$\begin{aligned}\displaystyle\nabla\cdotsymbol{\tau}\equiv\cfrac{\partial s_{ij}}{\partial x_{i}}&\displaystyle=\mu\left[\cfrac{\partial}{\partial x_{i}}\left(\cfrac{\partial u_{i}}{\partial x_{j}}+\cfrac{\partial u_{j}}{\partial x_{i}}\right)\right]+\lambda~{}\left[\cfrac{\partial}{\partial x_{i}}\left(\cfrac{\partial u_{k}}{\partial x_{k}}\right)\right]\delta_{ij}\\ &\displaystyle=\mu~{}\cfrac{\partial^{2}u_{i}}{\partial x_{i}\partial x_{j}}+\mu~{}\cfrac{\partial^{2}u_{j}}{\partial x_{i}\partial x_{i}}+\lambda~{}\cfrac{\partial^{2}u_{k}}{\partial x_{k}\partial x_{j}}\\ &\displaystyle=(\mu+\lambda)~{}\cfrac{\partial^{2}u_{i}}{\partial x_{i}\partial x_{j}}+\mu~{}\cfrac{\partial^{2}u_{j}}{\partial x_{i}^{2}}\\ &\displaystyle\equiv(\mu+\lambda)~{}\nabla(\nabla\cdot\mathbf{u})+\mu~{}\nabla^{2}\mathbf{u}~{}.\end{aligned}$


$\int_{2}^{5}x^{2}\,dx=F(5)-F(2)=\frac{5^{3}}{3}-\frac{2^{3}}{3}=\frac{125}{3}-\frac{8}{3}=\frac{117}{3}=39.$


$\gamma_{\alpha\theta}\cos{\alpha}+\gamma_{\theta\beta}\cos{\beta}+\gamma_{\alpha\beta}\ =0$


$\displaystyle\frac{\mathrm{d}}{\mathrm{d}x}\left(\frac{1}{g(x)}\right)=\frac{\mathrm{d}}{\mathrm{d}x}\left(\frac{f(x)}{g(x)}\right)$


$\begin{vmatrix}b_{1}&a_{12}\\ b_{2}&a_{22}\end{vmatrix}=\begin{vmatrix}a_{11}x_{1}&a_{12}\\ a_{21}x_{1}&a_{22}\end{vmatrix}=x_{1}\begin{vmatrix}a_{11}&a_{12}\\ a_{21}&a_{22}\end{vmatrix}$


${\underbrace{\partial h\over\partial t}}_{\begin{smallmatrix}\,\text{Change}\\ \,\text{in mass}\\ \,\text{over time}\end{smallmatrix}}+\underbrace{{\partial\overline{hu}\over\partial x}+{\partial\overline{hv}\over\partial y}}_{\begin{smallmatrix}\,\text{Total spatial}\\ \,\text{variation of}\\ \,\text{x,y mass fluxes}\end{smallmatrix}}=0$


$ax^{2}+bx+c=0,\,\text{ with }a\neq 0,$


$\frac{1}{1+\frac{1^{2}}{2+\frac{3^{2}}{2}}}=\frac{13}{15}=1-\frac{1}{3}+\frac{1}{5}.$


$E_{ij}:=\frac{1}{2}\left[\frac{\partial u_{i}}{\partial x_{j}}+\frac{\partial u_{j}}{\partial x_{i}}+\frac{\partial u_{k}}{\partial x_{i}}\,\frac{\partial u_{k}}{\partial x_{j}}\right]\,.$


$x,y\in V,(x,y)\in E$


$[\mathrm{Fe}/\mathrm{H}]=\log_{10}{\left(\frac{N_{\mathrm{Fe}}}{N_{\mathrm{H}}}\right)_{\mathrm{star}}}-\log_{10}{\left(\frac{N_{\mathrm{Fe}}}{N_{\mathrm{H}}}\right)_{\mathrm{sun}}}$


$f[x_{0},\dots,x_{n}]=\sum_{j=0}^{n}\frac{f(x_{j})}{q^{\prime}(x_{j})}.$


$\rm atm\,$


$f_{X\mid Y=y}(x)={f_{X}(x)L_{X\mid Y=y}(x)\over{\int_{-\infty}^{\infty}f_{X}(x)L_{X\mid Y=y}(x)\,dx}}$


${n\choose m}=\frac{n!}{m!\,(n-m)!}$


$\hat{symbol\varphi}\cdot\mathbf{\hat{r}}=-\sin\varphi\cos\varphi+\cos\varphi\sin\varphi=0$


$v_{C}(t)=V_{0}+{1\over C}\int_{0}^{t}i_{C}(\tau)\,d\tau\iff i_{L}(t)=I_{0}+{1\over L}\int_{0}^{t}v_{L}(\tau)\,d\tau$


$O(mr)\,$


$\frac{\partial}{\partial t}(\rho_{0}+\rho_{0}s)+\frac{\partial}{\partial x}(\rho_{0}u+\rho_{0}su)=0$


$A+B\rightarrow S$


$\begin{bmatrix}1&0\\ 1&3\\ \end{bmatrix}\begin{bmatrix}a&b\\ c&d\\ \end{bmatrix}=\begin{bmatrix}a&b\\ c&d\\ \end{bmatrix}\begin{bmatrix}x&0\\ 0&y\\ \end{bmatrix}$


$\forall x,y\in Nx<y\Rightarrow\exists z\in Nx+z=y$


$L(s)\equiv\sum_{n=0}^{\infty}\frac{(-1)^{n}}{(2n+1)^{s}}\qquad\qquad L(1-s)=L(s)\Gamma(s)2^{s}\pi^{-s}\sin\frac{\pi s}{2},$


$\operatorname{B}^{\bullet}:A(K)\stackrel{d}{\longrightarrow}\wedge^{2}K^{*}$


$\pi=\cfrac{4}{1+\cfrac{1^{2}}{3+\cfrac{2^{2}}{5+\cfrac{3^{2}}{7+\ddots}}}}\!$


$O(m\sqrt{n}\log n)$


$\mathrm{Ox}+ne^{-}\rightarrow\mathrm{Red},$


$\mathbf{k}=\mathbf{u}^{T}M=(M^{T}\mathbf{v})^{T}$


$\mathfrak{g}=\mathfrak{k}\oplus\mathfrak{p}$


$\beta(s)=\sum_{n=0}^{\infty}\frac{(-1)^{n}}{(2n+1)^{s}},$


$\int_{-x}^{x}(x^{2}-y^{2})^{n}\cos(y)\,dy.$


$Q_{D}\left(n\right)=O(Q_{S}\left(n\right)\log\left(n\right))\,\!$


$\omega_{P}=\sqrt{\frac{ne^{2}}{{\varepsilon_{0}}m^{*}}}$


$\frac{\int_{\Omega}f(x)\ w(x)dx}{\int_{\Omega}w(x)\ dx}$


$\frac{577}{408}=1+\cfrac{1}{2+\cfrac{1}{2+\cfrac{1}{2+\cfrac{1}{2+\cfrac{1}{2+\cfrac{1}{2+\cfrac{1}{2}}}}}}}.$


$\hat{\mathbf{x}}_{2}=[-\cos\beta\sin\gamma_{2}\,,\,\cos\gamma_{2}\,,\,\sin\beta\sin\gamma_{2}]$


$D=y\oplus x$


$L(s,\chi)=\sum_{n=1}^{\infty}\frac{\chi(n)}{n^{s}}.$


$0\rightarrow A^{\prime}\rightarrow A\rightarrow A^{\prime\prime}\rightarrow 0$


$a=d\neq b=c,\alpha=\zeta=120^{\circ},\beta=\gamma=\delta=\epsilon=90^{\circ},$


$C:=\{b\in B:\exists\,p(x)\in A[x]\,,\hbox{ which is monic and such that }p(b)=0\}\,.$


$s_{xy}=\frac{1}{N}\sum_{n=1}^{N}(x_{n}-\bar{x})(y_{n}-\bar{y}).$


$1-{1\over 2}+{1\over 3}-{1\over 4}+{1\over 5}-\cdots=\sum\limits_{n=1}^{\infty}{(-1)^{n+1}\over n}$


$\{0,1\dots p-1\}$


$\displaystyle\underbrace{1+\cdots\cdots\cdots+1}_{n\,\text{ terms}}$


${{\nu}_{e}}+n\to p+e^{-}$


$X=\begin{bmatrix}0&1\\ 1&0\end{bmatrix}$


$\log^{*}n:=\begin{cases}0&\mbox{if }~{}n\leq 1;\\ 1+\log^{*}(\log n)&\mbox{if }~{}n>1\end{cases}$


$αβ+βγ+γα=−2$


${{\int_{0}^{t}\mathrm{ROI}(\tau)d\tau}\over\mathrm{ROI}(t)}$


$A_{ij}=\frac{\partial\beta_{i}}{\partial\theta_{j}}=\delta_{ij}-\frac{\partial\alpha_{i}}{\partial\theta_{j}}=\delta_{ij}-\frac{\partial^{2}\psi}{\partial\theta_{i}\partial\theta_{j}}$


$\begin{matrix}\,\text{Total}\\ \,\text{resources}\\ \,\text{available}\end{matrix}\ \ \ =\ \ \ \begin{matrix}\,\text{Resources}\\ \,\text{at start}\\ \,\text{of innings}\end{matrix}\ \ \ \ -\ \ \ \ \ \ \ \ \ \ \ \ \ \begin{matrix}\,\text{Resources lost by}\\ \,\text{first interruption}\end{matrix}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ -\ \ \ \ \ \ \ \ \ \ \ \begin{matrix}\,\text{Resources lost by}\\ \,\text{second interruption}\end{matrix}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ -\ \ \ \ \ \ \ \ \ \ \ \ \begin{matrix}\,\text{Resources lost by}\\ \,\text{third interruption}\end{matrix}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ -\ \ \ \ \ \,\text{etc}...$


$\left(\frac{\partial\mathbf{u}}{\partial x}\right)^{\top}$


$\mathrm{variance}=s^{2}=\displaystyle\frac{\sum_{i=1}^{n}(x_{i}-\bar{x})^{2}}{n-1}\!$


$0\to L\to M\to N\to 0$


$\ell(y)=\begin{cases}\frac{1}{2}-ty&\,\text{if}~{}~{}ty\leq 0,\\ \frac{1}{2}(1-ty)^{2}&\,\text{if}~{}~{}0<ty\leq 1,\\ 0&\,\text{if}~{}~{}1\leq ty\end{cases}$


$T=e^{-\sum_{i=1}^{N}\sigma_{i}\int_{0}^{\ell}n_{i}(z)\mathrm{d}z}=10^{-\sum_{i=1}^{N}\varepsilon_{i}\int_{0}^{\ell}c_{i}(z)\mathrm{d}z},$


${}^{2}_{1}\mathrm{D}+\,^{3}_{1}\mathrm{T}\rightarrow\,^{4}_{2}\mathrm{He}\left(3.5\,\mathrm{MeV}\right)+\,^{1}_{0}\mathrm{n}\left(14.1\,\mathrm{MeV}\right)$


$\langle v_{e}\rangle=\frac{\int_{v_{e0}}^{\infty}f(v_{x})\,v_{x}\,dv_{x}}{\int_{-\infty}^{\infty}f(v_{x})\,dv_{x}}$


$O(E\log V)$


$\lim_{\lambda\to\infty}\int_{0}^{\lambda}\left(1-\frac{x}{\lambda}\right)^{\alpha}f(x)\,dx$


$(A\oplus B)\oplus A=A\oplus(A\oplus B)$


$\left(\frac{\,\text{volume of}\ o}{\,\text{volume of largest ball enclosed in}\ o}\right)^{1/d}$


$1\rightarrow K\stackrel{i}{\rightarrow}G\stackrel{\pi}{\rightarrow}H\rightarrow 1$


$x_{j}=\left(\frac{\partial\Phi}{\partial y_{j}}\right)_{\{y_{i\neq j}\}}$


$n=\begin{bmatrix}0&1\\ 1&0\end{bmatrix}$


$\left.\frac{\partial^{2}S(z)}{\partial z_{i}\partial z_{j}}\right|_{z=0}=2h_{ij}(0);$


$\mathrm{NPV}(i,N)=\sum_{t=0}^{N}\frac{R_{t}}{(1+i)^{t}}$


$0=\dot{\sigma}=a_{11}\mathord{\overbrace{e_{1}}^{{}=0}}+A_{12}\mathbf{e}_{2}-\mathord{\overbrace{v_{\,\text{eq}}}^{v(\sigma)}}=A_{12}\mathbf{e}_{2}-v_{\,\text{eq}}.$


$\pi=\frac{4}{1}-\frac{4}{3}+\frac{4}{5}-\frac{4}{7}+\frac{4}{9}-\frac{4}{11}+\frac{4}{13}\cdots.$


$\mathbf{C}=\sum_{i=1}^{n}(\mathbf{x_{i}}-\mathbf{\bar{x}})(\mathbf{x_{i}}-\mathbf{\bar{x}})^{T}$


$\rightarrow\ \ldots\,$


$\underline{\underline{\epsilon}}$


$\frac{1}{k}+\frac{1}{k}=\frac{1}{k}+\frac{1}{k+1}+\frac{1}{k(k+1)}.$


$SBD=\sqrt{\frac{b+c}{a+b+c+d}}$


$\varphi=\frac{1-\sqrt{5}}{2}=-0.6180\,339887\dots$


$X_{*}\mathfrak{P}=\mathfrak{P}\big(X^{-1}(\cdot)\big).$


$\langle X\rangle_{\infty}$


$f(n)=2(n-2^{\lfloor\log_{2}(n)\rfloor})+1$


$y_{2i}=\begin{cases}y_{2i}^{*}&\textrm{if}\;y_{1i}^{*}>0\\ 0&\textrm{if}\;y_{1i}^{*}\leq 0.\end{cases}$


$=A_{c}\cos\left(2\pi f_{c}t+2\pi f_{\Delta}\int_{0}^{t}x_{m}(\tau)d\tau\right)$


$\forall x,y\in X,x\leq y\wedge x\in S\Rightarrow y\in S$


$\left[\mathbf{S}\right]=\begin{bmatrix}S_{11}&S_{12}\\ S_{12}&S_{11}\end{bmatrix}$


$H\left(\frac{du}{dt}\right)=\frac{d}{dt}H(u)$


$V=\left(\frac{kba}{3}\right)\left(\frac{Tn}{P}\right)$


$A^{+}+B\to A+B^{+}$


$t\text{score}=\frac{(\widehat{\beta}-\beta_{0})\sqrt{n-2}}{\sqrt{\,\text{SSR}/\sum_{i=1}^{n}\left(x_{i}-\overline{x}\right)^{2}}}.$


${\left({{p}\atop{n}}\right)}=\frac{p!}{n!(p-n)!}.$


$\frac{\partial\bar{u_{i}}}{\partial t}+\bar{u_{j}}\frac{\partial\bar{u_{i}}}{\partial x_{j}}=-\frac{1}{\rho}\frac{\partial\bar{p}}{\partial x_{i}}+\nu\frac{\partial^{2}\bar{u_{i}}}{\partial x_{j}\partial x_{j}}-\frac{\partial\tau_{ij}}{\partial x_{j}}.$


$\underbrace{\,\text{Var}(R_{P})}_{\equiv\sigma^{2}_{P}}=\mathbb{E}[R_{P}-\mathbb{E}[R_{P}]]^{2}$


$(14.a)\quad\nabla^{2}\psi=\,(\nabla\psi)^{2}$


$Y_{i}=\begin{cases}1&\,\text{if }Y_{i}^{1\ast}>Y_{i}^{0\ast},\\ 0&\,\text{otherwise.}\end{cases}$


$\varphi=1+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{1+\;\;\ddots\,}}}$


$\forall a,b\in E,(a,b)\not\in R$


$\frac{1}{2}-\frac{1}{4}+\frac{1}{8}-\frac{1}{16}+\cdots=\frac{1/2}{1-(-1/2)}=\frac{1}{3}.$


$\Theta(mnp)$


$A^{*}\otimes B^{*}\to(B\otimes A)^{*}$


$X_{n}\ \xrightarrow{p}\ X$


$x,\,y\in K$


$\sin G=\cos L\cos D\cos R-\sin L\sin R=-\cos L\cos D\sin I+\sin L\cos I$


$A\oplus B=B\oplus A$


$q=\left\lceil y-0.5\right\rceil=-\left\lfloor-y+0.5\right\rfloor\,$


$x=\sum_{n=0}^{\infty}\frac{a_{n}}{10^{n}}.$


$\pi=3+\cfrac{1}{7+\cfrac{1}{15+\cfrac{1}{1+\cfrac{1}{292+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{2+\cfrac{1}{1+\cfrac{1}{3+\cfrac{1}{1+\ddots}}}}}}}}}}}$


$\forall x,y\in S,$


$\,{}_{p}F_{q}(a_{1},\ldots,a_{p};b_{1},\ldots,b_{q};z)=\sum_{n=0}^{\infty}\frac{(a_{1})_{n}\dots(a_{p})_{n}}{(b_{1})_{n}\dots(b_{q})_{n}}\,\frac{z^{n}}{n!}$


$\sum_{i=1}^{n}(X_{i}-\bar{X})^{2}=\begin{Vmatrix}X_{1}-\bar{X}\\ \vdots\\ X_{n}-\bar{X}\end{Vmatrix}^{2}.$


$\sigma_{ij}=\left(\frac{\partial f}{\partial\varepsilon_{ij}}\right)_{T},$


$\operatorname{Li}_{2}\left(\frac{1}{3}\right)-\frac{1}{6}\operatorname{Li}_{2}\left(\frac{1}{9}\right)=\frac{{\pi}^{2}}{18}-\frac{\ln^{2}3}{6}$


$\mbox{Mortgage Yield: ri such that P}~{}=\sum_{n=1}^{N}\frac{C(t)}{(1+ri/1200)^{t-1}}$


$\frac{\pi}{4}=\cfrac{1}{1+\cfrac{1^{2}}{2+\cfrac{3^{2}}{2+\cfrac{5^{2}}{2+\ddots}}}}=1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+-\dots$


$\nabla^{2}\mathbf{E}-\mu_{0}\epsilon_{0}\frac{\partial^{2}\mathbf{E}}{\partial t^{2}}=0.$


$L=\left\lceil\frac{\lfloor\log_{2}n\rfloor+1}{2}\right\rceil$


$\gamma\text{chir}=\gamma_{0}\gamma_{1}=\sigma_{3}=\gamma\text{chir}^{\dagger}~{}.$


$x,y\in\Omega$


$\,{}^{238}_{92}\mathrm{U}\ +\,^{24}_{12}\mathrm{Mg}\to\,^{259}_{104}\mathrm{Rf}\ +3\,^{1}_{0}\mathrm{n}$


$\text{ Enrichment ratio}=\left(\frac{\text{Protein concentration in the foam}}{\text{Protein concentration in the initial feed}}\right)$


$6=\underbrace{3+3}_{\,\text{2 parts}}.$


$C=\frac{\hat{\mu}_{4}}{\hat{\sigma}^{4}}=\frac{\frac{1}{n}\sum_{i=1}^{n}(x_{i}-\bar{x})^{4}}{\left(\frac{1}{n}\sum_{i=1}^{n}(x_{i}-\bar{x})^{2}\right)^{2}},$


$\stackrel{\mathrm{def}}{=}\ \int_{-\infty}^{\infty}f(\tau)\,g(t-\tau)\,d\tau$


$[\wp^{\prime}(z)]^{2}|_{z=0}\sim\frac{4}{z^{6}}-\frac{24}{z^{2}}\sum\frac{1}{(m\omega_{1}+n\omega_{2})^{4}}-80\sum\frac{1}{(m\omega_{1}+n\omega_{2})^{6}}$


$L(x)=\sum_{i}l_{i}x^{p^{i}},l_{i}\in\mathrm{GF}(p^{m}).$


$\displaystyle g_{1}=\frac{m_{3}}{m_{2}^{3/2}}=\frac{\frac{1}{n}\sum_{i=1}^{n}\left(x_{i}-\bar{x}\right)^{3}}{\left(\frac{1}{n}\sum_{i=1}^{n}\left(x_{i}-\bar{x}\right)^{2}\right)^{3/2}}\ ,$


$\rm Si+O_{2}\rightarrow SiO_{2}\,$


$\forall\epsilon>0\ \exists\delta>0\ \forall y\in X\ \left[d(x,y)<\delta\Rightarrow\forall n\in\mathbf{N}\ d\left(f^{n}(x),f^{n}(y)\right)<\epsilon\right].$


$A+n\to B^{*}+\gamma$


$\cos(A+B)=\cos(A)\cos(B)+\sin(A)\sin(B)\,$


$0\rightarrow B\rightarrow E\rightarrow A\rightarrow 0$


$S_{\nu}(x)=\sum_{k=0}^{\infty}\frac{\sin((2k+1)\pi x)}{(2k+1)^{\nu}}$


$p_{n}(k)=\frac{n!}{(n-k)!k!}p^{k}(1-p)^{n-k}.$


$A\oplus B.$


$\forall x\forall y\exists z(x\in z\land y\in z).$


$\frac{a_{0}}{2}-\frac{\Delta a_{0}}{4}+\frac{\Delta^{2}a_{0}}{8}-\frac{\Delta^{3}a_{0}}{16}+\cdots=\frac{1}{2}-\frac{1}{4}+\frac{1}{8}-\frac{1}{16}+\cdots.$


$\frac{1}{IA^{2}}+\frac{1}{IC^{2}}=\frac{1}{IB^{2}}+\frac{1}{ID^{2}}=\frac{1}{r^{2}}$


$\sin\left(a+b\right)=\sin a\cos b+\cos a\sin b$


$Z=\alpha X+\beta Y+\gamma[X,Y],$


$\displaystyle{T={1\over 2}I-{1\over 2}\int_{0}^{\infty}e^{-t}T(t)\,dt.}$


$1+\cfrac{1}{2+\cfrac{1}{5+\cfrac{1}{5+\cfrac{1}{4+\ddots}}}}$


$T=e^{-\int_{0}^{\ell}\mu(z)\mathrm{d}z}=10^{-\int_{0}^{\ell}\mu_{10}(z)\mathrm{d}z},$


