Markdown+itex2MML Sandbox
Play around below . Your changes will, periodically, be rolled back.
Some examples
(1) min w h p h + w r p r + w l p l
\mathop{min} w_h p_h + w_r p_r + w_l p_l
(2) { ∇ × B → − 1 c ∂ E → ∂ t = 4 π c j → ∇ ⋅ E → = 4 π ρ ∇ × E → + 1 c ∂ B → ∂ t = 0 → ∇ ⋅ B → = 0
\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
(3) ∫ − ∞ ∞ e − a x 2 / 2 d x = 2 π a
{\int_{-\infty}^\infty e^{-a x^2/2} \mathrm{d}x} = \sqrt{\frac{2\pi}{a}}
which we can later refer1 (3) .
Aligned equations:
(4) a + b = b + a a + ( b + c ) = ( a + b ) + c \begin{aligned}
a+b &= b+a \\
a+(b+c) &= (a+b)+c
\end{aligned}
The Dirac equation (boxed):
( i D + m ) ψ = 0
\boxed{(i\slash{D}+m)\psi = 0}
Here’s the table of Clifford2 ℝ \mathbb{R}
j j 0 0 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 𝒞 ℓ j − \mathcal{C}\ell_{j}^- ℝ \mathbb{R} ℂ \mathbb{C} ℍ \mathbb{H} ℍ ⊕ ℍ \mathbb{H}\oplus\mathbb{H} ℍ ( 2 ) \mathbb{H}(2) ℂ ( 4 ) \mathbb{C}(4) ℝ ( 8 ) \mathbb{R}(8) ℝ ( 8 ) ⊕ ℝ ( 8 ) \mathbb{R}(8)\oplus\mathbb{R}(8) ℝ ( 16 ) \mathbb{R}(16) 𝒞 ℓ j + \mathcal{C}\ell_{j}^+ ℝ \mathbb{R} ℝ ⊕ ℝ \mathbb{R}\oplus\mathbb{R} ℝ ( 2 ) \mathbb{R}(2) ℂ ( 2 ) \mathbb{C}(2) ℍ ( 2 ) \mathbb{H}(2) ℍ ( 2 ) ⊕ ℍ ( 2 ) \mathbb{H}(2)\oplus\mathbb{H}(2) ℍ ( 4 ) \mathbb{H}(4) ℂ ( 8 ) \mathbb{C}(8) ℝ ( 16 ) \mathbb{R}(16)
where the generators of 𝒞 ℓ j ± \mathcal{C}\ell_{j}^\pm
γ i γ j + γ j γ i = ± 2 δ i j
\gamma_i\gamma_j +\gamma_j \gamma_i =\pm 2\delta_{i j}
and 𝒞 ℓ n + 8 ± = 𝒞 ℓ n ± ⊗ ℝ ( 16 ) \mathcal{C}\ell_{n+8}^\pm = \mathcal{C}\ell_n^\pm \otimes \mathbb{R}(16)
(5) lim n → ∞ ∑ k = 1 n 1 k 2 = π 2 6
\lim_{n \to \infty}
\sum_{k=1}^n \frac{1}{k^2}
= \frac{\pi^2}{6}
(6) V 1 × V 2 = ∣ i j k ∂ X ∂ u ∂ Y ∂ u 0 ∂ X ∂ v ∂ Y ∂ v 0 ∣
\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
(7) ∇ × E → = − ∂ B → ∂ t \nabla \times \vec{E} = - \frac {\partial \vec{B}}{\partial t} (8) ∮ B ⋅ d l = μ 0 I enc
\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 ( 𝒵 , 𝒪 ( − k ) ) H^1(\mathcal{Z}, \mathcal{O}(-k)) G = ( V , E ) G=(V,E) w : V → [ 0 , 1 ] w:V\to [0,1]
(9) { Q i , Q j } = δ ij ℋ . \{Q_i, Q_j\} = \delta_{ij}\mathcal{H}. Theorems
Definition
Let H H G G left coset of H H G G G G x H x H x ∈ G x \in G x H = { x h : h ∈ H } x H = \{ x h : h \in H \}
Similarly a right coset of H H G G G G H x H x H x = { h x : h ∈ H } H x = \{ h x : h \in H \}
Lemma
Let H H G G x x y y G G x H ∩ y H x H \cap y H x H = y H x H = y H
Proof
