harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\hat{H}= \frac{1}{2}\hat{p}_{0}^{2}+ \frac{\omega_{0}^{2}}{2}\hat{q}_{0}^{2}+ \frac{1}{2}\sum_{k = 1}^{N}\left[ \hat{p}_{k}^{2}+ \omega_{k}^{2}\hat{q}_{k}^{2}+ 2 \lambda_{k}( t ) \hat{q}_{0}\hat{q}_{k}\right] \; .
2\pi\cdot|K(u,v(t))|=\frac{|(v(t)-u)\cdot n(0)|}{|v(t)-u|^{2}}\le\frac{2|\lambda|+C_{1}t^{2}}{\lambda^{2}+t^{2}}\le\frac{2|\lambda|}{\lambda^{2}+t^{2}}+C_{1}.
\left<{\cal O}\left[ \vec{\xi}{\,}\right] \right>_{\vec{\xi}}= \frac{\int_{\vec{\xi}( 0 ) = \vec{\xi}( 1 )}^{}{\cal D}\vec{\xi}e^{- S}{\cal O}\left[ \vec{\xi}{\,}\right]}{\int_{\vec{\xi}( 0 ) = \vec{\xi}( 1 )}^{}{\cal D}\vec{\xi}e^{- S}}.
\Gamma_{U}^{\prime}= \frac{\delta}{\delta U}i \mathrm{I m}\Gamma = - 2 \int_{0}^{\infty}\frac{d T}{T}\int [ D X_{\mu}] [ D \bar{X}] [ D X^{\prime}] ( - 1 )^{F}\Omega_{U}e^{- S_{W L}}
\int_{\mathbb{R}^n}\frac{v(x)^{q}}{|x|^{\sigma_1}}(x\cdot \nabla v(x))\,dx + \int_{\mathbb{R}^n}\frac{u(x)^{p}}{|x|^{\sigma_2}}(x\cdot \nabla u(x))\,dx = -(n-\alpha)\int_{\mathbb{R}^n}\frac{v(x)^{q+1}}{|x|^{\sigma_1}}\,dx.
R_c\simeq \frac{1}{4}\ \bigg[ \frac{8\, ( J^2_x-J_z^2)}{J_x(J_x-J_y)}\bigg]^{\frac{1}{2-2\eta}}\, \quad\qquad \Big(\eta=\frac{1}{\pi}\, \arccos(J_z/J_x)\, \Big)
u_\alpha-1 =\sum_{i=1}^sa_i\left (\sum_{j=1}^k w_{ij}X_j\right )+\sum_{i=1}^t b_iX_{j_i}\equiv\sum_{i=1}^sa_i\left (\prod_{j=1}^k g_j^{w_{ij}}-1\right )+\sum_{i=1}^tb_i(g_{j_i}-1))(\mod J^2) =\sum_{i=1}^sa_i(f_i-1)+\sum_{i=1}^tb_i(g_{j_i}-1),
\frac{\partial \widetilde{R}_{1,k}}{\partial\lambda}= \frac{\alpha\tau}{T}\cdot\frac{1}{ln2}\cdot\frac{-(\frac{\alpha}{\lambda \tau}+ \frac{f}{(1-\lambda)})+\frac{\alpha}{\lambda^2 \tau}}{\frac{(1-\lambda)\alpha}{\lambda \tau}+ f}+\Big(1-\frac{\tau}{T}\Big)\cdot\frac{1}{ln2}\cdot \frac{\frac{\alpha}{\lambda^2 \tau}}{\frac{(1-\lambda)\alpha}{\lambda \tau}+ f},
v_{A(3)}= -2\Omega_{A(3)}&= -16[B_1^+,B_3^+]^2(B_2^+)^2v_0 -16[B_2^+,B_3^+]^2(B_1^+)^2v_0 &\qquad-16[B_1^+,B_3^+][B_2^+,B_3^+]B_1^+B_2^+v_0-16[B_1^+,B_3^+][B_2^+,B_3^+]B_2^+B_1^+v_0 &\qquad-16[B_2^+,B_3^+][B_1^+,B_3^+]B_1^+B_2^+v_0-16[B_2^+,B_3^+][B_1^+,B_3^+]B_2^+B_1^+v_0.
\epsilon^{r_1\cdots r_9m}\epsilon^{s_1\cdots s_9}{}_m\,W_{r_1r_2s_1s_2}W_{r_3r_4s_3s_4}W_{r_5r_6s_5s_6}\,\Big( H_{r_7r_8r_9}H_{s_7s_8s_9}-3\,H_{r_7r_8s_9}H_{s_7s_8r_9}\Big)=0\, .
\ln P(z,e^{2\pi i\frac{h}{k}-w}) = \sum_{l=1}^\infty \sum_{m=1}^\infty \frac{z^l}{l}a_m e^{2\pi i \frac{hlm}{k}}\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty}\Gamma(s) ( l m w)^{-s}ds. &=\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty}\Gamma(s) \sum_{l=1}^\infty \sum_{m=1}^\infty \frac{z^l a_m e^{2\pi i \frac{hlm}{k}}}{l^{s+1}m^{s}}w^{-s}ds.
