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train/img_00000100.png | \begin{align*} \sum_{n=2}^\infty \frac{1}{a_n(\gamma_n^-+\gamma_n^+)} \le \sum_{n=2}^\infty (|v_{n-1}||v_{n+1}|+|v_{n+1}|^2) \le 2 \sum_{n=1}^\infty |v_n|^2. \end{align*} |
train/img_00000101.png | \begin{align*} S_{\mathrm{main}}=\sum_{L=1}^{\infty}L\sum_{U,\, V <\frac{D}{L}} \tau (U)\tau(V)\,x_{UL}\bar{x}_{VL}\sum_{T|L}\frac{\mu(T)}{T}\sum_{A=1}^{\infty}\frac{G_k(A^2T^2UV\slash p)}{A}, \end{align*} |
train/img_00000102.png | \begin{align*}\|f\|_{C^{0,\mu}H_\xi^s}:=\|f\|_{L^\infty H_\xi^s} +\sup_{\xi\neq \xi'\in Q_\ell^*}\frac{\|f(\xi,\cdot,\cdot)-f(\xi',\cdot,\cdot)\|_{H^s(Q_\ell\times Q_\ell)}}{|\xi-\xi'|^\mu}.\end{align*} |
train/img_00000103.png | \begin{align*}\|\mathcal{S}_k (\mathcal{A})\| = \varphi^k \| \mathcal{S}_k (U) \| \leq \varphi^k (1+\underbrace{\frac{k(k-1)}{2n} (1+ C)}_{\vartriangle_{k}})= \varphi^k (1+\vartriangle_{k})\end{align*} |
train/img_00000104.png | \begin{align*}\dot x(t)=(A-BKC)x(t)+BKC\int_{t-\tau(t)}^t\dot x(s)\,ds,\end{align*} |
train/img_00000105.png | \begin{align*}P_{tot} = \sum_{k=1}^ G \mathbf w_k \mathbf w_k^\dag= \mathrm{Trace\left( \mathbf {WW}^\dag\right)},\end{align*} |
train/img_00000106.png | \begin{align*}[\varphi(a\otimes x),b\otimes f] &= \varphi(a)b \cdot f(x) = \varphi( a\varphi^{-1}(b) \cdot f(x)) = ap^{-1}\varphi^{-1}(b)f(x)\\&= [a\otimes x, p^{-1}\varphi^{-1}(b \otimes f)]. \end{align*} |
train/img_00000107.png | \begin{align*}D_{s\eta^j}(\mathfrak{u})=\mathfrak{f}.\end{align*} |
train/img_00000108.png | \begin{align*}\partial \bar{\partial }\mbox{ log }\Delta _j=\frac{\Delta _{j+1}\cdot \Delta _{j-1}}{\Delta _j^2},\end{align*} |
train/img_00000109.png | \begin{align*}\operatorname{Re}\psi_2(i\rho,i\tau,\xi+i\eta)=-\alpha\rho^2-\beta\tau\eta\leqslant 0,\end{align*} |
train/img_00000110.png | \begin{align*}F_H(v)=0\Longrightarrow v=0.\end{align*} |
train/img_00000111.png | q _ { ( 1 ) } = q _ { i } \delta _ { i } ^ { j } \qquad \bar { q } _ { ( 1 ) } = \bar { q } _ { i } \delta _ { i } ^ { j } \qquad T _ { ( 1 ) } = T _ { i } \delta _ { i } ^ { j } |
train/img_00000112.png | \begin{align*} &\alpha -2A_1>0,\ \ \beta -2A_2>0,\ \ \gamma -2A_3>0,\\ &2\omega -\frac{|\mathbf{c}|^2}{8A_1}>0,\ \ \omega -\frac{|\mathbf{c}|^2}{8A_2}>0,\ \ \omega -\frac{|\mathbf{c}|^2}{8A_3}>0.\end{align*} |
train/img_00000113.png | \begin{align*} I_{1,3}= \log\sqrt{2\pi} \ x+ O(1). \end{align*} |
