...ll y \exists z (x \le z \& z \le y), \quad \exists z(1 \le z \& z \le 4), \quad 1 \le 2\]
\[\exists z(x \le z \& z \le y), \quad z \le 1\]
1 KB (203 words) - 05:16, 21 June 2013
$$f^{(k)}(x_0)=P_n^{(k)}(x_0),\quad k=0,\dotsc,n.$$
...style="width:94%;text-align:center;">$f(x)-P_n(x)=o\left((x-x_0)^n\right),\quad x\to x_0,$</td> <td valign="top" style="width:5%;text-align:right;">(*)</td
1 KB (233 words) - 17:01, 16 March 2013
$$ \phi_i(t) = \{ \phi_i^1(t), \ldots , \phi_i^n(t) \}, \quad i = 1, \ldots, n \tag{1} \label{eq-1} $$
W ( \phi _{1} (t), \dots, \phi _{n} (t) ) \quad \equiv \quad 0,\quad\quad
6 KB (889 words) - 12:21, 19 March 2023
$$f(x_0+h)-f(x_0)=o\left(\frac{1}{\log1/|h|}\right),\quad|h|\to+0,$$
$$a_n,b_n=O(n^{-\delta}),\quad n\to+\infty,$$
861 bytes (143 words) - 17:25, 19 September 2014
$$\cos z=\frac{e^{iz}+e^{-iz}}{2},\quad\sin z=\frac{e^{iz}-e^{-iz}}{2i}.$$
$$\cos x=\frac{e^{ix}+e^{-ix}}{2},\quad\sin x=\frac{e^{ix}-e^{-ix}}{2i}$$
1 KB (171 words) - 12:50, 10 August 2014
$$(R(g)f)(g_1)=f(g_1g),\quad g,g_1\in G,\quad f\in E,$$
$$(L(g)f)(g_1)=f(g^{-1}g_1),\quad g,g_1\in G,\quad f\in E,$$
2 KB (386 words) - 07:53, 30 November 2014
B(p,q) = \frac{q-1}{p+q-1}B(p,q-1), \quad q > 1.
B(p,1-p) = \frac{\pi}{\sin p\pi}, \quad 0 < p < 1.
1,009 bytes (161 words) - 17:31, 11 November 2023
$$0\leq x\leq1,\quad a>0,\quad b>0,$$
$$I_x(a,a)=\frac12I_{1-y}\left(a,\frac12\right),\quad y=4\left(x-\frac12\right)^2,\quad0<x\leq\frac12.$$
2 KB (393 words) - 15:50, 31 March 2017
$$\phi_{n+1}(x)=o(\phi_n(x)),\quad x\to x_0,\quad x\in M,$$
1,018 bytes (175 words) - 19:31, 13 April 2024
$$\dot x=f(x),\quad f\in C^1(G),\quad G\subset\mathbf R^n.$$
1 KB (167 words) - 15:27, 22 September 2014
$$x'=x-v_xt,\quad y'=y-v_yt,\quad z'=z-v_zt,\quad t'=t,$$
2 KB (317 words) - 16:38, 30 July 2014
$$x=f(t),\quad y=g(t),\quad z=h(t),$$
946 bytes (164 words) - 17:20, 30 July 2014
\text{div } \mathbf{v} = 0 \quad \text{or} \quad \frac{\partial v_x}{\partial x} + \frac{\partial v_y}{\partial y} + \frac{\
1 KB (175 words) - 12:28, 30 December 2018
$$\phi(x_1,\dots,x_n)=0,\quad\psi(x_1,\dots,x_n)=0$$
\begin{equation}\dot x_i(s)=Q_{\xi_i},\quad\dot\xi_i=-Q_{x_i},\quad i=1,\dots,n.\label{*}\end{equation}
3 KB (419 words) - 22:32, 10 December 2018
$$x=-f'(t),\quad y=-tf'(t)+f(t),\quad a<t<b;\label{2}\tag{2}$$
$$y=Cx+f(C),\quad C\in(a,b),\label{3}\tag{3}$$
3 KB (414 words) - 17:31, 14 February 2020
$$x=x(u),\quad y=y(u),\quad z=z(u),$$
1 KB (233 words) - 16:51, 8 April 2023
$$x=x(u),\quad y=y(u),\quad z=z(u),$$
1 KB (215 words) - 09:43, 15 August 2023
$$\{x_i\}_0^{n+1}\subset Q,\quad a\leq x_0<x_1<\ldots<x_{n+1}\leq b,$$
2 KB (297 words) - 18:38, 26 September 2014
x ^ {( k ) } = x ( x - 1 ) \dots ( x - k + 1 ) , \quad k \geq 1,
} , \quad n \geq k \geq 1,
3 KB (372 words) - 22:15, 5 June 2020
$$x=f_1(t),\quad y=f_2(t),\quad z=f_3(t),$$
1 KB (238 words) - 17:50, 13 August 2023
$$x=a\cos t,\quad y=a\sin t,\quad z=ht,$$
$$x=ce^{mt}\cos t,\quad y=ce^{mt}\sin t,\quad z=ce^{mt}\operatorname{cotan}\alpha,$$
3 KB (537 words) - 15:42, 29 September 2014
(AB)_{ik} = \sum_{j=1}^n a_{ij} b_{jk}, \quad i=1,\ldots,m;\ \ j=1,\ldots,p.
(A \circ B)_{ij} = a_{ij} b_{ij}, \quad i=1,\ldots,m;\ \ j=1,\ldots,n.
2 KB (289 words) - 19:22, 11 November 2023
{\dot{x} } = f ( x ) , \quad x \in \mathbf R ^ {n} ,
{\dot{y} } = g ( y ) , \quad y \in \mathbf R ^ {n}
3 KB (414 words) - 19:37, 5 June 2020
$$\frac{1}{(q+1)^{n+1}}\sum_{k=0}^n\binom nkq^{n-k}S_k,\quad S_k=\sum_{n=0}^ka_n,$$
188 bytes (34 words) - 14:11, 23 July 2014
\[L(x_1)=f(x_1),\quad L(x_2)=f(x_2).\]
\[ f(x)-L(x)=\frac{f''(\xi)}{2}(x-x_1)(x-x_2),\quad \xi\in [x_1,x_2].\]
1 KB (234 words) - 21:21, 15 July 2012
$$f(x)=a(x-x_0)^n+o((x-x_0)^n),\quad a\neq0,$$
$$g(x)=b(x-x_0)^m+o((x-x_0)^m),\quad b\neq0;$$
2 KB (420 words) - 18:01, 3 August 2014
$$\lambda_{k+1}-\lambda_k>\theta\lambda_k,\quad k=0,1,\ldots,\quad \theta>0,$$
1 KB (203 words) - 06:24, 22 April 2012
$$\sin\alpha=\frac{PQ}{OP},\quad\cos\alpha=\frac{OQ}{OP},\quad\tan\alpha=\frac{PQ}{OQ}.$$
1 KB (229 words) - 14:42, 14 February 2020
n! \approx \sqrt{2\pi n}\; n^n e^{-n}, \quad n \rightarrow \infty,
\Gamma(z+1) \approx \sqrt{2\pi z}\; z^z e^{-z}, \quad
1 KB (205 words) - 15:21, 14 February 2020
\rho ( u ) = 1 \quad ( 0 \leq u \leq 1 ) ,
u \rho ^ \prime ( u ) = - \rho ( u - 1 ) \quad ( u > 1 ) .