\mathrm{d}Y^{(N)}_i(t)=\frac{\beta}{2}\,\sum_{\substack{1\le j\le N j\neq i}}\frac{1}{Y^{(N)}_i(t)-Y^{(N)}_j(t)}\,\mathrm{d}t-\frac{Y^{(N)}_i(t)}{2}\,\mathrm{d}t+\mathrm{d}B^{(N)}_i(t),
-\frac{\partial L_{d}D_{\alpha\beta}(p\|q)}{\partial q_{j}}=\underbrace{\frac{Z_{A}}{\alpha}p^{\alpha}_{j}q^{\beta-2}_{j}}_{U_{j}}-\underbrace{\frac{Z_{B}}{\alpha}q^{\alpha+\beta-2}_{j}}_{V_{j}}
v_{s}(t,r) = \frac{\lambda''(s)}{r}\int_{0}^{s-t}\frac{\rho d\rho}{\sqrt{(s-t)^{2}-\rho^{2}}}\left(1+\frac{r^{2}-1-\rho^{2}}{\sqrt{(1+r^{2}+\rho^{2})^{2}-4 r^{2}\rho^{2}}}\right)
|\frac{1}{r}\int_{t}^{\infty}ds \int_{0}^{s-t}\frac{\rho d\rho \lambda''(s)}{(s-t)\sqrt{(s-t)^{2}-\rho^{2}}}\left(\frac{-1-\rho^{2}+r^{2}}{\sqrt{(1+\rho^{2}-r^{2})^{2}+4r^{2}}}+ 1 + F_{3}(r,\rho,\lambda(s))-1 \right)| &\leq C \frac{r}{t^{2}\log^{b+1}(t)}+ C \frac{r}{t^{2}\log^{1+b\alpha}(t)}
\lim_{m^{2}\to 0}m^{2}\int_{0}^{\infty}d t \, e^{- t m^{2}}\int d^{2}r \int_{0}^{M}d k \, k \, e^{- k^{2}t}\int_{0}^{2 \pi}d \Theta \, ( | \psi^{( + )}({\bf k , r}) |^{2}- | \psi_{0}({\bf k , r}) |^{2}) = 0 \, .
g^{2}\Phi_{i}^{2}={\frac{m ( z = h_{i}; \mu )}{a^{\prime}( z = h_{i})}}={\frac{\displaystyle \prod_{j = 1}^{N - 1}( h_{i}- \mu_{j})}{\displaystyle \prod_{j = 1 , j \neq i}^{N}( h_{i}- h_{j})}}\cdotp
\frac{d^2}{dt^2}\lambda(ug+h,dK) \leq \int_M \langle \frac{1}{2}\triangle h,h \rangle- \langle\mathring{R}h,h \rangle-\mu |h|^2+\frac{3}{2}|\Lambda|^2 dV_g &=\int_M \langle \frac{1}{2}\triangle h,h \rangle+3 \langle\mathring{R}h,h \rangle+\mu |h|^2 dV_g=\int_M \langle \triangle_G h,h \rangle dV_g\leq 0
|F_{4}(t,r)| \leq C \frac{\left(1-\chi_{\geq 1}(\frac{2r}{\log^{N}(t)})\right)}{(r^{2}+\lambda(t)^{2})^{2}}\left(\frac{r}{t^{2}\log^{3b+1-2b\alpha}(t)}\right) &+C\frac{\left(1-\chi_{\geq 1}(\frac{4r}{t})\right)}{(r^{2}+\lambda(t)^{2})^{2}}\frac{r}{t^{2}\log^{5b+2N-1}(t)}
\frac{\delta{\cal W}}{\delta \psi^{\alpha}}= \frac{\delta \Gamma}{\delta \psi^{\alpha}}\, , \frac{\delta{\cal W}}{\delta \phi_{A a}^{*}}= \frac{\delta \Gamma}{\delta \phi_{A a}^{*}}\, , \frac{\delta{\cal W}}{\delta \bar{\phi}_{A}}= \frac{\delta \Gamma}{\delta \bar{\phi}_{A}}\, ,
\int_0^T|w_{n+j}|^2\,dt \le D(T) \Big(\sum_{i=1}^n e_{1+i-n}(W_i)(0) +\sum_{k=1}^j e_{1}(W_{n+k})(0)\Big) &\le D(T) \Big(\sum_{i=1}^n e_{i-n}(U_i)(0) +\sum_{k=1}^q e_0(U_{n+k})(0)\Big) \le D(T) \Big(\sum_{i=1}^n e_{1+i-n}(U_i)(0) +\sum_{k=1}^q e_{1}(U_{n+k})(0)\Big) \,,\forall \j \in \{1, \ldots, q\}\,.
( 6 6 ) H ( z , \bar{z}) = \frac{\mu}{4}l ( l + 2 ) + l ( 1 + z \bar{z})^{- 1}[ \xi^{1}( z + \bar{z}) - i \xi^{2}( z - \bar{z}) + \xi^{3}( 1 - z \bar{z}) ] .
