${ax^k\over (1+r)^{4k-3}} = {a(1+r)^3 ({x\over(1+r)^4})^k}$


$\Bbb R^n-\Bbb R^m$


$\left(  1+x/n\right)^n\leq e^x.$


$ \int_0^\pi f (x) \sin x=\int_0^\pi f (x) \cos x=0$


$l_2: ax+by=d$


$[x,y]=x$


$ \sum_{n=-\infty}^\infty |f(n)|^2 = \sum_{n=-\infty}^\infty e^{-n^2} =\vartheta_3(0,e^{-1}) \approx 1.772637 $


$a=\frac{l-bc}{b+c}=\frac{m}{bc}$


$\hat{\gamma}=\{ \gamma(t):0\leq t\leq 1 \} .$


$U_1 \subset U_2 \subset ... \subset U_i \subset U_{i+1} \subset ...$


$I=\int_0^1\int_0^1 \frac{\sin^{-1}(xy)}{xy} \,\mathrm dx\,\mathrm dy.$


$n!=\sqrt{2\pi n}\left(\frac{n}{e}\right)^{n}\left(1+\frac{1}{12n}+O\left(\frac{1}{n^{2}}\right)\right).$


$\sum_{k=1}^{n-1}{\sin{\frac{2\pi k}{n}}} = 0$


$E|X-a|^{r} \leq 2^{r} (E|X|^{r}+|a|^{r})$


$\sum\limits_{k=1}^n r^k = \frac {r - r^{n+1}}{1-r} \implies \sum\limits_{k=1}^{n+1} r^k = \frac {r-r^{n+1}}{1-r}$


$2\,{\frac {\sin \left( x \right) \cos \left( x \right)  \left( \cos  \left( x \right) -\sin \left( x \right)  \right) }{ \left( 2\,  \left( \cos \left( x \right)  \right) ^{2}-1 \right)  \left( \cos  \left( x \right) +\sin \left( x \right)  \right) }} $


$\left\|A\right\|_{2} \ge \sqrt{\rho(A^{*}A)}$


$\lim_{n\to \infty} \frac1{2^n}\sum_{k=1}^n \frac1{\sqrt k}\binom nk$


$ \mathbb{E}|X|^r\le\bigl(\mathbb{E}|X|^s\bigr)^{r/s} $


$ T[\gamma] = \lambda\gamma \quad \Rightarrow \quad (\gamma,\beta)\gamma = \lambda\gamma. $


$\begin{cases} x=t \\ y=6-t \\ z= 1\end{cases}$


$ \begin{align} \sum_{m=1}^\infty\sum_{n=1}^\infty\frac1{mn(m+n)^2} &=\sum_{m=1}^\infty\sum_{n=1}^\infty\left(\color{#C00}{\frac1{nm^2(n+m)}}-\color{#090}{\frac1{m^2(n+m)^2}}\right)\tag{3a}\\ &=\color{#C00}{\frac{\pi^4}{72}}-\color{#090}{\frac{\pi^4}{120}}\tag{3b}\\[3pt] &=\frac{\pi^4}{180}\tag{3c} \end{align} $


$c_n = \frac{1}{3}(-1)^n + \frac{2}{3}2^n$


$ \gamma ~:=~\det\gamma_{\mu\nu}, \qquad  \gamma^{-1} ~=~\det\gamma^{\mu\nu}.\tag{E} $


$(-1)^n{2\pi p^n\over1-p^2}$


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


$\frac{1}{\pi + \cos(z)}  = \frac{2\gamma}{1+\gamma^2 + 2\gamma\cos(z)} = \frac{2\gamma}{(\gamma + e^{iz})(\gamma + e^{-iz})} = \frac{\gamma}{\gamma^2-1}\left[\frac{\gamma - e^{iz}}{\gamma + e^{iz}} + \frac{\gamma-e^{-iz}}{\gamma + e^{-iz}}\right] $


$f'(x)=(x+1)^{m-1} (x-)^{n-1} (m(x-1)+n(x+1))$


$\left[\begin{array}{ccc|c}1 & 3 & -1 & 4\\4 & -1 & 2 & 8\\2 & -7 & 4 & -3\end{array}\right].$


$  \lim_{x\to 0^+} f'(x) \ne \lim_{x\to0^-} f'(x) $


$F \subseteq V \subseteq \overline{V} \subseteq U$


$\log_a {\frac{b}{c}} = \log_a b -\log_a c$


$ \left\lfloor\frac{\left\lfloor\frac ab\right\rfloor}c\right\rfloor=  \left\lfloor\frac{qc+q'}c\right\rfloor=q$


$|S \times T| = mn$


$ \left[ \begin{array}{ccc|c}   1&2&-1&-3\\   3&5&k&-4\\   9&k+13&6&9\\ \end{array} \right] $


$I_n=\int^{\frac{\pi}{2}}_0 \frac{\sin(2nx)}{\sin(x)}dx$


$\begin{pmatrix} A & B\\ C & D\\ \end{pmatrix}$


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


$\{\,\sqrt1, \sqrt2, \ldots, \sqrt n\,\}$


$ \sum_{n=1}^{\infty}\frac{\eta(2n+1)}{2^{2n+1}}  \quad = \frac{1}{2}\int_{0}^{\infty}\frac{\cosh(x/2)-1}{e^{x}+1}\,dx = \frac{1}{2}\left(1-\log2\right) \tag{1} \\[6mm] $


$(p_1-1)(p_2-1)\mid p_1p_2$


$f_3(n) = n \log_2 n$


$T^{0,1}_pM\times T^{0,1}_pM$


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


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


$=2+\frac1{1+\frac1{\frac{26}9}}=2+\frac1{1+\frac1{2+\frac89}}=2+\frac1{1+\frac1{2+\frac1{\frac98}}}$


$\mbox{Tr}_{F/K}(a) = \sigma_1(a) + \ldots + \sigma_n(a)$


$c_1, c_0$


$\sin(\arccos(p))$


$\beta(a,b)=\gamma(a+b)-\gamma a -\gamma b, a,b\in A$


$|f_n(z)|\leq M|f_{n-1}(z)|$


$ \frac{Z^{m/2}-Z^{-m/2}}{Z^{1/2}-Z^{-1/2}} = \frac{(Z^{m/2}-Z^{-m/2})(Z^{1/2}+Z^{-1/2})}{(Z^{1/2}-Z^{-1/2})(Z^{1/2}+Z^{-1/2})} = \frac{Z^{(m+1)/2}+Z^{(m-1)/2}-Z^{(1-m)/2}-Z^{-1-m)/2}}{Z-Z^{-1}} = U_{(m-1)/2}\left(\frac{X}2\right) + U_{(m-3)/2}\left(\frac{X}2\right),$


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


$V\oplus N=F$


$\frac 1r[(1+r)^n -1]$


$ \eqalign{ \alpha^0&=&&=1,\\ \alpha^1&=&&=\alpha,\\ \alpha^2&=&&=\alpha^2,\\ \alpha^3&=&&=1+\alpha,\\ \alpha^4&=&\alpha\cdot\alpha^3=\alpha(1+\alpha)&=\alpha+\alpha^2,\\ \alpha^5&=&\alpha\cdot\alpha^4=\alpha(\alpha+\alpha^2)=\alpha^2+\alpha^3=\alpha^2+(1+\alpha)&=1+\alpha+\alpha^2,\\ \alpha^6&=&\alpha\cdot\alpha^5=\alpha(1+\alpha+\alpha^2)=\alpha+\alpha^2+\alpha^3= \alpha+\alpha^2+(1+\alpha)&=1+\alpha^2,\\ \alpha^7&=&\alpha\cdot\alpha^6=\alpha(1+\alpha^2)=\alpha+\alpha^3=\alpha+(1+\alpha)&=1. }$


$\langle a, \alpha a + \gamma c\rangle = \alpha\langle a, a\rangle + \gamma\langle a, c\rangle=0$


$\frac{x}{2} + \frac{x}{y} - \frac{3}{2} = \frac{10}{y}$


$\gamma = (\gamma^1, \gamma^2)$


$\frac{1}{2} \int_{0}^{\pi} \frac{1}{1+\sin^2 x} \, dx = \frac{1}{2} \int_{0}^{\pi} r^2 \, d\theta = \frac{\pi}{2\sqrt{2}} \implies \int_{0}^{\pi} \frac{1}{1+\sin^2 x} \, dx= \frac{\pi}{\sqrt{2}}$


$\tau=\dfrac{\det(\dot\gamma,\ddot\gamma,\dddot\gamma)}{\|\dot\gamma\times\ddot\gamma\|^2}$


$\lim_{N \to \infty} \sum_{k=1}^N \frac{1}{N+k} = \ln(2)$


$ B = \{x_1,x_2,x_3,\dots\}. $


$\gcd(a+b, a-b) = d$


$a^{d}a^{s-d}\equiv a^{s}\equiv -1\pmod{m}.$


$\mu( \lim \sup A_{n}) \geq \lim \inf _{n} \mu (A_{n})$


$(A\otimes B)^{\bot\bot}\vdash A^{\bot\bot}\otimes B^{\bot\bot}$


$x-m\in M^{\perp\perp}$


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


$M = P\oplus Q$


$f(x^2 + yf(z)) = xf(x) + zf(y)$


$(a+b,a-b) = 1$


$ \lim_{y\to 0} \frac{(1+y)^{P+1}-(1+y)(P+1)+P}{y^2} = \lim_{y\to 0} \frac{(1+y)^{P+1}-1-(P+1)y}{y^2} $


$\lim_{n \to \infty} \sum_{n=0}^\infty\frac{\left \lfloor \frac{n}{2} \right \rfloor}{n^{2}}$


$\frac{1}{2h}\int_a^b\mu(A\cap(x-h,x+h))\,\text{d}x\le \mu(A).$


$(1/\sqrt1 + \sqrt2) + (1/\sqrt2 + \sqrt3) +\cdots + (1/\sqrt{99} + \sqrt{100})$


$\mathcal A[x/a]$


$\|A\|=\sup\limits_{\|x\|=1}\|Ax\|=\sup\limits_{\|x\|=1}\|Bx\|=\|B\|=\|A^{-1}\|$


$f(x) + f(1/x) = f(1/(x+1)) + f(x/(x+1))$


$\left(\frac{\pm p\cos\,\varphi}{1\mp\varepsilon\cos\varphi},\frac{\pm p\sin\,\varphi}{1\mp\varepsilon\cos\,\varphi}\right)$


$d=\inf_{m\in M}\|x-m\|$


$f_{Z}(z)=\frac{1}{\pi}\frac{\sqrt{1-\rho^{2}}}{(1-\rho^{2})+(z-\rho)^{2}}$


$\displaystyle F(x)=\int_a^x f(t)^2dt$


$f(x) = (x-a)^{n-1}h(x)$


$X \subset Y \subset \overline{X}\subset \overline{A}$


$g(z)=(z-a)f(z)-(z-a)\bar{a}$


$E\tau=|ab|$


$|G|=pqr$


$F(x) = \int_a^x f(t)\ dt = x^3 - 2x^2 - x - a$


$x^3-1= (x-1)(x^2+x+1)=(x-1)[(x-1)^2-3x]$


$\omega = \alpha’ = \alpha$


$X_{k}= +/- 1$


$\lim_{n\to\infty}\inf a_n = -\lim_{n\to\infty}\sup b_n,$


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


$\sum_{n=1}^{\infty} \left(1\over2^{n-1}\right) $


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


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


$   0 = f(0) = f \left( f(y)^2 - f(y) f(y) \right) = -f(y) f(y - f(y)) $


$I_n J_n = J_n I_n$


$\forall n (n\geq m\implies (P(n)\implies P(n+1))$


$|f(z_{0})|\geq |f(z)|$


$ \mu(A)= \lim_{n\rightarrow \infty} \mu(A_n).$


$n![z^n](e^z)^s = s^n$


$s(\gamma)h(s(\gamma)^{-1}ks(\gamma))-h(k)=kf(s(\gamma))-f(s(\gamma))$


$2^{\alpha} = \alpha^+$


$\mu(A)=\lim_{n\to\infty} \mu_n(A\cap (-n,n))$


$\sum_{n=1}^{\infty}\mu(A_n)\geq\mu(A_m)=1$


$\frac{dx}{dt} = \mu x - x^3, \mu \in \Bbb R, x \in \mathbb R$


$\lim_{n\to \infty} (1 - \frac{1}{4})(1 - \frac{1}{9})(1 - \frac{1}{16}) \cdots (1 - \frac{1}{(n+1)^2})$


$1+2+3+4+\ldots = -1/2$


$(x+a)(x-a)$


$\{A+B,A-B\}$


$\widetilde{\gamma}\gamma' = \gamma$


$ \lambda(v^*v)=v^*(\lambda v)=v^*Av=v^*A^*v=(Av)^*v=(\lambda v)^*v=\lambda^*v^*v $


$\left(1+\frac xn\right)^n\leqslant e^x$


$ \lim_{n \to \infty} \sum\limits_{k=1}^n \frac{1}{k 2^k}? $


$\int\limits_0^x f(t)dt=0$


$ \begin{cases}  3t=0 \\ 1-t=0\\ 2-2t=s \end{cases} $


$s=\sum _{m=1}^{\infty } \sum _{n=1}^{\infty } \frac{1}{m n \left(m^2+n^2\right)}\tag{1}$


$S_i = (a_i, a_i+d_i, a_i+2d_i, ...)$


$|AB|=mn$


$f(-\pi) = f(\pi) = -f(-\pi) = 0$


$y(n) = \binom{n+2}{2} u(n)$


$D = (a+b, a-b)$


$\int\limits_0^\infty \frac{x^{s-1}}{e^x-1}dx=\zeta(s)\Gamma(s)$


$ f(x-f(y))=f(f(y))+xf(y)+f(x)-1$


$\mathbb{R} / \{0\}$


$k(f(x)+ g(x)) = kf(x) + kg(x)$


$\|A\|_2=\sqrt{\rho(A^TA)}\leq\sqrt{\|A^TA\|_{\infty}}$


$f(x) = 500 (1 - e^{-x/500})$


$\lim_{n\rightarrow\infty}n\ln(a)=\ln(a)\lim_{n\rightarrow\infty}n=-\infty$


$(x^2 + 5) = (x^2 + 5)$


$\begin{align} \int\frac{x}{(x^2+2)^2}dx&=\frac12\int  \frac{1}{(u+2)^2}\,du\\\\ &=-\frac12 \frac{1}{u+2}\\\\ &=-\frac12 \frac{1}{x^2+2} \tag 2 \end{align}$


$ \int \frac{x^3}{\left(x^2+1\right)^2}\sin^2 x dx $


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


$T \oplus F = F \oplus T = F$


$A= \left[ {\matrix{
    x_1 & x_2 \cr
    x_3 & x_4 \cr

  } } \right]$


$|ab|=$


$S=\sum_{k=1}^{n} 4^k\left(3-\frac{1}{k+2}\right) +\sum_{k=1}^{n}\frac{(2^{2k+1}-2^{2k})}{(k+1)(k+2)}$


$\int_0^\infty x \frac{M}{c} e^{(\frac{-x}{c})} (1-e^{\frac{-x}{c}})^{M-1} \,dx = c \sum_{k=1}^M \frac{1}{k}.$


$\begin{vmatrix}1&a&a^2\\1&b&b^2\\1&c&c^2\end{vmatrix}$


$  \mathcal{M} := \left\{f \in L^2([0,\pi]): \int_0^\pi f(x)\cos x dx =  \int_0^\pi f(x)\sin x dx = 1\right\}. $


$\zeta(x) = \sum_{1}^{\infty}\frac{1}{r^x}= \frac{1}{\Gamma(x)}\sum_{1}^{\infty} \int_{0}^{\infty}u^{x-1}e^{ru}du$


$\tan\theta =\frac{x}{\sqrt{1-x^2}}$


$\mathbb H^{+}$


$\sqrt[!]{x}=\sqrt{x\sqrt{(x-1)\sqrt{(x-2)\sqrt{\ddots\sqrt{2\sqrt1}}}}}$


$\frac{1}{24} (n^6 + 6 n^3 + 3 n^4 + 8 n^2 + 6 n^3) = \frac{1}{24} (n^6 + 3 n^4 + 12 n^3 + 8 n^2) \\= \frac{1}{24} n^6 + \frac{1}{8} n^4 + \frac{1}{2} n^3 + \frac{1}{3} n^2.$


$x^2 - x, x - 1, x$


$k!+2,k!+3, \ldots, 2k!-1$


$f(x,p)= \frac{1-p^2}{2\pi(1+p^2)-2p\cos(x)}$


$\omega^{+}$


$\int_{-\infty}^xf_n(t)\phi(t)dt\to \int_{-\infty}^xf(t)\phi(t)dt$


$f(1+x) = f(-x-1) = f(1-x) $


$\log_a(b) + \log_a(c) = \log_a(b \cdot c)$


$\{a+y, a+2y, a+3y,\dots\}$


$[S_n,S_n]\subseteq A_n$


$P(X_n\leq x) \to P(X\leq x)$


$x<y,x=y,x>y</y,x=y,x>$


$\exists\;\gamma_1,\gamma_2\in \Bbb R $


$ \eqalign{   & \left( \matrix{   x \cr    f(x) \cr}  \right)\quad \left| \matrix{   \;{\rm real }x \hfill \cr    \; - 1 < {\rm real }f(x) \hfill \cr}  \right.\;\quad  = {1 \over {2\,\pi }}\int_{\, - \,\pi \;}^{\,\pi \,} {e^{\, - \,i\,\,t\,f(x)} \left( {1 + e^{\,\,i\,\,t} } \right)^{\,x} \;d\,t}  =   \cr    &  = {1 \over {2\,\pi }}\int_{\, - \,\pi \;}^{\,\pi \,} {\sum\limits_{0\, \le \,j} {\left( \matrix{   x \cr    j \cr}  \right)e^{\,\,i\,\,t\,\left( {j - f(x)} \right)} } \;d\,t}  = {1 \over {2\,\pi }}\sum\limits_{0\, \le \,j} {{{x^{\,\underline {\,j\,} } } \over {j!}}\int_{\, - \,\pi \;}^{\,\pi \,} {e^{\,\,i\,\,t\,\left( {j - f(x)} \right)} \;d\,t} }  \cr}  $


$rb^{r-1} \leq r(a+b)^{r-1}$


$A\cong A_1\times\ldots\times A_n?$


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


$\mathbb{P}(\sup_{t \geq 0} M_t > x \mid \mathcal{F}_0)= 1 \wedge \frac{M_0}{x}$


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


$\sum_{i=2}^\infty\log(1-x^i)=-\sum_{n=1}^\infty\sum_{i=2}^\infty\frac{x^{in}}n=-\sum_{n=1}^\infty\frac{x^n}n\color{red}{\frac{x^n}{1-x^n}}\geqslant-\sum_{n=1}^\infty\frac{x^n}n=\log(1-x)$


$\chi(K_n + I) = (-1)^n x^{n-1} (x - n).$


$\int_0^\infty f_-(x)\,dx$


$y = \ln\frac{1+x}{1-x}$


$\mathbb C_R$


$f(n) = n^{2} + 2n + 1$


$x∘(y∘z) = x⊗y⊗z + x⊗z⊗y + y⊗z⊗x + z⊗y⊗x$


$ f_n(a) = \frac n2 $


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


$rn={n\choose 2}$


$\begin{cases}x_1+x_2=k\\1\leq x_1\\ 1\leq x_2\end{cases}$


$G_1\subseteq G_2\subseteq G_3\subseteq \dots\subseteq G_n \subseteq G_{n+1}\subseteq \cdots$


$\int \frac{x}{(x^2+4)^6}dx$


$ |z| = \sqrt[n]{|w|}[\cos (\frac{2κπ +φ}{n}) + i\sin (\frac{2κπ +φ}{n})] = \sqrt[n]{|w|}e^{\frac{2κπ +φ}{n}i} , κ \in \mathbb Z, κ=0,1,...,n-1$


$z^{\frac{1}{n}}=\sqrt[n]{r}\left(e^{i\left(\frac{\theta}{n}+\frac{2k\pi}{n} \right)}\right)\, $


$\sum_{n\ge 1}x_n=\sum_{n\ge 1}\left\lfloor\frac{x}{2^n}\right\rfloor+\beta(x)\;.$


$f_3(n)=n \sqrt{\log{n}}$


$3 \mid x^3 - x$


$\sum_{n\geq1}\frac1{n^2}\sum_{dq=n}\mu(d) =\sum_{n\geq1}\frac1{n^2}\sum_{d|n}\mu(d) =\sum_{n\geq1}\frac1{n^2}I(n)$


