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/>";}i:153;a:2:{i:0;s:39:"\oint_{(x,y)\in C} x^3\, dx + 4y^2\, dy";i:1;s:156:"<img class="mwe-math-fallback-png-inline tex" alt="\oint_{(x,y)\in C} x^3\, dx + 4y^2\, dy" src="/images/math/d/6/c/d6c5bf8e05426a4b56804937b9ffb559.png" />";}i:154;a:2:{i:0;s:20:"\bigcap_{i=_1}^n E_i";i:1;s:137:"<img class="mwe-math-fallback-png-inline tex" alt="\bigcap_{i=_1}^n E_i" src="/images/math/8/3/d/83d87c98d958c7c2db86180b49230b65.png" />";}i:155;a:2:{i:0;s:20:"\bigcup_{i=_1}^n E_i";i:1;s:137:"<img class="mwe-math-fallback-png-inline tex" alt="\bigcup_{i=_1}^n E_i" src="/images/math/e/6/4/e6409717ec9d63567a34f9d1173ce2ae.png" />";}i:156;a:2:{i:0;s:15:"\frac{2}{4}=0.5";i:1;s:132:"<img class="mwe-math-fallback-png-inline tex" alt="\frac{2}{4}=0.5" src="/images/math/4/6/d/46dc0b34e0ab4e944a437720a4431d6c.png" />";}i:157;a:2:{i:0;s:18:"\tfrac{2}{4} = 0.5";i:1;s:135:"<img class="mwe-math-fallback-png-inline tex" alt="\tfrac{2}{4} = 0.5" src="/images/math/2/8/4/284667fc4a92790093aa59b61b3667a0.png" />";}i:158;a:2:{i:0;s:72:"\dfrac{2}{4} = 0.5 \qquad \dfrac{2}{c + \dfrac{2}{d + \dfrac{2}{4}}} = a";i:1;s:189:"<img class="mwe-math-fallback-png-inline tex" alt="\dfrac{2}{4} = 0.5 \qquad \dfrac{2}{c + \dfrac{2}{d + \dfrac{2}{4}}} = a" src="/images/math/5/a/3/5a37ae94a95c7dd603c20cd4fbe8d9e9.png" />";}i:159;a:2:{i:0;s:46:"\cfrac{2}{c + \cfrac{2}{d + \cfrac{2}{4}}} = a";i:1;s:163:"<img class="mwe-math-fallback-png-inline tex" alt="\cfrac{2}{c + \cfrac{2}{d + \cfrac{2}{4}}} = a" src="/images/math/6/d/0/6d099c02b3faf73f9320656217415906.png" />";}i:160;a:2:{i:0;s:60:"\cfrac{x}{1 + \cfrac{\cancel{y}}{\cancel{y}}} = \cfrac{x}{2}";i:1;s:177:"<img class="mwe-math-fallback-png-inline tex" alt="\cfrac{x}{1 + \cfrac{\cancel{y}}{\cancel{y}}} = \cfrac{x}{2}" src="/images/math/f/a/0/fa001cd2dd438152e45a44591f235148.png" />";}i:161;a:2:{i:0;s:12:"\binom{n}{k}";i:1;s:129:"<img class="mwe-math-fallback-png-inline tex" alt="\binom{n}{k}" src="/images/math/6/b/2/6b2be63a1b8e310465d1b538e2d7d71b.png" />";}i:162;a:2:{i:0;s:13:"\tbinom{n}{k}";i:1;s:130:"<img class="mwe-math-fallback-png-inline tex" alt="\tbinom{n}{k}" src="/images/math/8/4/8/8482b29cc0af31fa35ff6bf04200b265.png" />";}i:163;a:2:{i:0;s:13:"\dbinom{n}{k}";i:1;s:130:"<img class="mwe-math-fallback-png-inline tex" alt="\dbinom{n}{k}" src="/images/math/c/4/4/c44601a9dbb85dfb88868c14dc54c8ef.png" />";}i:164;a:2:{i:0;s:42:"\begin{matrix} x & y \\ z & v
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\end{Vmatrix}";i:1;s:173:"<img class="mwe-math-fallback-png-inline tex" alt="\begin{Vmatrix} x &amp; y \\ z &amp; v&#10;\end{Vmatrix}" src="/images/math/b/b/a/bba5bfd11057dbb202307584eed8f2dc.png" />";}i:167;a:2:{i:0;s:90:"\begin{bmatrix} 0 & \cdots & 0 \\ \vdots
& \ddots & \vdots \\ 0 & \cdots &
