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a:449:{i:0;a:2:{i:0;s:21:"e^{i \pi} + 1 = 0\,\!";i:1;s:138:"<img class="mwe-math-fallback-png-inline tex" alt="e^{i \pi} + 1 = 0\,\!" src="/images/math/9/e/9/9e9a547076c6820b95e439dd1a5d6a32.png" />";}i:1;a:2:{i:0;s:21:"e^{i \pi} + 1 = 0\,\!";i:1;s:138:"<img class="mwe-math-fallback-png-inline tex" alt="e^{i \pi} + 1 = 0\,\!" src="/images/math/9/e/9/9e9a547076c6820b95e439dd1a5d6a32.png" />";}i:2;a:2:{i:0;s:67:"\definecolor{red}{RGB}{255,0,0}\pagecolor{red}e^{i \pi} + 1 = 0\,\!";i:1;s:184:"<img class="mwe-math-fallback-png-inline tex" alt="\definecolor{red}{RGB}{255,0,0}\pagecolor{red}e^{i \pi} + 1 = 0\,\!" src="/images/math/6/7/a/67aca9e0de80ac6ab651ed1097b49fe2.png" />";}i:3;a:2:{i:0;s:10:"\text{abc}";i:1;s:127:"<img class="mwe-math-fallback-png-inline tex" alt="\text{abc}" src="/images/math/4/6/0/46045b1f6fa9dc10a3112ba360d4d9d7.png" />";}i:4;a:2:{i:0;s:10:"\alpha\,\!";i:1;s:127:"<img class="mwe-math-fallback-png-inline tex" alt="\alpha\,\!" src="/images/math/4/b/c/4bc6c42bbabe567d1f2516326e52b775.png" />";}i:5;a:2:{i:0;s:15:" f(x) = x^2\,\!";i:1;s:132:"<img class="mwe-math-fallback-png-inline tex" alt=" f(x) = x^2\,\!" src="/images/math/3/a/5/3a5f0f03603148035120a3cba993e54f.png" />";}i:6;a:2:{i:0;s:8:"\sqrt{2}";i:1;s:125:"<img class="mwe-math-fallback-png-inline tex" alt="\sqrt{2}" src="/images/math/e/f/5/ef5590434a387b3c4427e09d5b08baaf.png" />";}i:7;a:2:{i:0;s:14:"\sqrt{1-e^2}\!";i:1;s:131:"<img class="mwe-math-fallback-png-inline tex" alt="\sqrt{1-e^2}\!" src="/images/math/0/4/c/04c93cf9f0a7cf697add9a2d4173a9e9.png" />";}i:8;a:2:{i:0;s:14:"\sqrt{1-z^3}\!";i:1;s:131:"<img class="mwe-math-fallback-png-inline tex" alt="\sqrt{1-z^3}\!" src="/images/math/1/0/8/108d6aa70c84fddabbbd3ec97f3d3ff8.png" />";}i:9;a:2:{i:0;s:1:"x";i:1;s:118:"<img class="mwe-math-fallback-png-inline tex" alt="x" src="/images/math/9/d/d/9dd4e461268c8034f5c8564e155c67a6.png" />";}i:10;a:2:{i:0;s:42:"\dot{a}, \ddot{a}, \acute{a}, \grave{a} \!";i:1;s:159:"<img class="mwe-math-fallback-png-inline tex" alt="\dot{a}, \ddot{a}, \acute{a}, \grave{a} \!" src="/images/math/c/0/9/c096beaae99e2d37b4050c4ccf30fbf8.png" />";}i:11;a:2:{i:0;s:43:"\check{a}, \breve{a}, \tilde{a}, \bar{a} \!";i:1;s:160:"<img class="mwe-math-fallback-png-inline tex" alt="\check{a}, \breve{a}, \tilde{a}, \bar{a} \!" src="/images/math/e/f/3/ef387ac79f18651dd3105d2c584b3c95.png" />";}i:12;a:2:{i:0;s:32:"\hat{a}, \widehat{a}, \vec{a} \!";i:1;s:149:"<img class="mwe-math-fallback-png-inline tex" alt="\hat{a}, \widehat{a}, \vec{a} \!" src="/images/math/e/a/e/eaededf26bb201c699ef1597902383c3.png" />";}i:13;a:2:{i:0;s:37:"\exp_a b = a^b, \exp b = e^b, 10^m \!";i:1;s:154:"<img class="mwe-math-fallback-png-inline tex" alt="\exp_a b = a^b, \exp b = e^b, 10^m \!" src="/images/math/1/9/9/199ac36bc19f7951df5041aedc1e2525.png" />";}i:14;a:2:{i:0;s:37:"\ln c, \lg d = \log