$R = \lbrace (x,y): a < x < b, c < y < d\rbrace$


$ \{v + \alpha(v) \mid v \in T^{0,1} M\} \subset T_{\mathbb{C}} M $


$B= \{b_1, b_2, b_3,.....\}$


$\forall \in \Bbb N^+$


$7\mid n^7-n$


$\{\emptyset, a, bc, abc\}$


$[a]+[b]=[a+b]=[b+a]=[b]+[a]$


$x^\frac{1}{(\log_a x)} = a^{log_a(x^\frac{1}{(\log_a x)})}$


$x^{m} \in <x^{d}>$


$P(k): 0^3+1^3+⋯+k^3=(k(k+1)/2)^2$


$(1+x/n)^{n} \leq e^{x}$


$f'(h)= \frac{d}{dh}\langle \gamma(s+h) - \gamma(s) , \gamma(s+h) - \gamma(s) \rangle = \frac{d}{dh} [\langle \gamma(s+h), \gamma(s+h) \rangle + \langle \gamma(s), \gamma(s) \rangle -2 \langle \gamma(s+h), \gamma(s) \rangle ] = 2[\langle \frac{d}{dh}\gamma(s+h), \gamma(s+h) \rangle - \langle \frac{d}{dh}\gamma(s+h), \gamma(s) \rangle] = 2[\langle \frac{d}{dh}\gamma(s+h), \gamma(s+h) - \gamma(s) \rangle ]$


$\displaystyle\implies \sum_{0\le k\le r-1}\cos\frac{2k\pi}r=0$


$\displaystyle \int_0^\infty f(x)dx=+\infty$


$C_n := \int_{-\pi}^\pi \lvert D_n(x)\rvert\,dx = \frac{2}{\pi}\int_0^{\pi/2} \frac{\lvert\sin ((2n+1)y)\rvert}{\sin y}\,dy.$


$2^{n\gamma}\le t^{-\gamma} \le 2^{(n+1)\gamma}$


$x^2 + x = y^2 + y = z^2 + z = x + y  + z$


$x^3 - x^2, x - x^2$


$P(A\mid B)\leq \frac{a+b-1}{b}$


$\mathscr{E} =\{ \{1\}\}$


$A = \left(\begin{array}{crc}  1 &  2 &  -1 & -1\\ 1 & 0 & 1 &  1\\  2 &  -4 &  6 & 2\\  \end{array}\right)$


$k|P = A \left(\frac{(1+i)^n-1}{i(1+i)^n}\right) \left(\frac{1}{(1+i)^k} \right)$


$ \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} $


$\mid \frac{\mid y-p \mid}{\mid y-p \mid} \cdot \frac{1}{\mid x-p \mid} - \frac{1}{\mid y-p \mid} \cdot \frac{\mid x-p \mid}{\mid x-p \mid} \mid < \epsilon$


$|f(z_n)|\to |f(z)|$


$\left | AB \right |=c$


$f_2(n) = n^2$


$ \frac{\lvert \langle x, y \rangle \rvert}{\lVert y \rVert} =\lVert x \rVert ~\overset{1/\lVert y \rVert}{\implies}~\frac{\lvert \langle x, y \rangle \rvert}{\lVert y \rVert^2} =\frac{\lVert x \rVert}{\lVert y \rVert} =\lvert a \rvert.$


$a,a',b,b',c,c',d,d'\in\mathbb R$


$\mathscr{L}(\zeta,\,\mathscr{L}(\gamma,\,x))(t)=\mathscr{L}(\zeta\,\gamma,\,x)(t);\quad\zeta,\,\gamma \in \mathrm{SL}(2,\,\mathbb{R})$


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


$Q_n(x)=\frac{(1-x)^{n+1}-(1+x)^{n+1}+2x(n+1)(1+x)^{n+1}}{2x^2},$


$\,r,s\mid k\,\Rightarrow\,rs\mid k\,$


$J_1=\int_{-\pi/2}^{\pi/2} \frac{\gamma (1+\sin^2 x)  \cos^4 x~dx}{\left(\cos^4 x+(\gamma~ cos^2 x-\sin x)^2 \right)^{7/4}}$


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


$\tan \theta = \frac{x}{1},$


$\int_0^1f(t)\ dt=0$


$f_{3}(n) = n + 1$


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


$x^2,x^3,x^4\in [0,1]$


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


$x^2+x, x\in F$


$ \langle x - \gamma y, x - \gamma y\rangle = \langle x, y\rangle - \gamma\langle y,x\rangle - \bar\gamma\langle x, y\rangle + |\gamma|^2\langle y, y\rangle $


$\lim_{x\to a}f_1(x)=\lim_{x\to a}f_2(x)=...=\lim_{x\to a}f_n(x)=N$


$A=\frac {\Delta f}{\Delta x} .$


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


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


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


$E(\hat{\theta^2}) = E\left(\frac{1}{n^2}\sum_{i=1}^n X_i^2\right) =E(X^2) = \int_0^\infty f_\theta dx = ... =  \frac{2\theta^2}{n^2} $


$f_X(x|\theta) = \frac{1}{\theta}\ \ \ \text{for $x\in [0,\theta]$}$


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


$ \theta = \aleph_{\theta} = \aleph_{\aleph_{\aleph_{\aleph_{\ddots}}}} = \Theta = \kappa = \aleph_\kappa = \beth_\kappa = \beth_{\beth_{\beth_{\beth_{\ddots}}}} $


$ \tan \theta_1 = \frac{l}{2L} = \frac{\tan \theta}{2} $


$x=\langle D|M^{\perp}\rangle $


$q^{n+1} - q^n$


$n \geq N \implies\ |\frac{x_1 + x_2 + ... + x_n}{n} - L|< \epsilon \quad (♫)$


$[X,Y] = XY - YX \in A$


$ \dfrac{f(x)-f(a)}{x-a} = \dfrac{1}{x-a} \sum_{n=1}^{+\infty} c_n[na^{n-1}(x-a)+\frac{1}{2}n(n-1)(\xi_n)^{n-2}(x-a)^2] $


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


$V=\{\langle x, y\rangle \in A\times A \mid x\le y\}.$


$L(s,\chi)=\sum_{n=1}^\infty \frac{\chi(n)}{n^s}=\int_1^\infty \frac{S(x)}{x^{s+1}}\; dx \tag 2$


$\textbf{m}= \gamma^2 \textbf{p}+\gamma\ |\textbf{p}|\ \textbf{P} \nabla_n \gamma$


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


$=\begin{vmatrix} x &x^2  &1 \\  y &y^2  &1 \\  z &z^2  &1  \end{vmatrix}+\begin{vmatrix} x &x^2  &x^3 \\  y &y^2  &y^3 \\  z &z^2  &z^3  \end{vmatrix}$


$     (a, a+b \bmod N, a+2b \bmod N, a+3b \bmod N, \dots, a + (N-1)b \bmod N) $


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


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


$d(x,A) \leq d(x,a) \leq d(x,y)+d(y,a).$


$A=\begin{pmatrix}1 &3 & 9& 27\\3 & 3 & 9& 27\\ 9 & 9 & 9& 27 \\ 27 & 27 &27& 27\end{pmatrix}$


$\mathbb R^m - \mathbb R^n$


$ E_1\subset  E_2\subset\cdots \subset E_i\subset E_{i+1}\subset\cdots $


$\frac{d}{dx} ( (x-a)^{n+1} p^{(n)}(x) ) = (x-a)^n q^{(n+1)}$


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


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


$t = p \oplus q$


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


$k=\left\lfloor \frac{n}{p}\right\rfloor+\left\lfloor \frac{n}{p^2}\right\rfloor+\left\lfloor \frac{n}{p^3}\right\rfloor+\ldots$


$f(a) > k > f(b)$


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


$ f(x,y)=\frac{1}{2\pi\sqrt{1-p^2}} \mathrm e^{-\frac{(y-px)^2}{2(1-p^2)}-\frac{x^2}{2}}=\frac{1}{\sqrt{2\pi}\sqrt{1-p^2}}\exp\left(-\frac{(y-px)^2}{2(1-p^2)}\right)\times \frac{1}{\sqrt{2\pi}}\exp\left(-\frac{x^2}{2}\right) $


$f_X(x;\theta) = \frac{1}{2\theta}$


$p_1p_2\mid \chi(1)$


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


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


$|f(z)| \leq k|z|^2$


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


$[(f_1f_2)f_3]'=(f_1f_2)'f_3+(f_1f_2)f_3'$


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


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


$ ax+by=dn$


$\lim\limits_{t\to-\infty}x = 0^-$


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


$ |E(\gamma_k \mid \gamma_1,\dots,\gamma_{k-1})|\le 2^{-k+1} $


$E = (e_1,e_2....e_m)$


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


$(w,x,y,z) = (1,1,1,2)$


$f^{-1}(f(x))=x \to (f^{-1}(f(x)))' = f'(x)(f^{-1}(f(x)))^{(1)}=1 \to (f^{-1}(f(x)))^{(1)} = \frac{1}{f'(x)}$


$z_1=\sqrt 2e^{\frac {5i\pi}{12}}$


$(a + b, a − b) = 1$


$20/a + 12/b = 1$


$f(\#1, \#2, \dots)\&$


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


$\lim_{x\to 0}\frac{\sqrt{1-x^2}}{2}=\frac{1}{2}$


$\{ e_1,e_2,..,e_{\vert H\vert \vert G \vert}  \}$


$N(z),N(w)$


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


$P\oplus Q=F\cong F^\ast =P^\ast \oplus Q^\ast$


$ \begin{align} \prod_{k=1}^{n-1}\cos\left(\frac{2k\pi}n\right) &=\frac{\cos(n\pi/2)-(-1)^n}{2^{n-1}}\\[3pt] &=\frac{\cos(n\pi/2)-\cos(n\pi)}{2^{n-1}}\\[6pt] &=\frac{\sin(n\pi/4)\sin(3n\pi/4)}{2^{n-2}} \end{align} $


$(\det M)^2=\det \begin{pmatrix} A & -B \\ B & A \end{pmatrix}.$


$\frac{g}{||f||+||g||}$


$G=A_1 \times ...\times A_n$


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


$\frac{1}{x} + \frac{1}{y} = \frac{4}{3} \tag{2}$


$\forall y (xRy \lor yRx)$


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


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


$f_{Y}(y:θ) = \frac{2y}{θ^{2}}$


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


$S(x) = \frac{x^2}{x^4+1}$


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


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


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


$\int_0 ^\infty \frac{|sin(x)|}{(x^2+1)^n}$


$ \lim _{n \to \infty} \sum_{r=1}^n\frac1r $


$\int_0^1 x(t)f(t)\,\mathrm dt = 0$


$\Vert x_{n+1}-x_n\Vert$


$R(x-a):= -r(x-a)(x-a)$


$\{1,x, y, xy, x^2, y^2, x^2y, xy^2, \cdots\},$


$\lim_{n \to \infty} \dfrac{\sqrt 1 + \sqrt 2 + \dots + \sqrt{n}}{n\sqrt{n}} =\lim_{n\to \infty} \dfrac1{n}\sum^{n}_{k = 1} \sqrt{\dfrac k n} $


$\left(\begin{array}{cc|c} 1 & 0 & 35 \\ 0 & 1 & 99 \end{array}\right) \longrightarrow \left(\begin{array}{cc|c} 17 & -6 & 1 \\ -14 & 5 & 5 \end{array}\right)$


$\langle x^3-Q(x), x \rangle =0$


$y=\displaystyle{\frac{ax}{bx+c}}$


$f(x)=\frac{61}{50}~e^{\frac{3}{20}x}-\frac{11}{50}e^{-6x}-3$


$\left(1+\frac xn\right)^n=e^x\left(1+O\left(\frac{x^2}n\right)\right)$


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


$|2^A|\leq |B^A| \leq \mathcal{P}(A\times B) \leq \mathcal{P}(A^2) = \mathcal{P}(A) = |2^A|.$


$\det(A-I\gamma)=0 \Rightarrow \begin{cases}\gamma=4 \\ \gamma=2 \end{cases}$


$\lim\sup \mu(A_n)\le \mu(\lim\sup A_n)$


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


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


$D = \{(x, y):1 \le x^2 + y^2 \le 4, x \le y \le \sqrt3x, x \ge 0 \}$


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


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


$A_1 \times A_2 \times \ldots \times A_n \times \ldots$


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


$\Big\lfloor{x}/({y}\cdot{z})\Big\rfloor=\Big\lfloor{\lfloor{x}/{y}\rfloor}/{z}\Big\rfloor$


$ (x_{1} x_{2} \cdots x_{n})^{1/n} \leq \frac{x_{1} + x_{2} + \cdots + x_{n}}{n}. $


$ \bbox[lightyellow] {   \sum\nolimits_{\;x} {x^{\,\overline {\,n\,} } }  = \sum\nolimits_{\;x} {\left( {x + n - 1} \right)^{\,\underline {\,n\,} } }  = \left\{ {\matrix{    {{1 \over {n + 1}}\;\left( {x - 1} \right)^{\,\overline {\,n + 1\,} }  + c} & { - 1 \ne n}  \cr     {\psi (x - 1) + c} & { - 1 = n}  \cr   } } \right.  } \tag{2.b}$


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


$|f_{n}(x)| \leq k(x-a)^{n-1} \leq k(b-a)^{n-1}$


$rs - r - s$


$A=\{\{q\}:q\in \mathbb Q\}.$


$\begin{cases} x_1=x_3\\x_2=-x_3\\x_3\ \text{is free}\end{cases}$


$ P(X>x) = \frac{(1-p)^x p}{1-(1-p)} = (1-p)^x, $


$ \zeta(s) = \frac{1}{\Gamma(s)}  \int_0^1 \frac{x^{s-1}}{e^{x}-1} dx + \frac{1}{\Gamma(s)}  \int_1^\infty \frac{x^{s-1}}{e^{x}-1} dx $


$X = \pmatrix{-1\\3}$


$α+β+γ+δ+2(\sqrt{αβ}+\sqrt{αγ}+\sqrt{αδ}+\sqrt{βγ}+\sqrt{βδ}+\sqrt{γδ})=0$


$P_1 P_2= P_1P_2=0$


$\mathbb R[X]/(X^{4}+1)$


$\tilde \gamma (s) = \int \limits _a ^s \dot {\tilde \gamma} (t) \ \Bbb d t + \tilde \gamma (a) = \int \limits _a ^s M_0 \dot \gamma (t) \ \Bbb d t + \tilde \gamma (a) = M_0 \big( \gamma (s) - \gamma (a) \big) + \tilde \gamma (a) = \\ M_0 \gamma (s) + \big( \tilde \gamma (a) - M_0 \gamma (a) \big) .$


$(k + 1) - 3 ≥ 14$


$\text{FV}=\text{PV}(1+r)^{n+1}=\frac {C(1+r)}r\left[(1+r)^n-1\right]$


$T({{1}_{[a,b]}})(s)=\left\{ \begin{matrix}    0\,\,\,if\,\,s<a  \\    \frac{1}{s}(s-a)\,\,\,if\,\,\,a\le s<b  \\    \frac{1}{s}(b-a)\,\,\,if\,\,\,s\ge b  \\ \end{matrix} \right. $


$ P(U_{K_t}<x\sqrt{K_t})\leq P\left(K_{t}\notin ((1-\epsilon) a(t),(1+\epsilon) a(t))\right)+P\left(U_{a_t}<x\sqrt{(1+\epsilon)a(t)}+m\cdot \sqrt{\epsilon a(t))}\right)+ P\left(\sup_{s\in ((1-\epsilon)a(t),(1+\epsilon)a(t))}|U_{s}-U_{a(t)}|>m\cdot \sqrt{\epsilon a(t))}\right) $


$f(x)=\int_a^x g(t)\ \mathbb{d}t$


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


$4n + 2, 4n + 1, \dots, 2n + 2$


$ \sum_{n=0}^\infty\frac{(2\sqrt a)^n}{n!}\sqrt \pi= \sqrt\pi \;\exp(2\sqrt a)$


${\displaystyle \|A\|_{2}={\sqrt {\lambda _{\max }(A^{^{T}}A)}}}$


$\beta<\alpha^{+}$


$\int(2\pi y dx)$


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


$U=\langle u_1,u_2,u_3,....,u_n \rangle$


$f(p(t)) = p(t) + p'(t) + p''(t), \tag{1}$


$ \left\{ \matrix{   x = 2u + 2n - 1 \hfill \cr    y = 1 - u - 2n \hfill \cr    z = n \hfill \cr}  \right.\quad  \Rightarrow \quad \left\{ \matrix{   0 \le x + y = u \hfill \cr    u \le 1 - 2n \hfill \cr    0 \le n \hfill \cr}  \right.\quad  \Rightarrow \quad \left\{ \matrix{   u_{\,\min }  = 1 \hfill \cr    n = 0 \hfill \cr    x = 1,\;y = 0,z = 0 \hfill \cr}  \right. $


$\sum_{n \geq 1} \frac{1}{n(n+1)} = \underbrace{\sum_{n \geq 1} \frac{1}{n^2 + n} \leq \sum_{n \geq 1} \frac{1}{n^2}}_{\text{Every element is smaller}} = \frac{\pi^2}{6}$


$\eta(s)=\frac1{\Gamma(s)}\int_{0}^{+\infty}\frac{x^{s-1}}{e^x+1}dx$


$y=a(x-A)(x-B)$


$S = \{e_1, e_2, \cdots, e_n\}$


$\varphi_n(x) = \sum_{j=1}^n \frac{1}{j} \chi_{A_j}(x)$


$ar=b$


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


$d(x_0,M)=\bigg|\frac {1}{||g||}g(x_0)\bigg|$


$|f(a)−f(b)|≤K|a−b$


$e_i + 1 \le 2^{e_i}$


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


$F=P\oplus Q$


$x!\approx \sqrt{2\pi x}\left(\frac xe \right)^x\implies \log(x!)\approx (x+\frac 12)\log(x)-x+\log(\sqrt{2\pi})$