train/img_00000114.png | \breve { R } ^ { \alpha \beta } ( x ) = \left( \begin{array} { c c c c } { { 1 } } & { { 0 } } & { { 0 } } & { { 0 } } \\ { { 0 } } & { { - ( [ \alpha ] _ { q } [ \beta ] _ { q } ) ^ { 1 / 2 } q ^ { ( \alpha + \beta ) / 2 } \cdot \frac { q - q ^ { - 1 } } { x - q ^ { \alpha + \beta } } } } & { { \frac { x q ^ { \beta } - q ^ { \alpha } } { x - q ^ { \alpha + \beta } } } } & { { 0 } } \\ { { 0 } } & { { \frac { x q ^ { \alpha } - q ^ { \beta } } { x - q ^ { \alpha + \beta } } } } & { { - ( [ \alpha ] _ { q } [ \beta ] _ { q } ) ^ { 1 / 2 } q ^ { ( \alpha + \beta ) / 2 } \cdot \frac { x ( q - q ^ { - 1 } ) } { x - q ^ { \alpha + \beta } } } } & { { 0 } } \\ { { 0 } } & { { 0 } } & { { 0 } } & { { \frac { 1 - x q ^ { \alpha + \beta } } { x - q ^ { \alpha + \beta } } } } \\ \end{array} \right) |
train/img_00000115.png | \begin{align*}\bigl\|x\pm\frac{y}{\|y\|}\bigr\|&\leq \bigl\|x-\sum_{j=1}^m\alpha_j\, m_{a_j,b_j}\bigr\|+\bigl\|\sum_{j=1}^m\alpha_j\, m_{a_j,b_j}\pm y\bigr\|+\bigl|\|y\|-1 \bigr|\\&< \frac{\varepsilon}{5}+\sum_{j=1}^m\alpha_j \|m_{a_j,b_j}\pm y_j\|+\frac{2\varepsilon}{5}\leq 1+\varepsilon.\end{align*} |
train/img_00000116.png | \begin{align*}2i\chi_D(\lambda)= (m_4+m_3-m_2-m_1)\vert_{(1,\lambda)}.\end{align*} |
train/img_00000117.png | \begin{align*} dK+Kd=I \end{align*} |
train/img_00000118.png | \begin{align*}\lambda = \frac{\eta\,J(B|M)_{\hat{\rho}_{BM}}}{\eta\,J(B|M)_{\hat{\rho}_{BM}} + |1-\eta|\,J(A|M)_{\hat{\rho}_{AM}}}\;,\end{align*} |
train/img_00000119.png | \left( w \sigma f ^ { \prime } e ^ { - 2 \Phi } \right) ^ { \prime } + \frac { \sigma f ( 1 - f ^ { 2 } ) e ^ { - 2 \Phi } } { r ^ { 2 } } = 0 \; , |
train/img_00000120.png | \begin{align*}\widehat J_n(h)=\frac{1}{n^2}\sum_{i=1}^n K_h^2(x_0-X_i).\end{align*} |
train/img_00000121.png | \begin{align*}P_{e_{2}} =&E_{H}\big[P(\xi_{1})P_{e\vert\xi_{1}}+P(\xi_{2})P_{e\vert\xi_{2}}+ P(\xi_{3})P_{e\vert\xi_{3}}+P(\xi_{4})P_{e\vert\xi_{4}}\big],\end{align*} |
train/img_00000122.png | \phi \rightarrow e ^ { 2 i \alpha } \phi , |
train/img_00000123.png | \begin{align*}\log \frac{1}{\|X_0(\varphi(re^{i\theta}))\|} = \log \sqrt{1 + \sum_{i=1}^n|\varphi_i(re^{i\theta})|^2} \le \sum_{i=1}^n\log^+|\varphi_i(re^{i\theta})| + \log(\sqrt{1+n})\end{align*} |
train/img_00000124.png | \begin{align*}x'=_Xx=_Xy+\gamma\gamma^{-1}(x-y)=_Xy+\gamma'\gamma^{-1}(x-y).\end{align*} |
train/img_00000125.png | \begin{align*}\partial_a \alpha_b(\zeta) - \partial_b \alpha_a(\zeta) = 0.\end{align*} |
train/img_00000126.png | \begin{align*} D_{\alpha}=\pi\left[ e^{\alpha d_{\mathcal{W}}(\cdot, x_0)^2}\right] < \infty \end{align*} |