Let z z x H ∩ y H x H \cap y H z = x a z = x a a ∈ H a \in H z = y b z = y b b ∈ H b \in H h h H H a h ∈ H a h \in H a − 1 h ∈ H a^{-1}h \in H H H G G z h = x ( a h ) z h = x(a h) xh = z ( a − 1 h ) xh = z(a^{-1}h) h ∈ H h \in H z H ⊂ x H z H \subset x H x H ⊂ z H x H \subset z H x H = z H x H = z H y H = z H y H = z H x H = y H x H = y H
Lemma
Let H H G G H H G G H H
Proof
Let H = { h 1 , h 2 , … , h m } H = \{ h_1, h_2,\ldots, h_m\} h 1 , h 2 , … , h m h_1, h_2,\ldots, h_m x x G G x H x H x h j x h_j j = 1 , 2 , … , m j = 1,2,\ldots,m j j k k 1 1 m m x h j = x h k x h_j = x h_k h j = x − 1 ( x h j ) = x − 1 ( x h k ) = h k h_j = x^{-1} (x h_j) = x^{-1} (x h_k) = h_k j = k j = k h 1 , h 2 , … , h m h_1, h_2,\ldots, h_m x h 1 , x h 2 , … , x h m x h_1, x h_2,\ldots, x h_m H H x H x H m m
Theorem
(Lagrange’s Theorem). Let G G H H G G H H G G
Proof
Each element x x G G H H G G x H x H H H G G 1 ). Therefore every element of G G H H H H | H | |H| 2 ). Therefore | G | = n | H | |G| = n |H| n n H H G G
Corollary
Let x x G G x x G G
Theorem
Let f : Δ ⟶ Δ , f : \Delta \longrightarrow \Delta, Δ = { z ∈ ℂ : | z | < 1 } \Delta=\{z\in\mathbb{C}: \vert z \vert \lt 1\} a ∈ Δ a \in \Delta
| f ( z ) − f ( a ) 1 − f ( a ) ¯ f ( z ) | ≤ | z − a 1 − a ¯ z |
\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 | z | ≤ 1 \vert z \vert \le 1
| f ′ ( a ) | 1 − | f ( a ) | 2 ≤ 1 1 − | a | 2 .
\frac{\vert f'(a)\vert}{1-\vert f(a)\vert^2}\le \frac{1}{1-\vert a \vert^2}.
Furthermore, equality holds iff f f Δ \Delta
Proof
Let w = z − a 1 − a ¯ z w=\frac{z-a}{1-\overline{a}z} ϕ ( w ) = f ( z ) − f ( a ) 1 − f ( a ) ¯ f ( z ) \phi(w)=\frac{f(z)-f(a)}{1-\overline{f(a)}f(z)} abs b < 1 \abs{b}\lt 1 C b ( z ) = z − b 1 − b ¯ z . C_b(z)=\frac{z-b}{1-\overline{b}z}. Δ \Delta b b 0 0 C b ( z ) e i γ C_b(z)e^{i\gamma} γ ∈ [ 0 , 2 π ] . \gamma\in[0,2\pi]. ϕ ( w ) = C f ( a ) ∘ f ∘ C a − 1 ( w ) , \phi(w)=C_{f(a)}\circ f \circ C_a^{-1}(w), C a − 1 C_a^{-1} C a C_a C a ( z ) C_a(z) ϕ ( 0 ) = C f ( a ) ∘ f ∘ C a − 1 ( 0 ) = C f ( a ) ( f ( a ) ) = 0 \phi(0)=C_{f(a)}\circ f \circ C_a^{-1}(0)=C_{f(a)}(f(a))=0 C a − 1 : Δ ⟶ Δ C_a^{-1}: \Delta \longrightarrow \Delta f : Δ ⟶ Δ f : \Delta \longrightarrow \Delta C f ( a ) : Δ ⟶ Δ , C_{f(a)}: \Delta \longrightarrow \Delta, | ϕ ( w ) | < 1 \vert\phi(w)\vert\lt 1 | w | < 1 . \vert w\vert \lt 1 . | ϕ ( w ) | ≤ | w | \vert\phi(w)\vert\le \vert w \vert | w | ≤ 1 \vert w \vert \le 1 f ( z ) = e i γ ′ z f(z)=e^{i\gamma'} z γ ′ ∈ [ 0 , 2 π ] \gamma'\in [0,2\pi]
(10) | f ( z ) − f ( a ) 1 − f ( a ) ¯ f ( z ) | ≤ | z − a 1 − a ¯ z |
\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 | z | ≤ 1 . \vert z \vert\le 1.
| f ( z ) − f ( a ) z − a | ≤ | 1 − f ( a ) ¯ f ( z ) 1 − a ¯ z | .