0\end{bmatrix} ";i:1;s:239:"<img class="mwe-math-fallback-png-inline tex" alt="\begin{bmatrix} 0 &amp; \cdots &amp; 0 \\ \vdots&#10;&amp; \ddots &amp; \vdots \\ 0 &amp; \cdots &amp;&#10;0\end{bmatrix} " src="/images/math/8/1/a/81a12a09ac84853e3d25323b8643c630.png" />";}i:168;a:2:{i:0;s:44:"\begin{Bmatrix} x & y \\ z & v
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\end{pmatrix}";i:1;s:173:"<img class="mwe-math-fallback-png-inline tex" alt="\begin{pmatrix} x &amp; y \\ z &amp; v&#10;\end{pmatrix}" src="/images/math/4/4/4/444df88e616def4e275b4e920c7b872e.png" />";}i:170;a:2:{i:0;s:63:"
\bigl( \begin{smallmatrix}
a&b\\ c&d
\end{smallmatrix} \bigr)
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\begin{cases}
n/2, & \text{if }n\text{ is even} \\
3n+1, & \text{if }n\text{ is odd}
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\begin{align}
f(x) & = (a+b)^2 \\
& = a^2+2ab+b^2 \\
\end{align}
";i:1;s:211:"<img class="mwe-math-fallback-png-inline tex" alt="&#10;\begin{align}&#10;f(x) &amp; = (a+b)^2 \\&#10;&amp; = a^2+2ab+b^2 \\&#10;\end{align}&#10;" src="/images/math/2/c/5/2c50960e8bcfd9e86527a123a0c43aa2.png" />";}i:173;a:2:{i:0;s:73:"
\begin{alignat}{2}
f(x) & = (a-b)^2 \\
& = a^2-2ab+b^2 \\
\end{alignat}
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z & = & a \\
f(x,y,z) & = & x + y + z
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z & = & a \\
f(x,y,z) & = & x + y + z
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\begin{array}{|c|c||c|} a & b & S \\
\hline
0&0&1\\
0&1&1\\
1&0&1\\
1&1&0\\
\end{array}
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\right)";i:1;s:181:"<img class="mwe-math-fallback-png-inline tex" alt="2 = \left(&#10;\frac{\left(3-x\right) \times 2}{3-x}&#10;\right)" src="/images/math/8/9/4/894f312e78ebc09a4e78c11b79cf4a8c.png" />";}i:381;a:2:{i:0;s:67:"S_{\text{new}} = S_{\text{old}} - \frac{ \left( 5-T \right) ^2} {2}";i:1;s:184:"<img class="mwe-math-fallback-png-inline tex" alt="S_{\text{new}} = S_{\text{old}} - \frac{ \left( 5-T \right) ^2} {2}" src="/images/math/a/a/0/aa0dc58e7114c5b91f6113130dcbc1d5.png" />";}i:382;a:2:{i:0;s:67:"S_{\text{new}} = S_{\text{old}} - \frac{ \left( 5-T \right) ^2} {2}";i:1;s:184:"<img class="mwe-math-fallback-png-inline tex" alt="S_{\text{new}} = S_{\text{old}} - \frac{ \left( 5-T \right) ^2} {2}" src="/images/math/a/a/0/aa0dc58e7114c5b91f6113130dcbc1d5.png" />";}i:383;a:2:{i:0;s:61:"\int_a^x \!\!\!\int_a^s f(y)\,dy\,ds = \int_a^x f(y)(x-y)\,dy";i:1;s:178:"<img class="mwe-math-fallback-png-inline tex" alt="\int_a^x \!\!\!\int_a^s f(y)\,dy\,ds = \int_a^x f(y)(x-y)\,dy" src="/images/math/4/4/6/4465ba032469b775777205effe6cdc0f.png" />";}i:384;a:2:{i:0;s:61:"\int_a^x \!\!\!\int_a^s f(y)\,dy\,ds