train/img_00000127.png | \begin{align*}E^2=-\frac{r}{pq}=-Z^2X^2=Y^2=0.\end{align*} |
train/img_00000128.png | \begin{align*}\left [ \frac{\delta C(\alpha)}{\delta \alpha} , \alpha \right ]^\dagger =0.\end{align*} |
train/img_00000129.png | \begin{align*}\tilde{A_{i}}\left(T_{j}\right):=\left[\tilde{A}\left(t_{i+1},T_{j}\right),\mathbb{E}^{\mathbb{P}}\left[\tilde{A}\left(t_{i+2},T_{j}\right)\left|\mathcal{F}_{i}\right.\right],...,\mathbb{E}^{\mathbb{P}}\left[\tilde{A}\left(t_{\max\left\{ I_{j}\right\} },T_{j}\right)\left|\mathcal{F}_{i}\right.\right]\right]^{\top}\end{align*} |
train/img_00000130.png | \Pi _ { k l } a _ { l } \star f _ { 0 } ( x , p ) = f _ { 0 } ( x , p ) \star \Pi _ { k l } a _ { l } ^ { \dagger } = 0 , ~ ~ \mathrm { f o r } ^ { \forall } k . |
train/img_00000131.png | \begin{align*}W_\phi(z+1) = \phi(z)W_\phi(z), W_\phi(1) = 1.\end{align*} |
train/img_00000132.png | \begin{align*}\left\{\det \sum_\nu \left( 2-u_\nu^0-u_\nu^{0\dagger} \right)\right\}^{({-5+1})} \left(\det \left[-\frac14 \left( u_\lambda^0-u_\lambda^{0\dagger} \right)^2 \right] \right)^{4}. \end{align*} |
train/img_00000133.png | \begin{align*}C(z) = \begin{pmatrix}0 & 1 \\-\mathfrak a^2(L,s) & \frac{z + s + 1}L\end{pmatrix} + \frac{1}{z + \frac{1}2}\begin{pmatrix}\mathfrak a(L,s)\varphi_+(L,s-1)\varphi_-(L,s) & -\varphi_+(L,s-1)\varphi_-(L,s-1) \\\mathfrak a^2(L,s)\varphi_+(L,s)\varphi_-(L,s) & -\mathfrak a(L,s)\varphi_+(L,s)\varphi_-(L,s-1)\end{pmatrix}\end{align*} |
train/img_00000134.png | \begin{align*} 2\hat\beta(l_n,\xi)-2\hat\beta(l_m,\xi)=h(l_n \xi,l_n \eta, l_n \omega)-h(l_m \xi,l_m \eta, l_m \omega).\end{align*} |
train/img_00000135.png | \begin{align*}c_{n}=\frac{\alpha }{\sqrt{2}}\left( n+\gamma +\frac{1}{2}-D\right)\end{align*} |
train/img_00000136.png | \begin{align*} T_k^i &:= \inf \{n \geq T_{k-1}^i : Q_n^i = q^i_0\} i=1,2 \,, \\ T_k^\Delta & := \inf \{n \geq T_{k-1}^\Delta : Q_n^1 = q^1_0, Q_n^2 = q^2_0\} \,. \end{align*} |
train/img_00000137.png | \begin{align*}(1-it)^{-z} &=\frac{y}{y+z} \,_2F_1(z,y+z;y+z+1;it) + \frac{z}{y+z}(1-it)^{-z} \,_2F_1(1,y;y+z+1;it)\\&= \frac{y}{y+z} (1-it)^{1-z}\,_2F_1(1,y+1;y+z+1;it)\\& + \frac{z}{y+z}(1-it)^{-z} \,_2F_1(1,y;y+z+1;it),\end{align*} |
train/img_00000138.png | \begin{align*}ds_E^2=\breve{g}_{\mu\nu}dx^\mu dx^\nu=(1+r^2\breve{\Omega}^2)d\tau^2-2\breve{\Omega} r^2d\tau d\tilde{\varphi}+r^2d\tilde{\varphi}^2+dr^2+dz^2~.\end{align*} |
train/img_00000139.png | \begin{align*} d_A(n) = \frac{|A\cap[n]|}{n + 1}.\end{align*} |
