-          <share href="#S0.Ex1.m1.sh1.cmml"/>
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-                <cn type="integer" xml:id="S0.Ex1.m1.sh2e.cmml">2</cn>
+          <share/>
+          <apply>
+            <plus encoding=""/>
+            <apply>
+              <times encoding=""/>
+              <cn encoding="" base="10" type="integer">4</cn>
+              <apply>
+                <csymbol encoding="">superscript</csymbol>
+                <ci encoding="">t</ci>
+                <cn encoding="" base="10" type="integer">2</cn>
               </apply>
             </apply>
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-              <times xml:id="S0.Ex1.m1.sh2h.cmml"/>
-              <cn type="integer" xml:id="S0.Ex1.m1.sh2i.cmml">8</cn>
-              <ci xml:id="S0.Ex1.m1.sh2j.cmml">t</ci>
+            <apply>
+              <times encoding=""/>
+              <cn encoding="" base="10" type="integer">8</cn>
+              <ci encoding="">t</ci>
             </apply>
-            <cn type="integer" xml:id="S0.Ex1.m1.sh2l.cmml">4</cn>
+            <cn encoding="" base="10" type="integer">4</cn>
           </apply>
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       </apply>
     </annotation-xml>
     <annotation-xml encoding="application/x-latexml">
       <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
         <XMApp>
-          <XMTok meaning="multirelation"/>
+          <XMTok/>
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-            <XMTok meaning="plus" role="ADDOP">+</XMTok>
+            <XMTok>+</XMTok>
             <XMApp>
-              <XMTok meaning="times" role="MULOP">⁢</XMTok>
-              <XMTok meaning="4" role="NUMBER">4</XMTok>
+              <XMTok>⁢</XMTok>
+              <XMTok>4</XMTok>
               <XMApp>
-                <XMTok role="SUPERSCRIPTOP" scriptpos="post2"/>
-                <XMTok role="UNKNOWN" font="italic">t</XMTok>
-                <XMTok meaning="2" role="NUMBER">2</XMTok>
+                <XMTok/>
+                <XMTok>t</XMTok>
+                <XMTok>2</XMTok>
               </XMApp>
             </XMApp>
             <XMApp>
-              <XMTok meaning="times" role="MULOP">⁢</XMTok>
-              <XMTok meaning="4" role="NUMBER">4</XMTok>
-              <XMTok role="UNKNOWN" font="italic">t</XMTok>
+              <XMTok>⁢</XMTok>
+              <XMTok>4</XMTok>
+              <XMTok>t</XMTok>
             </XMApp>
-            <XMTok meaning="1" role="NUMBER">1</XMTok>
+            <XMTok>1</XMTok>
           </XMApp>
-          <XMTok meaning="less-than" role="RELOP">&lt;</XMTok>
-          <XMApp xml:id="S0.Ex1.m1.sh1.xm" fragid="S0.Ex1.m1.sh1.xm">
-            <XMTok meaning="plus" role="ADDOP">+</XMTok>
+          <XMTok>&lt;</XMTok>
+          <XMApp>
+            <XMTok>+</XMTok>
             <XMApp>
-              <XMTok meaning="times" role="MULOP">⁢</XMTok>
-              <XMTok meaning="4" role="NUMBER">4</XMTok>
+              <XMTok>⁢</XMTok>
+              <XMTok>4</XMTok>
               <XMApp>
-                <XMTok role="SUPERSCRIPTOP" scriptpos="post2"/>
-                <XMTok role="UNKNOWN" font="italic">t</XMTok>
-                <XMTok meaning="2" role="NUMBER">2</XMTok>
+                <XMTok/>
+                <XMTok>t</XMTok>
+                <XMTok>2</XMTok>
               </XMApp>
             </XMApp>
             <XMApp>
-              <XMTok mathstyle="display" meaning="divide" role="MULOP"/>
+              <XMTok/>
               <XMApp>
-                <XMTok meaning="times" role="MULOP">⁢</XMTok>
-                <XMTok meaning="20" role="NUMBER">20</XMTok>
-                <XMTok role="UNKNOWN" font="italic">t</XMTok>
+                <XMTok>⁢</XMTok>
+                <XMTok>20</XMTok>
+                <XMTok>t</XMTok>
               </XMApp>
-              <XMTok meaning="3" role="NUMBER">3</XMTok>
+              <XMTok>3</XMTok>
             </XMApp>
             <XMApp>
-              <XMTok mathstyle="display" meaning="divide" role="MULOP"/>
-              <XMTok meaning="7" role="NUMBER">7</XMTok>
-              <XMTok meaning="3" role="NUMBER">3</XMTok>
+              <XMTok/>
+              <XMTok>7</XMTok>
+              <XMTok>3</XMTok>
             </XMApp>
           </XMApp>
-          <XMTok meaning="less-than" role="RELOP">&lt;</XMTok>
-          <XMApp xml:id="S0.Ex1.m1.sh2.xm" fragid="S0.Ex1.m1.sh2.xm">
-            <XMTok meaning="plus" role="ADDOP">+</XMTok>
+          <XMTok>&lt;</XMTok>
+          <XMApp>
+            <XMTok>+</XMTok>
             <XMApp>
-              <XMTok meaning="times" role="MULOP">⁢</XMTok>
-              <XMTok meaning="4" role="NUMBER">4</XMTok>
+              <XMTok>⁢</XMTok>
+              <XMTok>4</XMTok>
               <XMApp>
-                <XMTok role="SUPERSCRIPTOP" scriptpos="post2"/>
-                <XMTok role="UNKNOWN" font="italic">t</XMTok>
-                <XMTok meaning="2" role="NUMBER">2</XMTok>
+                <XMTok/>
+                <XMTok>t</XMTok>
+                <XMTok>2</XMTok>
               </XMApp>
             </XMApp>
             <XMApp>
-              <XMTok meaning="times" role="MULOP">⁢</XMTok>
-              <XMTok meaning="8" role="NUMBER">8</XMTok>
-              <XMTok role="UNKNOWN" font="italic">t</XMTok>
+              <XMTok>⁢</XMTok>
+              <XMTok>8</XMTok>
+              <XMTok>t</XMTok>
             </XMApp>
-            <XMTok meaning="4" role="NUMBER">4</XMTok>
+            <XMTok>4</XMTok>
           </XMApp>
         </XMApp>
       </XMath>
     </annotation-xml>
     <annotation encoding="application/x-tex">4t^{2}+4t+1&lt;4t^{2}+\frac{20t}{3}+\frac{7}{3}&lt;4t^{2}+8t+4</annotation>
   </semantics>
 </math>

Formula 46

( show / hide formula code)

Original

2 ⁢ t + 1 < u < 2 ⁢ t + 2 2 t 1 u 2 t 2 + ⁢ 2 t 1 < u < + ⁢ 2 t 2 2t+1<u<2t+2

(show/hide code)
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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="p5.m4" class="ltx_Math" alttext="2t+1&lt;u&lt;2t+2" display="inline">
  <semantics>
    <mrow>
      <mrow>
        <mrow>
          <mn>2</mn>
          <mo>⁢</mo>
          <mi>t</mi>
        </mrow>
        <mo>+</mo>
        <mn>1</mn>
      </mrow>
      <mo>&lt;</mo>
      <mi>u</mi>
      <mo>&lt;</mo>
      <mrow>
        <mrow>
          <mn>2</mn>
          <mo>⁢</mo>
          <mi>t</mi>
        </mrow>
        <mo>+</mo>
        <mn>2</mn>
      </mrow>
    </mrow>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <and/>
        <apply>
          <lt/>
          <apply>
            <plus/>
            <apply>
              <times/>
              <cn type="integer">2</cn>
              <ci>t</ci>
            </apply>
            <cn type="integer">1</cn>
          </apply>
          <ci xml:id="p5.m4.sh1.cmml">u</ci>
        </apply>
        <apply>
          <lt/>
          <share href="#p5.m4.sh1.cmml"/>
          <apply xml:id="p5.m4.sh2f.cmml">
            <plus xml:id="p5.m4.sh2.cmml"/>
            <apply xml:id="p5.m4.sh2d.cmml">
              <times xml:id="p5.m4.sh2a.cmml"/>
              <cn type="integer" xml:id="p5.m4.sh2b.cmml">2</cn>
              <ci xml:id="p5.m4.sh2c.cmml">t</ci>
            </apply>
            <cn type="integer" xml:id="p5.m4.sh2e.cmml">2</cn>
          </apply>
        </apply>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok meaning="multirelation"/>
          <XMApp>
            <XMTok meaning="plus" role="ADDOP">+</XMTok>
            <XMApp>
              <XMTok meaning="times" role="MULOP">⁢</XMTok>
              <XMTok meaning="2" role="NUMBER">2</XMTok>
              <XMTok role="UNKNOWN" font="italic">t</XMTok>
            </XMApp>
            <XMTok meaning="1" role="NUMBER">1</XMTok>
          </XMApp>
          <XMTok meaning="less-than" role="RELOP">&lt;</XMTok>
          <XMTok role="UNKNOWN" font="italic" xml:id="p5.m4.sh1.xm" fragid="p5.m4.sh1.xm">u</XMTok>
          <XMTok meaning="less-than" role="RELOP">&lt;</XMTok>
          <XMApp xml:id="p5.m4.sh2.xm" fragid="p5.m4.sh2.xm">
            <XMTok meaning="plus" role="ADDOP">+</XMTok>
            <XMApp>
              <XMTok meaning="times" role="MULOP">⁢</XMTok>
              <XMTok meaning="2" role="NUMBER">2</XMTok>
              <XMTok role="UNKNOWN" font="italic">t</XMTok>
            </XMApp>
            <XMTok meaning="2" role="NUMBER">2</XMTok>
          </XMApp>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">2t+1&lt;u&lt;2t+2</annotation>
  </semantics>
</math>

Canonicalized

2 t + 1 < u < 2 t + 2 2 t 1 u 2 t 2 + ⁢ 2 t 1 < u < + ⁢ 2 t 2 2t+1<u<2t+2

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics encoding="">
    <mrow>
      <mrow>
        <mrow>
          <mn>2</mn>
          <mi>t</mi>
        </mrow>
        <mo>+</mo>
        <mn>1</mn>
      </mrow>
      <mo>&lt;</mo>
      <mi>u</mi>
      <mo>&lt;</mo>
      <mrow>
        <mrow>
          <mn>2</mn>
          <mi>t</mi>
        </mrow>
        <mo>+</mo>
        <mn>2</mn>
      </mrow>
    </mrow>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <and/>
        <apply>
          <lt/>
          <apply>
            <plus/>
            <apply>
              <times/>
              <cn encoding="" base="10" type="integer">2</cn>
              <ci encoding="">t</ci>
            </apply>
            <cn encoding="" base="10" type="integer">1</cn>
          </apply>
          <ci encoding="">u</ci>
        </apply>
        <apply>
          <lt/>
          <share/>
          <apply>
            <plus encoding=""/>
            <apply>
              <times encoding=""/>
              <cn encoding="" base="10" type="integer">2</cn>
              <ci encoding="">t</ci>
            </apply>
            <cn encoding="" base="10" type="integer">2</cn>
          </apply>
        </apply>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok/>
          <XMApp>
            <XMTok>+</XMTok>
            <XMApp>
              <XMTok>⁢</XMTok>
              <XMTok>2</XMTok>
              <XMTok>t</XMTok>
            </XMApp>
            <XMTok>1</XMTok>
          </XMApp>
          <XMTok>&lt;</XMTok>
          <XMTok>u</XMTok>
          <XMTok>&lt;</XMTok>
          <XMApp>
            <XMTok>+</XMTok>
            <XMApp>
              <XMTok>⁢</XMTok>
              <XMTok>2</XMTok>
              <XMTok>t</XMTok>
            </XMApp>
            <XMTok>2</XMTok>
          </XMApp>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">2t+1&lt;u&lt;2t+2</annotation>
  </semantics>
</math>

Diff

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@@ -1,88 +1,86 @@
 <?xml version="1.0" encoding="UTF-8"?>
-<math xmlns="http://www.w3.org/1998/Math/MathML" id="p5.m4" class="ltx_Math" alttext="2t+1&lt;u&lt;2t+2" display="inline">
-  <semantics>
+<math xmlns="http://www.w3.org/1998/Math/MathML">
+  <semantics encoding="">
     <mrow>
       <mrow>
         <mrow>
           <mn>2</mn>
-          <mo>⁢</mo>
           <mi>t</mi>
         </mrow>
         <mo>+</mo>
         <mn>1</mn>
       </mrow>
       <mo>&lt;</mo>
       <mi>u</mi>
       <mo>&lt;</mo>
       <mrow>
         <mrow>
           <mn>2</mn>
-          <mo>⁢</mo>
           <mi>t</mi>
         </mrow>
         <mo>+</mo>
         <mn>2</mn>
       </mrow>
     </mrow>
     <annotation-xml encoding="MathML-Content">
       <apply>
         <and/>
         <apply>
           <lt/>
           <apply>
             <plus/>
             <apply>
               <times/>
-              <cn type="integer">2</cn>
-              <ci>t</ci>
+              <cn encoding="" base="10" type="integer">2</cn>
+              <ci encoding="">t</ci>
             </apply>
-            <cn type="integer">1</cn>
+            <cn encoding="" base="10" type="integer">1</cn>
           </apply>
-          <ci xml:id="p5.m4.sh1.cmml">u</ci>
+          <ci encoding="">u</ci>
         </apply>
         <apply>
           <lt/>
-          <share href="#p5.m4.sh1.cmml"/>
-          <apply xml:id="p5.m4.sh2f.cmml">
-            <plus xml:id="p5.m4.sh2.cmml"/>
-            <apply xml:id="p5.m4.sh2d.cmml">
-              <times xml:id="p5.m4.sh2a.cmml"/>
-              <cn type="integer" xml:id="p5.m4.sh2b.cmml">2</cn>
-              <ci xml:id="p5.m4.sh2c.cmml">t</ci>
+          <share/>
+          <apply>
+            <plus encoding=""/>
+            <apply>
+              <times encoding=""/>
+              <cn encoding="" base="10" type="integer">2</cn>
+              <ci encoding="">t</ci>
             </apply>
-            <cn type="integer" xml:id="p5.m4.sh2e.cmml">2</cn>
+            <cn encoding="" base="10" type="integer">2</cn>
           </apply>
         </apply>
       </apply>
     </annotation-xml>
     <annotation-xml encoding="application/x-latexml">
       <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
         <XMApp>
-          <XMTok meaning="multirelation"/>
+          <XMTok/>
           <XMApp>
-            <XMTok meaning="plus" role="ADDOP">+</XMTok>
+            <XMTok>+</XMTok>
             <XMApp>
-              <XMTok meaning="times" role="MULOP">⁢</XMTok>
-              <XMTok meaning="2" role="NUMBER">2</XMTok>
-              <XMTok role="UNKNOWN" font="italic">t</XMTok>
+              <XMTok>⁢</XMTok>
+              <XMTok>2</XMTok>
+              <XMTok>t</XMTok>
             </XMApp>
-            <XMTok meaning="1" role="NUMBER">1</XMTok>
+            <XMTok>1</XMTok>
           </XMApp>
-          <XMTok meaning="less-than" role="RELOP">&lt;</XMTok>
-          <XMTok role="UNKNOWN" font="italic" xml:id="p5.m4.sh1.xm" fragid="p5.m4.sh1.xm">u</XMTok>
-          <XMTok meaning="less-than" role="RELOP">&lt;</XMTok>
-          <XMApp xml:id="p5.m4.sh2.xm" fragid="p5.m4.sh2.xm">
-            <XMTok meaning="plus" role="ADDOP">+</XMTok>
+          <XMTok>&lt;</XMTok>
+          <XMTok>u</XMTok>
+          <XMTok>&lt;</XMTok>
+          <XMApp>
+            <XMTok>+</XMTok>
             <XMApp>
-              <XMTok meaning="times" role="MULOP">⁢</XMTok>
-              <XMTok meaning="2" role="NUMBER">2</XMTok>
-              <XMTok role="UNKNOWN" font="italic">t</XMTok>
+              <XMTok>⁢</XMTok>
+              <XMTok>2</XMTok>
+              <XMTok>t</XMTok>
             </XMApp>
-            <XMTok meaning="2" role="NUMBER">2</XMTok>
+            <XMTok>2</XMTok>
           </XMApp>
         </XMApp>
       </XMath>
     </annotation-xml>
     <annotation encoding="application/x-tex">2t+1&lt;u&lt;2t+2</annotation>
   </semantics>
 </math>

Formula 47

( show / hide formula code)

Original

t t t t

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="p5.m5" class="ltx_Math" alttext="t" display="inline">
  <semantics>
    <mi>t</mi>
    <annotation-xml encoding="MathML-Content">
      <ci>t</ci>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMTok role="UNKNOWN" font="italic">t</XMTok>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">t</annotation>
  </semantics>
</math>

Canonicalized

t t t t

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics encoding="">
    <mi>t</mi>
    <annotation-xml encoding="MathML-Content">
      <ci encoding="">t</ci>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMTok>t</XMTok>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">t</annotation>
  </semantics>
</math>

Diff

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@@ -1,15 +1,15 @@
 <?xml version="1.0" encoding="UTF-8"?>
-<math xmlns="http://www.w3.org/1998/Math/MathML" id="p5.m5" class="ltx_Math" alttext="t" display="inline">
-  <semantics>
+<math xmlns="http://www.w3.org/1998/Math/MathML">
+  <semantics encoding="">
     <mi>t</mi>
     <annotation-xml encoding="MathML-Content">
-      <ci>t</ci>
+      <ci encoding="">t</ci>
     </annotation-xml>
     <annotation-xml encoding="application/x-latexml">
       <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
-        <XMTok role="UNKNOWN" font="italic">t</XMTok>
+        <XMTok>t</XMTok>
       </XMath>
     </annotation-xml>
     <annotation encoding="application/x-tex">t</annotation>
   </semantics>
 </math>

Formula 48

( show / hide formula code)

Original

u u u u

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="p5.m6" class="ltx_Math" alttext="u" display="inline">
  <semantics>
    <mi>u</mi>
    <annotation-xml encoding="MathML-Content">
      <ci>u</ci>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMTok role="UNKNOWN" font="italic">u</XMTok>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">u</annotation>
  </semantics>
</math>

Canonicalized

u u u u

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics encoding="">
    <mi>u</mi>
    <annotation-xml encoding="MathML-Content">
      <ci encoding="">u</ci>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMTok>u</XMTok>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">u</annotation>
  </semantics>
</math>

Diff

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@@ -1,15 +1,15 @@
 <?xml version="1.0" encoding="UTF-8"?>
-<math xmlns="http://www.w3.org/1998/Math/MathML" id="p5.m6" class="ltx_Math" alttext="u" display="inline">
-  <semantics>
+<math xmlns="http://www.w3.org/1998/Math/MathML">
+  <semantics encoding="">
     <mi>u</mi>
     <annotation-xml encoding="MathML-Content">
-      <ci>u</ci>
+      <ci encoding="">u</ci>
     </annotation-xml>
     <annotation-xml encoding="application/x-latexml">
       <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
-        <XMTok role="UNKNOWN" font="italic">u</XMTok>
+        <XMTok>u</XMTok>
       </XMath>
     </annotation-xml>
     <annotation encoding="application/x-tex">u</annotation>
   </semantics>
 </math>

Formula 49

( show / hide formula code)

Original

m m m m

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="ThmLem2.p1.m1" class="ltx_Math" alttext="m" display="inline">
  <semantics>
    <mi>m</mi>
    <annotation-xml encoding="MathML-Content">
      <ci>m</ci>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMTok role="UNKNOWN">m</XMTok>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">m</annotation>
  </semantics>
</math>

Canonicalized

m m m m

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics encoding="">
    <mi>m</mi>
    <annotation-xml encoding="MathML-Content">
      <ci encoding="">m</ci>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMTok>m</XMTok>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">m</annotation>
  </semantics>
</math>

Diff

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@@ -1,15 +1,15 @@
 <?xml version="1.0" encoding="UTF-8"?>
-<math xmlns="http://www.w3.org/1998/Math/MathML" id="ThmLem2.p1.m1" class="ltx_Math" alttext="m" display="inline">
-  <semantics>
+<math xmlns="http://www.w3.org/1998/Math/MathML">
+  <semantics encoding="">
     <mi>m</mi>
     <annotation-xml encoding="MathML-Content">
-      <ci>m</ci>
+      <ci encoding="">m</ci>
     </annotation-xml>
     <annotation-xml encoding="application/x-latexml">
       <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
-        <XMTok role="UNKNOWN">m</XMTok>
+        <XMTok>m</XMTok>
       </XMath>
     </annotation-xml>
     <annotation encoding="application/x-tex">m</annotation>
   </semantics>
 </math>

Formula 50

( show / hide formula code)

Original

f m subscript f m f m f_{m}

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="ThmLem2.p1.m2" class="ltx_Math" alttext="f_{m}" display="inline">
  <semantics>
    <msub>
      <mi>f</mi>
      <mi>m</mi>
    </msub>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <csymbol cd="ambiguous">subscript</csymbol>
        <ci>f</ci>
        <ci>m</ci>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok role="SUBSCRIPTOP" scriptpos="post3"/>
          <XMTok role="UNKNOWN">f</XMTok>
          <XMTok role="UNKNOWN">m</XMTok>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">f_{m}</annotation>
  </semantics>
</math>

Canonicalized

f m subscript f m f m f_{m}

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics encoding="">
    <msub>
      <mi>f</mi>
      <mi>m</mi>
    </msub>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <csymbol encoding="">subscript</csymbol>
        <ci encoding="">f</ci>
        <ci encoding="">m</ci>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok/>
          <XMTok>f</XMTok>
          <XMTok>m</XMTok>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">f_{m}</annotation>
  </semantics>
</math>

Diff

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@@ -1,26 +1,26 @@
 <?xml version="1.0" encoding="UTF-8"?>
-<math xmlns="http://www.w3.org/1998/Math/MathML" id="ThmLem2.p1.m2" class="ltx_Math" alttext="f_{m}" display="inline">
-  <semantics>
+<math xmlns="http://www.w3.org/1998/Math/MathML">
+  <semantics encoding="">
     <msub>
       <mi>f</mi>
       <mi>m</mi>
     </msub>
     <annotation-xml encoding="MathML-Content">
       <apply>
-        <csymbol cd="ambiguous">subscript</csymbol>
-        <ci>f</ci>
-        <ci>m</ci>
+        <csymbol encoding="">subscript</csymbol>
+        <ci encoding="">f</ci>
+        <ci encoding="">m</ci>
       </apply>
     </annotation-xml>
     <annotation-xml encoding="application/x-latexml">
       <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
         <XMApp>
-          <XMTok role="SUBSCRIPTOP" scriptpos="post3"/>
-          <XMTok role="UNKNOWN">f</XMTok>
-          <XMTok role="UNKNOWN">m</XMTok>
+          <XMTok/>
+          <XMTok>f</XMTok>
+          <XMTok>m</XMTok>
         </XMApp>
       </XMath>
     </annotation-xml>
     <annotation encoding="application/x-tex">f_{m}</annotation>
   </semantics>
 </math>

Formula 51

( show / hide formula code)

Original

1 < D < m 1 / 12 1 D superscript m 1 12 1 < D < m / 1 12 1<D<m^{1/12}

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="ThmLem2.p1.m3" class="ltx_Math" alttext="1&lt;D&lt;m^{1/12}" display="inline">
  <semantics>
    <mrow>
      <mn mathvariant="normal">1</mn>
      <mo mathvariant="normal">&lt;</mo>
      <mi>D</mi>
      <mo mathvariant="normal">&lt;</mo>
      <msup>
        <mi>m</mi>
        <mrow>
          <mn mathvariant="normal">1</mn>
          <mo mathvariant="normal">/</mo>
          <mn mathvariant="normal">12</mn>
        </mrow>
      </msup>
    </mrow>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <and/>
        <apply>
          <lt/>
          <cn type="integer">1</cn>
          <ci xml:id="ThmLem2.p1.m3.sh1.cmml">D</ci>
        </apply>
        <apply>
          <lt/>
          <share href="#ThmLem2.p1.m3.sh1.cmml"/>
          <apply xml:id="ThmLem2.p1.m3.sh2f.cmml">
            <csymbol cd="ambiguous" xml:id="ThmLem2.p1.m3.sh2.cmml">superscript</csymbol>
            <ci xml:id="ThmLem2.p1.m3.sh2a.cmml">m</ci>
            <apply xml:id="ThmLem2.p1.m3.sh2e.cmml">
              <divide xml:id="ThmLem2.p1.m3.sh2b.cmml"/>
              <cn type="integer" xml:id="ThmLem2.p1.m3.sh2c.cmml">1</cn>
              <cn type="integer" xml:id="ThmLem2.p1.m3.sh2d.cmml">12</cn>
            </apply>
          </apply>
        </apply>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok meaning="multirelation"/>
          <XMTok meaning="1" role="NUMBER" font="upright">1</XMTok>
          <XMTok meaning="less-than" role="RELOP" font="upright">&lt;</XMTok>
          <XMTok role="UNKNOWN" xml:id="ThmLem2.p1.m3.sh1.xm" fragid="ThmLem2.p1.m3.sh1.xm">D</XMTok>
          <XMTok meaning="less-than" role="RELOP" font="upright">&lt;</XMTok>
          <XMApp xml:id="ThmLem2.p1.m3.sh2.xm" fragid="ThmLem2.p1.m3.sh2.xm">
            <XMTok role="SUPERSCRIPTOP" scriptpos="post3"/>
            <XMTok role="UNKNOWN">m</XMTok>
            <XMApp>
              <XMTok mathstyle="inline" meaning="divide" role="MULOP" font="upright">/</XMTok>
              <XMTok meaning="1" role="NUMBER" font="upright">1</XMTok>
              <XMTok meaning="12" role="NUMBER" font="upright">12</XMTok>
            </XMApp>
          </XMApp>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">1&lt;D&lt;m^{1/12}</annotation>
  </semantics>
</math>

Canonicalized

1 < D < m 1 / 12 1 D superscript m 1 12 1 < D < m / 1 12 1<D<m^{1/12}

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics encoding="">
    <mrow>
      <mn mathvariant="normal">1</mn>
      <mo mathvariant="normal">&lt;</mo>
      <mi>D</mi>
      <mo mathvariant="normal">&lt;</mo>
      <msup>
        <mi>m</mi>
        <mrow>
          <mn mathvariant="normal">1</mn>
          <mo mathvariant="normal">/</mo>
          <mn mathvariant="normal">12</mn>
        </mrow>
      </msup>
    </mrow>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <and/>
        <apply>
          <lt/>
          <cn encoding="" base="10" type="integer">1</cn>
          <ci encoding="">D</ci>
        </apply>
        <apply>
          <lt/>
          <share/>
          <apply>
            <csymbol encoding="">superscript</csymbol>
            <ci encoding="">m</ci>
            <apply>
              <divide encoding=""/>
              <cn encoding="" base="10" type="integer">1</cn>
              <cn encoding="" base="10" type="integer">12</cn>
            </apply>
          </apply>
        </apply>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok/>
          <XMTok>1</XMTok>
          <XMTok>&lt;</XMTok>
          <XMTok>D</XMTok>
          <XMTok>&lt;</XMTok>
          <XMApp>
            <XMTok/>
            <XMTok>m</XMTok>
            <XMApp>
              <XMTok>/</XMTok>
              <XMTok>1</XMTok>
              <XMTok>12</XMTok>
            </XMApp>
          </XMApp>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">1&lt;D&lt;m^{1/12}</annotation>
  </semantics>
</math>

Diff

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@@ -1,63 +1,63 @@
 <?xml version="1.0" encoding="UTF-8"?>
-<math xmlns="http://www.w3.org/1998/Math/MathML" id="ThmLem2.p1.m3" class="ltx_Math" alttext="1&lt;D&lt;m^{1/12}" display="inline">
-  <semantics>
+<math xmlns="http://www.w3.org/1998/Math/MathML">
+  <semantics encoding="">
     <mrow>
       <mn mathvariant="normal">1</mn>
       <mo mathvariant="normal">&lt;</mo>
       <mi>D</mi>
       <mo mathvariant="normal">&lt;</mo>
       <msup>
         <mi>m</mi>
         <mrow>
           <mn mathvariant="normal">1</mn>
           <mo mathvariant="normal">/</mo>
           <mn mathvariant="normal">12</mn>
         </mrow>
       </msup>
     </mrow>
     <annotation-xml encoding="MathML-Content">
       <apply>
         <and/>
         <apply>
           <lt/>
-          <cn type="integer">1</cn>
-          <ci xml:id="ThmLem2.p1.m3.sh1.cmml">D</ci>
+          <cn encoding="" base="10" type="integer">1</cn>
+          <ci encoding="">D</ci>
         </apply>
         <apply>
           <lt/>
-          <share href="#ThmLem2.p1.m3.sh1.cmml"/>
-          <apply xml:id="ThmLem2.p1.m3.sh2f.cmml">
-            <csymbol cd="ambiguous" xml:id="ThmLem2.p1.m3.sh2.cmml">superscript</csymbol>
-            <ci xml:id="ThmLem2.p1.m3.sh2a.cmml">m</ci>
-            <apply xml:id="ThmLem2.p1.m3.sh2e.cmml">
-              <divide xml:id="ThmLem2.p1.m3.sh2b.cmml"/>
-              <cn type="integer" xml:id="ThmLem2.p1.m3.sh2c.cmml">1</cn>
-              <cn type="integer" xml:id="ThmLem2.p1.m3.sh2d.cmml">12</cn>
+          <share/>
+          <apply>
+            <csymbol encoding="">superscript</csymbol>
+            <ci encoding="">m</ci>
+            <apply>
+              <divide encoding=""/>
+              <cn encoding="" base="10" type="integer">1</cn>
+              <cn encoding="" base="10" type="integer">12</cn>
             </apply>
           </apply>
         </apply>
       </apply>
     </annotation-xml>
     <annotation-xml encoding="application/x-latexml">
       <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
         <XMApp>
-          <XMTok meaning="multirelation"/>
-          <XMTok meaning="1" role="NUMBER" font="upright">1</XMTok>
-          <XMTok meaning="less-than" role="RELOP" font="upright">&lt;</XMTok>
-          <XMTok role="UNKNOWN" xml:id="ThmLem2.p1.m3.sh1.xm" fragid="ThmLem2.p1.m3.sh1.xm">D</XMTok>
-          <XMTok meaning="less-than" role="RELOP" font="upright">&lt;</XMTok>
-          <XMApp xml:id="ThmLem2.p1.m3.sh2.xm" fragid="ThmLem2.p1.m3.sh2.xm">
-            <XMTok role="SUPERSCRIPTOP" scriptpos="post3"/>
-            <XMTok role="UNKNOWN">m</XMTok>
+          <XMTok/>
+          <XMTok>1</XMTok>
+          <XMTok>&lt;</XMTok>
+          <XMTok>D</XMTok>
+          <XMTok>&lt;</XMTok>
+          <XMApp>
+            <XMTok/>
+            <XMTok>m</XMTok>
             <XMApp>
-              <XMTok mathstyle="inline" meaning="divide" role="MULOP" font="upright">/</XMTok>
-              <XMTok meaning="1" role="NUMBER" font="upright">1</XMTok>
-              <XMTok meaning="12" role="NUMBER" font="upright">12</XMTok>
+              <XMTok>/</XMTok>
+              <XMTok>1</XMTok>
+              <XMTok>12</XMTok>
             </XMApp>
           </XMApp>
         </XMApp>
       </XMath>
     </annotation-xml>
     <annotation encoding="application/x-tex">1&lt;D&lt;m^{1/12}</annotation>
   </semantics>
 </math>

Formula 52

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Original

p p p p

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="p6.m1" class="ltx_Math" alttext="p" display="inline">
  <semantics>
    <mi>p</mi>
    <annotation-xml encoding="MathML-Content">
      <ci>p</ci>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMTok role="UNKNOWN" font="italic">p</XMTok>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">p</annotation>
  </semantics>
</math>

Canonicalized

p p p p

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics encoding="">
    <mi>p</mi>
    <annotation-xml encoding="MathML-Content">
      <ci encoding="">p</ci>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMTok>p</XMTok>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">p</annotation>
  </semantics>
</math>

Diff

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@@ -1,15 +1,15 @@
 <?xml version="1.0" encoding="UTF-8"?>
-<math xmlns="http://www.w3.org/1998/Math/MathML" id="p6.m1" class="ltx_Math" alttext="p" display="inline">
-  <semantics>
+<math xmlns="http://www.w3.org/1998/Math/MathML">
+  <semantics encoding="">
     <mi>p</mi>
     <annotation-xml encoding="MathML-Content">
-      <ci>p</ci>
+      <ci encoding="">p</ci>
     </annotation-xml>
     <annotation-xml encoding="application/x-latexml">
       <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
-        <XMTok role="UNKNOWN" font="italic">p</XMTok>
+        <XMTok>p</XMTok>
       </XMath>
     </annotation-xml>
     <annotation encoding="application/x-tex">p</annotation>
   </semantics>
 </math>

Formula 53

( show / hide formula code)

Original

p ≡ 1 ( mod 4 ⁢ D ) p annotated 1 pmod 4 D ≡ p 1 ⁢ 4 D mod p\equiv 1\;\;(\mathop{{\rm mod}}4D)

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="p6.m2" class="ltx_Math" alttext="p\equiv 1\;\;(\mathop{{\rm mod}}4D)" display="inline">
  <semantics>
    <mrow>
      <mi>p</mi>
      <mo>≡</mo>
      <mrow>
        <mn>1</mn>
        <mspace width="veryverythickmathspace"/>
        <mrow>
          <mo>(</mo>
          <mrow>
            <mo>mod</mo>
            <mrow xml:id="p6.m2.xm1c" xref="p6.m2.xm1.cmml">
              <mn xml:id="p6.m2.xm1" xref="p6.m2.xm1.cmml">4</mn>
              <mo xml:id="p6.m2.xm1a" xref="p6.m2.xm1.cmml">⁢</mo>
              <mi xml:id="p6.m2.xm1b" xref="p6.m2.xm1.cmml">D</mi>
            </mrow>
          </mrow>
          <mo>)</mo>
        </mrow>
      </mrow>
    </mrow>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <equivalent/>
        <ci>p</ci>
        <apply>
          <csymbol cd="latexml">annotated</csymbol>
          <cn type="integer">1</cn>
          <apply>
            <ci>pmod</ci>
            <apply xml:id="p6.m2.xm1c.cmml" xref="p6.m2.xm1">
              <times xml:id="p6.m2.xm1.cmml" xref="p6.m2.xm1"/>
              <cn type="integer" xml:id="p6.m2.xm1a.cmml" xref="p6.m2.xm1">4</cn>
              <ci xml:id="p6.m2.xm1b.cmml" xref="p6.m2.xm1">D</ci>
            </apply>
          </apply>
        </apply>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok meaning="equivalent-to" name="equiv" role="RELOP">≡</XMTok>
          <XMTok role="UNKNOWN" font="italic">p</XMTok>
          <XMApp>
            <XMTok meaning="annotated"/>
            <XMTok meaning="1" role="NUMBER">1</XMTok>
            <XMDual role="MODIFIER">
              <XMApp role="MODIFIER">
                <XMTok name="pmod" role="UNKNOWN"/>
                <XMApp xml:id="p6.m2.xm1.xm" fragid="p6.m2.xm1.xm">
                  <XMTok meaning="times" role="MULOP">⁢</XMTok>
                  <XMTok meaning="4" role="NUMBER">4</XMTok>
                  <XMTok role="UNKNOWN" font="italic">D</XMTok>
                </XMApp>
              </XMApp>
              <XMApp close=")" open="(">
                <XMTok role="BIGOP">mod</XMTok>
                <XMRef idref="p6.m2.xm1.xm"/>
              </XMApp>
            </XMDual>
          </XMApp>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">p\equiv 1\;\;(\mathop{{\rm mod}}4D)</annotation>
  </semantics>
</math>

Canonicalized

p ≡ 1 ( mod 4 D ) p annotated 1 pmod 4 D ≡ p 1 ⁢ 4 D mod p\equiv 1\;\;(\mathop{{\rm mod}}4D)

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics encoding="">
    <mrow>
      <mi>p</mi>
      <mo>≡</mo>
      <mrow>
        <mn>1</mn>
        <mrow>
          <mo>(</mo>
          <mrow>
            <mo>mod</mo>
            <mrow>
              <mn>4</mn>
              <mi>D</mi>
            </mrow>
          </mrow>
          <mo>)</mo>
        </mrow>
      </mrow>
    </mrow>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <equivalent/>
        <ci encoding="">p</ci>
        <apply>
          <csymbol encoding="">annotated</csymbol>
          <cn encoding="" base="10" type="integer">1</cn>
          <apply>
            <ci encoding="">pmod</ci>
            <apply>
              <times encoding=""/>
              <cn encoding="" base="10" type="integer">4</cn>
              <ci encoding="">D</ci>
            </apply>
          </apply>
        </apply>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok>≡</XMTok>
          <XMTok>p</XMTok>
          <XMApp>
            <XMTok/>
            <XMTok>1</XMTok>
            <XMDual>
              <XMApp>
                <XMTok/>
                <XMApp>
                  <XMTok>⁢</XMTok>
                  <XMTok>4</XMTok>
                  <XMTok>D</XMTok>
                </XMApp>
              </XMApp>
              <XMApp>
                <XMTok>mod</XMTok>
                <XMRef/>
              </XMApp>
            </XMDual>
          </XMApp>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">p\equiv 1\;\;(\mathop{{\rm mod}}4D)</annotation>
  </semantics>
</math>