$f(x)=\begin{cases}1,&\mathrm{if}~{}|x|<\tfrac{1}{2},\\ 0,&\mathrm{otherwise.}\end{cases}$


$\cos(A\pm B)=\cos A\ \cos B\mp\sin A\ \sin B$


$\alpha=\tan^{-1}\left(\frac{\sin\theta}{\cos\theta+\lambda}\right)$


$\frac{\,\text{radius of circle}}{\,\text{length of chord}}=\frac{R}{2R\sin\frac{\theta}{2}}=\frac{1}{2\sin\frac{\theta}{2}}\approx\frac{1}{2\theta/2}=\frac{1}{\theta}$


$U\rightarrow V$


$\alpha_{s}=\arctan\left(\frac{b}{R}\right)-\arctan\left(\frac{b}{R+r_{s}}\right)$


$H^{2n}(t)]=\sum\limits_{k=0}^{2n}{B_{k}^{2n}(t)[H_{k}]},$


$O(m+\log n)$


$Pxy\leftrightarrow\forall z[Ozx\rightarrow Ozy].$


$(A\oplus B)$


$d=-\left\lfloor\frac{m}{2}\right\rfloor$


$(\forall z:z\not\in FV(y)z\not\in BV(b))\to\operatorname{beta-redex}[\lambda x.b\ y]=b[x:=y]$


$\sum_{k=0}^{\infty}\frac{(-1)^{k}}{(z+k)^{m+1}}=\frac{1}{(-2)^{m+1}m!}\left[\psi^{(m)}\left(\frac{z}{2}\right)-\psi^{(m)}\left(\frac{z+1}{2}\right)\right]$


$\Delta V\Delta{k}=(2\pi)^{3}$


$\mathbf{J}_{\mathbf{F}}(x_{1},x_{2},x_{3})=\begin{bmatrix}\dfrac{\partial y_{1}}{\partial x_{1}}&\dfrac{\partial y_{1}}{\partial x_{2}}&\dfrac{\partial y_{1}}{\partial x_{3}}\\ \dfrac{\partial y_{2}}{\partial x_{1}}&\dfrac{\partial y_{2}}{\partial x_{2}}&\dfrac{\partial y_{2}}{\partial x_{3}}\\ \dfrac{\partial y_{3}}{\partial x_{1}}&\dfrac{\partial y_{3}}{\partial x_{2}}&\dfrac{\partial y_{3}}{\partial x_{3}}\\ \dfrac{\partial y_{4}}{\partial x_{1}}&\dfrac{\partial y_{4}}{\partial x_{2}}&\dfrac{\partial y_{4}}{\partial x_{3}}\end{bmatrix}=\begin{bmatrix}1&0&0\\ 0&0&5\\ 0&8x_{2}&-2\\ x_{3}\cos x_{1}&0&\sin x_{1}\end{bmatrix}.$


$\Omega\left(\frac{\log n}{\log(Sw/n)}\right)$


$\mathrm{L}:\quad\begin{smallmatrix}4&&1\\ &\swarrow&\\ 2&\rightarrow&3\end{smallmatrix}\qquad\mathrm{U}:\quad\begin{smallmatrix}1&&4\\ \downarrow&&\uparrow\\ 2&\rightarrow&3\end{smallmatrix}\qquad\mathrm{X}:\quad\begin{smallmatrix}1&&4\\ &\searrow\!\!\!\!\!\!\nearrow&\\ 3&&2\end{smallmatrix}$


$\langle\ ,\ \rangle_{k}$


$\mathbb{R}^{n}$


$a+\cfrac{1}{b+\cfrac{1}{c+\cfrac{1}{\ddots}}}$


$\gamma_{\rm CMB}+p\rightarrow\Delta^{+}\rightarrow n+\pi^{+}.$


$\forall A,\exists B,\forall C,\forall D,P(C,D)\rightarrow(C\in A\rightarrow D\in B).$


$\frac{\partial\mathbf{y}}{\partial x}=\begin{bmatrix}\frac{\partial y_{1}}{\partial x}\\ \frac{\partial y_{2}}{\partial x}\\ \vdots\\ \frac{\partial y_{m}}{\partial x}\\ \end{bmatrix}.$


$O(pn(m+n\log n))$


$\scriptstyle{\underline{\underline{Q}}}$


$\underline{\#(\theta)}$


$Z=\frac{1}{N!h^{3N}}\int\,\exp[-\beta H(p_{1}\cdots p_{N},x_{1}\cdots x_{N})]\;d^{3}p_{1}\cdots d^{3}p_{N}\,d^{3}x_{1}\cdots d^{3}x_{N}$


$O(lgn)$


$f(z)=\sum_{n=0}^{\infty}(z-a)^{n}{1\over 2\pi i}\int_{C}{f(w)\over(w-a)^{n+1}}\,\mathrm{d}w.$


$\gamma=\alpha+j\beta\,$


$M=\begin{bmatrix}A&B\\ C&D\end{bmatrix}$


$\frac{\partial^{2}u}{\partial x_{i}\partial y_{j}}-\frac{\partial^{2}u}{\partial y_{i}\partial x_{j}}=0$


$\underbrace{A_{i}\land\dots\land A_{i}}_{i}$


$x\stackrel{*}{\rightarrow}y$


$q_{xy}=\frac{\sum(x-\bar{x})(y-\bar{y})w(x-\bar{x},y-\bar{y})I(x,y)}{\sum w(x-\bar{x},y-\bar{y})I(x,y)}$


$\int_{0}^{t}x_{m}(\tau)d\tau=\frac{A_{m}\cos(2\pi f_{m}t)}{2\pi f_{m}}\,$


$\begin{array}[]{c}2x^{2}+7x+6\,=\,0\\ x^{2}+\tfrac{7}{2}x+3\,=\,0\\ \left(x+\tfrac{7}{4}\right)^{2}-\tfrac{1}{16}\,=\,0\\ \left(x+\tfrac{7}{4}\right)^{2}\,=\,\tfrac{1}{16}\\ x+\tfrac{7}{4}=\tfrac{1}{4}\quad\,\text{or}\quad x+\tfrac{7}{4}=-\tfrac{1}{4}\\ x=-\tfrac{3}{2}\quad\,\text{or}\quad x=-2.\end{array}$


$f^{\star}\left(x^{*}\right)=\begin{cases}0,&\left|x^{*}\right|\leq 1\\ \infty,&\left|x^{*}\right|>1.\end{cases}$


$A^{c}\cup B^{c}\subseteq(A\cap B)^{c}$


$L_{6}\lambda=-\frac{2}{1+\lambda}+\frac{60}{2+\lambda}-\frac{420}{3+\lambda}+\frac{1120}{4+\lambda}-\frac{1260}{5+\lambda}+\frac{504}{6+\lambda}.$


$\frac{h\nu^{\prime}}{c}=\gamma\frac{h\nu}{c}-\gamma\beta\left\|p\right\|\cos\theta=\gamma\frac{h\nu}{c}-\gamma\beta\frac{h\nu}{c}\cos\theta$


$\mbox{absolute margin of victory}~{}=\begin{cases}0;&w\leq\frac{c}{2}\\ w-\max\{r,\frac{c}{2}\};&w>\frac{c}{2}\end{cases}$


$\therefore\mathrm{rect}(t)=\Pi(t)=\lim_{n\rightarrow\infty,n\in\mathbb{(}Z)}\frac{1}{(2t)^{2n}+1}=\begin{cases}0&\mbox{if }~{}|t|>\frac{1}{2}\\ \frac{1}{2}&\mbox{if }~{}|t|=\frac{1}{2}\\ 1&\mbox{if }~{}|t|<\frac{1}{2}.\\ \end{cases}$


$\delta_{ij}=\begin{cases}0&\,\text{if }i\neq j,\\ 1&\,\text{if }i=j.\end{cases}$


$A\triangle B=(A\cup B)-(A\cap B)$


$\omega_{p}=\sqrt{\frac{ne^{2}}{\epsilon_{0}m}}$


$a^{n}=\underbrace{a\times a\times\cdots\times a}_{n}$


$H_{ij}=\begin{bmatrix}{\partial^{2}V_{ij}\over\partial x_{i}\partial x_{j}}&{\partial^{2}V_{ij}\over\partial x_{i}\partial y_{j}}&{\partial^{2}V_{ij}\over\partial x_{i}\partial z_{j}}\\ {\partial^{2}V_{ij}\over\partial y_{i}\partial x_{j}}&{\partial^{2}V_{ij}\over\partial y_{i}\partial y_{j}}&{\partial^{2}V_{ij}\over\partial y_{i}\partial z_{j}}\\ {\partial^{2}V_{ij}\over\partial z_{i}\partial x_{j}}&{\partial^{2}V_{ij}\over\partial z_{i}\partial y_{j}}&{\partial^{2}V_{ij}\over\partial z_{i}\partial z_{j}}\end{bmatrix}$


$1=\frac{1}{2}+\frac{1}{3}+\frac{1}{7}+\frac{1}{43}+\frac{1}{1807}+\cdots.$


$\begin{pmatrix}B\\ C\end{pmatrix}=\begin{pmatrix}S_{11}&S_{12}\\ S_{21}&S_{22}\end{pmatrix}\begin{pmatrix}A\\ D\end{pmatrix}.$


$f_{X_{(k)}}(x)=\frac{n!}{(k-1)!(n-k)!}[F_{X}(x)]^{k-1}[1-F_{X}(x)]^{n-k}f_{X}(x)$


$y_{1i}=\begin{cases}y_{1i}^{*}&\textrm{if}\;y_{1i}^{*}>0\\ 0&\textrm{if}\;y_{1i}^{*}\leq 0.\end{cases}$


${\stackrel{\triangledown}{\mathbf{A}}}$


$\begin{aligned}\displaystyle e^{\pi\sqrt{19}}&\displaystyle\approx(5x)^{3}-6.000\,010\dots\\ \displaystyle e^{\pi\sqrt{43}}&\displaystyle\approx(5x)^{3}-6.000\,000\,010\dots\\ \displaystyle e^{\pi\sqrt{67}}&\displaystyle\approx(5x)^{3}-6.000\,000\,000\,061\dots\\ \displaystyle e^{\pi\sqrt{163}}&\displaystyle\approx(5x)^{3}-6.000\,000\,000\,000\,000\,034\dots\end{aligned}$


$1+\frac{1}{4+\cfrac{1}{1+\cfrac{1}{18+\cfrac{1}{\ddots\qquad{}}}}}$


$\frac{\partial^{2}H}{\partial u^{2}}>0$


$ax+by+cz+d=\begin{cases}>0&\,\text{front facing}\\ =0&\,\text{parallel}\\ <0&\,\text{back facing}\end{cases}$


$M_{0}=\begin{bmatrix}1&0&2&1\end{bmatrix}$


$\mbox{NPV}~{}=\sum_{t=0}^{n}\frac{C_{t}}{(1+r)^{t}}=0$


$i=0\ldots n-1$


$\mathrm{Var}(x)=\int_{-\infty}^{\infty}(x-\langle x\rangle)^{2}P_{n}(x)\,dx=\frac{L^{2}}{12}\left(1-\frac{6}{n^{2}\pi^{2}}\right)$


$M\text{an}\text{aniso}=M\text{s}\frac{\int_{0}^{\pi}\!e^{0.5(E(1)+E(2))}\sin(\theta)\cos(\theta)\,d\theta}{\int_{0}^{\pi}\!e^{0.5(E(1)+E(2))}\sin(\theta)\,d\theta}$


$e=\frac{\frac{1}{\psi}-\frac{1}{p}}{1-\frac{1}{p}}$


$\frac{\partial^{2}C}{\partial S^{2}}$


$0\to A\to B\to C\to 0\;$


$\forall a\,\forall s\,[(Ma\and\forall x\,[x\in s\leftrightarrow\exists y\,(x\in y\and y\in a)])\rightarrow Ms].$


$\begin{aligned}\displaystyle a_{n+1}&\displaystyle=\frac{a_{n}+b_{n}}{2},\\ \displaystyle\par  b_{n+1}&\displaystyle = \sqrt{a_{n} b_{n}},\\ \displaystyle
 t_{n+1}&\displaystyle = t_{n} - p_{n}(a_{n}-a_{n+1})^{2},\\ \displaystyle
 p_{n+1}&\displaystyle = 2p_{n}.
\end{aligned}$


$O(N\,\log N)$


$O(n\log n\log\log n)$


$\frac{\partial^{2}\epsilon_{y}}{\partial z^{2}}+\frac{\partial^{2}\epsilon_{z}}{\partial y^{2}}=2\frac{\partial^{2}\epsilon_{yz}}{\partial y\partial z}\,\!$


$C_{k}=\frac{\int zS_{k}(z)\;dz}{\int S_{k}(z)\;dz}$


$U:C\to D$


$\displaystyle\sum_{i=1}^{n}\left(y_{i}-\bar{y}\right)\left(x_{i}-\bar{x}\right)-\hat{\beta}\sum_{i=1}^{n}\left(x_{i}-\bar{x}\right)^{2}=0$


$O(VE\log V)$


$\langle X,Z,Y\rangle_{D}$


$\lambda(T,W)=\frac{\int_{-W}^{W}{\|U(f)\|}^{2}\,df}{\int_{-1/2}^{1/2}{\|U(f)\|}^{2}\,df}.$


$O(N\log{N})$


$\tanh\frac{1}{2}=\frac{e-1}{e+1}=0+\cfrac{1}{2+\cfrac{1}{6+\cfrac{1}{10+\cfrac{1}{14+\cfrac{1}{\ddots}}}}}$


$\underline{\lnot\psi\quad\quad\quad}\,\!$


$\mathrm{ZnO+SO_{3}\rightarrow ZnSO_{4}}$


$Y_{0}=\sqrt{\frac{G+j\omega C}{R+j\omega L}}$


${n\choose k}=\frac{n!}{k!(n-k)!}.$


$\exp\left(\frac{1}{\hbox{Vol}(U)}\int_{U}\log f\right).$


$1+\frac{1}{4}+\frac{1}{4^{2}}+\cdots+\frac{1}{4^{n}}=\frac{1-\left(\frac{1}{4}\right)^{n+1}}{1-\frac{1}{4}}.$


$e=1+\cfrac{2}{1+\cfrac{1}{6+\cfrac{1}{10+\cfrac{1}{14+\cfrac{1}{18+\cfrac{1}{22+\cfrac{1}{26+\ddots\,}}}}}}}.$


$\begin{bmatrix}V_{1}\\ V_{0}\end{bmatrix}=\begin{bmatrix}z(j\omega)_{11}&z(j\omega)_{12}\\ z(j\omega)_{21}&z(j\omega)_{22}\end{bmatrix}\begin{bmatrix}I_{1}\\ I_{0}\end{bmatrix}$


$\displaystyle\frac{d}{dx}\left(\frac{f(x)}{g(x)}\right)$


$\nabla\times\mathbf{B}=\frac{1}{\epsilon_{0}c^{2}}\mathbf{J}+\frac{1}{c^{2}}\frac{\partial\mathbf{E}}{\partial t}$


$X_{j}=\left.\frac{\partial g}{\partial\xi_{j}}\right|_{\xi_{j}=0}$


$E(t)=\frac{C(t)}{\int_{0}^{\infty}C(t)\ dt}$


$h_{i}=\left|\frac{\partial\mathbf{r}}{\partial q_{i}}\right|$


$symbol{\nabla\times}\left(symbol{\nabla\times B}\right)=\mu_{0}\epsilon_{0}\frac{\partial}{\partial t}symbol{\nabla\times E}\ .$


$V^{\otimes n}=\underbrace{V\otimes V\otimes\cdots\otimes V}_{n\,\text{ times}}$


$\begin{bmatrix}f\\ g\end{bmatrix}=\begin{bmatrix}A&B\\ C&D\end{bmatrix}\begin{bmatrix}x\\ y\end{bmatrix}$


$P_{j}(d_{j})=\begin{cases}0&\,\text{if }d_{j}\leq 0\\ 1&\,\text{if }d_{j}>0\end{cases}$


$n^{O(\log\log n)}$


$\scriptstyle 1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\cdots=\frac{\pi}{4}=0.7853981\ldots$


$\begin{aligned}\displaystyle ds^{2}&\displaystyle=\begin{bmatrix}du&dv\end{bmatrix}\begin{bmatrix}E&F\\ F&G\end{bmatrix}\begin{bmatrix}du\\ dv\end{bmatrix}\\ &\displaystyle=\begin{bmatrix}du^{\prime}&dv^{\prime}\end{bmatrix}\begin{bmatrix}\frac{\partial u}{\partial u^{\prime}}&\frac{\partial u}{\partial v^{\prime}}\\ \frac{\partial v}{\partial u^{\prime}}&\frac{\partial v}{\partial v^{\prime}}\end{bmatrix}^{\mathrm{T}}\begin{bmatrix}E&F\\ F&G\end{bmatrix}\begin{bmatrix}\frac{\partial u}{\partial u^{\prime}}&\frac{\partial u}{\partial v^{\prime}}\\ \frac{\partial v}{\partial u^{\prime}}&\frac{\partial v}{\partial v^{\prime}}\end{bmatrix}\begin{bmatrix}du^{\prime}\\ dv^{\prime}\end{bmatrix}\\ &\displaystyle=\begin{bmatrix}du^{\prime}&dv^{\prime}\end{bmatrix}\begin{bmatrix}E^{\prime}&F^{\prime}\\ F^{\prime}&G^{\prime}\end{bmatrix}\begin{bmatrix}du^{\prime}\\ dv^{\prime}\end{bmatrix}\\ &\displaystyle=(ds^{\prime})^{2}.\end{aligned}$


$\int_{\Delta}K\,dA=\alpha+\beta+\gamma-\pi.$


$pq=(qp)^{\prime}$


$p(k)={n\choose k}p^{k}(1-p)^{n-k},$


$\alpha,\alpha^{\prime},\beta,\gamma,\gamma^{\prime}$


$\pi_{k}=\begin{cases}\pi_{0}\dfrac{(c\rho)^{k}}{k!},&\mbox{if }~{}0<k<c\\ \pi_{0}\dfrac{\rho^{k}c^{c}}{c!},&\mbox{if }~{}c\leq k\end{cases}$


$T_{i,j}(m)=\begin{bmatrix}1&&&&&&&\\ &\ddots&&&&&&\\ &&1&&&&&\\ &&&\ddots&&&&\\ &&m&&1&&\\ &&&&&&\ddots&\\ &&&&&&&1\end{bmatrix}$


$\begin{pmatrix}b_{1}\\ a_{1}\end{pmatrix}=\begin{pmatrix}T_{11}&T_{12}\\ T_{21}&T_{22}\end{pmatrix}\begin{pmatrix}a_{2}\\ b_{2}\end{pmatrix}\,$


$\omega+i\delta\to 0$


$\Omega={\left(q+N^{\prime}-1\right)!\over q!(N^{\prime}-1)!}$


$\tan\gamma_{4}=\cos\beta/\tan\gamma_{2}=1/\tan\gamma_{1}=\tan(\gamma_{1}+\pi/2)\,$


$\forall x\,(x\neq\emptyset\rightarrow\exists y\in x\,(y\cap x=\emptyset))$


$\mathfrak{P}(\{d\})$


$Pr\begin{cases}Ds\begin{cases}Sp(\pi)\begin{cases}Va:\\ S^{0},\cdots,S^{T},O^{0},\cdots,O^{T}\\ Dc:\\ \begin{cases}&P\left(S^{0}\wedge\cdots\wedge O^{T}|\pi\right)\\ =&\left[\begin{array}[]{c}P\left(S^{0}\wedge O^{0}|\pi\right)\\ \prod_{t=1}^{T}\left[P\left(S^{t}|S^{t-1}\wedge\pi\right)\times P\left(O^{t}|S^{t}\wedge\pi\right)\right]\end{array}\right]\end{cases}\\ Fo:\\ \begin{cases}P\left(S^{t}\mid S^{t-1}\wedge\pi\right)\equiv G\left(S^{t},A\bullet S^{t-1},Q\right)\\ P\left(O^{t}\mid S^{t}\wedge\pi\right)\equiv G\left(O^{t},H\bullet S^{t},R\right)\end{cases}\end{cases}\\ Id\end{cases}\\ Qu:\\ P\left(S^{T}\mid O^{0}\wedge\cdots\wedge O^{T}\wedge\pi\right)\end{cases}$


$\wedge^{\hbox{even}}\,T^{*}M$


$\begin{aligned}\displaystyle x&\displaystyle=\int_{0}^{L}\cos\theta\,ds\\ &\displaystyle=\int_{0}^{L}\cos\left[(as)^{2}\right]ds\end{aligned}$


$O(ckN)$


$a=\arccos\left(\frac{\cos\alpha+\cos\beta\cos\gamma}{\sin\beta\sin\gamma}\right)$


$\alpha=2:\quad\operatorname{E}\left[-\frac{1}{N}\frac{\partial^{2}\ln\mathcal{L}(\alpha,\beta,a,c|Y)}{\partial\alpha\partial a}\right]={\mathcal{I}}_{\alpha,a}$


$\forall^{\mathrm{st}}x\,\exists^{\mathrm{st}}y\,\forall^{\mathrm{st}}t\,(t\in y\leftrightarrow(t\in x\land\phi(t,u_{1},\dots,u_{n})))$


$T=\begin{bmatrix}Q_{1}&\\ &Q_{2}\end{bmatrix}\left(\begin{bmatrix}D_{1}&\\ &D_{2}\end{bmatrix}+\beta zz^{T}\right)\begin{bmatrix}Q_{1}^{T}&\\ &Q_{2}^{T}\end{bmatrix}$


$\sin\alpha\sin\beta\cos\beta\cos\gamma+\sin\alpha\cos^{2}\beta\sin\gamma+\cos\alpha\sin^{2}\beta\cos\gamma+\cos\alpha\sin\beta\cos\beta\sin\gamma$


$Re(a b)=Re(b a)$


$g^{M}_{p}(u,v)=g^{N}_{f(p)}(df(u),df(v))\qquad\forall p\in M,\forall u,v\in T_{p}M.$


