$d(x, z) ≤ d(x, y) + d(y, z)$


$f(x)=\sum_{n=1}^{N}b_{n}\chi_{B_{n}}(x)$


$\lim_{n\to \infty} \frac{1+2\sqrt2+3\sqrt3+\ldots+n\sqrt n}{n^2 \sqrt{n}}= \text{?}$


$|x^2 + y| + |y^2 + z| + |z^2 + x| < |x| + |y| + |z|$


$ \dfrac 1{\sqrt1} + \dfrac{1}{\sqrt2} +....+\dfrac 1{\sqrt{k}} $


$\forall\epsilon\gt0,\exists\,\delta\gt0:\forall x,y\in\mathbb{R},|x-y|\le\delta\implies|f(x)-f(y)|\le\epsilon\tag{1}$


$1.2.20, \operatorname{ord(a)} = n \ldots$


$x^3-3x^2+3x+r, \  \ r\in\mathbb R.$


$y_n = \frac{(x_1 + x_2 + ... + x_n)}{n}$


$E_i=Span\{e_1,e_2, ...., e_i\}$


$\left(\frac{1}{k}\right)^{n - 2}$


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


$\{g, g+N, g+2N, \dots \}$


$A_1\subseteq A_2\subseteq A_3\subseteq\cdots\text{ and }A:=\bigcup_{n=1}^{\infty}A_n$


$\tan(\angle\text{A})=\frac{\text{a}}{\text{b}}\space\space\wedge\space\space\tan(\angle\text{B})=\frac{\text{b}}{\text{a}}$


$f(K)⊂[f(a),f(b)]$


$\lim_{n \rightarrow \infty} \sum_{i=1}^\infty |x^{(n)}_i| = 0$


$
 \sum_{n = 1}^{\infty} e^{-n^2 \pi x} = \frac{1}{2} \sum_{n \in \mathbb{Z}} e^{-n^2 \pi x } - \frac{1}{2} := \frac{1}{2} f(x) - \frac{1}{2}.
 $


$e_i=c_1e_1'+....+c_ie_i'+........c_ne_n' \ $


$d=ax+by,$


$2x \sqrt{1-x^2} = 2\sin(t) \cos(t) = \sin(2t)$


$\sum_{n=1}^{\infty}\frac{n^{11}}{e^{n\pi}-1}-16^3\sum_{n=1}^{\infty}\frac{n^{11}}{e^{4n\pi}-1}=\frac{691}{16}$


$f(n) = \frac{(n+3)(n+2)(n+1)(n)(n-1)!}{24(n-1)!} = \frac{(n+3)(n+2)(n+1)(n)}{24}$


$\forall \epsilon > 0, \ \exists \ \delta > 0: \ 0 < |x - a| < \delta \implies |f(x) - l| < \epsilon$


$\frac{40}{\sqrt{69}}\approx4.82\text{ ft/s}$


$p^n - p^{n-1}$


$\int_{-\infty}^\infty \psi(t)\cos(t)\; dt = \int_{-\infty}^\infty \psi(t) \sin(t)\; dt = 0$


$Y,Y,Y,Y,Y,Y$


$P(a) \rightarrow P(a+1)$


$ F(x):=\int_a^xf(t)dt $


$\lfloor{\frac x m}\rfloor=\lfloor\frac n m\rfloor$


$\vartheta(z) = e^{p(z)}$


$0=\lim_{x\to \infty}f'(x)=\lim_{x\to \infty}f'(c_x) =\lim_{x\to \infty}[f(x)-f(x+1)] $


$P\oplus Q=F$


$f'(x) = \frac{5^x}{5^x + 1} ~~~~~~~~~~~ \to ~~~~~ f(x) = \frac{\ln(5^x + 1)}{\ln(5)}$


$f(x)=\frac{e^{-x}(x+e^{-x-1})}{(1-e^{-x-1})^2}$


$|f(z)|<K|\phi (z)|$


$\chi_{\mathcal{A}}=(x-1)(x-(n-1))^{n-1}$


$I_n=\int_0^{\frac {\pi}{2}} \frac {\sin ((2n+1)x)dx}{\sin x}$


$p_1\times p_2=0$


$ \sum_{n=-\infty}^\infty e^{i n x} = 2 \pi \sum_{n=-\infty}^\infty \delta(x - 2 \pi n) $


$f(x) = \frac{4^x}{x^3}$


$a\,b=|BD|$


$\,\displaystyle \left(1+\frac{x}{s}\right)^s\approx e^x$


$\tilde{\mathbb{R}}_{\infty} \subset \tilde{\mathbb{R}}_{\infty}^{\tilde{\mathbb{R}}_{\infty}}$


$F(x) = \int_a^{x^2} f(t) \, dt - \int_a^{x} f(t) \, dt$


$\gcd(a,\,a')$


$x = \{ \{ x \} \}$


$\int(2\pi y ds)$


$x R y \Rightarrow yRx$


$f_X(x)=\int_{0}^{x} 8xy \, dy$


$p\in V\subseteq\overline V\subseteq U$


$I_n = \int_{0}^{\pi} \frac{\sin nx}{(1+2^x)\sin x}dx + \int_{0}^{\pi} \frac{\sin nx}{(1+2^{-x})\sin x}dx =\int_{0}^{\pi} \frac{\sin nx}{\sin x}dx$


$f^{-1}(f(x) + f(y)) = \sqrt{x^2 + y^2}$


$ \implies A_0 = \frac{1}{2\pi} \int_{0}^{2\pi} (2\pi x -x^{2}) dx $


$P(14)\implies P(18)$


$\sum_{k=1}^n\frac1{k^p}=\frac1{\Gamma(p)}\int_0^\infty\frac{1-e^{-nx}}{e^x-1}x^{p-1}~{\rm d}x$


$[-R,R] \cup C_R$


$ 1 - P(E^c \cup F^c)  = 1 -[P(E^c) + P(F^c)].$


$N = P \oplus Q$


$d(x,y)=||x||+||y||$


$G = \gamma_1(G) \ge \gamma_2(G) \ge \cdots,$


$ \frac{x_1 + x_2 + \dots + x_n}{n} \ge \sqrt[n]{x_1 x_2 \dots x_n} $


$1< \left(\frac{1}{n-2}+1+\frac{1}{(n-1)^2} \right)^{\frac{1}{n-2}}<1+\frac{1}{(n-2)^2}+\frac{1}{(n-2)(n-1)^2}$


$k^2\left\{ \begin{array}{c}n\\k\end{array} \right\}^2 +2k\left\{ \begin{array}{c}n\\k\end{array} \right\}\left\{ \begin{array}{c}n\\k-1\end{array} \right\}+\left\{ \begin{array}{c}n\\k-1\end{array} \right\}^2-(k-1)(k+1)\left\{ \begin{array}{c}n\\k-1\end{array} \right\}\left\{ \begin{array}{c}n\\k+1\end{array} \right\}-(k-1)\left\{ \begin{array}{c}n\\k-1\end{array} \right\}\left\{ \begin{array}{c}n\\k\end{array} \right\}-(k+1)\left\{ \begin{array}{c}n\\k-2\end{array} \right\}\left\{ \begin{array}{c}n\\k+1\end{array} \right\}-\left\{ \begin{array}{c}n\\k-2\end{array} \right\}\left\{ \begin{array}{c}n\\k\end{array} \right\}>0$


$F_{ 2n-1 }={ F }_{ n-1 }^{ 2 }+{ F }_{ n }^{ 2 }>{ F }_{ n }^{ 2 }>{ F }_{ n }^{ 2 }-{ F }_{ n-2 }^{ 2 }=F_{ 2n-2 }$


$\|A\|_2=\sqrt{\rho(A^TA)}=\rho\left(\begin{bmatrix}0&A\\A^T&0\end{bmatrix}\right)$


$f\left ( f(x)^{2} \right )=x^{3}f(x).$


$\Rightarrow P(\text{B wins})=P(B \text{ picks a red ball}|\text{A picks a white ball})P(\text{A picks a white ball})=\frac3{11}\frac{9}{12}$


$P(m) - P(m-1)$


$f(x)= 4^x+9^x+25^x$


$\forall \epsilon>0, \exists \delta_{g,\epsilon} >0, |x| \leq \delta \implies |g(x)-g(0)| \leq \epsilon$


$F'(x) = \frac{x + 8}{x^3 - 9}$


$=\left\lfloor \{a\}\lfloor b\rfloor+\{b\}\lfloor a\rfloor+\{a\}\{b\}\right\rfloor\text{ (1)}$


$s_1 \supseteq s_2 \supseteq \dots \supseteq s_n \supseteq s_{n+1} \supseteq\ \dots$


$\int_0^{\infty}f_1(x) \, dx,$


$\hspace{.2 in} e^{a/(a-1)}<\frac{a}{a-1}e^{1/(a-1)}\iff \frac{a-1}{a}<e^{-1}\iff 1-\frac{1}{a}<\frac{1}{e}\iff a<\frac{e}{e-1},\;$


$c = \frac{a+b}{a b-1}$


$\large{\int_0^\infty \frac{1}{(x^2+1)(x^a+1)}\ dx=\frac{\pi}{4}}$


$d(x,y) = d(A,B)$


$(1, 1, 1, z)$


$\Delta(\gamma\alpha, \gamma\beta) = \gamma\alpha\gamma'\beta' - \gamma'\alpha'\gamma\beta = \gamma\gamma'(\alpha\beta' - \alpha'\beta) = N(\gamma)\Delta(\alpha, \beta)$


$[a,c - \gamma]\subset [a,x] \subset X$


$|f(z^2)| \le |f(z)|$


$X=c_1e_1+\dots c_ne_n$


$xa+yb=d$


$TF\{\sum_{n=-\infty}^{\infty} \delta (t-nT) = \sum_{n=-\infty}^{\infty} e^{-jnT\omega} = \dfrac{2\pi}{T} \sum_{n=-\infty}^{\infty} \delta (\omega-n\dfrac{2 \pi n}{T})$


$\Sigma^+$


$\lim_{n\to \infty} \frac {1 + \sqrt[3]{2} + \sqrt[3]{3} +\cdots+ \sqrt[3]{n}}{n^{4/3}}$


$\displaystyle \sum_{n=-\infty}^{\infty}c_ne^{inx}=\sum_{n=-\infty}^{\infty}c_n\cos(nx)+\sum_{n=-\infty}^{\infty}ic_n\sin(nx)$


$a_1\cdot \dots \cdot a_n$


$\delta (a - x) = \delta(-(x - a)) = \delta(x - a)$


$kx\frac{d^2y}{dx^2}=\sqrt{1+\left(\frac{dy}{dx}\right)^2}$


$\lim_{n\rightarrow \infty}\sqrt{2+\sqrt{2+\sqrt{...+\sqrt{2}}}}$


$z^n =  re^{i\varphi} \implies z_k = r^{\frac{1}{n}}e^{i\left( \frac{\varphi}{n} + \frac{2\pi k}{n}\right)} \qquad k=0,\dots,n-1$


$\left(y+\frac{dy}{dx}\right)^2+\frac{dy}{dx}=0$


$\tan\left(\frac{2\tan^{-1}\left(\frac{1}{2}\right)}{2}\right)=\tan\left(\tan^{-1}\left(\frac{1}{2}\right)\right)=\frac{1}{2}$


$\int_{-\pi/2}^{\pi/2} \frac{\sin^{2012}{x}}{\left(1+ \alpha^x\right)\left(\sin^{2012} {x}+\cos^{2012}{x}\right)}\;{dx} $


$\mathbb{E}([X + Y]^3) = \mathbb{E}(X^3) + \mathbb{E}(Y^3)$


$Cov(X,Y)=Cov(X,g(Y))$


$\alpha < \alpha^{\omega}$


$\lim_{n \to \infty}{\displaystyle\sum_{k=1}^{n} \frac{1}{k^2}} $


$y-1/|x| = 8 +1/3.$


$(\Bbb{R}[X])^*$


$E[1_{X\ge x}]=P[X\ge x]$


$\diamond ^+$


$\int_0^\infty f(t)h(t)dt = 0$


$ u = \frac{\gamma(b) - \gamma(a)}{\|\gamma(b) - \gamma(a)\|} $


$\lim\limits_{n\to\infty} f(x_n)=\liminf\limits_{x\to 0} f(x)$


$-\frac{945\ 2^{n+6} \tau ^5 \,    _{12}F_{11}\left(\frac{n}{2},\frac{n}{2},\frac{n}{2},\frac{n}{2},\frac{n}{2},\frac{n}{2},\frac{n}{2},\frac{n}{2},\frac{n}{2},\frac{n}{2},\frac{n}    {2},n;\frac{n}{2}+1,\frac{n}{2}+1,\frac{n}{2}+1,\frac{n}{2}+1,\frac{n}{2}+1,\frac{n}{2}+1,\frac{n}{2}+1,\frac{n}{2}+1,\frac{n}{2}+1,\frac{n}{2}+1    ,\frac{n}{2}+1;-1\right)}{\pi ^5 n^{11}}+\frac{105\ 2^{n+5} \tau ^4 \,    _{10}F_9\left(\frac{n}{2},\frac{n}{2},\frac{n}{2},\frac{n}{2},\frac{n}{2},\frac{n}{2},\frac{n}{2},\frac{n}{2},\frac{n}{2},n;\frac{n}{2}+1,\frac{n    }{2}+1,\frac{n}{2}+1,\frac{n}{2}+1,\frac{n}{2}+1,\frac{n}{2}+1,\frac{n}{2}+1,\frac{n}{2}+1,\frac{n}{2}+1;-1\right)}{\pi ^4 n^9}-\frac{15\ 2^{n+4}    \tau ^3 \,    _8F_7\left(\frac{n}{2},\frac{n}{2},\frac{n}{2},\frac{n}{2},\frac{n}{2},\frac{n}{2},\frac{n}{2},n;\frac{n}{2}+1,\frac{n}{2}+1,\frac{n}{2}+1,\frac{    n}{2}+1,\frac{n}{2}+1,\frac{n}{2}+1,\frac{n}{2}+1;-1\right)}{\pi ^3 n^7}+\frac{3\ 2^{n+3} \tau ^2 \,    _6F_5\left(\frac{n}{2},\frac{n}{2},\frac{n}{2},\frac{n}{2},\frac{n}{2},n;\frac{n}{2}+1,\frac{n}{2}+1,\frac{n}{2}+1,\frac{n}{2}+1,\frac{n}{2}+1;-1    \right)}{\pi ^2 n^5}-\frac{2^{n+2} \tau  \,    _4F_3\left(\frac{n}{2},\frac{n}{2},\frac{n}{2},n;\frac{n}{2}+1,\frac{n}{2}+1,\frac{n}{2}+1;-1\right)}{\pi  n^3}+\frac{2^{n+1} \,    _2F_1\left(\frac{n}{2},n;\frac{n+2}{2};-1\right)}{n}$


$\mathscr{A}=\big\{\{x\}:x\in Y\big\}$


$\implies\tan\theta=\dfrac x5$


$\lim_{x\to5^+}f'(x)= \lim_{x\to5^-}f'(x)=1$


$\int_{1}^{10}\frac{1}{(x^{2}+1)^{10}}dx\approx .00009843725636$


$f(a + b) = f(a) \cdot f(b)$


$P : \overline{\mathbb{R}} \times \overline{\mathbb{R}} \to \overline{\mathbb{R}}$


$\left \| A \right \| _2=\sqrt{\lambda_{\text{max}}(A^{^*}A)}$


$\underset{1 \times 2}{x^+}$


$\alpha (\gamma) = (\gamma - \gamma , \gamma -\gamma, \dots) = (0,0, \dots ) = 0 \gamma$


$y'''+y'=y'+y$


$ |f(z)| \le (1+|z|) |g(z)|$


$\forall\varepsilon\gt 0,\,\exists \delta\gt 0 :|h(x)-h(y)|\lt\varepsilon$


$\dfrac{\sqrt{1-x^2}}{\sqrt{1-x^2}+\sqrt{1 - 2x^2}} + \dfrac{2\sqrt{x}}{1+\sqrt{1 - 2x^2}}> 0$


$\mathbb{R}^{n+m}=\mathbb{R}^{n}\oplus \mathbb{R}^{m}$


$\int_0^\infty f'(x)\,dx$


$e^x - (1+\frac{x}{n})^n \rightarrow 0$


$\begin{align} \left|e^x-\left(1+\frac xn\right)^n\right|&=\left|e^x-e^{n\log\left(1+\frac xn\right)}\right|\\\\ &\le \left|e^x-e^{\frac{x}{1+x/n}}\right|\\\\ &=e^x\,\left|1-e^{-x^2/(x+n)}\right|\\\\ &\le e^x\,\left|\frac{x^2}{x+n}\right|\\\\ &\le e^{b}\frac{|\max^2(a,b)|}{n+a}\\\\ &<\epsilon \end{align}$


$\int_0^{+\infty}f(x) = \int_0^{1}f(x) + \int_1^{+\infty}f(x)$


$T(n) = \dfrac{n(n+1)}{2}$


$\left\lfloor{x\over 2^n}\right\rfloor =\left\lfloor{\lfloor x\rfloor\over 2^n}\right\rfloor\ .\tag{1}$


$\sum_{k=1}^{\infty} \sum_{n=1}^{\infty} \frac{1}{n^2k^2(n+k)^2}=  \frac{1}{3}\zeta(6)$


$N! + 2, ..., N! + N$


$ \sum_{n \geq 1} \frac{n}{n^3 + 1} < \sum_{n \geq 1} \frac{n}{n^3} = \sum_{n \geq 1} \frac{1}{n^2}.$


$f(n) = n^2\!+3n+2\,$


$\sum_{k}|\langle f,\phi_p\rangle |^2= A||f||^2$


$X=\overline{X}={\mathbb R}^2$


$\frac{1}{2^{N-2}} < \varepsilon$


$ \int \frac{\sin(x)}{1+\sin^2(x)}\, dx $


$\int_1^\infty\frac{\sin^2(x)}{x}\mathrm d x$


$ax+by = d. (\dagger)$


$\frac{ax + b}{cx} = n$


$\implies\sum_{k=0}^n\cos\dfrac{2\pi k}n=0+\cos\dfrac{2\pi n}n=?$


$ 6^{x+1}-6^x = 3^{x+4}-3^x $


$\displaystyle J_n=\int_0^{\pi}\left(\frac{\sin nx}{\sin x}\right)^2\mathrm{d}x$


$|\theta-\sin \theta| < |2\tan\frac\theta 2 - \sin\theta|$


$\frac{1}{a} \biggl( 1 - \frac{x-a}{a} + \frac{(x-a)^2}{a^2} - \frac{(x-a)^3}{a^3} \pm \cdots \biggr) $


$\mathbb{E}|X-Y| \leq 1/2$


$E(X^2-2XE(X)+E(X)^2)= E(X^2)-2E(X)^2+E(X)^2= E(X^2)-E(X)^2.$


$"1+2+3+4+5+...=-{1\over 12}."$


$\frac{X_1+X_2+ \ldots +X_n}{n} \to 0$


$ \int_{\gamma -\ic\infty}^{\gamma + \ic\infty} {s\expo{st} \over \root{\pars{s + a}^{3}}}\,{\dd s \over 2\pi\ic} = \pars{1 + 2a\,\partiald{}{a}}{\expo{-at} \over \root{\pi t}} = {1 \over \root{\pi}}\,{\pars{1 - 2at}\expo{-at} \over \root{t}} $


$\lim_{n\to\infty} \inf(x_n) \leq \lim_{k\to\infty} \inf(x_{n_k})$


$\begin{cases} A = 1 \\ B = -6 \\ C = -30 \\ D = -48 \end{cases}$


$\sum_{k=0}^n \cos\left(\frac{k\pi}{n}\right) = 0$


$\sum\limits_{k=1}^n\frac k{(1+r)^k}$


$\cdot^+$


$\rho(A)< \|A\|_{\infty}$


$x^3, x^7, x^9$


$\forall \epsilon \gt 0 \exists \delta \gt 0  : |x-a| \lt \delta \implies |f(x)-f(a)|\lt \epsilon$


$\infty\d a=\dfrac 1 a$


$[\gamma_1,...,\gamma_{\ell}] \in \Pi(\gamma,\ell)$


$ \displaystyle \int_{-\infty}^{+\infty} \delta'(x-3)e^{x^2}dx $


$\sum_{m=1}^{\infty} \sum_{n=1}^{\infty} \frac{1}{m^2+n^2}.$


$5 \mid n^5-n$


$ 1+2\sum_{n=1}^\infty e^{-\pi n^2t} =\frac1{\sqrt{t}}+\frac2{\sqrt{t}}\sum_{n=1}^\infty e^{-\pi n^2/t}\tag{14} $


$\int_0^{\infty}x dF(x)$


$c = {a \over {a + b} }$


$\alpha \Omega + \beta \Omega \subset (\alpha+ \beta) \Omega$


$9^x+6^x< 2^{(x+1)}$


$A_n=\{a_1,a_2,\cdots, a_n\}$


$P_X(x) = \frac{1}{a}e^{\frac{-x}{a}}u(x)$


$|f^{(n)}(x)| \leq \alpha^{n-1}|f(x)|$


$(P(s)\implies P(s+1))$


$f_X(x) = F_X'(x) = \theta x^{-(\theta+1)}, \quad x > 1, \; \theta > 0$


$T^{1,0}_e M \subset T_e M \otimes \mathbb{C}$


$d = o( \sqrt{p})$


$\beta b - \alpha a = (\beta + \gamma + \delta) a' - (\alpha + \gamma + \delta) b'\\ \beta b + (\alpha + \gamma + \delta) b' = \alpha a + (\beta + \gamma + \delta) a'\\ \frac{\beta b + (\alpha + \gamma + \delta) b'}{\alpha + \beta + \gamma + \delta} = \frac{\alpha a + (\beta + \gamma + \delta) a'}{\alpha + \beta + \gamma + \delta} = e$


$(2a+b)^2-(2a-b)^2= (a+2b)^2-(a-2b)^2$


$S_1 = 1+2+3+4+5+\cdots = -\frac{1}{12}$


$\lim_{n \rightarrow \infty}\sum_{i=1}^{\infty}\mu_n(E_i)=\sum_{i=1}^{\infty}\lim_{n \rightarrow \infty}\mu_n(E_i)$


