MathML templates.

Last Update 23.02.2021

To use MathML write math symbols inside of <math> tag. By attribute display make a block. MathML in the page is styled by CSS.

Variable plus number:

x + 1


<mi>x</mi><mo>+</mo><mn>1</mn>

Two parameter Fifth degree polynomial with constant coefficients:

P (x, y) = A + 5 x - 3 y + B x y - x^{2} + C y^{2} + D x^{2} y^{2} E x^{3} + F y^{3} + 90 x^{3} y^{3} + G x^{4} + H y^{4} - 17 x^{4} y^{4} + x^{5} + y^{5}

<mi mathvariant=bold>P</mi>
<mo>&ApplyFunction;</mo>
<mo fence="true">(</mo>
  <mi>x</mi>
  <mo separator="true">,</mo>
  <mi>y</mi>
<mo fence="true">)</mo>
<mo>=</mo>
<mi>A</mi>
<mo>+</mo>
<mn>5</mn><mi>x</mi>
<mo>-</mo>
<mn>3</mn><mi>y</mi>
<mo>+</mo>
<mi>B</mi>
<mi>x</mi>
<mi>y</mi>
<mo>-</mo>
<msup><mi>x</mi><mi>2</mi></msup>
<mo>+</mo>
<mi>C</mi>
<msup><mi>y</mi><mi>2</mi></msup>
<mo>+</mo>
<mi>D</mi>
<msup><mi>x</mi><mi>2</mi></msup>
<msup><mi>y</mi><mi>2</mi></msup>
<mi>E</mi>
<msup><mi>x</mi><mi>3</mi></msup>
<mo>+</mo>
<mi>F</mi>
<msup><mi>y</mi><mi>3</mi></msup>
<mo>+</mo>
<mn>90</mn>
<msup><mi>x</mi><mi>3</mi></msup>
<msup><mi>y</mi><mi>3</mi></msup>
<mo>+</mo>
<mi>G</mi>
<msup><mi>x</mi><mi>4</mi></msup>
<mo>+</mo>
<mi>H</mi>
<msup><mi>y</mi><mi>4</mi></msup>
<mo>-</mo>
<mn>17</mn>
<msup><mi>x</mi><mi>4</mi></msup>
<msup><mi>y</mi><mi>4</mi></msup>
<mo>+</mo>
<msup><mi>x</mi><mi>5</mi></msup>
<mo>+</mo>
<msup><mi>y</mi><mi>5</mi></msup>

Recursive defition of sequence:

x_{n + 1} = {\begin{matrix} x_{n} - 2 x_{n - 5} & , n > 0 \land n mod 8 = 0 \\ x_{n} + x_{n - 2} & , n mod 3 = 0 \land n mod 8 \neq 0 \land n > 0 \\ 5 & , n = 2 \\ 1 & , n = 1 \\ 8 & , n = 0 \end{matrix}

<msub>
  <mi>x</mi>
  <mrow>
    <mi>n</mi>
    <mo>+</mo>
    <mn>1</mn>
  </mrow>
</msub>
<mo>=</mo>
<mrow><mo fence="true">{</mo><mtable>
  <mtr columnalign="left">
    <mtd>
      <msub><mi>x</mi><mi>n</mi></msub>
      <mo>-</mo>
      <mn>2</mn>
      <msub>
        <mi>x</mi>
        <mrow>
          <mi>n</mi>
          <mo>-</mo>
          <mn>5</mn>
        </mrow>
      </msub>
    </mtd>
    <mtd>
      <mtext>,</mtext>
      <mi>n</mi>
      <mo>&gt;</mo>
      <mn>0</mn>
      <mo>&and;</mo>
      <mi>n</mi>
      <mo mathvariant=bold>mod</mo>
      <mn>8</mn>
      <mo>=</mo>
      <mn>0</mn>
    </mtd>
  </mtr>
  <mtr columnalign="left">
    <mtd>
      <msub><mi>x</mi><mi>n</mi></msub>
      <mo>+</mo>
      <msub>
        <mi>x</mi>
        <mrow>
          <mi>n</mi>
          <mo>-</mo>
          <mn>2</mn>
        </mrow>
      </msub>
    </mtd>
    <mtd>
      <mtext>,</mtext>
      <mi>n</mi>
      <mo mathvariant=bold>mod</mo>
      <mn>3</mn><mo>=</mo>
      <mn>0</mn>
      <mo>&and;</mo>
      <mi>n</mi>
      <mo mathvariant=bold>mod</mo>
      <mn>8</mn>
      <mo>&ne;</mo>
      <mn>0</mn>
      <mo>&and;</mo>
      <mi>n</mi>
      <mo>></mo>
      <mn>0</mn>
    </mtd>
  </mtr>
  <mtr columnalign="left">
    <mtd>
      <mn>5</mn>
    </mtd>
    <mtd>
      <mtext>,</mtext>
      <mi>n</mi><mo>=</mo><mn>2</mn>
    </mtd>
  </mtr>
  <mtr columnalign="left">
    <mtd>
      <mn>1</mn>
    </mtd>
    <mtd>
      <mtext>,</mtext>
      <mi>n</mi><mo>=</mo><mn>1</mn>
    </mtd>
  </mtr>
  <mtr columnalign="left">
    <mtd>
      <mn>8</mn>
    </mtd>
    <mtd>
      <mtext>,</mtext>
      <mi>n</mi><mo>=</mo><mn>0</mn>
    </mtd>
  </mtr>
</mtable></mrow>

