$\frac{\partial}{\partial x}\left(\frac{f(x)}{g(x)}\right)\geq 0$


$\prod(x):=\begin{cases}1,&\mbox{if }~{}{|x|\leq 1/2},\\ 0&\mbox{if }~{}\mbox{otherwise}~{}.\\ \end{cases}$


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


$\int_{-\infty}^{\infty}f(r)\,dx=2\int_{0}^{\infty}f(r)\,dx,\;\;r=\sqrt{x^{2}+y^{2}}.$


$\frac{1}{\kappa}=\frac{\int_{0}^{\infty}\kappa_{\nu}^{-1}u(\nu,T)d\nu}{\int_{0}^{\infty}u(\nu,T)d\nu}$


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


$\begin{cases}\dot{\mathbf{x}}=A\mathbf{x}+B\mathbf{u}\\ y=\begin{bmatrix}1&0&0&\cdots&\end{bmatrix}\mathbf{x}=x_{1}\end{cases}$


$x,y\in A^{\ast}$


$\displaystyle\underbrace{x\diamondsuit\cdots\diamondsuit x}_{k\ \mathrm{factors}}.\,$


$\left(A+BK\left(I-DK\right)^{-1}C\right)$


$\begin{cases}\\ \\ \end{cases}$


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


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


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


$\int_{0}^{t}\mathrm{ROI}(\tau)\,d\tau=(-\mathbf{U}_{n}^{T}\mathbf{K}^{-1}\mathbf{Q}+V_{p})\int_{0}^{t}C_{p}(\tau)\,d\tau+\mathbf{U}_{n}^{T}\mathbf{K}^{-1}\mathbf{A}$


$\exp y=1+y+{1\over 2!}y^{2}+{1\over 3!}y^{3}+\dots=\lim_{N\to\infty}\sum_{r=0}^{N}{N!\over r!(N-r)!}({y\over N})^{r}=\lim_{N\to\infty}(1+{y\over N})^{N}.$


$\forall x,y,z\,\left(xFy\wedge xFz\to y=z\right)$


$I_{B}=\frac{\log(N!)-\sum_{i=1}^{K}(\log(n_{i}!))}{N}$


$\tau_{5}=\frac{\beta-\alpha}{60^{\circ}}\approx-0.01667\alpha+0.01667\beta$


$\begin{bmatrix}x_{1}\\ x_{2}^{*}\end{bmatrix}=\begin{bmatrix}h_{1}&-h_{2}\\ h_{2}^{*}&h_{1}^{*}\end{bmatrix}\begin{bmatrix}S_{1}\\ S_{2}^{*}\end{bmatrix}+\begin{bmatrix}n_{1}\\ n_{2}^{*}\end{bmatrix}$


$Y^{\prime}=Y\oplus N$


$\epsilon=(\log\log n)^{c}$


$\begin{aligned}\displaystyle\mathbb{S}&\displaystyle=\,\begin{pmatrix}S_{xx}&S_{xy}\\ S_{yx}&S_{yy}\end{pmatrix}\,=\,\mathbb{I}\,\left(\frac{c_{g}}{c_{p}}-\frac{1}{2}\right)\,E\,+\,\frac{1}{k^{2}}\,\begin{pmatrix}k_{x}\,k_{x}&k_{x}\,k_{y}\\ k_{y}\,k_{x}&k_{y}\,k_{y}\end{pmatrix}\,\frac{c_{g}}{c_{p}}\,E,\\ \displaystyle\mathbb{I}&\displaystyle=\,\begin{pmatrix}1&0\\ 0&1\end{pmatrix}\quad\,\text{and}\\ \displaystyle\nabla symbol{U}&\displaystyle=\,\begin{pmatrix}\displaystyle\frac{\partial U_{x}}{\partial x}&\displaystyle\frac{\partial U_{y}}{\partial x}\\ \displaystyle\frac{\partial U_{x}}{\partial y}&\displaystyle\frac{\partial U_{y}}{\partial y}\end{pmatrix},\end{aligned}$


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


$\gamma=\alpha+i\beta$


$O(KNM)$


$\left[\begin{matrix}\varepsilon_{xx}&\varepsilon_{xy}&\varepsilon_{xz}\\ \varepsilon_{yx}&\varepsilon_{yy}&\varepsilon_{yz}\\ \varepsilon_{zx}&\varepsilon_{zy}&\varepsilon_{zz}\\ \end{matrix}\right]=\left[\begin{matrix}\frac{\partial u_{x}}{\partial x}&\frac{1}{2}\left(\frac{\partial u_{x}}{\partial y}+\frac{\partial u_{y}}{\partial x}\right)&\frac{1}{2}\left(\frac{\partial u_{x}}{\partial z}+\frac{\partial u_{z}}{\partial x}\right)\\ \frac{1}{2}\left(\frac{\partial u_{y}}{\partial x}+\frac{\partial u_{x}}{\partial y}\right)&\frac{\partial u_{y}}{\partial y}&\frac{1}{2}\left(\frac{\partial u_{y}}{\partial z}+\frac{\partial u_{z}}{\partial y}\right)\\ \frac{1}{2}\left(\frac{\partial u_{z}}{\partial x}+\frac{\partial u_{x}}{\partial z}\right)&\frac{1}{2}\left(\frac{\partial u_{z}}{\partial y}+\frac{\partial u_{y}}{\partial z}\right)&\frac{\partial u_{z}}{\partial z}\\ \end{matrix}\right]\,\!$


$-\log(t)\,$


$\frac{1}{25}+\frac{1}{15}+\frac{1}{75}+\frac{1}{200}=\frac{1}{8}$


$\pi\oplus\sigma=\operatorname{rev}(\operatorname{rev}(\sigma)\ominus\operatorname{rev}(\pi))$


$\begin{bmatrix}|U_{e1}|^{2}&|U_{e2}|^{2}&|U_{e3}|^{2}\\ |U_{\mu 1}|^{2}&|U_{\mu 2}|^{2}&|U_{\mu 3}|^{2}\\ |U_{\tau 1}|^{2}&|U_{\tau 2}|^{2}&|U_{\tau 3}|^{2}\end{bmatrix}=\begin{bmatrix}\frac{2}{3}&\frac{1}{3}&0\\ \frac{1}{6}&\frac{1}{3}&\frac{1}{2}\\ \frac{1}{6}&\frac{1}{3}&\frac{1}{2}\end{bmatrix}.$


$O(\log{n})$


$H[n]=\begin{cases}0,&n<0,\\ 1/2,&n=0,\\ 1,&n>0,\end{cases}$


$\mathbf{a}_{i}=\begin{cases}1&\,\text{if }w_{i}>0,\\ 0&\,\text{if }w_{i}<0.\end{cases}$


$\begin{bmatrix}\dfrac{\partial x_{1}}{\partial x}&\dfrac{\partial x_{1}}{\partial y}&\dfrac{\partial x_{1}}{\partial z}\\ \dfrac{\partial x_{2}}{\partial x}&\dfrac{\partial x_{2}}{\partial y}&\dfrac{\partial x_{2}}{\partial z}\\ \dfrac{\partial x_{3}}{\partial x}&\dfrac{\partial x_{3}}{\partial y}&\dfrac{\partial x_{3}}{\partial z}\end{bmatrix}$


$g=\left(1,0,1,-\frac{1}{2},\frac{1}{3},-\frac{1}{4},\frac{1}{5},-\frac{1}{6},\cdots\right)$


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


$A=\begin{bmatrix}4&1\\ 6&3\end{bmatrix}$


$B\to(A\lor B)$


$[0.x_{1}\ldots x_{m}]=\sum_{k=1}^{m}x_{k}2^{-k}.$


$\mathrm{{}^{238}_{\ 92}U\ +\ ^{10}_{\ 5}B\ \longrightarrow\ ^{242}_{\ 97}Bk\ +\ 6\ ^{1}_{0}n\quad;\quad^{232}_{\ 90}Th\ +\ ^{15}_{\ 7}N\ \longrightarrow\ ^{242}_{\ 97}Bk\ +\ 5\ ^{1}_{0}n}$


$A(w)B(xw)=\sum_{n=0}^{\infty}P_{n}(x)w^{n}.$


$\lambda\text{piv}=\sqrt{\frac{\int e(\lambda)d\lambda}{\int e(\lambda)\lambda^{-2}d\lambda}}$


$\textstyle{f(x)=\sum_{n=0}^{\infty}{s(2^{n}x)\over 2^{n}}}$


$\forall x,y\in A,x\neq y\Rightarrow f(x)\neq f(y).$


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


$RMD(S)=\frac{\sum_{i=1}^{n}\sum_{j=1}^{n}|y_{i}-y_{j}|}{(n-1)\sum_{i=1}^{n}y_{i}}$


$A=\begin{bmatrix}\alpha&\beta\\ \gamma&\delta\end{bmatrix}$


$\scriptstyle b-2\,x-e\;\sim\;0.$


$\left(\frac{\partial E}{\partial v}\right)_{k}$


$\Delta_{n}(s)=\mathbb{P}\left\{x_{n}\leq s\right\}-\log_{2}(1+s),$


$x,y\in A_{1}$


$\gamma=\left[\begin{matrix}a&b\\ c&d\end{matrix}\right]$


$\,{}^{254}_{99}\mathrm{Es}+\,^{48}_{20}\mathrm{Ca}\to\,^{302}_{119}\mathrm{Uue}^{*}\to\ \ no\ atoms$


$\cos c=\cos a\,\cos b+\sin a\,\sin b\,\cos\gamma.$


$H_{HPF}(f)=1-\mathrm{rect}\left(\frac{f}{2B_{H}}\right).$


$e^{\pi\sqrt{58}}=396^{4}-104.000000177\dots.$


$V=U\oplus W$


$Z(\lambda,\nu)=\sum_{j=0}^{\infty}\frac{\lambda^{j}}{(j!)^{\nu}}.$


$x<y\rightarrow\exists z[x<z<y].$


$M=\frac{1}{1-\int_{0}^{L}\alpha(x)\,dx}$


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


$e^{-}+A\rightarrow A^{*}+e^{-}$


$g(s)=\sum_{n=1}^{\infty}\frac{a(n)}{n^{s}}$


$C_{V}=\left({\partial U\over\partial T}\right)_{V}.$


$D4\sigma=4\sigma=4\sqrt{\frac{\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}I(x,y)(x-\bar{x})^{2}\,dx\,dy}{\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}I(x,y)\,dx\,dy}}$