$\overline{V} \subset W \subset \overline{W}$


$c = ax + by$


$f(m) = f(n-1) = {n-1 \choose 2}$


$O(nlog_2 (b))$


$ L = \lim_{x \to 0^-} f(x) = \lim_{x \to 0^+} f(-x) = \lim_{x \to 0^+} f(x) = L. $


$R_2(n) = n^2 - n + 2$


$\lambda \begin {vmatrix} x & y & 1\\ x_1 & y_1 & 1\\ x_2 & y_2 & 1\end {vmatrix} $


$f'(x)= 9/2 \sqrt{x} (x^{3/2}+1)^2 ->f'(1)=18$


$\sum_{m=1}^\infty \sum_{n=1}^\infty \frac{1}{m^2+n^2}  = \sum_{m=1}^\infty \sum_{n=1}^\infty \frac{1}{(2m)^2+(2n)^2} + \sum_{m=1}^\infty \sum_{n=1}^\infty \frac{1}{(2m-1)^2+(2n)^2} $


$\Delta f=A\Delta x+o(\Delta x)$


$ \forall \epsilon>0 , \exists \delta > 0, if |x -a| < \delta \implies |f(x) - c| < \epsilon $


$x \in V, \overline{V} \subseteq U$


$1/\left|x\right| < 1$


$    \lVert A\rVert_{(1,\infty)} \text{ and } \lVert A\rVert_{(2,\infty)} \text{ and sketch the proof} \text{ ( A is a Matrix Norm)} $


$a=br=ars$


$|{\frac{1}{\gamma_2!} x^{\gamma_2+\alpha} e^{\langle{s,x}\rangle}}| \overset{(1)}{\leq} \frac{1}{\gamma_2!} (\gamma_2+\alpha)! \delta^{-|{\gamma}_2| - |{\alpha}|} e^C = \frac{(\gamma_2+\alpha)!}{\gamma_2!} \delta^{-|{\gamma_2}|} \cdot \delta^{-|{\alpha}|} e^C$


$=\left[\begin{array}{ccc|c}2&-4&4&0\\2&k&-1&3\\0&-1&-1&5\end{array}\right]$


$\frac{(1+n)^{k+1} - n -1}{n}C$


$log_{a} y = b$


$cov(X,Y) = 0)$


$x \to x^3-x-3$


$\zeta(2):=\sum_{k=1}^\infty{1\over k^2}=\sum_{k\geq 1\ {\rm even}}^\infty{1\over k^2}+\sum_{k\geq 1\ {\rm odd}}^\infty{1\over k^2}={1\over4}\zeta(2)+{\pi^2\over 8}\ ,$


$\alpha \text{ and }\gamma \text{ injective (surjective) }\implies \beta \text{ injective (surjective)}.$


$S : \overline{\mathbb{R}} \times \overline{\mathbb{R}} \to \overline{\mathbb{R}}$


$x^* \in (M^\perp)^\perp=M$


$ \mathcal{X} := \{ \{ x \} : x \in X \} $


$\lfloor{\frac{a}{b+1}}\rfloor = \lfloor \frac{a}{b} \rfloor  - 1 $


$rn^{n - r - 1}$


$[X_{p}^{k},X_{p}^{l}] \subset X_{p}^{max(-1,k+l)}$


$\displaystyle -\frac{dx}{dy} = \frac{\sqrt{1-y^2}}{y}$


$ y (x' - x) = y x' - y x = y' x - y x = (y' - y) x $


$E|X+Y|^r \leq E|X|^r+E|Y|^r$


$F(x) = \int_a^x f(x) \, dx$


$ (\forall m<n' \ P(m) \implies P(n'))$


$Var(X) = \mathbb{E}(X-\mathbb{E}(X))^2 = \mathbb{E}(X^2+\mathbb{E}(X)^2-2X*\mathbb{E}(X)) = \mathbb{E}(X^2)+\mathbb{E}(X)^2-2\mathbb{E}(X)^2 = \mathbb{E}(X^2)-\mathbb{E}(X)^2$


$[X, Y] = XY - YX,$


$ f_+'(c) = \lim_{x\to c^+}\frac{f(x) - f(c)}{x - c} = \lim_{x\to c^+}f'(c_x) = \lim_{x\to c}f'(x). $


$F(y) - F(x) = \lim_{k\to \infty} F_{n_k}(y) - F_{n_k}(x) = \lim_{k\to \infty} \int_0^1 f_n(t)\chi_{[x,y]}(t)\,dt = \int_0^1 f(t)\chi_{[x,y]}(t)\,dt,$


$\lim_{N\to \infty} ||\sum _{n=0}^{N} \frac{(if)^{n}}{n!}||\leq e^{r}$


$\sum_{a \in A^+} \frac{1}{a^2} < \sum_{n=1}^{\infty} \frac{1}{n^2}.$


$L\ = {\dfrac{(1+x)^n -1}{x(1+x)^n} }(Monthly\ payment)$


$A = \{\{0\}\}$


$R/M \cong \left( \mathbb{Z}[x]/(x^2-2) \right)/(5)$


$y=x^2-4x-5 \\\frac{dy}{dx}=\operatorname{slope}(x)=2x-4\\\text{ @ (-2, 7):}\\\frac{dy}{dx}\Big|_{x=-2,\;y=7}=-8=m\\\text{"normal" = perpendicular (}\perp\text{)}\\m_{\perp\xi}=-\frac1{m_\xi}\\\text{(slope of the line }\xi_\perp\text{, the line perpendicular to } \xi\text{ is the negative reciprocal of }\xi\text{'s slope)}\\\frac{dy_\perp}{dx}\Big|_{x=-2,\;y=7}=\frac18=m_\perp\\y_\perp=m_\perp x+b=\frac{x}8+b\\\text{If they meet at (-2,7):}\\7=\require{cancel}-\cancelto{\frac14}{\frac28}+b \\ \Rightarrow b=\frac{29}{4}$


$A\subseteq U,B\subseteq V$


$f(x)=2b\gamma\left(a-b\frac{1}{2}\left(\gamma-\frac{1}{\gamma}\right)\right)=-b^2\left(\gamma^2-2\frac{a}{b}\gamma-1\right)$


$L = \left(x + \dfrac{d}{dx} \right)$


$\begin{cases}2y+\frac 1y=3\\y=\frac 1y\end{cases}\iff \begin{cases}x=2\\y=1\end{cases}$


$e^x =(1+\frac{x}{n})^n $


$F = E + V$


$(\mathbb F_{p}[x]/(x^m))^{\times}$


$\sqrt1+\sqrt2+\dots+\sqrt n$


$ E(U^s)=\sqrt[3]{\frac{(1+\epsilon)^{s+1}-(1-\epsilon)^{s+1}}{2(s+1)\epsilon}}. $


$|\int_\gamma f | \leq M l(\gamma)$


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


$ K = (\frac{\gamma + ax}{\alpha + ax})^{2a} = (1 + \frac{\gamma -\alpha}{\alpha + ax})^{2a}  $


$\int_{-\pi}^{\pi}\frac {\sin nx}{(1+2^x)\sin x}dx = \int_{0}^{\pi}\frac {(1+2^x)\sin nx}{(1+2^x)\sin x}dx = \int_{0}^{\pi}\frac {\sin nx}{\sin x}dx$


$n! \simeq  \sqrt{2\pi n} \left(\frac{n}{e}\right)^n$


$\#_2 A$


$\sum\limits_{n=1}^\infty\frac{-1}{n^2}\leq\sum\limits_{n=1}^K \frac{-1}{n^2}\leq \sum\limits_{n=1}^K (-1)^n\leq \sum\limits_{n=1}^K\frac{1}{n^2}\leq\sum\limits_{n=1}^\infty \frac{1}{n^2}$


$ds = \sqrt{(dx)^2+(dy)^2}$


$ \{x, y, z\}, \{x, z, y\}, \{y, x, z\}, \{y, z, x\}, \{z, x, y\}, \{z, y, x\}. $


$(k(1) \,\, k(3) \, \, k(5)) \,\,\,\, (k(2) \,\, k(4)) = (2 \,\, 4 \,\, 6) \,\,\, (1\,\, 3)$


$t_x x + (1-t_x)y$


$f_{X,Y}(x,y)= \frac{e^{\frac{-x}{\beta}}}{\beta}\frac{e^{\frac{-y}{\beta}}}{\beta}=\frac1{\beta^2}e^{\frac{-x-y}{\beta}}$


$\mathbb{R}[x]/(x^2+1) \cong \mathbb{R}[x]$


$\det(a_{ij})_{i,j\in H}$


$\lim_{n\to \infty } \mu (A_n)=\mu (X)$


$ \begin{cases} \mu   &= \rho\frac{\sin(\gamma + \phi)}{\sin\gamma},\\ 1-\mu &= \rho\frac{\sin(\gamma-\phi)}{\sin\gamma} \end{cases} \quad\implies\quad 1 = \frac{\rho}{\sin\gamma}(\sin(\gamma + \phi) + \sin(\gamma - \phi)) = 2\rho\cos\phi$


$k^3*1/3 + k^2 + k + 1/3 + k^2/2 + k + 1/2 + k/6 + 1/6 = $


$P_1(h=k) = \frac16, \;\forall\ k \in\{1,2,3,4,5,6\}$


$n! = \sqrt{2\pi{n}}(\frac{n}{e})^n + r_1(n)$


$\left\lfloor\frac{\lfloor x\rfloor}{n}\right\rfloor\ge\left\lfloor\frac{x}{n}\right\rfloor$


$ F'(F^{-1}(x))(F^{-1}(x))' = 1 \implies\\ f(F^{-1}(x))(F^{-1}(x))'  = 1$


$\lim_{x\to-\infty}f'(x)=\infty=\lim_{x\to\infty}f'(x)$


$ \lim_{n\to\infty}\left( \sum_{r=1}^n \dfrac 1 {\sqrt{n^2 + r}} \right) = 1$


$\sin(t) = \frac x2$


$xRy \implies yRx$


$\dfrac{b - 2a\cos{\gamma}  }{a\sin{\gamma}}=\dfrac{\sin{\beta}}{\sin{\alpha}\sin{\gamma}}-\dfrac{2\cos{\gamma}}{\sin{\gamma}}$


$\lim\limits_{n\to\infty} \frac{1}{n} \sum_{k=1}^{n}{\frac{1}{k}}$


$\forall \epsilon > 0, \ \exists \ \delta > 0 \ s.t \ if\ |x-a| < \delta \Rightarrow  |f(x) - f(a)| < \epsilon$


$(x+y,x-y) \le \|x+y\|\|x-y\| \le 1$


$\,\, \forall \epsilon >0\,\,\exists \delta >0 : \forall a,x\in S\,\, |x-a|<\delta \implies |f(x)-f(a)|<\epsilon $


$h(x_n)=\sum y_m$


$n!=\sqrt{2\pi n}\left(\frac{n}{e}\right)^n\left(1+O\left(\frac{1}{n}\right)\right)$


$\vartheta(u,-i\tau)$


$Ax-b, x \in \mathbb{R^n}$


$\displaystyle I_n = \int_0^\pi \frac{\sin 2k x}{\sin x} \mathrm{d}x$


$=3\lim_{N\rightarrow \infty}\sum_{n=1}^{N} \frac{1}{N}-3\lim_{N\rightarrow \infty}\sum_{n=1}^{N}\frac{1}{N+1} -\lim_{N\rightarrow \infty}\sum_{n=1}^{N} \frac{1}{2N+3}+\lim_{N\rightarrow \infty}\sum_{n=1}^{N}\frac{1}{2N+5} $


$ \bbox[lightyellow] {   \left\{ \matrix{   \theta  \in \left\{ {\pi /2 + k\pi  - \alpha ,\;k\pi } \right\} \hfill \cr    \varphi  = \arcsin \left( {\sin \alpha /\sin \left( {\alpha  + \theta } \right)} \right) \hfill \cr    a = A\cos \varphi  \hfill \cr    b = A\sin \varphi  \hfill \cr}  \right. } \tag{2}$


$ \begin{vmatrix} 1 & 1 & 1 \\ x_0 & x_1 & x_2 \\ x_0^2 & x_1^2 & x_2^2 \end{vmatrix}. $


$d=xa+by$


$x^2-ax,x^3-ax\in\mathbb Q$


$\displaystyle \frac{n(n-1)}{2}=m$


$A_1 \wedge A_2 \wedge \ldots \wedge A_n \implies A_n$


$f(x)= \dfrac{2 x}{r^2}$


$\frac{\gamma u}{(1+\gamma)c} = \sqrt{\frac{\gamma-1}{\gamma+1}}$


$\int_\gamma V = \int_0^1 \langle V(\gamma(s)), \gamma'(s)\rangle ds.$


$S=\big\{\langle x,y\rangle:0\le x\le 1\text{ and }-1\le y\le 0\big\}$


$\Bbb R[X]/{\left<X^2+1\right>}$


$\int_0^{2\pi}f^2(t)dt=\int_0^{2\pi}f(t)F'(t)dt$


$Gl(n, F_3)$


$ \begin{pmatrix} 0 & -1 \\ 0 & 0 \end{pmatrix} $


$C_n(X) = 0$


$\int_0^{\infty} g(t)\phi(t)dt\leq \int_0^{\infty} f(t)\phi(t)dt.$


$ f(n) = 2*f(n-1)-f(n-2) + 2 $


$ \sum_{n=1}^{\infty}\mu(A_n)=\lim_{N\to\infty}\sum_{n=1}^N\mu(A_n)\leq \mu(A) $


$\partial, \bar{\partial}$


$d(0,r) \leqslant d(0,y) + d(y,r).$


$\lim_{\epsilon \to 0}\int_0^{2\pi}\lvert\frac{\epsilon e^{i \theta} i d\theta}{\sqrt[\alpha]{\epsilon} e^{i \theta/\alpha}(\epsilon e^{i \theta}+1)}\rvert\le$


$\left[\begin{array}{ccc|c}1&1&k&1\\k&1&1&1\\0&k-1&1-k&0\end{array}\right]$


$F(\gamma (i))=\int_i^{\gamma(i)}f(s)ds=\int_i^{\gamma(i)}f(\gamma_2(s))d\gamma_2(s)=\int_{\gamma_2(i)}^{\gamma_2(\gamma(i))}f(s)ds =F(\gamma_2(\gamma (i)))-F(\gamma_2 (i))$


$g: \begin{cases} x=3s \\ y=0 \\ z=1-4s \end{cases}$


$\sqrt w=\sqrt re^{it/2}$


$m! \approx \sqrt{2 \pi m} \left(\frac{m}{e}\right)^m,$


$n! = \sqrt{2 \pi n}  \left(\frac{n}{e}\right)^n e^{\lambda_n}$


$ a_n = \sqrt {1!\sqrt {2!\cdots\sqrt {n!} } }, \quad n\in\mathbb N. $


$d(x,V_1)=\frac 1 2$


$ \bbox[lightyellow] {   F(x) = {{a_{\,0} \left( {1 - x} \right)^{\,4}  + x\left( {1 + 4x + x^{\,2} } \right)} \over {\left( {1 - x} \right)^{\,5} }}\quad  \Leftrightarrow \quad a_{\,n}  = a_{\,0}  + \left( {{{n\left( {n + 1} \right)} \over 2}} \right)^{\,2} \quad  \Leftrightarrow \quad \left\{ {\matrix{    {a_{\,0}  = a_{\,0} }  \cr     {a_{\,n}  - a_{\,n - 1}  = n^{\,3} }  \cr     {a_{\,n + 1}  - a_{\,n}  = \left( {n + 1} \right)^{\,3} }  \cr   } } \right.  }$


$\int_0^\infty f(x)\,dx<\infty,$


$2 \mid b^2 + b$


$\frac{3}{a}+ \frac{2}{b}=1$


$  T_n = 1+2+\cdots+n=\frac{n(n+1)}{2} $


$f(x;\lambda)= \frac{1}{\lambda}e^\frac{-x}{\lambda}\chi\{x>0\}.$


$=\left\lfloor (\lfloor a\rfloor+\{a\})(\lfloor b\rfloor+\{b\})\right\rfloor-\lfloor a\rfloor\lfloor b\rfloor$


$\sum_{n=-\infty}^\infty \frac{1}{an^2+bn+c}=\frac{1}{a}\sum_{n=-\infty}^{\infty}\frac{1}{\left(n-z_{0}\right)\left(n-z_{1}\right)}.$


$\mathbf Z[i]/(q)\simeq (\mathbf Z[x]/(x^2+1))/q(\mathbf Z[x]/(x^2+1))\simeq  \mathbf (Z/q\mathbf Z)[x]/(x^2+1)$


$n! + 2, \dots, n!+n$


$\lim_{n \rightarrow \infty} \frac{1}{n} \sum_{k=1}^{n-1} \frac{1}{(1 + (k/n))^2}$


$(K[X]/(h))^m$


$x' y z + x y' z + x y z' + x' y z' + x y' z' + x' y' z = x y' + y z' + z x'$


$y_n= \dfrac{x_1+x_2+\cdots+x_n}{n}$


$(X_n(\omega_1))=(1,-1,-1,1,-1,1,-1,\dots)$


$\le_{\overline{\mathbb R}}\;\subseteq \overline{\mathbb R}\times\overline{\mathbb R}$


$\int_0^x f(t)\,dt = -\int_0^y g(t)\,dt = u$


$M=\left(\begin{smallmatrix} 0 & 1 & -1 \\ -1 & 0 & \alpha \\ 2 & -\alpha & 0 \end{smallmatrix}\right)$


$z \in \mathbb{R}[x]/(x^2+1)$


$\tan u = \frac{x}{2},$


$\int_0^{\infty}f(x) dx$


$\tau=\{\emptyset,\mathbb{Z}\}\cup\bigl\{\{a,a+1,a+2,\ldots\}\,|\,a\in\mathbb Z\bigr\}.$


$ds=\sqrt{1+\left( \frac{dy}{dx}\right) ^{2}}$


$z=\sqrt[4]{w}e^{\frac{\pi ki}{2}}$


$ f(x +f(y) + yf(x)) = y +f(x) + xf(y) ,\forall x,y \in \mathbb R$


$ \sqrt{x^2+y^2}[1-a(1-2x)(1-2y)]\to r\left(1-a(1-2r\cos \theta)(1-2r\sin \theta)\right) $


$|f(z)|\leq 2$


$\frac{(1-p)^2 p}{1-(1-p)^3}.$


$\lim\limits_{n\to\infty} f(x_n')=\limsup\limits_{x\to 0} f(x)$


$[\gamma_{n+1}(G),\gamma_l(G)] = [[G,\gamma_n(G)],\gamma_{l}(G)]\leq [\gamma_n(G),[G,\gamma_{l}(G)]][G,[\gamma_{n}(G),\gamma_l(G)]]\leq[\gamma_n(G),\gamma_{l+1}(G)][G,\gamma_{n+l}(G)]\leq \gamma_{n+1+l}(G)$


$\frac{\zeta'(1-s)}{\zeta(1-s)}=log(2)+log(\pi)+\frac{\pi}{2}tan(\frac{\pi s}{2})-\frac{\Gamma'(s)}{\Gamma(s)}-\frac{\zeta'(s)}{\zeta(s)}$


$\exists C_0, P_0$


$u=\arcsin(\sqrt{x})$


$F(x) := \int_{a}^{x}f(t) dt.$


$\langle f,\cos(t) \rangle = \int_{0}^{\pi} f \cos(t) dt = 0$


$\vert PQ \vert = rsin(\theta)$


$G = \{\ e_1\ ,\  e_2\ ,\  e_1e_2\  ,\ e_2e_1\  ,\ e_1e_2e_1\ \ , ... \}$


$f(x) = \int_a^x f(t)dt,$


$\gamma'(t) = \lim_{t \to a} \frac{\|\gamma(t) - \gamma(a)\|}{\|t-a\|}= 1$


$f(a+b)>f(a)+f(b)$


$\forall \epsilon, \exists \delta > 0 s.t. |x-c| < \delta, x \in D \implies |f(x) - f(c)| < \epsilon$


$\langle x,z\rangle\in R^2,$


$|RS| = r-s$


$n!+2, n!+3, n!+4, ..., n!+n$


$1+2+3+\cdots+n = \frac{(n+1)n}2 $


$u(x,v)=|\langle v,\nu_x\rangle|$


$\frac{1}{P(S_2 = j | S_0 = i)}\sum_{R^{*}_{1}=0}^r\sum_{l=1}^N P(S_1=l|S_0=i)I(R_{il}=R^*_1)P(S_2=j|S_1=l)I(R_{lj}=r-R^{*}_{1})$


$\forall x\in\mathbb{R}, \forall\varepsilon>0,\exists r\in \mathbb{Q}, |x-r|<\varepsilon $


$r=[(rs)(r^{2}s)]^{-1}=(r^{2}s)^{-1}(rs)^{-1}=(r^{2}s)(rs)$


$f(x+y)=f(x)+f(y)$


$ T^k_l(V) := V^{\otimes l} \otimes (V^\ast)^{\otimes k} \cong (V^{\ast\ast})^{\otimes l} \otimes (V^\ast)^{\otimes k} \cong ((V^\ast)^{\otimes l} \otimes V^{\otimes k})^\ast \cong T^l_k(V)^\ast. $


$A = \bigcup_{x\in A} B(x,r) $


$\left\lfloor \frac {\lfloor px \rfloor}{p} \right\rfloor=\lfloor x\rfloor$


$a^{log_b n} = n^{log_b a}$


$f(a+b)=f(a)+f(b)\in J$


$\mathop {\lim }\limits_{N \to \infty }N \sum\limits_{m = 1}^N {\frac{{{a^{N - m}}}}{m}} $


$P[X_2\leq x_2 |X_1=u_1]$


$ P =  \begin{bmatrix} 1 & 2 & 1 & 2 & 0 \\ 1 & -1 & -1 & 1 & 0 \\ 2 & -5 & -4 & 1 & 0 \\ 4 & 2 & 0 & 6 & 0 \end{bmatrix} $