= \int_a^x f(y)(x-y)\,dy";i:1;s:182:"<img class="mwe-math-fallback-png-inline tex" alt="\int_a^x \!\!\!\int_a^s f(y)\,dy\,ds&#10;= \int_a^x f(y)(x-y)\,dy" src="/images/math/4/4/6/4465ba032469b775777205effe6cdc0f.png" />";}i:385;a:2:{i:0;s:38:"\det(\mathsf{A}-\lambda\mathsf{I}) = 0";i:1;s:155:"<img class="mwe-math-fallback-png-inline tex" alt="\det(\mathsf{A}-\lambda\mathsf{I}) = 0" src="/images/math/6/9/1/691187249f1e86a2e459362d66b5a743.png" />";}i:386;a:2:{i:0;s:38:"\det(\mathsf{A}-\lambda\mathsf{I}) = 0";i:1;s:155:"<img class="mwe-math-fallback-png-inline tex" alt="\det(\mathsf{A}-\lambda\mathsf{I}) = 0" src="/images/math/6/9/1/691187249f1e86a2e459362d66b5a743.png" />";}i:387;a:2:{i:0;s:18:"\sum_{i=0}^{n-1} i";i:1;s:135:"<img class="mwe-math-fallback-png-inline tex" alt="\sum_{i=0}^{n-1} i" src="/images/math/9/c/3/9c3090bae1d9eccd9e1747ecc51eaece.png" />";}i:388;a:2:{i:0;s:18:"\sum_{i=0}^{n-1} i";i:1;s:135:"<img class="mwe-math-fallback-png-inline tex" alt="\sum_{i=0}^{n-1} i" src="/images/math/9/c/3/9c3090bae1d9eccd9e1747ecc51eaece.png" />";}i:389;a:2:{i:0;s:78:"\sum_{m=1}^\infty\sum_{n=1}^\infty\frac{m^2\,n}{3^m\left(m\,3^n+n\,3^m\right)}";i:1;s:195:"<img class="mwe-math-fallback-png-inline tex" alt="\sum_{m=1}^\infty\sum_{n=1}^\infty\frac{m^2\,n}{3^m\left(m\,3^n+n\,3^m\right)}" src="/images/math/5/c/d/5cd6041b50d619f041f121baea301898.png" />";}i:390;a:2:{i:0;s:79:"\sum_{m=1}^\infty\sum_{n=1}^\infty\frac{m^2\,n}
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|(\bar{z})^n| = |z|^n,
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= \frac{1}{4\pi^2\kappa^2} \int_0^\infty \frac{\sin(\kappa R)}{\kappa R} \frac{\partial}{\partial R} \left[R^2\frac{\partial D_n(R)}{\partial R}\right]\,dR";i:1;s:291:"<img class="mwe-math-fallback-png-inline tex" alt="\phi_n(\kappa)&#10;= \frac{1}{4\pi^2\kappa^2} \int_0^\infty \frac{\sin(\kappa R)}{\kappa R} \frac{\partial}{\partial R} \left[R^2\frac{\partial D_n(R)}{\partial R}\right]\,dR" src="/images/math/7/f/b/7fb11db1e8b5890998b2f0f59f0e3d60.png" />";}i:398;a:2:{i:0;s:170:"\phi_n(\kappa) =
\frac{1}{4\pi^2\kappa^2} \int_0^\infty
\frac{\sin(\kappa R)}{\kappa R}
\frac{\partial}{\partial R}
\left[R^2\frac{\partial D_n(R)}{\partial R}\right]\,dR";i:1;s:303:"<img class="mwe-math-fallback-png-inline tex" alt="\phi_n(\kappa) =&#10;\frac{1}{4\pi^2\kappa^2} \int_0^\infty&#10;\frac{\sin(\kappa R)}{\kappa R}&#10;\frac{\partial}{\partial R}&#10;\left[R^2\frac{\partial D_n(R)}{\partial R}\right]\,dR" src="/images/math/7/f/b/7fb11db1e8b5890998b2f0f59f0e3d60.png" />";}i:399;a:2:{i:0;s:86:"\phi_n(\kappa) = 0.033C_n^2\kappa^{-11/3},\quad \frac{1}{L_0}\ll\kappa\ll\frac{1}{l_0}";i:1;s:203:"<img class="mwe-math-fallback-png-inline tex" alt="\phi_n(\kappa) = 0.033C_n^2\kappa^{-11/3},\quad \frac{1}{L_0}\ll\kappa\ll\frac{1}{l_0}" src="/images/math/8/f/7/8f72d606f5f91bd51583a0a08b36eed9.png" />";}i:400;a:2:{i:0;s:86:"\phi_n(\kappa) =
0.033C_n^2\kappa^{-11/3},\quad
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f(x) =
\begin{cases}
1 & -1 \le x < 0 \\
\frac{1}{2} & x = 0 \\
1 - x^2 & \text{otherwise}
\end{cases}