train/img_00000140.png | \begin{align*}v^i \to v'^i = {R^i}_jv^j;\;\;\;\forall\;R\in SO(3).\end{align*} |
train/img_00000141.png | \begin{gather*}(z^t,z^x,z^y,z^p,z^u,z^v,z^\omega,z^\phi,z^T)=(t,x,y,p,u,v,\omega,\phi,T),\\(\tilde z^t,\tilde z^x,\tilde z^y,\tilde z^p,\tilde z^u,\tilde z^v,\tilde z^\omega,\tilde z^\phi,\tilde z^T)=(\tilde t,\tilde x,\tilde y,\tilde p,\tilde u,\tilde v,\tilde \omega,\tilde \phi,\tilde T).\end{gather*} |
train/img_00000142.png | \begin{align*}N=\frac{1}{2}\sum_{n>0} n\sum_{R=1}^{D-1}[(\alpha^R_n)^{\dag}\alpha^R_n+(\tilde{\alpha}^R_n)^{\dag}\tilde{\alpha}^R_n],\end{align*} |
train/img_00000143.png | \begin{align*} \frac{\partial^{n-1}t}{\partial u^{n-1}}(u,v)=e^{-K(u,v)}\left\{\frac{d^{n-1}h}{du^{n-1}}(u)-\int_0^ve^{K(u,v')}R_{u,n-1}(u,v')dv'\right\},\end{align*} |
train/img_00000144.png | \begin{align*}\sum_{k=0}^{\lfloor{n}/{2}\rfloor}\binom{n}{2k} (5F_j^2)^k\left(\frac{F_{j(n-2k+1)}}{n-2k+1}{B_{2k}}-\frac{F_jL_j^{n-2k}}{2^n}\right) = 0\,,\end{align*} |
train/img_00000145.png | \begin{align*}[ \lambda_m , \lambda_n ] = (m-n) \lambda_{m+n},\end{align*} |
train/img_00000146.png | \begin{align*}u''-{\tilde\Lambda^2\over z^2}u=\tilde f(z,u),\qquad\tilde\Lambda\to\infty,\end{align*} |
train/img_00000147.png | \begin{align*} \aligned K(t,u) & = \frac{1}{2} (|t|+|u|-|t-u|) = \begin{cases}~u, & ~t>0,~u>0,~t-u>0, \\~t, & ~t>0,~u>0,~t-u<0, \\~0, & ~t<0,~u>0,~t-u<0, \\~-u, & ~t<0,~u<0,~t-u<0, \\~-t, & ~t<0,~u<0,~t-u>0, \\~0, & ~t>0,~u<0,~t-u>0, \\\end{cases}\endaligned\end{align*} |
train/img_00000148.png | \begin{align*} K_{\bar j}(\zeta,\bar\eta)=\langle\langle \cdot\bar\eta_{\bar j}, \cdot\bar\zeta\rangle\rangle, \ \ K_{j}(\zeta,\bar\eta)=\langle\langle \cdot\bar\eta, \cdot\bar\zeta_{\bar j}\rangle\rangle.\end{align*} |
train/img_00000149.png | \begin{align*}\nu_j = \frac{q^{j-1} }{j} \mbox{for } j \in \mathbb{N}_{+}.\end{align*} |
train/img_00000150.png | \begin{align*}\lim_n\, \|\,\rho^n \cdot x\,\|\,=\,0\end{align*} |
train/img_00000151.png | \begin{align*} a\,f(b,c,d) -f(ab,c,d) + f(a,bc,d) -f(a,b,cd) +f(a,b,c)\, d = 0\end{align*} |
train/img_00000152.png | \begin{align*}(\omega(g)\phi,\phi)=&~\|\phi\|^2 (\bar{\xi}\xi')^{\nu_2+\frac{1}{2}}(\bar{\eta}\eta'\bar{\zeta}\zeta')^{\nu_1+\frac{1}{2}}(\bar{\gamma}\gamma')^{\beta-\frac{1}{2}}(\cosh t)^{-\nu_2-\beta-3} \\&\times ( \cos \theta \cos \theta' +\cosh^{-1}t \zeta\bar{\zeta}'\sin \theta \sin \theta')^r. \end{align*} |