$s = \{v_1, v_2, v_3, ... , v_n\}$


$T: p\mapsto f(p) - f(-p)$


$ f(x) = 1/|x|, $


$Y=\{\langle x,y\rangle\in\Bbb R^2:y=3\}$


$au + bv = d.$


$x \to x^3-x$


$\int\int\int\int\int\int\int dx$


$\lim_{n \to \infty} \inf (s_n) = s_*$


$I = \displaystyle\int_{-\pi}^\pi\dfrac{\sin^2(x)}{1+a^x}dx $


$ ...  \qquad =  - \lfloor a \rfloor f(a)   + \lfloor a \rfloor f\left(\lfloor a\rfloor+1\right)\\ +  \left(\lfloor a \rfloor + 1 \right)  \left[-f\left(\lfloor a\rfloor+1\right)+f\left(\lfloor a\rfloor+2 \right)\right]\\ + \left(\lfloor a \rfloor + 2 \right)  \left[-f\left(\lfloor a\rfloor+2\right)+f\left(\lfloor a\rfloor+3 \right)\right]\\ + ... \\ + \left(\lfloor b \rfloor -1 \right)  \left[-f\left(\lfloor b\rfloor-1\right)+f\left(\lfloor b\rfloor\right)\right]\\ + \lfloor b\rfloor~\left[f(b)-f\left( \lfloor b \rfloor\right) \right]\\[5mm]   = \quad - \lfloor a \rfloor f(a)  - f\left(\lfloor a\rfloor+1\right) - f\left(\lfloor a\rfloor+2 \right) ... - f\left(\lfloor b\rfloor\right) + \lfloor b\rfloor~f(b) $


$b_n=\int_{-\pi}^{\pi}\frac{1-a\cos(x)}{1-2a\cos(x)+a^2}\sin(nx)dx$


$\frac{1}{2} \begin{vmatrix} x_{1} & y_{1} & 1   \\  x_{3} & y_{3} & 1  \\ x_{2} & y_{2} & 1  \\   \end{vmatrix}$


$Cov(X,Y) < 0$


$\sum_n |\langle f, x_n \rangle |^2 = ||f||^2$


$\frac75=1+\frac25=1+\frac1{\frac52}=1+\frac1{2+\frac12}=1+\frac1{2+\frac1{1+1}}$


$\mathcal{X}(\mathcal{A})$


$\sum\limits_{n=-\infty }^{\infty }{{{e}^{i\pi {{n}^{2}}\tau +i2nz}}}=\frac{1}{\sqrt{-i\tau }}\sum\limits_{n=-\infty }^{\infty }{{{e}^{{{\left( z+n\pi  \right)}^{2}}/\pi i\tau }}}$


$(\gamma_1\wedge\gamma_2)((\gamma_2 + \gamma_3)\wedge(\gamma_4 + \gamma_1))\\  =\gamma_1\gamma_2((\gamma_2+\gamma_3)(\gamma_4+\gamma_1)-(\gamma_2+\gamma_3)\cdot(\gamma_4+\gamma_1))\\ \gamma_1\gamma_2(\gamma_2\gamma_4+\gamma_2\gamma_1+\gamma_3\gamma_4+\gamma_3\gamma_1-0)\\ =\gamma_1\gamma_2\gamma_2\gamma_4+\gamma_1\gamma_2\gamma_2\gamma_1+\gamma_1\gamma_2\gamma_3\gamma_4+\gamma_1\gamma_2\gamma_3\gamma_1\\ =\gamma_1\gamma_4+1+\gamma_1\gamma_2\gamma_3\gamma_4+\gamma_2\gamma_3$


$n! = \sqrt{2\pi n} (\frac{n}{e})^n \big(1 + \mathcal{O}(\frac{1}{n})\big)$


$f(x^2)+f(xy)=f(x)\,f(y)+y\,f(x)+x\,f(x+y)\,.$


$\left|\delta(x^{n-1}+...+a^{n-1})\right|$


$ \delta x= -A^{-1}\delta A (x+\delta x), $


$W'+ (\Phi^n)$


$p_1p_2\in P$


$\int_0^1 f(t)h(t)dt = \int_{0}^1 g(t)h(t)dt.$


$1/|x| = |1/x|$


$\gamma_1 \geq \gamma_2 \geq \cdots \geq \gamma_n$


$Q(a,b,c)=Q(b,a,c)=Q(a,c,d)=Q(c,b,a)$


$\| x_n - x_{n-1} \|$


$\sum\limits_{n=-\infty}^{\infty} e^{-\pi n^2 /t}e^{2\pi i n x} =t^{1/2}e^{-\pi t x^2}\sum\limits_{n=-\infty}^{\infty} e^{-\pi n^2 t}e^{-2\pi n xt}. $


$M:=\begin{bmatrix} A & -B \\ B & A\end{bmatrix}$


$V_1 = \pi\int_{0}^{1}\left(x^{3/2}-1\right)^2dx = 9/20\pi$


$T_\Bbb R = \begin{pmatrix} A & -B \\ B & A \end{pmatrix}$


$A_{0}\subseteq A_{1}\subseteq A_{2}\subseteq A_{3}\cdots\subseteq\Bbb N$


$\vec x = \left[\matrix{1\cr -1\cr 2\cr 1\cr 1\cr}\right]$


$1 = \int_0^\infty f(x) \, dx$


$s(x)=\displaystyle\sum_{i=1}^k{c_iI_{A_i}(x)}$


$\sqrt r e^{\frac {i \theta}{2}+i\pi}$


$\sum_{k=0}^{n-1}\cos\dfrac{2\pi k}n=0$


$\int_{0}^{\pi} \frac{x}{1+\sin^2 x} \, dx = \frac{\pi}{2} \int_{0}^{\pi} \frac{1}{1+\sin^2 x} \, dx $


$a_n = \frac{n(n+1)}{2}$


$\forall \epsilon>0, \exists \delta>0: |x-y|<\delta \implies |f(x)-f(y)|<\epsilon, $


$((A^+)^+)^+=(A^+)^+\cup \{(A^+)^+\}=\{\emptyset,\{\emptyset\}\}\cup\{\{\emptyset,\{\emptyset\}\}\}=$


$|2\tan\frac\theta 2 - \sin\theta| = |\sin\theta \frac{\sin^2\theta}{4\cos^4\frac{\theta}2}|$


$\liminf\mu(A_n)\ge\mu(A)\ge\limsup\mu(A_n).$


$\sqrt{\sqrt2+\sqrt3}-\sqrt{\sqrt2-\sqrt3}>0$


$\sum_{n=1}^\infty e^{-(n+x)^2}=e^{-x^2} \sum_{n=1}^\infty e^{-n^2} e^{-2nx}$


$\mathbb{E}[X^r] - \mathbb{E}[Y^r] + \sum_{i=1}^{r-1}c_{r,i}(\mathbb{E}[X^i] - \mathbb{E}[Y^i]) = 0$


$ \zeta(s) = \pi^{- \frac{1}{2} + s} \frac{\Gamma(\frac{1}{2} - \frac{s}{2})}{\Gamma(\frac{s}{2})} \zeta(1 - s).$


$x^n + x^{n+1}$


$\|A\|_2  = \sqrt{\lambda_\max(A^*A)}$


$\exists x \forall y Rxy$


$\displaystyle \sum_{n=1}^\infty \frac{1}{n}=\sum_{n=1}^\infty \frac{1}{n^2}+\sum_{n\ne\text{square}}^\infty \frac{1}{n}$


$ S_N-T_N=\sum_{n=1}^N\frac1{n^2} \to \sum_{n=1}^\infty\frac1{n^2}=\frac{\pi^2}6<\infty. $


$a^{\log_a b} = b$


$\OR_a:\MX\to\Bbb{Z}\cup\{\infty\},\quad \OR_a(f)=\begin{cases}0,&\text{ if } f \text{ is nonzero and holomorphic at }a\\k&,\text{ if f has a zero of order }k\text{ at a}\\-k,&\text{ if f has a pole of order k at a}\\ \infty,&\text{ if f is identically zero at a}\end{cases}$


${(\mathbb Z}[x]/(x + 3))/(x^2+1) \cong {\mathbb Z}/(10)$


$[40, 10]$


$E(X+Y)^p \le 2^p (E(X^p)+E(Y^p)).$


$  k^3 = 6 \; \left( \begin{array}{c} k \\ 3 \end{array} \right) + 6 \; \left( \begin{array}{c} k \\ 2 \end{array} \right) + \left( \begin{array}{c} k \\ 1 \end{array} \right),  $


$|x^n-a^n| < \delta$


$gcd(a, a+2)$


$v_1\cdot v_2 = \sum_{k=1}^d\cos\frac{(k-1)\pi}d\sin\frac{(k-1)\pi}d = \frac12\sum_{k=1}^d\sin\frac{(k+1)2\pi}{d} = 0$


$f(x)=f(x\cdot 1)=f(x)\,f(1)-f(x+1)+1=f(x)-f(x+1)+1$


$\tag{*}\{\,\ldots, a-2n, a-n,a,a+n,a+2n,\ldots\,\}.$


$a\equiv e\mod s$


$\mathbb{P}(X_{n+1}=X_n/2|\mathcal{F}_n)=1-X_n$


$\lvert DC \rvert = kb$


$\Phi(x)=\int_a^x f(t)dt$


$\begin{bmatrix}1&-3&0&5\\-1&1&5&2\\-1&1&5&3\end{bmatrix}$


$R=\{\langle x,y\rangle:s\le x\le t\text{ and }0\le y\le 1\}\;;$


$\chi_A=x^{n-1}(x-\tr A)$


$m \times \mu(A\times B)=m(A)\mu(B)$


$\sum_{j=1}^\infty\sum_{k=1}^\infty \frac{1}{j^2(j+k)^2}$


$f(a^k)=b^k$


$=|x+y-z|+|x-y+z|+|x-y+z|+|-x+y+z|+|-x+y+z|+|x+y-z|$


$x,y \in A \rightarrow xRy \lor yRx$


$f(z) = z^2-z-2$


$\left\lfloor \frac{1}{b} \left\lfloor \frac{a}{b} \right\rfloor \right\rfloor = \left\lfloor \frac{a}{b^2} \right\rfloor.$


$\alpha+$


$ \left\{ \begin{array}{c} a=\frac{1}{2}\\ b=1\\ c=0\\ d=1\\ e=-2 \end{array} \right.$


$m = m' + m''$


$d(x, A) \leq d(x, z) + d(z, A). \tag{1}$


$p_1 \mid p_2$


$ P(k) : k^3 < 3^k $


$0=f(t,\gamma+\delta\gamma,\dot\gamma+\delta\dot\gamma,\ddot\gamma+\delta\ddot\gamma,T+\delta T)$


$n = \left\lfloor\frac{M}{\sqrt{a}}\right\rfloor\left\lfloor\frac{N}{\sqrt{a}}\right\rfloor$


$x!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!$


$[a]=\{\dots,a-2b,a-b,a,a+b,a+2b,\dots\}.$


$I_n = \int_{-\pi}^{\pi} \frac{\sin (nx)}{\sin x} dx$


$B_n = \frac {n(n+1)}2$


$z=y^k+x\,, x\in \mathbb N$


$\cos(2x) - \cos(x) = (1-2\sin^2(x)) - (1-2\sin^2(x/2)) = 2\left(\sin^2(x/2) - \sin^2(x)\right)$


$B=P^{-1}A P=\left[\matrix{1&0&0\cr 0&0&-\sqrt{2}\cr 0&\sqrt{2}&0\cr}\right]\ .$


$f(x)=\frac {9^x}{x!}=\frac {9^x}{\Gamma(x+1)}$


$f=\sum f_j=\sum(-1)^j\chi_{I_j}.$


$\lim_{x\to a}f(x)=f(a)\Leftrightarrow \forall\varepsilon_1>0,\exists\delta_1>0,|x-a|<\delta_1\Rightarrow|f(x)-f(a)|<\varepsilon_1$


$ \int_{b}^{a} 2\pi \,f(x) \,dx $


$F=I\oplus K$


$\overline{\mathbb{R}} = \mathbb{R}$


$\sum_{n\geq 1}\frac{x}{n(x+n)} = \sum_{n\geq 1} \frac{x}{n^2} \frac{1}{1+\frac{x}{n}}$


$[F(n+2)-F(n)]^2  =  F(n+2)^2 - 2[F(n+2)F(n)] + F(n)^2$


$\mathcal A:=\{\{x\},x\in\mathbb N\}$


$\Phi (x)=\underset{n\to \infty }{\mathop{\lim }}\,{{f}_{n}}(x)=0$


$(x,y,z)\cdot (1,1,1)<0$


$\sum_{i=1}^n\cos\left(i\pi\frac kn\right)=0$


$F(x)=\int_a^x f(t)\,dx, $


$x:=(x,y)$


$\begin{bmatrix}A& -B^t\\B& A^t \end{bmatrix}$


$xRy \leftrightarrow yRx$


$ P_{0,a}^\alpha(\gamma)=P_{\gamma(\nu^\alpha (a)),\gamma(a)}...P_{\gamma(\alpha),\gamma(2\alpha)}P_{\gamma(0),\gamma(\alpha)},$


$ \lim_{n \rightarrow \infty} \frac{1}{n} \sum_{i=1}^n 1 =\lim_{n\rightarrow \infty} \frac{1}{n} n = 1$


$T = \begin{pmatrix}A & -\overline{B} \\ B & \overline{A} \end{pmatrix},$


$p(n) \implies p(n+1)$


$                 f(0) = 0,\;\; f(\pi)=0 $


$ \begin{vmatrix} a &   a+b & a+2b\\ a+2b & a  &  a+b\\ a+b & a+2b &  a \end{vmatrix}  =  9b^2 (a+b) $


$
 \begin{align*}
 xy \cdot
 \left|
 \begin{array}{ccc}
 1 & 1 & 1 \\
 x & y & 1 \\
 x^2 & y^2 & 1 \\
 \end{array} \right|
 &= xy \cdot
 \left|
 \begin{array}{ccc}
 0 & 0 & 1 \\
 x - 1 & y -1 & 1 \\
 x^2 -1 & y^2 - 1 & 1 \\
 \end{array} \right| \\
 &= xy \cdot (x-1)(y-1)
 \left|
 \begin{array}{ccc}
 0 & 0 & 1 \\
 1 & 1 & 1 \\
 x + 1 & y + 1 & 1
 \end{array}
 \right| & \text{because } x^2 - 1^2 = (x-1)(x+1)\\
 &=xy \cdot (x-1)(y-1) \cdot 1 \cdot(1 \cdot (y+1) - 1 \cdot (x + 1)) & \text{Laplace with row 1}\\
 &= xy \cdot (x-1)(x+1)(y-x)
 \end{align*}
 $


$L=\lim_{x\to a^-}f(x)=\lim_{x\to a^+}f(x)$


$ |ab| =ab = |a||b| $


$f(x) = \frac{x(x-1)}{2} = \binom{x}{2}$


$\lim_{x\to a} f(x) g(x) =\lim_{x\to a} f(x) \cdot \lim_{x\to a} g(x) =L\lim_{x\to a} g(x) $


$\left\Vert x_{n+1}-x_{n}\right\Vert \leq r_{n}-r_{n+1}$


$\;a^b\;$


$\int\frac{\sin^3(x)}{(\cos^4(x)+3\cos^2(x)+1)\cdot\arctan(\sec(x)+\cos(x))}$


$ \int_{0}^{\infty}{\sin^{n}\left(x\right)\over  x}\,\mathrm{d}x $


$I_n = n\pi$


$S = \{ v,e_1, e_2,\ldots,e_n\}$


$\displaystyle\lim_{h \to 0} f'(x) = \displaystyle\lim_{x \to c} f'(x) = d$


$ \left(1 + \sqrt{2} + \sqrt{3} + \dots + \sqrt{n} \right) - \frac{2}{3} n \sqrt{n}$


$=\frac{(x+y+z)(y-z)^2(x-y)(x^2(y+z)^2-y^2(x+z)^2)}{(x+z)^2(y+z)^2}=$


$ K\subset D \subset\overline{D}\subset A. $


$A=\{ \left(x,y \right)\in \mathbb{R}^2: 0 \le x \le \pi, 0 \le y \le \pi\}$


$g[n] = \delta[n] - A \delta[n-1]$


$\lim_{x\to c^+}f(x)=\lim_{x\to c^-} f(x)$


$f(x;3)=\frac{1}{3} e^{-x/2} x \left(e^{3 x/2}-2 \sin \left(\frac{1}{6} \left(3 \sqrt{3}    x+\pi \right)\right)\right)$


$p(e) = p(e')$


$\lim_{n \rightarrow \infty} \sum_{i=1}^\infty a^{(n)}_i =\sum_{i=1}^\infty a_i$


$=3+\frac1{1+\frac2{95}}=3+\frac1{1+\frac1{\frac{95}2}}=3+\frac1{1+\frac1{47+\frac12}}$


$\det\begin{pmatrix} 1 & s & s^2 \\ 1 & t & t^2 \\ 1 & u & u^2\end{pmatrix}$


$y'(x)={dy\over dx}={dy\over dt}\cdot{dt\over dx}={2a\cos(2t)\over 2a\left({\cos(t)\over \sin(t)}-2\sin(t)\cos(t)\right)}=\tan(t)$


$g(y)=\int_y^{\sqrt y } 15y \, dx$


$\Bbb R^n\times\Bbb R=\Bbb R^{n+1}$


$A_1 \times A_2 \times ... \times~ A_k$


$\mathbb{R}[x]/((x^2 +1)^2 )$


$l = \begin{cases} x=-2t+1 \\  y=3t-2 \\  z=t+4 \end{cases} $


$\text{ } \text{ } \text{ } \text{ } \text{ } \text{ } \text{ } \color{red}{p_{1}p_{2}}p_{3}$


$ J=\int_{-\pi}^{\pi}\frac{\sin^2\left(x\right)}{1+a^{x}}\text{d}x $


$\mathcal{P}(n) \implies \mathcal{P}(n+1)$


$\eta(s)  = \frac{1}{\Gamma(s)}\int_{0}^{\infty} \frac{x^{s-1}}{1+e^{x}} \, dx , \quad \text{Re}(s) >0,\tag{1}$


$\left\lfloor \frac{y}{b}\right\rfloor = \left\lfloor \frac{\lfloor y \rfloor}b\right\rfloor$


$\frac{2 + bx}{b + x}$


$a_n = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) \cos(nx) \, dx $


$\lim_{r \to \infty} \frac 1 r$


$\sum _{n=1}^{ \infty}\sum _{k=1}^{\infty }\frac{1}{k^2\left(n^2+\left(\frac{\pi }{x}\right)^2k^2\right)}$


$g[x0] = x1+ zg$


$40 + 40 = x + 40 + 10$


$(p_1p_2\cdots p_n)+1$


$\lim_{n \rightarrow \infty} \frac{n}{\sqrt[n]{n!}}$


$\{a + \delta,\ a+2\delta,\ a+3\delta,\ \dots\} \subseteq S$


$l: \begin{cases} x=1 \\ y=z \end{cases}$


$\left\lfloor \frac{r + b}{c}\right\rfloor = \left\lfloor \frac{\lfloor r \rfloor + b}{c}\right\rfloor$


$A=\{\langle x,y\rangle\in\Bbb I\times(\Bbb R\setminus\Bbb N):xy>1\}$


$|AB|=|AB|$


$f_D(d) = \dfrac{2d}{R^2}.$


$ \sum_{n=1}^{\infty}\frac{n^n}{(n!)^2}$


$\displaystyle\phi(x) = \frac{1}{\sqrt{2\pi}}e^{-\dfrac{1}{2}x^2}$


$\Sigma = \{\emptyset, \mathbb{N}, \mathcal{P}(\mathbb{N}), \mathbb{R}, \mathbb{R}-\mathbb{N}, \mathbb{R}-\mathcal{P}(\mathbb{N}),(\mathbb{R}-\mathcal{P}(\mathbb{N})) \cup \mathbb{N}, (\mathbb{R}-\mathbb{N}) \cup (\mathbb{R}-\mathcal{P}(\mathbb{N})) \}$


$\displaystyle f(1-x)=\frac{2}{4^{1-x}+2}=\frac{2\cdot 4^x}{4+2\cdot 4^x}=\frac{4^x}{2+ 4^x}$


$e^x=\lim_{n\to\infty}\left(1+\frac xn\right)^n$


$I_n=[2n \pi,2n\pi+\pi]$


$\alpha + \omega > \alpha$


${x\over 2}=\tan\theta$


$(a, b)+_\mathbb{C}(c, d)=(a+_\mathbb{R}c, b+_\mathbb{R}d)=(c+_\mathbb{R}a, d+_\mathbb{R}b)=(c, d)+_\mathbb{C}(a, b).$


$\gamma(a) = \gamma(0) = \gamma(2\pi) = \gamma(b).  \tag{2}$


$\left(\begin{array}{ccc|c}3&1&-1&0\\1&-1&2&0\\2&2&-3&0\end{array}\right)$


$f((a,b)+(h,k)) - f((a,b)) = f(a+h, b+k) - f(a,b) = f(a,b)+f(h,k)-f(a,b) = f(h,k)$


$Cov(x,y)=E(xy)-E(x)E(y).$


$\sum\limits_{r=0}^{n+1} \frac{{(n+1)}!}{r!(n+1-r)!} =2^{n+1}.$


$ \frac1{\sqrt{N}}\,\frac{\sqrt1+\sqrt2+\cdots+\sqrt{N}}N $


$\sum_{k=1}^\infty\frac{\cos 2\pi k}{k}$


$ \det\begin{bmatrix} 1 & 1 & 1 \\ x & y & z \\ x^2 & y^2 & z^2 \end{bmatrix} = \det\begin{bmatrix} 1 & 1 & 1 \\ 0 & y-x & z-x \\ 0 & y^2-x^2 & z^2-x^2 \end{bmatrix} = \det\begin{bmatrix} 1 & 1 & 1 \\ 0 & y-x & z-x \\ 0 & 0 & z^2-x^2-(z-x)(y+x) \end{bmatrix} $


$(P\oplus Q)\oplus P=P\oplus Q\oplus P=P\oplus P\oplus Q=0\oplus Q=Q$


$\langle\alpha+\beta,\gamma\rangle=\langle\alpha,\gamma\rangle+\langle\beta,\gamma\rangle$


