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train/img_00000000.png | h _ { \parallel } ( \beta _ { T } \mu ) \sim - \ln ( \beta _ { T } \mu ) > 0 |
train/img_00000001.png | \begin{align*}\|T_hf\|_p^p&=\int_X \Bigl|\int_X f(y)\> K_h(x,dy)\Bigr|^p\> d\omega_X(x) \\&\le\int_X\Bigl(\int_X |f(y)|^p\> K_h(x,dy)\Bigr) \Bigl(\int_X 1\> K_h(x,dy)\Bigr)^{p/q}d\omega_X(x) =\|f\|_p^p.\end{align*} |
train/img_00000002.png | \begin{align*}j(q^{20};q^{32}) = j(q^{10};q^{16})j(-q^{10};q^{16})\frac{(q^{32};q^{32})_\infty}{(q^{16};q^{16})_{\infty}^2},\\j(q^{4};q^{32}) = j(q^{2};q^{16})j(-q^{2};q^{16})\frac{(q^{32};q^{32})_\infty}{(q^{16};q^{16})_{\infty}^2}.\end{align*} |
train/img_00000003.png | P _ { 0 } = \frac \kappa 2 \, \left( 1 - e ^ { - 2 p _ { 0 } / \kappa } + \frac { \vec { p } \, { } ^ { 2 } } { \kappa ^ { 2 } } \right) . |
train/img_00000004.png | \begin{align*}C^{-1}&=\frac{1}{3}\begin{pmatrix}2&1\\ 1&2\end{pmatrix}& C(z)&=\begin{pmatrix}z+z^{-1}&-1\\ -1&z+z^{-1}\end{pmatrix}& \widetilde{C}(z)&=\frac{1}{z^{2}+1+z^{-2}}\begin{pmatrix}z+z^{-1}&1\\ 1&z+z^{-1}\end{pmatrix}.\end{align*} |
train/img_00000005.png | \begin{align*} \mathcal{I}\left(a, m, 0, \frac{\pi}{2}\right) = \frac{1}{(1+a)^{m}} F_{1}\left( \frac{1}{2}, \frac{1}{2} - m, m, \frac{3}{2}, 1, \frac{1}{1+a} \right)\end{align*} |
train/img_00000006.png | \begin{align*}\mathcal S_{\boldsymbol\nu,\boldsymbol\mu}=O\left(\frac{f_{k_1}(z_i)^Nf_{k_2}(z_o)^N}{z_i^M z_o^M}\left(\frac{M}{eN}\right)^{2M}\prod_{j=1}^3\nu_j^{\mu_{j,\boldsymbol\cdot}+\mu_{\boldsymbol\cdot,j}}\right).\end{align*} |
train/img_00000007.png | \begin{align*}s=\frac{r}{\chi-r} (<1),\end{align*} |
train/img_00000008.png | \begin{align*}p^{-1} R^{-p} & = p^{-1} \left( \frac{\sqrt{\frac{\kappa}{CA} +\left( \frac{c_d'|\nabla f(a)|}{2CA} \right)^2} + \frac{c_d'|\nabla f(a)|}{2CA}}{\frac\kappa{CA}} \right)^p \\& = \frac{c_{d,p}}{\kappa^p} \left( \sqrt{ \tfrac{CA}{c_d'^2} \kappa +\left( \tfrac12 |\nabla f(a)| \right)^2} + \tfrac12|\nabla f(a)| \right)^p.\end{align*} |
train/img_00000009.png | M _ { p } = 2 \left( \begin{array} { c } { { 2 p } } \\ { { p - 1 } } \\ \end{array} \right) = \frac { 2 p } { p + 1 } \, N _ { p } \ . |
train/img_00000010.png | \left( \begin{array} { c } { { { a } } } \\ { { { b } } } \\ \end{array} \right) \to U \cdot \left( \begin{array} { c } { { a } } \\ { { b } } \\ \end{array} \right) , |