$\mathbb{E}\sup_{t\leq T}|X_t|^p\leq Ce^{CT}(1+\mathbb{E}|X_0|^p)$


$\zeta(s) = \dfrac{1}{\Gamma(s)} \int_{0}^{\infty} \dfrac{x ^ {s-1}}{{\mathrm e} ^ x - 1}\mathrm{d}x,$


$n(\gamma,a)=-n(-\gamma,a)$


$\gcd(a+b, a-b) = 2$


$= (X/4+X/6) + (X/4+X/6) + X/6$


$j= \lfloor N/I \rfloor \mod J;\quad k= \lfloor\lfloor N/I \rfloor/J\rfloor=\lfloor N/(IJ)\rfloor$


$f(P)=\int_0^{2\pi}\frac1{\pi}P\,dP\,d\phi=2P\,dP$


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


$x+\mathcal a = x' + \mathcal a$


$\left\{|\mathbb{N}|,|\mathbb{R}|,|\mathbb{R}^\mathbb{R}|,\left|(\mathbb{R}^\mathbb{R})^{(\mathbb{R}^\mathbb{R})}\right|,\dots\right\}$


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


$ \big( A[x]/(x^2 - d) \big)^\times \simeq A^\times \times A^\times  $


$f(f(x))=xf(x)+a$


$X = (x,y), A = (a,b)$


$h(x) = \int_a^x{f(t)dt}$


$\vartheta(\tau)$


$\forall x[P(x) \vee Q(x)]$


$A_1 A_2 A_3$


$\lim_ {n\to \infty} \mu^*(E_n) < \mu^*(E) $


$ f(0)=0=f(\pi)  \} $


$\int_{0}^\pi l \, dφ=\int_0^\pi r\cdot \sin(φ) \, dφ=r\int_0^\pi \sin(φ) \, dφ = r (-\cos(\pi) +\cos(0))=2r$


$B^{\gamma,\gamma}_{n-\mu} + B^{\gamma,\gamma+1}_{n-\mu} + \ ... + \, B^{\gamma,\mu}_{n-\mu}$


$\mathbb{R}^{n+m}\cong\mathbb{R}^n\times\mathbb{R}^n$


$f(x)=\dfrac{a^x}{b^{x^c}} $


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


$|y-b|+|x-a|=b-y+a-x=(b-x)+(a-y)=(b-x)-(y-a)\\=(b-x)-|y-a|\le|x-b|+|y-a|$


$N! = \sqrt{2 \pi N} \cdot (\frac{N}{e})^{N}$


$[\alpha, \beta, \gamma, \delta][\beta, \gamma] - [\alpha, \beta, \gamma] \cdot [\beta, \gamma, \delta] = 1$


$\gcd(a, -a) = |a|$


$f_R(r) = (n-1)e^{-r}(1 - e^{-r})^{n-2}$


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


$a, a+b, a +2b, $


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


$E_i = \{X \le 1/i\}$


$g:\mathbb{R}^{n+1} \to \mathbb{R}^n$


$a_n = z^n + w^n + i \pi(z^n - w^n) $


$L_1 :\begin{cases} x = 1 + 2t\\ y = 2-2t\\ z = 3+4t\end{cases}$


$\varphi''(x)=-\frac{d}{dx}\int_0^1t f(t)\sin(xt)dt =-\int_0^1\frac{\partial}{\partial x}t f(t)\sin(xt)dt =-\int_0^1t^2 f(t)\cos(xt)dt$


$\lim_{n\rightarrow \infty}X_n(\omega)=X(\omega)<\infty$


$s_n =\frac {a_1 +a_2 +...+a_n} {n}$


$\Omega^{+}$


$\forall\varepsilon\gt0,\exists \delta\gt0,\quad\text{such that}\ \space |~x~| \lt \delta \Rightarrow ~|f(x)| \lt \varepsilon \tag{1}$


$|f(y)| \leq \frac{1}{2} |f(c)|$


$ I=\pi^{-s/2}\Gamma(s/2)\sum_{n=1}^\infty\frac{1}{n^s}=\pi^{-s/2}\Gamma(s/2)\zeta(s) $


$ax+by=n$


$B=\{d,e,..,H,G\}$


$[x,y]=x\circ y-y\circ x$


$\gamma(t) = (\gamma_x(t),\gamma_y(t)), \; \gamma(0) = (x_0, y_0), \; \gamma(1) = (x, y); \tag{10}$


$\int_0^\infty  {\frac{x^{2n - 1}}{{\left( x^2 + 1 \right)}^{n + 3}}\,dx}\quad\quad n\geq 1\;,\; n\in\mathbb{Z}$


$\color{blue}{\sqrt z= \sqrt r e^{i\varphi/2}}$


$(a+b)f=af+bf$


$\lim_{x\to0}\left(\frac{\sin^{-1}(x)}{x} \right) = 1 $


$\beta+\gamma\geq \alpha,\ \gamma+\alpha\geq \beta,\ \wedge\ \alpha+\beta\geq \gamma.$


$\mathcal{A}+\mathcal{X}=\mathcal{B}$


$ \mathbf{1.}\ (\forall m<n \ P(m) \implies P(n))$


$\frac{1}{(1+m)^{1/n}} + \frac{1}{(1+n)^{1/m}} \ge \frac{1}{1+\frac{m}{n}} + \frac{1}{1+\frac{n}{m}} = \frac{n}{n+m} + \frac{m}{n+m} = 1$


$A_{l_1} \times ... \times A_{l_m} \subseteq A_1 \times ... \times A_n = \textrm{Soc}(S_1)$


$\lim \limits_{n \to \infty}\frac{2}{n}\left\lfloor \frac{n}{3}\right\rfloor$


${\vartheta _n} = \int_{ - \infty }^{ + \infty } {\frac{1}{{{{\left( {1 + {x^2}} \right)}^n}}}} \frac{{dx}}{{1 + {x^2}}}$


$ \dfrac{2}{x} + \dfrac {3}{y} = 1 $


$\widetilde{\gamma}(\sum a_s s)=\sum a_s\gamma(s)\quad\text{and}\quad\overline{c}(s)=c(1s).$


$f(n) = [(1^{n-1}) + (-1^{n-1})] u(n-1)$


$T_x(U) = T_x(X)$


$
 d(x, \{a\}^{\perp})=\frac{\left|\langle x,a\rangle \right|}{\left\|a\right\|}.
 $


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


$\mathbb E|X|\mathbb E|Y|=1$


$\det M=(k-\lambda)^{n-1}\left(\lambda(n-1)+k\right)$


$[\gamma]\le[\gamma']\iff\gamma\le\gamma',$


$[\frac{1}{9}, \frac{1}{9}+\frac{1}{27}] \cup [\frac{2}{9}-\frac{1}{27}, \frac{2}{9}] \cup [\frac{1}{3}, \frac{1}{3}+\frac{1}{27}]\cup[\frac{2}{3}-\frac{1}{27}, \frac{2}{3}]\cup [\frac{7}{9},\frac{7}{9}+\frac{1}{27}]\cup [\frac{8}{9}-\frac{1}{27},\frac{8}{9}].$


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


$\left[   \begin{array}{cccc|c}   1 & 2 & -1 & 3  & 3 \\   2 & 4 &  4 & 3  & 9 \\   3 & 6 & -1 & 8  & 10 \\   \end{array}   \right]$


$({\frac {1} {2^{s-1}})}^j$


$A_n \supset A_{n+1} \supset \ldots$


$ f(a, b, c) = a^\alpha + b^\beta + c^\gamma, g(a, b, c) = abc - 1 = 0 \\ \nabla f = \left\langle\alpha a^{\alpha - 1}, \beta b^{\beta - 1}, \gamma c^{\gamma - 1}\right\rangle = \lambda \nabla g = \lambda \langle bc, ac, ab\rangle \\ \alpha a^\alpha = \lambda abc = \lambda \rightarrow a = \left(\frac{\lambda}{\alpha}\right)^\frac{1}{\alpha} \\ \beta b^\beta = \lambda abc = \lambda \rightarrow b = \left(\frac{\lambda}{\beta}\right)^\frac{1}{\beta}\\ \gamma c^\gamma = \lambda abc = \lambda \rightarrow c = \left(\frac{\lambda}{\gamma}\right)^\frac{1}{\gamma} \\ abc = 1 = \frac{\lambda^{\frac{1}{\alpha} + \frac{1}{\beta} + \frac{1}{\gamma}}}{\alpha^\frac{1}{\alpha}\beta^\frac{1}{\beta}\gamma^\frac{1}{\gamma}} \rightarrow \lambda = \left(\alpha^\frac{1}{\alpha}\beta^\frac{1}{\beta}\gamma^\frac{1}{\gamma}\right)^\frac{\alpha\beta\gamma}{\alpha\beta + \alpha\gamma + \beta\gamma} = \left(\alpha^{\beta\gamma}\beta^{\alpha\gamma}\gamma^{\alpha\beta}\right)^\frac{1}{\alpha\beta + \alpha\gamma + \beta\gamma} $


$\pmatrix{0\cr -1\cr -1\cr 1\cr}$


$7\mid a^7-a$


$F := P \oplus Q$


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


$T_X = T_X^1$


$f^n(x)=\dfrac{ax+b}{cx+d}$


$A_{n}\cdots A_{1}$


$S \equiv $


$\tag{1}\dfrac{(1+r)^k-(1-r)^k}{(1+r)^k+(1-r)^k}=-\frac {\sqrt{b}}a$


$a^{log b}$


$\sum_{i=1}^n \frac{1}{2^i}$


$ r = |\sin( \varphi) \cos( \varphi) \cos( \psi) \sin^2 (\psi)| $


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


$\{\gamma,-\gamma\}=\{\beta,-\beta\}$


$\mu(\lim \inf A_n) \leq \lim \inf \mu(A_n)\leq \lim \sup \mu(A_n) \leq \mu(\lim \sup A_n)$


$|X\times Y|=mn$


$x = (x_{1},x_{2}....,x_{n})$


$L_2 :\begin{cases} x = 2+2t\\ y = 1\\ z = t\end{cases}$


$\mathcal{O}_{\mathbb{R}^{m+n} | \mathbb{R}^n}$


$\sum_{n=1}^\infty\frac{n+2-2}{(n+2)!}=\sum_{n=1}^\infty\frac1{(n+1)!}-2\sum_{n=1}^\infty\frac1{(n+2)!}$


$[x,y]=z$


$\sum _{i=1}^n r^{i-1}=\frac{r^n-1}{r-1}$


$ \| \gamma(a)-\gamma(b)\| \le  \int_a^b \| \gamma '\|$


$\int_{0}^{2\pi} f(t) \cos(pt)\,\mathrm{d}t=\int_{0}^{2\pi} f(t) \sin(pt)\,\mathrm{d}t=0,$


$\zeta(s)={1\over \Gamma(s)}\int_{0}^{\infty}{x^{s-1}\over e^x-1}\mathrm dx\tag5$


$\left[\begin{smallmatrix}A&B\\C&D\end{smallmatrix}\right]$


$A = \{a_1,a_2,a_3,..,a_N\} $


$n! = \sqrt{2\pi n}(n/e)^n (1+o(1))$


$U \subset V \subset X,$


$\sum x_n e_n$


$\gcd(a,a) = |a|$


$ \frac{e^{-n}-e^{-n-1}}{n+1} \leq \int_{n}^{n+1} \frac{e^{-x}}{x} \ dx  \leq \frac{e^{-n}-e^{-n-1}}{n}, $


$ \frac{a+b}{b+x}\in\mathbb{Z}\tag{3} $


$ \dim S = \dim V-\dim S^\perp=\dim S^\perp+\dim S^{\perp\perp}-\dim S^\perp=\dim S^{\perp\perp}\tag{1} $


$\left[\begin{array}{ccc|c}1 & 3 & -1 & 4\\4 & -1 & 2 & 8\\2 & -7 & 4 & 0\end{array}\right],$


$
  \lim_{n\rightarrow\infty} \mu(A_n) =\mu (A)=1.
 $


$A_{1} \subset A_{2} \subset A_{3} \subset \dots \subset A_{n} \subset A_{n + 1} \subset \dots$


$f_2(x) = \dfrac{2}{(1-x)^{3}}$


$\tan (-\theta)=\frac {sin (-\theta)}{cos (-\theta)}\frac {-sin (\theta)}{cos (\theta)}=-tan (\theta)$


$\overline{[S]}\subset\overline{S^{\bot\bot}}=S^{\bot\bot}\Rightarrow \overline{[S]}^{\bot}\supset S^{\bot\bot\bot}\supset S^{\bot}$


$ (x-2)[(x-2)^{n-1}-n+1] $


$\int_0^1 f(t) dt = \int_0^1 g(t) dt.$


$=\frac a\gamma\int_{0}^x\frac{\gamma e^{\gamma u}du}{e^{\gamma u}-1+a\gamma}-a\int_0^x\frac{du}{e^{\gamma u}-1+a\gamma}$


$\int_0^\infty f(y)\,dy$


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


$\;\Bbb Z/(11)\;$


$\displaystyle{\lim_{x \to c}f(x)=L}$


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


$P\{\sup_{t\geq 0} M_t > x \,| \,F_0\} \leq 1\wedge \frac{M_0}{x}$


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


$\lim_{n\to\infty}\frac{\sqrt{n+1}+\sqrt{n+2}+\dots+\sqrt{2n}}{\sqrt{1}+\sqrt{2}+\dots+\sqrt{n}}$


$P^*(4) = P^*(5) = P^*(6) = P^*(7)=P^*(8)=P^*(9)= P^*(10) = 0$


$F = x+y$


$\begin{vmatrix} 1&1 &1 \\ x_1& x_2 &x \\ y_1& y_2& y \end{vmatrix}=0$


$D=\{(x,y)\in\mathbb R^2|0\leq x \leq m,0 \leq y \leq n\}$


$         \begin{pmatrix}          A & B^T \\          -B & A^\dagger \\         \end{pmatrix}  $


$f(x) = \sum_{j=1}^{\infty} \frac{1}{j}\chi_{A_{j}}(x).$


$1/\vert x \vert^4$


$e^x > \bigg(1+ \frac xn\bigg)^n $


$\|\dot\gamma\|^2=\dot\gamma\cdot\dot\gamma=1 \Rightarrow (\dot\gamma\cdot\dot\gamma)'=0 \Rightarrow \ddot\gamma\cdot\dot\gamma+\dot\gamma\cdot\ddot\gamma=0 \Rightarrow 2\ddot\gamma\cdot\dot\gamma=0 \Rightarrow \ddot\gamma\cdot\dot\gamma=0$


$\sum_{n=N}^\infty \frac1{n^2}+\sum_{n=N}^\infty\frac1{n^3}<\epsilon$


$S=\{ (x,y) \in \mathbb{R^2}\: | \:0 \leq x \leq 1, 0 \leq y \leq 1\}$


$d(x,a)\le d(x,b)+d(b,a)$


${Let S = \{x_1, x_2, x_3, ..., x_n\}}$


$p\mid a_1a_2$


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


$ f : \mathbb{R}^2 \rightarrow \mathbb{R}, f((x, y)) = x+y $


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


$\forall \epsilon > 0,  \exists \delta > 0 \text{ such that } \forall x, y \in I,  |x-y| < \delta \Rightarrow |f(x)-f(y)| < \epsilon$


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


$X_M^c\subset [X_M^c,X_M^c]$


$\int_{-\infty}^\infty \frac{1}{(x^2 + 1)(x^4+4)^2} dx$


$[x,y]=xy-yx$


$\frac{\pi^2}{4} \frac{\sin^2(\pi p/2)}{\cos(\pi p /2)}.$


$Cov(X,Z+Y)=Cov(X,Z)+Cov(X,Y)$


$\{x^2, y^2, z^2, xy, xz, yz\}$


$  \left[\begin{array}{rr|r}7.5 & -5 & 0 \\k + 7.5 & 0 & 0 \\\end{array}\right]$


$A=\frac {Pr(1+r)^n}{(1+r)^n-1}$


$a,b,c,d\in\mathbb{R},a\leq b,c\leq d$


$L(x)\vert E(x)$


$\mathbb Z\oplus T$


$\mathbb R^n\times\mathbb R^n\to\mathbb R^n\times\mathbb R^n$


$\int_0^\infty \sum_{n=-\infty}^\infty e^{-\pi{n^2}/ x}  dx $


$\Gamma(s) = \sum_{n=1}^{\infty} \frac{\mu(n)}{n^s} \int_0^\infty\frac{x^{s-1}}{e^x-1} dx .$


$\det{\alpha \gamma} = d^4 = (\det{\alpha}) (\det{\gamma}) = (\det{\alpha})(\det{\beta}) = dD.$


$Cov(X+Y, Z+W)=Cov(X,Z)+Cov(X,W)+Cov(Y,Z)+Cov(Y,W)$


$\det(A) = x (x-a)^{n-1}$


$g(x)=\frac{a+bx}{b+ax},$


$ \color{red}{\sum\limits_{k=1}^n\cos\left(\frac{2k-1}{2n+1}\pi\right)=\frac12}. $


$\vert ab\vert = \operatorname{lcm}(m,n)$


$\begin{vmatrix} 1 & a & a^3 \\ 1 & b & b^3 \\ 1 & c & c^3 \\ \end{vmatrix}$


$\mu(A)= \int_{a}^{b} 1~dx$


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


$\left\lfloor{\frac{x}{n}}\right\rfloor=\left\lfloor{\frac{\lfloor{x}\rfloor}{n}}\right\rfloor,$


$[-m, m] = [-m, m] + [-m, m]$


$\displaystyle \tan \theta = \frac{x}{20}$


$ \sqrt {11} = \frac {a+b}{b}$


$\int_f \leq \lim_{n \to \infty}\inf    \int f_n\ d\mu.$


$\gcd(a, b) = ax+by = d$


$\int_\gamma\frac{dz}{z+i}=\int_{\tilde\gamma}\frac{dz}{z+i}-\int_{\gamma_0}\frac{dz}{z+i}=\int_{\tilde\gamma}\frac{dz}{z+i}+\ln(\sqrt2)-\tfrac14\pi i,$


$\alpha=\omega\alpha$


$(k+6)^{1/3} > k^{1/3}$


$Cov(X,Y) = 0$


$\sum_{n\ge 1}\left(\frac3{n(n+3)}+\frac1{2^n}\right)=\sum_{n\ge 1}\frac3{n(n+3)}+\sum_{n\ge 1}\frac1{2^n}$


$ \left[ \begin{array}{cccc|c}   1&2&-1&-5&0\\   -1&0&3&5&0\\ 1&1&a&b&0 \end{array} \right] $


$E((X+Y)^p)\leq 2^p\left( E(X^p)+E(Y^p)\right)$


$\frac{246}{217}=1+\frac{29}{217}=1+\frac1{\frac{217}{29}}=1+\frac1{7+\frac{14}{29}}=1+\frac1{7+\frac1{\frac{29}{14}}}=1+\frac1{7+\frac1{2+\frac1{14}}}$


$log_{a}x=b $


$f(x,y)={{x^3}\over{y^2}} e^{{-x}\over{y}}$


$\lim_{n\to\infty}\sup (-a_n)=-\lim_{n\to\infty}\inf a_n.$


$ f_X(x) = \frac{1}{\sqrt{2\pi}}e^\frac{-x^2}{2} \; \forall x\in \mathbb{R} $


$ \int_{\gamma} \frac{dz}{z^2+1} = -i\int_{\gamma} \frac{dz}{z-i}  + i\int_{\gamma} \frac{dz}{z+i}.   $


$[X,Y]= X\wedge Y$


$F(a) + F(b) = F(a+b)$


$\left\vert\begin{matrix}1&1&1\\x&y&z\\f(x)&f(y)&f(z)\end{matrix}\right\vert\not=0$


$ \sum^{\infty}_{n=1}\left\|f_n\right\|<\infty, $


$\sum\Vert x_n-y_n\Vert <1/(2K)$


$T_pM\otimes\mathbb C=T^{1,0}_pM\oplus T^{0,1}_pM$


$S_3(n+1)=\binom n2=\frac{n(n-1)}2=\sum_{k=1}^{n-1}k.$


$[x, y]=x$


$e^{-x} - (1 - x/n)^n = e^{-x}[ 1 - e^{x}(1- x/n)^n] \leqslant e^{-x}[ 1 - (1 +x/n)^n(1 - x/n)^n] = e^{-x}[ 1 - (1 -x^2/n^2)^n] \\ \leqslant\frac{e^{-x}x^2}{n} .$