Diff

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@@ -1,70 +1,68 @@
 <?xml version="1.0" encoding="UTF-8"?>
-<math xmlns="http://www.w3.org/1998/Math/MathML" id="p6.m2" class="ltx_Math" alttext="p\equiv 1\;\;(\mathop{{\rm mod}}4D)" display="inline">
-  <semantics>
+<math xmlns="http://www.w3.org/1998/Math/MathML">
+  <semantics encoding="">
     <mrow>
       <mi>p</mi>
       <mo>≡</mo>
       <mrow>
         <mn>1</mn>
-        <mspace width="veryverythickmathspace"/>
         <mrow>
           <mo>(</mo>
           <mrow>
             <mo>mod</mo>
-            <mrow xml:id="p6.m2.xm1c" xref="p6.m2.xm1.cmml">
-              <mn xml:id="p6.m2.xm1" xref="p6.m2.xm1.cmml">4</mn>
-              <mo xml:id="p6.m2.xm1a" xref="p6.m2.xm1.cmml">⁢</mo>
-              <mi xml:id="p6.m2.xm1b" xref="p6.m2.xm1.cmml">D</mi>
+            <mrow>
+              <mn>4</mn>
+              <mi>D</mi>
             </mrow>
           </mrow>
           <mo>)</mo>
         </mrow>
       </mrow>
     </mrow>
     <annotation-xml encoding="MathML-Content">
       <apply>
         <equivalent/>
-        <ci>p</ci>
+        <ci encoding="">p</ci>
         <apply>
-          <csymbol cd="latexml">annotated</csymbol>
-          <cn type="integer">1</cn>
+          <csymbol encoding="">annotated</csymbol>
+          <cn encoding="" base="10" type="integer">1</cn>
           <apply>
-            <ci>pmod</ci>
-            <apply xml:id="p6.m2.xm1c.cmml" xref="p6.m2.xm1">
-              <times xml:id="p6.m2.xm1.cmml" xref="p6.m2.xm1"/>
-              <cn type="integer" xml:id="p6.m2.xm1a.cmml" xref="p6.m2.xm1">4</cn>
-              <ci xml:id="p6.m2.xm1b.cmml" xref="p6.m2.xm1">D</ci>
+            <ci encoding="">pmod</ci>
+            <apply>
+              <times encoding=""/>
+              <cn encoding="" base="10" type="integer">4</cn>
+              <ci encoding="">D</ci>
             </apply>
           </apply>
         </apply>
       </apply>
     </annotation-xml>
     <annotation-xml encoding="application/x-latexml">
       <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
         <XMApp>
-          <XMTok meaning="equivalent-to" name="equiv" role="RELOP">≡</XMTok>
-          <XMTok role="UNKNOWN" font="italic">p</XMTok>
+          <XMTok>≡</XMTok>
+          <XMTok>p</XMTok>
           <XMApp>
-            <XMTok meaning="annotated"/>
-            <XMTok meaning="1" role="NUMBER">1</XMTok>
-            <XMDual role="MODIFIER">
-              <XMApp role="MODIFIER">
-                <XMTok name="pmod" role="UNKNOWN"/>
-                <XMApp xml:id="p6.m2.xm1.xm" fragid="p6.m2.xm1.xm">
-                  <XMTok meaning="times" role="MULOP">⁢</XMTok>
-                  <XMTok meaning="4" role="NUMBER">4</XMTok>
-                  <XMTok role="UNKNOWN" font="italic">D</XMTok>
+            <XMTok/>
+            <XMTok>1</XMTok>
+            <XMDual>
+              <XMApp>
+                <XMTok/>
+                <XMApp>
+                  <XMTok>⁢</XMTok>
+                  <XMTok>4</XMTok>
+                  <XMTok>D</XMTok>
                 </XMApp>
               </XMApp>
-              <XMApp close=")" open="(">
-                <XMTok role="BIGOP">mod</XMTok>
-                <XMRef idref="p6.m2.xm1.xm"/>
+              <XMApp>
+                <XMTok>mod</XMTok>
+                <XMRef/>
               </XMApp>
             </XMDual>
           </XMApp>
         </XMApp>
       </XMath>
     </annotation-xml>
     <annotation encoding="application/x-tex">p\equiv 1\;\;(\mathop{{\rm mod}}4D)</annotation>
   </semantics>
 </math>

Formula 54

( show / hide formula code)

Original

( p , m ) = 1 p m 1 = p m 1 (p,m)=1

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="p6.m3" class="ltx_Math" alttext="(p,m)=1" display="inline">
  <semantics>
    <mrow>
      <mrow>
        <mo>(</mo>
        <mrow>
          <mi>p</mi>
          <mo>,</mo>
          <mi>m</mi>
        </mrow>
        <mo>)</mo>
      </mrow>
      <mo>=</mo>
      <mn>1</mn>
    </mrow>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <eq/>
        <apply>
          <interval closure="open"/>
          <ci>p</ci>
          <ci>m</ci>
        </apply>
        <cn type="integer">1</cn>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok meaning="equals" role="RELOP">=</XMTok>
          <XMApp>
            <XMTok argclose=")" argopen="(" meaning="open-interval" role="FENCED" separators=","/>
            <XMTok role="UNKNOWN" font="italic">p</XMTok>
            <XMTok role="UNKNOWN" font="italic">m</XMTok>
          </XMApp>
          <XMTok meaning="1" role="NUMBER">1</XMTok>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">(p,m)=1</annotation>
  </semantics>
</math>

Canonicalized

( p , m ) = 1 p m 1 = p m 1 (p,m)=1

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics encoding="">
    <mrow>
      <mrow>
        <mo>(</mo>
        <mrow>
          <mi>p</mi>
          <mo>,</mo>
          <mi>m</mi>
        </mrow>
        <mo>)</mo>
      </mrow>
      <mo>=</mo>
      <mn>1</mn>
    </mrow>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <eq/>
        <apply>
          <interval closure="open"/>
          <ci encoding="">p</ci>
          <ci encoding="">m</ci>
        </apply>
        <cn encoding="" base="10" type="integer">1</cn>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok>=</XMTok>
          <XMApp>
            <XMTok/>
            <XMTok>p</XMTok>
            <XMTok>m</XMTok>
          </XMApp>
          <XMTok>1</XMTok>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">(p,m)=1</annotation>
  </semantics>
</math>

Diff

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@@ -1,43 +1,43 @@
 <?xml version="1.0" encoding="UTF-8"?>
-<math xmlns="http://www.w3.org/1998/Math/MathML" id="p6.m3" class="ltx_Math" alttext="(p,m)=1" display="inline">
-  <semantics>
+<math xmlns="http://www.w3.org/1998/Math/MathML">
+  <semantics encoding="">
     <mrow>
       <mrow>
         <mo>(</mo>
         <mrow>
           <mi>p</mi>
           <mo>,</mo>
           <mi>m</mi>
         </mrow>
         <mo>)</mo>
       </mrow>
       <mo>=</mo>
       <mn>1</mn>
     </mrow>
     <annotation-xml encoding="MathML-Content">
       <apply>
         <eq/>
         <apply>
           <interval closure="open"/>
-          <ci>p</ci>
-          <ci>m</ci>
+          <ci encoding="">p</ci>
+          <ci encoding="">m</ci>
         </apply>
-        <cn type="integer">1</cn>
+        <cn encoding="" base="10" type="integer">1</cn>
       </apply>
     </annotation-xml>
     <annotation-xml encoding="application/x-latexml">
       <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
         <XMApp>
-          <XMTok meaning="equals" role="RELOP">=</XMTok>
+          <XMTok>=</XMTok>
           <XMApp>
-            <XMTok argclose=")" argopen="(" meaning="open-interval" role="FENCED" separators=","/>
-            <XMTok role="UNKNOWN" font="italic">p</XMTok>
-            <XMTok role="UNKNOWN" font="italic">m</XMTok>
+            <XMTok/>
+            <XMTok>p</XMTok>
+            <XMTok>m</XMTok>
           </XMApp>
-          <XMTok meaning="1" role="NUMBER">1</XMTok>
+          <XMTok>1</XMTok>
         </XMApp>
       </XMath>
     </annotation-xml>
     <annotation encoding="application/x-tex">(p,m)=1</annotation>
   </semantics>
 </math>

Formula 55

( show / hide formula code)

Original

p < D C p superscript D C < p D C p<D^{C}

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="p6.m4" class="ltx_Math" alttext="p&lt;D^{C}" display="inline">
  <semantics>
    <mrow>
      <mi>p</mi>
      <mo>&lt;</mo>
      <msup>
        <mi>D</mi>
        <mi>C</mi>
      </msup>
    </mrow>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <lt/>
        <ci>p</ci>
        <apply>
          <csymbol cd="ambiguous">superscript</csymbol>
          <ci>D</ci>
          <ci>C</ci>
        </apply>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok meaning="less-than" role="RELOP">&lt;</XMTok>
          <XMTok role="UNKNOWN" font="italic">p</XMTok>
          <XMApp>
            <XMTok role="SUPERSCRIPTOP" scriptpos="post2"/>
            <XMTok role="UNKNOWN" font="italic">D</XMTok>
            <XMTok role="UNKNOWN" font="italic">C</XMTok>
          </XMApp>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">p&lt;D^{C}</annotation>
  </semantics>
</math>

Canonicalized

p < D C p superscript D C < p D C p<D^{C}

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics encoding="">
    <mrow>
      <mi>p</mi>
      <mo>&lt;</mo>
      <msup>
        <mi>D</mi>
        <mi>C</mi>
      </msup>
    </mrow>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <lt/>
        <ci encoding="">p</ci>
        <apply>
          <csymbol encoding="">superscript</csymbol>
          <ci encoding="">D</ci>
          <ci encoding="">C</ci>
        </apply>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok>&lt;</XMTok>
          <XMTok>p</XMTok>
          <XMApp>
            <XMTok/>
            <XMTok>D</XMTok>
            <XMTok>C</XMTok>
          </XMApp>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">p&lt;D^{C}</annotation>
  </semantics>
</math>

Diff

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@@ -1,38 +1,38 @@
 <?xml version="1.0" encoding="UTF-8"?>
-<math xmlns="http://www.w3.org/1998/Math/MathML" id="p6.m4" class="ltx_Math" alttext="p&lt;D^{C}" display="inline">
-  <semantics>
+<math xmlns="http://www.w3.org/1998/Math/MathML">
+  <semantics encoding="">
     <mrow>
       <mi>p</mi>
       <mo>&lt;</mo>
       <msup>
         <mi>D</mi>
         <mi>C</mi>
       </msup>
     </mrow>
     <annotation-xml encoding="MathML-Content">
       <apply>
         <lt/>
-        <ci>p</ci>
+        <ci encoding="">p</ci>
         <apply>
-          <csymbol cd="ambiguous">superscript</csymbol>
-          <ci>D</ci>
-          <ci>C</ci>
+          <csymbol encoding="">superscript</csymbol>
+          <ci encoding="">D</ci>
+          <ci encoding="">C</ci>
         </apply>
       </apply>
     </annotation-xml>
     <annotation-xml encoding="application/x-latexml">
       <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
         <XMApp>
-          <XMTok meaning="less-than" role="RELOP">&lt;</XMTok>
-          <XMTok role="UNKNOWN" font="italic">p</XMTok>
+          <XMTok>&lt;</XMTok>
+          <XMTok>p</XMTok>
           <XMApp>
-            <XMTok role="SUPERSCRIPTOP" scriptpos="post2"/>
-            <XMTok role="UNKNOWN" font="italic">D</XMTok>
-            <XMTok role="UNKNOWN" font="italic">C</XMTok>
+            <XMTok/>
+            <XMTok>D</XMTok>
+            <XMTok>C</XMTok>
           </XMApp>
         </XMApp>
       </XMath>
     </annotation-xml>
     <annotation encoding="application/x-tex">p&lt;D^{C}</annotation>
   </semantics>
 </math>

Formula 56

( show / hide formula code)

Original

C C C C

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="p6.m5" class="ltx_Math" alttext="C" display="inline">
  <semantics>
    <mi>C</mi>
    <annotation-xml encoding="MathML-Content">
      <ci>C</ci>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMTok role="UNKNOWN" font="italic">C</XMTok>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">C</annotation>
  </semantics>
</math>

Canonicalized

C C C C

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics encoding="">
    <mi>C</mi>
    <annotation-xml encoding="MathML-Content">
      <ci encoding="">C</ci>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMTok>C</XMTok>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">C</annotation>
  </semantics>
</math>

Diff

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@@ -1,15 +1,15 @@
 <?xml version="1.0" encoding="UTF-8"?>
-<math xmlns="http://www.w3.org/1998/Math/MathML" id="p6.m5" class="ltx_Math" alttext="C" display="inline">
-  <semantics>
+<math xmlns="http://www.w3.org/1998/Math/MathML">
+  <semantics encoding="">
     <mi>C</mi>
     <annotation-xml encoding="MathML-Content">
-      <ci>C</ci>
+      <ci encoding="">C</ci>
     </annotation-xml>
     <annotation-xml encoding="application/x-latexml">
       <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
-        <XMTok role="UNKNOWN" font="italic">C</XMTok>
+        <XMTok>C</XMTok>
       </XMath>
     </annotation-xml>
     <annotation encoding="application/x-tex">C</annotation>
   </semantics>
 </math>

Formula 57

( show / hide formula code)

Original

D D D D

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="p6.m6" class="ltx_Math" alttext="D" display="inline">
  <semantics>
    <mi>D</mi>
    <annotation-xml encoding="MathML-Content">
      <ci>D</ci>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMTok role="UNKNOWN" font="italic">D</XMTok>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">D</annotation>
  </semantics>
</math>

Canonicalized

D D D D

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics encoding="">
    <mi>D</mi>
    <annotation-xml encoding="MathML-Content">
      <ci encoding="">D</ci>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMTok>D</XMTok>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">D</annotation>
  </semantics>
</math>

Diff

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@@ -1,15 +1,15 @@
 <?xml version="1.0" encoding="UTF-8"?>
-<math xmlns="http://www.w3.org/1998/Math/MathML" id="p6.m6" class="ltx_Math" alttext="D" display="inline">
-  <semantics>
+<math xmlns="http://www.w3.org/1998/Math/MathML">
+  <semantics encoding="">
     <mi>D</mi>
     <annotation-xml encoding="MathML-Content">
-      <ci>D</ci>
+      <ci encoding="">D</ci>
     </annotation-xml>
     <annotation-xml encoding="application/x-latexml">
       <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
-        <XMTok role="UNKNOWN" font="italic">D</XMTok>
+        <XMTok>D</XMTok>
       </XMath>
     </annotation-xml>
     <annotation encoding="application/x-tex">D</annotation>
   </semantics>
 </math>

Formula 58

( show / hide formula code)

Original

C = 5.5 C 5.5 = C 5.5 C=5.5

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="p6.m7" class="ltx_Math" alttext="C=5.5" display="inline">
  <semantics>
    <mrow>
      <mi>C</mi>
      <mo>=</mo>
      <mn>5.5</mn>
    </mrow>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <eq/>
        <ci>C</ci>
        <cn type="float">5.5</cn>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok meaning="equals" role="RELOP">=</XMTok>
          <XMTok role="UNKNOWN" font="italic">C</XMTok>
          <XMTok meaning="5.5" role="NUMBER">5.5</XMTok>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">C=5.5</annotation>
  </semantics>
</math>

Canonicalized

C = 5.5 C 5.5 = C 5.5 C=5.5

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics encoding="">
    <mrow>
      <mi>C</mi>
      <mo>=</mo>
      <mn>5.5</mn>
    </mrow>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <eq/>
        <ci encoding="">C</ci>
        <cn encoding="" base="10" type="float">5.5</cn>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok>=</XMTok>
          <XMTok>C</XMTok>
          <XMTok>5.5</XMTok>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">C=5.5</annotation>
  </semantics>
</math>

Diff

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@@ -1,27 +1,27 @@
 <?xml version="1.0" encoding="UTF-8"?>
-<math xmlns="http://www.w3.org/1998/Math/MathML" id="p6.m7" class="ltx_Math" alttext="C=5.5" display="inline">
-  <semantics>
+<math xmlns="http://www.w3.org/1998/Math/MathML">
+  <semantics encoding="">
     <mrow>
       <mi>C</mi>
       <mo>=</mo>
       <mn>5.5</mn>
     </mrow>
     <annotation-xml encoding="MathML-Content">
       <apply>
         <eq/>
-        <ci>C</ci>
-        <cn type="float">5.5</cn>
+        <ci encoding="">C</ci>
+        <cn encoding="" base="10" type="float">5.5</cn>
       </apply>
     </annotation-xml>
     <annotation-xml encoding="application/x-latexml">
       <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
         <XMApp>
-          <XMTok meaning="equals" role="RELOP">=</XMTok>
-          <XMTok role="UNKNOWN" font="italic">C</XMTok>
-          <XMTok meaning="5.5" role="NUMBER">5.5</XMTok>
+          <XMTok>=</XMTok>
+          <XMTok>C</XMTok>
+          <XMTok>5.5</XMTok>
         </XMApp>
       </XMath>
     </annotation-xml>
     <annotation encoding="application/x-tex">C=5.5</annotation>
   </semantics>
 </math>

Formula 59

( show / hide formula code)

Original

D 0 subscript D 0 D 0 D_{0}

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="p6.m8" class="ltx_Math" alttext="D_{0}" display="inline">
  <semantics>
    <msub>
      <mi>D</mi>
      <mn>0</mn>
    </msub>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <csymbol cd="ambiguous">subscript</csymbol>
        <ci>D</ci>
        <cn type="integer">0</cn>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok role="SUBSCRIPTOP" scriptpos="post2"/>
          <XMTok role="UNKNOWN" font="italic">D</XMTok>
          <XMTok meaning="0" role="NUMBER">0</XMTok>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">D_{0}</annotation>
  </semantics>
</math>

Canonicalized

D 0 subscript D 0 D 0 D_{0}

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics encoding="">
    <msub>
      <mi>D</mi>
      <mn>0</mn>
    </msub>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <csymbol encoding="">subscript</csymbol>
        <ci encoding="">D</ci>
        <cn encoding="" base="10" type="integer">0</cn>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok/>
          <XMTok>D</XMTok>
          <XMTok>0</XMTok>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">D_{0}</annotation>
  </semantics>
</math>

Diff

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@@ -1,26 +1,26 @@
 <?xml version="1.0" encoding="UTF-8"?>
-<math xmlns="http://www.w3.org/1998/Math/MathML" id="p6.m8" class="ltx_Math" alttext="D_{0}" display="inline">
-  <semantics>
+<math xmlns="http://www.w3.org/1998/Math/MathML">
+  <semantics encoding="">
     <msub>
       <mi>D</mi>
       <mn>0</mn>
     </msub>
     <annotation-xml encoding="MathML-Content">
       <apply>
-        <csymbol cd="ambiguous">subscript</csymbol>
-        <ci>D</ci>
-        <cn type="integer">0</cn>
+        <csymbol encoding="">subscript</csymbol>
+        <ci encoding="">D</ci>
+        <cn encoding="" base="10" type="integer">0</cn>
       </apply>
     </annotation-xml>
     <annotation-xml encoding="application/x-latexml">
       <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
         <XMApp>
-          <XMTok role="SUBSCRIPTOP" scriptpos="post2"/>
-          <XMTok role="UNKNOWN" font="italic">D</XMTok>
-          <XMTok meaning="0" role="NUMBER">0</XMTok>
+          <XMTok/>
+          <XMTok>D</XMTok>
+          <XMTok>0</XMTok>
         </XMApp>
       </XMath>
     </annotation-xml>
     <annotation encoding="application/x-tex">D_{0}</annotation>
   </semantics>
 </math>

Formula 60

( show / hide formula code)

Original

D > D 0 D subscript D 0 > D D 0 D>D_{0}

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="p6.m9" class="ltx_Math" alttext="D&gt;D_{0}" display="inline">
  <semantics>
    <mrow>
      <mi>D</mi>
      <mo>&gt;</mo>
      <msub>
        <mi>D</mi>
        <mn>0</mn>
      </msub>
    </mrow>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <gt/>
        <ci>D</ci>
        <apply>
          <csymbol cd="ambiguous">subscript</csymbol>
          <ci>D</ci>
          <cn type="integer">0</cn>
        </apply>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok meaning="greater-than" role="RELOP">&gt;</XMTok>
          <XMTok role="UNKNOWN" font="italic">D</XMTok>
          <XMApp>
            <XMTok role="SUBSCRIPTOP" scriptpos="post2"/>
            <XMTok role="UNKNOWN" font="italic">D</XMTok>
            <XMTok meaning="0" role="NUMBER">0</XMTok>
          </XMApp>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">D&gt;D_{0}</annotation>
  </semantics>
</math>

Canonicalized

D > D 0 D subscript D 0 > D D 0 D>D_{0}

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics encoding="">
    <mrow>
      <mi>D</mi>
      <mo>&gt;</mo>
      <msub>
        <mi>D</mi>
        <mn>0</mn>
      </msub>
    </mrow>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <gt/>
        <ci encoding="">D</ci>
        <apply>
          <csymbol encoding="">subscript</csymbol>
          <ci encoding="">D</ci>
          <cn encoding="" base="10" type="integer">0</cn>
        </apply>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok>&gt;</XMTok>
          <XMTok>D</XMTok>
          <XMApp>
            <XMTok/>
            <XMTok>D</XMTok>
            <XMTok>0</XMTok>
          </XMApp>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">D&gt;D_{0}</annotation>
  </semantics>
</math>

Diff

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@@ -1,38 +1,38 @@
 <?xml version="1.0" encoding="UTF-8"?>
-<math xmlns="http://www.w3.org/1998/Math/MathML" id="p6.m9" class="ltx_Math" alttext="D&gt;D_{0}" display="inline">
-  <semantics>
+<math xmlns="http://www.w3.org/1998/Math/MathML">
+  <semantics encoding="">
     <mrow>
       <mi>D</mi>
       <mo>&gt;</mo>
       <msub>
         <mi>D</mi>
         <mn>0</mn>
       </msub>
     </mrow>
     <annotation-xml encoding="MathML-Content">
       <apply>
         <gt/>
-        <ci>D</ci>
+        <ci encoding="">D</ci>
         <apply>
-          <csymbol cd="ambiguous">subscript</csymbol>
-          <ci>D</ci>
-          <cn type="integer">0</cn>
+          <csymbol encoding="">subscript</csymbol>
+          <ci encoding="">D</ci>
+          <cn encoding="" base="10" type="integer">0</cn>
         </apply>
       </apply>
     </annotation-xml>
     <annotation-xml encoding="application/x-latexml">
       <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
         <XMApp>
-          <XMTok meaning="greater-than" role="RELOP">&gt;</XMTok>
-          <XMTok role="UNKNOWN" font="italic">D</XMTok>
+          <XMTok>&gt;</XMTok>
+          <XMTok>D</XMTok>
           <XMApp>
-            <XMTok role="SUBSCRIPTOP" scriptpos="post2"/>
-            <XMTok role="UNKNOWN" font="italic">D</XMTok>
-            <XMTok meaning="0" role="NUMBER">0</XMTok>
+            <XMTok/>
+            <XMTok>D</XMTok>
+            <XMTok>0</XMTok>
           </XMApp>
         </XMApp>
       </XMath>
     </annotation-xml>
     <annotation encoding="application/x-tex">D&gt;D_{0}</annotation>
   </semantics>
 </math>

Formula 61

( show / hide formula code)

Original

p < m 1 / 2 / 6 p superscript m 1 2 6 < p / m / 1 2 6 p<m^{1/2}/6

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="p6.m10" class="ltx_Math" alttext="p&lt;m^{1/2}/6" display="inline">
  <semantics>
    <mrow>
      <mi>p</mi>
      <mo>&lt;</mo>
      <mrow>
        <msup>
          <mi>m</mi>
          <mrow>
            <mn>1</mn>
            <mo>/</mo>
            <mn>2</mn>
          </mrow>
        </msup>
        <mo>/</mo>
        <mn>6</mn>
      </mrow>
    </mrow>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <lt/>
        <ci>p</ci>
        <apply>
          <divide/>
          <apply>
            <csymbol cd="ambiguous">superscript</csymbol>
            <ci>m</ci>
            <apply>
              <divide/>
              <cn type="integer">1</cn>
              <cn type="integer">2</cn>
            </apply>
          </apply>
          <cn type="integer">6</cn>
        </apply>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok meaning="less-than" role="RELOP">&lt;</XMTok>
          <XMTok role="UNKNOWN" font="italic">p</XMTok>
          <XMApp>
            <XMTok mathstyle="inline" meaning="divide" role="MULOP">/</XMTok>
            <XMApp>
              <XMTok role="SUPERSCRIPTOP" scriptpos="post2"/>
              <XMTok role="UNKNOWN" font="italic">m</XMTok>
              <XMApp>
                <XMTok mathstyle="inline" meaning="divide" role="MULOP">/</XMTok>
                <XMTok meaning="1" role="NUMBER">1</XMTok>
                <XMTok meaning="2" role="NUMBER">2</XMTok>
              </XMApp>
            </XMApp>
            <XMTok meaning="6" role="NUMBER">6</XMTok>
          </XMApp>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">p&lt;m^{1/2}/6</annotation>
  </semantics>
</math>

Canonicalized

p < m 1 / 2 / 6 p superscript m 1 2 6 < p / m / 1 2 6 p<m^{1/2}/6

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics encoding="">
    <mrow>
      <mi>p</mi>
      <mo>&lt;</mo>
      <mrow>
        <msup>
          <mi>m</mi>
          <mrow>
            <mn>1</mn>
            <mo>/</mo>
            <mn>2</mn>
          </mrow>
        </msup>
        <mo>/</mo>
        <mn>6</mn>
      </mrow>
    </mrow>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <lt/>
        <ci encoding="">p</ci>
        <apply>
          <divide/>
          <apply>
            <csymbol encoding="">superscript</csymbol>
            <ci encoding="">m</ci>
            <apply>
              <divide/>
              <cn encoding="" base="10" type="integer">1</cn>
              <cn encoding="" base="10" type="integer">2</cn>
            </apply>
          </apply>
          <cn encoding="" base="10" type="integer">6</cn>
        </apply>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok>&lt;</XMTok>
          <XMTok>p</XMTok>
          <XMApp>
            <XMTok>/</XMTok>
            <XMApp>
              <XMTok/>
              <XMTok>m</XMTok>
              <XMApp>
                <XMTok>/</XMTok>
                <XMTok>1</XMTok>
                <XMTok>2</XMTok>
              </XMApp>
            </XMApp>
            <XMTok>6</XMTok>
          </XMApp>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">p&lt;m^{1/2}/6</annotation>
  </semantics>
</math>

Diff

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@@ -1,62 +1,62 @@
 <?xml version="1.0" encoding="UTF-8"?>
-<math xmlns="http://www.w3.org/1998/Math/MathML" id="p6.m10" class="ltx_Math" alttext="p&lt;m^{1/2}/6" display="inline">
-  <semantics>
+<math xmlns="http://www.w3.org/1998/Math/MathML">
+  <semantics encoding="">
     <mrow>
       <mi>p</mi>
       <mo>&lt;</mo>
       <mrow>
         <msup>
           <mi>m</mi>
           <mrow>
             <mn>1</mn>
             <mo>/</mo>
             <mn>2</mn>
           </mrow>
         </msup>
         <mo>/</mo>
         <mn>6</mn>
       </mrow>
     </mrow>
     <annotation-xml encoding="MathML-Content">
       <apply>
         <lt/>
-        <ci>p</ci>
+        <ci encoding="">p</ci>
         <apply>
           <divide/>
           <apply>
-            <csymbol cd="ambiguous">superscript</csymbol>
-            <ci>m</ci>
+            <csymbol encoding="">superscript</csymbol>
+            <ci encoding="">m</ci>
             <apply>
               <divide/>
-              <cn type="integer">1</cn>
-              <cn type="integer">2</cn>
+              <cn encoding="" base="10" type="integer">1</cn>
+              <cn encoding="" base="10" type="integer">2</cn>
             </apply>
           </apply>
-          <cn type="integer">6</cn>
+          <cn encoding="" base="10" type="integer">6</cn>
         </apply>
       </apply>
     </annotation-xml>
     <annotation-xml encoding="application/x-latexml">
       <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
         <XMApp>
-          <XMTok meaning="less-than" role="RELOP">&lt;</XMTok>
-          <XMTok role="UNKNOWN" font="italic">p</XMTok>
+          <XMTok>&lt;</XMTok>
+          <XMTok>p</XMTok>
           <XMApp>
-            <XMTok mathstyle="inline" meaning="divide" role="MULOP">/</XMTok>
+            <XMTok>/</XMTok>
             <XMApp>
-              <XMTok role="SUPERSCRIPTOP" scriptpos="post2"/>
-              <XMTok role="UNKNOWN" font="italic">m</XMTok>
+              <XMTok/>
+              <XMTok>m</XMTok>
               <XMApp>
-                <XMTok mathstyle="inline" meaning="divide" role="MULOP">/</XMTok>
-                <XMTok meaning="1" role="NUMBER">1</XMTok>
-                <XMTok meaning="2" role="NUMBER">2</XMTok>
+                <XMTok>/</XMTok>
+                <XMTok>1</XMTok>
+                <XMTok>2</XMTok>
               </XMApp>
             </XMApp>
-            <XMTok meaning="6" role="NUMBER">6</XMTok>
+            <XMTok>6</XMTok>
           </XMApp>
         </XMApp>
       </XMath>
     </annotation-xml>
     <annotation encoding="application/x-tex">p&lt;m^{1/2}/6</annotation>
   </semantics>
 </math>

Formula 62

( show / hide formula code)

Original

p p p p

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="p6.m11" class="ltx_Math" alttext="p" display="inline">
  <semantics>
    <mi>p</mi>
    <annotation-xml encoding="MathML-Content">
      <ci>p</ci>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMTok role="UNKNOWN" font="italic">p</XMTok>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">p</annotation>
  </semantics>
</math>

Canonicalized

p p p p

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics encoding="">
    <mi>p</mi>
    <annotation-xml encoding="MathML-Content">
      <ci encoding="">p</ci>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMTok>p</XMTok>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">p</annotation>
  </semantics>
</math>

Diff

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@@ -1,15 +1,15 @@
 <?xml version="1.0" encoding="UTF-8"?>
-<math xmlns="http://www.w3.org/1998/Math/MathML" id="p6.m11" class="ltx_Math" alttext="p" display="inline">
-  <semantics>
+<math xmlns="http://www.w3.org/1998/Math/MathML">
+  <semantics encoding="">
     <mi>p</mi>
     <annotation-xml encoding="MathML-Content">
-      <ci>p</ci>
+      <ci encoding="">p</ci>
     </annotation-xml>
     <annotation-xml encoding="application/x-latexml">
       <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
-        <XMTok role="UNKNOWN" font="italic">p</XMTok>
+        <XMTok>p</XMTok>
       </XMath>
     </annotation-xml>
     <annotation encoding="application/x-tex">p</annotation>
   </semantics>
 </math>

Formula 63

( show / hide formula code)

Original

p p p p

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="p6.m12" class="ltx_Math" alttext="p" display="inline">
  <semantics>
    <mi>p</mi>
    <annotation-xml encoding="MathML-Content">
      <ci>p</ci>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMTok role="UNKNOWN" font="italic">p</XMTok>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">p</annotation>
  </semantics>
</math>

Canonicalized

p p p p

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics encoding="">
    <mi>p</mi>
    <annotation-xml encoding="MathML-Content">
      <ci encoding="">p</ci>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMTok>p</XMTok>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">p</annotation>
  </semantics>
</math>

Diff

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@@ -1,15 +1,15 @@
 <?xml version="1.0" encoding="UTF-8"?>
-<math xmlns="http://www.w3.org/1998/Math/MathML" id="p6.m12" class="ltx_Math" alttext="p" display="inline">
-  <semantics>
+<math xmlns="http://www.w3.org/1998/Math/MathML">
+  <semantics encoding="">
     <mi>p</mi>
     <annotation-xml encoding="MathML-Content">
-      <ci>p</ci>
+      <ci encoding="">p</ci>
     </annotation-xml>
     <annotation-xml encoding="application/x-latexml">
       <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
-        <XMTok role="UNKNOWN" font="italic">p</XMTok>
+        <XMTok>p</XMTok>
       </XMath>
     </annotation-xml>
     <annotation encoding="application/x-tex">p</annotation>
   </semantics>
 </math>

Formula 64

( show / hide formula code)

Original

n n n n

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="p6.m13" class="ltx_Math" alttext="n" display="inline">
  <semantics>
    <mi>n</mi>
    <annotation-xml encoding="MathML-Content">
      <ci>n</ci>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMTok role="UNKNOWN" font="italic">n</XMTok>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">n</annotation>
  </semantics>
</math>

Canonicalized

n n n n

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics encoding="">
    <mi>n</mi>
    <annotation-xml encoding="MathML-Content">
      <ci encoding="">n</ci>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMTok>n</XMTok>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">n</annotation>
  </semantics>
</math>

Diff

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@@ -1,15 +1,15 @@
 <?xml version="1.0" encoding="UTF-8"?>
-<math xmlns="http://www.w3.org/1998/Math/MathML" id="p6.m13" class="ltx_Math" alttext="n" display="inline">
-  <semantics>
+<math xmlns="http://www.w3.org/1998/Math/MathML">
+  <semantics encoding="">
     <mi>n</mi>
     <annotation-xml encoding="MathML-Content">
-      <ci>n</ci>
+      <ci encoding="">n</ci>
     </annotation-xml>
     <annotation-xml encoding="application/x-latexml">
       <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
-        <XMTok role="UNKNOWN" font="italic">n</XMTok>
+        <XMTok>n</XMTok>
       </XMath>
     </annotation-xml>
     <annotation encoding="application/x-tex">n</annotation>
   </semantics>
 </math>

Formula 65

( show / hide formula code)

Original

x - 1 + u ⁢ D 2 x 1 u D 2 - x + 1 ⁢ u D 2 x-\frac{1+u\sqrt{D}}{2}

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="p6.m14" class="ltx_Math" alttext="x-\frac{1+u\sqrt{D}}{2}" display="inline">
  <semantics>
    <mrow>
      <mi>x</mi>
      <mo>-</mo>
      <mfrac>
        <mrow>
          <mn>1</mn>
          <mo>+</mo>
          <mrow>
            <mi>u</mi>
            <mo>⁢</mo>
            <msqrt>
              <mi>D</mi>
            </msqrt>
          </mrow>
        </mrow>
        <mn>2</mn>
      </mfrac>
    </mrow>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <minus/>
        <ci>x</ci>
        <apply>
          <divide/>
          <apply>
            <plus/>
            <cn type="integer">1</cn>
            <apply>
              <times/>
              <ci>u</ci>
              <apply>
                <root/>
                <ci>D</ci>
              </apply>
            </apply>
          </apply>
          <cn type="integer">2</cn>
        </apply>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok meaning="minus" role="ADDOP">-</XMTok>
          <XMTok role="UNKNOWN" font="italic">x</XMTok>
          <XMApp>
            <XMTok mathstyle="text" meaning="divide" role="MULOP"/>
            <XMApp>
              <XMTok meaning="plus" role="ADDOP">+</XMTok>
              <XMTok meaning="1" role="NUMBER">1</XMTok>
              <XMApp>
                <XMTok meaning="times" role="MULOP">⁢</XMTok>
                <XMTok role="UNKNOWN" font="italic">u</XMTok>
                <XMApp>
                  <XMTok meaning="square-root"/>
                  <XMTok role="UNKNOWN" font="italic">D</XMTok>
                </XMApp>
              </XMApp>
            </XMApp>
            <XMTok meaning="2" role="NUMBER">2</XMTok>
          </XMApp>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">x-\frac{1+u\sqrt{D}}{2}</annotation>
  </semantics>
</math>

Canonicalized

x - 1 + u D 2 x 1 u D 2 - x + 1 ⁢ u D 2 x-\frac{1+u\sqrt{D}}{2}

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics encoding="">
    <mrow>
      <mi>x</mi>
      <mo>-</mo>
      <mfrac>
        <mrow>
          <mn>1</mn>
          <mo>+</mo>
          <mrow>
            <mi>u</mi>
            <msqrt>
              <mi>D</mi>
            </msqrt>
          </mrow>
        </mrow>
        <mn>2</mn>
      </mfrac>
    </mrow>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <minus/>
        <ci encoding="">x</ci>
        <apply>
          <divide/>
          <apply>
            <plus/>
            <cn encoding="" base="10" type="integer">1</cn>
            <apply>
              <times/>
              <ci encoding="">u</ci>
              <apply>
                <root/>
                <ci encoding="">D</ci>
              </apply>
            </apply>
          </apply>
          <cn encoding="" base="10" type="integer">2</cn>
        </apply>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok>-</XMTok>
          <XMTok>x</XMTok>
          <XMApp>
            <XMTok/>
            <XMApp>
              <XMTok>+</XMTok>
              <XMTok>1</XMTok>
              <XMApp>
                <XMTok>⁢</XMTok>
                <XMTok>u</XMTok>
                <XMApp>
                  <XMTok/>
                  <XMTok>D</XMTok>
                </XMApp>
              </XMApp>
            </XMApp>
            <XMTok>2</XMTok>
          </XMApp>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">x-\frac{1+u\sqrt{D}}{2}</annotation>
  </semantics>
</math>

Diff

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@@ -1,70 +1,69 @@
 <?xml version="1.0" encoding="UTF-8"?>
-<math xmlns="http://www.w3.org/1998/Math/MathML" id="p6.m14" class="ltx_Math" alttext="x-\frac{1+u\sqrt{D}}{2}" display="inline">
-  <semantics>
+<math xmlns="http://www.w3.org/1998/Math/MathML">
+  <semantics encoding="">
     <mrow>
       <mi>x</mi>
       <mo>-</mo>
       <mfrac>
         <mrow>
           <mn>1</mn>
           <mo>+</mo>
           <mrow>
             <mi>u</mi>
-            <mo>⁢</mo>
             <msqrt>
               <mi>D</mi>
             </msqrt>
           </mrow>
         </mrow>
         <mn>2</mn>
       </mfrac>
     </mrow>
     <annotation-xml encoding="MathML-Content">
       <apply>
         <minus/>
-        <ci>x</ci>
+        <ci encoding="">x</ci>
         <apply>
           <divide/>
           <apply>
             <plus/>
-            <cn type="integer">1</cn>
+            <cn encoding="" base="10" type="integer">1</cn>
             <apply>
               <times/>
-              <ci>u</ci>
+              <ci encoding="">u</ci>
               <apply>
                 <root/>
-                <ci>D</ci>
+                <ci encoding="">D</ci>
               </apply>
             </apply>
           </apply>
-          <cn type="integer">2</cn>
+          <cn encoding="" base="10" type="integer">2</cn>
         </apply>
       </apply>
     </annotation-xml>
     <annotation-xml encoding="application/x-latexml">
       <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
         <XMApp>
-          <XMTok meaning="minus" role="ADDOP">-</XMTok>
-          <XMTok role="UNKNOWN" font="italic">x</XMTok>
+          <XMTok>-</XMTok>
+          <XMTok>x</XMTok>
           <XMApp>
-            <XMTok mathstyle="text" meaning="divide" role="MULOP"/>
+            <XMTok/>
             <XMApp>
-              <XMTok meaning="plus" role="ADDOP">+</XMTok>
-              <XMTok meaning="1" role="NUMBER">1</XMTok>
+              <XMTok>+</XMTok>
+              <XMTok>1</XMTok>
               <XMApp>
-                <XMTok meaning="times" role="MULOP">⁢</XMTok>
-                <XMTok role="UNKNOWN" font="italic">u</XMTok>
+                <XMTok>⁢</XMTok>
+                <XMTok>u</XMTok>
                 <XMApp>
-                  <XMTok meaning="square-root"/>
-                  <XMTok role="UNKNOWN" font="italic">D</XMTok>
+                  <XMTok/>
+                  <XMTok>D</XMTok>
                 </XMApp>
               </XMApp>
             </XMApp>
-            <XMTok meaning="2" role="NUMBER">2</XMTok>
+            <XMTok>2</XMTok>
           </XMApp>
         </XMApp>
       </XMath>
     </annotation-xml>
     <annotation encoding="application/x-tex">x-\frac{1+u\sqrt{D}}{2}</annotation>
   </semantics>
 </math>

Formula 66

( show / hide formula code)

Original

p p p p

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="p6.m15" class="ltx_Math" alttext="p" display="inline">
  <semantics>
    <mi>p</mi>
    <annotation-xml encoding="MathML-Content">
      <ci>p</ci>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMTok role="UNKNOWN" font="italic">p</XMTok>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">p</annotation>
  </semantics>
</math>