train/img_00000011.png | \begin{align*}\omega^{\sigma_{\cal F}} = -i(\omega_G - p^* \omega_{WP})~.\end{align*} |
train/img_00000012.png | \begin{align*} h_{ij}^X= & \frac{h_{ij}^Y}{\sqrt{(1-|Y|^2)(1-\langle N,Y\rangle^2)}},\end{align*} |
train/img_00000013.png | \begin{align*}&\mathbb{P}\left( \gamma_{\mathrm{u},ik} \geq \theta \right) \approx 1 - {2\beta \theta} = 1 - \frac{2 K \theta}{N}, \\ &\mathbb{P}\left( \gamma_{\mathrm{a},ik} \geq \theta \right) \approx 1 - \left( {2 \beta \theta}\right)^2 = 1 - \frac{4 K^2 \theta^2}{N^2}.\end{align*} |
train/img_00000014.png | \begin{align*}(\Phi^h)^m &= i^m(-1)^{\frac{m(m-1)}{2}}m!\:h\:\mathcal{Z}^1\wedge\dots\wedge\mathcal{Z}^m\wedge\mathcal{Z}^{\bar{1}}\wedge\dots\wedge\mathcal{Z}^{\bar{m}},\\(\Phi^v)^m &= i^m(-1)^{\frac{m(m-1)}{2}}m!\:h\:\delta\mathcal{V}^1\wedge\dots\wedge\delta\mathcal{V}^m\wedge\delta\mathcal{V}^{\bar{1}}\wedge\dots\wedge\delta\mathcal{V}^{\bar{m}},\end{align*} |
train/img_00000015.png | \mu \to \frac { d } { d \lambda } ~ , \quad \lambda \to \lambda ~ . |
train/img_00000016.png | \begin{align*}P_M^R(t)=\dfrac{P_M^S(t)}{1-t(P_R^S(t)-1)}.\end{align*} |
train/img_00000017.png | \begin{align*}\xi = \inf_{X\neq 0}\left\{ \dfrac{\sum_{\mu=1}^{M_A} \sum_{\nu=1}^{M_B} \left\vert \langle g_\mu^A \vert X \vert f_\nu^B \rangle \right\vert^2}{\sum_{\mu=1}^{N_v} \sum_{\nu=1}^{N_v} \left\vert \langle g_\mu^A \vert X \vert f_\nu^B \rangle \right\vert^2} \right\} \ .\end{align*} |
train/img_00000018.png | \begin{align*}f_{2}(a)=B_{+}\geq-e_{2}>0. \end{align*} |
train/img_00000019.png | \begin{align*}\Lambda_k^2=1+r_k^w(x,D_x),\end{align*} |
train/img_00000020.png | \begin{align*}I(t,\alpha,n)= \int_0^1e^{n(it\frac{p}{\sqrt{\alpha(1-\alpha)n}}+\alpha {\rm log}p + (1-\alpha) {\rm log}(1-p))}{\rm d} p.\end{align*} |
train/img_00000021.png | \begin{align*}A_{\mu}=e_{\mu}^{\;a}P_{a}+\omega_{\mu}^{\;a}J_{a}+\chi_{\mu}^{\;\alpha}Q_{\alpha}+\xi_{\mu}^{\;\alpha}Q'_{\alpha}\end{align*} |
train/img_00000022.png | \begin{align*}{\bf{P}}_{1} := a^{3} \, \sum_{n} \frac{1}{1 + \eta \, \lambda_{n}^{(3)}} \int_{B}e_n^{(3)}(y)\,dy\otimes\int_{B}e_n^{(3)}(y)\,dy = a^{3} \, \sum_{n} \frac{1}{1 + \eta \, \lambda_{n}^{(3)}} \, \sum_{m} \int_{B}e_{n,m}^{(3)}(y)\,dy\otimes\int_{B}e_{n,m}^{(3)}(y)\,dy, \end{align*} |