e, \log_{10} f \!";i:1;s:154:"<img class="mwe-math-fallback-png-inline tex" alt="\ln c, \lg d = \log e, \log_{10} f \!" src="/images/math/d/5/8/d58edc12e2750302cfcdfd47f7674607.png" />";}i:15;a:2:{i:0;s:48:"\sin a, \cos b, \tan c, \cot d, \sec e, \csc f\!";i:1;s:165:"<img class="mwe-math-fallback-png-inline tex" alt="\sin a, \cos b, \tan c, \cot d, \sec e, \csc f\!" src="/images/math/0/d/e/0de90ca439db043c53360a81e56e2543.png" />";}i:16;a:2:{i:0;s:34:"\arcsin h, \arccos i, \arctan j \!";i:1;s:151:"<img class="mwe-math-fallback-png-inline tex" alt="\arcsin h, \arccos i, \arctan j \!" src="/images/math/d/4/f/d4f41532d2a06150554f27d52b3c9479.png" />";}i:17;a:2:{i:0;s:37:"\sinh k, \cosh l, \tanh m, \coth n \!";i:1;s:154:"<img class="mwe-math-fallback-png-inline tex" alt="\sinh k, \cosh l, \tanh m, \coth n \!" src="/images/math/2/d/4/2d460f19d2addae865a78806e3a3afd8.png" />";}i:18;a:2:{i:0;s:91:"\operatorname{sh}\,k, \operatorname{ch}\,l, \operatorname{th}\,m, \operatorname{coth}\,n \!";i:1;s:208:"<img class="mwe-math-fallback-png-inline tex" alt="\operatorname{sh}\,k, \operatorname{ch}\,l, \operatorname{th}\,m, \operatorname{coth}\,n \!" src="/images/math/7/f/3/7f37a94f008e914726d78b52bf7e3ff4.png" />";}i:19;a:2:{i:0;s:76:"\operator
\end{matrix}";i:1;s:171:"<img class="mwe-math-fallback-png-inline tex" alt="\begin{matrix} x & y \\ z & v \end{matrix}" src="/images/math/b/9/9/b99890966e1b997497211428f8e3419d.png" />";}i:165;a:2:{i:0;s:44:"\begin{vmatrix} x & y \\ z & v
\end{vmatrix}";i:1;s:173:"<img class="mwe-math-fallback-png-inline tex" alt="\begin{vmatrix} x & y \\ z & v \end{vmatrix}" src="/images/math/9/2/b/92b8f0e57848a80b4babd2ba93775370.png" />";}i:166;a:2:{i:0;s:44:"\begin{Vmatrix} x & y \\ z & v
\end{Vmatrix}";i:1;s:173:"<img class="mwe-math-fallback-png-inline tex" alt="\begin{Vmatrix} x & y \\ z & v \end{Vmatrix}" src="/images/math/b/b/a/bba5bfd11057dbb202307584eed8f2dc.png" />";}i:167;a:2:{i:0;s:90:"\begin{bmatrix} 0 & \cdots & 0 \\ \vdots
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& \ddots & \vdots \\ 0 & \cdots &
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0\end{bmatrix} ";i:1;s:239:"<img class="mwe-math-fallback-png-inline tex" alt="\begin{bmatrix} 0 & \cdots & 0 \\ \vdots & \ddots & \vdots \\ 0 & \cdots & 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
\end{Bmatrix}";i:1;s:173:"<img class="mwe-math-fallback-png-inline tex" alt="\begin{Bmatrix} x & y \\ z & v \end{Bmatrix}" src="/images/math/b/f/7/bf7244e2842c8a7d55892e229560d5c1.png" />";}i:169;a:2:{i:0;s:44:"\begin{pmatrix} x & y \\ z & v
\end{pmatrix}";i:1;s:173:"<img class="mwe-math-fallback-png-inline tex" alt="\begin{pmatrix} x & y \\ z & v \end{pmatrix}" src="/images/math/4/4/4/444df88e616def4e275b4e920c7b872e.png" />";}i:170;a:2:{i:0;s:63:"
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\bigl( \begin{smallmatrix}
a&b\\ c&d
\end{smallmatrix} \bigr)