2 KB (376 words) - 08:55, 10 November 2023
$$\{x_i\}_0^{n+1}\subset Q,\quad x_0<\dotsb<x_{n+1},$$
997 bytes (159 words) - 12:37, 14 February 2020
$$(\pm1,0,0),\quad(0,\pm1,0)\quad(0,0,\pm1);$$
2 KB (244 words) - 14:08, 30 July 2014
$$p(x)=\frac{(n\mu)^n}{\Gamma(n)}x^{n-1}e^{-n\mu x},\quad x>0,$$
$$\frac{\alpha^n}{\Gamma(n)}x^{n-1}e^{-\alpha x},\quad x>0.$$
2 KB (279 words) - 19:37, 4 December 2016
$$ \begin{equation} (Lu)(x,y)=Mu-Nu=0, \quad x \in \mathbf{R}^n, \quad y \in \mathbf{R}^m, \label{1}\tag{1} \end{equation} $$
$$Mv+\lambda v=0,\quad Nw+\lambda w=0.\label{3}\tag{3}$$
4 KB (653 words) - 15:15, 14 February 2020
...e t \le 1} |f(t) - S_n(t, f)| \le 8 \omega \left( \frac{1}{n}, f \right), \quad n=1, 2, \dots
...mega \left( \frac{1}{2^m}, f \right), \quad n=2^m+k,\quad k=1, \dots, 2^m,\quad m=0, 1, \dots,
5 KB (754 words) - 21:07, 1 June 2013
\chi_1(t) \equiv 1\quad \text{ on } [0,1];
\chi_n = \begin{cases} \sqrt{2^m} \quad &&\text{ for } t\in\left(\frac{2k-2}{2^{m+1}}, \frac{2k-1}{2^{m+1}} \right
6 KB (1,003 words) - 05:01, 23 July 2018
\quad \text{and} \quad
2 KB (275 words) - 15:05, 14 February 2017
$$u_{xx}-y^mu_{yy}=0,\quad y>0,\quad0<m=\mathrm{const}<1,$$
$$u_{xx}-y^mu_{yy}+au_x+bu_y+cu=0,\quad y>0,\quad m>0,$$
3 KB (535 words) - 22:45, 10 December 2018
...1 } } { \nu _ { 2 } } x \right) ^ { ( \nu _ { 1 } + \nu _ { 2 } ) / 2 } , \quad x > 0, \end{equation*}
\begin{equation*} \frac { \nu _ { 2 } } { \nu _ { 2 } - 2 } \quad \text { for } \nu _ { 2 } > 2 \end{equation*}
7 KB (873 words) - 07:45, 27 January 2024
$$\sum_{k=0}^\infty u_k=s(P),\quad\lim s_n=s(P),\quad P-\lim s_n=s,$$
$$\sum_{k=0}^\infty u_k\quad\text{and}\quad\sum_{k=0}^\infty v_k$$
4 KB (617 words) - 17:36, 14 February 2020
$$|(z-z_1)\dotsm(z-z_n)|=r^n,\quad r>0,\quad z=x+iy.$$
2 KB (304 words) - 06:49, 7 April 2023
$$\frac{dx^i}{dt}=v^i,\quad\frac{dv^i}{dt}=f^i(x^1,\dots,x^n,v^1,\dots,v^n),\quad i=1,\dots,n,$$
$$f^i=\sum\Gamma_{jk}^i(x^1,\dots,x^n)v^jv^k,\quad\Gamma_{jk}^i=\Gamma_{kj}^i.\label{*}\tag{*}$$
3 KB (468 words) - 15:19, 14 February 2020
$$\gamma\geq0,\quad-\infty<\beta<\infty,$$
$$\phi_j(t)=\exp(-\gamma_jt^2+i\beta t),\quad\gamma_j\geq0,\quad-\infty<\beta<\infty.$$
4 KB (647 words) - 19:21, 24 March 2023
$$\def\rk{\textrm{rk}\;}\rk(f-E) = 1\quad\textrm{and}\quad \textrm{Im}(f - E)\subseteq \ker(f-E),$$
2 KB (271 words) - 20:19, 5 March 2012
$$x=\frac{x_1+\lambda x_2}{1+\lambda},\quad y=\frac{y_1+\lambda y_2}{1+\lambda}.$$
457 bytes (76 words) - 09:24, 13 April 2014
...a\in\Theta_0}\beta^*(\theta)=\alpha,\quad\beta^*(\theta)\geq\beta(\theta),\quad\theta\in\Theta_1,$$
2 KB (319 words) - 11:19, 13 August 2014
...bers $ a_i, b_i, \quad i = 1, \dots, n $. For complex numbers $ a_i, b_i, \quad i = 1, \dots, n$, it reads
2 KB (248 words) - 09:09, 31 May 2016
e _ {i} e _ {j} + e _ {j} e _ {i} = 0, \quad i \neq j,
e _ {i} ^ {2} = 1, \quad i = 1 \dots p,
9 KB (1,297 words) - 17:44, 4 June 2020
$$\left(\frac{-1}{p}\right)=(-1)^{(p-1)/2}\quad\text{and}\quad\left(\frac2p\right)=(-1)^{(p^2-1)/8}.$$
2 KB (295 words) - 17:43, 19 December 2014
$$S=\int\limits_{P_0}^{P_1}\sqrt{2(U+h)}ds,\quad ds^2=\sum_{i,j=1}^na_{ij}dq_idq_j$$
$$T=\frac12\sum_{i,j=1}^na_{ij}\dot q_i\dot q_j,\quad\dot q_i\equiv\frac{dq_i}{dt}.$$
2 KB (288 words) - 09:49, 5 August 2014
$$\sup_{x\in X}\inf_{y\in Y}F(x,y),\quad\max_{x\in X}\min_{y\in Y}F(x,y),\quad\text{etc.}\label{*}\tag{*}$$
2 KB (318 words) - 17:19, 14 February 2020
} \right ] , \quad t \geq 0,
F ( t \mid S ) = F ( a ( t ) \mid S _ {u} ) , \quad t \geq 0.
6 KB (853 words) - 21:12, 4 April 2020
$$AB=I_m,\quad BA=I_n,\quad m\neq n,$$
2 KB (351 words) - 12:24, 14 February 2020
$$A(x)\phi(x)+\int\limits_DK(x,s)\phi(s)ds=f(x),\quad x\in D.$$
378 bytes (72 words) - 16:46, 7 July 2014
$$\left\lbrace\sqrt\frac2\pi\cos nx\right\rbrace,\quad n=0,1,\ldots,$$
500 bytes (88 words) - 18:23, 14 August 2014
$$X'=A(t)X,\quad t\in\mathbf R,\label{1}\tag{1}$$
$$x'=F(t)x+G(t)u,\quad x\in\mathbf R^n,\quad u\in\mathbf R^m,$$
4 KB (628 words) - 17:38, 14 February 2020
$$\left(\frac pq\right)\quad\text{and}\quad\left(\frac qp\right)$$
2 KB (304 words) - 19:26, 14 August 2014
\sigma_n^k = \frac{S_n^k}{A_n^k} \rightarrow S, \quad n \rightarrow \infty,
\sum_{n=0}^\infty A_n^k x^n = \frac{1}{(1-x)^{k+1}}, \quad
3 KB (519 words) - 15:44, 7 May 2012
$$u_{rr}=\frac{\partial u_r}{\partial r},\quad u_{\theta\theta}=\frac1r\frac{\partial u_\theta}{\partial\theta}+\frac{u_r}
...ad u_{\phi\phi}=\frac1r\frac{\partial u_\phi}{\partial\phi}+\frac{u_r}{r},\quad u_{zz}=\frac{\partial u_z}{\partial z},$$
4 KB (616 words) - 15:37, 14 February 2020
$$\dot x=F(x),\quad x\in\mathbf R^n,\quad F(0)=0,\label{*}\tag{*}$$
2 KB (299 words) - 15:31, 14 February 2020
$$\def\a{\alpha}\def\b{\beta}(\a\b)(g) = \a(g)\b(g),\quad g\in G,\quad \a,\b\in X(G),$$
$$X(T)\simeq \Z,\quad X(\Z)\simeq T,\quad X(\R) \simeq \R,\quad X(G)\simeq G $$
5 KB (750 words) - 20:22, 17 August 2015
$$\cos\frac{2\pi}{b-a}kx,\quad\sin\frac{2\pi}{b-a}kx,\quad k=0,\dots,N-1.$$
2 KB (295 words) - 17:35, 24 March 2018
$$a_n=2r_n\tan\frac\pi n,\quad S_n=nr_n^2\tan\frac\pi n.$$
474 bytes (96 words) - 15:30, 13 October 2014
B(p,q) = \int_0^1 x^{p-1}(1-x)^{q-1}\rd x, \quad p,q > 0,
431 bytes (67 words) - 21:21, 29 April 2012
\| f \|_p = \left[ \int_E |f(x)|^p \; dx \right]^{1/p}, \quad f \in L_p(E)
491 bytes (78 words) - 21:36, 29 May 2016
\eta(\xi) = \Phi(x)y(x),\quad
\xi = \int_\alpha^x \sqrt{r(t)}\,\mathrm{d}t, \quad
2 KB (276 words) - 00:23, 30 July 2012
\{x_k\} \sim \{y_k\} \quad \iff \quad \lim_{k\to\infty} d (x_k, y_k)= 0\, .