train/img_00000023.png | \begin{align*}S_nQ_{n+1}-S_{n+1}Q_n=(-1)^nw\end{align*} |
train/img_00000024.png | \begin{align*}\pi_{n}=\frac{\lambda_1\cdots\lambda_{n-1}}{\mu_1\cdots \mu_n},\;\textup{for}\; n\geq 2\end{align*} |
train/img_00000025.png | \begin{align*}S(Q^{A},P_{A};N,N^{\alpha}) = {\int} dt(P_{A}{\dot Q}^{A}-N.H-N^{\alpha}.H_{\alpha})\ \longrightarrow\ stat.\end{align*} |
train/img_00000026.png | \begin{align*}G_{0}(X(\gamma),Y(\gamma))=\int_0^1g(X(\gamma;t),Y(\gamma;t))dt.\end{align*} |
train/img_00000027.png | g _ { 1 1 } = g _ { 2 2 } = 1 + ( \partial _ { 1 } A ) ^ { 2 } + ( \partial _ { 2 } A ) ^ { 2 } + ( \partial _ { 1 } C ) ^ { 2 } + ( \partial _ { 2 } C ) ^ { 2 } , |
train/img_00000028.png | \begin{align*} |-t_k^{j,1}\log p_r -\theta_r^j -2\pi q|<2^{-k}; j=1,\dots,N r=1,\dots,k. \end{align*} |
train/img_00000029.png | \begin{align*}\beta(n)=\frac{\beta(1)}{\beta(0)}\beta(n-1)=\left(\frac{\beta(1)}{\beta(0)}\right)^2\beta(n-2)=\cdots=\frac{\beta(1)^n}{\beta(0)^{n-1}}=(-\lambda)^{1-n}\beta(1)^n.\end{align*} |
train/img_00000030.png | \begin{align*}a^*_{1,1}=\frac{\mu_R}{3},a^*_{0,1}=\frac{2}{3}-\mu_R-\mu_T,a^*_{0,2}=\mu_T-\frac{1}{3},\end{align*} |
train/img_00000031.png | \begin{align*}\phi(t_0) = u(t_0), \phi'(t_0)= u'(t_0), \dots, \phi^{(n-1)}(t_0)= u^{(n-1)}(t_0).\end{align*} |
train/img_00000032.png | \begin{align*}\begin{aligned}2 i k z b(k)=& \Psi^+_{11}(0 ; z)\left(z \Psi^-_{21}(0;z)-\widehat{\Psi}^-_{21}(0)\right)-\Psi^-_{11}(0;z)\left(z \Psi^+_{21}(0; z)-\widehat{\Psi}^+_{21}(0)\right) \\&+\widehat{\Psi}^-_{21}(0)\left(\Psi^+_{11}(0;z)-e^{ic_+(0)}\right)-\widehat{\Psi}^+_{21}(0)\left(\Psi^-_{11}(0;z)-e^{ic_-(0)}\right)\end{aligned}\end{align*} |
train/img_00000033.png | X _ { j } = 2 \, a _ { j } \exp \left( \xi _ { j } ( { t , x } ) \right) |
train/img_00000034.png | \begin{align*}f(K)=f(K\cap H_i) + f(K \cap A) - f(H_i\cap A\cap K)\geq f(K\cap H_i)\end{align*} |
train/img_00000035.png | \begin{align*}\epsilon^{ij}\partial_i b_j=-{\textstyle\frac{1}{2(2\pi\alpha^\prime)}}\epsilon^{ij}(i_k C^{(3)})_{ij}\, .\end{align*} |
train/img_00000036.png | \begin{align*}\|f\|_{L^{p}(w)} \le C([\vec w]_{A_{\vec p}}) \prod_{i=1}^m\|f_i\|_{L^{p_i}(w_i)}\end{align*} |