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";i:1;s:204:"<img class="mwe-math-fallback-png-inline tex" alt=" \bigl( \begin{smallmatrix} a&b\\ c&d \end{smallmatrix} \bigr) " src="/images/math/c/d/4/cd49bbc188dce0f93fef57312af5a106.png" />";}i:171;a:2:{i:0;s:104:"f(n) =
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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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\end{cases} ";i:1;s:245:"<img class="mwe-math-fallback-png-inline tex" alt="f(n) = \begin{cases} n/2, & \text{if }n\text{ is even} \\ 3n+1, & \text{if }n\text{ is odd} \end{cases} " src="/images/math/9/8/5/98503cc6876b22f5900297971fdd42ed.png" />";}i:172;a:2:{i:0;s:66:"
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\begin{align}
f(x) & = (a+b)^2 \\
& = a^2+2ab+b^2 \\
\end{align}
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";i:1;s:211:"<img class="mwe-math-fallback-png-inline tex" alt=" \begin{align} f(x) & = (a+b)^2 \\ & = a^2+2ab+b^2 \\ \end{align} " src="/images/math/2/c/5/2c50960e8bcfd9e86527a123a0c43aa2.png" />";}i:173;a:2:{i:0;s:73:"
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\begin{alignat}{2}
f(x) & = (a-b)^2 \\
& = a^2-2ab+b^2 \\
\end{alignat}
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";i:1;s:218:"<img class="mwe-math-fallback-png-inline tex" alt=" \begin{alignat}{2} f(x) & = (a-b)^2 \\ & = a^2-2ab+b^2 \\ \end{alignat} " src="/images/math/f/e/4/fe45a0df3e20bc5caf718e5333678d08.png" />";}i:174;a:2:{i:0;s:68:"\begin{array}{lcl}
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z & = & a \\
f(x,y,z) & = & x + y + z
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\end{array}";i:1;s:213:"<img class="mwe-math-fallback-png-inline tex" alt="\begin{array}{lcl} z & = & a \\ f(x,y,z) & = & x + y + z \end{array}" src="/images/math/9/b/f/9bf19115bb27237fa997ca93b94ad217.png" />";}i:175;a:2:{i:0;s:68:"\begin{array}{lcr}
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z & = & a \\
f(x,y,z) & = & x + y + z
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\end{array}";i:1;s:213:"<img class="mwe-math-fallback-png-inline tex" alt="\begin{array}{lcr} z & = & a \\ f(x,y,z) & = & x + y + z \end{array}" src="/images/math/0/2/a/02ae32735e1e21ba3b05984289fd2763.png" />";}i:176;a:2:{i:0;s:9:"f(x) \,\!";i:1;s:126:"<img class="mwe-math-fallback-png-inline tex" alt="f(x) \,\!" src="/images/math/8/d/f/8dfae20000a042d8e9047aad1d7e171e.png" />";}i:177;a:2:{i:0;s:28:"= \sum_{n=0}^\infty a_n x^n ";i:1;s:145:"<img class="mwe-math-fallback-png-inline tex" alt="= \sum_{n=0}^\infty a_n x^n " src="/images/math/6/6/3/6633d51d63b35281d030755a6b0aebb1.png" />";}i:178;a:2:{i:0;s:24:"= a_0+a_1x+a_2x^2+\cdots";i:1;s:141:"<img class="mwe-math-fallback-png-inline tex" alt="= a_0+a_1x+a_2x^2+\cdots" src="/images/math/f/e/3/fe3e268382fd486e8572daf895bd4c9d.png" />";}i:179;a:2:{i:0;s:9:"f(x) \,\!";i:1;s:126:"<img