Canonicalized

p p p p

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics encoding="">
    <mi>p</mi>
    <annotation-xml encoding="MathML-Content">
      <ci encoding="">p</ci>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMTok>p</XMTok>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">p</annotation>
  </semantics>
</math>

Diff

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@@ -1,15 +1,15 @@
 <?xml version="1.0" encoding="UTF-8"?>
-<math xmlns="http://www.w3.org/1998/Math/MathML" id="p6.m15" class="ltx_Math" alttext="p" display="inline">
-  <semantics>
+<math xmlns="http://www.w3.org/1998/Math/MathML">
+  <semantics encoding="">
     <mi>p</mi>
     <annotation-xml encoding="MathML-Content">
-      <ci>p</ci>
+      <ci encoding="">p</ci>
     </annotation-xml>
     <annotation-xml encoding="application/x-latexml">
       <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
-        <XMTok role="UNKNOWN" font="italic">p</XMTok>
+        <XMTok>p</XMTok>
       </XMath>
     </annotation-xml>
     <annotation encoding="application/x-tex">p</annotation>
   </semantics>
 </math>

Formula 67

( show / hide formula code)

Original

p p p p

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="p6.m16" class="ltx_Math" alttext="p" display="inline">
  <semantics>
    <mi>p</mi>
    <annotation-xml encoding="MathML-Content">
      <ci>p</ci>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMTok role="UNKNOWN" font="italic">p</XMTok>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">p</annotation>
  </semantics>
</math>

Canonicalized

p p p p

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics encoding="">
    <mi>p</mi>
    <annotation-xml encoding="MathML-Content">
      <ci encoding="">p</ci>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMTok>p</XMTok>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">p</annotation>
  </semantics>
</math>

Diff

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@@ -1,15 +1,15 @@
 <?xml version="1.0" encoding="UTF-8"?>
-<math xmlns="http://www.w3.org/1998/Math/MathML" id="p6.m16" class="ltx_Math" alttext="p" display="inline">
-  <semantics>
+<math xmlns="http://www.w3.org/1998/Math/MathML">
+  <semantics encoding="">
     <mi>p</mi>
     <annotation-xml encoding="MathML-Content">
-      <ci>p</ci>
+      <ci encoding="">p</ci>
     </annotation-xml>
     <annotation-xml encoding="application/x-latexml">
       <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
-        <XMTok role="UNKNOWN" font="italic">p</XMTok>
+        <XMTok>p</XMTok>
       </XMath>
     </annotation-xml>
     <annotation encoding="application/x-tex">p</annotation>
   </semantics>
 </math>

Formula 68

( show / hide formula code)

Original

n 2 + n - m superscript n 2 n m - + n 2 n m n^{2}+n-m

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="p6.m17" class="ltx_Math" alttext="n^{2}+n-m" display="inline">
  <semantics>
    <mrow>
      <msup>
        <mi>n</mi>
        <mn>2</mn>
      </msup>
      <mo>+</mo>
      <mi>n</mi>
      <mo>-</mo>
      <mi>m</mi>
    </mrow>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <minus/>
        <apply>
          <plus/>
          <apply>
            <csymbol cd="ambiguous">superscript</csymbol>
            <ci>n</ci>
            <cn type="integer">2</cn>
          </apply>
          <ci>n</ci>
        </apply>
        <ci>m</ci>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok meaning="minus" role="ADDOP">-</XMTok>
          <XMApp>
            <XMTok meaning="plus" role="ADDOP">+</XMTok>
            <XMApp>
              <XMTok role="SUPERSCRIPTOP" scriptpos="post2"/>
              <XMTok role="UNKNOWN" font="italic">n</XMTok>
              <XMTok meaning="2" role="NUMBER">2</XMTok>
            </XMApp>
            <XMTok role="UNKNOWN" font="italic">n</XMTok>
          </XMApp>
          <XMTok role="UNKNOWN" font="italic">m</XMTok>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">n^{2}+n-m</annotation>
  </semantics>
</math>

Canonicalized

n 2 + n - m superscript n 2 n m - + n 2 n m n^{2}+n-m

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics encoding="">
    <mrow>
      <msup>
        <mi>n</mi>
        <mn>2</mn>
      </msup>
      <mo>+</mo>
      <mi>n</mi>
      <mo>-</mo>
      <mi>m</mi>
    </mrow>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <minus/>
        <apply>
          <plus/>
          <apply>
            <csymbol encoding="">superscript</csymbol>
            <ci encoding="">n</ci>
            <cn encoding="" base="10" type="integer">2</cn>
          </apply>
          <ci encoding="">n</ci>
        </apply>
        <ci encoding="">m</ci>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok>-</XMTok>
          <XMApp>
            <XMTok>+</XMTok>
            <XMApp>
              <XMTok/>
              <XMTok>n</XMTok>
              <XMTok>2</XMTok>
            </XMApp>
            <XMTok>n</XMTok>
          </XMApp>
          <XMTok>m</XMTok>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">n^{2}+n-m</annotation>
  </semantics>
</math>

Diff

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@@ -1,48 +1,48 @@
 <?xml version="1.0" encoding="UTF-8"?>
-<math xmlns="http://www.w3.org/1998/Math/MathML" id="p6.m17" class="ltx_Math" alttext="n^{2}+n-m" display="inline">
-  <semantics>
+<math xmlns="http://www.w3.org/1998/Math/MathML">
+  <semantics encoding="">
     <mrow>
       <msup>
         <mi>n</mi>
         <mn>2</mn>
       </msup>
       <mo>+</mo>
       <mi>n</mi>
       <mo>-</mo>
       <mi>m</mi>
     </mrow>
     <annotation-xml encoding="MathML-Content">
       <apply>
         <minus/>
         <apply>
           <plus/>
           <apply>
-            <csymbol cd="ambiguous">superscript</csymbol>
-            <ci>n</ci>
-            <cn type="integer">2</cn>
+            <csymbol encoding="">superscript</csymbol>
+            <ci encoding="">n</ci>
+            <cn encoding="" base="10" type="integer">2</cn>
           </apply>
-          <ci>n</ci>
+          <ci encoding="">n</ci>
         </apply>
-        <ci>m</ci>
+        <ci encoding="">m</ci>
       </apply>
     </annotation-xml>
     <annotation-xml encoding="application/x-latexml">
       <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
         <XMApp>
-          <XMTok meaning="minus" role="ADDOP">-</XMTok>
+          <XMTok>-</XMTok>
           <XMApp>
-            <XMTok meaning="plus" role="ADDOP">+</XMTok>
+            <XMTok>+</XMTok>
             <XMApp>
-              <XMTok role="SUPERSCRIPTOP" scriptpos="post2"/>
-              <XMTok role="UNKNOWN" font="italic">n</XMTok>
-              <XMTok meaning="2" role="NUMBER">2</XMTok>
+              <XMTok/>
+              <XMTok>n</XMTok>
+              <XMTok>2</XMTok>
             </XMApp>
-            <XMTok role="UNKNOWN" font="italic">n</XMTok>
+            <XMTok>n</XMTok>
           </XMApp>
-          <XMTok role="UNKNOWN" font="italic">m</XMTok>
+          <XMTok>m</XMTok>
         </XMApp>
       </XMath>
     </annotation-xml>
     <annotation encoding="application/x-tex">n^{2}+n-m</annotation>
   </semantics>
 </math>

Formula 69

( show / hide formula code)

Original

f m subscript f m f m f_{m}

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="p6.m18" class="ltx_Math" alttext="f_{m}" display="inline">
  <semantics>
    <msub>
      <mi>f</mi>
      <mi>m</mi>
    </msub>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <csymbol cd="ambiguous">subscript</csymbol>
        <ci>f</ci>
        <ci>m</ci>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok role="SUBSCRIPTOP" scriptpos="post2"/>
          <XMTok role="UNKNOWN" font="italic">f</XMTok>
          <XMTok role="UNKNOWN" font="italic">m</XMTok>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">f_{m}</annotation>
  </semantics>
</math>

Canonicalized

f m subscript f m f m f_{m}

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics encoding="">
    <msub>
      <mi>f</mi>
      <mi>m</mi>
    </msub>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <csymbol encoding="">subscript</csymbol>
        <ci encoding="">f</ci>
        <ci encoding="">m</ci>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok/>
          <XMTok>f</XMTok>
          <XMTok>m</XMTok>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">f_{m}</annotation>
  </semantics>
</math>

Diff

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@@ -1,26 +1,26 @@
 <?xml version="1.0" encoding="UTF-8"?>
-<math xmlns="http://www.w3.org/1998/Math/MathML" id="p6.m18" class="ltx_Math" alttext="f_{m}" display="inline">
-  <semantics>
+<math xmlns="http://www.w3.org/1998/Math/MathML">
+  <semantics encoding="">
     <msub>
       <mi>f</mi>
       <mi>m</mi>
     </msub>
     <annotation-xml encoding="MathML-Content">
       <apply>
-        <csymbol cd="ambiguous">subscript</csymbol>
-        <ci>f</ci>
-        <ci>m</ci>
+        <csymbol encoding="">subscript</csymbol>
+        <ci encoding="">f</ci>
+        <ci encoding="">m</ci>
       </apply>
     </annotation-xml>
     <annotation-xml encoding="application/x-latexml">
       <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
         <XMApp>
-          <XMTok role="SUBSCRIPTOP" scriptpos="post2"/>
-          <XMTok role="UNKNOWN" font="italic">f</XMTok>
-          <XMTok role="UNKNOWN" font="italic">m</XMTok>
+          <XMTok/>
+          <XMTok>f</XMTok>
+          <XMTok>m</XMTok>
         </XMApp>
       </XMath>
     </annotation-xml>
     <annotation encoding="application/x-tex">f_{m}</annotation>
   </semantics>
 </math>

Formula 70

( show / hide formula code)

Original

f m ⁢ ( n ) = ± p subscript f m n plus-or-minus p = ⁢ f m n ± p f_{m}(n)=\pm p

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="p6.m19" class="ltx_Math" alttext="f_{m}(n)=\pm p" display="inline">
  <semantics>
    <mrow>
      <mrow>
        <msub>
          <mi>f</mi>
          <mi>m</mi>
        </msub>
        <mo>⁢</mo>
        <mrow>
          <mo>(</mo>
          <mi>n</mi>
          <mo>)</mo>
        </mrow>
      </mrow>
      <mo>=</mo>
      <mrow>
        <mo>±</mo>
        <mi>p</mi>
      </mrow>
    </mrow>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <eq/>
        <apply>
          <times/>
          <apply>
            <csymbol cd="ambiguous">subscript</csymbol>
            <ci>f</ci>
            <ci>m</ci>
          </apply>
          <ci>n</ci>
        </apply>
        <apply>
          <csymbol cd="latexml">plus-or-minus</csymbol>
          <ci>p</ci>
        </apply>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok meaning="equals" role="RELOP">=</XMTok>
          <XMApp>
            <XMTok meaning="times" role="MULOP">⁢</XMTok>
            <XMApp>
              <XMTok role="SUBSCRIPTOP" scriptpos="post2"/>
              <XMTok possibleFunction="yes" role="UNKNOWN" font="italic">f</XMTok>
              <XMTok role="UNKNOWN" font="italic">m</XMTok>
            </XMApp>
            <XMTok close=")" open="(" role="UNKNOWN" font="italic">n</XMTok>
          </XMApp>
          <XMApp>
            <XMTok meaning="plus-or-minus" name="pm" role="ADDOP">±</XMTok>
            <XMTok role="UNKNOWN" font="italic">p</XMTok>
          </XMApp>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">f_{m}(n)=\pm p</annotation>
  </semantics>
</math>

Canonicalized

f m ( n ) = ± p subscript f m n plus-or-minus p = ⁢ f m n ± p f_{m}(n)=\pm p

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics encoding="">
    <mrow>
      <mrow>
        <msub>
          <mi>f</mi>
          <mi>m</mi>
        </msub>
        <mrow>
          <mo>(</mo>
          <mi>n</mi>
          <mo>)</mo>
        </mrow>
      </mrow>
      <mo>=</mo>
      <mrow>
        <mo>±</mo>
        <mi>p</mi>
      </mrow>
    </mrow>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <eq/>
        <apply>
          <times/>
          <apply>
            <csymbol encoding="">subscript</csymbol>
            <ci encoding="">f</ci>
            <ci encoding="">m</ci>
          </apply>
          <ci encoding="">n</ci>
        </apply>
        <apply>
          <csymbol encoding="">plus-or-minus</csymbol>
          <ci encoding="">p</ci>
        </apply>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok>=</XMTok>
          <XMApp>
            <XMTok>⁢</XMTok>
            <XMApp>
              <XMTok/>
              <XMTok>f</XMTok>
              <XMTok>m</XMTok>
            </XMApp>
            <XMTok>n</XMTok>
          </XMApp>
          <XMApp>
            <XMTok>±</XMTok>
            <XMTok>p</XMTok>
          </XMApp>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">f_{m}(n)=\pm p</annotation>
  </semantics>
</math>

Diff

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@@ -1,63 +1,62 @@
 <?xml version="1.0" encoding="UTF-8"?>
-<math xmlns="http://www.w3.org/1998/Math/MathML" id="p6.m19" class="ltx_Math" alttext="f_{m}(n)=\pm p" display="inline">
-  <semantics>
+<math xmlns="http://www.w3.org/1998/Math/MathML">
+  <semantics encoding="">
     <mrow>
       <mrow>
         <msub>
           <mi>f</mi>
           <mi>m</mi>
         </msub>
-        <mo>⁢</mo>
         <mrow>
           <mo>(</mo>
           <mi>n</mi>
           <mo>)</mo>
         </mrow>
       </mrow>
       <mo>=</mo>
       <mrow>
         <mo>±</mo>
         <mi>p</mi>
       </mrow>
     </mrow>
     <annotation-xml encoding="MathML-Content">
       <apply>
         <eq/>
         <apply>
           <times/>
           <apply>
-            <csymbol cd="ambiguous">subscript</csymbol>
-            <ci>f</ci>
-            <ci>m</ci>
+            <csymbol encoding="">subscript</csymbol>
+            <ci encoding="">f</ci>
+            <ci encoding="">m</ci>
           </apply>
-          <ci>n</ci>
+          <ci encoding="">n</ci>
         </apply>
         <apply>
-          <csymbol cd="latexml">plus-or-minus</csymbol>
-          <ci>p</ci>
+          <csymbol encoding="">plus-or-minus</csymbol>
+          <ci encoding="">p</ci>
         </apply>
       </apply>
     </annotation-xml>
     <annotation-xml encoding="application/x-latexml">
       <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
         <XMApp>
-          <XMTok meaning="equals" role="RELOP">=</XMTok>
+          <XMTok>=</XMTok>
           <XMApp>
-            <XMTok meaning="times" role="MULOP">⁢</XMTok>
+            <XMTok>⁢</XMTok>
             <XMApp>
-              <XMTok role="SUBSCRIPTOP" scriptpos="post2"/>
-              <XMTok possibleFunction="yes" role="UNKNOWN" font="italic">f</XMTok>
-              <XMTok role="UNKNOWN" font="italic">m</XMTok>
+              <XMTok/>
+              <XMTok>f</XMTok>
+              <XMTok>m</XMTok>
             </XMApp>
-            <XMTok close=")" open="(" role="UNKNOWN" font="italic">n</XMTok>
+            <XMTok>n</XMTok>
           </XMApp>
           <XMApp>
-            <XMTok meaning="plus-or-minus" name="pm" role="ADDOP">±</XMTok>
-            <XMTok role="UNKNOWN" font="italic">p</XMTok>
+            <XMTok>±</XMTok>
+            <XMTok>p</XMTok>
           </XMApp>
         </XMApp>
       </XMath>
     </annotation-xml>
     <annotation encoding="application/x-tex">f_{m}(n)=\pm p</annotation>
   </semantics>
 </math>

Formula 71

( show / hide formula code)

Original

f m subscript f m f m f_{m}

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="p6.m20" class="ltx_Math" alttext="f_{m}" display="inline">
  <semantics>
    <msub>
      <mi>f</mi>
      <mi>m</mi>
    </msub>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <csymbol cd="ambiguous">subscript</csymbol>
        <ci>f</ci>
        <ci>m</ci>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok role="SUBSCRIPTOP" scriptpos="post2"/>
          <XMTok role="UNKNOWN" font="italic">f</XMTok>
          <XMTok role="UNKNOWN" font="italic">m</XMTok>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">f_{m}</annotation>
  </semantics>
</math>

Canonicalized

f m subscript f m f m f_{m}

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics encoding="">
    <msub>
      <mi>f</mi>
      <mi>m</mi>
    </msub>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <csymbol encoding="">subscript</csymbol>
        <ci encoding="">f</ci>
        <ci encoding="">m</ci>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok/>
          <XMTok>f</XMTok>
          <XMTok>m</XMTok>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">f_{m}</annotation>
  </semantics>
</math>

Diff

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@@ -1,26 +1,26 @@
 <?xml version="1.0" encoding="UTF-8"?>
-<math xmlns="http://www.w3.org/1998/Math/MathML" id="p6.m20" class="ltx_Math" alttext="f_{m}" display="inline">
-  <semantics>
+<math xmlns="http://www.w3.org/1998/Math/MathML">
+  <semantics encoding="">
     <msub>
       <mi>f</mi>
       <mi>m</mi>
     </msub>
     <annotation-xml encoding="MathML-Content">
       <apply>
-        <csymbol cd="ambiguous">subscript</csymbol>
-        <ci>f</ci>
-        <ci>m</ci>
+        <csymbol encoding="">subscript</csymbol>
+        <ci encoding="">f</ci>
+        <ci encoding="">m</ci>
       </apply>
     </annotation-xml>
     <annotation-xml encoding="application/x-latexml">
       <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
         <XMApp>
-          <XMTok role="SUBSCRIPTOP" scriptpos="post2"/>
-          <XMTok role="UNKNOWN" font="italic">f</XMTok>
-          <XMTok role="UNKNOWN" font="italic">m</XMTok>
+          <XMTok/>
+          <XMTok>f</XMTok>
+          <XMTok>m</XMTok>
         </XMApp>
       </XMath>
     </annotation-xml>
     <annotation encoding="application/x-tex">f_{m}</annotation>
   </semantics>
 </math>

Formula 72

( show / hide formula code)

Original

n n n n

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="p6.m21" class="ltx_Math" alttext="n" display="inline">
  <semantics>
    <mi>n</mi>
    <annotation-xml encoding="MathML-Content">
      <ci>n</ci>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMTok role="UNKNOWN" font="italic">n</XMTok>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">n</annotation>
  </semantics>
</math>

Canonicalized

n n n n

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics encoding="">
    <mi>n</mi>
    <annotation-xml encoding="MathML-Content">
      <ci encoding="">n</ci>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMTok>n</XMTok>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">n</annotation>
  </semantics>
</math>

Diff

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@@ -1,15 +1,15 @@
 <?xml version="1.0" encoding="UTF-8"?>
-<math xmlns="http://www.w3.org/1998/Math/MathML" id="p6.m21" class="ltx_Math" alttext="n" display="inline">
-  <semantics>
+<math xmlns="http://www.w3.org/1998/Math/MathML">
+  <semantics encoding="">
     <mi>n</mi>
     <annotation-xml encoding="MathML-Content">
-      <ci>n</ci>
+      <ci encoding="">n</ci>
     </annotation-xml>
     <annotation-xml encoding="application/x-latexml">
       <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
-        <XMTok role="UNKNOWN" font="italic">n</XMTok>
+        <XMTok>n</XMTok>
       </XMath>
     </annotation-xml>
     <annotation encoding="application/x-tex">n</annotation>
   </semantics>
 </math>

Formula 73

( show / hide formula code)

Original

p < m 1 / 2 / 6 p superscript m 1 2 6 < p / m / 1 2 6 p<m^{1/2}/6

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="p6.m22" class="ltx_Math" alttext="p&lt;m^{1/2}/6" display="inline">
  <semantics>
    <mrow>
      <mi>p</mi>
      <mo>&lt;</mo>
      <mrow>
        <msup>
          <mi>m</mi>
          <mrow>
            <mn>1</mn>
            <mo>/</mo>
            <mn>2</mn>
          </mrow>
        </msup>
        <mo>/</mo>
        <mn>6</mn>
      </mrow>
    </mrow>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <lt/>
        <ci>p</ci>
        <apply>
          <divide/>
          <apply>
            <csymbol cd="ambiguous">superscript</csymbol>
            <ci>m</ci>
            <apply>
              <divide/>
              <cn type="integer">1</cn>
              <cn type="integer">2</cn>
            </apply>
          </apply>
          <cn type="integer">6</cn>
        </apply>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok meaning="less-than" role="RELOP">&lt;</XMTok>
          <XMTok role="UNKNOWN" font="italic">p</XMTok>
          <XMApp>
            <XMTok mathstyle="inline" meaning="divide" role="MULOP">/</XMTok>
            <XMApp>
              <XMTok role="SUPERSCRIPTOP" scriptpos="post2"/>
              <XMTok role="UNKNOWN" font="italic">m</XMTok>
              <XMApp>
                <XMTok mathstyle="inline" meaning="divide" role="MULOP">/</XMTok>
                <XMTok meaning="1" role="NUMBER">1</XMTok>
                <XMTok meaning="2" role="NUMBER">2</XMTok>
              </XMApp>
            </XMApp>
            <XMTok meaning="6" role="NUMBER">6</XMTok>
          </XMApp>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">p&lt;m^{1/2}/6</annotation>
  </semantics>
</math>

Canonicalized

p < m 1 / 2 / 6 p superscript m 1 2 6 < p / m / 1 2 6 p<m^{1/2}/6

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics encoding="">
    <mrow>
      <mi>p</mi>
      <mo>&lt;</mo>
      <mrow>
        <msup>
          <mi>m</mi>
          <mrow>
            <mn>1</mn>
            <mo>/</mo>
            <mn>2</mn>
          </mrow>
        </msup>
        <mo>/</mo>
        <mn>6</mn>
      </mrow>
    </mrow>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <lt/>
        <ci encoding="">p</ci>
        <apply>
          <divide/>
          <apply>
            <csymbol encoding="">superscript</csymbol>
            <ci encoding="">m</ci>
            <apply>
              <divide/>
              <cn encoding="" base="10" type="integer">1</cn>
              <cn encoding="" base="10" type="integer">2</cn>
            </apply>
          </apply>
          <cn encoding="" base="10" type="integer">6</cn>
        </apply>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok>&lt;</XMTok>
          <XMTok>p</XMTok>
          <XMApp>
            <XMTok>/</XMTok>
            <XMApp>
              <XMTok/>
              <XMTok>m</XMTok>
              <XMApp>
                <XMTok>/</XMTok>
                <XMTok>1</XMTok>
                <XMTok>2</XMTok>
              </XMApp>
            </XMApp>
            <XMTok>6</XMTok>
          </XMApp>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">p&lt;m^{1/2}/6</annotation>
  </semantics>
</math>

Diff

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@@ -1,62 +1,62 @@
 <?xml version="1.0" encoding="UTF-8"?>
-<math xmlns="http://www.w3.org/1998/Math/MathML" id="p6.m22" class="ltx_Math" alttext="p&lt;m^{1/2}/6" display="inline">
-  <semantics>
+<math xmlns="http://www.w3.org/1998/Math/MathML">
+  <semantics encoding="">
     <mrow>
       <mi>p</mi>
       <mo>&lt;</mo>
       <mrow>
         <msup>
           <mi>m</mi>
           <mrow>
             <mn>1</mn>
             <mo>/</mo>
             <mn>2</mn>
           </mrow>
         </msup>
         <mo>/</mo>
         <mn>6</mn>
       </mrow>
     </mrow>
     <annotation-xml encoding="MathML-Content">
       <apply>
         <lt/>
-        <ci>p</ci>
+        <ci encoding="">p</ci>
         <apply>
           <divide/>
           <apply>
-            <csymbol cd="ambiguous">superscript</csymbol>
-            <ci>m</ci>
+            <csymbol encoding="">superscript</csymbol>
+            <ci encoding="">m</ci>
             <apply>
               <divide/>
-              <cn type="integer">1</cn>
-              <cn type="integer">2</cn>
+              <cn encoding="" base="10" type="integer">1</cn>
+              <cn encoding="" base="10" type="integer">2</cn>
             </apply>
           </apply>
-          <cn type="integer">6</cn>
+          <cn encoding="" base="10" type="integer">6</cn>
         </apply>
       </apply>
     </annotation-xml>
     <annotation-xml encoding="application/x-latexml">
       <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
         <XMApp>
-          <XMTok meaning="less-than" role="RELOP">&lt;</XMTok>
-          <XMTok role="UNKNOWN" font="italic">p</XMTok>
+          <XMTok>&lt;</XMTok>
+          <XMTok>p</XMTok>
           <XMApp>
-            <XMTok mathstyle="inline" meaning="divide" role="MULOP">/</XMTok>
+            <XMTok>/</XMTok>
             <XMApp>
-              <XMTok role="SUPERSCRIPTOP" scriptpos="post2"/>
-              <XMTok role="UNKNOWN" font="italic">m</XMTok>
+              <XMTok/>
+              <XMTok>m</XMTok>
               <XMApp>
-                <XMTok mathstyle="inline" meaning="divide" role="MULOP">/</XMTok>
-                <XMTok meaning="1" role="NUMBER">1</XMTok>
-                <XMTok meaning="2" role="NUMBER">2</XMTok>
+                <XMTok>/</XMTok>
+                <XMTok>1</XMTok>
+                <XMTok>2</XMTok>
               </XMApp>
             </XMApp>
-            <XMTok meaning="6" role="NUMBER">6</XMTok>
+            <XMTok>6</XMTok>
           </XMApp>
         </XMApp>
       </XMath>
     </annotation-xml>
     <annotation encoding="application/x-tex">p&lt;m^{1/2}/6</annotation>
   </semantics>
 </math>

Formula 74

( show / hide formula code)

Original

m 1 / 2 superscript m 1 2 m / 1 2 m^{1/2}

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="p6.m23" class="ltx_Math" alttext="m^{1/2}" display="inline">
  <semantics>
    <msup>
      <mi>m</mi>
      <mrow>
        <mn>1</mn>
        <mo>/</mo>
        <mn>2</mn>
      </mrow>
    </msup>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <csymbol cd="ambiguous">superscript</csymbol>
        <ci>m</ci>
        <apply>
          <divide/>
          <cn type="integer">1</cn>
          <cn type="integer">2</cn>
        </apply>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok role="SUPERSCRIPTOP" scriptpos="post2"/>
          <XMTok role="UNKNOWN" font="italic">m</XMTok>
          <XMApp>
            <XMTok mathstyle="inline" meaning="divide" role="MULOP">/</XMTok>
            <XMTok meaning="1" role="NUMBER">1</XMTok>
            <XMTok meaning="2" role="NUMBER">2</XMTok>
          </XMApp>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">m^{1/2}</annotation>
  </semantics>
</math>

Canonicalized

m 1 / 2 superscript m 1 2 m / 1 2 m^{1/2}

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics encoding="">
    <msup>
      <mi>m</mi>
      <mrow>
        <mn>1</mn>
        <mo>/</mo>
        <mn>2</mn>
      </mrow>
    </msup>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <csymbol encoding="">superscript</csymbol>
        <ci encoding="">m</ci>
        <apply>
          <divide/>
          <cn encoding="" base="10" type="integer">1</cn>
          <cn encoding="" base="10" type="integer">2</cn>
        </apply>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok/>
          <XMTok>m</XMTok>
          <XMApp>
            <XMTok>/</XMTok>
            <XMTok>1</XMTok>
            <XMTok>2</XMTok>
          </XMApp>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">m^{1/2}</annotation>
  </semantics>
</math>

Diff

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@@ -1,38 +1,38 @@
 <?xml version="1.0" encoding="UTF-8"?>
-<math xmlns="http://www.w3.org/1998/Math/MathML" id="p6.m23" class="ltx_Math" alttext="m^{1/2}" display="inline">
-  <semantics>
+<math xmlns="http://www.w3.org/1998/Math/MathML">
+  <semantics encoding="">
     <msup>
       <mi>m</mi>
       <mrow>
         <mn>1</mn>
         <mo>/</mo>
         <mn>2</mn>
       </mrow>
     </msup>
     <annotation-xml encoding="MathML-Content">
       <apply>
-        <csymbol cd="ambiguous">superscript</csymbol>
-        <ci>m</ci>
+        <csymbol encoding="">superscript</csymbol>
+        <ci encoding="">m</ci>
         <apply>
           <divide/>
-          <cn type="integer">1</cn>
-          <cn type="integer">2</cn>
+          <cn encoding="" base="10" type="integer">1</cn>
+          <cn encoding="" base="10" type="integer">2</cn>
         </apply>
       </apply>
     </annotation-xml>
     <annotation-xml encoding="application/x-latexml">
       <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
         <XMApp>
-          <XMTok role="SUPERSCRIPTOP" scriptpos="post2"/>
-          <XMTok role="UNKNOWN" font="italic">m</XMTok>
+          <XMTok/>
+          <XMTok>m</XMTok>
           <XMApp>
-            <XMTok mathstyle="inline" meaning="divide" role="MULOP">/</XMTok>
-            <XMTok meaning="1" role="NUMBER">1</XMTok>
-            <XMTok meaning="2" role="NUMBER">2</XMTok>
+            <XMTok>/</XMTok>
+            <XMTok>1</XMTok>
+            <XMTok>2</XMTok>
           </XMApp>
         </XMApp>
       </XMath>
     </annotation-xml>
     <annotation encoding="application/x-tex">m^{1/2}</annotation>
   </semantics>
 </math>

Formula 75

( show / hide formula code)

Original

n n n n

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="p6.m24" class="ltx_Math" alttext="n" display="inline">
  <semantics>
    <mi>n</mi>
    <annotation-xml encoding="MathML-Content">
      <ci>n</ci>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMTok role="UNKNOWN" font="italic">n</XMTok>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">n</annotation>
  </semantics>
</math>

Canonicalized

n n n n

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics encoding="">
    <mi>n</mi>
    <annotation-xml encoding="MathML-Content">
      <ci encoding="">n</ci>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMTok>n</XMTok>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">n</annotation>
  </semantics>
</math>

Diff

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@@ -1,15 +1,15 @@
 <?xml version="1.0" encoding="UTF-8"?>
-<math xmlns="http://www.w3.org/1998/Math/MathML" id="p6.m24" class="ltx_Math" alttext="n" display="inline">
-  <semantics>
+<math xmlns="http://www.w3.org/1998/Math/MathML">
+  <semantics encoding="">
     <mi>n</mi>
     <annotation-xml encoding="MathML-Content">
-      <ci>n</ci>
+      <ci encoding="">n</ci>
     </annotation-xml>
     <annotation-xml encoding="application/x-latexml">
       <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
-        <XMTok role="UNKNOWN" font="italic">n</XMTok>
+        <XMTok>n</XMTok>
       </XMath>
     </annotation-xml>
     <annotation encoding="application/x-tex">n</annotation>
   </semantics>
 </math>

Formula 76

( show / hide formula code)

Original

f m subscript f m f m f_{m}

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="p6.m25" class="ltx_Math" alttext="f_{m}" display="inline">
  <semantics>
    <msub>
      <mi>f</mi>
      <mi>m</mi>
    </msub>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <csymbol cd="ambiguous">subscript</csymbol>
        <ci>f</ci>
        <ci>m</ci>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok role="SUBSCRIPTOP" scriptpos="post2"/>
          <XMTok role="UNKNOWN" font="italic">f</XMTok>
          <XMTok role="UNKNOWN" font="italic">m</XMTok>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">f_{m}</annotation>
  </semantics>
</math>

Canonicalized

f m subscript f m f m f_{m}

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics encoding="">
    <msub>
      <mi>f</mi>
      <mi>m</mi>
    </msub>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <csymbol encoding="">subscript</csymbol>
        <ci encoding="">f</ci>
        <ci encoding="">m</ci>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok/>
          <XMTok>f</XMTok>
          <XMTok>m</XMTok>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">f_{m}</annotation>
  </semantics>
</math>

Diff

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@@ -1,26 +1,26 @@
 <?xml version="1.0" encoding="UTF-8"?>
-<math xmlns="http://www.w3.org/1998/Math/MathML" id="p6.m25" class="ltx_Math" alttext="f_{m}" display="inline">
-  <semantics>
+<math xmlns="http://www.w3.org/1998/Math/MathML">
+  <semantics encoding="">
     <msub>
       <mi>f</mi>
       <mi>m</mi>
     </msub>
     <annotation-xml encoding="MathML-Content">
       <apply>
-        <csymbol cd="ambiguous">subscript</csymbol>
-        <ci>f</ci>
-        <ci>m</ci>
+        <csymbol encoding="">subscript</csymbol>
+        <ci encoding="">f</ci>
+        <ci encoding="">m</ci>
       </apply>
     </annotation-xml>
     <annotation-xml encoding="application/x-latexml">
       <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
         <XMApp>
-          <XMTok role="SUBSCRIPTOP" scriptpos="post2"/>
-          <XMTok role="UNKNOWN" font="italic">f</XMTok>
-          <XMTok role="UNKNOWN" font="italic">m</XMTok>
+          <XMTok/>
+          <XMTok>f</XMTok>
+          <XMTok>m</XMTok>
         </XMApp>
       </XMath>
     </annotation-xml>
     <annotation encoding="application/x-tex">f_{m}</annotation>
   </semantics>
 </math>

Formula 77

( show / hide formula code)

Original

p D ≡ 1 ( mod 4 ⁢ D ) subscript p D annotated 1 pmod 4 D ≡ p D 1 ⁢ 4 D mod p_{D}\equiv 1\;\;(\mathop{{\rm mod}}4D)

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="p7.m1" class="ltx_Math" alttext="p_{D}\equiv 1\;\;(\mathop{{\rm mod}}4D)" display="inline">
  <semantics>
    <mrow>
      <msub>
        <mi>p</mi>
        <mi>D</mi>
      </msub>
      <mo>≡</mo>
      <mrow>
        <mn>1</mn>
        <mspace width="veryverythickmathspace"/>
        <mrow>
          <mo>(</mo>
          <mrow>
            <mo>mod</mo>
            <mrow xml:id="p7.m1.xm1c" xref="p7.m1.xm1.cmml">
              <mn xml:id="p7.m1.xm1" xref="p7.m1.xm1.cmml">4</mn>
              <mo xml:id="p7.m1.xm1a" xref="p7.m1.xm1.cmml">⁢</mo>
              <mi xml:id="p7.m1.xm1b" xref="p7.m1.xm1.cmml">D</mi>
            </mrow>
          </mrow>
          <mo>)</mo>
        </mrow>
      </mrow>
    </mrow>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <equivalent/>
        <apply>
          <csymbol cd="ambiguous">subscript</csymbol>
          <ci>p</ci>
          <ci>D</ci>
        </apply>
        <apply>
          <csymbol cd="latexml">annotated</csymbol>
          <cn type="integer">1</cn>
          <apply>
            <ci>pmod</ci>
            <apply xml:id="p7.m1.xm1c.cmml" xref="p7.m1.xm1">
              <times xml:id="p7.m1.xm1.cmml" xref="p7.m1.xm1"/>
              <cn type="integer" xml:id="p7.m1.xm1a.cmml" xref="p7.m1.xm1">4</cn>
              <ci xml:id="p7.m1.xm1b.cmml" xref="p7.m1.xm1">D</ci>
            </apply>
          </apply>
        </apply>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok meaning="equivalent-to" name="equiv" role="RELOP">≡</XMTok>
          <XMApp>
            <XMTok role="SUBSCRIPTOP" scriptpos="post2"/>
            <XMTok role="UNKNOWN" font="italic">p</XMTok>
            <XMTok role="UNKNOWN" font="italic">D</XMTok>
          </XMApp>
          <XMApp>
            <XMTok meaning="annotated"/>
            <XMTok meaning="1" role="NUMBER">1</XMTok>
            <XMDual role="MODIFIER">
              <XMApp role="MODIFIER">
                <XMTok name="pmod" role="UNKNOWN"/>
                <XMApp xml:id="p7.m1.xm1.xm" fragid="p7.m1.xm1.xm">
                  <XMTok meaning="times" role="MULOP">⁢</XMTok>
                  <XMTok meaning="4" role="NUMBER">4</XMTok>
                  <XMTok role="UNKNOWN" font="italic">D</XMTok>
                </XMApp>
              </XMApp>
              <XMApp close=")" open="(">
                <XMTok role="BIGOP">mod</XMTok>
                <XMRef idref="p7.m1.xm1.xm"/>
              </XMApp>
            </XMDual>
          </XMApp>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">p_{D}\equiv 1\;\;(\mathop{{\rm mod}}4D)</annotation>
  </semantics>
</math>

Canonicalized

p D ≡ 1 ( mod 4 D ) subscript p D annotated 1 pmod 4 D ≡ p D 1 ⁢ 4 D mod p_{D}\equiv 1\;\;(\mathop{{\rm mod}}4D)

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics encoding="">
    <mrow>
      <msub>
        <mi>p</mi>
        <mi>D</mi>
      </msub>
      <mo>≡</mo>
      <mrow>
        <mn>1</mn>
        <mrow>
          <mo>(</mo>
          <mrow>
            <mo>mod</mo>
            <mrow>
              <mn>4</mn>
              <mi>D</mi>
            </mrow>
          </mrow>
          <mo>)</mo>
        </mrow>
      </mrow>
    </mrow>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <equivalent/>
        <apply>
          <csymbol encoding="">subscript</csymbol>
          <ci encoding="">p</ci>
          <ci encoding="">D</ci>
        </apply>
        <apply>
          <csymbol encoding="">annotated</csymbol>
          <cn encoding="" base="10" type="integer">1</cn>
          <apply>
            <ci encoding="">pmod</ci>
            <apply>
              <times encoding=""/>
              <cn encoding="" base="10" type="integer">4</cn>
              <ci encoding="">D</ci>
            </apply>
          </apply>
        </apply>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok>≡</XMTok>
          <XMApp>
            <XMTok/>
            <XMTok>p</XMTok>
            <XMTok>D</XMTok>
          </XMApp>
          <XMApp>
            <XMTok/>
            <XMTok>1</XMTok>
            <XMDual>
              <XMApp>
                <XMTok/>
                <XMApp>
                  <XMTok>⁢</XMTok>
                  <XMTok>4</XMTok>
                  <XMTok>D</XMTok>
                </XMApp>
              </XMApp>
              <XMApp>
                <XMTok>mod</XMTok>
                <XMRef/>
              </XMApp>
            </XMDual>
          </XMApp>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">p_{D}\equiv 1\;\;(\mathop{{\rm mod}}4D)</annotation>
  </semantics>
</math>