2 KB (340 words) - 15:24, 18 October 2014
...ac{1}{k_n}\int\limits_S\nu(y)\frac{\partial}{\partial n_x}E_n(x,y)dS(y)=0,\quad x\in S,\label{1}\tag{1}$$
$$E_2(x,y)=\ln\frac{1}{|x-y|},\quad E_n=\frac{1}{|x-y|^{n-2}}$$
5 KB (913 words) - 15:34, 14 February 2020
$$u=\frac{4x}{x^2+y^2},\quad v=\frac{4y}{x^2+y^2}$$
698 bytes (110 words) - 15:19, 17 July 2014
..._1}{\b_1}\cdots\binom{\a_n}{\b_n}=\frac{\a!}{\b!(\a-\b)!},\qquad \text{if}\quad \a\geqslant\b.
...partial}{\partial x_1},\dots,\frac{\partial}{\partial x_n}\biggr)=\partial\quad\text{if the choice of $x$ is clear from context.}
3 KB (512 words) - 06:16, 13 June 2022
$$k=0,\ldots,n-1,\quad\phi=\arg\left(-\frac ba\right).$$
524 bytes (94 words) - 17:30, 23 December 2014
$$x=\operatorname{ci}(t),\quad y=\operatorname{si}(t),$$
910 bytes (138 words) - 19:17, 11 January 2024
$$u_{tt}-\Delta u+\frac{\lambda(t,x)}{t}u_t=0,\quad t\neq0,$$
$$u(t,x)\bigg|_{t=0}=\phi(x),\quad u_t(t,x)\bigg|_{t=0}=0,$$
3 KB (478 words) - 11:53, 20 April 2012
x = v \quad y = c \cosh \frac{v}{c} \sin u \quad z = c \cosh \frac{v}{c} \cos u\, .
2 KB (303 words) - 20:07, 12 November 2023
$$x''+p(t)x'+q(t)x=0,\quad p\in C^1,\quad q\in C,\label{2}\tag{2}$$
2 KB (337 words) - 17:29, 14 February 2020
$$A\times_C B,\quad A\times_{\a,\b}B, \quad \textrm{ or }\quad A\prod_{\a,\b} B.$$
3 KB (575 words) - 10:30, 23 November 2013
$$v=\frac{q^{n+1}-1}{q-1},\quad k=\frac{q^n-1}{q-1},\quad\lambda=\frac{q^{n-1}-1}{q-1}.$$
$$v=p^r=4t-1,\quad k=2t-1,\quad\lambda=t-1.$$
5 KB (832 words) - 10:37, 15 November 2014
...right) \Gamma \left(n+1 - \textstyle{\frac{m}{2}}\right)} - {\bf H}_n (x) \quad \mbox{for }\; n\geq 0
...1}}{\Gamma \left(m + \textstyle{\frac{3}{2}}\right)} - {\bf H}_{-n} (x)\, \quad \mbox{for }\; n > 0\, ,
5 KB (721 words) - 08:49, 22 February 2014
$$\pi(x)>\operatorname{li}x-ax\ln^{-m}x,\quad\pi(x)<\operatorname{li}x+ax\ln^{-m}x$$
$$\theta(x)=\sum_{p\leq x}\ln p,\quad\psi(x)=\sum_{p^m\leq x}\ln p,$$
3 KB (542 words) - 16:58, 14 February 2020
\frac{x}{a}\left(1+\frac{x}{a}\right) = 1 \quad \text{or} \quad x = \frac{a}{1+\frac{x}{a}}
2 KB (325 words) - 21:24, 3 April 2015
$$u(x)=\phi(x),\quad x\in S,$$
810 bytes (116 words) - 14:55, 16 October 2014
x = \rho \cos t \quad y = \rho \sin t \quad z = \rho \arctan \alpha + kt\, .
2 KB (310 words) - 19:56, 12 November 2023
$$\inf[a,b]=a,\quad\sup[a,b]=b;$$
$$\inf(a,b)=a,\quad\sup(a,b)=b;$$
4 KB (722 words) - 19:45, 7 January 2015
$$v_x=\frac{\partial\psi}{\partial y},\quad v_y=-\frac{\partial\psi}{\partial x}.$$
903 bytes (145 words) - 15:08, 30 July 2014
$$f,g\in L^\#,\quad x\in L,\quad \l\in k.$$
$$x=\sum_{i=1}^n\l_{\nu_i}e_{\nu_i},\quad \l_{\nu_i}\in k,\quad
5 KB (903 words) - 21:31, 3 January 2021
$$x^{k+1} = x^k - \def\a{\alpha} \a_k f'(x^k),\quad k = 0,1,\dots ,$$
$$x^{k+1} = x^k - \a_k(Ax^k -f), \quad k = 0,1,\dots ,$$
3 KB (519 words) - 19:10, 6 April 2012
$$\dot z=A(t)z,\quad z\in\CC^n,\ A(t)=\|a_{ij}(t)\|_{i,j=1}^n$$
|y_*(t)|\le C|t-t_*|^{-d},\quad\text{resp.,}\quad \|z_*(t)\|\le C|t-t_*|^{-d},\qquad 0<C,d<+\infty
5 KB (815 words) - 21:42, 31 July 2015
$$x=\pi n,\quad n=0,\pm1,\pm2,\mathinner{\ldotp\ldotp\ldotp\ldotp}$$
824 bytes (127 words) - 14:15, 14 February 2020
$$\mathbf{E}X_j=0,\quad \mathbf{E}X_j^2=\sigma^2>0,\quad\mathbf{E}\lvert X_j\rvert^3<\infty.$$
2 KB (303 words) - 11:49, 23 November 2023
$$\sum_{j=1}^na_{ij}x_j=b_i,\quad i=1,\dots,m,$$
$$x_1\geq1,\quad x_2\geq1,\quad x_3\geq1,\quad t\geq3,$$
5 KB (700 words) - 19:16, 4 November 2014
F_x+F_y=F_{x+y};\quad\lambda F_x=F_{\lambda x},\quad\lambda\in K.
x=\sum_{\alpha}\lambda_\alpha x_\alpha,\quad\lambda_\alpha\in K,
14 KB (2,558 words) - 11:28, 21 June 2016
\[ g_k(\delta) = \frac{1}{\Gamma(1 + \delta k)}, \quad \delta > 0, \quad k = 0, 1, \dots, \]
2 KB (377 words) - 02:56, 2 June 2013
$$\rho=a\phi^2-l,\quad l\geq0.$$
750 bytes (121 words) - 19:26, 16 March 2023
...geq \frac { c _ { 1 } } { \operatorname { diam } \Omega } \cdot \omega , \quad c_ { 1 } = \frac { 2 \pi \alpha ( n - 1 ) } { \alpha ( n ) }, \end{equati
...{Vol} ( \Omega ) ^ { n - 1 } } \geq c _ { 2 } \cdot \omega ^ { n + 1 } , \quad c _ { 2 } = \frac { \alpha ( n - 1 ) ^ { n } } { \left( \frac { \alpha ( n
3 KB (378 words) - 16:56, 1 July 2020
$$\frac{\partial u(x)}{\partial n}=\phi(x),\quad x\in S,$$
885 bytes (129 words) - 07:13, 23 August 2014
$$(\Ad a)X = aXa^{-1}, \quad X\in T_e(G) = \g\subset \textrm{End}(V).$$
$$(\ad x)y = [x,y],\quad x,y\in \g,$$
3 KB (407 words) - 16:57, 27 March 2012
ax^2+bx+c=0,\quad a\ne0.
x_1+x_2=-\frac{b}{a},\quad x_1x_2=\frac{c}{a}.