$U(n)={n\choose 2}$


$\sum\limits_{n=0}^{\infty}xe^{-n^2x} = x\sum\limits_{n=0}^{\infty}e^{-n^2x}$


$\displaystyle\;T_n = \frac{n(n+1)(n+2)}{6}$


$\bigcup_{\gamma<\delta}\lambda_{\gamma}\times{\{\gamma\}}$


$w=\sqrt re^{i\theta/2}$


$c = b^{\log_b(c)}$


$[a]+([b]+[c])=[a]+[b+c]=[a+(b+c)]=[(a+b)+c]=([a]+[b])+[c]$


$\:\mathrm{d}s=\sqrt{1+\frac{\mathrm{d}^{2}y}{\mathrm{d}x^{2}}}\:\mathrm{d}x$


$A_1\times\dots A_n$


$2 + 2 + 2 + 2 + ... = -\frac12$


$|x^2-a^2|=|x-a||x+a| \leq \epsilon |x+a| \leq \epsilon (2|a|+\epsilon)$


$\text{gcd}(a+b, a+b) = 1$


$[x,y] = \textrm{lcm}[x,y]$


$\overline{Z_0} \subset \overline{Z_1} \subset \dots \subset \overline{Z_n}$


$V\subseteq\overline{V}\subseteq V_1\cap V_2$


$+++−−+−−=1+1+1-\frac12-\frac14+\frac18-\frac1{16}-\frac1{32}=2+\frac14+\frac1{32}$


$z=\sqrt[4]{2}e^{\frac{5\pi i}{3}}$


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


$K[T^2,T^3]\subset K[T]$


$} \mapsto \lbinom{n}{k+1} \right) = \left( \text{int $


$xRx\lor xRx$


$y_n = (x_1+ x_2+ \dots + x_n)/n$


$ f(t_x)=0\quad  \forall \;t_x\in(0,1). $


$\lfloor (a+b+c)/d\rfloor+\lfloor (a+b+d)/c\rfloor+\lfloor (a+d+c)/b\rfloor+\lfloor (d+b+c)/a\rfloor$


$\left\lfloor\frac{\lfloor x \rfloor + j}{m}\right\rfloor = \left\lfloor\frac{x + j}{m}\right\rfloor$


$F(n)F(n+2)-F(n+1)^2=F(n)^2-F(n-1)F(n+1)$


$f(x)=\sum_{j=1}^n c_j\lfloor x\rfloor \chi_{[j,j+1)}(x)$


$P(E(X\mid \mathcal F_0)\in C)=1$


$A= \int \int _D dS = \int \int _D 3/2 dA = 3/2(16\pi ) = 24\pi $


$|X_n|^p \leq (|X_n-X|+|X|)^p \leq |X_n-X|^p + |X|^p$


$[a]+[n-a]=[a+(n-a)]=[a+(-a+n)]=[(a-a)+n]=[n]=[0]$


$\|UAZ\|_2^2=\rho(Z^*A^*U^*UAZ)=\rho(Z^*A^*AZ)=\rho(AZZ^*A^*)=\rho(AA^*)=\rho(A^*A)=\|A\|_2^2$


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


$x,y \in \mathbb{R} ,f(x+y)=f(x)f(y)$


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


$\int_{-\pi}^\pi f^2=\int_{-\pi}^\pi f^2-\int_{-\pi}^\pi fp=\int_{-\pi}^\pi f(f-p)$


$z^{\frac{1}{n}}=\sqrt{n}e^{i\frac{\theta}{n}}$


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


$(x+y)^*(x+y)  = x^*x + x^*y + y^*x + y^*y$


$2^{n/2}/(100 \sqrt{n})$


$x\in V\subseteq\cl_YV\subseteq U$


$(x^y-y^x) + (y^z -z^y) + (z^x - x^z) = 0$


$P_1\mid P_2 \mid\cdots\mid P_k$


$d/dx(2 g(x)/(x^2 + 1))$


$\displaystyle \frac{|\langle u,v\rangle|}{\Vert u\Vert}\leq\Vert v\Vert$


$L(\gamma)=\int_\gamma \ ds =\int_a^b \gamma^*ds=\int_a^b\frac{|\gamma'(t)|}{1+|\gamma(t)|^2}dt. $


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


$P(m)=P(n)$


$x_1=\sqrt[10]{\frac43}e^{\frac{i\theta}5}$


$A_n\subseteq...\subseteq A_2\subseteq A_1$


$\sum_{n\geq1}\frac{e^{2\pi inx}}{n}=\sum_{n\geq1}\frac{1}{n}=\infty  $


$  (\delta, \gamma)^{(\overline k, 1)} = (\delta^{\overline k(\gamma)}, \gamma^1)   = (\delta^k, \gamma) = (\delta, \gamma). $


$a_0=0;\\a_1=1+a_0=1+0=1;\\a_2=2+a_1=2+1=3;\\a_3=3+a_2=3+3=6;\\a_4=4+a_3=4+6=10; \\a_5=5+a_4=5+10=15;\\a_6=6+a_5=6+15=21;$


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


$\mathrm{P}\left[\left.X_t\in A\space\right|\mathcal{F}_s\right]=\mathrm{P}\left[\left.X_t\in A\space\right|X_s\right]$


$(a+b,a-b)=(c,d)$


$\lim_{n \rightarrow \infty}\mu(B_n) \le \lim_{n \rightarrow \infty}(\inf_{k \ge n}\mu(A_k))=\liminf_{n \rightarrow \infty}\mu(A_n)$


$[x,y]=f$


$R(x) = o((x - a)^{n - 1})$


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


$\|A\|_2 \leq \sqrt{m} \|A\|_{\infty}$


$P(X_n \le x)$


$\gamma(t)=(\gamma_1(t),\gamma_2(t),\gamma_3(t)),0\le t \le 1$


$P(\omega \in \Omega : lim_{n \rightarrow \infty} X_n(\omega) = X(\omega))= 1 $


$(\beta+\gamma-2), (-\alpha + 3\gamma - 5)$


$\gcd(a,a)=a$


$μ(x,t)=μ·x$


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


$F=P \oplus M$


$f'(x+f(y)+xf(y))\cdot (f'(y)+xf'(y)) = 1+f(x)$


$\frac{{(1+\sqrt{1+4x})}^{n+1}-{(1-\sqrt{1+4x})}^{n+1}}{2^{n+1}\sqrt{1+4x}}$


$\phi_{max}=\arctan \frac{\sqrt{1-k^2}}{k}$


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


$\mbox{Cow} \approx \#^g_{i=1}\mathbb T^2 ,  $


$ \mathbb{E}\|X\|_{\infty}^r\le C_{d,r}\mathbb{E}\|Y\|_{\infty}^r $


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


$|c_n (-2)^n|=|c_n 2^n|=|c_n 4^n|\cdot \frac{1}{2^n}\leq \frac{1}{2^n} $


$B=\left\{\langle x,y\rangle\in\Bbb R^2:x>0\text{ and }y=\frac1x\right\}$


$f(e_i)=e_i,\ p+1\leq i\leq n$


$\phi = \begin{bmatrix} 1 & x & y & z & xy & xz & yz & x^2 & y^2 & z^2 \end{bmatrix}$


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


$\lim\limits_{x \rightarrow x_{0}^{-}} f(x) = \lim\limits_{x \rightarrow x_{0}^{+}} f(x) = \lim\limits_{x \rightarrow x_0}f(x)$


$ \sum_{n=1}^{\infty} \|x_n-y_n\|^2 \geq 1 $


$a_1 = g(f(a_1)$


$(3)^{\frac{1}{3}} (9)^{\frac{1}{9}} (27)^{\frac{1}{27}}$


$\int_a^x f(t)dt = F(x)-F(a).$


$\int_0^\infty \frac{\sin^2(x)}{x}dx$


$q(X)= X^2 + X + 2$


$\sigma(T^0,T^1)\subset T$


$I(s)=\displaystyle\int_{0}^{\infty}\dfrac{x^{s-1}}{e^x+1}\,dx=\eta(s)\Gamma(s) \tag*{} $


$A_{n+1} \subset A_1 = A_n \subset A_{n+1}$


$A \subseteq S \subseteq \bar A$


$ \zeta(2) = \sum_{n\geq 1}\frac{1}{n^2} = 2\sum_{n\geq 1}\frac{(-1)^{n+1}}{n^2} = 2\,\eta(2),$


$\int_0^{2\pi} f(t)p(t)\,dt=0$


$\sum_{n \leq x} \frac{1}{n} \left\lfloor\frac{x}{n}\right\rfloor = \sum_{n \leq x} \frac{1}{n} \left[ \frac{x}{n} + O(1) \right] = x \sum_{n \leq x} \frac{1}{n^2} + O(1) \sum_{n \leq x} \frac{1}{n}$


$f_Y(y)= \frac{1}{2}e^{\frac{-y}{2}}$


$\left(\begin{array}{cc} A & B \\ B^T & D\end{array}\right)$


$\forall\epsilon\gt 0,\exists \delta\gt 0,|x-c|\lt \delta , x\in D\Rightarrow |f(x)-f(c)|\lt\epsilon$


$11!+2,\,11!+3,\,11!+4,\,\ldots,\,11!+11$


$\phi = \sum_{i=0}^{n} a_i \chi_{A_i}$


$\begin{bmatrix}1&1&-2&3\\0&1&1&-1\\-2&-1&5&-7\end{bmatrix}$


$P(n) = \binom{n - 2}{3}$


$\neg\neg\phi\lor\neg\neg\neg\phi\tag{axiom 12}$


$(m, m, m, ...)$


$f\left ( f(x)^{2}y \right )=x^{3}f(xy) \tag{1}.$


$\Large{a^{\log_a(b)}=b\qquad\qquad a^{b/c}=(a^b)^{1/c}}\qquad\qquad a^{b-c}=\frac{a^b}{a^c}$


$p_1p_2\mid |G|$


$\mathcal{F}=\lbrace  \lbrace a \rbrace, \lbrace b\rbrace, \lbrace c\rbrace , X , \emptyset\rbrace.$


$ \det (a)=a\quad\text{and}\quad\det\begin{pmatrix}a&b\\c&d\end{pmatrix}=ad-bc $


$ax+by=n.$


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


$\begin{cases}x=4+1t\\y=0+0t\\z=1-2t\end{cases}$


$\gamma + \gamma \: < \: \Gamma(0,\gamma) + \gamma + \gamma +1 \: = \: \Gamma(0,\gamma+1) \: < \: \Gamma(0,\alpha) \: = \: \alpha$


$P[X \leq x-\epsilon] - P[|X_n - X| \geq \epsilon] \leq P[X_n \leq x] \leq P[X\leq x+\epsilon] + P[|X_n - x| \geq \epsilon]$


$d\in M^\perp$


$A+U \subseteq V$


$\prod\mathscr{A} = A_1 \times \cdots \times A_n$


${3\over a}+{5\over b}+{6\over c}=1\ .\qquad(1)$


$\mid f(z)\mid \leq \mid f(z^2)\mid\leq \mid z^2 \mid\leq \mid z\mid$


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


$\nabla_{\dot\gamma}\dot\gamma - \frac{g(\nabla_{\dot\gamma}\dot\gamma,\dot\gamma)}{g(\dot\gamma,\dot\gamma)}\dot\gamma$


$n\sum_{k=1}^{n-1}\cos\frac{2k\pi}{n} -[ \sum_{k=1}^{n-1} cos\frac{2k\pi}{n}] -[\sum_{k=2}^{n-1} cos\frac{2k\pi}{n}]-[\sum_{k=3}^{n-1} cos\frac{2k\pi}{n}].........-[\sum_{k=(n-2)}^{n-1} cos\frac{2k\pi}{n}]-cos\frac{2\pi}{n}$


$\int_0^{100} (1-e^{\frac{-x}{1000}})  $


$F(k)=\sum_{n=1}^{\infty}\frac{n^{4k-1}}{e^{n\pi}-1}-16^k\sum_{n=1}^{\infty}\frac{n^{4k-1}}{e^{4n\pi}-1}$


$\frac{m(x-\delta ,x+\delta )}{2\delta }=\frac{f(x+\delta )-f(x-\delta )}{2\delta }$


$\sum_{n=-\infty}^\infty|c_n|^2=\dfrac{1}{2\pi}\int_{-\pi}^\pi|\hat{f}(x)|^2dx,$


$\frac{1}{\sqrt{x}}e^{-\frac{n^2\pi}{x}} = \int_{-\infty}^\infty e^{-m^2\pi x -2\pi i m n} dm.$


$\left\lfloor\frac{\lfloor\frac{a}{b}\rfloor}{c}\right\rfloor = \left\lfloor\frac{\lfloor\frac{a}{c}\rfloor}{b}\right\rfloor = \left\lfloor\frac{a}{bc}\right\rfloor$


$\begin{cases}x=u\\y=v\\ z=w(1-u-v)\end{cases}$


$a \pmod 6 \in \{0,1,2,4,5\}$


$ e^{\ln(a)}=a \implies b^{\log_b(a)}=a $


$2(4^x) + 3(9^x)=5(6^x)$


$P(15)\implies P(19)$


$\tag 1\lim_{x\to \infty}f(x) = L\, \iff\, \lim_{x\to 0^+}f(1/x) = L.$


$\sum\left\|e_{n}-f_{n}\right\|^{2}<\infty$


$P(E(X|{\mathcal F}_0)=0)=0$


$1 =\frac{4}{10} * x$


$=8+\frac1{\frac{31}{22}}=8+\frac1{1+\frac1{\frac{22}9}}=8+\frac1{1+\frac1{2+\frac49}}=8+\frac1{1+\frac1{2+\frac1{\frac94}}} =8+\frac1{1+\frac1{2+\frac1{2+\frac14}}}$


$\neg\neg B\lor \neg B$


$\lim_{n \to \infty}\dfrac{\sqrt{1}+\sqrt{2}+\sqrt{3}+...+\sqrt{n-1}}{n\sqrt{n}} = \lim_{n \to \infty} \frac{1}{n} \sum_{k=0}^{n-1} \sqrt{\frac{k}{n}} = \int_0^1\sqrt{x}\mathrm{d}x$


$A = \begin{pmatrix} 1 & x & x^2 & x^3 \\ 1 & y & y^2 & y^3 \\ 1 & z & z^2 & z^3\\ 1 & w & w^2 & w^3 \\ \end{pmatrix} $


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


$(1+r)^{-N} = 1 + r(-N) + r^2\frac{(-N)(-N-1)}{2!}  ...$


$\lim_{n\to\infty} h(x_n)=\inf h(x)$


$\lim_{n\to\infty} \mu(A_n)\geq 1$


$ f(x) = {2x-1\over x^2} $


$\frac{dy}{dx}=(1+\frac{dy}{dx})(-\sin(x+y))$


$ \begin{align} \int_0^1\left(\frac{\sin^{-1}(x)}{x}\right)^2\,\mathrm{d}x &=\int_0^{\pi/2}\left(\frac{t}{\sin(t)}\right)^2\,\mathrm{d}\sin(t)\tag{14}\\ &=-\int_0^{\pi/2}t^2\,\mathrm{d}\frac1{\sin(t)}\tag{15}\\ &=-\frac{\pi^2}4+2\int_0^{\pi/2}\frac{t}{\sin(t)}\,\mathrm{d}t\tag{16}\\ &=-\frac{\pi^2}4-2\int_0^{\pi/2}t\,\mathrm{d}\log(\csc(x)+\cot(x))\tag{17}\\ &=-\frac{\pi^2}4+2\int_0^{\pi/2}\log(\csc(x)+\cot(x))\,\mathrm{d}t\tag{18}\\ &=-\frac{\pi^2}4+2\int_0^{\pi/2}\big(\log(1+\cos(x))-\log(\sin(x))\big)\,\mathrm{d}t\tag{19}\\ &=-\frac{\pi^2}4+2\left(2\mathrm{G}-\frac\pi2\log(2)+\frac\pi2\log(2)\right)\tag{20}\\ &=\bbox[5px,border:2px solid #C0A000]{-\frac{\pi^2}4+4\mathrm{G}}\tag{21} \end{align} $


$ \begin{align} \int \frac{1}{x^2+1}dx&=\arctan x +C \tag3 \end{align} $


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


$f(f(x) + x) = xf(x^2 + 1)$


$\{(x,x),(y,y)\},\{(x,x),(y,y),(x,y)\},\{(x,x),(y,y),(y,x)\},\{(x,x),(y,y),(x,y),(y,x)\}$


$\begin{cases}
 x = 1\\
 y = 0\\
 z = 0\\
 \end{cases}$


$P(X_1>X_2 | X_2=x) = 1-F_{X}(x)$


$d(x,p)\leq d(x,y)+d(y,p)  $


$\dfrac{1}{2\pi i}\int_\gamma \dfrac{2i}{z^2+1}=\int_\gamma \dfrac{2i}{z^2+1}=\dfrac{1}{2\pi i}\bigg(\int_\gamma\dfrac{1}{z-i}-\int_\gamma\dfrac{1}{z+i}\bigg)$


$(K[X]/(f))^\times$


$\displaystyle \left[ \begin{array}{rrr|r} 1 & 2 & 1 & 3 \\ 1 & 3 & -1 & 1 \\ 1 & 2 & a^2-8 & a \\ \end{array} \right] $


$x^3 - x^2 - x$


$A_1 \subseteq A_2 \subseteq \ldots \subseteq A_K \subseteq S $


$\forall m,[\forall m'<m,\forall n,P(m',n)]\to\forall n,[\forall n'<n,P(m,n')]\to P(m,n)\implies\forall xy,P(x,y)$


$K \subseteq M \subseteq L$


$\lim_{n \to \infty} \sup_{A \in \mathcal F}|\mu_n (A) - \mu(A)| \to 0,$


$ s(x)=\sum_{i=1}^mc_i\chi_{A_i}(x)\ . $


$\int_{-\infty}^{\infty}\frac{1}{(x^2+x+1)^a}\,\mathrm dx, a > 0$


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


$\qquad \small \begin{array} {rrrrrrr}  1 & -1 & 1/2 & -1/6 & 1/24 & -1/120 & 1/720 & -1/5040 \\  . & 1 & -3/2 & 1 & -5/12 & 1/8 & -7/240 & 1/180 \\  . & . & 1 & -11/6 & 35/24 & -17/24 & 35/144 & -23/360 \\  . & . & . & 1 & -25/12 & 15/8 & -49/48 & 7/18 \\  . & . & . & . & 1 & -137/60 & 203/90 & -967/720 \\  . & . & . & . & . & 1 & -49/20 & 469/180 \\  . & . & . & . & . & . & 1 & -363/140 \\  . & . & . & . & . & . & . & 1  \end{array} $


$f:=f_{1},...,f_{n}\in L^{p}(\mu)$


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


$\int_0^\infty\int_0^\infty f(x,y)dxdy$


$\mathcal{F}: \mathbb{R}^n\times\mathbb{R}^n\to \mathbb{R}$


$\frac{4}{x}+\frac{5}{y}\geq 23$


${\mathcal A}:X = X:{\mathcal A} = X$


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


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


$[\begin{smallmatrix}a & -b\\ b & \hphantom{-}a\end{smallmatrix}]$


$A_{n+1} \subset A_n \subset A_{n-1} \subset \ldots$


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


$\lim_{ n \to \infty }f(a_n) \ \ ?$


$ a \leq x_2 \leq b, \qquad c \leq y_2 \leq d$


$3|n^3 - n$


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


${C_r}: |z| = r$


$\det\begin{pmatrix}A& 0\\ C& B\end{pmatrix} = \det\begin{pmatrix}A& D\\ 0& B\end{pmatrix} = \det(A) \det(B) . $


$\bigcup_{k=0}^nG(k)=\bigl\{\langle x,y\rangle\in\Bbb N\times\Bbb N:x\leq n,y\leq n\bigr\}$


$P_3P_2 = P_1P_7$


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


$x_k= \sqrt[n]{a}e^{\frac{i(2k+1)\pi}{n}}$


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


$ s(n) =  \sum_{k=0}^n|\cos k| - \frac{2}{\pi}n  = \sum_{k=0}^{n^*}|\cos k| - \frac{2}{\pi} n^*  +  \sum_{k=n^* +1}^n|\cos k| - \frac{2}{\pi}(n-n^*) \\ = s(n^*) + \sum_{k=n^* +1}^n|\cos k| - \frac{2}{\pi}(n-n^*) $


$\sum_{n\geq1}\frac{\log^{2}\left(n\right)}{n^{2}}=\sum_{n=1}^{N}\frac{\log^{2}\left(n\right)}{n^{2}}+\sum_{n\geq N+1}\frac{\log^{2}\left(n\right)}{n^{2}}\leq  $


$d(x,z) \leq d(x,y)  + d(y,u)  + d(u,z)$


$\to E[X_{T \wedge n} | \mathscr{F_0}] =/ \le X_{T \wedge 0} \ \text{by choosing m = 0}$