$B+X\rightarrow Y+D$


$\gamma_{1}\!\left(\!\frac{1}{\,2\,}\!\right)=-2\gamma\ln 2-\ln^{2}\!2+\gamma_{1}\,=\,-1.353459680\ldots$


$\frac{\mbox{total}~{}\;\mbox{votes}~{}}{\mbox{total}~{}\;\mbox{seats}~{}}$


$F(y)=\frac{\int_{x_{0}}^{x_{1}}y(x)\;\mathrm{d}x}{\int_{x_{0}}^{x_{1}}(1+[y(x)]^{2})\;\mathrm{d}x}$


$M^{\downarrow Y}(X,Y)=\left[{\begin{array}[]{*{20}c}{\mu_{2}}\\ {\Sigma_{22}}\\ \end{array}}\right]$


$\frac{\partial^{2}y}{\partial x_{i}\partial x_{j}}=\sum_{k}\left(\frac{\partial y}{\partial u_{k}}\frac{\partial^{2}u_{k}}{\partial x_{i}\partial x_{j}}\right)+\sum_{k,\ell}\left(\frac{\partial^{2}y}{\partial u_{k}\partial u_{\ell}}\frac{\partial u_{k}}{\partial x_{i}}\frac{\partial u_{\ell}}{\partial x_{j}}\right).$


$\nu+\bar{\nu}\rightarrow Z\rightarrow\,\text{hadrons}$


$\Delta_{K}=\left\{\begin{array}[]{ll}d&\,\text{if }d\equiv 1\;\;(\mathop{{\rm mod}}4)\\ 4d&\,\text{if }d\equiv 2,3\;\;(\mathop{{\rm mod}}4).\\ \end{array}\right.$


$\frac{p!}{k!(p-k)!},$


$(X\ Y)[x:=y]=X[x:=y]\ Y[x:=y]$


$\mathrm{Glucose}+\mathrm{Alkaline\ copper\ tartarate}\xrightarrow{\mathrm{Reduction}}\mathrm{Cuprous\ oxide}$


$L(\theta;A,B)=\frac{(A+B)!}{A!B!}\theta^{A}(1-\theta)^{B},$


$R(D)=\left\{\begin{matrix}\frac{1}{2}\log_{2}(\sigma_{x}^{2}/D),&\mbox{if }~{}0\leq D\leq\sigma_{x}^{2}\\ \\ 0,&\mbox{if }~{}D>\sigma_{x}^{2}.\end{matrix}\right.$


$\zeta(s,t)=\sum_{n=1}^{\infty}\frac{H_{n,t}}{(n+1)^{s}}$


$\frac{S}{kN}=\ln\left(\frac{VT^{\hat{c}_{V}}}{N\Phi}\right)$


$\mathbf{M}=\begin{bmatrix}a&b\\ c&d\\ \end{bmatrix}$


$\begin{cases}\,\text{if }x\begin{cases}\,\text{if }y,3\\ \,\text{if }\neg y,2\end{cases}\\ \,\text{if }\neg x\begin{cases}\,\text{if }y\,\text{ , }1\\ \,\text{if }\neg y\,\text{ , }0\end{cases}\end{cases}$


$a_{0}+\cfrac{1}{a_{1}+\cfrac{1}{a_{2}+\cfrac{1}{\ddots+\cfrac{1}{a_{n}}}}},$


$\begin{matrix}a\ \underbrace{\uparrow\uparrow\!\!\dots\!\!\uparrow}_{n}\ b=\underbrace{a\ \underbrace{\uparrow\!\!\dots\!\!\uparrow}_{n-1}\ (a\ \underbrace{\uparrow\!\!\dots\!\!\uparrow}_{n-1}\ (\dots\ \underbrace{\uparrow\!\!\dots\!\!\uparrow}_{n-1}\ a))}_{b\,\text{ copies of }a}\end{matrix}$


$1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\cdots=\sum_{n=1}^{\infty}\frac{1}{n}.$


$L_{W}=L_{p}+10\log_{10}\!\left(\frac{4\pi r^{2}}{A_{0}}\right)\!~{}\mathrm{dB},$


$\delta^{a}_{b}=\begin{cases}1&\mbox{if }~{}a=b,\\ 0&\mbox{if }~{}a\neq b.\end{cases}$


$r=r_{xy}=\frac{\sum^{n}_{i=1}(x_{i}-\bar{x})(y_{i}-\bar{y})}{\sqrt{\sum^{n}_{i=1}(x_{i}-\bar{x})^{2}}\sqrt{\sum^{n}_{i=1}(y_{i}-\bar{y})^{2}}}$


$\forall x,y\in H:\qquad\lnot(x\vee y)=\lnot x\wedge\lnot y.$


$J_{F}=\left|\begin{matrix}\frac{\partial f_{1}}{\partial X_{1}}&\cdots&\frac{\partial f_{1}}{\partial X_{N}}\\ \vdots&\ddots&\vdots\\ \frac{\partial f_{N}}{\partial X_{1}}&\cdots&\frac{\partial f_{N}}{\partial X_{N}}\end{matrix}\right|,$


$\frac{1}{p!}\Delta_{Q}^{p}f(p,q,r)=\frac{1}{(n-p)!}\Delta_{P}^{n-p}f(a,b,c).$


$A+B\to C+D$


$\mathcal{Q}=\mathfrak{P}\left(\sum_{q\geq 1}\mathfrak{C}_{=q}(\mathcal{Z})\times\sum_{p=1}^{q}{q\choose p}\mathcal{U}^{p}\right).$


$(j c)b=j(b c)$


$\delta\Delta-\Delta\delta=-\bar{\nu}D+(\tau-\bar{\alpha}-\beta)\Delta+(\mu-\gamma+\bar{\gamma})\delta+\bar{\lambda}\bar{\delta}\,,$


$\left(A\cup B\right)^{c}=A^{c}\cap B^{c}.$


$L_{n}(x)=\frac{e^{x}}{n!}\frac{d^{n}}{dx^{n}}\left(e^{-x}x^{n}\right)=\frac{1}{n!}\left(\frac{d}{dx}-1\right)^{n}x^{n},$


$\cos C=-\cos A\cos B+\sin A\sin B\cosh c,$


$206.835-1.015\left(\frac{\,\text{total words}}{\,\text{total sentences}}\right)-84.6\left(\frac{\,\text{total syllables}}{\,\text{total words}}\right).$


$\left[\frac{\partial D}{\partial n}\right]_{\,\text{x,t}}=-P_{\,\text{x,t}}$


$U:C\to\,\textbf{Set}$


$D_{\tau(\mathfrak{P})}=\tau D_{\mathfrak{P}}\tau^{-1}$


$f(t)=\begin{cases}t^{1/3}&\,\text{if }t>(\frac{6}{29})^{3}\\ \frac{1}{3}\left(\frac{29}{6}\right)^{2}t+\frac{4}{29}&\,\text{otherwise}\end{cases}$


$\sqrt{\langle v^{2}\rangle}=\left(\int_{0}^{\infty}v^{2}\,f(v)\,dv\right)^{1/2}=\sqrt{\frac{3kT}{m}}=\sqrt{\frac{3RT}{M}}=\sqrt{\frac{3}{2}}v_{p}$


$\omega_{p}=\sqrt{\frac{ne^{2}}{\gamma m\epsilon_{0}}}$


$\theta_{N}(x)=\begin{cases}0&\,\text{if }x\leq N/2,\\ 1&\,\text{if }x>N/2.\end{cases}$


$O(\log a\log b)$


$(A\ominus B)\ominus C=A\ominus(B\oplus C)$


$\frac{n!}{(n-k)!k!}\left({\frac{w}{d}}\right)^{k-2}$


$W(e^{1+z})=1+\frac{z}{2}+\frac{z^{2}}{16}-\frac{z^{3}}{192}-\frac{z^{4}}{3072}+\frac{13z^{5}}{61440}-\frac{47z^{6}}{1474560}-\frac{73z^{7}}{41287680}+\frac{2447z^{8}}{1321205760}+O(z^{9}).$


$\nabla u(x,y)\cdot\nabla^{\perp}u(x,y)=0,\rVert\nabla u\rVert=\rVert\nabla^{\perp}u\rVert$


$\operatorname{Im}\left(\frac{\tan(x+iy)}{x+iy}\right)$


$\displaystyle 2x-2x-y$


$O(\log\log y)$


$\tau_{\,\text{rms}}=\sqrt{\frac{\int_{0}^{\infty}(\tau-\overline{\tau})^{2}A_{c}(\tau)d\tau}{\int_{0}^{\infty}A_{c}(\tau)d\tau}}$


${}^{3}_{2}\mathrm{He}+{}^{2}_{1}\mathrm{H}\to{}^{4}_{2}\mathrm{He}+\mathrm{p}$


$\mathrm{n}+{}_{\ 90}^{232}\mathrm{Th}\rightarrow{}_{\ 90}^{231}\mathrm{Th}+2\mathrm{n}\xrightarrow{\beta^{-}}{}_{\ 91}^{231}\mathrm{Pa}+\mathrm{n}\rightarrow{}_{\ 91}^{232}\mathrm{Pa}\xrightarrow{\beta^{-}}{}_{\ 92}^{232}\mathrm{U}$


$i=0\ldots n-2.$


$=\int_{0}^{\infty}G(\tau)F(t-\tau)\,d\tau$


$=\sum_{i=0}^{\infty}\sum_{j=0}^{\infty}a_{i}b_{j}(x-c)^{i+j}$


${n\choose i}=\frac{n!}{i!(n-i)!}$


$\lambda\wedge\theta\in\bigwedge^{n}V.$


$O\left(m^{g(k)}n^{k}\right)$


$\nu^{*n}=\underbrace{\nu*\cdots*\nu}_{n\,\text{ factors}}$


$\frac{\partial^{2}S}{\partial\beta_{j}\partial\beta_{k}}$


$F=\tfrac{{\displaystyle\sum_{j=1}^{k}n_{j}\left(\bar{y}_{j}-\bar{y}\right)^{2}}/{(k-1)}}{{\displaystyle{\sum_{j=1}^{k}}{\sum_{i=1}^{n_{j}}}\left(y_{ij}-\bar{y}_{j}\right)^{2}}/{(n-k)}}$


$1-\frac{1}{2}-\frac{1}{4}+\frac{1}{3}-\frac{1}{6}-\frac{1}{8}+\frac{1}{5}-\frac{1}{10}-\frac{1}{12}+\cdots$


$\begin{bmatrix}x_{1}\\ y_{1}\end{bmatrix}=\begin{bmatrix}E&F\\ G&H\end{bmatrix}\begin{bmatrix}f-By_{0}\\ g-Dy_{0}\end{bmatrix}$


$\sqrt{\frac{\Sigma(x-\bar{x})^{2}}{n}}=\sqrt{\frac{\Sigma\left[x-\left(\Sigma x\right)/n\right]^{2}}{n}}\ ,$


$\pi=3+\textstyle\frac{1}{7+\textstyle\frac{1}{15+\textstyle\frac{1}{1+\textstyle\frac{1}{292+\textstyle\frac{1}{1+\textstyle\frac{1}{1+\textstyle\frac{1}{1+\ddots}}}}}}}$


$2n+ 1 = 2(1) + 1 = 3$


$\pi=\cfrac{4}{1+\cfrac{1^{2}}{2+\cfrac{3^{2}}{2+\cfrac{5^{2}}{2+\cfrac{7^{2}}{2+\ddots}}}}}\,$


$p=-\rho{\partial\over\partial t}\Phi$


$\forall x\;\exists y\;\forall z\;(z\in y\leftrightarrow z\in x\wedge\phi(z))$


$(B/A)_{0}\ \left(\frac{\mu m}{hr}\right)$


$-1\cdot e$


$=\frac{(n+1)\,n}{(n-1)\,(n-2)\,(n-3)}\;\frac{\sum_{i=1}^{n}(x_{i}-\bar{x})^{4}}{k_{2}^{2}}-3\,\frac{(n-1)^{2}}{(n-2)(n-3)}$


$dp=-\rho\,d\phi$


$\underline{E\Gamma}$


$\gamma=1-\sum_{k=2}^{\infty}(-1)^{k}\frac{\lfloor\log_{2}k\rfloor}{k+1}.$


$\alpha<\beta\Rightarrow\gamma+\alpha<\gamma+\beta$


$\frac{1}{c_{0}^{2}}\frac{\partial^{2}p}{\partial t^{2}}-\nabla^{2}p=\rho_{0}\frac{\partial^{2}\hat{T}_{ij}}{\partial x_{i}\partial x_{j}},\quad\,\text{where}\quad\hat{T}_{ij}=v_{i}v_{j}.$


$k=\sqrt{\frac{a+b}{am+bn}}$


$P_{i}^{x}=\frac{N!}{n_{x}!(N-n_{x})!}p_{x}^{n_{x}}(1-p_{x})^{N-n_{x}}$


$\frac{\partial^{2}U}{\partial g^{2}}<0$


$=\frac{T\alpha}{T\beta}(\cos\phi+\epsilon\sin\phi)$


$\begin{array}[]{cccc}&&\begin{array}[]{ccccc}\sigma_{1}^{\prime}=+1&&&&\sigma_{1}^{\prime}=-1\end{array}\\ A&=&\left[\begin{array}[]{ccccccc}&&&|\\ &A_{+}&&|&&0\\ &&&|\\ -&-&-&|&-&-&-\\ &&&|\\ &0&&|&&A_{-}\\ &&&|\end{array}\right]&\begin{array}[]{c}\sigma_{1}=+1\\ \\ \\ \\ \sigma_{1}=-1\end{array}\end{array}$


$f:\mathbb{R}\to\mathbb{R};\qquad f(x)=\begin{cases}e^{-\frac{1}{x^{2}}}&x>0,\\ 0&x\leq 0.\end{cases}$


$M=\left(\begin{matrix}{{M}_{11}}&{{M}_{12}}\\ {{M}_{21}}&{{M}_{22}}\\ \end{matrix}\right)$


$\ell=\lfloor\log_{2}(\operatorname{depth}(v))\rfloor$


$\begin{aligned}\displaystyle\begin{bmatrix}v_{1}\\ v_{2}\end{bmatrix}&\displaystyle=R^{-1}\begin{bmatrix}v^{1}\\ v^{2}\end{bmatrix}\\ &\displaystyle=\begin{bmatrix}4&\sqrt{2}\\ \sqrt{2}&1\end{bmatrix}\begin{bmatrix}v^{1}\\ v^{2}\end{bmatrix}=\begin{bmatrix}6+2\sqrt{2}\\ 2+3/\sqrt{2}\end{bmatrix}\end{aligned}.$


$ct^{\prime}=\gamma ct-\gamma\beta x\,$


$\forall x,y\in G$


$\textstyle C_{n}=\sum_{i=1}^{n}(x_{i}-\bar{x}_{n})(y_{i}-\bar{y}_{n})$


$\begin{bmatrix}du\\ dv\end{bmatrix}=\begin{bmatrix}\frac{\partial u}{\partial u^{\prime}}&\frac{\partial u}{\partial v^{\prime}}\\ \frac{\partial v}{\partial u^{\prime}}&\frac{\partial v}{\partial v^{\prime}}\end{bmatrix}\begin{bmatrix}du^{\prime}\\ dv^{\prime}\end{bmatrix}$


$e^{-\delta\tau}$


$\gamma_{1}\,=\,\frac{\ln 2}{2}\sum_{k=2}^{\infty}\frac{(-1)^{k}}{k}\,\lfloor\log_{2}{k}\rfloor\cdot\big(2\log_{2}{k}-\lfloor\log_{2}{2k}\rfloor\big)$


$\mathrm{~{}^{14}_{6}C}\rightarrow\mathrm{~{}^{14}_{7}N}+e^{-}+\bar{\nu}_{e}$


$\log(x/y)=\log(x)-\log(y)$


$\displaystyle 2\Delta=(ab)^{2}$


$d=\sqrt{\frac{2b^{2}+2c^{2}-a^{2}}{4}}.$


$A\odot B=(A\ominus C)\cap(A^{c}\ominus D)$


$I=\frac{N}{\sum_{i}\sum_{j}w_{ij}}\frac{\sum_{i}\sum_{j}w_{ij}(X_{i}-\bar{X})(X_{j}-\bar{X})}{\sum_{i}(X_{i}-\bar{X})^{2}}$


$(g\cdot f)(x):=f(g^{-1}x)\qquad\forall x\in V,g\in G,f\in k[V].$


$\exp(z)=\sum_{n=0}^{\infty}\frac{z^{n}}{n!}.$


$\frac{d\vec{P}}{dt}=\int^{t}_{0}\mathbf{A}(t-\tau)\vec{P}(\tau)d\tau$


$p^{M}(1-p)^{{n\choose 2}-M}.$


${d\over dx}(\rho u\varphi)={d\over dx}\left({d\varphi\over dx}\right)$


$b^{n}=\underbrace{b\times\cdots\times b}_{n}$


$\Delta(x)=\int_{0}^{\infty}d\tau\int DXe^{-\int_{0}^{\tau}(\dot{x}^{2}/2+m^{2})d\tau^{\prime}}$


$C\cos{\mu}+L\sin{\mu}+T(\sin{\alpha}\sin{\mu}+\cos{\alpha}\cos{\mu}\sin{\beta})=\frac{W}{g}\frac{(V\cos{\gamma})^{2}}{R}\quad(x_{E}\,\text{-}y_{E}\,\text{ plane radial direction})$


$2+\cfrac{1}{4+\cfrac{1}{4+\cfrac{1}{4+\cfrac{1}{4+\ddots}}}}$


$\mathbf{P}=g(\mathbf{a})-g(\mathbf{b})$


$\textstyle\sinh x=\sum_{n=0}^{\infty}\frac{x^{2n+1}}{(2n+1)!}$


$\textstyle\mbox{Hess}~{}(F)=\begin{pmatrix}\frac{\partial^{2}F}{\partial x^{2}}&\frac{\partial^{2}F}{\partial x\partial y}&\frac{\partial^{2}F}{\partial x\partial z}\\ \frac{\partial^{2}F}{\partial x\partial y}&\frac{\partial^{2}F}{\partial y^{2}}&\frac{\partial^{2}F}{\partial y\partial z}\\ \frac{\partial^{2}F}{\partial x\partial z}&\frac{\partial^{2}F}{\partial y\partial z}&\frac{\partial^{2}F}{\partial z^{2}}\end{pmatrix}.$


$L^{*}=\prod_{n=1}^{N}{P_{i}^{Y_{i}}}\left(1-P_{i}\right)^{1-Y_{i}}$


$e=2+\cfrac{1}{1+\cfrac{1}{2+\cfrac{2}{3+\cfrac{3}{4+\cfrac{4}{5+\ddots}}}}}=2+\cfrac{2}{2+\cfrac{3}{3+\cfrac{4}{4+\cfrac{5}{5+\cfrac{6}{6+\ddots\,}}}}}$


$f(x)=\left\{\begin{matrix}1&\mbox{if }~{}x>0\\ 0&\mbox{otherwise}\end{matrix}\right.$


$M\setminus X=(M^{\ast}/x)^{\ast}$


$\displaystyle\forall x,y\in H:\quad\langle y,x\rangle$


$\frac{1}{r}=1-\Big(\frac{1}{p}-\frac{1}{q}\Big)$


$v_{x}{\partial c_{A}\over\partial x}+v_{y}{\partial c_{A}\over\partial y}=D_{AB}{\partial^{2}c_{A}\over\partial y^{2}}$


$r_{e}=\dfrac{\int\limits_{0}^{\infty}\pi\cdot r^{3}\cdot n(r)\,dr}{\int\limits_{0}^{\infty}\pi\cdot r^{2}\cdot n(r)\,dr}$


$s=\sqrt{\frac{\sum_{i=1}^{n}{\left(x_{i}-\bar{x}\right)}^{2}}{n-1}}$


$\mathit{c}_{p}=\left(\frac{\gamma\mathit{R}}{\gamma-1}\right)$


$(\Delta x)^{2}=\frac{\int_{-\infty}^{\infty}(x-\mu)^{2}f(x)f^{*}(x)\,dx}{\int_{-\infty}^{\infty}f(x)f^{*}(x)\,dx}$


$x^{n}-\tfrac{x}{2}-\tfrac{1}{2}-\left(s-\tfrac{1}{2}\right)(x+1)$


$\hat{H}_{\mathrm{lm}}=-\sum\mathcal{F}\,\hat{B}\hat{X}^{\dagger}+\mathrm{h.c.}\,,$


$\left(\frac{\partial u}{\partial T}\right)_{V}$


$H_{N,q,s}=\sum_{i=1}^{N}\frac{1}{(i+q)^{s}}$


$\begin{bmatrix}u\\ v\\ w\\ \end{bmatrix}=\begin{bmatrix}c_{11}&c_{12}&c_{13}\\ c_{21}&c_{22}&c_{23}\\ c_{31}&c_{32}&c_{33}\end{bmatrix}\begin{bmatrix}a_{1}\\ a_{2}\\ a_{3}\end{bmatrix}$


$\Omega_{conf}\,=\,\frac{N_{S}!}{N!(N_{S}-N)!}$


$=\int_{0}^{\infty}u^{\frac{1}{2}}e^{-\frac{u}{2}}\mathrm{d}u+\int_{0}^{\infty}u^{-\frac{1}{2}}e^{-\frac{u}{2}}\mathrm{d}u$


$\Psi(A)(B)=\mathrm{tr}(iAB)$


$\lfloor\log_{2}(x)\rfloor+2\lfloor\log_{2}(\lfloor\log_{2}(x)\rfloor+1)\rfloor+1$


$\stackrel{\alpha}{\omega}$


$1 –α/ 2$


$\begin{bmatrix}x^{\prime}\\ y^{\prime}\\ \end{bmatrix}=\begin{bmatrix}\cos\theta&-\sin\theta\\ \sin\theta&\cos\theta\\ \end{bmatrix}\begin{bmatrix}x\\ y\\ \end{bmatrix}$


$\nabla\times\mathbf{B}-\frac{1}{c^{2}}\frac{\partial\mathbf{E}}{\partial t}=\mu_{0}\mathbf{J}$