$ S = \{ E_1, E_2, E_3, E_4\}$


$\bigcup\mathcal A=X.$


$\tan \theta = \frac{x}{1}$


$K \subset V \subset \overline V \subset W$


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


$
 \left[\matrix{1&0&2&0\cr 0&1&-4&3\cr 0&0&-3&5\cr 0&0&0&7\cr }\right].
 $


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


$a(1,1,1,-1) + b(1,-1,0,1) = c(1,1,-1,1) + d(1,3,4,-5)$


$ b^{\log_b(x)} = x. $


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


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


$2\mid n^2-n$


$\displaystyle x= b^{\log_b(x)}$


$z_1^6 = 2^{3/2}e^{\frac{i\pi}{2}}$


$\det A_{a,b,n} = \det \begin{pmatrix}a & b\dotsc  b \\ 0 & A_{a-b,0,n-1}\end{pmatrix}$


$\begin{cases} x=6 \\ y=2 \\ z=8 \end{cases}$


$ \varphi(\delta)=\frac{f(x+\delta)-f(x)}{\delta}-a $


$I(a) = \int_{-\infty}^{\infty} \dfrac{\cos(ax)}{\pi(1+x^2)}dx$


$f(x) = \sin(1/|x|)$


$\zeta(2) = \frac{1}{\Gamma(2)}\int_0^\infty \frac{x}{e^x-1} dx$


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


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


$\left(\int_0^\infty f(x) \, dx\right)^s = C^s $


$\inf d(x,a)$


$\mathbb R^{n-1} \hookrightarrow \mathbb R^n$


$f(n) = (\frac{n^2-n}{(3300^2-3300)}*0.63)\quad,\quad n\lt3300$


$\sum_n \sum_m \mu(\{m\})\nu(\{n-m\})<\infty$


$x^n = (1+ \delta)^n > 1+ \delta n.$


$cRa\wedge aRc$


$\|x_n - x_m \| < \epsilon$


$\mathsf P(E)=\tfrac 1 4$


$\mathbb R^n\times \mathbb R = \mathbb R^{n+1},$


$ \|A^{\dagger}\|_2^2=\rho[A^\dagger(A^\dagger)^T]=\rho[(A^TA)^{-1}(A^TA)(A^TA)^{-1}]=\rho[(A^TA)^{-1}]=\|(A^TA)^{-1}\|_2. $


$S= \int 2 \pi x ds$


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


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


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


$P = \{ \{a\} , \{b, c\} \}$


$P[X_i \in A | \mathcal F] = P[ X_j \in A | \mathcal F]$


$S = \{s_1, s_2, s_3 .... s_n\}$


$ X = \{\, (2\cos{\varphi}, \sqrt{1+4\cos^2{\varphi}}, 2\sin{\varphi})\, |\, \varphi \in [0, 2\pi) \, \} \cup \{\, (2\cos{\varphi}, -\sqrt{1+4\cos^2{\varphi}}, 2\sin{\varphi})\, |\, \varphi \in [0, 2\pi) \, \}. $


$Payments=P \frac {R(1+R)^N}{(1+R)^N-1}$


$f(x)=\frac{x}{3} + 3$


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


$(40-x_1)+(40-x_2)+(40-x_3)+(40-x_4)=160-(x_1+x_2+x_3+x_4)$


$K \subseteq U \subseteq \overline{U} \subseteq O$


$d(X,Y)\leqslant d(X,Z) +d(Z,Y)$


$\wp(\Bbb N)\setminus\{\}=\wp(\Bbb N)\setminus\varnothing=\wp(\Bbb N)$


$\sum_{c=0}^{n-1}\cos(2\pi \frac{c}{n})=0$


$\tau_{a,ab}\equiv\{\emptyset,\{a\},\{a,b\},X\}$


$a_n = nc_n(x-a)^{n-1}$


$ \begin{align} \det\begin{bmatrix} x_1&y_1&z_1\\ x_2&y_2&z_2\\ x_3&y_3&z_3 \end{bmatrix} &= \begin{bmatrix} x_1\\ x_2\\ x_3 \end{bmatrix} \times \begin{bmatrix} y_1\\ y_2\\ y_3 \end{bmatrix} \cdot \begin{bmatrix} z_1\\ z_2\\ z_3 \end{bmatrix}\\ &= \begin{bmatrix} (x\times y)_1\\ (x\times y)_2\\ (x\times y)_3 \end{bmatrix} \cdot \begin{bmatrix} z_1\\ z_2\\ z_3 \end{bmatrix}\tag{1} \end{align} $


$\large x! \sim \sqrt{2\pi x}(\frac{x}{e})^x$


$z = \{ \{ u \} , \{ v \} \}$


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


$[x,y] = ax$


$\lim_{\text n \rightarrow \infty}\sum_{k=1}^\infty \frac{\text 1_k}{n}=1$


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


${1 \over F(n)} \le {1 \over 2^{n-1}}$


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


$\alpha\subseteq\gamma\subseteq\beta$


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


$z=be^{\frac{i\pi(1 + 4k)}{2n}}$


$\gamma':\gamma'\geq\beta,\gamma'\geq\gamma$


$\|A\| = \sqrt{\operatorname{tr}(A^TA)}$


$ax+by=c.$


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


$\lim_{n \to \infty}\left(\frac{\sqrt[n]a}{n+1}+\frac{\sqrt[n]{a^2}}{n+\frac12}+\frac{\sqrt[n]{a^3}}{n+\frac13}+\cdots+\frac{\sqrt[n]{a^n}}{n+\frac1n}\right)$


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


${\int\limits_0^{\pi /2} {dx} }$


$\overline{a_1\ldots a_n}$


$a_1 = f(a_1)$


$\int _{-1}^1\:\frac{\sqrt{1-x^2}}{x^2+1}$


$R: a\le x\le b,  c\le y\le d,$


$f:(\mathbb{R}^{n}, \mathcal{R}^{n})\to (\mathbb{R},\mathcal{R})$


$C_R(x) + r$


$E[T|C<c^*]$


$ \left[\begin{array}{ccc|c} 1 & 9 & -5 & 2 \\ 2 & -9 & -1 & -23 \\ 3 & 1 & 1 & 24 \end{array}\right] $


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


$\sum_{n\geq 1}\frac{1}{n^2}<2$


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


$A_1\subseteq A_2\subseteq A_3\subseteq ...\subset X$


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


$V_0 \subseteq V_1 \subseteq V_2 \subseteq V_3 \subseteq \cdots$


$\sum_{n=1}^\infty e^{i n x} = \frac{1}{2} \sum_{n=-\infty}^\infty e^{i n x} + \frac{1}{2} \sum_{n=-\infty}^\infty \mathop{\rm sgn}(n) e^{i n x} - \frac{1}{2}.$


$(\diamondsuit)<\sum_{n=1}^{+\infty}\|f_n-e_n\|<1\Rightarrow 1<1. $


$U\subseteq C\subseteq V$


$ \lim_{x\to\infty}\big(2x-x\big)=\infty-\infty=0 $


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


$\int_0 ^\infty \frac{sin(x)}{(x^2+1)^n} < \infty $


$\frac{1}{x}+\frac{1}{y}= \frac{1}{2007}$


$\{a\} \times \{a\} = \{\{\{a\}\}\}$


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


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


$as=r+b$


$v^+$


$\sum_{n\geq 1}\frac{1}{n^2}-\sum_{n=1}^{1000}\frac{1}{n^2}=\sum_{n\geq 1001}\frac{1}{n^2}\leq\frac{1}{1000+\frac{1}{2}}\tag{3} $


$x^2, \, x\in \mathbb{R}$


$\sqrt{1+\frac{d^2y}{dx^2}}=x+\frac{dy}{dx}$


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


$h'(\sum x_n)=\sum y_n$


$\ A^B$


$X^{a/b}=\{ x^\gamma \mid a\in \gamma, b\not\in\gamma,\gamma\in\mathcal R \}$


$ x_1=0+x_0=h\\ x_2=0+x_1=h\\ x_3=0+x_2=h\\ x_4=0+x_3=h\\ x_5=1+x_4=h+1\\ x_6=6+x_5=h+7 $


$ \alpha(\tau)=-\tfrac{1}{2}-i\tfrac{(2+2p-p^2) (1-2 p -2p^2) (1+4p+p^2)}{6 \sqrt{3} ~p(p+2) (2 p+1)(1-p^2)}=\tfrac1{4\sqrt{27}}\big(\lambda^3-\sqrt{27}\,\lambda^{-3}\big)^2 $


$\tan^2\theta-\sin^2\theta=(\tan\theta-\sin\theta)(\tan\theta+\sin\theta)\gt 0.$


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


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


$\partial^2 = \partial \bar\partial + \bar\partial \partial = \bar\partial^2 = 0.$


$||A||_F = \sqrt {Tr(A^TA)}$


$H_1(X;\mathbb{Z}/2\mathbb{Z}) = (\mathbb{Z}/2\mathbb{Z})^3$


$r(u,v)=\begin{cases}  x=u\\  y=v\\  z=2-u-v \end{cases}$


$E_2^{0,1} = E_\infty^{0,1} = F^0H^1$


$\gamma^{n}=\gamma\ast\cdots\ast\gamma$


$d(x,A)-d(x,y)\le d(y,A)$


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


$  \sum_{k=0}^n|\cos k|>\frac{2}{\pi}n.$


$f_n(x) = \dfrac{nx+x^2}{n^2}$


$\det(B) = \det(A-C)^{N-1}*\det(A+(N-1)C)$


$\begin{align} a x + b y = c\\  d x + e y = f \end{align}$


$p(m+N)=p'(m)$


$ax - by=d$


$\hat\gamma(t):=\begin{cases}\gamma(t)&0\le t\le 1\\ 2\gamma(1)-\gamma(t)&1\le t\le 2\\ 2\gamma(0)-\gamma(t)&-1\le t\le 0\end{cases} $


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


$I=\pi\int_0^{\pi/2}\frac{\sin x dx}{\sin x +1}=\frac{\pi^2}{2}-\pi\int_0^{\pi/2}\frac{dx}{1+\sin x}.$


$\vert d(x,z)-d(y,t)\vert\leq d(x,y)+d(z,t)$


$ {x}^2 \left(1-2 \sqrt{2}\, {x} \right) \sin ^2(\psi )-{y}^2    \left(1-2 \sqrt{2} \,{y} \right)=0 $


$   xf(x) = f(y) f(f(y)) = f(f(y)^2) = f \left( f(y)^2 + f(y) f \left( y - f(y) \right) \right) $


$T(n) = \sqrt1 + \sqrt2 + \sqrt{3} + \cdots + \sqrt n.$


$y=\alpha+\beta=\alpha+\beta+\gamma-\gamma=-\frac ba-\gamma\implies \gamma=-\left(y+\frac ba\right)$


$a^\gamma b^\gamma =(ab)^\gamma$


$\mathcal{N}_{a} = \{\{a\}\}$


$\int^1_0 \frac{\sin(x)}{x(1+x)^n} dx$


$e=e(P;p)$


$= Cov(2X, X) + Cov(2X, -3Y ) + Cov(Y , X) + Cov(Y ,  -3Y )$


$(A_1 \times A_2 \times ... \times A_{n-1}) \times A_n$


$\lim_{x\to c} \frac{f(x) - f(c)} {x-c} =f'(c) =L=\lim_{x\to c} f'(x)\tag{1} $


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


$\# R_X = \max (\# X, \aleph_0)$


$\frac{f}{||f||}$


$a(n) x^n + a(n-1) x^{n-1} + \dots + a(1) x + a(0)$


$a<y<b, c<x<d$


$\delta_v(A):=\max_i \ \frac{|\langle a_i,v\rangle|^2}{||v||_2^2}.$


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


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


$|f(x)|\leq M/|x|^{2+\epsilon}$


$ \begin{vmatrix} 0 & x & y & z \\ x & 0 & z^2 xy & y^2 xz \\ y & z^2 xy & 0 & x^2 yz\\ z & y^2 xz & x^2 yz & 0 \\ \end{vmatrix}$


$Z^+$


$(x+y)^r < x^r + y^r$


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


$e^N|\sum_{n=N+1}^{\infty}x^n| = e^N|x|^{N+1}\frac{1}{1-x}.$


$\beta'(s)=\frac{\langle\gamma'(s),\gamma(s)-\gamma(0)\rangle}{\beta(s)}=\frac{\langle\gamma'(s),\gamma(s)-\gamma(0)\rangle}{||\gamma(s)-\gamma(0)||}$


$\displaystyle\lim_{x\to c}f(x)=\displaystyle\lim_{x\to c}h(x)\implies \displaystyle\lim_{x\to c}f(x)=\displaystyle\lim_{x\to c}g(x)=\displaystyle\lim_{x\to c}h(x).$


$\zeta (3) = \frac{1}{\Gamma(3)}\int_{0}^{\infty} \frac{x^{3-1}}{e^x-1} dx \Rightarrow \int_{0}^{\infty} \frac{x^{2}}{e^x-1} dx = 2\zeta(3) $


$k! \sim (\frac{k}{e})^k \sqrt{2 \pi k}$


$\therefore log_ab^n=cn=n*log_ab$


$\left|\int_a^b f(x)g(x)d\alpha\right|\leq \int\left( \frac {1}{p}|f(x)|^p +\frac{1}{q}|g(x)|^q\right)d\alpha$


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


$\left(\frac{1}{3}\right)^{1/3} < \left(\frac{1}{2}\right)^{1/2}$


$|abc| = 1$


$\mathbb{E}(T_s\,|\, X_0=s)={1/\phi_s},$


$\int\frac{e^{-x}}{x}dx = \displaystyle\frac{1}{x}\cdot-e^{-x}-\int-e^{-x}\cdot\displaystyle\frac{-1}{x^{2}} dx = -\displaystyle\frac{e^{-x}}{x}-\int \displaystyle\frac{e^{-x}}{x^{2}} dx$


$ z=\frac{ab+c}{a+b+c},  $


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


$\begin{align}s= \int_0^2 \sqrt{1+(\frac{dy}{dx})^2}dt\end{align}$


$P = \lim_{n\rightarrow \infty}\sum_{r=0}^{n-1}\frac{r}{n^2+r^2}$


$\psi(x) = \sum \limits_{i=1}^n a_i \chi_{A_i} \left( x \right)$


$L = \mathbb{Q}(a,b,c,d) = \mathbb{Q}(a)$


$D(M) = P(M)$


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


$\overline{\Bbb R}^2$


$\displaystyle\lim_{n\to\infty}n\mu(A_n)=0$


$(m,m,m)$


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


$ A y = -(x-a)(x-b) $


$n!,n!+2,n!+3,n!+4....n!+n=n!,2(n!/2+1),3(n!/3+1)...$


$P(M) \implies P(\lambda x M)$


$||A||_2=\sqrt{r(A^TA)}\leq\sqrt{||A^TA||_\infty}\leq\sqrt{||A^T||_\infty||A||_\infty}=\sqrt{||A||_1||A||_\infty}$


$X_1\times...\times X_n$


$\sum_{n=1}^\infty n=1+2+3+4+\dots=-\frac 1{12}$


$f_Y(y) = \frac{\Gamma(a+b)}{\Gamma(a)\Gamma(b)} e^{-ay} (1-e^{-y})^{b-1}.$


$P(i) = 1/6$


$\large f_X(x)=\frac{2}{x^3}$


$\sqrt {1+\frac {dr}{dx}^2}$


$ x \sim y \iff x R_0 y \wedge y R_0 x. $


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


$Cov(U, V) = 0$


$\omega ^ {\omega ^ \alpha} = \alpha$


$n!=\sqrt{2\pi n}\left(\frac ne\right)^n e^{\frac{\theta(n)}{12n}}$


$\left(\begin{smallmatrix}a &-b\\b&a\end{smallmatrix}\right)$


$|f(z)|\le |f(z^{2^n})|$


$\forall n,m,p\in\Nt:n<m,\wedge,m<p\implies n<p$


$\sum_{\gamma<\alpha} a_\gamma = \sup_{\beta<\alpha} \sum_{\gamma<\beta} a_\gamma \le \sup_{\beta<\alpha} \prod_{\gamma<\beta} a_\gamma \le \prod_{\gamma<\alpha} a_\gamma.$


$f(a) < k < f(b)  \text{ or }  f(a) > k > f(b).$


$Cov(X,Z) = 0$


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


$(\mathbb{R}[X]/(X))/(2) = \mathbb{R}[X]/(2,X)$


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


$d(B,A)\leqslant d(B,C)+d(C,A),$


$\frac{478}{221}=2+\frac{36}{221}=2+\dfrac1{\frac{221}{36}}=2+\frac1{6+\frac5{36}}=2+\dfrac1{6+\dfrac1{\frac{36}{5}}}=2+\dfrac1{6+\dfrac1{7+\frac15}}$


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


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


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


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


$Q=(1-\tan^2(x)) \left(1-\tan^2 \left(\frac{x}{2}\right)\right)\cdots \left(1-\tan^2\left(\frac{x}{2^n}\right)\right)$


$ (a,b) \mapsto (a + b,a - b) $


$F(\theta, \gamma) = \{a + b\theta + c\gamma + d\theta\gamma : a,b,c,d \in F\}$


$f[x0] = x2 + zf$


$\Sigma ||e_n - f_n|| <1 $


$\displaystyle\int_1^\infty\dfrac{\sin^2(x)}{x}$


$\lim (f(x) \cdot f(y)) = \lim f(x) \cdot \lim f(y)$


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


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


$\begin{align*} 1+2+3+4+\cdots= \, -\frac{1}{12} \\ -1-2-3-\cdots= \; \: \, \, \frac{1}{12} \\ -1-2-3-\cdots= \; \: \, \, \frac{1}{12} \\ 1+2+\cdots =  -\frac{1}{12} \end{align*}$


$\ \overbrace{8n\!+\!6,\,6n\!+\!3,\,2n\!+\!3,\,{-}\color{#c00}6}^{\Large  a_{k-1} -\, j\ a_k\ =\ a_{k+1}}\,$