class="mwe-math-fallback-png-inline tex" alt="f(x) \,\!" src="/images/math/8/d/f/8dfae20000a042d8e9047aad1d7e171e.png" />";}i:180;a:2:{i:0;s:28:"= \sum_{n=0}^\infty a_n x^n ";i:1;s:145:"<img class="mwe-math-fallback-png-inline tex" alt="= \sum_{n=0}^\infty a_n x^n " src="/images/math/6/6/3/6633d51d63b35281d030755a6b0aebb1.png" />";}i:181;a:2:{i:0;s:25:"= a_0 +a_1x+a_2x^2+\cdots";i:1;s:142:"<img class="mwe-math-fallback-png-inline tex" alt="= a_0 +a_1x+a_2x^2+\cdots" src="/images/math/f/e/3/fe3e268382fd486e8572daf895bd4c9d.png" />";}i:182;a:2:{i:0;s:70:"\begin{cases} 3x + 5y + z \\ 7x - 2y + 4z \\ -6x + 3y + 2z \end{cases}";i:1;s:187:"<img class="mwe-math-fallback-png-inline tex" alt="\begin{cases} 3x + 5y + z \\ 7x - 2y + 4z \\ -6x + 3y + 2z \end{cases}" src="/images/math/6/3/4/6349be04b3562fc215c7a4e130422a96.png" />";}i:183;a:2:{i:0;s:89:"
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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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";i:1;s:278:"<img class="mwe-math-fallback-png-inline tex" alt=" \begin{array}{|c|c||c|} a & b & S \\ \hline 0&0&1\\ 0&1&1\\ 1&0&1\\ 1&1&0\\ \end{array} " src="/images/math/9/1/5/9151e94ef2bb52c18176dbe4c11921ed.png" />";}i:184;a:2:{i:0;s:15:"( \frac{1}{2} )";i:1;s:132:"<img class="mwe-math-fallback-png-inline tex" alt="( \frac{1}{2} )" src="/images/math/4/0/a/40ad9d3d1fc9a61e16d22d7e3f854fec.png" />";}i:185;a:2:{i:0;s:28:"\left ( \frac{1}{2} \right )";i:1;s:145:"<img class="mwe-math-fallback-png-inline tex" alt="\left ( \frac{1}{2} \right )" src="/images/math/2/8/b/28bcd5b82ce0e92b25e8a0b4bd5be215.png" />";}i:186;a:2:{i:0;s:28:"\left ( \frac{a}{b} \right )";i:1;s:145:"<img class="mwe-math-fallback-png-inline tex" alt="\left ( \frac{a}{b} \right )" src="/images/math/2/9/0/2905969500b40b2f2c7078206e7e0e81.png" />";}i:187;a:2:{i:0;s:75:"\left [ \frac{a}{b} \right ] \quad \left \lbrack \frac{a}{b} \right \rbrack";i:1;s:192:"<img class="mwe-math-fallback-png-inline tex" alt="\left [ \frac{a}{b} \right ] \quad \left \lbrack \frac{a}{b} \right \rbrack" src="/images/math/7/c/b/7cb5a74153ec87cdda6b92669ba685e1.png" />";}i:188;a:2:{i:0;s:77:"\left \{ \frac{a}{b} \right \} \quad \left \lbrace \frac{a}{b} \right \rbrace";i:1;s:194:"<img class="mwe-math-fallback-png-inline tex" alt="\left \{ \frac{a}{b} \right \} \quad \left \lbrace \frac{a}{b} \right \rbrace" src="/images/math/8/0/5/805b2e61cb380736d5366bccb844b1c7.png" />";}i:189;a:2:{i:0;s:40:"\left \langle \frac{a}{b} \right \rangle";i:1;s:157:"<img class="mwe-math-fallback-png-inline tex" alt="\left \langle \frac{a}{b} \right \rangle" src="/images/math/d/0/6/d06e733ce705ed26a7e048dbd2945371.png" />";}i:190;a:2:{i:0;s:72:"\left | \frac{a}{b} \right \vert \quad \left \Vert \frac{c}{d} \right \|";i:1;s:189:"<img class="mwe-math-fallback-png-inline tex" alt="\left | \frac{a}{b} \right \vert \quad \left \Vert \frac{c}{d} \right \|" src="/images/math/8/0/9/809fc4791f12abb16a5f9611a43469f9.png" />";}i:191;a:2:{i:0;s:85:"\left \lfloor \frac{a}{b} \right \rfloor \quad \left \lceil \frac{c}{d} \right \rceil";i:1;s:202:"<img class="mwe-math-fallback-png-inline tex" alt="\left \lfloor \frac{a}{b} \right \rfloor \quad \left \lceil \frac{c}{d} \right \rceil" src="/images/math/1/4/c/14c563a841b6c01dd13c5f3fa90845a1.png" />";}i:192;a:2:{i:0;s:37:"\left / \frac{a}{b} \right \backslash";i:1;s:154:"<img class="mwe-math-fallback-png-inline tex" alt="\left / \frac{a}{b} \right \backslash" src="/images/math/2/f/3/2f3c5907c0a4fc4fda69eb71890ce952.png" />";}i:193;a:2:{i:0;s:152:"\left \uparrow \frac{a}{b} \right \downarrow \quad \left \Uparrow \frac{a}{b} \right \Downarrow \quad \left \updownarrow \frac{a}{b} \right \Updownarrow";i:1;s:269:"<img class="mwe-math-fallback-png-inline tex" alt="\left \uparrow \frac{a}{b} \right \downarrow \quad \left \Uparrow \frac{a}{b} \right \Downarrow \quad \left \updownarrow \frac{a}{b} \right \Updownarrow" src="/images/math/d/e/7/de73c9252b269fb79408d6f791b5c3de.png" />";}i:194;a:2:{i:0;s:20:"\left [ 0,1 \right )";i:1;s:137:"<img class="mwe-math-fallback-png-inline tex" alt="\left [ 0,1 \right )" src="/images/math/a/3/8/a38771eae1778d0e214f6596a8dc1337.png" />";}i:195;a:2:{i:0;s:27:"\left \langle \psi \right |";i:1;s:144:"<img class="mwe-math-fallback-png-inline tex" alt="\left \langle \psi \right |" src="/images/math/d/a/2/da25fc177fd4c53a2c3399c25685dd4c.png" />";}i:196;a:2:{i:0;s:35:"\left . \frac{A}{B} \right \} \to X";i:1;s:152:"<img class="mwe-math-fallback-png-inline tex" alt="\left . \frac{A}{B} \right \} \to X" src="/images/math/b/7/1/b71d82a3ed5c1a72ded46efc19ecc582.png" />";}i:197;a:2:{i:0;s:57:"\big( \Big( \bigg( \Bigg( \dots \Bigg] \bigg] \Big] \big]";i:1;s:174:"<img class="mwe-math-fallback-png-inline tex" alt="\big( \Big( \bigg( \Bigg( \dots \Bigg] \bigg] \Big] \big]" src="/images/math/6/4/2/642a7988a93248dd92f1a53804cd40aa.png" />";}i:198;a:2:{i:0;s:85:"\big\{ \Big\{ \bigg\{ \Bigg\{ \dots \Bigg\rangle \bigg\rangle \Big\rangle \big\rangle";i:1;s:202:"<img class="mwe-math-fallback
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\frac{\left(3-x\right) \times 2}{3-x}
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\right)";i:1;s:181:"<img class="mwe-math-fallback-png-inline tex" alt="2 = \left( \frac{\left(3-x\right) \times 2}{3-x} \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 = \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}