";i:1;s:264:"<img class="mwe-math-fallback-png-inline tex" alt="&#10;f(x) =&#10;\begin{cases}&#10;1 &amp; -1 \le x &lt; 0 \\&#10;\frac{1}{2} &amp; x = 0 \\&#10;1 - x^2 &amp; \text{otherwise}&#10;\end{cases}&#10;" src="/images/math/3/e/3/3e3579f4c1c6a95f181f227fd3ede7de.png" />";}i:403;a:2:{i:0;s:122:"{}_pF_q(a_1,\dots,a_p;c_1,\dots,c_q;z) = \sum_{n=0}^\infty \frac{(a_1)_n\cdots(a_p)_n}{(c_1)_n\cdots(c_q)_n}\frac{z^n}{n!}";i:1;s:239:"<img class="mwe-math-fallback-png-inline tex" alt="{}_pF_q(a_1,\dots,a_p;c_1,\dots,c_q;z) = \sum_{n=0}^\infty \frac{(a_1)_n\cdots(a_p)_n}{(c_1)_n\cdots(c_q)_n}\frac{z^n}{n!}" src="/images/math/c/0/2/c02cbc6ec9c57aca74ebc3a0314dea79.png" />";}i:404;a:2:{i:0;s:123:"{}_pF_q(a_1,\dots,a_p;c_1,\dots,c_q;z)
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\frac{(a_1)_n\cdots(a_p)_n}{(c_1)_n\cdots(c_q)_n}
\frac{z^n}{n!}";i:1;s:252:"<img class="mwe-math-fallback-png-inline tex" alt="{}_pF_q(a_1,\dots,a_p;c_1,\dots,c_q;z)&#10;= \sum_{n=0}^\infty&#10;\frac{(a_1)_n\cdots(a_p)_n}{(c_1)_n\cdots(c_q)_n}&#10;\frac{z^n}{n!}" src="/images/math/c/0/2/c02cbc6ec9c57aca74ebc3a0314dea79.png" />";}i:405;a:2:{i:0;s:25:"\frac{a}{b}\ \tfrac{a}{b}";i:1;s:142:"<img class="mwe-math-fallback-png-inline tex" alt="\frac{a}{b}\ \tfrac{a}{b}" src="/images/math/5/4/e/54e172be623599fef29e40733c94895e.png" />";}i:406;a:2:{i:0;s:25:"\frac{a}{b}\ \tfrac{a}{b}";i:1;s:142:"<img class="mwe-math-fallback-png-inline tex" alt="\frac{a}{b}\ \tfrac{a}{b}" src="/images/math/5/4/e/54e172be623599fef29e40733c94895e.png" />";}i:407;a:2:{i:0;s:18:"S=dD\,\sin\alpha\!";i:1;s:135:"<img class="mwe-math-fallback-png-inline tex" alt="S=dD\,\sin\alpha\!" src="/images/math/3/8/5/385776efb87d3eb7fe18587efd484ef5.png" />";}i:408;a:2:{i:0;s:18:"S=dD\,\sin\alpha\!";i:1;s:135:"<img class="mwe-math-fallback-png-inline tex" alt="S=dD\,\sin\alpha\!" src="/images/math/3/8/5/385776efb87d3eb7fe18587efd484ef5.png" />";}i:409;a:2:{i:0;s:56:"V=\frac16\pi h\left[3\left(r_1^2+r_2^2\right)+h^2\right]";i:1;s:173:"<img class="mwe-math-fallback-png-inline tex" alt="V=\frac16\pi h\left[3\left(r_1^2+r_2^2\right)+h^2\right]" src="/images/math/6/2/4/624bfa733e479dff276edfdc7b1b8f6a.png" />";}i:410;a:2:{i:0;s:56:"V=\frac16\pi h\left[3\left(r_1^2+r_2^2\right)+h^2\right]";i:1;s:173:"<img class="mwe-math-fallback-png-inline tex" alt="V=\frac16\pi h\left[3\left(r_1^2+r_2^2\right)+h^2\right]" src="/images/math/6/2/4/624bfa733e479dff276edfdc7b1b8f6a.png" />";}i:411;a:2:{i:0;s:167:"\begin{align}
u & = \tfrac{1}{\sqrt{2}}(x+y) \qquad & x &= \tfrac{1}{\sqrt{2}}(u+v)\\
v & = \tfrac{1}{\sqrt{2}}(x-y) \qquad & y &= \tfrac{1}{\sqrt{2}}(u-v)
\end{align}";i:1;s:320:"<img class="mwe-math-fallback-png-inline tex" alt="\begin{align}&#10;u &amp; = \tfrac{1}{\sqrt{2}}(x+y) \qquad &amp; x &amp;= \tfrac{1}{\sqrt{2}}(u+v)\\&#10;v &amp; = \tfrac{1}{\sqrt{2}}(x-y) \qquad &amp; y &amp;= \tfrac{1}{\sqrt{2}}(u-v)&#10;\end{align}" src="/images/math/7/8/7/787eb92e00313cb866a89579fde92108.png" />";}i:412;a:2:{i:0;s:168:"\begin{align}
u & = \tfrac{1}{\sqrt{2}}(x+y) \qquad & x &= \tfrac{1}{\sqrt{2}}(u+v) \\
v & = \tfrac{1}{\sqrt{2}}(x-y) \qquad & y &= \tfrac{1}{\sqrt{2}}(u-v)
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