$\alpha \geq x_{2} \geq \beta \geq x_{1} \geq \gamma$


$  k = \left( \begin{array}{c} k \\ 1 \end{array} \right),  $


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


$ [F(a,b):K] = [F(a,b): F(a)][F(a): K] = [F(a,b): F(a)] \deg{(f)} $


$\int_0^\infty F_Z(x) \, dx$


$z = rs,\gcd(r,s) = 1$


$\{\langle{x,y}\rangle: 0\leq x\wedge 0\leq y\wedge x+y\leq 1\}$


$ f(x|y) = \frac{1}{2\pi\sqrt{1-y^2}}\cdot\mathbb{1}_{[-\sqrt{1-y^2},\sqrt{1-y^2}]}, $


$\sup_{\|[x]_B\|_{\mathbb R^m}=1}\|[Tx]_{B'}\|_{\mathbb R^n}=\sup_{\|[x]_B\|_{\mathbb R^m}=1}\|[T]_{B'B}[x]_B\|_{\mathbb R^n}.$


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


$\gamma(a),\gamma(b)$


$\begin{cases} a = 1 \\ b = 0 \\ c = -3 \\ d = -4 \\ \end{cases}$


$\neg A \vee \neg \neg A$


$N = \mathbb R^{n+1}$


$ = \frac{1}{2 \pi (1 - 2s)} \int_0^{2 \pi} \int_0^{\infty} e^{-u} \> du \> d\theta$


$\begin{pmatrix} 1 & -1 & 0 \\ 1 & 1 & -1\\ 1 & 2 & -1    \end{pmatrix} $


$(e^{-100\lambda})^7 (1-e^{-100\lambda})^3$


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


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


$H_n = A_1^C \cap A_2^C \cap ... \cap A_{n-1}^C \cap A_n$


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


$F[t]/P ((P))$


$det\left((a+b-e)I -buu^\top\right)= \left(a-(n-1)b-e\right)(a+b-e)^{n-1}.$


$\Delta f = f'(a)\Delta x$


$\overline{p^{-1}(V)}\subseteq p^{-1}(\overline{V})\subseteq U$


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


$K\subset V\subset U$


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


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


$f_3(n - 9k)$


$|t|^{-n}$


$\pmatrix{a_1&a_2&a_3\\ b_1&b_2&b_3\\ c_1&b_2&c_3}$


$3^k + 3^{k+1} > k^3 + 3^{k+1}$


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


$k^3 *1/3 + k^2(3/2) + k(7/6) + 1/3 + 1/2 + 1/6 = $


$\sum_{n=1}^\infty \sum_{k=n}^\infty$


$\mu(\overline{\mathbb{R}}) = \overline{\mathbb{R}}$


$\mathbb R^n \otimes \mathbb R^n$


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


$s < 1/|x|$


$\frac{x}{x-4} = \frac{x-4+4}{x-4} = 1 + \frac{4}{x-4}$


$\quad \underbrace{+ \quad \underbrace{+ \quad a \quad \underbrace{ * \quad b \quad c}_{\color{red}{b \,\cdot\, c}}}_{\color{red}{a \,+\,} b \,\cdot\, c} \quad d}_{(a\,+\,b \,\cdot\, c) \color{red}{\,+\, d}}$


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


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


$z = \frac{2ab-c^2}{a^2+b^2}.$


$P(A) = 1/6$


$1+2+3+4+\cdots = +\infty$


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


$\gamma=\sqrt{ai}=\sqrt{a}\ \text{cis} \frac{\pi}{4}=\frac{\sqrt{2a}}{2}+\frac{\sqrt{2a}}{2}i$


$\displaystyle\sum\limits_{n\geq1}\frac{a_n}{n^\beta}=\sum\limits_{n\geq1}\frac{a_n}{n^\alpha}\times \frac{1}{n^{\beta-\alpha}}$


$\begin{cases} a_0=2\\ a_1=1\\a_{n+2}=a_{n+1}-2a_n\end{cases}$


$E \lvert X-Y\rvert^r = 0 $


$ \begin{cases} xy=a\\ x+y=a\\ 0\leq x\leq 1\\ 0\leq y\leq 1\\ \end{cases} $


$\mathscr B_{[0,1]}\otimes \mathscr B_{[0,1]}.$


$\sum\limits_{i=0}^n r^i = \frac{1-r^{n+1}}{1-r}. $


$r = \frac{(100a+10b+c)}{(a+b+c)}$


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


$\sum\limits_{n=1}^\infty\frac{(n!)^n}{n^{4n}}$


$\Bbb R^{n}× \Bbb R^{n}$


$\sin x + \sin y = 2 \sin \frac {x+y} 2 \cos \frac {x-y} 2$


$p_1 \cdot p_2 \cdots p_k \mid a$


$d(x,a)\leq d(x,y)+d(y,a)\iff$


$\begin{align} \left(\begin{array}{rrr|r} 1 & -1 & 3 & -5 \\ 5 & 2 & 6 & \alpha \\ 2 & -1 & \alpha & -6 \end{array}\right) &\leadsto \left(\begin{array}{rrr|r} 1 & -1 & 3 & -5 \\ 0 & 7 & -9 & \alpha + 25 \\ 2 & -1 & \alpha & -6 \end{array}\right) \\ &\leadsto \left(\begin{array}{rrr|r} 1 & -1 & 3 & -5 \\ 0 & 7 & -9 & \alpha+25 \\ 0 & 1 & \alpha-6 & 4 \end{array}\right) \\ &\leadsto \left(\begin{array}{rrr|r} 1 & 0 & \alpha + 3 & -1 \\ 0 & 7 & -9 & \alpha+25 \\ 0 & 1 & \alpha-6 & 4 \end{array}\right) \\ \end{align}$


$\phi^n(z)=(\phi(z))^n$


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


$A-B=(\mathbb{R}-\mathbb{Q})-[(\mathbb{R}-\mathbb{Q})-C]=C$


$ \mathcal{F}:= \{(x,y) \in \mathbb{R^2}:x \in A,0<y \leq 1 \}$


$|a(t)| = r\omega^2$


$\displaystyle \sum_{k=0}^{n-1} \cos\left( \dfrac{2\pi}{n} \cdot k\right) = 0$


$\frac{e^{\frac{\gamma}{1+\gamma}}}{1+\gamma} = \frac{e}{\gamma} + o\!\left(\frac{1}{\gamma}\right)$


$\{a, a + \frac{b-a}{n}, a + 2\frac{b-a}{n}, \dots, b\} =: \{x_0, \dots, x_{n}\}$


$\inf \sum_{j=1}^{\infty}|R_j|=\inf\left(\sum_{rational\ rect.}|R_j|+\sum_{irrational\ rect.}|R_j^{*}|\right) = \inf\left(\sum_{rational\ rect.}|R_j|-\sum_{j=1^\ }^\infty\frac{\epsilon}{2^j}\right)$


${\rm gcd}(a+bi,a-bi) = 1$


$S=2n\sum_{k=1}^m |y_k|^2\cos{2k\pi\over n}\ ,$


$A_1 \subset A_2 \subset, \ldots \subset A_{n-1} \subset A_n \subset A_{n+1}$


$\{a\}\cup\{\{a\}\}$


$\begin{cases}n_p |s\\n_p  \equiv 1 \mod p\end{cases}$


$aRb \land bRa$


$(x\in \{\varnothing\}\,\Leftrightarrow\, x=\varnothing\,\not\Rightarrow\, x\in\varnothing)$


$f_x(x;a) = \frac{2x}{a^2} \ \ \ \ , \ \ \ \ 0<x<a$


$E[P(X_t\leq x|\mathcal{F}_s);\{X_k\leq y\}]=E[P(X_t\leq x|X_s);\{X_k\leq y\}]$


${1\over x}+{1\over y}+{1\over z}=-{-40\over{40\over3}}=3$


$\det \gamma = \det \beta^T = \det \beta$


$\tan \theta = \frac x4$


$j_f(c)=0 \Leftrightarrow \lim_{x \to c-}f(x)=\lim_{x \to c+}f(x)=l$


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


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


$ \left[       \begin{array}{cccc|c}         1&1&4&2\\         0&3&12&6 \\         -1&2&8&4 \\         1&1&4&2       \end{array}     \right]$


$A \subseteq B \subseteq \overline{A} \subseteq X$


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


$\|e_m - e_n\| \ge 1$


$\mathbb{Z}^+ \cup$


$F_X(x|\theta) = \frac{x}{\theta}\ \ \ \text{for $x\in [0, \theta$]}$


$\Bbb R^{n+1}-\{O\}$


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


$\zeta(s)=\sum_{m=0}^\infty \sum_{k=1}^\infty \frac{1}{(k+m^s)\prod_{n=0}^{k-2}(n+m^s)}$


$|f'(z)| \lt 2|f'(a)|$


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


$ \lim_{n \to \infty}\sin \frac{n \theta}{2}\sin \frac{(n+1) \theta}{2}$


$x\cdot y = a^{log_ax} \cdot a^{log_ay} = a^{log_ax\;+\;log_ay}$


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


$\int^{\pi}_{0}f(x)dx=\int^{\pi}_{0}f(\pi+0-x)dx$


$P(\bar E) = 1/2$


$\int_0^\pi \int_0^\pi \int_0^26\rho^3sin(\phi)cos(\phi)drd\phi d\theta = 0 $


$|AB|=mk$


$ \sum_{k=1}^\infty\log\biggl(1+\frac1{2^k}\biggr)<\sum_{k=1}^\infty\frac1{2^k}=1. $


$x^3 - x \in \mathbb{Z}$


$ \mathrm{length}(\gamma_{|[s_1, s_2]}) = \int\limits_{s_1}^{s_2} \sqrt{\langle \gamma^\prime (s), \gamma^\prime (s)\rangle} = s_2 - s_1. $


$\frac{1-r^2}{1-2r\cos t+r^2}$


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


$ ||T_u \times T_v|| = \sqrt{ \left( \frac{\partial g}{\partial u} \right)^2+\left( \frac{\partial g}{\partial v} \right)^2+1} $


$[t_{j}, t_{j+1}] \subset J_{j}$


$\sum\limits_{i=1}^{\log_2 n} [(\frac{1}{2^{i-1}})^2  \times    \frac{1}{\frac{n}{2^{i-1}}-1}]$


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


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


$\sup _{\gamma \in A}|\gamma (x)-\gamma (y)|\leq \sup _{\gamma \in F}|\gamma (x)-\gamma (y)|.$


$\gamma(\mathbf{v})\gamma(\mathbf{u})=\gamma[\gamma(\mathbf{v})(\mathbf{u})]$


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


$V = W\oplus Z$


$\text{Ls}_{n \rightarrow \infty} A_n = \{x \in X | \lim_{n \rightarrow \infty}\inf d(x,A_n) = 0\};$


$\gamma_{A, B} \gamma_{B, A} = \text{id}$


$(a+b, a-b)=4$


$\ \gcd(a,0) = \gcd(a) =$


$|e^x-(1+{x \over n})^n| \le \epsilon $


$a( \alpha+ \gamma )=2\gamma \Rightarrow a(-3\gamma+ \gamma )=2\gamma \Rightarrow  -2\gamma  a =2\gamma$


$\left( \begin {matrix} 1 & -1 \\ 1 & 4 \end {matrix} \right)$


$\displaystyle n\log\left(1+\frac{1}{n}\right) - \log\left(1+\frac{1}{n+1}\right) < \log\left(1+\frac{1}{n+1}\right)$


$\frac{p}{2\pi}\int_{-\infty}^{+\infty}\frac{\sin xt}{t\cdot \sin\frac12pt}\sin([\frac xp]+\frac12) pt \, \mathrm dt=\cdots$


$y'(x) + a(x)y(x) = a^{-A(x)}d(a^{A(x)}y(x)).$


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


$\gamma_1\geq\gamma_2\geq\cdots\geq\gamma_{n-1}$


$\int_{-\infty}^{\infty} f(t)u(t)dt = \int_{0}^{\infty} f(t)dt$


$ \mathbf{P}\left(\left. X_n =  x \right| X_t, t \in {^\circ T}\right) = \mathbf{P}\left( X_n =  x \right), $


$u_x = v_y, v_x = -u_y,u_x=-v_y,v_x=u_y \implies u_x=v_x=u_y=v_y=f_x=f_y=f'(z)=0$


$\frac{d}{dt}g\circ\gamma=(\nabla g\circ\gamma)\cdot\gamma\,'=0.$


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


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


$1+2+(-3+4)+\dots+(-1999+2000)=1+2+1+\dots+1$


$n!+2,n!+3,\cdots ,n!+n$


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


$\sum_{n=1}^\infty \infty = \infty$


$|\hat{f}(\xi)|\leq C/|\xi|^{1+\alpha}$


$(z^2)'' = 2 = f''(f(z))f'(z) + f'(f(z))f''(z)$


$C = \{\{a\}, E\}$


$\tan a+\cot a=\frac{\sin a}{\cos a}+\frac{\cos a}{\sin a}$


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


$\langle x^3-Q(x), x^2 \rangle =0$


$ \implies s_n = \frac{n(n+1)}{2}. $


$\begin{align}\gamma\leq \psi(t)=\Big\|\frac{x}{\| x\|_0}\Big\|\implies \gamma \| x\|_0\leq \| x\|\leq \beta\| x \|_0, \;\;\text{for some} \;\;\gamma,\beta>0.\end{align}$


$p\cdot t$


$S=\displaystyle \sum_{n=1}^{\infty} \frac{1}{n^2} =\sum_{n=1}^{5}\frac{1}{n^2}+\sum_{n=6}^{\infty} \frac{1}{n^2} \leq \sum_{n=1}^{5}\frac{1}{n^2}+\sum_{n=6}^{\infty} \frac{1}{n(n-1)}$


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


$\lim_{n \to \infty} t_n$


$I(y) = 1/|y|^{n-1}$


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


$\forall x_1, x_2,..., x_n\in \mathbb{N}$


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


$\gamma_+ \cup\gamma_R \cup\gamma_- \cup\gamma_{\epsilon}$


$\Lambda f=\int_a^xf(t)dt$


$A_{1} \subseteq A_2 \subseteq \dots$


$[t_1,t_2] \subset T'$


$M = \begin{bmatrix} 0&0&0&1\\[0.3em] 0&0&0.5&0.5\\ 0 & (0.5)^2 & 2(0.5)^2 & (0.5)^2 \\[0.3em] (0.5)^3 &3(0.5)^3 &3(0.5)^3 &(0.5)^3\\      \end{bmatrix}$


$x(t)=|t|^{n}$


$\langle M,{\in^M}\rangle$


$b=a^{log_a b} $


$\lim_{n\to\infty}( \inf A_n) = \lim_{n\to\infty} n = \lim_{n\to\infty} x_n = \liminf_{n\to\infty} x_n = +\infty$


$\gamma \wedge \gamma = \gamma \wedge \gamma$


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


$n\#+2,n\#+3,\ldots, n\#+n$


$\frac{d}{ds}f(g(s)) =f'(g(s))g'(s)=\frac{\gamma(s)-\gamma(0)}{\Vert \gamma(s)-\gamma(0)\Vert}\gamma'(s) $


$x_1 + x_4 + x_5 = 40,$


$f(n) = {n \choose4} + {n \choose 2} + 1$


$as + mt =1$


$E[x_0] = \mathbf{0}, x_0 \perp v_t, x_0 \perp w_t$


$\int_{-\pi}^{\pi}f(x)\sin(2nx)dx= \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}g(t)\sin(2n(t+\frac{\pi}{2}))dt=0$


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


$dx+by=d$


$ mid = {P_1 + P_2 \over 2} $


$\begin{equation}   \begin{cases}     A = 1\\     B = -2\\ C = 1   \end{cases} \end{equation}$


$ \Bbb P[X_t=B_t]=1,\qquad\forall t\ge 0. $


$\int ds=\int \sqrt{dx^2+dy^2}.$


$E(A) = 1/6$


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


$f(x)=\sum_{i=1}^n a_i \chi_{A_i}(x)$


$\lim\limits_{n\to\infty} \frac{\sqrt[n]{3n-1}}{\sqrt2} = \frac1{\sqrt2}.$


$(x, 1)^n = (A, -B)$


$ 1 = \lim_{N \to \infty} \sum_{n=1}^N \frac 1 {2^n}.$


$\lim_{n \to \infty}\dfrac {x_{1}+x_{2}+\ldots +x_{n}} {n}=x$


$\sum_{n\leq x}\frac{\phi(n)}{n^{2}}=\sum_{n\leq x}\frac{1}{n}\sum_{d|n}\frac{\mu(d)}{d}.$


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


$\gamma=\frac{a+b}{a-b}$


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


$(1 + \frac{\gamma}{a})^n \cdot (\gamma +a) \ge n * \gamma + \gamma + a$


$d(a,H)=\frac{|f(a)|}{||f||}$


$\operatorname{dist}(x,M)=\frac {|f(x)|}{\| f \|}$


$\gamma J\subset \gamma A\subset R$


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


$P(X_1 = 2 | X_0 = 1) = 1/2$


$a^{(\log_ab)^2}=a^{\log_a b\cdot \log_a b}=(a^{\log_a b})^{\log_a b}=b^{\log_a b}=b^{\frac{1}{\log_b a}}.$


$K \subset \bigcup_{\gamma \in \Gamma} U_\gamma$


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


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


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


$\#_1 A$


$cov(X,Y) = 1$


$\min \{ -(b_1 \gamma_1+b_2 \gamma_2) | \gamma_1 \ge 0, \gamma_2 \ge 0, v_1\gamma_1+v_1 \gamma_2 \le -a, -v_2\gamma_1+-v_2 \gamma_2 \le -1 \}$


$\sum_{n=1}^{N}z^n=e^{\phi i}+e^{2\phi i}+...+e^{N\phi i}$


$d(x, z) ≤ d(x, y) + d(y, z).$


$[\alpha,\alpha+\omega]$


$d(x,y)=|f(x)−f(y)|$


$\alpha^{-}$


$f(0)=1=\lim_{x\to 0^+}f(x)=\lim_{x\to 0^-} f(x).........................(1) $


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


$\lim_{n\rightarrow \infty} \frac{\sqrt{1}+\sqrt{2}+\sqrt{3}+\cdots+\sqrt{n}}{n^{3/2}}$