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


$=\frac{\mu_{0}r^{2}N^{2}\pi}{l}\left(1-\frac{8w}{3\pi}+\frac{w^{2}}{2}-\frac{w^{4}}{4}+\frac{5w^{6}}{16}-\frac{35w^{8}}{64}+...\right)$


$\mathrm{{}^{238}_{\ 92}U+\,^{1}_{0}n\;\rightarrow\;^{239}_{\ 92}U\;\rightarrow\;^{239}_{\ 93}Np+\beta\;\rightarrow\;^{239}_{\ 94}Pu+\beta}$


$O(mn\log m)$


$P_{0}=\sum_{t=1}^{T}\frac{C_{t}}{(1+r_{t})^{t}}$


$q_{and}=k_{1}\land^{p}k_{2}\land^{p}....\land^{p}k_{t}$


$\wedge^{k}F$


$\forall x[\forall y(Rxy\rightarrow\exists z[Ryz])\rightarrow\exists y(Rxy\wedge\forall z[Ryz\rightarrow z=y])]$


$\langle B\rangle_{\mathcal{A}}$


$\begin{bmatrix}f_{1}\\ g_{1}\end{bmatrix}=\begin{bmatrix}A&B\\ C&D\end{bmatrix}\begin{bmatrix}0\\ y_{0}\end{bmatrix}$


$f(x)=\begin{cases}1&\mbox{if }~{}x<1,\\ 2&\mbox{if }~{}x=1,\\ 1/2&\mbox{if }~{}x>1,\end{cases}$


$F(s)=\sum_{n=1}^{\infty}\frac{f(n)}{n^{s}}$


$0\rightarrow X\rightarrow Y\rightarrow Z\rightarrow 0$


$\Bigg[{\partial{V}\over{\partial{\mathbf{r}}}}\Bigg]_{0}$


$dX_{t}=-aX_{t}dt+\sigma dW_{t}$


$q_{jk}=\frac{\sum_{i=1}^{N}w_{i}}{\left(\sum_{i=1}^{N}w_{i}\right)^{2}-\sum_{i=1}^{N}w_{i}^{2}}\sum_{i=1}^{N}w_{i}\left(x_{ij}-\bar{x}_{j}\right)\left(x_{ik}-\bar{x}_{k}\right).$


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


$M=\left[{\begin{array}[]{*{10}c}2&0&0&-2\\ 0&-2&2&0\\ 2&0&0&-2\\ -2&0&0&2\\ 0&2&-2&0\\ -2&0&0&2\\ \end{array}}\right]$


$\ A\vee B:=(A\rightarrow B)\rightarrow B.$


$M\stackrel{a}{\to}M$


$\operatorname{RMSD}=\sqrt{\frac{\sum_{t=1}^{n}(\hat{y}_{t}-y)^{2}}{n}}.$


$\cos C=-\cos A$


${N\choose n_{1}}{N-n_{1}\choose n_{2}}{N-n_{1}-n_{2}\choose n_{3}}...=\frac{N!}{(N-n_{1})!n_{1}!}\frac{(N-n_{1})!}{(N-n_{1}-n_{2})!n_{2}!}\frac{(N-n_{1}-n_{2})!}{(N-n_{1}-n_{2}-n_{3})!n_{3}!}....$


$\epsilon_{0}\mathbf{E}\cdot\frac{\partial\mathbf{E}}{\partial t}$


$P_{ij}(\nu)=\left\{\begin{array}[]{cc}{1\over 4}+{3\over 4}e^{-4\nu/3}&\mbox{ if }~{}i=j\\ {1\over 4}-{1\over 4}e^{-4\nu/3}&\mbox{ if }~{}i\neq j\end{array}\right.$


$0\to I\to L\to L^{\prime}\to 0$


$p(x)=a(y)p(x|y)+a(\overline{y})p(x|\overline{y})$


$R_{1}=\frac{1}{2!}f^{\prime\prime}(\xi_{n})(\alpha-x_{n})^{2}\,,$


$H(f,g)=\begin{bmatrix}0&\dfrac{\partial g}{\partial x_{1}}&\dfrac{\partial g}{\partial x_{2}}&\cdots&\dfrac{\partial g}{\partial x_{n}}\\ \dfrac{\partial g}{\partial x_{1}}&\dfrac{\partial^{2}f}{\partial x_{1}^{2}}&\dfrac{\partial^{2}f}{\partial x_{1}\,\partial x_{2}}&\cdots&\dfrac{\partial^{2}f}{\partial x_{1}\,\partial x_{n}}\\ \dfrac{\partial g}{\partial x_{2}}&\dfrac{\partial^{2}f}{\partial x_{2}\,\partial x_{1}}&\dfrac{\partial^{2}f}{\partial x_{2}^{2}}&\cdots&\dfrac{\partial^{2}f}{\partial x_{2}\,\partial x_{n}}\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ \dfrac{\partial g}{\partial x_{n}}&\dfrac{\partial^{2}f}{\partial x_{n}\,\partial x_{1}}&\dfrac{\partial^{2}f}{\partial x_{n}\,\partial x_{2}}&\cdots&\dfrac{\partial^{2}f}{\partial x_{n}^{2}}\end{bmatrix}$


$\ \cfrac{1}{15+\cfrac{1}{1+\cfrac{1}{102}}}$


$O(\log n)$


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


$x\oplus y=(x\vee y)\wedge\neg{(x\wedge y)}$


$\psi_{1}(z)=\sum_{n=0}^{\infty}\frac{1}{(z+n)^{2}},$


$\frac{23}{16}=1+\cfrac{1}{2+\cfrac{1}{3+\frac{1}{2}}}=[1;2,3,2],$


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


$\sigma_{jk}\cfrac{\partial\epsilon_{jk}}{\partial x_{1}}=\sigma_{jk}\cfrac{\partial^{2}u_{j}}{\partial x_{1}\partial x_{k}}$


$\mathbf{\varepsilon}=\begin{bmatrix}\dfrac{\partial v_{1}}{\partial S_{1}}&\cdots&\dfrac{\partial v_{1}}{\partial S_{m}}\\ \vdots&\ddots&\vdots\\ \dfrac{\partial v_{n}}{\partial S_{1}}&\cdots&\dfrac{\partial v_{n}}{\partial S_{m}}\end{bmatrix}.$


$\mu_{a}=\left(\frac{\partial G}{\partial N_{a}}\right)_{PTN}$


$\begin{bmatrix}K_{11}&K_{12}\\ K_{21}&K_{22}\end{bmatrix}\begin{bmatrix}x_{1}\\ x_{2}\end{bmatrix}=\begin{bmatrix}F_{1}\\ F_{2}\end{bmatrix}$


$W\left(\lambda_{1},\lambda_{2},\lambda_{3}\right)=\sum_{p=1}^{N}\frac{\mu_{p}}{\alpha_{p}}\left(\lambda_{1}^{\alpha_{p}}+\lambda_{2}^{\alpha_{p}}+\lambda_{3}^{\alpha_{p}}-3\right)$


$\ ax^{2}+bx+c=0$


$\theta=\sin^{-1}\left(\frac{\lambda\ \Delta\Phi}{2\pi d}\right)$


$S_{X}^{2}=\frac{1}{n-1}\sum_{i=1}^{n}\left(X_{i}-\overline{X}\right)^{2}\,\text{ and }S_{Y}^{2}=\frac{1}{m-1}\sum_{i=1}^{m}\left(Y_{i}-\overline{Y}\right)^{2}$


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


$\cos^{2}\alpha+\cos^{2}\beta+\cos^{2}\gamma+2\cos(\alpha)\cos(\beta)\cos(\gamma)=1,$


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


$\mathbf{x}^{{}^{\prime}}$


$\forall x,y\in A\;[x\neq y\rightarrow\neg\exists z\in X\;[z\leq x\land z\leq y]].$


$\forall xy\,[x\neq y\rightarrow\exists z\neq y\,C(x;yz)].$


$f(t)=2\int_{0}^{\infty}\int_{-\infty}^{\infty}f(\tau)\cos 2\pi\lambda(\tau-t)d\tau d\lambda.$


$\forall x\in t\ (\phi)\Leftrightarrow\forall x(x\in t\rightarrow\phi)$


$\rho\equiv\frac{\sum_{i\neq j}(a_{ij}-\bar{a})(a_{ji}-\bar{a})}{\sum_{i\neq j}(a_{ij}-\bar{a})^{2}}$


$U+W$


$f(x)=\begin{cases}\frac{1}{x}&\mbox{if }~{}x>0,\\ 5&\mbox{if }~{}x\leq 0.\end{cases}$


$H_{2n}-H_{n}\sim\log 2-\frac{1}{4n}+\frac{1}{16n^{2}}-\frac{1}{128n^{4}}+\frac{1}{256n^{6}}-\frac{17}{4096n^{8}}+\cdots,$


$0\rightarrow\ker h\rightarrow H^{n}(C;G)\stackrel{h}{\rightarrow}\,\text{Hom}(H_{n}(C),G)\rightarrow 0.$


$-j\infty\,$


$\zeta=\begin{cases}1&\mbox{if }~{}n\neq 0\\ 1/2&\mbox{if }~{}n=0\end{cases},\quad\xi=\begin{cases}1&\mbox{if }~{}m\neq 0\\ 1/2&\mbox{if }~{}m=0\end{cases}$


$\mathrm{NPV}=\sum_{n=0}^{N}\frac{C_{n}}{(1+r)^{n}}=0$


$\lambda^{\theta}$


$\mathbf{T}^{n}\begin{bmatrix}0\\ \vec{b}^{n-1}\\ \end{bmatrix}=\begin{bmatrix}t_{0}&\dots&t_{-n+2}&t_{-n+1}\\ \vdots&&&\\ t_{n-2}&&\mathbf{T}^{n-1}&\\ t_{n-1}&&&\end{bmatrix}\begin{bmatrix}\\ 0\\ \\ \vec{b}^{n-1}\\ \\ \end{bmatrix}=\begin{bmatrix}\epsilon_{b}^{n}\\ 0\\ \vdots\\ 0\\ 1\end{bmatrix}.$


$AB=G$


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


$\sqrt{2}=1+\cfrac{1}{2+\cfrac{1}{2+\cfrac{1}{2+\cfrac{1}{2+\cfrac{1}{2+\ddots\,}}}}}.$