Diff

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@@ -1,81 +1,79 @@
 <?xml version="1.0" encoding="UTF-8"?>
-<math xmlns="http://www.w3.org/1998/Math/MathML" id="p7.m1" class="ltx_Math" alttext="p_{D}\equiv 1\;\;(\mathop{{\rm mod}}4D)" display="inline">
-  <semantics>
+<math xmlns="http://www.w3.org/1998/Math/MathML">
+  <semantics encoding="">
     <mrow>
       <msub>
         <mi>p</mi>
         <mi>D</mi>
       </msub>
       <mo>≡</mo>
       <mrow>
         <mn>1</mn>
-        <mspace width="veryverythickmathspace"/>
         <mrow>
           <mo>(</mo>
           <mrow>
             <mo>mod</mo>
-            <mrow xml:id="p7.m1.xm1c" xref="p7.m1.xm1.cmml">
-              <mn xml:id="p7.m1.xm1" xref="p7.m1.xm1.cmml">4</mn>
-              <mo xml:id="p7.m1.xm1a" xref="p7.m1.xm1.cmml">⁢</mo>
-              <mi xml:id="p7.m1.xm1b" xref="p7.m1.xm1.cmml">D</mi>
+            <mrow>
+              <mn>4</mn>
+              <mi>D</mi>
             </mrow>
           </mrow>
           <mo>)</mo>
         </mrow>
       </mrow>
     </mrow>
     <annotation-xml encoding="MathML-Content">
       <apply>
         <equivalent/>
         <apply>
-          <csymbol cd="ambiguous">subscript</csymbol>
-          <ci>p</ci>
-          <ci>D</ci>
+          <csymbol encoding="">subscript</csymbol>
+          <ci encoding="">p</ci>
+          <ci encoding="">D</ci>
         </apply>
         <apply>
-          <csymbol cd="latexml">annotated</csymbol>
-          <cn type="integer">1</cn>
+          <csymbol encoding="">annotated</csymbol>
+          <cn encoding="" base="10" type="integer">1</cn>
           <apply>
-            <ci>pmod</ci>
-            <apply xml:id="p7.m1.xm1c.cmml" xref="p7.m1.xm1">
-              <times xml:id="p7.m1.xm1.cmml" xref="p7.m1.xm1"/>
-              <cn type="integer" xml:id="p7.m1.xm1a.cmml" xref="p7.m1.xm1">4</cn>
-              <ci xml:id="p7.m1.xm1b.cmml" xref="p7.m1.xm1">D</ci>
+            <ci encoding="">pmod</ci>
+            <apply>
+              <times encoding=""/>
+              <cn encoding="" base="10" type="integer">4</cn>
+              <ci encoding="">D</ci>
             </apply>
           </apply>
         </apply>
       </apply>
     </annotation-xml>
     <annotation-xml encoding="application/x-latexml">
       <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
         <XMApp>
-          <XMTok meaning="equivalent-to" name="equiv" role="RELOP">≡</XMTok>
+          <XMTok>≡</XMTok>
           <XMApp>
-            <XMTok role="SUBSCRIPTOP" scriptpos="post2"/>
-            <XMTok role="UNKNOWN" font="italic">p</XMTok>
-            <XMTok role="UNKNOWN" font="italic">D</XMTok>
+            <XMTok/>
+            <XMTok>p</XMTok>
+            <XMTok>D</XMTok>
           </XMApp>
           <XMApp>
-            <XMTok meaning="annotated"/>
-            <XMTok meaning="1" role="NUMBER">1</XMTok>
-            <XMDual role="MODIFIER">
-              <XMApp role="MODIFIER">
-                <XMTok name="pmod" role="UNKNOWN"/>
-                <XMApp xml:id="p7.m1.xm1.xm" fragid="p7.m1.xm1.xm">
-                  <XMTok meaning="times" role="MULOP">⁢</XMTok>
-                  <XMTok meaning="4" role="NUMBER">4</XMTok>
-                  <XMTok role="UNKNOWN" font="italic">D</XMTok>
+            <XMTok/>
+            <XMTok>1</XMTok>
+            <XMDual>
+              <XMApp>
+                <XMTok/>
+                <XMApp>
+                  <XMTok>⁢</XMTok>
+                  <XMTok>4</XMTok>
+                  <XMTok>D</XMTok>
                 </XMApp>
               </XMApp>
-              <XMApp close=")" open="(">
-                <XMTok role="BIGOP">mod</XMTok>
-                <XMRef idref="p7.m1.xm1.xm"/>
+              <XMApp>
+                <XMTok>mod</XMTok>
+                <XMRef/>
               </XMApp>
             </XMDual>
           </XMApp>
         </XMApp>
       </XMath>
     </annotation-xml>
     <annotation encoding="application/x-tex">p_{D}\equiv 1\;\;(\mathop{{\rm mod}}4D)</annotation>
   </semantics>
 </math>

Formula 78

( show / hide formula code)

Original

D ≤ D 0 D subscript D 0 ≤ D D 0 D\leq D_{0}

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="p7.m2" class="ltx_Math" alttext="D\leq D_{0}" display="inline">
  <semantics>
    <mrow>
      <mi>D</mi>
      <mo>≤</mo>
      <msub>
        <mi>D</mi>
        <mn>0</mn>
      </msub>
    </mrow>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <leq/>
        <ci>D</ci>
        <apply>
          <csymbol cd="ambiguous">subscript</csymbol>
          <ci>D</ci>
          <cn type="integer">0</cn>
        </apply>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok meaning="less-than-or-equals" name="leq" role="RELOP">≤</XMTok>
          <XMTok role="UNKNOWN" font="italic">D</XMTok>
          <XMApp>
            <XMTok role="SUBSCRIPTOP" scriptpos="post2"/>
            <XMTok role="UNKNOWN" font="italic">D</XMTok>
            <XMTok meaning="0" role="NUMBER">0</XMTok>
          </XMApp>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">D\leq D_{0}</annotation>
  </semantics>
</math>

Canonicalized

D ≤ D 0 D subscript D 0 ≤ D D 0 D\leq D_{0}

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics encoding="">
    <mrow>
      <mi>D</mi>
      <mo>≤</mo>
      <msub>
        <mi>D</mi>
        <mn>0</mn>
      </msub>
    </mrow>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <leq/>
        <ci encoding="">D</ci>
        <apply>
          <csymbol encoding="">subscript</csymbol>
          <ci encoding="">D</ci>
          <cn encoding="" base="10" type="integer">0</cn>
        </apply>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok>≤</XMTok>
          <XMTok>D</XMTok>
          <XMApp>
            <XMTok/>
            <XMTok>D</XMTok>
            <XMTok>0</XMTok>
          </XMApp>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">D\leq D_{0}</annotation>
  </semantics>
</math>

Diff

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@@ -1,38 +1,38 @@
 <?xml version="1.0" encoding="UTF-8"?>
-<math xmlns="http://www.w3.org/1998/Math/MathML" id="p7.m2" class="ltx_Math" alttext="D\leq D_{0}" display="inline">
-  <semantics>
+<math xmlns="http://www.w3.org/1998/Math/MathML">
+  <semantics encoding="">
     <mrow>
       <mi>D</mi>
       <mo>≤</mo>
       <msub>
         <mi>D</mi>
         <mn>0</mn>
       </msub>
     </mrow>
     <annotation-xml encoding="MathML-Content">
       <apply>
         <leq/>
-        <ci>D</ci>
+        <ci encoding="">D</ci>
         <apply>
-          <csymbol cd="ambiguous">subscript</csymbol>
-          <ci>D</ci>
-          <cn type="integer">0</cn>
+          <csymbol encoding="">subscript</csymbol>
+          <ci encoding="">D</ci>
+          <cn encoding="" base="10" type="integer">0</cn>
         </apply>
       </apply>
     </annotation-xml>
     <annotation-xml encoding="application/x-latexml">
       <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
         <XMApp>
-          <XMTok meaning="less-than-or-equals" name="leq" role="RELOP">≤</XMTok>
-          <XMTok role="UNKNOWN" font="italic">D</XMTok>
+          <XMTok>≤</XMTok>
+          <XMTok>D</XMTok>
           <XMApp>
-            <XMTok role="SUBSCRIPTOP" scriptpos="post2"/>
-            <XMTok role="UNKNOWN" font="italic">D</XMTok>
-            <XMTok meaning="0" role="NUMBER">0</XMTok>
+            <XMTok/>
+            <XMTok>D</XMTok>
+            <XMTok>0</XMTok>
           </XMApp>
         </XMApp>
       </XMath>
     </annotation-xml>
     <annotation encoding="application/x-tex">D\leq D_{0}</annotation>
   </semantics>
 </math>

Formula 79

( show / hide formula code)

Original

m > 6 ⁢ max ⁡ p D m 6 subscript p D > m ⁢ 6 max p D m>6\max p_{D}

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="p7.m3" class="ltx_Math" alttext="m&gt;6\max p_{D}" display="inline">
  <semantics>
    <mrow>
      <mi>m</mi>
      <mo>&gt;</mo>
      <mrow>
        <mn>6</mn>
        <mo>⁢</mo>
        <mrow>
          <mo>max</mo>
          <mo>⁡</mo>
          <msub>
            <mi>p</mi>
            <mi>D</mi>
          </msub>
        </mrow>
      </mrow>
    </mrow>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <gt/>
        <ci>m</ci>
        <apply>
          <times/>
          <cn type="integer">6</cn>
          <apply>
            <max/>
            <apply>
              <csymbol cd="ambiguous">subscript</csymbol>
              <ci>p</ci>
              <ci>D</ci>
            </apply>
          </apply>
        </apply>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok meaning="greater-than" role="RELOP">&gt;</XMTok>
          <XMTok role="UNKNOWN" font="italic">m</XMTok>
          <XMApp>
            <XMTok meaning="times" role="MULOP">⁢</XMTok>
            <XMTok meaning="6" role="NUMBER">6</XMTok>
            <XMApp>
              <XMTok meaning="maximum" role="LIMITOP" scriptpos="post">max</XMTok>
              <XMApp>
                <XMTok role="SUBSCRIPTOP" scriptpos="post2"/>
                <XMTok role="UNKNOWN" font="italic">p</XMTok>
                <XMTok role="UNKNOWN" font="italic">D</XMTok>
              </XMApp>
            </XMApp>
          </XMApp>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">m&gt;6\max p_{D}</annotation>
  </semantics>
</math>

Canonicalized

m > 6 max ⁡ p D m 6 subscript p D > m ⁢ 6 max p D m>6\max p_{D}

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics encoding="">
    <mrow>
      <mi>m</mi>
      <mo>&gt;</mo>
      <mrow>
        <mn>6</mn>
        <mrow>
          <mo>max</mo>
          <mo>⁡</mo>
          <msub>
            <mi>p</mi>
            <mi>D</mi>
          </msub>
        </mrow>
      </mrow>
    </mrow>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <gt/>
        <ci encoding="">m</ci>
        <apply>
          <times/>
          <cn encoding="" base="10" type="integer">6</cn>
          <apply>
            <max/>
            <apply>
              <csymbol encoding="">subscript</csymbol>
              <ci encoding="">p</ci>
              <ci encoding="">D</ci>
            </apply>
          </apply>
        </apply>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok>&gt;</XMTok>
          <XMTok>m</XMTok>
          <XMApp>
            <XMTok>⁢</XMTok>
            <XMTok>6</XMTok>
            <XMApp>
              <XMTok>max</XMTok>
              <XMApp>
                <XMTok/>
                <XMTok>p</XMTok>
                <XMTok>D</XMTok>
              </XMApp>
            </XMApp>
          </XMApp>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">m&gt;6\max p_{D}</annotation>
  </semantics>
</math>

Diff

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@@ -1,60 +1,59 @@
 <?xml version="1.0" encoding="UTF-8"?>
-<math xmlns="http://www.w3.org/1998/Math/MathML" id="p7.m3" class="ltx_Math" alttext="m&gt;6\max p_{D}" display="inline">
-  <semantics>
+<math xmlns="http://www.w3.org/1998/Math/MathML">
+  <semantics encoding="">
     <mrow>
       <mi>m</mi>
       <mo>&gt;</mo>
       <mrow>
         <mn>6</mn>
-        <mo>⁢</mo>
         <mrow>
           <mo>max</mo>
           <mo>⁡</mo>
           <msub>
             <mi>p</mi>
             <mi>D</mi>
           </msub>
         </mrow>
       </mrow>
     </mrow>
     <annotation-xml encoding="MathML-Content">
       <apply>
         <gt/>
-        <ci>m</ci>
+        <ci encoding="">m</ci>
         <apply>
           <times/>
-          <cn type="integer">6</cn>
+          <cn encoding="" base="10" type="integer">6</cn>
           <apply>
             <max/>
             <apply>
-              <csymbol cd="ambiguous">subscript</csymbol>
-              <ci>p</ci>
-              <ci>D</ci>
+              <csymbol encoding="">subscript</csymbol>
+              <ci encoding="">p</ci>
+              <ci encoding="">D</ci>
             </apply>
           </apply>
         </apply>
       </apply>
     </annotation-xml>
     <annotation-xml encoding="application/x-latexml">
       <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
         <XMApp>
-          <XMTok meaning="greater-than" role="RELOP">&gt;</XMTok>
-          <XMTok role="UNKNOWN" font="italic">m</XMTok>
+          <XMTok>&gt;</XMTok>
+          <XMTok>m</XMTok>
           <XMApp>
-            <XMTok meaning="times" role="MULOP">⁢</XMTok>
-            <XMTok meaning="6" role="NUMBER">6</XMTok>
+            <XMTok>⁢</XMTok>
+            <XMTok>6</XMTok>
             <XMApp>
-              <XMTok meaning="maximum" role="LIMITOP" scriptpos="post">max</XMTok>
+              <XMTok>max</XMTok>
               <XMApp>
-                <XMTok role="SUBSCRIPTOP" scriptpos="post2"/>
-                <XMTok role="UNKNOWN" font="italic">p</XMTok>
-                <XMTok role="UNKNOWN" font="italic">D</XMTok>
+                <XMTok/>
+                <XMTok>p</XMTok>
+                <XMTok>D</XMTok>
               </XMApp>
             </XMApp>
           </XMApp>
         </XMApp>
       </XMath>
     </annotation-xml>
     <annotation encoding="application/x-tex">m&gt;6\max p_{D}</annotation>
   </semantics>
 </math>

Formula 80

( show / hide formula code)

Original

m m m m

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="ThmLem3.p1.m1" class="ltx_Math" alttext="m" display="inline">
  <semantics>
    <mi>m</mi>
    <annotation-xml encoding="MathML-Content">
      <ci>m</ci>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMTok role="UNKNOWN">m</XMTok>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">m</annotation>
  </semantics>
</math>

Canonicalized

m m m m

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics encoding="">
    <mi>m</mi>
    <annotation-xml encoding="MathML-Content">
      <ci encoding="">m</ci>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMTok>m</XMTok>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">m</annotation>
  </semantics>
</math>

Diff

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@@ -1,15 +1,15 @@
 <?xml version="1.0" encoding="UTF-8"?>
-<math xmlns="http://www.w3.org/1998/Math/MathML" id="ThmLem3.p1.m1" class="ltx_Math" alttext="m" display="inline">
-  <semantics>
+<math xmlns="http://www.w3.org/1998/Math/MathML">
+  <semantics encoding="">
     <mi>m</mi>
     <annotation-xml encoding="MathML-Content">
-      <ci>m</ci>
+      <ci encoding="">m</ci>
     </annotation-xml>
     <annotation-xml encoding="application/x-latexml">
       <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
-        <XMTok role="UNKNOWN">m</XMTok>
+        <XMTok>m</XMTok>
       </XMath>
     </annotation-xml>
     <annotation encoding="application/x-tex">m</annotation>
   </semantics>
 </math>

Formula 81

( show / hide formula code)

Original

f m subscript f m f m f_{m}

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="ThmLem3.p1.m2" class="ltx_Math" alttext="f_{m}" display="inline">
  <semantics>
    <msub>
      <mi>f</mi>
      <mi>m</mi>
    </msub>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <csymbol cd="ambiguous">subscript</csymbol>
        <ci>f</ci>
        <ci>m</ci>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok role="SUBSCRIPTOP" scriptpos="post3"/>
          <XMTok role="UNKNOWN">f</XMTok>
          <XMTok role="UNKNOWN">m</XMTok>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">f_{m}</annotation>
  </semantics>
</math>

Canonicalized

f m subscript f m f m f_{m}

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics encoding="">
    <msub>
      <mi>f</mi>
      <mi>m</mi>
    </msub>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <csymbol encoding="">subscript</csymbol>
        <ci encoding="">f</ci>
        <ci encoding="">m</ci>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok/>
          <XMTok>f</XMTok>
          <XMTok>m</XMTok>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">f_{m}</annotation>
  </semantics>
</math>

Diff

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@@ -1,26 +1,26 @@
 <?xml version="1.0" encoding="UTF-8"?>
-<math xmlns="http://www.w3.org/1998/Math/MathML" id="ThmLem3.p1.m2" class="ltx_Math" alttext="f_{m}" display="inline">
-  <semantics>
+<math xmlns="http://www.w3.org/1998/Math/MathML">
+  <semantics encoding="">
     <msub>
       <mi>f</mi>
       <mi>m</mi>
     </msub>
     <annotation-xml encoding="MathML-Content">
       <apply>
-        <csymbol cd="ambiguous">subscript</csymbol>
-        <ci>f</ci>
-        <ci>m</ci>
+        <csymbol encoding="">subscript</csymbol>
+        <ci encoding="">f</ci>
+        <ci encoding="">m</ci>
       </apply>
     </annotation-xml>
     <annotation-xml encoding="application/x-latexml">
       <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
         <XMApp>
-          <XMTok role="SUBSCRIPTOP" scriptpos="post3"/>
-          <XMTok role="UNKNOWN">f</XMTok>
-          <XMTok role="UNKNOWN">m</XMTok>
+          <XMTok/>
+          <XMTok>f</XMTok>
+          <XMTok>m</XMTok>
         </XMApp>
       </XMath>
     </annotation-xml>
     <annotation encoding="application/x-tex">f_{m}</annotation>
   </semantics>
 </math>

Formula 82

( show / hide formula code)

Original

D ≥ m 1 / 12 D superscript m 1 12 ≥ D m / 1 12 D\geq m^{1/12}

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="ThmLem3.p1.m3" class="ltx_Math" alttext="D\geq m^{1/12}" display="inline">
  <semantics>
    <mrow>
      <mi>D</mi>
      <mo mathvariant="normal">≥</mo>
      <msup>
        <mi>m</mi>
        <mrow>
          <mn mathvariant="normal">1</mn>
          <mo mathvariant="normal">/</mo>
          <mn mathvariant="normal">12</mn>
        </mrow>
      </msup>
    </mrow>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <geq/>
        <ci>D</ci>
        <apply>
          <csymbol cd="ambiguous">superscript</csymbol>
          <ci>m</ci>
          <apply>
            <divide/>
            <cn type="integer">1</cn>
            <cn type="integer">12</cn>
          </apply>
        </apply>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok meaning="greater-than-or-equals" name="geq" role="RELOP" font="upright">≥</XMTok>
          <XMTok role="UNKNOWN">D</XMTok>
          <XMApp>
            <XMTok role="SUPERSCRIPTOP" scriptpos="post3"/>
            <XMTok role="UNKNOWN">m</XMTok>
            <XMApp>
              <XMTok mathstyle="inline" meaning="divide" role="MULOP" font="upright">/</XMTok>
              <XMTok meaning="1" role="NUMBER" font="upright">1</XMTok>
              <XMTok meaning="12" role="NUMBER" font="upright">12</XMTok>
            </XMApp>
          </XMApp>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">D\geq m^{1/12}</annotation>
  </semantics>
</math>

Canonicalized

D ≥ m 1 / 12 D superscript m 1 12 ≥ D m / 1 12 D\geq m^{1/12}

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics encoding="">
    <mrow>
      <mi>D</mi>
      <mo mathvariant="normal">≥</mo>
      <msup>
        <mi>m</mi>
        <mrow>
          <mn mathvariant="normal">1</mn>
          <mo mathvariant="normal">/</mo>
          <mn mathvariant="normal">12</mn>
        </mrow>
      </msup>
    </mrow>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <geq/>
        <ci encoding="">D</ci>
        <apply>
          <csymbol encoding="">superscript</csymbol>
          <ci encoding="">m</ci>
          <apply>
            <divide/>
            <cn encoding="" base="10" type="integer">1</cn>
            <cn encoding="" base="10" type="integer">12</cn>
          </apply>
        </apply>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok>≥</XMTok>
          <XMTok>D</XMTok>
          <XMApp>
            <XMTok/>
            <XMTok>m</XMTok>
            <XMApp>
              <XMTok>/</XMTok>
              <XMTok>1</XMTok>
              <XMTok>12</XMTok>
            </XMApp>
          </XMApp>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">D\geq m^{1/12}</annotation>
  </semantics>
</math>

Diff

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@@ -1,50 +1,50 @@
 <?xml version="1.0" encoding="UTF-8"?>
-<math xmlns="http://www.w3.org/1998/Math/MathML" id="ThmLem3.p1.m3" class="ltx_Math" alttext="D\geq m^{1/12}" display="inline">
-  <semantics>
+<math xmlns="http://www.w3.org/1998/Math/MathML">
+  <semantics encoding="">
     <mrow>
       <mi>D</mi>
       <mo mathvariant="normal">≥</mo>
       <msup>
         <mi>m</mi>
         <mrow>
           <mn mathvariant="normal">1</mn>
           <mo mathvariant="normal">/</mo>
           <mn mathvariant="normal">12</mn>
         </mrow>
       </msup>
     </mrow>
     <annotation-xml encoding="MathML-Content">
       <apply>
         <geq/>
-        <ci>D</ci>
+        <ci encoding="">D</ci>
         <apply>
-          <csymbol cd="ambiguous">superscript</csymbol>
-          <ci>m</ci>
+          <csymbol encoding="">superscript</csymbol>
+          <ci encoding="">m</ci>
           <apply>
             <divide/>
-            <cn type="integer">1</cn>
-            <cn type="integer">12</cn>
+            <cn encoding="" base="10" type="integer">1</cn>
+            <cn encoding="" base="10" type="integer">12</cn>
           </apply>
         </apply>
       </apply>
     </annotation-xml>
     <annotation-xml encoding="application/x-latexml">
       <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
         <XMApp>
-          <XMTok meaning="greater-than-or-equals" name="geq" role="RELOP" font="upright">≥</XMTok>
-          <XMTok role="UNKNOWN">D</XMTok>
+          <XMTok>≥</XMTok>
+          <XMTok>D</XMTok>
           <XMApp>
-            <XMTok role="SUPERSCRIPTOP" scriptpos="post3"/>
-            <XMTok role="UNKNOWN">m</XMTok>
+            <XMTok/>
+            <XMTok>m</XMTok>
             <XMApp>
-              <XMTok mathstyle="inline" meaning="divide" role="MULOP" font="upright">/</XMTok>
-              <XMTok meaning="1" role="NUMBER" font="upright">1</XMTok>
-              <XMTok meaning="12" role="NUMBER" font="upright">12</XMTok>
+              <XMTok>/</XMTok>
+              <XMTok>1</XMTok>
+              <XMTok>12</XMTok>
             </XMApp>
           </XMApp>
         </XMApp>
       </XMath>
     </annotation-xml>
     <annotation encoding="application/x-tex">D\geq m^{1/12}</annotation>
   </semantics>
 </math>

Formula 83

( show / hide formula code)

Original

m = 2 m 2 = m 2 m=2

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="p8.m1" class="ltx_Math" alttext="m=2" display="inline">
  <semantics>
    <mrow>
      <mi>m</mi>
      <mo>=</mo>
      <mn>2</mn>
    </mrow>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <eq/>
        <ci>m</ci>
        <cn type="integer">2</cn>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok meaning="equals" role="RELOP">=</XMTok>
          <XMTok role="UNKNOWN" font="italic">m</XMTok>
          <XMTok meaning="2" role="NUMBER">2</XMTok>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">m=2</annotation>
  </semantics>
</math>

Canonicalized

m = 2 m 2 = m 2 m=2

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics encoding="">
    <mrow>
      <mi>m</mi>
      <mo>=</mo>
      <mn>2</mn>
    </mrow>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <eq/>
        <ci encoding="">m</ci>
        <cn encoding="" base="10" type="integer">2</cn>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok>=</XMTok>
          <XMTok>m</XMTok>
          <XMTok>2</XMTok>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">m=2</annotation>
  </semantics>
</math>

Diff

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@@ -1,27 +1,27 @@
 <?xml version="1.0" encoding="UTF-8"?>
-<math xmlns="http://www.w3.org/1998/Math/MathML" id="p8.m1" class="ltx_Math" alttext="m=2" display="inline">
-  <semantics>
+<math xmlns="http://www.w3.org/1998/Math/MathML">
+  <semantics encoding="">
     <mrow>
       <mi>m</mi>
       <mo>=</mo>
       <mn>2</mn>
     </mrow>
     <annotation-xml encoding="MathML-Content">
       <apply>
         <eq/>
-        <ci>m</ci>
-        <cn type="integer">2</cn>
+        <ci encoding="">m</ci>
+        <cn encoding="" base="10" type="integer">2</cn>
       </apply>
     </annotation-xml>
     <annotation-xml encoding="application/x-latexml">
       <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
         <XMApp>
-          <XMTok meaning="equals" role="RELOP">=</XMTok>
-          <XMTok role="UNKNOWN" font="italic">m</XMTok>
-          <XMTok meaning="2" role="NUMBER">2</XMTok>
+          <XMTok>=</XMTok>
+          <XMTok>m</XMTok>
+          <XMTok>2</XMTok>
         </XMApp>
       </XMath>
     </annotation-xml>
     <annotation encoding="application/x-tex">m=2</annotation>
   </semantics>
 </math>

Formula 84

( show / hide formula code)

Original

ϵ m subscript ϵ m ϵ m \epsilon_{m}

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="p8.m2" class="ltx_Math" alttext="\epsilon_{m}" display="inline">
  <semantics>
    <msub>
      <mi>ϵ</mi>
      <mi>m</mi>
    </msub>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <csymbol cd="ambiguous">subscript</csymbol>
        <ci>ϵ</ci>
        <ci>m</ci>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok role="SUBSCRIPTOP" scriptpos="post2"/>
          <XMTok name="epsilon" role="UNKNOWN" font="italic">ϵ</XMTok>
          <XMTok role="UNKNOWN" font="italic">m</XMTok>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">\epsilon_{m}</annotation>
  </semantics>
</math>

Canonicalized

ε m subscript ε m ε m \epsilon_{m}

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics encoding="">
    <msub>
      <mi>ε</mi>
      <mi>m</mi>
    </msub>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <csymbol encoding="">subscript</csymbol>
        <ci encoding="">ε</ci>
        <ci encoding="">m</ci>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok/>
          <XMTok>ε</XMTok>
          <XMTok>m</XMTok>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">\epsilon_{m}</annotation>
  </semantics>
</math>

Diff

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@@ -1,26 +1,26 @@
 <?xml version="1.0" encoding="UTF-8"?>
-<math xmlns="http://www.w3.org/1998/Math/MathML" id="p8.m2" class="ltx_Math" alttext="\epsilon_{m}" display="inline">
-  <semantics>
+<math xmlns="http://www.w3.org/1998/Math/MathML">
+  <semantics encoding="">
     <msub>
-      <mi>ϵ</mi>
+      <mi>ε</mi>
       <mi>m</mi>
     </msub>
     <annotation-xml encoding="MathML-Content">
       <apply>
-        <csymbol cd="ambiguous">subscript</csymbol>
-        <ci>ϵ</ci>
-        <ci>m</ci>
+        <csymbol encoding="">subscript</csymbol>
+        <ci encoding="">ε</ci>
+        <ci encoding="">m</ci>
       </apply>
     </annotation-xml>
     <annotation-xml encoding="application/x-latexml">
       <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
         <XMApp>
-          <XMTok role="SUBSCRIPTOP" scriptpos="post2"/>
-          <XMTok name="epsilon" role="UNKNOWN" font="italic">ϵ</XMTok>
-          <XMTok role="UNKNOWN" font="italic">m</XMTok>
+          <XMTok/>
+          <XMTok>ε</XMTok>
+          <XMTok>m</XMTok>
         </XMApp>
       </XMath>
     </annotation-xml>
     <annotation encoding="application/x-tex">\epsilon_{m}</annotation>
   </semantics>
 </math>

Formula 85

( show / hide formula code)

Original

ℚ ⁢ ( D ) ℚ D ⁢ Q D \mathbb{Q}(\sqrt{D})

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="p8.m3" class="ltx_Math" alttext="\mathbb{Q}(\sqrt{D})" display="inline">
  <semantics>
    <mrow>
      <mi>ℚ</mi>
      <mo>⁢</mo>
      <mrow>
        <mo>(</mo>
        <msqrt>
          <mi>D</mi>
        </msqrt>
        <mo>)</mo>
      </mrow>
    </mrow>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <times/>
        <ci>ℚ</ci>
        <apply>
          <root/>
          <ci>D</ci>
        </apply>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok meaning="times" role="MULOP">⁢</XMTok>
          <XMTok possibleFunction="yes" role="UNKNOWN" font="blackboard">Q</XMTok>
          <XMApp close=")" open="(">
            <XMTok meaning="square-root"/>
            <XMTok role="UNKNOWN" font="italic">D</XMTok>
          </XMApp>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">\mathbb{Q}(\sqrt{D})</annotation>
  </semantics>
</math>

Canonicalized

Q ( D ) Q D ⁢ Q D \mathbb{Q}(\sqrt{D})

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics encoding="">
    <mrow>
      <mi>Q</mi>
      <mrow>
        <mo>(</mo>
        <msqrt>
          <mi>D</mi>
        </msqrt>
        <mo>)</mo>
      </mrow>
    </mrow>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <times/>
        <ci encoding="">Q</ci>
        <apply>
          <root/>
          <ci encoding="">D</ci>
        </apply>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok>⁢</XMTok>
          <XMTok>Q</XMTok>
          <XMApp>
            <XMTok/>
            <XMTok>D</XMTok>
          </XMApp>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">\mathbb{Q}(\sqrt{D})</annotation>
  </semantics>
</math>

Diff

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@@ -1,39 +1,38 @@
 <?xml version="1.0" encoding="UTF-8"?>
-<math xmlns="http://www.w3.org/1998/Math/MathML" id="p8.m3" class="ltx_Math" alttext="\mathbb{Q}(\sqrt{D})" display="inline">
-  <semantics>
+<math xmlns="http://www.w3.org/1998/Math/MathML">
+  <semantics encoding="">
     <mrow>
-      <mi>ℚ</mi>
-      <mo>⁢</mo>
+      <mi>Q</mi>
       <mrow>
         <mo>(</mo>
         <msqrt>
           <mi>D</mi>
         </msqrt>
         <mo>)</mo>
       </mrow>
     </mrow>
     <annotation-xml encoding="MathML-Content">
       <apply>
         <times/>
-        <ci>ℚ</ci>
+        <ci encoding="">Q</ci>
         <apply>
           <root/>
-          <ci>D</ci>
+          <ci encoding="">D</ci>
         </apply>
       </apply>
     </annotation-xml>
     <annotation-xml encoding="application/x-latexml">
       <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
         <XMApp>
-          <XMTok meaning="times" role="MULOP">⁢</XMTok>
-          <XMTok possibleFunction="yes" role="UNKNOWN" font="blackboard">Q</XMTok>
-          <XMApp close=")" open="(">
-            <XMTok meaning="square-root"/>
-            <XMTok role="UNKNOWN" font="italic">D</XMTok>
+          <XMTok>⁢</XMTok>
+          <XMTok>Q</XMTok>
+          <XMApp>
+            <XMTok/>
+            <XMTok>D</XMTok>
           </XMApp>
         </XMApp>
       </XMath>
     </annotation-xml>
     <annotation encoding="application/x-tex">\mathbb{Q}(\sqrt{D})</annotation>
   </semantics>
 </math>

Formula 86

( show / hide formula code)

Original

1 < | ϵ m | ≪ m 1 subscript ϵ m much-less-than m 1 < ϵ m ≪ m 1<|\epsilon_{m}|\ll m

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="p8.m4" class="ltx_Math" alttext="1&lt;|\epsilon_{m}|\ll m" display="inline">
  <semantics>
    <mrow>
      <mn>1</mn>
      <mo>&lt;</mo>
      <mrow>
        <mo fence="true">|</mo>
        <msub>
          <mi>ϵ</mi>
          <mi>m</mi>
        </msub>
        <mo fence="true">|</mo>
      </mrow>
      <mo>≪</mo>
      <mi>m</mi>
    </mrow>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <and/>
        <apply>
          <lt/>
          <cn type="integer">1</cn>
          <apply xml:id="p8.m4.sh1e.cmml">
            <abs xml:id="p8.m4.sh1.cmml"/>
            <apply xml:id="p8.m4.sh1d.cmml">
              <csymbol cd="ambiguous" xml:id="p8.m4.sh1a.cmml">subscript</csymbol>
              <ci xml:id="p8.m4.sh1b.cmml">ϵ</ci>
              <ci xml:id="p8.m4.sh1c.cmml">m</ci>
            </apply>
          </apply>
        </apply>
        <apply>
          <csymbol cd="latexml">much-less-than</csymbol>
          <share href="#p8.m4.sh1.cmml"/>
          <ci xml:id="p8.m4.sh2.cmml">m</ci>
        </apply>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok meaning="multirelation"/>
          <XMTok meaning="1" role="NUMBER">1</XMTok>
          <XMTok meaning="less-than" role="RELOP">&lt;</XMTok>
          <XMApp xml:id="p8.m4.sh1.xm" fragid="p8.m4.sh1.xm">
            <XMTok argclose="|" argopen="|" meaning="absolute-value" role="FENCED"/>
            <XMApp>
              <XMTok role="SUBSCRIPTOP" scriptpos="post2"/>
              <XMTok name="epsilon" role="UNKNOWN" font="italic">ϵ</XMTok>
              <XMTok role="UNKNOWN" font="italic">m</XMTok>
            </XMApp>
          </XMApp>
          <XMTok meaning="much-less-than" name="ll" role="RELOP">≪</XMTok>
          <XMTok role="UNKNOWN" font="italic" xml:id="p8.m4.sh2.xm" fragid="p8.m4.sh2.xm">m</XMTok>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">1&lt;|\epsilon_{m}|\ll m</annotation>
  </semantics>
</math>

Canonicalized

1 < | ε m | ≪ m 1 subscript ε m much-less-than m 1 < ε m ≪ m 1<|\epsilon_{m}|\ll m

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics encoding="">
    <mrow>
      <mn>1</mn>
      <mo>&lt;</mo>
      <mrow>
        <mo>|</mo>
        <msub>
          <mi>ε</mi>
          <mi>m</mi>
        </msub>
        <mo>|</mo>
      </mrow>
      <mo>≪</mo>
      <mi>m</mi>
    </mrow>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <and/>
        <apply>
          <lt/>
          <cn encoding="" base="10" type="integer">1</cn>
          <apply>
            <abs encoding=""/>
            <apply>
              <csymbol encoding="">subscript</csymbol>
              <ci encoding="">ε</ci>
              <ci encoding="">m</ci>
            </apply>
          </apply>
        </apply>
        <apply>
          <csymbol encoding="">much-less-than</csymbol>
          <share/>
          <ci encoding="">m</ci>
        </apply>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok/>
          <XMTok>1</XMTok>
          <XMTok>&lt;</XMTok>
          <XMApp>
            <XMTok/>
            <XMApp>
              <XMTok/>
              <XMTok>ε</XMTok>
              <XMTok>m</XMTok>
            </XMApp>
          </XMApp>
          <XMTok>≪</XMTok>
          <XMTok>m</XMTok>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">1&lt;|\epsilon_{m}|\ll m</annotation>
  </semantics>
</math>

Diff

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@@ -1,61 +1,61 @@
 <?xml version="1.0" encoding="UTF-8"?>
-<math xmlns="http://www.w3.org/1998/Math/MathML" id="p8.m4" class="ltx_Math" alttext="1&lt;|\epsilon_{m}|\ll m" display="inline">
-  <semantics>
+<math xmlns="http://www.w3.org/1998/Math/MathML">
+  <semantics encoding="">
     <mrow>
       <mn>1</mn>
       <mo>&lt;</mo>
       <mrow>
-        <mo fence="true">|</mo>
+        <mo>|</mo>
         <msub>
-          <mi>ϵ</mi>
+          <mi>ε</mi>
           <mi>m</mi>
         </msub>
-        <mo fence="true">|</mo>
+        <mo>|</mo>
       </mrow>
       <mo>≪</mo>
       <mi>m</mi>
     </mrow>
     <annotation-xml encoding="MathML-Content">
       <apply>
         <and/>
         <apply>
           <lt/>
-          <cn type="integer">1</cn>
-          <apply xml:id="p8.m4.sh1e.cmml">
-            <abs xml:id="p8.m4.sh1.cmml"/>
-            <apply xml:id="p8.m4.sh1d.cmml">
-              <csymbol cd="ambiguous" xml:id="p8.m4.sh1a.cmml">subscript</csymbol>
-              <ci xml:id="p8.m4.sh1b.cmml">ϵ</ci>
-              <ci xml:id="p8.m4.sh1c.cmml">m</ci>
+          <cn encoding="" base="10" type="integer">1</cn>
+          <apply>
+            <abs encoding=""/>
+            <apply>
+              <csymbol encoding="">subscript</csymbol>
+              <ci encoding="">ε</ci>
+              <ci encoding="">m</ci>
             </apply>
           </apply>
         </apply>
         <apply>
-          <csymbol cd="latexml">much-less-than</csymbol>
-          <share href="#p8.m4.sh1.cmml"/>
-          <ci xml:id="p8.m4.sh2.cmml">m</ci>
+          <csymbol encoding="">much-less-than</csymbol>
+          <share/>
+          <ci encoding="">m</ci>
         </apply>
       </apply>
     </annotation-xml>
     <annotation-xml encoding="application/x-latexml">
       <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
         <XMApp>
-          <XMTok meaning="multirelation"/>
-          <XMTok meaning="1" role="NUMBER">1</XMTok>
-          <XMTok meaning="less-than" role="RELOP">&lt;</XMTok>
-          <XMApp xml:id="p8.m4.sh1.xm" fragid="p8.m4.sh1.xm">
-            <XMTok argclose="|" argopen="|" meaning="absolute-value" role="FENCED"/>
+          <XMTok/>
+          <XMTok>1</XMTok>
+          <XMTok>&lt;</XMTok>
+          <XMApp>
+            <XMTok/>
             <XMApp>
-              <XMTok role="SUBSCRIPTOP" scriptpos="post2"/>
-              <XMTok name="epsilon" role="UNKNOWN" font="italic">ϵ</XMTok>
-              <XMTok role="UNKNOWN" font="italic">m</XMTok>
+              <XMTok/>
+              <XMTok>ε</XMTok>
+              <XMTok>m</XMTok>
             </XMApp>
           </XMApp>
-          <XMTok meaning="much-less-than" name="ll" role="RELOP">≪</XMTok>
-          <XMTok role="UNKNOWN" font="italic" xml:id="p8.m4.sh2.xm" fragid="p8.m4.sh2.xm">m</XMTok>
+          <XMTok>≪</XMTok>
+          <XMTok>m</XMTok>
         </XMApp>
       </XMath>
     </annotation-xml>
     <annotation encoding="application/x-tex">1&lt;|\epsilon_{m}|\ll m</annotation>
   </semantics>
 </math>

Formula 87

( show / hide formula code)