{3^m\left(m\,3^n+n\,3^m\right)}";i:1;s:200:"<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:391;a:2:{i:0;s:35:"u'' + p(x)u' + q(x)u=f(x),\quad x>a";i:1;s:170:"<img class="mwe-math-fallback-png-inline tex" alt="u'' + p(x)u' + q(x)u=f(x),\quad x>a" src="/images/math/d/7/b/d7b3799aedae667fcc79b43ba678b94a.png" />";}i:392;a:2:{i:0;s:35:"u'' + p(x)u' + q(x)u=f(x),\quad x>a";i:1;s:170:"<img class="mwe-math-fallback-png-inline tex" alt="u'' + p(x)u' + q(x)u=f(x),\quad x>a" src="/images/math/d/7/b/d7b3799aedae667fcc79b43ba678b94a.png" />";}i:393;a:2:{i:0;s:61:"|\bar{z}| = |z|, |(\bar{z})^n| = |z|^n, \arg(z^n) = n \arg(z)";i:1;s:178:"<img class="mwe-math-fallback-png-inline tex" alt="|\bar{z}| = |z|, |(\bar{z})^n| = |z|^n, \arg(z^n) = n \arg(z)" src="/images/math/2/e/a/2eac34dbc8ebbccb22ce8dfe9d5c1a86.png" />";}i:394;a:2:{i:0;s:61:"|\bar{z}| = |z|,
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|(\bar{z})^n| = |z|^n,
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\arg(z^n) = n \arg(z)";i:1;s:186:"<img class="mwe-math-fallback-png-inline tex" alt="|\bar{z}| = |z|, |(\bar{z})^n| = |z|^n, \arg(z^n) = n \arg(z)" src="/images/math/2/e/a/2eac34dbc8ebbccb22ce8dfe9d5c1a86.png" />";}i:395;a:2:{i:0;s:35:"\lim_{z\rightarrow z_0} f(z)=f(z_0)";i:1;s:152:"<img class="mwe-math-fallback-png-inline tex" alt="\lim_{z\rightarrow z_0} f(z)=f(z_0)" src="/images/math/0/2/1/02122c7e5ff915c4616fb457747c8bf4.png" />";}i:396;a:2:{i:0;s:35:"\lim_{z\rightarrow z_0} f(z)=f(z_0)";i:1;s:152:"<img class="mwe-math-fallback-png-inline tex" alt="\lim_{z\rightarrow z_0} f(z)=f(z_0)" src="/images/math/0/2/1/02122c7e5ff915c4616fb457747c8bf4.png" />";}i:397;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:291:"<img class="mwe-math-fallback-png-inline tex" alt="\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" src="/images/math/7/f/b/7fb11db1e8b5890998b2f0f59f0e3d60.png" />";}i:398;a:2:{i:0;s:170:"\phi_n(\kappa) =
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\frac{1}{4\pi^2\kappa^2} \int_0^\infty
\frac{\sin(\kappa R)}{\kappa R}
\frac{\partial}{\partial R}
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\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) = \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: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) =
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0.033C_n^2\kappa^{-11/3},\quad
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\frac{1}{L_0}\ll\kappa\ll\frac{1}{l_0}";i:1;s:211:"<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:401;a:2:{i:0;s:100:"f(x) = \begin{cases}1 & -1 \le x < 0 \\
\frac{1}{2} & x = 0 \\ 1 - x^2 & \text{otherwise}\end{cases}";i:1;s:236:"<img class="mwe-math-fallback-png-inline tex" alt="f(x) = \begin{cases}1 & -1 \le x < 0 \\ \frac{1}{2} & x = 0 \\ 1 - x^2 & \text{otherwise}\end{cases}" src="/images/math/3/e/3/3e3579f4c1c6a95f181f227fd3ede7de.png" />";}i:402;a:2:{i:0;s:104:"
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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}
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";i:1;s:264:"<img class="mwe-math-fallback-png-inline tex" alt=" f(x) = \begin{cases} 1 & -1 \le x < 0 \\ \frac{1}{2} & x = 0 \\ 1 - x^2 & \text{otherwise} \end{cases} " 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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= \sum_{n=0}^\infty