train/img_00000153.png | \begin{align*} w_{=2}(\zeta)= w_{2}(\zeta), \widehat{w}_{=2}(\zeta)=\widehat{w}_{2}(\zeta),\end{align*} |
train/img_00000154.png | \begin{align*}J_{UV}(A)=q_{UV}(p_{UV}^{-1}(A))\ .\end{align*} |
train/img_00000155.png | \begin{gather*}\frac{F^2}{w}(z) = \left(1 \mp \frac{i\pi}{w(z)} - \frac{\pi^2}{2w^2(z)} + O\left(\frac{1}{w^3(z)}\right)\right)\end{gather*} |
train/img_00000156.png | \begin{align*} |J'(\alpha,\alpha)|^2=q.\end{align*} |
train/img_00000157.png | \begin{align*}{1\over 2i}\,(T-T^*)={1\over 16\sqrt{s}}\,\int_0^{2\pi}\,{d\theta\over2\pi}\,|T(s,\theta)|^2,\quad 4\mu^2\le s<16\mu^2.\end{align*} |
train/img_00000158.png | \begin{align*}\sqrt{\sum_i \sqrt{IJ_i}}=\sqrt{I\sqrt{\sum J_i}}.\end{align*} |
train/img_00000159.png | \begin{align*} \begin{array}{c} p_{xx}+2q_{xy}+r_{yy}=0,\\ \ \\ m_x+n_y=0,\\ \ \\ m_z-qm_x-rm_y+(q_x+r_y)m=n_t-pn_x-qn_y+(p_x+q_y)n, \end{array} \end{align*} |
train/img_00000160.png | \begin{align*}\mathcal F(w)= \frac 12 w\cdot wJF^t =-\frac1 2 w F Jw^t\,,\end{align*} |
train/img_00000161.png | \begin{align*}\int_0^{+\infty} k \bar k^{-1}(\lambda) y V(y,d) \exp(-(k +\bar k(\lambda)) y)dy=2\bar D- W_k(\lambda, \nu, d),\end{align*} |
train/img_00000162.png | \begin{align*}W_1^a L_{-1} v^{(1)\pm} = L_1 W_{-1}^a v^{(1)\pm} = -\frac{1}{2} L_1 L_{-1} W^a_0 v^{(1)\pm},\end{align*} |
train/img_00000163.png | \begin{align*}[\beta,1-aT^i)=0.\end{align*} |
train/img_00000164.png | \begin{align*}{\tilde{X}^R}_a\psi=D^{(\epsilon)}(T_a)\psi\,\, ,\end{align*} |
train/img_00000165.png | \begin{align*}\sum_{j=1}^k n_j=n,n_j\geq 1\ .\end{align*} |
train/img_00000166.png | \begin{align*}\mathfrak{F}^2:= {\rm im}\,(\mathcal{W})\subset L^2(\mathbb{R}^{2d}) \ ,\end{align*} |
train/img_00000167.png | \begin{align*}\div |\beta|^2 g &= - d |\beta|^2= -2 g(\nabla \beta, \beta)\\ &=-2 \sum_{i,j} (\nabla_{e_i} \beta) (e_j)\beta(e_j)e_i\\&=-2 \sum_{i,j} ((\nabla_{e_j} \beta) (e_i)+d\beta(e_i,e_j))\beta(e_j)e_i\\&= - 2 \nabla_{\beta^\sharp}\beta + 2 \iota_{\beta^\sharp} d\beta.\end{align*} |
train/img_00000168.png | \begin{align*}u & =\frac{\xi^4}{\left(\sum_{r=0}^{t-1} \tilde{D}_{s,r}^2\right)^2}, \end{align*} |
train/img_00000169.png | f ( u ) \! \! = \! \! \prod _ { j = 1 } ^ { N } \frac { \sin ( u - \lambda \theta _ { j } ) } { \sin ( u - \lambda \theta _ { j } - \mu ) } , \qquad Q ( u ) \! \! = \! \! \prod _ { k = 1 } ^ { N / 2 } \sin \left( u - \frac { \mu } { 2 } - \lambda \alpha _ { k } \right) |
train/img_00000170.png | \begin{align*}b_\lambda(s) = \dfrac{1- q^{a_\lambda(s)} t^{l_\lambda(s)+1} }{1-q^{a_\lambda(s)+1} t^{l_\lambda(s)}}\end{align*} |