Y_\nu(x)\ = \ \sqrt{\frac{2}{\pi x}}\Big(\sin(x-\nu\pi/2-\pi/4)+O(\frac{1+\nu^2}{x})\Big), K_\nu(x)\ = \ \sqrt{\frac{2}{\pi x}}e^{-x}\Big(1+O(\frac{1+\nu^2}{x})\Big).
C_1(\lambda) := \frac{\vert \lambda \vert^2}{{\rm dist}\big( \lambda^2,\cup_{q = 0}^\infty \lbrace \Lambda_q + 1 \rbrace \big)}{\rm and}C_2(\lambda) := \frac{\vert \lambda \vert^2}{{\rm dist}\big( \lambda^2,\cup_{q = 0}^\infty \lbrace \Lambda_q + 2b + 1 \rbrace \big)},
\phi\big(|G_X(it)|^\frac{2}{s}\big) = \ \phi\Big(\big(X_2^{\frac{rs(1+it)}{2}}C_2^{-\frac{rs(1+it)}{2}}M^*MC_2^{-\frac{rs(1-it)}{2}}X_2^{\frac{rs(1-it)}{2}}\big)^\frac{1}{s}\Big) = \ \phi\Big(\big(MC_2^{-\frac{rs(1-it)}{2}}X_2^{rs}C_2^{-\frac{rs(1+it)}{2}}M^*\big)^\frac{1}{s}\Big).
g(\tau + 1) = \begin{pmatrix}\zeta_{60}^{-1}0 0 \zeta_{60}^{11}\end{pmatrix}g(\tau), g(-\frac{1}{\tau}) = \frac{2}{\sqrt{5}}\begin{pmatrix}\sin \frac{2 \pi}{5}\sin \frac{\pi}{5}\sin \frac{\pi}{5}-\sin \frac{2 \pi}{5}\end{pmatrix}g(z).
T\sum_{n=-\infty}^\infty \frac{1}{(((2n+1)\pi T)^{2}+m^{2})^{p}}\sim \frac{m^{1-2p}}{2\sqrt{\pi}}\frac{\Gamma (p-\frac{1}{2})}{\Gamma(p)}-\frac{(2mT)^{1-p}}{m\Gamma (p)}e^{-\frac{m}{T}}+ \cdots
q = x_1 r =\frac{\sqrt{x_{n+1}-\bigl(x_1x_2 +x_1\sum\limits_{j=3}^n x_j^2\bigr)}}{\sqrt{x_1}}t = \frac{x_1x_2}{x_{n+1}-\bigl(x_1x_2 +x_1\sum\limits_{j=3}^n x_j^2\bigr)}s_j = \frac{x_j\sqrt{x_1}}{\sqrt{x_{n+1}-\bigl(x_1x_2 +x_1\sum\limits_{j=3}^n x_j^2\bigr)}}, \, j=3,\ldots, n,
w\left(\frac{1}{X}\right) + \sum_{\alpha}''' w\left(\frac{N(\alpha)}{X}\right) \psi_n(\alpha) = \frac{1}{2\pi i}\int_{(2)}\sum_{\alpha}\frac{\nu(\alpha)\psi_n(\alpha)}{N(\alpha)^s}X^s \mathfrak{w}(s) ds.
\mathcal{D}\left( A\right) = V \times U,A\left(\begin{array}[c]{c}y z\end{array}\right) =\left(\begin{array}[c]{cc}0 I -\Lambda 0\end{array}\right) \left(\begin{array}[c]{c}y z\end{array}\right) ,\left(\begin{array}[c]{c}y z\end{array}\right) \in\mathcal{D}\left( A\right),
\ln(U_{np}^{-1}) =\ln \bigg[(\sum_{i}b_{ni}^{2})^{p/2}/\sum_{i}b_{ni}^{p}\bigg] &\gg\ln[n^{p(3-2\alpha)/2}L^{p}(n)/ (n^{1+p(1-\alpha)}L^p(n))] &\sim \frac{1}{2}(p-2)\ln n.
D_- = r\left(\frac{X_{Z_1}-1}{n}\right) - r\left(\frac{X_{Z_1}}{n}\right) D_+ = \begin{cases}r\left(\frac{X_{Z_1'}+1}{n}\right) - r\left(\frac{X_{Z_1'}}{n}\right) Z_1\neq Z_1' r\left(\frac{X_{Z_1'}}{n}\right) - r\left(\frac{X_{Z_1'}-1}{n}\right) Z_1= Z_1' \end{cases}.
{\rm sign}\left\langle Y_{2},i_{1},i_{1}+1,\dots,i_{r-1}+1,\widehat{n},n+1\right\rangle =-{\rm sign}\left\langle Y_{2},i_{1},i_{1}+1,\dots,i_{r-1}+1,n,\widehat{n+1}\right\rangle
f=\sum_{|J|=q,k,\lambda}f_{Ik\lambda}dz_N\wedge d\overline{z}_J\otimes e_k\otimes b_\lambda=\sum_k f_k \otimes e_k, \,\,f_k=\sum_{|J|=q,\lambda}f_{Ik\lambda}dz_N\wedge d\overline{z}_J\otimes b_\lambda.