$\mathbb{Q}[x]/(x^2+1)\cong \mathbb{Q}[i]$


$f'(x)= \frac{1}{3} ((x+3)^\frac{-2}{3} - x^\frac{-2}{3}) = 0$


$\vartheta^4(z+2) = \vartheta^4(z), \; \; \vartheta^4(-1/z) = -z^2 \vartheta^4(z).$


$x=a^{log_a x}$


$\left(\begin{array}{c}-1\\ -1\\1\end{array}\right)$


$2x! > \sqrt{4\pi{x}}(\frac{2x}{e})^{2x}$


$(x+y)^r=\sum_{k\ge 0}\binom{r}kx^{r-k}y^k\;,$


$\sum_{m=1}^\infty \sum_{n=1}^\infty \frac{1}{m^2+n^2}=+\infty$


$\int_0^x f(t)g(t)\,dt = \int_a^{x+a} f(t)g(t)\,dt$


$ \begin{bmatrix} 1 & -1 & -a & 1 \\ -2 & 2 & -1 & 2\\ 2 & 2 & b & -2 \end{bmatrix} $


$|AB| = 4$


$  \mathbb{P}(-X_n < x)\leq\mathbb{P}(-X_n \leq x) \leq \mathbb{P}(-X_n < x+\delta). $


$\gamma,\tilde{\gamma}:[a,b]\to M:\gamma(a)=\tilde{\gamma}(a),\gamma(b)=\tilde{\gamma}(b)$


$\frac{r}{2\pi(1-\rho^2)}\exp{\left({\frac{\rho r^2\sin(2\theta)}{(1-\rho^2)}-\frac{r^2}{2(1 -\rho^2)}}\right)}$


$x\rightarrow P(X\leq x)$


$A\subset B\subset \bar B\subset U,$


$N(A) = \lbrace x\in \mathbb R^q : A x = 0\rbrace$


$F_n(x) = \int_a^{x}f_n(t)dt, x\in [a,b]$


$T=\big\{\{a\},\{a,b\}\big\}$


$A=\{\{a\}\}$


$ \|x_n-x_m\|<\eps $


$\det(A-xI)=(-x+(n-1))(-x-1)^{n-1}\color{red }{\longrightarrow} x=n-1  ,-1$


$S \subseteq A_1 \times ... \times A_n$


$\alpha=\omega^\alpha$


$A_1 ......A_n$


$\begin{vmatrix} 1&x&x^{2} \\ 1&y&y^{2} \\ 1&z&z^{2} \end{vmatrix} = \begin{vmatrix} 1& 1 & 1 \\ x & y & z \\ x ^2 & y^2 & z^2\end{vmatrix} = (y-x)(z-x)(z-y).$


$\begin{align}\left(1+\frac{x}{n}\right)^n&\approx e^x\\ \end{align}$


$ S = \lbrace a_1,a_2,a_3,.............a_n \rbrace $


$\sqrt {(-x)^2}=|x|$


$\begin{bmatrix} \cos(\varphi) & -\sin(\varphi) \\ \sin(\varphi) & \cos(\varphi) \end{bmatrix}$


$A_1\subseteq A_2 \subseteq \dots \subseteq A_n \subseteq \dots$


$f(a)+f(b) \leq f(a+b)$


$b^{log_b{x}} = x$


$ \frac{df}{d\gamma} = {\frac {-2-\gamma\,s+2\,\gamma-\gamma\,n+2\,\sqrt { \left( 1+\gamma\,n -\gamma \right)  \left( 1+\gamma\,s-\gamma \right) }}{{\gamma}^{2} \sqrt { \left( 1+\gamma\,n-\gamma \right)  \left( 1+\gamma\,s-\gamma  \right) }}}$


$(A_1, A_2,\dots, A_r)$


$2I= \int_{-\pi}^{\pi} \dfrac{\sin nx}{\sin x} \dfrac{\pi^x+1}{1+\pi^x}dx$


$ \lim\limits_{n\rightarrow\infty} \left\lfloor  \frac{1}{n}\right\rfloor =?$


$ (\frac{(-1)^{n-1} x^{n-1} e^{-x}}{(n-1)!}) $


$U\subset S \implies U \subset X$


$\text{ Sum = } \lim_{n\to\infty} = \dfrac{n-1}{n+1} = 1$


$\gamma(x,y)=|\langle x,y\rangle|$


$\int_0^1 p'(t)f(t)\,dt = \int_0^1 p(t)g(t)\,dt$


$\{a,a+d,a+2d, \dots\}$


$\bigl([x]+[y]\bigr)+[z]=[x+y]+[z]=[x+y+z]=[x]+[y+z]=[x]+\bigl([y]+[z]\bigr)\ .$


$1\leq k\leq e_i$


$\lim\limits_{n\to\infty}(\sqrt2-\sqrt[3]2)\cdots(\sqrt2-\sqrt[n]2)$


$x \in \mathbb{R}^{n+1} - S^n$


$\displaystyle f(x) = \frac{x^3}{e^{x}+1}$


$F(x)=\int_a^x f(t)$


$as+b=r$


$ 1-x = 1-\frac{\gamma_1}{\gamma_1 + \gamma_2} = \frac{\gamma_1 + \gamma_2}{\gamma_1 + \gamma_2}-\frac{\gamma_1}{\gamma_1 + \gamma_2} = \frac{\gamma_2}{\gamma_1 + \gamma_2} $


$|M|+|M^∗|−|M|=|M^∗|$


$|\hat{f}(n)|\leq C/|n|^{k+a}$


$\int \left(\frac{\tan^{-1}x}{x-\tan^{-1}x}\right)^{2}dx$


$\theta_{\gamma_1\gamma_2,\gamma_3}\theta_{\gamma_1,\gamma_2}=\theta_{\gamma_1,\gamma_2\gamma_3}\theta_{\gamma_2,\gamma_3}.$


$\omega^{\gamma'} + \omega^{\gamma} = \omega^{\gamma} < \omega^{\gamma} + \omega^{\gamma'} = \omega^{\gamma'} \oplus \omega^{\gamma}$


$v = (1/12,1/12,-1/6,1/12,-1/6,1/12)$


$f(yf(x)+x)  = xf(y) +f(x)  $


$\frac{1}{1-x}\log\frac{1}{1-x}=\sum_{n=1}^{\infty}\left(1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{n}\right)x^n$


$\Delta^n(X^n)=\Delta^n(X^n)(a)=n!$


$A\cong A_1\times\dots\times A_r$


$d(x,z) + d(y,z) = d(x,y)$


$g(b) \in A_1 \times \dotsc \times A_n$


$\frac{a+bx}{c-dx}$


$n!\approx\sqrt{2\pi n}*(\frac{n}{e})^n$


$(2x \frac{dy}{dx})(y)+x^2$


$f(a+_Sb)= f(a)+_R f(b)$


$\lim_{x\to c} f'(x) = \lim_{x\to c} g'(x) = 0$


$\lim_{n\to\infty}\int_{E_n}f(x)d\mu=\int_E f(x)d\mu\,\,\,\,\,\,\text{ where }E=\cup_{n=1}^\infty E_n$


$\{X_1\otimes X_2|X_1\in\mathfrak{g}_1,X_2\in\mathfrak{g}_2\}$


$ y=\frac{a}{x-c}+c = \frac{a + cx - c^2}{ x - c}$


$(\cos^2\varphi + \sin^2\varphi) = 1 $


$ M\leq f(x) \leq \lim_{n\rightarrow\infty}\inf f(x_n). $


$\frac 1 2 < x <1 \implies \pi < \frac{\pi}{x} < 2 \pi \implies 0 < 1+ \sin \frac{\pi}x <1$


$x\in (a-\gamma, a+ \gamma)\setminus \{a\}$


$\|x^m_n - x_n\| < \epsilon/2$


$|f(z)|\leq M|\varphi^{-1} (z) |$


$\sum_{n=-\infty}^{+\infty}\delta(t-nT)=\frac{1}{T} \sum_{m=-\infty}^{+\infty} e^{\frac {-i2\pi m t}{T}}$


$\mathcal{A}^+$


$x R y\land y R x$


$\begin{bmatrix} A & I \\ A & -I \end{bmatrix}$


$\sum_n ||e_n - f_n||^2 < 1$


$\frac{2b(b-a) -b^2-b+a^2+a +2b-2a}{2b-2a+2}$


$A_1 \times A_2 \times \dots \times A_n \times Y \times Y \times \cdots.$


$\frac{x}{|x|} = -1$


$\vartheta (x) = \sum_{n\in \mathbb Z} e^{-\pi {n^2} x} = \sum_{k\in \mathbb Z} {e^{-\pi {k^2} {1\over x}}} \int_{-\infty}^{\infty} {e^{-\pi x \big( y+ i {k\over x}\big)^2}} \space dy$


$c_{i_{0}}\in C_{i_{0}} $


$\int_{-\infty}^{\infty}f(t)\exp(-st)dt=\int_{0}^{\infty}f(t)\exp(-st)dt+\int_{-\infty}^{0}f(t)\exp(-st)dt$


$\large f(n) = \frac{n(n+1)}{2}$


$ D=\{(x,y)\in\mathbb{R}^2:0\le x\le 1, 0\le y \le x\}. $


$\int_0^{\infty} f(x)$


$\int_{0}^{\pi} f^{\prime} \sin(t) dt = f \sin(t)$


$4^x+15^x=9^x+10^x(2^x-3^x)(2^x-3^x-5^x)$


$\liminf_{n \rightarrow \infty} \mu (A_n) \leq \limsup_{n \rightarrow \infty} \mu (A_n)$


$\sum_{n=-N}^Ne^{-n^2/2}\le\sum_{n=-\infty}^\infty e^{-n^2/2} \le\sum_{n=-N}^Ne^{-n^2/2}+\frac2Ne^{-N^2/2}.$


$\sum_{k=1}^{\infty} \frac 1k = 1 + (\frac 12 + \frac 13) + (\frac 14 + \frac 15 + \frac 16 + \frac 17) + ...... + (\frac 1{2^n} + \frac 1{2^n + 1} + .... + \frac 1{2^n + (2^n -1)}) + .....  > \frac 12 + (\frac 14 + \frac 14) + (\frac 18 + \frac 18 + \frac 18 + \frac 18) + ...... (\frac 1{2^{n+1}} + \frac 1{2^{n+1}} + .... + \frac 1{2^{n+1}}) + ... = \frac 12 + \frac 12 + .... + \frac 12 + .....$


$ \left( \begin{array}_ 0 & 1 & 4 & 9 & ... & (n-1)^2  \\ 1 & 0 & 1 & 4 & ... & (n-2)^2  \\ ... &... &... & ... & ... & ... \\ (n-1)^2  & (n-2)^2  & (n-3)^2  & (n-4)^2  & ... & 0  \end{array} \right)$


$z=y \ \ \text{or} \ \ x=-2\lambda-1$


$\int\limits_\gamma f dz = \int\limits_\gamma \frac{1}{g} dz = \int\limits_\gamma \frac{1}{(z + 1)(z + 2) \cdot \ldots \cdot (z + r)} dz$


$P(k+1)\implies P(k)$


$\forall x\in E,\quad ||x||^2=\sum_{i=1}^n\langle e_i,x\rangle^2$


$ J(a) = \frac{1}{2} \int_0^1 t^{(a-1)/2} (1-t)^{-1/2}\; dt = \frac{B((a+1)/2,1/2)}{2} = \frac{\Gamma(a+1/2) \Gamma(1/2)}{\Gamma(a+1)} $


$\sum_{n\leq x} \frac{\sigma (n)}{n} = \sum_{n \leq x} \frac{1}{n} \lfloor \frac{x}{n} \rfloor $


$f(1-x)=\dfrac2{2+4^{1-x}}=\dfrac{2\cdot4^x}{2\cdot4^x+4}=\dfrac{4^x}{4^x+2}=\dfrac{4^x+2-2}{4^x+2}=1-f(x)$


$[x,y,z]:=(x+y+z)(-x+y+z)(x-y+z)(x+y-z)$


$f(\perp)=g(\perp)=\perp$


$f = (1-x y) (1+x+z-x z) (1-y z) = 1+x-x y-x^2 y+z-x z-y z  \mathbf{-2}  x y z+x^2 y z+x y^2 z+x^2 y^2 z-y z^2+x y z^2+x y^2 z^2-x^2 y^2 z^2$


$E(n, p) = n^ 2 - n + p$


$ \begin{array}{llllcrrr} [1/9  -2/27, & 3/9, & 5/9 + 2/27 &]&=&[ 1/27, &9/27,& 17/27]\\ [1/9   , & 3/9 + 2/27, & 5/9 - 2/27 &]&=& [3/27, &11/27, &13/27]\\ [1/9  +2/27, & 3/9- 2/27, & 5/9  &]&=& [5/27, &7/27,& 15/27]\\ [3/9  -2/27, & 5/9 , & 1/9+ 2/27 &]&=& [7/27, &15/27,&5/27]\\ [3/9   , & 5/9 + 2/27, & 1/9 - 2/27 &]&=& [9/27, &17/27,& 1/27]\\ [3/9  +2/27, & 5/9  - 2/27, & 1/9 &]&=& [11/27, &13/27,& 3/27]\\ [5/9  -2/27, & 1/9 , & 3/9 + 2/27&]&=& [13/27, &3/27, &11/27]\\ [5/9  , & 1/9 + 2/27, & 3/9 - 2/27& ]&=& [15/27, &5/27,& 7/27]\\ [5/9  +2/27, & 1/9 - 2/27, & 3/9 &] &=& [17/27, &1/27,& 9/27]. \end{array} $


$|\gamma(1)-\gamma(t)+\gamma(t)-\gamma(0)|= |\gamma(1)-\gamma(t)|+|\gamma(t)-\gamma(0)|,  $


$\lim_{n \to \infty} \left(\sum_{r=1}^n \frac{1}{x_r}\right)$


$ \left(  \left(  R^{+}\right)  ^{\sigma}\right)  ^{=}=\left(  \left( R^{=}\right)  ^{+}\right)  ^{\sigma}=\left(  \left(  R^{+}\right) ^{=}\right)  ^{\sigma}=\bigcup_{n=-\infty}^{\infty}R^{n}% $


$ s(x)=\sum\limits_{j=1}^ma_j\chi_{E_j}(x). $


$A\subset B\subset C\subset A$


$P(n) \implies S(m)$


$\frac{1}{3^{n-2}}< \epsilon $


$f(a + b) \leq f(a) + f(b).$


$ \mathbb{E}[(X - \mathbb{E}[X])^2] = \mathbb{E}[X^2 - 2\mathbb{E}[X]\cdot X + \mathbb{E}[X]^2] = \mathbb{E}[X^2] - 2\mathbb{E}[X]\mathbb{E}[X] + \mathbb{E}[X]^2 =  \mathbb{E}[X^2] - \mathbb{E}[X]^2 $


$  \left\{ \begin{array}{l}  x=1+2t\\ y=1+7t\\ z=3t\\ \end{array} \right. $


$ \sum_{n\le x}\frac{1}{n^2}=\frac{\pi^2}{6}-\frac{1}{x}+O(x^{-2}), $


$\begin{align}\int_0^\infty f(x) \;dx &= \int_0^1f(x)\;dx + \int_1^\infty f(x)\;dx \\ &\le \ln(2) + 1  \end{align}$


$|\mathbb P\{X_n\leq x\}-\mathbb P\{X\leq x\}|\leq \mathbb P\{X_n\leq x,X>x\}+\mathbb P\{X_n>x,X\leq x\}$


$Y= \{a_1,a_2,...,n\}$


$f(f(0) + f(z)) = -f(z)$


$\lim\limits_{x \to 0} 2 \dfrac {(e^{\frac{x}{2}}-e^{\frac {-x}{2}})^{\frac {2} {x^2}}} {x^2}$


$ \dot x = x \ , \quad x \in \mathbb R $


$\lfloor{\frac{a}{b+1}}\rfloor = \lfloor \frac{a}{b} \rfloor  $


$\tag 2 n^2 - n + 2 \ge 0$


$\begin{array}{ccc}1&k&1\\0&(1-k^2)&0\\0&(-k^2-k-1)&-k\end{array}$


$\sum_{r=1}^{\infty} \frac{1}{2^r}=2$


$x^{-n}e^x>\frac{x^{-n}x^{n+1}}{(n+1)!}=\frac{x}{(n+1)!}$


$\alpha=\omega+\alpha.$


$P(n) \implies P(n+1)$


$a^+$


$\int_0^\pi f\left(\frac{v+k\pi}{n}\right)\cos(v)dv \sim \int_0^\pi f\left(\frac{k\pi}{n}\right)\cos(v)dv =0$


$\vartheta(z+1, \tau) = \vartheta(z, \tau)$


$\left[\begin{array}{cccccc|c} 1  & 2  &  0  &  0  & a     & 1   & -2 \\ 0 & 0 &  0  &  0  & -1  & 0  & 1  \\ 0 & 0 & -1 & 2  & a^{2}+2a &   2  & 3  \\ 0  & 0  & 1  & -2 & 2    & -2  & -4 \end{array}\right]$


$\begin{pmatrix} a & b \\ -b & a \end{pmatrix}$


$(\mathbb{R}^n, \mathcal{O})$


$\frac ax + \frac by = 1$


$|f(z)|\leq c|g(z)|$


$\frac{\pi^2}6=\sum_{n=1}^\infty\frac1{n^2}=\frac14\sum_{n=1}^\infty\frac1{n^2}+\sum_{n=1}^\infty\frac1{(2n-1)^2}\implies$


$\sum I_j)$


$\gamma_1 =S_4\ge \gamma_2=A_4\ge\gamma_3=A_4\ge \gamma_4 =A_4\ge...  $


$\exists c_0$


$a_k = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x)\cos(nx) dx,$


$\lim_{n\to\infty}\frac{1+\sqrt[n]{2}+\sqrt[n]{3}+\cdots+\sqrt[n]{n}}{n}$


$rs-r's'=(rs-rs')+(rs'-r's')=r(s-s') + s'(r-r') \lt 0\,$


$T\bigg(\sum x_n e_n\bigg) \mapsto \sum y_n x_n e_n.$


$\mathbb{E}|XY|\le\sqrt{\mathbb{E}X^2\mathbb{E}Y^2}$


$\lim_{n \to \infty}\dfrac{\sqrt{1}+\sqrt{2}+\sqrt{3}+\cdots+\sqrt{n-1}}{n\sqrt{n}}=\lim_{n \to \infty}\dfrac{\sqrt{\dfrac{1}{n}}+\sqrt{\dfrac{2}{n}}+\sqrt{\dfrac{3}{n}}+\cdots+\sqrt{\dfrac{n-1}{n}}}{n}\ge \lim_{n \to \infty}\bigg(\sqrt{\dfrac{1}{n}}.\sqrt{\dfrac{2}{n}}.\sqrt{\dfrac{3}{n}}\cdots\sqrt{\dfrac{n-1}{n}}\bigg )^\dfrac{1}{n-1}=\lim_{n \to \infty}\bigg(\dfrac{(n-1)!}{n^n}\bigg )^\dfrac{1}{2(n-1)}$


$ k\cdot\gcd(a,b)=\lambda ka+\gamma kb=\lambda 0+\gamma 0\in\mathbb Z_n $


$\theta_x=\frac\pi2-\arctan\left(\frac{\sqrt{1-x^2}}x\right)$


$\text{lcm}(a+c,b+c)=k$


$h = \left(h_1, h_2, h_3, .... ,h_n \right)$


$A_1\times \dots A_k$


$1+2+3+4+5+..... = -\frac{1}{12}$


$\binom{-n}{r} = \frac{(-n)(-n-1)(-n-2)…(-n-r+1)}{r!} = \frac{(-1)^rn(n+1)(n+2)…(n+r-1)}{r!} = (-1)^r \binom{n+r-1}{r}$


$\begin{array}\\ \sum_{n=1}^∞ \prod_{j=1}^n(\frac1{a} + \frac1{b j})  &= -\dfrac{(\frac{a - 1}{a})^{-a/b} (a (\frac{a - 1}{a})^{a/b} -  (\frac{a - 1}{a})^{a/b} - a)}{a - 1}\\ &= -\dfrac{a - 1 - a(\frac{a - 1}{a})^{-a/b}}{a - 1}\\ &= \dfrac{ a(\frac{a - 1}{a})^{-a/b}}{a - 1}-1\\ &= \dfrac{ a^{1+a/b}}{(a - 1)^{1+a/b}}-1\\ &= \left(\dfrac{ a}{a - 1}\right)^{1+a/b}-1\\ \end{array} $


$f(x) = \dfrac{3x^2}{\theta^3}$


$V = X \oplus Y$


$B\,'(p,q)=\left\{\langle x,y\rangle:a\le x<b\text{ and }c\le y<d\right\}\;;$


$f(n) = n^2$


$\lim\limits_{n\rightarrow\infty} \mu(A_n)=\mu(A)$


$\int_{0}^\frac{1}{2}\frac{\sin^{-1}x}{\sqrt{1-x^2}} dx $


$A = \{ \{m\}, \{x\}\}$


$\left| {TP} \right| = r\;\theta $


$\lim_{n\rightarrow \infty} \sum_{i=1}^n a_i$


$\frac{1}{2}\sqrt{\frac{\gamma}{mg}}\int\frac{1}{v - \sqrt{\frac{mg}{\gamma}}}- \frac{1}{v + \sqrt{\frac{mg}{\gamma}}}dv=\int \frac{\gamma}{m}dt$


$B = \{\{a, b\}\}$


$AP\cap AZ\cap PZ$


$0 \rightarrow A_1 \rightarrow A_2 \rightarrow ... \rightarrow A_{n-1} \rightarrow A_n \rightarrow 0$


$a_n= n^2, b_n = \frac{-1}{n}, c_n = n$


$5 \cdot -5 = 1 \bmod{26}$


$m+n=p+n \implies m=p$


$\frac{\ln(1+x)}{\arcsin(x)}=\frac{\sqrt{1-y^2}}{1+y}$


$ = \left[     \begin{array}{ccc|c}       1&-2&3&2\\       1&1&1&k\\       2&-1&4&k^2     \end{array} \right] $


