Sandbox in MAN Wiki

[[!include contents]]

## Markdown+itex2MML Sandbox## {: #sandbox}

Play around [below](#more). Your changes will, periodically, be rolled back.

### Some examples### {: #examples}

\[
\label{objftn}
  \mathop{min} w_h p_h + w_r p_r + w_l p_l
\]

\[
\left\{
\begin{aligned}
\nabla \times \vec{\mathbf{B}} - \frac{1}{c}\frac{\partial\vec{\mathbf{E}}}{\partial t} &= \frac{4\pi}{c}\vec{\mathbf{j}} \\
\nabla \cdot \vec{\mathbf{E}} &= 4 \pi \rho \\
\nabla \times \vec{\mathbf{E}}+\frac{1}{c}\frac{\partial\vec{\mathbf{B}}}{\partial t} &= \vec{\mathbf{0}} \\
\nabla \cdot \vec{\mathbf{B}} &= 0
\end{aligned}
\right.
\]

Here's an equation
\[
\label{gaussian}
  {\int_{-\infty}^\infty e^{-a x^2/2} \mathrm{d}x} = \sqrt{\frac{2\pi}{a}}
\]
which we can later refer[^nb] back to as (eq:gaussian).

[^nb]: You can also refer to it as \eqref{gaussian}. _Chacun à  son goût!_{: xml-lang="fr"}.

Aligned equations:
\[\begin{aligned}
  a+b &= b+a \\
  a+(b+c) &= (a+b)+c
\end{aligned}\]

The Dirac equation (boxed):
$$
   \boxed{(i\slash{D}+m)\psi = 0}
$$

Here's the table of Clifford[^moreInfo] algebras over $\mathbb{R}$:


[^moreInfo]: For more information, see [Wikipedia](http://en.wikipedia.org/wiki/Clifford_algebra).

{:r: scope="row"}

$j$|$0$|$1$|$2$|$3$|$4$|$5$|$6$|$7$|$8$
--:|:-:|:-:|:-:|:-:|:-:|:-:|:-:|:-:|:-:|
{:r}$\mathcal{C}\ell_{j}^-$|$\mathbb{R}$|$\mathbb{C}$|$\mathbb{H}$|$\mathbb{H}\oplus\mathbb{H}$|$\mathbb{H}(2)$|$\mathbb{C}(4)$|$\mathbb{R}(8)$|$\mathbb{R}(8)\oplus\mathbb{R}(8)$|$\mathbb{R}(16)$
{:r}$\mathcal{C}\ell_{j}^+$|$\mathbb{R}$|$\mathbb{R}\oplus\mathbb{R}$|$\mathbb{R}(2)$|$\mathbb{C}(2)$|$\mathbb{H}(2)$|$\mathbb{H}(2)\oplus\mathbb{H}(2)$|$\mathbb{H}(4)$|$\mathbb{C}(8)$|$\mathbb{R}(16)$
{: class="plaintable" style="text-align:center;margin-left:0;" summary="The Clifford Algebras"}

where the generators of $\mathcal{C}\ell_{j}^\pm$ satisfy
$$
  \gamma_i\gamma_j +\gamma_j \gamma_i =\pm 2\delta_{i j}
$$
and $\mathcal{C}\ell_{n+8}^\pm = \mathcal{C}\ell_n^\pm \otimes \mathbb{R}(16)$.

\[
\lim_{n \to \infty}
\sum_{k=1}^n \frac{1}{k^2}
= \frac{\pi^2}{6}
\]

\[
    \mathbf{V}_{1} \times \mathbf{V}_{2} =
    \begin{vmatrix}
        \mathbf{i} & \mathbf{j} & \mathbf{k} \\\\
        \frac{\partial X}{\partial u} & \frac{\partial Y}{\partial u} & 0 \\
        \frac{\partial X}{\partial v} & \frac{\partial Y}{\partial v} & 0 \\
    \end{vmatrix}
\]

###More Examples### {: #more}

\[\nabla \times \vec{E} = - \frac {\partial \vec{B}}{\partial t}\]

\[
\label{Ampere}
  \oint \mathbf{B}\cdot \mathrm{d}\mathbf{l} = \href{https://en.wikipedia.org/wiki/Amp%C3%A8re%27s_circuital_law}{\mu_0 I_\text{enc}}
\]

$H^1(\mathcal{Z}, \mathcal{O}(-k))$ 
Let $G=(V,E)$ be a graph, with $w:V\to [0,1]$ a weight function.

\[\{Q_i, Q_j\} = \delta_{ij}\mathcal{H}.\]

### Theorems ### {: #theorems}

+-- {: .un_defn}
###### Definition
Let $H$ be a subgroup of a group $G$.  A *left coset* of $H$ in $G$ is a subset of $G$ that is of the form $x H$, where $x \in G$ and $x H = \{ x h : h \in H \}$.

Similarly a *right coset* of $H$ in $G$ is a subset of $G$ that is of the form $H x$, where $H x = \{ h x : h \in H \}$.
=--

+-- {: .num_lemma #LeftCosetsDisjoint}
###### Lemma
Let $H$ be a subgroup of a group $G$, and let $x$ and $y$ be
elements of $G$. Suppose that $x H \cap y H$ is non-empty. Then $x H = y H$.
=--

+-- {: .proof}
###### Proof
Let $z$ be some element of $x H \cap y H$.  Then $z = x a$ for some $a \in H$, and $z = y b$ for some $b \in H$.  If $h$ is any element of $H$ then $a h \in H$ and $a^{-1}h \in H$, since $H$ is a subgroup of $G$. But $z h = x(a h)$ and $xh = z(a^{-1}h)$ for all $h \in H$. Therefore $z H \subset x H$ and $x H \subset z H$, and thus $x H = z H$. Similarly $y H = z H$, and thus $x H = y H$, as required.
=--

+-- {: .num_lemma #SizeOfLeftCoset}
###### Lemma
Let $H$ be a finite subgroup of a group $G$.  Then each left
coset of $H$ in $G$ has the same number of elements as $H$.
=--