2 KB (384 words) - 07:34, 18 December 2014
...}{\partial q},\quad\dot q=\frac{\partial H}{\partial p},\quad p=(p_1,p_2),\quad q=(q_1,q_2),\label{*}\end{equation}
4 KB (703 words) - 21:33, 23 November 2018
\rho = a\sqrt{\phi} + l,\quad l>0.
1 KB (165 words) - 05:44, 9 April 2023
$$\ln N=\frac{\lg N}{\lg e},\quad\lg N=\frac{\ln N}{\ln10},$$
$$\frac{1}{\lg e}=2.30258\dots,\quad\frac{1}{\ln10}=0.43429\dots.$$
3 KB (453 words) - 15:11, 19 August 2014
u_1 - u_2 + \cdots + (-1)^{n-1}u_n + \cdots, \quad u_k > 0.
865 bytes (138 words) - 17:34, 4 May 2012
$$\frac{n!}{n_1!\dotsm n_m!},\quad n_1+\dotsb+n_m=n,$$
971 bytes (155 words) - 13:33, 14 February 2020
α(n + kℤ) & \text{if}\quad (n, k) = 1;\\
0 & \text{if}\quad (n, k) ≠ 1.
3 KB (438 words) - 13:36, 23 July 2015
$$x\mapsto \frac{a_1x+b_1y+c_1}{a_2x+b_2y+c_2}, \quad y\mapsto \frac{a_3x+b_3y+c_3}{a_4x+b_4y+c_4}.$$
$$y\mapsto \frac{P(x)y+Q(x)}{R(x)y+S(x)},\quad P,Q,R,S\in k[x],$$
5 KB (821 words) - 11:12, 1 January 2021
$$\rho(M)=M-\sqrt{M^2-1},\quad M\geq1,$$
1 KB (215 words) - 13:50, 14 February 2020
$$\frac{x^2}{p}+\frac{y^2}{q}=2z,\quad p,q>0.$$
892 bytes (150 words) - 13:52, 29 April 2014
{f_{1}}(x_{0},\ldots,x_{n}) = 0, \quad \ldots \quad {f_{r}}(x_{0},\ldots,x_{n}) = 0.
3 KB (412 words) - 13:07, 17 April 2023
\sum_{n=1}^{\infty}(-1)^{n+1}a_n,\quad a_n>0,
866 bytes (131 words) - 20:16, 9 December 2013
$$a_1\leq\dotsb\leq a_n,\quad b_1\leq\dotsb\leq b_n$$
690 bytes (132 words) - 12:59, 14 February 2020
$$L(D,x)\equiv\sum_{|\alpha|\leq m}a_\alpha(x)D^\alpha,\quad x\in\mathbf R^n,$$
922 bytes (151 words) - 17:23, 26 September 2014
\forall \epsilon > 0, ~ \exists \delta > 0, ~ \forall B \in \mathfrak{B}: \quad
\mu(B) < \delta \quad \Longrightarrow \quad \left\| \int_{B} f ~ \mathrm{d}{\mu} \right\|_{X} < \epsilon.
7 KB (1,028 words) - 21:30, 26 April 2016
...}{\partial q},\quad\dot q=\frac{\partial H}{\partial p},\quad p=(p_1,p_2),\quad q=(q_1,q_2),\label{*}\end{equation}
4 KB (662 words) - 21:17, 23 November 2018
\zeta(s)=\sum_{n=1}^\infty\frac{1}{n^s},\quad s=\sigma+it,
1 KB (160 words) - 20:17, 18 October 2014
$$x'=f(t,x),\quad x=(x_1,\ldots,x_n)\in\mathbf R^n,$$
$$x=\phi(t,C_1,\ldots,C_n),\quad (C_1,\ldots,C_n)\in C\subset\mathbf R^n,\label{2}\tag{2}$$
4 KB (713 words) - 15:28, 14 February 2020
$$(A)\quad\int\limits_a^bf(x)dx.$$
940 bytes (157 words) - 11:01, 31 March 2017
$$ [F,F]\subseteq\Phi\subseteq F \quad and \quad F/\Phi=F(G/\Phi) , $$
3 KB (508 words) - 13:05, 12 December 2013
\left\lvert f^{(k)}(a)\right\rvert\leq k!\frac{M(r)}{r^k},\quad\lvert c_k\rvert\leq\frac{M(r)}{r^k},
...,r_n)}{r_1^{k_1}\cdots r_n^{k_n}},\quad a=(a_1,\ldots,a_n)\in\mathbb{C}^n,\quad k_1,\ldots,k_n=0,1,\ldots,
4 KB (654 words) - 11:56, 10 June 2016
...ation} \tag{a1} \frac { d u ( t ) } { d t } = A ( t ) u ( t ) + f ( t ) , \quad 0 < t \leq T, \end{equation}
...a _ { i } - 1 } , \lambda \in S _ { \theta _ { 0 } } \backslash \{ 0 \} , \quad 0 \leq s \leq t \leq T. \end{equation*}
13 KB (1,982 words) - 07:42, 4 February 2024
1 &\quad\mbox{if } 0\in E\\
0 &\quad\mbox{if } 0\not\in E\, .