$P(n)=P(m)$


$d = xa + yb$


$I=\int_{-\pi}^{\pi}\dfrac{\sin nx}{(1+\pi^{x})\sin x}dx$


$s(\omega) = \sum \limits_{i = 1}^{n} \alpha_{i} \chi_{A_{i}}(\omega)$


$\mathbb P(E) = \frac 14$


$F(x) = \int_{0}^x f(t) \sin t dt$


$\lim _ {n\rightarrow \infty }\inf $


$f(x;θ) =  \dfrac{2(θ−x)}{θ^2}$


$ R = \{(x,y) \in \mathbb{R}^2: a \leq x \leq b, c \leq y \leq d\}. $


$f^\#f_\# f^\# =f^\#$


$I=\int_{-\pi}^{\pi}\dfrac{\pi^x \sin n(x)}{(\pi^x+1)\sin x} dx$


$\dfrac{1 - x^{2}}{1 - 2x\cos t + x^{2}}$


$-5<\frac1x\le1\stackrel{\text{mult. by}\;x^2}\iff-5x^2<x\le x^2\iff\begin{cases}-5x^2<x\iff x(5x+1)>0\\{}\\\text{and}\\{}\\x\le x^2\iff x(x-1)\ge 0\end{cases}$


$[X,Y] = XY - YX?$


$P(T_x\ge N|X_0=x_0)\le p.$


$\int_0^{\infty}\frac{f(x)}{x}=\int_0^{\infty}F(s)~ds$


$ \frac{9}{27}=\frac{1}{3}$


$ \|A\|_2 = \sqrt{\text{largest eigen value of } A^{\ast}A} $


$|f(z)|\leq |f(z_0)|$


$\left[\begin{array}{ccc|c}1 & 2 & 3 & 0\\0 & -1 & -2 & 0\\0 & 0 & 0 & 0\end{array}\right],$


$\lim\limits_{n\to\infty} \mu(X_n) = \mu(X)$


$(a+b, a^2-ab+1)=1$


$\det \begin{pmatrix} A & -B \\ B & A \end{pmatrix}$


$u=\frac{1-r^2}{1-2r \cos \theta + r^2}$


$M' M = M M' + [M, M']$


$X^n \setminus X^{n-1}$


$\mathbb{E}[W|B \leq b^*]$


$F(x)=\int_a^x f(t)\ \mathsf dt$


$N\oplus T$


$A_1 \times A_2 \times ...\times A_p$


$|A\times B|=mn$


$\alpha^+ < \beta$


$\gcd(a, a+2)$


$P(E) = |E| / |S|$


$ \begin{cases} x = \alpha \\ y = \beta \\ z = - \alpha - \beta \end{cases} $


$ \det(M)=\det(U^*U)=\det(U^*)\det(U)=\overline{\det(U)}\det(U)=|\det(U)|^2. $


$P(E) = 1/52$


$aRc\wedge cRa$


$|f'(z)| \leq |z^2+1|$


$P(k)\implies P(k+1).$


$ a =   \begin{cases}     x=-2+s       \\     y=2+s     \\     z=1-s   \end{cases} $


$ \int_\gamma \mathrm{d} s = \int_\gamma \sqrt{\mathrm{d}x ^2 + \mathrm{d}y^2 }.  $


$|f(z)| \le |\exp(z)|$


$\displaystyle {\frac{{dx}}{{ax + b}}}$ = $\displaystyle {\frac{{1}}{{a}}}$


$v=\frac{a+bx}{c+dx}$


$\vartheta(-u,-i\tau)=\vartheta(u,-i\tau)=\vartheta(u+1,-i\tau)$


$sin\frac{(sin\theta + tan\theta)}{2}=\frac{(sin\theta + tan\theta)}{2}$


$\displaystyle\sum_{n=1}^\infty \frac{1}{n^2}$


$\lim_{n \rightarrow \infty} t_n = \infty$


$T = A \oplus B$


$f(x) \, \delta''(x-a)=f(a)\delta''(x-a)-f'(a)\delta'(x-a)+f''(a)\delta(x-a)$


$\{x_\gamma\}_{\gamma\in\Gamma}\subseteq A$


$ \lim \limits_{k \rightarrow \infty } \int \limits_{-\pi}^\pi f(x)\cos(kx)dx = 0 \textit{ and } \lim \limits_{k \rightarrow \infty } \int \limits_{-\pi}^\pi f(x)\sin(kx)dx = 0 $


$\frac{ a_{n+1}}{a_n} = \frac{\left(1+\frac 1n\right)\ln\left(1+\frac{2}{2n+1}\right)-\frac 1n}{\ln\left(1+\frac{2}{2n-1}\right)-\frac 1n}$


$\sum_{n\geq N}\frac{1}{n^{\log\left(\log\left(n\right)\right)}}\leq\sum_{n\geq N}\frac{1}{n^{2}}<\infty.  $


$z=\sqrt[4]{2}e^{\frac{2\pi i}{3}}$


$\mathbb{P}[X_N = 1 | X_0 = 1]$


$\inf\{s_\gamma\mid \gamma<\alpha\}=\inf\{F_\gamma(0)\mid \gamma<\alpha\}=0$


$S=\sum_{n=-\infty}^{\infty}\sum_{m=-\infty}^{\infty}e^{-\sqrt{n^2+m^2}}$


$e^{-x}(\frac{x^{n+1}-x^{n}}{(1+x^{n+1})(1+x^{n})})$


$\gamma\geq\beta,\gamma\geq\alpha$


$\begin{align*} \sum_{n=2}^\infty \frac{\zeta(n)}{k^n} &= \sum_{n=2}^\infty \frac{1}{k^n \Gamma(n)}\int_0^\infty \frac{u^{n-1}}{e^u-1}du\\ &=\int_0^\infty\frac{1}{u(e^u-1)}\left( \sum_{n=2}^\infty \frac{1}{(n-1)!}\left(\frac{u}{k}\right)^n\right)du \\ &= \frac{1}{k}\int_0^\infty \frac{e^{\frac{u}{k}}-1}{e^u-1}du \tag{4} \end{align*}$


$P(2) \implies P(3)$


$f(x)=\sum_{k=1}^n a_k {\mathbf 1}_{A_k}(x)$


$\frac{(1+x)^{r}(1-(1+x)^{2(k+1)})}{1-(1+x)^2}$


$|f(z)| \le \max |f(z+r e^{it})|$


$ \left\{ \matrix{   f(0) = a \hfill \cr    f'(0) = \,\,b\omega  + \rho a\quad  \Rightarrow \quad b = \;{1 \over \omega }\left( {f'(0) - \,\,\rho f(0)} \right) \hfill \cr}  \right. $


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


$\displaystyle J = 3\int\frac{1}{(x^2+1)^2}dx\;,$


$0 \rightarrow A_1 \rightarrow A_2 \rightarrow ... \rightarrow A_{n-1} \rightarrow A_n $


$d(x,y)+d(y,s)\leqslant d(x,s)$


$G=(\{v\},\{e\})$


$n>3, n \in \Bbb N \;\;\text{and}\;\; A = \{1,2,3,...,n\}$


$27 +9 + 3 + (3-1) = 27 + 9 + (9-3) - 1 = 27 + 27 -9 -3 -1$


$(a-\gamma, a+ \gamma)\subset I$


$1-\frac{(1-r^2)(1-\rho^2)}{1+r^2\rho^2-2r\rho\cos\theta'}.$


$\displaystyle \zeta(s)-\frac{1}{s-1} = \sum_{n=1}^{\infty} \int_{n}^{n+1} (\frac{1}{n^s}-\frac{1}{x^s}) dx$


$\Bbb R^{n-1}\to\Bbb R^n$


$A^+$


$X_1\times\dots\times X_n$


$A_1 \times \dotsc \times A_n \to B$


$dL = \sqrt{1 + \left(\frac{dy}{dx}\right)^2} dx$


$a\leq x\leq b, ~~c\leq y\leq d$


$ a_m - a_n = (1 + \frac{1}{2} + \ldots + \frac{1}{m}) - (1+\frac{1}{2} + \ldots + \frac{1}{n}) \\ = (1 + \frac{1}{2} + \ldots + \frac{1}{n} + \frac{1}{n+1} + \ldots + \frac{1}{2n}) - (1+\frac{1}{2} + \ldots + \frac{1}{n}) \\ = (\frac{1}{n+1} + \frac{1}{n+2} + \ldots + \frac{1}{2n}) \geq  \frac{1}{2n} + \frac{1}{2n} + \ldots + \frac{1}{2n} = \frac{n}{2n} = \frac{1}{2} = \epsilon) $


$\sup_{x>a} |(1+\frac{x}{n})^n-e^x| $


$\lVert u \rVert ^2 = \lVert u_v \rVert ^2 +\lVert u_{\perp v} \rVert ^2= \lvert \frac{\langle v,u\rangle}{\langle v,v \rangle} \rvert ^2 \cdot \lVert v\rVert^2+\lVert u_{\perp v} \rVert ^2 \ge \frac{\lvert\langle v,u\rangle \rvert ^2}{\lVert v\rVert ^2}$


$\sum_{k=0}^n\frac{1}{k!}\geq \left(1+\frac{1}{n}\right)^n$


$f(a+h,b+k) = f((a,b)+(h,k)) = f((a,b)) + f((h,k))$


$f_0(n) = n\bmod 2$


$f([a]_6 + [b]_6) = f(a) + f(b)$


$\sum_{n=1}^\infty(-1)^na_n=\sum_{n=1}^\infty-\frac1n=-\sum_{n=1}^\infty\frac1n=-\infty$


$Cov(X + Y, X − Y) = Cov(X, X-Y) + Cov(Y, X-Y) = Var(X)- Cov(X,Y) + Cov(Y,X)- Var(Y)$


$ \| M \|_2 = \sqrt{\lambda _{max} (MM^T)}. $


$\frac{\sin\alpha}{a}=\frac{\sin \gamma}{c} \Rightarrow c\sin \alpha=a\sin \gamma \Rightarrow 18\cdot \frac{1}{2}=a\sin \gamma \Rightarrow a\sin \gamma=9 \ \ \ \ (2)$


$ M= \frac {Ar(1+r)^n}{(1+r)^n-1}$


$p_j \mid (p_1p_2\cdots p_n)$


$A_1\cap A_2\cap\dots\cap A_k\cap A_{k+1}$


$\mathbb{E}|X_n|^p \rightarrow \mathbb{E}|X|^p.$


$\lim_{n\to\infty}\frac{\sum\limits_{r=1}^{n}r^0}{\sum\limits_{r=1}^{n}r}$


$f(ab) = f(a) + \rho_V(a)f(b)$


$(y'+y)' = y'' + y' = y + y'$


$|f(z)+f(-z)|\leq|f(z)|+|f(-z)|\leq2|z|^2$


$[x,y] = [e_i,0]$


$S=\sum_{k=1}^{n-1}(n-k)\cos\frac{2\pi k}{n} = \sum_{k=1}^{n-1}k\cos\frac{2\pi k}{n}$


$ (x', y') = (x, y) + (a, b) \\ (x'-x , y' - y) = (a, b) \\ (x'-x) + (y'-y) = a + b $


$f(f(y)+f(x))=f(x)f(f(y))+f(f(x))+f(f(y))-xf(y).$


$U_0=\mathbb R^{N+1}\times \mathbb R^N$


$f(x_{2\space periodic})= \begin{equation} \begin{cases} &f(x), x<n\\ &f(x-n),n<x<2n\\ &f(x-2n), 2n< x < 3n\\ &... \end{cases} \end{equation} $


$y = \frac {a+b-x} {ab}$


$1 + 2 + 3 + \cdots = -\frac{1}{12},$


$d'(x,z) \leqslant d'(x,y) + d'(y,z)$


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


$ \frac{a-x}{x} = \frac{x}{b-x}  \;\Rightarrow\; x = \frac{ab}{a+b}.$


$∫_{0}^\infty xdf(x)$


$f(a + (-a))=f(a)+f(-a)$


$\begin{bmatrix} 2 &-m&-3&1 \\ 1 &-1&2&2\\ -1&3&m+1&1 \end{bmatrix} $


$\{\langle x,y\rangle\mid 1 \leq x \leq 2, y = 0\}$


$\lim_{n \rightarrow \infty}\frac{1}{n} \sum_{i=1}^{n}\frac{1}{i}=0$


$\frac{|x-27|}{x^{2/3} + 3x^{1/3} + 9} \leq \frac{|x-27|}{9}. $


$ \sum_{m=1}^{+ \infty} \sum_{n=1}^{+ \infty} \frac{1}{n^{\alpha}+ m^{\alpha }} = \sum_{m=1}^{+ \infty} O \left( \frac{1}{m^{\alpha -1}} \right)$


$\gamma\implies G\leq \frac{\min\left(\lfloor \frac{R}{\gamma}\rfloor,N\right)}{k}$


$\left\lfloor\frac{\lfloor 2a\rfloor}{2}\right\rfloor = \lfloor a\rfloor$


$a^{x\log_a b}=b^x$


$\int_0^{\infty}e^{-x^2}\frac{\sin(2nx)}{\sin x}{\rm d}x$


$ 2S=2+4+8+\dots+2^{n+1}+\dotsb=S-1 $


$e^{\frac{2ik\pi}{p}}\sqrt[p]t\notin\mathbb F_p(t)$


$\left[\begin{array}{ccc|c} 1 & 2 & -1 & a \\ 2 & 1 & 3 & b \\ 1 & -4 & 9 & 2b-3a \\ \end{array}\right].$


$|\mu * \lambda|(\mathbb{R}) \leq |\mu \times \lambda|(\mathbb{R}^2)$


$\mu(A_n)\xrightarrow{n\to\infty}\mu(A)\tag1$


$\gamma < a \implies \exists x\in E_1 < \gamma$


$=\sqrt2-\sqrt1+\sqrt3-\sqrt2+\sqrt4-\sqrt3+...+\sqrt9-\sqrt8=\sqrt9-\sqrt1=3-1=2$


$\sum\Vert x_n-y_n\Vert\le 1/2K$


$\lim_{n\to\infty}t_{n}\\=\lim_{n\to\infty}\frac{1}{n}\sum_{r=1}^{n}\frac{1}{\sqrt{r}}$


$ \int\frac{1}{\sqrt{x^2+1}}\,dx$


$\gamma^{4}+\gamma+1=(\gamma +b)(\gamma^{3}+d\gamma^{2}+e\gamma+\gamma)$


$ F(x):=\int_a^xf(t)\,\mathrm dt$


$\dot \gamma, \ddot \gamma, \dots, \gamma^{(m)}$


$y*x=y*(x*y)*x=(y*x)*(y*x)$


$\Delta{y} = f'(a) \Delta{x} + \epsilon \Delta{x} $


$\frac{\pi}{2z^2} = \sum_{m=0}^\infty \sum_{n=0}^\infty e^{-z\sqrt{m^2+n^2}}-\frac{1}{2}\sum_{n=0}^\infty e^{-zn}+\frac{1}{4}$


$A\times A\times A$


$ \sum_{n=1}^\infty \frac 1 {n^3} = 1 + \sum_{n=2}^\infty \frac 1 {n^3} \le 1 + \sum_{n=2}^\infty \int_{n-1}^n \frac 1 {x^3}\,dx = 1 + \int_1^\infty \frac 1 {x^3} \, dx = 1 + \frac 1 2.   $


$\gamma(s)=(\gamma_1,\gamma_2,\gamma_3)$


$\frac{1}{n}\sum_{i=1}^n X_i^k = E\left(x\right)$


$s:\left\{\begin{array}{ll} x\mapsto y\\ y\mapsto x\\ \lambda\mapsto \lambda \text{ for } \lambda\not\in\{x,y\} \end{array}\right.$


$\frac{\bigl|\langle v,u\rangle\bigr|}{\|u\|}$


$f(n)=\binom n2$


$A_1 \times A_2$


$\mathbb{R}[x]/(x^2+1) \cong \mathbb{C}[x]$


$\;d=gcd(x,m)\iff \begin{cases}x=x'd\\{}\\m=m'd\end{cases}\implies a=x-mk=(x'-m'k)d\implies d\mid a$


$\left\{\begin{pmatrix} a & -b \\ b & a \end{pmatrix} \,\middle |\, a, b \in F\right\} \subset M_2(F)$


$ \int_a^bf(t)dt = F(a) - F(b) \\ \int_a^xf(t)dt = F(x) - F(a) \\ \int_x^bf(t)dt = F(b) - F(x) \\ $


$I_n=\int_0^{\frac{\pi}{2}}\frac{\sin(nx)}{\sin x}dx$


$ f(x)= \left(\frac{11}{12}+O(e^{-x})\right)^{1/3}. $


$P(X ≤ 11) = 1/6$


$\begin{cases}\dot\gamma(t)=\tilde{X}(\gamma(t))=a-\langle{a,\gamma(t)}\rangle\gamma(t),\\\gamma(0)=p.\end{cases}$


$\frac{ds}{dx} = \sqrt{1 + \left(\frac{dy}{dx}\right)^2}$


$a_n = \dfrac{(n+2)(n+1)}{2}$


$\delta|x^{n-1}+...+a^{n-1})|<\delta M<\epsilon$


$-y''(y'-y'')^3 + y'''(y''-y''') = y''''(y'-y'')$


$P(X_t\in A|\mathcal{F}_s)=P(X_t\in A|X_s)$


$ \begin{cases} y=22+(-x-2z)\\ x=3\\ z=7 \end{cases}\Longleftrightarrow $


$x^2 +y^2 +ax +by = c$


$\vec{m} = \begin{cases} x = 0 \\ y = 0 \\ z = m \end{cases}$


$F_i \subseteq V_i \subseteq \overline{V_i} \subseteq U_i$


$f(x+f(y)) = f(f(x)) + f(y)$


$\frac{f}{||f||} = \frac{g}{||g||},$


$\int_{0}^{\pi}f(t) dt=2\int_{0}^{{\pi}/{2}} f(t) dt.$


$S_n=\frac{X_1 + X_2 + . . . + X_n}{n}-n$


$\int_0^\pi f(x) \sin(x) dx= \int_0^\pi f(x) \cos(x) dx=1.$


$\nabla f(x,y,z)\equiv(1,1,1)$


$f_X(x)=\frac{1}{2\sqrt{2\pi}}e^{\frac{-(x-1)^2}{8}} \text{for all x real numbers}$


$x^2-x\sin x, x \in [0,\pi/2]$


$ax'+by'=d$


$ \lim_{N \to \infty} \sum_{n=1}^N \frac 1n. $


$ \text{Card}(A) > 2 \implies \text{Card}(A) \geq \text{Card}(\mathbb{R}).$


$ \begin{matrix}a&-b\\a&b\end{matrix}$


$y \in W, x \in V \subseteq \overline V \subseteq U$


$\gamma = \pm \delta \left[\int \frac{d p d\phi_p}{(2\pi)^2}qp^2(1-\cos(\phi_q -\phi_p))^2 \pi a^2 e^{- (q^2+ p^2 -2 qp \cos(\phi_q -\phi_q)) a^2/4}\right]^{1/2}.$


$\lim_n \frac{1}{n} \sum_{k=1}^{[\frac{n}{2}]} \cos (\frac{k \pi}{n}) $


$ x^3~ \sin (nx),~~n \in  (\mathbb{N}) $


$f(c^*) = f(a) + f(b)$


$ \lim_{n\to\infty}\frac1n\sum_{k=1}^n\frac1k=0 $


$gcd(a+b, a-b)\geq gcd(a,b)$


$[x,y]=y$


$\mathbb{R}^n\longrightarrow \mathbb{R}^n$


$(6/9)^x + (4/9)^x$


$\forall \epsilon > 0, ~ \exists \delta > 0 ~ : ~ |x-x_0|<\delta \Rightarrow |f(x) - f(x_0)| < \epsilon$


$\frac{11!}{4!} - \frac{10!}{4!} = \frac{11!-10!}{4!}$


$\gamma[1,0,1]+\delta[0,1,1]=[\gamma,\delta,\gamma+\delta]$


$a\,b=|BE|$


$P[X \geq x]$


$H(x) = \int^{x}_{a} f(t)dt$


$x=\{x,t\}$


$f_X(x;\theta)=\theta\cdot x$


$ f(x,y,z) = \text{det} \begin{bmatrix} x^2&y^2&z^2\\ x&y&z\\ 1&1&1 \end{bmatrix},$


$b_n = \left(\frac 1 2 \right)^{n/2}$


$I=\int_{0}^{\pi}\frac{(\pi-x)\sin xdx}{1+\cos^2 x}$


$\langle x \rangle ^{\perp} = \langle y \rangle ^{\perp} \Rightarrow \left( \langle x \rangle ^{\perp} \right)^\perp = \langle x \rangle = \left( \langle y \rangle ^{\perp} \right)^\perp =  \langle y \rangle, $


$ (\gamma^3+1)(\gamma^2+1)=\gamma^{14}\cdot\gamma^8=\gamma^{22}=\gamma^7=\gamma^3+\gamma+1. $


$\alpha+1\mapsto\begin{cases}\{\beta,A\},&\beta=\gamma\\\{\beta+1,\gamma\},&\beta<\gamma\end{cases}$


$P\times T$


$\epsilon^+$


$   \gamma(s)^t=\gamma(s)\Rightarrow\gamma(s)^t\gamma(s)^{-1}=id $


$\sum (b_n-a_n)<1$


$P(m) := \forall n \ (0 < m \to [(p(m) < p(n)) \leftrightarrow (m < n)])$


$\forall \epsilon \gt 0, \exists \delta_n \gt 0$


$x = a, x = b, y = a$


$ \left \lfloor{\frac{\left \lfloor{\frac{n}{2}}\right \rfloor}{2}}\right \rfloor= \left \lfloor{\frac{n}{2^2}}\right \rfloor$


$ax+by=N.$


$\mbox{cov}[X+Y,X-Y] = \mbox{cov}[X,X]+\mbox{cov}[Y,X]-\mbox{cov}[X,Y]-\mbox{cov}[Y,Y].$