+-- {: .proof}
###### Proof
Let $H = \{ h_1, h_2,\ldots, h_m\}$, where $h_1, h_2,\ldots, h_m$ are distinct, and let $x$ be an element of $G$.  Then the left coset $x H$ consists of the elements $x h_j$ for $j = 1,2,\ldots,m$. Suppose that $j$ and $k$ are integers between $1$ and $m$ for which $x h_j = x h_k$.  Then $h_j = x^{-1} (x h_j) = x^{-1} (x h_k) = h_k$, and thus $j = k$, since $h_1, h_2,\ldots, h_m$ are distinct.  It follows that the elements $x h_1, x h_2,\ldots, x h_m$ are distinct. We conclude that the subgroup $H$ and the left
coset $x H$ both have $m$ elements, as required.
=--

+-- {: .num_theorem #Lagrange}
###### Theorem
**(Lagrange's Theorem).** Let $G$ be a finite group, and let $H$ be a subgroup of $G$. Then the order of $H$ divides the order of $G$.
=--

+-- {: .proof}
###### Proof
Each element $x$ of $G$ belongs to at least one left coset of $H$ in $G$ (namely the coset $x H$), and no element can belong to two distinct left cosets of $H$ in $G$ (see Lemma \ref{LeftCosetsDisjoint}).  Therefore every element of $G$ belongs to exactly one left coset of $H$. Moreover each left coset of $H$ contains $|H|$ elements (Lemma \ref{SizeOfLeftCoset}).  Therefore $|G| = n |H|$, where $n$ is the number of left cosets of $H$ in $G$. The result follows.
=--

+-- {: .num_cor #OrderDivides}
###### Corollary
Let $x$ be an element of a finite group $G$. Then the order of $x$ divides the order of $G$.
=--

+-- {: .num_theorem}
###### Theorem
Let $f : \Delta \longrightarrow \Delta,$ where $\Delta=\{z\in\mathbb{C}: \vert z \vert \lt 1\}$, be analytic with $a \in \Delta$. Then 
$$ 
\left\vert\frac{f(z)-f(a)}{1-\overline{f(a)}f(z)}\right\vert\le \left\vert\frac{z-a}{1-\overline{a}z}\right\vert 
$$
for all $\vert z \vert \le 1$ and 
$$
\frac{\vert f'(a)\vert}{1-\vert f(a)\vert^2}\le \frac{1}{1-\vert a \vert^2}.
$$  
Furthermore, equality holds iff $f$ realizes a conformal mapping of $\Delta$ onto itself.
=--
+-- {: .proof}
###### Proof
Let $w=\frac{z-a}{1-\overline{a}z}$ and put $\phi(w)=\frac{f(z)-f(a)}{1-\overline{f(a)}f(z)}$. Define for $\abs{b}\lt 1$ $C_b(z)=\frac{z-b}{1-\overline{b}z}.$  All conformal maps from $\Delta$ to itself, sending $b$ to $0$, are of the form $C_b(z)e^{i\gamma}$ for $\gamma\in[0,2\pi].$ In this notation, $\phi(w)=C_{f(a)}\circ f \circ C_a^{-1}(w),$ where $C_a^{-1}$ is the inverse of $C_a$ as a function. Note that $C_a(z)$ is conformal, so it has an inverse. It is clear that $\phi(0)=C_{f(a)}\circ f \circ C_a^{-1}(0)=C_{f(a)}(f(a))=0$. Since $C_a^{-1}: \Delta \longrightarrow \Delta$ and $f : \Delta \longrightarrow \Delta$ and $C_{f(a)}: \Delta \longrightarrow \Delta,$ then $\vert\phi(w)\vert\lt 1$ for $\vert w\vert \lt 1 .$  Applying Schwarz's lemma, we obtain $\vert\phi(w)\vert\le \vert w \vert$ for $\vert w \vert \le 1$.  Furthermore, if equality holds, then $f(z)=e^{i\gamma'} z$ for $\gamma'\in [0,2\pi]$. Therefore, 
\[
\label{eqn1}
\left\vert\frac{f(z)-f(a)}{1-\overline{f(a)}f(z)}\right\vert\le \left\vert\frac{z-a}{1-\overline{a}z}\right\vert
\]
for all $\vert z \vert\le 1.$  Rearranging, we obtain
$$
\left\vert\frac{f(z)-f(a)}{z-a}\right\vert\le\left\vert\frac{1-\overline{f(a)}f(z)}{1-\overline{a}z}\right\vert.
$$  
If we take the limit as $z$ tends to $a$, we obtain 
$$
\left\vert f'(a)\right\vert \le \left\vert\frac{1-\vert f(a)\vert ^2}{1-\vert a \vert^2}\right\vert=\frac{1-\vert f(a)\vert^2}{1-\vert a \vert^2},
$$ 
or 
$$
\frac{\vert f'(a)\vert}{1-\vert f(a)\vert^2}\le \frac{1}{1-\vert a \vert^2}.
$$
As said above, if equality holds in (eq:eqn1), then Schwarz's lemma tells us that $\phi(w)=e^{i\gamma'}w$.  Thus, $\phi(w)=C_{f(a)}\circ f \circ C_a^{-1}(w)=e^{i\gamma'}w,$ so $f(z)=C_{f(a)}^{-1}(e^{i\gamma'}C_a(z))$. Since $e^{i\gamma'}C_a(z)$ is conformal, $C_{f(a)}^{-1},$ the inverse function of $C_{f(a)}$, is conformal, and a composition of conformal maps is conformal, then $f$ is a conformal map of $\Delta$ onto itself.  Conversely, if $f$ is a conformal map of $\Delta$ onto itself, then $\phi(w)=C_{f(a)}\circ f \circ C_a^{-1}(w)=e^{i\gamma}C_b(w),$ since a composition of conformal maps is conformal and because all conformal maps from $\Delta$ onto itself are of the form $e^{i\gamma}C_b(w).$  We also know that $\phi(0)=0,$ so $b=0$. Therefore, 
$$
\phi(w)=e^{i\gamma}C_0(w)=e^{i\gamma}w \Leftrightarrow \vert\phi(w)\vert=\vert w \vert \Leftrightarrow \left\vert\frac{f(z)-f(a)}{1-\overline{f(a)}f(z)}\right\vert=\left\vert\frac{z-a}{1-\overline{a}z}\right\vert
$$ 
for all $\vert z \vert\le 1$. In sum, equality holds in (eq:eqn1) iff $f$ is a conformal map from $\Delta$ to itself.
=--
+-- {: .num_remark}
###### Remark
Someone needs to code \abs, i.e. \left\vert # \right\vert.  This is terribly annoying. Also, when using $$, why must one enter a line before and after.  
=--