Differential equation:

\frac{\partial^{2} F (t)}{\partial^{2} t} = 15 \cdot \frac{\partial F (t)}{\partial t} + 2 F (t)

<mfrac>
  <mrow>
    <msup>
      <mo>&part;</mo><mn>2</mn>
    </msup>
    <mi mathvariant=bold>F</mi>
    <mo>&ApplyFunction;</mo>
    <mo fence="true">(</mo>
      <mi>t</mi>
    <mo fence="true">)</mo>
  </mrow>
  <mrow>
    <msup>
      <mo>&part;</mo>
      <mn>2</mn>
    </msup>
    <mi>t</mi>
  </mrow>
</mfrac>
<mo>=</mo>
<mn>15</mn>
<mo>&sdot;</mo>
<mfrac>
  <mrow>
    <mo>&part;</mo>
    <mi mathvariant=bold>F</mi>
    <mo>&ApplyFunction;</mo>
    <mo fence="true">(</mo>
      <mi>t</mi>
    <mo fence="true">)</mo>
  </mrow>
  <mrow><mo>&part;</mo><mi>t</mi></mrow>
</mfrac>
<mo>+</mo>
<mn>2</mn>
<mi mathvariant=bold>F</mi>
<mo>&ApplyFunction;</mo>
<mo fence="true">(</mo>
  <mi>t</m>
<mo fence="true">)</mo>

Set equation:

X \times Y = {(x, y) | x \in Y^{c} \land (y \in Y \Rightarrow \exists k \in N (y + k \in Y))} \cap N^{2}

<mi>X</mi><mo>&times;</mo><mi>Y</mi>
<mo>=</mo>
<mo fence="true">{</mo>
  <mo fence="true">(</mo>
    <mi>x</mi>
    <mo>,</mo>
    <mi>y</mi>
  <mo fence="true">)</mo>
  <mo>|</mo>
  <mi>x</mi>
  <mo>&in;</mo>
  <msup><mi>Y</mi><mo>c</mo></msup>
  <mo>&and;</mo>
  <mo fence="true">(</mo>
    <mi>y</mi><mo>&in;</mo><mi>Y</mi>
    <mo>&Rightarrow;</mo>
    <mi>&exist;</mi>
    <mi>k</mi>
    <mo>∈</mo>
    <mi mathvariant="double-struck">N</mi>
    <mo fence="true">(</mo>
      <mi>y</mi><mo>+</mo><mi>k</mi>
      <mo>&in;</mo>
      <mi>Y</mi>
    <mo fence="true">)</mo>
  <mo fence="true">)</mo>
<mo fence="true">}</mo>
<mo>&cap;</mo>
<msup>
  <mi mathvariant="double-struck">N</mi>
  <mn>2</mn>
</msup>

Integral of sum:

\int_{0}^{10} \sum_{n = 0}^{⎿ N ⏌} (n + 5) \partial N = 6 + \int_{1}^{10} \sum_{n = 0}^{⎿ N ⏌} (n + 5) \partial N