$\vert f(t)\phi(\theta)\vert\le M\vert f(t)\vert$


$[x,y]=$


$G(t,s)=\begin{cases} s(1-t)~ 0\leq s\leq t\leq 1\\ t(1-s)~ 0\leq t\leq s\leq 1\end{cases}$


$d(x,A) \leq d(x,y) + d(y,z)$


$\mathbb{R}^n \times \mathbb{R}^n$


$γ˜=M(\gamma)=P\gamma+a$


$ \sum_{n=1}^\infty ||e_n - f_n||^2 < 1 $


$\sum_{n=1}^{\infty}(-1)^n < \infty$


$x^2-x\in I$


$d(x,z)\preceq d(x,y)+d(y,z)$


$L=\lim_{x \rightarrow c^{+}} f'(x)=\lim_{\epsilon \rightarrow 0} f'(c_2(\delta(\epsilon), f))=\lim_{\epsilon \rightarrow 0} \frac{f(c+\delta(\epsilon))-f(c)}{\delta}=\lim_{x \rightarrow c^{-}} \frac{f(x) - f(c)}{x-c}$


$=n\sum_{k=1}^{n-1}\cos\frac{2k\pi }n-\sum_{k=1}^{n-1}k\cos\frac{2k\pi }n$


$F(x) = \int_a^x g(t) dt $


$ \sum_{n=1}^{\infty}\frac{{(-1)}^{n-1}}{{n}^{x}}=\frac{1}{\Gamma(x)}\int_0^\infty\frac{u^{x-1}}{e^u+1} \mathrm{d}u, \quad x>1. $


$X = \left\{x_1,x_2,\cdots,x_n\right\}$


$z = \sqrt[n] r e^{{(i\theta+2\pi m)}/n}$


$K^+$


$y^{y^{y^{y^{.^{.^.}}}}}$


$\forall x\in X \forall y\in X :(xRy \vee yRx)$


$xRy \rightarrow yRx$


$ \left\|\pmatrix{A_1x\\ A_2x} \right\| \geq \max \{\|A_1x\|,\|A_2 x\|\} $


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


$(xRy \vee x=y)$


$\mathsf{AD}^+$


$n!=\sqrt{2\pi n}\,\left(\frac{n}{e}\right)^n\!\left[1+\mathcal{O}\left(\frac{1}{n}\right)\right],\quad n\rightarrow\infty.$


$ \begin{split} \sin(e^u)&=(D+b)^{-1}\bigl[e^u\cos(e^u)+b\sin(e^u)\bigr]\qquad(**)\\ &=e^u(D+b+1)^{-1}\cos(e^u)+b(D+b)^{-1}\sin(e^u). \end{split} $


$dv = \int_a^x f(t)\,dt$


$ \lim_{n\rightarrow\infty}\|x_n\|_{\ell^1}=\lim_{n\rightarrow\infty}\sum_{i=1}^\infty |x_{i,n}|=0\qquad ? $


$f(x)= \dfrac{e^x}{1+9e^x}$


$\gamma(s) = (\gamma^0(s), \gamma^1(s), \gamma^2(s), \gamma^3(s))$


$h \mid g_1 g_2$


$T_{\mathbb{C}}M = T^{1,0}M \oplus T^{0,1}M$


$\mathbf{T} = ({dx \over ds}; {dy \over ds})$


$ \mathbb{P}\{X \geq 2\} \leq \left(\frac{e^\gamma}{(1+\gamma)^{1+\gamma}}\right)^{\frac{2}{1+\gamma}} = \left(\frac{e^{\frac{\gamma}{1+\gamma}}}{1+\gamma}\right)^{2} = \Theta\!\left(\frac{1}{\gamma^2}\right) = O\!\left(\frac{1}{C^2}\right) $


$Y = \mathbb{R}[x]/(x^2+1)$


$S = \{a_1,a_2,...., a_n\}$


$\sum_{n=-\infty}^{-1}\frac{1}{n^2}=\sum_{n=1}^{\infty}\frac{1}{n^2}$


$Cov(x, y) = E(XY)$


$f(n) = n^2 + n$


$ Y := \left\{ (x,y) \in \mathbb{R}^2 \; : \; a \leq x \leq b, \; R \leq y \leq f(x) \right\} $


$p_1q_1 \mid m$


$T_a = \{(x,y)\in\Bbb R^2\mid 0\leq x\leq a, 0\leq y\leq a-x\}$


$\{\emptyset,\{a\}\}=\{\{a\}\}$


$Cov(X+Y,Y+Z) = Cov(X,Y)+Cov(X,Z)+Cov(Y,Y)+Cov(Y,Z)$


$\phi(n) = n^{-s}$


$ x\mapsto F(x) =\int_a^x f(t)dt ~~~~ $


$ 1-\cos (xy) \le\int_0^xf(t) \sin {(tf(t))}dt + \int_0^y f^{-1}(t) \sin{(tf^{-1}(t))} dt .$


$(-sin(\varphi), cos(\varphi))$


$\sum_{n=0}^\infty \frac{1}{an^2+bn+c}= \frac{1}{a}\sum_{n=0}^\infty \frac{1}{(n-z_0)(n-z_1)} $


$Â(z) = z\sum_{j\ge0}(e^z - e^{(1 - 2^{-j})z})$


$(1 - \cos A)^2 + \sin^2 A \over \sin A(1 - \cos A)$


$xRy \land yRz$


$f(f(y))+f(x-y)=f(xf(y)-x)$


$\lim_{n\rightarrow\infty} \int_{-\pi}^\pi x^2 \frac{\sin(2nx)}{\sin x} dx$


$ \sum_{n\geq 1}\sum_{k\geq 1}\frac{1}{n^2+k^2}=\infty. $


$\sum_{k=1}^{N} \frac1{k^2} = \frac{\pi^2}{6} - \sum_{k=1}^{K} \frac{a_k}{N^k} +  O\left (\frac1{N^{K+1}}\right )$


$\int_0^1 x^5 (\sin^{-1}x) \, dx$


$f_{Y_{(5)}}(y) = 5y^4/\theta^5, \quad 0 \le y \le \theta.$


$d(x,A) - d(x,y) \leq d(y,A).$


$I=\int_{-\pi}^{\pi}\frac{\cos x}{1+2^x}dx$


$\lim_{k \to \infty}|\frac{{(k+1)^3}(x-3)^{(k+1)!}3^{k!}}{k^3(x-3)^{k!}3^{(k+1)!}}|$


$\frac{m}{d} \in D^{-1}M$


$f(x) = \sum_{j=1}^k c_j 1_{A_j}(x)$


$  = \frac {log_{c}x}{nlog_{c} b}$


$\begin{eqnarray}   (1)\quad& F(m,k,b) &=& 0 \quad \text{if } b < k-1 \text{ or } m < \tfrac{k(k-1)}{2} \text{ or } m > \tfrac{b(b+1) - (b-k)(b-k+1)}{2} \\   (2)\quad& F(m,k,b) &=& 1 \quad \text{if } b \geq m \text{ and } k=1 \\   (3)\quad& F(m,k,b) &=& 1 \quad \text{if } b \geq k-1 \text{ and } m = \tfrac{k(k-1)}{2} \text{.} \end{eqnarray} $


$\{e_1,\, e_2,\, e_3,\, e_1e_2,\, e_2e_3,\, e_1e_3,\, e_1e_2e_3\}$


$f(x)=\frac{2x}{\theta^2}, 0\leq x \leq \theta$


$(*) \,\,\,\,\,\,(\gamma - a) \cdot (\gamma - a) = r^{2} $


$d(x,z)\leq d(x,y) + d(y,x)$


$F(x)=\int_{a}^{x}f(t)dt\int_{a}^{b}f(t)dt$


$[T]_D=\begin{bmatrix}T_{1,1} & T_{1,2} \\ T_{2,1} & T_{2,2}\end{bmatrix}$


$ \lim\limits_{n\to\infty}\mu(B_n)\leq  \lim\limits_{n\to\infty}\inf\limits_{k\geq n}\mu(A_k)=\liminf\limits_{n\to\infty}\mu(A_n)\tag{1} $


$\operatorname{frac}(x+y)=\operatorname{frac}\big(\operatorname{frac}(x)+\operatorname{frac}(y)\big)$


$ax+by=d,$


$\text{im}(\gamma)\subseteq A$


$y=(40, -60, 40, -60, -10, -10, -10,...)$


$e^x\ge \left(1+\frac xn\right)^n\ge 1+x$


$(a_n)^2 = \left(\dfrac{(-1)^n}{n^{1/2}}\right)^2 = \dfrac{1}{n}$


$\lim_{n\to\infty}n\sum_{i=1}^n\frac{1}{(n+i)^2}=\frac{1}{2}.$


$f(a+f(b))=f(a+b)+f(b)$


$\bigg\lfloor\frac{[x]}{m}\bigg\rfloor=\bigg\lfloor\frac{x}{m}\bigg\rfloor$


$\sum_{r=1}^{n} \frac{r}{2^r}$


$f(x) = \frac{2^x}{2^x - 8} - 2$


$ X''(x)Y(y)=X(x)[Y(y)-Y''(y)] $


$\sum_{n=1}^{\infty} \|x_n -y_n\| < 1$


$\sum_{m=1}^k\cos{4m\pi\over k}=\sum_{m=1}^k\mathfrak{R}(e^{4im\pi\over k})=\mathfrak{R}\left(\sum_{m=1}^ke^{4im\pi\over k}\right)$


$E[X_{(1)}]=\frac{\alpha}{\alpha+\beta}=\frac{1}{6} $


$|f(x_0)|\le 2x_0|f(x_2)|$


$\displaystyle\sum_1^n\sin\frac{2k\pi}n=0$


$=\frac{1}{2\sqrt 2}\lim_{n\to \infty}\sum_{r=1}^{n}\frac{1}{\sqrt {nr}}$


$\int_0^\infty f(x) dx=\int_0^\infty f(x)^2 dx$


$\big([x,y,z], (x^2+y^2+z^2)\big)$


$P^\mu[X_{S+t}\in\Gamma|\mathcal F_{S+}]=0$


$P[X_n=x] \leq P[X_n\leq x] - P[X_n \leq y_i] $


$K_1 = \frac{\gamma + \beta - r}{2\gamma}\\K_2 = \frac{\gamma - \beta + r}{2\gamma}\\w(i) = \frac{(\beta + \gamma - r)(\beta + \gamma)^i - (\beta - \gamma - r)(\beta - \gamma)^i}{2\gamma}$


$4 ^{x} +6 ^{x ^{2}} =5 ^{x} +5 ^{x ^{2}}$


$|G|=rs$


$\int{1\over{(1+x^2)^n}}dx, n\in \mathbb{Z}^+$


$M=N\oplus T$


$f_Y(y) = \frac{2}{y^3}$


$I_n = \pi^{-1}A_n(\pi^{-2})   $


$ A^T = A^* = (UDU^*)^* = UD^*U^* = UDU^* = A$


$x = \pmatrix{1\cr 1\cr-2\cr}$


$I_r = \{s\in R\mid rs = 0\}$


$b=\{\{\varnothing\}\}$


$d_1 \mid a$


$A_1 \ldots A_p$


$\int_{-x}^{1-x}f(x+t)Q_n(t)dt = \int_{0}^{1}f(t)Q_n(t-x)dt $


$\mu(A) = \lim_{n \to \infty} \mu(A_n)$


$\sum_{n\in \mathbb{Z}}e^{-n^2\pi a}=\frac{1}{\sqrt{a}}\sum_{k\in \mathbb{Z}}e^{-k^2 \pi / a}.$


$z^+$


$\int _0 ^\infty f(x) dx=0$


$P(n - 1) \implies P(n)$


$B = \{B_1, B_2, B_3,\cdots, B_n\}$


$p(V)\subseteq p(U)$


$\sum_{k\geq 0}\frac{1}{(k+a)^s} \stackrel{\mathcal{L}^{-1}}{=}\frac{1}{\Gamma(s)}\int_{0}^{+\infty}\frac{e^{(1-a)u}u^{s-1}}{e^u-1}\,du $


$S=E-P$


$
 a_n=-3+{n(n+1)\over 2},
 $


$x\mathcal{R}y \land y\mathcal{R}x\implies x=y$


$\alpha^+\cdot\beta^+=\alpha^+\cdot\beta+\alpha^+$


$K: \begin{cases} x=t+1  \\ y=-t+2s-1 \\ z=s-1 \end{cases}$


$L^+$


$\lim_{N\to\infty}\sum_{k=1}^N\dfrac{k^r}{N^{r+1}}=\lim_{N\to\infty}\dfrac1N\sum_{k=1}^N\left(\dfrac kN\right)^r$


$P(1)\implies P(2)$


$xRy ∧ yRx$


$y'/y = y'''/y''$


$\int_0^x f_1(t) dt = \int_0^{Y(x)} f_2(t) \sin(t) dt$


$A=\begin{bmatrix}1&1&1\\1&\omega^2&\omega\\1&\omega&\omega^2\end{bmatrix},$


$ab = |ab|$


$W^{1,1}\cap W^{1,\infty}$


$\mathbb{R}^{n+2}\rightarrow \mathbb{R}^n$


$\rm\:p_1 p_2.$


$\inf\{F_\gamma(x)\mid\gamma\le x<\alpha\}=\inf(\{t_\gamma\}\cup\{s_\gamma\mid\gamma<\alpha\})=0$


$\frac{ax+b}{x+c}$


$\begin{align}xRy, \;yRz&\implies x-y,y-z\in\mathbb{Q} \\ &\implies (x-y)-(y-z)\in\mathbb{Q} \\ &\implies x-z\in \mathbb{Q} \\ &\implies xRz.\end{align}$


$0 \to F_{1,n-1} \to F_{0,n}  \to E_{\infty}^{0,n} \subset E_2^{0,n}$


$P(E_2) = 1/6$


$ \begin{cases} x=t\\ y=0\\ z=0 \end{cases} $


$ B=\{A_1,A_2,...,A_{n-1},Ab\}\subset\mathbb R^{n-1}, $


$a_n =2\left(\frac{\binom{2^{n-1}}{2^{n-2}}}{\binom{2^n}{2^{n-1}}}\right)^{1/2^{n-1}}$


$\vartheta, \vartheta'$


$X=\frac{\frac{c(1+r)^{T+1}+(1-c)r^2}{r^2(1+r)^{T+1}}}{\frac{(c-r)(1+r)^{T}+(1-c)r}{r(1+r)^{T}}}$


$\begin{pmatrix} {1}, {0}, {0}  \\ {0}, {0}, {1}\end{pmatrix}$


$\displaystyle\sum _{k=1}^{n-1} (n-k)\cos\frac{2k\pi}{n}= $


$ \lim_{n \to \infty} \frac {\sqrt{n+1}+\sqrt{n+2}+...+\sqrt{2n}}{n^{3/2}}$


$P(S_t\gt x\mid\mathcal F_0)=M_0/x$


$\lim_{n\to\infty}n\sum_{r=2}^{n+1}\frac1{(n+r)^2}$


$\sum_{k=1}^{\frac{n-1}2} \frac{1}{\cos^2 \frac{k\pi}{n}}$


$a_n=\frac{n^2}{2}+\frac{n}{2}=\frac{n(n+1)}{2}$


$R_3 = \left(\dfrac{1}{k^2-1}\right)R_3$


$9^x - 6^x - 2\cdot 4^x = 0 $


$\lim_{n\rightarrow \omega}\ |x_n-p|=\lim_{n\rightarrow \omega}\ |x_n-q|=0$


$ 1+2+3+4+5+\cdots=-\frac{1}{12}.$


$\sum_{n\in \Bbb N}=1+2+3+4+\dots =-\frac 1{12}$


$ \begin{align} \sum_{j=1}^\infty\sum_{k=1}^\infty\frac1{j^2+k^2} &\ge\sum_{j=1}^\infty\sum_{k=1}^\infty\frac1{(j+k)^2-(j+k)}\\ &=\sum_{j=1}^\infty\sum_{k=1}^\infty\left(\frac1{j+k-1}-\frac1{j+k}\right)\\ &=\sum_{j=1}^\infty\frac1j\tag1 \end{align} $


$ a \le x \le b \\ c \le y \le d $


$\displaystyle F(x+k)=\int^{x}_{a}f(t)dt+\int^{x+k}_{x}f(t)dt$


$(a+b,a-b).$


$\lim_{x\rightarrow c^{+}} f(x)=\lim_{x\rightarrow c^{-}} f(x)=f(c)$


$a^k\equiv a_s^k \equiv a_s^{k\ \bmod\ S}\not\equiv 1\pmod s$


$\int_0^\pi f(t) \sin t dt=-f(t)\cos t + \int_0^\pi f'(t) \cos t dt\\ =-f(t)\cos t + f'(t)\sin t - \int_0^\pi f''(t) \sin t dt $


$\gamma^{4} + \gamma+1=(\gamma)(\gamma^{3}+d\gamma^{2}+e\gamma+\gamma)$


$V_1\subseteq V_2^c\subseteq U$


$f_R(r)=F'(r)={2r\over a^2}\ ,$


$f(xy)=xf(y)+f(x).$


$P(X_{16}=2 \mid X_0=0)$


$\mathbb{R}^n \rightarrow \mathbb{R}^{n+1}$


$p_1 p_2 = p_3 p_4$


$ \limsup_{n\to\infty}  \left|\int_0^\pi f(x+t)F_n(t)\,dt - \int_0^\delta f(x+t)F_n(t)\,dt\right| \leq \limsup_{n\to\infty}\int_\delta^\pi|f(x+t)|F_n(t)\,dt = 0 $


$p_1 p_2 p_3 \cdots p_{n-1}$


$ \left[\begin{array}{ccc|c}  1 & 2 & 1 &  3\\ 1 & 1 & 1 &  1/5\\ 0 & 0 & 1 &  -k+42/5 \end{array}\right] $


$\sum_{n \leq x} \frac{1}{n} \lfloor \frac{x}{n} \rfloor = \sum_{n \leq x} \frac{x}{n^2}  + O(1) = x \sum_{n =1}^{\infty} \frac{1}{n^2}  + \sum_{n \leq x} O(1) - \sum_{x}^{\infty} \frac {1}{x} =  x \space \zeta(2)  + \sum_{n \leq x} O(1) - O(\log (x))$


$\mathcal{B}= \{(x,y): a<x\leq b, c<y\leq d\}$


$=\int_0^\pi(\pi-t)f(\sin t)dt$


$\gamma \wedge \gamma = - \gamma \wedge \gamma$


$p\mid k_1k_2$


$ \begin{align} \int_0^1\frac{\tan^{-1}(x)}{1+x}\,\mathrm{d}x &=\int_0^1\frac{\frac\pi4-\tan^{-1}(y)}{\frac2{1+y}}\frac{2\mathrm{d}y}{(1+y)^2}\\ &=\int_0^1\frac{\frac\pi4-\tan^{-1}(y)}{1+y}\,\mathrm{d}y\\[4pt] &=\frac\pi4\log(2)-\int_0^1\frac{\tan^{-1}(y)}{1+y}\,\mathrm{d}y\\ \end{align} $


$\sum\limits_{r=1}^n \cot \frac{r\pi}{n+1}=0$


$X = \{ \{a \} \} = \{ \{c\} \}$


$P(X_t\in A\mid\mathcal{F}_s)=P(X_t\in A\mid X_s)$


$.2, .4, .4 = \frac{2}{10}, \frac{4}{10}, \frac{4}{10}$


$\nabla_{\dot{\gamma}} \dot{\gamma} - \frac{g(\nabla_{\dot{\gamma}}\dot{\gamma}, \dot{\gamma} )}{g(\dot{\gamma}, \dot{\gamma})}\dot{\gamma} = 0.$


$C_R \subseteq C_R'$


$F(x)=\int_{\infty}^x f(t) dt$


$\lambda = \frac{\alpha}{||F||}$


$i=\lbrace a+b,a+2b,...,a+nb\rbrace$


$\lim_{n\to\infty} \frac{\sqrt{n+1}+\sqrt{2(n+2)}+...+\sqrt{n(n+n)}}{n^2}$


$f(t)=(t^p-x^p)(t^p-y^p)$


$ \det \begin{pmatrix} A & B\\B& A\end{pmatrix}=\det \begin{pmatrix} A & B\\B-A&  A-B\end{pmatrix}\\ =\det \begin{pmatrix} A+B & B\\0&  A-B\end{pmatrix} =\det(A-B)\det(A+B) $


$f: \mathbb{R}^n \rightarrow \mathbb{R}^{n+1}$


$ r(t)= \begin{cases} x=t\\ y=1-t\\ z=2t^2-2t+1 \end{cases}\quad \text{where }t\in[-1,1] $


$f(x-y+f(y))=f(x)+f(y)$


$\mathcal{C}^{+}$


$\frac 1{2^{i+2}}$


$\frac{1}{a+(a^{2}-1)^{1/2}}=\frac{(a-(a^{2}-1)^{1/2})}{(a-(a^{2}-1)^{1/2})(a+(a^{2}-1)^{1/2})}=\frac{a-(a^{2}-1)^{1/2}}{a^{2}-(a^{2}-1)}=a-(a^{2}-1)^{1/2}$


$x^4+ \frac{1}{x^4} +y^4+ \frac{1}{y^4}$


$a_{k_n}=(2+1)\frac{2n}{2n+1}=6\frac{1}{2+\frac1n}\to\frac{6}{2}=3$


$[x, y]^3=[x, y]$


$(M^{s})^{a}\equiv M^{sa}\equiv M^{sa}\times 1\equiv M^{sa}M^{x(p-1)}\equiv M^{sa+x(p-1)}\equiv M^{1}\equiv M(\mod p)$


$ \left( \begin{array}{cc} AB & 0\\ B & 0\\ \end{array} \right) $


$ f(2f(x)+f(y)) = 2x+f(y) $


$\Gamma(x) = \sqrt{2\pi(x-1)}\left(\frac{x-1}{e}\right)^{x-1}$


$\frac{3}{x} +\frac{4}{y} +\frac{5}{z} =6$


$g(x)=\frac{1}{2} e^{-2}(x-e^2+1)+1.$


$M^{n}=M*M*M*...*M$


$\langle f,g\rangle = \int_{-1}^{1}f(t)g(t)dt = \int_{-1}^{1}f(t)\cdot\left(\delta(t-t_0)+\delta(t+t_o)\right)dt$