Original

m = t 2 m superscript t 2 = m t 2 \displaystyle m=t^{2}

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="S0.Ex2.m2" class="ltx_Math" alttext="\displaystyle m=t^{2}" display="inline">
  <semantics>
    <mrow>
      <mi>m</mi>
      <mo>=</mo>
      <msup>
        <mi>t</mi>
        <mn>2</mn>
      </msup>
    </mrow>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <eq/>
        <ci>m</ci>
        <apply>
          <csymbol cd="ambiguous">superscript</csymbol>
          <ci>t</ci>
          <cn type="integer">2</cn>
        </apply>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok meaning="equals" role="RELOP">=</XMTok>
          <XMTok role="UNKNOWN" font="italic">m</XMTok>
          <XMApp>
            <XMTok role="SUPERSCRIPTOP" scriptpos="post7"/>
            <XMTok role="UNKNOWN" font="italic">t</XMTok>
            <XMTok meaning="2" role="NUMBER">2</XMTok>
          </XMApp>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">\displaystyle m=t^{2}</annotation>
  </semantics>
</math>

Canonicalized

m = t 2 m superscript t 2 = m t 2 \displaystyle m=t^{2}

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics encoding="">
    <mrow>
      <mi>m</mi>
      <mo>=</mo>
      <msup>
        <mi>t</mi>
        <mn>2</mn>
      </msup>
    </mrow>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <eq/>
        <ci encoding="">m</ci>
        <apply>
          <csymbol encoding="">superscript</csymbol>
          <ci encoding="">t</ci>
          <cn encoding="" base="10" type="integer">2</cn>
        </apply>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok>=</XMTok>
          <XMTok>m</XMTok>
          <XMApp>
            <XMTok/>
            <XMTok>t</XMTok>
            <XMTok>2</XMTok>
          </XMApp>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">\displaystyle m=t^{2}</annotation>
  </semantics>
</math>

Diff

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@@ -1,38 +1,38 @@
 <?xml version="1.0" encoding="UTF-8"?>
-<math xmlns="http://www.w3.org/1998/Math/MathML" id="S0.Ex2.m2" class="ltx_Math" alttext="\displaystyle m=t^{2}" display="inline">
-  <semantics>
+<math xmlns="http://www.w3.org/1998/Math/MathML">
+  <semantics encoding="">
     <mrow>
       <mi>m</mi>
       <mo>=</mo>
       <msup>
         <mi>t</mi>
         <mn>2</mn>
       </msup>
     </mrow>
     <annotation-xml encoding="MathML-Content">
       <apply>
         <eq/>
-        <ci>m</ci>
+        <ci encoding="">m</ci>
         <apply>
-          <csymbol cd="ambiguous">superscript</csymbol>
-          <ci>t</ci>
-          <cn type="integer">2</cn>
+          <csymbol encoding="">superscript</csymbol>
+          <ci encoding="">t</ci>
+          <cn encoding="" base="10" type="integer">2</cn>
         </apply>
       </apply>
     </annotation-xml>
     <annotation-xml encoding="application/x-latexml">
       <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
         <XMApp>
-          <XMTok meaning="equals" role="RELOP">=</XMTok>
-          <XMTok role="UNKNOWN" font="italic">m</XMTok>
+          <XMTok>=</XMTok>
+          <XMTok>m</XMTok>
           <XMApp>
-            <XMTok role="SUPERSCRIPTOP" scriptpos="post7"/>
-            <XMTok role="UNKNOWN" font="italic">t</XMTok>
-            <XMTok meaning="2" role="NUMBER">2</XMTok>
+            <XMTok/>
+            <XMTok>t</XMTok>
+            <XMTok>2</XMTok>
           </XMApp>
         </XMApp>
       </XMath>
     </annotation-xml>
     <annotation encoding="application/x-tex">\displaystyle m=t^{2}</annotation>
   </semantics>
 </math>

Formula 88

( show / hide formula code)

Original

: normal-: : \displaystyle:

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="S0.Ex2.m3" class="ltx_Math" alttext="\displaystyle:" display="inline">
  <semantics>
    <mo>:</mo>
    <annotation-xml encoding="MathML-Content">
      <ci>normal-:</ci>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMTok name="colon" role="METARELOP">:</XMTok>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">\displaystyle:</annotation>
  </semantics>
</math>

Canonicalized

/ normal-: : \displaystyle:

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics encoding="">
    <mo>/</mo>
    <annotation-xml encoding="MathML-Content">
      <ci encoding="">normal-:</ci>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMTok>:</XMTok>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">\displaystyle:</annotation>
  </semantics>
</math>

Diff

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@@ -1,15 +1,15 @@
 <?xml version="1.0" encoding="UTF-8"?>
-<math xmlns="http://www.w3.org/1998/Math/MathML" id="S0.Ex2.m3" class="ltx_Math" alttext="\displaystyle:" display="inline">
-  <semantics>
-    <mo>:</mo>
+<math xmlns="http://www.w3.org/1998/Math/MathML">
+  <semantics encoding="">
+    <mo>/</mo>
     <annotation-xml encoding="MathML-Content">
-      <ci>normal-:</ci>
+      <ci encoding="">normal-:</ci>
     </annotation-xml>
     <annotation-xml encoding="application/x-latexml">
       <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
-        <XMTok name="colon" role="METARELOP">:</XMTok>
+        <XMTok>:</XMTok>
       </XMath>
     </annotation-xml>
     <annotation encoding="application/x-tex">\displaystyle:</annotation>
   </semantics>
 </math>

Formula 89

( show / hide formula code)

Original

  ⁢ ϵ m = 2 ⁢ t + 4 ⁢ m + 1 normal-  subscript ϵ m 2 t 4 m 1 = ⁢   ϵ m + ⁢ 2 t + ⁢ 4 m 1 \displaystyle\quad\epsilon_{m}=2t+\sqrt{4m+1}

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="S0.Ex2.m4" class="ltx_Math" alttext="\displaystyle\quad\epsilon_{m}=2t+\sqrt{4m+1}" display="inline">
  <semantics>
    <mrow>
      <mrow>
        <mi mathvariant="normal"> </mi>
        <mo>⁢</mo>
        <msub>
          <mi>ϵ</mi>
          <mi>m</mi>
        </msub>
      </mrow>
      <mo>=</mo>
      <mrow>
        <mrow>
          <mn>2</mn>
          <mo>⁢</mo>
          <mi>t</mi>
        </mrow>
        <mo>+</mo>
        <msqrt>
          <mrow>
            <mrow>
              <mn>4</mn>
              <mo>⁢</mo>
              <mi>m</mi>
            </mrow>
            <mo>+</mo>
            <mn>1</mn>
          </mrow>
        </msqrt>
      </mrow>
    </mrow>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <eq/>
        <apply>
          <times/>
          <ci>normal- </ci>
          <apply>
            <csymbol cd="ambiguous">subscript</csymbol>
            <ci>ϵ</ci>
            <ci>m</ci>
          </apply>
        </apply>
        <apply>
          <plus/>
          <apply>
            <times/>
            <cn type="integer">2</cn>
            <ci>t</ci>
          </apply>
          <apply>
            <root/>
            <apply>
              <plus/>
              <apply>
                <times/>
                <cn type="integer">4</cn>
                <ci>m</ci>
              </apply>
              <cn type="integer">1</cn>
            </apply>
          </apply>
        </apply>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok meaning="equals" role="RELOP">=</XMTok>
          <XMApp>
            <XMTok meaning="times" role="MULOP">⁢</XMTok>
            <XMTok role="UNKNOWN"> </XMTok>
            <XMApp>
              <XMTok role="SUBSCRIPTOP" scriptpos="post7"/>
              <XMTok name="epsilon" role="UNKNOWN" font="italic">ϵ</XMTok>
              <XMTok role="UNKNOWN" font="italic">m</XMTok>
            </XMApp>
          </XMApp>
          <XMApp>
            <XMTok meaning="plus" role="ADDOP">+</XMTok>
            <XMApp>
              <XMTok meaning="times" role="MULOP">⁢</XMTok>
              <XMTok meaning="2" role="NUMBER">2</XMTok>
              <XMTok role="UNKNOWN" font="italic">t</XMTok>
            </XMApp>
            <XMApp>
              <XMTok meaning="square-root"/>
              <XMApp>
                <XMTok meaning="plus" role="ADDOP">+</XMTok>
                <XMApp>
                  <XMTok meaning="times" role="MULOP">⁢</XMTok>
                  <XMTok meaning="4" role="NUMBER">4</XMTok>
                  <XMTok role="UNKNOWN" font="italic">m</XMTok>
                </XMApp>
                <XMTok meaning="1" role="NUMBER">1</XMTok>
              </XMApp>
            </XMApp>
          </XMApp>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">\displaystyle\quad\epsilon_{m}=2t+\sqrt{4m+1}</annotation>
  </semantics>
</math>

Canonicalized

ε m = 2 t + 4 m + 1 normal- subscript ε m 2 t 4 m 1 = ⁢ ε m + ⁢ 2 t + ⁢ 4 m 1 \displaystyle\quad\epsilon_{m}=2t+\sqrt{4m+1}

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics encoding="">
    <mrow>
      <mrow>
        <mi mathvariant="normal"> </mi>
        <msub>
          <mi>ε</mi>
          <mi>m</mi>
        </msub>
      </mrow>
      <mo>=</mo>
      <mrow>
        <mrow>
          <mn>2</mn>
          <mi>t</mi>
        </mrow>
        <mo>+</mo>
        <msqrt>
          <mrow>
            <mn>4</mn>
            <mi>m</mi>
          </mrow>
          <mo>+</mo>
          <mn>1</mn>
        </msqrt>
      </mrow>
    </mrow>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <eq/>
        <apply>
          <times/>
          <ci encoding="">normal- </ci>
          <apply>
            <csymbol encoding="">subscript</csymbol>
            <ci encoding="">ε</ci>
            <ci encoding="">m</ci>
          </apply>
        </apply>
        <apply>
          <plus/>
          <apply>
            <times/>
            <cn encoding="" base="10" type="integer">2</cn>
            <ci encoding="">t</ci>
          </apply>
          <apply>
            <root/>
            <apply>
              <plus/>
              <apply>
                <times/>
                <cn encoding="" base="10" type="integer">4</cn>
                <ci encoding="">m</ci>
              </apply>
              <cn encoding="" base="10" type="integer">1</cn>
            </apply>
          </apply>
        </apply>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok>=</XMTok>
          <XMApp>
            <XMTok>⁢</XMTok>
            <XMTok> </XMTok>
            <XMApp>
              <XMTok/>
              <XMTok>ε</XMTok>
              <XMTok>m</XMTok>
            </XMApp>
          </XMApp>
          <XMApp>
            <XMTok>+</XMTok>
            <XMApp>
              <XMTok>⁢</XMTok>
              <XMTok>2</XMTok>
              <XMTok>t</XMTok>
            </XMApp>
            <XMApp>
              <XMTok/>
              <XMApp>
                <XMTok>+</XMTok>
                <XMApp>
                  <XMTok>⁢</XMTok>
                  <XMTok>4</XMTok>
                  <XMTok>m</XMTok>
                </XMApp>
                <XMTok>1</XMTok>
              </XMApp>
            </XMApp>
          </XMApp>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">\displaystyle\quad\epsilon_{m}=2t+\sqrt{4m+1}</annotation>
  </semantics>
</math>

Diff

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@@ -1,106 +1,101 @@
 <?xml version="1.0" encoding="UTF-8"?>
-<math xmlns="http://www.w3.org/1998/Math/MathML" id="S0.Ex2.m4" class="ltx_Math" alttext="\displaystyle\quad\epsilon_{m}=2t+\sqrt{4m+1}" display="inline">
-  <semantics>
+<math xmlns="http://www.w3.org/1998/Math/MathML">
+  <semantics encoding="">
     <mrow>
       <mrow>
-        <mi mathvariant="normal"> </mi>
-        <mo>⁢</mo>
+        <mi mathvariant="normal"> </mi>
         <msub>
-          <mi>ϵ</mi>
+          <mi>ε</mi>
           <mi>m</mi>
         </msub>
       </mrow>
       <mo>=</mo>
       <mrow>
         <mrow>
           <mn>2</mn>
-          <mo>⁢</mo>
           <mi>t</mi>
         </mrow>
         <mo>+</mo>
         <msqrt>
           <mrow>
-            <mrow>
-              <mn>4</mn>
-              <mo>⁢</mo>
-              <mi>m</mi>
-            </mrow>
-            <mo>+</mo>
-            <mn>1</mn>
+            <mn>4</mn>
+            <mi>m</mi>
           </mrow>
+          <mo>+</mo>
+          <mn>1</mn>
         </msqrt>
       </mrow>
     </mrow>
     <annotation-xml encoding="MathML-Content">
       <apply>
         <eq/>
         <apply>
           <times/>
-          <ci>normal- </ci>
+          <ci encoding="">normal- </ci>
           <apply>
-            <csymbol cd="ambiguous">subscript</csymbol>
-            <ci>ϵ</ci>
-            <ci>m</ci>
+            <csymbol encoding="">subscript</csymbol>
+            <ci encoding="">ε</ci>
+            <ci encoding="">m</ci>
           </apply>
         </apply>
         <apply>
           <plus/>
           <apply>
             <times/>
-            <cn type="integer">2</cn>
-            <ci>t</ci>
+            <cn encoding="" base="10" type="integer">2</cn>
+            <ci encoding="">t</ci>
           </apply>
           <apply>
             <root/>
             <apply>
               <plus/>
               <apply>
                 <times/>
-                <cn type="integer">4</cn>
-                <ci>m</ci>
+                <cn encoding="" base="10" type="integer">4</cn>
+                <ci encoding="">m</ci>
               </apply>
-              <cn type="integer">1</cn>
+              <cn encoding="" base="10" type="integer">1</cn>
             </apply>
           </apply>
         </apply>
       </apply>
     </annotation-xml>
     <annotation-xml encoding="application/x-latexml">
       <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
         <XMApp>
-          <XMTok meaning="equals" role="RELOP">=</XMTok>
+          <XMTok>=</XMTok>
           <XMApp>
-            <XMTok meaning="times" role="MULOP">⁢</XMTok>
-            <XMTok role="UNKNOWN"> </XMTok>
+            <XMTok>⁢</XMTok>
+            <XMTok> </XMTok>
             <XMApp>
-              <XMTok role="SUBSCRIPTOP" scriptpos="post7"/>
-              <XMTok name="epsilon" role="UNKNOWN" font="italic">ϵ</XMTok>
-              <XMTok role="UNKNOWN" font="italic">m</XMTok>
+              <XMTok/>
+              <XMTok>ε</XMTok>
+              <XMTok>m</XMTok>
             </XMApp>
           </XMApp>
           <XMApp>
-            <XMTok meaning="plus" role="ADDOP">+</XMTok>
+            <XMTok>+</XMTok>
             <XMApp>
-              <XMTok meaning="times" role="MULOP">⁢</XMTok>
-              <XMTok meaning="2" role="NUMBER">2</XMTok>
-              <XMTok role="UNKNOWN" font="italic">t</XMTok>
+              <XMTok>⁢</XMTok>
+              <XMTok>2</XMTok>
+              <XMTok>t</XMTok>
             </XMApp>
             <XMApp>
-              <XMTok meaning="square-root"/>
+              <XMTok/>
               <XMApp>
-                <XMTok meaning="plus" role="ADDOP">+</XMTok>
+                <XMTok>+</XMTok>
                 <XMApp>
-                  <XMTok meaning="times" role="MULOP">⁢</XMTok>
-                  <XMTok meaning="4" role="NUMBER">4</XMTok>
-                  <XMTok role="UNKNOWN" font="italic">m</XMTok>
+                  <XMTok>⁢</XMTok>
+                  <XMTok>4</XMTok>
+                  <XMTok>m</XMTok>
                 </XMApp>
-                <XMTok meaning="1" role="NUMBER">1</XMTok>
+                <XMTok>1</XMTok>
               </XMApp>
             </XMApp>
           </XMApp>
         </XMApp>
       </XMath>
     </annotation-xml>
     <annotation encoding="application/x-tex">\displaystyle\quad\epsilon_{m}=2t+\sqrt{4m+1}</annotation>
   </semantics>
 </math>

Formula 90

( show / hide formula code)

Original

m = t 2 + t ± 1 m plus-or-minus superscript t 2 t 1 = m ± + t 2 t 1 \displaystyle m=t^{2}+t\pm 1

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="S0.Ex3.m2" class="ltx_Math" alttext="\displaystyle m=t^{2}+t\pm 1" display="inline">
  <semantics>
    <mrow>
      <mi>m</mi>
      <mo>=</mo>
      <mrow>
        <msup>
          <mi>t</mi>
          <mn>2</mn>
        </msup>
        <mo>+</mo>
        <mi>t</mi>
        <mo>±</mo>
        <mn>1</mn>
      </mrow>
    </mrow>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <eq/>
        <ci>m</ci>
        <apply>
          <csymbol cd="latexml">plus-or-minus</csymbol>
          <apply>
            <plus/>
            <apply>
              <csymbol cd="ambiguous">superscript</csymbol>
              <ci>t</ci>
              <cn type="integer">2</cn>
            </apply>
            <ci>t</ci>
          </apply>
          <cn type="integer">1</cn>
        </apply>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok meaning="equals" role="RELOP">=</XMTok>
          <XMTok role="UNKNOWN" font="italic">m</XMTok>
          <XMApp>
            <XMTok meaning="plus-or-minus" name="pm" role="ADDOP">±</XMTok>
            <XMApp>
              <XMTok meaning="plus" role="ADDOP">+</XMTok>
              <XMApp>
                <XMTok role="SUPERSCRIPTOP" scriptpos="post7"/>
                <XMTok role="UNKNOWN" font="italic">t</XMTok>
                <XMTok meaning="2" role="NUMBER">2</XMTok>
              </XMApp>
              <XMTok role="UNKNOWN" font="italic">t</XMTok>
            </XMApp>
            <XMTok meaning="1" role="NUMBER">1</XMTok>
          </XMApp>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">\displaystyle m=t^{2}+t\pm 1</annotation>
  </semantics>
</math>

Canonicalized

m = t 2 + t ± 1 m plus-or-minus superscript t 2 t 1 = m ± + t 2 t 1 \displaystyle m=t^{2}+t\pm 1

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics encoding="">
    <mrow>
      <mi>m</mi>
      <mo>=</mo>
      <mrow>
        <msup>
          <mi>t</mi>
          <mn>2</mn>
        </msup>
        <mo>+</mo>
        <mi>t</mi>
        <mo>±</mo>
        <mn>1</mn>
      </mrow>
    </mrow>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <eq/>
        <ci encoding="">m</ci>
        <apply>
          <csymbol encoding="">plus-or-minus</csymbol>
          <apply>
            <plus/>
            <apply>
              <csymbol encoding="">superscript</csymbol>
              <ci encoding="">t</ci>
              <cn encoding="" base="10" type="integer">2</cn>
            </apply>
            <ci encoding="">t</ci>
          </apply>
          <cn encoding="" base="10" type="integer">1</cn>
        </apply>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok>=</XMTok>
          <XMTok>m</XMTok>
          <XMApp>
            <XMTok>±</XMTok>
            <XMApp>
              <XMTok>+</XMTok>
              <XMApp>
                <XMTok/>
                <XMTok>t</XMTok>
                <XMTok>2</XMTok>
              </XMApp>
              <XMTok>t</XMTok>
            </XMApp>
            <XMTok>1</XMTok>
          </XMApp>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">\displaystyle m=t^{2}+t\pm 1</annotation>
  </semantics>
</math>

Diff

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@@ -1,60 +1,60 @@
 <?xml version="1.0" encoding="UTF-8"?>
-<math xmlns="http://www.w3.org/1998/Math/MathML" id="S0.Ex3.m2" class="ltx_Math" alttext="\displaystyle m=t^{2}+t\pm 1" display="inline">
-  <semantics>
+<math xmlns="http://www.w3.org/1998/Math/MathML">
+  <semantics encoding="">
     <mrow>
       <mi>m</mi>
       <mo>=</mo>
       <mrow>
         <msup>
           <mi>t</mi>
           <mn>2</mn>
         </msup>
         <mo>+</mo>
         <mi>t</mi>
         <mo>±</mo>
         <mn>1</mn>
       </mrow>
     </mrow>
     <annotation-xml encoding="MathML-Content">
       <apply>
         <eq/>
-        <ci>m</ci>
+        <ci encoding="">m</ci>
         <apply>
-          <csymbol cd="latexml">plus-or-minus</csymbol>
+          <csymbol encoding="">plus-or-minus</csymbol>
           <apply>
             <plus/>
             <apply>
-              <csymbol cd="ambiguous">superscript</csymbol>
-              <ci>t</ci>
-              <cn type="integer">2</cn>
+              <csymbol encoding="">superscript</csymbol>
+              <ci encoding="">t</ci>
+              <cn encoding="" base="10" type="integer">2</cn>
             </apply>
-            <ci>t</ci>
+            <ci encoding="">t</ci>
           </apply>
-          <cn type="integer">1</cn>
+          <cn encoding="" base="10" type="integer">1</cn>
         </apply>
       </apply>
     </annotation-xml>
     <annotation-xml encoding="application/x-latexml">
       <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
         <XMApp>
-          <XMTok meaning="equals" role="RELOP">=</XMTok>
-          <XMTok role="UNKNOWN" font="italic">m</XMTok>
+          <XMTok>=</XMTok>
+          <XMTok>m</XMTok>
           <XMApp>
-            <XMTok meaning="plus-or-minus" name="pm" role="ADDOP">±</XMTok>
+            <XMTok>±</XMTok>
             <XMApp>
-              <XMTok meaning="plus" role="ADDOP">+</XMTok>
+              <XMTok>+</XMTok>
               <XMApp>
-                <XMTok role="SUPERSCRIPTOP" scriptpos="post7"/>
-                <XMTok role="UNKNOWN" font="italic">t</XMTok>
-                <XMTok meaning="2" role="NUMBER">2</XMTok>
+                <XMTok/>
+                <XMTok>t</XMTok>
+                <XMTok>2</XMTok>
               </XMApp>
-              <XMTok role="UNKNOWN" font="italic">t</XMTok>
+              <XMTok>t</XMTok>
             </XMApp>
-            <XMTok meaning="1" role="NUMBER">1</XMTok>
+            <XMTok>1</XMTok>
           </XMApp>
         </XMApp>
       </XMath>
     </annotation-xml>
     <annotation encoding="application/x-tex">\displaystyle m=t^{2}+t\pm 1</annotation>
   </semantics>
 </math>

Formula 91

( show / hide formula code)

Original

: normal-: : \displaystyle:

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="S0.Ex3.m3" class="ltx_Math" alttext="\displaystyle:" display="inline">
  <semantics>
    <mo>:</mo>
    <annotation-xml encoding="MathML-Content">
      <ci>normal-:</ci>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMTok name="colon" role="METARELOP">:</XMTok>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">\displaystyle:</annotation>
  </semantics>
</math>

Canonicalized

/ normal-: : \displaystyle:

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics encoding="">
    <mo>/</mo>
    <annotation-xml encoding="MathML-Content">
      <ci encoding="">normal-:</ci>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMTok>:</XMTok>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">\displaystyle:</annotation>
  </semantics>
</math>

Diff

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@@ -1,15 +1,15 @@
 <?xml version="1.0" encoding="UTF-8"?>
-<math xmlns="http://www.w3.org/1998/Math/MathML" id="S0.Ex3.m3" class="ltx_Math" alttext="\displaystyle:" display="inline">
-  <semantics>
-    <mo>:</mo>
+<math xmlns="http://www.w3.org/1998/Math/MathML">
+  <semantics encoding="">
+    <mo>/</mo>
     <annotation-xml encoding="MathML-Content">
-      <ci>normal-:</ci>
+      <ci encoding="">normal-:</ci>
     </annotation-xml>
     <annotation-xml encoding="application/x-latexml">
       <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
-        <XMTok name="colon" role="METARELOP">:</XMTok>
+        <XMTok>:</XMTok>
       </XMath>
     </annotation-xml>
     <annotation encoding="application/x-tex">\displaystyle:</annotation>
   </semantics>
 </math>

Formula 92

( show / hide formula code)

Original

  ⁢ ϵ m = 2 ⁢ t + 1 + 4 ⁢ m + 1 2 normal-  subscript ϵ m 2 t 1 4 m 1 2 = ⁢   ϵ m + ⁢ 2 t 1 + ⁢ 4 m 1 2 \displaystyle\quad\epsilon_{m}=\frac{2t+1+\sqrt{4m+1}}{2}

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="S0.Ex3.m4" class="ltx_Math" alttext="\displaystyle\quad\epsilon_{m}=\frac{2t+1+\sqrt{4m+1}}{2}" display="inline">
  <semantics>
    <mrow>
      <mrow>
        <mi mathvariant="normal"> </mi>
        <mo>⁢</mo>
        <msub>
          <mi>ϵ</mi>
          <mi>m</mi>
        </msub>
      </mrow>
      <mo>=</mo>
      <mstyle displaystyle="true">
        <mfrac>
          <mrow>
            <mrow>
              <mn>2</mn>
              <mo>⁢</mo>
              <mi>t</mi>
            </mrow>
            <mo>+</mo>
            <mn>1</mn>
            <mo>+</mo>
            <msqrt>
              <mrow>
                <mrow>
                  <mn>4</mn>
                  <mo>⁢</mo>
                  <mi>m</mi>
                </mrow>
                <mo>+</mo>
                <mn>1</mn>
              </mrow>
            </msqrt>
          </mrow>
          <mn>2</mn>
        </mfrac>
      </mstyle>
    </mrow>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <eq/>
        <apply>
          <times/>
          <ci>normal- </ci>
          <apply>
            <csymbol cd="ambiguous">subscript</csymbol>
            <ci>ϵ</ci>
            <ci>m</ci>
          </apply>
        </apply>
        <apply>
          <divide/>
          <apply>
            <plus/>
            <apply>
              <times/>
              <cn type="integer">2</cn>
              <ci>t</ci>
            </apply>
            <cn type="integer">1</cn>
            <apply>
              <root/>
              <apply>
                <plus/>
                <apply>
                  <times/>
                  <cn type="integer">4</cn>
                  <ci>m</ci>
                </apply>
                <cn type="integer">1</cn>
              </apply>
            </apply>
          </apply>
          <cn type="integer">2</cn>
        </apply>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok meaning="equals" role="RELOP">=</XMTok>
          <XMApp>
            <XMTok meaning="times" role="MULOP">⁢</XMTok>
            <XMTok role="UNKNOWN"> </XMTok>
            <XMApp>
              <XMTok role="SUBSCRIPTOP" scriptpos="post7"/>
              <XMTok name="epsilon" role="UNKNOWN" font="italic">ϵ</XMTok>
              <XMTok role="UNKNOWN" font="italic">m</XMTok>
            </XMApp>
          </XMApp>
          <XMApp>
            <XMTok mathstyle="display" meaning="divide" role="MULOP"/>
            <XMApp>
              <XMTok meaning="plus" role="ADDOP">+</XMTok>
              <XMApp>
                <XMTok meaning="times" role="MULOP">⁢</XMTok>
                <XMTok meaning="2" role="NUMBER">2</XMTok>
                <XMTok role="UNKNOWN" font="italic">t</XMTok>
              </XMApp>
              <XMTok meaning="1" role="NUMBER">1</XMTok>
              <XMApp>
                <XMTok meaning="square-root"/>
                <XMApp>
                  <XMTok meaning="plus" role="ADDOP">+</XMTok>
                  <XMApp>
                    <XMTok meaning="times" role="MULOP">⁢</XMTok>
                    <XMTok meaning="4" role="NUMBER">4</XMTok>
                    <XMTok role="UNKNOWN" font="italic">m</XMTok>
                  </XMApp>
                  <XMTok meaning="1" role="NUMBER">1</XMTok>
                </XMApp>
              </XMApp>
            </XMApp>
            <XMTok meaning="2" role="NUMBER">2</XMTok>
          </XMApp>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">\displaystyle\quad\epsilon_{m}=\frac{2t+1+\sqrt{4m+1}}{2}</annotation>
  </semantics>
</math>

Canonicalized

ε m = 2 t + 1 + 4 m + 1 2 normal- subscript ε m 2 t 1 4 m 1 2 = ⁢ ε m + ⁢ 2 t 1 + ⁢ 4 m 1 2 \displaystyle\quad\epsilon_{m}=\frac{2t+1+\sqrt{4m+1}}{2}

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics encoding="">
    <mrow>
      <mrow>
        <mi mathvariant="normal"> </mi>
        <msub>
          <mi>ε</mi>
          <mi>m</mi>
        </msub>
      </mrow>
      <mo>=</mo>
      <mfrac>
        <mrow>
          <mrow>
            <mn>2</mn>
            <mi>t</mi>
          </mrow>
          <mo>+</mo>
          <mn>1</mn>
          <mo>+</mo>
          <msqrt>
            <mrow>
              <mn>4</mn>
              <mi>m</mi>
            </mrow>
            <mo>+</mo>
            <mn>1</mn>
          </msqrt>
        </mrow>
        <mn>2</mn>
      </mfrac>
    </mrow>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <eq/>
        <apply>
          <times/>
          <ci encoding="">normal- </ci>
          <apply>
            <csymbol encoding="">subscript</csymbol>
            <ci encoding="">ε</ci>
            <ci encoding="">m</ci>
          </apply>
        </apply>
        <apply>
          <divide/>
          <apply>
            <plus/>
            <apply>
              <times/>
              <cn encoding="" base="10" type="integer">2</cn>
              <ci encoding="">t</ci>
            </apply>
            <cn encoding="" base="10" type="integer">1</cn>
            <apply>
              <root/>
              <apply>
                <plus/>
                <apply>
                  <times/>
                  <cn encoding="" base="10" type="integer">4</cn>
                  <ci encoding="">m</ci>
                </apply>
                <cn encoding="" base="10" type="integer">1</cn>
              </apply>
            </apply>
          </apply>
          <cn encoding="" base="10" type="integer">2</cn>
        </apply>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok>=</XMTok>
          <XMApp>
            <XMTok>⁢</XMTok>
            <XMTok> </XMTok>
            <XMApp>
              <XMTok/>
              <XMTok>ε</XMTok>
              <XMTok>m</XMTok>
            </XMApp>
          </XMApp>
          <XMApp>
            <XMTok/>
            <XMApp>
              <XMTok>+</XMTok>
              <XMApp>
                <XMTok>⁢</XMTok>
                <XMTok>2</XMTok>
                <XMTok>t</XMTok>
              </XMApp>
              <XMTok>1</XMTok>
              <XMApp>
                <XMTok/>
                <XMApp>
                  <XMTok>+</XMTok>
                  <XMApp>
                    <XMTok>⁢</XMTok>
                    <XMTok>4</XMTok>
                    <XMTok>m</XMTok>
                  </XMApp>
                  <XMTok>1</XMTok>
                </XMApp>
              </XMApp>
            </XMApp>
            <XMTok>2</XMTok>
          </XMApp>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">\displaystyle\quad\epsilon_{m}=\frac{2t+1+\sqrt{4m+1}}{2}</annotation>
  </semantics>
</math>

Diff

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@@ -1,123 +1,116 @@
 <?xml version="1.0" encoding="UTF-8"?>
-<math xmlns="http://www.w3.org/1998/Math/MathML" id="S0.Ex3.m4" class="ltx_Math" alttext="\displaystyle\quad\epsilon_{m}=\frac{2t+1+\sqrt{4m+1}}{2}" display="inline">
-  <semantics>
+<math xmlns="http://www.w3.org/1998/Math/MathML">
+  <semantics encoding="">
     <mrow>
       <mrow>
-        <mi mathvariant="normal"> </mi>
-        <mo>⁢</mo>
+        <mi mathvariant="normal"> </mi>
         <msub>
-          <mi>ϵ</mi>
+          <mi>ε</mi>
           <mi>m</mi>
         </msub>
       </mrow>
       <mo>=</mo>
-      <mstyle displaystyle="true">
-        <mfrac>
+      <mfrac>
+        <mrow>
           <mrow>
+            <mn>2</mn>
+            <mi>t</mi>
+          </mrow>
+          <mo>+</mo>
+          <mn>1</mn>
+          <mo>+</mo>
+          <msqrt>
             <mrow>
-              <mn>2</mn>
-              <mo>⁢</mo>
-              <mi>t</mi>
+              <mn>4</mn>
+              <mi>m</mi>
             </mrow>
             <mo>+</mo>
             <mn>1</mn>
-            <mo>+</mo>
-            <msqrt>
-              <mrow>
-                <mrow>
-                  <mn>4</mn>
-                  <mo>⁢</mo>
-                  <mi>m</mi>
-                </mrow>
-                <mo>+</mo>
-                <mn>1</mn>
-              </mrow>
-            </msqrt>
-          </mrow>
-          <mn>2</mn>
-        </mfrac>
-      </mstyle>
+          </msqrt>
+        </mrow>
+        <mn>2</mn>
+      </mfrac>
     </mrow>
     <annotation-xml encoding="MathML-Content">
       <apply>
         <eq/>
         <apply>
           <times/>
-          <ci>normal- </ci>
+          <ci encoding="">normal- </ci>
           <apply>
-            <csymbol cd="ambiguous">subscript</csymbol>
-            <ci>ϵ</ci>
-            <ci>m</ci>
+            <csymbol encoding="">subscript</csymbol>
+            <ci encoding="">ε</ci>
+            <ci encoding="">m</ci>
           </apply>
         </apply>
         <apply>
           <divide/>
           <apply>
             <plus/>
             <apply>
               <times/>
-              <cn type="integer">2</cn>
-              <ci>t</ci>
+              <cn encoding="" base="10" type="integer">2</cn>
+              <ci encoding="">t</ci>
             </apply>
-            <cn type="integer">1</cn>
+            <cn encoding="" base="10" type="integer">1</cn>
             <apply>
               <root/>
               <apply>
                 <plus/>
                 <apply>
                   <times/>
-                  <cn type="integer">4</cn>
-                  <ci>m</ci>
+                  <cn encoding="" base="10" type="integer">4</cn>
+                  <ci encoding="">m</ci>
                 </apply>
-                <cn type="integer">1</cn>
+                <cn encoding="" base="10" type="integer">1</cn>
               </apply>
             </apply>
           </apply>
-          <cn type="integer">2</cn>
+          <cn encoding="" base="10" type="integer">2</cn>
         </apply>
       </apply>
     </annotation-xml>
     <annotation-xml encoding="application/x-latexml">
       <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
         <XMApp>
-          <XMTok meaning="equals" role="RELOP">=</XMTok>
+          <XMTok>=</XMTok>
           <XMApp>
-            <XMTok meaning="times" role="MULOP">⁢</XMTok>
-            <XMTok role="UNKNOWN"> </XMTok>
+            <XMTok>⁢</XMTok>
+            <XMTok> </XMTok>
             <XMApp>
-              <XMTok role="SUBSCRIPTOP" scriptpos="post7"/>
-              <XMTok name="epsilon" role="UNKNOWN" font="italic">ϵ</XMTok>
-              <XMTok role="UNKNOWN" font="italic">m</XMTok>
+              <XMTok/>
+              <XMTok>ε</XMTok>
+              <XMTok>m</XMTok>
             </XMApp>
           </XMApp>
           <XMApp>
-            <XMTok mathstyle="display" meaning="divide" role="MULOP"/>
+            <XMTok/>
             <XMApp>
-              <XMTok meaning="plus" role="ADDOP">+</XMTok>
+              <XMTok>+</XMTok>
               <XMApp>
-                <XMTok meaning="times" role="MULOP">⁢</XMTok>
-                <XMTok meaning="2" role="NUMBER">2</XMTok>
-                <XMTok role="UNKNOWN" font="italic">t</XMTok>
+                <XMTok>⁢</XMTok>
+                <XMTok>2</XMTok>
+                <XMTok>t</XMTok>
               </XMApp>
-              <XMTok meaning="1" role="NUMBER">1</XMTok>
+              <XMTok>1</XMTok>
               <XMApp>
-                <XMTok meaning="square-root"/>
+                <XMTok/>
                 <XMApp>
-                  <XMTok meaning="plus" role="ADDOP">+</XMTok>
+                  <XMTok>+</XMTok>
                   <XMApp>
-                    <XMTok meaning="times" role="MULOP">⁢</XMTok>
-                    <XMTok meaning="4" role="NUMBER">4</XMTok>
-                    <XMTok role="UNKNOWN" font="italic">m</XMTok>
+                    <XMTok>⁢</XMTok>
+                    <XMTok>4</XMTok>
+                    <XMTok>m</XMTok>
                   </XMApp>
-                  <XMTok meaning="1" role="NUMBER">1</XMTok>
+                  <XMTok>1</XMTok>
                 </XMApp>
               </XMApp>
             </XMApp>
-            <XMTok meaning="2" role="NUMBER">2</XMTok>
+            <XMTok>2</XMTok>
           </XMApp>
         </XMApp>
       </XMath>
     </annotation-xml>
     <annotation encoding="application/x-tex">\displaystyle\quad\epsilon_{m}=\frac{2t+1+\sqrt{4m+1}}{2}</annotation>
   </semantics>
 </math>

Formula 93

( show / hide formula code)

Original

m = t 2 + t ± 2 ⁢ t + 1 3 m plus-or-minus superscript t 2 t 2 t 1 3 = m ± + t 2 t + ⁢ 2 t 1 3 \displaystyle m=t^{2}+t\pm\frac{2t+1}{3}

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="S0.Ex4.m2" class="ltx_Math" alttext="\displaystyle m=t^{2}+t\pm\frac{2t+1}{3}" display="inline">
  <semantics>
    <mrow>
      <mi>m</mi>
      <mo>=</mo>
      <mrow>
        <msup>
          <mi>t</mi>
          <mn>2</mn>
        </msup>
        <mo>+</mo>
        <mi>t</mi>
        <mo>±</mo>
        <mstyle displaystyle="true">
          <mfrac>
            <mrow>
              <mrow>
                <mn>2</mn>
                <mo>⁢</mo>
                <mi>t</mi>
              </mrow>
              <mo>+</mo>
              <mn>1</mn>
            </mrow>
            <mn>3</mn>
          </mfrac>
        </mstyle>
      </mrow>
    </mrow>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <eq/>
        <ci>m</ci>
        <apply>
          <csymbol cd="latexml">plus-or-minus</csymbol>
          <apply>
            <plus/>
            <apply>
              <csymbol cd="ambiguous">superscript</csymbol>
              <ci>t</ci>
              <cn type="integer">2</cn>
            </apply>
            <ci>t</ci>
          </apply>
          <apply>
            <divide/>
            <apply>
              <plus/>
              <apply>
                <times/>
                <cn type="integer">2</cn>
                <ci>t</ci>
              </apply>
              <cn type="integer">1</cn>
            </apply>
            <cn type="integer">3</cn>
          </apply>
        </apply>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok meaning="equals" role="RELOP">=</XMTok>
          <XMTok role="UNKNOWN" font="italic">m</XMTok>
          <XMApp>
            <XMTok meaning="plus-or-minus" name="pm" role="ADDOP">±</XMTok>
            <XMApp>
              <XMTok meaning="plus" role="ADDOP">+</XMTok>
              <XMApp>
                <XMTok role="SUPERSCRIPTOP" scriptpos="post7"/>
                <XMTok role="UNKNOWN" font="italic">t</XMTok>
                <XMTok meaning="2" role="NUMBER">2</XMTok>
              </XMApp>
              <XMTok role="UNKNOWN" font="italic">t</XMTok>
            </XMApp>
            <XMApp>
              <XMTok mathstyle="display" meaning="divide" role="MULOP"/>
              <XMApp>
                <XMTok meaning="plus" role="ADDOP">+</XMTok>
                <XMApp>
                  <XMTok meaning="times" role="MULOP">⁢</XMTok>
                  <XMTok meaning="2" role="NUMBER">2</XMTok>
                  <XMTok role="UNKNOWN" font="italic">t</XMTok>
                </XMApp>
                <XMTok meaning="1" role="NUMBER">1</XMTok>
              </XMApp>
              <XMTok meaning="3" role="NUMBER">3</XMTok>
            </XMApp>
          </XMApp>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">\displaystyle m=t^{2}+t\pm\frac{2t+1}{3}</annotation>
  </semantics>
</math>