\alpha_{n}=\sqrt{2}\frac{\pi}{\varpi}\frac{(-1)^{(n-1)/2}}{n!}\sum_{k=1}^{\infty}\frac{(2k\pi/\varpi)^{n+1}}{coshk\pi},|\alpha_{n}|\sim2^{n+5/2}\frac{n+1}{\varpi^{n+2}}
i^{3,3}(H^3(Y;\mathbb{Q})) = 1, i^{2,2}(H^3(Y;\mathbb{Q})) = ph(Y,w) - 3, i^{2,1}(H^3(Y;\mathbb{Q})) = i^{1,2}(H^3(Y;\mathbb{Q})) = h^{1,2}(Z).
M^{d, n}= (M^{d, n}_s)_{s\geq 0}=(\mathbb E(\xi_n^d|\mathcal F_{s}))_{s\geq 0}, M^{c,n}= (M^{c, n}_s)_{s\geq 0}=(\mathbb E(\xi_n^c|\mathcal F_{s}))_{s\geq 0}
V_{C}\left( x_{1}, \ldots , x_{N}; y_{1}, \ldots , y_{K}\right) = \sum_{1 \leq i < j \leq N}v \left( x_{i}, y_{j}\right) - \sum_{i = 1 , \ldots , N \atop l = 1 , \ldots , K}Z_{l}v \left( x_{i}, v_{l}\right) + \sum_{1 \leq l < k \leq K}Z_{l}Z_{k}v \left( y_{l}, y_{k}\right) \ .
D^0_{\mu\nu}(k)_=-{1\over{k^2+i\epsilon}}\bigl(g_{\mu\nu}-{k_\mu\eta_\nu+k_\nu\eta_\mu\over\eta\cdot k}+k_\mu k_\nu{\lambda k^2+\eta^2\over{(\eta\cdot k)^2}}\biggr),
\xi_j(\mu) = \frac{1}{f_j}\left( \frac{g_j}{f_j}\frac{\partial f_j}{\partial y}- \left( \frac{\partial f_j}{\partial x}+ \frac{\partial g_j}{\partial y}\right)+ \frac{f_j}{g_j}\frac{\partial g_j}{\partial x}\middle) \right|_{x = y = 0},
\frac{Z(h)}{{\sigma_n}}=\frac{\frac{1}{2}\mathbf{h}_\ast^\top\boldsymbol{\Sigma}_{1\ast}^{-1}\mathbf{h}_\ast h}{\sqrt{\omega_1 \mathbf{h}_\ast^\top\boldsymbol{\Sigma}^{-1}_{1\ast}\mathbf{h}_\ast h^2+\omega_2 \mathbf{h}_\ast^\top\boldsymbol{\Sigma}^{-1}_{1\ast}\boldsymbol{\Sigma}_{2\ast}\boldsymbol{\Sigma}^{-1}_{1\ast}\mathbf{h}_\ast}}.
S [ A , \phi ] = \sum_{j = 1}^{2}\int_{S^{1}\times U_{j}}{\cal L}[ A^{( j )}, \phi^{( j )}] d^{3}x + \mu i \oint_{S^{1}\times \partial U_{1}}f_{1 2}^{- 1}d f_{1 2}\wedge A^{( 1 )}
I\left(\{X_i\}_{i=0}^q\right) \longrightarrow I(\{X_i\}_{i=0}^q)+\varepsilon\mathrm{tr}\frac{1}{z-X_0}\frac{1}{z'-M}(M-X_0)+\mathcal{O}(\varepsilon^2)\ , \mathrm{d}X_0 \longrightarrow \mathrm{d}X_0\left(1+\varepsilon\mathrm{tr}\frac{1}{z-X_0}\mathrm{tr}\frac{1}{z-X_0}\frac{1}{z'-M}+\mathcal{O}(\varepsilon^2)\right)\ ,
\left|\frac{x'(r)}{x(r)}\right| =\left|\frac{(d/dr)\Psi(re^{if(r)})}{\Psi(re^{if(r)})}\right| \ge\frac{f(r)^{2}}{2\beta}\frac{\sqrt{1+r^{2}f'(r)^{2}}}{r},
\begin{aligned}\Delta \tilde H^\pm =0, \mbox{in}\ \Omega^\pm, \tilde H^\pm =0, \mbox{on}\ \partial\Omega^\pm\cap\partial\Omega, \tilde H^\pm =2\log |\partial_nH_\gamma^\pm|-\log h_\gamma, \mbox{on}\ \gamma.\end{aligned}
\hat{H}' = \frac{1}{4 \Omega_G^2}\Bigl(\frac{1}{4}\hat{a}^{\dagger 4}_0+ \hat{a}^{\dagger 3}_0 \hat{a}_0 + \frac{3}{2}\hat{a}^{\dagger2}_0 \hat{a}^2_0 + \hat{a}^{\dagger}_0 \hat{a}^3_0+ \frac{1}{4}\hat{a}^4_0 \Bigr).