$O(mn)$


$s_{p}^{2}=\frac{\sum_{i=1}^{k}(n_{i}-1)s_{i}^{2}}{\sum_{i=1}^{k}(n_{i}-1)}$


$\pi_{T}=\left(\frac{\partial U}{\partial V}\right)_{T}$


$=\begin{vmatrix}\!symbol{i}&\!symbol{j}&\!symbol{k}\\ 0&0&\omega\\ v\cos\alpha&v\sin\alpha&\\ \;+\omega t\ v\sin\alpha&\;-\omega t\ v\cos\alpha&0\end{vmatrix}\ \ ,$


$0\to A\to B\to C\to 0$


$A(x)B(y)=\sum_{i}c_{i}(x-y)^{i}C_{i}(y)$


$\left\langle X\right\rangle_{0}$


$\forall a((\forall x\in a\;\exists y\;\phi(x,y))\to\exists b\;(\forall x\in a\;\exists y\in b\;\phi(x,y))\wedge(\forall y\in b\;\exists x\in a\;\phi(x,y)))$


$\,\text{M}+\,\text{IE}\text{M}\to\,\text{M}^{+}+\,\text{e}^{-}$


$\begin{bmatrix}V_{1}\\ 0\end{bmatrix}=\begin{bmatrix}R_{1}+R_{2}&-R_{2}\\ -R_{2}&R_{2}+R_{3}\end{bmatrix}\begin{bmatrix}I_{1}\\ I_{2}\end{bmatrix}$


$A(w)e_{q}(zw)=\sum_{n=0}^{\infty}\frac{p_{n}(z)}{[n]_{q}!}w^{n}$


$x=1+\cfrac{1}{1+\left(1+\cfrac{1}{1+x}\right)}=1+\cfrac{1}{2+\cfrac{1}{1+x}}.\,$


$\nabla\times\mathbf{B}=\mu_{0}\mathbf{J}+\frac{1}{c^{2}}\frac{\partial\mathbf{E}}{\partial t}$


$G(q,p)=\frac{\int_{0}^{p}t^{q-1}e^{-t}\,\,\text{d}t}{\int_{0}^{\infty}t^{q-1}e^{-t}\,\,\text{d}t}$


$\begin{bmatrix}V_{1}\\ I_{2}\end{bmatrix}=\begin{bmatrix}0&n\\ -n&0\end{bmatrix}\begin{bmatrix}I_{1}\\ V_{2}\end{bmatrix}$


$G=1-\frac{1}{\mu}\int_{0}^{\infty}(1-F(y))^{2}dy=\frac{1}{\mu}\int_{0}^{\infty}F(y)(1-F(y))dy$


$\mathbf{\nabla}\times\mathbf{B}=\mu_{0}\mathbf{J}+\underbrace{\mu_{0}\epsilon_{0}\frac{\partial}{\partial t}\mathbf{E}}_{\mathrm{Maxwell^{\prime}s\ term}}$


$M\stackrel{f}{\to}N\stackrel{\omega}{\to}T^{*}N\stackrel{(Df)^{*}}{\longrightarrow}T^{*}M.$


$S_{\sigma^{2}}=\sum_{i=1}^{m}(X_{i}-\overline{X})^{2},\,\text{ where }\overline{X}=\frac{S_{\mu}}{m}$


$\left\{{n\atop k}\right\}=\sum_{j=1}^{k}(-1)^{k-j}\frac{j^{n-1}}{(j-1)!(k-j)!}=\frac{1}{k!}\sum_{j=0}^{k}(-1)^{k-j}{k\choose j}j^{n}.$


$\begin{bmatrix}r\\ \theta\\ z\end{bmatrix}=\begin{bmatrix}\sqrt{x^{2}+y^{2}}\\ \operatorname{arctan}(y/x)\\ z\end{bmatrix},\ \ \ 0\leq\theta<2\pi,$


$O(\log\log n)\,$


$A\cap B=\overline{\overline{A}\cup\overline{B}}$


$U:\mathbf{C}\to\mathbf{Set}$


$O(\lambda_{\delta}(n)\log n)$


$\nu_{e}+n\rightarrow e^{-}+p$


${\rm det}\left(\beta I-B\right)=\beta^{3}-3\beta-{\rm det}(B)=0.$


$\frac{4}{\pi}=1+\cfrac{1^{2}}{2+\cfrac{3^{2}}{2+\cfrac{5^{2}}{2+\cfrac{7^{2}}{2+\cfrac{9^{2}}{2+\ddots}}}}}$


$A+B\rightarrow A^{\prime}+B^{\prime}$


$x=\left[\frac{\sqrt{5}\,\eta(5\tau)}{\eta(\tau)}\right]^{6}$


$u_{t}-\alpha u-\beta u^{2}+\gamma u^{3}=0$


$x=a_{0}+\cfrac{1}{a_{1}+\cfrac{1}{a_{2}+\cfrac{1}{a_{3}+\cfrac{1}{\ddots\,}}}}$


$=\mathbf{P}(n-1)\mathbf{r}_{dx}(n-1)-\mathbf{g}(n)\mathbf{x}^{T}(n)\mathbf{P}(n-1)\mathbf{r}_{dx}(n-1)+d(n)\mathbf{g}(n)$


$\forall z[z<x\rightarrow z<y]\rightarrow x\leq y.$


$\mathbf{J}_{\mathbf{F}}(r,\theta,\varphi)=\begin{bmatrix}\dfrac{\partial x}{\partial r}&\dfrac{\partial x}{\partial\theta}&\dfrac{\partial x}{\partial\varphi}\\ \dfrac{\partial y}{\partial r}&\dfrac{\partial y}{\partial\theta}&\dfrac{\partial y}{\partial\varphi}\\ \dfrac{\partial z}{\partial r}&\dfrac{\partial z}{\partial\theta}&\dfrac{\partial z}{\partial\varphi}\end{bmatrix}=\begin{bmatrix}\sin\theta\cos\varphi&r\cos\theta\cos\varphi&-r\sin\theta\sin\varphi\\ \sin\theta\sin\varphi&r\cos\theta\sin\varphi&r\sin\theta\cos\varphi\\ \cos\theta&-r\sin\theta&0\end{bmatrix}.$


$\sum_{i=1}^{n}(X_{i}-\overline{X})^{2}$


$AB=e^{\ln(A)+\ln(B)}.\,$


$=\begin{bmatrix}0.9901&0\\ 0&0.9901\end{bmatrix}\begin{bmatrix}1\\ 1\end{bmatrix}=\begin{bmatrix}0.9901\\ 0.9901\end{bmatrix}$


$\begin{aligned}\displaystyle\Delta_{0}&\displaystyle=b^{2}-3ac\\ \displaystyle\Delta_{1}&\displaystyle=2b^{3}-9abc+27a^{2}d\end{aligned}$


$y(u,v)=-\cos\theta\,\sinh v\,\cos u+\sin\theta\,\cosh v\,\sin u$


$\scriptstyle O(N\log N)$


$y=px-f^{\star}(p)$


$u_{2}=\begin{bmatrix}0\\ 1\end{bmatrix}$


$x,y\in A$


$0\to 2\pi i\mathbb{Z}\to\mathbf{O}\xrightarrow{\exp}\mathbf{O}^{*}\to 0$


$\begin{pmatrix}b_{1}\\ b_{2}\end{pmatrix}=\begin{pmatrix}S_{11}&S_{12}\\ S_{21}&S_{22}\end{pmatrix}\begin{pmatrix}a_{1}\\ a_{2}\end{pmatrix}\,$


$\mathrm{{}^{238}_{\ 92}U\ \xrightarrow[-2\ \beta^{-}]{+\ 6\ (n,\gamma)}\ ^{244}_{\ 94}Pu}$


$=\int\limits_{-\infty}^{\infty}f(\tau)\delta(\tau-(t-T))\,d\tau$


$\frac{\partial\mathbf{\phi}(\mathbf{X})}{\partial\mathbf{X}}=\begin{bmatrix}\frac{\partial\mathbf{\phi}}{\partial x_{1,1}}&\cdots&\frac{\partial\mathbf{\phi}}{\partial x_{1,q}}\\ \vdots&\ddots&\vdots\\ \frac{\partial\mathbf{\phi}}{\partial x_{n,1}}&\cdots&\frac{\partial\mathbf{\phi}}{\partial x_{n,q}}\\ \end{bmatrix}$


$O(mnp)$


$\frac{1071}{462}=2+\cfrac{1}{3+\cfrac{1}{7}}=[2;3,7]$


$O(m\log n)$


$\tfrac{2}{3}-\tfrac{1}{2}=\tfrac{4}{6}-\tfrac{3}{6}=\tfrac{1}{6}$


$p(x,y|\theta)=p(y|\theta)p(x|y,\theta)$


$x$


$f_{\mathrm{img}}=\begin{cases}f+2f_{\mathrm{IF}},&\mbox{if }~{}f_{\mathrm{LO}}>f\mbox{ (high side injection)}\\ f-2f_{\mathrm{IF}},&\mbox{if }~{}f_{\mathrm{LO}}<f\mbox{ (low side injection)}\end{cases}$


$\nu_{t}=\left|\frac{\partial u}{\partial y}\right|l_{m}^{2}$


$=\sum_{k=0}^{\infty}\left[(k+r-1)(k+r)+p(z)(k+r)+q(z)\right]A_{k}z^{k+r}$


$\forall x,y\in U,~{}x\neq y:~{}~{}\Pr_{h\in H}[h(x)=h(y)]\leq\frac{1}{m}$


$|x|=\begin{cases}x&\mbox{if }~{}1\leq x\leq\frac{p-1}{2},\\ p-x&\mbox{if }~{}\frac{p+1}{2}\leq x\leq p-1.\end{cases}$


$A\text{pfd}=$


$\cos(\alpha-\beta)=\cos\alpha\cos\beta+\sin\alpha\sin\beta$


$\scriptstyle{\underline{\underline{q}}}$


$c_{h}=\frac{1}{N}\sum_{t=1}^{N-h}\left(Y_{t}-\bar{Y}\right)\left(Y_{t+h}-\bar{Y}\right)$


$\eta(s)=\sum_{n=1}^{\infty}{(-1)^{n-1}\over n^{s}}=\frac{1}{1^{s}}-\frac{1}{2^{s}}+\frac{1}{3^{s}}-\frac{1}{4^{s}}+\cdots$


$V=2(\frac{1}{r_{12}}-\frac{1}{r_{1a}}-\frac{1}{r_{1b}}-\frac{1}{r_{2a}}-\frac{1}{r_{2b}}+\frac{1}{r_{ab}})$


$I^{p,\bullet}=J\oplus K$


$\tan\gamma_{2}=\cos\beta\,\tan\gamma_{1}\qquad\tan\gamma_{4}=\cos\beta\,\tan\gamma_{3}$


$=\sqrt{\frac{\sum{(y_{i}-mx_{i}-b)}^{2}}{n-2}}$


$\hat{\theta}_{n}\ \xrightarrow{p}\ \theta_{0}$


$\lambda_{ij}=\frac{\partial^{2}f}{\partial z_{i}\partial\bar{z}_{j}}$


$W=\phi(b)-\phi(a)$


$a,b,c,α,β,γ,α′,β′,γ′$


$L(\chi,s)=\sum_{n=1}^{\infty}\frac{\chi(n)}{n^{s}}$


$Alt^{p}(V)=(\wedge^{p+1}V^{*})\otimes V$


$\tilde{f}_{i}(b\otimes b^{\prime})=\begin{cases}\tilde{f}_{i}b\otimes b^{\prime},&\,\text{if }\phi_{i}(b)>\epsilon_{i}(b^{\prime}),\\ b\otimes\tilde{f}_{i}b^{\prime},&\,\text{if }\phi_{i}(b)\leq\epsilon_{i}(b^{\prime}).\end{cases}$


$\frac{1}{\tau_{C}}=\frac{1}{\tau_{U}}+\frac{1}{\tau_{M}}+\frac{1}{\tau_{B}}+\frac{1}{\tau_{ph-e}}$


$\begin{bmatrix}V_{1}\\ I_{2}\end{bmatrix}=\begin{bmatrix}h_{11}&h_{12}\\ h_{21}&h_{22}\end{bmatrix}\begin{bmatrix}I_{1}\\ V_{2}\end{bmatrix}$


$\operatorname{Re}\left(\frac{\tanh\left(x+iy\right)}{x+iy}\right)$


$\nabla^{2}E-\mu_{0}\,\varepsilon_{0}\,\frac{\partial^{2}E}{\partial t^{2}}=0.$


$J\ \xrightarrow{p}\ \infty$


$e^{-n\lambda}\lambda^{(x_{1}+x_{2}+\cdots+x_{n})}\cdot{1\over x_{1}!x_{2}!\cdots x_{n}!}\,$


$g$


$v_{3}(t)=K_{1}\int_{t_{0}}^{t}i_{1}(\tau)d\tau+K_{2}\int_{t_{0}}^{t}i_{2}(\tau)d\tau,$


$\mathbf{f}+\epsilon_{0}\mu_{0}\frac{\partial\mathbf{S}}{\partial t}\,=\nabla\cdot\mathbf{\sigma}$


$=(\frac{k\rho C}{\mu H})^{\gamma}$


$\,\sin(x+y)=\sin x\cos y+\cos x\sin y,\,$


$d=\sqrt{\int_{-\infty}^{t}(t-r)^{2}f(r)\,dr}$


$f(x)=\begin{cases}x&\mbox{if }~{}x>0\\ 0.01x&\mbox{otherwise}\end{cases}$


$\sin(\alpha-\beta)=\sin\alpha\cos-\beta+\cos\alpha\sin-\beta$


$\left(\frac{\partial^{2}S}{\partial y\partial x}\right)=\left(\frac{\partial^{2}S}{\partial x\partial y}\right):\left(\frac{\partial^{2}V}{\partial y\partial x}\right)=\left(\frac{\partial^{2}V}{\partial x\partial y}\right)$


$\textstyle\frac{3\sqrt{3}}{4}\left(1-\frac{1}{2^{2}}+\frac{1}{4^{2}}-\frac{1}{5^{2}}+\frac{1}{7^{2}}-\frac{1}{8^{2}}+\frac{1}{10^{2}}\pm\cdots\right)$


$\stackrel{\mathfrak{p}}{}$


$H(X_{1},...,X_{n})=-\sum_{x_{1}}...\sum_{x_{n}}P(x_{1},...,x_{n})\log_{2}[P(x_{1},...,x_{n})]\!$


$\left[\frac{\partial q}{\partial\theta}\right]_{j}=\left[\frac{\partial z}{\partial t}\right]_{j}.$


$0\rightarrow M^{\prime}\rightarrow M\rightarrow M^{\prime\prime}\rightarrow 0$


$\begin{aligned}\displaystyle L&\displaystyle=T-U=\frac{1}{2}M\dot{\mathbf{R}}^{2}+\left(\frac{1}{2}\mu\dot{\mathbf{r}}^{2}-U(r)\right)\\ &\displaystyle=L_{\mathrm{cm}}+L_{\mathrm{rel}}\end{aligned}$


$-\eta n_{\eta}(\xi)$


$\cos\gamma=\cos\theta\cos\theta^{\prime}+\sin\theta\sin\theta^{\prime}\cos(\varphi-\varphi^{\prime}).$


$\frac{n!}{(n-k)!k!}p^{k}(1-p)^{n-k}\rightarrow e^{-\lambda}\frac{\lambda^{k}}{k!}.$


$v^{\odot k}=\underbrace{v\odot v\odot\cdots\odot v}_{k\,\text{ times}}=\underbrace{v\otimes v\otimes\cdots\otimes v}_{k\,\text{ times}}=v^{\otimes k}.$


$m_{Moon}=0.25+2.5\log_{10}{\left(\frac{3}{2}0.00257^{2}\right)}=-12.26\!\,$


$\phi_{cm}(r)=\left\{\begin{array}[]{ll}\frac{r\left(3r+1\right)}{\left(r+1\right)^{2}},\quad r>0,\quad\lim_{r\rightarrow\infty}\phi_{cm}(r)=3\\ 0\quad\quad\,,\quad r\leq 0\end{array}\right.$


$\begin{bmatrix}x^{\prime}\\ y^{\prime}\end{bmatrix}=\begin{bmatrix}1&0\\ k&1\end{bmatrix}\begin{bmatrix}x\\ y\end{bmatrix}$


$\cos(\alpha+\beta)=\cos\alpha\cos\beta-\sin\alpha\sin\beta$


$\frac{\pi}{4}=1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\cdots$


$H(f)=\left[\frac{\partial^{2}f}{\partial x_{i}\,\partial x_{j}}\right].$


$V_{\,\text{out}}=\begin{cases}V_{\,\text{S}+}&\,\text{if }V_{1}>V_{2},\\ V_{\,\text{S}-}&\,\text{if }V_{1}<V_{2},\\ 0&\,\text{if }V_{1}=V_{2},\end{cases}$


$M=\left\lfloor\frac{-1}{\log_{2}(1-p)}\right\rfloor$


$O(Tm)=O(n^{2}m\log n)$


$g_{2}=\frac{m_{4}}{m_{2}^{2}}-3=\frac{\tfrac{1}{n}\sum_{i=1}^{n}(x_{i}-\overline{x})^{4}}{\left(\tfrac{1}{n}\sum_{i=1}^{n}(x_{i}-\overline{x})^{2}\right)^{2}}-3$


$\nabla^{2}\mathbf{E}=\mu_{0}\epsilon_{0}\frac{\partial^{2}\mathbf{E}}{\partial t^{2}}$


$\mathbf{J}=\begin{bmatrix}\cfrac{\partial x_{1}}{\partial q^{1}}&\cfrac{\partial x_{1}}{\partial q^{2}}&\cfrac{\partial x_{1}}{\partial q^{3}}\\ \cfrac{\partial x_{2}}{\partial q^{1}}&\cfrac{\partial x_{2}}{\partial q^{2}}&\cfrac{\partial x_{2}}{\partial q^{3}}\\ \cfrac{\partial x_{3}}{\partial q^{1}}&\cfrac{\partial x_{3}}{\partial q^{2}}&\cfrac{\partial x_{3}}{\partial q^{3}}\\ \end{bmatrix},\quad\mathbf{J}^{-1}=\begin{bmatrix}\cfrac{\partial q^{1}}{\partial x_{1}}&\cfrac{\partial q^{1}}{\partial x_{2}}&\cfrac{\partial q^{1}}{\partial x_{3}}\\ \cfrac{\partial q^{2}}{\partial x_{1}}&\cfrac{\partial q^{2}}{\partial x_{2}}&\cfrac{\partial q^{2}}{\partial x_{3}}\\ \cfrac{\partial q^{3}}{\partial x_{1}}&\cfrac{\partial q^{3}}{\partial x_{2}}&\cfrac{\partial q^{3}}{\partial x_{3}}\\ \end{bmatrix}$


$\displaystyle\begin{matrix}+2a+b-c=a_{1}&-2a+2b+2c=b_{1}&-2a+b+3c=c_{1}&\quad\to\left[\,\text{ }a_{1},\,\text{ }b_{1},\,\text{ }c_{1}\right]\end{matrix}$


$\scriptstyle X_{n}\xrightarrow{p}X$


$I_{\mathrm{rms}}=\sqrt{\frac{1}{T}\int_{0}^{T}\left[I\left(t\right)\right]^{2}\mathrm{d}t}\,\!$


$GWP\left(x\right)=\frac{\int_{0}^{TH}a_{x}\cdot\left[x(t)\right]dt}{\int_{0}^{TH}a_{r}\cdot\left[r(t)\right]dt}$


$\sum_{n=0}^{\infty}u_{n}=\sum_{n=0}^{\infty}\frac{p(n)}{q(n)},$


${\mathbf{W}}=\frac{1}{n-1}\sum_{i=1}^{n}(\mathbf{x}_{i}-\overline{\mathbf{x}})(\mathbf{x}_{i}-\overline{\mathbf{x}})^{\prime}$


$\log 2=\log(1+1)=\cfrac{1}{1+\cfrac{1}{2+\cfrac{1}{3+\cfrac{2}{2+\cfrac{2}{5+\cfrac{3}{2+\ddots}}}}}}=\cfrac{2}{3-\cfrac{1^{2}}{9-\cfrac{2^{2}}{15-\cfrac{3^{2}}{21-\ddots}}}}$


$\mathrm{rect}(t)=\Pi(t)=\begin{cases}0&\mbox{if }~{}|t|>\frac{1}{2}\\ \frac{1}{2}&\mbox{if }~{}|t|=\frac{1}{2}\\ 1&\mbox{if }~{}|t|<\frac{1}{2}.\\ \end{cases}$


${\left({{N}\atop{N_{R}}}\right)}=\frac{N!}{N_{R}!(N-N_{R})!}$


$z_{1}=-0.4121345\ldots+i0.5978119\ldots$


$X=\begin{bmatrix}A&B\\ C&D\end{bmatrix}$


$e=H(x)-H(\hat{x})$


$C_{2}=\left[\begin{array}[]{rrr}1&0\\ \\ 0&1\end{array}\right]-\frac{1}{2}\left[\begin{array}[]{rrr}1&1\\ \\ 1&1\end{array}\right]=\left[\begin{array}[]{rrr}\frac{1}{2}&-\frac{1}{2}\\ \\ -\frac{1}{2}&\frac{1}{2}\end{array}\right]$


$y_{i}=\begin{cases}y_{i}^{*}&\textrm{if}\;y_{i}^{*}>y_{L}\\ y_{L}&\textrm{if}\;y_{i}^{*}\leq y_{L}.\end{cases}$


$\sigma^{2}_{X}=\frac{\sum_{i=1}^{n}(X_{i}-\bar{X})^{2}\,{}}{n}.$


$a=\arccos\left(\frac{\cos\alpha+\cos\beta\cos\gamma}{\sin\beta\sin\gamma}\right),$


$\forall x,y,z\,(xRy\wedge yRz\rightarrow xRz)$


$\mathbb{C}^{n}$


$E\begin{bmatrix}\phi\\ \chi\end{bmatrix}=\begin{bmatrix}m\mathbf{I}&\vec{\sigma}\vec{p}\\ \vec{\sigma}\vec{p}&-m\mathbf{I}\end{bmatrix}\begin{bmatrix}\phi\\ \chi\end{bmatrix}\,$