Canonicalized

m = t 2 + t ± 2 t + 1 3 m plus-or-minus superscript t 2 t 2 t 1 3 = m ± + t 2 t + ⁢ 2 t 1 3 \displaystyle m=t^{2}+t\pm\frac{2t+1}{3}

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics encoding="">
    <mrow>
      <mi>m</mi>
      <mo>=</mo>
      <mrow>
        <msup>
          <mi>t</mi>
          <mn>2</mn>
        </msup>
        <mo>+</mo>
        <mi>t</mi>
        <mo>±</mo>
        <mfrac>
          <mrow>
            <mrow>
              <mn>2</mn>
              <mi>t</mi>
            </mrow>
            <mo>+</mo>
            <mn>1</mn>
          </mrow>
          <mn>3</mn>
        </mfrac>
      </mrow>
    </mrow>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <eq/>
        <ci encoding="">m</ci>
        <apply>
          <csymbol encoding="">plus-or-minus</csymbol>
          <apply>
            <plus/>
            <apply>
              <csymbol encoding="">superscript</csymbol>
              <ci encoding="">t</ci>
              <cn encoding="" base="10" type="integer">2</cn>
            </apply>
            <ci encoding="">t</ci>
          </apply>
          <apply>
            <divide/>
            <apply>
              <plus/>
              <apply>
                <times/>
                <cn encoding="" base="10" type="integer">2</cn>
                <ci encoding="">t</ci>
              </apply>
              <cn encoding="" base="10" type="integer">1</cn>
            </apply>
            <cn encoding="" base="10" type="integer">3</cn>
          </apply>
        </apply>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok>=</XMTok>
          <XMTok>m</XMTok>
          <XMApp>
            <XMTok>±</XMTok>
            <XMApp>
              <XMTok>+</XMTok>
              <XMApp>
                <XMTok/>
                <XMTok>t</XMTok>
                <XMTok>2</XMTok>
              </XMApp>
              <XMTok>t</XMTok>
            </XMApp>
            <XMApp>
              <XMTok/>
              <XMApp>
                <XMTok>+</XMTok>
                <XMApp>
                  <XMTok>⁢</XMTok>
                  <XMTok>2</XMTok>
                  <XMTok>t</XMTok>
                </XMApp>
                <XMTok>1</XMTok>
              </XMApp>
              <XMTok>3</XMTok>
            </XMApp>
          </XMApp>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">\displaystyle m=t^{2}+t\pm\frac{2t+1}{3}</annotation>
  </semantics>
</math>

Diff

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@@ -1,97 +1,94 @@
 <?xml version="1.0" encoding="UTF-8"?>
-<math xmlns="http://www.w3.org/1998/Math/MathML" id="S0.Ex4.m2" class="ltx_Math" alttext="\displaystyle m=t^{2}+t\pm\frac{2t+1}{3}" display="inline">
-  <semantics>
+<math xmlns="http://www.w3.org/1998/Math/MathML">
+  <semantics encoding="">
     <mrow>
       <mi>m</mi>
       <mo>=</mo>
       <mrow>
         <msup>
           <mi>t</mi>
           <mn>2</mn>
         </msup>
         <mo>+</mo>
         <mi>t</mi>
         <mo>±</mo>
-        <mstyle displaystyle="true">
-          <mfrac>
+        <mfrac>
+          <mrow>
             <mrow>
-              <mrow>
-                <mn>2</mn>
-                <mo>⁢</mo>
-                <mi>t</mi>
-              </mrow>
-              <mo>+</mo>
-              <mn>1</mn>
+              <mn>2</mn>
+              <mi>t</mi>
             </mrow>
-            <mn>3</mn>
-          </mfrac>
-        </mstyle>
+            <mo>+</mo>
+            <mn>1</mn>
+          </mrow>
+          <mn>3</mn>
+        </mfrac>
       </mrow>
     </mrow>
     <annotation-xml encoding="MathML-Content">
       <apply>
         <eq/>
-        <ci>m</ci>
+        <ci encoding="">m</ci>
         <apply>
-          <csymbol cd="latexml">plus-or-minus</csymbol>
+          <csymbol encoding="">plus-or-minus</csymbol>
           <apply>
             <plus/>
             <apply>
-              <csymbol cd="ambiguous">superscript</csymbol>
-              <ci>t</ci>
-              <cn type="integer">2</cn>
+              <csymbol encoding="">superscript</csymbol>
+              <ci encoding="">t</ci>
+              <cn encoding="" base="10" type="integer">2</cn>
             </apply>
-            <ci>t</ci>
+            <ci encoding="">t</ci>
           </apply>
           <apply>
             <divide/>
             <apply>
               <plus/>
               <apply>
                 <times/>
-                <cn type="integer">2</cn>
-                <ci>t</ci>
+                <cn encoding="" base="10" type="integer">2</cn>
+                <ci encoding="">t</ci>
               </apply>
-              <cn type="integer">1</cn>
+              <cn encoding="" base="10" type="integer">1</cn>
             </apply>
-            <cn type="integer">3</cn>
+            <cn encoding="" base="10" type="integer">3</cn>
           </apply>
         </apply>
       </apply>
     </annotation-xml>
     <annotation-xml encoding="application/x-latexml">
       <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
         <XMApp>
-          <XMTok meaning="equals" role="RELOP">=</XMTok>
-          <XMTok role="UNKNOWN" font="italic">m</XMTok>
+          <XMTok>=</XMTok>
+          <XMTok>m</XMTok>
           <XMApp>
-            <XMTok meaning="plus-or-minus" name="pm" role="ADDOP">±</XMTok>
+            <XMTok>±</XMTok>
             <XMApp>
-              <XMTok meaning="plus" role="ADDOP">+</XMTok>
+              <XMTok>+</XMTok>
               <XMApp>
-                <XMTok role="SUPERSCRIPTOP" scriptpos="post7"/>
-                <XMTok role="UNKNOWN" font="italic">t</XMTok>
-                <XMTok meaning="2" role="NUMBER">2</XMTok>
+                <XMTok/>
+                <XMTok>t</XMTok>
+                <XMTok>2</XMTok>
               </XMApp>
-              <XMTok role="UNKNOWN" font="italic">t</XMTok>
+              <XMTok>t</XMTok>
             </XMApp>
             <XMApp>
-              <XMTok mathstyle="display" meaning="divide" role="MULOP"/>
+              <XMTok/>
               <XMApp>
-                <XMTok meaning="plus" role="ADDOP">+</XMTok>
+                <XMTok>+</XMTok>
                 <XMApp>
-                  <XMTok meaning="times" role="MULOP">⁢</XMTok>
-                  <XMTok meaning="2" role="NUMBER">2</XMTok>
-                  <XMTok role="UNKNOWN" font="italic">t</XMTok>
+                  <XMTok>⁢</XMTok>
+                  <XMTok>2</XMTok>
+                  <XMTok>t</XMTok>
                 </XMApp>
-                <XMTok meaning="1" role="NUMBER">1</XMTok>
+                <XMTok>1</XMTok>
               </XMApp>
-              <XMTok meaning="3" role="NUMBER">3</XMTok>
+              <XMTok>3</XMTok>
             </XMApp>
           </XMApp>
         </XMApp>
       </XMath>
     </annotation-xml>
     <annotation encoding="application/x-tex">\displaystyle m=t^{2}+t\pm\frac{2t+1}{3}</annotation>
   </semantics>
 </math>

Formula 94

( show / hide formula code)

Original

: normal-: : \displaystyle:

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="S0.Ex4.m3" class="ltx_Math" alttext="\displaystyle:" display="inline">
  <semantics>
    <mo>:</mo>
    <annotation-xml encoding="MathML-Content">
      <ci>normal-:</ci>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMTok name="colon" role="METARELOP">:</XMTok>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">\displaystyle:</annotation>
  </semantics>
</math>

Canonicalized

/ normal-: : \displaystyle:

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics encoding="">
    <mo>/</mo>
    <annotation-xml encoding="MathML-Content">
      <ci encoding="">normal-:</ci>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMTok>:</XMTok>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">\displaystyle:</annotation>
  </semantics>
</math>

Diff

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@@ -1,15 +1,15 @@
 <?xml version="1.0" encoding="UTF-8"?>
-<math xmlns="http://www.w3.org/1998/Math/MathML" id="S0.Ex4.m3" class="ltx_Math" alttext="\displaystyle:" display="inline">
-  <semantics>
-    <mo>:</mo>
+<math xmlns="http://www.w3.org/1998/Math/MathML">
+  <semantics encoding="">
+    <mo>/</mo>
     <annotation-xml encoding="MathML-Content">
-      <ci>normal-:</ci>
+      <ci encoding="">normal-:</ci>
     </annotation-xml>
     <annotation-xml encoding="application/x-latexml">
       <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
-        <XMTok name="colon" role="METARELOP">:</XMTok>
+        <XMTok>:</XMTok>
       </XMath>
     </annotation-xml>
     <annotation encoding="application/x-tex">\displaystyle:</annotation>
   </semantics>
 </math>

Formula 95

( show / hide formula code)

Original

  ⁢ ϵ m = 6 ⁢ t + 3 ± 2 + 3 ⁢ 4 ⁢ m + 1 2 normal-  subscript ϵ m plus-or-minus 6 t 3 2 3 4 m 1 2 = ⁢   ϵ m + ± + ⁢ 6 t 3 2 ⁢ 3 + ⁢ 4 m 1 2 \displaystyle\quad\epsilon_{m}=\frac{6t+3\pm 2+3\sqrt{4m+1}}{2}

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="S0.Ex4.m4" class="ltx_Math" alttext="\displaystyle\quad\epsilon_{m}=\frac{6t+3\pm 2+3\sqrt{4m+1}}{2}" display="inline">
  <semantics>
    <mrow>
      <mrow>
        <mi mathvariant="normal"> </mi>
        <mo>⁢</mo>
        <msub>
          <mi>ϵ</mi>
          <mi>m</mi>
        </msub>
      </mrow>
      <mo>=</mo>
      <mstyle displaystyle="true">
        <mfrac>
          <mrow>
            <mrow>
              <mn>6</mn>
              <mo>⁢</mo>
              <mi>t</mi>
            </mrow>
            <mo>+</mo>
            <mn>3</mn>
            <mo>±</mo>
            <mn>2</mn>
            <mo>+</mo>
            <mrow>
              <mn>3</mn>
              <mo>⁢</mo>
              <msqrt>
                <mrow>
                  <mrow>
                    <mn>4</mn>
                    <mo>⁢</mo>
                    <mi>m</mi>
                  </mrow>
                  <mo>+</mo>
                  <mn>1</mn>
                </mrow>
              </msqrt>
            </mrow>
          </mrow>
          <mn>2</mn>
        </mfrac>
      </mstyle>
    </mrow>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <eq/>
        <apply>
          <times/>
          <ci>normal- </ci>
          <apply>
            <csymbol cd="ambiguous">subscript</csymbol>
            <ci>ϵ</ci>
            <ci>m</ci>
          </apply>
        </apply>
        <apply>
          <divide/>
          <apply>
            <plus/>
            <apply>
              <csymbol cd="latexml">plus-or-minus</csymbol>
              <apply>
                <plus/>
                <apply>
                  <times/>
                  <cn type="integer">6</cn>
                  <ci>t</ci>
                </apply>
                <cn type="integer">3</cn>
              </apply>
              <cn type="integer">2</cn>
            </apply>
            <apply>
              <times/>
              <cn type="integer">3</cn>
              <apply>
                <root/>
                <apply>
                  <plus/>
                  <apply>
                    <times/>
                    <cn type="integer">4</cn>
                    <ci>m</ci>
                  </apply>
                  <cn type="integer">1</cn>
                </apply>
              </apply>
            </apply>
          </apply>
          <cn type="integer">2</cn>
        </apply>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok meaning="equals" role="RELOP">=</XMTok>
          <XMApp>
            <XMTok meaning="times" role="MULOP">⁢</XMTok>
            <XMTok role="UNKNOWN"> </XMTok>
            <XMApp>
              <XMTok role="SUBSCRIPTOP" scriptpos="post7"/>
              <XMTok name="epsilon" role="UNKNOWN" font="italic">ϵ</XMTok>
              <XMTok role="UNKNOWN" font="italic">m</XMTok>
            </XMApp>
          </XMApp>
          <XMApp>
            <XMTok mathstyle="display" meaning="divide" role="MULOP"/>
            <XMApp>
              <XMTok meaning="plus" role="ADDOP">+</XMTok>
              <XMApp>
                <XMTok meaning="plus-or-minus" name="pm" role="ADDOP">±</XMTok>
                <XMApp>
                  <XMTok meaning="plus" role="ADDOP">+</XMTok>
                  <XMApp>
                    <XMTok meaning="times" role="MULOP">⁢</XMTok>
                    <XMTok meaning="6" role="NUMBER">6</XMTok>
                    <XMTok role="UNKNOWN" font="italic">t</XMTok>
                  </XMApp>
                  <XMTok meaning="3" role="NUMBER">3</XMTok>
                </XMApp>
                <XMTok meaning="2" role="NUMBER">2</XMTok>
              </XMApp>
              <XMApp>
                <XMTok meaning="times" role="MULOP">⁢</XMTok>
                <XMTok meaning="3" role="NUMBER">3</XMTok>
                <XMApp>
                  <XMTok meaning="square-root"/>
                  <XMApp>
                    <XMTok meaning="plus" role="ADDOP">+</XMTok>
                    <XMApp>
                      <XMTok meaning="times" role="MULOP">⁢</XMTok>
                      <XMTok meaning="4" role="NUMBER">4</XMTok>
                      <XMTok role="UNKNOWN" font="italic">m</XMTok>
                    </XMApp>
                    <XMTok meaning="1" role="NUMBER">1</XMTok>
                  </XMApp>
                </XMApp>
              </XMApp>
            </XMApp>
            <XMTok meaning="2" role="NUMBER">2</XMTok>
          </XMApp>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">\displaystyle\quad\epsilon_{m}=\frac{6t+3\pm 2+3\sqrt{4m+1}}{2}</annotation>
  </semantics>
</math>

Canonicalized

ε m = 6 t + 3 ± 2 + 3 4 m + 1 2 normal- subscript ε m plus-or-minus 6 t 3 2 3 4 m 1 2 = ⁢ ε m + ± + ⁢ 6 t 3 2 ⁢ 3 + ⁢ 4 m 1 2 \displaystyle\quad\epsilon_{m}=\frac{6t+3\pm 2+3\sqrt{4m+1}}{2}

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics encoding="">
    <mrow>
      <mrow>
        <mi mathvariant="normal"> </mi>
        <msub>
          <mi>ε</mi>
          <mi>m</mi>
        </msub>
      </mrow>
      <mo>=</mo>
      <mfrac>
        <mrow>
          <mrow>
            <mn>6</mn>
            <mi>t</mi>
          </mrow>
          <mo>+</mo>
          <mn>3</mn>
          <mo>±</mo>
          <mn>2</mn>
          <mo>+</mo>
          <mrow>
            <mn>3</mn>
            <msqrt>
              <mrow>
                <mn>4</mn>
                <mi>m</mi>
              </mrow>
              <mo>+</mo>
              <mn>1</mn>
            </msqrt>
          </mrow>
        </mrow>
        <mn>2</mn>
      </mfrac>
    </mrow>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <eq/>
        <apply>
          <times/>
          <ci encoding="">normal- </ci>
          <apply>
            <csymbol encoding="">subscript</csymbol>
            <ci encoding="">ε</ci>
            <ci encoding="">m</ci>
          </apply>
        </apply>
        <apply>
          <divide/>
          <apply>
            <plus/>
            <apply>
              <csymbol encoding="">plus-or-minus</csymbol>
              <apply>
                <plus/>
                <apply>
                  <times/>
                  <cn encoding="" base="10" type="integer">6</cn>
                  <ci encoding="">t</ci>
                </apply>
                <cn encoding="" base="10" type="integer">3</cn>
              </apply>
              <cn encoding="" base="10" type="integer">2</cn>
            </apply>
            <apply>
              <times/>
              <cn encoding="" base="10" type="integer">3</cn>
              <apply>
                <root/>
                <apply>
                  <plus/>
                  <apply>
                    <times/>
                    <cn encoding="" base="10" type="integer">4</cn>
                    <ci encoding="">m</ci>
                  </apply>
                  <cn encoding="" base="10" type="integer">1</cn>
                </apply>
              </apply>
            </apply>
          </apply>
          <cn encoding="" base="10" type="integer">2</cn>
        </apply>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok>=</XMTok>
          <XMApp>
            <XMTok>⁢</XMTok>
            <XMTok> </XMTok>
            <XMApp>
              <XMTok/>
              <XMTok>ε</XMTok>
              <XMTok>m</XMTok>
            </XMApp>
          </XMApp>
          <XMApp>
            <XMTok/>
            <XMApp>
              <XMTok>+</XMTok>
              <XMApp>
                <XMTok>±</XMTok>
                <XMApp>
                  <XMTok>+</XMTok>
                  <XMApp>
                    <XMTok>⁢</XMTok>
                    <XMTok>6</XMTok>
                    <XMTok>t</XMTok>
                  </XMApp>
                  <XMTok>3</XMTok>
                </XMApp>
                <XMTok>2</XMTok>
              </XMApp>
              <XMApp>
                <XMTok>⁢</XMTok>
                <XMTok>3</XMTok>
                <XMApp>
                  <XMTok/>
                  <XMApp>
                    <XMTok>+</XMTok>
                    <XMApp>
                      <XMTok>⁢</XMTok>
                      <XMTok>4</XMTok>
                      <XMTok>m</XMTok>
                    </XMApp>
                    <XMTok>1</XMTok>
                  </XMApp>
                </XMApp>
              </XMApp>
            </XMApp>
            <XMTok>2</XMTok>
          </XMApp>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">\displaystyle\quad\epsilon_{m}=\frac{6t+3\pm 2+3\sqrt{4m+1}}{2}</annotation>
  </semantics>
</math>

Diff

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@@ -1,151 +1,143 @@
 <?xml version="1.0" encoding="UTF-8"?>
-<math xmlns="http://www.w3.org/1998/Math/MathML" id="S0.Ex4.m4" class="ltx_Math" alttext="\displaystyle\quad\epsilon_{m}=\frac{6t+3\pm 2+3\sqrt{4m+1}}{2}" display="inline">
-  <semantics>
+<math xmlns="http://www.w3.org/1998/Math/MathML">
+  <semantics encoding="">
     <mrow>
       <mrow>
-        <mi mathvariant="normal"> </mi>
-        <mo>⁢</mo>
+        <mi mathvariant="normal"> </mi>
         <msub>
-          <mi>ϵ</mi>
+          <mi>ε</mi>
           <mi>m</mi>
         </msub>
       </mrow>
       <mo>=</mo>
-      <mstyle displaystyle="true">
-        <mfrac>
+      <mfrac>
+        <mrow>
           <mrow>
-            <mrow>
-              <mn>6</mn>
-              <mo>⁢</mo>
-              <mi>t</mi>
-            </mrow>
-            <mo>+</mo>
-            <mn>3</mn>
-            <mo>±</mo>
-            <mn>2</mn>
-            <mo>+</mo>
-            <mrow>
-              <mn>3</mn>
-              <mo>⁢</mo>
-              <msqrt>
-                <mrow>
-                  <mrow>
-                    <mn>4</mn>
-                    <mo>⁢</mo>
-                    <mi>m</mi>
-                  </mrow>
-                  <mo>+</mo>
-                  <mn>1</mn>
-                </mrow>
-              </msqrt>
-            </mrow>
+            <mn>6</mn>
+            <mi>t</mi>
           </mrow>
+          <mo>+</mo>
+          <mn>3</mn>
+          <mo>±</mo>
           <mn>2</mn>
-        </mfrac>
-      </mstyle>
+          <mo>+</mo>
+          <mrow>
+            <mn>3</mn>
+            <msqrt>
+              <mrow>
+                <mn>4</mn>
+                <mi>m</mi>
+              </mrow>
+              <mo>+</mo>
+              <mn>1</mn>
+            </msqrt>
+          </mrow>
+        </mrow>
+        <mn>2</mn>
+      </mfrac>
     </mrow>
     <annotation-xml encoding="MathML-Content">
       <apply>
         <eq/>
         <apply>
           <times/>
-          <ci>normal- </ci>
+          <ci encoding="">normal- </ci>
           <apply>
-            <csymbol cd="ambiguous">subscript</csymbol>
-            <ci>ϵ</ci>
-            <ci>m</ci>
+            <csymbol encoding="">subscript</csymbol>
+            <ci encoding="">ε</ci>
+            <ci encoding="">m</ci>
           </apply>
         </apply>
         <apply>
           <divide/>
           <apply>
             <plus/>
             <apply>
-              <csymbol cd="latexml">plus-or-minus</csymbol>
+              <csymbol encoding="">plus-or-minus</csymbol>
               <apply>
                 <plus/>
                 <apply>
                   <times/>
-                  <cn type="integer">6</cn>
-                  <ci>t</ci>
+                  <cn encoding="" base="10" type="integer">6</cn>
+                  <ci encoding="">t</ci>
                 </apply>
-                <cn type="integer">3</cn>
+                <cn encoding="" base="10" type="integer">3</cn>
               </apply>
-              <cn type="integer">2</cn>
+              <cn encoding="" base="10" type="integer">2</cn>
             </apply>
             <apply>
               <times/>
-              <cn type="integer">3</cn>
+              <cn encoding="" base="10" type="integer">3</cn>
               <apply>
                 <root/>
                 <apply>
                   <plus/>
                   <apply>
                     <times/>
-                    <cn type="integer">4</cn>
-                    <ci>m</ci>
+                    <cn encoding="" base="10" type="integer">4</cn>
+                    <ci encoding="">m</ci>
                   </apply>
-                  <cn type="integer">1</cn>
+                  <cn encoding="" base="10" type="integer">1</cn>
                 </apply>
               </apply>
             </apply>
           </apply>
-          <cn type="integer">2</cn>
+          <cn encoding="" base="10" type="integer">2</cn>
         </apply>
       </apply>
     </annotation-xml>
     <annotation-xml encoding="application/x-latexml">
       <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
         <XMApp>
-          <XMTok meaning="equals" role="RELOP">=</XMTok>
+          <XMTok>=</XMTok>
           <XMApp>
-            <XMTok meaning="times" role="MULOP">⁢</XMTok>
-            <XMTok role="UNKNOWN"> </XMTok>
+            <XMTok>⁢</XMTok>
+            <XMTok> </XMTok>
             <XMApp>
-              <XMTok role="SUBSCRIPTOP" scriptpos="post7"/>
-              <XMTok name="epsilon" role="UNKNOWN" font="italic">ϵ</XMTok>
-              <XMTok role="UNKNOWN" font="italic">m</XMTok>
+              <XMTok/>
+              <XMTok>ε</XMTok>
+              <XMTok>m</XMTok>
             </XMApp>
           </XMApp>
           <XMApp>
-            <XMTok mathstyle="display" meaning="divide" role="MULOP"/>
+            <XMTok/>
             <XMApp>
-              <XMTok meaning="plus" role="ADDOP">+</XMTok>
+              <XMTok>+</XMTok>
               <XMApp>
-                <XMTok meaning="plus-or-minus" name="pm" role="ADDOP">±</XMTok>
+                <XMTok>±</XMTok>
                 <XMApp>
-                  <XMTok meaning="plus" role="ADDOP">+</XMTok>
+                  <XMTok>+</XMTok>
                   <XMApp>
-                    <XMTok meaning="times" role="MULOP">⁢</XMTok>
-                    <XMTok meaning="6" role="NUMBER">6</XMTok>
-                    <XMTok role="UNKNOWN" font="italic">t</XMTok>
+                    <XMTok>⁢</XMTok>
+                    <XMTok>6</XMTok>
+                    <XMTok>t</XMTok>
                   </XMApp>
-                  <XMTok meaning="3" role="NUMBER">3</XMTok>
+                  <XMTok>3</XMTok>
                 </XMApp>
-                <XMTok meaning="2" role="NUMBER">2</XMTok>
+                <XMTok>2</XMTok>
               </XMApp>
               <XMApp>
-                <XMTok meaning="times" role="MULOP">⁢</XMTok>
-                <XMTok meaning="3" role="NUMBER">3</XMTok>
+                <XMTok>⁢</XMTok>
+                <XMTok>3</XMTok>
                 <XMApp>
-                  <XMTok meaning="square-root"/>
+                  <XMTok/>
                   <XMApp>
-                    <XMTok meaning="plus" role="ADDOP">+</XMTok>
+                    <XMTok>+</XMTok>
                     <XMApp>
-                      <XMTok meaning="times" role="MULOP">⁢</XMTok>
-                      <XMTok meaning="4" role="NUMBER">4</XMTok>
-                      <XMTok role="UNKNOWN" font="italic">m</XMTok>
+                      <XMTok>⁢</XMTok>
+                      <XMTok>4</XMTok>
+                      <XMTok>m</XMTok>
                     </XMApp>
-                    <XMTok meaning="1" role="NUMBER">1</XMTok>
+                    <XMTok>1</XMTok>
                   </XMApp>
                 </XMApp>
               </XMApp>
             </XMApp>
-            <XMTok meaning="2" role="NUMBER">2</XMTok>
+            <XMTok>2</XMTok>
           </XMApp>
         </XMApp>
       </XMath>
     </annotation-xml>
     <annotation encoding="application/x-tex">\displaystyle\quad\epsilon_{m}=\frac{6t+3\pm 2+3\sqrt{4m+1}}{2}</annotation>
   </semantics>
 </math>

Formula 96

( show / hide formula code)

Original

ϵ D > 1 subscript ϵ D 1 > ϵ D 1 \epsilon_{D}>1

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="p8.m5" class="ltx_Math" alttext="\epsilon_{D}&gt;1" display="inline">
  <semantics>
    <mrow>
      <msub>
        <mi>ϵ</mi>
        <mi>D</mi>
      </msub>
      <mo>&gt;</mo>
      <mn>1</mn>
    </mrow>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <gt/>
        <apply>
          <csymbol cd="ambiguous">subscript</csymbol>
          <ci>ϵ</ci>
          <ci>D</ci>
        </apply>
        <cn type="integer">1</cn>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok meaning="greater-than" role="RELOP">&gt;</XMTok>
          <XMApp>
            <XMTok role="SUBSCRIPTOP" scriptpos="post2"/>
            <XMTok name="epsilon" role="UNKNOWN" font="italic">ϵ</XMTok>
            <XMTok role="UNKNOWN" font="italic">D</XMTok>
          </XMApp>
          <XMTok meaning="1" role="NUMBER">1</XMTok>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">\epsilon_{D}&gt;1</annotation>
  </semantics>
</math>

Canonicalized

ε D > 1 subscript ε D 1 > ε D 1 \epsilon_{D}>1

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics encoding="">
    <mrow>
      <msub>
        <mi>ε</mi>
        <mi>D</mi>
      </msub>
      <mo>&gt;</mo>
      <mn>1</mn>
    </mrow>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <gt/>
        <apply>
          <csymbol encoding="">subscript</csymbol>
          <ci encoding="">ε</ci>
          <ci encoding="">D</ci>
        </apply>
        <cn encoding="" base="10" type="integer">1</cn>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok>&gt;</XMTok>
          <XMApp>
            <XMTok/>
            <XMTok>ε</XMTok>
            <XMTok>D</XMTok>
          </XMApp>
          <XMTok>1</XMTok>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">\epsilon_{D}&gt;1</annotation>
  </semantics>
</math>

Diff

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@@ -1,38 +1,38 @@
 <?xml version="1.0" encoding="UTF-8"?>
-<math xmlns="http://www.w3.org/1998/Math/MathML" id="p8.m5" class="ltx_Math" alttext="\epsilon_{D}&gt;1" display="inline">
-  <semantics>
+<math xmlns="http://www.w3.org/1998/Math/MathML">
+  <semantics encoding="">
     <mrow>
       <msub>
-        <mi>ϵ</mi>
+        <mi>ε</mi>
         <mi>D</mi>
       </msub>
       <mo>&gt;</mo>
       <mn>1</mn>
     </mrow>
     <annotation-xml encoding="MathML-Content">
       <apply>
         <gt/>
         <apply>
-          <csymbol cd="ambiguous">subscript</csymbol>
-          <ci>ϵ</ci>
-          <ci>D</ci>
+          <csymbol encoding="">subscript</csymbol>
+          <ci encoding="">ε</ci>
+          <ci encoding="">D</ci>
         </apply>
-        <cn type="integer">1</cn>
+        <cn encoding="" base="10" type="integer">1</cn>
       </apply>
     </annotation-xml>
     <annotation-xml encoding="application/x-latexml">
       <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
         <XMApp>
-          <XMTok meaning="greater-than" role="RELOP">&gt;</XMTok>
+          <XMTok>&gt;</XMTok>
           <XMApp>
-            <XMTok role="SUBSCRIPTOP" scriptpos="post2"/>
-            <XMTok name="epsilon" role="UNKNOWN" font="italic">ϵ</XMTok>
-            <XMTok role="UNKNOWN" font="italic">D</XMTok>
+            <XMTok/>
+            <XMTok>ε</XMTok>
+            <XMTok>D</XMTok>
           </XMApp>
-          <XMTok meaning="1" role="NUMBER">1</XMTok>
+          <XMTok>1</XMTok>
         </XMApp>
       </XMath>
     </annotation-xml>
     <annotation encoding="application/x-tex">\epsilon_{D}&gt;1</annotation>
   </semantics>
 </math>

Formula 97

( show / hide formula code)

Original

ℚ ⁢ ( D ) ℚ D ⁢ Q D \mathbb{Q}(\sqrt{D})

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="p8.m6" class="ltx_Math" alttext="\mathbb{Q}(\sqrt{D})" display="inline">
  <semantics>
    <mrow>
      <mi>ℚ</mi>
      <mo>⁢</mo>
      <mrow>
        <mo>(</mo>
        <msqrt>
          <mi>D</mi>
        </msqrt>
        <mo>)</mo>
      </mrow>
    </mrow>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <times/>
        <ci>ℚ</ci>
        <apply>
          <root/>
          <ci>D</ci>
        </apply>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok meaning="times" role="MULOP">⁢</XMTok>
          <XMTok possibleFunction="yes" role="UNKNOWN" font="blackboard">Q</XMTok>
          <XMApp close=")" open="(">
            <XMTok meaning="square-root"/>
            <XMTok role="UNKNOWN" font="italic">D</XMTok>
          </XMApp>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">\mathbb{Q}(\sqrt{D})</annotation>
  </semantics>
</math>

Canonicalized

Q ( D ) Q D ⁢ Q D \mathbb{Q}(\sqrt{D})

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics encoding="">
    <mrow>
      <mi>Q</mi>
      <mrow>
        <mo>(</mo>
        <msqrt>
          <mi>D</mi>
        </msqrt>
        <mo>)</mo>
      </mrow>
    </mrow>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <times/>
        <ci encoding="">Q</ci>
        <apply>
          <root/>
          <ci encoding="">D</ci>
        </apply>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok>⁢</XMTok>
          <XMTok>Q</XMTok>
          <XMApp>
            <XMTok/>
            <XMTok>D</XMTok>
          </XMApp>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">\mathbb{Q}(\sqrt{D})</annotation>
  </semantics>
</math>

Diff

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@@ -1,39 +1,38 @@
 <?xml version="1.0" encoding="UTF-8"?>
-<math xmlns="http://www.w3.org/1998/Math/MathML" id="p8.m6" class="ltx_Math" alttext="\mathbb{Q}(\sqrt{D})" display="inline">
-  <semantics>
+<math xmlns="http://www.w3.org/1998/Math/MathML">
+  <semantics encoding="">
     <mrow>
-      <mi>ℚ</mi>
-      <mo>⁢</mo>
+      <mi>Q</mi>
       <mrow>
         <mo>(</mo>
         <msqrt>
           <mi>D</mi>
         </msqrt>
         <mo>)</mo>
       </mrow>
     </mrow>
     <annotation-xml encoding="MathML-Content">
       <apply>
         <times/>
-        <ci>ℚ</ci>
+        <ci encoding="">Q</ci>
         <apply>
           <root/>
-          <ci>D</ci>
+          <ci encoding="">D</ci>
         </apply>
       </apply>
     </annotation-xml>
     <annotation-xml encoding="application/x-latexml">
       <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
         <XMApp>
-          <XMTok meaning="times" role="MULOP">⁢</XMTok>
-          <XMTok possibleFunction="yes" role="UNKNOWN" font="blackboard">Q</XMTok>
-          <XMApp close=")" open="(">
-            <XMTok meaning="square-root"/>
-            <XMTok role="UNKNOWN" font="italic">D</XMTok>
+          <XMTok>⁢</XMTok>
+          <XMTok>Q</XMTok>
+          <XMApp>
+            <XMTok/>
+            <XMTok>D</XMTok>
           </XMApp>
         </XMApp>
       </XMath>
     </annotation-xml>
     <annotation encoding="application/x-tex">\mathbb{Q}(\sqrt{D})</annotation>
   </semantics>
 </math>

Formula 98

( show / hide formula code)

Original

ℚ ⁢ ( D ) ℚ D ⁢ Q D \mathbb{Q}(\sqrt{D})

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="p8.m7" class="ltx_Math" alttext="\mathbb{Q}(\sqrt{D})" display="inline">
  <semantics>
    <mrow>
      <mi>ℚ</mi>
      <mo>⁢</mo>
      <mrow>
        <mo>(</mo>
        <msqrt>
          <mi>D</mi>
        </msqrt>
        <mo>)</mo>
      </mrow>
    </mrow>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <times/>
        <ci>ℚ</ci>
        <apply>
          <root/>
          <ci>D</ci>
        </apply>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok meaning="times" role="MULOP">⁢</XMTok>
          <XMTok possibleFunction="yes" role="UNKNOWN" font="blackboard">Q</XMTok>
          <XMApp close=")" open="(">
            <XMTok meaning="square-root"/>
            <XMTok role="UNKNOWN" font="italic">D</XMTok>
          </XMApp>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">\mathbb{Q}(\sqrt{D})</annotation>
  </semantics>
</math>

Canonicalized

Q ( D ) Q D ⁢ Q D \mathbb{Q}(\sqrt{D})

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics encoding="">
    <mrow>
      <mi>Q</mi>
      <mrow>
        <mo>(</mo>
        <msqrt>
          <mi>D</mi>
        </msqrt>
        <mo>)</mo>
      </mrow>
    </mrow>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <times/>
        <ci encoding="">Q</ci>
        <apply>
          <root/>
          <ci encoding="">D</ci>
        </apply>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok>⁢</XMTok>
          <XMTok>Q</XMTok>
          <XMApp>
            <XMTok/>
            <XMTok>D</XMTok>
          </XMApp>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">\mathbb{Q}(\sqrt{D})</annotation>
  </semantics>
</math>

Diff

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@@ -1,39 +1,38 @@
 <?xml version="1.0" encoding="UTF-8"?>
-<math xmlns="http://www.w3.org/1998/Math/MathML" id="p8.m7" class="ltx_Math" alttext="\mathbb{Q}(\sqrt{D})" display="inline">
-  <semantics>
+<math xmlns="http://www.w3.org/1998/Math/MathML">
+  <semantics encoding="">
     <mrow>
-      <mi>ℚ</mi>
-      <mo>⁢</mo>
+      <mi>Q</mi>
       <mrow>
         <mo>(</mo>
         <msqrt>
           <mi>D</mi>
         </msqrt>
         <mo>)</mo>
       </mrow>
     </mrow>
     <annotation-xml encoding="MathML-Content">
       <apply>
         <times/>
-        <ci>ℚ</ci>
+        <ci encoding="">Q</ci>
         <apply>
           <root/>
-          <ci>D</ci>
+          <ci encoding="">D</ci>
         </apply>
       </apply>
     </annotation-xml>
     <annotation-xml encoding="application/x-latexml">
       <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
         <XMApp>
-          <XMTok meaning="times" role="MULOP">⁢</XMTok>
-          <XMTok possibleFunction="yes" role="UNKNOWN" font="blackboard">Q</XMTok>
-          <XMApp close=")" open="(">
-            <XMTok meaning="square-root"/>
-            <XMTok role="UNKNOWN" font="italic">D</XMTok>
+          <XMTok>⁢</XMTok>
+          <XMTok>Q</XMTok>
+          <XMApp>
+            <XMTok/>
+            <XMTok>D</XMTok>
           </XMApp>
         </XMApp>
       </XMath>
     </annotation-xml>
     <annotation encoding="application/x-tex">\mathbb{Q}(\sqrt{D})</annotation>
   </semantics>
 </math>

Formula 99

( show / hide formula code)

Original

k k k k

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="p8.m8" class="ltx_Math" alttext="k" display="inline">
  <semantics>
    <mi>k</mi>
    <annotation-xml encoding="MathML-Content">
      <ci>k</ci>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMTok role="UNKNOWN" font="italic">k</XMTok>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">k</annotation>
  </semantics>
</math>

Canonicalized

k k k k

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics encoding="">
    <mi>k</mi>
    <annotation-xml encoding="MathML-Content">
      <ci encoding="">k</ci>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMTok>k</XMTok>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">k</annotation>
  </semantics>
</math>

Diff

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@@ -1,15 +1,15 @@
 <?xml version="1.0" encoding="UTF-8"?>
-<math xmlns="http://www.w3.org/1998/Math/MathML" id="p8.m8" class="ltx_Math" alttext="k" display="inline">
-  <semantics>
+<math xmlns="http://www.w3.org/1998/Math/MathML">
+  <semantics encoding="">
     <mi>k</mi>
     <annotation-xml encoding="MathML-Content">
-      <ci>k</ci>
+      <ci encoding="">k</ci>
     </annotation-xml>
     <annotation-xml encoding="application/x-latexml">
       <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
-        <XMTok role="UNKNOWN" font="italic">k</XMTok>
+        <XMTok>k</XMTok>
       </XMath>
     </annotation-xml>
     <annotation encoding="application/x-tex">k</annotation>
   </semantics>
 </math>

Formula 100

( show / hide formula code)

Original

ϵ m = ϵ D k subscript ϵ m superscript subscript ϵ D k = ϵ m ϵ D k \epsilon_{m}=\epsilon_{D}^{k}