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


$(1+i)^{n+1}-\left (1+\frac{A}{P} \right ) (1+i)^{n} + \frac{A}{P}=0$


$\begin{array}{c|c|c}\#1&\#2&\#3&\mbox{Result at }\#1&\mbox{result otherwise}\\\hline c&g&g&win&lose\\\hline g&c&g&lose&win\\\hline g&g&c&lose&win\end{array}$


$|x-a|n(|a|+1)^{n-1}  < \delta$


$\frac{1}{48} \cdot (e^{it}+e^{-it})^3$


$\mathop {\lim }\limits_{n \to \infty } \left( \frac{\sqrt[2]{2}+\sqrt[4]{4}+...+\sqrt[2n]{2n}}{1+\sqrt[3]{3}+...\sqrt[2n-1]{2n-1}} \right) ^n$


$ 1+2+3+\cdots = -\frac{1}{12}\qquad\mathbf{FALSE} $


$\frac a x + \frac by=\frac 1 z$


$\lim_{n\rightarrow\infty}\mathbb{P}(A_{n+1})$


$P\left({m \over 2}-1\right) \implies P(m - 2) \implies P(m+2)$


$\frac {6p^3}{(1-p)(1-2p)}$


$ \int_a^b f(t) \;dt =\int_a^b F'(t) \;dt =F(b)-F(a), $


$E = \{e_1,..,e_n,... \}$


$\frac3x - \frac4y = 1$


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


$ x= \begin{bmatrix}   T^{n+1}_1      \\   T^{n+1}_2     \\   T^{n+1}_3  \\   T^{n+1}_4   \end{bmatrix} $


$X=\frac{a+b}{b+c}$


$A(x)=\int_{a}^{x} f(t)dt,$


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


$A=\left[\matrix{0&1&1&1\cr 1&0&0&0\cr 1&0&0&-1\cr 1&0&-1&0\cr}\right]\ .$


$f\left( a+b\right) =f\left( a\right) \oplus f\left(b\right)$


$\lim_\limits{x\to a^+} f'(x) \ne  \lim_\limits{x\to a^-} f'(x)$


$z = 2\ e^{\frac{i\pi}{4}}$


$Y=\{\{a\}\}$


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


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


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


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


$ v_{n+1} = v_1 (1+r)^n + T \sum_{k=1}^n (1+r)^k = T \frac{1+r}{r}\left((1+r)^{n+1}-1\right) ~~. $


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


$\det(\gamma,\gamma',\gamma'')=\det(\gamma,T,\kappa N)=\kappa\det(\gamma,T,N)$


$\,\Bbb Z[x]/(x^2+1)\cong \Bbb Z[i],\,$


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


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


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


$c = \frac{P_0  (1+r)^{N-1}r}{(1+r)^N-1} $


$\chi(n)=\left(\frac{n}{p}\right)$


$\begin{equation} f_{X_n}(x)=\frac{2x}{R^2}, \ \ \ H\leq x\leq \sqrt{H^2+R^2}, \end{equation}$


$\sum\limits_{k=1}^n \frac{1}{3^k}$


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


$ \sqrt{21} = 5 - \frac{4}{10-\frac{4}{10-\frac{4}{10-\frac{4}{10-\frac{4}{10-\ddots}}}}} $


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


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


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


$ (\frac{1}{2^{k/2}})_{j \in \mathbb{N}} $


$\lfloor \frac{\lfloor\frac{a}{b}\rfloor}{c} \rfloor = q_2 = \lfloor \frac{a}{b \times c} \rfloor$


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


$d(x, z) \leqslant d(x, y) + d(y, z),\ \forall x, y, z \in X.$


$0, \frac{1}{\pm N}, \frac{2}{\pm N}\cdots\frac{N - 2}{\pm N}, \frac{N - 1}{\pm N}, \pm1, \frac{\pm N}{N-1}, \frac{\pm N}{N-2}\cdots\frac{\pm N}{2}, \frac{\pm N}{1}, \infty$


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


$ {\left(1 + \frac{x}{n}\right)}^n \leq e^x\mbox{.} \tag{2} $


$n+0, n+1,\dots ,n+n$


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


$\vartheta(\gamma \tau) = (c\tau+d)^{12}\vartheta(\tau)$


$\det A = \prod \lambda_k = \left[\lambda+(n-1)\mu\right]\left[\lambda-\mu\right]^{n-1}$


$A_1\times\cdots\times A_n$


$\gcd(A)=A$


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


$n\mid p_1p_2$


$P(X_n \leq X, |X_n - X|< \epsilon) \leq P(X \leq X + \epsilon)$


$|f(x)|\leq |f^2(x)|$


$\begin{cases} b_1+b_2+\dots+b_{k+1} = n\\ 1\leq b_1\\ 1\leq b_2\\ \vdots\\ 1\leq b_k\\0\leq b_{k+1}\end{cases}$


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


$2^{x}+4^{x}+9^{x}=3^{x}+2 \cdot 6^{x}$


$\left(\begin{vmatrix}x_2&x_3\\y_2&y_3\end{vmatrix},\begin{vmatrix}x_1&x_3\\y_1&y_3\end{vmatrix},\begin{vmatrix}x_1&x_2\\y_1&y_2\end{vmatrix}\right)$


$ X(\pi/2)T(t) = \frac{\pi}{2}t \implies  X'(0) = 2\frac{X(\frac{\pi}{2})}{\pi} $


$\tag{1}\sum_{n=1}^{N} \frac{1}{n} \leq \sum_{n=1}^{N} \frac{1}{n^2}$


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


$   \theta_g (f)   := \frac{|\langle f', g \rangle|}{\|g\|_{L^2}}    = \frac{|\langle f, g' \rangle|}{\|g\|_{L^2}} \, . $


$\forall \epsilon \gt 0 \ ,\exists \delta \gt 0 \  ( |x-a|\lt \delta \Rightarrow |f(x)-L| \lt \epsilon)$


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


$|f(x_1)| \leq \frac{1}{2} |f(x_0)|.$


$\sum_{n=1}^\infty\sum_{k=1}^\infty\frac{\mu(n)\mu(k)}{n^k k^n}$


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


$\frac 1x+\frac 1y=\frac 1n$


$\alpha = 1/|x|$


$\phi = \sum_{i=1}^Nc_i\chi_{E_i}$


$\frac{\pi p}{2(p^2-1)\sqrt{p^2-1}}$


$K \subseteq V\subseteq \bar{V}\subseteq U $


$f^{'}(a)<k<f^{'}(b)$


$ \sum_{n=0}^{\infty}\frac{B_n(n+k)!}{k!n!}=\frac{1}{k!}\int_{0}^{\infty}\frac{x^{k+1}e^{-x}}{e^x-1}dx=(k+1)(\zeta(k+2)-1) $


$ \mathcal{B}(X)=\left\{\mu \in \Omega^{0,1}(X,K_X^{-1}) : |\mu|<1\right\} $


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


$ \zeta(3)=\sum_{n\geq 1}\frac{1}{n^3}=1+\sum_{n\geq 1}\frac{1}{(n+1)^3}\color{red}{\leq}  1+\sum_{n\geq 1}\frac{1}{n(n+1)(n+2)}=1+\frac{1}{4}=\color{red}{\frac{5}{4}}.\tag{1}$


$\Phi(x) = e^{-\gamma x^2} \left[1, \sqrt{\frac{2\gamma}{1!}}x,\sqrt{\frac{(2\gamma)^2}{2!}}x^2, \sqrt{\frac{(2\gamma)^3}{3!}}x^3, \ldots\right]$


$\nabla_{\dot{\alpha}} \dot{\alpha}=\phi''(s)\cdot \dot \gamma(\phi(s)) + \big(\phi'(s)\big)^2 \cdot \nabla_{\dot{\gamma}} \dot{\gamma} (\phi(s))= \big(\phi'(s)\big)^2 \cdot \Big( \big(\nabla_{\dot{\gamma}} \dot{\gamma} - \frac{g(\nabla_{\dot{\gamma}}\dot{\gamma}, \dot{\gamma} )}{g(\dot{\gamma}, \dot{\gamma})}\dot{\gamma} \big) \circ \phi \Big) =0$


$\lfloor\lfloor a/b \rfloor /c \rfloor = \lfloor a/bc \rfloor$


$ \frac 1 {2^m} \frac {1 - e^{2\pi i \delta 2^m}} {1 - e^{2\pi i \delta }} \geqslant  \frac 1 {2^m} \frac {4 \delta 2^m} {2 \pi \delta} = \frac 2 {\pi}. $


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


$F_{1/2}(x)=\int\frac{dx}{\left(1+x^{1/2}\right)^{1/2}}=\frac43\left(x^{1/2}-2\right) \left(1+x^{1/2}\right)^{1/2}$


$\gamma(0) = id \quad \text{and} \quad \gamma(1) = \text{Legendre transform}$


$\lim_{n\to \infty} \dfrac{\sqrt[n]{e}+\sqrt[n]{e^2}+\sqrt[n]{e^3}+...+\sqrt[n]{e^n}}{n}$


$\begin{cases}x=t,\\y=t,\\z=t\end{cases}$


$F(x) = \int _a^x f'(t) \; dt$


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


$f(x;\theta) = \frac1\theta e^{\frac{-x^2}{(2\theta)}}$


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


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


$ \begin{cases} t\equiv r\pmod{m}\\ t\equiv s\pmod{n} \end{cases} $


$\lim_{n\rightarrow\infty}=\left(1+\sqrt{2}+\sqrt[3]{3}+...+\sqrt[n]{n}\right) \ln\frac{2n+1}{n}$


$(\det \alpha)(\det \gamma) = (\det \alpha)(\det \beta)$


$x-c, c \in \mathbb{R}$


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


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


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


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


$\gamma_3(G)/\gamma_5(G) \le Z(\gamma_2(G)/\gamma_5(G)$


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


$d(\gamma,\gamma') = \max_{0 \leq t \leq 1} \rho(\gamma(t), \gamma'(t))$


$I=\int_0^\infty \frac{1}{(x^2+p)^{n+1}}dx$


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


$z=s^{\frac 1 n}e^{\frac{i(\phi+2k\pi)}n} $


$\begin{equation} \int_0^{2\pi} f(x) dx = \int_0^{2\pi} f(x) \sin xdx= \int_0^{2\pi} f(x) \cos xdx=...= \int_0^{2\pi} f(x) \sin nx dx=\int_0^{2\pi} f(x) \cos nx dx=0  \end{equation}$


$z_k = r^{1\over n} e^{{i \varphi + i 2 \pi k \over n}}$


$\overline{N^{n-1}_\epsilon(y)}\subseteq V\subseteq \overline{V}\subseteq U$


$\int_{-\infty}^{\infty}\sin^2(x)/x^2=\pi$


$\displaystyle (1+x)^{n+1}-(1+x)^r = \binom{n+1}{r+1}$


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


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


$|f(z^{2})|\leq e^{|z|}$


$x \delta^{(n)}(x) = - n \delta^{(n-1)}(x)$


$\frac{\frac{\sin \theta}{\theta}}{\frac{4\theta}{\theta} +\frac{\sin \theta}{\theta\cos\theta}} = \frac{\frac{\sin \theta}{\theta}}{4 +\frac{\sin \theta}{\theta\cos\theta}}$


$ \alpha + \beta +\gamma+\delta=-\frac{b}{a}$


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


$1*1!+2*2!+...+n*n! + (n+1)(n+1)! = (n+2)! - 1 \\(n+1)! -1 + (n+1)(n+1)! = (n+2)! -1   \\(n+1)! + (n+1)(n+1)! = (n+2)! \\(n+1)! + (n+1)(n+1)! = (n+1)!(n+2) \\n+2=n+2$


$\Phi(n)=S(n)$


$\langle Df, Dg\rangle_{C^\prime}=\int_0^1 (Df)^\prime(t) (Dg)^\prime(t) dt = \int_0^1 f(t) g(t) dt = \langle f,g\rangle_{L^2[0,1]}$


$2 \pi R sin(\varphi)$


$P(m)=P(k)$


$\le x \le b, c \le y\le d$


$4x-x^4 \leq 3, x \in \Bbb R$


$\sum_i \mu(A_i)\geq \mu(A_{i^*})=1=\mu\Big(\bigcup_i{A_i}\Big).$


$\mu/|x|$


$f(\theta^{(p)}(t)) = f(p)+t$


$[x, y] = \emptyset$


$\partial\circ\overline{\partial}$


$6 \cdot 4^x - 13 \cdot 6^x + 6\cdot 9^x = 0 /:4^x$


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


$\lfloor\frac{x-1}{y}\rfloor=\lfloor\frac{x}{y}\rfloor$


$h=ax+by$


$G_2:=\{x\in H_2: y\in H_1, x=2y\}.$


$d(x,y) = \arccos(\langle x, y \rangle)$


$f(x)=\frac12+e^{x-\frac12}-e^{\frac12-x}$


$w^+$


$ \Vert XYZ \Vert = \sup_{\Vert x \Vert = 1} \Vert XYZ x\Vert = \sup_{\Vert x \Vert = 1} \Vert YZ x\Vert = \sup_{z = Zx, \Vert z \Vert = 1} \Vert Y z\Vert = \sup_{\Vert x \Vert = 1} \Vert Y x\Vert = \Vert Y \Vert.$


$d(x,\ker(f))=\frac{|f(x)|}{\|f\|}$


$\xi\in\kappa_\gamma\subseteq \lambda_\gamma$


$\frac{1}{x}+\frac{1}{y}=15 \tag{II}$


$\displaystyle\int_0^\pi f(t) dt=0$


$\sqrt{A(ABCD)} =\sqrt{A(ABE)}+ \sqrt{A(CDE)}$


$B(u,v)=\begin{pmatrix}  u^1&u^2  &u^3  \end{pmatrix}\begin{pmatrix} {v^1+v^2+v^3}\\ {v^1+2v^2} \\{v^1+3v^3} \end{pmatrix}$


$0\oplus T$


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


$\lfloor a\rfloor+\lfloor b\rfloor+\lfloor c\rfloor$


$\sum\limits_{n=1}^\infty \left\lfloor \frac{N}{p^n}\right\rfloor$


$A_1\times A_2\times \ldots \times A_{n-1}\times A_{n}=X\times A_{n}$


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


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


$[256] = [220 + 36] = [220] + [36] = [0] + [36] = [36] = [33] + [3] = [3]$


$x/|x|$


$\mathbb R^2,\emptyset\in \tau$


$g^1\gamma - \gamma = \partial(H\gamma) + H(\partial\gamma).$


$a \lfloor \frac{b}{c} \rfloor \leq \lfloor \frac{ab}{c} \rfloor$


$p(E|S) = p(E \& S)/p(S) = p(E).p(T) / (1 - (1 -p(E)).p(T) + p(E).(1 - p(T))$


$\mathrm{Li}_{s}(z)=\frac{z}{\Gamma\left(s\right)}\int_{0}^{\infty} \frac{x^{s-1}}{e^{x}-z}dx \ge \frac{z}{\Gamma\left(s\right)}\int_{0}^{\infty} \frac{x^{s-1}}{e^{x}}dx = z $


$\gamma' \subset \gamma$


$x/2 = \sec \theta$


$d_L(h)=|\langle h,e\rangle|$


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


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


$\frac{d}{dx} \sum_{n=0}^\infty e^{-nx} = -\sum_{n=0}^\infty n e^{-nx}$


$x = (ax+b)/(cx+d),$


$\binom {n}{2}=n(n-1)/2 =f(n).$


$\tan\left((n+1)\frac{x}{2}\right) = (n+1)\tan\left(\frac{x}{2}\right).$


$\frac{a^{-3}}{(1-a^{-2})}=\frac{\frac{1}{a^3}}{(1-\frac{1}{a^2})}=\frac{a^2}{a^3(a^2-1)}=\frac{1}{a(a^2-1)}$


$p'(t) = f(t, p(t))$


$\{a, \gamma\}$


$p'(t)=f(t)\ p(t)$


$F = \{e_0, e_1, e_2, ..., e_n\}$


$\cos\;\theta=\frac{x}{r}$


$f_n(e^{-2/n}) = \frac{4e^{-2}}{n^2}$


$\left\{\begin{array}{l}x=1+3t\\ y=-2+t\\z=t  \end{array} \right., t\in\mathbb{R}.$


$g = \sum_{j=1}^n c_j 1_{A_j}$


$|f(x)g(x)-f(y)g(y)| \leq |g(x)||f(x)+f(y)|+|f(y)||g(x)+g(y)| $


$ n≥3L \tag{$*$} $


$R = \{(x,y) \in \mathbb{R}^2\colon\, 0 \leq x \leq a ,\, -b \leq y \leq b \}$


$\left\lfloor\frac{a+b}{n}\right\rfloor=\left\lfloor\frac{a}{n}\right\rfloor + \left\lfloor\frac{b}{n}\right\rfloor?$


$\sum_{\gamma\le T}\log |\gamma| > \sum_{ f(T) < \gamma \le T}\log |\gamma| > [ N(T)-N(f(T)) ] \log f(T).$


$(\mathbb{Z}[x]/(x^2+1))_{(5)}=\mathbb{Z}_{(5)}[x]/(x^2+1)=\mathbb{Z}_{(5)}[i]$


$ \frac{23}{12}(1,2,-1,0) + \frac{48}{12}(1,0,-1,1)  $


$x! < \sqrt{w\pi{x}}(\frac{x}{e})^x$


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


$I_n=\int \frac 1 {(x^2-1) ^n} \Bbb d x$


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


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


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


$\sin ^{2}\varphi +\cos ^{2}\varphi =1\iff\dfrac{\sin ^{2}\varphi }{\cos ^{2}\varphi }+1=\dfrac{1}{\cos ^{2}\varphi }$