####SVG graphics, created in SVG-Edit:

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$$
=--

+-- {: #CommutativeCube}
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  <marker refX="8" orient="auto" markerHeight="5" markerWidth="5" markerUnits="strokeWidth" refY="5" id="se_arrow_83381_fw5" viewBox="0 0 10 10">
   <path fill="#000" d="m0,0l10,5l-10,5l5,-5l-5,-5z" id="svg_83381_3"/>
  </marker>
  <linearGradient id="svg_83381_1" x1="0" y1="0" x2="1" y2="1">
   <stop offset="0.25" stop-color="#888"/>
   <stop offset="1" stop-color="#000"/>
  </linearGradient>
 </defs>
 <g class="layer">
  <title>Layer 1</title>
  <foreignObject y="3" x="2" width="20" height="24" font-size="16" id="svg_83381_4">
   <math xmlns="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" display="inline">
    <semantics>
     <mrow>
      <msub>
       <mi>A</mi>
       <mn>0</mn>
      </msub>
     </mrow>
     <annotation encoding="application/x-tex">A_0</annotation>
    </semantics>
   </math>
  </foreignObject>
  <foreignObject y="3" x="156" width="20" height="24" font-size="16" id="svg_83381_5">
   <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML">
    <semantics>
     <mrow>
      <msub>
       <mi>C</mi>
       <mn>0</mn>
      </msub>
     </mrow>
     <annotation encoding="application/x-tex">C_0</annotation>
    </semantics>
   </math>
  </foreignObject>
  <foreignObject y="160" x="2" width="20" height="24" font-size="16" id="svg_83381_6">
   <math xmlns="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" display="inline">
    <semantics>
     <mrow>
      <msub>
       <mi>A</mi>
       <mn>1</mn>
      </msub>
     </mrow>
     <annotation encoding="application/x-tex">A_1</annotation>
    </semantics>
   </math>
  </foreignObject>
  <foreignObject y="160" x="156" width="20" height="24" font-size="16" id="svg_83381_7">
   <math xmlns="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" display="inline">
    <semantics>
     <mrow>
      <msub>
       <mi>C</mi>
       <mn>1</mn>
      </msub>
     </mrow>
     <annotation encoding="application/x-tex">C_1</annotation>
    </semantics>
   </math>
  </foreignObject>
  <foreignObject y="80" x="80" width="20" height="24" font-size="16" id="svg_83381_8">
   <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML">
    <semantics>
     <mrow>
      <msub>
       <mi>B</mi>
       <mn>0</mn>
      </msub>
     </mrow>
     <annotation encoding="application/x-tex">B_0</annotation>
    </semantics>
   </math>
  </foreignObject>
  <foreignObject y="80" x="234" width="20" height="24" font-size="16" id="svg_83381_9">
   <math xmlns="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" display="inline">
    <semantics>
     <mrow>
      <msub>
       <mi>D</mi>
       <mn>0</mn>
      </msub>
     </mrow>
     <annotation encoding="application/x-tex">D_0</annotation>
    </semantics>
   </math>
  </foreignObject>
  <foreignObject y="237" x="80" width="20" height="24" font-size="16" id="svg_83381_10">
   <math xmlns="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" display="inline">
    <semantics>
     <mrow>
      <msub>
       <mi>B</mi>
       <mn>1</mn>
      </msub>
     </mrow>
     <annotation encoding="application/x-tex">B_1</annotation>
    </semantics>
   </math>
  </foreignObject>
  <foreignObject y="237" x="234" width="20" height="24" font-size="16" id="svg_83381_11">
   <math xmlns="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" display="inline">
    <semantics>
     <mrow>
      <msub>
       <mi>D</mi>
       <mn>1</mn>
      </msub>
     </mrow>
     <annotation encoding="application/x-tex">D_1</annotation>
    </semantics>
   </math>
  </foreignObject>
  <line marker-end="url(#se_arrow_83381_fw4)" y2="160" x2="10" y1="30" x1="10" fill="none" stroke-width="2" stroke="#666" id="svg_83381_12"/>
  <line marker-end="url(#se_arrow_83381_fw4)" y2="172" x2="150" y1="172" x1="25" fill="none" stroke-width="2" stroke="#666" id="svg_83381_13"/>
  <line marker-end="url(#se_arrow_83381_fw4)" y2="14" x2="150" y1="14" x1="25" fill="none" stroke-width="2" stroke="#666" id="svg_83381_14"/>
  <line marker-end="url(#se_arrow_83381_fw4)" y2="160" x2="165" y1="30" x1="165" fill="none" stroke-width="2" stroke="#666" id="svg_83381_15"/>
  <line marker-end="url(#se_arrow_83381_fw5)" y2="235" x2="90" y1="105" x1="90" fill="none" stroke-width="2" stroke="#000" id="svg_83381_16"/>
  <line marker-end="url(#se_arrow_83381_fw5)" y2="248" x2="225" y1="248" x1="105" fill="none" stroke-width="2" stroke="#000" id="svg_83381_17"/>
  <line marker-end="url(#se_arrow_83381_fw5)" y2="90" x2="225" y1="90" x1="105" fill="none" stroke-width="2" stroke="#000" id="svg_83381_18"/>
  <line marker-end="url(#se_arrow_83381_fw5)" y2="235" x2="243" y1="105" x1="243" fill="none" stroke-width="2" stroke="#000" id="svg_83381_19"/>
  <line marker-end="url(#se_arrow_83381_fw5)" y2="80" x2="78" y1="25" x1="20" fill="none" stroke-width="2" stroke="url(#svg_83381_1)" id="svg_83381_20"/>
  <line marker-end="url(#se_arrow_83381_fw5)" y2="80" x2="233" y1="25" x1="175" fill="none" stroke-width="2" stroke="url(#svg_83381_1)" id="svg_83381_21"/>
  <line marker-end="url(#se_arrow_83381_fw5)" y2="240" x2="78" y1="185" x1="20" fill="none" stroke-width="2" stroke="url(#svg_83381_1)" id="svg_83381_22"/>
  <line marker-end="url(#se_arrow_83381_fw5)" y2="240" x2="233" y1="185" x1="175" fill="none" stroke-width="2" stroke="url(#svg_83381_1)" id="svg_83381_23"/>
 </g>
</svg>
\end{svg}
$$
=--