{{\mathrm{\boldmath ~ \sigma ~}}\cdot{\bf p}}{{\mathrm{\boldmath ~ \sigma ~}}\cdot{\bf n}}{\frac{f ( r )}{r}}{\phi^{l}}_{j m}= -{\frac{i}{r}}({\frac{d f}{d r}}-{\frac{\kappa f}{r}}){\phi^{l}}_{j m},
\left(\int_{N^{-((d-1)+\alpha)}\leq \lvert t \rvert \leq N^{\beta}}\lvert \hat{f}\rvert^{2q'}\lvert t \rvert^{q'(1-2s)}dt\right)^{1/q'}\ll N^{\left((d-1)+\alpha\right)(q'-1)/q'}&=N^{\left((d-1)+\alpha\right)/q}.
D\Phi((\overline{x},\overline{\lambda},\overline{v})|(0,0))(\Delta x,\Delta\lambda,\Delta v) =\Big\{(\Delta \xi,\Delta\eta)\in\mathbb{X}\times\mathbb{Y}\ |\ \Phi'((\overline{x},\overline{\lambda},\overline{v});(\Delta x,\Delta\lambda,\Delta v)) =(\Delta \xi,\Delta\eta)\Big\}.
\widetilde{h}_\epsilon(y) = f_{\tiny \mbox{BEP}}^{\delta=0}(y) (1+ \gamma_1 - \gamma_2)+ f_{\tiny \mbox{BEP}}^{\delta=2}(y)(\gamma_2- 2 \gamma_1)+ f_{\tiny \mbox{BEP}}^{\delta=4}(y)\gamma_1,
\begin{array}{l}{{{\cal D}_{- k}{\cal X}^{- k}= 1 + q^{- 2}{\cal X}^{- k}{\cal D}_{- k}~ , ~ ~ \mathrm{f o r}~ k > 0 ~ ,}}\\{{{\cal D}_{+ k}{\cal X}^{+ k}= 1 + q^{2}{\cal X}^{+ k}{\cal D}_{+ k}~ ,}}\\{{[{\cal D}_{i},{\cal D}_{j}] = 0 ~ , ~ ~ [{\cal X}^{i},{\cal X}^{j}] = 0 ~ ,}}\\{{{\cal D}_{i}{\cal X}^{j}={\cal X}^{j}{\cal D}_{i}~ , ~ ~ \mathrm{f o r}~ i \not = j ~ .}}\\ \end{array}
q(r,V)\equiv\frac{V_{r}^{2}}{V_{0}^{2}(1-r^{2}/r_{0}^{2})}+(\frac{V_{\theta}^{2}}{V_{0}^{2}}+\frac{V_{\varphi}^{2}}{V_{0}^{2}})\equiv u_{r}^{2}+u_{\theta}^{2}+u_{\varphi}^{2}\le1,
|\alpha k^3 - k_1^3 -(n-k_1)^3 - \alpha(k-n)^3 |= |\alpha k^3 - (d_1k + \delta)^3 -[(1 - d_1 + \mu)k - \delta]^3 + \alpha \mu^3 k^3 | = \left|[ \alpha - 1 + 3d_1 - 3d_1^2 + \mathcal{O}(\epsilon)]k^3 + \mathcal{O}(\epsilon)k^2 + \mathcal{O}(\epsilon^2)k + \mathcal{O}(\epsilon^3) \right| \gtrsim |k^3|.
\theta_i^2\frac{Z_i^*(u)}{e_i^*(u)}+\frac{T_i^{\infty}}{e_i^*(u)^2}=\theta_i^2\frac{Z_i(u)}{\theta_i}\left( \frac{\theta_i}{e^{\rho_i(u)}}-\frac{\theta_iZ_i(u)}{T_i^{\infty}}\right)+T_i^{\infty}\left(\frac{\theta_i}{e^{\rho_i(u)}}-\frac{\theta_iZ_i(u)}{T_i^{\infty}}\right)^2 &=\theta_i^2\left(\frac{T_i^{\infty}}{e^{2\rho_i(u)}}-\frac{Z_i(u)}{e^{\rho_i(u)}}\right).
\begin{aligned}\mathbf{y}_{\mathcal{Q}_{k_{i}}}=\mathbf{y}_{\mathcal{Q}}-\mathbf{A}\left(\sum_{j=1}^{i-1}(\mathbf{H})_{k_{j}}\tilde{x}_{k_{j}}\right) &=\mathbf{A}\left(\mathbf{Hx}-\sum_{j=1}^{i-1}(\mathbf{H})_{k_{j}}\tilde{x}_{k_{j}}\right)+\mathbf{An}+\mathbf{n}_q\end{aligned}
D^{prop}_{\mu \nu}= \left[ -g^{\perp}_{\mu\nu}+\frac{n_{\mu}\partial^{\perp}_{\nu}+n_{\nu}\partial^{\perp}_{\mu}}{(n\partial)}-\frac{n_{\mu}n_{\nu}\partial^2_{\perp}}{(n\partial)^2}\right]\frac{1}{\partial^2_{\mu}+ i \varepsilon}.