$(1+2^{\frac n2})^2$


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


$\equiv \; s \vee r$


$\left(\begin{array}{cc}
 A&B\\
 C&D
 \end{array}\right).$


$\mathrm{Cov}(X,Y)=\mathrm{Cov}(-X,Y)=-\mathrm{Cov}(X,Y)$


$I_n = \pi n$


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


$\mu(A_n)\geq \mu(A)$


$ \|M\|_2 = \sigma_1(M) = \sqrt{\lambda_{max}(M^TM)} $


$T_x(X) \times T_x(X).$


$M_1 = \begin{bmatrix} AB &  0\\ B  &  0\\ \end{bmatrix}$


$B_n= { 2^n\choose 2^{n-1}}$


$\zeta(3)=\lim_{N\to \infty}{1\over N}\sum_{k=1}^{N}{1\over k^{\phi^2}\ln{\left(1+{k^{\phi^{-2}}\over N}\right)}}$


$|\{\alpha \in A: \alpha < \omega\}|<\omega$


$I=\log (x) \sin ^{-1}(x)-\int \frac{\sin ^{-1}(x)}{x}\,dx$


$ \begin{align} \left(1 + \frac{x}{n}\right)^n & = 1 + x + \frac{n(n-1)}{2!n^2}x^2 + \frac{n(n-1)(n-2)}{3! n^3}x^3 + \cdots \\\\ & = \frac{x^0}{0!} + \frac{x^1}{1!} + \left(\frac{n-1}{n}\right)\frac{x^2}{2!} + \left(\frac{(n-1)(n-2)}{n^2}\right)\frac{x^3}{3!} + \cdots  \end{align} $


$P(1) \implies P(2),$


$ \mu\left(\liminf\limits_{n\to\infty}A_n\right)\leq \liminf\limits_{n\to\infty}\mu(A_n) $


$\left[\begin{array}{cc}1&0&0\\0&(x^1)^2&0\\0&0&(x^1 \sin x^2)^2\end{array}\right]$


$x-\delta < f(x+\delta) < x + \delta$


$a_n = \|x_{n+1} - x_n\|$


$y = \frac{2+kz}{k-1}.$


$ \lim_{n \to \infty} \frac {\sqrt{n+1}+\sqrt{n+2}+...+\sqrt{2n}}{n^{3/2}}. $


$H(\omega) = 2\pi(\cos(\omega)-1 - i\sin(\omega))\frac{\delta(\omega)}{2-2\cos(\omega)}.$


$R = \mathbb Z_4[x]/((x^2+1)\mathbb Z_4[x])$


$ mn = |S \times T|$


$Y_0=\{a,a+2,a+4,\cdots\}, \quad Y_1=\{b,b+2,b+4,\cdots\}.$


$\alpha \subset \Omega$


$\\n! + 2, \\n! + 3, \\n! + 4, \\n! + 5, \\\dots \\n + k$


$\int_{1}^{-1}dx\sqrt{1+\frac{dy}{dx}^2}$


$(sin(\varphi), -cos(\varphi))$


$\int_0^\infty f(x)dx = 3$


$p_1p_2...p_k \mid a$


$(x+y)^r\leq x^r+ry^r\leq x^r+y^r\mathrm{.}$


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


$\mathrm{gcd}(a+b,a-b)\leq 2$


$R=S\oplus T$


$f\in W_0^{1,2}((0,R),r^{N-1})$


$1-P(A^c\cup B^c)=1-(P(A^c)+P(B^c))$


$3\int_0^1 \frac{\tan^{-1}(x)}{x}-2\int_0^{1/2} \frac{\tan^{-1}(x)}{x}-\int_0^{1/3} \frac{\tan^{-1}(x)}{x}-\frac 12 \int_0^{3/4} \frac{\tan^{-1}(x)}{x}=\frac{\pi\log 2}{2}$


$(\overline {\mathbb{R}})^2$


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


$\frac{(1 + r)^{1 - T_1} - (1 + r)^{-T_2}}{r}$


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


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


$ \sum |x_k|^s \leq n^{\frac{r-s}{r}} \left(\sum |x_k|^r \right)^{s/r}. $


$K_1\subset K_2\subset\dots$


$\displaystyle\;f_1(n) = \binom{m+n}{m}\;$


$j_1\mid j_2$


$\lim_{x\rightarrow 1^{+}} \frac{\sqrt{1-x^2}}{1-x^2}$


$1/x+1/y+1/z = 4$


$r(X+Y)\subseteq r(r(X)+r(Y))$


$\{e_1\}\subset \{e_1,e_2\}\subset \dots \subset \{e_1,e_2,e_3,\dots ,e_n\}$


$(*,*,x,y),(*,x,*,y),(*,x,y,*),(x,*,*,y),(x,*,y,*),(x,y,*,*) $


$\lim_{n\to\infty}\mu(A_n^{\complement})=\mu(A^{\complement})$


$1=\dfrac{2}{x^2}+\dfrac{3}{y^2}+\dfrac{4}{z^2}$


$d \in \langle m,n\rangle$


$\lim_{n\to\infty}\frac{\sqrt1+\cdots+\sqrt n}{n\sqrt n}=\lim_{n\to\infty}\frac{\sqrt n}{n\sqrt n-(n-1)\sqrt{n-1}}=\lim_{n\to\infty}\frac{n^2\left(1+\left(1-\frac1n\right)\sqrt{1-\frac1n}\right)}{n^3-(n-1)^3}=\frac23$


$x_1 - x_0, \ldots, x_k - x_0 \in \mathbb{R}^n$


$\frac{1}{(1+r)}+\frac{1}{(1+r)^2}+\dots+\frac{1}{(1+r)^{n-1}}+\frac{1}{(1+r)^n}$


$|f'(x)| \le 2|f(x)|$


$ \sum_{n\geq 1}\frac{1}{n^2}=\sum_{n\geq 1}\frac{3}{n^2\binom{2n}{n}}\tag{3} $


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


$f_{x_{(4)}} =  20e^{-2x}(1-e^{-x})^3  $


$E|X_n|^r\to E|X|^r<\infty$


$A_1\subseteq A_2\subseteq...$


$ \sum_{n=-\infty}^\infty e^{-n^2\pi x}=\frac1{\sqrt{x}}\sum_{n=-\infty}^\infty e^{-\frac{n^2\pi}x}, $


$J_n=\int x^n \tan^{-1}(x)\,dx=\frac{x^{n+1} \left((n+2) \tan ^{-1}(x)-x \,    _2F_1\left(1,\frac{n}{2}+1;\frac{n}{2}+2;-x^2\right)\right)}{(n+1) (n+2)}$


$\int_1^\infty \frac{1}{(1+x^2)^\alpha}\,dx$


$x> A=N\delta$


$\forall \epsilon>0\quad\exists\delta>0\quad\forall y\quad|x-y|<\delta\Rightarrow|f(x)-f(y)|<\epsilon$


$\mu\left(\bigcup_N \bigcap_{n=N}^{\infty} A_n \right) = \mu\left(\bigcup_N B_N\right) = \lim_{N \to \infty} \mu(B_N) \leq \lim_{N \to \infty} \inf_{n \geq N} \mu(A_n) = \lim \inf \mu(A_n)$


$\mathcal B = \{\; \{x\}\;\}.$


$ \displaystyle \lim \limits_{n \rightarrow \infty}\hspace{2mm}\lim \limits_{h \rightarrow 0}\sum \limits_{i = 1}^n\frac{1}{n} f(x+\frac{h}{n}i)]=$


$\operatorname{cl}V\subseteq B\subseteq U$


$\int_0^{2\pi} \left(a\cos x+\sqrt{1-a^2\sin^2(x)}\right)^{2k} dt=2\pi\sum_{m=1}^k \binom{k}{m} \binom{k-1}{m-1}a^{2(k-m)}$


$W(\gamma, 0) = \text{deg}(\gamma/|\gamma|)$


$c=\{\{\varnothing\}\}=\{\{a\}\}=\{b\}$


$|\langle x, e_n \rangle |^2 = |x_i|^2$


$P(S_t\geqslant x\mid\mathcal F_0)=M_0/x$


$\eqalign{   & S \equiv y,\,\,\,\,\,\,\,\,\,s \equiv x  \cr    & P(s) = {1 \over {12}}  \cr    & Q(s) = {k \over {48}}f(s)  \cr    & A(s) = \int_a^s {P(t)dt}  = \int_a^s {{1 \over {12}}dt}  = {1 \over {12}}(s - a) \cr}\tag{7} $


$e^x \leq  \left(1+\frac{x}{n}\right)^{n+1}$


$\text{add}(\mathsf{nonstat}_\gamma)=\text{cov}(\mathsf{nonstat}_\gamma)=\kappa$


$(a+b, a)$


$\{(x,y) \in \mathbb{R}^n |\ a \le x \le b, \ g(x) \le y \le h(x) \}$


$ f(r) = \frac{r^m - \left(\frac{r}{1+r}\right)^m + \left(\frac{r^2}{1+r}\right)^m - r^{m+1}}{r^m} \implies\\ f(r) = \frac{[r(1+r)]^m - r^m + r^{2m} - r[r(1 + r)]^m}{[r(1+r)]^m} $


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


$2^n=(1+1)^n=1+n+\frac{n(n-1)}{2}+\frac{n(n-1)(n-2)}{6}+\frac{n(n-1)(n-2)(n-3)}{24}+\cdots.$


$\int_0^{\pi} f(t)\sin(nt)~dt=0$


$cov(X,X) - cov(X,Y) cov(Y,Y)^{-1} cov(Y,X)$


$\therefore \boxed{f(x)=9^x-3^x}$


$\Rightarrow \cdots < A < A\oplus B<(A\oplus B)\oplus [(A\oplus B )\oplus \left\{(A\oplus B)\oplus B\right\}] <(A\oplus B )\oplus \left\{(A\oplus B)\oplus B\right\}<(A\oplus B)\oplus B < B < \cdots$


$(\sec\varphi - \tan\varphi)^2 = \left(\frac{1}{\cos \varphi}-\frac{\sin \varphi}{\cos \varphi}\right)^2 = \frac{(1 - \sin \varphi)^2}{\cos^2 \varphi}$


$ax+by=dz$


$101!+2, 101!+3,...,101!+101$


$ \sqrt{z}=\sqrt{r}e^{i\varphi/2} $


$\lim_{n \to \infty} \frac{1}{n} \sum_{k=1}^n \frac{1}{k/n+1}.$


$P \in V \subseteq U \subseteq X$


$n \in F_3$


$ \lim_{n \to\infty}\{x_n\} = \lim_{n \to \infty} \inf \{x_n\} $


$b_n = \dfrac{x_1 + x_2 +...+ x_n}{n}$


$S:=\begin{cases}  x^{2}+y^{2}=1  \\   z=1   \end{cases}$


$[a]+\left( [b]+[c]\right)=[a]+[b+c]=[a+(b+c)]=[(a+b)+c]=[a+b]+[c]=\left( [a]+[b]\right)+[c]$


$ 1 + 2 + 3 + 4 + \cdots + n \; {“ \;=\; ”} - \frac{1}{12} . $


$\begin{array}{}
 \frac{1}{i} &= \frac{1}{i} \cdot \frac{-i}{-i} \\
 &= \frac{-i}{i \cdot (-i)} \\
 &= \frac{-i}{-(i \cdot i)} \\
 &= \frac{-i}{-(-1)} \\
 &= -i.
 \end{array}$


$\forall \gamma(a),\gamma(b)\in G$


$\forall x,y \in A: xRy \wedge yRx \Rightarrow x=y$


$(r,s) = rs$


$ \left\vert \frac{f(x)-f(x_{0})-H(x_{0})\cdot(x-x_{0})}{\Vert x-x_{0\Vert}% }\right\vert \leq4\varepsilon $


$
 \begin{align*}
 \int_0^{2\pi}\frac{\sin x}{x}\,dx&=\int_0^{\pi}\frac{\sin x}{x}\,dx+\int_\pi^{2\pi}\frac{\sin x}{x}\,dx\\
 &=\int_0^{\pi}\frac{\sin x}{x}\,dx+\int_0^{\pi}\frac{\sin(x+\pi)}{x+\pi}\,dx\\
 &=\int_0^{\pi}\Bigl(\frac{1}{x}-\frac{1}{x+\pi}\Bigr)\sin x\,dx\\
 &=\pi\int_0^{\pi}\frac{\sin x}{x(x+\pi)}\,dx\\
 &>0
 \end{align*}
 $


$P(E)=\frac{\operatorname{elements in E}}{\operatorname{elements in S}}$


$\mathbb{R}^{n+1} - M$


$P(X \in A_0|H_0)=1$


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


$\sum_{n=m+1}^\infty \|e_n-f_n\|^2 <1$


$\{i\gamma:-1\le \gamma \le 1\}$


$ T=(T^*)^*=(T^*T)^*=T^*T=T^*. $


$\gamma + a * \gamma + \frac{\gamma ^ 2 * n}{a} + \gamma * n \ge n*\gamma + \gamma + a$


$f^+$


$d(x,e)\leqslant d(x,y)+d(y,e)<\varepsilon.$


$f(\mathbf{a}+\mathbf{b})=f(\mathbf{a})+f(\mathbf{b})$


$|ab|=k$


$f \left( x \right) = \frac{9^x}{3+9^x}$


$\alpha=\alpha^\omega$


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


$|P_1P_2| = |P_1||P_2|$


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


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


$f(x)=\lim_{n\to\infty}[2x+4x^3+\cdots+2nx^{2n-1}]$


$E(X_1+X_2+X_3+X_4)=E(X_1)+E(X_2)+E(X_3)+E(X_4)=E(40)=40.$


$(sa)b = a(sb)$


$C = \{(x,y) \in \mathbb{R^2} : 0\leq x\leq 1, 0\leq y\leq 1\}$


$\frac{1}{(1 + r)^1}$


$\lim_{n\rightarrow\infty}|X_n(\omega)-E(X_n)|=0,$


$\sum _{n=1}^{\infty } \frac{1}{n^2+1}$


$Z=\frac{\lambda}a+\frac1{(a+3)^2}=\frac{\lambda}b+\frac1{(b+3)^2}=\frac{\lambda}c+\frac1{(c+3)^2}=\frac{\lambda}d+\frac1{(d+3)^2}$


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


$\det(A)\det(C)=\det(A)\det(C^T)=\det(AC^T)=\det((\det A)I)=(\det A)^n$


$t^+$


$ \left[ \begin{array}{ccccc} 1 & -1 & 5 & -2 & 2 \\ 2 & -2 & -2 & 5 & 1 \\ 0 &0 &  -12 & 9 & -3 \\ -1 & 1 &  7 & -7 & 1 \\ \end{array}\right] $


$|f(z)|\,|\Im(z)|^2\leq 1$


$f'(a)<k<f'(b)$


$P(max\space M)-P(max \space x)$


$\int_0^1 f(t)\,dt = \int_1^2 f(t)\,dt = 1$


$P(X_{n+N} \notin S|\mathcal{F}_n) \geq \epsilon 1_{X_n \in S}$


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


$[x,y]=xy-yx\in L$


$f(x) = 2e^{\frac{x}{3}} - e^{-\frac{x}{2}}$


$d(x,A) =  Inf(d(x,y): y in A)$


$ y = |\frac{b+cx}{a+x}| $


$f(a+b)< f(a) + f(b)$


$S=(e_1, e_2, e_3)$


$(\Sigma_{\gamma\in \tau}a_{\gamma}t^\gamma) +_{N} (\Sigma_{\gamma\in \tau}b_{\gamma}t^\gamma) =\Sigma_{\gamma\in \tau}(a_{\gamma}+b_{\gamma})t^\gamma$


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


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


$e^x\ge \left(1+\frac x2\right)^2 \tag 2$


$\lim_{n \rightarrow \infty}(x_n)(1/t_n)=$


$\int \frac{\tan{x}}{1+\tan{x}}= \int 1-\frac{1}{1+\tan{x}}$


$ a^{log_a u} = \,u $


$\Bbb R\setminus[a,b)=\bigcup_{x<a}[x,a)\cup\bigcup_{x>b}[b,x)$


$\lim_{n\rightarrow\infty}\sum_{i=1}^nE[X_i^+]=\infty$


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


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


$\|x_m - x_n \| < \varepsilon$


$m' = P \frac{r}{1-(1+r)^{-12 y}} - f \frac{r}{(1+r)^{12}-1}$


$\gamma<\eta\implies\gamma<\sup X\implies\gamma<\gamma'$


$|f(z)|\leq {\rm Max}_{\theta}|f(\exp(i\theta))|$


$ \int_0^\infty f(x) < \int_0^\infty g(x) $


$\int_{0}^\infty {\sin^3(x)\over x}dx$


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


$ax+by = A\tag{1}$


$P(\{s\}) = \frac{1}{6}$


$p_1p_2 \mid p-1$


$ \frac a {1-r} = \frac{(1-p)^n p}{1-(1-p)} = (1-p)^n. $


$=\frac12\cdot\frac{(1+r+r^2)-(1-r+r^2)}{(1+r+r^2)(1-r+r^2)}$


$\int_0^\infty dx \ f(x) = \int_0^\infty dx \ g(x) \rightarrow \int_0^\infty dx \ f(x)h(x) = \int_0^\infty dx \ g(x)h(x) $


$\psi_n(x)=a^{-1/2}e^{i\pi nx/a}$


$I(a_1) = a_1$


$\frac{1}{x} + \frac{1}{y} + \frac{4}{z} = 4$


$\ell(x,v)=2R|\langle v,\nu_x\rangle|$


$\!\!\!\! \overbrace{\begin{align}\color{#0a0}{n^2}\equiv(\pm1)^2\equiv \color{#0a0}1\\  n^{\color{#c00}{1+2k}}\!\equiv n(\color{#0a0}{n^2})^k\!\equiv\color{#c00} n\end{align}}^{\Large\qquad\quad n\ \ \equiv\ \ \pm 1\ \ {\rm if}\ \  \color{#a0f}{n\, \not \equiv\, 0}}\!\! $


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


$=\sigma_w^2*(\phi^{2(t-1)}+\phi^{2(t-2)}+...+\phi^{2*0})=\sigma_w^2\sum_{i=0}^{t-1}(\phi^2)^i=\sigma_w^2[\dfrac{1-(\phi^2)^t}{1-\phi^2}] $


$xRy \wedge yRx \Longrightarrow xRx$


$|ab| = 1,$


$[0] + [u] = [0+u] = [u] = [u+0] = [u] + [0]$


$\cos\theta = \dfrac{x}{2}$


$p_1p_2\ldots p_i2^j$


$A = \begin{bmatrix} 1 & a & a^2 \\ 1 & b & b^2 \\ 1 & c & c^2 \\ \end{bmatrix}$


$\begin{cases}x=0 \\ y=0 \end{cases}$


$|f(z)| \le k|g(z)|$


$\begin{align} \int\frac{1}{(x^2+2)^2}dx&=\frac{\sqrt{2}}{4}\int  \frac{\sec^2 \theta}{\sec^4 \theta}\,d\theta\\\\ &=\frac{\sqrt{2}}{4}\int  \cos^2 \theta \,d\theta\\\\ &=\frac{\sqrt{2}}{4}\int  \left(\frac{1+\cos 2x}{2}\right) \,d\theta\\\\ &=\frac{\sqrt{2}}{8} \left(\theta +\frac12 \sin 2\theta\right)\\\\ &=\frac{\sqrt{2}}{8} \left(\theta + \sin \theta \cos \theta \right)\\\\ &=\frac{\sqrt{2}}{8} \left(\arctan(x/\sqrt{2}) + \frac{\sqrt{2}x}{x^2+2} \right) \tag 3\\\\ \end{align}$


$ \begin{align} \left(1+\frac xn\right)^n &=\sum_{k=0}^n\binom{n}{k}\frac{x^k}{n^k}\\ &=\sum_{k=0}^n\left(\frac{n}{n}\frac{n-1}{n}\frac{n-2}{n}\dots\frac{n-k+1}{n}\right)\frac{x^k}{k!}\\ &\le\sum_{k=0}^n\frac{x^k}{k!} \end{align} $


$Perimeter=2\int_{-a}^{a}\sqrt{1+\frac{dy}{dx}^2}dx$


$p_1p_2\ldots $


$\int_{-\infty}^\infty \frac{1}{\left(x^2+1\right)^2}\,dx$


$x^2-x,y^2-y\in Z(A)$


$\displaystyle\sum_{n=1}^{\infty} b_n=\sum_{n=1}^{\infty} b_n^{+}-\sum_{n=1}^{\infty} b_n^{-}=\sum_{n=1}^{\infty} a_n^{+}-\sum_{n=1}^{\infty} a_n^{-}=\sum_{n=1}^{\infty} a_n$


$\sum_{n\geq0}\frac{1}{1+n^{2}}=\frac{1}{2}\left(1+\pi\coth\left(\pi\right)\right)\approx2.0767.$