+-- {: #box_diagram style="text-align:center; overflow: auto;"}
<svg width="480" height="230" xmlns="http://www.w3.org/2000/svg" xmlns:svg="http://www.w3.org/2000/svg" xmlns:se="http://svg-edit.googlecode.com" xmlns:math="http://www.w3.org/1998/Math/MathML" se:nonce="94113">
 <desc>Box diagram</desc>
 <defs>
  <marker id="se_arrow_94113_fw" markerHeight="5" markerUnits="strokeWidth" markerWidth="5" orient="auto" refX="5" refY="5" viewBox="0 0 10 10">
   <path d="m0,0l10,5l-10,5l5,-5l-5,-5z" fill="#000000" id="svg_94113_5"/>
  </marker>
 </defs>
 <g class="layer">
  <foreignObject font-size="16" height="22" id="svg_94113_6" width="32" x="229" y="193">
   <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML">
    <semantics>
     <mrow>
      <msup>
       <mi>W</mi>
       <mo>+</mo>
      </msup>
     </mrow>
     <annotation encoding="application/x-tex">W^+</annotation>
    </semantics>
   </math>
  </foreignObject>
  <foreignObject font-size="16" height="22" id="svg_94113_7" width="32" x="230" y="17">
   <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML">
    <semantics>
     <mrow>
      <msup>
       <mi>W</mi>
       <mo>−</mo>
      </msup>
     </mrow>
     <annotation encoding="application/x-tex">W^-</annotation>
    </semantics>
   </math>
  </foreignObject>
  <foreignObject font-size="16" height="22" id="svg_94113_8" width="16" x="4" y="177">
   <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML">
    <semantics>
     <mrow>
      <mover>
       <mi>s</mi>
       <mo>¯</mo>
      </mover>
     </mrow>
     <annotation encoding="application/x-tex">\overline{s}</annotation>
    </semantics>
   </math>
  </foreignObject>
  <foreignObject font-size="16" height="20" id="svg_94113_9" width="20" x="460" y="177">
   <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML">
    <semantics>
     <mrow>
      <mover>
       <mi>d</mi>
       <mo>¯</mo>
      </mover>
     </mrow>
     <annotation encoding="application/x-tex">\overline{d}</annotation>
    </semantics>
   </math>
  </foreignObject>
  <title>Layer 1</title>
  <foreignObject font-size="16" height="20" id="svg_94113_1" width="20" x="460" y="28">
   <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML">
    <semantics>
     <mrow>
      <mi>s</mi>
     </mrow>
     <annotation encoding="application/x-tex">s</annotation>
    </semantics>
   </math>
  </foreignObject>
  <foreignObject font-size="16" height="20" id="svg_94113_2" width="20" x="4" y="28">
   <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML">
    <semantics>
     <mrow>
      <mi>d</mi>
     </mrow>
     <annotation encoding="application/x-tex">d</annotation>
    </semantics>
   </math>
  </foreignObject>
  <path d="m20,40l75,0l75,0" fill="none" id="svg_94113_10" marker-mid="url(#se_arrow_94113_fw)" stroke="#000000" stroke-width="2"/>
  <path d="m170,40l0,70l0,80" fill="none" id="svg_94113_11" marker-mid="url(#se_arrow_94113_fw)" stroke="#000000" stroke-width="2"/>
  <path d="m170,190l-75,0l-75,0" fill="none" id="svg_94113_12" marker-mid="url(#se_arrow_94113_fw)" stroke="#000000" stroke-width="2"/>
  <foreignObject font-size="16" height="20" id="svg_94113_3" width="52" x="110" y="104">
   <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML">
    <semantics>
     <mrow>
      <mi>u</mi>
      <mo>,</mo>
      <mspace width="0.16667em"/>
      <mi>c</mi>
      <mo>,</mo>
      <mspace width="0.16667em"/>
      <mi>t</mi>
     </mrow>
     <annotation encoding="application/x-tex">u,\, c,\, t</annotation>
    </semantics>
   </math>
  </foreignObject>
  <path d="m170,40c10,10 10,-10 20,0s10,-10 20,0s10,-10 20,0s10,-10 20,0s10,-10 20,0s10,-10 20,0s10,-10 20,0" fill="none" id="svg_94113_13" stroke="#000000" stroke-width="2"/>
  <path d="m460,190l-75,0l-75,0" fill="black" id="svg_94113_14" marker-mid="url(#se_arrow_94113_fw)" stroke="#000000" stroke-width="2"/>
  <path d="m310,190l0,-70l0,-80" fill="black" id="svg_94113_15" marker-mid="url(#se_arrow_94113_fw)" stroke="#000000" stroke-width="2"/>
  <path d="m310,40l75,0l75,0" fill="black" id="svg_94113_16" marker-mid="url(#se_arrow_94113_fw)" stroke="#000000" stroke-width="2"/>
  <path d="m170,190c10,10 10,-10 20,0s10,-10 20,0s10,-10 20,0s10,-10 20,0s10,-10 20,0s10,-10 20,0s10,-10 20,0" fill="none" id="svg_94113_17" stroke="#000000" stroke-width="2"/>
  <foreignObject font-size="16" height="20" id="svg_94113_4" width="52" x="315" y="104">
   <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML">
    <semantics>
     <mrow>
      <mi>u</mi>
      <mo>,</mo>
      <mspace width="0.16667em"/>
      <mi>c</mi>
      <mo>,</mo>
      <mspace width="0.16667em"/>
      <mi>t</mi>
     </mrow>
     <annotation encoding="application/x-tex">u,\, c,\, t</annotation>
    </semantics>
   </math>
  </foreignObject>
 </g>
</svg><br />
$K^0\overline{K}^0$ Mixing
=--