<msubsup>
  <mi mathsize=1.5em>&int;</mi>
  <mn>0</mn>
  <mn>10</mn>
</msubsup>
<mrow>
  <munderover>
    <mi mathsize=1.5em>&sum;</mi>
    <mrow>
      <mi>n</mi>
      <mo>=</mo>
      <mn>0</mn>
    </mrow>
    <mrow>
      <mo fence="true" lspace=0em rspace=0em>&#x23bf;</mo>
      <mi rspace=0em lspace=0em>N</mi>
      <mo fence="true" lspace=0em rspace=0em>&#x23cc;</mo>
    </mrow>
  </munderover>
  <mrow>
    <mo fence="true">(</mo>
      <mi>n</mi>
      <mo>+</mo>
      <mn>5</mn>
    <mo fence="true">)</mo>
  </mrow>
  <mo rspace=0em>&part;</mo>
  <mi>N</mi>
</mrow>
<mo>=</mo>
<mn>6</mn>
<mo>+</mo>
<msubsup>
  <mi mathsize=1.5em>&int;</mi>
  <mn>1</mn>
  <mn>10</mn>
</msubsup>
<mrow>
  <munderover>
    <mi mathsize=1.5em>&sum;</mi>
    <mrow>
      <mi>n</mi>
      <mo>=</mo>
      <mn>0</mn>
    </mrow>
    <mrow>
      <mo fence="true" lspace=0em rspace=0em>&#x23bf;</mo>
      <mi rspace=0em lspace=0em>N</mi>
      <mo fence="true" lspace=0em rspace=0em>&#x23cc;</mo>
    </mrow>
  </munderover>
  <mrow>
    <mo fence="true">(</mo>
      <mi>n</mi>
      <mo>+</mo>
      <mn>5</mn>
    <mo fence="true">)</mo>
  </mrow>
  <mo rspace=0em>&part;</mo>
  <mi>N</mi>
</mrow>

Solving X of second degree polynomial:

\begin{matrix} a x^{2} + b x + c & = & 0 & | - c \\ a x^{2} + b x & = & -c & | \cdot 4 a \\ {(2 a x)}^{2} + 4 a b x & = & -4 a c & | + b^{2} \\ {(2 a x)}^{2} + 4 a b x + b^{2} & = & b^{2} - 4 a c \\ {(2 a x + b)}^{2} & = & b^{2} - 4 a c & | \pm \sqrt{} \\ 2 a x + b & = & \pm \sqrt{b^{2} - 4 a c} & | - b \\ 2 a x & = & -b \pm \sqrt{b^{2} - 4 a c} & | \cdot 2 a \\ x & = & \frac{-b \pm \sqrt{b^{2} - 4 a c}}{2 a} \end{matrix}