$\tan\theta = \frac{x}{d}$


$|f(z)|\le\Re(f(z))+\Im(f(z))$


$\sum_{i = 1}^k \frac{1}{10^i}$


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


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


$F=G\oplus H$


$ (\widetilde{\gamma'} \widetilde{\gamma})\gamma' = \widetilde{\gamma'}\gamma = \gamma' \ . $


$\delta_{\alpha \gamma}=\sum_\beta (-1)^{\alpha+\gamma} \binom{\alpha}{\alpha-\beta, \beta-\gamma, \gamma}=\pm \binom{\alpha}{\alpha-\gamma, 0, \gamma}\pm \binom{\alpha}{\alpha-\gamma-1, 1, \gamma}\pm \cdots$


$ \operatorname{Var}(X)=E[(X-E[X])^2]=E[X^2-2XE[X]+E[X]^2]=E[X^2]-E[X]^2. $


$3(x^2+y^2+z^2)\ge x(x+y+z)+y(y+z+x)+z(z+x+y) = (x+y+z)^2$


$b=log_a c$


$\lim_{x\to -3^-} f(x)=\lim_{x\to 4^+} f(x)=+\infty,$


$\Bbb P[X=Y|\mathcal G]=1$


$ \begin{align} &\lim_{n\to \infty} \frac{1+2\sqrt{2}+...+n\sqrt{n}}{n^2 \sqrt{n}}\\ =&\lim_{n\to \infty} \frac{(1+2\sqrt{2}+...+(n+1)\sqrt{n+1})-(1+2\sqrt{2}+...+n\sqrt{n})}{(n+1)^2 \sqrt{n+1}-n^2\sqrt{n}}\\ =&\lim_{n\to \infty} \frac{(n+1)\sqrt{n+1}}{(n+1)^2 \sqrt{n+1}-n^2\sqrt{n}}\\ =&\lim_{n\to \infty} \frac{(n+1)\sqrt{n+1}((n+1)^2 \sqrt{n+1}+n^2\sqrt{n})}{((n+1)^2 \sqrt{n+1}-n^2\sqrt{n})((n+1)^2 \sqrt{n+1}+n^2\sqrt{n})}\\ =&\lim_{n\to \infty} \frac{(n+1)\sqrt{n+1}((n+1)^2 \sqrt{n+1}+n^2\sqrt{n})}{(n+1)^5-n^5}\\ =&\lim_{n\to \infty} \frac{(n+1)^4+(n+1)n^2\sqrt{n(n+1)}}{5n^4+10n^3+10n^2+5n+1}\\ =&\lim_{n\to \infty} \frac{(1+n^{-1})^4+(1+n^{-1})\sqrt{1+n^{-1}}}{5+10n^{-1}+10n^{-2}+5n^{-3}+n^{-4}}\\ =&\frac{2}{5} \end{align} $


$\left[\begin{array}{ccc|c}1 & 3 & -1 & -4\\0 & -13 & 6 & 19\\0 & -7 & -1 & 9\end{array}\right]$


$a_0 = (-1)[x^n + a_{n-1}x^{n-1} +...+a_1x]$


$\lim_{n \to\infty}\inf (\sin^n(x))$


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


$X^3, X^2, X, X^0$


$f(a) < k < f(b)$


$3\mid u^3-u.$


$1/|x| < \delta$


$N_1\subsetneq N_2\subsetneq \cdots\subsetneq N_m\subsetneq N_{m+1}\subsetneq \cdots.$


$\sum_{k=1}^n \sqrt{n^4+k}\ \sin\frac{2k\pi}{n}=-\frac{1}{4 \pi }+\frac{\pi }{12 n^2}+\frac{1}{16 \pi  n^3}+O\left(\left(\frac{1}{n}\right)^4\right)$


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


$\lim_{x\to c}|f(x)|= |L|$


$\frac{(10+4)!}{4!10!}=\frac{14!}{10!4!}$


$G=([n],E)$


$(\widetilde{\mathcal L}_{x_0},\widetilde{\mathcal L}_{x_1})$


$p_1p_2\dots p_{k-1}p_k$


$i=\frac{C_1-C_0}{C_0}=\frac{C_1-C_0}{C_0}\cdot 100\% $


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


$s_n(x) = \sum_{j=1}^{n} c_j \chi_{K_j}(x)$


$\sigma(a)\,\theta\,\sigma(b)$


$\mathbb R\times\mathbb R^{n-1}=\mathbb R^n$


$ b^x = \left(1+\frac xn\right)^n. $


$T=\{\{x\},\emptyset\}$


$d(x,y) = 1/2^n$


$\det A = \alpha \beta^{n-1} = [(n-1)a +b] (b-a)^{n-1}$


$ f(z)= \frac{1}{z}{e^z}{(1- e^{-z})^{-1}} $


$ \int \frac{1}{\cos^2(x)}\frac{(1 + \tan x)^n}{(1 - \tan x)^{n + 2}} dx,$


$p_G(x) = x(x-1)^{n-1}$


$\int_{a}^{b} f(t) \dot{h}(t) dt = \int_{a}^{b} g(t) h(t) dt$


$117!+2, 117!+3, 117!+4, \ldots, 117!+117$


$[X,Y] = XY - YX$


$\exists n \forall m P(m, n) \implies \forall m \exists n P(m, n)$


$P(2,n)=\frac{2^n}{2^n+2},\\ P(3,n)=\frac{3^n}{3^n+3\cdot 2^n+3}, \\P(4,n)=\frac{4^n}{4^n+4\cdot 3^n+6\cdot 2^n+4}.$


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


$\gamma[\gamma_1 \vert \gamma_2 \vert \cdots \vert \gamma_n]\equiv \gamma \otimes \gamma_1 \otimes \cdots \otimes \gamma_n$


$ \begin{align} \int_{-\infty}^\infty\frac{\sin(ax)}{x(1+x^2)}\,\mathrm{d}x &=\color{#C00000}{\frac1{2i}\int_{\gamma_+}\frac{e^{iaz}}{z(1+z^2)}\,\mathrm{d}z} -\color{#00A000}{\frac1{2i}\int_{\gamma_-}\frac{e^{-iaz}}{z(1+z^2)}\,\mathrm{d}z}\\ \end{align} $


$\lim_{x\to y^{+}} f(x) = \lim_{x\to y^{-}} f(x), \qquad \forall y \in \mathbb{R}m $


$ \left[     \begin{array}{cccc|c}       1&-1&3&2&b1\\       -2&1&5&1&b2\\-3&2&2&-1&b3\\4&-3&1&3&b4     \end{array} \right] $


$\frac{\sqrt[4]{x}}{x}=\frac{x^{\frac{1}{4}}}{x^1}=x^{\frac{1}{4}-1}=x^{\frac{1}{4}-\frac{4}{4}}=x^{\frac{1-4}{4}}=x^{-\frac{3}{4}}=\frac{1}{x^{\frac{3}{4}}}$


$A_1 \supset A_2 \supset \ldots \supset A_n \supset \ldots$


$\lim_{n \rightarrow \infty} \sum_{k=n}^\infty 1=\lim_{n \rightarrow \infty} \infty=\infty.$


$1/2\int_{0}^{1/2}\frac{\sin^{-1}(t)}{t}dt$


$\sqrt{2} \neq \sqrt{1} + \sqrt{1} = 2$


$ \alpha,\beta=1\pm\sqrt{3}\\ x_n=\frac{(1+\sqrt{3})^{n+1}-(1-\sqrt{3})^{n+1}}{2\sqrt{3}} $


$F: \mathbb{R}^{n+1} \rightarrow \mathbb{R}^{n+1}$


$nx+1, nx+2, nx+3, \ldots, nx+n$


$f_Y(y) = \frac{1}{2\pi}e^\frac{-(y)^2}{2}$


$a=\{\{\varnothing\}\}$


$(x, y, z, w)=(1, 1, 1, 0)$


$           \int_{0}^{\infty}f(t)\cos(st)dt,\;\; \int_{0}^{\infty}f(t)\sin(st)dt. $


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


$f(x)=e^{20 x/29} \left(1-e^{-x}\right)-y$


$\{ ( \gamma , \alpha ] = \{ \xi : \gamma < \xi \leq \alpha \} : \gamma < \alpha \}$


$A_1 \subseteq A_2 \subseteq \cdots \subseteq A_n \subseteq \cdots$


$\gcd(A')=\gcd(A)$


$\Im(f)=-\frac{\sin(\theta)(-2\cos(\theta)bcd+abcd+2a\cos(\theta)-ab-ac-ad+bc+bd+cd-1)}{(2\cos(\theta)b-b^2-1)(2\cos(\theta)c-c^2-1)(2\cos(\theta)d-d^2-1)}$


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


$\lim_{x\to inf}|r^{n+1}|=\lim_{n\to inf}|r^{n}|$


$\left(1+\frac{x}{n}\right)^n = e^{n\log\left(1+\dfrac{x}{n}\right)}$


$\lim_{n \rightarrow \infty}\frac{\pi+\sqrt{\pi}+\cdots+\sqrt[n]{\pi}}{n}$


$a R b \land b R a$


$f(re^{i\theta};n) = f(\sqrt[n]{r}e^{i\theta/n}\,;1)^n \qquad (n \geq 1)$


$ \begin{align} \int_0^\infty f(x)\,dx &\leq \sum_{n = 2}^\infty \int_{n^2 - 2}^{n^2 + 2}{dx\over 1 + x}  \leq \sum_{n = 2}^\infty{4\over n^2 - 1}. \end{align} $


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


$\begin{vmatrix}x_1&y_1&z_1\\x_2&y_2&z_2\\x_3&y_3&z_3\end{vmatrix}\geqslant0$


$\sin\frac{\theta}{2}=\frac{\frac{\sin\theta}{2}}{\sqrt{\sin\theta}},$


$\frac{2}{a^2} + \frac{4}{b^2} = 1.$


$\left(1+\frac{x}{n}\right)^n < e^x.$


$\;\phi_{\frac1a,-\frac ba}\;$


$\tan(u) = \frac{x}{1}$


$f(Z^2) = f(Z)^2.$


$\sum_{\gamma_1,\gamma_2} \binom{\beta}{\gamma_1} s^{\beta-\gamma_1} h^{\gamma_1} \cdot \frac{1}{\gamma_2!} x^{\gamma_2+\alpha} e^{\langle{s,x}\rangle} h^{\gamma_2} = \sum_{\gamma_2} (s+h)^\beta \cdot \frac{1}{\gamma_2!} x^{\gamma_2+\alpha} e^{\langle{s,x}\rangle} h^{\gamma_2}$


$x!\sim_{+\infty}\left(\frac{x}{e}\right)^x\sqrt{2\pi x}$


$cov (\bar{X},\bar{Y}) = 0$


$n! + 2, n! + 3, \dots, n! + (n - 1)$


$\frac1x\int_0^x\frac t{e^t-1}\ dt=\frac1x\left(\frac{\pi^2}6-\sum_{n=1}^\infty\frac{e^{-nx}}{n^2}\right)+\ln(1-e^{-x})$


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


$=[a+(n-1)b](a-b)^{n-1} $


$S=A_1 \times A_2 \times ...\times A_n$


$K=\mathbb{Q} [x]/(x^{4}+1)$


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


$(P ∨ ¬P) = (T ∨ F) = T$


$\therefore \frac{dF}{dL} = - \gamma K x^{\gamma} L^{-\gamma - 1} - (1 - \gamma)x^\gamma L^{-\gamma}$


$\frac{a^{-1}-b^{-1}}{a^{-1/2}-b^{-1/2}}=\frac{(a^{-1/2}-b^{-1/2})(a^{-1/2}+b^{-1/2})}{a^{-1/2}-b^{-1/2}}=a^{-1/2}+b^{-1/2}=\frac{\sqrt a}a+\frac{\sqrt b}b$


$|f(x)| \leq \frac{1}{3}|f(y)|$


$\int\pi (r_1^2 - r_2^2)\,dx = \int\pi r_1^2 \,dx - \int\pi r_2^2\,dx.$


$\int\dfrac{dx}{(x^2+1)^n}=\int\dfrac1{2x}\dfrac{2x}{(x^2+1)^n}dx$


$K\subset A\subset \overline{A}\subset U$


$\,p_1 \mid q_1$


$|\alpha|=|\alpha+\omega|$


$\zeta(s)\Gamma(\frac s2)=\pi^{\frac12-s} \frac{\Gamma(\frac{1-s}{2})}{\Gamma(\frac{s}{2})}\zeta(1-s)$


$(x,y) \mapsto [x,y]=xy-yx$


$S(n,d) = \phi(n)S(n',d')/\phi(n')$


$\lvert\dot{\gamma}(t)\rvert \leqslant \lvert t\dot{\gamma}(t)\rvert$


$d(x,A) = \inf \{d(x, y) : y ∈ A\}$


$\lim\limits_{n\to\infty} \frac{3+\sqrt{3}+\sqrt[3]{3}+\dots+\sqrt[n]{3}-n}{\ln n}.$


$E(X|{\mathcal F}_0)$


$ \mu (X_{n} - A_{n}) < {1}/{2^{n}} $


$\int_0^\pi f(\sin x)\cos x\ dx=\int_0^\pi(F(\sin x))'\ dx=[F(\sin x)]_0^\pi=F(0)-F(0)=0$


$ (x-y)^2 (x+y) + (y-z)^2 (y+z) + (z-x)^2 (z+x) \geq  \frac{(y-z)^2}{(x+y+z)^2} (x+y+z)^3 $


$\mathscr O_{X,x}\subset \widehat{\mathscr O_{X,x}}$


$\gamma, - \gamma, i \gamma, -i \gamma, \bar{\gamma}, - \bar{\gamma}, i \bar{\gamma}, -i \bar{\gamma}$


$\displaystyle\lim_{x\to c^+}f'\big(d(x)\big)=\displaystyle\lim_{x\to c^+}f'(x)=A$


$f(x,y)= \frac 1 {2\pi \sqrt{1-p^2}} e^{ \frac {-(x^2 - 2pxy + y^2)}{2(1-p^2)}}$


$\int_0^{2\pi} g(t) S_n(t) \ dt \to \int_0^{2\pi} g(t) f(t) \ dt$


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


$\textbf{x}=(x,y)$


$\sum_{n\ge k}\mu(A_n)\ge\mu\left(\bigcup_{n\ge k}A_n\right)\ge\mu\left(\limsup A_n\right)$


$R=\{(x,y):a\le{x}\le{b}, c\le{y}\le{d}\}$


$a_n \geq (1+\frac{1}{n})^n$


$\sqrt{\gamma^2}=\begin{cases}\gamma & \gamma\geq 0\\-\gamma & \gamma<0,\end{cases}$


$alog(b)=log(b^a)$


$P_1 P_2,$


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


$ 1-\frac{N}{N+RN}=1-\frac{N}{N(1+R)}=1-\frac{1}{1+R}=\frac{1+R-1}{1+R}=\frac{R}{1+R}. $


$[x,y]=[y,x]$


$ \bbox[lightyellow] {   \left\{ \matrix{   q_{\,k}  = {{n^{\,\underline {\,k\,} } } \over {n^{\,k} }}\quad \left| {\;0 \le k} \right. \hfill \cr    p_{\,k}  = q_{\,k - 1}  - q_{\,k}  = q_{\,k - 1} {{k - 1} \over n} = {{\left( {k - 1} \right)\,n^{\,\underline {\,k - 1\,} } } \over {n^{\,k} }}\quad \left| {\;1 \le k} \right. \hfill \cr}  \right. }$


$yRx$


$\frac{1}{4}\left(e^{x/20}+e^{-x/20}\right)^2.$


$ X:= \{x = (x_i)_{i\in\mathbb{N}} \in \mathbb{R}^\mathbb{N} \ | \ \exists \ n \in \mathbb{N} \ \forall \ i \in \mathbb{N} \ : \ i \ge n \implies x_i =0 \}, $


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


$A\subset E \subset tU$


$ a_n = 2^{-n}  \\ b_n = \frac{n^2}{n^3 -10} \\ c_n = 1 +(-1)^n \\ d_n = 2^{1/n}$


$F(x) = \int_{c}^{x} f(t)\; dt$


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


$ \gamma\sigma\gamma^{-1} = \gamma(s_{1}\quad s_{2}\quad\ldots\quad s_{k})\gamma^{-1} = \big(\gamma(s_{1})\quad \gamma(s_{2})\quad\ldots\quad \gamma(s_{k})\big). $


$\underline{~b~}~\underline{~a~}~\underline{~a~}~\underline{~b~}$


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


$B = \{q \in \mathbb{Q} \mid q>0 \land q^2 >2\}$


$\begin{cases} x\equiv\theta\\x\equiv\pi-\theta \end{cases}\mod2\pi$


$\lim_{n\rightarrow\infty}\sum_{i=1}^nE[X_i^-]=\infty$


$\chi(n)=(\frac{n}{q})$


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


$\int_R \dfrac{1}{(1+x^2)^t}dx $


$D = \begin{pmatrix} A & B \\ -B & A \end{pmatrix}$


$\sum_{n\ge1}{\frac{1}{\sqrt{n}}} = \sum_{n\ge1}{\frac{1}{n^{\frac12}}}$


$0 \to A_1 \xrightarrow{f_1} A_2 \xrightarrow{f_2} \ldots \xrightarrow{f_{n-2}} A_{n-1} \xrightarrow{f_{n-1}} A_n \to 0$


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


$(A[x]/(x-1))_{(x-1)} = k$


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


$ D=\{(x,y)\in\mathbb{R}^2|0\leq x\leq1,0\leq y\leq1\} $


$B = \{ \gamma(1) \mid \gamma \text{ is a geodesic}, \gamma(0)=p, \text{ and } \langle \gamma'(0), \gamma'(0) \rangle \, \lt \epsilon \} .$


$E_1E_2\mid F$


$\mathbb E\lvert X\rvert^p\leqslant \frac 12\mathbb E\lvert X-Y\rvert^p.$


$\int_0^{2\pi} f(t) \sin(nt)dt$


$ \frac{a^2-b^2-c^2}{bc} = \frac{c(\gamma-\alpha)}{b(\beta-\alpha)} + \frac{b(\beta-\alpha)}{c(\gamma-\alpha)}\\ \frac{b^2-a^2-c^2}{ac} = \frac{a(\alpha-\beta)}{c(\gamma-\beta)} + \frac{c(\gamma-\beta)}{a(\alpha-\beta)}\\ \frac{c^2-a^2-b^2}{ba} = \frac{b(\beta-\gamma)}{a(\alpha - \gamma)} + \frac{a(\alpha - \gamma)}{b(\beta-\gamma)} $


$ax+by=a$


$f_\theta(x) = \frac{8x^7}{\theta^8}$


$P(a) \implies P(x)$


$m(M) = m^*(M).$


$P(E)=P(S\cup E)-P(S)+P(S\cap E)$


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


$F(x)=\int_a^{x} f(t)\,dt \rightarrow {d\over dx} F(x)=f(x)$


$(p_1p_2\cdots p_k)+1$


$\det [a_1, \cdots, ca_r, \cdots, a_n] = c \det A$


$[1,1,0,0],[1,2,-1,1],[0,0,1,1],[2,1,2,-1]$


$\vert f(x)\vert \leq  K^n\vert f(z)\vert$


$h[n]=(\frac{1}{2^{n+2}}-\frac{1}{2^{2n+3}})U[n]$


$E[P(X_t\leq x|\mathcal{F}_s);\{X_k\leq y\}]=P[\{X_t\leq x\}\cap\{X_k\leq y\}]=P[\{(t-\tau)^+\leq x\}\cap\{(k-\tau)^+\leq y\}]$


$\left[\begin{array}{cccccc|c} 1  & 2  &  0  &  0  & a     & 1   & -2 \\ -1 & -2 &  0  &  0  & -1-a  & -1  & 3  \\ -2 & -4 & -1 & 2  & a^{2} &   0  & 7  \\ 1  & 2  & 1  & -2 & a+2    & -1  & -6 \end{array}\right]$


$ \int_0^\infty \frac{\sin^2(x)}{x^2}dx=-\lim\limits_{x\to \infty}\frac{\sin^2(x)}{x}+\lim\limits_{x\to 0}\frac{\sin^2(x)}{x}+\int_0^\infty \frac{2\sin(x)\cos(x)}{x}dx $


$ax+by=ax'+by'=d$


$|f(x)||x|^{\sigma-1} \leq 1/|x|^2$


$\mu^{-1}(0) = \mathbb{R}^{n+1} \times S^n$


$\left[\begin{array}{ccc|c} 1 & -17 & 0 & 3\\ 0 & 17 & -6 & -2\end{array}\right]$