\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 z a a
| f ′ ( a ) | ≤ | 1 − | f ( a ) | 2 1 − | a | 2 | = 1 − | f ( a ) | 2 1 − | a | 2 ,
\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
| f ′ ( a ) | 1 − | f ( a ) | 2 ≤ 1 1 − | a | 2 .
\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 (10) , then Schwarz’s lemma tells us that ϕ ( w ) = e i γ ′ w \phi(w)=e^{i\gamma'}w ϕ ( w ) = C f ( a ) ∘ f ∘ C a − 1 ( w ) = e i γ ′ w , \phi(w)=C_{f(a)}\circ f \circ C_a^{-1}(w)=e^{i\gamma'}w, f ( z ) = C f ( a ) − 1 ( e i γ ′ C a ( z ) ) f(z)=C_{f(a)}^{-1}(e^{i\gamma'}C_a(z)) e i γ ′ C a ( z ) e^{i\gamma'}C_a(z) C f ( a ) − 1 , C_{f(a)}^{-1}, C f ( a ) C_{f(a)} f f Δ \Delta f f Δ \Delta ϕ ( w ) = C f ( a ) ∘ f ∘ C a − 1 ( w ) = e i γ C b ( w ) , \phi(w)=C_{f(a)}\circ f \circ C_a^{-1}(w)=e^{i\gamma}C_b(w), Δ \Delta e i γ C b ( w ) . e^{i\gamma}C_b(w). ϕ ( 0 ) = 0 , \phi(0)=0, b = 0 b=0
ϕ ( w ) = e i γ C 0 ( w ) = e i γ w ⇔ | ϕ ( w ) | = | w | ⇔ | f ( z ) − f ( a ) 1 − f ( a ) ¯ f ( z ) | = | z − a 1 − a ¯ z |
\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 | z | ≤ 1 \vert z \vert\le 1 (10) iff f f Δ \Delta
SVG graphics, created in SVG-Edit:
Layer 1
E
Σ
E_\Sigma
E
Σ
out
E_{\Sigma_{\text{out}}}
E
in
E_{\text{in}}
[
Σ
,
X
]
[\Sigma, X]
[
Σ
out
,
X
]
[\Sigma_{\text{out}}, X]
[
Σ
in
,
X
]
[\Sigma_{\text{in}}, X]
Bord
(
X
)
Bord(X)
V
V
E
out
E_{\text{out}}
in
\text{in}
out
\text{out}
exp
(
S
∇
)
|
Σ
in
\exp(S_\nabla)\vert_{\Sigma_{\text{in}}}
exp
(
S
∇
)
\exp(S_\nabla)
exp
(
S
∇
)
|
Σ
out
\exp(S_\nabla)\vert_{\Sigma_{\text{out}}}
E
in
E_{\text{in}}
\begin{svg}<svg width="450" height="400" 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="67492">
<defs>
<marker refX="8" orient="auto" markerHeight="5" markerWidth="5" markerUnits="strokeWidth" refY="5" id="se_arrow_67492_fw1" viewBox="0 0 10 10">
<path id="svg_67492_23" fill="#000" d="m0,0l10,5l-10,5l5,-5l-5,-5z"/>
</marker>
</defs>
<g class="layer">
<title>Layer 1</title>
<polyline se:connector="svg_67492_7 svg_67492_8" marker-end="url(#se_arrow_67492_fw1)" fill="none" stroke-width="2" stroke="#000" points="296,279 345.5,316.125 395,353.25" id="svg_67492_22"/>
<polyline se:connector="svg_67492_5 svg_67492_7" marker-end="url(#se_arrow_67492_fw1)" fill="none" stroke-width="2" stroke="#000" points="276.99,178 278.208,214 279.425,250" id="svg_67492_21"/>
<polyline se:connector="svg_67492_6 svg_67492_7" marker-end="url(#se_arrow_67492_fw1)" fill="none" stroke-width="2" stroke="#000" points="179,268.156 212,267.779 245,267.401" id="svg_67492_20"/>
<polyline se:connector="svg_67492_4 svg_67492_7" marker-end="url(#se_arrow_67492_fw1)" fill="none" stroke-width="2" stroke="#000" points="163.529,174 211.119,212 258.71,250" id="svg_67492_18"/>
<polyline se:connector="svg_67492_4 svg_67492_5" marker-end="url(#se_arrow_67492_fw1)" fill="none" stroke-width="2" stroke="#000" points="173,162.287 206.5,162.68 240,163.072" id="svg_67492_17"/>
<polyline se:connector="svg_67492_4 svg_67492_6" marker-end="url(#se_arrow_67492_fw1)" fill="none" stroke-width="2" stroke="#000" points="148.556,174 148.732,211.5 148.908,249" id="svg_67492_16"/>
<polyline se:connector="svg_67492_1 svg_67492_3" marker-end="url(#se_arrow_67492_fw1)" fill="none" stroke-width="2" stroke="#000" points="70.1165,92 70.4757,129 70.835,166" id="svg_67492_15"/>