####Yet More examples

* SVG:

<div style="margin: 10px 10px 10px 10em;width:6em;height:6em;">
<svg xmlns="http://www.w3.org/2000/svg" width="100%" height="100%"  viewBox="0 0 100 100">
  <g stroke-width="5" fill="none">

    <g stroke="#fa3">
      <circle r="6" cx="24" cy="16"/>
      <circle r="6" cx="51" cy="88"/>
      <path d="M35,39a16,16 0,1,0 4-5l-11-13M50,62l1,20"/>
    </g>
    <g stroke="#a38">
      <circle r="6" cx="72" cy="13"/>
      <path d="M37,23a27,27 0,0,1 39,24M68,18l-5,7"/>
    </g>

    <g stroke="#e33">
      <circle r="6" cx="49" cy="46"/>
      <circle r="6" cx="11" cy="62"/>
      <circle r="6" cx="89" cy="58"/>
      <path d="M16,60l8-4M56,73a27,27 0,0,0 19-19l8,2M45,73a27,27 0,0,1-17-43"/>
    </g>
  </g>
</svg>
</div>

* Animated SVG
{: #SVGanimation}

<svg width="8cm" height="3cm" viewBox="0 0 800 300"
     xmlns="http://www.w3.org/2000/svg" version="1.1">
  <desc>Example anim01 - demonstrate animation elements</desc>
  <rect x="1" y="1" width="798" height="298" 
        fill="none" stroke="blue" stroke-width="2" />
  <rect id="RectElement" x="300" y="100" width="300" height="100"
        fill="rgb(255,255,0)"  >
    <animate id="fooanim" attributeName="x" attributeType="XML"
             begin="0s" dur="9s" fill="freeze" from="300" to="0" repeatCount="indefinite"/>
    <animate attributeName="y" attributeType="XML"
             begin="0s" dur="9s" fill="freeze" from="100" to="0" repeatCount="indefinite"/>
    <animate attributeName="width" attributeType="XML"
             begin="0s" dur="9s" fill="freeze" from="300" to="800" repeatCount="indefinite"/>
    <animate attributeName="height" attributeType="XML"
             begin="0s" dur="9s" fill="freeze" from="100" to="300" repeatCount="indefinite"/>
  </rect>
  <g transform="translate(100,100)" >
    <g id="TextElement">
      <foreignObject markdown='1' x="0" y="0" width="186"
           height="49" font-size="20">
       ${\int_{-\infty}^{\infty}e^{-a x^2}d x}=\sqrt{\tfrac{\pi}{a}}$
      </foreignObject>
      <animateTransform attributeName="transform" attributeType="XML"
           type="scale" additive="sum" keyTimes="0;.333;1" values="1;1;3"
           begin="0s" dur="9s" fill="freeze" repeatCount="indefinite" />
      <animate attributeName="visibility" attributeType="CSS"
           keyTimes="0;.333;1" values="hidden;visible;hidden"
           begin="0s" dur="9s" fill="freeze" repeatCount="indefinite" />
      <animateMotion value="0,0;0,0;100,40" keyTimes="0;.333;1"
           begin="0s" dur="9s" fill="freeze" repeatCount="indefinite" />
      <animate attributeName="color" attributeType="CSS"
           values="#0000FF;#0000FF;#880000" keyTimes="0;.333;1"
           begin="0s" dur="9s" fill="freeze" repeatCount="indefinite" />
      <animateTransform attributeName="transform" attributeType="XML"
           type="rotate" values="-30;-30;0" keyTimes="0;.333;1" additive="sum"
           begin="0s" dur="9s" fill="freeze" repeatCount="indefinite" />
    </g>
  </g>
</svg>

* Complicated commutative diagrams (equations in SVG)