$A\oplus B=(A^{c}\ominus B^{s})^{c}$


$Peak\ Criteria\begin{cases}\mathrm{\left(\frac{\Delta Amplitude}{\Delta Frequency}\right)Cell(n-1)<0}\\ \mathrm{\left(\frac{\Delta Amplitude}{\Delta Frequency}\right)Cell(n+1)>0}\end{cases}$


$s_{m}=\sqrt{\frac{\sum(\Delta hR_{Fi}-\Delta hR_{Ft})^{2}}{n+1}}$


$s^{2}=\displaystyle\frac{\sum_{i=1}^{n}(x_{i}-K)^{2}-(\sum_{i=1}^{n}(x_{i}-K))^{2}/n}{n-1}.\!$


$v_{i,j}=\begin{cases}n^{-\frac{1}{2}}&j=1\\ \sqrt{\frac{2}{n}}\cos(\frac{\pi(j-1)(i-\frac{1}{2})}{n})&otherwise\end{cases}$


$\begin{bmatrix}\mathbf{N}\\ \mathbf{M}\end{bmatrix}=\begin{bmatrix}\mathbf{A}&\mathbf{B}\\ \mathbf{B}&\mathbf{D}\end{bmatrix}\begin{bmatrix}\varepsilon^{0}\\ \kappa\end{bmatrix}$


$a_{k+1}=\begin{cases}\left\lfloor a_{k}^{\frac{1}{2}}\right\rfloor,&\mbox{if }~{}a_{k}\mbox{ is even}\\ \\ \left\lfloor a_{k}^{\frac{3}{2}}\right\rfloor,&\mbox{if }~{}a_{k}\mbox{ is odd}~{}.\end{cases}$


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


$\begin{bmatrix}V_{1}\\ V_{2}\end{bmatrix}=\int_{-t/2}^{t/2}\begin{bmatrix}\sigma_{13}\\ \sigma_{23}\end{bmatrix}\,dx_{3}\,.$


$\hat{b}=\frac{\sum_{i=1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})}{\sum_{i=1}^{n}(x_{i}-\bar{x})^{2}},$


$\langle r\rangle_{\rm H}$


$\dot{y}=\frac{\partial y}{\partial x_{j}}\dot{x}_{j}+\frac{1}{2}\frac{\partial^{2}y}{\partial x_{k}\,\partial x_{l}}g_{km}g_{ml}.$


$\wedge^{p+1}M_{q-1}\rightarrow\wedge^{p}M_{q}\rightarrow\wedge^{p-1}M_{q+1}$


$\,{}^{238}_{92}\mathrm{U}+\,^{64}_{28}\mathrm{Ni}\to\,^{302}_{120}\mathrm{Ubn}^{*}\to\ \mathit{fission\ only}$


$\forall x\forall y[\forall z[z\in x\leftrightarrow z\in y]\rightarrow x=y].$


$\cos(c)=\cos(a)\cos(b)+\sin(a)\sin(b)\cos(C).\,$


$\frac{\partial y}{\partial\mathbf{X}}=\begin{bmatrix}\frac{\partial y}{\partial x_{11}}&\frac{\partial y}{\partial x_{12}}&\cdots&\frac{\partial y}{\partial x_{1q}}\\ \frac{\partial y}{\partial x_{21}}&\frac{\partial y}{\partial x_{22}}&\cdots&\frac{\partial y}{\partial x_{2q}}\\ \vdots&\vdots&\ddots&\vdots\\ \frac{\partial y}{\partial x_{p1}}&\frac{\partial y}{\partial x_{p2}}&\cdots&\frac{\partial y}{\partial x_{pq}}\\ \end{bmatrix}.$


$G=\begin{pmatrix}G_{11}&G_{12}\\ G_{21}&G_{22}\end{pmatrix}$


$\frac{1+{\scriptstyle\frac{1}{2}}z}{1-{\scriptstyle\frac{1}{2}}z}$


$\begin{aligned}\displaystyle ML&\displaystyle=\left[\begin{matrix}A&B\\ C&D\end{matrix}\right]\left[\begin{matrix}I_{p}&0\\ -D^{-1}C&I_{q}\end{matrix}\right]=\left[\begin{matrix}A-BD^{-1}C&B\\ 0&D\end{matrix}\right]\\ &\displaystyle=\left[\begin{matrix}I_{p}&BD^{-1}\\ 0&I_{q}\end{matrix}\right]\left[\begin{matrix}A-BD^{-1}C&0\\ 0&D\end{matrix}\right].\end{aligned}$


$H(f)=\begin{bmatrix}\frac{\partial^{2}f}{\partial x^{2}}&\frac{\partial^{2}f}{\partial x\,\partial y}&\frac{\partial^{2}f}{\partial x\,\partial z}\\ \\ \frac{\partial^{2}f}{\partial y\,\partial x}&\frac{\partial^{2}f}{\partial y^{2}}&\frac{\partial^{2}f}{\partial y\,\partial z}\\ \\ \frac{\partial^{2}f}{\partial z\,\partial x}&\frac{\partial^{2}f}{\partial z\,\partial y}&\frac{\partial^{2}f}{\partial z^{2}}\end{bmatrix},$


$1\,-\,\frac{1}{3}\,+\,\frac{1}{5}\,-\,\frac{1}{7}\,+\,\frac{1}{9}\,-\,\cdots\;=\;\frac{\pi}{4}.\!$


$\,\text{SSE}=\sum_{i=1}^{n}(X_{i}-\bar{X})^{2}+\sum_{i=1}^{n}(Y_{i}-\bar{Y})^{2}+\sum_{i=1}^{n}(Z_{i}-\bar{Z})^{2}$


$P(k_{t}=k)={K\choose k}p_{t}^{k}(1-p_{t})^{K-k},\,\quad\quad k=0,1,\dots,K.$


${\rm C}+{\rm O}_{2}\rightarrow{\rm CO}_{2}$


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


$\,\text{INR}=\left(\frac{\,\text{PT}\text{test}}{\,\text{PT}\text{normal}}\right)\text{ISI}$


$\frac{n!}{l_{1}!l_{2}!\cdots l_{k}!}\prod_{i<j}(l_{i}-l_{j})$


$\tau_{ij}=\mu\left(\frac{\partial v_{i}}{\partial x_{j}}+\frac{\partial v_{j}}{\partial x_{i}}-\frac{2}{3}\delta_{ij}\nabla\cdot\mathbf{v}\right)+\kappa\delta_{ij}\nabla\cdot\mathbf{v}$


$\epsilon(X^{n})=\begin{cases}1&\mbox{if }~{}n=0\\ 0&\mbox{if }~{}n>0\end{cases}$


$\displaystyle\widehat{symbol\Sigma}={1\over n}\sum_{j=1}^{n}\left(\mathbf{x}_{j}-\bar{\mathbf{x}}\right)\left(\mathbf{x}_{j}-\bar{\mathbf{x}}\right)^{T}$


$\mathbf{1}_{A}(x)=\begin{cases}1&\,\text{if }x\in A,\\ 0&\,\text{if }x\notin A.\end{cases}$


$0\to M^{\prime}\xrightarrow{f}M\to M^{\prime\prime}\to 0,$


$0\rightarrow B\rightarrow Y\rightarrow A\rightarrow 0$


$\begin{aligned}\displaystyle p_{i}&\displaystyle=\cos\Omega\cos\omega-\sin\Omega\cos i\sin\omega\\ \displaystyle p_{j}&\displaystyle=\sin\Omega\cos\omega+\cos\Omega\cos i\sin\omega\\ \displaystyle p_{k}&\displaystyle=\sin i\sin\omega\\ \displaystyle q_{i}&\displaystyle=-\cos\Omega\sin\omega-\sin\Omega\cos i\cos\omega\\ \displaystyle q_{j}&\displaystyle=-\sin\Omega\sin\omega+\cos\Omega\cos i\cos\omega\\ \displaystyle q_{k}&\displaystyle=\sin i\cos\omega\\ \displaystyle w_{i}&\displaystyle=\sin i\sin\Omega\\ \displaystyle w_{j}&\displaystyle=-\sin i\cos\Omega\\ \displaystyle w_{k}&\displaystyle=\cos i\end{aligned}$


$\psi:\mathbf{Gr}(r,V)\to\mathbf{P}\left(\wedge^{r}V\right).$


$j_{2A}\Big(\tfrac{\sqrt{-58}}{2}\Big)=396^{4},\qquad\quad e^{\pi\sqrt{58}}\approx 396^{4}-104.00000017\dots$


$1\,-\,\frac{1}{2}\,+\,\frac{1}{3}\,-\,\frac{1}{4}\,+\,\frac{1}{5}\,-\,\cdots\;=\;\ln 2.$


$b^{x}=\lim_{r(\in\mathbb{Q})\to x}b^{r}\quad(b\in\mathbb{R}^{+},\,x\in\mathbb{R})$


$\sum_{i=1}^{n}(x_{i}-\overline{x})(\theta-\overline{x})=0$


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


${}^{238}_{92}\,\text{U}\to{}^{234}_{90}\,\text{Th}$


$y\cos B-z\cos C=0.$


$\left(\tfrac{\Delta Q}{\Delta P}\right)\times\tfrac{P}{Q}$


$x(t)=\frac{1}{2}a_{0}(t)\ +\ \sum_{n=1}^{\infty}\left[a_{n}(t)\cos\left(2\pi n\int_{0}^{t}f_{0}(\tau)\,d\tau\right)-b_{n}(t)\sin\left(2\pi n\int_{0}^{t}f_{0}(\tau)\,d\tau\right)\right]$


$O(Ns\kappa)$


${}_{2}F_{1}(a,b;c;z)=\sum_{n=0}^{\infty}\frac{(a)_{n}(b)_{n}}{(c)_{n}}\frac{z^{n}}{n!}.$


$\displaystyle f(x)=\sum_{k=1}^{\infty}\frac{\sin(2^{k}x)}{\sqrt{2}^{k}}$


$O(KN\log N)$


$\begin{aligned}\displaystyle\sigma_{ij}&\displaystyle=-\begin{pmatrix}p&0&0\\ 0&p&0\\ 0&0&p\end{pmatrix}+\mu\begin{pmatrix}2\frac{\partial u}{\partial x}&\frac{\partial u}{\partial y}+\frac{\partial v}{\partial x}&\frac{\partial u}{\partial z}+\frac{\partial w}{\partial x}\\ \frac{\partial v}{\partial x}+\frac{\partial u}{\partial y}&2\frac{\partial v}{\partial y}&\frac{\partial v}{\partial z}+\frac{\partial w}{\partial y}\\ \frac{\partial w}{\partial x}+\frac{\partial u}{\partial z}&\frac{\partial w}{\partial y}+\frac{\partial v}{\partial z}&2\frac{\partial w}{\partial z}\end{pmatrix}\\ &\displaystyle=-pI+\mu(\nabla\mathbf{v}+(\nabla\mathbf{v})^{T})\end{aligned}$