<polyline se:connector="svg_67492_1 svg_67492_2" marker-end="url(#se_arrow_67492_fw1)" fill="none" stroke-width="2" stroke="#000" points="82,80.0462 128.5,80.225 175,80.4038" id="svg_67492_14"/>
<polyline se:connector="svg_67492_2 svg_67492_5" marker-end="url(#se_arrow_67492_fw1)" fill="none" stroke-width="2" stroke="#000" points="211.982,93.5 235.255,118.75 258.527,144" id="svg_67492_13"/>
<polyline se:connector="svg_67492_1 svg_67492_4" marker-end="url(#se_arrow_67492_fw1)" fill="none" stroke-width="2" stroke="#000" points="81.4878,92 106.857,118.5 132.226,145" id="svg_67492_11"/>
<polyline se:connector="svg_67492_3 svg_67492_6" marker-end="url(#se_arrow_67492_fw1)" fill="none" stroke-width="2" stroke="#000" points="81.9474,195 106.579,222 131.211,249" id="svg_67492_10"/>
<foreignObject height="24" width="24" font-size="16" id="svg_67492_1" y="68" x="58">
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow>
<msub>
<mi>E</mi>
<mi>Σ</mi>
</msub>
</mrow>
<annotation encoding="application/x-tex">E_\Sigma</annotation>
</semantics>
</math>
</foreignObject>
<foreignObject height="26" width="40" font-size="16" id="svg_67492_2" y="67.5" x="180">
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow>
<msub>
<mi>E</mi>
<mrow>
<msub>
<mi>Σ</mi>
<mtext>out</mtext>
</msub>
</mrow>
</msub>
</mrow>
<annotation encoding="application/x-tex">E_{\Sigma_{\text{out}}}</annotation>
</semantics>
</math>
</foreignObject>
<foreignObject height="24" width="24" font-size="16" id="svg_67492_3" y="171" x="59">
<math display="inline" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow>
<msub>
<mi>E</mi>
<mtext>in</mtext>
</msub>
</mrow>
<annotation encoding="application/x-tex">E_{\text{in}}</annotation>
</semantics>
</math>
</foreignObject>
<foreignObject height="24" width="49" font-size="16" id="svg_67492_4" y="150" x="124">
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow>
<mo stretchy="false">[</mo>
<mi>Σ</mi>
<mo>,</mo>
<mi>X</mi>
<mo stretchy="false">]</mo>
</mrow>
<annotation encoding="application/x-tex">[\Sigma, X]</annotation>
</semantics>
</math>
</foreignObject>
<foreignObject height="29" width="63" font-size="16" id="svg_67492_5" y="149" x="245">
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow>
<mo stretchy="false">[</mo>
<msub>
<mi>Σ</mi>
<mtext>out</mtext>
</msub>
<mo>,</mo>
<mi>X</mi>
<mo stretchy="false">]</mo>
</mrow>
<annotation encoding="application/x-tex">[\Sigma_{\text{out}}, X]</annotation>
</semantics>
</math>
</foreignObject>
<foreignObject height="29" width="60" font-size="16" id="svg_67492_6" y="254" x="119">
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow>
<mo stretchy="false">[</mo>
<msub>
<mi>Σ</mi>
<mtext>in</mtext>
</msub>
<mo>,</mo>
<mi>X</mi>
<mo stretchy="false">]</mo>
</mrow>
<annotation encoding="application/x-tex">[\Sigma_{\text{in}}, X]</annotation>
</semantics>
</math>
</foreignObject>
<foreignObject height="24" width="60" font-size="16" id="svg_67492_7" y="255" x="250">
<math display="inline" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow>
<mi>Bord</mi>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo stretchy="false">)</mo>
</mrow>
<annotation encoding="application/x-tex">Bord(X)</annotation>
</semantics>