train/img_00000037.png | \begin{align*}\frac{\ddot T}{T |T|^2} = \frac{1}{2} X_{11} \bar X + \frac{1}{2}|X|^2 - \frac{1}{2} X{\bar{X}}_{11} -\frac{1}{2}(X_{1})^2\frac{\bar{X}}{X} = \frac{A}{2}\end{align*} |
train/img_00000038.png | \begin{align*}I_{k}:= \int_{\mathbb{R}^{N}}\int_{\mathbb{R}^{N}}\frac{(|\tilde{u}_k(y)|^{p(x,y)}+|\tilde{v}_k(y)|^{p(x,y)})| \phi (x)- \phi (y)|^{p(x,y)}}{\vartheta_{k}^{p(x,y)}|x-y|^{N+sp(x,y)}} \,dy\,dx.\end{align*} |
train/img_00000039.png | \begin{align*}(f_1\cup f_2)\bullet f_3-(f_1\bullet f_3)\cup f_2-(-1)^{(m_1-p_1)(m_3-p_3-1)}f_1\cup (f_2\bullet f_3)=0.\end{align*} |
train/img_00000040.png | \begin{align*}\Gamma_k := \bigg\{ f\in L^p([0,T],H) : \int_0^{T-\delta_k} \Vert f(t+\delta_k) - f(t) \Vert_Y^p dt \leq \frac{1}{k} \bigg\} .\end{align*} |
train/img_00000041.png | \begin{align*}[f_1,\cdots,f_M]_*\;:=\;\sum_{\sigma \;\in\; S_M}\;({\rm sign}\;\sigma)\;(f_{\sigma\,1}\cdots f_{\sigma\,M})_*\end{align*} |
train/img_00000042.png | \begin{align*}\mathbf v=a_1\mathbf v_1+\sigma a_2\mathbf v_2+\alpha(\delta_2/g\mathbf v_1-\sigma\delta_1/g\mathbf v_2),\end{align*} |
train/img_00000043.png | \begin{align*}X^3+\frac{1}{3}\alpha^2X^2+\alpha X+1=(X-x_1)(X-x_2)(X-x_3).\end{align*} |
train/img_00000044.png | \begin{align*}e^{i\Gamma[A]}=\int{\cal D}\psi{\cal D}\overline\psi e^{i\int d^2x\overline\psi iD\!\!\!/\psi},\end{align*} |
train/img_00000045.png | \begin{align*} u^f_{n,t}=\prod_{k=0}^{n-1}a_kf_{t-n}+\sum_{s=n}^{t-1}w_{n,s}f_{t-s-1},n,t\in\mathbb{N}.\end{align*} |
train/img_00000046.png | \frac { d } { d z } \left( \begin{array} { r r } { { \chi _ { 1 } ( z ) } } \\ { { \chi _ { 2 } ( z ) } } \\ \end{array} \right) = |
train/img_00000047.png | \begin{align*}\lambda_{1,e}(B_\infty^n) = \frac{n}{n-1} .\end{align*} |
train/img_00000048.png | \begin{gather*}[\lambda] = \big\{ (i,j) \in \mathbb{Z}^2 \colon 1\leq i \leq \ell(\lambda),\ 1 \leq j \leq \lambda_i \big\}.\end{gather*} |
train/img_00000049.png | \sigma _ { ~ ~ ; \alpha } ^ { ; \alpha } = 2 - \frac { 1 } { 6 } g _ { \gamma \delta } R \sigma ^ { ; \gamma } \sigma ^ { ; \delta } + \frac { 1 } { 2 4 } g _ { \gamma \delta } R _ { ; \rho } \sigma ^ { ; \gamma } \sigma ^ { ; \delta } \sigma ^ { ; \rho } + O ( \sigma ^ { 2 } ) ~ . |
train/img_00000050.png | \begin{align*} dz_t = \Pi_{\mathrm{ker}A}(B(y_t,y_t)-Ay_t) dt + \Pi_{\mathrm{ker}A}\sigma dW_t. \end{align*} |