$y' = -y+\frac{10}{4}\frac{xy}{1+x}$


$g\in W_0^{\alpha,1}(0,T)$


$ax+b\color{red}{y}=d$


$\exists f_1,f_2\in M \forall d\in M$


$aX+bY = d$


$\lim_{n\rightarrow\infty} X_n(\omega)\rightarrow X(\omega)$


$ P(E\cap E) = P(E)P(E)$


$\sum_{p}\frac{1}{p^{2s}}\leq \sum_{n}\frac{1}{n^2}$


$ 1 + \sum_{n \ge 1} \frac{1}{n^2}. $


$S^n \subset \mathbb R^{n+1}$


$\Vert e_n-c_n\Vert^2=r_n^2$


$\sec\theta = h/2$


$m<n,f^{m}\in \langle x \rangle$


$\Gamma(x)\zeta(x) = \sum_{n=1}^\infty\frac{\Gamma(x)}{n^x} = \sum_{n=1}^\infty\int_0^\infty \frac{t^{x-1}}{e^{nt}}dt = \int_0^\infty t^{x-1}\sum_{n=1}^\infty\frac{1}{(e^{t})^n}dt = \int_0^\infty\frac{t^{x-1}}{e^{t} - 1}dt$


$f_3(n) = n^3$


$\zeta (s) = \frac{\Pi(- s)}{2 \pi i} \int_{+ \infty}^{+ \infty} \frac{(- x)^s}{e^x - 1}\frac{dx}{x}$


$\frac{1}{x} + \frac{1}{y} = \frac{1}{ab}$


$\int_\gamma\omega=\int_{\partial\gamma} f=f(\gamma(1))-f(\gamma(0))$


$\int \frac{1}{(1+x^2)^2}dx$


$\alpha=\{\langle{x},y\rangle\in{\mathbb{R}\times\mathbb{R}}: x+2y=1, 0\lt{x}\le{1},0\le{y}\lt{1/2}\}$


$a^{\log_a(x)} = x$


$\gamma(a \wedge b) = \gamma(a) \vee \gamma(b)$


$\displaystyle \left|e^x-\left(1+\frac xn\right)^n\right|<\epsilon$


$b^{\log_b(a)} = x$


$\{a, a+d, a+2d, a+3d\}$


$\tan\theta = \frac{x}{2} => x = 2\tan\theta => dx = 2\sec^2\theta d\theta$


$\begin{align} [X, Y] = XY − YX. \end{align}$


$\sin(\theta)=\frac{\sin^2(\theta)}{\sin(\theta)}=\sin(\theta)$


$ \lim_{n \to \infty} \frac{\sqrt{1} + \sqrt{2} + ... + \sqrt{n}}{\sqrt{n}}\frac1n= \lim_{n \to \infty} \left(\sqrt{\frac1n} + \sqrt{\frac2n} + \sqrt{\frac3n} +\cdots+\sqrt{\frac{n}{ n}} \right) \frac1n $


$A = V \oplus W$


$x = c_1e_1 + \ldots + c_ne_n$


$\partial_i\gamma\partial_i\gamma=\partial_i\partial_i\gamma\gamma$


$\left[\begin{array}{ccc|c} 1 & 0 & 1 & 1 \\ 1 & -1 & 1 & 1 \\ -2 & 1 & -2 & k \\ \end{array}\right]$


$Cov(X,Y) = 3$


$A \cap B = \emptyset: \mu(A \cup B) = \mu(A) + \mu(B)$


$\alpha_{m}\le\dfrac{(n-1)^{m-1}+(-1)^m}{n(n-1)^{(m/2)-1}}$


$r_3 > 2^{k-1} $


$f(x) = \dfrac{(1+x)^3}{x \, (1+x^2)}$


$ \left[     \begin{array}{ccc|c}       1&-3&0&6\\       1&0&3&-3\\       2&k&3-k&1     \end{array} \right] $


$ \begin{array}{ccccc} \text{order}    & 1& 3& 5& 15 \\ \# \text{elems} & 1& 2& 4& 8  \end{array}$


$cI = \gamma'\gamma I \subset \gamma'R \subset R$


$f(f(x)) = x f(x) - F(x)$


$S_n=\sqrt1+\sqrt2+\cdots+\sqrt n$


$\int\tan^nx\ dx=\frac{\tan^{n-1}x}{n-1}-\int\tan^{n-2}x\ dx$


$\lim_{n\rightarrow\infty} \frac{1}{\sqrt{n}}\sum_{k=1}^n\frac{1}{\sqrt{n+k}}$


$\sum\limits_{n = 1}^\infty \sum\limits_{m = 1}^\infty \frac{m^2}{(n^2 + m^2)^4}$


$ P(t^*>t|H_0) $


$\lim_{x\to c}f'(x)=L$


$(1+a(n-1))(1+a) \ge 1+a(n)$


$\mathbb P\left[X_n\to x \text{ as } n\to\infty\right]=1$


$f(y+f(x))=f(x)f(y)+f(f(x))+f(y)-xy.$


$|f(z)| \le M |T(z)|$


$\begin{align}    \binom{5 + 5}{5, 5} = \binom{10}{5} = 252 \end{align}$


$f(n)={n\choose{\log{n}}}$


$S_n = \frac{(n+1)n}{2}$


$f(a*b) = f(a) + f(b)$


$\textbf{G}(w_1,x_{\gamma},y_{\gamma},z_{\gamma})=0\\ F_2(x_{\gamma},y_{\gamma},z_{\gamma})=0\\ F_3(x_{\gamma},y_{\gamma},z_{\gamma})=0$


$\begin{array}\\ t(m) &=1+2a+3a^2+...+(m-1)a^{m-2}\\ &=((m-1) a^{m+1}-m a^m+a)/((a-1)^2 a)\\ &=(ma^m(a-1)-a^{m+1}+a)/((a-1)^2 a)\\ \end{array} $


$\mathbb{R}^{n+m} \to \mathbb{R}^n$


$T^{1,0}M\cap  T^{0,1}M\not=(0)$


$d(x,\alpha)\leqslant d(x,y)+d(y,\alpha)<\frac{2}{m}$


$B \subseteq A \subseteq N$


$p(E) = 1/2$


$e^{\sqrt{\frac{m}{2}}(1+i)}-e^{-\sqrt{\frac{m}{2}}(1+i)}=e^{\sqrt{\frac{m}{2}}}[\cos \sqrt{\frac{m}{2}}+i \sin\sqrt{\frac{m}{2}}] -e^{-\sqrt{\frac{m}{2}}}[\cos \sqrt{\frac{m}{2}}-i \sin\sqrt{\frac{m}{2}}]$


$(a_n, b_n) \subset U_n$


$\det ((a-b)I + b e e^T) = (a-b)^n \times \left( 1 + n \dfrac{b}{a-b} \right) = (a-b)^{n-1} (a+(n-1)b)$


$   \mathbb{E}f\left(\frac{|X+Y|}{a+b} \right)  \le \frac{a}{a+b}\mathbb{E}f\left(\frac{|X|}{a}\right) + \frac{b}{a+b}\mathbb{E}f\left(\frac{|Y|}{b}\right). $


$\rm\: \  \lfloor x/(mn)\rfloor\ =\ \lfloor{\lfloor x/m\rfloor}/n\rfloor\ \  $


$|x-a|<\delta\implies -\delta<x-a<\delta\implies-\delta+a<x<\delta+a$


$L = L^2= \lim_{x \to 0} f(x) \lim_{x \to 0}f(-x) = \lim_{x \to 0} f(x)f(-x) = \lim_{x \to 0} f(x-x) = \lim_{x \to 0} f(0) = f(0)$


$\frac{1}{\sin x}$


$\text{trace}(\gamma) = \text{trace}(\gamma^{-1})$


$f_{X}(x)= \frac{1}{x^{2}}$


$(a,\gamma)\mapsto \gamma(1)$


$e^{x} > (1 + x/n)^n $


$f(n) = f(n-1) * f(n-2)\\ f(n) = (f(n-2))^2 * (f(n-3))^1\\ f(n) = (f(n-3))^3 * (f(n-3))^2\\ f(n) = (f(n-3))^5 * (f(n-4))^3$


$e^x\ge \left(1+\frac xm\right)^m$


$\left\|A \right\| = \sqrt{\lambda_{max}(A^TA)}$


$A\not R B \land B \not R A$


$\frac{n}{2(n+1)(n+2)}=\frac{1}{2}(\frac{-1}{n+1}+\frac{2}{n+2})$


$|x^3-x_0^3|\le|x-x_0||x+x_0|^2\le\delta(2|x_0|+\delta)^2$


$ax+by=h$


$ T = \{(x,y)\in\mathbb{R}^2: 0\leq x\leq 1,0\leq y\leq x\},$


$\frac{1}{2}\left(e^{x/20}-e^{-x/20}\right)$


$E_1^{0,n} = \frac{\ker (E_0^{0,n} \to E_0^{0,n+1})}{\text{im}(E_0^{0,n-1} \to E_0^{0,n})}$


$\sqrt[k]{\frac{a_{1}^{k}+a_{2}^{k}+\ldots +a_{n}^{k}}{n}}\geq \frac{a_{1}+a_{2}+\ldots + a_{n}}{n}$


$\frac{1}{\pi}\sum_{n = -\infty}^{\infty}\frac{a}{a^2 + n^2} = \sum_{n = -\infty}^{\infty}e^{-2\pi a |n|}$


$\sum_{\gamma\in \Gamma} a_\gamma X^\gamma$


$\begin{cases}x=-t-z\\y=tz\\t=-u-q\\z=uq \\u=-x-y\\q=xy\end{cases}$


$\mathbb E\lvert X-Y\rvert^p= \mathbb E\lvert X\rvert^p+p(p-1)\int_0^1(1-s)\mathbb E\left[Y^2\lvert X+sY\rvert^{p-2}      \right] \mathrm ds.           $


$reg(\alpha) = \alpha^+$


$ \mathbb{E}\|X\|_2^r\le C_r'\left(\mathbb{E}\|X\|_2^2\right)^{r/2}. $


$\zeta(s) = \displaystyle\sum_{n=1}^{\infty} \displaystyle\int_n^{n+1} \frac{1}{n^s} - \frac{1}{x^s} dx + \frac{1}{s-1}$


$ \begin{align} a_n & =\frac{1}{\pi}\int_{-\pi}^{\pi}{f(x)\cos(nx)dx} =\frac{2}{\pi}\int_{0}^{\pi}{f(x)\cos(nx)dx} =\frac{2}{\pi}\int_{0}^{\pi}{(e^x+e^{-x})\cos(nx)dx}\\ & =\frac{2}{\pi}\int_{0}^{\pi}{e^x\cos(nx)dx}   +\frac{2}{\pi}\int_{0}^{\pi}{e^{-x}\cos(nx)dx} =I+J \end{align} $


$1+2+3+4+5.... = -\frac{1}{12}$


$f_X(x)=\theta e^{-\theta x}$


$d(x,y) \le d(x,x_{n_k}) + d(x_{n_k}, y_{n_k}) + d( y_{n_k}, y)$


$\int_0^\infty \frac{\sin^3(x)}{x^2}\,dx$


$\sum_{n=1}^\infty\sum_{m=1}^\infty{a_{nm}}.$


$\alpha R\beta\lor\beta R\alpha$


$\operatorname{True} \implies P(x).$


$\tag{1}\sum_{n=-\infty}^\infty e^{-s\pi(n+x)^2}=s^{-1/2} \sum_{n=-\infty}^\infty e^{-\pi n^2/s}e^{2\pi i n x}$


$\ker \Phi = T \oplus F$


$\frak \overline{a^+}$


$\sum_{n=-\infty}^{\infty}\frac{1}{(n\!-\!a)^2\!-\!b^2} = \frac{1}{2b}\sum_{n=-\infty}^{\infty}\frac{a-b}{n\!-\!(a\!-\!b)^2}\!+\!\frac{a+b}{n\!-\!(a\!+\!b)^2} =\frac{\pi}{2b}\left[\cot\pi (a\!-\!b)\!-\!\cot\pi (a\!+\!b)\right]\tag{1}$


$r^2 \equiv s^2 \equiv 1$


$\mathrm{d}(a,L)=\frac{ |\langle a, \alpha \rangle-c|} {\| \alpha\|}$


$\forall \epsilon \gt 0, \forall n \ge N, |c_n-x| \lt \epsilon$


$\int \frac{\sin^{-1}\sqrt{x}}{\sqrt{1-x}}\,dx$


$\bigg\lfloor\dfrac{\lfloor x \rfloor }{m}\bigg\rfloor = y = \bigg\lfloor\dfrac{x }{m}\bigg\rfloor$


$\mathrm{x} = (x, y)$


$ \frac14 \sum_{m=1}^\infty \frac1m \frac{1}{\sinh^2 \frac{m\alpha}2} = - \sum_{n=1}^\infty n\log (1-q^n)$


$\lim\limits_{n\to\infty} \sum\limits_{r=0}^{n-1}\frac{1}{n}f(\frac{r}{n})$


$E_i \subset (a_i, b_i)$


$ \left(1+\frac xn\right)^n \le e^x \le \left(1-\frac xn\right)^{-n} $


$|m'-m|=|m''-m'|=|m-m''|$


$g'(x, y)=\left(\dfrac{\sqrt{\gamma (1-\gamma)}y}{\left(\sqrt{\gamma}x+\sqrt{1-\gamma}y\right)^2}, -\dfrac{\sqrt{\gamma (1-\gamma)}x}{\left(\sqrt{\gamma}x+\sqrt{1-\gamma}y\right)^2} \right)^t$


$F_n=\int_0^1\frac{\sin(nx)}{2+\cos(nx)}f_n(x)dx,\:\:\:n\geq 1.$


$I = \int_0^\infty f(x) ~ dx$


$0 = V_0 \subseteq V_1 \subseteq V_2 \subseteq ... \subseteq V_r \subseteq V$


$ F(\alpha) = a_2 (\alpha-\beta) (\alpha-\gamma) \\  F(\beta) = a_3 (\beta-\gamma) (\beta-\alpha) \\ F(\gamma) = a_1 (\gamma-\alpha) (\gamma-\beta)$


$A = A_{1} \times \dots \times A_{n}$


$a_n = \frac{n(n+1)}{2} * (\frac{(2n+4)}{6})$


$\alpha(f)=\frac1{1/T}\sum^\infty_{m=-\infty}c_ne^{-i2\pi nf/(1/T)}=\sum^\infty_{m=-\infty}e^{-i2\pi nfT}$


$\sum\limits_{i = 1}^{n} r^{i - 1} = \sum\limits_{i = 0}^{n - 1} r^{i} = \frac{1 - r^n}{1 - r}$


$\Phi(s \otimes n) = s\varphi(n)$


$xRy\Rightarrow \neg yRx$


$H(z,\alpha)\restriction \beta=\{(\gamma,H(z,\alpha)(\gamma))\mid \gamma<\beta\}=\{(\gamma,H(z,\gamma+1)(\gamma))\mid \gamma<\beta\}=$


$\mathbb{E}|X_n|^p \to \mathbb{E}|X|^p$


$a^{(\log_ab)^2}=a^{\log_a(b)\times \log_a(b)}=b^{\log_a(b)}$


$ \begin{matrix}   & 1 & (12) & (123)\\ \hline  \chi_{S^2(V)} & 3 & 1 & 0 \end{matrix}$


$f_X ( x ; \theta ) = (1/\theta) x \exp (-x/\theta).$


$\left(\begin{array}{rrr|r} 1 & 1 & 2 & 7\\ -2 & -2 & k & -14\\ 3 & 3 & 6 & 14 \end{array}\right)$


$\sum_{k=1}^{10} (\sin \frac{2 \pi k}{11}-i \cos \frac{2 \pi k}{11})=?$


$\int_{-\infty}^{\infty}\frac{1}{(x^2+1)(2-2x+x^2)}dx$


$\left[\begin{array}{ccc|c}       1&   0&   2&  1\\       0&   1&  -1&  2\\       1&  -2& k+4&  5     \end{array}\right]$


$(x - y)^z + (x - z)^y + (y - x)^z + (y - z)^x + (z - x)^y + (z - y)^x$


$[\forall m,P(m,0)]\land[\forall n,P(0,n)]$


$(-)\cdot(-)=(-)\cdot(+)=(+)\cdot(-)=(-)$


$A_1\subseteq A_2\subseteq \cdots \subseteq A_{n+1} $


$|x_i + y_i|^p \leq (\alpha+ \beta)^p \bigg( \frac{\alpha}{\alpha + \beta} |\hat{x}_i|^p + \frac{\beta}{\alpha + \beta} |\hat{y}_i|^p \bigg). $


$\geq{1\over 2^{n-2}}$


$f(a)\oplus f(b)=f(a+b)\ .$


$a, b, c, d ∈ \mathbb{Q}$


$\gamma^\prime = \gamma + \frac{a-\gamma}{2} <a \implies \gamma^\prime \in E_1$


$2\pi \int\int y\  dy\ dx$


$\frac1{\sqrt{n}}\|A\|_\infty = \|A\|_2 = \sqrt{m}\|A\|_\infty$


$\xrightarrow{R_2-R_1\to R_2} \left[     \begin{array}{ccc|c}       1&-3&0&6\\       1&3&3&-9\\       0&k&24-6k&-5     \end{array} \right] $


$\forall \epsilon >0, \exists \delta>0,\text{ s.t. }|x-a|< \delta \implies |f(x)-f(a)| <\epsilon$


$\qquad =(a-b)^{n-1}(a+(n-1)b)$


$A = \begin{bmatrix} 1 &-1  & 2 & -1\\  -1 & 0 & -1 &2 \\  2 &-4  &6  & 0 \end{bmatrix}$


$f(n) = n^2 - 7n + 12$


$\exists \epsilon >0, \forall \delta >0, \exists x \in \mathbb R, s.t. |f(x)-f(a)|> \epsilon, |x-a|< \delta$


$ \frac{k^3}{3}+k^2<\frac{(k+1)^3}{3} $


$A(M(\gamma))=\frac{1}{2}\int_0^T  ||(\gamma+a) \times \gamma'||$


$f(a+\Delta x)-f(a-\Delta x)$


$(6,1,-6,2) = \alpha(1,-1,-1,0) + \beta(-1,0,1,1) +\gamma(1,1,-1,1).$


$s > 1/|x|$


$\mathcal C^n-\mathcal C^{n+1}$


$\exists! h':C_0' \to C_1'$


$\displaystyle \int \limits _{\gamma (n)}\varphi=\int \limits _{\gamma _1(n)}\varphi +\int \limits_{\gamma _2(n)}\varphi \tag {*}$


$q!+2,\ldots, q!+q-1$


$x \neq y \rightarrow ( x \not\mathrel{R} y \vee y \not\mathrel{R} x )$


$E[Y\mid X = x_0] = 1 + x_0$


$P_\theta[\Theta] = \frac{2r(\theta)}{R^2} = \frac{2d}{R^2 \tan(\theta)}$


$\|AB\|_2 = \sqrt{\rho((AB)^TAB)} = \sqrt{\rho((AB)^2)} = \rho(AB)$


$U \subseteq V \subseteq \operatorname{Spec} A$


$\mathrm{dist}(\gamma_\varepsilon(t),\gamma[0,1]) = \inf_{s \in [0,1]}|\gamma_\varepsilon(t) - \gamma(s)| \leq |\gamma_\varepsilon(t) - \gamma(t)| \leq \varepsilon \|f\|_\infty$


$P(H...TT) = 1/6$


$g(x)=\frac{a+bx}{c+dx}$


$(\forall n[  \forall m<n, P(m)\implies P(n)]) \implies \forall n P(n))$


$\sum_{n\geq1}\frac{\log\left(n\right)}{n^{2}}\leq C + \sum_{n\geq N}\frac{1}{n^{2-\alpha}}<\infty.$


$ \int_{0}^{1}\frac{\tan^{-1}(x)}{1+x}dx$


$k_1 = 2+(n)!, k_2 =3+(n)!,....$


$\sum_{k=1}^{n}\cot\left(\frac{k\pi}{n+1}\right)=0$


$|R|=m=rs$


$\mathbb R^{2n} - \mathbb R^{n}$


$\int_0^1\frac{\sin^{-1}x}x dx.$


$\mathbb{Q} [x]/(x^{4}+1) \simeq \mathbb{Q}(i,\sqrt{2})$


$aRc \lor cRa$


$P(m)\to P(b)$


$|\omega^{\alpha}| = |\alpha|$


$\lim_{(x,y)\to(0,0)} f(x,y)=L\ne \lim_{x\to 0} \lim_{y=0} f(x,y) \ \ \ ?$


$ 2x^2-8x,  x>2 $


$(1,x,y,x^2, xy, y^2, x^3, x^2y,xy^2, y^3)$


$\frac{|\langle f,x\rangle|}{\|f\|}\le d(x,M).$


$a,b,c,A,B,C,D\in\mathbb{R}$


$\sum_{n = 0}^\infty \sum_{m = 0}^\infty \frac{x^n y^m}{n! m!}.$


$5\mid k^5-k$


$\lbrace a \rbrace \cup\lbrace b \rbrace= \lbrace a,b\rbrace\not\in \mathcal{F}$


$\sqrt[]{1 + {\frac {dy} {dx}}^2}$


$\lim_{x\rightarrow1^-}f'(x)\neq\lim_{x\rightarrow1^+}f'(x)$


$\frac {b}{a} = -\alpha - \beta  - \gamma\\ \frac {c}{a} = \alpha \beta + \beta\gamma + \gamma\alpha\\ \frac {d}{a} = -\alpha \beta\gamma\\ -\frac {b}{a}\frac {c}{a} + 3\frac {d}{a} = (\alpha^2 \beta + \beta^2\gamma + \gamma^2\alpha)+(\alpha \beta^2 + \beta\gamma^2 + \gamma\alpha^2)$


$\lim_{n\to\infty}\sum_{k=1}^n\sin\frac{k\pi}{n}=\lim_{n\to\infty}\frac{2n}{\pi}=\infty.$