\frac{(a_1)_n\cdots(a_p)_n}{(c_1)_n\cdots(c_q)_n}
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\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) = \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: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}
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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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\end{align}";i:1;s:320:"<img class="mwe-math-fallback-png-inline tex" alt="\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}" src="/images/math/7/8/7/787eb92e00313cb866a89579fde92108.png" />";}i:412;a:2:{i:0;s:168:"\begin{align}
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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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\end{align}";i:1;s:321:"<img class="mwe-math-fallback-png-inline tex" alt="\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}" src="/images/math/7/8/7/787eb92e00313cb866a89579fde92108.png" />";}i:413;a:2:{i:0;s:172:" with a thumbnail- we don't render math in the parsertests by default, so math is not stripped and turns up as escaped <math> tags. [[Image:foobar.jpg|thumb|<math>2+2";i:1;s:259:"<strong class='error texerror'>Failed to parse (syntax error): with a thumbnail- we don't render math in the parsertests by default, so math is not stripped and turns up as escaped &lt;math&gt; tags. [[Image:foobar.jpg|thumb|<math>2+2</strong>
";}i:414;a:2:{i:0;s:66:" with a thumbnail- math enabled [[Image:foobar.jpg|thumb|<math>2+2";i:1;s:189:"<img class="mwe-math-fallback-png-inline tex" alt=" with a thumbnail- math enabled [[Image:foobar.jpg|thumb|<math>2+2" src="/images/math/4/b/1/4b1d6eacd0bcc60a0aadf0d34626ee74.png" />";}i:415;a:2:{i:0;s:41:"<script>alert(document.cookies);</script>";i:1;s:170:"<img class="mwe-math-fallback-png-inline tex" alt="<script>alert(document.cookies);</script>" src="/images/math/5/9/f/59f1117d63b4ce95a694d44b588f0840.png" />";}i:416;a:2:{i:0;s:11:"\widehat{x}";i:1;s:128:"<img class="mwe-math-fallback-png-inline tex" alt="\widehat{x}" src="/images/math/9/9/8/998309e831dfb051f233c92b4b8a825b.png" />";}i:417;a:2:{i:0;s:13:"\widetilde{x}";i:1;s:130:"<img class="mwe-math-fallback-png-inline tex" alt="\widetilde{x}" src="/images/math/e/9/e/e9e91996778a6d6f5cdf4cc951955206.png" />";}i:418;a:2:{i:0;s:9:"\euro 200";i:1;s:126:"<img class="mwe-math-fallback-png-inline tex" alt="\euro 200" src="/images/math/1/8/8/18867d4c568a19ae7b2388346e504ec3.png" />";}i:419;a:2:{i:0;s:8:"\geneuro";i:1;s:125:"<img class="mwe-math-fallback-png-inline tex" alt="\geneuro" src="/images/math/9/8/b/98b63c235ee187a38267e0e170b10e9d.png" />";}i:420;a:2:{i:0;s:14:"\geneuronarrow";i:1;s:131:"<img class="mwe-math-fallback-png-inline