train/img_00000051.png | \begin{align*}L(x)^{q^i}=\sum_{k=1}^mc_{k-i}^{q^i}\,x^{q^k},\end{align*} |
train/img_00000052.png | \begin{align*}\sum_{i=0}^{r}d_i\sigma(x)^i = a(x)P_\alpha(x) + b(x)Q_\alpha(x)+ c(x)T(x),\end{align*} |
train/img_00000053.png | \begin{align*}a_1=\alpha_1, a_2=\alpha_2, {a_3} =\frac{2\,\alpha_1+3\,\alpha_3}{8}\end{align*} |
train/img_00000054.png | \begin{align*}\binom{\l +q+1}{m+n+1}=\sum_{0\le k \le \l} \binom{\l -k}{m}\binom{q+k}{n} (\l, m \ge 0, n\ge q\ge 0).\end{align*} |
train/img_00000055.png | \begin{align*}M_{12}(\lambda)=\begin{bmatrix}\lambda P_4-P_3 & \lambda P_3 \end{bmatrix} \mbox{and} M_{22}(\lambda)=\begin{bmatrix}\lambda P_3-P_2 & \lambda P_2 \\\lambda P_2 & \lambda P_1+P_0 \end{bmatrix}.\end{align*} |
train/img_00000056.png | { \Delta _ { 1 } u } ^ { \alpha ( i , j ) } = \frac { u ^ { \alpha ( i + 1 , j ) } - u ^ { \alpha ( i , j ) } } { x _ { 1 } ^ { ( i + 1 ) } - x _ { 1 } ^ { ( i ) } } , \qquad { \Delta _ { 2 } u } ^ { \alpha ( i , j ) } = \frac { u ^ { \alpha ( i , j + 1 ) } - u ^ { \alpha ( i , j ) } } { x _ { 2 } ^ { ( j + 1 ) } - x _ { 2 } ^ { ( j ) } } . |
train/img_00000057.png | i \gamma ^ { \mu } \partial _ { \mu } \left( \begin{array} { c } { { \psi _ { L } } } \\ { { \psi _ { R } } } \\ { { \psi _ { \nu } } } \\ \end{array} \right) = \left( \begin{array} { c c c } { { 0 } } & { { \phi ^ { 2 } } } & { { 0 } } \\ { { \overline { { { \phi ^ { 2 } } } } } } & { { 0 } } & { { \overline { { { \phi ^ { 1 } } } } } } \\ { { 0 } } & { { \phi ^ { 1 } } } & { { 0 } } \\ \end{array} \right) \left( \begin{array} { c } { { \psi _ { L } } } \\ { { \psi _ { R } } } \\ { { \psi _ { \nu } } } \\ \end{array} \right) . |
train/img_00000058.png | \begin{align*}u_x=\frac{1}{\kappa}[s_1X_x\tilde{u}_{\tilde{x}}-r_2X_x\tilde{v}_{\tilde{x}}+r_2(s_{1x}v+s_{2x}u+s_{3x})-s_1(r_{1x}u+r_{2x}v+r_{3x})],\end{align*} |
train/img_00000059.png | \begin{align*}&\frac12\frac{d}{dt}\sum_{q\geq -1}\left(\lambda_q^{2s}\|u_q\|_2^2+\lambda_q^{2s}\|b_q\|_2^2\right)\\\leq &-\nu\sum_{q\geq-1}\lambda_q^{2s+2}\|u_q\|_2^2-\mu\sum_{q\geq-1}\lambda_q^{2s+2\alpha}\|b_q\|_2^2+ I_1+I_2+I_3+I_4+I_5,\end{align*} |
train/img_00000060.png | \begin{align*}(I-\Delta)^{-\alpha} u = G_{2\alpha}\star u\end{align*} |
train/img_00000061.png | \begin{align*} \upsilon_R'(r)= -\frac{i}{\sqrt\pi}r^{-1-\frac k2} \int_{\partial D_r}e^{-\frac{i}{2}\theta_{\mathbf e}}(i\nabla+A_{\mathbf e})w_R\cdot\nu\, \psi_k\, ds.\end{align*} |