$p(E|S) = p(E \& S)/p(S)$


$x^n\left(1+\dfrac xn\right)^n$


$P(E)=P(E\cap S)+P(E\cap S^c)$


$f_Y(y)=\int_{y}^{1} 8xy \, dx$


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


$ [a+(n-1)d] r^{n-1} $


$\alpha^+\in C$


$E(X) = \int_{-2}^2 x f_x(x)\, dx = 0$


$\rm\ \mathbb N_+\: $


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


$f: \mathbb R \rightarrow \mathbb R, f(x+y)=f(x)+f(y), \forall x,y \in \mathbb R$


$\sum_{k=1}^{n-1}\lfloor \frac{mk}{n} \rfloor$


$\int_{0}^{\pi}cos2nxdx=0=\int_{0}^{\pi}sin2nxdx,n\in\mathbb{N}$


$ \int{1\over(x^2+1)^n}\mathrm{d}x$


$||z|| = \frac{1}{||f||}$


$d_n={1\over 2^{n-2}}\ .$


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


$S \subseteq \cup_{\gamma \in F}\ U_\gamma$


$\sum_{k=0}^{M-1}\cos\frac{2\pi kj}{M}=0$


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


$Pxy \land Pyx$


$R_M(x):=\frac{\langle Mx,x\rangle}{\lVert x\rVert^2}$


$p_{\theta}(x)= \frac{2x}{\theta^{2}}1_{0 \leq x \leq \theta}.$


$r+s-rs$


$K\subset V\subset \bar{V}\subset U.$


$I_n=\int \tan ^{n}(x)\,dx=\frac{\tan ^{n+1}(x)}{n+1} \, _2F_1\left(1,\frac{n+1}{2};\frac{n+3}{2};-\tan   ^2(x)\right)$


$T^*M \otimes \mathbb{C} = {(T^{1,0})}^*M \oplus {(T^{0,1})}^*M$


$\mathbb{P}[X_n \geq b_n \text{ infinitely often}] = 1$


$f(x)=\int_0^\infty\frac{e^{-t}}t\sin (xt)dt=\int_0^\infty h(x,t)dt$


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


$f_Y(y)=\frac{4y}{\sqrt{2\pi}}e^\frac{-y^4}{2}=2y{\sqrt{\frac2\pi}}e^\frac{-y^4}{2}.$


$d(x_0,a) \leq  d(x_0,y)+d(y,a)$


$[x,y]= s^{2^{n-1}}$


$(p_1,p_2) \to p_1 + p_2 - p_1p_2$


$\sum_{n\geqslant N} \|f_n-e_n\|^2 < 1$


$\{f((0,0)), f((0,\pi)), f((\pi,0)), f((\pi,\pi))\}$


$\int_{0}^{2\pi} f(t) \sin ^2 (t-\theta) dt = g(\theta)$


$\int_0^\infty \frac{\sin^{2n-1}x}{x}dx $


$\mathcal{E} = \left(e_1 = \frac{x - y}{\|x - y\|}, e_2, e_3, \ldots, e_N\right).$


$c_0 \subset \overline{c_{00}}$


$\begin{cases}x-1=0\\ \dfrac{y-3}1=\dfrac{t-4}{-1}\\z-2=0\end{cases}\quad\text{whence}\quad\begin{cases}x=1\\z=2\\y+t=7\end{cases}$


$\gamma(st)=r_2\gamma(s)=\gamma(t)\gamma(s)$


$ \sum\limits_{n=1}^N \cos(2\pi n/N)= 0 $


$ \forall \alpha\in(-\omega,\omega), |\tan(\alpha)-\alpha|<\epsilon |\alpha| $


$f_{X,Y}(x,y) = \frac{\exp\left(-\frac{1}{1-\rho^2}\left(\frac{x^2}{2}-\frac{\rho x y}{\sigma}+\frac{y^2}{2\sigma^2}\right)\right)}{2\pi\sqrt{1-\rho^2}\sigma}.$


$F(x) = \int_2^x f(t)dt$


$a: \begin{cases} y=m_1 x\\ z=0\\ t=0\end{cases}$


$(a+b,a-b)$


$\left\{ \left( \begin{matrix} a & b \\ -b & a \end{matrix} \right) \right\}.$


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


$ 105! + 2, 105!+3, 105!+4, \dots, 105! + 101 $


$C^{î,*}(D)=C^{î+1,*}(D_0) \oplus C^{î,*}(D_{1})$


$\gamma \subseteq \omega(\gamma)$


$g_n=\frac{a}{b^2}\big((b+1)^{n+1}-(b+1)-bn\big)$


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


$\begin{vmatrix} 1 & x & x^2 & x^3 \\ 1 & y & y^2 & y^3 \\ 1 & z & z^2 & z^3 \\ 1 & x+y+z & xy+yz+zx & xyz \end{vmatrix}$


$\sin\theta + \sin\theta \tan^2\theta$


$\det(\gamma)=\det(\gamma') = 1$


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


$ \langle R(\dot\gamma,\gamma')\dot\gamma,\gamma'\rangle  =  - \langle R(\dot\gamma,\gamma')\gamma',\dot\gamma\rangle . $


$\gamma\beta\in\gamma J\subseteq R$


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


$\left(a,b\right)\mapsto\left(a-\lfloor a\rfloor,b-\lfloor b\rfloor\right)$


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


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


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


$C=\lim_{x\to0}\frac{360^{\circ}}{x^{\circ}}\times r\times\sin(x^{\circ})=360^{\circ}r\lim_{x\to0}\frac{\sin(x^{\circ})}{x^{\circ}}=360^{\circ}r$


$[\mathfrak{g}_{\mathbb{C}},\mathfrak{g}_{\mathbb{C}}] \subset \mathfrak{g}_{\mathbb{C}}$


$\begin{cases} x^{,} = y - x \\ y^{,} = z - x - y \end{cases}$


$x_n = e_n$


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


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


$ \sum_{n=1}^{+\infty}\frac1{n^2}=\frac{\pi^2}{6}\;\;. $


$f(x) = \sum_{m=1}^\infty\alpha_m\chi_{A_m}(x)$


$n+1,n+2,\ldots, n+50$


$\int_0^x f(t)dt = \int_x^1 f(t)dt$


$ \begin{pmatrix}  11 & -1 & 27 & 0 \\ -2 & 1 & -12 & -1\\ 1 & 16 & -4 & a\\ 9 & 5 & 5 & 0 \end{pmatrix}$


$(2^m m)m < m^m$


$b=a+c;a,b,c,d\in \mathbb{R} $


$\vartheta(-1/\tau) = \tau^{12} \vartheta(\tau)$


$L=\lim_{x \rightarrow c^{-}} f'(x)=\lim_{\epsilon \rightarrow 0} f'(c_1(\delta(\epsilon), f))=\lim_{\epsilon \rightarrow 0} \frac{f(c) - f(c-\delta(\epsilon))}{\delta}=\lim_{x \rightarrow c^{-}} \frac{f(c) - f(x)}{c-x}$


$\sum^\infty_{k=1}{\sum ^\infty _{n=1}\frac{a_n}{n^2+k^2}}=\sum^\infty_{n=1}{a_n\sum ^\infty _{k=1}\frac{1}{n^2+k^2}}.$


$X, Y \in \Gamma(M, T^{0,1}M)$


$(1 + \frac {k}{N})^N < e^k$


$\sum\limits^\infty_{k=1}{\sum\limits^\infty _{n=1}\frac{a_n}{n^2+k^2}}$


$x \in U \subseteq \overline {U} \subseteq C^c$


$\dfrac{2}{m} = \dfrac{1}{2^{n-1}}$


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


$\int_{\gamma} \omega = \int_{\gamma} d\theta = \theta (\gamma(1)) - \theta(\gamma(0)). $


$ \mathbb{E}|X - \mathbb{E}X|^r \leq \mathbb{E} |Y|^r, $


$p(E|S) = p(E \& S)/p(S) = 1$


$\forall x,y \, (xRy \land yRx \Rightarrow x=y )$


$\mu^+$


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


$gcd(a,a) = a$


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


$|x_n| \leq \| x \|_{p}$


$\Delta_n= \begin{vmatrix} a +(n-1)b& b & b & \ldots & b\\ 0 & a-b & 0 &  \ldots & 0 \\ 0 & 0 & a-b &  \ldots & 0 \\ \vdots&&&&\vdots\\ 0 & 0 & 0 &  \ldots & a -b\end{vmatrix}=[a+(n-1)b](a-b)^{n-1}.$


$ \log_a x = log_{a^2}(x+6) $


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


$E[X_{T \wedge n} - X_0 | \mathscr{F_m}] =/ \le X_{T \wedge m} - X_0 \ \forall m < n$


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


$P(k): 1^3+2^3+3^3+....+k^3=[\frac {k(k+1)}{2}]^2$


$\sin x=\frac{x}{2}$


$X_1 \subseteq X_2 \subseteq$


$f(x)=\sum_{n=-\infty}^\infty e^{-(x-n)^2} =\sum_{n=-\infty}^\infty e^{-(x+n)^2} = \sqrt{\pi}+2\sqrt{\pi}\sum_{k=1}^\infty e^{-k^2 \pi^2}\cos 2\pi kx$


$gcd(a,b) = gcd(a, b-a) = gcd(a, [b-a]-a) = ...$


$x^\gamma \implies (x^\gamma)^\gamma*(x^\gamma)^2=1/x \implies x^{\gamma^2+2\gamma}=x^{-1}\implies x^{\gamma^2+2\gamma+1}=1 \implies x^{(\gamma+1)^2}=1 \implies \gamma=-1\implies f(x)=1/x$


$\frac{\partial l(\theta)}{\partial \theta} = -\frac{1}{\theta} + \frac{|x|}{\theta^2}$


$B = \{ e_1, e_2,\dots, \}$


$G(x)=\int_a^x g(t)dt$


$S=\{1, 2, 3, ...., n\}$


$f(x)=\sum_{q_n<x}\frac1{n^2}=\sum_{n\in Q_x}\frac1{n^2}\;.$


$ax+by=d.$


${\bf M} = \left[\begin{array}{cccc} 1&0&0&0\\ 1&1&0&0\\ 1&1&1&0\\ 1&1&1&1 \end{array}\right] , {\bf M}^2 = \left[\begin{array}{cccc} 1&0&0&0\\ 2&1&0&0\\ 3&2&1&0\\ 4&3&2&1 \end{array}\right], \\{\bf M}^3 =  \left[\begin{array}{cccc} 1&0&0&0\\ 3&1&0&0\\ 6&3&1&0\\ 10&6&3&1 \end{array}\right] , {\bf M}^4 = \left[\begin{array}{cccc} 1&0&0&0\\ 4&1&0&0\\ 10&4&1&0\\ 20&10&4&1 \end{array}\right] $


$f\left( y,\pm\frac{\sqrt{1-y^2}}{2}\right)$


$\frac {10^4-1}{10^4}\,\frac {10^4-2}{10^4-1}\,\frac 1{10^4-2}=\frac {1}{10^4}$


$ \begin{align} \lim_{n\to\infty}\sum_{k=1}^{\lfloor n/2\rfloor}\frac{n}{n^2-k^2} &=\lim_{n\to\infty}\sum_{k=1}^{\lfloor n/2\rfloor}\frac1{1-\frac{k^2}{n^2}}\frac1n\\ &=\int_0^{1/2}\frac{\mathrm{d}x}{1-x^2}\\ &=\frac12\int_0^{1/2}\left(\frac1{1-x}+\frac1{1+x}\right)\mathrm{d}x\\ &=\frac12\left[\log\left(\frac{1+x}{1-x}\right)\right]_0^{1/2}\\[6pt] &=\frac12\log(3) \end{align} $


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


$g(\text{A N A}) = f(A)+f(A)+1 = 2f(A)+1 =f(\text{A N A})+1 \leq 2f(\text{A N A})+1$


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


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


$\overline{L}(\gamma) \ge \Vert \gamma(1) - \gamma(a) \Vert + \Vert \gamma(a) - \gamma(0) \Vert > \Vert \gamma(1) - \gamma(0) \Vert$


$ 9 x^4 (x^2 - 2x + 1) = 9 x^6 - 18 x^5 + 9 x^4;  $


$a_0 = \root 6 \of 3 e^{\frac{{i\pi }} {6}}  $


$T=\mbox{Th}_\tau (\cal M) = \lbrace\gamma|\gamma\mbox{ is a } \tau \mbox{- sentence and }\cal M\models\gamma\rbrace.$


$(s_{1,0} s_{0,1})$


$\displaystyle \int_0^x \sin(t^2) f(t) \; dt = \displaystyle \int_0^1 k(t,x) f(t) \; dt$


$\frac{-\frac{x-4}{4(x+4)}}{x-4}=-\frac{x-4}{4(x+4)(x-4)}=-\frac{1}{4(x+4)}$


$\left[ \begin{array}{ccc|c} 1 & 1 & 1 & 6 \\  1 & 2 & 3 & 10 \\  1 & 2 & \lambda & \mu  \end{array} \right]$


$\sum_{k=0}^N e^{\frac{2\pi k}{N+1}}=0\Rightarrow \sum_{k=0}^N\cos{\frac{2\pi k}{N+1}}=0\Rightarrow \sum_{k=1}^N\cos{\frac{2\pi k}{N+1}}=-1.$


$(a-b)\mid(a^{rs}-b^{rs})$


$\lim_{L \rightarrow \infty} (\ln{L})^2\left[1-\left(1-e^{-\frac{1}{2}(\sqrt{L}-2)}\right)^L\right]$


$\log 3 \log(1+\sqrt{x}) =\log 2 \log x $


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


$ \det A=(l-b)^{n-1}(l+(n-1)b). $


$f_1,f_2 \in \mathbb{N}^\mathbb{N}$


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


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


$\lim_{x \to c}f(x)g(x) = \lim_{x \to c}f(x) \times \lim_{x \to c }g(x)= L \times \infty = \infty$


$d(x,y)\leq d(x,x_n)+d(x_n,y_n)+d(y_n,y)$


$\dfrac{x_1+x_2+\cdots+x_n}{n} \ge \sqrt[n]{x_1x_2\cdots x_n}$


$ \int_{--\frac{b^2}{a}}^{\frac{b^2}{a}} \sqrt{1+ {\frac{dy}{dx}}^{2}} dx = \int_{--\frac{b^2}{a}}^{\frac{b^2}{a}} \sqrt{1+\frac{a^{2}x^{2}}{b^{2}(b^{2}+x^{2})}} \ dx $


$A_1\subseteq A_2\subseteq\cdots \subseteq A_k\subseteq \{1,2,3,\cdots ,n\}$


$[X,Y] = X$


$N_1\times\dots\times N_n$


$ \zeta(s) = \frac{1}{\Gamma(s)}\int_{0}^{+\infty}\frac{x^{s-1}}{e^x-1}\,dx. $


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


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


$ A_1 \subseteq A_2 \subseteq \dots \subseteq A_{k+1} \subseteq \lbrace 1, 2, \dots, n \rbrace $


$ \begin{equation} \left\{ \begin{array}{lr}  -(\tilde{a}(x)u_{x})_x + \tilde{c}(x) u = \tilde{f}, 0<x<L \\ u(0) = u(L) = 0 \end{array} \right. \end{equation} $


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


$\lfloor\alpha\rfloor+\lfloor\beta\rfloor+\lfloor\gamma\rfloor=9-(\{\alpha\}+\{\beta\}+\{\gamma\})=\color{red}{7,8}$


$\,p\mid d\mid p_1 p_2\Rightarrow\,p\mid p_1p_2\,$


$ {\rm Hess} f (\dot \gamma, \dot \gamma) := \dot \gamma (\dot \gamma f)-(\nabla _{\dot \gamma} \dot \gamma)f=\dot \gamma (\dot \gamma f).$


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


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


$(a_{\varepsilon},b_{\varepsilon}) \subset B_{\varepsilon}$


$A\subset U\subset K\subset B\subset Y?$


$ \lim_{n\to \infty}J(u_n) = \inf_{u\ne 0}J(u). $


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


$\sigma^{+}$


$\sum\limits_{n\geq 1}\frac{1}{n^s}=\sum\limits_{n\geq 1}n\left(\frac{1}{n^s}-\frac{1}{(n+1)^s}\right)$


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


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


$1+2+3+\dots n+\dots=-\frac1{12}$


$ f(x)^2+f(y)^2=f(x+y)(f(f(y))+f(x)) $


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


$\displaystyle \lim\limits_{n\to \infty} \sum\limits_{m=1}^{n} \log\left(1+\log\left(1+\frac{m}{n^2}\right)\right) = \lim\limits_{n\to \infty} \sum\limits_{m=1}^{n} \log\left(1+\frac{m}{n^2}\right) = \lim\limits_{n\to \infty} \sum\limits_{m=1}^{n} \frac{m}{n^2} = \frac{1}{2}$


$p = \frac {1}{2\pi(1 - \cos\theta)}$


$\Lambda^+$


$ y + \lambda = C \sqrt{1+(\frac{dy}{dx})^2} $


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


$\{e_1,e_2,e_3,\cdots, e_n\}$


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


$ Lie(g) = \{ \dot{\gamma}(0) | \gamma:(-\epsilon, \epsilon) \rightarrow g, \gamma \in C^1,  \gamma(0) =  \mathbb{I} \} $


$S=(x_1,x_2,x_3,\cdots , x_n)$


$  f = \frac{ (1-f)^N \lbrack 1 - (1-f)^2 \rbrack }{ 1- (1-f)^{N} } $


$\frac{1}{x}+\frac{1}{y}=\frac{1}{2}+\frac{1}{z}$


$f(a+b)=f(a)+f(b) , \forall a \in A , b\in B$


$x^3, x^2, x$


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


$v((x-a)^mp(x)+(x-a)^nq(x))=v((x-a)^k[(x-a)^{m-k}p(x)+(x-a)^{n-k}q(x)])\geq v((x-a)^k)=k,$


$\sin\left(\frac{\pi}{4}+x\right)\left(1-2\sin(x)\cos(x)\right)=\sin\left(\frac{\pi}{4}+x\right)\left(1-2\sin(x)\cos(x)\right)$