<svg xmlns="http://www.w3.org/2000/svg" width="18.75em" height="17.5em" viewBox="-30 -25 270 255">
 <desc>Complicated commutative diagram, realized in SVG</desc>
 <defs>
  <marker id="svg295arrowhead" viewBox="0 0 10 10" refX="0" refY="5" markerUnits="strokeWidth" markerWidth="8" markerHeight="5" orient="auto">
   <path d="M 0 0 L 10 5 L 0 10 z"/>
  </marker>
  <marker id="svg296arrowhead" viewBox="0 0 10 10" refX="0" refY="5" markerUnits="strokeWidth" markerWidth="6" markerHeight="4" orient="auto">
   <path d="M 0 0 L 10 5 L 0 10 z"/>
  </marker>
 </defs>
 <g font-size="14">
  <foreignObject markdown='1' x="53"  y="-11" width="16" height="22">$1$</foreignObject>
  <foreignObject markdown='1' x="155" y="-11" width="16" height="22">$1$</foreignObject>
  <foreignObject markdown='1' x="53"  y="192" width="16" height="22">$1$</foreignObject>
  <foreignObject markdown='1' x="155" y="192" width="16" height="22">$1$</foreignObject>
  <foreignObject markdown='1' x="220"  y="90" width="16" height="22">$Id$</foreignObject>
  <foreignObject markdown='1' x="-18"  y="90" width="16" height="22">$Id$</foreignObject>
  <foreignObject markdown='1' x="52"   y="90" width="16" height="22">$A$</foreignObject>
  <foreignObject markdown='1' x="155"  y="90" width="16" height="22">$B$</foreignObject>
  <foreignObject markdown='1' x="115"  y="90" width="16" height="22">$\rho$</foreignObject>
  <foreignObject markdown='1' x="97"  y="-19" width="16" height="22">$H$</foreignObject>
  <foreignObject markdown='1' x="97"  y="204" width="16" height="22">$H$</foreignObject>
  <foreignObject markdown='1' x="104"  y="46" width="16" height="22">$K$</foreignObject>
  <foreignObject markdown='1' x="100" y="137" width="16" height="22">$K'$</foreignObject>
  <foreignObject markdown='1' x="92"   y="21" width="18" height="22">$\phi_1$</foreignObject>
  <foreignObject markdown='1' x="110" y="159" width="18" height="22">$\phi_2$</foreignObject>
  <foreignObject markdown='1' x="31"   y="36" width="22" height="22">$N_A$</foreignObject>
  <foreignObject markdown='1' x="163"  y="36" width="22" height="22">$N_B$</foreignObject>
  <foreignObject markdown='1' x="32"  y="134" width="24" height="24">$N^\vee_A$</foreignObject>
  <foreignObject markdown='1' x="162" y="134" width="24" height="24">$N^\vee_B$</foreignObject>
 </g>
 <g fill="none" stroke="#000">
  <g marker-end="url(#svg295arrowhead)">
   <path d="M64.3,1H137"/>
   <path d="M159,11v67"/>
   <path d="M159,112.3v68"/>
   <path d="M56.3,112v68"/>
   <path d="M65,204.3h72"/>
   <path d="M56.3,11.3v67"/>
   <path d="M168.3,9c0,0,46,39.3,46,92.7s-36.7,85-36.7,85"/>
   <path d="M48,8.7c0,0-47,44.3-47,94s37.3,84,37.3,84"/>
   <path d="M63.3,112c0,0,16,27.3,44.7,27s39-20.6,39-20.6"/>
   <path d="M63.3,93.3C63.3,93.3,84,66,108,66s39,20.6,39,20.6"/>
  </g>
  <g stroke-width="2">
   <path stroke-width="4" d="M43,102.7H19.3"/>
   <path stroke="#FFF" d="M43,102.7H19.3" marker-end="url(#svg296arrowhead)"/>
   <path stroke-width="4" d="M205,102.7h-22"/>
   <path stroke="#FFF" d="M205,102.7h-22" marker-end="url(#svg296arrowhead)"/>
   <path stroke-width="4" d="M146,15.5l-72,55"/>
   <path stroke="#FFF" d="M146,15.5l-72,55" marker-end="url(#svg296arrowhead)"/>
   <path stroke-width="4" d="M147,126l-72,65"/>
   <path stroke="#FFF" d="M147,126l-72,65" marker-end="url(#svg296arrowhead)"/>
   <path stroke-width="4" d="M108,75v46"/>
   <path stroke="#FFF" d="M108,75v46" marker-end="url(#svg296arrowhead)"/>
  </g>
 </g>
</svg>

* SVG in equations.

 In $SU(3)$, $\begin{svg}
<svg xmlns="http://www.w3.org/2000/svg" width="30" height="16" viewBox="0 0 30 16">
  <desc>Rank-2 Symmetric Tensor Representation</desc>
  <g transform="translate(5,5)" fill="#FCC" stroke="#000" stroke-width="2">
    <rect width="10" height="10"/>
    <rect width="10" height="10" x="10"/>
  </g>
</svg>
\end{svg}\includegraphics[width=2em]{young1}
 \otimes
\begin{svg}
<svg xmlns="http://www.w3.org/2000/svg" width="20" height="16" viewBox="0 0 20 16">
  <desc>Fundamental Representation</desc>
  <g transform="translate(5,5)" fill="#FCC" stroke="#000" stroke-width="2">
    <rect width="10" height="10"/>
  </g>
</svg>
\end{svg}\includegraphics[width=1em]{young2}
 =
\begin{svg}
<svg xmlns="http://www.w3.org/2000/svg" width="30" height="26" viewBox="0 0 30 26">
  <desc>Adjoint Representation</desc>
  <g transform="translate(5,5)" fill="#FCC" stroke="#000" stroke-width="2">
    <rect width="10" height="10"/>
    <rect width="10" height="10" x="10"/>
    <rect width="10" height="10" y="10"/>
  </g>
</svg>
\end{svg}\includegraphics[width=2em]{young3}
 \oplus
\begin{svg}
<svg xmlns="http://www.w3.org/2000/svg" width="40" height="16" viewBox="0 0 40 16">
  <desc>Rank-3 Symmetric Tensor Representation</desc>
  <g transform="translate(5,5)" fill="#FCC" stroke="#000" stroke-width="2">
    <rect width="10" height="10"/>
    <rect width="10" height="10" x="10"/>
    <rect width="10" height="10" x="20"/>
  </g>
</svg>
\end{svg}\includegraphics[width=3em]{young4}$.

* Cases:

$$ r_{a+1} = \begin{cases}
    0 & \text{with prob.}\quad \exp(-\theta r_a) \\
    \max \lbrace \delta r_a, z \rbrace & \text{with prob.}\quad 1 - \exp(-\theta r_a)
  \end{cases} $$

* Stretchy Brackets:

$$q_a(z) = \sigma_a^{-1} \exp{\left[ -\frac{\gamma + z}{\sigma_a} \right]}$$

* Linearity of Quadrature Rules

$$\sum_{i = 1}^N {\left( {\alpha f(x_i ) + \beta g(x_i )} \right)w_i }  = \alpha \sum_{i = 1}^N {f(x_i )w_i }  + \beta \sum_{i = 1}^N {g(x_i )w_i } $$

* Linearity of Integrals

$${\int_a^b {\left( {\alpha f(x)\, + \beta g(x)} \right)dx = } \alpha \int_a^b {f(x)\,dx}  + \beta \int_a^b {g(x)\,dx} }$$

* Can we talk about $x_i^2$ inline?  What about $\int_a^b x^2\,dx$?  Inline fractions $\frac{x-x_2}{x_1-x_2}$?