train/img_00000171.png | ( \partial _ { \mu } - \beta _ { \nu } \beta _ { \mu } \partial _ { \nu } ) \psi ( x ) = 0 ; \quad { \bar { \psi } } ( x ) ( \partial _ { \mu } - \beta _ { \nu } \beta _ { \mu } \partial _ { \nu } ) = 0 |
train/img_00000172.png | \begin{align*}N_{m}&=N_{m-1}+N_{m-3}\\&=(6N_{m-6}+5N_{m-11}+N_{m-16})+(6N_{m-8}+5N_{m-13}+N_{m-18})\\&=6(N_{m-6}+N_{m-8})+5(N_{m-11}+N_{m-13})+(N_{m-16}+N_{m-18}) \\&=6N_{m-5}+5N_{m-10}+N_{m-15},\end{align*} |
train/img_00000173.png | \begin{align*} (D_{E_\pm}\mathcal{J})E_\mp =0, (D_{E_\pm}\mathcal{J})E_\pm \subset E_\pm .\end{align*} |
train/img_00000174.png | \begin{align*}p(x) = y_1 \delta_{1}(x) + y_2 \delta_2(x) + \cdots + y_t \delta_{t}(x), \end{align*} |
train/img_00000175.png | \begin{align*}y_1 = \rho_0^2 \mathrm{exp}\left( 1-{A\over B}\right) \, ,\, y_{\chi}=\sqrt{e}y_1 \, , \,\chi=1 +{B\over 2 \rho_0^4}\mathrm{exp}\left( {2A\over B}-3\right)~.\end{align*} |
train/img_00000176.png | \begin{align*}f_{N}(x) &:= 2\log|p_{N}(x)|-2\mathbb{E}\log|p_{N}(x)| \sim 2\log|\det(e^{-(x^{2}-1/2-\log(2))}(xI-\mathcal{H}))|\end{align*} |
train/img_00000177.png | \begin{align*}S_{K}={\rm tr}_k\,\chi_a{\bf L}\chi_a - 4\pi i {\rm tr}_{k}\,\chi_{AB}\Lambda^{AB}\ .\end{align*} |
train/img_00000178.png | \begin{align*}\tau = \frac{\tau_0 + k }{2N- 2}, ~k=0,\cdots 2N-1,\end{align*} |
train/img_00000179.png | \begin{align*}L= \left(\begin{array}{cc|c} -2 & \phantom{-}1 & \phantom{-}2 \\ \phantom{-}1 & -2 & \phantom{-}2 \\ \hline \phantom{-}1 & \phantom{-}1 & -4 \end{array}\right), \widehat L= \left(\begin{array}{cc|c} -1 & \phantom{-}2 & -2 \\ \phantom{-}2 & -1 & -2 \\ \hline -1 & -1 & \phantom{-}4 \end{array}\right). \end{align*} |
train/img_00000180.png | \begin{align*}\mathbf{u}(t,x)=(x/r)u(t,r), \rho(t,x)=\rho(t,r), S(t,x)=S(t,r), r=|x|,\end{align*} |
train/img_00000181.png | \begin{align*}\chi _{P,a,eff}^{1,desc}\left( \mathbf{V};\mathcal{G}\right) =\frac{I_{a}(A)}{\pi_{a}(\mathbf{O}_{min};P) }\left\{ Y-b_{a}\left( \mathbf{O;}P\right) \right\}.\end{align*} |
train/img_00000182.png | D \omega _ { k \, d i v } = ( - 1 ) ^ { k } \sigma \omega _ { k + 1 \, d i v } . |
train/img_00000183.png | \begin{array} { l } { { \epsilon ( k _ { i } ) = 1 ~ , ~ \epsilon ( X _ { i } ^ { - } ) = 0 = \epsilon ( X _ { i } ^ { + } ) ~ , } } \\ { { S ( k _ { i } ) = k _ { i } ^ { - 1 } ~ , S ( X _ { i } ^ { - } ) = - X _ { i } ^ { - } k _ { i } ^ { - 1 } ~ , S ( X _ { i } ^ { + } ) = - k _ { i } X _ { i } ^ { + } ~ . } } \\ \end{array} |