3 KB (370 words) - 17:22, 20 November 2016
...( z ) = \frac { | a | } { a } \frac { z - a } { 1 - \overline { a } z } , \quad | a | < 1, \end{equation*}
...infty } \frac { | a | } { a } \frac { z - a } { 1 - \overline { a } z } , \quad \sum ( 1 - | a _ { j } | ) < \infty . \end{equation*}
3 KB (432 words) - 20:48, 22 January 2024
\begin{equation}\lim_{r\to\infty}u(M)=0,\quad r=|OM|.\label{1}\end{equation}
\begin{equation}u(M)=O(1),\quad r\to\infty.\label{2}\end{equation}
5 KB (853 words) - 00:31, 24 December 2018
$$p(\rho)=\frac{1}{2\pi}\int\limits_0^{2\pi}n(\rho e^{i\theta})d\theta,\quad\rho>0.$$
$$R_1=(h+2)2^{h-1}\mu_h,\quad R_2=\max_{|z|=r}|f(z)|,\quad0<r<1,$$
4 KB (608 words) - 19:06, 26 March 2023
$$h(x)=\frac{h_0(x)}{\sqrt{1-x^2}},\quad x\in(-1,1),\label{1}\tag{1}$$
$$|P_n(x)|\leq C_1,\quad x\in A\subset[-1,1],\label{2}\tag{2}$$
6 KB (876 words) - 15:49, 14 February 2020
$$[f(y)x+g(y)x^\alpha]y'=h(y),\quad\alpha\neq0,1,$$
1 KB (175 words) - 18:48, 9 January 2024
$$r_d=\sum_{g\in R_d}g,\quad c_d=\sum_{g\in C_d}\epsilon(g)g,$$
961 bytes (180 words) - 13:01, 10 August 2014
$$\dot z=A(t)z,\qquad z\in\CC^n,\quad t\in (\CC,t_*)\tag {LS}$$
\dot w=t^{-1}(A_0+tA_1+\dots+t^d A_d)w, \quad AS_0,A_1,\dots,A_d\in\operatorname{Mat}(n,\CC),\tag{NF}
7 KB (1,237 words) - 11:48, 23 November 2023
$$\operatorname{Re}\left[\frac{zf'(z)}{f(z)}\right]>0,\quad|z|<1.$$
$$\operatorname{Re}\left[\frac{zf'(z)}{f(z)}\right]>\alpha,\quad|z|,\alpha<1,$$
3 KB (519 words) - 14:43, 30 July 2014
$$x_n\equiv x_{n-1} \mod p^n,\
quad n\ge 1$$
$$x=\sum_{k=k_0}^\infty a_kp^k,\
quad 0\le a_k <p,\label{*}$$
6 KB (1,089 words) - 20:29, 9 April 2017
$$\int\limits_a^bQ_n^2(x)\rho(x)dx,\quad\rho(x)>0,\label{2}\tag{2}$$
$$a=-1,\quad b=1,\quad\rho(x)=(1-x)^\alpha(1+x)^\beta,$$
3 KB (552 words) - 15:05, 14 February 2020
$$\eta-y=\frac{dy}{dx}(\xi-x),\quad F(\xi,\eta)=0,$$
1 KB (199 words) - 11:15, 26 March 2023
$$|f(z)-g(z)|<\epsilon(|z|),\quad z\in E,$$
2 KB (243 words) - 18:27, 13 August 2015
$$r_n(x)=o\left((x-x_0)^n\right),\quad x\to x_0.$$
$$0<\theta<1,\quad x\in(x_0-\delta,x_0+\delta);$$
3 KB (536 words) - 09:04, 27 December 2013
$$x_k = y_k\quad (k\ne j),\quad x_j = y_j + cy_i,$$
$$x_k = y_k\quad (k\ne i),\quad x_i = -y_i,$$
12 KB (2,085 words) - 22:05, 5 March 2012
\expect X_j=0,\quad
\expect X_j^2=b_j,\quad
5 KB (873 words) - 17:30, 24 July 2012
f (x_0) \geq \limsup_{x\to x_0}\; f(x) \qquad \left(\mbox{resp. }\quad f(x_0)\leq \liminf_{x\to x_0}\; f(x)\right)\, .
\{f< a\} \quad (\mbox{resp.}\;\; \{f>a\}) \quad \mbox{ is a neighborhood of } x_0 \mbox{ for all } a> f(x_0) \mbox{ (r
8 KB (1,260 words) - 15:29, 5 January 2017
...partial}{\partial y^{i}} \left[ \sqrt{g} a^{i} \right], \quad \text{with} \quad
g \stackrel{\text{df}}{=} \det([g_{ij}]) \quad \text{and} \quad g_{ij} \stackrel{\text{df}}{=} \sum_{\alpha = 1}^{n} \frac{\partial y^{\alp
6 KB (941 words) - 05:04, 11 December 2016
$$f(v-2n(n,v),x,t)=f(v,x,t),\quad x\in\partial\Omega,\quad v\in\mathbf R^3,$$
3 KB (545 words) - 15:08, 14 February 2020
$$\frac{n(n+1)(n+2)}{6},\quad n=1,2,\dotsc.$$
$$n+(k-2)\frac{n(n-1)}{2},\quad n=1,2,\dotsc.$$
4 KB (566 words) - 07:28, 26 March 2023
e^{-1/x^2} \quad &\mbox{if } x\neq 0\\
0 \quad &\mbox{otherwise.}
4 KB (710 words) - 06:13, 13 June 2022
$$\langle X,Y\rangle=g_{ij}(p)X^iY^j,\quad X=\{X^i\},\quad Y=\{Y^j\},\quad0\leq i,j\leq n.$$
3 KB (539 words) - 17:02, 14 February 2020
$$P_{-1}(x)=0,\quad P_0=1,$$
1 KB (188 words) - 20:27, 15 November 2023
...htarrow{\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } - \infty.} \end{cases} \end{equation*}
18 KB (2,677 words) - 19:56, 4 February 2024
$$\phi(y,x) = \phi(x,y)^J,\quad x,y\in X.$$
$$\psi(y,x) = \e\psi(x,y)^J,\quad x,y\in X.$$
5 KB (831 words) - 17:13, 9 October 2016
} {\dot \partial } _ {i} {\dot \partial } _ {j} F ^ {2} , \quad i,j = 1 \dots n,
} + \gamma _ {jk } ^ {i} ( x,b ) y ^ {j} y ^ {k} = 0, \quad {
10 KB (1,411 words) - 10:58, 29 May 2020
$$|h_0|,|h_1|\leq H,\quad |h_0|+|h_1|\neq0.$$
1 KB (189 words) - 21:10, 14 December 2014
$$\begin{equation} \left.\begin{aligned} a_{ii}=2;\quad a_{ij}\le 0 \ &\textrm{and}\ a_{ij} \in \Z\textrm{ for }i\ne j,\\
..._i,f_j]=\d_{ij}h_i;\quad [h_i,e_j]=a_{ij}e_j;\quad [h_i,f_j] = -a_{ij}f_j;\quad [h_i,h_j] = 0.\tag{2}$$
6 KB (1,055 words) - 08:14, 22 May 2012
$$z_j=\{a_j+(z_j^0-a_j)e^{i\phi_j}\},\quad0\leq\phi_j\leq2\pi,\quad j=1,2,$$
1,005 bytes (162 words) - 14:23, 30 July 2014
...d} \quad \hat{r}_{i} = r_{i}; \qquad \hat{p}_{i} = p_{i} \quad \text{and} \quad \hat{r}_{i} = - \frac{\hbar}{i} \frac{\partial}{\partial p_{i}}; \qquad i \
6 KB (937 words) - 09:15, 14 December 2016
V_1(x) &= x,\\ V_n(x) &= x V_{n-1}(x)+V_{n-2}(x),\quad n = 2,3,\dots
\end{align*}\quad\right\}\tag{a1}$$
7 KB (1,115 words) - 16:05, 22 January 2021
$$x=\frac{x'-Vt'}{\sqrt{1-V^2/c^2}};\quad y=y'\quad z=z';\quad t=\frac{t'-Vx'/c^2}{\sqrt{1-V^2/c^2}}.$$
$$\sinh\psi=-\frac{V/c}{\sqrt{1-V^2/c^2}},\quad\cosh\psi=\frac{1}{\sqrt{1-V^2/c^2}},$$
6 KB (942 words) - 13:30, 10 August 2014
...z ) = m ! \sum _ { 0 \leq n \leq m - 1 } b _ { m } ( n + 1 ) z ^ { n } , \quad z \in \mathbf{C}, \end{equation*}
...n*} w \rightarrow \frac { ( z - 1 ) e ^ { w } } { z ( z - e ^ { w } ) } , \quad z \in \mathbf{C}, \end{equation*}
5 KB (742 words) - 07:27, 25 January 2024
$$x=\int\limits_0^t\cos\frac{s^2}{2a}\,ds,\quad y=\int\limits_0^t\sin\frac{s^2}{2a}\,ds,$$
1 KB (205 words) - 10:58, 26 March 2023
$$u(x,0)=\tau(x),\quad u_t(x,0)=\nu(x),$$
$$u=\phi\quad\text{if }x-x_0=t-t_0,$$
5 KB (891 words) - 20:40, 22 December 2018
...j,n} = \sum {\mathsf P} ( A _ {i _ 1 } \cap \dots \cap A _ { i _ j } ) , \quad j \geq 1,
\right ) \right ] , \quad k \geq 0.