* Big fractions

$$p_3 (x) = \left( {\frac{1}{2}} \right)\frac{{\left( {x - \frac{1}{2}} \right)\left( {x - \frac{3}{4}} \right)\left( {x - 1} \right)}}{{\left( {\frac{1}{4} - \frac{1}{2}} \right)\left( {\frac{1}{4} - \frac{3}{4}} \right)\left( {\frac{1}{4} - 1} \right)}} + \left( {\frac{1}{2}} \right)\frac{{\left( {x - \frac{1}{2}} \right)\left( {x - \frac{3}{4}} \right)\left( {x - 1} \right)}}{{\left( {\frac{1}{4} - \frac{1}{2}} \right)\left( {\frac{1}{4} - \frac{3}{4}} \right)\left( {\frac{1}{4} - 1} \right)}} + \left( {\frac{1}{2}} \right)\frac{{\left( {x - \frac{1}{2}} \right)\left( {x - \frac{3}{4}} \right)\left( {x - 1} \right)}}{{\left( {\frac{1}{4} - \frac{1}{2}} \right)\left( {\frac{1}{4} - \frac{3}{4}} \right)\left( {\frac{1}{4} - 1} \right)}} + \left( {\frac{1}{2}} \right)\frac{{\left( {x - \frac{1}{2}} \right)\left( {x - \frac{3}{4}} \right)\left( {x - 1} \right)}}{{\left( {\frac{1}{4} - \frac{1}{2}} \right)\left( {\frac{1}{4} - \frac{3}{4}} \right)\left( {\frac{1}{4} - 1} \right)}}
$$


* Diagram

$$
\begin{matrix}
     P_1(Y) &\to& P_1(X) \\
     \downarrow &\Downarrow\mathrlap{\sim}& \downarrow \\
     T' &\to& T
\end{matrix}
$$

* \mathcal{} versus \mathscr{}
$$
\begin{gathered}
\mathcal{ABCDEFGHIJKLMNOPQRSTUVWXYZ}\\
\text{versus}\\
\mathscr{ABCDEFGHIJKLMNOPQRSTUVWXYZ}
\end{gathered}
$$

***

\[\label{quiver}
\array{\arrayopts{\align{center}}
\begin{svg}
<svg width="108" height="122" xmlns="http://www.w3.org/2000/svg" xmlns:svg="http://www.w3.org/2000/svg" xmlns:se="http://svg-edit.googlecode.com" xmlns:math="http://www.w3.org/1998/Math/MathML" se:nonce="91165">
 <desc>A_n Quiver</desc>

 <g>
  <title>Layer 1</title>
  <g fill="none" stroke="black" id="svg_91165_1">
   <path d="m55.888885,27l25,20l0,30l-25,20l-24.999996,-20l0,-30l24.999996,-20z" id="svg_91165_2"/>
   <g stroke-dasharray="2" id="svg_91165_3">
    <path d="m55.888885,2l0,25" id="svg_91165_4"/>
    <path d="m80.888885,47l25,-20" id="svg_91165_5"/>
    <path d="m80.888885,77l25,20" id="svg_91165_6"/>
    <path d="m55.888885,97l0,25" id="svg_91165_7"/>
    <path d="m30.888889,77l-25,20" id="svg_91165_8"/>
    <path d="m30.888889,47l-25,-20" id="svg_91165_9"/>
   </g>
  </g>
  <g fill="red" id="svg_91165_10">
   <circle cx="55.888889" cy="27" r="4" id="svg_91165_11"/>
   <circle cx="80.888889" cy="47" r="4" id="svg_91165_12"/>
   <circle cx="80.888889" cy="77" r="4" id="svg_91165_13"/>
   <circle cx="55.888889" cy="97" r="4" id="svg_91165_14"/>
   <circle cx="30.888889" cy="77" r="3.999999" id="svg_91165_15"/>
   <circle cx="30.888889" cy="47" r="3.999999" id="svg_91165_16"/>
  </g>
  <foreignObject font-size="16" x="39.638889" y="0" width="16" height="26" id="svg_91165_17">
   <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML">
    <semantics>
     <mrow>
      <msub>
       <mi>v</mi>
       <mn>1</mn>
      </msub>
     </mrow>
     <annotation encoding="application/x-tex">v_1</annotation>
    </semantics>
   </math>
  </foreignObject>
  <foreignObject font-size="16" x="75.888889" y="15.25" width="16" height="26" id="svg_91165_18">
   <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML">
    <semantics>
     <mrow>
      <msub>
       <mi>v</mi>
       <mn>2</mn>
      </msub>
     </mrow>
     <annotation encoding="application/x-tex">v_2</annotation>
    </semantics>
   </math>
  </foreignObject>
  <foreignObject x="0.888889" y="27" width="20" height="29" id="svg_91165_19" font-size="16">
   <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline">
    <semantics>
     <mrow>
      <msub>
       <mi>v</mi>
       <mrow>
        <msub>
         <mi>n</mi>
         <mn>1</mn>
        </msub>
       </mrow>
      </msub>
     </mrow>
     <annotation encoding="application/x-tex">v_{n_1}</annotation>
    </semantics>
   </math>
  </foreignObject>
 </g>
</svg>
\end{svg}
} \equiv \left({U(k)}^{n_1},\{v_i\}\right)
\]

"This is **my text**", says Anymouse.

Ruby code example:

~~~~~~~~~~ {: lang=ruby}
    class Person
      attr_reader :name, :age
      def initialize(name, age)
        @name, @age = name, age
      end
      def <=>(person) # Comparison operator for sorting
        @age <=> person.age
      end
      def to_s
        "#@name (#@age)"
      end
    end
 
    group = [
      Person.new("Bob", 33), 
      Person.new("Chris", 16), 
      Person.new("Ash", 23) 
    ]
 
    puts group.sort.reverse
~~~~~~~~~~

A Python example:
{:  #python}

~~~~~~{: lang=python}
ListOfStrings(
              title = _("A List of some strings"),
              help  = _("A List of strings"),
              orientation = "horizontal"
              )
~~~~~~

## Tikz Pictures ## {: #tikz}

A new [[Tikz|feature]] to play around with. Requires an [additional install](https://github.com/distler/tex2svg) (in addition to Instiki) to render the Tikz code into SVG.