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="p8.m9" class="ltx_Math" alttext="\epsilon_{m}=\epsilon_{D}^{k}" display="inline">
  <semantics>
    <mrow>
      <msub>
        <mi>ϵ</mi>
        <mi>m</mi>
      </msub>
      <mo>=</mo>
      <msubsup>
        <mi>ϵ</mi>
        <mi>D</mi>
        <mi>k</mi>
      </msubsup>
    </mrow>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <eq/>
        <apply>
          <csymbol cd="ambiguous">subscript</csymbol>
          <ci>ϵ</ci>
          <ci>m</ci>
        </apply>
        <apply>
          <csymbol cd="ambiguous">superscript</csymbol>
          <apply>
            <csymbol cd="ambiguous">subscript</csymbol>
            <ci>ϵ</ci>
            <ci>D</ci>
          </apply>
          <ci>k</ci>
        </apply>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok meaning="equals" role="RELOP">=</XMTok>
          <XMApp>
            <XMTok role="SUBSCRIPTOP" scriptpos="post2"/>
            <XMTok name="epsilon" role="UNKNOWN" font="italic">ϵ</XMTok>
            <XMTok role="UNKNOWN" font="italic">m</XMTok>
          </XMApp>
          <XMApp>
            <XMTok role="SUPERSCRIPTOP" scriptpos="post2"/>
            <XMApp>
              <XMTok role="SUBSCRIPTOP" scriptpos="post2"/>
              <XMTok name="epsilon" role="UNKNOWN" font="italic">ϵ</XMTok>
              <XMTok role="UNKNOWN" font="italic">D</XMTok>
            </XMApp>
            <XMTok role="UNKNOWN" font="italic">k</XMTok>
          </XMApp>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">\epsilon_{m}=\epsilon_{D}^{k}</annotation>
  </semantics>
</math>

Canonicalized

ε m = ε D k subscript ε m superscript subscript ε D k = ε m ε D k \epsilon_{m}=\epsilon_{D}^{k}

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics encoding="">
    <mrow>
      <msub>
        <mi>ε</mi>
        <mi>m</mi>
      </msub>
      <mo>=</mo>
      <msup>
        <msub>
          <mi>ε</mi>
          <mi>D</mi>
        </msub>
        <mi>k</mi>
      </msup>
    </mrow>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <eq/>
        <apply>
          <csymbol encoding="">subscript</csymbol>
          <ci encoding="">ε</ci>
          <ci encoding="">m</ci>
        </apply>
        <apply>
          <csymbol encoding="">superscript</csymbol>
          <apply>
            <csymbol encoding="">subscript</csymbol>
            <ci encoding="">ε</ci>
            <ci encoding="">D</ci>
          </apply>
          <ci encoding="">k</ci>
        </apply>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok>=</XMTok>
          <XMApp>
            <XMTok/>
            <XMTok>ε</XMTok>
            <XMTok>m</XMTok>
          </XMApp>
          <XMApp>
            <XMTok/>
            <XMApp>
              <XMTok/>
              <XMTok>ε</XMTok>
              <XMTok>D</XMTok>
            </XMApp>
            <XMTok>k</XMTok>
          </XMApp>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">\epsilon_{m}=\epsilon_{D}^{k}</annotation>
  </semantics>
</math>

Diff

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@@ -1,58 +1,60 @@
 <?xml version="1.0" encoding="UTF-8"?>
-<math xmlns="http://www.w3.org/1998/Math/MathML" id="p8.m9" class="ltx_Math" alttext="\epsilon_{m}=\epsilon_{D}^{k}" display="inline">
-  <semantics>
+<math xmlns="http://www.w3.org/1998/Math/MathML">
+  <semantics encoding="">
     <mrow>
       <msub>
-        <mi>ϵ</mi>
+        <mi>ε</mi>
         <mi>m</mi>
       </msub>
       <mo>=</mo>
-      <msubsup>
-        <mi>ϵ</mi>
-        <mi>D</mi>
+      <msup>
+        <msub>
+          <mi>ε</mi>
+          <mi>D</mi>
+        </msub>
         <mi>k</mi>
-      </msubsup>
+      </msup>
     </mrow>
     <annotation-xml encoding="MathML-Content">
       <apply>
         <eq/>
         <apply>
-          <csymbol cd="ambiguous">subscript</csymbol>
-          <ci>ϵ</ci>
-          <ci>m</ci>
+          <csymbol encoding="">subscript</csymbol>
+          <ci encoding="">ε</ci>
+          <ci encoding="">m</ci>
         </apply>
         <apply>
-          <csymbol cd="ambiguous">superscript</csymbol>
+          <csymbol encoding="">superscript</csymbol>
           <apply>
-            <csymbol cd="ambiguous">subscript</csymbol>
-            <ci>ϵ</ci>
-            <ci>D</ci>
+            <csymbol encoding="">subscript</csymbol>
+            <ci encoding="">ε</ci>
+            <ci encoding="">D</ci>
           </apply>
-          <ci>k</ci>
+          <ci encoding="">k</ci>
         </apply>
       </apply>
     </annotation-xml>
     <annotation-xml encoding="application/x-latexml">
       <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
         <XMApp>
-          <XMTok meaning="equals" role="RELOP">=</XMTok>
+          <XMTok>=</XMTok>
           <XMApp>
-            <XMTok role="SUBSCRIPTOP" scriptpos="post2"/>
-            <XMTok name="epsilon" role="UNKNOWN" font="italic">ϵ</XMTok>
-            <XMTok role="UNKNOWN" font="italic">m</XMTok>
+            <XMTok/>
+            <XMTok>ε</XMTok>
+            <XMTok>m</XMTok>
           </XMApp>
           <XMApp>
-            <XMTok role="SUPERSCRIPTOP" scriptpos="post2"/>
+            <XMTok/>
             <XMApp>
-              <XMTok role="SUBSCRIPTOP" scriptpos="post2"/>
-              <XMTok name="epsilon" role="UNKNOWN" font="italic">ϵ</XMTok>
-              <XMTok role="UNKNOWN" font="italic">D</XMTok>
+              <XMTok/>
+              <XMTok>ε</XMTok>
+              <XMTok>D</XMTok>
             </XMApp>
-            <XMTok role="UNKNOWN" font="italic">k</XMTok>
+            <XMTok>k</XMTok>
           </XMApp>
         </XMApp>
       </XMath>
     </annotation-xml>
     <annotation encoding="application/x-tex">\epsilon_{m}=\epsilon_{D}^{k}</annotation>
   </semantics>
 </math>

Formula 101

( show / hide formula code)

Original

ϵ D < m subscript ϵ D m < ϵ D m \epsilon_{D}<m

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="p8.m10" class="ltx_Math" alttext="\epsilon_{D}&lt;m" display="inline">
  <semantics>
    <mrow>
      <msub>
        <mi>ϵ</mi>
        <mi>D</mi>
      </msub>
      <mo>&lt;</mo>
      <mi>m</mi>
    </mrow>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <lt/>
        <apply>
          <csymbol cd="ambiguous">subscript</csymbol>
          <ci>ϵ</ci>
          <ci>D</ci>
        </apply>
        <ci>m</ci>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok meaning="less-than" role="RELOP">&lt;</XMTok>
          <XMApp>
            <XMTok role="SUBSCRIPTOP" scriptpos="post2"/>
            <XMTok name="epsilon" role="UNKNOWN" font="italic">ϵ</XMTok>
            <XMTok role="UNKNOWN" font="italic">D</XMTok>
          </XMApp>
          <XMTok role="UNKNOWN" font="italic">m</XMTok>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">\epsilon_{D}&lt;m</annotation>
  </semantics>
</math>

Canonicalized

ε D < m subscript ε D m < ε D m \epsilon_{D}<m

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics encoding="">
    <mrow>
      <msub>
        <mi>ε</mi>
        <mi>D</mi>
      </msub>
      <mo>&lt;</mo>
      <mi>m</mi>
    </mrow>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <lt/>
        <apply>
          <csymbol encoding="">subscript</csymbol>
          <ci encoding="">ε</ci>
          <ci encoding="">D</ci>
        </apply>
        <ci encoding="">m</ci>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok>&lt;</XMTok>
          <XMApp>
            <XMTok/>
            <XMTok>ε</XMTok>
            <XMTok>D</XMTok>
          </XMApp>
          <XMTok>m</XMTok>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">\epsilon_{D}&lt;m</annotation>
  </semantics>
</math>

Diff

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@@ -1,38 +1,38 @@
 <?xml version="1.0" encoding="UTF-8"?>
-<math xmlns="http://www.w3.org/1998/Math/MathML" id="p8.m10" class="ltx_Math" alttext="\epsilon_{D}&lt;m" display="inline">
-  <semantics>
+<math xmlns="http://www.w3.org/1998/Math/MathML">
+  <semantics encoding="">
     <mrow>
       <msub>
-        <mi>ϵ</mi>
+        <mi>ε</mi>
         <mi>D</mi>
       </msub>
       <mo>&lt;</mo>
       <mi>m</mi>
     </mrow>
     <annotation-xml encoding="MathML-Content">
       <apply>
         <lt/>
         <apply>
-          <csymbol cd="ambiguous">subscript</csymbol>
-          <ci>ϵ</ci>
-          <ci>D</ci>
+          <csymbol encoding="">subscript</csymbol>
+          <ci encoding="">ε</ci>
+          <ci encoding="">D</ci>
         </apply>
-        <ci>m</ci>
+        <ci encoding="">m</ci>
       </apply>
     </annotation-xml>
     <annotation-xml encoding="application/x-latexml">
       <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
         <XMApp>
-          <XMTok meaning="less-than" role="RELOP">&lt;</XMTok>
+          <XMTok>&lt;</XMTok>
           <XMApp>
-            <XMTok role="SUBSCRIPTOP" scriptpos="post2"/>
-            <XMTok name="epsilon" role="UNKNOWN" font="italic">ϵ</XMTok>
-            <XMTok role="UNKNOWN" font="italic">D</XMTok>
+            <XMTok/>
+            <XMTok>ε</XMTok>
+            <XMTok>D</XMTok>
           </XMApp>
-          <XMTok role="UNKNOWN" font="italic">m</XMTok>
+          <XMTok>m</XMTok>
         </XMApp>
       </XMath>
     </annotation-xml>
     <annotation encoding="application/x-tex">\epsilon_{D}&lt;m</annotation>
   </semantics>
 </math>

Formula 102

( show / hide formula code)

Original

log ⁡ ( h ⁢ ( ℚ ⁢ ( D ) ) ⁢ log ⁡ | ϵ D | ) ∼ log ⁡ D similar-to h ℚ D subscript ϵ D D ∼ log ⁢ h ⁢ Q D log ϵ D log D \log(h(\mathbb{Q}(\sqrt{D}))\log|\epsilon_{D}|)\sim\log\sqrt{D}

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="p8.m11" class="ltx_Math" alttext="\log(h(\mathbb{Q}(\sqrt{D}))\log|\epsilon_{D}|)\sim\log\sqrt{D}" display="inline">
  <semantics>
    <mrow>
      <mrow>
        <mi>log</mi>
        <mo>⁡</mo>
        <mrow>
          <mo>(</mo>
          <mrow>
            <mi>h</mi>
            <mo>⁢</mo>
            <mrow>
              <mo>(</mo>
              <mrow>
                <mi>ℚ</mi>
                <mo>⁢</mo>
                <mrow>
                  <mo>(</mo>
                  <msqrt>
                    <mi>D</mi>
                  </msqrt>
                  <mo>)</mo>
                </mrow>
              </mrow>
              <mo>)</mo>
            </mrow>
            <mo>⁢</mo>
            <mrow>
              <mi>log</mi>
              <mo>⁡</mo>
              <mrow>
                <mo fence="true">|</mo>
                <msub>
                  <mi>ϵ</mi>
                  <mi>D</mi>
                </msub>
                <mo fence="true">|</mo>
              </mrow>
            </mrow>
          </mrow>
          <mo>)</mo>
        </mrow>
      </mrow>
      <mo>∼</mo>
      <mrow>
        <mi>log</mi>
        <mo>⁡</mo>
        <msqrt>
          <mi>D</mi>
        </msqrt>
      </mrow>
    </mrow>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <csymbol cd="latexml">similar-to</csymbol>
        <apply>
          <log/>
          <apply>
            <times/>
            <ci>h</ci>
            <apply>
              <times/>
              <ci>ℚ</ci>
              <apply>
                <root/>
                <ci>D</ci>
              </apply>
            </apply>
            <apply>
              <log/>
              <apply>
                <abs/>
                <apply>
                  <csymbol cd="ambiguous">subscript</csymbol>
                  <ci>ϵ</ci>
                  <ci>D</ci>
                </apply>
              </apply>
            </apply>
          </apply>
        </apply>
        <apply>
          <log/>
          <apply>
            <root/>
            <ci>D</ci>
          </apply>
        </apply>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok meaning="similar-to" name="sim" role="RELOP">∼</XMTok>
          <XMApp>
            <XMTok argclose=")" argopen="(" meaning="logarithm" role="OPFUNCTION">log</XMTok>
            <XMApp>
              <XMTok meaning="times" role="MULOP">⁢</XMTok>
              <XMTok possibleFunction="yes" role="UNKNOWN" font="italic">h</XMTok>
              <XMApp close=")" open="(">
                <XMTok meaning="times" role="MULOP">⁢</XMTok>
                <XMTok possibleFunction="yes" role="UNKNOWN" font="blackboard">Q</XMTok>
                <XMApp close=")" open="(">
                  <XMTok meaning="square-root"/>
                  <XMTok role="UNKNOWN" font="italic">D</XMTok>
                </XMApp>
              </XMApp>
              <XMApp>
                <XMTok meaning="logarithm" role="OPFUNCTION">log</XMTok>
                <XMApp>
                  <XMTok argclose="|" argopen="|" meaning="absolute-value" role="FENCED"/>
                  <XMApp>
                    <XMTok role="SUBSCRIPTOP" scriptpos="post2"/>
                    <XMTok name="epsilon" role="UNKNOWN" font="italic">ϵ</XMTok>
                    <XMTok role="UNKNOWN" font="italic">D</XMTok>
                  </XMApp>
                </XMApp>
              </XMApp>
            </XMApp>
          </XMApp>
          <XMApp>
            <XMTok meaning="logarithm" role="OPFUNCTION">log</XMTok>
            <XMApp>
              <XMTok meaning="square-root"/>
              <XMTok role="UNKNOWN" font="italic">D</XMTok>
            </XMApp>
          </XMApp>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">\log(h(\mathbb{Q}(\sqrt{D}))\log|\epsilon_{D}|)\sim\log\sqrt{D}</annotation>
  </semantics>
</math>

Canonicalized

log ( h ( Q ( D ) ) log | ( ε D | ) ) ∼ log ( D ) similar-to h Q D subscript ε D D ∼ log ⁢ h ⁢ Q D log ε D log D \log(h(\mathbb{Q}(\sqrt{D}))\log|\epsilon_{D}|)\sim\log\sqrt{D}

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics encoding="">
    <mrow>
      <mrow>
        <mi>log</mi>
        <mrow>
          <mo>(</mo>
          <mrow>
            <mi>h</mi>
            <mrow>
              <mo>(</mo>
              <mrow>
                <mi>Q</mi>
                <mrow>
                  <mo>(</mo>
                  <msqrt>
                    <mi>D</mi>
                  </msqrt>
                  <mo>)</mo>
                </mrow>
              </mrow>
              <mo>)</mo>
            </mrow>
            <mrow>
              <mi>log</mi>
              <mrow>
                <mo>|</mo>
                <mrow>
                  <mo>(</mo>
                  <mrow>
                    <msub>
                      <mi>ε</mi>
                      <mi>D</mi>
                    </msub>
                    <mo>|</mo>
                  </mrow>
                  <mo>)</mo>
                </mrow>
              </mrow>
            </mrow>
          </mrow>
          <mo>)</mo>
        </mrow>
      </mrow>
      <mo>∼</mo>
      <mrow>
        <mi>log</mi>
        <mrow>
          <mo>(</mo>
          <msqrt>
            <mi>D</mi>
          </msqrt>
          <mo>)</mo>
        </mrow>
      </mrow>
    </mrow>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <csymbol encoding="">similar-to</csymbol>
        <apply>
          <log/>
          <apply>
            <times/>
            <ci encoding="">h</ci>
            <apply>
              <times/>
              <ci encoding="">Q</ci>
              <apply>
                <root/>
                <ci encoding="">D</ci>
              </apply>
            </apply>
            <apply>
              <log/>
              <apply>
                <abs/>
                <apply>
                  <csymbol encoding="">subscript</csymbol>
                  <ci encoding="">ε</ci>
                  <ci encoding="">D</ci>
                </apply>
              </apply>
            </apply>
          </apply>
        </apply>
        <apply>
          <log/>
          <apply>
            <root/>
            <ci encoding="">D</ci>
          </apply>
        </apply>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok>∼</XMTok>
          <XMApp>
            <XMTok>log</XMTok>
            <XMApp>
              <XMTok>⁢</XMTok>
              <XMTok>h</XMTok>
              <XMApp>
                <XMTok>⁢</XMTok>
                <XMTok>Q</XMTok>
                <XMApp>
                  <XMTok/>
                  <XMTok>D</XMTok>
                </XMApp>
              </XMApp>
              <XMApp>
                <XMTok>log</XMTok>
                <XMApp>
                  <XMTok/>
                  <XMApp>
                    <XMTok/>
                    <XMTok>ε</XMTok>
                    <XMTok>D</XMTok>
                  </XMApp>
                </XMApp>
              </XMApp>
            </XMApp>
          </XMApp>
          <XMApp>
            <XMTok>log</XMTok>
            <XMApp>
              <XMTok/>
              <XMTok>D</XMTok>
            </XMApp>
          </XMApp>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">\log(h(\mathbb{Q}(\sqrt{D}))\log|\epsilon_{D}|)\sim\log\sqrt{D}</annotation>
  </semantics>
</math>

Diff

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@@ -1,134 +1,138 @@
 <?xml version="1.0" encoding="UTF-8"?>
-<math xmlns="http://www.w3.org/1998/Math/MathML" id="p8.m11" class="ltx_Math" alttext="\log(h(\mathbb{Q}(\sqrt{D}))\log|\epsilon_{D}|)\sim\log\sqrt{D}" display="inline">
-  <semantics>
+<math xmlns="http://www.w3.org/1998/Math/MathML">
+  <semantics encoding="">
     <mrow>
       <mrow>
         <mi>log</mi>
-        <mo>⁡</mo>
         <mrow>
           <mo>(</mo>
           <mrow>
             <mi>h</mi>
-            <mo>⁢</mo>
             <mrow>
               <mo>(</mo>
               <mrow>
-                <mi>ℚ</mi>
-                <mo>⁢</mo>
+                <mi>Q</mi>
                 <mrow>
                   <mo>(</mo>
                   <msqrt>
                     <mi>D</mi>
                   </msqrt>
                   <mo>)</mo>
                 </mrow>
               </mrow>
               <mo>)</mo>
             </mrow>
-            <mo>⁢</mo>
             <mrow>
               <mi>log</mi>
-              <mo>⁡</mo>
               <mrow>
-                <mo fence="true">|</mo>
-                <msub>
-                  <mi>ϵ</mi>
-                  <mi>D</mi>
-                </msub>
-                <mo fence="true">|</mo>
+                <mo>|</mo>
+                <mrow>
+                  <mo>(</mo>
+                  <mrow>
+                    <msub>
+                      <mi>ε</mi>
+                      <mi>D</mi>
+                    </msub>
+                    <mo>|</mo>
+                  </mrow>
+                  <mo>)</mo>
+                </mrow>
               </mrow>
             </mrow>
           </mrow>
           <mo>)</mo>
         </mrow>
       </mrow>
       <mo>∼</mo>
       <mrow>
         <mi>log</mi>
-        <mo>⁡</mo>
-        <msqrt>
-          <mi>D</mi>
-        </msqrt>
+        <mrow>
+          <mo>(</mo>
+          <msqrt>
+            <mi>D</mi>
+          </msqrt>
+          <mo>)</mo>
+        </mrow>
       </mrow>
     </mrow>
     <annotation-xml encoding="MathML-Content">
       <apply>
-        <csymbol cd="latexml">similar-to</csymbol>
+        <csymbol encoding="">similar-to</csymbol>
         <apply>
           <log/>
           <apply>
             <times/>
-            <ci>h</ci>
+            <ci encoding="">h</ci>
             <apply>
               <times/>
-              <ci>ℚ</ci>
+              <ci encoding="">Q</ci>
               <apply>
                 <root/>
-                <ci>D</ci>
+                <ci encoding="">D</ci>
               </apply>
             </apply>
             <apply>
               <log/>
               <apply>
                 <abs/>
                 <apply>
-                  <csymbol cd="ambiguous">subscript</csymbol>
-                  <ci>ϵ</ci>
-                  <ci>D</ci>
+                  <csymbol encoding="">subscript</csymbol>
+                  <ci encoding="">ε</ci>
+                  <ci encoding="">D</ci>
                 </apply>
               </apply>
             </apply>
           </apply>
         </apply>
         <apply>
           <log/>
           <apply>
             <root/>
-            <ci>D</ci>
+            <ci encoding="">D</ci>
           </apply>
         </apply>
       </apply>
     </annotation-xml>
     <annotation-xml encoding="application/x-latexml">
       <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
         <XMApp>
-          <XMTok meaning="similar-to" name="sim" role="RELOP">∼</XMTok>
+          <XMTok>∼</XMTok>
           <XMApp>
-            <XMTok argclose=")" argopen="(" meaning="logarithm" role="OPFUNCTION">log</XMTok>
+            <XMTok>log</XMTok>
             <XMApp>
-              <XMTok meaning="times" role="MULOP">⁢</XMTok>
-              <XMTok possibleFunction="yes" role="UNKNOWN" font="italic">h</XMTok>
-              <XMApp close=")" open="(">
-                <XMTok meaning="times" role="MULOP">⁢</XMTok>
-                <XMTok possibleFunction="yes" role="UNKNOWN" font="blackboard">Q</XMTok>
-                <XMApp close=")" open="(">
-                  <XMTok meaning="square-root"/>
-                  <XMTok role="UNKNOWN" font="italic">D</XMTok>
+              <XMTok>⁢</XMTok>
+              <XMTok>h</XMTok>
+              <XMApp>
+                <XMTok>⁢</XMTok>
+                <XMTok>Q</XMTok>
+                <XMApp>
+                  <XMTok/>
+                  <XMTok>D</XMTok>
                 </XMApp>
               </XMApp>
               <XMApp>
-                <XMTok meaning="logarithm" role="OPFUNCTION">log</XMTok>
+                <XMTok>log</XMTok>
                 <XMApp>
-                  <XMTok argclose="|" argopen="|" meaning="absolute-value" role="FENCED"/>
+                  <XMTok/>
                   <XMApp>
-                    <XMTok role="SUBSCRIPTOP" scriptpos="post2"/>
-                    <XMTok name="epsilon" role="UNKNOWN" font="italic">ϵ</XMTok>
-                    <XMTok role="UNKNOWN" font="italic">D</XMTok>
+                    <XMTok/>
+                    <XMTok>ε</XMTok>
+                    <XMTok>D</XMTok>
                   </XMApp>
                 </XMApp>
               </XMApp>
             </XMApp>
           </XMApp>
           <XMApp>
-            <XMTok meaning="logarithm" role="OPFUNCTION">log</XMTok>
+            <XMTok>log</XMTok>
             <XMApp>
-              <XMTok meaning="square-root"/>
-              <XMTok role="UNKNOWN" font="italic">D</XMTok>
+              <XMTok/>
+              <XMTok>D</XMTok>
             </XMApp>
           </XMApp>
         </XMApp>
       </XMath>
     </annotation-xml>
     <annotation encoding="application/x-tex">\log(h(\mathbb{Q}(\sqrt{D}))\log|\epsilon_{D}|)\sim\log\sqrt{D}</annotation>
   </semantics>
 </math>

Formula 103

( show / hide formula code)

Original

f m subscript f m f m f_{m}

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="p8.m12" class="ltx_Math" alttext="f_{m}" display="inline">
  <semantics>
    <msub>
      <mi>f</mi>
      <mi>m</mi>
    </msub>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <csymbol cd="ambiguous">subscript</csymbol>
        <ci>f</ci>
        <ci>m</ci>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok role="SUBSCRIPTOP" scriptpos="post2"/>
          <XMTok role="UNKNOWN" font="italic">f</XMTok>
          <XMTok role="UNKNOWN" font="italic">m</XMTok>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">f_{m}</annotation>
  </semantics>
</math>

Canonicalized

f m subscript f m f m f_{m}

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics encoding="">
    <msub>
      <mi>f</mi>
      <mi>m</mi>
    </msub>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <csymbol encoding="">subscript</csymbol>
        <ci encoding="">f</ci>
        <ci encoding="">m</ci>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok/>
          <XMTok>f</XMTok>
          <XMTok>m</XMTok>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">f_{m}</annotation>
  </semantics>
</math>

Diff

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@@ -1,26 +1,26 @@
 <?xml version="1.0" encoding="UTF-8"?>
-<math xmlns="http://www.w3.org/1998/Math/MathML" id="p8.m12" class="ltx_Math" alttext="f_{m}" display="inline">
-  <semantics>
+<math xmlns="http://www.w3.org/1998/Math/MathML">
+  <semantics encoding="">
     <msub>
       <mi>f</mi>
       <mi>m</mi>
     </msub>
     <annotation-xml encoding="MathML-Content">
       <apply>
-        <csymbol cd="ambiguous">subscript</csymbol>
-        <ci>f</ci>
-        <ci>m</ci>
+        <csymbol encoding="">subscript</csymbol>
+        <ci encoding="">f</ci>
+        <ci encoding="">m</ci>
       </apply>
     </annotation-xml>
     <annotation-xml encoding="application/x-latexml">
       <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
         <XMApp>
-          <XMTok role="SUBSCRIPTOP" scriptpos="post2"/>
-          <XMTok role="UNKNOWN" font="italic">f</XMTok>
-          <XMTok role="UNKNOWN" font="italic">m</XMTok>
+          <XMTok/>
+          <XMTok>f</XMTok>
+          <XMTok>m</XMTok>
         </XMApp>
       </XMath>
     </annotation-xml>
     <annotation encoding="application/x-tex">f_{m}</annotation>
   </semantics>
 </math>

Formula 104

( show / hide formula code)

Original

h ⁢ ( ℚ ⁢ ( D ) ) = 1 h ℚ D 1 = ⁢ h ⁢ Q D 1 h(\mathbb{Q}(\sqrt{D}))=1

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="p8.m13" class="ltx_Math" alttext="h(\mathbb{Q}(\sqrt{D}))=1" display="inline">
  <semantics>
    <mrow>
      <mrow>
        <mi>h</mi>
        <mo>⁢</mo>
        <mrow>
          <mo>(</mo>
          <mrow>
            <mi>ℚ</mi>
            <mo>⁢</mo>
            <mrow>
              <mo>(</mo>
              <msqrt>
                <mi>D</mi>
              </msqrt>
              <mo>)</mo>
            </mrow>
          </mrow>
          <mo>)</mo>
        </mrow>
      </mrow>
      <mo>=</mo>
      <mn>1</mn>
    </mrow>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <eq/>
        <apply>
          <times/>
          <ci>h</ci>
          <apply>
            <times/>
            <ci>ℚ</ci>
            <apply>
              <root/>
              <ci>D</ci>
            </apply>
          </apply>
        </apply>
        <cn type="integer">1</cn>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok meaning="equals" role="RELOP">=</XMTok>
          <XMApp>
            <XMTok meaning="times" role="MULOP">⁢</XMTok>
            <XMTok possibleFunction="yes" role="UNKNOWN" font="italic">h</XMTok>
            <XMApp close=")" open="(">
              <XMTok meaning="times" role="MULOP">⁢</XMTok>
              <XMTok possibleFunction="yes" role="UNKNOWN" font="blackboard">Q</XMTok>
              <XMApp close=")" open="(">
                <XMTok meaning="square-root"/>
                <XMTok role="UNKNOWN" font="italic">D</XMTok>
              </XMApp>
            </XMApp>
          </XMApp>
          <XMTok meaning="1" role="NUMBER">1</XMTok>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">h(\mathbb{Q}(\sqrt{D}))=1</annotation>
  </semantics>
</math>

Canonicalized

h ( Q ( D ) ) = 1 h Q D 1 = ⁢ h ⁢ Q D 1 h(\mathbb{Q}(\sqrt{D}))=1

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics encoding="">
    <mrow>
      <mrow>
        <mi>h</mi>
        <mrow>
          <mo>(</mo>
          <mrow>
            <mi>Q</mi>
            <mrow>
              <mo>(</mo>
              <msqrt>
                <mi>D</mi>
              </msqrt>
              <mo>)</mo>
            </mrow>
          </mrow>
          <mo>)</mo>
        </mrow>
      </mrow>
      <mo>=</mo>
      <mn>1</mn>
    </mrow>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <eq/>
        <apply>
          <times/>
          <ci encoding="">h</ci>
          <apply>
            <times/>
            <ci encoding="">Q</ci>
            <apply>
              <root/>
              <ci encoding="">D</ci>
            </apply>
          </apply>
        </apply>
        <cn encoding="" base="10" type="integer">1</cn>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok>=</XMTok>
          <XMApp>
            <XMTok>⁢</XMTok>
            <XMTok>h</XMTok>
            <XMApp>
              <XMTok>⁢</XMTok>
              <XMTok>Q</XMTok>
              <XMApp>
                <XMTok/>
                <XMTok>D</XMTok>
              </XMApp>
            </XMApp>
          </XMApp>
          <XMTok>1</XMTok>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">h(\mathbb{Q}(\sqrt{D}))=1</annotation>
  </semantics>
</math>

Diff

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@@ -1,67 +1,65 @@
 <?xml version="1.0" encoding="UTF-8"?>
-<math xmlns="http://www.w3.org/1998/Math/MathML" id="p8.m13" class="ltx_Math" alttext="h(\mathbb{Q}(\sqrt{D}))=1" display="inline">
-  <semantics>
+<math xmlns="http://www.w3.org/1998/Math/MathML">
+  <semantics encoding="">
     <mrow>
       <mrow>
         <mi>h</mi>
-        <mo>⁢</mo>
         <mrow>
           <mo>(</mo>
           <mrow>
-            <mi>ℚ</mi>
-            <mo>⁢</mo>
+            <mi>Q</mi>
             <mrow>
               <mo>(</mo>
               <msqrt>
                 <mi>D</mi>
               </msqrt>
               <mo>)</mo>
             </mrow>
           </mrow>
           <mo>)</mo>
         </mrow>
       </mrow>
       <mo>=</mo>
       <mn>1</mn>
     </mrow>
     <annotation-xml encoding="MathML-Content">
       <apply>
         <eq/>
         <apply>
           <times/>
-          <ci>h</ci>
+          <ci encoding="">h</ci>
           <apply>
             <times/>
-            <ci>ℚ</ci>
+            <ci encoding="">Q</ci>
             <apply>
               <root/>
-              <ci>D</ci>
+              <ci encoding="">D</ci>
             </apply>
           </apply>
         </apply>
-        <cn type="integer">1</cn>
+        <cn encoding="" base="10" type="integer">1</cn>
       </apply>
     </annotation-xml>
     <annotation-xml encoding="application/x-latexml">
       <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
         <XMApp>
-          <XMTok meaning="equals" role="RELOP">=</XMTok>
+          <XMTok>=</XMTok>
           <XMApp>
-            <XMTok meaning="times" role="MULOP">⁢</XMTok>
-            <XMTok possibleFunction="yes" role="UNKNOWN" font="italic">h</XMTok>
-            <XMApp close=")" open="(">
-              <XMTok meaning="times" role="MULOP">⁢</XMTok>
-              <XMTok possibleFunction="yes" role="UNKNOWN" font="blackboard">Q</XMTok>
-              <XMApp close=")" open="(">
-                <XMTok meaning="square-root"/>
-                <XMTok role="UNKNOWN" font="italic">D</XMTok>
+            <XMTok>⁢</XMTok>
+            <XMTok>h</XMTok>
+            <XMApp>
+              <XMTok>⁢</XMTok>
+              <XMTok>Q</XMTok>
+              <XMApp>
+                <XMTok/>
+                <XMTok>D</XMTok>
               </XMApp>
             </XMApp>
           </XMApp>
-          <XMTok meaning="1" role="NUMBER">1</XMTok>
+          <XMTok>1</XMTok>
         </XMApp>
       </XMath>
     </annotation-xml>
     <annotation encoding="application/x-tex">h(\mathbb{Q}(\sqrt{D}))=1</annotation>
   </semantics>
 </math>

Formula 105

( show / hide formula code)

Original

log ⁡ | ϵ D | ≤ log ⁡ | ϵ m | < log ⁡ m ≤ 12 ⁢ log ⁡ D , subscript ϵ D subscript ϵ m m 12 D log ϵ D ≤ log ϵ m < log m ≤ ⁢ 12 log D \log|\epsilon_{D}|\leq\log|\epsilon_{m}|<\log m\leq 12\log D,

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="S0.Ex5.m1" class="ltx_Math" alttext="\log|\epsilon_{D}|\leq\log|\epsilon_{m}|&lt;\log m\leq 12\log D," display="block">
  <semantics>
    <mrow>
      <mrow>
        <mrow>
          <mi>log</mi>
          <mo>⁡</mo>
          <mrow>
            <mo fence="true">|</mo>
            <msub>
              <mi>ϵ</mi>
              <mi>D</mi>
            </msub>
            <mo fence="true">|</mo>
          </mrow>
        </mrow>
        <mo>≤</mo>
        <mrow>
          <mi>log</mi>
          <mo>⁡</mo>
          <mrow>
            <mo fence="true">|</mo>
            <msub>
              <mi>ϵ</mi>
              <mi>m</mi>
            </msub>
            <mo fence="true">|</mo>
          </mrow>
        </mrow>
        <mo>&lt;</mo>
        <mrow>
          <mi>log</mi>
          <mo>⁡</mo>
          <mi>m</mi>
        </mrow>
        <mo>≤</mo>
        <mrow>
          <mn>12</mn>
          <mo>⁢</mo>
          <mrow>
            <mi>log</mi>
            <mo>⁡</mo>
            <mi>D</mi>
          </mrow>
        </mrow>
      </mrow>
      <mo>,</mo>
    </mrow>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <and/>
        <apply>
          <leq/>
          <apply>
            <log/>
            <apply>
              <abs/>
              <apply>
                <csymbol cd="ambiguous">subscript</csymbol>
                <ci>ϵ</ci>
                <ci>D</ci>
              </apply>
            </apply>
          </apply>
          <apply xml:id="S0.Ex5.m1.sh1g.cmml">
            <log xml:id="S0.Ex5.m1.sh1.cmml"/>
            <apply xml:id="S0.Ex5.m1.sh1f.cmml">
              <abs xml:id="S0.Ex5.m1.sh1a.cmml"/>
              <apply xml:id="S0.Ex5.m1.sh1e.cmml">
                <csymbol cd="ambiguous" xml:id="S0.Ex5.m1.sh1b.cmml">subscript</csymbol>
                <ci xml:id="S0.Ex5.m1.sh1c.cmml">ϵ</ci>
                <ci xml:id="S0.Ex5.m1.sh1d.cmml">m</ci>
              </apply>
            </apply>
          </apply>
        </apply>
        <apply>
          <lt/>
          <share href="#S0.Ex5.m1.sh1.cmml"/>
          <apply xml:id="S0.Ex5.m1.sh2b.cmml">
            <log xml:id="S0.Ex5.m1.sh2.cmml"/>
            <ci xml:id="S0.Ex5.m1.sh2a.cmml">m</ci>
          </apply>
        </apply>
        <apply>
          <leq/>
          <share href="#S0.Ex5.m1.sh2.cmml"/>
          <apply xml:id="S0.Ex5.m1.sh3e.cmml">
            <times xml:id="S0.Ex5.m1.sh3.cmml"/>
            <cn type="integer" xml:id="S0.Ex5.m1.sh3a.cmml">12</cn>
            <apply xml:id="S0.Ex5.m1.sh3d.cmml">
              <log xml:id="S0.Ex5.m1.sh3b.cmml"/>
              <ci xml:id="S0.Ex5.m1.sh3c.cmml">D</ci>
            </apply>
          </apply>
        </apply>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp punctuation=",">
          <XMTok meaning="multirelation"/>
          <XMApp>
            <XMTok meaning="logarithm" role="OPFUNCTION">log</XMTok>
            <XMApp>
              <XMTok argclose="|" argopen="|" meaning="absolute-value" role="FENCED"/>
              <XMApp>
                <XMTok role="SUBSCRIPTOP" scriptpos="post2"/>
                <XMTok name="epsilon" role="UNKNOWN" font="italic">ϵ</XMTok>
                <XMTok role="UNKNOWN" font="italic">D</XMTok>
              </XMApp>
            </XMApp>
          </XMApp>
          <XMTok meaning="less-than-or-equals" name="leq" role="RELOP">≤</XMTok>
          <XMApp xml:id="S0.Ex5.m1.sh1.xm" fragid="S0.Ex5.m1.sh1.xm">
            <XMTok meaning="logarithm" role="OPFUNCTION">log</XMTok>
            <XMApp>
              <XMTok argclose="|" argopen="|" meaning="absolute-value" role="FENCED"/>
              <XMApp>
                <XMTok role="SUBSCRIPTOP" scriptpos="post2"/>
                <XMTok name="epsilon" role="UNKNOWN" font="italic">ϵ</XMTok>
                <XMTok role="UNKNOWN" font="italic">m</XMTok>
              </XMApp>
            </XMApp>
          </XMApp>
          <XMTok meaning="less-than" role="RELOP">&lt;</XMTok>
          <XMApp xml:id="S0.Ex5.m1.sh2.xm" fragid="S0.Ex5.m1.sh2.xm">
            <XMTok meaning="logarithm" role="OPFUNCTION">log</XMTok>
            <XMTok role="UNKNOWN" font="italic">m</XMTok>
          </XMApp>
          <XMTok meaning="less-than-or-equals" name="leq" role="RELOP">≤</XMTok>
          <XMApp xml:id="S0.Ex5.m1.sh3.xm" fragid="S0.Ex5.m1.sh3.xm">
            <XMTok meaning="times" role="MULOP">⁢</XMTok>
            <XMTok meaning="12" role="NUMBER">12</XMTok>
            <XMApp>
              <XMTok meaning="logarithm" role="OPFUNCTION">log</XMTok>
              <XMTok role="UNKNOWN" font="italic">D</XMTok>
            </XMApp>
          </XMApp>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">\log|\epsilon_{D}|\leq\log|\epsilon_{m}|&lt;\log m\leq 12\log D,</annotation>
  </semantics>
</math>

Canonicalized

log | ( ε D | ) ≤ log | ( ε m | ) < log ( m ) ≤ 12 log ( D ) , subscript ε D subscript ε m m 12 D log ε D ≤ log ε m < log m ≤ ⁢ 12 log D \log|\epsilon_{D}|\leq\log|\epsilon_{m}|<\log m\leq 12\log D,