$-\frac{x}{4} = \frac{-x}{4} = \frac{x}{-4}$


$\implies \gamma'=5\gamma $


$[(x,u)]=x$


$\int_a^b 1(x)\, dx$


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


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


$\sum_{n=1}^\infty\frac 1n\sum_{m=1}^\infty\frac 1m=\sum_{n,m=1}^\infty\frac 1{nm}=\sum_{n=1}^\infty\frac1{n^2}+\sum_{n<m}\frac2{nm}.$


$F_0=Q\oplus P$


$\mathcal{A} = \{e_1, e_2, e_3, ..., e_n \}$


$\;\begin{cases}x=t\\y=t\\z=t\end{cases}\enspace(0\le t\le 1)$


$F\oplus T$


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


$||A||_{2}= \sqrt{\rho(A^{*}A)} <1$


$\frac {50}{x} + \frac {126}{y} = 1$


$3\mid n^3 - n$


$ \int_{0}^{2\pi}\frac{\sin(nx)}{\sin(x)} \mathrm dx $


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


$p|r_1 r_2$


$\tan \theta = \frac{x}{\sqrt{3}}$


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


$y = \frac{-1 - 2x}{1 - x}$


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


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


$P(E)=P(E)$


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


$\begin{align*} &\; \int_0^1 \frac{\log^2(x)\tanh^{-1}(x)}{1+x^2}dx \\ &= -2\int_0^1 \frac{\log(x)\tan^{-1}(x)\tanh^{-1}(x)}{x}dx-\int_0^1 \frac{\log^2(x)\tan^{-1}(x)}{1-x^2}dx \tag{1} \end{align*}$


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


$\forall x,y \in X: xIy \leftrightarrow xRy \land yRx$


$\lim_y\to2^- f(y)=+\infty$


$\sum_{n=1}^\infty\left\|x_n\right\|_X\left\|y_n\right\|_Y<\pi(u)+\epsilon\tag1$


$\sum_{n\ge 1} \frac{1}{n^2}<\infty$


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


$E(x) = \int_{a}^{b}3x^2dx$


$y[n-1] = (n-1)(x[n-1] - 5)^2$


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


$ra + sp = 1$


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


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


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


$ I_n =\int_{-1}^{1}\frac{\sqrt{n}f(x)}{1+n x^2}dx $


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


$a_1 \cdot \dotsc \cdot a_n$


$\cfrac{15}{x}+ \cfrac{12}{x}+\cfrac{6}{x}+\cfrac{3}{x}=1$


$V(X):=E((X-E(X))^2)=E(X^2-2E(X)X+E(X)^2) = E(X^2)-E(X)^2$


$d(x,a)\geq d(y,a)-d(x,y)$


$f(x) = \lim_{n\to \infty} \sum_{r=1}^n \frac{\lfloor2rx\rfloor}{n^2}$


$cov(\mathbf{e}, \mathbf{b}) = 0$


$p(E|S) = p(E \& S)/p(S) = 0$


$1=x+y\leq x^r+y^r$


$A,B,X,Y < C < D$


$A_1\times\cdots\times A_n\subseteq k^n$


$P(2)\implies P(3)\implies P(4)\implies\cdots\implies P(18)\implies P(19)$


$d(x,y) = \dfrac{|x-y|}{1+|x-y|}$


$P[X \geq k] = 1 - F_n(k)$


$\displaystyle\sum_{n=1}^{\infty} a_{n}^{-}=\frac{1}{2}\sum_{n=1}^{\infty}\left(a_n-\left|a_n\right|\right)$


$n^{s_{0}}=\sqrt{n}e^{it_{0}\log\left(n\right)}  $


$f(n) = n^2 + n + 41$


$ T^{1,0}M \times T^{0,1}M \subset TM^{\Cpx} \times TM^{\Cpx}, $


$P\left[X_{1}=1\wedge X_{2}=0\right]=P\left[X_{2}=0\mid X_{1}=1\right]P\left[X_{1}=1\right]=\frac{g}{r+g+c}\frac{r}{r+g}$


$\Delta (x' + y') \leq \epsilon(\Delta| x| + \Delta |y| + |x| + |y|)$


$A \subseteq V = \overline{V} \subseteq U$


$2|n^2 + n$


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


$f(n) = (\frac{n}{q})=\prod_{p^k \| q} (\frac{n}{p})^k$


$ \|x_n - x_m \|<{1\over 2}$


$\rho \circ \gamma \circ \rho \subseteq \gamma \circ \rho \circ \gamma$


$(y-x) \begin{vmatrix} x&1&z\\ x^2&y+x&z^2\\ x^3&y^2+xy+x^2&z^3\\ \end{vmatrix}$


$T: P(F) → P(F),$


$E[Y\mid X=x_0] = 3x_0/2$


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


$A_1\times\dots\times A_n.$


$\matrix{1\cr 2&3\cr 4&5&6\cr 7&8&9&10\cr 11&12&13&14&15\cr   \vdots&\vdots&\vdots&\vdots&\vdots&\ddots\cr}$


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


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


$T\gamma = b.(\gamma - Sd\gamma - T\partial\gamma)                          = b.\gamma -b.Sd\gamma - b.T\partial\gamma $


$\lbrace T_x : x \in [0,1]\rbrace$


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


$[t_N^k,t_N^{k + 1}]$


$\underline{~a~}~\underline{~b~}~\underline{~c~}~\underline{~d~}~\underline{~e~}~\underline{~d~}~\underline{~c~}~\underline{~b~}~\underline{~a~}$


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


$\gamma''= (\gamma'')^\perp = - \gamma + h(\gamma', \gamma') \nu$


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


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


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


$\inf_{g \in X} || y -g|| = \sup_{(f,X)=0} \frac{(y,f)}{||f||},$


$\lim_{n\to \infty} \frac{\sqrt[n]e + \sqrt[n] {e^2} + \sqrt[n]{e^3}+...+\sqrt[n]{e^{2n}}}{n}$


$3^x + 10^x = 4^x + 9^x$


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


$\frac{\frac{x+6}{x+4}-\frac{10}{8}}{x-4}$


$(\alpha, \gamma), (\beta, \gamma) \in \Delta$


$[\frac52] - [\frac57] + [\frac{-2}7] - [\frac23] + [\frac53] - [\frac{-5}2]$


$Y_n = \frac{1}{n^2} \sum_{k=1}^{n} X_k \stackrel{d}{=} \frac{1}{n^2} \sum_{k=1}^{n} \bigg( \sum_{i=1}^{k} W_i \bigg) = \frac{n-1}{n^2} \sum_{k=1}^{n} W_k - \frac{1}{n^2} \sum_{k=1}^{n} k W_k,$


$\|x_n - x_{n+1}\| < \frac{1}{2^n}$


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


$0=(-A)+\lim_{x\to p}f(x)=\lim_{x\to p}(-A)+f(x)=\lim_{x\to p}f_1(x),$


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


$ar + bs$


$\frac{ax + b}{c-x} = y$


$\sqrt{\frac{\pi }{t}}\sum\limits_{n=-\infty }^{\infty }{{{e}^{-{{\left( x/2+n\pi  \right)}^{2}}/t}}}=\sum\limits_{n=-\infty }^{\infty }{{{e}^{-{{n}^{2}}t+inx}}}$


$\frac{k_{n+1}}{k_n}\le\frac{(1+\varepsilon)^{n+1}}{\frac{1}{2}(1+\varepsilon)^n}=2+2\varepsilon$


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


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


$M = \left(\begin{matrix} A & B \\ B^T & C \end{matrix}\right)$


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


$\sum_{m=1}^k \sum_{n=1}^{\infty} \frac{1}{m^2+n^2} \approx \frac{\pi \ln k}{2} + \frac{6\gamma \pi-\pi^2}{12}$


$(x_1,y_1)=(x_2,y_2)=(x_1^2+y_1^2,x_2^2+y_2^2)=1$


$ (4n)!+3, (4n)!+7,......,(4n)!+3+4n$


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


$ \sum_{n\geq 1}\frac{(-1)^{n+1}}{n^s} = \frac{1}{\Gamma(s)}\int_{0}^{+\infty}\frac{x^{s-1}}{e^x+1}\,dx $


$1/|x| >1$


$g(x) = 3x^2 - x + 2$


$ \left\{ \matrix{   y < 0 \hfill \cr    a < x < b \hfill \cr    {{b - z\left( {1 - y} \right)} \over y} < a < x < b < {{a - z\left( {1 - y} \right)} \over y} \hfill \cr}  \right. $


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


$[x,y] = ax + by$


$F: (\mathbb{R}^n,\mathcal{O}_{\mathbb{R}^n}) \to (F(\mathbb{R}^n),\mathcal{O}_{\mathbb{R}^{m+n} | \mathbb{R}^n})$


$L = \overline{p_{1}p_{2}}$


$M_{0,1}\to M_{1,1}$


$\frac{16}{a^2} + \frac{9}{b^2} = 1$


$\phi = (a_1,a_2,a_3,....,a_n)$


$f(a_1)=f(a_1^\prime) \implies a_1=a_1^\prime$


$\{\omega\in\Omega| \lim_{n\rightarrow\infty} X_n(\omega)=X(\omega)\}\in\mathcal{F}$


$I(s) = \int_0^\infty \frac{\log^{s-1}(1+x)}{x(x+1)} dx = \int_0^\infty \frac{u^{s-1}du}{e^u-1} = \zeta(s)\Gamma(s)$


$|f(a_1)|\le \frac{1}{2}|f(a_0)|$


$D(x)M\in\langle P\rangle$


$\int_0^{+\infty}\int_0^{+\infty}F(x,t)dxdt=\int_0^{+\infty}\int_0^{+\infty}F(x,t)dtdx.$


$A_1\subset A_2\subset\ldots\subset A_n\subset \ldots\subset X$


$f(n)=(2n-3) f(n-1) -x^2 f(n-2)$


$F=M\oplus N$


$\begin{pmatrix} 1&0&0\\ 0&r^2&0\\ 0&0&r^2\sin(\xi)^2 \end{pmatrix}$


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


$ \lim_{n\to \infty}\frac{2n}{n}=\lim_{n\to \infty}\frac{3n}{n}=\frac{\infty}{\infty}=1 $


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


$P+N=F$


$\gamma,\gamma' < \kappa$


$\,p_i\mid a\, \Rightarrow\ p_1\cdots p_n\mid a\ \ \ $


$\root n \of re^{i\theta/n}$


$w = \frac{(ab - d) }{c - a - b}$


$\int_{-\pi}^{\pi}=\int_{-\pi}^{0}+\int_{0}^{\pi}$


$\Delta f(x)= f'(x)\Delta x + \epsilon (\Delta x) \Delta x$


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


$a_n = \frac{n(n+1)(n+2)}{6}$


$[x, y, z]\succ  [1, 1, 1]$


$COV (X,Y) = 0$


$x =c_1e_1+c_2e_2+..c_ne_n$


$36^x=9^x \times 4^x=3^{2x}\times 4^x$


$\frac{1}{x}+\frac{1}{y} = \frac{2}{40}$


$z = \sqrt[n]{r} e^{i(\theta + 2 \pi k)/n} \ \text{ for } \ k = 0, 1,..., n-1$


$dC=\sqrt{\bigg(\frac{dx}{d\theta}\bigg)^2+\bigg(\frac{dy}{d\theta}\bigg)^2}\,d\theta$


$\frac{19}7=2+\frac57=2+\frac1{\frac75}=2+\frac1{1+\frac25}=2+\frac1{1+\frac1{\frac52}}=2+\frac1{1+\frac1{2+\frac12}}$


$|f'(z)| \leq 2|f'(p)|$


$z = \sqrt[4]{8} e^{\frac{5\pi}{8}i}$


$(\gamma(i),\gamma(j))_{\gamma(0)} - d(\gamma(0),x) \leq (\gamma(i),\gamma(j))_x$


$G=\mathbb{Z}\oplus \mathbb{Z}$


$ \left|\begin{matrix} 1 & l & l^2 \\ 1 & b & b^2 \\ 1 & h & h^2 \end{matrix}\right| = 0 $


$|f_n (x)| \leq 2 |f(x)|$


$c/|x|$


$h^+$


$ \sum_{n = 1}^{\infty}\sum_{k = 1}^{\infty} {1 \over \left(k^{2} + n^{2}\right)^{\left(1+\epsilon\right)/2}} $


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


$\frac{1}{n^2}\sum_{k=1}^nk H_k  = \frac{1}{n^2}H_n \sum_{k=1}^n k + \frac{1}{n^2}\sum_{k=1}^{n-1}\left(\sum_{j=1}^kj \right)(H_k - H_{k+1}) \\ = \frac{H_n}{n^2}\frac{n(n+1)}{2} - \frac{1}{4}\left(1 - \frac{1}{n}\right)  $


$ \left\{ \matrix{   \left( {2a - b} \right)x = \left( {2a - b} \right)d + 2\left( {u - ad} \right) + 2\left( {3b - 2c} \right)n \hfill \cr    \left( {2a - b} \right)y =  - \left( {u - ad} \right) + \left( {c - 3a} \right)n \hfill \cr    z = n \hfill \cr}  \right. $


$f(n) = n^2 + n + 3$


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


$\int \kappa\,ds \geq 2\pi$


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


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


$\operatorname {Cov}(X,Y)=0.$


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


$\frac{1}{n!}\le \frac{1}{2^{n-1}}$


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


$(\Delta\Delta a)(n-1) = -1\;.$


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


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


$\mathbb{A} = \{a,a+b,a+2b,...\}$


$S \oplus T$


$\Bbb R[x]/(x^2+1)\cong \Bbb C$


$(\cal X, \cal A)$


$t\langle \ddot{\gamma}(s) \times \gamma(s),\, \dot{\gamma}(s)\rangle/t^2,\text{ or }(1/t)\det(\dot{\gamma}(s)\text{ }\ddot{\gamma}(s)\text{ }\gamma(s)).$


$\int_{0}^{\pi} \frac{\cos(nx)}{(p+\cos(x))^2+q^2}\ \mathrm dx$


$1+\sum_n (x_n+y_n) =1+\frac{b}{1-\frac{1+\sqrt{5}}4} + \frac{c}{1-\frac{1-\sqrt{5}}{4}} = 1+b(3+\sqrt{5}) + c(3-\sqrt{5})$


$\vert \vert A \vert \vert_2 = \sqrt{ \lambda_{max} (A^T A)} $


$\cdots \subseteq V_{-1} \subseteq V_0 \subseteq V_1 \subseteq V_2 \subseteq \cdots$


$s \oplus t$


$ =(1-q)^2\int_1^\infty\frac{dx}{x(x-q)^{n+1}}(x-1)^{n-1}\ . $


$x^3, x^5, x^7,\ldots$


$\sum\limits_{N\geq 1} \frac{1}{N^2}$


$y=\dfrac{a+c}{b+d}$


$\text{int}A\subseteq A\subseteq\overline{A}$


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


$\sum_{n=1}^{+\infty}\frac{1}{n(2n+1)}=2\sum_{n=1}^{+\infty}\int_{0}^{1}\left(x^{2n-1}-x^{2n}\right)\,dx = 2\int_{0}^{1}\frac{x}{1+x}\,dx = 2-\log 4.$


$P \frac {R}{1-(1+R)^{-N}}$


$\sigma_1 = \sqrt{\lambda_{max}(A^TA)}$


$A(z)-z = z A(z) + \frac{1}{2} (A(z)^2 - 2z A(z) + A(z^2)) = \frac{1}{2} (A(z)^2 + A(z^2)).$


$2n , 2n+1 , ... , 3n$


$\Vert x_n-x_m\Vert>1$


$f\big(xf(y)+f(x)\big) = 2f(x)+xy.$


$G \cong A_{1} \times ... \times A_{n}$


$|a+b|=s-t$


$ \sum_{i=1}^n k\frac{3i}{n}=1. $


$R(M,v):={\langle v, Mv\rangle \over |v|^2}$


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


$A = 2\pi\int\rho\, ds,$


$y = \dfrac{ax^2 + bx + c}{dx + e}$


$X\sim Exp(\lambda) \rightarrow f(x) = \frac{1}{\lambda} e^{\frac{-x}{\lambda}}$


$A=\pmatrix{1&2&3\cr 4&5&6\cr 2&1&0\cr 1&0&0\cr 0&1&0\cr 0&0&1\cr }$


$\gamma=\gamma+ \beta\le \gamma+\beta$


$L(\gamma,\dot{\gamma},\ddot{\gamma})=\left|\ddot{\gamma}\right|^{2}+K(\gamma,\dot{\gamma})$


$\frac{\delta J}{\delta \gamma_i} = \frac{\partial F}{\partial \gamma_i} - \frac{d}{dt}\frac{\partial F}{\partial \gamma_i'} = \frac{\partial }{\partial \gamma_i}H(\gamma(t)) - \frac{d}{dt}\frac{\partial }{\partial \gamma_i'} \sum_j\gamma_j'(t)^2 = \frac{\partial }{\partial \gamma_i}H(\gamma(t)) - 2\gamma_i''(t)$


$g(t)=F(p+t)-F(p-t)-tf(p-t)-tf(p+t)$


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


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


$[a,b] / [c,d] = \mathbb{R}$


$E(X) = 1/6$


$xRyR=yRxR$


$x = y = z;\ x = y > z;\ x > y = z;\ x > y > z;\ x > z > y;\ x = z > y;\ y > x = z;\ y > x > z;\ y > z > x;\ y = z > x;\ z > x = y;\ z > x > y;\  z > y > x.$


$e(P_3\mid P_1)=e(P_3\mid P_2)\cdot e(P_2\mid P_1)$


$x = e^{\frac{i\pi}{n}} t$


$\mathbb R^n / O(n) = \mathbb R^n$


$2 \ln |x| = \ln |x|^2 = \ln (x^2)$


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


$I_1\supseteq I_2\supseteq\dots\supseteq I_n\supseteq I_{n+1}\supseteq\dots$


$P(S \mid E) = \frac{P(E \mid S)~P(S)}{P(E)}$


$\displaystyle I_m = \int \frac1{(1+t^2)^m}dt$


$y= \frac{-Ax + Bx +C}{ B +A} $


$\begin{cases}x_1+x_2+x_3=k\\1\leq x_1\\1\leq x_2\\1\leq x_3\end{cases}$


$N(\gamma(t))=\gamma''-\left\langle \gamma'',\,\frac{\gamma'}{||\gamma'||}\right\rangle\frac{\gamma'}{||\gamma'||}=\gamma''-\frac1{||\gamma'||^2}\langle\gamma'',\gamma'\rangle\gamma'$