$\forall x\exist y[Pyx\and(Czy\rightarrow Ozx)\and\lnot(Pxy\and(Czx\rightarrow Ozy))].$


$r_{c}=\left\{\frac{\partial V}{\partial J}\right\}_{V=0}$


$ds^{2}=-\left(\alpha^{2}-\beta_{i}\beta^{i}\right)\,dt^{2}+2\beta_{i}\,dx^{i}\,dt+\gamma_{ij}\,dx^{i}\,dx^{j}$


$e=2+\cfrac{1}{1+\cfrac{2}{5+\cfrac{1}{10+\cfrac{1}{14+\cfrac{1}{18+\ddots\,}}}}}=1+\cfrac{2}{1+\cfrac{1}{6+\cfrac{1}{10+\cfrac{1}{14+\cfrac{1}{18+\ddots\,}}}}}$


$\mathrm{length}(c_{k})\approx-\log_{2}\left(p_{k}\right)$


$\cdots\to X^{-1}\xrightarrow{d^{-1}}X^{0}\xrightarrow{d^{0}}X^{1}\xrightarrow{d^{1}}X^{2}\to\cdots$


$f(x)=\begin{cases}1&-1\leq x<0\\ \frac{1}{2}&x=0\\ 1-x^{2}&\,\text{otherwise}\end{cases}$


$\alpha=\arctan\left({R\over L}\right)-\arccos\left({\sqrt{L^{2}+R^{2}}\over 2\rho}\right)$


$H_{2N}=\begin{bmatrix}H_{N}\otimes[1,1]\\ I_{N}\otimes[1,-1]\end{bmatrix}$


$n+{1\over 2}$


$O(MST)$


$\rm Succinate+Q\rightarrow Fumarate+QH_{2}\!$


$U[]\to\epsilon$


$T=\begin{bmatrix}T_{11}&T_{12}\\ 0&T_{22}\end{bmatrix}$


$\alpha+\alpha^{\prime}+\beta+\beta^{\prime}+\gamma+\gamma^{\prime}=1.$


$G^{\wedge}$


$p_{i}=\left(\frac{\partial E}{\partial V}\right)_{T}\,$


$\ v_{i}=\sqrt{\frac{2GMd}{r^{2}}}$


$C=\sum\frac{(-1)^{i}}{s_{i}-1}=\frac{1}{1}-\frac{1}{2}+\frac{1}{6}-\frac{1}{42}+\frac{1}{1806}-\cdots\approx 0.64341054629.$


$\begin{aligned}\displaystyle A_{mn}&\displaystyle=\frac{4}{ab}\int_{0}^{a}\int_{0}^{b}\varphi(x_{1},x_{2})\sin\frac{m\pi x_{1}}{a}\sin\frac{n\pi x_{2}}{b}dx_{1}dx_{2}\\ \displaystyle B_{mn}&\displaystyle=\frac{4}{ab\omega_{mn}}\int_{0}^{a}\int_{0}^{b}\psi(x_{1},x_{2})\sin\frac{m\pi x_{1}}{a}\sin\frac{n\pi x_{2}}{b}dx_{1}dx_{2}\,.\end{aligned}$


$D=h-h^{\prime}=\frac{1}{\frac{1}{\lambda}-\frac{1}{\lambda^{\prime}}}$


$w(n)=\sin^{2}\left(\frac{\pi n}{N-1}\right)$


$\mu_{JT}=\left(\frac{\partial T}{\partial p}\right)_{H}$


$\displaystyle 1\;\;+\;\;\frac{1}{2}\;\;+\;\;\frac{1}{3}\,+\,\frac{1}{4}\;\;+\;\;\frac{1}{5}\,+\,\frac{1}{6}\,+\,\frac{1}{7}\,+\,\frac{1}{8}\;\;+\;\;\frac{1}{9}\,+\,\cdots$


$\begin{bmatrix}Z^{\prime}_{11}&Z^{\prime}_{12}\\ Z^{\prime}_{21}&Z^{\prime}_{22}\end{bmatrix}=\begin{bmatrix}Z_{11}&Z_{12}\\ Z_{21}&Z_{22}\end{bmatrix}^{-1}$


$\gamma=\alpha+i\beta\,\!$


$D\varphi=\begin{bmatrix}\frac{\partial\varphi^{1}}{\partial x^{1}}&\frac{\partial\varphi^{1}}{\partial x^{2}}&\dots&\frac{\partial\varphi^{1}}{\partial x^{n}}\\ \frac{\partial\varphi^{2}}{\partial x^{1}}&\frac{\partial\varphi^{2}}{\partial x^{2}}&\dots&\frac{\partial\varphi^{2}}{\partial x^{n}}\\ \vdots&\vdots&\ddots&\vdots\\ \frac{\partial\varphi^{m}}{\partial x^{1}}&\frac{\partial\varphi^{m}}{\partial x^{2}}&\dots&\frac{\partial\varphi^{m}}{\partial x^{n}}\\ \end{bmatrix}.$


$A+R\rightarrow\ C$


$\frac{1}{2}a_{0}-\frac{1}{4}\Delta a_{0}+\frac{1}{8}\Delta^{2}a_{0}-\cdots=\frac{1}{2}-\frac{1}{4}.$


$\mathbf{C}=\begin{pmatrix}\mathbf{C}_{11}&\mathbf{C}_{12}\\ \mathbf{C}_{21}&\mathbf{C}_{22}\\ \end{pmatrix}=\begin{pmatrix}\mathbf{A}_{11}&\mathbf{A}_{12}\\ \mathbf{A}_{21}&\mathbf{A}_{22}\\ \end{pmatrix}\begin{pmatrix}\mathbf{B}_{11}&\mathbf{B}_{12}\\ \mathbf{B}_{21}&\mathbf{B}_{22}\\ \end{pmatrix}=\mathbf{A}\mathbf{B}$


$\bullet\overrightarrow{\to}\bullet$


$\ a_{0}+\cfrac{1}{a_{1}+\cfrac{1}{a_{2}+\cfrac{1}{a_{3}}}}$


$A+h\nu\rightarrow A^{*}$


$A(x)=\lfloor x\rfloor\,$


$\,\text{Subclutter Visibility}=\left(\tfrac{\,\text{Dynamic Range}}{\,\text{CFAR Detection Threshold}}\right)$


$L=\begin{pmatrix}1&&&&&0\\ -l_{2,1}&\ddots&&&&\\ &&1&&&\\ \vdots&&-l_{n+1,n}&\ddots&&\\ &&\vdots&&1&\\ -l_{N,1}&&-l_{N,n}&&-l_{N,N-1}&1\end{pmatrix}.$


$\sum_{i=1}^{n}(X_{i}-\bar{X})^{2}\sim\sigma^{2}\chi^{2}_{n-1},\quad$


$r=\frac{\sum^{N}_{i=1}(X_{i}-\bar{X})(Y_{i}-\bar{Y})}{\sqrt{\sum^{N}_{i=1}(X_{i}-\bar{X})^{2}}\sqrt{\sum^{N}_{i=1}(Y_{i}-\bar{Y})^{2}}}$


$\,\text{gyr}[\mathbf{u},\mathbf{v}]\mathbf{w}=\ominus(\mathbf{u}\oplus\mathbf{v})\oplus(\mathbf{u}\oplus(\mathbf{v}\oplus\mathbf{w}))$


$z_{2}=-1.4455692\ldots+i0.6992608\ldots$


$\mbox{E}~{}\left[T^{k}\right]=\begin{cases}\left(\frac{\nu}{2}\right)^{\frac{k}{2}}\frac{\Gamma\left(\frac{\nu-k}{2}\right)}{\Gamma\left(\frac{\nu}{2}\right)}\mbox{exp}~{}\left(-\frac{\mu^{2}}{2}\right)\frac{d^{k}}{d\mu^{k}}\mbox{exp}~{}\left(\frac{\mu^{2}}{2}\right),&\mbox{if }~{}\nu>k;\\ \mbox{Does not exist}~{},&\mbox{if }~{}\nu\leq k.\\ \end{cases}$


$\begin{bmatrix}z\\ v\end{bmatrix}=\mathbf{P}(s)\,\begin{bmatrix}w\\ u\end{bmatrix}=\begin{bmatrix}P_{11}(s)&P_{12}(s)\\ P_{21}(s)&P_{22}(s)\end{bmatrix}\,\begin{bmatrix}w\\ u\end{bmatrix}$


$\mathcal{B}=\mathfrak{P}(\mathcal{U}\;\mathfrak{P}_{\geq 1}(\mathcal{Z})).$


$\sin\alpha_{\mathrm{s}}=\cos h\cos\delta\cos\varphi+\sin\delta\sin\varphi$


$a(n)=e^{\int_{0}^{n}\delta_{t}\,dt}\ ,$


$\left\{\begin{aligned}\displaystyle\frac{\partial}{\partial z_{1}}&\displaystyle=\frac{1}{2}\left(\frac{\partial}{\partial x_{1}}-i\frac{\partial}{\partial y_{1}}\right)\\ &\displaystyle\qquad\qquad\vdots\\ \displaystyle\frac{\partial}{\partial z_{n}}&\displaystyle=\frac{1}{2}\left(\frac{\partial}{\partial x_{n}}-i\frac{\partial}{\partial y_{n}}\right)\\ \end{aligned}\right.\quad,\quad$


$O(n\log n)$


$\ -3+\cfrac{1}{2+\cfrac{1}{18}}$


$\textstyle\Q(k;p)={N-2\choose k-1}p^{(k-1)}(1-p)^{(N-k-1)}$


$P_{\ell}^{-m}=(-1)^{m}\frac{(\ell-m)!}{(\ell+m)!}P_{\ell}^{m}$


$(a+b)(c+d)=\underbrace{ac}_{\mathrm{first}}+\underbrace{ad}_{\mathrm{outside}}+\underbrace{bc}_{\mathrm{inside}}+\underbrace{bd}_{\mathrm{last}}$


$\forall x\forall y\exists z\forall w[w\in z\leftrightarrow(w=xw=y)].$


$T_{\mathrm{amortized}}(m)=O(m\log n)$


$[w]^{-1}\begin{bmatrix}s\\ t\end{bmatrix}[w]=\begin{bmatrix}s\\ -ws+t\end{bmatrix}=\begin{bmatrix}1&0\\ -w&1\end{bmatrix}\begin{bmatrix}s\\ t\end{bmatrix}$


$\frac{d^{\alpha_{i}}}{dx_{i}^{\alpha_{i}}}x_{i}^{\beta_{i}}=\frac{\beta_{i}!}{(\beta_{i}-\alpha_{i})!}x_{i}^{\beta_{i}-\alpha_{i}}$


$C=\left\{\begin{array}[]{ c l }C_{0}&\textrm{if}\ |\xi|<1\\ C_{0}\xi^{2}&\textrm{if}\ |\xi|>1.\end{array}\right.$


$\mu_{n}=\int_{-\infty}^{\infty}(x-c)^{n}\,f(x)\,dx.$


$O(VE\log V\log(VC))$


$\frac{-\log(2)}{\log\left(\displaystyle\frac{1-\gamma}{2}\right)}$


$q_{xy}=\frac{\sum(x-\bar{x})(y-\bar{y})I(x,y)}{\sum I(x,y)}$


$V_{ij}=\frac{\partial^{2}V}{\partial x_{i}\partial x_{j}}.$


$\lambda_{c}=\sqrt{\frac{8\pi^{2}\kappa}{f^{\prime\prime}}}.$


$L(s,E)=\sum_{n=1}^{\infty}\frac{a_{n}}{n^{s}}.$


$\pi=\cfrac{4}{1+\cfrac{1^{2}}{2+\cfrac{3^{2}}{2+\cfrac{5^{2}}{2+\cfrac{7^{2}}{2+\ddots}}}}}.\,$


$A\Lambda=\begin{bmatrix}1&-3\\ 1&2\end{bmatrix}\begin{bmatrix}\lambda_{1}\\ \lambda_{2}\end{bmatrix}.\,\!$


$\sigma^{2}=\frac{\sum_{i=1}^{N}w_{i}(x_{i}-\overline{x}^{\,*})^{2}}{\sum_{i=1}^{N}w_{i}}$


$\nabla\cdot\mathbf{S}+\epsilon_{0}\mathbf{E}\cdot\frac{\partial\mathbf{E}}{\partial t}+\frac{\mathbf{B}}{\mu_{0}}\cdot\frac{\partial\mathbf{B}}{\partial t}+\mathbf{J}\cdot\mathbf{E}=0,$


$\begin{bmatrix}v\\ F\end{bmatrix}=\begin{bmatrix}z_{11}&z_{12}\\ z_{21}&z_{22}\end{bmatrix}\begin{bmatrix}i\\ u\end{bmatrix}$


$\begin{bmatrix}x^{\prime}\\ y^{\prime}\end{bmatrix}=\begin{bmatrix}\cos\theta&-\sin\theta\\ \sin\theta&\cos\theta\end{bmatrix}\begin{bmatrix}x\\ y\end{bmatrix}$


$O(\log\log n)$


$\mathrm{Ran}\;e_{i}(T)=\mathrm{Ker}(T-\lambda_{i})^{\nu_{i}}.$


$\delta=\frac{1}{4\pi}\iint_{R_{st}}\frac{\begin{vmatrix}x&y&z\\ \dfrac{\partial x}{\partial s}&\dfrac{\partial y}{\partial s}&\dfrac{\partial z}{\partial s}\\ \dfrac{\partial x}{\partial t}&\dfrac{\partial y}{\partial t}&\dfrac{\partial z}{\partial t}\end{vmatrix}}{(x^{2}+y^{2}+z^{2})\sqrt{x^{2}+y^{2}+z^{2}}}dsdt.$


$\mu_{i}=\left(\frac{\partial G}{\partial N_{i}}\right)_{T,P}$


$\cos\alpha=-\cos\beta\cos\gamma+\sin\beta\sin\gamma\cosh\frac{a}{k},\,$


$\mu_{0}\mathbf{J}=\nabla\times\mathbf{B}$


$\displaystyle 7.48181818\ldots$


$T=k\left(k-1\right)\frac{\sum\limits_{j=1}^{k}\left(X_{\bullet j}-\frac{N}{k}\right)^{2}}{\sum\limits_{i=1}^{b}X_{i\bullet}\left(k-X_{i\bullet}\right)}$


$O(V\log V)$


$\mathbf{R}^{n}$


$\underbrace{\underbrace{\mathrm{root+suffix}}_{\mathrm{stem}}+\mathrm{ending}}_{\mathrm{word}}$


$C_{0}^{N}=(1+r)^{-N}\sum_{n=0}^{N}\frac{N!}{n!(N-n)!}q^{n}{(1-q)}^{N-n}{[S_{0}{(1+b)}^{n}{(1+a)}^{N-n}-K]}^{+}$


$\sin\delta=\sin\beta\cos\varepsilon+\cos\beta\sin\varepsilon\sin\lambda$


$\tfrac{1}{1+\tfrac{1}{0+\tfrac{1}{8+\tfrac{1}{4+\tfrac{1}{1+\tfrac{1}{0+1{/\cdots}}}}}}}$


$N=\left\lfloor 0.5-\log_{2}\left(\frac{\,\text{Frequency of this item}}{\,\text{Frequency of most common item}}\right)\right\rfloor$


$v_{g}=\left(\frac{\partial(\operatorname{Re}k)}{\partial\omega}\right)^{-1}.$


$0=\underbrace{1+e_{1}^{2}+\cdots+e_{n-1}^{2}}_{=:a}+\underbrace{e_{n}^{2}+\cdots+e_{s}^{2}}_{=:b}\;.$


$f(x)-P_{n-1}(x)=\frac{f^{(n)}(\xi)}{n!}\prod_{i=1}^{n}(x-x_{i})$


$E\ \rightarrow^{\!\!\!\!\!\!d}\ E\wedge E\ \rightarrow^{\!\!\!\!\!\!d}\ \bigwedge^{3}E\rightarrow^{\!\!\!\!\!\!d}\ \dots$


$P^{\prime}(A,B)=A\oplus B$


$\frac{\partial u}{\partial\varphi}=-\frac{\partial u}{\partial x}r\sin\varphi+\frac{\partial u}{\partial y}r\cos\varphi=-y\frac{\partial u}{\partial x}+x\frac{\partial u}{\partial y}.$


$Y_{a,b}=\begin{cases}(-1)^{b}\sqrt{\frac{2a+1}{4\pi}\frac{(a-b)!}{(a+b)!}}P_{a}^{b}(\cos\theta)e^{ib\phi}&\,\text{if }a>0\\ Y_{-a-1,b}&\,\text{if }a<0\end{cases}$


$\exists y\forall x(x\in y\iff P(x))$


$O(m+n\log n)$


$O\left(M(m)\log^{2}m\right)=O\left(M(n)\log n\right).\,$


$G^{u}H/H=(G/H)^{u}$


$O(\log\log N)$


$p\cdot q=\textstyle\frac{1}{2}(p^{*}q+q^{*}p)=\textstyle\frac{1}{2}(pq^{*}+qp^{*}).$


$s_{n-1}=\sqrt{\frac{1}{n-1}\sum_{i=1}^{n}(X_{i}-\overline{X})^{2}}.$


$\begin{bmatrix}\sigma_{11}\\ \sigma_{22}\\ \sigma_{12}\end{bmatrix}=\begin{bmatrix}C_{11}&C_{12}&C_{13}\\ C_{12}&C_{22}&C_{23}\\ C_{13}&C_{23}&C_{33}\end{bmatrix}\begin{bmatrix}\varepsilon_{11}\\ \varepsilon_{22}\\ \varepsilon_{12}\end{bmatrix}$


$\sum_{n=1}^{\infty}\frac{1}{n{2n\choose n}}=1-\frac{1}{2}+\frac{1}{4}-\frac{1}{5}+\frac{1}{7}-\frac{1}{8}+\cdots$


$Q=\frac{-\kappa A}{\mu}\left(\frac{dp}{dx}\right)$


$\alpha\ominus\beta:=\alpha\oplus(\ominus\beta)$


$\forall x\exists y[x\in y\wedge\forall z\in y(\mathcal{P}(z)\subseteq y\wedge\mathcal{P}(z)\in y)\wedge\forall z\in\mathcal{P}(y)(\neg z\approx y\to z\in y)]$


$Z_{0}=\sqrt{\frac{R+j\omega L}{G+j\omega C}}$


$\nabla\times\mathbf{B}=\mu_{0}\mathbf{J}+\mu_{0}\varepsilon_{0}\frac{\partial\mathbf{E}}{\partial t},$


$f(k)={n\choose k}p^{k}(1-p)^{n-k}$


$\forall A\;([\forall x\in A\;\exists y\;\phi(x,y)]\to\exists B\;\forall x\in A\;\exists y\in B\;\phi(x,y))$


$\mathrm{{}^{235}_{\ 92}U\ +\ ^{1}_{0}n\ \longrightarrow\ ^{236}_{\ 92}U_{m}\ \xrightarrow[120\ ns]{}\ ^{236}_{\ 92}U\ +\ \gamma}$


$\frac{\partial^{2}\Pi}{\partial x_{i}\partial x_{j}}$


$\Pi(t)\ \stackrel{\,\text{def}}{=}\ \begin{cases}1&\,\text{if }|t|<\frac{1}{2},\\ 0&\,\text{if }|t|>\frac{1}{2}.\end{cases}$


$\overline{\left[\left(\tau_{s}-\bar{\tau}_{s}\right)^{2}\right]}^{\frac{1}{2}}=1.4\sqrt{s},$


$A(r)B(r)\,=K$


$\alpha y_{\xi\xi}-2\beta y_{\xi\eta}+\gamma y_{\eta\eta}=-I^{2}(Py_{\xi}+Qy_{\eta})$


$O(\eta*\log{n})$


$\beta_{1}\log\alpha_{1}+\cdots+\beta_{n-1}\log\alpha_{n-1}=\log\alpha_{n}$


$\mathbf{v}−2·B(\mathbf{v},\mathbf{u})/Q(\mathbf{u}) ·\mathbf{u}$


$\mathbf{\rho}_{i},\mathbf{\rho}_{j}\in\mathit{S}\big(i\neq j\big)$


$\exists_{x}\forall_{y}(x\neq y\Rightarrow G(x,y)).$


$Df(x,y)=\begin{bmatrix}\frac{\partial u}{\partial x}&\frac{\partial u}{\partial y}\\ \frac{\partial v}{\partial x}&\frac{\partial v}{\partial y}\end{bmatrix}$


$M(n)=\left\{\begin{matrix}n-10,&\mbox{if }~{}n>100\mbox{ }\\ M(M(n+11)),&\mbox{if }~{}n\leq 100\mbox{ }\end{matrix}\right.$


$\alpha_{1}=a+b+e;\quad\beta_{1}=a+b+c+d;$


$\frac{n!}{m!(n-m)!}$


$\mathbf{\hat{r}}=\sin\theta\cos\varphi\mathbf{\hat{x}}+\sin\theta\sin\varphi\mathbf{\hat{y}}+\cos\theta\mathbf{\hat{z}}$


$\hat{\lambda}\text{MM}=\frac{\sum_{i=1}^{t}c_{i}\bar{Y}_{i}}{\sqrt{\sum_{i=1}^{t}c_{i}^{2}s_{i}^{2}}}.$


$s=\sqrt{\frac{1}{n-1}\sum_{i=1}^{n}(x_{i}-\overline{x})^{2}}\,,$


$e^{-2\alpha t}$


$\left(\frac{\partial T}{\partial V}\right)_{S}$


$\begin{matrix}&\underbrace{\left\{A,B,C\right\}}&&\underbrace{\left\{X,Y\right\}}\\ \mathrm{To}\ \mathrm{choose}\ \mathrm{one}\ \mathrm{of}&\mathrm{these}&\mathrm{AND}\ \mathrm{one}\ \mathrm{of}&\mathrm{these}\end{matrix}$


$\frac{n!}{k_{1}!k_{2}!\cdots k_{m-1}!K!}\frac{K!}{k_{m}!k_{m+1}!}=\frac{n!}{k_{1}!k_{2}!\cdots k_{m+1}!}.$