$a^{b+2a}\mid a^{a+2b}=b^{b+2a}$


$=\frac {Tan^2\theta - sin^2\theta}{sin^2\theta}$


$||x+y|^t-|x|^t-|y|^t|\leq C(|x|^{t-1}|y|+|x||y|^{t-1})$


$1+\frac {dy}{dx}y=f(x)$


$0 \begin{pmatrix} 1 & 1 & -2  \\ 3 & 1 &-6 \\ -1 & 8 & 7   \end{pmatrix}$


$\displaystyle \sum_{i = 1}^n \sum_{j = 1}^n \frac1{i^2+j^2}$


$ f(xy) = f(x) + f(y) $


$ \begin{pmatrix} 0 & -1\\1 & 0 \end{pmatrix}$


$\int_{-\pi}^\pi \overline{f}(x)\sin(nx) dx =2 \int_{-\pi}^\pi \overline{f}(x)\sin(nx)dx=2\int_0^\pi f(x)\sin(nx)d=0$


$\lim_{n \to \infty} \frac{\sqrt{a(4^0n)}+\sqrt{a(4^1n)}+\cdots+\sqrt{a(4^{10}n)}}{\sqrt{a(2^0n)}+\sqrt{a(2^1n)}+\cdots+\sqrt{a(2^{10}n)}}$


$ \int^{\infty}_{-\infty}\frac{1}{\sqrt{x^{2}+1}} dx $


$\left\lfloor\sum_{k = 1}^{n}{\varphi^{3k}}\right\rfloor$


$\tan \theta = \frac y{100}$


$\dfrac{1}{\sqrt{2\pi ep(1-p)}}$


$\gamma \implies N(\gamma) \mid 21 \implies N(\gamma) = 1,3$


$ \sin(x)=x^2-x, \quad x\in (1,2).  $


$\;\;\fbox{X}\fbox{X}\fbox{2}\fbox{X}\fbox{1}\fbox{X}\fbox{1}\fbox{2}\fbox{X}\fbox{X}\;$


$f_X(x \mid \gamma) = \begin{cases} \frac{1}{2\log \gamma} \frac{1}{x} \mathbb 1(1/\gamma \le x \le \gamma), & \gamma \ge 1 \\ \frac{1}{-2 \log \gamma} \frac{1}{x} \mathbb 1(\gamma \le x \le 1/\gamma), & \gamma < 1. \end{cases}$


$ \begin{align} F(x)=\int_a^xf(t)dt, &&x\in[a,b] \end{align} $


$\bigl|f\bigl(\gamma(t)\bigr)\gamma'(t)\bigr|\leqslant\sup|f|.\bigl|\gamma'(t)\bigr|,$


$\mathbb{B}=\left\{ \left\{ a\right\} ,\left\{ b\right\} \right\} $


$f(x) = 1/(1+x), \\ f(f(x)) = 1/(1 + (1/(1+x)), \\ f(f(f(x))) = 1/(1+1/(1 + (1/(1+x)))  , ...$


$\Leftrightarrow \sin \gamma\frac{\tan a +\tan \gamma}{1-\tan a \tan \gamma}+\cos \gamma = \frac{b}{c}$


$ \begin{align} \frac{(1+x)^{n+1}-(1+y)^{n+1}}{x-y} &=\frac{(1+x)^{n+1}-(1+y)^{n+1}}{(1+x)-(1+y)}\\ &=\sum_{k=0}^n(1+x)^k(1+y)^{n-k}\\ &=\sum_{k=0}^n\sum_{i=0}^k\binom{k}{i}x^i\sum_{j=0}^{n-k}\binom{n-k}{j}y^j\\ &=\sum_{i=0}^k\sum_{j=0}^{n-k}x^iy^j\sum_{k=0}^n\binom{k}{i}\binom{n-k}{j}\tag{6} \end{align} $


$\int_{-\infty}^{x} f(x) dx=1+\frac{-1}2xe^{\frac {-x}2}-e^{\frac {-x}2}=\frac 12$


$\gamma_s(a) = \gamma(a)$


$ J_0\supset J_1\supset\cdots\supset J_n\supset J_{n+1}\supset\cdots. $


$R=P\oplus P$


$\lim_{x \to 0} f(x) = L \ \Longleftrightarrow \ \lim_{x \to \infty} f(1/x) = \lim_{x \to -\infty} f(1/x) = L$


$d(a,X)=\inf|x-a|$


$\{(1,-1,-1,0), (-1,0,1,1), (1,1,-1,1)\}$


$\int_{0}^{\infty}\frac{\sin^{2n+1}(x)}{x} \mathrm {d}x$


$\lim_{y\to\infty}\sum_{x=1}^y\frac{1}{2^x},$


$\sum_{k=1}^\infty |\langle x,x_k\rangle|^2$


$\gamma^{-1}(\gamma(x * y)) = \gamma^{-1}(\gamma(x) \circ \gamma(y)) \implies (x * y) = \gamma^{-1}(\gamma(x) \circ \gamma(y))$


$\displaystyle\int_0^{\pi} f'(t)^2dt\geq\int_0^{\pi}f(t)^2dt$


$ar + bs = 1$


$\dfrac 1x+\dfrac 2y-\dfrac3z=1 $


$\frac{\left|\langle f,v \rangle \right|}{\|v\|_{H^1_0}}=\|u_f\|_{L^2}$


$ x\delta^{(n)} = -n \delta^{(n-1)}. $


$d(x,y) + d(y,z) \geq d(x,z)$


$\frac{ax+b}{bx+c}=\frac{b-cx}{xb-a}$


$P(E) = 1/2$


$(2a+b)-(a+b) = 2a+b-a-b = a$


$|\tilde \gamma'|\le |\gamma'|$


$M×M →M$


$C\subset V\subset \overline{V} \subset U$


$ J_1 =\int_0^\infty f(x)dx $


$(x_1, y_2, z_3)=(1, 1, 1)$


$\{(x,y)\mid 0\leq x \leq 2, 1\leq y\leq 3\}$


$f(z) = z^2-z+1$


$f(n)=\binom n2=\frac{n(n-1)}2$


$\mathbb{A} = \mathbb{A}_{\mathbb{C}/\mathbb{Q}}$


$\mathcal{H}=V^\perp \oplus (V^\perp)^\perp = \{0\} \oplus \overline{V}=\overline{V}$


$\gamma(1)=1,\gamma(2)=\gamma(3)=2,\gamma(4)=\gamma(5)=\gamma(6)=3$


$y=e^{\gamma x}\implies\left(e^{\gamma x}\right)''-e^{\gamma x}=e^{\gamma x}(\gamma^2-1)=0\implies\gamma=\pm1$


$[x,y] = e$


$10y=x-4 \Rightarrow y=\frac{x}{10}-\frac{4}{10}$


$|f(y)| \leq \frac{1}{2}|f(x)|$


$\sin\gamma=\sqrt{\frac{a+b}{a+b+c}}.$


$\rho(2) = \frac{\gamma_{2}}{\gamma_{0}} = \frac{\varphi_{1}\gamma(1) + \varphi_{2}\gamma(0)}{\gamma(0)} = \frac{\varphi_{1}\gamma(1)}{\gamma(0)} + \frac{\varphi_{2}\gamma(0)}{\gamma(0)}$


$ \mu\left(\lim_{n\rightarrow \infty} \inf E_n\right) \leq \lim_{n\rightarrow \infty} \inf ~\mu\left(E_n\right).$


$ \begin{vmatrix} x&y&z\\ x^2&y^2&z^2\\ x^3&y^3&z^3\\ \end{vmatrix}$


$2^\gamma=2^{\gamma^2}=(2^\gamma)^\gamma \ge \gamma^\gamma \ge \kappa^\gamma$


$U \subseteq P(V)$


$\displaystyle \frac{(1+x)^{n+1}-(1+x)^r}{x}$


$ 1+x+x^2+x^3 + \cdots = \frac{1}{1-x} $


$(\mathbb{Z}[x] / (x^2-x))_U\simeq\mathbb Z_U\times\mathbb Z_U$


$\lim_{x\to0^-}f'(x)=\lim_{x\to0^+}f'(x)=1$


$ f(c)=\lim_{k\to \infty}f(c_k)=m=\inf f, $


$\lim_{x\to c^+} = L = \lim_{x \to c^-}$


$(n!)+2, (n!)+3, ..., (n!)+n.$


$(a+b)^r\leqslant 2^{r-1}(a^r+b^r)\mbox{ if }r\gt1.$


$|ab| = 2.$


$\lim_{n\rightarrow \infty} \frac{\sqrt{1}+\sqrt{2}+\sqrt{3}+\cdots+\sqrt{n}}{n^{3/2}}=\lim_{n\rightarrow \infty} \frac{\sqrt{n}}{n^{3/2}-(n-1)^{\frac{3}{2}}}=\lim_{n\rightarrow \infty} \frac{\sqrt{n}\left(n^{3/2}+(n-1)^{\frac{3}{2}}\right)}{n^3-(n-1)^3}=$


$( (N-m+Nr)(1+r)^x -m((1+r)^x-1)/r )r $


$(a+b,a^2-ab+b^2) = 1$


$ \frac{-a_2}{x} + \frac{b_2}{y} = 1 $


$ \displaystyle s(x)=\int_a^x f(t) \, dt $


$\alpha^+\cdot\beta+\alpha$


$B = \{B_1,B_2,\cdots,B_n\}$


$\left[         \begin{array}{ccc|c}          4 & -4 & 2 & 20\\         -3 & -1 & 3 &-18\\         1 & -5 & 5 &k\\         \end{array} \right] $


$    \det(A) = (x-2)^{n-1} (x + 2(n-1)). $


$x_1 + x_2 + \cdots + x_n < m$


$\left( 1+ \frac{x}{p} \right) ^{p}  < \left( 1+ \frac{x}{q} \right)  ^{q} $


$P(X_1 = 1 | X_0 = 0) = 1$


$\frac{10^4}{x} - \frac{1}{x}$


$\tilde{A} = \cases{a \text{ if } \hat A \leq a\\\hat A \text{ if } a< \hat A <b\\{b \text{ if } \hat A \geq b}}{}$


$x = \pm 1, y = 0, z = 0, \lambda = 0$


$\overline{\mathbb{R}}^X$


$\ f(n)=n^3-n+1$


$(x-a,y-b) = (x-a)$


$\lim_{x \to c^{+}}f'(x) = A, \lim_{x \to c^{-}}f'(x) = B$


$\sqrt z=\sqrt r e^{i\theta/2}$


$d(y,a) \leq d(x,a) + d(y,x)$


$\left\|A\right\|_{2} \le \sqrt{\rho(A^{*}A)}$


$ x = f(f^{-1}(x)) \Rightarrow \\ 1 = f'(f^{-1}(x)) (f^{-1})'(x) \Rightarrow \\ (f^{-1})'(x) = 1 / f'(f^{-1}(x)) $


$\displaystyle J_{1}=\int_{0}^{\infty}f(x) dx$


$\sum_{-\infty}^\infty |J_n(x)|^2$


$|(\omega(z))^{\varepsilon}f(z)|\leq M$


$f(x) = f(a+b) = f(a) + f(b) = f(a) + 0 = f(a)$


$[E_C, E_C] \subseteq E_C$


$\sum_{k=1}^{\infty}\sum_{n=1}^{\infty}\frac{1}{k^{2}+n^{1/\gamma}}$


$3^x + 10^x = 4^x + 9^x.$


$\{\langle{x , y}\rangle: 0\leq x\wedge 0\leq y\wedge x+y\leq z\}$


$\cos^2 x=\frac{1+\cos 2x}{2}\Rightarrow2\sum_{k=1}^N\cos^2\frac{k\pi}{N+1} =N+\sum_{k=1}^N\cos\frac{2k\pi}{N+1}=N-1,$


$F = P  \oplus T$


$E(p) =  \frac{1-a}{a (n-2)}\left(1-(1-a)^{n-2}\right)$


$f(x) + f(y) = f(x+y)$


$\frac{1}{x} + \frac{1}{y} + \frac{1}{z} > \frac{1}{10}$


$ I_n=\int_{0}^{\infty} \frac{\sin(\frac{x}{n})}{x(1+x^2)}dx\ ? $


$\displaystyle\lim_{x\to c^+}f'(x)=A\not= B= \displaystyle\lim_{x\to c^-}f'(x)$


$f_n=\sum a_j\chi_{I_j}$


$\dfrac{1}{2\pi} \cdot \dfrac{1-r^2}{1-2r \cos(\varphi) + r^2 } $


$\int \frac{\sin (\pi x)}{|x|^a + 1} dx$


$\sum_{n=1}^{\infty}\sum_{k=1}^{\infty}\frac{1}{n^2k^2(n+k)^2}$


$f(a + b) \ne f(a) + f(b)$


$\begin{align}(1 - \cos A)^2 + \sin^2 A \over \sin A(1 - \cos A)  & = \dfrac{1 - 2 \cos A + \cos^2 A + \sin^2 A}{\sin A(1 - \cos A)} \\ \\ & = {1 \color{blue}{\bf + \cos^2 A} -2\cos A + 1 \color{blue}{\bf - \cos^2A} \over \sin A(1-\cos A)} \\ \\ & = \dfrac{2 - 2\cos A}{\sin A(1 - \cos A)}\\ \\ & = \dfrac{2\color{red}{\bf (1-\cos A)}}{\sin A\color{red}{\bf (1 - \cos A)}}\\ \\ & = \frac{2}{\sin A} \\ \\ & = 2 \csc A \end{align}$


$r:\begin{cases}x=t \\ y=2t+1\\z=t-2 \end {cases}$


$\nabla(x,y,z) = (1,1,1)$


$\int_{0}^{\infty}f^2(x)dx< \infty$


$\frac{x_1+x_2+\cdots +x_n}{n}\in X$


$ \left( \matrix{   x \cr    m \cr}  \right) = \left\{ {\matrix{    {{{x^{\,\underline {\,m\,} } } \over {m!}} = {1 \over {m!}}\prod\limits_{0\, \le \,k\, \le \,m - 1} {\left( {x - k} \right)} } & {0 \le m \in Z}  \cr     0 & {m < 0\; \vee \;m \notin Z}  \cr   } } \right. $


$\begin{cases} t\equiv y \\ u\equiv x\end{cases}$


$\sum_{j=1}^n c_j X_{t_j}$


$f(x)[f'(x-1)+f'(x+1)]-f'(x)[f(x-1)+f(x+1)]=0$


$\hat f (n) = \frac{1}{2\pi}\int_{-\pi}^\pi f(x)e^{-inx}\,dx = \frac{1}{\pi}\int_{0}^\pi f(x)\cos (nx)\,dx = 0$


$\gamma(s)\gamma(t)=\gamma(s+t)=\gamma(t)\gamma(s)$


$A = (\{a\}, \{(a, a)\})$


$ xy=xz^*z=(xz^*)z=z(xz^*)=(zx)z^*=z^*zx=yx. $


$\lbrace$


$ LHS = \frac{n}{n-1} e^{\frac{-2}{n}}(\frac{1 - e^{\frac{2k}{n}}}{1 - e^{\frac{-2}{n}}}) $


$ p\mid r_1r_2 $


$P(X_n \leq x) \to P(X \leq x)$


$\sum_p \frac1{p^s} = \log \frac1{s - 1} + O(1) \tag2$


$\pi^+$


$f(A_1) = A_1$


$\bar  X=\frac{X_1+X_2+...+X_n}{n}$


$f(z) = e^{i \theta} \prod_{j=1}^{k} {{z-a_j} \over {1- \overline a_jz}}$


$  x^n+1 \geq (1+\frac1n \frac{x^n}{x^n+1})^n $


$\frac{x!}{2^x}={\sqrt {2\pi x}}\left({\frac {x}{2e}}\right)^{x}.$


$\hat{f}(k)={n\choose k}.$


$A:= \frac{x_1+\ldots+x_n}{n}$


$\mathbb R[X]/((X^2+1)^2)$


$f(x)^2+f(y)^2=f(x+y)(f(y)+f(x) + a)$


$\begin{cases}{x^2 +y^2 −z(x+y)=2\\ y^2 +z^2 −x(y+z)=4\\ z^2 +x^2 −y(z+x)=8}\end{cases}$


$Cov(X,Y) = 0.0007$


$ a_0=\color{lightgrey}{\binom 02}+\binom 20=0+1=1\\ a_1=\color{lightgrey}{\binom 12}+\binom 21=0+2=2\\ a_2=\binom 22+\binom 22=1+1=2\\ a_3=\binom 32+\color{lightgrey}{\binom 23}=3+0=3\\ a_4=\binom 42+\color{lightgrey}{\binom 24}=6+0=6\\ \vdots$


$x*(y*y)=y*(y*y)=y*y=y$


$\begin{pmatrix}c&-s\\s&c\end{pmatrix}$


$ O_X(F) \to A_X^{0,0,r}(F) \to A_X^{0,1,r-1}(F)\to \dots \to A_X^{0,n,r-n}(F) \to 0 $


$\mathrm ds = \sqrt{r^2 + \left(\frac{\mathrm dr}{\mathrm dθ}\right)^2}$


$|f(x_0)|\le x_0|f(y_0)|$


$ \zeta(-1) =  \sum \limits_{n=1}^{\infty} \frac{1}{n^{-1}}= 1+2+3+4 + ... = -\frac{1}{12}$


$ \lim\limits_{x\to \infty  } \lim\limits_{n\to \infty} \sum\limits_{r=1}^n \frac{r^2(\sin x)^x}{n^3}$


$t=0 \implies \frac x 2=k\pi\implies x=2k\pi$


$\log_a b$


$b_0\in V\subseteq \overline{V}\subseteq p(U)$


$\mathbb{R}^{n+1} \subset \mathbb{R}^{n+2}$


$\sum_{k=2}^{n}\frac{\cos\frac{\pi}{n+1}}{\cos\frac{\pi}{n+1}-\cos\frac{k\pi}{n+1}}$


$A\subset U \subset \overline U\subset O$


$y=\frac{c+d}{b-a}$


$\tan \theta=\frac12$


$f(x,\gamma)=\frac{2\gamma+1}{2\gamma}x^{\frac{1}{2\gamma}},\quad 0\leq x \leq 1,\gamma>0,$


$a_n = \frac{1+\sqrt2+\sqrt 3+ \cdots +\sqrt n}{n\sqrt n}$


$\cos(k\pi)=\frac{2k\sin(\pi k)}{2\pi k}+\sum_{n=1}^{\infty}\frac{2k\sin(\pi k)(-1)^n}{k^2-n^2}\cos(n\pi)$


$P(X\leq x) = P(X_1\leq x,X_2\leq x,\ldots,X_n\leq x).$


$40 - x_1 - x_2 -x_4$


$\lim_{n\rightarrow \infty}\inf J(u_n)\geq J(u)$


$x^{3}, x^{2}, x, x^{0}$


$ E|X|^p\le E|X+y-y|^p\le E(|X+y|+|y|)^p\le C_p(E|X+y|^p+|y|^p)<\infty. $


$b\cdot c = \cfrac {2A} {\sin Â}\\(b+c)^2 = b^2 +c^2 +2bc = k^2 \implies b^2+c^2=k^2-2bc\\ a^2=k^2-2bc(1+\cosÂ)=k^2- 4A \cfrac {1+\cos Â} {\sin Â}=k^2- 4A \cdot\cot \cfrac Â2 \implies \tan \cfrac Â2=\cfrac {4A} {k^2-a^2}$


$ = \frac{P(A=t|B=t,F=t) P(B=t) P(F=t) +P(A=t|B=f,F=t)P(B=f)P(F=t)} {P(F=t)}$


$\forall \epsilon>0,\ \exists\delta>0,\ \forall|x-y|<\delta,\ |f(x)-f(y)|<\epsilon$


$t_n = n(n+1)/2$


$\det\begin{pmatrix} A & B \\ B & A\end{pmatrix}=\det(A-B)\det(A+B)$


$I_n=\int_{0}^{\pi/2}\frac{\sin \left ( ax \right )}{\sin x+\cos x}\, {\rm d}x$


$\sum_k |\langle x, e_k \rangle |^2 = ||x||^2$


$\frac{\frac{(n+2)^2}{(n+1)^2}}{\frac{((n+1)+2)^2}{((n+1)+1)^2}}<1$


$\mathscr{D}=\{\{a\}:a\in A\}$


$\lim_{n\to\infty}{\sqrt{5+\sqrt{5+...+\sqrt 5}}}$


$\begin{align*} &(1)\quad \left\lceil \frac{\left\lceil \frac{x}{a} \right\rceil} {b}\right\rceil = \left\lceil {\frac{x}{ab}}\right\rceil\\\\ &(2)\quad\left\lfloor \frac{\left\lfloor \frac{x}{a} \right\rfloor} {b}\right\rfloor = \left\lfloor {\frac{x}{ab}}\right\rfloor\\\\ &(3)\quad \left\lceil {\frac{a}{b}} \right\rceil \le \frac{a + b - 1}{b}\\\\ &(4)\quad \left\lfloor {\frac{a}{b}} \right\rfloor \le \frac{a - b + 1}{b} \end{align*}$


$\begin{cases} x=s\\ y=s\\ z=0 \end{cases}$


$P\circ T$


$E(n) = (\frac{n}{2})^{(n-1-\lfloor\frac{n}{2}\rfloor)}$


$X = \bigcup \mathcal{A}$


$h+x = (-x_n)+(x_n) = (-x_n+x_n) = (0) = 0$


$Cov(X,Y) = Cov(X,Z) = 0$


$2^\gamma\le\alpha^\gamma\le(2^\gamma)^\gamma=2^{\gamma^2}=2^\gamma$


$\ell^+$


$|OX| = ab$


$f(0) = f(\pi) = 0$


$\vartheta(u,-i\tau)=\frac 1{\sqrt{-i\tau}}e^{i\pi\tau u^2}\vartheta(\tau u,\frac i\tau)$


$\gamma+\bar\gamma=\gamma\bar\gamma=1, {\gamma}^2=\gamma-1, |\gamma|=1$


$\mathbb{E}|X_1|^r < \infty$


$U\subseteq B\subseteq \overline{A}\subseteq\overline{V}$