<mtable 
  rowspacing=0em
  columnspacing=0em
  groupalign="{right left left right}"
>
  <mtr>
    <mtd>
      <maligngroup/>
      <mi>a</mi>
      <msup>
        <mi>x</mi>
        <mn>2</mn>
      </msup>
      <mo>+</mo>
      <mi>b</mi><mi>x</mi>
      <mo>+</mo>
      <mi>c</mi>
    </mtd>
    <mtd><maligngroup/><mo>=</mo></mtd>
    <mtd>
      <maligngroup/>
      <mn>0</mn>
    </mtd>
    <mtd>
      <maligngroup/>
      <mspace width=2em />
      <mtext>|</mtext>
      <mo>-</mo>
      <mi>c</mi>
    </mtd>
  </mtr>
  <mtr>
    <mtd>
      <maligngroup/>
      <mi>a</mi>
      <msup>
        <mi>x</mi><mn>2</mn>
      </msup>
      <mo>+</mo>
      <mi>b</mi><mi>x</mi>
    </mtd>
    <mtd><maligngroup/><mo>=</mo></mtd>
    <mtd>
      <maligngroup/>
      <mi>-c</mi>
    </mtd>
    <mtd>
      <maligngroup/>
      <mspace width=2em />
      <mtext>|</mtext>
      <mo>&sdot;</mo>
      <mn>4</mn>
      <mi>a</mi>
    </mtd>
  </mtr>
  <mtr>
    <mtd>
      <maligngroup/>
      <msup>
        <mrow>
          <mo fence=true>(</mo>
            <mn>2</mn>
            <mi>a</mi>
            <mi>x</mi>
          <mo fence=true>)</mo>
        </mrow>
        <mn>2</mn>
      </msup>
      <mo>+</mo>
      <mn>4</mn>
      <mi>a</mi>
      <mi>b</mi>
      <mi>x</mi>
    </mtd>
    <mtd><maligngroup/><mo>=</mo></mtd>
    <mtd>
      <maligngroup/>
      <mn>-4</mn>
      <mi>a</mi>
      <mi>c</mi>
    </mtd>
    <mtd>
      <maligngroup/>
      <mspace width=2em />
      <mtext>|</mtext>
      <mo>+</mo>
      <msup><mi>b</mi><mn>2</mn></msup>
    </mtd>
  </mtr>
  <mtr>
    <mtd>
      <maligngroup/>
      <msup>
        <mrow>
          <mo fence=true>(</mo>
            <mn>2</mn>
            <mi>a</mi>
            <mi>x</mi>
          <mo fence=true>)</mo>
        </mrow>
        <mn>2</mn>
      </msup>
      <mo>+</mo>
      <mn>4</mn>
      <mi>a</mi>
      <mi>b</mi>
      <mi>x</mi>
      <mo>+</mo>
      <msup><mi>b</mi><mn>2</mn></msup>
    </mtd>
    <mtd><maligngroup/><mo>=</mo></mtd>
    <mtd>
      <maligngroup/>
      <msup><mi>b</mi><mn>2</mn></msup>
      <mo>-</mo>
      <mn>4</mn>
      <mi>a</mi>
      <mi>c</mi>
    </mtd>
  </mtr>
  <mtr>
    <mtd>
      <maligngroup/>
      <msup>
        <mrow><mo fence=true>(</mo>
          <mn>2</mn>
          <mi>a</mi>
          <mi>x</mi>
          <mo>+</mo>
          <mi>b</mi>
        <mo fence=true>)</mo></mrow>
        <mn>2</mn>
      </msup>
    </mtd>
    <mtd><maligngroup/><mo>=</mo></mtd>
    <mtd>
      <maligngroup/>
      <msup><mi>b</mi><mn>2</mn></msup>
      <mo>-</mo>
      <mn>4</mn>
      <mi>a</mi>
      <mi>c</mi>
    </mtd>
    <mtd>
      <maligngroup/>
      <mspace width=2em />
      <mtext>|</mtext>
      <mo>&plusmn;</mo>
      <msqrt><mspace width=1em /></msqrt>
    </mtd>
  </mtr>
  <mtr>
    <mtd>
      <maligngroup/>
      <mn>2</mn><mi>a</mi><mi>x</mi>
      <mo>+</mo>
      <mi>b</mi>
    </mtd>
    <mtd><maligngroup/><mo>=</mo></mtd>
    <mtd>
      <maligngroup/>
      <mo>&plusmn;</mo>
      <msqrt>
        <msup><mi>b</mi><mn>2</mn></msup>
        <mo>-</mo>
        <mn>4</mn>
        <mi>a</mi>
        <mi>c</mi>
      </msqrt>
    </mtd>
    <mtd>
      <maligngroup/>
      <mspace width=2em />
      <mtext>|</mtext>
      <mo>-</mo>
      <mi>b</mi>
    </mtd>
  </mtr>
  <mtr>
    <mtd>
      <maligngroup/>
      <mn>2</mn><mi>a</mi><mi>x</mi>
    </mtd>
    <mtd><maligngroup/><mo>=</mo></mtd>
    <mtd>
      <maligngroup/>
      <mi>-b</mi>
      <mo>&plusmn;</mo>
      <msqrt>
        <msup><mi>b</mi><mn>2</mn></msup>
        <mo>-</mo>
        <mn>4</mn>
        <mi>a</mi>
        <mi>c</mi>
      </msqrt>
    </mtd>
    <mtd>
      <maligngroup/>
      <mspace width=2em />
      <mtext>|</mtext>
      <mo>&sdot;</mo>
      <mn>2</mn>
      <mi>a</mi>
    </mtd>
  </mtr>
  <mtr>
    <mtd>
      <maligngroup/>
      <mi>x</mi>
    </mtd>
    <mtd><maligngroup/><mo>=</mo></mtd>
    <mtd>
      <mfrac>
        <mrow>
          <maligngroup/>
          <mi>-b</mi>
          <mo>&plusmn;</mo>
          <msqrt>
            <msup><mi>b</mi><mn>2</mn></msup>
            <mo>-</mo>
            <mn>4</mn>
            <mi>a</mi>
            <mi>c</mi>
          </msqrt>
        </mrow>
        <mrow><mn>2</mn><mi>a</mi></mrow>
      </mfrac>
    </mtd>
  </mtr>
</mtable>