$\alpha=a+\sqrt{D}\,\text{ to }\alpha^{\prime}=a-\sqrt{D}\,$


$\sin(\alpha+\beta)=\sin\alpha\cos\beta+\cos\alpha\sin\beta$


$\frac{\partial^{2}\rho}{\partial t^{2}}-c^{2}_{0}\nabla^{2}\rho=\frac{\partial^{2}T_{ij}}{\partial x_{i}\partial x_{j}},\quad(*)$


$\mathfrak{P}$


$y^{\prime}=\underbrace{\begin{pmatrix}1&-1/x\\ 1+x&-1\end{pmatrix}}_{=\,A(x)}y$


$0\rightarrow M\rightarrow\mathfrak{h}\rightarrow\mathfrak{g}\rightarrow 0$


$y_{i}=\begin{cases}y_{i}^{*}&\textrm{if}\;y_{i}^{*}>0\\ 0&\textrm{if}\;y_{i}^{*}\leq 0\end{cases}$


$\cos(\theta-\alpha)=\cos(\theta)\cos(\alpha)+\sin(\theta)\sin(\alpha)$


$\mathfrak{P}(\mathfrak{V})$


$+\underbrace{E_{\mathrm{sig}}E_{\mathrm{LO}}\cos((\omega_{\mathrm{sig}}-\omega_{\mathrm{LO}})t+\varphi)}_{beat\;component}.$


$0=-\frac{\mbox{d}~{}P}{\mbox{d}~{}x}+\mu\left(\frac{\mbox{d}~{}^{2}u}{\mbox{d}~{}y^{2}}\right)$


$0\to C^{0}\stackrel{d_{0}}{\longrightarrow}C^{1}\stackrel{d_{1}}{\longrightarrow}C^{2}\stackrel{d_{2}}{\longrightarrow}\cdots\stackrel{d_{n-1}}{\longrightarrow}C^{n}\longrightarrow 0.$


$\mathcal{A}\stackrel{F}{\rightarrow}\mathcal{B}\stackrel{G}{\rightarrow}\mathcal{C},\,$


$p(r)=\frac{r^{2}}{2\pi^{2}}\int_{0}^{\infty}I(q)\frac{\sin qr}{qr}q^{2}dq.$


$\ ax^{2}+bx=c$


$(\xi)_{+}=\begin{cases}\xi,&\mbox{if }~{}\xi>0\\ 0,&\mbox{otherwise}~{}.\end{cases}$


$r_{t}=\sum\limits_{k=1}^{n}\underbrace{\left(r_{k,t}^{\,\text{base}}+r_{k,t}^{\,\text{crcy}}\right)}_{=r_{k,t}}$


$O\left(mnr^{2}\log\frac{1}{\epsilon}\right)$


$\,\text{similarity}=\cos(\theta)={A\cdot B\over\|A\|\|B\|}=\frac{\sum\limits_{i=1}^{n}{A_{i}\times B_{i}}}{\sqrt{\sum\limits_{i=1}^{n}{(A_{i})^{2}}}\times\sqrt{\sum\limits_{i=1}^{n}{(B_{i})^{2}}}}$


$pB=(\prod P_{j})^{e}\ .$


$\mathbf{J}=\begin{bmatrix}A&0\\ B&C\\ \end{bmatrix}$


$AB=\begin{bmatrix}2&3\\ 3&4\end{bmatrix}\begin{bmatrix}1&0\\ 1&2\end{bmatrix}\stackrel{?}{=}\begin{bmatrix}6&5\\ 8&7\end{bmatrix}=C.$


$\displaystyle\frac{d}{dr}\left(\frac{1}{\rho}\frac{dP}{dr}\right)$


$M=\begin{bmatrix}1.44&0\\ 0&2.89\end{bmatrix}$


$N=\lfloor u^{2}\rfloor$


$O\left(\frac{\log n}{\log\log n}\right)$


$\lim_{k\to\infty}c_{2}(k)=-2.157782996659\ldots.$


$y=\arctan\left(\frac{\tan(\phi)}{\cos(\lambda)}\right)$


$O(mn\log p)=O(n\log n)$


$\left[{\begin{array}[]{*{20}{c}}1&1\\ 2&3\end{array}}\right]\leftarrow\,\text{parent}$


$X=\frac{1}{2}\left[\left(\frac{T}{\beta}\right)^{0.5}-\left(\frac{T}{\beta}\right)^{-0.5}\right]$


$r=\frac{\frac{1}{n-1}\sum_{i=1}^{n}(X_{i}-\overline{X})(Y_{i}-\overline{Y})}{s_{X}s_{Y}}$


${\rm blanc}(x)=\sum_{n=0}^{\infty}{s(2^{n}x)\over 2^{n}},$


$\operatorname{E}(X\mid Y=y)=\int_{\mathcal{X}}xf_{X\mid Y}(x\mid y)\,dx$


$Y_{1}=\begin{cases}1&\,\text{if }Y^{*}_{1}>0,\\ 0&\,\text{otherwise},\end{cases}$


$\sum_{a,b}P\left({{a,b}{|}{A,B}}\right)=1\quad\forall{A,B}$


$g=\begin{cases}-\sqrt{\frac{5}{9(x^{2}+1)}}\cdot S,&\mbox{if}~{}~{}T>\frac{1}{2}\\ \sqrt{\frac{5}{9(x^{2}+1)}}\cdot S,&\mbox{if}~{}~{}T<\frac{1}{2}\\ 0,&\mbox{if}~{}~{}T=0\\ \end{cases}$


$\tfrac{n!}{m!(n-m)!}$


$X=\begin{vmatrix}x_{11}&x_{12}&x_{13}&\cdots&x_{1n}\\ x_{12}&x_{22}&x_{23}&\cdots&x_{2n}\\ x_{13}&x_{23}&x_{33}&\cdots&x_{3n}\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ x_{1n}&x_{2n}&x_{3n}&\cdots&x_{nn}\end{vmatrix},D=\begin{vmatrix}2\frac{\partial}{\partial x_{11}}&\frac{\partial}{\partial x_{12}}&\frac{\partial}{\partial x_{13}}&\cdots&\frac{\partial}{\partial x_{1n}}\\ \frac{\partial}{\partial x_{12}}&2\frac{\partial}{\partial x_{22}}&\frac{\partial}{\partial x_{23}}&\cdots&\frac{\partial}{\partial x_{2n}}\\ \frac{\partial}{\partial x_{13}}&\frac{\partial}{\partial x_{23}}&2\frac{\partial}{\partial x_{33}}&\cdots&\frac{\partial}{\partial x_{3n}}\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ \frac{\partial}{\partial x_{1n}}&\frac{\partial}{\partial x_{2n}}&\frac{\partial}{\partial x_{3n}}&\cdots&2\frac{\partial}{\partial x_{nn}}\end{vmatrix}$


$\mu_{t}=\begin{cases}{\mu_{t}}\text{inner}&\mbox{if }~{}y\leq y\text{crossover}\\ {\mu_{t}}\text{outer}&\mbox{if }~{}y>y\text{crossover}\end{cases}$


$\zeta(s)=\sum_{n=1}^{\infty}\frac{1}{n^{s}},$


$\underbrace{a\diamondsuit\cdots\diamondsuit a}_{k\,\text{ factors}}.\,$


$\begin{bmatrix}4&3\\ 6&3\end{bmatrix}=\begin{bmatrix}l_{11}&0\\ l_{21}&l_{22}\end{bmatrix}\begin{bmatrix}u_{11}&u_{12}\\ 0&u_{22}\end{bmatrix}.$


$\frac{d^{2}y}{dx^{2}}=\frac{d}{dx}\left(\frac{dy}{dx}\right).$


$\begin{bmatrix}\hat{p}_{i}\\ \hat{q}_{i}\end{bmatrix}=\begin{bmatrix}A&B\\ C&D\end{bmatrix}\begin{bmatrix}\hat{p}_{o}\\ \hat{q}_{o}\end{bmatrix}$


$AB=BA$


$i:A\to B\to D$


$\forall a,b,c\in X\,(b\,R\,a\land c\,R\,a\to b\,R\,c).$


$\psi=\frac{1-\sqrt{5}}{2}=1-\varphi=-{1\over\varphi}\approx-0.61803\,39887\cdots$


$A\bullet B=(A\oplus B)\ominus B,\,$


$\forall x\forall y\forall s[(\langle x,s\rangle\in F\and\langle y,s\rangle\in F)\rightarrow x=y])].$


$J=\begin{bmatrix}\dfrac{\partial\Delta P}{\partial\theta}&\dfrac{\partial\Delta P}{\partial|V|}\\ \dfrac{\partial\Delta Q}{\partial\theta}&\dfrac{\partial\Delta Q}{\partial|V|}\end{bmatrix}$


$\pi\in G^{\wedge}$


$\alpha^{m}$


$\forall x,y\in V,$


$\sigma^{j}$


$f(r|H=h,T=t)=\frac{(h+t+1)!}{h!\,\,t!}\;r^{h}\,(1-r)^{t}.\!$


$kT\left(\frac{\partial\rho}{\partial p}\right)$


$f_{x^{*}}(x)=\begin{cases}Ae^{-Be^{Cx}+CDx}(e^{Cx}+E)^{-F},&\,\text{if}\ d>0\\ Ae^{-Bx^{2}+Cx}&\,\text{if}\ d=0\end{cases}$


$\underline{A\quad\quad\quad}\,\!$


$\,\text{angle}=\arctan\left(\frac{\,\text{slope}}{100\%}\right)$


$y(t)=\int_{0}^{T}x(t-\tau)\,h(\tau)\,d\tau$


$\mathrm{{}^{238}_{\ 92}U\ +\ ^{2}_{1}D\ \longrightarrow\ ^{238}_{\ 93}Np\ +\ 2\ ^{1}_{0}n\quad;\quad^{238}_{\ 93}Np\ \xrightarrow[2.117\ d]{\beta^{-}}\ ^{238}_{\ 94}Pu}$


$\mathbf{m}=-\gamma\mathbf{L}$


$=\frac{(n+1)\,n\,(n-1)}{(n-2)\,(n-3)}\;\frac{\sum_{i=1}^{n}(x_{i}-\bar{x})^{4}}{\left(\sum_{i=1}^{n}(x_{i}-\bar{x})^{2}\right)^{2}}-3\,\frac{(n-1)^{2}}{(n-2)\,(n-3)}$


$A\oplus B=\{z\in E\mid(B^{s})_{z}\cap A\neq\varnothing\}$


$\frac{\sqrt{n}(\bar{X}-\mu_{0})}{\sqrt{\sum\limits_{i=1}^{n}(X_{i}-\bar{X})^{2}/(n-1)}}$


$k=\begin{cases}-j-\tfrac{1}{2}&\,\text{if }j=\ell+\tfrac{1}{2}\\ j+\tfrac{1}{2}&\,\text{if }j=\ell-\tfrac{1}{2}\end{cases}$


$\frac{m}{n}=-2eV_{1}\left(\frac{t}{f}\right)^{2}$


$r=\frac{\bar{y}}{\bar{x}}=\frac{\sum_{i=1}^{n}y}{\sum_{i=1}^{n}x}$


$n\frac{n}{− 1}$


$\sum\limits_{n=1}^{\infty}{(-1)^{n+1}\over n}=1-{1\over 2}+{1\over 3}-{1\over 4}+{1\over 5}-\cdots$


$\mathbf{H}_{\alpha}(x)=\frac{2\left(\frac{x}{2}\right)^{\alpha}}{\sqrt{\pi}\Gamma\left(\alpha+\frac{1}{2}\right)}\int_{0}^{\frac{\pi}{2}}\sin(x\cos\tau)\sin^{2\alpha}(\tau)d\tau.$


$f(x,y)=\begin{bmatrix}\ 0.00&\ 0.00\\ 0.00&\ 0.16\end{bmatrix}\begin{bmatrix}\ x\\ y\end{bmatrix}$


$m_{\rm star}=m_{\odot}-2.5\log_{10}\left[\frac{L_{\rm star}}{L_{\odot}}\left(\frac{d_{\odot}}{d_{\rm star}}\right)^{2}\right]$


$Y\stackrel{\pi}{\to}X$


$\theta_{n}(x)=\frac{n!}{(-2)^{n}}\,L_{n}^{-2n-1}(2x)$


$u\equiv_{S}v\Leftrightarrow\forall x,y\in M(xuy\in S\Leftrightarrow xvy\in S).$


$1+{1\over 2}+{1\over 3}+{1\over 4}+{1\over 5}+\cdots=\sum_{n=1}^{\infty}{1\over n}.$


$\omega=d\alpha+\delta\beta+\gamma$


$H_{1}=\begin{bmatrix}2&1\\ 1&1\end{bmatrix}$


$1,1,2,3,1,-2,-3,-5,-2,-3,\ldots\,\text{ for the signs }+,+,+,-,-,+,-,-,\ldots.$


$\begin{aligned}\displaystyle m(\varphi)&\displaystyle=\int_{0}^{\varphi}M(\varphi)\,d\varphi=a(1-e^{2})\int_{0}^{\varphi}\bigl(1-e^{2}\sin^{2}\varphi\bigr)^{-3/2}\,d\varphi.\end{aligned}$


$\pi=\sum_{k=0}^{\infty}\left[\frac{1}{16^{k}}\left(\frac{4}{8k+1}-\frac{2}{8k+4}-\frac{1}{8k+5}-\frac{1}{8k+6}\right)\right]$


$y=\begin{cases}0&\,\text{if }y^{*}\leq\mu_{1},\\ 1&\,\text{if }\mu_{1}<y^{*}\leq\mu_{2},\\ 2&\,\text{if }\mu_{2}<y^{*}\leq\mu_{3},\\ \vdots\\ N&\,\text{if }\mu_{N}<y^{*}.\end{cases}$


$\nabla^{2}\varphi+{{\partial}\over\partial t}\left(\nabla\cdot\mathbf{A}\right)=-{\rho\over\varepsilon_{0}}\,,$


$\mathrm{{}^{236}_{\ 92}U\ +\ ^{1}_{0}n\ \longrightarrow\ ^{237}_{\ 92}U\ \xrightarrow[6.75\ d]{\beta^{-}}\ ^{237}_{\ 93}Np}$


$0\to X\to E\to Y\to 0\,$


${\nabla({\partial\Phi\over\partial p})}$


$\frac{dy}{dx}\left.{\!\!\frac{}{}}\right|_{x=a}=\frac{dy}{dx}(a).$


$C=\frac{\int xg(x)\;dx}{\int g(x)\;dx}$


$\mathrm{d}H=\begin{bmatrix}\frac{\partial H}{\partial q^{i}}\\ \frac{\partial H}{\partial p_{i}}\end{bmatrix}.$


$\Phi(z,s,q)=\sum_{k=0}^{\infty}\frac{z^{k}}{(k+q)^{s}}\!$


$\begin{bmatrix}\vec{y}\\ 1\end{bmatrix}=\left[\begin{array}[]{ccc|c}&A&&\vec{b}\\ 0&\ldots&0&1\end{array}\right]\begin{bmatrix}\vec{x}\\ 1\end{bmatrix}$


$\frac{dy}{dx}=\mathrm{st}\left(\frac{\Delta y}{\Delta x}\right)$


$L(s,a)=\sum^{\infty}_{n=1}\frac{a(n)}{n^{s}}=\prod_{p}\biggl(1-\frac{a(p)}{p^{s}}\biggr)^{-1},$


$\ O(n\log n)$


$APY=\left(1+\frac{i\text{nom}}{N}\right)^{N}-1$


$\frac{d}{dt}\left(\frac{dt}{d\tau}\right)\approx 0$


$A\cup B=\left(A^{C}\cap B^{C}\right)^{C}$


$\exp(\nu)=\sum_{n=0}^{\infty}{\nu^{*n}\over n!}$


$\Delta^{1}_{\rm LAT}=111132.954-559.822\cos 2\phi+1.175\cos 4\phi$


$A=\{x\in X|(\exists y\in Y)\langle x,y\rangle\in B\}.$


$\frac{1}{2}\,+\,\frac{1}{4}\,+\,\frac{1}{8}\,+\,\frac{1}{16}\,+\,\cdots\;=\;1$


$Y_{n}\ \xrightarrow{p}\ Y$


$\begin{bmatrix}V_{2}\\ I^{\prime}_{2}\end{bmatrix}=\begin{bmatrix}A^{\prime}&B^{\prime}\\ C^{\prime}&D^{\prime}\end{bmatrix}\begin{bmatrix}V_{1}\\ I_{1}\end{bmatrix}$


${1\over 1}-{1\over 2}+{1\over 4}-{1\over 8}+{1\over 16}-{1\over 32}+\cdots={2\over 3}.$


${n\choose p}=\frac{n!}{p!(n-p)!}$


$\frac{1}{2}\cdot\frac{1}{2}\cdot\frac{1}{2}\cdot\frac{1}{2}=\frac{1}{16}.$


$\cos(x-y)=\cos x\cos y+\sin x\sin y\,$


$\mathbf{G}=\begin{bmatrix}\frac{\Delta t^{2}}{2}\\ \Delta t\end{bmatrix}$


$\pi=\frac{4}{1}-\frac{4}{3}+\frac{4}{5}-\frac{4}{7}+\frac{4}{9}-\frac{4}{11}+\frac{4}{13}-\cdots$


$A\triangle B=A^{c}\triangle B^{c}$


$(1-x^{2})^{1/3}=1-\frac{x^{2}}{3}-\frac{x^{4}}{9}-\frac{5x^{6}}{81}\cdots$


$x_{n}\xrightarrow{d}\ \mu_{n}$


$\scriptstyle\zeta^{\prime}(2)\,\,\text{= Derivative of }\zeta(2)=-\sum\limits_{n=2}^{\infty}\frac{\ln n}{n^{2}}=-0.9375482543\ldots$


$\exist r\forall y[A(y)\rightarrow Byr]\rightarrow\exist x\forall y[y\in x\leftrightarrow A(y)]\,.$


$\begin{aligned}\displaystyle Var(\beta_{1})&\displaystyle=\frac{\sigma^{2}}{\sum_{i=1}^{n}{(x_{i}-\bar{x})^{2}}}.\end{aligned}$


$\log_{b}\!\left(\begin{matrix}\frac{x}{y}\end{matrix}\right)=\log_{b}(x)-\log_{b}(y)$


$\begin{bmatrix}y_{1,t}\\ y_{2,t}\end{bmatrix}=\begin{bmatrix}c_{1}\\ c_{2}\end{bmatrix}+\begin{bmatrix}A_{1,1}&A_{1,2}\\ A_{2,1}&A_{2,2}\end{bmatrix}\begin{bmatrix}y_{1,t-1}\\ y_{2,t-1}\end{bmatrix}+\begin{bmatrix}e_{1,t}\\ e_{2,t}\end{bmatrix},$


$X_{n}\ \xrightarrow{p}\ Y$


$\,\text{Attenuation (dB)}=10\times\log_{10}\left(\frac{\,\text{Input intensity(W)}}{\,\text{Output intensity (W)}}\right)$


$S+C\rightarrow P+C$


$O(n\cdot\log{n})$


$\,\text{Minimize}=\begin{cases}f_{1}\left(x\right)&=\begin{cases}-x,&\,\text{if }x\leq 1\\ x-2,&\,\text{if }1<x\leq 3\\ 4-x,&\,\text{if }3<x\leq 4\\ x-4,&\,\text{if }x>4\\ \end{cases}\\ f_{2}\left(x\right)&=\left(x-5\right)^{2}\\ \end{cases}$


$D_{n,0}=\left[{n!\over e}\right]$


$\sum_{i=1}^{n}(x_{i}-\mu)^{2}=\sum_{i=1}^{n}(x_{i}-\bar{x})^{2}+n(\bar{x}-\mu)^{2}$


$\hat{\beta}=\frac{\tfrac{1}{T}\sum_{t=1}^{T}(x_{t}-\bar{x})(y_{t}-\bar{y})}{\tfrac{1}{T}\sum_{t=1}^{T}(x_{t}-\bar{x})^{2}}\,,$


$A\oplus B=\mathcal{N}.$


$\overline{\left(A_{t}-\overline{A}\right)^{2}}\equiv\lim_{\tau\to\infty}\frac{1}{\tau}\int_{0}^{\tau}\left(A_{t}-\overline{A}\right)^{2}dt,$


$c_{n}\ \stackrel{\mathrm{def}}{=}\ \begin{cases}\frac{A_{n}}{2i}e^{i\phi_{n}}=\frac{1}{2}(a_{n}-ib_{n})&\,\text{for }n>0\\ \frac{1}{2}a_{0}&\,\text{for }n=0\\ c_{|n|}^{*}&\,\text{for }n<0.\end{cases}$


$\forall x,y\in N$


$G(x,z;\lambda)=\sum_{i=1}^{n}\frac{e_{i}(x)f_{i}^{*}(z)}{\lambda-\lambda_{i}}.$


$\delta^{{}^{\prime}}$


$\,\text{Surplus Transfer Value}=\left({{\,\text{Total value of Candidate's votes}-\,\text{Quota}}\over\,\text{Total value of Candidate's votes}}\right)\times\,\text{Value of each vote}$


$\sum_{n=0}^{\infty}\frac{P(n)}{\alpha^{n}}=\frac{\alpha^{2}(\alpha+1)}{\alpha^{3}-\alpha-1}.$


$A\bullet B=(A^{c}\circ B^{s})^{c}$


$1\to K\stackrel{i^{\prime}}{\rightarrow}G^{\prime}\stackrel{\pi^{\prime}}{\rightarrow}H\rightarrow 1$


$A\circ B=(AB+BA)/2$


$\forall x\,(x\neq\varnothing\rightarrow\exists y\in x\,(y\cap x=\varnothing))$


$B_{k}(n)=\begin{cases}{1\over 2}N_{k}(n)+{1\over 4}(k+1)k^{n/2}&\,\text{if }n\,\text{ is even}\\ \\ {1\over 2}N_{k}(n)+{1\over 2}k^{(n+1)/2}&\,\text{if }n\,\text{ is odd}\end{cases}$