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


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


$Q=\int_{0}^{t}I(\tau)\ d\tau$


$\mathcal{L}=-\frac{1}{2}\mathrm{tr}\left(G_{\alpha\beta}G^{\alpha\beta}\right)+\bar{\psi}\left(iD_{\mu}\right)\gamma^{\mu}\psi$


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


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


$H_{\frac{1}{4}}=4-\tfrac{\pi}{2}-3\ln{2}$


$-\omega\delta$


$\mathrm{{}^{232}_{\ 90}Th\ +\ ^{1}_{0}n\ \longrightarrow\ ^{233}_{\ 90}Th\ \xrightarrow[22.3\ min]{\beta^{-}}\ ^{233}_{\ 91}Pa\ \xrightarrow[26.967\ d]{\beta^{-}}\ ^{233}_{\ 92}U}$


$V_{\mathrm{rms}}=\sqrt{\frac{1}{T}\int_{0}^{T}{[v(t)]^{2}dt}}.$


$\overline{A\oplus B}$


$\sigma_{y}^{2}(\tau)=\int_{0}^{\infty}S_{y}(f)\frac{2\sin^{4}\pi\tau f}{(\pi\tau f)^{2}}\,df$


$M_{1}=M\oplus P=(M\setminus P)\cup(P\setminus M)$


$A\cup B=S$


$\mathrm{nMe+mC\rightarrow Me_{n}C_{m}},$


$V^{\otimes n}\;\overset{\mathrm{def}}{=}\;\underbrace{V\otimes\cdots\otimes V}_{n}.$


$f(p)={(n+1)!\over s!(n-s)!}p^{s}(1-p)^{n-s}.$


$\forall x\forall y[\forall z(z\in x\Leftrightarrow z\in y)\Rightarrow\forall z(x\in z\Leftrightarrow y\in z)]$


$\frac{1-e^{-\lambda\theta}}{1-e^{-2\pi\lambda}}$


$d_{1}=\begin{bmatrix}x&y\\ \end{bmatrix}$


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


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


$\Phi(x,z,t)=\beta x-\gamma t+\varphi(x-ct,z),$


$x,y\in I$


$(\pi+20)^{i}=-0.9999999992\ldots-i\cdot 0.000039\ldots\approx-1$


$\mathrm{{}_{20}^{40}Ca}+\mathrm{{}_{2}^{4}He}\rightarrow\mathrm{{}_{22}^{44}Ti}+\gamma$


$\,\ -\sin x$


$O(n\,{\log|\Sigma|})$


$\forall x,y[x\neq y\rightarrow[xRy\vee yRx]]$


$y_{i}=\alpha+\beta x_{i}+\gamma z_{i}+u_{i}$


$\begin{cases}x^{*}\log(x^{*})-x^{*}&\,\text{if }x^{*}>0\\ 0&\,\text{if }x^{*}=0\end{cases}$


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


${I_{1}\choose I_{2}}=\begin{pmatrix}Y_{11}&Y_{12}\\ Y_{21}&Y_{22}\end{pmatrix}{V_{1}\choose V_{2}}$


$P^{2}=\begin{bmatrix}0&0\\ \alpha&1\end{bmatrix}\begin{bmatrix}0&0\\ \alpha&1\end{bmatrix}=\begin{bmatrix}0&0\\ \alpha&1\end{bmatrix}=P.$


$\mathbf{\begin{bmatrix}\hat{a}\\ \hat{b}\\ \hat{c}\\ \end{bmatrix}=\begin{bmatrix}\frac{1}{a}&-\frac{\cos(\gamma)}{a\sin(\gamma)}&\frac{\cos(\alpha)\cos(\gamma)-\cos(\beta)}{av\sin(\gamma)}\\ 0&\frac{1}{b\sin(\gamma)}&\frac{\cos(\beta)\cos(\gamma)-\cos(\alpha)}{bv\sin(\gamma)}\\ 0&0&\frac{\sin(\gamma)}{cv}\\ \end{bmatrix}}\begin{bmatrix}x\\ y\\ z\\ \end{bmatrix}$


$\widehat{symbol\Sigma}={1\over n}\sum_{i=1}^{n}({\mathbf{x}}_{i}-\overline{\mathbf{x}})({\mathbf{x}}_{i}-\overline{\mathbf{x}})^{T}$


$\tfrac{1}{2}(0-\tfrac{6}{4})=-\tfrac{3}{4}$


$\zeta(s,q)=\sum_{k=0}^{\infty}\frac{1}{(k+q)^{s}}$


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


$\frac{1}{w}=\frac{1}{a+1}+\frac{1}{b+1}+\frac{1}{c+1}+\frac{1}{d+1}$


$\widetilde{\Gamma}^{\alpha}_{\beta\gamma}=\Gamma^{\alpha}_{\beta\gamma}+S^{\alpha}_{\beta\gamma}$


$(\Delta k)^{2}=\frac{\int_{-\infty}^{\infty}(k-k_{0})^{2}F(k)F^{*}(k)\,dk}{\int_{-\infty}^{\infty}F(k)F^{*}(k)\,dk}$


$J_{\alpha}(x)=\frac{1}{\pi}\int_{0}^{\pi}\cos(\alpha\tau-x\sin\tau)\,d\tau-\frac{\sin(\alpha\pi)}{\pi}\int_{0}^{\infty}e^{-x\sinh(t)-\alpha t}\,dt.$


$f^{\prime}(x)={\rm st}\Bigg(\frac{\Delta y}{\Delta x}\Bigg)$


$O(L^{2}\log L)$


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


$A+B\rightleftharpoons|A\cdots B|^{\ddagger}\rightarrow P$


$\forall w_{1},\ldots,w_{n}\,\forall A\,\exists B\,\forall x\in A\,[\exists y\phi(x,y,w_{1},\ldots,w_{n})\Rightarrow\exists y\in B\,\phi(x,y,w_{1},\ldots,w_{n})]$


$\nabla\times\mathbf{B_{1}}=\mu_{0}\mathbf{J_{1}}-i\omega\epsilon_{0}\mu_{0}\mathbf{E_{1}}$


$\frac{1}{\lambda}\sum_{i>c}\frac{c!}{i!}\left(\frac{\lambda}{\mu}\right)^{i-c}$


$\left(\frac{\operatorname{d}v}{\operatorname{d}t}\right)_{s}$


$\forall x,y\in A,f(x+y)\leq f(x)+f(y).$


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


$\begin{array}[]{cc}P_{j}(d_{j})=\left\{\begin{array}[]{lll}0&\,\text{if}&|d_{j}|\leq q_{j}\\ \\ \frac{1}{2}&\,\text{if}&q_{j}<|d_{j}|\leq p_{j}\\ \\ 1&\,\text{if}&|d_{j}|>p_{j}\\ \end{array}\right.\end{array}$


$\begin{aligned}\displaystyle d\text{avg}&\displaystyle:=8\int_{0}^{1\over 2}\int_{0}^{x}\sqrt{x^{2}+y^{2}}\,dy\,dx\\ &\displaystyle=8\int_{0}^{1\over 2}{1\over 2}x^{2}(\ln(1+\sqrt{2})+\sqrt{2})\,dx\\ &\displaystyle=4P\int_{0}^{1\over 2}x^{2}\,dx\\ &\displaystyle={P\over 6}.\end{aligned}$


$\frac{c!}{s!(c-s)!}$


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


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


$\textstyle U:X\rightarrow S$


$x,y\in\mathbb{N}$


$\left(\frac{d\mathbf{L}}{dt}\right)_{\mathrm{relative}}$


$O(d\log{n})$


${S\over n}={1\over n}\sum_{i=1}^{n}(X_{i}-\overline{X})(X_{i}-\overline{X})^{\mathrm{T}}$


$\frac{da}{dt}=-2a^{2}b\quad\,\text{and }\frac{db}{dt}=0.$


$x_{+}^{\alpha}=\begin{cases}x^{\alpha}&\,\text{if }x>0\\ 0&\,\text{otherwise}\end{cases}$


$\exists y[My\and\varnothing\in y\and\forall z(z\in y\rightarrow\exists x[x\in y\and\forall w(w\in x\leftrightarrow[w=zw\in z])])].$


$\mathrm{PWV}=\sqrt{\dfrac{\operatorname{d}\!P\cdot V}{\rho\cdot\operatorname{d}\!V}}$


$\ 2x^{2}+3x+1=0$


$\cos(\varphi)=\sum_{n=0}^{\infty}\frac{(-\varphi^{2})^{n}}{(2n)!},$


$A\cap B=\varnothing$


$\frac{\zeta(2s)}{\zeta(s)}=\sum_{n=1}^{\infty}\frac{\lambda(n)}{n^{s}}.$


$0.\underbrace{999\ldots 9}_{H}\,$


$\frac{1}{n}=\sum_{r=1}^{\infty}\frac{1}{(n+1)^{r}}.$


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


$z\in L\implies\Pr\nolimits_{x}[\exists y.\phi(x,y,z)]\geq\tfrac{2}{3}$


${}^{\ast}\mathbb{R}$


$\,{}^{238}_{92}\mathrm{U}+\,^{66}_{30}\mathrm{Zn}\to\,^{304}_{122}\mathrm{Ubb}^{*}\to\ \mbox{no atoms}~{}.$


$T_{0}+T_{0}\rightarrow S^{*}+S_{0}+\,\text{phonons}$


$\mathfrak{P}=\{U_{p}(b)\,|\,p,b\in\mathbf{Z}^{+},p\,\text{ is prime}\}$


${}^{\omega}\omega$


$A=\begin{bmatrix}a&c\\ c&b\end{bmatrix}$


$\mathbf{\nabla}\cdot\mathbf{A}^{\prime}=-\mu_{0}\varepsilon_{0}\frac{\partial\varphi^{\prime}}{\partial t}$


$\langle\cdot,\cdot\rangle_{\mathcal{V}}$


$\cfrac{1}{3+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{3+\cfrac{1}{9+\ddots}}}}}$