tex" alt="\geneuronarrow" src="/images/math/a/a/4/aa4a1ed370f4ee705c6930384bf89502.png" />";}i:421;a:2:{i:0;s:12:"\geneurowide";i:1;s:129:"<img class="mwe-math-fallback-png-inline tex" alt="\geneurowide" src="/images/math/4/4/0/4404468e6187fb04e4f7e1f15e550825.png" />";}i:422;a:2:{i:0;s:13:"\officialeuro";i:1;s:130:"<img class="mwe-math-fallback-png-inline tex" alt="\officialeuro" src="/images/math/d/7/0/d708de0eed23dbd6f02b99ea9073547b.png" />";}i:423;a:2:{i:0;s:8:"\digamma";i:1;s:125:"<img class="mwe-math-fallback-png-inline tex" alt="\digamma" src="/images/math/2/f/0/2f057b6e514c8ca2d9cf9a3e549f8865.png" />";}i:424;a:2:{i:0;s:21:"\Coppa\coppa\varcoppa";i:1;s:138:"<img class="mwe-math-fallback-png-inline tex" alt="\Coppa\coppa\varcoppa" src="/images/math/8/3/0/8308ee5003aa36112414cad8ef874f85.png" />";}i:425;a:2:{i:0;s:8:"\Digamma";i:1;s:125:"<img class="mwe-math-fallback-png-inline tex" alt="\Digamma" src="/images/math/5/c/f/5cfd6e5df6c87798542dca2e22c1e7cb.png" />";}i:426;a:2:{i:0;s:12:"\Koppa\koppa";i:1;s:129:"<img class="mwe-math-fallback-png-inline tex" alt="\Koppa\koppa" src="/images/math/5/2/5/52593a0cdac178d165985ac014788b97.png" />";}i:427;a:2:{i:0;s:12:"\Sampi\sampi";i:1;s:129:"<img class="mwe-math-fallback-png-inline tex" alt="\Sampi\sampi" src="/images/math/e/9/d/e9dabb19e4c27bf23d3c2a3629474562.png" />";}i:428;a:2:{i:0;s:24:"\Stigma\stigma\varstigma";i:1;s:141:"<img class="mwe-math-fallback-png-inline tex" alt="\Stigma\stigma\varstigma" src="/images/math/7/b/9/7b9233276816994a33a5e968202cef6e.png" />";}i:429;a:2:{i:0;s:17:"\text{next years}";i:1;s:134:"<img class="mwe-math-fallback-png-inline tex" alt="\text{next years}" src="/images/math/6/6/9/6691dbc0b36631a68b78dd5ace256d80.png" />";}i:430;a:2:{i:0;s:18:"\text{next year's}";i:1;s:140:"<img class="mwe-math-fallback-png-inline tex" alt="\text{next year's}" src="/images/math/2/3/6/236fd262b1976d04bc0e7a969d61aede.png" />";}i:431;a:2:{i:0;s:18:"\text{`next' year}";i:1;s:140:"<img class="mwe-math-fallback-png-inline tex" alt="\text{`next' year}" src="/images/math/0/5/8/05854b483a833f067cb6ae72319b44bc.png" />";}i:432;a:2:{i:0;s:6:"\sin x";i:1;s:123:"<img class="mwe-math-fallback-png-inline tex" alt="\sin x" src="/images/math/c/d/b/cdba58911c590ced3e2435dfa39f6873.png" />";}i:433;a:2:{i:0;s:7:"\sin(x)";i:1;s:124:"<img class="mwe-math-fallback-png-inline tex" alt="\sin(x)" src="/images/math/3/e/2/3e21673ce6c9b09f9ec50b7237248576.png" />";}i:434;a:2:{i:0;s:7:"\sin{x}";i:1;s:124:"<img class="mwe-math-fallback-png-inline tex" alt="\sin{x}" src="/images/math/f/b/5/fb5551723991d4dcb974a23c162ae813.png" />";}i:435;a:2:{i:0;s:9:"\sin x \,";i:1;s:126:"<img class="mwe-math-fallback-png-inline tex" alt="\sin x \," src="/images/math/7/6/a/76a8e1a01bd233c1e4e
";}i:448;a:2:{i:0;s:1:" ";i:1;s:161:"<strong class='error texerror'>Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): </strong>
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";}}