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics encoding="">
    <mrow>
      <mrow>
        <mrow>
          <mi>log</mi>
          <mrow>
            <mo>|</mo>
            <mrow>
              <mo>(</mo>
              <mrow>
                <msub>
                  <mi>ε</mi>
                  <mi>D</mi>
                </msub>
                <mo>|</mo>
              </mrow>
              <mo>)</mo>
            </mrow>
          </mrow>
        </mrow>
        <mo>≤</mo>
        <mrow>
          <mi>log</mi>
          <mrow>
            <mo>|</mo>
            <mrow>
              <mo>(</mo>
              <mrow>
                <msub>
                  <mi>ε</mi>
                  <mi>m</mi>
                </msub>
                <mo>|</mo>
              </mrow>
              <mo>)</mo>
            </mrow>
          </mrow>
        </mrow>
        <mo>&lt;</mo>
        <mrow>
          <mi>log</mi>
          <mrow>
            <mo>(</mo>
            <mi>m</mi>
            <mo>)</mo>
          </mrow>
        </mrow>
        <mo>≤</mo>
        <mrow>
          <mn>12</mn>
          <mrow>
            <mi>log</mi>
            <mrow>
              <mo>(</mo>
              <mi>D</mi>
              <mo>)</mo>
            </mrow>
          </mrow>
        </mrow>
      </mrow>
      <mo>,</mo>
    </mrow>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <and/>
        <apply>
          <leq/>
          <apply>
            <log/>
            <apply>
              <abs/>
              <apply>
                <csymbol encoding="">subscript</csymbol>
                <ci encoding="">ε</ci>
                <ci encoding="">D</ci>
              </apply>
            </apply>
          </apply>
          <apply>
            <log encoding=""/>
            <apply>
              <abs encoding=""/>
              <apply>
                <csymbol encoding="">subscript</csymbol>
                <ci encoding="">ε</ci>
                <ci encoding="">m</ci>
              </apply>
            </apply>
          </apply>
        </apply>
        <apply>
          <lt/>
          <share/>
          <apply>
            <log encoding=""/>
            <ci encoding="">m</ci>
          </apply>
        </apply>
        <apply>
          <leq/>
          <share/>
          <apply>
            <times encoding=""/>
            <cn encoding="" base="10" type="integer">12</cn>
            <apply>
              <log encoding=""/>
              <ci encoding="">D</ci>
            </apply>
          </apply>
        </apply>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok/>
          <XMApp>
            <XMTok>log</XMTok>
            <XMApp>
              <XMTok/>
              <XMApp>
                <XMTok/>
                <XMTok>ε</XMTok>
                <XMTok>D</XMTok>
              </XMApp>
            </XMApp>
          </XMApp>
          <XMTok>≤</XMTok>
          <XMApp>
            <XMTok>log</XMTok>
            <XMApp>
              <XMTok/>
              <XMApp>
                <XMTok/>
                <XMTok>ε</XMTok>
                <XMTok>m</XMTok>
              </XMApp>
            </XMApp>
          </XMApp>
          <XMTok>&lt;</XMTok>
          <XMApp>
            <XMTok>log</XMTok>
            <XMTok>m</XMTok>
          </XMApp>
          <XMTok>≤</XMTok>
          <XMApp>
            <XMTok>⁢</XMTok>
            <XMTok>12</XMTok>
            <XMApp>
              <XMTok>log</XMTok>
              <XMTok>D</XMTok>
            </XMApp>
          </XMApp>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">\log|\epsilon_{D}|\leq\log|\epsilon_{m}|&lt;\log m\leq 12\log D,</annotation>
  </semantics>
</math>

Diff

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@@ -1,146 +1,161 @@
 <?xml version="1.0" encoding="UTF-8"?>
-<math xmlns="http://www.w3.org/1998/Math/MathML" id="S0.Ex5.m1" class="ltx_Math" alttext="\log|\epsilon_{D}|\leq\log|\epsilon_{m}|&lt;\log m\leq 12\log D," display="block">
-  <semantics>
+<math xmlns="http://www.w3.org/1998/Math/MathML">
+  <semantics encoding="">
     <mrow>
       <mrow>
         <mrow>
           <mi>log</mi>
-          <mo>⁡</mo>
           <mrow>
-            <mo fence="true">|</mo>
-            <msub>
-              <mi>ϵ</mi>
-              <mi>D</mi>
-            </msub>
-            <mo fence="true">|</mo>
+            <mo>|</mo>
+            <mrow>
+              <mo>(</mo>
+              <mrow>
+                <msub>
+                  <mi>ε</mi>
+                  <mi>D</mi>
+                </msub>
+                <mo>|</mo>
+              </mrow>
+              <mo>)</mo>
+            </mrow>
           </mrow>
         </mrow>
         <mo>≤</mo>
         <mrow>
           <mi>log</mi>
-          <mo>⁡</mo>
           <mrow>
-            <mo fence="true">|</mo>
-            <msub>
-              <mi>ϵ</mi>
-              <mi>m</mi>
-            </msub>
-            <mo fence="true">|</mo>
+            <mo>|</mo>
+            <mrow>
+              <mo>(</mo>
+              <mrow>
+                <msub>
+                  <mi>ε</mi>
+                  <mi>m</mi>
+                </msub>
+                <mo>|</mo>
+              </mrow>
+              <mo>)</mo>
+            </mrow>
           </mrow>
         </mrow>
         <mo>&lt;</mo>
         <mrow>
           <mi>log</mi>
-          <mo>⁡</mo>
-          <mi>m</mi>
+          <mrow>
+            <mo>(</mo>
+            <mi>m</mi>
+            <mo>)</mo>
+          </mrow>
         </mrow>
         <mo>≤</mo>
         <mrow>
           <mn>12</mn>
-          <mo>⁢</mo>
           <mrow>
             <mi>log</mi>
-            <mo>⁡</mo>
-            <mi>D</mi>
+            <mrow>
+              <mo>(</mo>
+              <mi>D</mi>
+              <mo>)</mo>
+            </mrow>
           </mrow>
         </mrow>
       </mrow>
       <mo>,</mo>
     </mrow>
     <annotation-xml encoding="MathML-Content">
       <apply>
         <and/>
         <apply>
           <leq/>
           <apply>
             <log/>
             <apply>
               <abs/>
               <apply>
-                <csymbol cd="ambiguous">subscript</csymbol>
-                <ci>ϵ</ci>
-                <ci>D</ci>
+                <csymbol encoding="">subscript</csymbol>
+                <ci encoding="">ε</ci>
+                <ci encoding="">D</ci>
               </apply>
             </apply>
           </apply>
-          <apply xml:id="S0.Ex5.m1.sh1g.cmml">
-            <log xml:id="S0.Ex5.m1.sh1.cmml"/>
-            <apply xml:id="S0.Ex5.m1.sh1f.cmml">
-              <abs xml:id="S0.Ex5.m1.sh1a.cmml"/>
-              <apply xml:id="S0.Ex5.m1.sh1e.cmml">
-                <csymbol cd="ambiguous" xml:id="S0.Ex5.m1.sh1b.cmml">subscript</csymbol>
-                <ci xml:id="S0.Ex5.m1.sh1c.cmml">ϵ</ci>
-                <ci xml:id="S0.Ex5.m1.sh1d.cmml">m</ci>
+          <apply>
+            <log encoding=""/>
+            <apply>
+              <abs encoding=""/>
+              <apply>
+                <csymbol encoding="">subscript</csymbol>
+                <ci encoding="">ε</ci>
+                <ci encoding="">m</ci>
               </apply>
             </apply>
           </apply>
         </apply>
         <apply>
           <lt/>
-          <share href="#S0.Ex5.m1.sh1.cmml"/>
-          <apply xml:id="S0.Ex5.m1.sh2b.cmml">
-            <log xml:id="S0.Ex5.m1.sh2.cmml"/>
-            <ci xml:id="S0.Ex5.m1.sh2a.cmml">m</ci>
+          <share/>
+          <apply>
+            <log encoding=""/>
+            <ci encoding="">m</ci>
           </apply>
         </apply>
         <apply>
           <leq/>
-          <share href="#S0.Ex5.m1.sh2.cmml"/>
-          <apply xml:id="S0.Ex5.m1.sh3e.cmml">
-            <times xml:id="S0.Ex5.m1.sh3.cmml"/>
-            <cn type="integer" xml:id="S0.Ex5.m1.sh3a.cmml">12</cn>
-            <apply xml:id="S0.Ex5.m1.sh3d.cmml">
-              <log xml:id="S0.Ex5.m1.sh3b.cmml"/>
-              <ci xml:id="S0.Ex5.m1.sh3c.cmml">D</ci>
+          <share/>
+          <apply>
+            <times encoding=""/>
+            <cn encoding="" base="10" type="integer">12</cn>
+            <apply>
+              <log encoding=""/>
+              <ci encoding="">D</ci>
             </apply>
           </apply>
         </apply>
       </apply>
     </annotation-xml>
     <annotation-xml encoding="application/x-latexml">
       <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
-        <XMApp punctuation=",">
-          <XMTok meaning="multirelation"/>
+        <XMApp>
+          <XMTok/>
           <XMApp>
-            <XMTok meaning="logarithm" role="OPFUNCTION">log</XMTok>
+            <XMTok>log</XMTok>
             <XMApp>
-              <XMTok argclose="|" argopen="|" meaning="absolute-value" role="FENCED"/>
+              <XMTok/>
               <XMApp>
-                <XMTok role="SUBSCRIPTOP" scriptpos="post2"/>
-                <XMTok name="epsilon" role="UNKNOWN" font="italic">ϵ</XMTok>
-                <XMTok role="UNKNOWN" font="italic">D</XMTok>
+                <XMTok/>
+                <XMTok>ε</XMTok>
+                <XMTok>D</XMTok>
               </XMApp>
             </XMApp>
           </XMApp>
-          <XMTok meaning="less-than-or-equals" name="leq" role="RELOP">≤</XMTok>
-          <XMApp xml:id="S0.Ex5.m1.sh1.xm" fragid="S0.Ex5.m1.sh1.xm">
-            <XMTok meaning="logarithm" role="OPFUNCTION">log</XMTok>
+          <XMTok>≤</XMTok>
+          <XMApp>
+            <XMTok>log</XMTok>
             <XMApp>
-              <XMTok argclose="|" argopen="|" meaning="absolute-value" role="FENCED"/>
+              <XMTok/>
               <XMApp>
-                <XMTok role="SUBSCRIPTOP" scriptpos="post2"/>
-                <XMTok name="epsilon" role="UNKNOWN" font="italic">ϵ</XMTok>
-                <XMTok role="UNKNOWN" font="italic">m</XMTok>
+                <XMTok/>
+                <XMTok>ε</XMTok>
+                <XMTok>m</XMTok>
               </XMApp>
             </XMApp>
           </XMApp>
-          <XMTok meaning="less-than" role="RELOP">&lt;</XMTok>
-          <XMApp xml:id="S0.Ex5.m1.sh2.xm" fragid="S0.Ex5.m1.sh2.xm">
-            <XMTok meaning="logarithm" role="OPFUNCTION">log</XMTok>
-            <XMTok role="UNKNOWN" font="italic">m</XMTok>
+          <XMTok>&lt;</XMTok>
+          <XMApp>
+            <XMTok>log</XMTok>
+            <XMTok>m</XMTok>
           </XMApp>
-          <XMTok meaning="less-than-or-equals" name="leq" role="RELOP">≤</XMTok>
-          <XMApp xml:id="S0.Ex5.m1.sh3.xm" fragid="S0.Ex5.m1.sh3.xm">
-            <XMTok meaning="times" role="MULOP">⁢</XMTok>
-            <XMTok meaning="12" role="NUMBER">12</XMTok>
+          <XMTok>≤</XMTok>
+          <XMApp>
+            <XMTok>⁢</XMTok>
+            <XMTok>12</XMTok>
             <XMApp>
-              <XMTok meaning="logarithm" role="OPFUNCTION">log</XMTok>
-              <XMTok role="UNKNOWN" font="italic">D</XMTok>
+              <XMTok>log</XMTok>
+              <XMTok>D</XMTok>
             </XMApp>
           </XMApp>
         </XMApp>
       </XMath>
     </annotation-xml>
     <annotation encoding="application/x-tex">\log|\epsilon_{D}|\leq\log|\epsilon_{m}|&lt;\log m\leq 12\log D,</annotation>
   </semantics>
 </math>

Formula 106

( show / hide formula code)

Original

12 ⁢ log ⁡ D > D 1 / 2 + o ⁢ ( 1 ) 12 D superscript D 1 2 o 1 > ⁢ 12 log D D + / 1 2 ⁢ o 1 12\log D>D^{1/2+o(1)}

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="S0.Ex6.m1" class="ltx_Math" alttext="12\log D&gt;D^{1/2+o(1)}" display="block">
  <semantics>
    <mrow>
      <mrow>
        <mn>12</mn>
        <mo>⁢</mo>
        <mrow>
          <mi>log</mi>
          <mo>⁡</mo>
          <mi>D</mi>
        </mrow>
      </mrow>
      <mo>&gt;</mo>
      <msup>
        <mi>D</mi>
        <mrow>
          <mrow>
            <mn>1</mn>
            <mo>/</mo>
            <mn>2</mn>
          </mrow>
          <mo>+</mo>
          <mrow>
            <mi>o</mi>
            <mo>⁢</mo>
            <mrow>
              <mo>(</mo>
              <mn>1</mn>
              <mo>)</mo>
            </mrow>
          </mrow>
        </mrow>
      </msup>
    </mrow>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <gt/>
        <apply>
          <times/>
          <cn type="integer">12</cn>
          <apply>
            <log/>
            <ci>D</ci>
          </apply>
        </apply>
        <apply>
          <csymbol cd="ambiguous">superscript</csymbol>
          <ci>D</ci>
          <apply>
            <plus/>
            <apply>
              <divide/>
              <cn type="integer">1</cn>
              <cn type="integer">2</cn>
            </apply>
            <apply>
              <times/>
              <ci>o</ci>
              <cn type="integer">1</cn>
            </apply>
          </apply>
        </apply>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok meaning="greater-than" role="RELOP">&gt;</XMTok>
          <XMApp>
            <XMTok meaning="times" role="MULOP">⁢</XMTok>
            <XMTok meaning="12" role="NUMBER">12</XMTok>
            <XMApp>
              <XMTok meaning="logarithm" role="OPFUNCTION">log</XMTok>
              <XMTok role="UNKNOWN" font="italic">D</XMTok>
            </XMApp>
          </XMApp>
          <XMApp>
            <XMTok role="SUPERSCRIPTOP" scriptpos="post2"/>
            <XMTok role="UNKNOWN" font="italic">D</XMTok>
            <XMApp>
              <XMTok meaning="plus" role="ADDOP">+</XMTok>
              <XMApp>
                <XMTok mathstyle="inline" meaning="divide" role="MULOP">/</XMTok>
                <XMTok meaning="1" role="NUMBER">1</XMTok>
                <XMTok meaning="2" role="NUMBER">2</XMTok>
              </XMApp>
              <XMApp>
                <XMTok meaning="times" role="MULOP">⁢</XMTok>
                <XMTok possibleFunction="yes" role="UNKNOWN" font="italic">o</XMTok>
                <XMTok close=")" meaning="1" open="(" role="NUMBER">1</XMTok>
              </XMApp>
            </XMApp>
          </XMApp>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">12\log D&gt;D^{1/2+o(1)}</annotation>
  </semantics>
</math>

Canonicalized

12 log ( D ) > D 1 / 2 + o ( 1 ) 12 D superscript D 1 2 o 1 > ⁢ 12 log D D + / 1 2 ⁢ o 1 12\log D>D^{1/2+o(1)}

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics encoding="">
    <mrow>
      <mrow>
        <mn>12</mn>
        <mrow>
          <mi>log</mi>
          <mrow>
            <mo>(</mo>
            <mi>D</mi>
            <mo>)</mo>
          </mrow>
        </mrow>
      </mrow>
      <mo>&gt;</mo>
      <msup>
        <mi>D</mi>
        <mrow>
          <mrow>
            <mn>1</mn>
            <mo>/</mo>
            <mn>2</mn>
          </mrow>
          <mo>+</mo>
          <mrow>
            <mi>o</mi>
            <mrow>
              <mo>(</mo>
              <mn>1</mn>
              <mo>)</mo>
            </mrow>
          </mrow>
        </mrow>
      </msup>
    </mrow>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <gt/>
        <apply>
          <times/>
          <cn encoding="" base="10" type="integer">12</cn>
          <apply>
            <log/>
            <ci encoding="">D</ci>
          </apply>
        </apply>
        <apply>
          <csymbol encoding="">superscript</csymbol>
          <ci encoding="">D</ci>
          <apply>
            <plus/>
            <apply>
              <divide/>
              <cn encoding="" base="10" type="integer">1</cn>
              <cn encoding="" base="10" type="integer">2</cn>
            </apply>
            <apply>
              <times/>
              <ci encoding="">o</ci>
              <cn encoding="" base="10" type="integer">1</cn>
            </apply>
          </apply>
        </apply>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok>&gt;</XMTok>
          <XMApp>
            <XMTok>⁢</XMTok>
            <XMTok>12</XMTok>
            <XMApp>
              <XMTok>log</XMTok>
              <XMTok>D</XMTok>
            </XMApp>
          </XMApp>
          <XMApp>
            <XMTok/>
            <XMTok>D</XMTok>
            <XMApp>
              <XMTok>+</XMTok>
              <XMApp>
                <XMTok>/</XMTok>
                <XMTok>1</XMTok>
                <XMTok>2</XMTok>
              </XMApp>
              <XMApp>
                <XMTok>⁢</XMTok>
                <XMTok>o</XMTok>
                <XMTok>1</XMTok>
              </XMApp>
            </XMApp>
          </XMApp>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">12\log D&gt;D^{1/2+o(1)}</annotation>
  </semantics>
</math>

Diff

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@@ -1,100 +1,101 @@
 <?xml version="1.0" encoding="UTF-8"?>
-<math xmlns="http://www.w3.org/1998/Math/MathML" id="S0.Ex6.m1" class="ltx_Math" alttext="12\log D&gt;D^{1/2+o(1)}" display="block">
-  <semantics>
+<math xmlns="http://www.w3.org/1998/Math/MathML">
+  <semantics encoding="">
     <mrow>
       <mrow>
         <mn>12</mn>
-        <mo>⁢</mo>
         <mrow>
           <mi>log</mi>
-          <mo>⁡</mo>
-          <mi>D</mi>
+          <mrow>
+            <mo>(</mo>
+            <mi>D</mi>
+            <mo>)</mo>
+          </mrow>
         </mrow>
       </mrow>
       <mo>&gt;</mo>
       <msup>
         <mi>D</mi>
         <mrow>
           <mrow>
             <mn>1</mn>
             <mo>/</mo>
             <mn>2</mn>
           </mrow>
           <mo>+</mo>
           <mrow>
             <mi>o</mi>
-            <mo>⁢</mo>
             <mrow>
               <mo>(</mo>
               <mn>1</mn>
               <mo>)</mo>
             </mrow>
           </mrow>
         </mrow>
       </msup>
     </mrow>
     <annotation-xml encoding="MathML-Content">
       <apply>
         <gt/>
         <apply>
           <times/>
-          <cn type="integer">12</cn>
+          <cn encoding="" base="10" type="integer">12</cn>
           <apply>
             <log/>
-            <ci>D</ci>
+            <ci encoding="">D</ci>
           </apply>
         </apply>
         <apply>
-          <csymbol cd="ambiguous">superscript</csymbol>
-          <ci>D</ci>
+          <csymbol encoding="">superscript</csymbol>
+          <ci encoding="">D</ci>
           <apply>
             <plus/>
             <apply>
               <divide/>
-              <cn type="integer">1</cn>
-              <cn type="integer">2</cn>
+              <cn encoding="" base="10" type="integer">1</cn>
+              <cn encoding="" base="10" type="integer">2</cn>
             </apply>
             <apply>
               <times/>
-              <ci>o</ci>
-              <cn type="integer">1</cn>
+              <ci encoding="">o</ci>
+              <cn encoding="" base="10" type="integer">1</cn>
             </apply>
           </apply>
         </apply>
       </apply>
     </annotation-xml>
     <annotation-xml encoding="application/x-latexml">
       <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
         <XMApp>
-          <XMTok meaning="greater-than" role="RELOP">&gt;</XMTok>
+          <XMTok>&gt;</XMTok>
           <XMApp>
-            <XMTok meaning="times" role="MULOP">⁢</XMTok>
-            <XMTok meaning="12" role="NUMBER">12</XMTok>
+            <XMTok>⁢</XMTok>
+            <XMTok>12</XMTok>
             <XMApp>
-              <XMTok meaning="logarithm" role="OPFUNCTION">log</XMTok>
-              <XMTok role="UNKNOWN" font="italic">D</XMTok>
+              <XMTok>log</XMTok>
+              <XMTok>D</XMTok>
             </XMApp>
           </XMApp>
           <XMApp>
-            <XMTok role="SUPERSCRIPTOP" scriptpos="post2"/>
-            <XMTok role="UNKNOWN" font="italic">D</XMTok>
+            <XMTok/>
+            <XMTok>D</XMTok>
             <XMApp>
-              <XMTok meaning="plus" role="ADDOP">+</XMTok>
+              <XMTok>+</XMTok>
               <XMApp>
-                <XMTok mathstyle="inline" meaning="divide" role="MULOP">/</XMTok>
-                <XMTok meaning="1" role="NUMBER">1</XMTok>
-                <XMTok meaning="2" role="NUMBER">2</XMTok>
+                <XMTok>/</XMTok>
+                <XMTok>1</XMTok>
+                <XMTok>2</XMTok>
               </XMApp>
               <XMApp>
-                <XMTok meaning="times" role="MULOP">⁢</XMTok>
-                <XMTok possibleFunction="yes" role="UNKNOWN" font="italic">o</XMTok>
-                <XMTok close=")" meaning="1" open="(" role="NUMBER">1</XMTok>
+                <XMTok>⁢</XMTok>
+                <XMTok>o</XMTok>
+                <XMTok>1</XMTok>
               </XMApp>
             </XMApp>
           </XMApp>
         </XMApp>
       </XMath>
     </annotation-xml>
     <annotation encoding="application/x-tex">12\log D&gt;D^{1/2+o(1)}</annotation>
   </semantics>
 </math>

Formula 107

( show / hide formula code)

Original

D D D D

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="p8.m14" class="ltx_Math" alttext="D" display="inline">
  <semantics>
    <mi>D</mi>
    <annotation-xml encoding="MathML-Content">
      <ci>D</ci>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMTok role="UNKNOWN" font="italic">D</XMTok>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">D</annotation>
  </semantics>
</math>

Canonicalized

D D D D

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics encoding="">
    <mi>D</mi>
    <annotation-xml encoding="MathML-Content">
      <ci encoding="">D</ci>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMTok>D</XMTok>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">D</annotation>
  </semantics>
</math>

Diff

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@@ -1,15 +1,15 @@
 <?xml version="1.0" encoding="UTF-8"?>
-<math xmlns="http://www.w3.org/1998/Math/MathML" id="p8.m14" class="ltx_Math" alttext="D" display="inline">
-  <semantics>
+<math xmlns="http://www.w3.org/1998/Math/MathML">
+  <semantics encoding="">
     <mi>D</mi>
     <annotation-xml encoding="MathML-Content">
-      <ci>D</ci>
+      <ci encoding="">D</ci>
     </annotation-xml>
     <annotation-xml encoding="application/x-latexml">
       <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
-        <XMTok role="UNKNOWN" font="italic">D</XMTok>
+        <XMTok>D</XMTok>
       </XMath>
     </annotation-xml>
     <annotation encoding="application/x-tex">D</annotation>
   </semantics>
 </math>

Formula 108

( show / hide formula code)

Original

m ≤ D 12 m superscript D 12 ≤ m D 12 m\leq D^{12}

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="p8.m15" class="ltx_Math" alttext="m\leq D^{12}" display="inline">
  <semantics>
    <mrow>
      <mi>m</mi>
      <mo>≤</mo>
      <msup>
        <mi>D</mi>
        <mn>12</mn>
      </msup>
    </mrow>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <leq/>
        <ci>m</ci>
        <apply>
          <csymbol cd="ambiguous">superscript</csymbol>
          <ci>D</ci>
          <cn type="integer">12</cn>
        </apply>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok meaning="less-than-or-equals" name="leq" role="RELOP">≤</XMTok>
          <XMTok role="UNKNOWN" font="italic">m</XMTok>
          <XMApp>
            <XMTok role="SUPERSCRIPTOP" scriptpos="post2"/>
            <XMTok role="UNKNOWN" font="italic">D</XMTok>
            <XMTok meaning="12" role="NUMBER">12</XMTok>
          </XMApp>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">m\leq D^{12}</annotation>
  </semantics>
</math>

Canonicalized

m ≤ D 12 m superscript D 12 ≤ m D 12 m\leq D^{12}

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics encoding="">
    <mrow>
      <mi>m</mi>
      <mo>≤</mo>
      <msup>
        <mi>D</mi>
        <mn>12</mn>
      </msup>
    </mrow>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <leq/>
        <ci encoding="">m</ci>
        <apply>
          <csymbol encoding="">superscript</csymbol>
          <ci encoding="">D</ci>
          <cn encoding="" base="10" type="integer">12</cn>
        </apply>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok>≤</XMTok>
          <XMTok>m</XMTok>
          <XMApp>
            <XMTok/>
            <XMTok>D</XMTok>
            <XMTok>12</XMTok>
          </XMApp>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">m\leq D^{12}</annotation>
  </semantics>
</math>

Diff

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@@ -1,38 +1,38 @@
 <?xml version="1.0" encoding="UTF-8"?>
-<math xmlns="http://www.w3.org/1998/Math/MathML" id="p8.m15" class="ltx_Math" alttext="m\leq D^{12}" display="inline">
-  <semantics>
+<math xmlns="http://www.w3.org/1998/Math/MathML">
+  <semantics encoding="">
     <mrow>
       <mi>m</mi>
       <mo>≤</mo>
       <msup>
         <mi>D</mi>
         <mn>12</mn>
       </msup>
     </mrow>
     <annotation-xml encoding="MathML-Content">
       <apply>
         <leq/>
-        <ci>m</ci>
+        <ci encoding="">m</ci>
         <apply>
-          <csymbol cd="ambiguous">superscript</csymbol>
-          <ci>D</ci>
-          <cn type="integer">12</cn>
+          <csymbol encoding="">superscript</csymbol>
+          <ci encoding="">D</ci>
+          <cn encoding="" base="10" type="integer">12</cn>
         </apply>
       </apply>
     </annotation-xml>
     <annotation-xml encoding="application/x-latexml">
       <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
         <XMApp>
-          <XMTok meaning="less-than-or-equals" name="leq" role="RELOP">≤</XMTok>
-          <XMTok role="UNKNOWN" font="italic">m</XMTok>
+          <XMTok>≤</XMTok>
+          <XMTok>m</XMTok>
           <XMApp>
-            <XMTok role="SUPERSCRIPTOP" scriptpos="post2"/>
-            <XMTok role="UNKNOWN" font="italic">D</XMTok>
-            <XMTok meaning="12" role="NUMBER">12</XMTok>
+            <XMTok/>
+            <XMTok>D</XMTok>
+            <XMTok>12</XMTok>
           </XMApp>
         </XMApp>
       </XMath>
     </annotation-xml>
     <annotation encoding="application/x-tex">m\leq D^{12}</annotation>
   </semantics>
 </math>

Formula 109

( show / hide formula code)

Original

m m m m

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="p8.m16" class="ltx_Math" alttext="m" display="inline">
  <semantics>
    <mi>m</mi>
    <annotation-xml encoding="MathML-Content">
      <ci>m</ci>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMTok role="UNKNOWN" font="italic">m</XMTok>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">m</annotation>
  </semantics>
</math>

Canonicalized

m m m m

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics encoding="">
    <mi>m</mi>
    <annotation-xml encoding="MathML-Content">
      <ci encoding="">m</ci>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMTok>m</XMTok>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">m</annotation>
  </semantics>
</math>

Diff

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@@ -1,15 +1,15 @@
 <?xml version="1.0" encoding="UTF-8"?>
-<math xmlns="http://www.w3.org/1998/Math/MathML" id="p8.m16" class="ltx_Math" alttext="m" display="inline">
-  <semantics>
+<math xmlns="http://www.w3.org/1998/Math/MathML">
+  <semantics encoding="">
     <mi>m</mi>
     <annotation-xml encoding="MathML-Content">
-      <ci>m</ci>
+      <ci encoding="">m</ci>
     </annotation-xml>
     <annotation-xml encoding="application/x-latexml">
       <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
-        <XMTok role="UNKNOWN" font="italic">m</XMTok>
+        <XMTok>m</XMTok>
       </XMath>
     </annotation-xml>
     <annotation encoding="application/x-tex">m</annotation>
   </semantics>
 </math>

Formula 110

( show / hide formula code)

Original

m 0 subscript m 0 m 0 m_{0}

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="p9.m1" class="ltx_Math" alttext="m_{0}" display="inline">
  <semantics>
    <msub>
      <mi>m</mi>
      <mn>0</mn>
    </msub>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <csymbol cd="ambiguous">subscript</csymbol>
        <ci>m</ci>
        <cn type="integer">0</cn>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok role="SUBSCRIPTOP" scriptpos="post2"/>
          <XMTok role="UNKNOWN" font="italic">m</XMTok>
          <XMTok meaning="0" role="NUMBER">0</XMTok>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">m_{0}</annotation>
  </semantics>
</math>

Canonicalized

m 0 subscript m 0 m 0 m_{0}

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics encoding="">
    <msub>
      <mi>m</mi>
      <mn>0</mn>
    </msub>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <csymbol encoding="">subscript</csymbol>
        <ci encoding="">m</ci>
        <cn encoding="" base="10" type="integer">0</cn>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok/>
          <XMTok>m</XMTok>
          <XMTok>0</XMTok>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">m_{0}</annotation>
  </semantics>
</math>

Diff

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@@ -1,26 +1,26 @@
 <?xml version="1.0" encoding="UTF-8"?>
-<math xmlns="http://www.w3.org/1998/Math/MathML" id="p9.m1" class="ltx_Math" alttext="m_{0}" display="inline">
-  <semantics>
+<math xmlns="http://www.w3.org/1998/Math/MathML">
+  <semantics encoding="">
     <msub>
       <mi>m</mi>
       <mn>0</mn>
     </msub>
     <annotation-xml encoding="MathML-Content">
       <apply>
-        <csymbol cd="ambiguous">subscript</csymbol>
-        <ci>m</ci>
-        <cn type="integer">0</cn>
+        <csymbol encoding="">subscript</csymbol>
+        <ci encoding="">m</ci>
+        <cn encoding="" base="10" type="integer">0</cn>
       </apply>
     </annotation-xml>
     <annotation-xml encoding="application/x-latexml">
       <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
         <XMApp>
-          <XMTok role="SUBSCRIPTOP" scriptpos="post2"/>
-          <XMTok role="UNKNOWN" font="italic">m</XMTok>
-          <XMTok meaning="0" role="NUMBER">0</XMTok>
+          <XMTok/>
+          <XMTok>m</XMTok>
+          <XMTok>0</XMTok>
         </XMApp>
       </XMath>
     </annotation-xml>
     <annotation encoding="application/x-tex">m_{0}</annotation>
   </semantics>
 </math>

Formula 111

( show / hide formula code)

Original

m > m 0 m subscript m 0 > m m 0 m>m_{0}

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="p9.m2" class="ltx_Math" alttext="m&gt;m_{0}" display="inline">
  <semantics>
    <mrow>
      <mi>m</mi>
      <mo>&gt;</mo>
      <msub>
        <mi>m</mi>
        <mn>0</mn>
      </msub>
    </mrow>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <gt/>
        <ci>m</ci>
        <apply>
          <csymbol cd="ambiguous">subscript</csymbol>
          <ci>m</ci>
          <cn type="integer">0</cn>
        </apply>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok meaning="greater-than" role="RELOP">&gt;</XMTok>
          <XMTok role="UNKNOWN" font="italic">m</XMTok>
          <XMApp>
            <XMTok role="SUBSCRIPTOP" scriptpos="post2"/>
            <XMTok role="UNKNOWN" font="italic">m</XMTok>
            <XMTok meaning="0" role="NUMBER">0</XMTok>
          </XMApp>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">m&gt;m_{0}</annotation>
  </semantics>
</math>

Canonicalized

m > m 0 m subscript m 0 > m m 0 m>m_{0}

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics encoding="">
    <mrow>
      <mi>m</mi>
      <mo>&gt;</mo>
      <msub>
        <mi>m</mi>
        <mn>0</mn>
      </msub>
    </mrow>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <gt/>
        <ci encoding="">m</ci>
        <apply>
          <csymbol encoding="">subscript</csymbol>
          <ci encoding="">m</ci>
          <cn encoding="" base="10" type="integer">0</cn>
        </apply>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok>&gt;</XMTok>
          <XMTok>m</XMTok>
          <XMApp>
            <XMTok/>
            <XMTok>m</XMTok>
            <XMTok>0</XMTok>
          </XMApp>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">m&gt;m_{0}</annotation>
  </semantics>
</math>

Diff

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@@ -1,38 +1,38 @@
 <?xml version="1.0" encoding="UTF-8"?>
-<math xmlns="http://www.w3.org/1998/Math/MathML" id="p9.m2" class="ltx_Math" alttext="m&gt;m_{0}" display="inline">
-  <semantics>
+<math xmlns="http://www.w3.org/1998/Math/MathML">
+  <semantics encoding="">
     <mrow>
       <mi>m</mi>
       <mo>&gt;</mo>
       <msub>
         <mi>m</mi>
         <mn>0</mn>
       </msub>
     </mrow>
     <annotation-xml encoding="MathML-Content">
       <apply>
         <gt/>
-        <ci>m</ci>
+        <ci encoding="">m</ci>
         <apply>
-          <csymbol cd="ambiguous">subscript</csymbol>
-          <ci>m</ci>
-          <cn type="integer">0</cn>
+          <csymbol encoding="">subscript</csymbol>
+          <ci encoding="">m</ci>
+          <cn encoding="" base="10" type="integer">0</cn>
         </apply>
       </apply>
     </annotation-xml>
     <annotation-xml encoding="application/x-latexml">
       <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
         <XMApp>
-          <XMTok meaning="greater-than" role="RELOP">&gt;</XMTok>
-          <XMTok role="UNKNOWN" font="italic">m</XMTok>
+          <XMTok>&gt;</XMTok>
+          <XMTok>m</XMTok>
           <XMApp>
-            <XMTok role="SUBSCRIPTOP" scriptpos="post2"/>
-            <XMTok role="UNKNOWN" font="italic">m</XMTok>
-            <XMTok meaning="0" role="NUMBER">0</XMTok>
+            <XMTok/>
+            <XMTok>m</XMTok>
+            <XMTok>0</XMTok>
           </XMApp>
         </XMApp>
       </XMath>
     </annotation-xml>
     <annotation encoding="application/x-tex">m&gt;m_{0}</annotation>
   </semantics>
 </math>

Formula 112

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Original

f m subscript f m f m f_{m}

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="p9.m3" class="ltx_Math" alttext="f_{m}" display="inline">
  <semantics>
    <msub>
      <mi>f</mi>
      <mi>m</mi>
    </msub>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <csymbol cd="ambiguous">subscript</csymbol>
        <ci>f</ci>
        <ci>m</ci>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok role="SUBSCRIPTOP" scriptpos="post2"/>
          <XMTok role="UNKNOWN" font="italic">f</XMTok>
          <XMTok role="UNKNOWN" font="italic">m</XMTok>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">f_{m}</annotation>
  </semantics>
</math>

Canonicalized

f m subscript f m f m f_{m}

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics encoding="">
    <msub>
      <mi>f</mi>
      <mi>m</mi>
    </msub>
    <annotation-xml encoding="MathML-Content">
      <apply>
        <csymbol encoding="">subscript</csymbol>
        <ci encoding="">f</ci>
        <ci encoding="">m</ci>
      </apply>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMApp>
          <XMTok/>
          <XMTok>f</XMTok>
          <XMTok>m</XMTok>
        </XMApp>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">f_{m}</annotation>
  </semantics>
</math>

Diff

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@@ -1,26 +1,26 @@
 <?xml version="1.0" encoding="UTF-8"?>
-<math xmlns="http://www.w3.org/1998/Math/MathML" id="p9.m3" class="ltx_Math" alttext="f_{m}" display="inline">
-  <semantics>
+<math xmlns="http://www.w3.org/1998/Math/MathML">
+  <semantics encoding="">
     <msub>
       <mi>f</mi>
       <mi>m</mi>
     </msub>
     <annotation-xml encoding="MathML-Content">
       <apply>
-        <csymbol cd="ambiguous">subscript</csymbol>
-        <ci>f</ci>
-        <ci>m</ci>
+        <csymbol encoding="">subscript</csymbol>
+        <ci encoding="">f</ci>
+        <ci encoding="">m</ci>
       </apply>
     </annotation-xml>
     <annotation-xml encoding="application/x-latexml">
       <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
         <XMApp>
-          <XMTok role="SUBSCRIPTOP" scriptpos="post2"/>
-          <XMTok role="UNKNOWN" font="italic">f</XMTok>
-          <XMTok role="UNKNOWN" font="italic">m</XMTok>
+          <XMTok/>
+          <XMTok>f</XMTok>
+          <XMTok>m</XMTok>
         </XMApp>
       </XMath>
     </annotation-xml>
     <annotation encoding="application/x-tex">f_{m}</annotation>
   </semantics>
 </math>

Formula 113

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Original

L L L L

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML" id="bib.bib3.m1" class="ltx_Math" alttext="L" display="inline">
  <semantics>
    <mi>L</mi>
    <annotation-xml encoding="MathML-Content">
      <ci>L</ci>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMTok role="UNKNOWN">L</XMTok>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">L</annotation>
  </semantics>
</math>

Canonicalized

L L L L

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<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics encoding="">
    <mi>L</mi>
    <annotation-xml encoding="MathML-Content">
      <ci encoding="">L</ci>
    </annotation-xml>
    <annotation-xml encoding="application/x-latexml">
      <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
        <XMTok>L</XMTok>
      </XMath>
    </annotation-xml>
    <annotation encoding="application/x-tex">L</annotation>
  </semantics>
</math>

Diff

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@@ -1,15 +1,15 @@
 <?xml version="1.0" encoding="UTF-8"?>
-<math xmlns="http://www.w3.org/1998/Math/MathML" id="bib.bib3.m1" class="ltx_Math" alttext="L" display="inline">
-  <semantics>
+<math xmlns="http://www.w3.org/1998/Math/MathML">
+  <semantics encoding="">
     <mi>L</mi>
     <annotation-xml encoding="MathML-Content">
-      <ci>L</ci>
+      <ci encoding="">L</ci>
     </annotation-xml>
     <annotation-xml encoding="application/x-latexml">
       <XMath xmlns="http://dlmf.nist.gov/LaTeXML">
-        <XMTok role="UNKNOWN">L</XMTok>
+        <XMTok>L</XMTok>
       </XMath>
     </annotation-xml>
     <annotation encoding="application/x-tex">L</annotation>
   </semantics>
 </math>