${n!\over(k-1)!(n-k)!}u^{k-1}(1-u)^{n-k}\,du+O(du^{2}),$


$\left\lfloor\frac{(n-1)}{m}\right\rfloor+1=\left\lfloor\frac{(7-1)}{4}\right\rfloor+1=\left\lfloor\frac{6}{4}\right\rfloor+1=1+1=2$


$\Omega=\begin{vmatrix}\frac{\partial}{\partial x_{11}}&\cdots&\frac{\partial}{\partial x_{1n}}\\ \vdots&\ddots&\vdots\\ \frac{\partial}{\partial x_{n1}}&\cdots&\frac{\partial}{\partial x_{nn}}\end{vmatrix}.$


$\underbrace{a+\cdots+a}_{n\,\text{ summands}}=0$


$M=\quad\left[\frac{1+\frac{C}{D}}{\frac{C}{D}+\frac{R}{D}}\right]B$


$Δ(s)=s⊗1 + 1⊗s$


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


$O\left(\frac{N(\log\log N)^{5}}{\log N}\right)$


$\,\text{Cl}_{2m}\left(\frac{q\pi}{p}\right)=$


$|x_{1}x_{2}\cdots x_{N};S\rangle=\frac{\prod_{j}n_{j}!}{N!}\sum_{p}|x_{p(1)}\rangle|x_{p(2)}\rangle\cdots|x_{p(N)}\rangle$


$w_{\mathbf{\xi}}=\frac{1}{n(n+1)}\cdot\begin{cases}\frac{1}{2}\,\text{ if }\mathbf{\xi}\,\text{ is a vertex point}\\ 1\,\text{ if }\mathbf{\xi}\,\text{ is an edge point}\\ 2\,\text{ if }\mathbf{\xi}\,\text{ is an interior point.}\end{cases}$


$y_{2}=\left.\frac{\partial y_{b}}{\partial c}\right|_{c=0}$


$0\to\mathbb{Z}\to\mathcal{O}\to\mathcal{O}^{*}\to 0$


$\textstyle v=\frac{1}{k}\sum_{i=1}^{k}(m_{i}-\bar{m})^{2}$


$\Omega(r)=\frac{\int_{r}^{\infty}(1-F(x))\,dx}{\int_{-\infty}^{r}F(x)dx}$


$R\stackrel{i}{\to}R$


$\mu_{r}^{\prime}=\bigg({\frac{1}{2}}\bigg)\sum_{k=0}^{r}\bigg[{\frac{r!}{k!(r-k)!}}b^{k}\mu^{(r-k)}k!\{1+(-1)^{k}\}\bigg]$


$\lambda(n)=\begin{cases}\;\;\varphi(n)&\mbox{if }~{}n=2,3,4,5,6,7,9,10,11,13,14,17,19,22,23,25,26,27,29\dots\\ \tfrac{1}{2}\varphi(n)&\,\text{if }n=8,16,32,64,128,256\dots\end{cases}$


$\mathbf{Q}={1\over{N-1}}\sum_{i=1}^{N}(\mathbf{x}_{i}-\mathbf{\bar{x}})(\mathbf{x}_{i}-\mathbf{\bar{x}})^{\mathrm{T}},$


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


$\begin{bmatrix}T_{1}\\ T_{2}\\ T_{3}\end{bmatrix}=\begin{bmatrix}\sigma_{11}&\sigma_{21}&\sigma_{31}\\ \sigma_{12}&\sigma_{22}&\sigma_{32}\\ \sigma_{13}&\sigma_{23}&\sigma_{33}\end{bmatrix}\begin{bmatrix}n_{1}\\ n_{2}\\ n_{3}\end{bmatrix}$


$\left\{\begin{array}[]{l}f_{1}(Z)=(1-cP)+bQ+1,\\ f_{2}(Z)=aP-lQ+1.\end{array}\right.$


$\chi_{A}(x)=\begin{cases}1&\mbox{if }~{}x\in A,\\ 0&\mbox{if }~{}x\notin A.\\ \end{cases}$


$\operatorname{var}(\widehat{\varepsilon}_{i})=\sigma^{2}\left(1-\frac{1}{n}-\frac{(x_{i}-\bar{x})^{2}}{\sum_{j=1}^{n}(x_{j}-\bar{x})^{2}}\right).$


$\log^{*}(n)=\begin{cases}0,&\,\text{if }n\leq 1\\ 1+\log^{*}(\log n),&\,\text{if }n>1\end{cases}$


$0\;\rightarrow\;A\;\rightarrow\;B\;\rightarrow\;C\;\rightarrow\;0$


$\begin{array}[]{lll}\displaystyle\frac{\partial f}{\partial x}(0,r_{o})=0,&\displaystyle\frac{\partial^{2}f}{\partial x^{2}}(0,r_{o})=0,&\displaystyle\frac{\partial^{3}f}{\partial x^{3}}(0,r_{o})\neq 0,\\ \displaystyle\frac{\partial f}{\partial r}(0,r_{o})=0,&\displaystyle\frac{\partial^{2}f}{\partial r\partial x}(0,r_{o})\neq 0.\end{array}$


$\begin{bmatrix}c_{11}&c_{12}&c_{13}\\ c_{21}&c_{22}&c_{23}\\ c_{31}&c_{32}&c_{33}\end{bmatrix}\begin{bmatrix}x_{1}\\ x_{2}\\ x_{3}\end{bmatrix}=\begin{bmatrix}y_{1}\\ y_{2}\\ y_{3}\end{bmatrix}$


$\Delta_{n}(s)=\mathbb{P}\left\{x_{n}\leq s\right\}-\log_{2}(1+s)$


$\begin{aligned}\displaystyle\Rightarrow I(n)&\displaystyle=\int_{0}^{\pi}\sin^{n}xdx=\int_{0}^{\pi}udv=uv|_{x=0}^{x=\pi}-\int_{0}^{\pi}vdu\\ &\displaystyle=-\sin^{n-1}x\cos x|_{x=0}^{x=\pi}-\int_{0}^{\pi}-\cos x(n-1)\sin^{n-2}x\cos xdx\\ &\displaystyle=0-(n-1)\int_{0}^{\pi}-\cos^{2}x\sin^{n-2}xdx,n>1\\ &\displaystyle=(n-1)\int_{0}^{\pi}(1-\sin^{2}x)\sin^{n-2}xdx\\ &\displaystyle=(n-1)\int_{0}^{\pi}\sin^{n-2}xdx-(n-1)\int_{0}^{\pi}\sin^{n}xdx\\ &\displaystyle=(n-1)I(n-2)-(n-1)I(n)\\ &\displaystyle=\frac{n-1}{n}I(n-2)\\ \displaystyle\Rightarrow\frac{I(n)}{I(n-2)}&\displaystyle=\frac{n-1}{n}\\ \displaystyle\Rightarrow\frac{I(2n-1)}{I(2n+1)}&\displaystyle=\frac{2n+1}{2n}\end{aligned}$


$-z\frac{dX(z)}{dz}$


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


$(A\times B)^{c}=(A^{c}\times B^{c})\cup(A^{c}\times B)\cup(A\times B^{c}).$


$\theta^{\prime}(0)=-\frac{\ln\pi+\gamma+\pi/2+3\ln 2}{2}=-2.6860917\ldots$


${}_{\ 92}^{238}\mathrm{U}+\mathrm{n}\rightarrow{}_{\ 92}^{239}\mathrm{U}+\gamma\xrightarrow{\beta^{-}}{}_{\ 93}^{239}\mathrm{Np}\xrightarrow{\beta^{-}}{}_{\ 94}^{239}\mathrm{Pu}$


$f^{*}(x^{*})=\begin{cases}2x^{*}-4,&x^{*}<4\\ \frac{{x^{*}}^{2}}{4},&4\leqslant x^{*}\leqslant 6,\\ 3x^{*}-9,&x^{*}>6\end{cases}$


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


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


$=\frac{d}{dx}\left[p(x)\left(u\frac{dv}{dx}-v\frac{du}{dx}\right)\right],$


$\rightleftharpoons$


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


$\mathrm{{}^{238}_{\ 92}U\ \xrightarrow{+\ 15n,7\beta^{-}}\ ^{253}_{\ 99}Es}$


$\sigma_{t,T}=\sqrt{\frac{T\sigma_{0,T}^{2}-t\sigma_{0,t}^{2}}{T-t}}$


$S^{1}\wedge S^{1}\to S^{1}\wedge S^{1}$


$(-1)^{s-m-w}\frac{(m+v)!(n+w)!(2s-2n)!s!}{(m-v)!(s-l)!(s-m)!(s-n)!(2s+1)!}$


$A+\gamma\rightarrow A^{*}$


$\forall x\exists y\forall z[z\subseteq x\Rightarrow z\in y].$


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


$\begin{bmatrix}\\ \end{bmatrix}$


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


$0\to V^{\prime}\to V\to V^{\prime\prime}\to 0,$


$W(n,N)={N\choose n}={{N!}\over{n!(N-n)!}}$


$f(k;r,p)=\frac{\Gamma(k+r)}{k!\cdot\Gamma(r)}p^{k}(1-p)^{r}=\frac{\lambda^{k}}{k!}\cdot\frac{\Gamma(r+k)}{\Gamma(r)\;(r+\lambda)^{k}}\cdot\frac{1}{\left(1+\frac{\lambda}{r}\right)^{r}}$


$\Box A=\{x\in F;\,\forall y\in F\,(x\,R\,y\to y\in A)\}$


$\rho\frac{Du_{i}}{Dt}=-\frac{\partial p}{\partial x_{i}}+\mu\left(\frac{\partial^{2}u_{i}}{\partial x_{j}\partial x_{j}}\right),$


$\frac{1}{\kappa}=\frac{\int_{0}^{\infty}(\kappa_{\nu,{\rm es}}+\kappa_{\nu,{\rm ff}})^{-1}u(\nu,T)d\nu}{\int_{0}^{\infty}u(\nu,T)d\nu}$


$\Delta_{r}H=\left(\frac{\partial H}{\partial\xi}\right)_{p,T}$


$\psi(t)=\begin{cases}1&0\leq t<\frac{1}{2},\\ -1&\frac{1}{2}\leq t<1,\\ 0&\mbox{otherwise.}\end{cases}$


$A(z)=z•r$


$\cos\theta_{1}\cos\theta_{2}+\sin\theta_{1}\sin\theta_{2}=0$


${\left({{n}\atop{k}}\right)}p^{k}(1-p)^{n-k}$


$F(x)=\begin{cases}1&\,\text{if}\ \|x\|\leq\lambda\\ 0&\,\text{if}\ \|x\|>\lambda\\ \end{cases}$