train/img_00000062.png | \begin{align*}\bar{L}=L_{0}-\frac{\partial L^{i}}{\partial x^{i}}-y_{i}^{\alpha }\frac{\partial L^{i}}{\partial y^{\alpha }},\end{align*} |
train/img_00000063.png | \begin{align*} S=\int L \,,L=-\Big(\frac14 F_{\mu\nu}F^{\mu\nu}+C\partial^\mu A_\mu^\ast\Big)d^4x\,.\end{align*} |
train/img_00000064.png | \begin{align*}\sup_{x\in D_{k+2}}\frac{u^k_{k+1}(x)}{v^k_{k+1}(x)}=(1+c_1\delta\rho\zeta^k)\inf_{x\in D_{k+2}}\frac{u^k_{k+1}(x)}{v^k_{k+1}(x)}\end{align*} |
train/img_00000065.png | \begin{align*}V_{ij}={n-j \choose n-i}(-1)^{i-j}I.\end{align*} |
train/img_00000066.png | \begin{align*} P_{k,\ell,p}(j) = \sum_{k'=2}^{k} \sum_{\ell'=2}^{\ell} \binom{k-k'+j-1}{k-k'} \binom{\ell-\ell'+j-1}{\ell-\ell'} \Big( \Big( \frac{p}{p-1} \Big)^{k'-1} + \Big( \frac{p}{p-1} \Big)^{\ell'-1}-1 \Big)\end{align*} |
train/img_00000067.png | \begin{align*}\det(B)\ne0\mathrm{ind}(B)=\Phi(D,k')-(k'-1).\end{align*} |
train/img_00000068.png | \begin{align*}B_{{\mathbf x}}(w)=\displaystyle\frac{1}{2}(0,b_{21}^2{\mathbf x}_2w_1+b_{22}^2{\mathbf x}_2w_2).\end{align*} |
train/img_00000069.png | F ^ { I _ { 3 } I _ { 2 } I _ { 1 } I _ { 4 } } F ^ { I _ { 1 } I _ { 2 } I _ { 3 } I _ { 4 } } \ = \ 1 \ |
train/img_00000070.png | \begin{align*}K_2=(K_1\cap K_2)(K_2\cap K_3).\end{align*} |
train/img_00000071.png | ( d s ) ^ { 2 } = - \frac { 1 } { \sqrt { \Lambda } } \left( 1 + \frac { 2 } { 3 } \delta \mathrm { c o s } R \right) \mathrm { s i n } ^ { 2 } R d ^ { 2 } T + \frac { 1 } { \Lambda } \left( 1 - \frac { 2 } { 3 } \delta \mathrm { c o s } R \right) d ^ { 2 } R + \frac { 1 } { \Lambda } \left( 1 - 2 \delta \mathrm { c o s } \right) d ^ { 2 } \Omega _ { ( 2 ) } |
train/img_00000072.png | \begin{align*}F(x) = - \frac{(2x+2)^2 -e^{-4}}{4x + 4}, x \ne -1.\end{align*} |
train/img_00000073.png | \begin{align*} \begin{cases} \dot S = -a(-)SI +c(-)(N-S-I)\\ \dot I = a(-)SI - b(-)I.\end{cases}\end{align*} |
train/img_00000074.png | \Psi _ { \mathrm { { 0 } } } ( r ^ { \prime } , \theta ^ { \prime } ; 0 ) = { \frac { 1 } { \sqrt { 2 \pi } \xi } } \exp \left\{ i k r ^ { \prime } \cos \theta ^ { \prime } - { \frac { 1 } { 4 \xi ^ { 2 } } } ( r ^ { 2 } + r _ { 0 } ^ { 2 } + 2 r r ^ { \prime } \cos \theta ^ { \prime } ) \right\} . |