+--{.centered}
\begin{tikzpicture}[decoration={markings,
mark=at position .5 with {\arrow{>}}}]
\usetikzlibrary{arrows,shapes,decorations.markings}
\begin{scope}[scale=2.0]
\node[Bl,scale=.75] (or1) at (8,3) {};
\node[scale=1] at (8.7,2.9) {$D3$ brane};
\node[draw,diamond,fill=yellow,scale=.3] (A1) at (7,0) {}; 
\draw[dashed] (A1) -- (7,-.7);
\node[draw,diamond,fill=yellow,scale=.3] (A2) at (7.5,0) {}; 
\draw[dashed] (A2) -- (7.5,-.7);
\node[draw,diamond,fill=yellow,scale=.3] (A3) at (8,0) {}; 
\draw[dashed] (A3) -- (8,-.7);
\node[draw,diamond,fill=yellow,scale=.3] (A4) at (8.5,0) {}; 
\draw[dashed] (A4) -- (8.5,-.7);
\node[draw,diamond,fill=yellow,scale=.3] (A5) at (9,0) {};
\draw[dashed] (A5) -- (9,-.7);
\node[draw,circle,fill=aqua,scale=.3] (B) at (9.5,0) {};
\draw[dashed] (B) -- (9.5,-.7);
\node[draw,regular polygon,regular polygon sides=5,fill=purple,scale=.3] (C1) at (10,0) {}; 
\draw[dashed] (C1) -- (10,-.7);
\node[draw,regular polygon,regular polygon sides=5,fill=purple,scale=.3] (C2) at (10.5,0) {};
\draw[dashed] (C2) -- (10.5,-.7);
\draw (6.8,-.7) -- (6.8,-.9) to (9.2,-.9) to (9.2,-.7);
\draw (9.8,-.7) -- (9.8,-.9) to (10.7,-.9) to (10.7,-.7);
\draw[->-=.75] (C2) to (10.2,.35);
\draw[->-=.75] (C1) to (10.2,.35);
\node[scale=.6] at (9.9,.35) {$(2,2)$};
\draw[->-=.7] (B) to (9.6,.7);
\draw (10.2,.35) to (9.6,.7);
\node[scale=.6] at (9.35,.9) {$(4,0)$};
\draw[->-=.5] (9.1,.8) to (A5);
\draw (9.6,.7) to (9.1,.8) to (A5);
\draw (9.1,.8) to [out=170,in=280] (8.3,1.45);
\draw[dashed] (8.3,1.45) to (8.1,2.5);
\draw[->-=.5] (8.1,2.5) to (or1);
\node[scale=.75] at (7.7,2.7) {$(3,0)$};
%\draw (11.4,2.4) to [out=180,in=90] (6.2,-.5) to [out=90,in=0] (or1) -- cycle;
\node[scale=.75] at (8,-1.1) {A-type};
\node[scale=.75] at (9.5,-1.1) {B-type};
\node[scale=.75] at (10.25,-1.1) {C-type};
\draw[dashed] (8.7,.6) to [out=180,in=90] (6.2,-.55) to [out=270,in=180] (8.7,-1.6) to [out=0,in=270] (11.2,-.55) to [out=90,in=0] (8.7,.6) -- cycle;
\node[scale=1] at (12,.6) {$E_6$ singularity};
\end{scope}
\end{tikzpicture}
=--

And two more:

+--{.centered}
+--{: style="float:left"}
\begin{tikzpicture}
\useasboundingbox (-4,-3) rectangle (4,4.75);
\draw[thick] (-3.46410,-2) -- (0,4) -- ( 3.46410,-2) -- (-3.46410,-2);
\draw[thick] (0,0) circle (2);
\draw[thick] (0,4) -- (0,-2);
\draw[thick] (-3.46410,-2) -- ( 1.73205,1);
\draw[thick] ( 3.46410,-2) -- (-1.73205,1);
\filldraw (-1.73205,1) circle (3pt) node[anchor=south east, scale=1.5] {$e_1$};
\filldraw ( 1.73205,1) circle (3pt) node[anchor=south west, scale=1.5] {$e_2$};
\filldraw (0,-2) circle (3pt) node[anchor=north, scale=1.5] {$e_3$};
\filldraw (0,0) circle (3pt) node[anchor=west, xshift=7pt, scale=1.5] {$e_4$};
\filldraw (3.46410,-2) circle (3pt) node[anchor=north, scale=1.5] {$e_5$};
\filldraw (-3.46410,-2) circle (3pt) node[anchor=north, scale=1.5] {$e_6$};
\filldraw (0,4) circle (3pt) node[anchor=south, scale=1.5] {$e_7$};
\end{tikzpicture}
=--
+--{: style="float:left; margin-left:3em; margin-top: 3em;"}
\begin{tikzpicture}[every tqft/.style={fill=orange,fill opacity=0.25},
tqft/every boundary component/.style={fill=yellow,fill opacity=.5},
tqft/every upper boundary component/.style={draw,thin,blue},
tqft/every lower boundary component/.style={draw,dashed,thin,blue}]
\usetikzlibrary{tqft}
\begin{scope}[tqft/every incoming boundary component/.style={fill=green, fill opacity= .25}]
\pic[tqft/pair of pants,draw,name=a,at={(0,0)}];
\pic[tqft/reverse pair of pants,draw,anchor=incoming boundary 2,name=e,at={(8,0)}];
\end{scope}
\begin{scope}[tqft/every outgoing boundary component/.style={draw,thin,blue,fill=yellow}]
\pic[tqft/reverse pair of pants,draw,anchor=incoming boundary 1,name=b,at=(a-outgoing boundary 2)];
\end{scope}
\begin{scope}[tqft/every incoming boundary component/.style={fill=green, fill opacity= .25}]
\pic[tqft/cylinder,draw,anchor=outgoing boundary 1,name=d,at=(b-incoming boundary 2)];
\end{scope}
\path (b-incoming boundary 2) ++(1.5,0) node[font=\Huge] {\(=\)};
\begin{scope}[tqft/every outgoing boundary component/.style={draw,thin,blue,fill=yellow}]
\pic[tqft/cylinder,draw,anchor=incoming boundary 1,name=c,at=(a-outgoing boundary 1)];
\pic[tqft/pair of pants,draw,anchor=incoming boundary 1,name=f,at=(e-outgoing boundary 1)];
\end{scope}
\end{tikzpicture}
=--
=--
+--{: style="clear:left"}
=--

\begin{tikzpicture}\definecolor{mycolor}{RGB}{255,51,76}\draw[color=mycolor] (0.23,0.86) circle (4ex);\end{tikzpicture}