train/img_00000184.png | \begin{align*}\P[A_u] = \int_{A_u} D(u) d\widehat{\P},\end{align*} |
train/img_00000185.png | \begin{align*}\left(\Pi_{\lambda_1,\lambda_2}g\right)^{\lambda_1,\lambda_2}_{i+\frac{1}{2},j+\frac{1}{2}} = g^{\lambda_1,\lambda_2}_{i+\frac{1}{2},j+\frac{1}{2}}, \mbox{if} \left(i,j\right) \in (\Bbbk_2^+,\Bbbk_1^+)\end{align*} |
train/img_00000186.png | \eta _ { \nu } ^ { \rho } \, \nabla _ { \rho } \, T ^ { \mu \nu } = 0 |
train/img_00000187.png | { \frac { 1 } { L } } E _ { g s } = - { \frac { 1 } { \pi } } h ^ { 2 } \ln ( h / m ) |
train/img_00000188.png | \begin{align*}&\mathbb P \left(\,s\in [t,t+\Delta t]\cap S(\mathbf{X}): a_s> t+\Delta t,~b_s> T-t \right)\\&=\left|\{s\in S(g)\cap [0,1): a_s> t+\Delta t,~b_s> T-t\}\right | \cdot \Delta t,\end{align*} |
train/img_00000189.png | \begin{align*}\nu\geq 0, \alpha_1\geq 0,\alpha_1+\alpha_2=0.\end{align*} |
train/img_00000190.png | \begin{align*}\zeta(z;\alpha,\beta,\gamma)=\frac{3-\alpha}{2}-\frac{2z-1}{2\sqrt{z^2-z+1}}-\frac{(1-\alpha)^2 z+\alpha\gamma-\beta}{2\sqrt{[(\gamma-\beta)z+\alpha+\beta]^2-4\gamma\beta z(1-z)}}\end{align*} |
train/img_00000191.png | \begin{align*}C_{a,b}(x,y,z):&=\dfrac{a}{2}(x^2 +y^2+ z^2)+b xy^{\lambda},\\H_{c,d}(x,y,z):&=\dfrac{c}{2}(x^2 +y^2 +z^2)+d xy^{\lambda}, ~ \forall (x,y,z)\in\Omega,\end{align*} |
train/img_00000192.png | \begin{align*}\sum_it^{h_i-1}\overline{t}^{\overline{h_i}-1}b_0^{(1)}\overline{b}_0^{(1)}|\Phi_i\rangle^{(1)}|\Phi^i\rangle^{(2)}\end{align*} |
train/img_00000193.png | \begin{align*}(c-K)\Delta K-|\nabla K|^2-4K(c-K)^2=0;\end{align*} |
train/img_00000194.png | \langle I | c _ { 0 } = \langle I | { \frac { ( - 1 ) ^ { n } } { 2 } } ( c _ { 2 n } + c _ { 2 n } ^ { \dagger } ) \not = 0 |
train/img_00000195.png | \begin{align*}{F}^{\infty}_{\mathrm{ren}} = - \frac1{8\pi^2} \int\! {\mathrm d}^3 x \, {g}^{1/2} {\mathrm tr}\Big\{ R_{i j } \gamma_1(-\triangle) R^{i j }+ R \gamma_2(-\triangle) R +{\mathrm{O}}[\Re^3]\Big\}. \end{align*} |
train/img_00000196.png | \begin{align*}\pi^{-1}(D) = G\times F_0.\end{align*} |
train/img_00000197.png | \begin{gather*}f_{M}((1,0)) =\alpha _{M}f_{M}((0,1)) =\beta _{M}^{-1}.\end{gather*} |
train/img_00000198.png | \begin{align*}\nabla_Y \mathcal{P}(X)= - \mathcal{P}(B^*(Y,X)) - B(Y,s(X)) + B^*(Y,t(X)).\end{align*} |
train/img_00000199.png | e x p ( - W [ g , G , B ] ) = \int [ d \eta ] \, e x p ( - I [ x + \eta , G ( x + \eta ) , B ( x + \eta ) , C ( x + \eta ) , g ] ) . |
Subsets and Splits