train/img_00000075.png | \begin{align*} d X_t = \xi_t \;d t + \sqrt{2\nu} \;d W_t \forall t\in [0,T],\end{align*} |
train/img_00000076.png | \begin{align*}\begin{array}{l}\|\tilde{\mathcal{G}}_{\tilde{e}_N}^M(s)\tilde\Phi_M-\tilde{\mathcal{G}}_{\tilde{e}}^M(s)\tilde\Phi_M\|_M\leq\\\sum\limits_{i,j=1}^N\|\tilde{e}_N^i(s)-\tilde{e}_i(s)\|_\infty\|f\circ\sigma_{ij}-f\|_{\infty}\leq\\2(N-1)\sum\limits_{i=1}^N\|\tilde{e}_N^i(s)-\tilde{e}_i(s)\|_\infty\|f\|_{\infty},\end{array}\end{align*} |
train/img_00000077.png | \begin{align*} \pi_2\left(gB_0g^{-1},gB_1g^{-1},..., g(FB_0)g^{-1}\right)= \left(gB_{r+1}g^{-1},F(gB_1g^{-1}),...,F(gB_{r+1}g^{-1})\right).\end{align*} |
train/img_00000078.png | \begin{align*}a(u,v)=(f,v)~v\in H^1_0(D),\end{align*} |
train/img_00000079.png | \begin{align*} {\cal O}^{(k)} = u_{i_1} \cdots u_{i_k} \mbox{Tr}(\phi^{i_1} \cdots \phi^{i_k}) \, .\end{align*} |
train/img_00000080.png | \begin{align*} p(t,g)=&\mathcal{F}^{-1}_{\xi}[{\hat{p}(0,e^{-t}(\xi-\mu)+\mu)e^{-H_t(\xi-\mu)}}]\\ &=e^{i\mu g}\mathcal{F}^{-1}_{\xi}[{\hat{p}(0,e^{-t}\xi+\mu)e^{-H_t(\xi)}}]\\ &=e^{i\mu g}\mathcal{F}^{-1}_{\xi}[\hat{p}(0,e^{-t}\xi+\mu)]*(\frac{1}{\sqrt{2\pi}}\mathcal{F}^{-1}_{\xi}[e^{-H_t(\xi)}]). \end{align*} |
train/img_00000081.png | \begin{align*}a_{\mathrm{pw}}(\phi_{\mathrm{nc}}({j}),v_{\mathrm{nc}})=\lambda_h(j) b(\phi_{\mathrm{pw}}({j}),v_{\mathrm{nc}})\phi_{\mathrm{pw}}({j})-\phi_{\mathrm{nc}}({j})=\lambda_h({j})\kappa_{m}^2h_{\mathcal{T}}^{2m}\phi_{\mathrm{pw}}({j}). \end{align*} |
train/img_00000082.png | \begin{align*}|\delta e|^2 = \int d \lambda ({\hat e} (\lambda; l ))^{-1} (\delta e(\lambda;l))^2 = \int d \lambda [ - {\hat e}(\lambda) (\delta f) {{d^2}\over{d \lambda^2}} (\delta f) + {{(\delta \rho [f(\lambda)])^2}\over{{\hat e}}} ] , \end{align*} |
train/img_00000083.png | \begin{align*}\hat\phi(0)-\phi(0)+\int_{-\infty}^{\infty}\hat\phi(x)|x|dx=\int_{-\infty}^{\infty}\phi(x)W_{\rm U}^1(x)dx=\int_{-\infty}^{\infty}\phi(x)\left(1-\frac{\sin^2(\pi x)}{(\pi x)^2}\right)dx.\end{align*} |
train/img_00000084.png | \begin{align*}\begin{array}{l}\displaystyle [e_{n+1},e_1]=-e_3,[e_{n+1},e_{2}]=-e_2,[e_{n+1},e_i]=-e_i-\epsilon e_{i+2}-\sum_{k=i+3}^n{b_{k-i-2}e_k},\\\displaystyle (\epsilon=0,\pm1,3\leq i\leq n,5\leq j\leq n);DS=[n+1,n-1,0],LS=[n+1,n-1,n-1,...].\end{array} \end{align*} |