4 KB (657 words) - 12:53, 6 May 2023
\quad \text{and} \quad
M(s) = \sum_{d\le u} \mu(d) d^{-s}, \quad F(s) = \sum_{e \le v} \Lambda(e) e^{-s}.
10 KB (1,642 words) - 12:01, 13 February 2024
$$ A(x)=\sum_{\substack{a\in A\\0<a\leq x}}1,\quad x\geq 1.$$
1 KB (180 words) - 13:38, 25 November 2023
$$S_i=-[q_1p_1(x)r_1i+\dotsb+q_kp_k(x)r_{ki}],\quad i=1,\dotsc,k,$$
1 KB (185 words) - 12:24, 14 February 2020
(x_1,\dotsc,x_n),\quad x_1\leq\dotsb\leq x_n\,,
1 KB (196 words) - 12:46, 14 February 2020
$$\lim_{n\to\infty}a_n=\infty,\quad\lim_{n\to\infty}\frac{a_n}{n^2}=0,$$
$$h(x)\leq c\exp(-\alpha|x|),\quad\alpha>0,\quad x\in(-\infty,\infty).$$
7 KB (1,066 words) - 15:42, 14 February 2020
$$a_{mn},\quad m,n=1,2,\ldots.$$
1 KB (180 words) - 11:33, 29 June 2014
{\dot{x} } = f ( x, \alpha ) , \quad x \in \mathbf R ^ {n} , \alpha \in \mathbf R ^ {1} ,
{\dot{x} } = f ( x, \alpha ) , \quad x \in \mathbf R ^ {1} , \alpha \in \mathbf R ^ {1} ,
6 KB (872 words) - 08:12, 6 June 2020
b(u, v) = l(v), \quad \forall v \in V,
|b(u,v)| \le M \|u\|. \|v\|, \quad u,v \in V
5 KB (784 words) - 03:17, 15 February 2024
$$f(c_i)=g(c_i)h(c_i),\quad i=1,\dots,k+1,$$
$$g(c_i)=d_i,\quad i=1,\dots,k+1.$$
3 KB (574 words) - 18:14, 14 June 2023
$$\frac{df}{dl}=\sum_{i=1}^nl_i(P)\frac{\partial f}{\partial x_i},\quad P=(x_1,\ldots,x_n),$$
1 KB (187 words) - 17:57, 30 July 2014
$$\sigma x_1=\xi,\quad \sigma x_2=\eta,$$
$$\sigma x_3=\frac{1-(\xi^2-\eta^2)}{2},\quad\sigma x_4=\frac{1+(\xi^2+\eta^2)}{2}.$$
4 KB (675 words) - 14:01, 17 March 2023
x_{in} = \sum_{j=1}^{n}\alpha_{jn} K(s_{in}, s_{jn})x_{jn} + y(s_{in}), \quad i = 1, \dots, n,
...epsilon_n \le \max_{1\le i \le n}|x_{in} - x(s_{in})| \le c_2 \epsilon_n, \quad n \ge n_0, \]
3 KB (528 words) - 06:14, 2 June 2013
...\sqrt{\pi}} \frac{\Gamma((N-n+1)/2)}{\Gamma((N-n)/2)}(1-x^2)^{(N-n-2)/2}, \quad -1<x<1$$
$$t=\sqrt{N-n}\frac{r}{\sqrt{1-r^2}},\quad \textrm{where}\ r = r_{12;3\dots n},$$
4 KB (537 words) - 20:05, 6 April 2012
2^{\aleph_{\xi}} = \aleph_{\xi + 1} \quad \text{for all} \quad \xi < \omega_{1},
4 KB (603 words) - 09:05, 2 January 2021
$$G(x,y)=o(e^{-a|x-y|}),\quad\frac{\partial G}{\partial x_i}=o(e^{-a|x-y|})$$
1 KB (190 words) - 08:35, 31 October 2014
$$\inf_{y\in Y}\sup_{x\in X}F(x,y),\quad\min_{y\in Y}\max_{x\in X}F(x,y),$$
1 KB (217 words) - 18:20, 14 August 2014
\lambda e^{-\lambda x}, \quad & x \geq 0 , \\
0, \quad & x < 0.
3 KB (485 words) - 19:23, 10 April 2024
$$w=v\exp(t^2/4),\quad t=z\sqrt2$$
1 KB (190 words) - 19:23, 16 October 2023
$$x=\frac\pi2(2n+1),\quad n=0,\pm1,\pm2,\mathinner{\ldotp\ldotp\ldotp\ldotp}\label{*}\tag{*}$$
1 KB (202 words) - 15:13, 14 February 2020
$$\sigma_n=\sum_{k=0}^\infty a_{nk}s_k,\quad n=0,1,\mathinner{\ldotp\ldotp\ldotp\ldotp}$$
$$\sum_{n=0}^\infty|b_{nk}|\leq M,\quad\sum_{n=0}^\infty b_{nk}=1,\quad k=0,1,\dots$$
5 KB (850 words) - 15:41, 14 February 2020
$$r=\{u\cos v,u\sin v,f(u)+hv\},\quad h=\text{const},$$
1 KB (200 words) - 20:09, 5 March 2024
$$b_0 = 1,\quad P_{-2} = 0,\quad P_{-1} = 1,\quad Q_{-2} = 1,\quad Q_{-1} = 0,$$
$$\d_0 = a_0,\quad \d_1 = a_0+{b_1\over a_1} \quad \d_2 = a_0+{b_1\over {a_1+{b_2\over a_2}}}, \cdots,$$
11 KB (1,649 words) - 11:41, 4 September 2013
$$x\wedge(x\vee y)=x,\quad x\vee(x\wedge y)=x,$$
1 KB (218 words) - 18:42, 21 September 2017
...} P ( t ) = \prod _ { m = 1 } ^ { n } ( t - t _ { m } ) ^ { r _ { m } } ; \quad q _ { i } ( t ) = \left\{ \frac { ( t - t _ { i } ) ^ { r _ { i } } } { P (
...) \prod _ { m = 1 , m \neq i } ^ { n } ( t - t _ { m } ) ^ { r _ { m } } \quad ( i = 1 , \ldots , n ). \end{equation*}
10 KB (1,502 words) - 22:40, 12 December 2020
$$\dfrac{P_1,P_2}{P_3}\quad\dfrac{\dfrac{P_4,\dfrac{P_1,P_2}{P_3}}{P_5}}{P_6}$$
1 KB (203 words) - 19:15, 16 April 2014
$$K(r,s)=\sum a_{ij}r_is_j,\quad r_i\in R,\quad s_j\in S.$$
3 KB (567 words) - 15:23, 29 September 2014
$\quad \quad \quad Y \Subset X $
$\quad \quad \quad \quad$ if $X \subseteq Y$ and $X \ \vdash_\mathcal{S} \ A$, then $Y \ \vdash_\mat
33 KB (5,175 words) - 19:12, 7 December 2023
$$I(\omega_m)=\sum_{n=1}^NC_n\omega_m(P_n),\quad m=1,\ldots,M.$$
$$g(a_1,\ldots,a_M,P_n)=f(P_n),\quad n=1,\ldots,N.$$
4 KB (621 words) - 11:59, 12 August 2014
...ation} \tag{a1} \frac { d u ( t ) } { d t } = A ( t ) u ( t ) + f ( t ) , \quad 0 < t \leq T, \end{equation}
...c { \partial U ( t , s ) } { \partial t } \| \leq \frac { C } { t - s } , \quad s , t \in [ 0 , T ], \end{equation*}
14 KB (2,104 words) - 18:47, 26 January 2024
{\dot{n} } = \lambda [ 1 - n ( t - 1 ) ] n, \quad \lambda > 0.