nR-n\epsilon_n \leq I(W;Y^n|E^n) &= \sum_{t=1}^{n}I(W;Y_t|Y^{t-1},E^n) &= \sum_{t=1}^{n}I(X_t;Y_t|Y^{t-1},E^n) &= \sum_{t=1}^{n}[H(Y_t|Y^{t-1},E^n)-h_2(\alpha)] &= H(Y^n|E^n)-nh_2(\alpha) &=\sum_{e^n}p(e^n)H(Y^n|E^n=e^n)-nh_2(\alpha).
\begin{aligned}\partial_t v_i - d_i\Delta v_i = L_i v + f_i(v+u_{\infty}) - \nabla f_i(u_{\infty})\cdot v =: L_i v + g_i(v), x\in\Omega, \nabla v_i \cdot \nu = 0, x\in\partial\Omega, v_i(x,0) = u_{i,0}(x) - u_{i,\infty}, x\in\Omega\end{aligned}
\tilde{\omega}^a{}_{\mu b}=(K^{-1})^a{}_c\omega^c{}_{\mu d}K^d{}_{b}-K^c{}_{b}\partial_\mu (K^{-1})^a{}_{c},\tilde{\phi}^a{}_{\mu b}= (K^{-1})^a{}_{c}\phi^c{}_{\mu d}K^d{}_b.
g_{2j-1}(x) = v_j^\prime(L_j - x) + u_j^{\prime\prime}(L_j - x) = - \frac{d}{dx}\left(v_j(L_j - x) + u_j^{\prime}(L_j - x)\right) = - f_{2j-1}^\prime(x), g_{2j}(x) = v_j^\prime(x) - u_j^{\prime\prime}(x) = \frac{d}{dx}\left(v_j(x) - u_j^\prime(x)\right) = -f_{2j}^\prime(x),
\langle ( \varphi^{\dagger}\varphi )_{b J}^{a I}\rangle_{\mathrm{F T}}^{\mathrm{1 - l o o p , W Z}}\sim g^{2}\delta_{J}^{I}\delta_{b}^{a}p^{2}\int \frac{d^{4}k}{( 2 \pi )^{4}}\frac{1}{\left( k + \frac{p}{2}\right)^{2}\left( k - \frac{p}{2}\right)^{2}}.
c_{i}= \frac{\beta_{i}\sqrt{g_{v}}}{\mid J ( \frac \phi v )_{p_{i}}\mid}= \frac{\beta_{i}\eta_{i}\sqrt{g_{v}}}{J ( \frac \phi v )_{p_{i}}}, \, \, \, \, g_{v}= \operatorname*{d e t}( g_{A B}) ,
q^{n^{2}/2}S_{n}\left(xq^{-n-1/2};q\right)=\frac{1}{\sqrt{\pi\log q^{-2}}}\int_{-\infty}^{\infty}\frac{\left(xe^{iy};q\right)_{n}}{\left(q;q\right)_{n}}\exp\left(\frac{y^{2}}{\log q^{2}}-iny\right)dy.
\phi_1 = \left( \begin{array}{cc}s - ib_{12}b_{13}-ib_{23}-b_{13}-ib_{23}s+ib_{12}\end{array}\right) , \quad\phi_2 = \left( \begin{array}{cc}-b_{03}+i p -b_{01}+ib_{02}-b_{01}- ib_{02}b_{03}+ i p\end{array}\right) .
A_1=PU(\tau)^{-1}PR(\zeta, \tau)PU(\tau)P, &A_2=PU(\tau)^{-1}(I-P)R(\zeta, \tau)PU(\tau)P, &A_3=PU(\tau)^{-1}PR(\zeta, \tau)(I-P)U(\tau)P, &\quad A_4=PU(\tau)^{-1}(I-P)R(\zeta, \tau)(I-P)U(\tau)P.
\mu_\ell^{(\infty)}\bigl( Q_m^{-1}(B) \setminus A_k^{(\infty)}\bigr) = \mu_\ell^{(\infty)}\bigl( Q_m^{-1}(B \setminus A_k^{(m)}) \bigr) = \mu_\ell^{(m)}(B \setminus A_k^{(m)}) = \mu_k^{(m)}(B) = \mu_k^{(\infty)}(Q_m^{-1}(B)),
\frac{5}{6}= \frac{5 \times 3}{6 \times 3}= \frac{1 5}{1 8}= \frac{1 5 \times 2}{1 8 \times 2}= \frac{3 0}{3 6}= \frac{3 0 \times 2}{3 6 \times 2}= \frac{6 0}{7 2}
\begin{aligned}\sum_{{\substack{i_1,i_2,i_3\geq0 i_1+i_2+i_3=n}}}\frac{n!}{i_{1}!i_{2}!i_{3}!}B_{N,i_1}(x_1)B_{N,i_2}(x_2)B_{N,i_3}(x_3) &=\frac1{2N^2}\biggl[(N-n)(2N-n)B_{N,n}(x) &\quad+n((2N-n)(x-1)+(x-2)(N-n+1))B_{N,n-1}(x) &\quad+n(n-1)(x-1)(x-2)B_{N,n-2}(x)\biggl] &\quad(x=x_1+x_2+x_3 n\geq2).\end{aligned}
H_{\lambda,K,x,r}^+: = \left\{y \in \mathbb R^n \colon ||y||_K > r,\,\frac{|f(x)-f(y)|}{||x-y||_K^{\frac{n}{p}}}\geq\lambda\right\}&=\left\{y\in \mathbb R^n \colon ||y||_K > r, \,\frac{|f(x)|}{||x-y||_K^{\frac{n}{p}}}\geq\lambda\right\}.