$  \int k_g ds + \int\int K dA = 2 \pi \tag2 $


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


$\det \begin{bmatrix} x_1&y_1&0\\x_2&y_2&0\\x_3&y_3&1\end{bmatrix}$


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


$\kappa(s) = \|\gamma''(s)\| = |\langle \gamma''(s),N(s)\rangle| = |\frac{d}{ds} \langle \gamma'(s),N(s)\rangle - \langle \gamma'(s),\frac{d}{ds}N(s)\rangle| = |\langle \gamma'(s),\frac{d}{ds} N(s)\rangle|.$


$\delta^+$


$\sin{2\varphi}=2\sin{\varphi}\cos{\varphi}.$


$||A||_2 = \sqrt{\rho (A^T A)}=\sqrt{\rho (I)}=1$


$\lim_{N \rightarrow \infty} \frac{1}{N} \sum_{n=1}^N f(x_n) = \int_0^1 f$


$ f(\delta) = \delta - (n - 1)\delta^{n-1} + n\delta^{n-1} - n\delta + n - 2 = \delta(1- n) + n - 2 + \delta^{n-1} = \delta^{n-1} - 1 + (n-1)(1 - \delta) = (\delta - 1)(\delta^{n-2} + \dots + \delta + 1 - n + 1) > 0, $


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


$K^{\times}=\left(\mathbb{F}_3[X]/(X^3-X^2+1)\right)^{\times}$


$\large\sqrt z=re^{\frac{\theta}{2}i}$


$\sum_{n\geq1}\frac{\sin^{2}\left(n\right)}{n^{2}}=\frac{1}{2}\left(\sum_{n\geq1}\frac{1}{n^{2}}-\sum_{n\geq1}\frac{\cos\left(2n\right)}{n^{2}}\right)$


$d(x,A)\leqslant d(x,y)+d(y,a)$


$\mathbb{Z}_p=\{a,a+\beta, a+2\beta,\dots , a+(p-1)\beta\}\subseteq A $


$\ldots <_M \{3\} <_M \{2\} <_M \{1\} <_M \{0\}$


$x, y, z \in S, d(x, z) \le d(x, y) + d(y, z)$


$\lim_{n \to \infty} \frac{\sqrt{1} + \sqrt{2} +\ldots + \sqrt{n}}{n\sqrt{n}}=\lim_{n \to \infty}\frac{1}{n}\sum_{k=1}^n\sqrt{\frac{k}{n}}=\int_0^1\sqrt{x}\;dx=\frac{2}{3}.$


$rx = r e^{\frac{2\pi i}{n}}$


$P(X \geq x \mid H_{0})$


$F_n \ge (\frac{1+\sqrt{5}}{2})^{n-2}$


$3^x+4^x+5^x = 1+x^2$


$\zeta(A+B) = \lim_{N\to\infty}\frac{1}{N}\sum_{k=1}^N \frac{1}{k^{A}\log\left(1 + \frac{k^{B}}{N}\right)}$


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


$f(t)= \int_0^1 \frac{\sin(xt)}{x}\:dx$


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


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


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


$\displaystyle f(k)=\frac{9^k}{9^k+3},$


$\displaystyle tan\theta = \frac{z}{2}$


$\lim\; \inf\; a_n = \lim_{n\to\infty} \inf(\bigcup_{k=n}^\infty a_k)$


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


$a\,x+b\,y=u$


$\tan^2 \theta = \dfrac{x}{y}$


$a \cdot b = |A \times B|$


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


$\sum\|e_n-f_n\| < 1$


$\det\begin{pmatrix} A & I\\B & O\end{pmatrix}=\det B$


$ \begin{align} &\sum_{n=1}^{\infty} \frac{(-1)^{n+1} }{n(n+1)} \, \eta(n) \\ &=  \sum_{n=1}^{\infty}\frac{(-1)^{n+1} }{n(n+1)}\frac{1}{\Gamma(n)} \int_{0}^{\infty} \frac{x^{n-1}}{1+e^{x}} \, dx \\ &= \int_{0}^{\infty} \frac{1}{1+e^{x}} \frac{1}{x^{2}} \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n!} \, \frac{x^{n+1}}{n+1} \, dx \\ &= \int_{0}^{\infty} \frac{1}{1+e^{x}} \frac{1}{x^{2}} \left(e^{-x}+x-1 \right) \, dx \\ &= \lim_{s \to -1^{+}} \int_{0}^{\infty} \frac{x^{s-1}}{1+e^{x}} \left(e^{-x}+x-1 \right) \, dx \\ &= \lim_{s \to -1^{+}} \left[2^{-s} \,  \Gamma(s) \left( \zeta(s) - \zeta \left(s, \frac{3}{2} \right)\right) + \Gamma(s+1) \eta(s+1 ) - \Gamma(s) \eta(s) \right] \\&= \lim_{s \to -1^{+}} \left[2^{-s} \, \Gamma(s) \left(\zeta(s) - \zeta \left(s, \frac{1}{2} \right) +2^{s}\right) + \Gamma(s+1) \eta(s+1 ) - \Gamma(s) \eta(s) \right] \tag{3} \\&= \lim_{s \to -1^{+}} \left[2^{-s} \, \Gamma(s) \left(\zeta(s) - \left(2^{s}-1\right)\zeta (s)  +2^{s}\right) + \Gamma(s+1) \eta(s+1 ) - \Gamma(s) \eta(s) \right] \tag{4} \\ &= \lim_{s \to -1^{+}} \left[\left(2^{1-s}-1 \right) \Gamma(s) \zeta(s) + \Gamma(s) + \Gamma(s+1) \eta(s+1 ) - \Gamma(s) \eta(s)\right] \\&= \lim_{s \to -1^{+}} \left[ \Gamma(s) + \Gamma(s+1) \eta(s+1) -2\Gamma(s) \eta(s) \right] \tag{5}. \end{align}$


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


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


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


$\sum_{k=1}^\color{red}{n} \frac1{2^k}$


$\begin{align} d(\alpha, A)-d(x,A) &= \inf\{ d(\alpha, a):a\in A\}-\inf\{d(x,a): a \in A\}\\ &= \inf\{ d(\alpha, a) - d(x,a): a \in A\}\\ &\leqslant \inf\{d(\alpha, x): a \in A\} = d(x,\alpha) \end{align}$


$\mathcal A x = \mathcal A$


$\int_0^1 \frac{\tanh^{-1}(x)\ln x}{x(1-x^2)} \, dx $


$\tau = \{ \emptyset, A_1, A_2, \dots \} $


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


$\forall \epsilon > 0, \exists z  \neq 0, |z - 0| < \epsilon \land z \in \{0\}$


$m=\Omega(\sqrt{n})$


$\sum_{n=1}^{\infty}\frac{1}{n^2}I_n \ge \frac{1}{N^2}I_N - \sum_{n\ne N}\frac{1}{n^2}|I_n| \ge \frac{N^3}{N^2} -  \pi^2/6,$


$0=f(\sigma)+\sigma f(\sigma)$


$\sqrt {(3-x)^2 + 2^2} $


$f(x) = b + \dfrac{a-b^2}{x+b} = \dfrac{bx + a}{x+b}$


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


$ \bar{a}r + \bar{b}s = 1 $


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


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


$ GCD(a,b) = GCD(a, b-a) = GCD (a, r_b) $


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


$d_2 \mid a$


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


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


$L = \lim_{n\to \infty } \sup(n) = \lim_{n\to \infty } \inf(n)$


$P(S\cup E)=P(S)+P(E)-P(S\cap E)$


$-y''(y'-y'')^2 + \dfrac{y'''}{y'-y''}(y''-y''') = y''''$


$[\Lambda, \partial] = i\bar{\partial}^*$


$f: P \to T, P(T) = P(P)$


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


$ P( E | S ) = \dfrac{P(S | E) \times P(E)}{P(S)} $


$\displaystyle\sum_{n=1}^{\infty}|\langle f, e_n \rangle|^2 = ||f||^2,$


$b_n=(\frac{1}{n}-\frac{1}{n^2})<a_n <\frac{1}{n}=c_n$


$(e_i)_{1\leq i\leq n}$


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


$X \subseteq Y \subseteq \overline{X}$


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


$f_r(r)= \frac{2r}{R^{2}}$


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


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


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


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


$ \frac{1}{x} + \frac{1}{y} = \frac{1}{2} $


$e_n := \|X_n\|$


$ f(x)=f(a)+\int_a^xf'(t)dt=f(a)+\int_a^xf(t-a)dt=f(a)+\int_0^{x-a}f(t)dt, $


$(k+1)^3-(k-1)^3-2^3=6(k-1)(k+1)=6k^2-6$


$ \begin{bmatrix} x & y\\ -y & x \end{bmatrix} $


$JM(A,B)J=M(A,-B),$


$\lim _{ x\rightarrow 9 }{ \frac { (9)^{ 1/2 }+(9)-6 }{ (9)^{ 3/2 }-27 }  } =\frac { 3+9-6 }{ 27-27 } =\frac { 6 }{ 0 } =\infty $


$\limsup f_n(x)\le \lim_{m\rightarrow \infty} f(y_m)=f(x)$


$\frac{dx}{dt} =  \sqrt{ {150}^2 - \frac{dy}{dt}^2 }$


$\sum_{k=1}^{\infty} \dfrac{1-e^{-k}(k+1)}{k^2} = \sum_{k=1}^{\infty} \dfrac1{k^2} - \sum_{k=1}^{\infty} \dfrac{e^{-k}}{k} - \sum_{k=1}^{\infty} \dfrac{e^{-k}}{k^2}=\zeta(2) + \log(1-1/e) - \text{Li}_2(1/e)$


$2^{|A \setminus \{1,2\}|} = 2^{|\{3,\ldots,n\}|} = 2^{n - 2}$


$A_1\subseteq A_2\subseteq \cdots$


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


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


$\sqrt{1-\sin^2\varphi} = \cos\varphi$


$\sum_{n=1}^\infty \|x_n-e_n\|^2<1$


$ \int_{-\pi}^\pi \frac{\sin^2(nx/2)}{2\sin^2(x/2)}\,\mathrm{d}x =\int_{-\pi/2}^{\pi/2} \frac{\sin^2(n u)}{\sin^2(u)}\,\mathrm{d}u \tag{2} $


$ A_1 \times ... \times A_N, $


$\sigma:P\rightarrow F=P\oplus Q$


$S \equiv R$


$ \left\{ \matrix{   \left( {m + 2} \right)a + 2mb = 1 \hfill \cr    \left( {m + 2} \right)b + 2c + 2md = 0 \hfill \cr    2a + \left( {m + 1} \right)b = 0 \hfill \cr    c = 1 \hfill \cr    2b + c + \left( {m + 1} \right)d = 0 \hfill \cr}  \right.\quad  \Rightarrow \quad \left\{ \matrix{   a = \left( {m + 1} \right)/\left( {m^{\,2}  - m + 2} \right) \hfill \cr    b =  - 2/\left( {m^{\,2}  - m + 2} \right) \hfill \cr    c = 1 \hfill \cr    d = \left( {2 - m} \right)/\left( {m^{\,2}  - m + 2} \right) \hfill \cr}  \right. $


$f(A+B)\subset f(A)+f(B)$


$aRb \lor bRa$


$\lvert ab\rvert = \lvert a\rvert\cdot \lvert b\rvert$


$COV(X,Y) = 1$


$\left[\begin{array}{ccc|c} 2 & 2 & 2 & -1 \\ 0 & 4 & -12 & 5 \\ 7 & 5 & 13 & -6 \end{array}\right]$


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


$P(n), P(m)\implies P({n\over m})$


$S_k = \{v_1, v_2, ..., v_k\}$


$\frac1T \int_0^T\left(\sum_{k=-\infty}^{\infty}c_ke^{j{\frac{2\pi kt}{T}}}\right)^2dt= \sum_{k=-\infty}^{\infty}|c_k|^2$


$\mathcal{A}=\{\{n\}:n\in \mathbb{Z}\}$


$G=P\oplus H$


$ \bbox[lightyellow] {   N_b (s,r,q + 1) = {\rm No}{\rm .}\,{\rm of}\,{\rm solutions}\,{\rm to}\;\left\{ \matrix{   {\rm 0} \le {\rm integer}\;x_{\,j}  \le r \hfill \cr    x_{\,1}  + x_{\,2}  + \; \cdots \; + x_{\,q + 1}  = s \hfill \cr}  \right.  }$


$ax + by = f$


$\gamma=a\otimes \gamma=\gamma a+\lambda a+\lambda \gamma +\mu$


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


$\mathrm{d}x\sqrt{1+\left(\frac{\mathrm{d}y}{\mathrm{d}x}\right)^2}$


$f({}n) = \frac{n{}}{2}$


$ \left[\begin{array}{rrr|r} 1&3&3&6\\2&2&1&4\\-1&-1&2&3\end{array}\right] $


$\begin{cases}x\equiv r\mod a\\x\equiv s\mod b\end{cases}$


$\frac4x + \frac9y + \frac{16}z$


$\int_0^\pi \frac{\sin(nx)}{n^2} dx$


$\sum_{m=0}^{\infty}\sum_{n=1}^{\infty}\frac{1}{3^{mn}}$


$\sqrt{1-(1-\frac{\gamma}{\beta})\cos(\theta)^2} = \sqrt{1-(1-\frac{\gamma}{\beta})(1- \sin(\theta)^2)}$


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


$w=e^{i/n}z$


$\sum_{k=1}^5\cos\frac{2k\pi}{11}=-\frac{1}{2}$


$\lim \limits_{x \to x_0+}f(x)g(x) = \lim \limits_{x \to x_0+}f(x)*\lim \limits_{x \to x_0+}g(x)$


$\int \frac{x^2}{\sqrt{x^3+2}}\, dx$


$|f(x_0)|\le x_0^n|f(z_n)|$


$c_n = \sqrt1 + \sqrt2 + \cdots + \sqrt{n}$


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


$\langle_{\mathbb{R}}=\{\langle x,y \rangle \in \mathbb{R}^2: \text{ x is smaller that y}\}$


$3^x + 9^x = 27^x$


$\int_\gamma \omega = \int_{\gamma} df = f(\gamma(2\pi)) - f(\gamma(0)) = 0.$


$T\left[ \begin{array}{}    1  \\    x  \\    x^2  \\   \end{array}  \right]$


$\mathbb{E}(\overline{X}(1-\overline{X})) = \mathbb{E}(\overline{X}-(\overline{X})^2)= \mathbb{E}\overline{X}-\mathbb{E}(\overline{X})^2$


$V\subseteq \overline{V}\subseteq W$


$ \zeta(a)=\frac{1}{\Gamma(a)}\int_{0}^{+\infty}\frac{x^{a-1}}{e^{x}-1}\,dx$


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


$\lim_{x\to c^-} f'(x) = L_1, \lim_{x\to c^+} f'(x) = L_2,$


$P[X \geq \lambda] = 1/2$


$x_\omega =\lim_{n\rightarrow \omega}\ x_n$


$A(x+y,x) = A(x,x)+ A(y,x)$


$(x,-y),(-x,-y),(-x,y),(y,x),(y,-x),(-y,x)$


$M=R\oplus T$


$\lim \limits_{n \to \infty} \sum_{k=1}^{\lfloor n/2 \rfloor} \frac {n^2}{n^2-k^2}$


$\lim_{n\to\infty}\sum_{r=1}^{n}\frac{1}{(r+2)r!}\:$


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


$ 3 > 2 \\ 3 \cdot -1 = -3 \\ 2 \cdot -1 = -1 \\ -3 < -1 \\ 3 \cdot -1 < 2 \cdot -1 $


$\int \int dS$


$ \mathbb R^n \mathrm x \mathbb R^n$


$(x,x,y,y)+(x',x',y',z')=(x+x',x+x',y+y',y+z')=(x'',x'',y'',z'').$


$i(f)=\displaystyle\binom n2$


$\frac{\sin ^{-1}(x)}{x}$


$d(x,z)\le d(x,y)+d(y,z)\;.$


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


$A\subseteq U,\ B\subseteq V,$


$Cov(a,a)=Cov(bt,bt)=Cov(a,X)=Cov(bt,X)=0,$


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


$ar + bs =1.$


$ x = \frac{ \ell^2 - a b }{ a + b - 2 \ell }, $


$6! + 2, 6!+3, 6!+4, 6!+5$


$x! \approx (\frac xe)^x \sqrt {2 \pi x}$


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


$A = \begin{bmatrix}x&y&z&1\\x_1&y_1&z_1&1\\x_2&y_2&z_2&1\\x_3&y_3&z_3&1\end{bmatrix}$


$\begin{pmatrix}x&-y\\y&x\end{pmatrix}$


$\;\color{blue}{\bf a} + \dfrac{1}{\color{red}{\bf b}+1/\color{green}{\bf c}} = \dfrac{37}{16} =  2 + \dfrac 5{16} = 2 + \dfrac 1{\frac{16}{5}} = \color{blue}{\bf 2} + \dfrac 1{\color{red}{\bf 3} + 1/\color{green}{\bf 5}}$


$x = a^{\log_a (x)}.$


$\mathbb{P}(\sup_{t}M_{t}= x \mid\mathcal{F}_{0})$


$x^*, x_n \in \mathbb{R}$


$A_n \subseteq A_{n+1}$


$1 = \frac 1x + \frac 2y - \frac 3z \le \frac 3y - \frac 3z$


$\int_0^\infty f(x)\,\mathrm{d}x$


$ (x-a)^n  \overset{\alpha}{\mapsto}  (x-a)[n(x-a)^{n-1} - 0] - 2[(x-a)^n - 0] = (n-2)(x-a)^n $