$\omega=\begin{cases}\sqrt{D}&\mbox{if }~{}D\equiv 2,3\;\;(\mathop{{\rm mod}}4)\\ {{1+\sqrt{D}}\over 2}&\mbox{if }~{}D\equiv 1\;\;(\mathop{{\rm mod}}4)\end{cases}$


$\forall x,y\in S,\quad\alpha(x,y)+\alpha(y,x)=0$


$C=A\oplus B$


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


$\left.\begin{matrix}G&=&3\underbrace{\uparrow\uparrow\cdots\cdots\cdots\cdots\cdots\uparrow}3\\ &&3\underbrace{\uparrow\uparrow\cdots\cdots\cdots\cdots\uparrow}3\\ &&\underbrace{\qquad\;\;\vdots\qquad\;\;}\\ &&3\underbrace{\uparrow\uparrow\cdots\cdot\cdot\uparrow}3\\ &&3\uparrow\uparrow\uparrow\uparrow 3\end{matrix}\right\}\,\text{64 layers}$


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


$L=\sum_{i=1}^{n}b_{i}(x)\frac{\partial}{\partial x_{i}}+\sum_{i,j=1}^{n}a_{ij}(x)\frac{\partial^{2}}{\partial x_{i}\,\partial x_{j}},$


$\sin a=\sin\phi_{o}\sin\delta+\cos\phi_{o}\cos\delta\cos h$


$\operatorname{build-param-lists}[o\ x\ y,D,V,[]]\and D[g]=[x,\_,\_]::[o,\_,\_]::[y,\_,\_]::[]$


$\zeta^{\prime}(-1)=\frac{1}{12}-\ln A\approx-0.1654211437\ldots$


$A\equiv B\quad=\quad(A\multimap B)\&(B\multimap A)$


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


$\dot{e}_{2}=h_{3}(\hat{x})-m_{2}(\hat{x})\operatorname{sgn}(e_{2})$


$\begin{pmatrix}\frac{1}{c}\frac{\partial\phi}{\partial t^{\prime}}&\frac{\partial\phi}{\partial x^{\prime}}&\frac{\partial\phi}{\partial y^{\prime}}&\frac{\partial\phi}{\partial z^{\prime}}\end{pmatrix}=\begin{pmatrix}\frac{1}{c}\frac{\partial\phi}{\partial t}&\frac{\partial\phi}{\partial x}&\frac{\partial\phi}{\partial y}&\frac{\partial\phi}{\partial z}\end{pmatrix}\begin{pmatrix}\gamma&-\beta\gamma&0&0\\ -\beta\gamma&\gamma&0&0\\ 0&0&1&0\\ 0&0&0&1\end{pmatrix}\,.$


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


$\forall z\exists y\forall x[x\in y\leftrightarrow(x\in z\land\phi(x))].$


$U\overset{j}{\to}V\overset{g}{\to}Y$


$=(ac-bd)+(bc+ad)i$


$\alpha_{2}=\arctan\left(\frac{\sin\alpha}{-\sin U_{1}\sin\sigma+\cos U_{1}\cos\sigma\cos\alpha_{1}}\right)\,$


$A\,\Delta\,B=(A\setminus B)\cup(B\setminus A).$


$\underbrace{\qquad\qquad\qquad\qquad\qquad\qquad}$


$\mathrm{{}^{7}_{4}Be}+\mathrm{e{}^{-}}\ \xrightarrow{\ \mathrm{53.22d}}\ \mathrm{{}^{7}_{3}Li}$


$=\int_{0}^{h}\pi r^{2}\,dy$


$\mathrm{{}^{226}_{\ 88}Ra\ +\ ^{1}_{0}n\ \longrightarrow\ ^{227}_{\ 88}Ra\ \xrightarrow[42.2\ min]{\beta^{-}}\ ^{227}_{\ 89}Ac}$


$\therefore x^{2}+R^{2}-2Ry+y^{2}=R^{2}$


$v=\sqrt{\frac{2\mu_{0}^{2}}{3\lambda}}$


${1\over 1}+{1\over 2}+{1\over 3}+{1\over 4}+{1\over 5}+{1\over 6}+\cdots\rightarrow\infty.$


$i=1-(1-\frac{1}{2})\alpha=1-\frac{\alpha}{2}$


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


$O(\log^{2}n\log\log n)$


$z^{2}+c\to z$


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


$\widehat{\theta}_{F}(z)=\sum_{k=0}^{\infty}R_{F}(k)\exp(2\pi ikz),$


$(\forall x)Eq(x,x)\wedge(\forall x,y,z)[Eq(x,y)\rightarrow(Eq(x,z)\rightarrow Eq(y,z))]$


$-\log_{10}\left(\frac{1+2d}{365}\right)$


$=\alpha+\beta+\gamma-\pi,$


$\textstyle d\nu=\left(\frac{dR}{d\nu}\right)^{-1}$


$\forall n\in\mathbb{Z}(\exists m\in\mathbb{Z}[m>n\wedge P(m)])$


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


$\begin{bmatrix}\dot{x}\\ \dot{e}\\ \end{bmatrix}=\begin{bmatrix}A-BK&BK\\ 0&A-LC\\ \end{bmatrix}\begin{bmatrix}x\\ e\\ \end{bmatrix}.$


$\frac{\Delta E_{i}}{T}=\ln\left(\frac{p\text{i=on}}{1-p\text{i=on}}\right)$


$\chi=\alpha-\beta=k\ln\xi+\mathrm{const},\,$


${\ \atop{\ }}{{\underbrace{a^{a^{\cdot^{\cdot^{a}}}}}}\atop n}$


$Y_{n}=\begin{cases}1,&\,\text{if }U_{n}>0,\\ 0,&\,\text{if }U_{n}\leq 0\end{cases}$


$\sum_{i=1}^{n}(x_{i}-\mu)(x_{i}-\mu)^{\mathrm{T}}=\sum_{i=1}^{n}(x_{i}-\bar{x})(x_{i}-\bar{x})^{\mathrm{T}}=S$


$|C_{k}|\leq c^{\frac{k!}{n!(k-n)!}}$


${22\over 7}-{1\over 630}<\pi<{22\over 7}-{1\over 1260},$


$\mathbf{P}=\begin{bmatrix}\mathbf{P}_{11}&\mathbf{P}_{12}\\ \mathbf{P}_{21}&\mathbf{P}_{22}\end{bmatrix}.$


$C_{P}=\left(\frac{\partial H}{\partial T}\right)_{P}$


$M_{d,k}=\begin{cases}1+d\frac{(d-1)^{k}-1}{d-2}&\,\text{ if }d>2\\ 2k+1&\,\text{ if }d=2\end{cases}$


$\begin{bmatrix}t^{\prime}\\ x^{\prime}\end{bmatrix}=\begin{bmatrix}\gamma&\delta\\ \beta&\alpha\end{bmatrix}\begin{bmatrix}t\\ x\end{bmatrix},$


$Y_{2}=\begin{cases}1&\,\text{if }Y^{*}_{2}>0,\\ 0&\,\text{otherwise},\end{cases}$


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


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


$c=-\frac{1}{6}\cdot\frac{1}{2!}=-\frac{1}{12}.$


$A^{x}+B^{y}=C^{z}$


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


${1}+{1\over 3}+\cdots+{1\over 2a-1}-{1\over 2}-{1\over 4}-\cdots-{1\over 2b}+{1\over 2a+1}+\cdots+{1\over 4a-1}-{1\over 2b+2}-\cdots$


${\textstyle\left({{n}\atop{r}}\right)}=\tfrac{n!}{r!(n-r)!}$


$w=\begin{cases}w^{*}&\mbox{if }~{}w^{*}>\frac{1}{2},\\ \frac{1}{2}&\mbox{if }~{}w^{*}\leq\frac{1}{2}.\\ \end{cases}$


$\pi={3+\cfrac{1^{2}}{6+\cfrac{3^{2}}{6+\cfrac{5^{2}}{6+\cfrac{7^{2}}{6+\ddots\,}}}}}$


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


$H(X|Y)=-\sum_{x,~{}y}P_{X,Y}(x,~{}y)\log P_{X|Y}(x|y).$


$\begin{array}[]{rcl}\frac{1}{N}\log W&=&\frac{1}{N}\log\frac{N!}{n_{1}!\,n_{2}!\,\cdots\,n_{m}!}\\ \\ &=&\frac{1}{N}\log\frac{N!}{(Np_{1})!\,(Np_{2})!\,\cdots\,(Np_{m})!}\\ \\ &=&\frac{1}{N}\left(\log N!-\sum_{i=1}^{m}\log((Np_{i})!)\right).\end{array}$


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


$\mathrm{n}+{}_{\ 90}^{232}\mathrm{Th}\rightarrow{}_{\ 90}^{233}\mathrm{Th}\xrightarrow{\beta^{-}}{}_{\ 91}^{233}\mathrm{Pa}\xrightarrow{\beta^{-}}{}_{\ 92}^{233}\mathrm{U}$


$x\neq z\to M[x:=y][z]=M[z]$


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


$\cos B=-\cos C\,\cos A+\sin C\,\sin A\,\cos b,$


$A(o)=o^{\prime}$


$\int_{C}{\sqrt{z}\over z^{2}+6z+8}\,dz=2\int_{0}^{\infty}{\sqrt{x}\over x^{2}+6x+8}\,dx.$


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


$\operatorname{E}(X)=\mu+\gamma\beta,$


$=\sum{n!\over j_{1}!j_{2}!\cdots j_{n-k+1}!}\left({x_{1}\over 1!}\right)^{j_{1}}\left({x_{2}\over 2!}\right)^{j_{2}}\cdots\left({x_{n-k+1}\over(n-k+1)!}\right)^{j_{n-k+1}},$


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


$\forall x,y:\neg\;(x<y\;\wedge\;y<x)$


$\tfrac{n!}{k!\,(n-k)!}$


$O(nk\log k)$


$x,y\in\,\textbf{Q}$


$\delta(x)=\frac{\int_{-\infty}^{\infty}{\theta f(x_{1}-\theta,\dots,x_{n}-\theta)d\theta}}{\int_{-\infty}^{\infty}{f(x_{1}-\theta,\dots,x_{n}-\theta)d\theta}}.$