train/img_00000085.png | p _ { i } = \varepsilon _ { i j } \frac { E _ { j } } { B } |
train/img_00000086.png | \begin{align*}\eta_k(r) : \begin{cases} =1 &\mbox{if } -\infty<r\leq k\\ 0<\eta_k(r)<1&\mbox{if } k< r< k+\delta_k\\ =0 &\mbox{if } k+\delta_k\leq r<\infty. \end{cases}\end{align*} |
train/img_00000087.png | \begin{align*}\tilde{O}({\bf R}, {\bf r})= O_{0}({\bf R}) + {\bf r} \cdot{\bf O}_{1}({\bf R}) +o(r^{2}).\end{align*} |
train/img_00000088.png | \begin{align*}\overline{W}^{(q)}(x) = 0,\overline{\overline{W}}^{(q)}(x) = 0,Z^{(q)}(x) = 1,\textrm{and} \overline{Z}^{(q)}(x) = x, x \leq 0. \end{align*} |
train/img_00000089.png | \begin{align*}c_{\alpha}d_{h-\alpha}\varepsilon(h, \alpha)=-\frac{1}{3}\end{align*} |
train/img_00000090.png | \begin{align*}\xi^\ast=\frac{\beta(\omega)(1-\omega)-1+\alpha}{2-\alpha}.\end{align*} |
train/img_00000091.png | \begin{align*}J_{k_U}(n, \rho) = O\left( 2^{-(2\epsilon+\epsilon^2)\log_2 n + O(\sqrt{\log n} \log\log n )} \right),\end{align*} |
train/img_00000092.png | \begin{align*}\forall y,y' \in I,\; \phi_H(u_{J(y)},u_{J(y')})=0\;,\end{align*} |
train/img_00000093.png | \begin{align*}\left(1 - \frac{\beta_{i}\pi_{i}}{\gamma_{i} + \delta m}\right)^{-1} \leq \mathbb{E} \left(\sum_{j=1}^{m} T_{ij}(\pi,Q) \right), \end{align*} |
train/img_00000094.png | \begin{align*} T_{R} = T_{R 33} e_3 \otimes e_3.\end{align*} |
train/img_00000095.png | \begin{align*}\tilde{\Gamma}^i_{0k}=\tilde{\Gamma}^i_{k0}&=\frac12 \tilde{g}^{im}\left(\tilde{g}_{m0,k}+\tilde{g}_{mk,0}-\tilde{g}_{0k,m}\right)\\&=\frac12 r^{-2}g^{im}\left(\tilde{g}_{mk,0}\right)\\&=\frac12 r^{-2}g^{im}\left(2rg_{mk}\right)\\&=r^{-1}\delta^i_k.\end{align*} |
train/img_00000096.png | \begin{align*}\widetilde{\gamma}^*\omega_n^{2n}=\langle\frac{de_n(s)}{ds},e_{2n}(s)\rangle ds=\kappa_n(s)ds.\end{align*} |
train/img_00000097.png | c _ { A B C } = ( \frac { \omega _ { B } - \omega _ { C } } { \omega _ { A } } - 1 ) d _ { A B C } . |
train/img_00000098.png | \begin{align*}Qv = \left(I + v (u'-u)\right) v= v + v\left(u'v - uv\right)= v\end{align*} |
train/img_00000099.png | \begin{align*} & \int x (q^2x^2;q^2)_\infty h_n(x;q) d_qx= \frac{(1-q)x(x^2;q^2)_\infty}{[n]_q-1} \left(\frac{q^n}{x}h_n(\frac{x}{q};q)-q^{n-1}[n]_q h_{n-1}(\frac{x}{q};q) \right), \end{align*} |
Subsets and Splits