{\dot{n} } = D \Delta n + \lambda [ 1 - n ( t - 1,x ) ] n, \quad \left . {
4 KB (552 words) - 22:11, 5 June 2020
$$y(a^2+x^2)=a^3,\quad a>0.$$
1 KB (189 words) - 20:28, 15 November 2023
$$z=l_1e^{\omega_1ti}+l_2e^{\omega_2ti},\quad z=x+iy.$$
2 KB (259 words) - 16:53, 8 April 2023
$$\max_x|R_n(f,x)|\leq(L_n+1)E_n(f),\quad n=1,2,\ldots,$$
2 KB (236 words) - 09:15, 1 August 2014
$$A=\| a_{nk}\|,\quad n,k=1,2,\ldots,$$
1 KB (205 words) - 10:25, 28 April 2012
...abel{a3}\def\ln{\mathrm{ln\;}}\phi = \ln\big(\frac{2u_t u_\tau}{u^2}\big),\quad \phi = \ln\big(\frac{2u_t u_\tau}{\cos^2 u}\big).\tag{a3}\end{equation}
$$\phi_t - u_t+ a e^{(\phi+u)/2} = 0,\quad \phi_\tau + u_\tau + \frac{2}{a} e^{(\phi-u)/2} = 0.$$
3 KB (518 words) - 12:51, 30 March 2024
$$\left\{1,\frac{1+\sqrt d}{2}\right\} \quad \text{when}~d \equiv 1 \pmod 4$$
$$\left\{1,\sqrt d\right\} \quad \text{when}~d \equiv 2,3 \pmod 4$$
5 KB (867 words) - 17:41, 12 November 2023
$$x=(m+1)C\int\limits_0^t\sin^{m+1}tdt,\quad y=C\sin^{m+1}t,$$
1 KB (243 words) - 20:35, 7 April 2015
g \circ f & = \mathsf{Id}_{X} \quad \text{on} ~ M, \qquad (1) \\
f \circ g & = \mathsf{Id}_{Y} \quad \text{on} ~ f[M], \qquad (2)
4 KB (752 words) - 03:29, 9 January 2017
...c}{b\cdot d},\quad \frac{a}{b} : \frac{c}{d} = \frac{a\cdot d}{b\cdot c},\quad (c\ne 0). $$
4 KB (649 words) - 17:15, 9 December 2013
\begin{equation}\int\limits_a^x\frac{\phi(s)}{(x-s)^\alpha}ds=f(x),\quad a\leq x\leq b,\label{2}\end{equation}
\begin{equation}\int\limits_a^b\frac{\phi(s)}{|x-s|^\alpha}ds=f(x),\quad a\leq x\leq b,\label{3}\end{equation}
4 KB (633 words) - 19:23, 22 December 2018
$$G(t)=1-e^{-\lambda t},\quad t\ge 0,$$
1 KB (204 words) - 19:33, 28 October 2014
$$f(x)=a_0x^n+a_1x^{n-1}+\dotsb+a_n,\quad a_0>0;\label{*}\tag{*}$$
1 KB (229 words) - 15:40, 14 February 2020
$$\rho(g,f)=\left(\int\limits_a^b|g(x)-f(x)|^pdx\right)^{1/p},\quad p\geq1,$$
2 KB (277 words) - 18:41, 11 April 2023
$$W^+=U+iV,\quad W^-=U-iV$$
1 KB (235 words) - 10:40, 26 May 2016
\quad\text{and}\quad
4 KB (615 words) - 16:20, 24 November 2012
$$x^{-1/2}f(x)\in L(0,1),\quad x^{1/2}f(x)\in L(1,\infty),$$
1 KB (238 words) - 15:43, 11 August 2014
$$|f(x+\tau)-f(x)|<\epsilon,\quad-\infty<x<\infty.$$
2 KB (376 words) - 10:22, 24 August 2014
$$\begin{equation} \left.\begin{aligned} a_{ii}=2;\quad a_{ij}\le 0 \ &\textrm{and}\ a_{ij} \in \Z\textrm{ for }i\ne j,\\
[h_i,h_j] = 0,\quad [e_i,f_i] = h_i, \quad [e_i,f_j] = 0\quad &\textrm{ if }i\ne j,\\
15 KB (2,514 words) - 20:27, 15 November 2017
\sum_j p_{ij} = 1 \quad \text{for all $i$.}
\quad \text{for all}\ j\,,
6 KB (982 words) - 09:11, 1 October 2023
$$\|f\| = \max_{x\in X}\left|f(x)\right|, \quad f\in C(X),$$
2 KB (269 words) - 13:13, 12 December 2013
$$x=R\cos^3\frac t4,\quad y=R\sin^3\frac t4.$$
2 KB (264 words) - 07:29, 26 March 2023
...{\beta} = x^{0} y^{0} - \sum_{k = 1}^{3} x^{k} y^{k}, \qquad \text{where} \quad
1 KB (187 words) - 23:29, 14 December 2016
$$\xi=\frac x{px-y+q},\quad\eta=\frac y{px-y+q},$$
2 KB (254 words) - 19:27, 31 March 2017
$$\frac{dx^i}{dt}=\pm\sum_jg^{ij}\frac{\partial f}{\partial x^j},\quad i=1,\dots,n,$$
2 KB (250 words) - 10:12, 24 August 2014
x^2y'' + xy'+(x^2-\nu^2)y = 0,\quad \nu = {\rm const},\label{(1)}
$$y=C_1J_\nu(x) + C_2 Y_\nu(x),\quad y=C_1 H_\nu^{(1)}(x) + C_2 H_\nu^{(2)}(x).$$
7 KB (1,130 words) - 20:00, 22 March 2024
g _ {2} : x - y = 1, \quad g _ {3} : x + y = 0
u = p ( s ) \textrm{ on } g _ {1} , \quad u = q ( x ) \textrm{ on } g _ {3} ,
6 KB (792 words) - 10:59, 29 May 2020
$$h=c+ab-\frac{\partial a}{\partial u},\quad k=c+ab-\frac{\partial b}{\partial v}, $$
2 KB (242 words) - 13:06, 12 December 2013
$$\rho=a^\phi,\quad a>0.$$
2 KB (265 words) - 11:22, 26 July 2014
$$\tan x=x+\frac{x^3}{3}+\frac{2x^5}{15}+\frac{17x^7}{315}+\dotsb,\quad|x|<\frac\pi2.$$
1 KB (237 words) - 14:16, 14 February 2020
$$y'(x)=f(x,y(x)),\quad x_0\leq x\leq X,\quad y(x_0)=y_0,\tag{1}$$
k_1 = h f(x_0,y_0), \quad y_0 = y(x_0),\\
7 KB (1,053 words) - 17:13, 14 February 2020
\begin{equation*} R [ K ( x _ { \nu } , . ) ] = 0 , \quad \nu = 1 , \dots , n, \end{equation*}
\begin{equation*} R [ K _ { x } ( x _ { \nu } , . ) ] = 0 , \quad \nu = 2 , \dots , n - 1, \end{equation*}
5 KB (662 words) - 20:04, 23 January 2021
f ( x , k ) = e ^ { i k x } + o ( 1 ) , \quad x \rightarrow \infty ,
g ( x , k ) = e ^ { - i k x } + o ( 1 ) , \quad x \rightarrow - \infty.
12 KB (1,851 words) - 04:43, 15 February 2024
...\cos \frac{\phi + 2 \pi k}{n} + i \sin \frac{\phi + 2 \pi k}{n} \right), \quad k = 0, 1, \dots, \]
1 KB (241 words) - 10:03, 4 June 2013
$$w_n(\omega;H)>c_3H^{-n-\epsilon},\quad c_3=c_3(\omega;n,\epsilon)$$
2 KB (342 words) - 12:47, 20 December 2014
$$\sum_{i=1}^\infty\phi_i(x),\quad x\in[a,b],\label{1}\tag{1}$$
2 KB (254 words) - 17:26, 14 February 2020
p(\lambda x) = |\lambda|p(x),\quad p(x+y)\leq p(x) + p(y)
2 KB (258 words) - 07:13, 5 December 2012
\omega\in\varLambda^2(M),\qquad \rd \omega=0,\qquad \forall v\in T_p M\quad \exists w\in T_p M:\ \omega_p(v,w)\ne0.