$ \lim_{n \to \infty} \nu(A_n) \leq c \lim_{n \to \infty}\mu(A_n) = 0, $


$z=\left( 1+\frac {a} {m}\right) e^{\frac {2k\pi i} {m}}$


$a_s\equiv a(s)$


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


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


$A^{p+1}(V) \subseteq A^p(V)$


$p_1p_2\leq x$


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


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


$H(\alpha)\restriction \beta=\{(\gamma,H(\alpha)(\gamma))\mid \gamma<\beta\}=\{(\gamma,H(\gamma+1)(\gamma))\mid \gamma<\beta\}=$


$\lim_{n \to \infty} \left \lfloor \frac{an+b}{cn+d}\right\rfloor  = \left \lfloor{a\over c} \right \rfloor-1 $


$\zeta(2)=\sum_{n=1}^\infty\frac{1}{n^2}=\int_0^1\int_0^1\frac{1}{1-xy}dA$


$(a, a+1, a+2, \dots, b)$


$\det  \begin{pmatrix}    A & C  \\     {0} & B  \end{pmatrix}=\det A\cdot \det B $


$\displaystyle \sum_{n=1}^{\infty}\frac{1}{n(e^{\pi n}+1)}-\frac{1}{2}\sum_{n=1}^{\infty}\frac{1}{n(e^{2\pi n}+1)}+\sum_{n=1}^{\infty}\frac{1}{n(e^{\pi n}-1)}-\frac{1}{2}\sum_{n=1}^{\infty}\frac{1}{n(e^{2\pi n}-1)}=\frac{log(2)}{8}$


$\operatorname{Var}[\hat\theta] = \operatorname{E}[\hat \theta^2 - 2\hat\theta \operatorname{E}[\hat \theta] + \operatorname{E}[\hat\theta]^2] = \operatorname{E}[\hat\theta^2] -  \operatorname{E}[\hat\theta]^2,$


$\prod\limits_{k=1}^{n}\cos\left(\frac{k\pi}{n}\right)$


$\lim_{n \to \infty} \int_0^\pi x^{1/n} \sin(x) dx = \int_0^\pi \lim_{n \to \infty} x^{1/n} \sin(x) dx = \int_0^\pi \sin(x)dx = 2. $


$\Bbb{R}^{n+1} - {0}$


$\ \underbrace{b/a = p}_{\large b\ =\ ap}$


$\lim_{k\rightarrow\infty} \mu(B_k) = \mu(A)$


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


$y=\frac{c+x}{1-\ln x}$


$E = \{e_1,e_2,e_3,e_4,e_5,e_6\}, X = \{\max x, ?, ?, ?, ?, ? \}$


$\mathbb{P}[X_\tau=n\mid X_0=k]=P_k[X_\tau=n]=k/n$


$\lim_{n\to\infty}\sum_{k=1}^n\frac{k^r}{n^{r+1}}.$


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


$\zeta(s)=\frac1{(s-1)\Gamma(s)}+\frac1{\Gamma(s)}\sum_{n=1}^\infty\frac{B_n}{n!(n+s-1)}+\frac1{\Gamma(s)}\int_1^\infty\frac{t^{s-1}}{\mathrm e^t-1} \mathrm dt$


$y = 10 + 8x + x^2 - x^3,\; x \geq 0$


$|\mathbb E[\gamma_k\mid \gamma_1,\dots,\gamma_{k-1}]|\leqslant  |\mathbb E[D_k(\gamma_k)\mid \gamma_1,\dots,\gamma_{k-1}]|+|\mathbb E[D_k(\gamma_k)-\gamma_k\mid \gamma_1,\dots,\gamma_{k-1}]|\leqslant 2\cdot 2^{-k}.$


$y-\frac{b-c}{2} = ax + \frac{b+c}{2}$


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


$I = -\int_0^{\infty} \dfrac{\sin^2(x)}{x^2}dx$


$\sum_{k=0}^{n-1}\cos\left(\frac{2\pi k}{n}+\phi\right)=0$


$\delta(\gamma)=\sup_{s,t\in S^1}\frac{d_\gamma(\gamma(s),\gamma(t))}{|\gamma(s)-\gamma(t)|}$


$i_j (x) = (x,j)$


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


$ (\gamma ^\mu )^*=\gamma _\mu=\begin{cases}(-\gamma ^0,\vec{\gamma}) & \text{for signature }(-,+,\cdots ,+) \\ (\gamma ^0,-\vec{\gamma}) & \text{for signature }(+,-,\cdots ,-)\end{cases}, $


$2\sum_{n\geq1}\frac{\left(-1\right)^{n}}{4n^{2}-1}=\sum_{n\in\mathbb{Z}}\frac{\left(-1\right)^{n}}{4n^{2}-1}+1.$


$\int_{0}^{\infty} {f(x)} dx = C$


$ \left[       \begin{array}{cccc|c}         1&1&4&2\\         0&3&12&6 \\         -1&2&8&4 \\         0&1&4&2       \end{array}     \right]$


$\frac{1}{2} + \frac{2}{2^2} +  \ldots +\frac{n-1}{2^{n-1}} + \frac{n}{2^{n}} <2-\frac{n+1}{2^{n-1}}<2$


$ f(x) = \frac{e^x}{1+ \lceil x\rceil } $


$A_1 ... A_c$


$p_1 \mid q_2 \cdot (q_3 \ldots q_n)$


$T_x(X)\times T_x(X)$


$|f(z)|\leq 1$


$\text{span}(v_1,v_2,\ldots,v_n,\ldots)=\text{span}(e_1,e_2,e_3,\ldots,e_n,\ldots)$


$ f(a_1) = g(a_1) - g(-a_1) $


$P\cfrac{(1+r/12)^{n}-(1+r/12)^{k}}{(1+r/12)^{12}-1}$


$\sum_{n=-\infty}^\infty e^{-\pi n^2 z}=\frac{1}{\sqrt{z}}\sum_{n=-\infty}^{\infty}e^{-\pi n^2/z}.$


$A = A_1 \times \ldots \times A_{n+1}$


$\lim_{n \rightarrow\infty}\inf f_n(k)$


$i_1 = A, i_2 = B, i_3 = C, i_4=i_A = D, i_5 = E, i_T=i_6 = F, i_7=G, i_8 = H, i_9=I.$


$x^n\delta^{(n)}(x)=(-1)^n\, n! \, \delta(x)$


$G = \left\{\begin{bmatrix} \alpha & \beta \\ . & \gamma \end{bmatrix} : \alpha,\beta,\gamma \in K, \alpha\gamma = 1 \right\} \leq B = \left\{\begin{bmatrix} \alpha & \beta \\ . & \gamma \end{bmatrix} : \alpha,\beta,\gamma \in K, \alpha\gamma \neq 0\right\}$


$ [1,-1,-1,1,1,-1,-1,1,1,-1, \ldots]$


$ab = r p$


$M,M',M''$


$[t_{i-1}^n, t_{i}^n]$


$\left[ \begin{array} {rrr|r}  1 &  0 &  2 &  3 \\ -1 & -1 &  1 & -6 \\  0 &  1 & -2 &  1 \\ \end{array}\right]$


$xRy \wedge yRx \rightarrow x = y$


$y(x)=(x-a)/(b-a)$


$f(x) = x^3-x+2$


$ \gcd\{a,b\}=\gcd\{|a|,b\}=\gcd\{a,|b|\}=\gcd\{|a|,|b|\} . $


$\iff Cov(X,Y)=0$


$\vartheta (z;\tau)$


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


$A = \{ \langle x , y \rangle \in \mathbb{R} \times \mathbb{R} : y > 0 \text{ or } x = 0 \},$


$(x,y,z)=(\log 2)\cdot(1,1,1)$


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


$\|A\|_2 = [\operatorname{Tr}(A^TA)]^{1/2}$


$\sum_{m=-\infty}^{\infty}e^{-\pi m^{2}/x}=\lim_{N\rightarrow\infty}\sum_{n=-N}^{N}\int_{-\infty}^{\infty}e^{-\left(\pi t^{2}/x\right)-2\pi int}dt=\lim_{N\rightarrow\infty}\sum_{n=-N}^{N}\sqrt{x}e^{\pi n^{2}x}=  $


$λ(m) = [40, 10] = 40$


$\Delta = |Â| = -m^2$


$\sum_{m=0}^\infty \sum_{k=1}^1 \frac{1}{(k+m^2)\prod_{n=0}^{k-2}(n+m^2)}=\frac{1}{2} (\pi  \coth (\pi )-1)$


$\langle_{\mathbb{R}}=\{\langle x,y \rangle \in \mathbb{R}^2: \text{ x is greater that y}\}$


$a:\left\{  \begin{array}{c} x=2\lambda+3\mu  \\ y=1\lambda-2\mu\\ z=-1\lambda-\mu\\ t=3\lambda+\mu \end{array} \right. $


$D=\{(x,y)\in\Bbb R^2: a\le x\le b, l(x)\le y\le u(x)\},$


$ \begin{cases} x+2y=z\\ x+2y=2\\ x+z=4 \end{cases}\implies\begin{cases} z=2\\ x=2\\ y=0 \end{cases}$


$A\subset  U\subset X$


$ (1+2+3+4+5+6+7+.....)=-\frac{1}{12} $


$\mathbf{x} = (x,y).$


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


$d(\alpha)=\alpha^+$


$x \in y \vee y \in x$


$\|x_n - x_{n-1} \| < \frac{1}{3^n}$


$x\ge 1, x \in \mathbb N$


$\det (\mathrm A) = \left(1 + \frac{n b}{a-b}\right) (a-b)^n = (a + (n-1) b) \, (a-b)^{n-1}$


$x^3 - x - x + 1$


$\left(a+b\right)^r\leqslant 2^{r-1}\left(a^r+b^r\right)$


$\mu (A)=\lim_{n\to \infty}\mu(A_n)+G(n)=\lim_{n\to \infty}\mu(A_n)=\sup_{n\in \Bbb N}\mu(A_n).$


$|ab|=-ab$


$\|A'\|_2^2=\rho(A'^TA)\leq\rho(A^TA)=\|A\|_2^2$


$F^{++}$


$ \lim_{n\to\infty} \frac{1}{\log n}\sum_{m=1}^n \frac{1}{m^2} = 0, $


$*M*M*M*M*M*M*$


$\Jac \left( \mathbb{Q}[x] / (x^8-1) \right) = \{0\}$


$b_n=\int_{-\pi}^{\pi} f(x)\,\sin(nx)dx$


$\sum_{n=-\infty}^\infty e^{-n^2\pi x}=\vartheta _3\left(0,e^{-\pi  x}\right)$


$I=\lim_{t\to 0}\dfrac{\dfrac{(t+1)^{t+1}-(t+1)}{\ln{(t+1)}-t}+2}{t}=\lim_{t\to  0}\dfrac{(t+1)^{t+1}-(t+1)+2\ln{(t+1)}-2t}{t(\ln{(t+1)}-t)}=\lim_{t\to 0}\dfrac{(t+1)^{t+1}+2\ln{(t+1)}-3t-1}{-\dfrac{t^3}{2}}$


$ G=\{ e,x,y,y^2,y^3,y^4,z,z^2,z^3,z^4 \}$


$\frac{(\gamma)_k}{(\gamma+1)_k} = \frac{\gamma}{\gamma+k}$


$r^+$


$|\gamma'(t)|=|X(\gamma(t))| \le C|\gamma(t)|.$


$\dfrac{\pi}{\sin{p\pi}}$


$f_Y(y,\theta) = 2y/\theta^2$


$c_0(I, C(X_i)) = C_0(X)$


$f(x+f(y)+xf(y)) = y+f(x)+yf(x)\forall x,y \in \mathbb{R}-\{-1\}\;,$


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


$P(n) \land P(m) \rightarrow P(mn)$


$\left(\begin{array}{ccc|c} 1 & 0 & -3 & 10 \\ -1 & 1 & 2 & 7 \\ 2 & 2 & -5 & -8  \end{array}\right) = \left(\begin{array}{ccc|c} 1 & 0 & -3 & 10 \\ 0 & 1 & -1 & 17  \\ 0 & 3 & 0 & -11  \end{array}\right) \begin{array}{l}  \\  R_1 + R_2 \\  R_3 + R_2 - R_1  \end{array}= \left(\begin{array}{ccc|c} 1 & 0 & -3 & 10 \\ 0 & 1 & -1 & 17  \\ 0 & 0 & 3 & -62  \end{array}\right) \begin{array}{l}  \\  \\  R_3 - 3R_2  \end{array}= \left(\begin{array}{ccc|c} 1 & 0 & 0 & -52 \\ 0 & 1 & 0 & -7  \\ 0 & 0 & 1 & -24  \end{array}\right) \begin{array}{l}  R_1 + R_3 \\  R_2 + R_3/3 \\  R_3/3  \end{array}$


$\sum_{n=-\infty}^\infty e^{-\pi n^2x}=x^{-1/2}\sum_{n=-\infty}^\infty e^{-\pi n^2/x}.$


$4a(x)[(2-x)a(x)-2a(x)^2]=-xa(x)+x(1-x)a'(x)-x^2a''(x)$


$\|A\|_F = \sqrt{\operatorname{tr}(A^\dagger A)}$


$[x,y]=xu=yv$


$\int_{-\infty}^{\infty} e^{-|x+1|} dx$


$\{\gamma,-\gamma\}=\{\alpha,-\alpha\}$


$|a-b| +|b| = -a + 2b = -a + 2a = a$


$ A = \left[\begin{array}{rrr|r} 1 & 1 & a & -1 \\ 3 & (a+1) & (a-1) &-1\\ a&2&1&0  \end{array}\right] $


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


$F=A\oplus B$


$\mathbb P(Y_n=0|\mathcal{F_n})=1-X_n$


$W_0\subseteq W_1\subseteq W_2\subseteq\ldots~$


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


$
 F(x)=\int_a^xf(t)dt\Rightarrow F'(x)=f(x)
 $


$\sum^{\infty}_{n=1}\left\|f_n\right\|<\infty$


$\lim_n \frac{x_1+x_2+...x_n}{n}=0$


$A_1 \subseteq A_2 \subseteq A_3 \subseteq \cdots$


$ k^2 = \begin{vmatrix}a,b\\-b,a\\\end{vmatrix} $


$E(G) = \{e_1, e_2, ..., e_m\}$


$\langle u^\top xx^\top u\rangle-\langle u^\top x\rangle\langle x^\top u\rangle=u^\top\langle xx^\top\rangle u-u^\top\langle x\rangle\langle x^\top\rangle u=u^\top\left(\langle xx^\top\rangle-\langle x\rangle\langle x^\top\rangle\right)u\;.$


$B^* = (AA^*)^* = (A^*)^* A^* = AA^* = B$


$|f(x) - f(a)| < \epsilon, |x - a| < \delta$


$ \sum_{k=1}^{n-1}\sin\frac{\pi k}{n} = \cot \frac{\pi}{2 n} $


$\frac{K}{\pi}=\frac{1}{2\pi}\int_0^{\pi/2} \frac{x}{\sin(x)} \ dx=\frac{1}{2}\int_0^{1/2} \frac{\pi x}{\sin(\pi x)} \ dx$


$\;\displaystyle\phi =  \color{red}{1 + \frac{1}{1 + \frac{1}{1 + \frac{1}{...}}}} = \color{blue}1 + \frac{1}{\color{red}{1 + \frac{1}{1 + \frac{1}{...}}}} = \color{blue}{1}+\frac{1}{\color{red}{\phi}}\,$


$X=[X, y]$


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


$\mathcal{A}(X)$


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


$\frac{1}{\Gamma(s)} \int_0^{\infty} \frac {x^{s-1}}{e^x -1} \ dx = \zeta(s)$


$\frac 14 (k + 1)((k + 1) + 1)((k + 1) + 2)((k + 1) + 3)  = \frac 14(k + 1)(k + 2)(k + 3)(k + 4)$


$\frac{ds}{dx} = \sqrt{1 + {\frac{dy}{dx}}^2}$


$f(k) = \binom{n}{k}$


$ (y - 4) = \frac{4}{10}(x - 10) \\ y = \frac{2x}{5} \\ $


$ \begin{align} \int_{0}^{\infty} \frac{\sin^{2n+1} (x)}{x} \ dx &= \frac{1}{2} \int_{-\infty}^{\infty} \frac{\sin^{2n+1} (x)}{x} \ dx = \frac{1}{2} \int_{-\infty}^{\infty} \frac{\sin x}{x} \sin^{2n} (x) \ dx \\ &= \frac{1}{2} \int_{0}^{\pi} \sin^{2n} (x) \ dx = \int_{0}^{\frac{\pi}{2}} \sin^{2n} (x) \ dx \\ &= \frac{\pi}{2^{2n+1}} \binom{2n}{n}. \tag{1} \end{align}$


$|f(z^3)| \leq 1 + |z|$


$ \ x^4 -6\, x^2 + 9\ =\, -4\, x^2,\ $


$d=ax+yb$


$ [T_x,T_y] = 0$


$B=\{\{X\}\}$


$\frac 1{2^{n-2}} \le 1$


$f'(x) = \dfrac{1}{5}+ e^{25x} (25) - e^3 -\dfrac{7}{3}\sqrt[3]{x^4}$


$\lim_{N \to \infty}\sum\limits_{n=1}^{N} \frac{n^2+1}{2^n}$


$p_1 \mid q_{1} \cdot (q_{2}\cdots q_{n}) \implies p_1 \mid q_1\,$


$f(x) = \dfrac1\theta e^{\frac{-x}{\theta}}$


$f(x) = \sum_{i=1}^k x_ic_i$


$x\in{\langle m\rangle}$


$ X''(x)Y(y) + X(x)Y''(y)=X(x)Y(y) $


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


$\dfrac{\sqrt{1-x^2}\cdot(1-x^2-1+2x^2)}{\sqrt{1-x^2}+\sqrt{1 - 2x^2}}+\dfrac{(1-1+2x^2)\sqrt{x}}{1+\sqrt{1 - 2x^2}} = 0$


$=\frac{1}{a}(1-e^{\frac{-1}{a}})e^{\frac{-1}{a}}\Big(0-(\frac{-a}{2})\Big)=\frac{1}{2}(1-e^{\frac{-1}{a}})e^{\frac{-1}{a}(u)}$


$2\pi\int \rho\,ds,$


$\mu(\lim\inf A_n)\le \lim\inf\mu(A_n)$


$\lim_{x \to c} = L$


$p_1 p_2=lcm(p_1,p_2) \mid (aX-1)$


$\lim_{n \to \infty }\mu(E_n) = \mu(E)$


$T_x H + T_x Y = T_x X$


$\begin{vmatrix}x_1&y_1&z_1\\x_2&y_2&z_2\\x_3&y_3&z_3\end{vmatrix}$


$b^{\mathrm{log}_b(x)} = x$


$L \equiv \begin{cases} x+z=16\\ 2y-3z=-30 \end{cases}$


$\mathbb {\bar{R}^+}$


$ \int_{0}^{2\pi}\frac{\sin(x)}{x} = \int_0^{\pi} \frac{\sin(x)}{x}\; dx + \int_{\pi}^{2\pi} \frac{\sin(x)}{x}\; dx = \dots $


$X \supset X_1 \supset X_2 \supset \cdots \supset X_\alpha \supset X_{\alpha+1} \supset \cdots$


$\{e_1,e_2,e_3,..., e_n\}$


$x_{n+1}=2018^{1+[2/n(n-1)]}x_1^{-1/n(n-1)}.$


$1+2+3+\dots = {-1\over 12}$


$ P\left( \sup_{t \geq 0 } M_t > x \mid \mathcal{F}_0 \right) = 1 \wedge \frac{M_0}{x}. $


$\operatorname{H}^{0,1}(X)=\Omega ^{0,1}(X)/ \bar{\partial}( \Omega^0(X)  ) $


$\sum_{n=1}^\infty \frac{1}{n2^n(3n-1)}=\frac{1}{\sqrt[3]{2}}\int_{0}^{\frac{1}{\sqrt[3]{2}}}\frac{1}{x^2}\log\frac{1}{1-x^3}dx$


$P(2) \implies P(n)$


$((a+b),(a-b))=e$


$\sum\limits_{k=1}^n \cos \left(\frac {2 \pi k}{n} \right) =0= \sum \limits_{k=1}^n \sin \left(\frac {2 \pi k}{n} \right)$


$ =\left(1+\frac{\gamma}{x+\alpha}\right)^{ x+\alpha+1/2} (1+\beta/x)^{\gamma}\left(\frac{x}{e}\right)^{\gamma}  \approx e^\gamma (x/e)^\gamma = x^\gamma.  $


$ A_1 \subset A_2 \subset \cdots \subset A_i \subset A_{i+1} \subset \cdots $


$f^2(x) = \frac x9 + 1 + 3 = \frac x9 + 4$


$S(a)=\frac12\int_0^1u^{\frac{a-1}2}(1-u)^{-1/2}du$


$C_n = C_1 - 1 + \frac {n (n+1)} 2 = 3 + \frac {n (n+1)} 2$


$H(\alpha^+)$


$F(\Delta x) = \frac{f(a + \Delta x) - f(a - \Delta x)}{2\Delta x}$


$(x^r+y^r)/(x+y)^r$


$1/\lvert x\rvert$


$(a,b)  + (a,b)= (2a,2b)$


$f(a_1) = g(a_1)$


$I_n = [n\pi, (n+1)\pi]$


$\det\left(  I_{n}+A\right)  =\det\left( \begin{array} [c]{ccc} 1+a_{1,1} & a_{1,2} & a_{1,3}\\ a_{2,1} & 1+a_{2,2} & a_{2,3}\\ a_{3,1} & a_{3,2} & 1+a_{3,3} \end{array} \right)  $


$\frac{x_1+x_2+\dots+x_n}n \le \sqrt{\frac{x_1^2+x_2^2+\dots+x_n^2}n}.$


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


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


$-2i\sin(\pi s) \Gamma(s)\zeta(s)= -2i\sin(\pi s) \Gamma(s)\frac{\Gamma(1-s)}{\Gamma(1-s)}\zeta(s) = \frac {-2\pi i}{\Gamma(1-s)}\zeta(s) = \Large\int_{\gamma} \frac{(-z)^{s-1}}{e^z-1}dz$


$ \int_{0}^{x}f(x-t)g(t)dt = \int_{0}^{x}f(t)g(x-t)dt. $


$4\pi\int_0^{\pi/2}\sin(\varphi)\cos(\varphi)\,d\varphi$


$K = cov(X) - cov(X,Y) cov(Y)^{-1} cov(Y,X)$


$\forall a,b \in \Bbb N, f(ab)=f(a)+f(b)$


$\newcommand\cov{\operatorname{Cov}}\cov[X,E(Y|X)]=\cov[X,Y]$


$f(x) \equiv \frac{e^{x^2}}{10^x}$


$= \frac{(1+i)^{n+1} - (1+i)^n}{(1+i)^n-1}$


$\lim_{t\to 0} \frac{(1+t)\ln{(1+t)}-t}{t^2(1+t)+t^2-t(1+t)\ln{(1+t)}}=Taylor=$


$E[|XY|^2] \leq \displaystyle \sup_{\omega \in \Omega}|X(\omega)|^2 E[|Y|^2]$


$x^4 + 1 = 2y^2$


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


$ \begin{cases}   x\equiv 1\ [u] \\   x\equiv 1 \ [v] \end{cases}$


$f(x) = \frac{2x}{x^2 - 4}$


$ \sum_{n=1}^\infty |a_n| = \sum_{n=1}^\infty |b_n| + \sum_{n=1}^\infty |c_n| < \infty $


$X_\gamma=\{a+b\gamma: a, b\in\mathbb{Z}\}$


$\mathbb{E}[X|\mathcal{F}_0]:=P_{\mathcal{H}}(X)$


$\lim _{n \to \infty} \frac{1}{n} \sum_{r=1} \frac{1}{r/n}$


$A=\begin{pmatrix}1&1&1\\a&b&c\\a^2&b^2&c^2\end{pmatrix}$


$\int_0^{\pi} \Phi(\theta) \delta(\cos \theta-\cos \theta_0)\sin \theta d\theta=\int_0^{\pi} \Phi(\theta) \frac{\delta(\theta- \theta_0)}{|\sin \theta|}\sin \theta d\theta$


$d(x,A)\leqslant d(x,a)$


$T^{1,0}M\cap T^{0,1}M=(0)$


$(E(|X+Y|^r))^{1/r}\le (E(|X|^r))^{1/r}+(E(|Y|))^{1/r}$


$\tan(\alpha) = \frac{s}{2}.$


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


$K \subseteq D \subseteq \bar{D} \subseteq U$


$\bar {x}_n =\frac {x_1+x_2+...+x_n} n$


$\left( \begin{array}{ll}  k+1 & k-1\\  k-1 & k+1 \end{array}\right)^5$


$\lfloor\frac{\lfloor \frac{n}{s} \rfloor}{r}\rfloor = \lfloor \frac{n}{rs} \rfloor$


$\gamma(s+t)=\gamma(s)\gamma(t)$


$\lim \limits_{x \to c^+} f(x) = f(c)$


$\Bbb R[x]/(x^4+1)$


$[x,y] = \alpha$


$\begin{align} n^3 \frac{1}{n}\sum_{k=1}^n \sqrt{1+\frac{k}{n^4}}\sin\frac{2\pi k}{n} &=  n^3 \frac{1}{n}\sum_{k=1}^n \left(\sqrt{1+\frac{k}{n^4}}-1\right)\sin\frac{2\pi k}{n} + n^3\cdot\frac{1}{n}\sum_{k=1}^n\sin\frac{2\pi k}{n} \\ &\approx n^3 \frac{1}{n}\sum_{k=1}^n \frac{k}{2n^4}\sin\frac{2\pi k}{n} + n^3\cdot\frac{1}{n}\sum_{k=1}^n\sin\frac{2\pi k}{n} \\ &= \frac{1}{2}\cdot\frac{1}{n}\sum_{k=1}^n \frac{k}{n}\sin\frac{2\pi k}{n} + n^3\cdot\frac{1}{n}\sum_{k=1}^n\sin\frac{2\pi k}{n} \\ &\xrightarrow[n\to\infty]{} \frac{1}{2}\int_0^1 x\sin(2\pi x)\, dx+\frac{1}{2}\int_0^1 \sin(2\pi x)\, dx =  \frac{1}{2}\int_0^1 x\sin(2\pi x)\, dx \\&= \boxed{-\frac{1}{4\pi}} \end{align}$


$\lim_{n \to \infty} \frac{x_1+x_2+...+x_n}{n}=a$


$(n+1)!+2, (n+1)!+3,\ldots, (n+1)!+n$


$|AB|=r$


$ G=\{A\in M_{2\times2}(\Bbb{R})\mid \det A\neq0, XA=X\}. $


$L=P\oplus S$


$\ \sin\theta = \frac{x}{2}$


$\int_0^\pi{f(x)\sin xdx} = \int_0^\pi{f(x)\cos xdx} = 1.$


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


$I_n(\pi) = 0$


$f_{U}(u)=\int_{-\infty}^{\infty}f_{X}(u+v)\cdot f_{Y}(v)\cdot 1 \, dv=\int_{0}^{\infty}\frac{1}{a}e^{\frac{-1}{a}(u+v)}(1-e^{\frac{-1}{a}})e^{\frac{-1}{a}(v)}\, dv$


$\int_0^\infty f(x) = \int_0^\infty g(x) =\infty.$


$V=\{\langle x,y\rangle\in G\times G:xy\in U\}$


$ \sum_{k=1}^{\infty} \sum_{n=1}^{\infty} \frac{1}{k^2 + n^{1/\gamma}} \geq \sum_{k=1}^{\infty}\sum_{n=1}^{\infty} \frac{1}{k^2 + n^2}. $


$\lvert f(z)\rvert \leqslant m\lvert g(z)\rvert.$


$\sum_{i=1}^n \frac{1}{3^i}\tag{displayed}$


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


$a^{log_ax} = a^b$


$(x,y)\in D:=\{(x,y):a\leq x \leq b, c\leq y\leq d \}$


$f(z) = \sqrt r e^{\frac {i\theta}2}$


$\frac{\sin\theta}{\tan\theta}=\frac{\sin\theta}{\frac{\sin\theta}{\cos\theta}}$


$\lim_{n\to\infty}\inf_{k\geq n}\mu(A_k)$


$ \sum_{n \geq 1} \frac{1}{n^2},$


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


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


$C_r = C_r(0)$


$s=t+1, s=t+2, s=t+4, s=t+17, s=t+34, s=t+68$


$\langle a,b\rangle := \lim\limits_{N \to \infty} \frac{1}{N}\sum\limits_{i=1}^{N}a_i b_i$


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


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


$ \int_0^{2\pi} \frac{\sin x}{x} dx = \int_0^{\pi} \frac{\sin x}{x} dx  + \int_\pi^{2\pi} \frac{\sin x}{x} dx = \int_0^{\pi} \frac{\sin x}{x} dx + \int_0^{\pi} \frac{\sin (v+\pi)}{v+\pi} dv $


$\alpha^+ > \alpha$


$ d(x,z) \leq d(x,y)+d(y,z)?$


$(x R y\wedge y R x)\to x = y$


$\zeta(s)=\sum_{k=1}^N\frac{1}{k^s}+\frac{N^{1-s}}{s-1}-s\int_{N}^\infty \frac{x-\lfloor x \rfloor}{x^{s+1}} dx,$


$\mp 7= \color{blue}{\pm \mp} (\mp 7)=\begin{cases}+(-(-7)) & =+7\\ -(+(+7))  & =-7\end{cases}=\pm 7\ne \mp 7\tag{*}$


$\lambda =1/|x|$


$\displaystyle F(x)=\int_0^x f(t)dt$


$I_n = \dfrac1{2\pi} \displaystyle\int_{-\pi}^{\pi} \dfrac{\sin(nx)}{\sin(x)}dx$


$P[X_i=0\mid A]=1-(w/n)$


$A=\{\gamma\in\Bbb N:\gamma=0\text{ or }S\alpha\le\alpha+\gamma\}$


$a_n = \frac{x_1+x_2+\ldots + x_n}{n}$


$f_X(x; \theta) = (\theta +1)x^\theta$


$\sum_{n=-\infty}^\infty \int e^{-\pi{n^2}/ x}  dx $


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


$\displaystyle F(x) = \int \limits_a ^x f(t) dt \rightarrow \frac{dF(x)}{dx} = f(x)$


$\pi:\mathbb{R}^{n+m}\to \mathbb{R}^n$


$P(E(X|{\mathcal F}_0)=1)=0$


$\sum_{k=1}^n \cos(2 \pi j k/n) = n$


$\text{ind}_{\gamma}(1)=1\;\;\;\text{and}\;\;\;\text{ind}_{\gamma}(2)=2$


$g_i(x) = \sum_{j=1}^n c_{ij}f_j(x)$


$A = \int_0^\pi\frac{\cos x}{(x+2)^2}\,dx$


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


$\begin{bmatrix}z&w\\-\overline{w}&\overline{z}\end{bmatrix}$


$||x||^2 = \sum_{n=1}^{\infty} |\langle x, e_n \rangle |^2 $


$D\left[X\right] = E\left[(X- E\left[X\right])^2\right] = E\left[X^2- 2\cdot X \cdot E\left[X\right] + E\left[X\right]^2\right] = E\left[X^2\right] - 2 \left[X \cdot E\left[X\right]\right] + E\left[E\left[X\right]^2\right]$


$d_1\mid d_2$


$ \det\begin{pmatrix} A-B & B-A\\ B & A \end{pmatrix}=\det\begin{pmatrix} A-B & 0\\ B & A+B \end{pmatrix}.$


$\forall \epsilon_2>0, \exists \delta_2>0: |x-L|<\delta_2\implies |f(x) - f(L)|<\epsilon_2$


$ (0^x+4^x)<3^x+4^x<4^x+4^x $


$ \gamma = \frac{\gamma + \overline{\gamma}}{2} + \frac{\gamma - \overline{\gamma}}{2},$


$\lim_{n \to \infty} 2^{-n} \lim_{\delta \to 0} {\left( \cos(3^nx) ( \cos(3^n\delta) - 1 ) - \sin(3^nx)\sin(3^n\delta) \right) \over \delta}$


$ \sum_{r = 0}^n\frac1{r!} = 1 + \sum_{r = 1}^n\frac{1}{r!} \leq 1+\sum_{r = 1}^n\frac{1}{2^{r-1}}\\  = 1 + \sum_{r = 0}^{n-1}\frac{1}{2^r}\\ \leq 1 + \sum_{r = 0}^\infty\frac{1}{2^r}\\ = 1 + \frac{1}{1-\frac12} = 3 $


$V_{i,x} \subseteq \overline{V_{i,x}} \subseteq U_i$


$ W=\int_{t_0}^{t_1}\mathrm d W=\int_{t_0}^{t_1}F_T(t)v(t)\mathrm dt=\int_{0}^{\pi/\omega}m4B^2t\,\mathrm dt=\frac{m2B^2\pi^2}{\omega^2} $


$(c + d, c - d) = 1$


$|f(z)|\leq M |P(z)|$


$|\mathbb{R}^n|=|\mathbb{R}^{n-1}\times\mathbb{R}|=|\mathbb{R}\times\mathbb{R}|=|\mathbb{R}|.$


$\begin{align} \lim_{N\to \infty}\sum_{k=1}^N \frac{1}{k+N}&=\lim_{N\to \infty}\sum_{k=1}^{2N}\frac{(-1)^{k-1}}{k}\\\\ &=\sum_{k=1}^\infty \frac{(-1)^{k-1}}{k}\\\\ &=\log(2) \end{align}$


$f(a_1) = a_1$


$d(x,A)=\inf\{d(a,x):x∈A\}$


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


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


$\lim_{\gamma \rightarrow 1} \frac{x^{1-\gamma}-1}{1-\gamma} = \frac{\lim_{\gamma \rightarrow 1}\frac{\mathrm{d}}{\mathrm{d\gamma}}\left(x^{1-\gamma}-1\right)}{\lim_{\gamma \rightarrow 1}\frac{\mathrm{d}}{\mathrm{d\gamma}}\left(1-\gamma\right)} = \frac{\lim_{\gamma \rightarrow 1}-\log(x)x^{1-\gamma}}{\lim_{\gamma \rightarrow 1}-1} = \log(x)$


$\lim_{n\to \infty}\dfrac{\sum\limits_{k=2}^n k\cos\dfrac{π}{k}}{n^2} $


$\tan x=\frac{\sin x}{\cos x}\Rightarrow \cos x=\frac{\sin x}{\tan x}=\left(\frac{2\sqrt{ab}}{a+b}\right)/\left(\frac{2\sqrt{ab}}{a-b}\right)=\frac{a-b}{a+b}.$


$y_n = \frac{x_1 + x_2 + x_3 + x_4 + \ldots +x_n}{n} $


$\{a, a+1, b, b+2\}$


$g(n)=\binom{nr}{s}$


$(C_n)_{n\in N}\ with \ c_n = (-1)^n\frac{n^3 + 2}{n^2+1}$


$(\neg \neg A\vee \neg \neg B)$


$z_2=\sqrt 2e^{\frac {17i\pi}{12}}$


$J:=\{a\in B| \forall n \in \Bbb N: a(n+1)\geq a(n)\} $


$o(ab)=rs$


$
 f(x)=\sum_{j=i}^n c_j f(e_{i_j}).
 $


$P(X_1 = 0 | X_0 = 1) = 1/2$


$r=z^{\frac{1}{n}}e^{\frac{2i\pi k}{n}}$


$a_{m+n} = a_m + a_n + mn$


$xRy, yRx,$


$\lvert f(z)\rvert = \lim_{\epsilon \searrow 0} \lvert f_\epsilon(z)\rvert \leqslant 1$


$5Y^2=X^3-X, X\ne 0, \pm 1.$


$|ab|=m$


$F_3\times F_3$


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


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


$\,\gcd\{a,b\} = \gcd\{a\} = a.\,$


$X^3-X, X^2-1$


$\beta:T=\emptyset\rightarrow\emptyset$


$\mathbb{E}s[\mathbf{1}_{\{\tau=0\}\cup\{s\geq z\}}]=\mathbb{E}s-\gamma\mathbb{E}s\mathbf{1}_{\left(-\infty,z\right)}\left(s\right)=\mu_s-\gamma\int_{-\infty}^{z}xf_s(x)dx$


$\lim_{x \to a} f_1(x) = L_1, \lim_{x \to a} f_2(x) = L_2, ..., \lim_{x \to a} f_n(x) = L_n$


$A_1 \times A_2 \times \cdots \times A_n$


$     \left(  \begin{array}{rr}   A  &  pB  \\    B   &  A   \end{array}    \right)  ,   $


$A_1\subseteq A_2\subseteq \cdots \subseteq A_n $


$T((a,b)+(x,y))=T((a+x,b+y))=(a+b+x+y,a+x)=(a+b,a)+(x+y,x)=T((a,b))+T((x,y))$


$\;m=14\cdot x\;,\;\;x\in\Bbb N\;$


$\left\lfloor\frac{\left\lfloor\frac{N}{a}\right\rfloor}{b}\right\rfloor = \left\lfloor\frac{N}{a*b}\right\rfloor$


$\left(1+\frac xn\right)^n\le e^x$


$\overline{\frak a^+}$


$Cov(X,Y)=c$


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


$ \lim_{N \rightarrow \infty} \sum_{n=1}^N \frac{n^t}{e^n} = \sum_{n=1}^{N_0} \frac{n^t}{e^n} + \lim_{N \rightarrow \infty} \sum_{n={N_0}}^N \frac{n^t}{e^n} $


$c^Tx,x \in P$


$\int_0^{2\pi} f(x)dx = \int_0^{\pi} f(x)dx -\int_{0}^{\pi} f(t)dt=0 $


$P(0)\implies P(1)\implies P(2)\implies P(3)\implies P(4)$


$ mn = (\tan\theta+\sin\theta)(\tan\theta-\sin\theta) = \tan^2\theta-\sin^2\theta $


$(\gamma \circ \rho \circ \gamma)^n=\gamma \circ \rho \circ \gamma$


$ \beta_1\geq\gamma_1\geq\beta_2\geq\gamma_2\geq\cdots\geq\gamma_{n-1}\geq\beta_n. $


$\lim_{k \to \infty}\sum_{n=1}^{2^k-1}\frac{ \left\lfloor\sqrt{4^k-n^2}\right\rfloor\ } {2^{2k}}$


$p_1^2 p_2 p_3$


$ \frac{2}{x} + \frac{3}{y} = 1,\quad x, y \in  \mathbb{Z}^{*}$


$log_a(b) = \frac{log(b)}{log(a)}$


$B(n)=\{n!+2, n!+3,\cdots,n!+n\}$


$ y''' + y' - y'' -y =  \\ y''' -y'' + y' - y = 0 $


$\frac{1}{\eta(s)}=\frac1{(1-2^{1-s})\cdot\zeta(s)}=\frac1{1-2^{1-s}}\cdot\sum_{n=1}^{\infty}\frac{\mu(n)}{n^{s}}=\sum_{n=1}^{\infty}\frac{b(n)}{n^{s}}\iff b(n)=\frac{\mu(n)}{1-2^{1-s}}$


$\int_0^\infty g(x) \, dx$


$\lim_{n\to\infty} \frac 1n \sum_{r=1}^n\frac{\lfloor 2rx \rfloor} {n}=x $


$u= (\gamma (b) - \gamma (a))/|\gamma (b) - \gamma (a)|,$


$ \frac{6}{x}+\frac{-4}{y}+\frac{1}{z}=0$


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


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


$E((\xi-E\xi)^2)=E(\xi^2-2\xi E(\xi)+E(\xi)^2)=E(\xi^2)-2E(\xi)E(\xi)+E(\xi)^2=E(\xi^2)-E(\xi)^2=D\xi.$


$y=ax + \frac{b+c}{2} + b - \frac{b+c}{2}$


$\frac{1}{\gamma}-\frac{f(1/\gamma)}{f'(1/\gamma)}=\frac{1}{\gamma}-\frac{1}{2\gamma^3}+O\left(\frac{1}{\gamma^5}\right).$


$\zeta(s) = \sum_{n = 1}^{+\infty} \frac{1}{n^s} = s\int_1^{+\infty} \frac{\lfloor x\rfloor}{x^{s+1}}\,dx\,,$


$\gcd(a,b) = \gcd(a,4)$


$f(x) = \binom x2$


$\dfrac{2\pi(1-\cos(\theta))}{4\pi} = \dfrac{1-\cos(\theta)}{2}$


$ 2[x^2(x+y)(x+z)+y^2(x+y)(y+z)+z^2(x+z)(y+z)]\ge 3(x+y)(x+z)(y+z). $


$(k+1)-k=((k+1)^{\frac{1}{3}}-k^{\frac{1}{3}})((k+1)^{\frac{2}{3}}+(k+1)^{\frac{1}{3}}k^{\frac{1}{3}}+(k)^{\frac{2}{3}})$


$\sum_{a=0}^m \sum_{b=0}^a \sum_{c=0}^b,\sum_{a=0}^m \sum_{b=0}^a   \sum_{c=b}^a,  \sum_{a=0}^m \sum_{b=0}^a \sum_{c=a}^m,\\ \sum_{a=0}^m \sum_{b=a}^m \sum_{c=0}^a \sum_{a=0}^m \sum_{b=a}^m \sum_{c=a}^b, \sum_{a=0}^m \sum_{b=a}^m \sum_{c=b}^m  $


$Corr\left( x,y\right) = \dfrac {cov\left( xy\right) }{\sigma _{x}\sigma _{y}} $


$R=M\oplus T$


$\lim_{n \rightarrow \infty} \left\| x_{n+1} - x_n \right\| = 0$


$F(n, p)=\frac{\binom{n+1}{p+1 }}{\binom{n}{p }}=\frac{n+1}{p+1}$


$C=\frac{N-AB}{A+B}$


$R : a < x < b, \:\: c < y<d$


$p(M)\leq \lambda(M)$


$\text{a}+\left(\text{a}+1\right)+\dots+\left(\text{b}-1\right)+\text{b}=\sum_{\text{q}=\text{a}}^\text{b}\text{q}=\frac{\left(\text{a}+\text{b}\right)\left(1-\text{a}+\text{b}\right)}{2}$


$\lim_{n \to \infty} \frac1n \sum_{r=1}^n r^{\frac1r}$


$P[e^{sX} \ge e^{sa}] \le \dfrac{E(e^{sX})}{e^{sa}}=e^{-sa}\cdot M_X(s)$


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


$\frac{1}{x} + \frac{1}{y} = \frac{1}{-z}$


$\therefore \frac{dy}{dx}=\frac{2^{x}}{2^{x}+5}$


$xRy→yRx$


$\{\gamma\in[0,\infty] | r_{\gamma}\mathscr{U}\subseteq\mathscr{V}\}$


$\sin x+1=0\implies \sin x=-1 \implies x=\frac {3\pi}{2}+2n\pi$


$\lim_{x\to0^-}f'(x)=\lim_{x\to 0^+}f'(x)$


$\int_{0}^{\infty}\frac{\sin^{2n+1}(x)}{x} \ dx=\frac{\pi \binom{2n}{n}}{2^{2n+1}}$


$\boxed{\sqrt{1!\sqrt{2!\sqrt{\ldots\sqrt{n!}}}}<e^{1.097...}=2.996...<3}$


$A =\frac{αa + βb + γc}{α + β + γ}.$


$\lim\limits_{n \to \infty}\sum_{i=1}^{n}{i^r \over n^{r+1}}={1 \over 1+r}$


$ \frac{\sqrt{1-2 \epsilon  \log \left(\sqrt{2} l\right)}}{\sqrt{1-k\, l^2    \left(1-2 \epsilon  \log \left(\sqrt{2} l\right)\right)}} $


$f(xf(y)+x)=xy+f(x), \; \forall x,y \in \mathbb{R}.$


$P(X) = \frac{1}{6}$


$p_1p_2\cdots p_r\mid k$


$X = \sum^k_{j=1} I_j$


$x^3, x^6, x^9, x^{12}$


$\frac {21}{x} + \frac {70}{y} = n$


$p\in V \subset  \bar{V}\subset U$


$||x|| \geq \frac{\alpha}{||F||}$


$x^2 - 3x -5 \lt 0, x \in \mathbb{N}$


$r, r' \in R, c_r \neq c_r' $


$\{\langle x,y\rangle\in\Bbb R^2:x,y\ge 0\text{ and }x+y\le 1\}\;.$


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


$I_j := f^{-1}(a_j)$


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


$\sum_{j=1}^{m}c_j\chi_{S_j} = \sum_{j=1}^{m}c_j\chi_{K_j}$


$ \psi(x)=\frac{1}{x^{\frac32}}e^{\frac{-1}{4x^2}}$


$V \cong \mathbb{R}[x]/(x^4 - 1)$


$x \in (M,d)$


$ \sin(\theta)=\frac{\tan(\theta)}{\sqrt{1+\tan^2(\theta)}}. $


$ \sin\theta+\sin\theta\tan^2\theta=\sin\theta(1+\tan^2\theta)=\sin\theta\sec^2\theta=\sec\theta\tan\theta $


$\int_0^1 \frac{\sin(x)-x}{x^2} dx.$


$ E_n(x)=\frac M {(n+1)!}(x-a)^{n-1} $


$\|\overline{x}_n-x_n\| < 1/n$


$(xRy \wedge yRx) \implies x=y$


$\left(1+\frac{x}{n} \right)^n < e^x $


$\cdots=\frac{1}{n}\sum_{j=0}^{n-1}\omega^{-j\mu} \sum_{l=0}^\infty a_l(\omega^j x)^l=\frac{1}{n}\sum_{j=0}^{n-1}\omega^{-j\mu} f(\omega^j x).$


$(\gamma\gamma' )f(x)=f(x\cdot(\gamma\gamma'))=f((x\gamma')\gamma)=\gamma f(x\gamma')=\gamma(\gamma' f)(x)$


$|ab|=|c|$


$\overline{\Bbb R}_+$


$\gamma(a) = \gamma(\bar a) = a$


$F(x) = \int_{a}^{x}f(t)\,dt, G(x) = \int_{a}^{x}g(t)\,dt$


$        \begin{pmatrix}        2  & 0 &  1  \\        1 &-2 & 1   \\         1  &-1 &0 \\      2 &-1 &1\\         \end{pmatrix} $


$\,f(n) = 2^n-n^2\,$


$ \forall \epsilon \gt 0, \exists \delta \gt 0,\forall a \in D, \forall x \in D: |x-a| \lt \delta \Longrightarrow |f(x) - f(a)|\lt \epsilon  $


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


$|\tau(a)| = r(a)$


$f(\pi) + f(0) = \int_0^\pi  g(v) \sin(\pi - v) dv = \int_0^\pi  g(v) \sin v dv \ge 0$


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


$  S = \sqrt{1+\left(\dfrac{dy}{dx}\right)^2 + \left(\dfrac{dy}{dz}\right)^2} $


$J^{\ast}VE(M, \omega) \cong T^{0,1}M $


$s = \sum_{j=1}^n \alpha_j \ \chi_{A_j}$


$P(A \cap B)=1/6$


$\frac{1}{n^2}\sum_{i=1}^n\sum_{j=1}^n \mathbb E [X_i Y_j] = \frac{1}{n^2}\sum_{i=1}^n\sum_{j=1, i\ne j}^n \mathbb E [X_i]\mathbb E[Y_j] +\frac{1}{n^2}\sum_{i=1}^n \mathbb E [X_i Y_i]$


$b_n=\frac1\pi\int_{-\pi}^\pi g(x)\sin nx\ dx=\frac{2}{\pi}\int_0^\pi g(x)\sin nx\ dx$


$e^{log(|z|)}=r$


$\int_{0}^{1} (e^{\frac{-x}{a}}-a(1-e^{-\frac{1}{a}}))^2 dx$


$ E[s^N]=\frac{1-\sqrt{1-4p(1-p)s^2}}{2(1-p)s}, $


$\left\{ \matrix{   \hat p\left( \theta  \right) = \cos \theta \hat x + \sin \theta \hat y \hfill \cr    \hat \theta \left( \theta  \right) =  - \sin \theta \hat x + \cos \theta \hat y \hfill \cr}  \right.$


$[x,y]=l$


$\mathbb{R}^{n+m}-D^m$


$\alpha m + \beta n = g$


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


$x_1,x_2,x_3 \in K$


$\sum_{n=1}^\infty\frac{1}{\sqrt n(n+a)}=\frac{1}{\sqrt{a}}\int_0^\infty \frac{\text{erfi}(\sqrt{ar}) e^{-ar}}{e^{r}-1}  ~dr$


$ \int \sum \chi_{I_j}(F(t)) F'(t) =\sum \int \chi_{I_j}(F(t))F'(t) =\sum \lambda(I_j)<1/n, $


$(1+y)^r \sim 1 + r y$


$\Bbb Q(a,b,c,d)$


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


$\int_2^n {\frac{1}{{(x^2  + 1)^n }}\,{\rm d}x}  \le \frac{{n - 2}}{{5^n }}$


$\dfrac{x}{6}+\dfrac{x}{10}=\dfrac{4}{5}$


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


$\frac1x + \frac1y = \frac4{15}$


$\gamma\subset \omega(\gamma)$


$ \begin{align} \sum_{n=1}^\infty\frac1{n^2+z^2} &=\frac i{2z}\sum_{n=1}^\infty\left(\frac1{n+iz}+\frac1{-n+iz}\right)\\ &=\frac i{2z}\left[\mathrm{PV}\sum_{n\in\mathbb{Z}}\frac1{n+iz}-\frac1{iz}\right]\\ &=-\frac1{2z^2}+\frac{\pi i}{2z}\cot(\pi iz)\\[6pt] &=-\frac1{2z^2}+\frac\pi{2z}\coth(\pi z)\tag{2} \end{align} $


$Â=(Â^{*})^{t}$


$x^n=a^n+na^{n-1}(x-a)+\frac{1}{2}n(n-1)(\xi_n)^{n-2}(x-a)^2 \ \ \ \ (2)$


$\left\{
     \begin{array}{ll}
       \displaystyle\frac{\partial f}{\partial x}=0  \\
       \displaystyle\frac{\partial f}{\partial y}=0  \\
       \displaystyle\frac{\partial f}{\partial z}=0
     \end{array}
   \right.$


$\frac{(W-p)^{1-\gamma }}{1-\gamma}= \frac{1}{2}\frac{(W-x)^{1-\gamma}}{1-\gamma}+\frac{1}{2}\frac{(W+x)^{1-\gamma}}{1-\gamma}$


$ f(x) = \sum_{j=1}^n x_j g_j(x) $


${ \xi  }_{ n }=\frac { { x }_{ 1 }{ x }_{ 2 }+...+{ x }_{ n } }{ n } $


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


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


$\int_0^\infty[ f(x)+g(x)]\, dx=\int_0^\infty f(x)\, dx +\int_0^\infty g(x)\, dx $


$a=\{\{a\},a\}$


$\begin{align*} \intop_{0}^{-\infty}f(x)dx & =\intop_{0}^{\infty}f(-x)dx\\ \end{align*}$


$|a\times b | = \sqrt{3} $


$\ldots\subseteq A_5\subseteq A_3\subseteq A_1\subseteq A_2\subseteq A_4\subseteq A_6\subseteq\ldots\;,$


$ \mbox{angle }x = \arccos \frac{|\langle u,v\rangle|}{\Vert u \Vert \,\Vert v \Vert}. $


$a_1 a_2a_3...a_n \mid k$


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


$S\times A_1 \times A_2 \times ...\times A_n$


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


$\{I_\gamma\}_{\gamma\in\Gamma}\subseteq\mathcal{K}$


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


$\sin\theta = \frac{\tan\theta}{\sqrt{1+\tan^2\theta}}$


$ \int_{0}^{2\pi} \dfrac{1}{2\pi} \cdot \dfrac{1-r^2}{1-2r \cos(\varphi) + r^2 } d\varphi$


$\frac{\sin^2\theta}{\theta^2} = \frac{\sin\theta}{\theta}\cdot\frac{\sin\theta}{\theta}$


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


$P(M \text{and} H) = P(M) P(H) $


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


$U, V \subseteq \overline{A}$


$(a-b,a+b) =1$


$\mu = \frac{x_1 + x_2 + ... + x_n}{n}$


$\frac 2x + \frac 1y = \frac 17,$


$\tilde \gamma = \gamma \oplus [\gamma(1), \gamma_n(1)] \oplus \gamma_n^- \oplus [\gamma_n(0), \gamma(0)]$


$f_n(x)\xrightarrow[n\to\infty]{}f(x):=\begin{cases}0&,\;\;0\le x<1\\{}\\1&,\;\;x=1\end{cases}$


$\frac{1}{x} + \frac{1}{y} = \frac{x+y}{xy}$


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


$I_R(c_r) = r$


$W^{1,p}(\Omega)\subset C^{0,1-n/p}$


$\sum_{n\geq1}\sum_{k\geq1}\frac{1}{k^{a}n^{b}\left(n^{2}+k^{2}\right)}=\sum_{n\geq1}\sum_{k\geq1}\frac{1}{k^{b}n^{a}\left(n^{2}+k^{2}\right)}  $


$(X, T_x)$


$D=\{(x,y)\in\mathbb{R}^2|1\le x\le3, 0\le y\le \ln(x)\}$


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


$C_2\subseteq V'\subseteq\overline{V'}\subseteq U$


$x=c_0 y +c_1 z \; , \; c_0 \neq 1$


$\displaystyle\sum_{x=1}^\infty\sum_{y=1}^\infty\frac{(-1)^{x+y}}{x^2+y^2}$


$|G|=pqk$


$\gamma:[a, b] \to \Bbb C, \; \gamma(a) = \gamma(b) = z_0, \tag 5$


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


$3^k +1 < 3^k +3^k +3^k = 3*3^k = 3^{k+1}$


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


$y = (ax + b)/(cx + d)$


$T_1=\prod_{k=1}^{n-1} \cos\frac{k\pi}{2n}$


$\lim_{N\to\infty}2\sum_{k=1}^N\frac{\mu(k)}{k!},$


$x! \sim C \sqrt{x}\left(\dfrac{x}e\right)^x + \mathcal{O}(1/x)$


$u = [a_1, a_2, a_3, ...., a_n]$


$e^{log(X)} = X$


$P(\liminf{X_n\leq x})\leq \liminf P({X_n\leq x})\leq \limsup P({X_n\leq x})\leq P(\limsup{X_n≤x})$


$\{\langle x,y\rangle\in\Bbb R^2:a\le x+y<b\}\;;$


$y^A_\gamma = \begin{cases} x_\gamma, &\text{if }\gamma \in A \\ 0, &\text{otherwise.} \end{cases}$


$y=\frac{a+b x}{c+d x}$


$ \lim_{n\rightarrow\infty}\left\lfloor\sum_{k=1}^{n}9\cdot 10^{-k}\right\rfloor=0, $


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


$ G(x)=P[X\geq x]. $


$B = \{ (x,y) \in \mathbb{R}^2 : y \leq x \leq y^2,  0 \leq y \leq 1 \},  $


$\frac{d^2y}{dx^2}=-\frac{\frac{d^2x}{dy^2}}{(\frac{dx}{dy})^3}$


$\int_{0}^{1} \frac{\sin^{-1}(x)}{x} dx = \frac{\pi}{2}\ln2$


$ a u + b v = d $


$\int_0^1p_{n\geq M}(t^\lambda)f(t)dt=0=\int_0^1f(t)^2dt=0$


$p \not | q_1 q_2$


$p_1 \mid p_2^nb$


$(a,a) = \{\{a\}, \{a, a\}\} = \{\{a\}\}$


$\Rightarrow 0<|f(z)|<2|f(z_0)|$


$\lim_{k \to \alpha^+} \phi(k)$


$\liminf_{n \to \infty} \mu(B_1(x_n)) \ge \int \liminf_{n \to \infty} \chi_{x_n}(y)\ d\mu(y) \ge \mu(B_1(x))$


$M''=M/M'$


$\color{red}{\int \frac{1}{(x+1)^2}dx}$


$x^5-1=(x-1)\left(x-e^{\frac{2\pi i}5}\right)\left(x-e^{\frac{-2\pi i}5}\right)\left(x-e^{\frac{4\pi i}5}\right)\left(x-e^{\frac{-4\pi i}5}\right)$


$\overline{v}(\lnot \alpha) = \top$


$f_X(x) = \frac{1}{50}e^{\frac{-x}{50}}$


$\phi(G) = (a+b,a-b)$


$S = \{ a_1, a_2, a_3, \dots, a_n\}$


$f(x) = \left\{ {\matrix{    0 \hfill & {x = a} \hfill  \cr     {{x^2}} \hfill & {x \ne a,b} \hfill  \cr     0 \hfill & {x = b} \hfill  \cr   } } \right.$


$a(1-a)^{n-1} = s$


$|a-b| + |b|=a-2b = a-2a = -a$


$\frac{a}{b}=\frac{a+b}{a}$


$\varphi = \sum_{j=1}^{p}c_{j}\chi_{E_{j}}$


$ \left(- \frac{\ell}{2}\right)\frac{2}{\sqrt{-\gamma}} \frac {\delta S_{ct}}{\delta \gamma^{tt}}  ~=~\frac{2}{\sqrt{-\gamma}} \frac {\partial \sqrt{-\gamma}}{\partial \gamma^{tt}} ~=~\frac{1}{\gamma} \frac {\partial \gamma}{\partial \gamma^{tt}} ~=~ -\frac{1}{\gamma^{-1}} \frac {\partial \gamma^{-1}}{\partial \gamma^{tt}}$


$\varphi \colon K[T_1^{\pm 1}, \dots, T_r^{\pm 1}] \to \mathcal O_X(U), T_i \mapsto \varphi_i.$


$ a \leq x_1 \leq b, \qquad c \leq y_1 \leq d$


$d(y,a)\leq d(y,x)+d(x,a)$


$ \int_{-\pi}^{\pi}\frac{\sin(nx)}{x}\cos(mx)\,dx = 2\pi $


$\frac{1}{x} + \frac{1}{y} = c$


$\{t\gamma\} = t\gamma - \lfloor t\gamma\rfloor$


$ a\alpha -\beta  +a\gamma =0  \Rightarrow  a\alpha -2\gamma  +a\gamma =0  \Rightarrow  a( \alpha+ \gamma )=2\gamma$


$\gamma(t+s) = \gamma(t)\gamma(s)$


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


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


$ a+ar+ar^2+\cdots+ar^n=a\frac{r^{n+1}-1}{r-1}=a\frac{1-r^{n+1}}{1-r} $


$(M,M,M)$


$|A×B|= mn$


$ F(A)=Pr(a\le A)=\frac{\ln A-\gamma}{\ln x-\gamma}. $


$\frak p\oplus t$


$\frac{p(\cos(x) - p)}{1 - 2p\cos(x)+p^2} = \frac{1}{2} \cdot \left( \frac{1 - p^2}{1 - 2p\cos(x) + p^2} - 1 \right)$


$f_{x}(x)= \large \frac{1}{5} e^{\frac{-x}{5}}$


$ \lnot (\exists c_0, c_1) (\forall x) (x = c_0 \lor x = c_1) $


$\xi = 1-2p^{2}\sin^{2} \omega -ip\sin w$


$\;\pm\sin\varphi = \sqrt{1 - \cos^2\varphi}$


$f_n(x)={\lceil x\rceil\over n^2}$


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


$\displaystyle Y_n=\frac{X_1+X_2+...+X_n}{y_n}$


$ \theta(t)=\sum_{n=-\infty}^{\infty}e^{-\pi n^2 t}=1+2\sum_{n=1}^{\infty} e^{-\pi n^2 t} $


$ \frac{a}{1-a^2}\frac{a^{-2}-1}{a^{-1}}= \frac{a}{1-a^2}\frac{\dfrac{1}{a^2}-1}{\dfrac{1}{a}}= \frac{a}{1-a^2}\frac{1-a^2}{a^2}\,a=1 $


$\gcd(a,b)=\gcd(a,\gcd(a,b))=\gcd(b,\gcd(a,b))$


$f(a+b)=f(a)+f(b) + 2\sqrt{f(a)\cdot f(b)}$


$\tau=\vartheta$


$a\le x\le b,\qquad c\le y\le d,\qquad x\ge y$


$ \lim_{x\to b^-}\dfrac{f(x)-f(b)}{x-b}= \lim_{x\to b^-}f'(c_x)=\lim_{x \to b^-}f'(x) $


$p(n,m)=p(n,m-1)+p(n-m,m)\text{ if }m,n\ne 0\;.$


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


$\begin{cases}a_n=b_{n-1}^2+4a_{n-1}\\b_n=4b_{n-1}\\c_n=4\end{cases}$


$P_1\mid P_2\mid\cdots\mid P_r.$


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


$\tag 2 |g(z_1)| \lt |g(z_0)|$


$(a, a + d, a + 2d)$


$f(n) \leq {n \choose 2}$


$ \gamma\xi\gamma^{-1}(\gamma(a_k))= \gamma\xi(a_k)=\gamma(a_1) $


$ a \leq x_3 \leq b, \qquad c \leq y_3 \leq d$


$\frac{f(a+\Delta x)-f(a)}{(a+\Delta x) - a}=\frac{m(a+\Delta x)-ma}{\Delta x}=\frac{m\Delta x}{\Delta x}=m$


$\int_0^{2\pi} \int_0^\pi \frac{d}{dR}\int_0^R f(r,\theta,\phi)\;r^2 \sin(\phi)\,dr\,d\phi\,d\theta= \int_0^{2\pi} \int_0^\pi f(R,\theta,\phi)R^2 \sin(\phi)\,d\phi\,d\theta= \iint_{S(R)}f(R,\theta,\phi)R^2\;dS \;\;\;\text{ where }\;\;\; dS=\sin(\phi)\,d\phi\,d\theta,\;\; S(R) \text{ is the sphere of radius R.} $


$ab = ma + nb$


$\begin{cases}x=t,\\y=\pm{\sqrt{1-t^2}}, \\z=1-t^2,\end{cases}$