\bigl(S_{\gamma\beta}(\xi)\bigr)_{1\leq\gamma,\beta\leq M}:=\Bigl[{\rm Symb}_L(\xi)\Bigr]^{-1}\in{\mathbb{C}}^{\,M\times M},\qquad\forall\,\xi\in{\mathbb{R}}^n\setminus\{0\}.
\Psi(2i)=\dfrac{\pi^{1/4}(2-2^{1/2})^{1/2}\cdot e^{\pi/2}}{\Gamma(3/4)\cdot 2^2},\Psi(4i)=\dfrac{\pi^{1/4}(1-2^{-1/4})\cdot e^{\pi}}{\Gamma(3/4)\cdot 2^2}.
\int_0^\infty \bigl([u_0']^2 + [v_0']^2\bigr)\,dt = \liminf_{k \to \infty}\int_0^\infty (e^{-\gamma_k t}u_k)'^2 + (e^{-\gamma_k t}v_k)'^2 \,dt &\le \frac{2}{N}\lim_{k \to \infty}([u_k'(0)]^2 + [v_k'(0)]^2) = \frac{2}{N}\bigl([u_0'(0)]^2 + [v_0'(0)]^2\bigr)< \infty.
S_g(f) =S(g)^{-1}S(f+g'+(g-g'))= &=S(g)^{-1}S(f+g')S(g')^{-1}S(g'+(g-g'))= &=S(g)^{-1}S(g')S(g')^{-1}S(f+g')S_{g'}(g-g')= &=\mathrm{Ad}S_{g'}(g-g')^{-1}(S_{g'}(f))\ .
S_{0}( \phi ) = \frac{1}{2}\int \frac{d^{d}k}{( 2 \pi )^{d}}\left[ \sum_{\mu}\frac{4}{a^{d - 2}}\sin^{2}\left( \frac{a k_{\mu}}{2}\right) + m^{2}\right] \phi ( - k ) \phi ( k ) .
J_K^{2,-}u(x) := \{(p,X)\in \mathbb{R}^n \times \mathbb{S}^n| \u(y)\ge u(x) + p\cdot (y-x) + \frac{1}{2}X(y-x) \cdot (y-x) + o(|y-x|^2) \{\rm as}\y\in K\rightarrow x\},
I_j =\{k\in [1, \mu_j]: \eta_{jk}\in \Gamma_G\cdot \chi_{jl}\vartheta_j \textrm{for some}l \in [1, \lambda_j]\}, & I_j' =\{k\in [1, \mu'_j]: \eta'_{jk}\in \Gamma_G\cdot \chi'_{jl}\vartheta_j' \textrm{for some}l \in [1, \lambda'_j]\}.
{\bf E}[\beta_{1,0}(a, b, \bar{b})^q] = \frac{\Gamma(\frac{q+b_0}{a})}{\Gamma(\frac{b_0}{a})}\frac{\Gamma(1-\frac{q+b_0}{a})}{\Gamma(1-\frac{b_0}{a})}, = \frac{\sin(\frac{\pi b_0}{a})}{\sin(\frac{\pi (q+b_0)}{a})}.
\frac{I_{\alpha}}{(\alpha+1)!}:=-\frac{uA_{\alpha}^2}{(K+u+L)^{\alpha+1}}-\frac{A_{\alpha}^2}{K^{\alpha+1}}+\frac{(1+u)A_{\alpha}^2}{(K+u)^{\alpha+1}}&\quad-\frac{uB_{\alpha}^2}{(K+u+L)^{\alpha+1}}-\frac{2uA_{\alpha}B_{\alpha}}{(K+u+L)^{\alpha+1}}.
\bar{Y}_s^{1n}=g^1(X_T^{t,x})+\int_t^T C_h(1+|\varphi_n(X_r^{t,x})|^\gamma + |\bar Y_r^{1n}|)dr -\int_s^T \bar Z_r^{1n}dB_r^n, s\in [t,T].