$\cos A=-\cos B\cos C+\sin B\sin C\cosh a.\,$


${\rm R}+{\rm L}\to{\rm RL}$


$KP(y|x^{\ast})\leq KP(x,y)-KP(x)+O(1)$


$\operatorname{E}\left[F\right]=\begin{cases}\frac{\nu_{2}(\nu_{1}+\lambda)}{\nu_{1}(\nu_{2}-2)}&\nu_{2}>2\\ \,\text{Does not exist}&\nu_{2}\leq 2\\ \end{cases}$


$\underline{\lnot\varphi\quad\quad}\,\!$


$\stackrel{\nabla}{\mathbf{T}}$


$0\to G^{p+1}\to G^{p}\to K^{p,*}\to 0$


$m_{b}=\sqrt{\frac{2a^{2}+2c^{2}-b^{2}}{4}},$


$\wedge^{n}V$


$0\rightarrow\mathrm{Pic}(X)\rightarrow\mathbb{Z}\stackrel{\delta}{\rightarrow}\mathrm{Br}(K)\rightarrow\mathrm{Br}(K)(X)\rightarrow 0\ .$


$=\int_{-\infty}^{t}G(t-\tau)F(\tau)\,d\tau$


$\beta(x)=\sum_{k=0}^{\infty}\frac{(-1)^{k}}{(2k+1)^{x}}$


$w(n_{i},g_{i})=\frac{g_{i}!}{n_{i}!(g_{i}-n_{i})!}\ .$


$\frac{\sum_{k=1}^{n}(x_{k}-\bar{x})(y_{k}-\bar{y})}{\sqrt{\sum_{k=1}^{n}(x_{k}-\bar{x})^{2}\sum_{k=1}^{n}(y_{k}-\bar{y})^{2}}}$


$0\rightarrow F\rightarrow G\rightarrow H\rightarrow 0.$


$O(n\log h)$


$\cos a=\cos b\cos c+\sin b\sin c\cos\alpha$


$\log_{b}\!\left(\frac{x}{y}\right)=\log_{b}(x)-\log_{b}(y)$


$\alpha_{\mathrm{sun}}=\displaystyle\frac{\int_{0}^{\infty}\alpha_{\lambda}I_{\lambda\mathrm{sun}}(\lambda)\,d\lambda}{\int_{0}^{\infty}I_{\lambda\mathrm{sun}}(\lambda)\,d\lambda}$


$I_{|X|\geq K}=\begin{cases}1&\,\text{if }|X|\geq K,\\ 0&\,\text{if }|X|<K.\end{cases}$


$\Delta H=Q-P\,\Delta V+P\,\Delta V$


$Lu=a_{ij}(x)\frac{\partial^{2}u}{\partial x_{i}\partial x_{j}}+b_{i}(x)\frac{\partial u}{\partial x_{i}}+c(x)u,\qquad x\in\Omega.$


$f(x)=\begin{cases}\exp(-1/x)&\,\text{if }x>0,\\ 0&\,\text{if }x\leq 0,\end{cases}$


$\psi(x)=\ln(x)-\frac{1}{2x}-\frac{1}{12x^{2}}+\frac{1}{120x^{4}}-\frac{1}{252x^{6}}+\frac{1}{240x^{8}}-\frac{5}{660x^{10}}+\frac{691}{32760x^{12}}-\frac{1}{12x^{14}}+O\left(\frac{1}{x^{16}}\right)$


$O(n^{d-1}\log n)$


$\cos(\alpha+\beta)+\cos(\alpha-\beta)=\cos\alpha\cos\beta\ -\sin\alpha\sin\beta+\cos\alpha\cos\beta+\sin\alpha\sin\beta=2\cos\alpha\cos\beta$


$\forall x,y\in U,~{}x\neq y$


$\tilde{O}(m)$


$A\cup B=(A\,\triangle\,B)\,\triangle\,(A\cap B)$


$::\begin{aligned}\displaystyle(r\nleftarrow q)\nleftarrow p&\displaystyle=r^{\prime}q\nleftarrow p\qquad\qquad\qquad~{}~{}~{}~{}\,\text{(by definition)}\\ &\displaystyle=(r^{\prime}q)^{\prime}p\qquad\qquad\qquad~{}~{}~{}~{}~{}~{}\,\text{(by definition)}\\ &\displaystyle=(r+q^{\prime})p\qquad\qquad~{}~{}~{}~{}~{}~{}~{}~{}~{}\,\text{(De Morgan's laws)}\\ &\displaystyle=(r+r^{\prime}q^{\prime})p\qquad\qquad~{}~{}~{}~{}~{}~{}~{}\,\text{(Absorption law)}\\ &\displaystyle=rp+r^{\prime}q^{\prime}p\\ &\displaystyle=rp+r^{\prime}(q\nleftarrow p)\qquad~{}~{}~{}~{}~{}~{}~{}~{}\,\text{(by definition)}\\ &\displaystyle=rp+r\nleftarrow(q\nleftarrow p)\qquad~{}~{}~{}~{}\,\text{(by definition)}\\ \end{aligned}$


$\mathrm{~{}^{238}_{92}U}\rightarrow\mathrm{~{}^{234}_{90}Th}+{\alpha}$


$c^{2}=\left(\frac{\partial p}{\partial\rho}\right)_{S}$


$\ 0\to E^{\prime}\to E\to E^{\prime\prime}\to 0$


$I=\underbrace{\tau\int_{-\infty}^{0}\frac{H(\mu+\tau x)}{e^{x}+1}\,\mathrm{d}x}_{I_{1}}+\underbrace{\tau\int_{0}^{\infty}\frac{H(\mu+\tau x)}{e^{x}+1}\,\mathrm{d}x}_{I_{2}}\,.$


$A+B=C\,$


$\begin{bmatrix}FR_{1}\\ FR_{2}\end{bmatrix}=\begin{bmatrix}A_{11}&A_{12}\\ A_{21}&A_{22}\end{bmatrix}\begin{bmatrix}DP_{1}\\ DP_{2}\end{bmatrix}$


$=\int_{0}^{L}\frac{1}{2}u^{2}\rho(x)\,dx$


$0\rightarrow A\rightarrow B\rightarrow P\rightarrow 0\,$


$\begin{bmatrix}b_{1}\\ b_{2}\end{bmatrix}=\begin{bmatrix}S_{11}&S_{12}\\ S_{21}&S_{22}\end{bmatrix}\begin{bmatrix}a_{1}\\ a_{2}\end{bmatrix}$


$L(s,\Delta)=\sum_{n=1}^{\infty}\frac{a_{n}}{n^{s}}$


$H_{2}=\begin{bmatrix}1&1\\ 1&-1\end{bmatrix}.$


${1\over 1}-{1\over 2}+{1\over 3}-{1\over 4}+{1\over 5}\cdots=\ln(2)$


$f(x)=\begin{cases}2|x|,&\mbox{if }~{}x\leq 0\\ 2x-1,&\mbox{if }~{}x>0.\end{cases}$


$\frac{1}{r_{\min}}-\frac{1}{p}=\frac{1}{p}-\frac{1}{r_{\max}}$


$\langle I\rangle_{e}$


$+:R\times R\to R\,$


$\,q\overline{q}=-a^{2}+b^{2}+c^{2}+d^{2}$


$v_{i}=\left\{\begin{matrix}-1&\mbox{with prob. }1/2\\ 1&\mbox{with prob. }1/2\end{matrix}\right.$


$W\stackrel{f}{\ \to\ }X\stackrel{g}{\ \to\ }Y\stackrel{h}{\ \to\ }Z$


$\vec{\nabla}\times\vec{B}=\mu_{0}\vec{J},$


$R\oplus G=(\,\text{dom }G\ntriangleleft R)\cup G$


$\begin{aligned}\displaystyle SS_{\rm err}&\displaystyle=\sum_{i=1}^{N}\;(y_{i}-\widehat{y_{i}})^{2}\\ \displaystyle SS_{\rm tot}&\displaystyle=\sum_{i=1}^{N}\;(y_{i}-\bar{y})^{2}\\ \displaystyle SS_{\rm reg}&\displaystyle=\sum_{i=1}^{N}\;(\widehat{y_{i}}-\bar{y})^{2}\,\text{ and}\\ \displaystyle\bar{y}&\displaystyle=\frac{1}{N}\sum{}_{i=1}^{N}\;y_{i}.\end{aligned}$


$\left(-\frac{3}{2}\right)+\left(-\frac{1}{2}\right)+\frac{1}{2}=\frac{3\left(-\frac{3}{2}+\frac{1}{2}\right)}{2}=-\frac{3}{2}.$


$O(n\log\log n)$


${n\choose r}={{n!}\over{r!(n-r)!}}={52\choose 5}={{52!}\over{5!(52-5)!}}=2,598,960$


$\displaystyle\pi=2+\cfrac{4}{3+\cfrac{1\cdot 3}{4+\cfrac{3\cdot 5}{4+\cfrac{5\cdot 7}{4+\ddots}}}}$


$\pi=\cfrac{4}{1+\cfrac{1^{2}}{3+\cfrac{2^{2}}{5+\cfrac{3^{2}}{7+\cfrac{4^{2}}{9+\ddots}}}}}$


${n\choose k}p^{k}(1-p)^{n-k}$


$\mathbf{IC}=\frac{\displaystyle\sum_{i=1}^{c}n_{i}(n_{i}-1)}{N(N-1)/c}$


$L(\lambda,\alpha,s)=\sum_{n=0}^{\infty}\frac{\exp(2\pi i\lambda n)}{(n+\alpha)^{s}}.$


$0\ \rightarrow\mathcal{A}\ \stackrel{\phi}{\rightarrow}\ \mathcal{B}\ \stackrel{\psi}{\rightarrow}\ \mathcal{C}\ \rightarrow\ 0$


$\mathfrak{P}(\mathfrak{P}_{\geq 1}(\mathcal{Z})).$


$x=1+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{1+\ddots\,}}}}$


$\operatorname{Im}\left(\frac{\sinh(x+iy)}{x+iy}\right)$


$w_{i}=\sum_{j=2}^{n}\frac{a_{j}}{(i+1)^{j}},$


$\rm\stackrel{GCE}{\rightleftharpoons}$


$0\ 1\ 00\ 01\ 10\ 11\ 000\ 001\ldots$


$U=\frac{D^{2}}{12\mu}\left(\frac{-dP}{dx}\right)$


$\cos C=-\cos A\,\cos B+\sin A\,\sin B\,\cos c.$


$\widehat{\beta_{1}}=\frac{\sum(x_{i}-\bar{x})(y_{i}-\bar{y})}{\sum(x_{i}-\bar{x})^{2}}\,\text{ and }\hat{\beta_{0}}=\bar{y}-\widehat{\beta_{1}}\bar{x}$


$\mathbb{R}^{d}$


$\log\left(x\right)=-\log\left(\frac{1}{x}\right)$


$s_{f_{2}}(n)=O\left(M(n)\log n\right).$


$\zeta(3)=14\sum_{k=1}^{\infty}\frac{1}{k^{3}\sinh(\pi k)}-\frac{11}{2}\sum_{k=1}^{\infty}\frac{1}{k^{3}(e^{2\pi k}-1)}-\frac{7}{2}\sum_{k=1}^{\infty}\frac{1}{k^{3}(e^{2\pi k}+1)}.$


$D_{\gamma}(\gamma(b)||\gamma(a))=\int_{a}^{b}(b-s)g_{\gamma}(s)ds$


$\ Z_{t}\ =e^{\int_{0}^{t}X_{s}\,dW_{s}-\frac{1}{2}\int_{0}^{t}X_{s}^{2}\,ds},\quad 0\leq t\leq T$


$-i\Gamma$


$r=\lfloor 0.5+\mu_{2,l}\rfloor=4$


$x^{*n}=\underbrace{x*x*x*\cdots*x*x}_{n},\quad x^{*0}=\delta_{0}$


$T=\frac{\int_{x=0}^{\infty}x\ell(x)m(x)\,\mathrm{d}x}{\int_{x=0}^{\infty}\ell(x)m(x)\,\mathrm{d}x}$


$f^{(n)}(x)=\begin{cases}\displaystyle\frac{p_{n}(x)}{x^{2n}}\,f(x)&\,\text{if }x>0,\\ 0&\,\text{if }x\leq 0,\end{cases}$


$[\ ]_{+}$


$y_{3i}=\begin{cases}y_{3i}^{*}&\textrm{if}\;y_{1i}^{*}>0\\ 0&\textrm{if}\;y_{1i}^{*}\leq 0.\end{cases}$


$\mu_{3,1}=\mu_{3,1}-r=\frac{14}{3}-5=\frac{-1}{3}$


${}_{p}F_{q}(a_{1},\dots,a_{p};c_{1},\dots,c_{q};z)=\sum_{n=0}^{\infty}\frac{(a_{1})_{n}\cdots(a_{p})_{n}}{(c_{1})_{n}\cdots(c_{q})_{n}}\frac{z^{n}}{n!}$


$\Delta H=\underbrace{\Delta G}_{\textrm{elec.}}+\underbrace{T\Delta S}_{\textrm{heat}}$


$0\to H\stackrel{i}{{}\to{}}G\stackrel{\pi}{{}\to{}}G/H\to 0$


$O(n\log k)$


$\phi(T)=\begin{cases}1&\,\text{if }T>t_{0}\\ 0&\,\text{if }T<t_{0}\end{cases}$


$\mathbf{\nabla}\times\mathbf{B}=\mu_{0}\mathbf{J}$


$\partial^{2}u\over\partial t^{2}$


$\{x^{2}-y^{2}=0\}=\{(x+y)(x-y)=0\}=\{x+y=0\}\cup\{x-y=0\}.\,$


$\ln 2=\cfrac{1}{1+\cfrac{1}{2+\cfrac{1}{3+\cfrac{2}{2+\cfrac{2}{5+\cfrac{3}{2+\cfrac{3}{7+\cfrac{4}{2+\ddots}}}}}}}}=\cfrac{2}{3-\cfrac{1^{2}}{9-\cfrac{2^{2}}{15-\cfrac{3^{2}}{21-\ddots}}}}$


$\underline{\psi\quad\quad\ \ }\,\!$


$\forall x,y\in A:\|x\,y\|\ \leq\|x\|\,\|y\|$


$\sum_{k=1}^{\infty}\frac{(-1)^{k}}{s_{k}-1}=\frac{1}{1}-\frac{1}{2}+\frac{1}{6}-\frac{1}{42}+\frac{1}{1806}{\,\pm\cdots}$


$O(N\log N)$


$\mu(A)=\begin{cases}(1-\frac{1}{\lambda})\mathbf{1}_{0\in A}+\nu(A),&\,\text{if }\lambda>1\\ \nu(A),&\,\text{if }0\leq\lambda\leq 1,\end{cases}$


$\int_{0}^{\infty}\cos t^{2}\,\mathrm{d}t=\int_{0}^{\infty}\sin t^{2}\,\mathrm{d}t=\frac{\sqrt{2\pi}}{4}=\sqrt{\frac{\pi}{8}}.$


$Y=\begin{cases}0,&\mbox{if }~{}Y^{*}>0\\ 1,&\mbox{if }~{}Y^{*}<0.\end{cases}$


$\nabla\times\mathbf{B}=\mu_{0}\left(\mathbf{J}+\varepsilon_{0}\frac{\partial\mathbf{E}}{\partial t}\right)$


$\mathfrak{Poiss}$


$\begin{bmatrix}4&3\\ 6&3\end{bmatrix}=\begin{bmatrix}1&0\\ 1.5&1\end{bmatrix}\begin{bmatrix}4&3\\ 0&-1.5\end{bmatrix}.$


$m\text{AB}=-\frac{5}{2}\log_{10}\left(\frac{f_{\nu}}{\,\text{Jy}}\right)+8.90.$


$\ P={{\ 1\over\Delta R\sqrt{2\pi\,}\ }\exp[{\ (\ln{R}\ -\ln{\bar{R}}\ )^{2}\over\ 2\Delta R^{2}\ \ }\ }]$


$\begin{bmatrix}V_{1}\\ I_{1}\end{bmatrix}=\begin{bmatrix}A&B\\ C&D\end{bmatrix}\begin{bmatrix}V_{2}\\ -I_{2}\end{bmatrix}$


$S_{a}=\frac{E[R_{a}-R_{b}]}{\sigma_{a}}=\frac{E[R_{a}-R_{b}]}{\sqrt{\mathrm{var}[R_{a}-R_{b}]}},$


$=\frac{d}{dx}\left(\frac{dy}{dx}\right)$


$\mbox{votes needed to win}~{}=\left({{\rm\mbox{valid votes cast}~{}}\over{\rm\mbox{seats to fill}~{}}+1}\right)+1$


$S^{\lambda}\stackrel{\varphi}{\to}s_{\lambda},$


$s_{\mathrm{a}}(t)=\begin{cases}e^{j(\omega t+\theta)}\ \ =\ e^{j|\omega|t}\cdot e^{j\theta},&\,\text{if}\ \omega>0,\\ e^{-j(\omega t+\theta)}=\ e^{j|\omega|t}\cdot e^{-j\theta},&\,\text{if}\ \omega<0.\end{cases}$


$0\rightarrow\mathfrak{a}\rightarrow\mathfrak{e}\rightarrow\mathfrak{g}\rightarrow 0$


$\frac{2}{\frac{1}{60}+\frac{1}{40}}=48$


$\forall x,y\in I:x-y\in I$


$O(n\log m)$


$\forall z[z\in x\Leftrightarrow z\in y]\land\forall w[x\in w\Leftrightarrow y\in w].$


$\Gamma=\gamma_{1}+\gamma_{2}+\cdots+\gamma_{n}.$


$H_{2}=\langle x,G^{\prime}\rangle$


$\;\epsilon_{\mathrm{n}}=\epsilon-K_{\mathrm{p}}...........(16)$


$\bar{x}=\frac{\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}I(x,y)x\,dx\,dy}{\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}I(x,y)\,dx\,dy}$


$DR=\sqrt{\int_{-\infty}^{T}(T-r)^{2}f(r)\,dr}$


$R(t)=K\int_{0}^{t}C_{p}(\tau)\,d\tau+V_{0}C_{p}(t)$


$B^{c}\subseteq A^{c}$


$\forall x,y\in A\qquad\|xy\|\leq\|x\|\|y\|$


$S(\tau_{B})-S(\tau_{A})=\int_{A}^{B}dS=\int_{\tau_{A}}^{\tau_{B}}\frac{dS}{d\tau}d\tau=\int_{\tau_{A}}^{\tau_{B}}\frac{S(\tau+d\tau)-S(\tau)}{(\tau+d\tau)-\tau}d\tau$


$a_{i,0}=\left[\frac{n+[L/2]-1}{[L/2]}\right].$


$\underline{\underline{\mathsf{S}}}$


$Y=\left\{\begin{matrix}X&\,\text{if }\left|X\right|\leq c\\ -X&\,\text{if }\left|X\right|>c\end{matrix}\right.$


$d_{k}=\int_{b_{k-1}}^{b_{k}}(x-y_{k})^{2}f(x)dx$


$\underline{AB}$


$O((n+r)\log(n))$


$dG_{T,P}=-\sum_{k}\mathbb{A}_{k}\,d\xi_{k}+W^{\prime}.\,$


$\vec{n}+p\to d+\gamma$


$-\log(x)$


$\wedge^{*}A$


$\tilde{\theta}\,\xrightarrow{P}\,\theta$


$\begin{aligned}\displaystyle x^{(n+3)}&\displaystyle=x^{(n+2)}P=\left(x^{(n+1)}P\right)P\\ \\ &\displaystyle=x^{(n+1)}P^{2}=\left(x^{(n)}P^{2}\right)P\\ &\displaystyle=x^{(n)}P^{3}\\ &\displaystyle=\begin{bmatrix}0&1&0\end{bmatrix}\begin{bmatrix}0.9&0.075&0.025\\ 0.15&0.8&0.05\\ 0.25&0.25&0.5\end{bmatrix}^{3}\\ &\displaystyle=\begin{bmatrix}0&1&0\end{bmatrix}\begin{bmatrix}0.7745&0.17875&0.04675\\ 0.3575&0.56825&0.07425\\ 0.4675&0.37125&0.16125\\ \end{bmatrix}\\ &\displaystyle=\begin{bmatrix}0.3575&0.56825&0.07425\end{bmatrix}.\end{aligned}$


$n^{\wedge}A$


$\frac{n!}{k!(n-k)!}$


$0\to G^{\wedge}\stackrel{\pi^{\wedge}}{\to}X^{\wedge}\stackrel{\imath^{\wedge}}{\to}H^{\wedge}\to 0$


$f^{+}(x)=\max(f(x),0)=\begin{cases}f(x)&\mbox{ if }~{}f(x)>0\\ 0&\mbox{ otherwise.}\end{cases}$


$\mathcal{L}(\overline{x},\Sigma)\propto\det(\Sigma)^{-n/2}\exp\left(-{1\over 2}\sum_{i=1}^{n}(x_{i}-\overline{x})^{\mathrm{T}}\Sigma^{-1}(x_{i}-\overline{x})\right),$


$c_{2}I=c_{1}E-S_{e}^{-1}\left(\frac{E}{k_{e}-r_{e}E}\right)+P$


${}_{\ 90}^{232}\mathrm{Th}+\mathrm{n}\rightarrow{}_{\ 90}^{233}\mathrm{Th}+\gamma\ \xrightarrow{\beta^{-}}\ {}_{\ 91}^{233}\mathrm{Pa}\ \xrightarrow{\beta^{-}}\ {}_{\ 92}^{233}\mathrm{U}$


$S_{k}={\left({{m}\atop{k}}\right)}\frac{(m-k)!}{m!}=\frac{1}{k!}.$


$\textrm{pH}_{new}=\textrm{pK}_{a}+\log\left(\frac{1.5\textrm{n}}{0.5\textrm{n}}\right)$


$L_{p_{2}}=L_{p_{1}}+20\log_{10}\!\left(\frac{r_{1}}{r_{2}}\right)\!~{}\mathrm{dB}.$


$w=\begin{cases}w_{min},&\mbox{if }~{}w_{min}\geq\;w(L)\\ w(L),&\mbox{if }~{}w_{min}\leq\;w(L)\end{cases}\,\!$