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


$x-1-\frac{1}{2}-\frac{1}{4}-\frac{1}{5}-\frac{1}{6}-\frac{1}{9}-\cdots=1$


$\Psi_{1}=D\beta-\delta\varepsilon-(\alpha+\pi)\sigma-(\bar{\rho}-\bar{\varepsilon})\beta+(\mu+\gamma)\kappa+(\bar{\alpha}-\bar{\pi})\varepsilon\,,$


$\alpha_{\tau\tau}=-\frac{1}{2}\lambda^{2}e^{4\alpha},\ \beta_{\tau\tau}=\gamma_{\tau\tau}=\frac{1}{2}\lambda^{2}e^{4\alpha}$


$\sum_{n=0}^{\infty}{\left(\frac{(-1)^{n}}{2n+1}\right)}^{1}=\frac{1}{1}-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\frac{1}{9}-\cdots=\arctan{1}=\frac{\pi}{4}\!$


$f_{Y}(y)=\begin{cases}\tfrac{1}{2},&\,\text{if }y=1,\\ \\ \tfrac{1}{2},&\,\text{if }y=0,\\ \end{cases}$


$\mathfrak{P}\mid\mathfrak{p}$


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


$O(n\log{n})$


$R=(A+I)^{n}$


$H=\begin{bmatrix}H_{11}&H_{12}\\ H_{12}^{\ast}&H_{22}\end{bmatrix}$


$n$


$\underbrace{111\cdots 111}_{N}0$


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


$a=\begin{cases}a+\epsilon,&\,\text{if }a=0\\ a,&\,\text{if }a>0\end{cases}$


$1 + 1 − 1 + 1 + 1=3$


$N$


$\sqrt{11}=3+\cfrac{1}{3+\cfrac{1}{6+\cfrac{1}{3+\cfrac{1}{6+\cfrac{1}{3+\ddots}}}}}$


$\sigma_{x}=\frac{\partial^{2}\Phi_{yy}}{\partial z\partial z}+\frac{\partial^{2}\Phi_{zz}}{\partial y\partial y}-2\frac{\partial^{2}\Phi_{yz}}{\partial y\partial z}$


$h_{i}=\left|\frac{\partial{r}}{\partial q^{i}}\right|$


$0\rightarrow U\rightarrow V\rightarrow W\rightarrow 0$


$\pm 17.8455995405\ldots$


$\cot\omega=\cot\alpha+\cot\beta+\cot\gamma.\,$


$\prod_{n=1}^{\infty}\left(\frac{4n^{2}}{4n^{2}-1}\right)=\frac{2}{1}\cdot\frac{2}{3}\cdot\frac{4}{3}\cdot\frac{4}{5}\cdot\frac{6}{5}\cdot\frac{6}{7}\cdot\frac{8}{7}\cdot\frac{8}{9}\cdots$


$\scriptstyle B_{1}=-{1\over 2}$


$-Ay$


$\lambda(n)=\begin{cases}\;\;\phi(n)&\,\text{if }n=2,3,4,5,7,9,11,13,17,19,23,25,27,\dots\\ \tfrac{1}{2}\phi(n)&\,\text{if }n=8,16,32,64,\dots\end{cases}$


$\mathcal{|}a|=\begin{cases}a,&\textrm{if }a^{(p-1)/2}\equiv 1\bmod{p},\\ -a,&\textrm{if }a^{(p-1)/2}\equiv-1\bmod{p}.\end{cases}$


$\pi=\cfrac{4}{1+\cfrac{1^{2}}{3+\cfrac{2^{2}}{5+\cfrac{3^{2}}{7+\ddots}}}}=4-1+\frac{1}{6}-\frac{1}{34}+\frac{16}{3145}-\frac{4}{4551}+\frac{1}{6601}-\frac{1}{38341}+-\cdots$


$M=-6+12/\sqrt{2}-8/\sqrt{3}+6/2-24/\sqrt{5}+\cdots=-1.74756\dots.$


$n+p\rightarrow d+\gamma$


$\displaystyle\sum_{n}\frac{s_{n}(x)}{n!}t^{n}=\left(\frac{1+t}{1-t}\right)^{x}(1-t)^{-1}$


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


$\ C_{M}=-0.5\pi(A_{0}+A_{1}-A_{2}/2)$


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


$\frac{\Gamma(r+k)}{k!\,\Gamma(r)}\,p^{r}\,(1-p)^{k}\,$


$d\alpha=\beta\wedge\gamma,\,\,d\beta=\gamma\wedge\alpha,\,\,d\gamma=\alpha\wedge\beta.$


$\Phi(z,s,\alpha)=\sum_{n=0}^{\infty}\frac{z^{n}}{(n+\alpha)^{s}}.$


$\mu(x+y)+\mu(x)+\mu(y)\equiv\lambda(x,y)\;\;(\mathop{{\rm mod}}2)\;\forall\,x,y\in H_{k}(M;\mathbb{Z}_{2})$


$\int_{-\infty}^{\infty}w(\tau)\,d\tau=1.$


$a(n)=\begin{cases}\frac{2^{n+1}-2}{3},&\,\text{when }n\,\text{ is even,}\\ \frac{2^{n+1}-1}{3},&\,\text{when }n\,\text{ is odd.}\end{cases}$


$\begin{aligned}\displaystyle y&\displaystyle=mx+b\\ \displaystyle y&\displaystyle=\frac{(\Delta y)}{(\Delta x)}x+b\\ \displaystyle(\Delta x)y&\displaystyle=(\Delta y)x+(\Delta x)b\\ \displaystyle 0&\displaystyle=(\Delta y)x-(\Delta x)y+(\Delta x)b\end{aligned}$


$\begin{aligned}\displaystyle V_{abc}&\displaystyle=\begin{bmatrix}V_{0}\\ V_{0}\\ V_{0}\end{bmatrix}+\begin{bmatrix}V_{1}\\ \alpha^{2}V_{1}\\ \alpha V_{1}\end{bmatrix}+\begin{bmatrix}V_{2}\\ \alpha V_{2}\\ \alpha^{2}V_{2}\end{bmatrix}\\ &\displaystyle=\begin{bmatrix}1&1&1\\ 1&\alpha^{2}&\alpha\\ 1&\alpha&\alpha^{2}\end{bmatrix}\begin{bmatrix}V_{0}\\ V_{1}\\ V_{2}\end{bmatrix}\\ &\displaystyle=\,\textbf{A}V_{012}\end{aligned}$


$J=\begin{bmatrix}\frac{\partial u}{\partial u^{\prime}}&\frac{\partial u}{\partial v^{\prime}}\\ \frac{\partial v}{\partial u^{\prime}}&\frac{\partial v}{\partial v^{\prime}}\end{bmatrix}.$


$\forall x\forall y\forall z[(Rxy\land Ryz)\rightarrow Rxz]$


$\forall X\subseteq U_{p}[CUM_{p}(X)\Leftrightarrow\exists x,y[X(x)\,\wedge\,X(y)\,\wedge\,\neg(x=y)]\;\wedge\;\forall x,y[X(x)\,\wedge\,X(y)\Rightarrow X(x\,\oplus\,y)]]$


$\mathbf{R}(\alpha,\beta,\gamma)=\begin{pmatrix}\cos\alpha&-\sin\alpha&0\\ \sin\alpha&\cos\alpha&0\\ 0&0&1\end{pmatrix}\begin{pmatrix}\cos\beta&0&\sin\beta\\ 0&1&0\\ -\sin\beta&0&\cos\beta\\ \end{pmatrix}\begin{pmatrix}\cos\gamma&-\sin\gamma&0\\ \sin\gamma&\cos\gamma&0\\ 0&0&1\end{pmatrix}$


$A\cup B=\{1,...,N\}$


$a^{2}=\left[\frac{\partial p}{\partial\rho}\right]_{S}.$


$\mathbf{A}=\begin{bmatrix}\frac{\partial F_{1}(\vec{q})}{\partial p_{1}}&\frac{\partial F_{1}(\vec{q})}{\partial p_{2}}&\cdots&\frac{\partial F_{1}(\vec{q})}{\partial p_{n}}\\ \frac{\partial F_{2}(\vec{q})}{\partial p_{1}}&\cdots&\frac{\partial F_{2}(\vec{q})}{\partial p_{n-1}}&\frac{\partial F_{2}(\vec{q})}{\partial p_{n}}\\ \vdots&\frac{\partial F_{j}(\vec{q})}{\partial p_{i}}&\vdots&\vdots\\ \frac{\partial F_{m}(\vec{q})}{\partial p_{1}}&\frac{\partial F_{m}(\vec{q})}{\partial p_{2}}&\cdots&\frac{\partial F_{m}(\vec{q})}{\partial p_{n}}\\ \end{bmatrix}\!$


$0\rightarrow K\rightarrow X\rightarrow P^{\prime}\rightarrow 0,$


$0\rightarrow\mathbb{R}\rightarrow X\rightarrow V\rightarrow 0.\,\!$


$\forall x,y\in A,f(x)=f(y)\Rightarrow x=y.$


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


$\forall x,y.P(x,y)\vee Q(f(x))$


$-\Lambda\hbar$


$A\xrightarrow{\ \ p\ \ }B$


$0\to VP\to TP\xrightarrow{d\pi}\pi^{*}TM\to 0$


$H(f)=\begin{bmatrix}\frac{\partial^{2}f}{\partial x^{2}}&\frac{\partial^{2}f}{\partial x\,\partial y}\\ \frac{\partial^{2}f}{\partial y\,\partial x}&\frac{\partial^{2}f}{\partial y^{2}}\end{bmatrix}.$


$A=R\oplus V,$


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


$LPM/\sqrt{\,}\text{bar}$


$L(F)=\frac{\int_{0}^{F}x(F_{1})\,dF_{1}}{\int_{0}^{1}x(F_{1})\,dF_{1}}$


$|a|=\begin{cases}a,&\mbox{if }~{}a\geq 0\\ -a,&\mbox{if }~{}a\leq 0\end{cases}\;$


$g(s)=\sqrt{2/\pi}\int_{0}^{\infty}(st)^{1/2}\,K_{\nu}(st)\,f(t)\;dt,$


$\nabla\times\mathbf{B}=\mu\mathbf{J}+\mu\epsilon\frac{\partial\mathbf{E}}{\partial t}.$


$\gamma=\tanh\left(\frac{ze\psi_{0}}{4kT}\right)$


$\mu=\left(\frac{\partial E}{\partial N}\right)_{Z},$


$B=\alpha_{1}\alpha_{2}+\alpha_{1}\beta_{1}+\alpha_{1}\beta_{2}+\alpha_{2}\beta_{1}+\alpha_{2}\beta_{2}+\beta_{1}\beta_{2}\,$