...\implies\ \exists h\in\operatorname{Diff}(M,a):\quad h^*\omega_1=\omega_2,\quad h|_N\equiv\operatorname{id}.
5 KB (856 words) - 05:41, 24 February 2022
$$a_1x_1+\cdots+a_n x_n = b,\quad a_i,b\in k.\label{4}$$
$$a_{j1}x_1+\cdots+a_{jn} x_n = b_j,\quad j=1,\dots,m.$$
4 KB (759 words) - 14:13, 17 November 2011
x \sim y \iff \exists k \in\mathbf{Z} \,\psi^k(x) = y \quad\text{and}\quad \pi: x \mapsto [x]
4 KB (606 words) - 19:47, 24 November 2023
} \# \left \{ {1 \leq i \leq n } : {X _ {i} \in C } \right \} , \quad C \in {\mathcal S}.
\alpha _ {n} ( C ) = \sqrt n ( P _ {n} ( C ) - P ( C ) ) , \quad C \in {\mathcal C},
9 KB (1,327 words) - 19:37, 5 June 2020
$$z=\int\limits^u\frac{dz}{w},\quad w^2=4z^3-g_2z-g_3,$$
2 KB (250 words) - 22:06, 28 November 2014
{\dot{x} } = f ( x, \alpha ) , \quad x \in \mathbf R ^ {n} , \alpha \in \mathbf R ^ {1} ,
{\dot{x} } = f ( x, \alpha ) , \quad x \in \mathbf R ^ {2} , \alpha \in \mathbf R ^ {1} ,
9 KB (1,234 words) - 22:11, 5 June 2020
$$F\left(x,z,\frac{z'-\tan\alpha}{1+z'\tan\alpha}\right)=0,\quad F\left(x,z\frac{z'+\tan\alpha}{1-z'\tan\alpha}\right)=0.$$
2 KB (259 words) - 17:35, 14 February 2020
(2\pi)^{(m-1)/2}m^{(1/2)-mz}\Gamma(mz), \quad m = 2,3,\ldots
+ O\bigl(z^{-2m-1}\bigr), \quad m = 1,2,\ldots,
10 KB (1,513 words) - 09:50, 5 December 2012
$$f_i(x_0,\dots,x_n)=0,\quad i=1,\dots,n,\label{*}\tag{*}$$
2 KB (268 words) - 15:02, 14 February 2020
$$\mu(f,\phi)=\int\limits_a^b|f(t)-\phi(t)|^qd\sigma(t)\quad(q>0),$$
2 KB (331 words) - 08:01, 23 August 2014
$$K_1(x,s)=K(x,s),\quad K_n(x,s)=\int\limits_a^bK_{n-1}(x,t)K(t,s)dt.$$
2 KB (292 words) - 19:24, 9 October 2014
$$a(b+c)=ab+ac,\quad(b+c)a=ba+ca,$$
2 KB (297 words) - 17:54, 3 January 2016
$$\overline{G}\subset G_{n+1}\subset\overline{G}_{n+1}\subset G_n,\quad n=1,2,\ldots,$$
2 KB (269 words) - 10:07, 22 April 2012
x _ {_ j } \in \mathbf Z, \quad j \in N ^ \prime \subset \{ 1 \dots n \} ,
x _ {j} \in \mathbf Z, \quad j \in N ^ \prime ,
5 KB (757 words) - 22:15, 5 June 2020
$$a_{ik}=a_{ki},\quad i,k=1,\dots,n.$$
2 KB (288 words) - 22:15, 30 November 2018
$$\int\limits_a^b|f(t)|^2\,dt=\sum_{n=1}^\infty c_n^2,\quad c_n=\int\limits_a^bf(t)\phi_n(t)\,dt.$$
2 KB (290 words) - 14:27, 14 February 2020
...} x _ { j } ^ { \prime } = \sum _ { i , k } p _ { i k,j } x_i x _ { k } , \quad x _ { i } \geq 0 , \sum _ { i } x _ { i } = 1. \end{equation*}
\begin{equation*} p = x _ { 1 } + \frac { 1 } { 2 } x _ { 3 } , \quad q = x _ { 2 } + \frac { 1 } { 2 } x _ { 3 } \end{equation*}
9 KB (1,319 words) - 17:02, 1 July 2020
...ation} \tag{a3} f _ { j } = z _ { j } ^ { k _ { j } } + P _ { j } ( z ) , \quad j = 1 , \dots , n, \end{equation}
...sum _ { \text{l} = 1 } ^ { M } ( z _ { 1 } ^ { ( \text{l} ) } ) ^ { j } , \quad j = 1 , \ldots , M, \end{equation*}
7 KB (1,090 words) - 10:25, 16 March 2024
$$t_k(x)=\frac{\Delta x}{\Delta'(x)2\sin(x-x_k)/2},\quad\Delta(x)=\prod_{k=0}^{2n}2\sin\frac{x-x_k}{2}.$$
2 KB (269 words) - 15:19, 14 February 2020
...dots, x_{k+1} $ are marked on the abscissa, and the segments $ x_ix_{i+1} \quad (i = 1,\dots, k) $ are taken as the bases of rectangles with heights $ h_i/
3 KB (376 words) - 00:30, 11 June 2013
$$\eta_i(x)=a_{i1}\xi_1+\dots+a_{in}\xi_n+a_i\leq0,\quad i=1,\dots,m,$$
2 KB (296 words) - 16:59, 22 November 2018
$$k_1=\phi(s),\quad k_2=\psi(s),$$
2 KB (304 words) - 19:04, 26 January 2024
$$\left|\frac ab\right|_p=p^{\nu_p(b)-\nu_p(a)},\quad a,b\in\mathbf Z,$$
2 KB (283 words) - 17:19, 30 November 2014
$$\left(\frac{-1}{P}\right)=(-1)^{(P-1)/2},\quad\left(\frac 2P\right)=(-1)^{(P^2-1)/8}$$
2 KB (315 words) - 17:41, 11 November 2023
...imits_{\mathbf R^n}\exp\left(-\frac{|x-\xi|^2}{4a^2t}\right)\phi(\xi)d\xi,\quad t>0,$$
2 KB (322 words) - 19:05, 30 July 2014
$$(x+y)=(x\cap Cy)\cup(y\cap Cx),\quad x\cdot y=x\cap y,$$
2 KB (335 words) - 15:09, 13 August 2014
\rho(x,y) = 0 \text{ if } x=y \quad \text{and} \quad \rho(x,y) = 1 \text{ if } x\ne y.
5 KB (793 words) - 09:16, 7 June 2016
\phi ^ {2} = - I + \eta \otimes \xi, \quad \eta ( \xi ) = 1,
\eta \circ \phi = 0, \quad \phi ( \xi ) = 0,
6 KB (891 words) - 01:08, 8 May 2022
...alpha_{2} = \omega^{\omega}, \quad \alpha_{3} = \omega^{\omega^{\omega}}, \quad \ldots.
9 KB (1,404 words) - 18:33, 4 December 2017
...+ \\ \mathbf{x}^- \\ \mathbf{s} \end{pmatrix} = \mathbf{b} \quad\text{and}\quad \begin{pmatrix} \mathbf{x}^+ \\ \mathbf{x}^- \\ \mathbf{s} \end{pmatrix} \g
\begin{equation} \mathbf{b}^T\mathbf{y} < 0,\quad\mathbf{v}^T\mathbf{y}\geq 0 \end{equation}
12 KB (1,986 words) - 19:13, 31 March 2017