$\begin{bmatrix}\sigma_{1}\\ \sigma_{2}\\ \sigma_{3}\\ \sigma_{4}\\ \sigma_{5}\\ \sigma_{6}\end{bmatrix}=\begin{bmatrix}C_{11}&C_{12}&C_{13}&C_{14}&C_{15}&C_{16}\\ C_{12}&C_{22}&C_{23}&C_{24}&C_{25}&C_{26}\\ C_{13}&C_{23}&C_{33}&C_{34}&C_{35}&C_{36}\\ C_{14}&C_{24}&C_{34}&C_{44}&C_{45}&C_{46}\\ C_{15}&C_{25}&C_{35}&C_{45}&C_{55}&C_{56}\\ C_{16}&C_{26}&C_{36}&C_{46}&C_{56}&C_{66}\end{bmatrix}\begin{bmatrix}\varepsilon_{1}\\ \varepsilon_{2}\\ \varepsilon_{3}\\ \varepsilon_{4}\\ \varepsilon_{5}\\ \varepsilon_{6}\end{bmatrix}$


$k\propto\exp\left(\frac{-\Delta^{\ddagger}G^{\ominus}}{RT}\right)$


$f(n)=n+\left\lfloor\log_{2}\left(n+\log_{2}n\right)\right\rfloor.$


$\mathrm{NPV}(i)=\sum_{t=0}^{N}\frac{R_{t}}{(1+i)^{t}}$


$A\cdot B=\overline{\overline{A}+\overline{B}}$


$\frac{1}{18}=\frac{1}{2}-\frac{1}{3}-\frac{1}{3^{2}}.$


$L(M,V,s)=\sum_{\mu\in\{M-0\}/V}\frac{\operatorname{sign}N(\mu)}{|N(\mu)|^{s}}$


$\cos(\alpha-\beta)=\cos\alpha\cos\beta+\sin\alpha\sin\beta\,$


$ax^{2}+bx+c=0\ .$


$\hat{\sigma}=\sqrt{\frac{1}{N-1.5}\sum_{i=1}^{n}(x_{i}-\bar{x})^{2}},$


$\underline{\underline{\mathbf{A}}}$


$C_{i,m}=\left(\frac{\partial C}{\partial n}\right)$


$\nabla\times\mathbf{B}=\mu_{0}\mathbf{J}+\mu_{0}\epsilon_{0}\frac{\partial\mathbf{E}}{\partial t}$


$W_{ijkl}=\frac{\partial^{2}F_{ij}}{\partial x_{k}\partial x_{l}}+\frac{\partial^{2}F_{kl}}{\partial x_{i}\partial x_{j}}-\frac{\partial^{2}F_{il}}{\partial x_{j}\partial x_{k}}-\frac{\partial^{2}F_{jk}}{\partial x_{i}\partial x_{l}}$


$\alpha=\pi/2-\gamma,\ \mathrm{while}\ \beta=\pi/2-\gamma/2\ ,$


$(\underbrace{+,\cdots,+}_{k},\underbrace{-,\cdots,-}_{n})$


$\cos(A)=-\cos(B)\cos(C)+\sin(B)\sin(C)\cos(a)\,$


$A(c)=\alpha$


$\mathbf{S}(p)=\begin{bmatrix}s_{11}&s_{12}\\ s_{21}&s_{22}\end{bmatrix}$


$M=\log_{2}(R/G)=\log_{2}(R)-\log_{2}(G)$


$\forall r\forall x[Fxr\leftrightarrow[\forall y(y\in x\rightarrow Byr)\and\lnot Bxr]]\,.$


$\pi=\sum_{n=0}^{\infty}\left(\frac{4}{8n+1}-\frac{2}{8n+4}-\frac{1}{8n+5}-\frac{1}{8n+6}\right)\left(\frac{1}{16}\right)^{n}\!$


$\langle x\rangle_{0}$


$:\;\;4x^{4}{-}28x^{3}{-}7x^{2}{+}16x{+}16=0$


$\,\text{dBW}=10\log\left(\frac{\,\text{power out}}{1\,\mathrm{W}}\right)$


$M=T\oplus P$


$d=69+12\log_{2}\left(\frac{f}{440\ \mathrm{Hz}}\right).\,$


$\bar{s}_{i}=\sqrt{\frac{\sum_{j=1}^{n}\left(x_{ij}-\bar{x}_{i}\right)^{2}}{n-1}}$


$\left\langle i\right\rangle=\left(\frac{s}{q}\right)\frac{dq}{ds}$


$\mathbf{P}\mathbf{B}*\left(\frac{MR-PBR}{MR}\right)$


$\mathcal{I}_{j}=\frac{2}{\sqrt{-\mu_{j}}\sqrt{\lambda}}\int_{0}^{\infty}e^{-\frac{\xi^{2}}{2}}d\xi=\sqrt{\frac{2\pi}{\lambda}}(-\mu_{j})^{-\frac{1}{2}}.$


$\sum_{i=1}^{n}(X_{i}-\overline{X})(X_{i}-\overline{X})^{\mathrm{T}}\sim W_{p}(\Sigma,n-1).$


$=\lim_{N\to\infty}Nb\log_{2}\left(\frac{A}{NW}+1\right)$


$\tfrac{dS}{dT}=B-\beta SI-\mu S+\gamma I$


$\underline{\underline{A}}$


$O(\log{k}\ \log{\log{k}})$


$F_{total}=\mu mg\cos{\theta}+mg\sin{\theta}$


${\left({{n}\atop{k}}\right)}=\begin{cases}n^{\underline{k}}/k!&\,\text{if }\ k\leq\frac{n}{2}\\ n^{\underline{n-k}}/(n-k)!&\,\text{if }\ k>\frac{n}{2}\end{cases}.$


$0\rightarrow L\rightarrow M\rightarrow N\rightarrow 0$


$\left(\frac{\mbox{Net Income}~{}}{\mbox{Net Sales}~{}}\right)\left(\frac{\mbox{Net Sales}~{}}{\mbox{Total Assets}~{}}\right)$


$e_{i}^{2}=\gamma_{1}+\gamma_{2}z_{2i}+\dots+\gamma_{p}z_{pi}+\eta_{i}.$


$H_{2}=\begin{bmatrix}1&1\\ 1&-1\end{bmatrix},$


$Z=\sqrt{{R+j\omega L}\over{G+j\omega C}}$


$G=\left.\begin{matrix}3\underbrace{\uparrow\ldots\uparrow}3\\ \underbrace{\vdots}\\ 3\uparrow\uparrow\uparrow\uparrow 3\end{matrix}\right\}\,\text{64 layers}$


$w_{0}(n)=\mathrm{sinc}\left(\frac{2n}{N-1}\right)\,$


$0\to R\xrightarrow{\ d_{2}\ }R^{2}\xrightarrow{\ d_{1}\ }R\to 0,$


$\operatorname{Im}\left(\frac{\cosh(x+iy)}{x+iy}\right)$


$2\mathbf{E}\mathbf{E}^{T}\mathbf{E}-\operatorname{tr}(\mathbf{E}\mathbf{E}^{T})\mathbf{E}=0.$


$\begin{bmatrix}t^{\prime}\\ -vt^{\prime}\end{bmatrix}=\begin{bmatrix}\gamma&\delta\\ \beta&\alpha\end{bmatrix}\begin{bmatrix}t\\ 0\end{bmatrix},$


$y(x)=x\frac{dy}{dx}+f\left(\frac{dy}{dx}\right).$


$f(x)=\begin{cases}1&\,\text{if }w\cdot x+b>0\\ 0&\,\text{otherwise}\end{cases}$


$\langle A\rangle_{\sigma}$


$x\oplus y=(x+y)/2$


$0\rightarrow A\stackrel{q}{\longrightarrow}B\stackrel{r}{\longrightarrow}C\rightarrow 0\,$


$\gamma=\alpha+c\beta$


$\forall x,y\in A$


${\left({{n}\atop{k}}\right)}=\frac{n!}{k!(n-k)!},$


$\mathrm{n}+{}_{\ 90}^{232}\mathrm{Th}\rightarrow{}_{\ 90}^{233}\mathrm{Th}\xrightarrow{\beta^{-}}{}_{\ 91}^{233}\mathrm{Pa}\xrightarrow{\beta^{-}}{}_{\ 92}^{233}\mathrm{U}+\mathrm{n}\rightarrow{}_{\ 92}^{232}\mathrm{U}+2\mathrm{n}$


$V^{\infty}\wedge V^{\infty}\wedge Y\wedge Y$


$\lfloor n/2-1\rfloor=m$


$P^{\prime}=P\oplus\Delta$


$\langle A\rangle_{r}$


$\hat{{H}}_{\operatorname{PI}}=\begin{bmatrix}0.052&0.510\\ 0.510&8.882\end{bmatrix}.$


$b_{1}=\frac{m_{3}}{s^{3}}=\frac{\tfrac{1}{n}\sum_{i=1}^{n}(x_{i}-\overline{x})^{3}}{\left[\tfrac{1}{n-1}\sum_{i=1}^{n}(x_{i}-\overline{x})^{2}\right]^{3/2}}\ ,$


$\iff x=y\and(L\ z)[x:=y]$


$\mu(\sigma,\tau)=(-1)^{n-r}(2!)^{r_{3}}(3!)^{r_{4}}\cdots((n-1)!)^{r_{n}}$


$\frac{1}{4}\int_{0}^{1}\frac{x^{8}(1-x)^{8}}{1+x^{2}}\,dx=\pi-\frac{47\,171}{15\,015}$


$f(t)=\int_{0}^{\infty}h(\tau)s(t-\tau)\,d\tau$


$e_{q}^{x}=\sum_{n=0}^{\infty}\frac{x^{n}}{[n]_{q}!}.$


$J_{\mathrm{diffusion}}(x)=-D\frac{d\rho}{dx}$


$1+\cfrac{1}{2+\cfrac{1}{2+\cfrac{1}{2+\cfrac{1}{2+\ddots}}}}$


$\frac{\hat{p}+\frac{1}{2n}z^{2}}{1+\frac{1}{n}z^{2}}$


$\omega(z)=\sum_{n=0}^{+\infty}\frac{q_{n}(\omega_{a})}{(1+\omega_{a})^{2n-1}}\frac{(z-a)^{n}}{n!}$


$\frac{\partial^{2}V}{\partial S\partial\sigma}$


$\zeta(s,a)=\sum_{n=0}^{\infty}\frac{1}{(n+a)^{s}}\!$


$SE_{\widehat{\beta}}=\frac{\sqrt{\frac{1}{n-2}\sum_{i=1}^{n}(y_{i}-\widehat{y}_{i})^{2}}}{\sqrt{\sum_{i=1}^{n}(x_{i}-\overline{x})^{2}}}$


$O(m)$


$\frac{\partial^{2}f}{\partial\sigma^{2}}<0$


$\forall n\in\mathbb{N}\ \exists A\subseteq\mathbb{N}\ \forall x\in\mathbb{N}\ [x\in A\,\text{ iff }x\leq n].$


$\frac{\partial^{2}(A(x,y))}{\partial y^{2}}=0$


$Y(t)=\chi\text{i}X(t)+\int_{0}^{\infty}\Phi\text{d}(\tau)X(t-\tau)\,\mathrm{d}\tau,$


$AB=\frac{1}{2}\pi\left(2x+y\right)^{2}$


$\left([-1,1]+\frac{1}{2}\right)^{2}-\frac{1}{4}=\left[-\frac{1}{2},\frac{3}{2}\right]^{2}-\frac{1}{4}=\left[0,\frac{9}{4}\right]-\frac{1}{4}=\left[-\frac{1}{4},2\right]$


$T_{j}(x)U_{k}(x)=\begin{cases}\tfrac{1}{2}\left(U_{j+k}(x)+U_{k-j}(x)\right),&\,\text{if }k\geq j-1.\\ \tfrac{1}{2}\left(U_{j+k}(x)-U_{j-k-2}(x)\right),&\,\text{if }k\leq j-2.\end{cases}$


$\zeta(s)=\sum_{n=1}^{\infty}\frac{1}{n^{s}}.$


$0\to U\to M\to V\to 0$


$=2\sqrt{2}\int_{0}^{\infty}w^{\frac{1}{2}}e^{-w}\mathrm{d}w+\sqrt{2}\int_{0}^{\infty}w^{-\frac{1}{2}}e^{-w}\mathrm{d}w$


$\zeta(s,q)=\sum_{n=0}^{\infty}\frac{1}{(q+n)^{s}}.$


$y=\begin{cases}0~{}~{}\,\text{if}~{}~{}y^{*}\leq 0,\\ 1~{}~{}\,\text{if}~{}~{}0<y^{*}\leq\mu_{1},\\ 2~{}~{}\,\text{if}~{}~{}\mu_{1}<y^{*}\leq\mu_{2}\\ \vdots\\ N~{}~{}\,\text{if}~{}~{}\mu_{N-1}<y^{*}.\end{cases}$


$e=2+\cfrac{1}{1+\cfrac{1}{2+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{4+{}\ddots}}}}}$


$\,{}^{254}_{99}\mathrm{Es}+\,^{48}_{20}\mathrm{Ca}\to\,^{302}_{119}\mathrm{Uue}^{*}$


$\sigma_{2}=\cos\psi\,d\theta+\sin\psi\sin\theta\,d\phi$


$\frac{\partial^{2}\mathbf{U}^{-1}}{\partial x\partial y}=$


$\left\{y~{}\backepsilon~{}(y\succcurlyeq x)\land\lnot(x\succcurlyeq y)\right\}$


$M/X=(M^{\ast}\setminus x)^{\ast}$


$\eta_{\mathrm{PAR}}(T)=\frac{\int_{\lambda_{1}}^{\lambda_{2}}B(\lambda,T)\,d\lambda}{\int_{0}^{\infty}B(\lambda,T)\,d\lambda},$


$0\rightarrow A\rightarrow B\rightarrow C\rightarrow 0$


$\frac{1}{\lambda_{1}}+\frac{1}{\lambda_{2}}$


$\mathrm{A}^{*}+\mathrm{Q}\rightarrow\mathrm{A}+\mathrm{Q}^{*}$


$\frac{\int^{\infty}_{0}y_{\lambda}J_{\lambda}d\lambda}{\int^{\infty}_{0}J_{\lambda}d\lambda},$


$E_{n}(x)=\frac{n!}{\sqrt{\pi}}\int_{0}^{x}e^{-t^{n}}\,\mathrm{d}t=\frac{n!}{\sqrt{\pi}}\sum_{p=0}^{\infty}(-1)^{p}\frac{x^{np+1}}{(np+1)p!}\,.$


$A\stackrel{\rho}{\to}A^{\prime}$


$\begin{matrix}&\underbrace{a^{a^{{}^{.\,^{.\,^{.\,^{a}}}}}}}&\\ &b\mbox{ copies of }~{}a\end{matrix}$


$\sum_{k=0}^{\infty}a_{k}=\sum_{k=0}^{\infty}(-1)^{k}(k+1),$


$\displaystyle{\mathfrak{g}=\mathfrak{k}\oplus\mathfrak{p}}$


$\{*\}\xrightarrow{d}D$


$p(t)=\int_{0}^{\infty}h(\tau)q(t-\tau)\,d\tau$


$\forall w_{1},\ldots,w_{n}\,[(\forall x\,\exists\,y\phi(x,y,w_{1},\ldots,w_{n}))\Rightarrow\forall A\,\exists B\,\forall x\in A\,\exists y\in B\,\phi(x,y,w_{1},\ldots,w_{n})]$


$\displaystyle{M\rightarrow\mathfrak{k}^{*}\rightarrow\mathfrak{t}^{*}.}$


${n\choose k}=\frac{n!}{k!\,(n-k)!},$


$P=\begin{bmatrix}1&0\\ 0&0\end{bmatrix}$


$f_{Y}(y\mid X=x)f_{X}(x)=f_{X,Y}(x,y)=f_{X}(x\mid Y=y)f_{Y}(y).$


$i+4\rightarrow i$


$\langle n\rangle_{\,\text{th}}$


$\displaystyle\alpha(x^{2}+y^{2})-2\beta x-2\gamma y+\delta=0,$


$S+E\rightarrow P+E$


$f(x)=\sum_{i=0}^{\infty}{x\over 2^{i}}$


$1-\frac{1}{2}-\frac{1}{4}+\frac{1}{8}-\frac{1}{16}+\cdots=\frac{1}{3}.$


$\nabla\times\mathbf{B}=\mu_{0}\epsilon_{0}\frac{\partial\mathbf{E}}{\partial t}\qquad\quad\ (4)$


$A(BC)=(AB)C$


$\nabla\times\mathbf{B}=\mu_{0}\mathbf{J}$


$1\xrightarrow{\ \ \alpha\ }2$


$y=\begin{cases}1&\,\text{if }u\geq\theta\\ 0&\,\text{if }u<\theta\end{cases}$


$\mathrm{{}^{238}_{\ 92}U\ +\ ^{1}_{0}n\ \longrightarrow\ ^{239}_{\ 92}U\ \xrightarrow[23\ min]{\beta^{-}}\ ^{239}_{\ 93}Np\ \xrightarrow[2.355\ days]{\beta^{-}}\ ^{239}_{\ 94}Pu}$


$\begin{bmatrix}x^{\prime}\\ y^{\prime}\end{bmatrix}=\begin{bmatrix}1&k\\ 0&1\end{bmatrix}\begin{bmatrix}x\\ y\end{bmatrix}$


$O(k\log n)$


$\pi^{+}\to W^{*}\to\mu^{+}\nu_{\mu}$


$\textrm{cr}(K_{n})=\frac{1}{4}\left\lfloor\frac{n}{2}\right\rfloor\left\lfloor\frac{n-1}{2}\right\rfloor\left\lfloor\frac{n-2}{2}\right\rfloor\left\lfloor\frac{n-3}{2}\right\rfloor.$


$q\stackrel{a}{\rightarrow}q^{\prime}$


$\chi_{\nu}(z)=\sum_{k=0}^{\infty}\frac{z^{2k+1}}{(2k+1)^{\nu}}.$


${\mathit{l}\over\mathit{l}^{*}}={1\over 2}$


$\cos(\theta^{\prime}-\theta)=\cos\theta^{\prime}\cos\theta+\sin\theta\sin\theta^{\prime}$


$F_{n}=\sqrt{\frac{n\lambda d_{1}d_{2}}{d_{1}+d_{2}}}$


$0\to M^{\prime}\to M\to M^{\prime\prime}\to 0$


$x\in V\land\exists y(y\in x)\rightarrow\exists y(y\in x\land\lnot\exists z(z\in y\land z\in x)).$


$C_{p}=\left(\frac{\partial H}{\partial T}\right)_{p}$


$\nabla\times\mathbf{B}=\mu_{0}\mathbf{j}$


$\,\frac{e^{hk}p^{k}(1-p)^{n-k}}{1-p+pe^{h}}$


$x\xrightarrow{d}\ \mu$


$J=\Big|{\partial x^{a}\over\partial x^{{}^{\prime}b}}\Big|$


$P_{N}(N_{R})=\frac{N!}{2^{N}N_{R}!(N-N_{R})!}$


$\lambda\text{piv}=\sqrt{\frac{\int e(\lambda)\lambda d\lambda}{\int e(\lambda)\lambda^{-1}d\lambda}}$


$\mathbf{\delta}=\hat{\mathbf{h}}(n)-\mathbf{h}(n)$


$\forall x\Big(\forall y(y\in x\rightarrow P[y])\rightarrow P[x]\Big)\rightarrow\forall x\,P[x]$


$\sin(\alpha+\beta)\sin(\beta+\gamma)=\sin\alpha\sin\gamma+\sin\beta\sin(\alpha+\beta+\gamma)$


$\ Ax^{2}+Bx+C=0$


$\mathrm{{}^{238}_{92}U+{}^{1}_{0}n\ \xrightarrow{\ }\ {}^{239}_{92}U\ \xrightarrow[23.5\ min]{\beta^{-}}\ {}^{239}_{93}Np\ \xrightarrow[2.3\ days]{\beta^{-}}\ {}^{239}_{94}Pu\ \xrightarrow[2.4\cdot 10^{4}\ years]{\alpha}}$


$U\colon A\to{Set}$


$\frac{d\Bigl(\frac{d\left(\frac{dy}{dx}\right)}{dx}\Bigr)}{dx}=\left(\frac{d}{dx}\right)^{3}\bigl(f(x)\bigr)$


$H=\begin{bmatrix}\dfrac{\partial^{2}f}{\partial x_{1}^{2}}&\dfrac{\partial^{2}f}{\partial x_{1}\,\partial x_{2}}&\cdots&\dfrac{\partial^{2}f}{\partial x_{1}\,\partial x_{n}}\\ \dfrac{\partial^{2}f}{\partial x_{2}\,\partial x_{1}}&\dfrac{\partial^{2}f}{\partial x_{2}^{2}}&\cdots&\dfrac{\partial^{2}f}{\partial x_{2}\,\partial x_{n}}\\ \vdots&\vdots&\ddots&\vdots\\ \dfrac{\partial^{2}f}{\partial x_{n}\,\partial x_{1}}&\dfrac{\partial^{2}f}{\partial x_{n}\,\partial x_{2}}&\cdots&\dfrac{\partial^{2}f}{\partial x_{n}^{2}}\end{bmatrix}.$


$1\,-\,\frac{1}{3}\,+\,\frac{1}{5}\,-\,\frac{1}{7}\,+\,\frac{1}{9}\,-\,\cdots\;=\;\frac{\pi}{4},\!$


$\mathrm{pH}=\mathrm{p}K_{\mathrm{a}}+\log_{10}\left(\frac{[\mathrm{A}^{-}]}{[\mathrm{HA}]}\right)$


$Z(P,Q,s)=\sum_{n=1}^{\infty}\frac{f_{n}(P)f_{n}(Q)}{\lambda_{n}^{s}}$


$g(f)(x)=(\int_{0}^{\infty}|\nabla u(x,y)|^{2}ydy)^{\frac{1}{2}}$


$A^{*}A=(A^{*}A)^{\frac{1}{2}}(A^{*}A)^{\frac{1}{2}},$


$\begin{bmatrix}i\\ u\end{bmatrix}=\begin{bmatrix}y_{11}&y_{12}\\ y_{21}&y_{22}\end{bmatrix}\begin{bmatrix}v\\ F\end{bmatrix}$