$\mathrm{{}^{238}_{\ 92}U\ \xrightarrow{(n,\gamma)}\ ^{239}_{\ 92}U\ \xrightarrow[23.5\ min]{\beta^{-}}\ ^{239}_{\ 93}Np\ \xrightarrow[2.3565\ d]{\beta^{-}}\ ^{239}_{\ 94}Pu}$


$\langle\tau\rangle\equiv\int_{0}^{\infty}dt\,e^{-\left({t/\tau_{K}}\right)^{\beta}}={\tau_{K}\over\beta}\Gamma({1\over\beta})$


$=\lim_{h\to 0}\frac{f(x+2h)-2f(x+h)+f(x)}{h^{2}},$


$\underbrace{a\cdots a}_{n\,\text{ factors}},$


$p=\sqrt{\frac{ab^{2}-a^{2}b-ac^{2}+bd^{2}}{b-a}},$


$e^{2}=1+\cfrac{4}{0+\cfrac{2^{2}}{6+\cfrac{2^{2}}{10+\cfrac{2^{2}}{14+\ddots\,}}}}=7+\cfrac{2}{5+\cfrac{1}{7+\cfrac{1}{9+\cfrac{1}{11+\ddots\,}}}}$


$\begin{aligned}\displaystyle\mathrm{P}\{d=1\}&\displaystyle=\frac{1}{n}\\ \displaystyle\mathrm{P}\{d=k\}&\displaystyle=\frac{1}{k(k-1)}\qquad(k=2,3,\dots,n).\end{aligned}$


$\alpha=\left(\frac{D}{R}\right)\left(\frac{\partial f}{\partial y}\right)$


$k=\frac{2\pi}{\frac{1}{r_{AM}}-\frac{1}{r_{BM}}-\frac{1}{r_{AN}}+\frac{1}{r_{BN}}}$


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


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


$f(p)=\frac{(n+1)!}{k!(n-k)!}p^{k}(1-p)^{n-k}\,\text{ for }0\leq p\leq 1$


$\,\text{IP}=\Bigg(\frac{ID}{MT}\Bigg)$


$\begin{bmatrix}\dfrac{dx_{1}^{*}}{dp_{1}}\\ \dfrac{dx_{2}^{*}}{dp_{1}}\\ \dfrac{dy_{2}^{*}}{dp_{1}}\end{bmatrix}=\begin{bmatrix}\dfrac{\partial^{2}U_{x}}{\partial x_{1}\partial x_{1}}&\dfrac{\partial^{2}U_{x}}{\partial x_{1}\partial x_{2}}&\dfrac{\partial^{2}U_{x}}{\partial x_{1}\partial y_{2}}\\ \dfrac{\partial^{2}U_{x}}{\partial x_{1}\partial x_{2}}&\dfrac{\partial^{2}U_{x}}{\partial x_{2}\partial x_{2}}&\dfrac{\partial^{2}U_{x}}{\partial y_{2}\partial x_{2}}\\ \dfrac{\partial^{2}U_{y}}{\partial x_{1}\partial y_{2}}&\dfrac{\partial^{2}U_{y}}{\partial x_{2}\partial y_{2}}&\dfrac{\partial^{2}U_{y}}{\partial y_{2}\partial y_{2}}\end{bmatrix}^{-1}\begin{bmatrix}-\dfrac{\partial^{2}U_{x}}{\partial p_{1}\partial x_{1}}\\ -\dfrac{\partial^{2}U_{x}}{\partial p_{1}\partial x_{2}}\\ -\dfrac{\partial^{2}U_{y}}{\partial p_{1}\partial y_{2}}\end{bmatrix}$


$\langle Tx\mid y\rangle_{2}=\left\langle x\mid T^{*}y\right\rangle_{1},\qquad x\in D(T).$


$B_{0}^{\prime}=\left(\frac{\partial B}{\partial P}\right)_{P=0}$


$q_{xx}=\frac{\sum(x-\bar{x})^{2}I(x,y)}{\sum I(x,y)}$


$\mathrm{{}^{238}_{\ 92}U\ +\ ^{1}_{0}n\ \longrightarrow\ ^{239}_{\ 92}U\ \xrightarrow[23.5\ min]{\beta^{-}}\ ^{239}_{\ 93}Np\ \xrightarrow[2.3565\ d]{\beta^{-}}\ ^{239}_{\ 94}Pu}$


$H_{n}=\frac{1}{n!}\left[{n+1\atop 2}\right].$


$\log_{2}\!\left(\frac{1}{2}\right)=-1,\,$


$symbol\omega=(\dot{\alpha}\sin\beta\sin\gamma+\dot{\beta}\cos\gamma)I+(\dot{\alpha}\sin\beta\cos\gamma-\dot{\beta}\sin\gamma)J+(\dot{\alpha}\cos\beta+\dot{\gamma})K$


$\ln 2=\frac{1}{1}-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5}-\cdots.$


$-\otimes A$


$\sigma=\sqrt{\cfrac{2E\gamma}{\pi a}}$


$k_{0}\ \stackrel{\mathrm{def}}{=}\ \sqrt{\frac{3e^{2}\rho}{2\epsilon_{0}E_{F}}}=\sqrt{\frac{me^{2}k_{f}}{\epsilon_{0}\pi^{2}\hbar^{2}}}$


$-\mathrm{i}\nabla$


$\sigma^{2}=\int_{0}^{\infty}(t-\bar{t})^{2}\cdot E(t)\,dt$


$M(a,b,z)=\sum_{n=0}^{\infty}\frac{a^{(n)}z^{n}}{b^{(n)}n!}={}_{1}F_{1}(a;b;z),$


$\nabla\times\mathbf{E}=\hat{\mathbf{k}}\times\mathbf{E}_{0}f^{\prime}\left(\hat{\mathbf{k}}\cdot\mathbf{x}-c_{0}t\right)=-\frac{\partial\mathbf{B}}{\partial t}$


$\,\text{Concentration}_{t}=\sqrt{\frac{\sum_{i=1}^{N_{t}}(S_{it})^{2}-\frac{1}{N_{t}}}{1-\frac{1}{N_{t}}}}$


$t_{n}=\sum_{m=0}^{\infty}\frac{G_{mn}}{(m+1)(m+2)}.$


$\begin{pmatrix}\bar{x}_{1}\\ \bar{x}_{2}\\ \bar{x}_{3}\end{pmatrix}=\begin{pmatrix}\frac{\partial\bar{x}_{1}}{\partial x_{1}}&\frac{\partial\bar{x}_{1}}{\partial x_{2}}&\frac{\partial\bar{x}_{1}}{\partial x_{3}}\\ \frac{\partial\bar{x}_{2}}{\partial x_{1}}&\frac{\partial\bar{x}_{2}}{\partial x_{2}}&\frac{\partial\bar{x}_{2}}{\partial x_{3}}\\ \frac{\partial\bar{x}_{3}}{\partial x_{1}}&\frac{\partial\bar{x}_{3}}{\partial x_{2}}&\frac{\partial\bar{x}_{3}}{\partial x_{3}}\end{pmatrix}\begin{pmatrix}x_{1}\\ x_{2}\\ x_{3}\end{pmatrix}$


$y\,f\left({f(z)\over y}\right)=z\,f\left({f(y)\over z}\right)$


$\begin{aligned}\displaystyle p\vee q\\ \displaystyle\neg p\\ \displaystyle\therefore\overline{q\quad\quad\quad}\\ \end{aligned}$


$\iiint_{V}\left[\frac{\partial u_{i}}{\partial t}+\frac{\partial u_{i}u_{j}}{\partial x_{j}}\right]dV=\iiint_{V}\left[-\frac{\partial P}{\partial x_{i}}+\nu\frac{\partial^{2}u_{i}}{\partial x_{j}\partial x_{j}}+f_{i}\right]dV$


$O(mn)=O(n^{3}\log n)$


$\frac{\partial^{2}f}{\partial y\,\partial x}=\frac{\partial}{\partial y}\left(\frac{\partial f}{\partial x}\right)=(f_{x})_{y}=f_{xy}=\partial_{yx}f.$


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


$c_{2}=-\frac{1}{6}\quad;\quad c_{3}=-\frac{1}{108}\quad;\quad c_{4}=\frac{7}{3240}\quad;\quad c_{5}=-\frac{19}{48600}\ \dots$


$l_{x}(y)=xy,x\in\mathfrak{g},y\in U(\mathfrak{g})$


$\begin{bmatrix}t^{\prime}\\ 0\end{bmatrix}=\begin{bmatrix}\gamma&\delta\\ \beta&\alpha\end{bmatrix}\begin{bmatrix}t\\ vt\end{bmatrix},$


$x^{5}+a_{4}x^{4}+a_{3}x^{3}+a_{2}x^{2}+a_{1}x+a_{0}=0\,$


$abc\sqrt{1-\cos^{2}\alpha-\cos^{2}\beta-\cos^{2}\gamma+2\cos\alpha\cos\beta\cos\gamma}$


$A\cap B=\emptyset$


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


$\cos(x+y)=\cos{x}\cos{y}-\sin{x}\sin{y}$


$θ =kn+ 1$


$P-Q_{i}A_{i}=O((x-\lambda_{i})^{\nu_{i}})\qquad$


$J(f,g)=\frac{\int\min(f,g)d\mu}{\int\max(f,g)d\mu},$


$s_{x}=\frac{s_{y}}{|m|}\sqrt{\frac{1}{n}+\frac{1}{k}+\frac{(y_{unk}-\bar{y})^{2}}{m^{2}\sum{(x_{i}-\bar{x})^{2}}}}$


$0\to R\xrightarrow{r-s}X\to Y\to 0$


${V_{1}\choose V_{2}}=\begin{pmatrix}Z_{11}&Z_{12}\\ Z_{21}&Z_{22}\end{pmatrix}{I_{1}\choose I_{2}}$


$e=\sum_{k=0}^{\infty}\frac{3-4k^{2}}{(2k+1)!}$


$r=\frac{1}{Ns^{2}}\sum_{n=1}^{N}(x_{n,1}-\bar{x})(x_{n,2}-\bar{x})$


$x=1+\cfrac{1}{2+\cfrac{1}{2+\cfrac{1}{2+\cfrac{1}{2+\cfrac{1}{2+\ddots}}}}}=\sqrt{2}.\,$