</math>
</foreignObject>
<foreignObject height="24" width="24" font-size="16" id="svg_67492_8" y="351" x="396">
<math display="inline" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow>
<mi>V</mi>
</mrow>
<annotation encoding="application/x-tex">V</annotation>
</semantics>
</math>
</foreignObject>
<path marker-end="url(#se_arrow_67492_fw1)" stroke-width="2" stroke="#000000" fill="none" d="m303,177c37.30435,34.82927 93.82608,76.82927 104,168" id="svg_67492_35"/>
<path marker-end="url(#se_arrow_67492_fw1)" stroke-width="2" stroke="#000000" fill="none" d="m162,281c67,70 143,83 230,83" id="svg_67492_36"/>
<foreignObject x="115.5" y="55" id="svg_67492_99" font-size="16" width="35" height="24">
<math display="inline" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow>
<msub>
<mi>E</mi>
<mtext>out</mtext>
</msub>
</mrow>
<annotation encoding="application/x-tex">E_{\text{out}}</annotation>
</semantics>
</math>
</foreignObject>
<foreignObject height="24" width="31" font-size="16" id="svg_67492_98" y="197" x="123">
<math display="inline" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow>
<mtext>in</mtext>
</mrow>
<annotation encoding="application/x-tex">\text{in}</annotation>
</semantics>
</math>
</foreignObject>
<foreignObject x="193" y="142" font-size="16" width="31" height="24" id="svg_67492_9">
<math display="inline" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow>
<mtext>out</mtext>
</mrow>
<annotation encoding="application/x-tex">\text{out}</annotation>
</semantics>
</math>
</foreignObject>
<foreignObject x="179" y="328" id="svg_67492_12" font-size="16" width="85" height="29" transform="rotate(30.4258, 221.5, 342.5)">
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow>
<mi>exp</mi>
<mo stretchy="false">(</mo>
<msub>
<mi>S</mi>
<mo>∇</mo>
</msub>
<mo stretchy="false">)</mo>
<msub>
<mo stretchy="false">|</mo>
<mrow>
<msub>
<mi>Σ</mi>
<mtext>in</mtext>
</msub>
</mrow>
</msub>
</mrow>
<annotation encoding="application/x-tex">\exp(S_\nabla)\vert_{\Sigma_{\text{in}}}</annotation>
</semantics>
</math>
</foreignObject>
<foreignObject transform="rotate(37.6477, 338.5, 328.5)" height="29" width="66" font-size="16" id="svg_67492_38" y="314" x="305.49998">
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow>
<mi>exp</mi>
<mo stretchy="false">(</mo>
<msub>
<mi>S</mi>
<mo>∇</mo>
</msub>
<mo stretchy="false">)</mo>
</mrow>
<annotation encoding="application/x-tex">\exp(S_\nabla)</annotation>
</semantics>
</math>
</foreignObject>
<foreignObject id="svg_67492_19" transform="rotate(49.3987, 365, 218)" height="29" width="96" font-size="16" y="203.5" x="317">
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow>
<mi>exp</mi>
<mo stretchy="false">(</mo>
<msub>
<mi>S</mi>
<mo>∇</mo>
</msub>
<mo stretchy="false">)</mo>
<msub>
<mo stretchy="false">|</mo>
<mrow>
<msub>
<mi>Σ</mi>
<mtext>out</mtext>
</msub>
</mrow>
</msub>
</mrow>
<annotation encoding="application/x-tex">\exp(S_\nabla)\vert_{\Sigma_{\text{out}}}</annotation>
</semantics>
</math>
</foreignObject>
<foreignObject height="24" width="31" font-size="16" id="svg_67492_97" x="39" y="121">
<math display="inline" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow>
<msub>
<mi>E</mi>
<mtext>in</mtext>
</msub>
</mrow>