Logical statement:

φ \Leftrightarrow (α \land β) \lor (α \Rightarrow β) ⊻ ¬γ

<mi>&phi;</mi>
<mo>&hArr;</mo>
<mo fence=true>(</mo>
  <mi>&alpha;</mi>
  <mo>&and;</mo>
  <mi>&beta;</mi>
<mo fence=true>)</mo>
<mo>&or;</mo>
<mo fence=true>(</mo>
  <mi>&alpha;</mi>
  <mo>&rArr;</mo>
  <mi>&beta;</mi>
<mo fence=true>)</mo>
<mo>&#8891;</mo>
<mi>&not;&gamma;</mi>

A matrix with undefined size:

[\begin{matrix} x_{1, 1} & \dots & x_{1, n} \\ ⋮ & ⋱ & ⋮ \\ x_{m, 1} & \dots & x_{m, n} \end{matrix}]

<mrow><mo fence="true">[</mo><mtable>
  <mtr>
    <mtd>
      <msub>
        <mi>x</mi>
        <mrow>
          <mn>1</mn>
          <mo>,</mo>
          <mn>1</mn>
        </mrow>
      </msub>
    </mtd>
    <mtd>
      <mo>&hellip;</mo>
    </mtd>
    <mtd>
      <msub>
        <mi>x</mi>
        <mrow>
          <mn>1</mn>
          <mo>,</mo>
          <mn>n</mn>
        </mrow>
      </msub>
    </mtd>
  </mtr>
  <mtr>
    <mtd><mo>&vellip;</mo></mtd>
    <mtd><mo>&dtdot;</mo></mtd>
    <mtd><mo>&vellip;</mo></mtd>
  </mtr>
  <mtr>
    <mtd>
      <msub>
        <mi>x</mi>
        <mrow>
          <mn>m</mn>
          <mo>,</mo>
          <mn>1</mn>
        </mrow>
      </msub>
    </mtd>
    <mtd><mo>&hellip;</mo></mtd>
    <mtd>
      <msub>
        <mi>x</mi>
        <mrow>
          <mn>m</mn>
          <mo>,</mo>
          <mn>n</mn>
        </mrow>
      </msub>
    </mtd>
  </mtr>
</mtable><mo fence="true">]</mo></mrow>

Simple 2 times 4 matrix equation:

[\begin{matrix} 600 & 500 \\ 400 & 300 \end{matrix}] \overset{⇋}{H} = [\begin{matrix} 5 & 10 & 15 & 20 \\ 60 & 53 & 46 & 39 \end{matrix}]

<mrow><mo fence=true>[</mo><mtable>
  <mtr>
    <mtd><mn>600</mn></mtd>
    <mtd><mn>500</mn></mtd>
  </mtr>
  <mtr>
    <mtd><mn>400</mn></mtd>
    <mtd><mn>300</mn></mtd>
  </mtr>
</mtable><mo fence=true>]</mo></mrow>
<mover accent=true><mi>H</mi><mi>&lrhar;</mi></mover>
<mo>=</mo>
<mrow><mo fence=true>[</mo><mtable>
  <mtr>
    <mtd><mn>5</mn></mtd>
    <mtd><mn>10</mn></mtd>
    <mtd><mn>15</mn></mtd>
    <mtd><mn>20</mn></mtd>
  </mtr>
  <mtr>
    <mtd><mn>60</mn></mtd>
    <mtd><mn>53</mn></mtd>
    <mtd><mn>46</mn></mtd>
    <mtd><mn>39</mn></mtd>
  </mtr>
</mtable><mo fence=true>]</mo></mrow>

A vector:

\vec{V} = [\begin{matrix} x_{1} \\ ⋮ \\ x_{1} \end{matrix}]

<mover accent="true"><mi>V</mi><mi>&rarr;</mi></mover>
<mo>=</mo>
<mrow><mo fence="true">[</mo><mtable>
<mtr>
  <mtd>
    <msub><mi>x</mi><mn>1</mn></msub>
  </mtd>
</mtr>
<mtr>
  <mtd>
    <mo>&vellip;</mo>
  </mtd>
</mtr>
<mtr>
  <mtd>
    <msub><mi>x</mi><mn>1</mn></msub>
  </mtd>
</mtr>
</mtable><mo fence="true">]</mo></mrow>