$ \|B\|_2 = \sqrt{\lambda_\max(B^t B)} $


$x!=(x/e)^x\sqrt {2\pi x}\;(1+d(x))$


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


$\left(\mathbb{F}_3[X]/(X^3-X^2+1)\right)^{\times}$


$\tan \theta = \dfrac 5 2$


$ A = \begin{bmatrix} 1 & -1 & -7 \\ 1 & 3 & -1 \\ 2 & -1 & k \end{bmatrix} $


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


$f(x) = \dfrac{a+x}{b + cx}$


$|g'(a_1)|\leq |f'(a_1)|.$


$a<(a+b)/2<b\iff 2a<a+b<2b$


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


$4. d(x,z) \leq d(x,y)+d(y,z)$


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


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


$I_n=\int{\tan^{\frac 1n} (x)}\,dx=\frac{n \tan ^{\frac{n+1}{n}}(x)}{n+1}\,\,\, _2F_1\left(1,\frac{n+1}{2 n};\frac{3n+1}{2 n};-\tan    ^2(x)\right)$


$\| \triangle ABC \| = rs$


$\displaystyle I=\int_0^{2π}(4\cos(\theta)+4i\sin(\theta)+1)^3\,d\theta$


$
 \sum_{i=1}^n\cos\left(\frac{2\pi}{n}i\right)=0\tag{10a}
 $


$(\gamma_1,-\overline{\gamma_1}),
 (\gamma_2,-\overline{\gamma_2}),\ldots,(\gamma_k,-\overline{\gamma_k})$


$\partial_i\frac{\gamma}{\|\gamma\|}=\frac{1}{\|\gamma\|}(\partial_i\gamma)  -\frac{\gamma\cdot (\partial_i\gamma)}{\|\gamma\|^3}\gamma,$


$\cos \theta = \frac{x}{3}$


$g_n(x)=\sum_{j=1}^na_j\chi_{V_j}(x)>\alpha$


$\|B\|_2 = \sqrt{\rho(B^\top B)}$


$I_1=\int_{\frac{1}{2}}^1 \frac{\sinh^{-1}(x)}{x}\,dx$


$A \meta\to B = \quote(+A+\quote\to+B+\quote)$


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


$\int_0^\pi\sin(x)\cos(x)dx=2\int_0^\pi\sin(2x)dx=0$


$\begin{split}\{  \gamma  \in\tau : r(f(\gamma))\le  1-\epsilon\} &=  \{  \gamma  \in\tau : d(f(\gamma)) \ge  \phi(1-\epsilon)\}  \\ &\subseteq \{  \gamma  \in\tau : d( \gamma ) \ge  \delta\} \\  &\subseteq \{  \gamma  \in\tau : r( \gamma ) \le  1-\tau\}  \end{split}$


$A=\big\{2x - 1 \wedge y^2 \; | \; x \in \mathbb{N}, y \in \mathbb{N}, x \leq 10\}$


$n^{12}\equiv1\mod p$


$\vert f(z) \vert = r^{-n} \vert g(z) \vert, \tag{5}$


$x=b,\, y=t, c\leq t\leq d$


$A=\{\langle x,y\rangle\in S^2:x+y\in S\setminus\Bbb Z\}$


$\liminf f_n(x)\ge \lim_{m\rightarrow \infty} f(x_m)=f(x).$


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


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


$ax+by=c\ ,$


$d(x,A)\leq d(x,a)\leq d(x,y)+d(y,a)$


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


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


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


$[x,y] = [1766319049, 226153980].$


$Q(n) \implies P(n)$


$\sqrt[m]{\alpha}\mathcal{O}_L = \prod_{i=1}^s \mathfrak{p}_i^{mk_i / m} = \mathfrak{a}\mathcal{O}_L$


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


$A_1 \subset A_2 \subset \ldots \subset A_m \subset A_{m+1} \subset \ldots$


$\tan^2\theta=\frac ba$


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


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


$\frac {1-(1+R)^{T+1}}{R}$


$ z = e^{\frac{2i\pi}{5}} $


$||A||_2=\sigma_1(A)=\sqrt{\lambda_{max}(A^TA)}$


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


$\gamma( a \vee b) = \gamma(a) \wedge \gamma(b)$


$S\equiv R^n$


$x\ R\ y \Leftrightarrow y\ R\ x$


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


$g'(x)=2f(x)\int_0^xf(t)dt-f(x)^3=f(x)\left[2\int_0^xf(t)dt-f(x)^2\right].$


$\gamma \leq \gamma = \gamma+0$


$P[M_n > x \mid X_0]$


$\int_0^1\frac{x^n(x-1)^n}{n!}e^xdx=\int-\frac{x^n(x-1)^{n-1}}{(n-1)!}e^x-\frac{x^{n-1}(x-1)^n}{(n-1)!}e^xdx $


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


$f(x,y)=e^\frac{-x^2}{2t}-e^\frac{-y^2}{2t}$


$4^x = 3\cdot 6^x$


$\mathbb A=\{\{a\}:a\in A\}$


$[x,y]=(Ax,y)$


$\sum_{n=1}^{+\infty}\frac{(-1)^n}{n^4} = -\sum_{n=1}^{+\infty}\frac{1}{n^4}+2\sum_{m=1}^{+\infty}\frac{1}{(2m)^4} = \left(-1+\frac{2}{16}\right)\sum_{n=1}^{+\infty}\frac{1}{n^4}=-\frac{7}{8}\zeta(4) = -\frac{7\pi^4}{720}.$


$(1-2^{1-z})\zeta(z)=\sum\limits_{s=1}^\infty \frac{(-1)^s}{s^z}=\frac{1}{\Gamma(z)}\int\limits_{0}^\infty \frac{t^{z-1}}{e^t+1}dt$


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


$C=\{\{A\}\}$


$B = \{b_1, b_2, b_3, \ldots, b_n\}$


$2^{log_3 x}+x^{log_3 2}=4$


$\zeta (s)=\sum_{n=1}^\infty \frac {1}{n^s}=\frac {1}{\Gamma (s)} \int _0^{\infty}\frac { x^{s-1}}{e^x-1} dx$


$\sum_{n=1}^\infty \|e_n-f_n\|<\infty.$


$\lim_{n\to\infty}\left(\sum_{k=1}^{2n}\frac{1}{2n}\left\lfloor\frac{2n}{k}\right\rfloor-\sum_{k=1}^{n}\frac{1}{n}\left\lfloor\frac{n}{k}\right\rfloor\right)=L_1,$


$[x,_0\, y]=x$


$\mathbb{E}^+$


$d(x,M) = \delta > 0$


$\begin{align} \int_{-\pi/2}^{\pi/2} \frac{x^n \left(\cos x\right)^{n-2}}{\left(\sin x\right)^n} \, {\rm d}x &\stackrel{y=2x}{=} 2^{-n+1} \, i^n \int_{-\pi}^{\pi} y^n \, e^{-iy} \, \frac{\left(1+e^{-iy}\right)^{n-2}}{\left(1-e^{-iy}\right)^n} \, {\rm d}y \\ &\stackrel{z=e^{iy}}{=} -i \, 2^{-n+1} \int_\gamma \left(\log z\right)^n \, \frac{(z+1)^{n-2}}{(z-1)^n} \, {\rm d}z \end{align}$


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


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


$c:=-\max_{0\leq\theta\leq a}\mathrm{Re}(\gamma(\theta))\quad(>0)$


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


$ (*) \Leftrightarrow \tan(a+\gamma)=\frac{\frac{b}{c}-\cos\gamma}{\sin \gamma}$


$\sum_{k=1}^{\infty} \sum_{n=1}^{\infty} = \sum_{n=1}^{\infty} \sum_{k=1}^{\infty}$


$p_2 \mid p_1^k-1$


$[x,e]=x$


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


$\langle f,m\rangle\in\mathscr{D}$


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


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


$A= [a_1, a_2, a_3, ... , a_n]$


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


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


$\tan\theta=\frac x3$


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


$\begin{cases}x\equiv r\pmod a\\x\equiv s\pmod b\end{cases}\iff x\equiv sua+rvb\pmod{ab}.$


$\dim(M)+\dim(H)=\dim(M+H)=\dim(M+K)=\dim(M)+\dim(K)$


$\frac{1}{\sqrt1+\sqrt3}=\frac{\sqrt3-\sqrt1}{2}$


$[u,v] = [\gamma(a), \gamma(b)] \subseteq \gamma([a,b]) \subseteq I.$


$\hat{M} = \frac{X_{1} + X_{2} + ... + X_{n}}{n}$


$\int{\frac{1}{(x^4 -1)^2}}\, dx$


$\forall \epsilon > 0, \exists \delta >0 \text{ such that, } 0<|x-a|<\delta\implies |f(x) - L| < \epsilon$


$ \begin{bmatrix} z & -\bar{w} \\ w & \bar{z} \end{bmatrix} $


$\;a_1\cdot\ldots\cdot a_n\;$


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


$(M,M,M,m)$


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


$\bigcup\mathcal{P}(\mathcal{A}) = \bigcup_{{\mathcal X} \subseteq {\mathcal A}} {\mathcal X}.$


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


$F = n - k$


$p_1p_2 > x$


$\underset{n\to \infty }{\mathop{\lim }}\,\sum\limits_{k=1}^{n}{\frac{1}{{{2}^{k}}}}$


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


$\gamma_{3}(G)=[\gamma_{2}(G),G]=[\gamma_{3}(G),G]=\gamma_{4}(G)$


$1-p=\int_0^1(1-z^{2/3})^2dz\stackrel{z^2=t^3}=\int_0^1(1-t)^2\frac32t^{1/2}dt=\frac32\frac{\Gamma(3)\Gamma(3/2)}{\Gamma(9/2)}$


$\forall \varepsilon >0 \exists K>0: x>K \Rightarrow \left | f(x)-l \right |<\varepsilon$


$\sum_{k = -\infty}^{\infty}e^{-k^2\tfrac{\pi}{n}} = \sum_{k = -\infty}^{\infty}\sqrt{n}e^{-n\pi k^2}.$


$f(x) = 1/|x|^\alpha$


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


$f_x(X) = \frac{2}{x^3}$


$b^+$


$\begin{pmatrix}x&y\\-y&x\end{pmatrix}$


$x = \sin \theta + \tan \theta, y = \sin \theta - \tan \theta$


$[X,Y] = X\circ Y - Y\circ X$


$n!\delta(a)^n=\delta^n(a^n)$


$(a+\Delta x, f(a+\Delta x))$


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


$ Z := \left\{ (x,y) \in \mathbb{R}^2 \; : \; b \leq x \leq T, \; f(x) \leq y \leq R \right\} $


$e^x - x - 1, x>0$


$p_1^* = \frac{1/(1-f)}{1 + 2^{(H_2(f)/(1 - f))}}$


$\mathbb F_3^N$


$\vec r= \begin{cases}  x=u\\  y=a\cos v\\  z=a\sin v \end{cases}$


$a_j = B I_j$


$\overline{V}\subseteq p(U')$


$P(\{ \omega \}) = 1/6$


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


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


$|ab| = L$


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


$y!\approx \sqrt{2πy}(\frac ye)^y$


$F(s)L[e^{ax}]=\int_0^\infty e^{-sx}e^{ax}dx=\lim\limits_{R \to \infty} \int_0^R e^{(a-s)x}dx=\lim\limits_{R \to \infty}[\frac {e^{(a-s)}x} {a-s}]=\lim\limits_{R \to \infty}[\frac {e^{(a-s)^R} -1} {a-s}]=\frac {1}{s-a}(s>a)$


$ \sum_{h=0}^Nhr^h=r+2r^2+3r^3+...+Nr^N=r\frac{1-r^{N+1}}{(1-r)^2}-\frac{(N+1)r^{N+1}}{1-r}, \quad |r|<1, \tag2 $


$d(x, A) := \inf\{d(x, y): y ∈ A\}$


$\int_0^\infty P(X > t)dt = \int_0^\infty\int_t^\infty f(x)dxdt = \int_0^\infty f(x) \int_0^xdt dx = \int_0^\infty f(x)*x dx = E(X).$


$ \begin{align} \int_0^1\left(\frac{\sin^{-1}(x)}{x}\right)\mathrm{d}x &=\int_0^{\pi/2}\left(\frac{t}{\sin(t)}\right)\mathrm{d}\sin(t)\tag{22}\\ &=\int_0^{\pi/2}t\,\mathrm{d}\log(\sin(t))\tag{23}\\ &=-\int_0^{\pi/2}\log(\sin(t))\,\mathrm{d}t\tag{24}\\ &=\bbox[5px,border:2px solid #C0A000]{\frac\pi2\log(2)}\tag{25} \end{align} $


$Cov(X,Y)\neq 0$


$ \left[ \begin{array}{@{}ccc|c@{}} 1 & -2 & 0 & 2 \\ 0 & 1 & 2 & -2 \\ 5 & -6 & 8 & 6 \\ \end{array} \right] $


$a \notin \{\gamma \}$


$ax + by = d \implies d \in H$


$\lim_{N\rightarrow\infty}\sum_{k=1}^N\left(\frac{k-1}{N}\right)^N$


$p(T|A,F)=P(T|F)$


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


$p_2q_2 \mid n$


$\int_0^1 \dfrac{\sin(x)}{x} ~dx \approx 0.9459962252$


$f(a + b) = f(a) + f(b)\;\; \forall a, b \in \mathrm{dom}(f)$


$\sqrt[n]{a} = |\sqrt[n]{a}|e^{2\pi i k/n}$


$\Psi\begin{pmatrix}a&b\\b&a\end{pmatrix}=(a+b,a-b).$


$\lim_{N\to\infty}\sum_{k=1}^N \frac{1}{k+\alpha N}$


$  \operatorname {dim}k[x,y,z]/(x^3-1)=\operatorname {dim}_k( \frac {k[x]}{(x^3-1)})[y,z]=0+2=2$


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


$f(a,b,k) = f(0,b-a,k)$


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


$(14 - (k-1))^3= k^3$


$\sum_{m=1}^\infty \sum_{n=1}^\infty$


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


$\int_0^\pi + \int_\pi^{2\pi}$


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


$\lim_{n\to\infty}\inf \ f_n$


$\forall \epsilon > 0, \exists\delta>0, s.t. 0<|x-x_0|<\delta \implies |f(x)-f(x_o)|< \epsilon$


$ 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\} $


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


$\frac{(1+\sqrt{3})^{n+2}-(1-\sqrt{3})^{n+2}}{4\sqrt{3}}.$


$\sum_{n=0}^\infty a_n = \int_0^1\frac{\sin(\pi x)}{(1-x)^2}dx=\infty$


$\{ e_1,e_2,e_3,u \}$


$f_k(n) = \binom{n}{k}$


$ \int_{0}^{\pi}f(x)\cos kxdx=\int_{0}^{\pi}f(x)\sin kxdx=0 $


$S_n = \{x_1, x_2, x_3, \dots, x_n\}$


$x\mapsto x^3-x$


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


$\int\limits_{0}^{1} \frac{\tan^{-1}(x)}{x} \mathrm{d}x$


$ \dfrac{\sqrt{n}+1}{n\sqrt{n}+\sqrt{n}+1}\ge \frac{\sqrt{n}}{3n\sqrt{n}}=\frac{1}{3}\frac{1}{n} $


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


$t_1 \in [0, t_0],$


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


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


$\gamma[\lambda a(f)]:=\lambda\gamma[a(f)]\\ \gamma[a(f)+a(g)]:=\gamma[a(f)]+\gamma[a(g)]\\ \gamma[a(f)a(g)]:=\gamma[a(f)]\gamma[a(g)]\\ \gamma[a(f)^*]:=\gamma[a(f)]^*$


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


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


$N(A_1\dots A_n)$


$\mathbb{E}\left[\left| X + l^{r+1}Y^{r+1}\right|\right] \geq \mathbb{E}\left[\left| X \right|\right]$


$\lfloor a / b\rfloor = c$


$u_{n}=\sum\limits_{r=1}^{n} \frac{r}{2^r}$


$x R^T y \Leftrightarrow y R x$


$ \zeta(s)=\sum_{n\geq 1}\frac{1}{n^s} = \sum_{n\geq 1}\int_{0}^{+\infty}\frac{u^{s-1}}{\Gamma(s)}e^{-nu}\,du = \frac{1}{\Gamma(s)}\int_{0}^{+\infty}\frac{u^{s-1}}{e^u-1}\,du$


$p_1 p_2 \equiv p_3 p_4$


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


$ \left\{ \begin{array}{ll} \frac{\lambda x\sqrt{3}}{2}=1\\  \frac{\lambda y\sqrt{3}}{2}=1\\ \frac{x^2\sqrt{3}}{4}+\frac{y^2\sqrt{3}}{4}-2=0 \end{array} \right.$


$f'(x) = 5 f(x)$


$\frac{\gamma(t)-\gamma(s)}{|\gamma(t)-\gamma(s)|} =  \frac{\gamma(\tilde t)-\gamma(s)}{|\gamma(\tilde t)-\gamma(s)|} \to -\gamma'(0)$


$(x+y)^r = \sum^{\infty}_{k=0}\binom{r}{k}x^{r-k}y^{k}$


$\epsilon(a,\Delta x) = \frac{\Delta y}{\Delta x}-f'(a),$


$a \hat{x} + b \hat{y} = d$


$L = \lim_{x\to c} f_1(x)$


$a^{k*\log_a b}=b^k $


$\gamma(\phi(xy)) = \gamma(\phi(x)\phi(y)) = \gamma(x'y') = \gamma(x')\gamma(y') = [\gamma(\phi(x))][\gamma(\phi(y))]$


$x! \sim \sqrt{2\pi x} (x/e)^x$


$\det(A) = \det(A_{n, n-1}) + a_{n,n}\det(A_{n,n})$


$ \frac{\pi(1-\sin\theta_0\cos\varphi_0)}{\sin^2\theta_0\sin^2\varphi_0+\cos^2\theta_0} $


$\forall x \forall y (xRy \lor yRx)$