<annotation encoding="application/x-tex">E_{\text{in}}</annotation>
</semantics>
</math>
</foreignObject>
</g>
</svg>\end{svg}
Commutative cube
Layer 1
A
0
A_0
C
0
C_0
A
1
A_1
C
1
C_1
B
0
B_0
D
0
D_0
B
1
B_1
D
1
D_1
\begin{svg}
<svg width="260" height="260" 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="83381">
<desc>Commutative cube</desc>
<defs>
<marker refX="8" orient="auto" markerHeight="5" markerWidth="5" markerUnits="strokeWidth" refY="5" id="se_arrow_83381_fw4" viewBox="0 0 10 10">
<path fill="#666" d="m0,0l10,5l-10,5l5,-5l-5,-5z" id="svg_83381_2"/>
</marker>
<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
W
+
W^+
W
−
W^-
s
¯
\overline{s}
d
¯
\overline{d}
Layer 1
s
s
d
d
u
,
c
,
t
u,\, c,\, t
u
,
c
,
t
u,\, c,\, t
K 0 K ¯ 0 K^0\overline{K}^0
Yet More examples
Example anim01 - demonstrate animation elements
∫ − ∞ ∞ e − a x 2 d
x
=
π
a
{\int_{-\infty}^{\infty}e^{-a x^2}d x}=\sqrt{\tfrac{\pi}{a}}
Complicated commutative diagrams (equations in SVG)
Complicated commutative diagram, realized in SVG
1
1
1
1
1
1
1
1
Id
Id
Id
Id
A
A
B
B
ρ
\rho
H
H
H
H
K
K
K
'
K'
ϕ 1
\phi_1
ϕ 2
\phi_2
N A
N_A
N B
N_B
N A ∨
N^\vee_A
N B ∨
N^\vee_B
In SU ( 3 ) SU(3)
Rank-2 Symmetric Tensor Representation
⊗
Fundamental Representation
=
Adjoint Representation
⊕
Rank-3 Symmetric Tensor Representation
\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}
r a + 1 = { 0 with prob. exp ( − θ r a ) max { δ r a , z } with prob. 1 − exp ( − θ r a ) 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}
q a ( z ) = σ a − 1 exp [ − γ + z σ a ] q_a(z) = \sigma_a^{-1} \exp{\left[ -\frac{\gamma + z}{\sigma_a} \right]}
Linearity of Quadrature Rules
∑ i = 1 N ( α f ( x i ) + β g ( x i ) ) w i = α ∑ i = 1 N f ( x i ) w i + β ∑ i = 1 N g ( x i ) w i \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 } ∫ a b ( α f ( x ) + β g ( x ) ) dx = α ∫ a b f ( x ) dx + β ∫ a b g ( x ) dx {\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} } p 3 ( x ) = ( 1 2 ) ( x − 1 2 ) ( x − 3 4 ) ( x − 1 ) ( 1 4 − 1 2 ) ( 1 4 − 3 4 ) ( 1 4 − 1 ) + ( 1 2 ) ( x − 1 2 ) ( x − 3 4 ) ( x − 1 ) ( 1 4 − 1 2 ) ( 1 4 − 3 4 ) ( 1 4 − 1 ) + ( 1 2 ) ( x − 1 2 ) ( x − 3 4 ) ( x − 1 ) ( 1 4 − 1 2 ) ( 1 4 − 3 4 ) ( 1 4 − 1 ) + ( 1 2 ) ( x − 1 2 ) ( x − 3 4 ) ( x − 1 ) ( 1 4 − 1 2 ) ( 1 4 − 3 4 ) ( 1 4 − 1 ) 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)}}
P 1 ( Y ) → P 1 ( X ) ↓ ⇓ ∼ ↓ T ′ → T
\begin{matrix}
P_1(Y) &\to& P_1(X) \\
\downarrow &\Downarrow\mathrlap{\sim}& \downarrow \\
T' &\to& T
\end{matrix}
\mathcal{} versus \mathscr{}𝒜ℬ𝒞𝒟ℰℱ𝒢ℋℐ𝒥𝒦ℒℳ𝒩𝒪𝒫𝒬ℛ𝒮𝒯𝒰𝒱𝒲𝒳𝒴𝒵 versus 𝒜ℬ𝒞𝒟ℰℱ𝒢ℋℐ𝒥𝒦ℒℳ𝒩𝒪𝒫𝒬ℛ𝒮𝒯𝒰𝒱𝒲𝒳𝒴𝒵
\begin{gathered}
\mathcal{ABCDEFGHIJKLMNOPQRSTUVWXYZ}\\
\text{versus}\\
\mathscr{ABCDEFGHIJKLMNOPQRSTUVWXYZ}
\end{gathered}
(11)
A_n Quiver
Layer 1
v
1
v_1
v
2
v_2
v
n
1
v_{n_1}
≡ ( U ( k ) n 1 , { v i } )
\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:
class Person
attr_reader :name , :age
def initialize ( name , age )
@name , @age = name , age
end
def <=> ( person )
@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:
ListOfStrings(
title = _("A List of some strings"),
help = _("A List of strings"),
orientation = "horizontal"
)Tikz Pictures
A new feature to play around with. Requires an additional install (in addition to Instiki) to render the Tikz code into SVG.
And two more: