$\int_x^{x+2\pi}f(t)\cos(t)dt=\int_x^{x+2\pi}f(t)\sin(t)dt=0$


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


$\max (C_r)=r$


$r=\frac{1+\sqrt{1-4p^2}}{2p},\qquad s=\frac{1-\sqrt{1-4p^2}}{2p}.$


$d=ax+by.$


$|(f^3(x))'| =  3|f'(x)\cdot f'(f^2)(x)| \le \frac 34<1$


$\lim\limits_{n\to\infty} b_n = \lim\limits_{n\to\infty} (\sqrt{n+\sqrt{2n}}-(n+\frac{\sqrt2}2) ) - $


$\int_\gamma df(\partial/\partial\gamma)=f(\gamma(b))-f(\gamma(a)),$


$E(m)\implies E(m+1)$


$\Gamma(x+1)={\sqrt {2\pi x}}\left({\frac {x}{e}}\right)^{x}\left(1+\mathcal O\left({\frac {1}{x}}\right)\right)$


$\sum_{i=0}^x M(1+r)^{x-i} = M\sum_{i=0}^x (1+r)^{i} = M\frac{1-(1+r)^{x+1}}{1-(1+r)}.$


$ (x, y, z) \mapsto (1, x, y, z, \frac{1}{2} ( x^2 + y^2 + z^2 ) )$


$|\mathbb EX^b|\leq\mathbb E|X^b|=\mathbb E|X|^b\leq \mathbb E(1+|X|^a)=1+\mathbb E|X|^a<\infty$


$(\frac 1{2^n})$


$\varepsilon = \frac{a + bi \sqrt{3}}{2}$


$\alpha^+ = \alpha +1$


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


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


$A\in \mathbb{R}^{m,n} \Rightarrow \ \|A\|_2 = \sqrt{\rho(A^TA)}$


$ \sum_{m=-\infty}^\infty e^{-\pi m^2 x} = x^{-1/2} \sum_{n=-\infty}^\infty e^{-\pi n^2/x} $


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


$\|\frac{\langle u,v\rangle}{\|v\|^2}v\|^2=|\frac{\langle u,v\rangle}{\|v\|^2}\times \|v\||^2=\frac{|\langle u,v\rangle|^2}{\|v\|^2}$


$y = \frac{d x+k}{x+d}$


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


$\frac{100}{a} + \frac{100}{b} = 1$


$\mathbb E|X|^a=\mathbb E|X^a|<\infty$


$\lim_{n\to\infty} T_n=\infty$


$p\circ t$


$(\mathbb Z[x]/(x^2 - 3))/(1+2√3)$


$f_3=g_3=\frac n2$


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


$I(a, b) = \int_0^{2 \pi} \frac{1 -  a \sin(\theta) -  b \cos(\theta)}{(1 +a^2 +b^2 - 2 a \sin(\theta) - 2 b \cos(\theta) )^{3/2}} \mathrm{d} \theta, \quad 0 \le a, b < 1$


$A=\textrm{Spec} (\mathbf{R}[x]/(x^2+1))$


$ \int_0^1\frac{\tan^{-1}(x)}{1+x}\,\mathrm{d}x=\frac\pi8\log(2) $


$\int_{-\infty}^\infty  \sin(ax) \log\Big(\frac{x+1}{x-1} \Big) \,dx=\frac{2 \pi  (\sin (a)-i \cos (a))}{a}$


$\{a,a+d,a+2d, \cdots, a+(n-1)d \}$


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


$x^2+y^2=4 \Bigl(1-2\sin^2 (t)\cos^2 (t)\Bigr)$


$\bigg\lfloor\frac{\lfloor 4.5 \rfloor }{5}\bigg\rfloor \overset{?}=\bigg\lfloor\frac{4.5  }{5}\bigg\rfloor $


$σ(x,t)=σ·x$


$\dfrac 2a+\dfrac 2b=1$


$\limsup P[X_n< x]\leq P[X<x]$


$V=P\oplus L\oplus W$


$\lim_{n\rightarrow \infty}[(1+i)^n+(1-i)^n]$


$H = P \ltimes T$


$\displaystyle \sum_{n\geq1}\frac1{n^2}.$


$F(n) = F(n-2) + F(n-3) + 2n$


$\neg\alpha\lor \neg\neg\alpha$


$\gamma_0 \subseteq \gamma_1 \subseteq \gamma_2...$


$ \lim_{n\to\infty}F[u_{l_n}]=\liminf_{n\to\infty}F[u_n] $


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


$G/K = \mathbb{Z}_2^m \implies G \simeq \mathbb{Z}_2^{m+1}$


$16/\!/2 = 8$


$A = \{v_1,v_2,....,v_n\}$


$\mathbb{E}|\zeta_n|\le c_k[\mathbb{E}|X_n-X|^k+\mathbb{E}|\zeta|]$


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


$d(x,a)\leqslant d(x,y)+d(y,a)<d(x,y)+r$


$\Delta^+$


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


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


$ \mathbb E(|X_n^p-X^p|)\leqslant p\mathbb E(|X_n-X|\,|X_n|^{p-1})+p\mathbb E(|X_n-X|\,|X|^{p-1}). $


$ z_1 = e^{\frac{i \pi}{4}} $


$\cos k\frac{2\pi}{n}\cos (k+1)\frac{2\pi}{n}+\sin k\frac{2\pi}{n}\sin (k+1)\frac{2\pi}{n}=\cos[(k+1)-k]\left(\frac{2\pi}{n}\right)=\cos\frac{2\pi}{n}$


$\lim_{N\to \infty}(-\sqrt1 - \sqrt2 - ... -\sqrt{N} + \sqrt2 + \sqrt3 +...+\sqrt{N} + \sqrt{N+1})$


$\color{red}{\det \left(\begin{matrix}  A& -B\\B&A \end{matrix}\right)= \bigg|\det(A+iB)\bigg|^2 }$


$ P \oplus F = P \oplus(\underset{i \in \mathbb{N}}{\oplus}F_0) = P \underset{i \in \mathbb{N}} \oplus (P\oplus T ) =$


$P(m/p) \implies P(m)$


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


$|\alpha|^+$


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


$\log_a (BC) = \log_a (B) + \log_a (C)$


$\log_a b + \log_a c = \log_a bc$


$\overline{V_1}\subseteq V_2^c\subseteq U$


$x \notin y \lor y \notin x$


$ \zeta(s)=\left(1-\frac{2}{2^s}\right)^{-1}\eta(s)= \frac{2^s}{\Gamma(s)(2^s-2)}\int_{0}^{+\infty}\frac{x^{s-1}}{e^x+1}\,dx\tag{1}$


$\displaystyle L = \lim_{n\rightarrow \infty}\sum_{r=1}^{n}\frac{1}{2^r} = \lim_{n\rightarrow \infty}\frac{1}{2}+\frac{1}{2^2}+\cdots+\frac{1}{2^n}$


$\|f\|_K^2=\int_0^K f(t)^Tf(t) \, dt\leq \int_0^\infty f(t)^Tf(t) \, dt$


$\{1,-1,-2-n,1+n,1+2+3(k-l),-1-2-2(k-l-1)..\},l<k$


$(y-|y|,y+|y|)$


$ F_n(x) = \int_a^x f_n(t)dt, \hspace{2mm}  F(x) = \int_a^x f(t)dt $


$c_n=(-1)^n\frac{1}{n!}<b_n=\frac{1}{2^n} \text{ for $n>3$},$


$B \sim \left( \begin{array}{ccc} 1 & x & x^{2} \\ 1 & y & y^{2} \\ 1 & z & z^{2} \end{array} \right) \sim \left( \begin{array}{ccc} 1 & x & x^{2} \\ 0 & y - x & y^{2} - x^{2} \\ 0 & z - x & z^{2} - x^{2} \end{array} \right)$


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


$ \sum_{k=1}^{\infty}\mu(A_{k}):=\lim_{n\to\infty}\sum_{k=1}^{n}\mu(A_{k})\leq \mu(A). $


$ \tag{2} f(x)^2+f(y)^2 = f(x+y)(f(x)+f(f(y))) $


$2^x + 3^x = 6^x + 6$


$\overline f(a) \stackrel{\cdot}{+} \overline f(b) = \overline f(a+b)$


$A_0 \subset A_1 \subset \ldots \subset A_n$


$ \sum_{j=1}^\infty\sum_{k=1, k\neq j}^\infty \frac{1}{j^2-k^2}$


$\{f \in \mathbb{Z}^{\mathbb{N}}: \forall n \in \mathbb{N} \ \ f(2n) + f(2n+1) = 0\}$


$\lim_{n\rightarrow\infty} \sum_{k=1}^{n}\cos\left( \frac{2\pi k}{2n+1} \right)$


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


$\lim_{n\to\infty} inf A_n=\{0\}$


$f(x) =\sum_{j=1}^n a_j\chi_{E_j},$


$\sum_{n \geq 1} \frac{1}{p_n} < \sum_{n \geq 1} \frac{1}{n^2} = \frac{\pi^2}{6} < \infty.$


$ (x^a-y^a)(x^b-y^b)=x^{a+b}+y^{a+b} -x^{a}y^{b} -x^{b}y^a$


$|ab|=ab \tag{eq1}$


$(x_2,y_2,z_2) = (1,1,1)$


$\begin{align} \int_{0}^{\infty}f(x)dx \end{align}$


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


$\int_0^\infty f_n(x)dx=\infty$


$A_1^c \cap A_2^c \ldots A_{n-1}^c \cap A_n \subseteq A_n$


$\frac{2}{2^n} = \frac{1}{2^{n-1}}$


$P(X = k) = \binom {3}{k} p^k(1-p)^{3-k}$


$\frac{\sqrt{1+\frac xa}}{\sqrt{1}+\sqrt{1+\frac xa}}=\frac{\sqrt{1-\frac xa}}{\sqrt{1}-\sqrt{1-\frac xa}}$


$1\le n \leq e_i,$


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


$K=\frac{1}{1980}+\frac{1}{1981}+\frac{1}{1982}+........+\frac{1}{2012}<\frac{1}{1980}+\frac{1}{1980}+\frac{1}{1980}+........+\frac{1}{1980}=\frac{33}{1980}=\frac{1}{60}$


$\overline{V_1} \subseteq {V_2}^c \subseteq U$


$ \mathcal A'=\{a\setminus\{\gamma\}\mid a\in\mathcal A\mbox{ and }\min(a)=\gamma\}, $


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


$|x^3-x_0^3|\le\delta(\delta^2+3\delta|x_0|+2|x_0|^2)$


$\lim_{k\rightarrow \infty}  \frac{2}{3 (\sqrt{k+1} - \sqrt{k-1})(\sqrt{k+2} + \sqrt{k})} = \frac{1}{3}$


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


$\ds{\int_{\gamma -\ic\infty}^{\gamma + \ic\infty}      {s\expo{st} \over \root{\pars{s + a}^3}}\,{\dd s \over 2\pi\ic}      =\int_{\gamma -\ic\infty}^{\gamma + \ic\infty}      s\pars{s + a}^{-3/2}\expo{st}\,{\dd s \over 2\pi\ic}\,,\quad\gamma > \verts{a}\ \mbox{and}\ t > 0.\quad}$


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


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


$[-B, A]$


$ \left( \begin{array}{ccc|c} 1&1&b&1\\b&3&-1&-2\\3&4&1&c \end{array} \right) $


$\binom{k}{2} + \binom{n-k}{2} + k(n-k) = \binom{n}{2}$


$[L_i,L_\alpha]\subseteq L_i$


$T^{2, 0} M \otimes T^{0, 2} M \to T^{1, 1} \cong \operatorname{End} TM$


$\lim_{x \to 0^-} f'(x) = 0 = \lim_{x \to 0^+} f'(x) = (e^x - 1)\big |_{x =0}.$


$\int_{0}^{\infty}f(x)\text dx $


$g[x1] = x2 + zg$


$\left(\begin{smallmatrix}2&1&0&5\\-1&1&1&6\\5&1&-1&4\\5&1&3&0\end{smallmatrix}\right)$


$A_0 \times\dotsc\times A_k$


$\vartheta(z)$


$S_n:=\sum_{k=1}^n\cos\frac{k\pi}{n+1},$


$\Delta''= \begin{vmatrix} x&y&z\\ x^2&y^2&z^2\\ x^3&y^3&z^3\\ \end{vmatrix}$


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


$\sigma_1^2 = \frac{t}{2}$


$ \log \mathbb{E} |X|-\mathbb{E}\log |X|\leq\mathbb{E}\left[\frac{1}{2}\left(1+|X|\right)(\log|X|)^2\right]$


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


$\epsilon \equiv \frac{1}{4!} \epsilon^{abcd} \gamma_a \gamma_b \gamma_c \gamma_d = \gamma_0 \gamma_1 \gamma_2 \gamma_3 \implies \epsilon^{abcd} = \epsilon \cdot (\gamma^d \wedge \gamma^c \wedge \gamma^b \wedge \gamma^a)$


$\begin{align} \lim_{N\to \infty}\prod_{k=0}^N(1+z^{2^k})&=\lim_{N\to \infty}\sum_{k=0}^{2^{N+1}-1}z^k\\\\ &=\lim_{N\to \infty}\frac{1-z^{2^{N+1}}}{1-z}\\\\ &=\frac{1}{1-z} \end{align}$


$4^x+9^x=2\cdot 6^x$


$\int_0^ag(x)\cos(x)dx=\int_0^\infty g(x)\cos(x)dx=0$


$log_a (b)=c$


$\sum_{n\geq1} u_n=\frac12\sum_{n\geq1} u_{n-1}+\sum_{n\geq1}\frac1{2^n}$


$ f_x(x) = \frac {2x}{10^{(2)}} $


$A=A_0 \times ... \times A_n$


$E(x_n e_n) = E(x_n)E(e_n)=0$


$\sum_{n=1}^{\infty}\sum_{k=1}^{\infty}\frac{\cos(\frac{\pi k}{3})\cos(\frac{\pi n}{3})}{kn^{2}}$


$\lim_{n\to \infty} \mathbb{P}(A_n)=1$


$(a,\gamma) (b,\delta) = (ab^{\gamma_\alpha},\gamma \delta)$


$A_1\times...\times A_m$


$\left[Q[x]/(x^2+1)^2\right]^{n_1}\oplus \left[Q[x]/(x^2+1)\right]^{n_2}$


$\|x_n-x_m\| \leq 1$


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


$\int 2\pi y dx $


$\mathbf{x}:=(x,y)$


$\tan u = \frac{x}{1}.$


$x \in V_{i,x} \subseteq \overline{V_{i,x}} \subseteq U_i$


$ = \frac {1}{n} (\frac{log_{c}x}{log_{c} b})$


$\hat f_n(t)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}\frac{\sin x\sin nx}{x^2}e^{-ixt}dx$


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


$\tan \theta = \frac{x-y}2$


$\left\lfloor\frac{c}{ab}\right\rfloor$


$\sum_{n=1}^\infty \|f_n\|_\infty = \sum_{n=1}^\infty 1/n =\infty.  $


$\begin{cases}x=2\mp1,\\ y=2\pm1,\\ z=2,\end{cases}$


$P_1(-.6\mid 1.2),P_2(1\mid 2)$


$ 2a < a+b \\ a+b < 2b $


$ \lim_{n\rightarrow \infty} \frac{\sqrt{1}+\sqrt{2}+\sqrt{3}+.....+\sqrt{n}-\frac23n\sqrt{n}}{\sqrt{n}} = \frac12. $


$\|x_n - y_n \|< 2\epsilon_n$


$f(x)=1\wedge x$


$\lim_{N \to +\infty}\frac{1}{N}\sum_{k=1}^{N}\frac{\phi(k)}{k}=\frac{6}{\pi^2}$


$u_\epsilon(x) = \frac{2x}{x^2+\epsilon^2}$


$\|A\| = \sqrt{\mathrm{tr}(A^* A)}. $


$B\subset V\subset\overline{V}\subset U$


$dist(v,U)=\frac {\lvert \left\langle v,u \right\rangle\rvert}{\lVert u\rVert}$


$\begin{align}\int\frac{1}{\sqrt{x^2+1}}\,dx&=\int\frac{\sec^2 t}{\sec t}\,dt \\&=\int\sec t\,dt \\&=\ln(\tan t+\sec t)+C \\&=\ln\left(x+\frac{1}{\sqrt{x^2+1}}\right)+C\end{align}$


$\int\int z^2\,dS,$


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


$f(a + 1) = f(a) + f(1)$


$(a,b)_\mathbb{Q}=(a,c)_\mathbb{Q} \cup (c,b)_\mathbb{Q}.$


$2/\!/16 = 1$


$\alpha^{\sup_{\gamma<\beta}\{\gamma\}}=\sup_{\gamma<\beta}\{\alpha^\gamma\}$


$G(x)=P[T\geq x|H_{0}]$


$\gamma^q-\gamma=0 \implies \gamma^{-1}(\gamma^q-\gamma)=\gamma^{q-1}-\gamma^0=\gamma^{q-1}-1=0\implies \gamma^{q-1}=1 \implies (\gamma^{q-1})^{1/2}=\gamma^{(q-1)/2}=1^{1/2}=1$


$1/|\log x|$


$\lim_{n\to \infty}{\inf[a_n]}\le\lim_{n\to \infty}{\inf[b_n]}$


$a\in\mathbb{F}$


$(\mathrm{pr}_1(\tilde K)\cup \mathrm{pr}_2(\tilde K)) \subseteq V \subseteq \overline V \subseteq  \Omega$


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


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


$ \ \pi \ \int \ y^2 \ dx \ $


$D^{n+1}\times S^{m-n}$


$s(x)= \int \sqrt{1+ \Big(\frac{dy}{dx}\Big)^2} dx$


$a = \lim a_n = \sup a_n = \sup_{n}\inf X_n\\b=\lim b_n = \inf b_n = \inf_{n} \sup X_n$


$ \left[ \begin{array}{ccc|c}   1&5&-6&2\\   k&1&-1&3\\   5&-k&3&7 \end{array} \right] $


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


$\lim \limits_{n \to \infty}\left( 1 + \sqrt{2} + \sqrt[3]{3} + \dots \sqrt[n]{n} \right) \ln{2n+1 \over n}$


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


$\{e_1, e_2, \ldots, e_{n+1} \}$


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


$\frac{z}{xy}\,\frac{\displaystyle\prod_{p\text{ prime}}\,\left(1-\frac{1}{p^2}\right)}{\displaystyle \prod_{\substack{{p\text{ prime}}\\{p\mid \gcd(xy,z)}}}\,\left(1+\frac{1}{p}\right)}\gtrsim \frac{z}{\zeta(2)\,xy\,\ln(xy)}\gtrsim 1\,.$


$\log_aa^b=b,\log_a(ab)=\log_aa+\log_ab=1+\log_ab$


$|e_i| \le 1$


$\sum_{k=1}^n \sin(2 \pi j k/n) = 0$


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


$\begin{gathered}   \frac{{113}}{{50}} = \frac{{100}}{{50}} + \frac{{13}}{{50}} = 2 + \frac{1}{{\frac{{39}}{{13}} + \frac{{11}}{{13}}}} = 2 + \frac{1}{{3 + \frac{{11}}{{13}}}} \hfill \\    = 2 + \frac{1}{{3 + \frac{{11}}{{11 + 2}}}} = 2 + \frac{1}{{3 + \frac{1}{{1 + \frac{2}{{10 + 1}}}}}} \hfill \\    = 2 + \frac{1}{{3 + \frac{1}{{1 + \frac{1}{{5 + \frac{1}{2}}}}}}} = [2;3,1,5,2] \hfill \\  \end{gathered}$


$p_1 p_2 p$


$\langle Ax,x\rangle=\langle(-m+in)x,x\rangle= (-m+in)\langle x,x\rangle=(-m+in)\|x\|^2$


$\left\lfloor \frac{\left\lfloor\frac ab \right\rfloor}c \right\rfloor =\left\lfloor \frac{\left\lfloor\frac ac \right\rfloor}b \right\rfloor  $


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


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


$\begin{cases} x_1=1+x_2+2x_3 \\ x_2=s \\ x_3=t \end{cases}$


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


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


${N \choose r} \frac{1}{2^N}$


$\mu (\lim\inf_n A_n) \le \lim\inf_n \mu( A_n) \le  \lim\sup_n \mu( A_n) \le  \mu (\lim\sup_n A_n)$


$\delta(n,a) = \delta(n+1,a),$


$|ab|=\alpha$


$\lim_{n \to \infty} \frac{n \left (1-2^{-1-n} \right )}{n+1} \sum_{k=1}^n \frac{1}{2^k} = 1$


$\sum_{i=0}^{n-1} \cos\left(\frac{2\pi i}{n}\right) = 0 \tag{1} \label{1} $


$\left| A \right| = \left| {\matrix{    {x + y} & {xy} & 0 &  \cdots  &  \cdots  & 0  \cr     1 & {x + y} & {xy} &  \cdots  &  \cdots  & 0  \cr     0 & 1 & {x + y} &  \cdots  &  \cdots  & 0  \cr      \cdots  &  \cdots  &  \cdots  &  \cdots  &  \cdots  &  \vdots   \cr     0 &  \cdots  & 0 & 1 & {x + y} & {xy}  \cr     0 &  \cdots  & 0 & 0 & 1 & {x + y}  \cr   } } \right|$


$f(x)= f(a)+\int_a^x f'(t)dt$


$r \equiv s$


$A_1, A_2, \ldots,A_{\ell-1}, A_{\ell+1}\ldots \ldots, A_{m} $


$\mathbb{Z}^{+}$


$\begin{pmatrix}r&p\\p&-r\end{pmatrix}$


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


${\|A\|}_2 = \infty$


$|ab| = |ba|.$


$0 for (x,y) = 0$


$ \mathcal{R} = \begin{cases} 0\leq x \leq 60\\ 0 \leq y \leq 60 \\ y \leq x+15\\ y \geq x-15 \end{cases} $


$ax+by=N$


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


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


$(\Bbb Z[i]/(1+i)^3)^*$


$\mathscr{C} = \{\{a\},\{b\}\}$


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


$a=\frac{\langle x,y\rangle}{\lVert y \rVert^2}\overset{\color{pink} \heartsuit}{\implies}\lvert a \rvert=\frac{\lVert x \rVert}{\lVert y \rVert}$


$n=4: \dfrac{k(k^2 + 3 k + 14)}{6}$


$G = A_1 \times \cdots \times A_n$


$\rho\sigma\rho^{-1} = \left(\begin{array}{cccccccc} 1 & 2 & 3 & 4 & \dots & r-2 & r-1 & r\\ r & r-1 & r-2 & r-3 & \dots & 3 & 2 & 1\\ 1 & r & r-1 & r-2 & \dots & 4 & 3 & 2\\ r & 1 & 2 & 3 & \dots & r-3 & r-2 & r-1 \end{array}\right)=\sigma^{-1}  $


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


$\int_{x_0}^\infty f(x)\,dx$


$\left[  \begin{array}{cccc}    1 & -8 & 10 & 15\\    0 & 2 & -3 & -3\\    1 & -4 & 4 & d\\   \end{array} \right]$


$n^n=(s+(n-s))^n = \sum_{i=0}^n \binom{n}{i}s^i(n-s)^{n-i}\ge \binom{n}{s}s^s(n-s)^{n-s}\ge \binom{n}{s}$


$\frac{1}{|x|^n}=(1+\delta)^n>1+\delta^n+n\delta>n\delta>K$


$\frac{1-p^2}{1-2p\cos x+p^2}-\frac{1-2p\cos x+p^2}{1-2p\cos x+p^2}$


$\sin x + \cos x=t$


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


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


$\zeta(s)=\frac{1}{\Gamma(s)} \int_0^\infty \frac{x^{s-1}}{e^x-1}dx$


$A:=\{\langle x,y\rangle\mid0\leq x\leq1, y^2\leq x\}\subseteq\mathbb R^2$


$f(n) = {n \choose 4} $


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


$ \lim_{n \to \infty} \frac{\sqrt{n}(\sqrt{1} + \sqrt{2} + ... + \sqrt{n})}{n^2} $


$n^+ = 1 \Rightarrow n^+ = ∅^+ \Rightarrow n^+ = ∅^+ = \{∅\}$


$G_1 = A_1 \times \ldots \times A_t$


$ \ x \ \ne \ y \ , \ z \ = \ - 2 \lambda \ $


$ S_1=(e_1,e_2,..,e_n)$


$(a_{k},b_k)\subset I_k$


$T_k = \pmatrix{k^2 \sin(1/k) & k^2 \cos(1/k)\cr 0 & 1\cr}$


$\sqrt{1!\sqrt{2!\sqrt{3!\sqrt{\cdots\sqrt{n!}}}}} <2$


$f(\det\gamma(t),0)=\det\gamma(t),\qquad f(\det\gamma(t),1)=\det\gamma(0).$


$P(m+1) \Rightarrow P(m)$


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


$E[N | X_0=x] = 1/x$


$\mathfrak o^*/(1+2\mathfrak o)$


$B_{\text{alt}} = \{ \gamma(\epsilon) \mid \gamma \text{ is a geodesic}, \gamma(0)=p, \text{ and } \langle \gamma'(0), \gamma'(0) \rangle  \lt 1 \} .$


$x\in V\subset\overline{V}\subset U$


$(x+y)^r-x^r=ryc^{r-1}\leq ryx^{r-1}\leq ry^r$


$\phi(x) =\sum_{i=1}^n a_i \Large {\chi}_{A_i}$


$C \rightarrow \left( \begin{smallmatrix} A&-\bar B\\  B&\bar A \end{smallmatrix} \right)$


$s(t) = \sum_{j=1}^n c_j 1_{(a_j,b_j]}(t),$


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


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


$\sum_{q=0}^{n-x} p^{x+q}  (1-p)^{n-(x+q)}$


$P(X\leq x) = P(X_1\leq x,\ldots,X_n\leq x) = P(X_1\leq x)^n.$


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


$\frac {(k+1)(k+2)(k+3)}{(k+3)!}-\frac{1 }{(k+3)!} $


$ f(m) = {n \choose m}$


$\int_0^{\infty}\frac{x^n}{(x^2+1)^n}\,\mathrm dx$


$rs \mid r^2+s^2$


$k\equiv a+sm \equiv a \pmod m$


$A:=A_1\times\ldots\times A_n\in\bigotimes_{i=1}^{n}\mathcal{A}_i$


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


$I(m)=\frac12\int_0^1u^{\frac{m-1}2}(1-u)^{\frac{m-1}2}du$


$\frac{\left(2\pi i\right)\cdot\left(e^{\frac{x\pi i}{3}}\right)}{\left(1-e^{\frac{i2\pi x}{3}}\right)\cdot\left(3e^{\frac{2\pi i}{3}}\right)}=$


$(1+\frac{x}{n})^n < \frac1{1-x}$


$\exists! x(Px \mid Qx)$


$f(x) = \sum_{m=1}^{M}a_m \chi_{A_m}(x)$


$\rho^+$


$ab = rk$


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


$\lim\limits_{n\to\infty}{\dfrac{1}{x_n}}=\dfrac{1}{\lim\limits_{n\to\infty}{x_n}}.$


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


$ T^* = (VP)^* = (PV)^* = V^*P^* = V^*P, $


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


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


$(ab)^{rs}=a^{rs}b^{rs} =(a^r)^s (b^s)^r \equiv 1 \mod p $


$[x,y]=xy$


$ Z(\phi,s)  Z(\phi^*,s)  = \sum_{n,m} \frac{\phi(n) \phi^*(m)}{n^s m^s}$


$\left(y + \dfrac{dy}{dx} x\right)$


$A(t) = A^*(t) = A^{**}(t) = B^{**}(t) = B^*(t) = B(t)$


$\binom{n+1}{3} * \frac{(n-1)! + (n-2)!}{(n+1)!} = \frac{(n+1)!}{3!(n+1-3)!} * \frac{(n-1)! + (n-2)!}{(n+1)!} = \frac{(n+1)!}{3!(n-2)!} * \frac{(n-1)! + (n-2)!}{(n+1)!} = \frac{(n-1)! + (n-2)!}{3!(n-2)!} = \frac{(n-2)!((n-1) + 1)}{3!(n-2)!}=\frac{n}{3!} = \frac{n}{6}$


$ \begin{align} \int_0^1\frac{\sin^{-1}\left(\sqrt{x}\right)}{x^2-x+1}\,\mathrm{d}x &=\int_0^1\frac{\frac12\cos^{-1}(1-2x)}{x^2-x+1}\,\mathrm{d}x\tag{1}\\ &=\int_{-1}^1\frac{\cos^{-1}(x)}{3+x^2}\,\mathrm{d}x\tag{2}\\ &=\frac12\int_{-1}^1\frac{\cos^{-1}(x)+\cos^{-1}(-x)}{3+x^2}\,\mathrm{d}x\tag{3}\\ &=\frac\pi2\int_{-1}^1\frac1{3+x^2}\,\mathrm{d}x\tag{4}\\ &=\frac\pi{2\sqrt3}\int_{-1/\sqrt3}^{1/\sqrt3}\frac1{1+x^2}\,\mathrm{d}x\tag{5}\\ &=\frac\pi{\sqrt3}\tan^{-1}\left(\frac1{\sqrt3}\right)\tag{6}\\ &=\frac{\pi^2}{6\sqrt3}\tag{7} \end{align} $


$f(x)-f(x+1)-1 < f(x-1)-f(x+1) \le f(x)-f(x+1)$


$\sum\limits_{n = 1}^\infty \sum\limits_{m = 1}^\infty \frac{1}{(n^2 + m^2)^3}$


$\lim_{n\to\infty}\inf a_n =0$


$
 d(x,y) \geq \|p(x)\| = \left\|\frac{\langle x,a\rangle}{\|a\|^2} a\right \| = \frac{|\langle x,a\rangle|}{\|a\|^2} \|a\| = \frac{|\langle x,a\rangle|}{\|a\|}.
 $


$\lim_{N\rightarrow\infty}\sum_{n=-\infty}^\infty f_N(n)=\sum_{n=-\infty}^\infty f(n).$


$1\neq P_1\mid P_2$


$\int_{0}^{\infty}f(x)dx, $


$9^x = 6^x + 2.4^x$


$|AB|=|CD|=|EF|=r$


$P[X \ge (i+j) | X \ge j] = P[X \ge i]$


$ \Vert proj_{W}\mathbf{v}\Vert=\Vert\frac{\langle v,u\rangle}{\langle u,u\rangle}u\Vert=\frac{|\langle v,u\rangle|}{|\langle u,u\rangle|}\Vert u\Vert=\frac{|\langle v,u\rangle|}{||u||^{2}}\Vert u\Vert=\frac{|\langle u,v\rangle|}{\Vert u\Vert} $


$Cov (U,X) = 0$


$a\cdot v= \det(a,b,c),\ \forall a$


$ \sum_{j=1}^{M/2}\sum_{k=1}^{M-1}\cos{\left(\frac{2\pi}{M}kj\right)}=-M/2 $


$f(n) = {n\choose 2}+{11-n\choose 2} = n^2-11n+55$


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


$P(E(X|{\mathcal F}_0)\in C)=1$


$N!+2, N!+3,...,N!+N$


$ \begin{array}{r} &&1&1&2\\\hline 1&0&1&-1&3\\ 0&1&-1&2&-5\\ 5&3&2&1&0\\ \end{array}\tag{2} $


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


$(n+1)!+2, (n+1)!+3, \ldots ,(n+1)!+(n+1)$


$\gamma(v,w)=\gamma(v+0,w)=\gamma_{|u}(v,w)+0(0,w)=\gamma_{|u}(v,w)+0=\gamma_{|u}(v,w)$


$y'=-y,y'''=-y$


$\lim_{x \to0^{-}} f(x) = \lim_{x \to 0^{+}} f(x) = \lim_{x \to 0} f(x) = C,$


$t = |y|/|x| < 1$


$\begin{array}\\ \gamma\\ \gamma\ \cup\{\delta\}\implies \lnot\ \gamma\\ \gamma\ \cup\{\delta\}\implies \gamma\\ \delta\\ \hline \gamma\ \land \lnot\ \gamma \end{array}$


$\frac{\cot(x+110^\circ)}{\cot x} = \cot(x+60^\circ)\cdot \cot(x-60^\circ)$


$p_1p_2p_3\cdots p_n+1$


$c_1\geq c_0$


$\,f_2(n) = \dfrac{n(3n\!-\!1)}2\,$


$\lim_{x\to0} f(c(x))=L$


$\sum_{n=2}^\infty (\frac1{n}-\frac1{n^2})$


$\lim\limits_{x\to a^-}h(x)=\lim\limits_{x\to a^-}f(x)=f(a)=h(a)=g(a)=\lim\limits_{x\to a^+}g(x)=\lim\limits_{x\to a^+}h(x)$


$\lfloor \frac{a-1}{d}\rfloor = \lfloor \frac{(a-1)\gcd (ab,c)}{ab} \rfloor$


$Log_{a} (x^{n})=c $


$f(x+2)=f(1)-f(x+1)=f(1)-(f(1)-f(x))=f(x)$


$ S=2\pi\int x\ ds \\ V=\pi\int xy\ dx  $


$\sum_{n=1}^{\infty} \frac{1}{n^3} =\sum_{n=1}^k \frac{1}{n^3} + \sum_{n=k+1}^{\infty} \frac{1}{n^3}< \sum_{n=1}^k \frac{1}{n^3}+\int_{k}^{\infty} \frac{1}{x^3} dx =\sum_{n=1}^k \frac{1}{n^3}+\frac{1}{2 k^2}$


$0\le M-m+m^*-M^*=(M-M^*)+(m^*-m)\,,$


$\forall{x,y} \;\; (xRy \lor yRx)$


$S^a := \{f \in X^* : f(x) = 0 \ \ \forall x \in S\}.$


$|a+b|=s+t$


$c_0 = c_1$


$\alpha\alpha^\omega=\alpha^{1+\omega}=\alpha^\omega.$


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


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


$[x,y]=t$


$\to^+$


$ \operatorname{Ext}_\gamma=\{a\not \in \gamma : \operatorname{Ind}_\gamma a=0\} $


$\lim_{n \to ∞} \frac{\sqrt{1}+\sqrt{2}+\sqrt{3}+\sqrt{4}+...+\sqrt{n}}{n\sqrt{n}}$


$-\frac12\int_0^1\frac{\ln(1-x^2)}x\ dx=\int_0^1\sum_{n=0}^\infty\frac{x^{2n+1}}{2n+1}\ dx=\sum_{n=0}^\infty\frac1{(2n+1)^2}$


$[x,y]=-x$


$ \small{\sum_{k=1}^N \! \left( \cos \frac{\pi}{2k}-\cos \frac{\pi}{2(k+2)} \right)=\!\!\left(\cos \frac{\pi}{2}-\cos \frac{\pi}{2(N+1)}\right)\!+\!\left(\cos \frac{\pi}{4}-\cos \frac{\pi}{2(N+2)}\right)} $


$ F(x) := \int_a^x f(t) \, \mathrm{d}t. $


$f (x) =\sum_{k=1}^\infty r_k \chi_{A_k}(x)$


$b_n = \frac{a_1 + a_2 + \dots + a_n}{n}$


$\dfrac{\sqrt{1-t^2}}{2\pi}\displaystyle\int_{-\pi}^{\pi}\dfrac{g(\theta)}{1-t\cos\theta}\,d\theta$


$ \sqrt{ \frac{dx}{dt}^2 + \frac{dy}{dt}^2 }$


$\mathbf x = (x_1,x_2,....,x_{n-1},x_n)$


$\lim_{x\to 0}{\frac{\sin^{-1}(x)}{x}} =1$


$\forall \epsilon>0, \exists r>0:|u-0| <r \implies |f(u) - f(0)| < \epsilon$


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


$n \uparrow\uparrow 5 = n^{n^{n^{n^n}}} < (m^m)^{(m^m)^{(m^m)^{m^{m^m - m^{m-1}}}}} = m^{m^{m^{m^{m^m - m^{m-1} + 1} + 1} + 1}} < m^{m^{m^{m^{m^m - m^{m-1} + 1} + 2}}} < m^{m^{m^{m^{m^m - m^{m-1} + 2}}}}< m^{m^{m^{m^{m^m}}}}.$


$f_1(n) = (-1)^n$


$\mathbb{E}(Z) = \frac{1}{2}(\mathbb{E}(X)+\mathbb{E}(Y))$


$\sum_{k=1}^\infty\frac1{k+k^3}\text{ and }\sum_{k=1}^\infty\frac1{1+k^2}$


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


$\left [\begin{array}{cc} 1 & -1\\0 & 2\end{array}\right]$


$\sqrt{1}+\sqrt{2}+\sqrt{3}+...+\sqrt{k}\ge\frac{2}{3}k\sqrt{k}$


$\mid p_1 - p_2\mid \rightarrow 0$


$ \begin{pmatrix} 1 & 3 & 2 & 1 \\ 0 & 1 & 4 & -4 \\ 0 & -1 & -6 & 7 \\ 0 & 1 & 3 & 1 \\ \end{pmatrix} $


$e_n=x_n/|x_n|\in G$


$f(x) = \frac2 {\sqrt{2 \pi}}e^\frac{-x^2}{2} $


$\int_{-\pi}^\pi\frac{\sin(13x)}{\sin x}\cdot\frac1{1+2^x}\mathrm dx?$


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


$\mathbb{R}^+$


$F=\bigoplus_{i\in\mathbb{N}}(P\oplus T)$


$k!+2,\dots, k!+k$


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


$ \langle a\wedge b,c\wedge d\rangle=\det\begin{pmatrix} \langle a,c\rangle & \langle a,d\rangle \\ \langle b,c\rangle & \langle b,d\rangle \end{pmatrix}. $


$ n! + 2, .... n! + n$


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


$\Gamma(s) \zeta(s) =  \int_{0}^{\infty} \frac{u^{s-1}}{e^{u}-1} du$


$p_1 \mid p_2^n$


$f(x) =\dfrac{3x+5}{x^2+2} $


$d(x,v) \le d(x,y)+d(y,v)$


$F = \frac{1}{2} F_{ab} \gamma^a \gamma^b \implies F_{ab} = F \cdot (\gamma_b \wedge \gamma_a)$


$ \bar{A}=\{x \in \mathbb{R^n}\mid \forall r \in \mathbb{R^+}:B_r(x)\ \cap A \neq \phi \} $


$P(E_i) = 1/2$


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


$\delta = \gamma + 1 =\gamma \cup \{\gamma\}$


$C = (p-nT) \sum_\gamma (\partial_\gamma u_\gamma) + \sum_{\alpha,\beta} (A_{\alpha\beta} \partial_\alpha u_\beta) - (\zeta \sum_\gamma \partial_\gamma u_\gamma)\sum_\alpha(\partial_\alpha u_\alpha) + n^2 \sum_\gamma (\partial_\gamma u_\gamma) + K \left( -\frac12 (\sum_\gamma \partial_\gamma n) (\sum_\gamma \partial_\gamma n) (\sum_\alpha \partial_\alpha u_\alpha) - n(\sum_\gamma \partial_\gamma n)(\sum_{\alpha,\gamma} \partial_\gamma \partial_\alpha u_\alpha) - (\sum_\gamma \partial_\gamma n)(\sum_{\alpha,\gamma} \partial_\gamma u_\alpha)(\sum_\alpha \partial_\alpha n) \right)$


$\overline{p_1 p_2\ldots p_{12}}$


$Cov(\epsilon, \theta)=0$


$ det(A) = (\alpha - \beta)^{n-1}(\alpha + (n-1)\beta). $


$ \begin{cases}x=1.3\\ y=0\\ z=0.8912 \end{cases} $


$f_R(r) = 2r/R^2_{\max}$


$\lim\limits_{n\rightarrow \infty}\sum\limits_{r=1}^{n}\frac{1}{T_r}$


$S = A ( (1 + R)^{n + 1} - (1 + R) ) / R$


$\frac {2^{n-2}-1}{2^{n-2}} + \frac 13*\frac 1{2^{n-2}}(2 + 1)= \frac {2^{n-2}-1}{2^{n-2}}+\frac 1{2^{n-2}} = \frac {2^{n-2}}{2^{n-2}} = 1$


$\left\{\begin{array}{cc}x=3t \\ y=-5(t+1) \\ z=2t+3\end{array}\right.,\space t\in\Bbb{R}$


$s(x)=\sum_{k=1}^n a_n \chi_{A_n}(x)$


$||x|| = ||x − y + y|| ≤ ||x − y + y|| ⇒ ||x−y||≤||x − y||$


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


$f^+,$


$\lim_{y\to 1^-} \frac{\sqrt {1-y^2}}{y-1}$


$V \subseteq P$


$[\frac{p}{3^k}, \frac{p+1}{3^k}]$


$\forall \omega\in\Omega\setminus N:\lim_{n\rightarrow\infty}X_n(\omega)=X(\omega)$


$\textbf{P}_{\gamma} = \exp(\textbf G \gamma)$


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


$\gamma(0)=\gamma_1(0)=\gamma_2(1)=\gamma(2)$


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


$\displaystyle\sum_{n}\mu(A_{n})\geq\mu(A_{K})=\sqrt{3}=\mu\left(\displaystyle\bigcup_{n}A_{n}\right)$


$\forall a,b,c\in\mathbb{R},d(a,c)=0\leq d(a,b)+d(a,c)=0$


$(\mathbb{R}^{\mathbb{N}}, \mathcal{R}^{\mathbb{N}})$


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


$\vartheta(z)+\vartheta(z+1)=2\vartheta(4z)$


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


$\mathcal F=\{\{x\},\{y\}\}$


$y=\frac{-(-2)±\sqrt{(-2)^{2}-8(-220)}}{4}$


$ X \in \mathcal{F}_0, X \subseteq Y \Rightarrow Y \in \mathcal{F}_0.$


$(\mathbb Q^+,.)$


$P_0=\{\xi\colon \xi=x-x_0,\,x\in P\}$


$ dm = \left|ab\right| ,$


$f(x\mid\theta) = \dfrac{2x}{\theta^2}\mathbb{1}_{(0,\theta)}(x)$


$\phi = \tan ^{-1}\frac{\sqrt{1-\delta^2}}{\delta}$


$\forall n\in \mathbb{N}^*, \cases{ p_{2n}=9\times 10^{n-1} \\ p_{2n+1}=10\times p_{2n}=9\times10^n }$


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


$\Phi(f) = \lim_{n\to\infty} \Phi(f_n) = \sup \{\Phi(g) : g \in \mathcal{F}\},$


$h(x) = \sum_{j=1}^n a_j 1_{C_j}(x).$


$\sum_{n=-\infty}^\infty e^{-\pi\alpha n^2} = \frac{1}{\sqrt{\alpha}}\sum_{n=-\infty}^\infty e^{-\pi n^2/\alpha}.$


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


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


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


$cov(X, Y) = 0,$


$\displaystyle\sum_{n\ge1}\frac{n\cos(nx)}{n^3}=\sum_{n\ge1}\frac{\cos(nx)}{n^2}$


$f(a + (b + 1)) = f(a) + f(b + 1)$


$p\in V\subseteq U\subseteq X$


$f(v)=f(a)+f(b)=f(b)$


$\gamma(1,2,3,4,5)\gamma^{-1}=(1,\gamma(2),\gamma(3),\gamma(4),\gamma(5))$


$\lim_{n\to \infty}\int_{1}^{n}\frac{1}{(x^{2}+1)^{n}}dx\sim \frac{1}{n\cdot 2^{n}}?.$


$\left(1+\frac xn\right)^n=\mathrm e^x\left(1+O\left(\frac1n\right)\right)$


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


$\mathrm{cov}(X,Z) = 0$


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


$\displaystyle \int_0^{\pi/2}\frac{\sin(2n+1)x}{\sin x}dx=\frac\pi2$


$a\mathbf{u} + b\mathbf{v} = \mathbf{0}$


$ \begin{vmatrix} 0 & x & y & z \\ x & 0 & z & y \\ y & z & 0 & x \\ z & y & x & 0 \\ \end{vmatrix} = \begin{vmatrix} 0 & 1 & 1 & 1 \\ 1 & 0 & z^2 & y^2 \\ 1 & z^2 & 0 & x^2 \\ 1 & y^2 & x^2 & 0 \\ \end{vmatrix} $


$f (z/2) =f (z) /2$


$ \begin{align}    \int_{-1}^1\frac1{(x^2+1)^3}\:dx&=2\int_0^1\frac1{(x^2+1)^3}\:dx    \\\\&=2\int_0^{\pi/4}\frac{(\tan^2\theta+1)\:d\theta}{(\tan^2\theta+1)^3}    \\\\&=2\int_0^{\pi/4}\cos^4\!\theta \:d\theta \\\\&=\frac12\int_0^{\pi/4}(1+\cos    (2\theta))^2 \:d\theta \\\\&=\frac12\int_0^{\pi/4}(1+2\cos    (2\theta)+\cos^2(2\theta)) \:d\theta \end{align} $


$J=\int_0^a \frac{\sin ^{-1}(x)}{x}\,dx$


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


$e^{-x}\ge \left(1-\frac xn\right)^n$


$\text{ Deoes } \lim\limits_{n\to\infty}y_nx_n\geq\alpha?$


$\,f=\big\{(\{a\},a)\big\}$


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


$f'(n)=xn^{x-1}-x(n-1)^{x-1}=x(n-1)^{x-1}[(1+\frac{1}{n-1})^{x-1}-1]>0$


$\mid P_1P_2\mid $


$ \left(\begin{array}{ccc}         x_1^2 & x_1 & 1\\         x_2^2 & x_2 & 1\\         x_3^2 & x_3 & 1\\ \end{array}\right) % \left(\begin{array}{c}          a\\          b\\          c\\ \end{array}\right) % = \left(\begin{array}{c}          y_1\\          y_2\\          y_3\\ \end{array}\right) $


$M'=M+(M'-M)$


$\sqrt{z}=\sqrt{r}\: \mathrm{e}^{\frac{\psi \mathrm{i}}{2}}$


$\lVert XY\rVert_r \leq \lVert |X|^r\rVert_p^{1/r} \lVert|Y|^r\rVert_q^{1/r} ,$


$\lim\limits_{n\to\infty} \sqrt n a_n = \sqrt3$


$\mathcal{O}_X[T, T^{-1}] \to \mathcal{O}_X[T, T^{-1}] \otimes \mathcal{O}_X[T, T^{-1}]$


$ \begin{cases} x=3\\ y=5\\ z=7 \end{cases} $


$\tag{1}
 \sup_{\|w^\ast\|_{W^\ast} \leq 1}{
   \left|
     \langle w^{\ast\ast}, w^\ast\rangle_{W^{\ast\ast},W^\ast}
   \right|
 } =
 \|w^{\ast\ast}\|_{W^{\ast\ast}}
 \stackrel{\color{red}{(!!)}}{=}
 \|i^{\ast\ast}w^{\ast\ast}\|_{V^{\ast\ast}} =
 \sup_{\|v^{\ast}\|_{V^{\ast}} \leq 1}{
   \left|
     \langle i^{\ast\ast}w^{\ast\ast},v^{\ast}\rangle_{V^{\ast\ast},V^{\ast}}
   \right|
 }. \quad
 $


$\sum_{n=101}^\infty\frac1{n^2} >\sum_{n=101}^\infty\frac1{n^2+n}=\frac1{101}. $


$C = \{(x,y)\ |\ a \leq x \leq b , c \leq y \leq d \}$


$u(0) = \frac{1}{2\pi}\int_0^{2\pi} u(e^{i\varphi})\,d\varphi = \frac{1}{2\pi}\int_0^{2\pi} \cos(\varphi)e^{\cos\varphi}\cos (\sin\varphi) - \sin(\varphi)e^{\cos\varphi}\sin(\sin\varphi)\,d\varphi.$


$=\sqrt{1+\frac{\partial f}{\partial x}^2+\frac{\partial f}{\partial y}^2}$


$\bigcup\mathscr A = X$


$n^a - (n-3)^a= \frac{1}{\left(\frac{1}{n}\right)^a} -\frac{\left(1-\frac{3}{n}\right)^a}{\left(\frac{1}{n}\right)^a}\\ =\frac{1}{\left(\frac{1}{n}\right)^a}\left(1-\left(1-\frac{3}{n}\right)^a\right) $


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


$m : M ⊗ M → M$


$=\frac{(1-p)^{n-2}\cdot p}{1-(1-p)}=(1-p)^{n-2}$


$a_1\times\cdots\times a_n$


$ \vartheta(z + \alpha + \beta \tau, \tau ) = \vartheta(z, \tau) e^{- \pi i \beta^2 \tau - 2 \pi i \beta z} $


$\displaystyle\;\frac{\sqrt{1-z^2}}{1+z^2} \sim \frac{i}{z}$


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


$f(x,y,z) = (x^2, y^2, z^2)$


$x=y=z=-\lambda$


$\sum_{s=2}^\infty \frac{\zeta(s)-1}{s}=1-\gamma$


$\prod_{k=0}^{n-1} \cos \frac{2\pi k}{n} = \frac{1}{2^{n-1}}$


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


$\phi(n) = i^n$


$\left(\begin{array}{l}1&x&x^2\\1&y&y^2\\1&z&z^2\end{array}\right)$


$(x_n, e_n) \to 0$


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


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


$\{\{A\}\}$


$\int _0 ^{2 \pi} \sin x \cos x dx = 2 \int _0 ^\pi \sin x \cos x dx = 0,$


$\det(A)=(a+(n-1)b)\begin{vmatrix}
       1 & 1 & 1 & \cdots & 1 \\
       0 & a-b & 0 &\cdots & 0 \\
      0 & 0 & a-b &\cdots & 0 \\
       \vdots & \vdots & \vdots  & \ddots & \vdots \\
       0 & 0 & 0 & \ldots & a-b \\
     \end{vmatrix}=(a+(n-1)b)(a-b)^{n-1}.$


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


$\implies \frac{\pi}{x} < \frac{\pi}{2}$


$d=|f(Yn)-f(a)| \le ||f||*||Yn-a||$


$|\gamma| \ge d(\gamma(a),\gamma(b))$


$P(D) = 2/3$


$(2a,b),  (a + b, 2b + a),  (2b,2a)$


$ (1-r-\lambda)^{n-1}(1+(n-1)r-\lambda)=0 $


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


$\bigg\lfloor\frac{q_1}{a_2}\bigg\rfloor=\Bigg\lfloor\frac{\bigg\lfloor\frac{n}{a_1}\bigg\rfloor}{a_2}\Bigg\rfloor=\bigg\lfloor\frac{n}{a_1a_2}\bigg\rfloor$


$\lim_{N\to\infty}\sum_{n=1}^{N}\frac{1}{(n+1)}\sum_{i=1}^{n}\frac{1}{i(n+1-i)}$


$(1 + {x \over n})^n$


$ \begin{align} P(N \geq k \mid X_0 = x) &= \int_0^1 P(N \geq k \mid X_0=x,X_1=y)\,dy \\ & = \int_x^1P(N \geq k \mid X_1 = y)\,dy\\ & = \int_x^1P(N \geq k-1\mid X_0=y)\,dy \end{align} $


$\int_{-\infty}^\infty \frac{x^{2n}}{(x^2 + 1)^{n + 1}}\ dx = 2\pi i \operatorname{res}_i f.$


$\displaystyle\int_0^2 F(\gamma)\gamma ' +\displaystyle\int_2^3 F(\gamma)\gamma ' +\displaystyle\int_3^4 F(\gamma)\gamma '$


$\begin{cases} x+2y=0 \\ z=0 \\ 0=0\end{cases}$


$Cov(x,y)$


$a_1 \ldots a_n$


$Tr(-B)$


$r_2\mid r_1n$


$\dot x = x-x^3, x \in \mathbb{R}$


$T_N^{0,1}\subset T_M^{0,1}$


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


$\{a+b,ab,a^b,\dots\}$


$\int \limits_{0}^{\infty}\frac {\sin (x^n)} {x^n}dx$


$\frac{1}{\gamma}\log\left(\mathbb{E}[e^X]\right)^{-\gamma}\leq\lim_{k\rightarrow \infty}\frac{2^k}{\gamma}\log\mathbb{E}[e^{-\gamma \frac{X}{2^k}}]\leq \frac{1}{\gamma}\log \mathbb{E}[e^{-\gamma X}]$


$(a+bi,a-b)=(\alpha,\beta)$


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


$   \int_0^1 dx = \lim_{n \rightarrow \infty} \sum_{n=1}^n 1/n               = \lim_{n \rightarrow \infty} n.1/n               = \lim_{n \rightarrow \infty} 1 = 1 $


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


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


$ \int_0^\pi f(x)\sin(2\,n\,x)\,dx=\int_0^\pi f(\pi-x)\sin(2\,n\,\pi-2\,n\,x)\,dx=-\int_0^\pi f(x)\sin(n\,x)\,dx. $


$ XRY + YRX = 0$


$\sum\limits_{n=0}^\infty n^{2m}e^{-an^2}  =\left(-\frac{\partial}{\partial a}\right)^m \sum\limits_{n=0}^\infty e^{-an^2}$


$ \iiint_T 2x \,\,\, dxdydz=2\int_{0}^{1}\int_{-\pi}^{\pi}\int_{0}^{\pi/2} \rho^2 \sin(\theta)\cos(\sigma) d\theta d\sigma d\rho=\frac{2}{3}\int_{-\pi}^{\pi}\cos(\sigma)d\sigma $


$H^+$


$1+8^x+27^x = 2^x+12^x+9^x$


$\forall N >0, \; \exists M >0: \; x > M \Rightarrow f(x) > N$


$ r\colon \; \begin{cases} x=at\\ y=bt\\ z=ct \end{cases} $


$\sigma(a)<\sigma(b)$


${1 \over 6^{n-2}}$


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


$P[Y_n=1|\ Y_m=1]=1/2$


$ e_1 = \alpha+\beta+\gamma = -\frac{b}{a} \\ e_2 = \alpha\beta+\beta\gamma+\gamma\alpha = +\frac{c}{a} \\ e_3 = \alpha\beta\gamma = -\frac{d}{a} $


$\left \| A \right \|_F = \sqrt{\mathrm{trace} \left ( A^H A \right )}$


$F(x,y)= \sum_{n=-\infty}^{\infty} f(|n|x) e^{-2 i \pi |n|y}= \frac{1}{x} \sum_{n=-\infty}^{\infty} \hat{f}(\frac{n+y}{x})$


$log(b)/log(a) = n$


$\int_{0}^{\frac{\pi}{2}} \frac{\sin^2(x)}{\sin(x)+\cos(x)}$


$K\cong \Bbb Q(\gamma)=\{a+b\gamma+c\gamma^2: a,b,c\in\Bbb Q\}$


$ dS=\sqrt{\left(\frac{dx}{d\phi}\right)^2 + \left(\frac{dy}{d\phi}\right)^2} $


$\int_0^{\pi/2} \cos^n(t) \sin(nt) \,dt = \int_0^{\epsilon/n} \cos^n(t) \sin(nt) \,dt + \int_{\epsilon/n}^{\pi/2} \cos^n(t) \sin(nt) \,dt$


$ \int_{0}^{ \infty} f(x) dx $


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


${A_{ik}} = \left\{ \matrix{   1 - {1 \over P}\,\,\,\,\,\,\,\,\,\,\,\,i = k\, \hfill \cr   - {1 \over P}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,i \ne k\, \hfill \cr}  \right.\tag{4}$


$\gamma = (\gamma_1, \gamma_2, \ldots \gamma_n)$


$\underset{n=1}{\overset{\infty}{\sum}}\parallel z_{n}-x_{n}\parallel_{n}^{2}<\epsilon^{2}$


$rn = r(1)n$


$M=\pmatrix{a&b\\c&d}$


$4+\lim_{n \rightarrow \infty} \sum_{i=1}^n 2=\infty$


$(a(x)b(y))'=a'(x)b(y)+a(x)b'(y)$


$\rm {\bf Lemma}\quad\ \lfloor x/(mn)\rfloor = \lfloor{\lfloor x/m\rfloor}/n\rfloor\quad for\ \ \ n > 0 $


$\det \begin{pmatrix} A & -B \\ B & A \end{pmatrix} = \det{(A+iB)}\det{(A-iB)}$


$f'(x) =\frac{f(\pi) - f(0) }{ \pi - 0 }$


$[x,y]$


$f(x) = 1/|x|^{d-2}$


$6a(x)[(3-x)a(x)-3a(x)^2]=-xa(x)+x(2-x)a'(x)-x^2a''(x)$


$\sigma \subseteq \gamma \subseteq \tau.$


$\frac {d}{dt} \int_0^t (E i(x)-R i(x)^2)dx=E i(t)-R i(t)^2=(E-R i(t)) i(t).$


$(5,-2,-2,2,0) = x(3,-2,-1,1,-1) + y(1,0,-1,1,1)$


$\beta = \{A_1,A_2,A_3,...,A_n\}$


$\begin{align}\frac{ds}{dt}= \sqrt{(\frac{dy}{dt})^2+(\frac{dx}{dt})^2}\end{align}$


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


$e^x\geq \left(1+\frac{x}{6}\right)^6$


$  \int_{0}^{1} \frac{\sin(x)dx}{x} $


$(\alpha\beta)^{+}=\beta^{+}\alpha^{+}$


$3q^2p \equiv \begin{cases} 1 & p\equiv q \equiv 1\\                               0 & \text{otherwise} \end{cases}$


$p\oplus t$


$\int_a^b f(t)\sin(t)dt=0=\int_a^b f(t)\cos(t)dt$


$A_1 \subseteq \dots \subseteq A_r$


$X=\frac{\frac{c(1+r)^{T+1}+(1-c)r^2}{r(1+r)}}{\frac{(c-r)(1+r)^{T}+(1-c)r}1}$


$E(X^n) = E((X + 1)^{n−1})$


$\operatorname E|X_n|^r\to\operatorname E|X|^r$


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


$(1 + x/n)^n \approx e^x$


$\left[\begin{array}{cccc|c} 1&-1&2&1 & 2\\   2&-3&2&0 & 3\\ -1&1&2&3 & 6\\ -3&2&0&3 & 9\\ \end{array}\right]$


$\det(A_{x,i,j})=cofactor(a_{i,j})x+\det(A)$


$M = \{\{\emptyset\}\}$


$\mathbb{R}^2: (a,b) + (c,d) = (a+b, c+d)$


$\sum_{n=1}^r \frac{1}{1+nQ}\leq\sum_{t=1}^\infty\left(\sum_{n=k+1}^\infty\frac{1}{p_n}\right)^t$


$ \gamma+\beta\gamma^{-1}=\frac{\gamma^2+\beta}{\gamma}=\alpha\gamma^{-1} $


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


$\mathscr{A}_0 =\{a\}$


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


$P(X\geq 1 \vert X\geq 2) = 1$


$X=\frac{(c(1+r)^{T+1}+(1-c)r^2)1}{r(1+r)((c-r)(1+r)^{T}+(1-c)r)}$


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


$\begin{align*} && P(0)\implies P(1) &&&&\text{is true and}\\ && P(1)\implies P(2) &&&&\text{is true and}\\ && P(2)\implies P(3) &&&&\text{is true and}\\ && &&\vdots && \end{align*}$


$\left \lfloor \sum_{i=1}^{n}\{x_i\} \right \rfloor = L$


$\int_0^1 x^2\sqrt{1-x^2}\,dx = \int_0^{\pi/2} \sin^2\varphi \sqrt{\cos^2\varphi}\cos\varphi\,d\varphi = \int_0^{\pi/2}\sin^2\varphi\cos^2\varphi\,d\varphi.$


$\sqrt[n]{z} = \sqrt[n]{r}e^{\theta i/n}$


$(y-z)(y+z)+(z-x)(z+x)+(x-y)(x+y)=y^2-z^2+z^2-x^2+x^2-y^2=0$


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


$\,p_1p_2\,\,\mid\,\,p_1p_2\,=>\,p_1p_2\,\,\mid\,\,p_1\,\,\,\vee \,\,\,p_1p_2\,\,\mid\,\,p_2$


$T_k[f](x_n+e_n) \approx f(x_n + e_n) = 0$


$ \gamma \in K :\gamma B \subset R$


$L=\lim\limits_{x\to c}f(x)$


$\left(\begin{array}{ccc|c} 1 & 1 & 1 & 4\\ 0 & 5 & 4 & 2\\ 0 & 5 & 4 & 2\end{array}\right)\begin{array}{c} \\ R_2+R_3\\ R_3+R_2\end{array} \Leftrightarrow\left(\begin{array}{ccc|c} 1 & 0 & 1/5 & 18/5\\ 0 & 1 & 4/5 & 2/5\\ 0 & 0 & 0 & 0\end{array}\right)\begin{array}{c} R_1-R_2/5\\ R_2/5\\ R_2-R_3\end{array}$


$\begin{align*} \lim\limits_{n\to \infty}\dfrac{1}{\ln(n)}\sum\limits_{r=1}^{n}\dfrac{a_r}{r} &=\lim\limits_{n\to \infty}\dfrac{1}{n}\cdot\dfrac{n}{\ln(n)}\sum\limits_{r=1}^{n}\dfrac{a_r}{r}\\ &=\lim\limits_{n\to \infty}\dfrac{1}{n}\sum\limits_{r=1}^{n}\dfrac{a_r}{r/n}\cdot\dfrac{1}{\ln(n)} \end{align*}$


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


$\lim_{x \to c}f(x)k(x) = \lim_{x \to c}h(x)g(x)$


$ \lfloor \frac{a+b}{c}\rfloor +\lfloor \frac{b+c}{a} \rfloor+\lfloor \frac{c+a}{b} \rfloor $


$ d=\frac{a+bc}{a-1} $


$\{a\}\in\{\{a\}\}$


$\sum_{k \in A} |x_k + y_k|^2 \leq 4\left( \sum_{k \in A} |x_k|^2 + \sum_{k\in A} |y_k|^2 \right) < \infty,$


$\int |fg|^r\leq \left(\int |f|^{r\frac{p}{r}}\right)^{r/p}\left(\int |g|^{r\frac{q}{r}}\right)^{r/q}.$


$[y,x]=y$


$e\in V\subseteq \overline{V}\subseteq U$


$V=\ker(P)\oplus U$


$P(E)P(E')=P(E\cap E')$


$\vartheta(x):=\sum_{n=-\infty}^\infty e^{-n^2\pi x}={1\over\sqrt{x}}\vartheta\left({1\over x}\right)\ .$


$ Var(X^*_t | X^*_0 = x^*_0) = \frac{\sigma^2}{2 \mu},\quad \forall t > 0 $


$u(x)=1 - 1/|x|$


$V(a)\subseteq V(a)$


$4 \times2^x+3^x=5^x$


$\mu\left(\liminf A_{n}\right)=\lim_{n\to\infty}\mu\left(B_{n}\right)\leq\lim_{n\to\infty}\inf_{k\geq n} \mu A_{k}  =\liminf\mu A_{n}$


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


$\begin{cases} x=t \\ y=2-t \\ z= 5\end{cases}$


$a+b,a+c,a+d \in \mathbb{Q}$


$(x,y) = x$


$\mathop {\lim }\limits_{n \to \infty } \int_{ - \pi }^\pi  {{x^2}} \frac{{\sin \left( {2nx} \right)}}{{\sin x}}xdx =  - 8\pi \sum\limits_{k \geqslant 1} {\frac{1}{{{{\left( {2k - 1} \right)}^2}}}}  =  - 8\pi \frac{{{\pi ^2}}}{8} =  - {\pi ^3}$


$\lim_{n\rightarrow \infty}{1^n}=\lim_{n\rightarrow \infty}{1}=1$


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


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


$\frac{1}{|D_n|} f_n(2) = {n+2 \choose 2}$


$ \gamma'(t) = \left(\frac{\partial \gamma_1}{\partial t},\frac{\partial \gamma_2}{\partial t}\right)\approx\frac{\gamma(t+h)-\gamma(t)}{h}$


$ \begin{vmatrix} x & y & 1  \\  x1 & y1 & 1 \\ x2 & y2 & 1  \end{vmatrix} $


$ det\begin{pmatrix}A & -B\\ B & A \end{pmatrix}=|det(A+iB)|^2, $


$x_3,x_2\in \mathbb R$


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


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


$\Bigl(1+\dfrac1{n+1}\Bigr)^{\!n+1}-\Bigl(1+\dfrac1n\Bigr)^{\!n}<\mathrm e-2.$


$=\begin{vmatrix} x &x^2  &1 \\  y &y^2  &1 \\  z &z^2  &1  \end{vmatrix}+xyz\begin{vmatrix} 1 &x  &x^2 \\  1 &y  &y^2 \\  1 &z  &z^2  \end{vmatrix}=\ldots$


$ax+by=d$


$\text{for}\;\;1\le k\le n\;,\;\;\;i_k=\begin{cases}0\;,\;\;\text{if}\;\;k\notin X\\{}\\1\;,\;\;\text{if}\;\;k\in X\end{cases}$


$\cos(n\theta) = \sum_{p=0}^{\lfloor \frac{n}{2} \rfloor} (-1)^{p} \begin{pmatrix} n \\ 2p \end{pmatrix} \sin^{2p}(\theta)\cos^{n-2p}(\theta)$


$3\mid k^3-k$


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


$\mathcal{R} = \{\{0\}\}$


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


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


$|xy|=rs$


$z^n = |z|^n (\cos(n\phi) + i\sin(n\phi))$


$\;\mathrm e^{i\tfrac{(n+1)x}2}$


$e^{2 i \pi/(p-1)}$


$A_n=a_1+\ldots+a_n = \frac{n(n+1)}{2}$


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


$ \cos^4 \varphi - 6 \cos^2 \varphi \sin^2 \varphi + \sin^4 \varphi = (\cos^2 \varphi - \sin^2 \varphi)^2 - 4 \cos^2 \varphi \sin^2 \varphi = \cos^2 2\varphi -  \sin^2 2\varphi  $


$\frac 1r[(1+r)^n -1] = \sum_1^T (1+r)^{T-1}$


$ \frac{\langle x,v \rangle}{\vert\vert v \vert\vert} \leq \frac{\vert \langle x,v \rangle \vert}{\vert\vert v \vert\vert} \leq \vert\vert x \vert\vert $


$d_{g}(x,y) = \inf\{\ell(\gamma)\mid\gamma:[0,1]\to M,\gamma\text{ is a curve},\gamma(0)=x,\gamma(1)=y\}$


$y,y',y'',y'''$


$ P(k+1) : (k+1)^3 < 3^{k+1} $


$\forall m P(n,m) \Rightarrow \forall m P(n++,m)$


$\int_0^\infty\int_t^\infty f(x)dxdt = \int_0^\infty f(x) \int_0^xdt dx$


$ \left[\begin{array}{ccc|c} 1 & -1 & 1 & 1 \\ 0 & 0 & -1 & 0 \\ 0 & -6 & 4 & 5 \end{array}\right] $


$m=rs, a^m=a^{rs}=(a^r)^s\implies (a^r-b^r)\mid\{(a^r)^s- (b^r)^s\}$


$ax+by=f(a,b)$


${\sqrt[3]{1} + \sqrt[3]{2} + \sqrt[3]{3} + \dots + \sqrt[3]{n} \over \sqrt[3]{n^4}}$


$\frac{||Tf||}{||f||}\leq c <\infty$


$\left\lfloor\frac{x/a}b\right\rfloor = \left\lfloor\frac{\lfloor x/a\rfloor}b\right \rfloor$


$A_1 \subseteq A_2 \supseteq A_3 \subseteq A_4 \supseteq ...$


$R/2R\cong \mathbb Z_2[x]/(x^4 + 1)$


$\begin{bmatrix}1 & 3 & -1\\4 & -1 & 2\\2 & -1 & -3\end{bmatrix}$


$D = \lbrace (x,y) \in \Bbb{R}^2 | 0\le x \le 1, 0 \le y \le x \rbrace$


$P(s) = 1/6$


$\|A\|_2^2=\rho(A^TA)\leq\|A^TA\|_\infty$


$\gamma(a)\le\max\{\gamma(x_1),\gamma(x_2)\}$


$         \begin{pmatrix}         1 & 1 & 1 \\         a & b & c \\         a^2 & b^2 & c^2 \\         \end{pmatrix}^{-1}         \begin{pmatrix}         1  \\         t  \\         t^2 \\         \end{pmatrix} =        \begin{pmatrix}         x  \\         y  \\         z \\         \end{pmatrix} $


$\displaystyle \left\{(1+x)^{r}+(1+x)^{r+1}+(1+x)^{r+1}\cdot \cdot \cdot \cdot \cdot \cdot \cdot+(1+x)^{n}\right\} = \frac{(1+x)^{n+1}-(1+x)^r}{(1+x)-1}=\frac{(1+x)^{n+1}-(1+x)^r}{x}$


$\{f, 0_{[0,1]},f, 0_{[0,1]},\dots \}$


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


$xHx^{-1}, x\in G$


$ \sum_{n=2}^\infty x^{\ln n}=\sum_{n=2}^\infty \frac1{n^{-\ln x}} $


$b=as$


$R^{n+1} -  $


$0=B_0(1+r)^N - C\frac{(1+r)^N-1}{r}$


$ \begin{align} \left(1+\frac{x}n\right)^n &\le\lim_{n\to\infty}\left(1+\frac{x}n\right)^n\\[6pt] &=e^x\tag4 \end{align} $


$\|A\|_{2} = \sqrt{\rho(A^{T}A)}$


$ f(m+a)=f(m)+b$


$A_3 \subset A_2 \subset A_1$


$\lim_{n\rightarrow\infty}\sum_{i=1}^n p_i(1-p_i)=\infty$


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


$ \pi\,f=(f^*\,\pi^*)^*=(f\,\pi)^*=(\pi\,f\,\pi)^*=\pi^*\,f^*\,\pi^*=\pi\,f\,\pi=f\,\pi. $


$\alpha^{+}$


$AC\lor \bar BC\lor A\bar B$


$ \frac{2}{T}\frac{1}{\dot{\theta}(\theta)}=\frac{\sqrt{\left(1-e^2\right)^3}}{\pi\left(1+e\cos{\theta}\right)^2} $


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


${x}^{2}-z\,x-y\,x+{z}^{2}-y\,z+{y}^{2}=1$


$Cov(X,Y)=\rho$


$\begin{align} \int \tan^n x\,dx &= \int \tan^{n-2}x\,(\sec^2 x - 1)\,dx \\ &= \int \tan^{n-2}x\,\sec^2 x\,dx - \int \tan^{n-2}x\,dx \\ &= \int u^{n-2}\,\,du - \int \tan^{n-2}x\,dx \\ &= \frac{1}{n-1}u^{n-1} - \int \tan^{n-2}x\,dx \\ &= \frac{1}{n-1}\tan^{n-1} x - \int \tan^{n-2}x\,dx \end{align}$


$ n = m+1\\ \text{and}\\ P(n) \implies P(m+1) $


$1+\frac{1}{4}\left((e^{x/20})^2-2+(e^{-x/10})^2\right).$


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


$\frac{1}{x}+\frac{1}{y}=\frac{1}{x+y}.$


$x = \frac{u}{\gamma} - \frac{ay}{2\gamma^{2}}$


$\int_{0}^{\infty}\frac{\sin^2(x)}{x^3}\,dx$


$\int_{0}^{\infty} f(x)^2dx$


$\displaystyle\sum_{m=0}^{n-1}\cos\frac{2m\pi}n=\frac0{2^{n-1}}$


$\Bbb{E}(|X-Y|) \le \Bbb{E}(|X+Y|)$


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


$|x-a|<\delta\implies |f(x)-f(a)|<\epsilon$


$\int_C \cos z\ dz=\int_0^{\pi/2}f(z(t))\dot{z}(t)\ dt=\int_0^{\pi/2}\cos(\sqrt{2}\cos t+i\sin t)\cdot(-\sqrt{2}\sin t+i\cot t)\ dt.$


$f(x)=x^2\sum_{n\ge0}{\frac{(-1)^n}{x^n}}=\sum_{n\ge0}{\frac{(-1)^nx^2}{x^n}}=\sum_{n\ge0}{\frac{(-1)^n}{x^{n-2}}}=\sum_{n\ge0}{\frac{1}{(-x)^{n-2}}}$


$=\frac{1}{\gamma(a\gamma-1)}\int \frac{dt}{t+1}-\frac{1}{\gamma(a\gamma-1)}\int \frac{dt}{t+a\gamma}$


$ \liminf_{x\to 0}f(x)=\liminf_{|y|\to\infty}f\Big(\frac{1}{y}\Big)=\liminf_{y\to\pm\infty}f\Big(\frac{1}{y}\Big). $


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


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


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


$\lim_{n \rightarrow \infty} \mu^*(A_n) \leq \mu^*(A)$


$\frac {a^6 - (a^3-a) - (a^2-a) - a}{6} + \frac  {a^3-a}{3}+ \frac  {a^2-a}{2} + a$


$ \Vert A \Vert_2 =\sqrt{\lambda_\max (A^TA)}=\sqrt{\lambda_\max (D^2)}$


$\lim_{n\to\infty} \inf s_n \leq \lim_{n\to\infty} \inf t_n,$


${{\vartheta }_{3}}\left( z|\tau  \right)=\sum\limits_{n=-\infty }^{\infty }{{{e}^{i\pi {{n}^{2}}\tau }}{{e}^{2niz}}}$


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


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


$(a,x) *(b,y)=(a,x)$


$H = [a,b]\times[c,d] := \{(x,y)| a \le x \le b; c \le y \le d\}$


$\sum_{n=-\infty}^{\infty}\cos\frac{2n\pi}{3} = 0$


$M=\begin{pmatrix}A&B\\0&C\end{pmatrix}$


$F(x) = \int\limits_a^{b(x)} {f(t)dt} $


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


$T_X X$


$P(E_2) = 3/10.$


$K = \{k_1, k_2, k_3,....k_m \}$


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


$A\in\{A_1, A_2, \ldots\}$


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


$\mathbb{E}[X|X_{0} = x_{0}]$


$I_{n,a}=\int_{-\infty} ^{\infty} \frac{dx}{(1+\frac{x^2}{a})^n}$


$A_1 ... A_k$


$\sum_{n=0}^\infty \left \lvert \frac{(-1)^n}{n+x^2}\right \rvert = \sum^\infty_{n=0} \frac{1}{n+x^2} = \sum_{0\le n < x^2} \frac{1}{n+x^2} + \sum_{n \ge x^2} \frac{1}{n+x^2} \ge \sum_{0\le n < x^2} \frac{1}{n+x^2} + \sum_{n\ge x^2} \frac{1}{2n}.$


$\{a, aa, bbb, aab, z, Z\}$


$xRy, yRh$


$  p(k)=(\frac{2}{3})^{k-1}(\frac{1}{3}), $


$x_4 = 40$


$x(n(x+y)^{n-1} + x n(n-1)(x+y)^{n-2}) = \sum_{k=0}^n k^2 \binom{n}{k} x^k y^{n-k}$


$\tan u = x/2$


$S(m) \Rightarrow S(m+1)$


$ X_n \leq x \Rightarrow X \leq x + \epsilon$


$1 + 3 + 9 + 27 + ... = -\frac{1}{2}$


$\int_{0}^{\infty }\frac{x^{n-1}}{e^{x}-1}\, \mathrm{d}x=\sum_{k=1}^{\infty }\int_{0}^{\infty }x^{n-1}e^{-kx}\, \mathrm{d}x\overset{[1]}{=}\sum_{k=1}^{\infty }\frac{1}{k^{n}}\int_{0}^{\infty }x^{n-1}e^{-x}\, \mathrm{d}x=\zeta \left ( n \right )\Gamma \left ( n \right )$


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


$xRy \wedge yRx \Longrightarrow y=x$


$\dfrac{10!}{3!\cdot 7!}=120 \text{ ways.}$


$\{\gamma \in W(\omega_1): \gamma \leq a\}$


$
 \|M\|_2 = \sqrt{ \rho(M^T M) }
 $


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


$\gamma_n(t) = \gamma(\frac{1}{n}\lfloor n t \rfloor)$


$||A||_2 = \sqrt{\rho(A^HA)}$


$\gamma = \gamma_+ - \gamma_-$


$\gamma*(\gamma*\gamma) = (\gamma*\gamma)*\gamma$


$\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=\frac{3}{2014}$


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


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


$\mathbb{P}(X_1 = 1 | X_0 =n) > 0$


$\operatorname{det}Df(\varphi)=-\sin(\varphi)\cos(\varphi)$


$\frac{m}{d} \in [x,y]$


$\tan(E) = \frac{\sqrt{1-e^2} \sin(\theta)}{e + \cos(\theta)}$


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


$\displaystyle \frac {\partial} {\partial x} f(x,y(x))=\frac {\partial} {\partial x}(x^2+y(x)x+y(x)^2)=2x+y'(x)x+y(x)+2y'(x)y(x)$


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


$d(x,A)-d(x',A)\leqslant d(x,x')$


$\displaystyle I = \int_{0}^{\pi}\frac{\sin (884 x)\sin (1122x)}{2\sin x}dx$


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


$xn(x+y)^{n-1} + x^2 n(n-1)(x+y)^{n-2} = \sum_{k=0}^n k^2 \binom{n}{k} x^k y^{n-k}$


$\{a, a + b, \ldots, a + (N − 1)b\}$


$16^x = 12^x + 9^x$


$(1-r)(1+r+r^{2}+\ldots + r^{n})=1 - r^{n+1}$


$d{\bf g}(0,0)=\left[\matrix{1&0\cr 0&1\cr}\right]\ .$


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


$q!-q+1,\ldots, q!-2$


$d(x,A)=inf\{d(x,y):y \in A\}$


$\circ: \mathcal{C}'(\mathbb{R}^n,\mathbb{R}^n) \times \mathcal{C}'(\mathbb{R}^n,\mathbb{R}^n) \longrightarrow \mathcal{C}'(\mathbb{R}^n,\mathbb{R}^n)$


$\frac{t(1+r)^{t-1}}{(\sum_{k=1}^T (1+r)^{k-t})^2}$


$\displaystyle \int\frac{1}{(x^3+1)^2} \mathrm dx$


$10[(1+r)^{n-1}+(1+r)^{n-2}+\ldots (1+r)^1+1]=10\frac {(1+r)^n-1}{1+r}$


$\int_{-\pi}^\pi\frac{\sin(x)}{1+x^2}{\rm~d}x=0$


$\gcd(a/\gcd(a,b)$


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


$g_n(x) = (\frac{k-1}{2^{n-1}})^2.$


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


$ \int \frac{\sin^{-1}(x)}{\sqrt{1+x}} \; dx$


$T_x M = T_x X$


$Cov (Z,X) = 0$


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


$ a, a + d, a + 2d, \cdots $


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


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


$Cov(X,Y) = 1$


$\frac{dx}{dt} = \frac{t+3x}{t-x}$


$\int_{0}^{1/4}\frac{\sin^{-1}(2x)}{2x}dx$


$p = (a, a+k, a+2k,...)$


$\lim_{n\to\infty}\frac{x_1+x_2+...+x_n}{n}=\lim_{n\to\infty}x_n$


$F_{X_{(2)}|X_{(3)}}(x;y)=\frac{2x}{y^2}$


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


$\sum_{k=0}^{n-1}\cos\left(\frac{2\pi k}{n}\right)=0=\sum_{k=0}^{n-1}\sin\left(\frac{2\pi k}{n}\right)$


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


$P(F|T)=\frac{P(T|F)P(F)}{P(T|F)P(F)+P(T|F^c)P(F^c)}$


$\begin{cases}a_0=0\\b_0=1\\c_0=0\end{cases}\quad$


$x\rho y \: \wedge \: y \rho x$


$e^{+}$


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


$ a = b^{\log_b(a)} \tag{a}.$


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


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


$ A = \int_{0}^{1}(1-x^{2/3})^{3/2}\,dx = \frac{3}{2}\int_{0}^{1}z^{1/2}(1-z)^{3/2}\,dz=\frac{3}{2}\,B\left(\frac{3}{2},\frac{5}{2}\right)\tag{1}$


$\int\frac{\partial^2{\bar{u}}}{\partial\sigma\partial\gamma}d\sigma=g(\gamma)\implies \int\frac{\partial{\bar{u}}}{\partial\gamma}=\int g(\gamma)d\gamma$


$v(x)= ae^{\frac{x}{-A}}+be^\frac{-x}{-A}$


$1-i =\sqrt{2}e^{i\pi\over 4}$


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


$antilog_a (y) = x$


$(F_2[x]/(x^3+x+1))/(x^2+x+1) $


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


$\lim \limits_{x \to \alpha^+}{f(x)}$


$F(x) = \int_{a}^x f(t) \ dt = \frac{x-a}{b-a}$


$f(x) = \sum_{j=1}^n c_j 1_{A_j}(x).$


$\pi_1(\mathrm{Spec}(\mathbb{Z}[x]/(x^6-1))) \cong \widehat{\mathbb{Z}}$


$\lfloor\gamma\rfloor$


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


$f(x)\equiv -1$


$p_{1}p_{2} = a^2$


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


$\alpha=\sqrt 2e^\frac{2\pi i}{3}$


$A_1\times  A_2\times \cdots\times A_n \times A_{n+1}$


$f = \{\{a\}\}$


$A_i=\{a_i, a_i+d_i, a_i+2d_i, \ldots\}$


$\sum_{n=-\infty}^\infty e^{-n^2\pi x}= \frac{1}{\sqrt{x}}\sum_{n=-\infty}^\infty e^{-n^2\pi / x}$


$\sum_{n=-\infty}^\infty e^{-n^2\pi / x}=\vartheta _3\left(0,e^{-\frac{\pi }{x}}\right)$


$ \sum_{k=0}^{n}kr^k = r\frac{1-(n+1)r^n + nr^{n+1}}{ (1 - r)^2 } $


$\|u\|_{L^\infty(0, 1)} \le \epsilon\|u'\|_{L^p(0, 1)} + C\|u\|_{L^1(0, 1)}$


$a^2 b+ b^2 a= a^2 + b^2$


$f(K)=[f(a),f(b)]$


$-9(x + 9)^{\frac{2}{3}} +27(x +9)^{\frac{1}{3}} =-9-9 +27 $


$\sum\limits_{r=0}^{n+1} \frac{{(n+1)}!}{r!(n+1-r)!} =\sum\limits_{r=0}^{n+1} \Bigl(\frac{n!}{r!(n-r)!}+\frac{n!}{(r-1)!(n-r+1)!}\Bigr),$


$f_3(5n,\ 5k) = \binom{n}{k}$


$\lim_{\text x \rightarrow c}c(x)= L$


$\tan(\vartheta)=-\rho/\tau$


$(n+1)! + 2, (n + 1)! + 3, ..., (n + 1)! + n + 1$


$\cos(\frac{\pi}{k}) + \cos(\frac{2\pi}{k}) + \cos(\frac{3\pi}{k}) + ... + \cos(\frac{n\pi}{k})$


$(A^2＋B^2)(A-B)＝A^3-B^3＋B^2A-A^2B＝B^2A-A^2B$


$yb + xa = d$


$\mathbb{P}(X_1=1\vert X_0=2) = \mathbb{P}(X_1=2\vert X_0=3)=1.$


$\left(\sum^\infty_{i_j=1} \frac{x^{i_j}}{i_j!} \right)$


$ \tan (\gamma+a)=\frac{k-\cos(\gamma)}{\sin(\gamma)}$


$ \mathcal E = \{ e_1, e_2, \dots, e_n, \dots \} $


$xy-1,x\in P$


$|AB|=b$


$ \begin{align} 2^{n-1}\prod_{k=1}^{n-1}\cos\left(\frac{2k\pi}n\right) &=\prod_{k=1}^{n-1}\left(1+e^{4\pi ik/n}\right)e^{-2\pi ik/n}\\ &=(-1)^{n-1}\left[\prod_{k=1}^{n-1}\left(-z^2+e^{4\pi ik/n}\right)\right]_{\large z=i}\\ &=\left[\prod_{k=1}^{n-1}\left(z^2-e^{4\pi ik/n}\right)\right]_{\large z=i}\\ &=\left[\prod_{k=1}^{n-1}\left(z-e^{2\pi ik/n}\right)\prod_{k=1}^{n-1}\left(z+e^{2\pi ik/n}\right)\right]_{\large z=i}\\[3pt] &=\left[\frac{z^n-1}{z-1}\frac{z^n-(-1)^n}{z+1}\right]_{\large z=i}\\[6pt] &=\frac{(i^n-1)(i^n-(-1)^n)}{-2}\\[9pt] &=\frac{2(-1)^n-i^n(1+(-1)^n)}{-2}\\[12pt] &=\cos(n\pi/2)-(-1)^n \end{align} $


$w = e^{\frac{2\pi\iota}{n}}$


$\bigg\lfloor\frac{I+F}{c}\bigg\rfloor=\bigg\lfloor\frac{I}{c}\bigg\rfloor$


$[x+y]=[[x]+\{x\}+[y]+\{y\}]=[x]+[y]+[\{x\}+\{y\}]$


$t^2=\frac{\sqrt{2r^2(1-\cos(x))}+\sqrt{2R^2(1-\cos(x))}}{g\sin(x)}$


$\neg P(s(m))\implies \neg P(m)$


$ A_i \times \cdots \times A_n$


$(t^{\frac{j-1}{2}}-1)^2 \equiv 0 \pmod{p^2}$


$S=\sum_{n\geqslant1}\frac{1}{n^2}-\sum_{n\geqslant1}\frac{2}{(2n)^2}=\frac12\zeta(2).$


$\frac{36}{19}=1+\frac{17}{19}=1+\frac1{\frac{19}{17}}=1+\frac1{1+\frac2{17}}=1+\frac1{1+\frac1{\frac{17}2}}=1+\frac1{1+\frac1{8+\frac12}}$


$||x||_{E} \geq \frac{1}{||f||}$


$ \begin{cases} x = 2 \alpha \\ y = -\alpha \\ z = -\alpha \end{cases} $


$Pr\left(X_{n+1}=\frac{X_{n}}{2}|\mathcal{F}_{n}\right)=1-X_{n}$


$d(x,F)=\parallel y-x\parallel$


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


$p(x|a) = \frac{2x}{a^2}$


$L_2 = \begin{cases} x= 1+t \\ y=2-t \\ z=3+2t \end{cases}$


$T =T_1\oplus T'$


$\lVert\gamma(s)-\gamma(t)\rVert\lt\frac{\varepsilon}{4n}$


$\mathbb{E}[f(X_T)^2]\le\mathbb{E}[(2d)^{2T}]f(0)^2$


$\frac{(\gamma)_k}{(\gamma+1)_k} = \frac{\gamma(\gamma+1)\cdots(\gamma+k-1)}{(\gamma+1)(\gamma+2)\cdots(\gamma+k)} = \frac{\gamma}{\gamma+k}$


$\operatorname{dist}(x,A) \leqslant d(x,y) + \operatorname{dist}(y,A),\tag{2'}$


$\left[\frac{10}4,\frac{10}4,\frac{10}4,\frac{10}4\right].$


$1/x - 1/|x|$


$D_n=a(a-x)^{n-1}+x\left(\sum_{k=0}^{n-1}(a-x)^{k}(a-y)^{n-1-k}-(a-x)^{n-1}\right)$


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


$\begin{array}{c|c|c|c} & \begin{pmatrix}        0&1\\       1&0 \end{pmatrix} & \begin{pmatrix}        0&-i\\       i&0 \end{pmatrix} & \begin{pmatrix}       1&0\\       0&-1 \end{pmatrix} \\\hline \begin{pmatrix}         0&1\\       1&0 \end{pmatrix} & \begin{pmatrix}        1&0\\       0&1 \end{pmatrix} & \begin{pmatrix}       i&0\\       0&-i \end{pmatrix} & \begin{pmatrix}       0&-1\\       1&0 \end{pmatrix} \\\hline \begin{pmatrix}        0&-i\\       i&0 \end{pmatrix} & \begin{pmatrix}        -i&0\\       0&i \end{pmatrix} & \begin{pmatrix}       1&0\\       0&1 \end{pmatrix} & \begin{pmatrix}       0&i\\       i&0 \end{pmatrix} \\\hline \begin{pmatrix}        1&0\\       0&-1 \end{pmatrix} & \begin{pmatrix}        0&1\\       -1&0 \end{pmatrix} & \begin{pmatrix}       0&-i\\       -i&0 \end{pmatrix} & \begin{pmatrix}       1&0\\       0&1 \end{pmatrix} \\ \end{array}$


$\sup_{0\le |z|\le 1/2} |f(z)|\le\sup_{0\le |z|\le 1/2} |f(z^2)|=\sup_{0\le |z|\le 1/4} |f(z)|.$


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


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


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


$\mathbb{E}|\xi| \leq \lim_{p \downarrow 1} (\mathbb{E} |\xi|^p)^{1/p}$


$a^{y/x}=b-a^x$


$P(\lambda_0)=-\frac{|\langle x,y\rangle|^2}{\|y\|^2}$


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


$ \zeta(s) = \frac{1}{\Gamma(s)} \int_0^{\infty} \frac{x^{s-1}}{e^x-1} \, dx, $


$k= a^{log_b c}$


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


$\,2\mid n^2+n,\,$


$f(A+B)=f(A)+f(B)$


$\zeta\left(s\right)=\frac{1}{\Gamma\left(s\right)}\int_{0}^{\infty}\frac{u^{s-1}}{e^{u}-1}du,\,\mathrm{Re}\left(s\right)>1$


$K\subseteq G\subseteq \overline{G}\subseteq V\subseteq \overline{V}\subseteq U.$


$\sum_{n=1}^{\infty} \frac{\cos {\frac{n \pi}3}}n$


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


$ds = \sqrt{1 + \left(\frac{3x^2}{y}\right)^2}$


$\forall_{\epsilon > 0} \exists{\delta > 0}: |x-a| < \delta \implies |f(x) - f(a)| < \epsilon$


$[a_n,d_n) \subseteq U_n$


$(X+y+z)(x+Y+z)(x+y+Z)(X+y+z)(X+y+z)$


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


$1\equiv a^{n-1} \equiv a^{sd+r} \equiv a^r\mod{n}$


$\int_0^1 {\tan^{-1}\left(x\right) \over {1+x}}\,{\rm d}x$


$3^x + 6^x = 5^x + 4^x$


$ (1+r/n)^{rn} $


$ -(b^n + b^{n-1}) $


$\begin{cases} x=2,\\ y=2, \\z=\pm 1,-1. \end{cases}$


$\begin{pmatrix}     0.8 & -0.6 \\     0.6 & 0.8 \end{pmatrix}$


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


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


$\rm{dist}(f,Y)=\frac{|T(f)|}{||T||}.$


$C+E = \dfrac{P((1+r)^N-1)}{r(1+r)^N}$


$S_1\subset S_2\subset...\subset S_i\subset...$


$ E = \left( \begin{matrix} A & -B \\ B & A \end{matrix} \right) $


$\dim(M+H) = \dim M + \dim H = \dim M + \dim K = \dim(M+K)$


$f_n(x) = {(nx) \over n^2}$


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


$P[M_n \leq x \mid X_0](\omega) = I_{[X_0 \leq x]}(\omega)x^n$


$F(x) = \int_{-\infty}^{x}f(t)dt$


$   I(\gamma) = \int_0^\gamma \frac{1}{\sqrt{\pi}} \frac{1}{(\gamma^2+\delta)^2} \mathrm{d} \gamma = \frac{\gamma }{2 \sqrt{\pi } \delta  \left(\gamma ^2+\delta \right)}+\frac{1}{2 \sqrt{\pi } \delta ^{3/2}} \arctan\left(\frac{\gamma }{\sqrt{\delta }}\right) $


$ -\frac{\ln[1-(1-e^x)]}{1-e^x} = -\frac{(-\sum_{n=1}^{\infty} \frac{(1-e^x)^n}{n})}{1-e^x}=\frac{1}{1-e^x}.\sum_{n=1}^{\infty} \frac{(1-e^x)^n}{n}=\sum_{n=1}^{\infty} \frac{(1-e^x)^{n-1}}{n} $


$m\in M,m'\in M^{\perp}$


$ \mathbb{E}\|X\|_{\infty}^r \le C_r d^{r/2}\mathbb{E}\|Y\|_{\infty}^r, $


$\varphi=\varphi(s,t),\quad \varphi(s,a)=\gamma (a),\quad \varphi(s,b)=\gamma (b)\quad \text{and}\quad \varphi(0,t)=\gamma (t).$


$A_1\times...\times A_n$


$\{\{a\}\} \subset A$


$ U(\alpha,\gamma;z)=\frac{\Gamma(1-\gamma)}{\Gamma(\alpha-\gamma+1)}M(\alpha,\gamma;z) + \frac{\Gamma(\gamma-1)}{\Gamma(\alpha)}z^{1-\gamma}M(\alpha-\gamma+1,2-\gamma;z)  \tag{4} $


$\mathcal{R}^+$


$Pr(E2|E1)=\frac{Pr(E_2\cap E_1)}{Pr(E_1)}$


$E[min(|X|,|Y|)] = \frac{4}{\pi}\int_{x=0}^{\infty}e^{\frac{-x^{2}}{2}}(1-e^{\frac{-x^{2}}{2}})dx$


$x^2-x \in C$


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


$\lim\limits_{n}\mu_{M_0}(B_n)\geq \epsilon_2 >0$


$2\sum_{n\geq1}\frac{1}{\left(1+n^{2}\right)^{2}}=\frac{1}{2}\left(\pi^{2}\textrm{csch}^{2}\left(\pi\right)+\pi\coth\left(\pi\right)-2\right).$


$\eqalign{   J &= s:s \cr\cr  dJ &= 2s:ds \cr     &= 2s:Xda \cr     &= 2X^Ts:da \cr\cr \frac{\partial J}{\partial a} &= 2X^Ts \cr     &= 2X^T(Xa-y) \cr\cr } $


$ \Gamma(s) = \frac{1}{\zeta(s)} \int_0^{\infty}\frac{x^{s-1}}{e^x-1}dx $


$AB=\begin{pmatrix}A_1B_1 & A_1B_2\\A_2B_1 & A_2B_2 \end{pmatrix}$


$M\|A\|_\infty \le \|A\|_2 \le \|A\|_\infty$


$y\in V\subseteq\bar{V}\subseteq X\setminus\bar{U}$


$\frac{\frac{1}{2^{n-2}}}{\frac{1}{2^n}}=\frac{1}{4}$


$t_x\in [0,1]$


$ \sum_{n\geq 1}\frac{\log\left(1+\frac{1}{n}\right)}{n+1}<\sum_{n\geq 1}\frac{1}{n(n+1)}=1.$


$\frac 14 k(k + 1)(k + 2)(k + 3) $


$ P F P^T = F $


$\sum_{\gamma}a_\gamma t^{\gamma}+\sum_{\gamma}b_\gamma t^{\gamma}=\sum_\gamma (a_\gamma+b_\gamma)t^\gamma\text{,}$


$\sqrt[100]{\sqrt3 + \sqrt2} + \sqrt[100]{\sqrt3 - \sqrt2}$


$ S = \{s\space |\space s=\frac{lcm(a,\space a+d,\space a+2d,\space ...,\space a+10d)}{a+10d}\}$


$\left[\begin{array}{ccc|c} 1 & -17 & 0 & 3\\ 1 & 0 & -6 & 1\end{array}\right]$


$\left\lfloor \sum_{k=1}^{100}\frac{1}{\sqrt{k}}\right\rfloor$


$ax+by = g$


$ F'(x) = \int_a^x f(t)\ dt + xf(x) - xf(x) = \int_a^x f(t)\ dt $


$|a+b|^r\leq 2^{r-1}(|a|^r+|b|^r)$


$\frac{\hat \gamma}{a + \hat \gamma} A - \hat \gamma B$


$f(a+h)=f(a)+L(h)+r(h)$


$\{a,a+1,a+2,\ldots, b\}$


$\{\gamma:\exists\beta\in \alpha \cup \{\alpha\} :\gamma\in\beta\} = \{\gamma :\gamma\in\alpha\}$


$Q:=\{[a,b)\times[c,d): a,b,c,d \in \mathbb{R}; a < b,c < d\}$


$n+2, n+3, \dots, 2n+1$


$\gamma_\epsilon=\gamma|_{[\epsilon, 2\pi-\epsilon]}$


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


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


$ax+b=\frac{x}{a}-\frac{b}{a}$


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


$xy + ax + by = c$


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


$ (n+1)!+2, (n+1)!+3, (n+1)!+4, ... (n+1)!+(n+1) $


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


$\int_{0}^{\infty} f(x) dx \leq 1,$


$F_k=\{x\in \mathbb{R} | \ x\geq0,\  2\leq x^2\leq 2+1/k \}.$


$\rho\left(\left[\begin{matrix}0 & A \\ -A^{\dagger} & 0\end{matrix}\right]\right)=\sqrt{\rho\left(A^{\dagger} A\right)}$


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


$|D(u\circ \gamma)|\le L|D\gamma|$


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


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


$\cos\theta = \frac{a}{x}$


$(N+2)!+2 , (N+2)! + 3,(N+2)!+4,...(N+2)!+(N+2)$


$f''(x) = n(n-1)a_n(x-a)^{n-2}$


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


$ \forall x \in X: f(x)=x \iff R_f= \left\{ \left(T_\alpha,T_\beta\right)\subset \omega_X \times \omega_X | T_\beta \subset T_\alpha  \right\}$


$ k=\frac{e^{x+n-1+O(1/(x+n))}}{(x+n-1)^{1-1/(x+n)}}=\frac{e^{n-1+O(1/(x+n))}}{(x+n-1)^{1-1/(x+n)}}e^x $


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


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


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


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


$\mathcal{A}_{+} +\mathcal{A}_{−} = X$


$\frac{a+bx}{e^x}$


$S_n = \left\{a^{n!}_1, a^{n!}_2, a^{n!}_3, \dots, a^{n!}_n\right\}$


$B = \{b_1 , b_2, \ldots, b_n \} \subseteq A$


$|f(x+h)-f(x)| \leq 2|f(x)|$


$\Bbb{P}[X_t \in B | \mathcal{F}_s] = \Bbb{P}[X_t \in B | X_s]$


$\color{red}{y=\dfrac{a+b+c}{2\sqrt{x}}}$


$ \frac {a + bx}{1+x^2}$


$(a, a+r,a+2r,a+3r)$


$\left[{\partial^2f\over\partial X_i\partial X_k}(A)\right]_{i,\,k}=\left[\matrix{0&0&0&1\cr 0&0&-1&0\cr  0&-1&0&0\cr  1&0&0&0\cr}\right]=:H\ . $


$X^p=t\iff X=e^{\frac{2ik\pi}{p}}\sqrt[p]t, k=0,...,p-1$


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


$xRy\not =yRx$


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


$\sqrt[n]{Z}=\sqrt[n]{R}.e^{i\frac{\theta+2k\pi}{n}}$


$f_M(x)=F_M'(x)=3(1-\exp(-x/200))^2\frac{\exp(-x/200)}{200}$


$f_X(x) = \frac{2x}{10000}$


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


$e_N^*(x_n) \to 0$


$(a-b,a+b) \neq a-b$


$\det(\alpha_j \gamma) = \det(\alpha_j)$


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


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


$\lambda=\dfrac{\bigl\lvert\langle x,y\rangle\bigr\rvert}{\langle x,y\rangle}$


$[f(f^{-1}(x))]'=f'(f^{-1}(x))(f^{-1})'(x)=1$


$ a^n + a^{n-1} $


$y''y'=y'/y$


$|f(x)| \leq A\  |f(c)|(x-a)$


$d(x,A) \leq d(x,y) + d(y,A).$


$\begin{align} \int \frac{1}{(x+1)^2} \, dx & = \ln((x+1)^2) \div (2x+2) + c \\                              & = \frac{2\ln(x+1)}{2x+2} + c\\                              & = \frac{\ln(x+1)}{x+1} + c \end{align}$


$|A\times B|=ab$


$\int_0^\infty \frac{1}{(x^2+1)^{3/2}}\,dx=1$


$\|A\|=\sup_{\| \psi \| = 1} \|A \psi \| =\sup_{\psi=1}\left( \sup_{\|\phi\|=1}| \langle \phi | A | \psi \rangle| \right) =\sup_{\|\phi \| = \| \psi \| = 1} | \langle \phi | A | \psi \rangle|$


$\forall\varepsilon\gt0,\exists \delta_{1}\gt0, \text{ such that } |x| \lt \delta_{1} ,|f(x)| \leq \varepsilon \tag{2}$


$f(x) = \sum_{j=1}^n c_j 1_{A_j}(x)$


$f_{n}\in D(A_{n})=D(A)$


$a \leq log(b) \implies 10^a \leq 10^{log(b)} \implies 10^a \leq b$


$f^{\otimes p}\otimes (f^{\ast-1})^{\otimes q}: V^{\otimes p}\otimes_k (V^\ast)^{\otimes q}\to W^{\otimes p}\otimes_k (W^\ast)^{\otimes q}$


$x\to\infty\implies x!\sim \sqrt{2\pi x}\left(\frac{x}{e}\right)^x$


$\int_{0}^{1}\frac{\sin(x)\arctan(x)}{x}dx$


$N= \left\lfloor {\frac{{n\,m}}{{a\,b}}} \right\rfloor$


$ \int_0^{x+h}f(t)\,dt-\int_0^x f(t)\,dt= \int_x^0 f(t)\,dt+\int_0^{x+h}f(t)\,dt-\int_0^x f(t)\,dt= \int_{x}^{x+h}f(t)\,dt=hf(\xi) $


$(n+1)^{2}=n^{2}+2n+1\le n!+2n+1\le n!+n!+n!=3n!\le(n+1)n!=(n+1)!$


$(a;b;c;d) = (a;aγ;aγ^{2};aγ^{3}) $


$g(x) = (x-a)^{n+1}$


$d(x,\ker f) = \frac{|f(x)|}{\|f\|}$


$\|A\|_F = \sqrt{ \operatorname{Tr}(AA^*)^{\vphantom{l}}}.$


$\sum_{n = -\infty}^\infty e^{-\pi n^2 x} = \sum_{n = -\infty}^\infty f(n\sqrt{x}) = \sum_{n = -\infty}^\infty \frac{1}{\sqrt{x}}\hat{f}\left(\frac{n}{\sqrt{x}}\right) = \frac{1}{\sqrt{x}} \sum_{n = -\infty}^\infty e^{-\pi n^2/x} = \frac{1}{\sqrt{x}}\vartheta\left(\frac{1}{x}\right).$


$\color{red}{\det A} = [1 + n \beta (\alpha - \beta)^{-1}] (\alpha - \beta)^n \color{red}{ = (\alpha - \beta)^{n - 1}[\alpha + (n - 1) \beta]}.$


$f(K)\subseteq V\subseteq\overline{V}\subseteq g^{-1}(U)$


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


$p(E|S) = p(E \& S)/p(S) = p(E).1 / (1 - (1 -p(E).1) + p(E).(1 - 1) = p(E)/p(E) = 1 $


$\left|\int_0^\infty f(x)dx\right|=0$


$F(\pi) = F(0) = 0 $


$p^*=\inf_{x\in X}f_0(x)$


$\gamma\tau\gamma^{-1} = (\gamma(2), \gamma(4), \gamma(5))(\gamma(3), \gamma(1), \gamma(6)).$


$\color{red}{13}\mid n^{13}-n$


$M^{-1}=\left[\begin{matrix}\frac{\beta \gamma}{(\alpha-\beta) (\alpha-\gamma)} \ -\frac{\beta+\gamma}{(\alpha-\beta) (\alpha-\gamma)} \ \frac{1}{(\alpha-\beta) (\alpha-\gamma)} \\ \frac{\alpha \gamma}{(\beta-\alpha) (\beta-\gamma)} \ -\frac{\alpha+\gamma}{(\beta-\alpha) (\beta-\gamma)} \ \frac{1}{(\beta-\alpha) (\beta-\gamma)} \\ \frac{\alpha \beta}{(\gamma-\alpha) (\gamma-\beta)} \ -\frac{\alpha+\beta}{(\gamma-\beta) (\gamma-\beta)} \ \frac{1}{(\gamma-\alpha) (\gamma-\beta)}\end{matrix}\right]$


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


$ a_{\gamma},b_{\gamma} \in \mathbb C$


$\hat{y} = \frac{\hat{x}}{||f||}$


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


$f(x;\theta)=\frac{\beta^3}{2}e^{-\beta(x-\theta)}(x-\theta)^2\mathbf1_{x\ge\theta}$


$\log_a AB = \log_a A + \log_a B$


$F_1\subseteq F_2\subseteq\dotso\subseteq F_n\subseteq\dotso$


$x \to 0^+ \implies P \to \infty, x \to \infty  \implies P \to \infty$


$\int_{\pi/4}^{x}f(t)\,dt=\sin(x)f(0)-\cos(x)\int_{0}^{\pi/4}f(t)\,dt$


$\gamma(s+t)=\gamma(s)\gamma(t),\quad\gamma(0)=I,$


$K=\lim\limits_{x\to c^-} f(x)<\lim\limits_{x\to c^+} f(x)=L$


$\gamma_{a,b} + \gamma_{b,z} = \gamma_{a,b}$


$\sum^k_{j=1} I_j = \sum_{j \not = i} I_i I_j + \sum^k_{j=1}I_j I_j$


$\begin{align} & \alpha + \beta + \gamma = -\frac ba \\ & \alpha\beta + \beta\gamma + \alpha\gamma = \frac ca \\  & \alpha\beta\gamma = -\frac da \end{align} $


$\lim\limits_{n\to\infty}\frac1{n}\sum\limits_{k=1}^{\infty}\left\lfloor\frac{n}{3^k}\right\rfloor=\frac{1}{2}$


$P(V)\subseteq V$


$y = (2\sin(u) - 2\sin(u)\cos(u))(1 - v)^2$


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


$\int_{-\pi}^{\pi}F(x)\cos nx\,dx = 2\int_0^{\pi}f(x)\cos nx\,dx = 0$


$ \forall V\in\tau_{Y}\colon f^{-1}\left(V\right)\in\tau_{X}. $


$M=T\oplus P$


$\,\ln(a^b)=b\ln(a) \\a^{log_ax}=x \\ log_a(x)=b \implies a^b=x\\$


$\mathcal O(\mathbb R^n)$


$= \frac{1}{\sqrt{2^n}}\sum_{x}\bigg[\bigg[\sum_{y,z}\big(\left|z\right>\!\!\left<z\right|\otimes\left|y\oplus f(z)\right>\!\!\left<y\right|\big)\bigg]\left|x\right>\!\!\left|0^n\right>\bigg]$


$f_{\Phi}(\phi)=\frac{\sqrt{1-\rho^2}}{2\pi(1-\rho\sin2\phi)},\quad 0<\phi<2\pi$


$\lim_{x→c} f(x) = L$


$\begin{bmatrix}\sigma & -\omega\\\omega & \sigma\end{bmatrix}$


$xRy \not\Rightarrow yRx$


$e^A = I\gamma^0/0! + (A/\gamma)\gamma^1/1! + I\gamma^2/2! + (A/\gamma)\gamma^3/3!+...$


$ f'(x_0) = \lim_{x \to x_0^{-}} f'(x) = \lim_{x \to x_0^{+}} f'(x). $


$e^x\ge1+x\implies e^x\ge\left(1+\frac xk\right)^k$


$HCF(a+b, a-b) \ge HCF(a,b)$


$P(a<X\le b, c<Y\le d)$


$yRx \land xRy$


$f_X(x)=\theta x^{\theta-1},$


$f(n) = n^2-n-1>0$


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


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


$\sum_{l=-\infty}^{\infty}e^{j2\pi lNt/T}=\frac{T}{N}\sum_{m=-\infty}^{\infty}\delta\left(t-m\frac{T}{N}\right)\tag{2}$


$\displaystyle s_n(x) = \sum_{i=1}^N \alpha_i \chi_{Y_{i,n}}(x)$


$ (n+1)! + n! = (n+1)\cdot n! + n! = ... $


$d(x, z) \leq d(x, y) + d(y, z) < \epsilon + \epsilon = 2\epsilon$


$\vec{u} =  \begin{vmatrix}         \mathbf i & \mathbf j & \mathbf k \\         x_1 & y_1 & z_1 \\         x_2 & y_2 & z_2 \\ \end{vmatrix}\\ $


$(\gamma\gamma')f(x)=\gamma(\gamma' f)(x)=\gamma' f(\gamma x)=f(\gamma'\gamma x)$


$\varepsilon^+$


$d(x,A)-d(x,y)\leqslant d(y,A)$


$ \sqrt{2\sqrt[3]{3!\sqrt[4]{4!\sqrt[5]{5!..}}}}=2.2902182705436512868 $


$R_{0,1} \subset \dots \subset R_{0,r}$


$\frac{1}{16} \left(e^{-i x}-e^{i x}\right)^4$


$\lim\limits_{n\to \infty}\frac{1}{\ln(n)}\sum_{r=1}^{n}\frac{a_r}{r}$


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


$cx + ay = d$


$\lfloor\frac{E}{K}\rfloor =\lfloor \frac{E}{K + m}\rfloor$


$d(x,L)=\dfrac{\left| f(x) \right|}{\left|\left|f\right|\right|}$


$w:={z-z_1\over z-z_2}\ .$


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


$x^3-x, x^5-x, x^7-x, x^9-x$


$G=\gamma_1(G) \ge \gamma_2(G) \ge \cdots \ge \gamma_{k+1}(G)=1$


$\left|\begin{array}{r}1&x&1\\x&1&0\\0&1&x\end{array}\right|=1$


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


$Cov(x+y,x-y)$


$|ab|=p$


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


$ \{ x\in E \ \vert \ \forall m\in M : \langle x, m\rangle=0 \} = \{ x\in E \ \vert \ \forall m\in span(M) : \langle x, m\rangle=0 \}$


$\int_{-\pi}^{\pi} \frac{\cos(nx)}{2\pi} \ dx= \delta_{0,n}, n \in \mathbb{N}.$


$f_{r,t}(r,t) = 4 r (1- r \cos(t))(1- r \sin(t))$


$\forall E \in M, \lim_{n \to \infty }\mu_n(E)=\mu(E)< \infty $


$|f(z)| + |f(−z)| ≤ 2 |z^2| $


$\mathcal{M} = \langle D, \in \rangle$


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


$\frac{9}{m}+\frac{4}{r}+\frac{10}{h}= 7$


$3^x + 4^x + 5^x = 6^x? $


$\sum_{n=1}^{n}\frac{(-1)^n}{n^2}=\sum_{n \text{ even}}\frac{1}{n^2}-\sum_{n\text{ odd}}\frac{1}{n^2}=2\cdot\sum_{n \text{ even}}\frac{1}{n^2}-\sum_{n\geq 1}\frac{1}{n^2}=\frac{2}{4}\sum_{n\geq 1}\frac{1}{n^2}-\sum_{n\geq 1}\frac{1}{n^2}$


$\omega^\alpha=\alpha$


$E=A_1\times\ldots\times A_k$


$B=\{\{a\}\}$


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


$\gamma(t)=\frac{\gamma_2(t)-a}{\gamma_1(t)-a}+a$


$S^n \to \mathbb R^{n+1}$


$\|x_n - x_{n+1}\| \to 0$


$d(x,a)>d(x,b)+d(b,a)$


$\frac{z}{z*}= \frac{a+bi}{a-bi}$


$x^+$


$\left ( \begin{array} \\F_{k+2} \\ F_{k+1} \\ \end{array} \right ) = \left ( \begin{array} \\ 1 & 1\\1 & 0 \\ \end{array} \right )^k \left ( \begin{array} \\F_{2} \\ F_1 \\ \end{array} \right ) = \left ( \begin{array} \\ 1 & 1\\1 & 0 \\ \end{array} \right )^k \left ( \begin{array} \\1 \\ 1 \\ \end{array} \right )$


$\bigg\lfloor \lfloor \frac{b^2}{4} \rfloor / a \bigg\rfloor$


$\overline{X}(\mathbb{R})$


$\operatorname{Re}z^*x_0\ge\operatorname{Re}y^*x_0$


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


$(\gamma(i),\gamma(j))_{\gamma(0)} = \frac{1}{2}(d(\gamma(i), \gamma(0)) + d(\gamma(j), \gamma(0)) - d(\gamma(i), \gamma(j)))$


$\int_{0}^{\infty}\frac{\sin^{2n+1}(x)}{x} \ dx=\frac{\pi \binom{2n}{n}}{2^{2n+1}}.$


$\zeta(s) = \frac{1}{\Gamma(s)} \sum_{n=0}^\infty \frac{B_n}{n!} \int_0^\infty x^{s + n - 2} dx$


$V=P\bigoplus Q$


$\lim_{n\to\infty}\left\lfloor\sum_{k=1}^n\frac9{10^k}\right\rfloor=0\\ \left\lfloor\lim_{n\to\infty}\sum_{k=1}^n\frac9{10^k}\right\rfloor=1\\$


$s_n={\bigl(1+\sqrt{2}\bigr)^{n+1} +\bigl(1-\sqrt{2}\bigr)^{n+1}\over2}\qquad(n\geq0)\ .\tag{3}$


$P(m) \geq P(n)$


$\tau^+$


$\int_{0}^{\infty} \dfrac{\sin(x)}{x}  = \dfrac{\pi}{2}$


$\log_a(A\cdot B)=\log_aA+\log_aB$


$y\in V\subseteq \overline{V}\subseteq U$


$Q \equiv R \equiv S$


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


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


$bsa+brp=b$


$a_k = \frac{1}{2^{n-2}}$


$\displaystyle\sum_{n \le x}\cfrac{1}{n} = \log x + 1 -\int_1^\infty\cfrac{t-[t]}{t^2}dt + O(\cfrac{1}{x})$


$\int_0^{2\pi} f(x)\sin nx\,dx = \int_0^{2\pi} f(x)\cos nx\,dx = 0$


$P( \text{a child is a girl})= P( \text{a child is a boy})= \frac12$


$\ \color{#0a0}{tm\equiv 0}\,\Rightarrow\, 1\,\equiv\, sa+\color{#0a0}{tm}\,\equiv\, sa +\color{#0a0}0\equiv\, sa\ $


$\overline{d}(x, z) \leq \overline{d}(x, y) + \overline{d}(y, z)$


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


$ \frac{\|\alpha(v)\|^2}{\|v\|^2} = \frac{\sum_n s_n^2 |\langle v,e_n\rangle|^2 }{\sum_n  |\langle v,e_n\rangle|^2 } \ge \min_n s_n^2. $


$x[n] = \delta[n+1] + \delta[n-1]$


$P[X_{16}=2|X_0=0] $


$x\to-\infty, \cot y=\infty\implies y\to 0$


$P_{x}[X_{0}=x]=1\quad({}^{\forall} x\in E)$


$1-rsr=(rs)^{-1}(rs-r)=-(rs)^{-1}(1-s)r$


$J_n = \displaystyle \int_0^{\pi/2} \dfrac{\sin((2n+1)x)}{\sin(x)}dx$


$\alpha\cdot\sup_{\gamma<\beta}\{\gamma\}=\sup_{\gamma<\beta}\{\alpha\cdot\gamma\}$


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


$s_n(x) = \sum_j c_j \chi_{K_j}(x)$


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


$\dfrac1x + \dfrac1y= \dfrac1a$


$P(X_1=0\vert X_0=2)=0=P(X_1=4\vert X_0=2)$


$bRx \lor xRb$


${\int_{0}^{1}\frac{(1-3x^2)-\pi x(1-x^2)\cot(\pi x)}{2x^2(1-x^2)}\,dx}$


$\begin{vmatrix} 1 & 1 & 1\\ x & y & z\\ x^2 & y^2 & z^2\\ \end{vmatrix}$


$4^{2x}-1= 8^x+3.\;$


$\zeta(s)=\frac{1}{s-1}\sum_{n=1}^{\infty}\left(\frac{n}{(n+1)^s}-\frac{n-s}{n^s}\right)\forall \Re(s)>0$


$\int_0^\pi \sin(t)\cos(t)dt=0=\int_0^0 xdx$


$T_n={n(n+1)\over 2}.$


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


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


$|\cos \varphi + \ii\sin\varphi| = \sqrt{\cos^2 \varphi + \sin^2 \varphi} = 1$


$D(x,B)\leqslant d(x,y)+d(y,b)$


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


$\lim_{n \to \infty} \left \lfloor \frac{an+b}{cn+d}\right\rfloor  = \left \lfloor{a\over c} \right \rfloor $


$f^k(B)= A^k B$


$     \begin{bmatrix}     1 & x & x^2 \\     1 & y & y^2 \\     1 & z & z^2 \\     \end{bmatrix} $


$\text{lcm}(m, \frac{n}{m}) = \text{lcm}(p, p) = p$


$\lim_{n\to \infty} \mathbb P(A_n)=0$


$\ f(n) = n^2-3n+1$


$I_0\subset I_1\subset\dots \subset I_{n}\subset I_{n+1}\subset \dots,$


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


$f_X(x) = \theta\kappa^\theta/x^{\theta + 1},$


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


$f\left(f(x+1) + f\left( x + f(x)\right)\right) = x+2$


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


$\lim_{x\to a} f(x) =L_1 \land \lim_{x\to b} g(x) =L_2 \implies \lim_{x\to a} f(x) \pm \lim_{x\to b} g(x) =L_1 \pm L_2$


$ \begin{aligned} \cos(e^u)&=(D+a)^{-1}\bigl[-e^u\sin(e^u)+a\cos(e^u)\bigr]\qquad(*)\\ &=-e^u(D+a+1)^{-1}\sin(e^u)+a(D+a)^{-1}\cos(e^u) \end{aligned} $


$I=\int_{0}^{\frac{\pi}{2}}\frac{x dx}{\sin x+\cos x}=\int_{0}^{\frac{\pi}{2}}\frac{(\frac{\pi}{2}-x) dx}{\sin(\frac{\pi}{2}-x)+\cos (\frac{\pi}{2}-x)} \implies 2I=\frac{\pi}{2}\int_{0}^{\frac{\pi}{2}}\frac{ dx}{\sin x+\cos x}$


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


$ \exists \varepsilon > 0, \forall \delta > 0, \exists r_\delta\in(x-\delta,x+\delta)\text{ s.t. } |f(x)-f(r_\delta)|=f(x) > \varepsilon$


$\qquad A_1\times\cdots\times A_n$


$d(p,a) \leq d(p,q)+d(q,a),$


$f(n)=\binom{n}{a}$


$P(0)\implies P(1)$


$\{e_i\}_{1\leq i\leq n}$


$f(a+b) = f(a) + f(b).$


$\mathbb EY_1^k=\frac12(\mathbb EX_1^k+\mathbb EX_2^k)$


$F(x) = \sum_{i = 1}^{n}a_{i}\chi_{A_{i}}$


$=\int \frac{1+\tan^2 x-2\tan x}{1-\tan^2 x}dx=\int \frac{1+\tan^2 x}{1-\tan^2 x}dx-\int\frac{2\tan x}{1-\tan^2 x}dx$


$f_{z_0}^{-1}(re^{i\theta})=\sqrt[k]{r}e^{\frac{i\theta}k},$


$\mathbb P[X< x | D_1] = 1$


$aX + bY = c$


$\gamma(A) \le \gamma(A+K) \le \gamma(A) + \gamma(K) = \gamma(A)$


$= P(X_1 = 7 | X_0 = 3) P(X_1 = 3 | X_0 = 1)$


$ (f*g_{\varepsilon})(x) = \int\limits_{-\infty}^{\infty}f(t)g_{\varepsilon}(x-t)dt = \int\limits_{x-\varepsilon}^{x+\varepsilon}f(t)g_{\varepsilon}(x-t)dt = g_{\varepsilon}(x) $


$\sin\theta = \dfrac{x}{1}$


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


$2I=\int_{-\pi}^\pi \frac{\sin nx}{\sin x}dx$


$f(y+f(x))=f(x)f(y)+f(f(x))+f(y)-xy.\tag{$*$}$


$\int { \frac{1}{x + x^\gamma} dx}=\int {\left( \frac{x^{\gamma-1}+1}{x + x^\gamma } - \frac{x^{\gamma-1}}{x + x^\gamma } \right) dx}=f_\gamma(x)$


$\exists d_i,  d_s (\alpha a_{ss}+(1-\alpha) b_{ss}) \geq \sum_{j\not=s } ( d_s (\alpha a_{sj}+(1-\alpha) b_{sj} )  $


$\sum_{k = 0}^{n - 1} \cos \left(\frac{2\pi k}{n}\right) = 0$


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


$\begin{align} \frac1{e}-\left (\frac{x}{1+x}\right )^x &= \frac1{e}-\left ( 1+\frac1{x}\right)^{-x}\\&=\frac1{e}-e^{-x \log{\left(1+\frac1{x}\right)}}\\&\sim \frac1{e}-e^{-x \left (\frac1{x}-\frac1{2 x^2}\right )}\\ &= \frac1{e}-\frac1{e} e^{\frac1{2 x}}\\ &=\frac1{e} \left (1-e^{\frac1{2 x}} \right ) \\ &\sim -\frac1{2 e x} \end{align}$


$\lim \inf a_n+\lim \inf b_n=\lim_{n\to \infty}A_n+\lim_{n\to \infty} B_n=\lim_{n\to \infty}(A_n+B_n)\leq$


$\left\lfloor \left\lfloor \frac{x}{j} \right\rfloor \frac{1}{k}\right\rfloor=\left\lfloor \frac{x}{jk}\right\rfloor$


$v = (x,y,z)=(1,1,1)$


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


$\sum_{n=1}^{\infty}\frac{x}{(x + n^2)^2}-\frac{1}{2} \sum_{n=1}^{\infty} \frac{1}{x + n^2}=\frac{\pi ^2 x \text{csch}^2\left(\pi  \sqrt{x}\right)-1}{4 x}$


$\sqrt[4]{c}\:\mathrm e^{\tfrac{i\pi}4}$


$sec\ \theta = \frac{x}{3}$


$\sum_{n \ge 1} \frac{1}{n^2} = \sum_{n \le x} \frac{1}{n^2}+ \mathcal{O}(1/x)$


$ax+by = A$


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


$f(x) = \frac{4^x}{4^x+2},$


$ \mathbb{E}[X^2 + \mathbb{E}[X]^2 - 2X\mathbb{E}[X]] = \mathbb{E}[X^2] + \mathbb{E}[X]^2 - 2\mathbb{E}[X]^2 = \mathbb{E}[X^2] - \mathbb{E}[X]^2, $


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


$n!=\sqrt{2\pi n}\left(\frac ne\right)^n e^{\frac{\theta_n}{12n}}$


$d(x,y)=\frac{x}{y}$


$[a]+[b]=[a+b]=[b+a]=[b]+[a].$


$\gcd(a,n)=\gcd(a,a+b)=\gcd(a,b)$


$h[n] * (\delta[n] - A \delta[n-1])  = \delta[n]$


$100 + 1000 * (e^{\frac{-100}{1000}} - e^0)$


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


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


$\sqrt 1+\sqrt 2 +\sqrt 3 +\cdots +\sqrt {2009}$


$S_1\subseteq S_2\subseteq S_3\subseteq S_4\subseteq S_5\subseteq\ldots$


$118!+2, 118!+3, 118!+4, \ldots, 118!+118$


$x = \frac{z + ab\bar{z} -(a+b)}{a-b}$


$ \int_{\pi}^\pi f(x)\,\cos nx\,dx=\int_{\pi}^\pi f(x)\,\sin nx\,dx=0, \quad\text{for every}\,\, n\in\mathbb N, $


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


$\sum\limits_{n=0}^N r^n=\frac{r^{N+1}-1}{r-1}$


$(x+y+z)^2+(x+y-z)^2+(x-y+z)^2+(x-y-z)^2\\+(-x+y+z)^2+(-x+y-z)^2+(-x-y+z)^2+(-x-y-z)^2\\=8(x^2+y^2+z^2)$


$\begin{cases}x=0|-1\\y=1|\ \ \ \ \ 2\end{cases}$


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


$f(n) = f(2)^{F(n-2)} * f(1)^{F(n-3)}$


$V_1 \subseteq \cdots \subseteq V_n$


$x\mathrel Ry\wedge y\mathrel Rx\implies x=y.$


$\gamma = \min \{k, \lfloor \tfrac{n}{\mu} \rfloor \}$


$\lim_{a \to -\infty} +\infty = +\infty$


$ \mathbb{E} \left[ \frac{| \langle \gamma, Y \rangle |}{\| \gamma\| \| Y \| } \right] \leq \frac{1}{n}$


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


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


$ A u = b \iff \\ [ A \mid b ] = \\ \left[ \begin{array}{rrr|r} 1 & -2 & 5 & 2 \\ 1 &  1 & 1 & k \\ 2 & -1 & 6 & k^2 \end{array} \right] \to \left[ \begin{array}{rrr|r} 1 & -2 & 5 & 2 \\ 0 &  3 & -4 & k-2 \\ 0 &  3 & -4 & k^2-4 \end{array} \right] \to \\ \left[ \begin{array}{rrr|r} 1 & -2 &  5 & 2 \\ 0 &  3 & -4 & k-2 \\ 0 &  0 &  0 & k^2-k-2 \end{array} \right] \to \left[ \begin{array}{rrr|r} 1 & -2 &  5 & 2 \\ 0 &  1 & -4/3 & (k-2)/3 \\ 0 &  0 &  0 & k^2-k-2 \end{array} \right] $


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


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


$ s_n(x)=\sum\limits_{j=1}^{m+1}a_j\chi_{E_j^n}(x). $


$1+\sum_{r=1}^{r=n} r\cdot r! = (n+1)!$


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


$=61+\frac1{28+\dfrac49}=61+\frac1{28+\dfrac1{\dfrac94}}=61+\frac1{28+\dfrac1{2+\dfrac14}}$


$|AB| = |CD| = |EF| = r = 1$


$ax+by-d$


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


$p^T\eta p = F(p)^T \eta F(p)$


$(n+1)!+2, (n+1)!+3,...(n+1)!+n+1 $


$\;\begin{cases} x\equiv 0\mod4,\\x\equiv 1\mod 25.\end{cases}$


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


$\int_0^{x/3} f(t)dt =\int_0^xf(t)dt$


$n > 1/|x|$


$ \sum_{k=1}^n \sqrt{n^4+k}\sin\frac{2\pi k}{n} = n^3 \frac{1}{n}\sum_{k=1}^n \sqrt{1+\frac{k}{n^4}}\sin\frac{2\pi k}{n} $


$a>2\implies\gcd(a+2,a)=\gcd(a,a+2-a)=\gcd(a,2)\leq2$


$\left| \int_a^b dX_t \right| = |X_b-X_a| \leq c \left( \sum_{i=0}^n |X_{t_{i+1}}-X_{t_i}|^{1/\alpha} \right)^{\alpha}$


$d(x,A):=inf $


$\sum_{n=0}^{\infty}\frac{n^2}{n!}=\sum_{n=0}^{\infty}\frac1{(n-1)!}+\sum_{n=0}^{\infty}\frac1{n!}=\sum_{n=1}^{\infty}\frac1{(n-1)!}+\sum_{n=0}^{\infty}\frac1{n!}$


$T_X\cong T_X^{1,0}$


$D_n=a(a-x)^{n-1}+x\sum_{k=0}^{n-2}(a-x)^{k}(a-y)^{n-1-k}$


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


$ \begin{matrix}\\    & 1 &  x& x^2 &   & w & x w &   & w^2 &  & \\ 1  & 1 &  .& 1   &   & . & .   &   & .   &  & \\ y  & . & -2&     &   & . &     & \\ y^2& 1 &   &     &   &   &     & \\ \\ z  & . & . &     &   & . &     & \\ y z& . &   &\\ \\ z^2& . &   &\\ \end{matrix} $


$\sum\limits_{n=2}^\infty\frac{\cos(n\pi/3)}{n}$


$\sum_{k=m}^{k=n}\frac{1}{(1+r)^k}=\frac{(r+1)^{1-m}-(r+1)^{-n}}{r}$


$ =\frac{1}{r(\gamma)}[\dot{r}{(\gamma)\frac {\partial}{\partial \gamma} +r(\gamma) \frac{\partial^2}{\partial \gamma^2}+1/r(\gamma)\frac {\partial ^2}{\partial \theta^2}}]$


$P(m)\ne P(n)$


$1  / \vert x \vert \le 1/2$


$A = \n_{\dot \gamma} \dot \gamma$


$\sum_{n=1}^{\infty}\frac{\zeta(2n)}{4^{n-1}} =\sum_{n=1}^{\infty}\frac{\zeta(2n)}{2^{2n-2}}= 2\int_{0}^{\infty}\frac{\sinh(x/2)}{e^x-1}\,dx =\int_{0}^{\infty}e^{-x/2}\,dx = 2$


$\lim_{x \rightarrow c}{\vert f(x) \vert} = \lim_{x \rightarrow c^-}{\vert f(x) \vert} = \lim_{x \rightarrow c^+}{\vert f(x) \vert} = 1$


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


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


$ \begin{align} \sum_{m=1}^\infty\sum_{n=1}^\infty\frac1{m^2+n^2} &\ge\sum_{m=1}^\infty\sum_{n=1}^\infty\frac1{m^2+n^2+2mn+m+n}\\ &=\sum_{m=1}^\infty\sum_{n=1}^\infty\left(\frac1{m+n}-\frac1{m+n+1}\right)\\ &=\sum_{m=1}^\infty\frac1{m+1}\\[6pt] &=\infty \end{align} $


$a_n=\int_{-\pi}^{\pi}(1/\pi)|\sin(x)|\cos(nx)dx$


$ \|A\|_2 = \sqrt{\lambda_\max(A^t A)} = \sqrt{\lambda_\max(A^2)} = \sqrt{\lambda_\max(A)^2} = \lambda_\max(A) $


$ \begin{align} \lim_{n\to\infty}\frac n{2^n}\sum_{k=1}^n\frac1k &\le\lim_{n\to\infty}\frac{n^2}{2^n}\\ &=0 \end{align} $


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


$Y_a = \frac{1+aX}{a-X}$


$ \mathrm{dist}(x_0,M)=\frac{|F(x_0)|}{\Vert F\Vert} $


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


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


$F=P(H)\oplus \text{Coker P}$


$|(x-a)-(y-b)|+|(x-a)+(y-b)|=A$


$ |(1-\lambda)X_i+\lambda Y_i|^p\leq (1-\lambda)|X_i|^p+\lambda |Y_i|^p. $


$\{1,x,y,x^2,xy,y^2,x^3,x^2y,\cdots\}\subset\Lambda(\mathbb R^\infty)$


$\displaystyle\left(\frac{n}{2^{k/2}}\right)^k\le 1$


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


$y = f(a) + f(b)$


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


$\gamma^+$


$\det\begin{pmatrix}A & -B \\ B & A \end{pmatrix}=|\det(A+iB)|^{2}.$


$m={\rm ord}(a), n={\rm ord}(b)$


$\left|\frac{f(a)-f(b)}{1-\overline{f(b)}f(a)}\right|\le\left |\frac{a-b}{1-\bar{b}a}\right|$


$A_1\times\ldots\times A_n \cong A_1\oplus\ldots\oplus A_n$


$\sum_{n\ge 1} \frac{1}{n^2}$


$\mathbb{E}(X_{t}-X_{s})^{2}\leq c \vert t-s\vert^{2}$


$I_n < \pi < C_n$


$\eta(s) = \frac1{\Gamma(s)} \int_0^{\infty}\frac{x^{s-1}}{e^x+1}\mathrm dx$


$\bigg\lfloor\frac{\lfloor \frac{x}{m}\rfloor}{n}\bigg\rfloor=\bigg\lfloor \frac{x}{mn}\bigg\rfloor?$


$f_X(x) = { 6x(L-x) \over L^3}, where: 0 \le x \le L $


$\{ e_{1},e_{2},e_{3},\cdots\}$


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


$d(x',A)-d(x,A)\leqslant d(x',x)$


$\cos(\alpha) = \frac{|\langle x,y\rangle|}{||x|| ||y|| } = 1$


$x\in V\subseteq\cl V\subseteq U$


$\int_0^{\pi}{\sin x\over x}$


$f'(x) = \frac{e^x(x-n)}{x^{n+1}}>\frac{e^{n+1}}{n^{n+1}}, \quad (x > n+1)$


$\begin{cases}x\equiv 6 \pmod{13}\\ x\equiv 2 \pmod{17} \end{cases}\quad \begin{cases}x\equiv 6 \pmod{13}\\ x\equiv -2 \pmod{17} \end{cases}\\ \begin{cases}x\equiv -6 \pmod{13}\\ x\equiv 2 \pmod{17} \end{cases}\quad \begin{cases}x\equiv -6 \pmod{13}\\ x\equiv -2 \pmod{17} \end{cases}$


$0\le \sum_{n=1}^N \frac{1}{n^2+3n+2}\le \sum_{n=1}^N \frac{1}{n^2}\le \sum_{n=1}^\infty \frac{1}{n^2} \tag 1$


$   \left(  \begin{array}{cc}   a  &  b   \\   -b  &  a      \end{array}    \right)  , $


$\require{cancel} \frac{r (1-r)\cancel{(1+r)}}{\cancel{(1+r)}}$


$U\oplus T$


$[a+(n-1)\delta, a+n\delta]$


$ \binom{n}{\gamma n} \sim \left( 2 \pi \gamma (1 - \gamma) n \right)^{- 1/2} \left( \gamma^{\gamma} (1 - \gamma)^{1 - \gamma} \right)^{-n}.$


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


$m'(m'(-,-),-)=m'(-,m'(-,-))$


$f(a+b)=f(a)+f(b), f(ab)=f(a)f(b)$


$\gamma \wedge \gamma = (-1)^{n^2} \gamma \wedge \gamma$


$\lim\limits_{x\to a^-}k(x)=\lim\limits_{x\to a^-}g(x)=g(a)=k(a)=f(a)=\lim\limits_{x\to a^+}f(x)=\lim\limits_{x\to a^+}k(x)$


$\lim_n  \frac{6}{\pi^2}\sum_{k=1}^n\frac{1}{k^2} =1.$


$P(x) = 1/6$


$|g(z)| \leq C(|z|^{3/2}|z-1|^2+|z-1|^{1/2}), z\in\mathbb{C}-\{1\}$


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


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


$ \int_0^\pi v(t)\,dt = \int_0^\pi \cos (2t) \, dt = 0, $


$G=\left(\begin{array}{cc|cc} 1 & 0 &1 &1\\ 0 & 1 &1 &-1 \end{array}\right)$


$\int \frac{1}{(x^2+bx+c)^n}\, dx$


$A \times \ldots \times A$


$L_1 = \begin{cases} x= 1+as \\ y=-2 +bs \\ z=-1+cs \end{cases}$


$q(n) = \left(\frac{n}{10}\right)^{10}$


$ 1+2r+3r^2+...+nr^{n-1}=\sum_{k=1}^{n-1}kr^k=\frac{1-r^{n+1}}{(1-r)^2}+\frac{-(n+1)r^{n}}{1-r}, \quad |r|<1. \tag2 $


$ \sum_{n=1}^{+\infty} 1 = +\infty $


$\begin{cases} x=2 \\ y=0 \\ z=-2 \\ \end{cases}$


$\mathbb R[x]/(x^2)$


$\frac{\sum_{n=0}^{N-1}c_n^N}{c_N^N}=\frac{1-c_N^N}{c_N^N}=\frac{1-\dfrac1{2^N}}{\dfrac1{2^N}}=2^N-1\to\infty.$


$(X,\mathcal{A})$


$ 1/|x|^{k}$


$\int_0^\infty \frac{x^{2n - 1}}{(x^2 + 1)^{n + 3}}\,dx$


$\|A\|_2=\sqrt{\max_n{\lambda_n(AA^T)}}$


$f(n)={n\choose k}$


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


$\mathbb{Z}_{2}[X] / (x^4+1)$


$\begin{bmatrix} 3 & -5\\  5 & 3  \end{bmatrix}x$


$f(x)^2+f(y)^2=f(x+y)\Big(f(x)+f(y)+a\Big).\tag{1}$


$A_1 \times A_2 \times \ldots \times A_n$


$\overline{\sqrt{z}}= \begin{cases} \sqrt{|z|}e^{\frac{-i\phi}{2}} \\ \sqrt{|z|}e^{\frac{-i\phi}{2 + \pi}} \end{cases}$


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


$\int \frac1{(x^2+1)^{3/2}} dx $


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


$\frac{a + bc}{b}$


$\Gamma^+$


$p \ | \ n_1n_2$


$\Rightarrow \forall x\in X \exists y=x\in X :(xRy \vee yRx)$


$\frac{\mathbb{Z}_4 [x]}{(x^2 + 1)}$


$a_3(p^k) = \binom{k}{2}$


$\frac{1-2\sin\left(α\right)\cos\left(α\right)}{1-2\sin^2\left(α\right)}$


$\int_{-\pi}^{\pi}\frac {\sin nx}{(1+2^x)\sin x}dx \:\:\:  n \in  \mathbb{N}$


$\frac{8}{a}+\frac{2}{b} = 1.$


$\sum_{n \leq x} \frac{1}{n} \lfloor \frac{x}{n} \rfloor \ge \sum_{n \leq x} \frac{1}{n}  (\frac{x}{n}-1) = x\sum_{n \leq x} \frac{1}{n^2} -\sum_{n \leq x} \frac{1}{n}  = x\sum_{n \leq x} \frac{1}{n^2} -\ln(x)+O(1) $


$C^{î,*}(D)=C^{î,*}(D_0) \oplus C^{î-1,*}(D_{1})$


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


$ \gcd(a,b-a)=\gcd(-a,b-a)\leq \gcd(-a,b)=\gcd(a,b)$


$f_T(t) = \frac {1}{m} e^{\frac{-x}{m}}$


$\begin{align} a^n \equiv (a^s)^k \equiv a^k\pmod{s} \tag{2}\\ b^n\equiv b^k\pmod{s} \tag{3} \end{align}$


$f(t)=\dfrac{t}{t^{11} + 1}\tag{1a}$


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


$ds(t) = \sqrt{\left( \frac{d \, x(t)}{d \, t} \right)^2 + \left ( \frac{ d \, y(t) }{d \, t} \right)^2} $


$f_2(x) = x^2+x+2$


$\mathfrak{a}^+$


$\forall \epsilon \gt 0, \exists \delta \gt 0 \; \bigl| \; 0 \lt | x - a | \lt \delta \implies |\sum_{k=1}^n f_k(x) - \sum _{k=1}^n L_k| \lt \epsilon$


$ f(x) = \frac{2x}{\alpha^2}$


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


$ f(a^{-k}) M = f(b) \\ M = f(b) f(a^{k}) $


$g(f(x))=\frac{\sqrt{1-\sin{x}}}{1+\sqrt{1-\sin{x}}}$


$=\frac{1}{r(\gamma)}[\dot{r}{(\gamma)\frac {\partial}{\partial \gamma} +r(\gamma) \frac{\partial^2}{\partial \gamma^2}+r(\gamma)^3 \frac {\partial ^2}{\partial \theta^2}}] $


$A=\{\langle x,y\rangle\in\mathbb R^2\mid y>x+1\}$


$(P r)/(1-(1+r)^{-N})$


$C_n = \frac{1}{2 \pi} \int_{-\pi}^\pi (\cos(2x) +\sin(x)) e^{-inx} \, dx$


$2|\alpha|=|\beta-\gamma|\leq|\beta|+|\gamma|\leq 2|\alpha|$


$x = y = z = 2\lambda$


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


$\ln(\lfloor{x}\rfloor!) = \sqrt{2\pi{x}}(\frac{x}{e})^x + R_k(x)$


$\lim\inf \mu(A_n)$


$ b_{n+1}-b_n=c_n(\frac{c_n}{n+c_n}-\frac{1}{n+c_n}). $


$\color {green} {\text{Numbers}\,\, 20!+2,20!+3,\cdots ,20!+21 \,\,\text{will do the trick. The following result by Euclid has been known for more than 2000 years.}}$


$\lbrace\, a,a+b,a-b,a+2b,a-2b,\dots\,\rbrace$


$V\subseteq\ker(P[A])$


$\sum_{n=-\infty}^{\infty} e^{-(u+n a)^2} = \frac{\sqrt{\pi}}{a} \left [1+2 \sum_{n=1}^{\infty} e^{-n^2 \pi^2/a^2} \cos{\left (2 \pi n \frac{u}{a} \right )} \right ]$


$\left|\frac{\sin(x)}{|x|^{3/2}}\right|^2 \sim \frac{1}{|x|}$


$P(B) = 1/6$


$f(x) = \frac{(-3)^{x-1}}{4^x}$


$e^{-x} \geqslant \left(1 - \frac{x}{n}\right)^{n}.$


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


$f''(x)=(x-a)^{2n-1}(x-b)^{2m}[2n(x-b)+(2m+1)(x-a)]$


$x^2(z-y)+y^2(x-z)+z^2(y-x)=x^2(z-y)+x(y^2-z^2)+z^2y-y^2z=(z-y)(x^2-x(y+z)+zy)=(z-y)(x-y)(x-z)$


$\lim \inf \mu(A_{n}) >0$


$\alpha < \alpha^+$


$\{(x,y,z,w,m)\in \mathbb{R}^5 :x=y,z=w=m\}=\{(x,x,z,z,z): x,z\in \mathbb{R}\}$


$     \begin{vmatrix}     1 & x & x^2 \\     1 & y & y^2 \\     1 & z & z^2 \\     \end{vmatrix}             +            \begin{vmatrix}     a & b & c \\     d & e & f \\     g & h & i \\     \end{vmatrix}. $


$(a, a+k, a+2k...)$


$M_{n}= k(b-a)^{n-1}$


$f(x,y)=\frac{\langle x,y \rangle}{\|x\|\|y\|}.$


$S_{2k}$=$\ \lim_{k\to \infty}$


$bx-ay =d$


$C_R \cup C_R^+ \cup C_R^-$


$2\pi \mathrm i\;\delta'(\cos(\varphi))$


$ \begin{cases} x=2-t\\ y=t\\ z=3 \end{cases} $


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


$P(x) = \frac{2a}\pi \arcsin\big(\sin\big(\frac{2 \pi}p x\big) \big)$


$\frac{\sin \theta}{4\theta+\tan\theta} = \frac{\sin \theta}{4\theta+\frac{\sin \theta}{\cos\theta}}$


$x_2,x_3 \in\mathbb N$


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


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


$ \gcd(a,2b)=\gcd(a,2)\gcd(a,b)=\gcd(a,2). $


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


$ \quad \quad \quad \quad \implies s \equiv ar \equiv 1 \pmod p \implies s^p \equiv (ar)^p \equiv 1 \pmod {p^2}$


$(A_1\ldots A_n)$


$\ Cov(Y,Y)+Cov(Y,e^X)$


$\lim_{x\to 0}{\left[ 100\frac{\sin^{-1}(x)}{x}\right]} = 100$


$\int_0^{\pi} \sin(2t) \phi(t) dt = k \int_0^{\pi} \sin(2t) \sin(t)dt = 0$


$\lim_{n\rightarrow\infty}\sum_{i=1}^{n}\int_{I} \chi_{i}=\lim_{n\rightarrow\infty}\mu(T_{n}).$


$(x/y)^{x-y}=(x^{x-y}/y^{x-y})=(x^x/x^y)*(y^y/y^x)=(x^x*y^y)/(x^y*y^x/)$


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


$i : N \to \mathbb{R}^{n+1} \times \mathbb{R}^{n+1}$


$y,y',y'',y''',y''''$


$a+b^2-c=0 \implies  3^x+4^x=5^x$


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


$\lim_{n \to \infty} \frac{\sqrt{2^{n+1}+1}-\sqrt{2^{n+1}}}{\sqrt{2^n+1}-\sqrt{2^n}}=\lim_{n \to \infty} \frac{1}{(\sqrt{2^{n+1}+1}+\sqrt{2^{n+1}})(\sqrt{2^n+1}-\sqrt{2^n})}<1$


$\phi^+$


$ A_n = \frac{n^2(n+1)^2}4 $


$=\dfrac1{(\sqrt1+1)(\sqrt{1+\sqrt1}+\sqrt2)}$


$X_0=\Bbb E(X|\cal F_0)$


$RS = \{rs | r \in R, s \in S\}$


$\begin{align}\sum_{n=1}^{\infty} \frac{1}{n^2+a^2} &= \frac{1}{n^2} \frac{1}{1+ \frac{a^2}{n^2}}\\ &= \sum_{n=1}^{\infty} \frac{1}{n^2} \sum_{k=0}^{\infty} (-1)^k \left (\frac{a^2}{n^2}\right )^{k} \\ &=\sum_{k=0}^{\infty} (-1)^k a^{2 k}\sum_{n=1}^{\infty} \frac{1}{n^{2 k+2}} \\ &=\sum_{k=0}^{\infty} (-1)^k a^{2 k} \zeta(2 k+2)\\ &= \frac{1}{2 a^2}\sum_{k=1}^{\infty} \frac{B_{2 k} (2 \pi a)^{2 k}}{(2 k)!} \\ &= \frac{1}{2 a^2} ( \pi a \coth{\pi a} - 1)\\  \end{align}$


$\lim\limits_{n\to\infty}\sum\limits^n_{i=0}F_{2i}-\phi F_{2i-1}$


$\dots=[-2]_{\sim}=[-1]_{\sim}=[0]_{\sim}=[1]_{\sim}=[2]_{\sim}=\,\dots$


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


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


$ d(x,z) \le d(x,y) + d(y,z) \quad \Longrightarrow \quad f(x) \le d(x,y) + f(y) $


$F(x)= \int_a^x f(t)\, dt,\,\,  x\in [a,b].$


$\{\omega\in\Omega| \lim_{n\rightarrow\infty} X_n(\omega)$


$9^x + 6^x = 2× 4^x$


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


$\int  \frac { \tan { x }  }{ \left( \sin { x }  \right) ^{ 2 }+2\left( \cos { x }  \right) ^{ 2 } } dx=\int  \frac { \tan { x }  }{ \cos ^{ 2 }{ x } \left( 2+\tan ^{ 2 }{ x }  \right)  } dx=\int  \frac { \tan { x }  }{ \left( 2+\tan ^{ 2 }{ x }  \right)  } d\tan { x } \\ $


$f_X(x)=\frac{1}{\theta}1_{[-\theta, \theta]}(x)$


$A(V) \subseteq V$


$\displaystyle \sum\limits_{n=0}^{\infty} (-1)^{n}e^{-\alpha n^2} = \frac{1}{2} + \sqrt{\frac{\pi}{\alpha}}\sum\limits_{n=0}^{\infty} e^{-\frac{(2n+1)^2\pi^2}{4\alpha}}$


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


$ \left(\frac{1}{2}\right) > \left(\frac{1}{3}\right)^{3/2} $


$\gamma,\gamma'<\beta$


$ \Sigma = diag(\sigma_1, \sigma_2, \sigma_3,....,\sigma_i)$


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


$ \sum_{d|N} \frac1{d^2}\leq\sum_{n=1}^N\frac1{n^2}\leq 2 $


$P(S\cap E) = P(S| E)P(E)$


$ U_n=\frac{(1+\rho)^n-(1+\rho)^N}{1-(1+\rho)^N}$


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


$\mathcal F=\{\{\{1\}\}\}$


$H\ge \mathbb Z+(a/2^\gamma)\mathbb Z=\frac{(2^\gamma+a)\mathbb Z}{2^\gamma}=\frac{gcd(2^\gamma,a)\mathbb Z}{2^\gamma}$


$|ab| = |ba|$


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


$d(x,z)\le d(x,y)+d(y,z)<2/m_x=1/n_x,$


$\ds{{\rm g}\pars{t}      =\int_{\gamma - \infty\ic}^{\gamma + \infty\ic}       {\Gamma\pars{a + 1} \over s^{a + 1}}\,\expo{st}\,{\dd s \over 2\pi\ic}\,,       \qquad\gamma > 0}$


$\rm\ \lfloor \lfloor a/b \rfloor / c\rfloor =  \lfloor a/(bc)\rfloor. $


$\left(1+\frac xn\right)^n \geq 2^{x}$


$\langle \alpha ^-,\alpha^+\rangle$


$|\triangle ABC| = rs$


$d(x,A)=\inf _{y\in A}d(x,y)$


$E(X\mid\mathcal F_0)$


$\ f(x) = x^2 + x - 2\ $


$S/\equiv$


$t\gamma - \lfloor t\gamma \rfloor$


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


$p_1 p_2 + 1$


$                      p_j(M)=p_j(M)^{2}. $


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


$\Bbb P[X_{n+1}=j\,|\,\mathcal F_n] = p(n,i,j)$


$w={\normalsize e}^{\Large{\frac{2\pi i}{N}}}$


$(\frac{1}{3})^k * (\frac{2}{3})^{n-k}$


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


${\bf \hat{x}} = (\hat{x}, \hat{y})$


$\lim_{n\to\infty}f_n(0)=\lim_{n\to\infty} n^{1/p}= \infty.$


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


$d_1(x,z)\leqslant d_1(x,y)+d_1(y,z)$


$\sum_{m=-\infty}^\infty e^{-a \pi (x-m)^2} = f(x) = g(x) = \sum_{n=-\infty}^\infty a^{-1/2}e^{-\pi n^2 /a} e^{2 i \pi n x}$


$\sup_{n\geq n_0}|E|X_n|^{\alpha}-E|X|^{\alpha}|\leq 3\varepsilon$


$f\left( f(x)^2+f(y) \right)=xf(x)+y$


$\sum_{t=1}^T (1+r)^{T-t+1} = (1+r)^3 + (1+r)^2 + (1+r)^1.$


$\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n^{2}}=\sum_{n ~\text{odd}}\frac{1}{n^{2}} - \sum_{n ~\text{even}}\frac{1}{n^{2}}.$


$\det(\psi_\gamma(H))=\det(\psi_\beta(H))\det(Q)^2.$


$P[X_{0}=1|X_{3}=0]*P[X_{3}=0] $


$\ln (n+\gamma) + \gamma
 = \ln \ n + ln(1+\gamma/n)+\gamma
 = \ln \ n + \gamma + \gamma/n + O(1/n^2)
 $


$  \|A\|  = \sqrt{\lambda_{\mathrm{max}}(A^T A)}, $


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


$E=\{e_1, e_2=ie_1, e_3=ie_1, e_4=i e_2 \}$


$|AB| = a$


$p_X(x) = \dfrac {2^{x}}{3^{x+1}}$


$\sum_{k=1}^N\frac{1}{2^k} = 1 - \frac{1}{2^N}$


$ \left( \frac{1}{2} \right)^{1/2e} $


$\sum_{n=1}^{\infty} \frac1{\left (n+\frac{3}{2} \right )^2-\frac{5}{4}} = \frac12 \sum_{n=-\infty}^{\infty} \frac1{\left (n+\frac{3}{2} \right )^2-\frac{5}{4}} - 1 +1$


$\epsilon \Delta x = \Delta y - f'(a)\Delta x$


$\lim_{x \rightarrow k} f(x) = \lim_{x \rightarrow k^+} f(x) = \lim_{x \rightarrow k^-} f(x)$


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


$\lim_{N\to \infty}\sum_{n=1}^{N}\frac{1}{n^2} \to 0$


$\theta(x)\ :=\ \sum_{n=-\infty}^\infty e^{-n^2\pi x}\ .$


$|y| \le e^{-1/|x|}$


$\sec\theta= \frac x2$