\left( \begin{matrix}{X_{1}}\\{X_{2}}\\{X_{3}}\\{X_{4}}\\ \end{matrix}\right) = \frac{1}{2 r}\left( \begin{matrix}{3}&{1}&{1}\\{1}&{- 1}&{- 3}\\{- 1}&{- 1}&{3}\\{- 3}&{1}&{- 1}\\ \end{matrix}\right) \left( \begin{matrix}{\varphi^{( 1 )}}\\{\varphi^{( 2 )}}\\{\varphi^{( 3 )}}\\ \end{matrix}\right) ,
\widehat{|\cdot|\eta}(\xi) = \int_{-1}^1|x|e^{2\pi ix\xi}dx=\int_{-1}^{0}(-x)e^{2\pi i x\xi}dx+\int_{0}^{1}xe^{2\pi i x\xi}dx= 2\frac{\sin(2\pi\xi)}{2\pi\xi}-\frac{\sin^2(\pi\xi)}{(\pi\xi)^2}.
(\ast) = w_jC^\varphi_{kji}R^\varphi_{ik}+ \frac{1}{2}\langle \nabla w,\nabla |T^\varphi|^2 \rangle + \frac{1}{2}(1+w)|C^\varphi|^2 w_{ti}R^\varphi_{ik,tk}- \alpha w_{ti}R^\varphi_{ij}\varphi^a_j \varphi^a_t - \alpha w_t R^\varphi_{jk}\varphi^a_{jt}\varphi^a_k \, .
x_i[y_j - (\alpha_1 g_1 + \cdots + \alpha_r g_r)] = (\beta_1 g_1 + \cdots + \beta_r g_r) y_j y_j[x_i - (\beta_1 g_1 + \cdots + \beta_r g_r)] = (\alpha_1 g_1 + \cdots + \alpha_r g_r) x_i
\eta_{\alpha\alpha}\eta_T^2 - 2 \eta_{\alpha T}\eta_{\alpha}\eta_T + \eta_{TT}\eta_{\alpha}^2 = \frac{2(1+\alpha)}{\alpha^2(1-\alpha)^2}\frac{T_{\rm i}}{T}\left\{-\left[1 - 3\alpha - \frac{5}{2}\alpha^2(1 - \alpha)^2\right]\frac{T_{\rm i}}{T}+ 4\left[1 + \frac{5}{4}\alpha(1 - \alpha)\right]^{\! 2}\right\},
D^o=\{(d_1,d_2)\in R_{\geq 0}^2|\d_i\leq \min(M_i,N_i)\i=1,2\mbox{and}\frac{d_1}{r_1}+\frac{d_2}{r_2}\leq\frac{\min(N_1,M_1+M_2)}{r_1}\}.
\begin{aligned}\zeta\bigl(f\bigr)(s) = \operatorname{Ad}(u(s)) \Bigl( \operatorname{diag}\bigl[c(f(0)), \dots, c(f(0)), f(t_1(s)), \dots, f(t_k(s)), c(f(1)), \dots, c(f(1)) \bigr] \Bigr), \end{aligned}
\langle \theta_{2}, \theta_{1}| T_{\mu}^{\mu}| \theta_{1}, \theta_{2}\rangle_{\mathrm{s y m}}= 8 \pi m^{2}\left( \frac{2 \sin \pi \alpha \cosh \theta_{1 2}}{\sinh^{2}\theta_{1 2}+ \sin^{2}\pi \alpha}\right) \cosh^{2}\frac{\theta_{1 2}}{2}\, \, \, .
\frac{d\mu}{dx}(x) =\frac{1}{2\pi i x}\left(w_{-}(x)-w_{+}(x)\right) = \frac{1}{\pi x}\Im (w_{-}(x)) &=\frac{1}{2\pi^2 x}\Re \int\limits_{\gamma_{\alpha}}\log\left(1-x\frac{Q(t)}{P(t)}\right) dt.
\dot{W}_{0,}^{1,N}\hookrightarrow L^{\infty, N}(\log L)^{-1}\hookrightarrow L^{\infty, q}(\log L)^{-1+ \frac{1}{N}- \frac{1}{q}}\hookrightarrow L^{\infty, \infty}(\log L)^{-1+\frac{1}{N}}={\rm Exp L}^{\frac{N}{N-1}}.
\sum_{l=1}^{p-1}\binom{p-1}{l}\sum_{r=1}^{l}\binom{p}{r}H_{r-1}\equiv_{p^2}-p\sum_{r=1}^{p-1}\frac{(-1)^rH_{r-1}}{r}\sum_{r=l}^{p-1}(-1)^l &=-\frac{p}{2}\left(H_{p-1}(1,-1;-1)+H_{p-1}(1,1)\right)\equiv_{p^2}-\frac{pq_p^2(2)}{2}.
X_{kk}= \sum_{i=1}^{\epsilon_k+1}\left[\begin{array}{c}1 \lambda_{ki}\lambda_{ki}^2 \vdots \lambda_{ki}^{\epsilon_k}\end{array}\right] \left[\begin{array}{c}1 \lambda_{ki}\lambda_{ki}^2 \vdots \lambda_{ki}^{\epsilon_k}\end{array}\right]^H
\lim_{t\to \infty}\big(\Vert \rho(t)-1\Vert_{\widetilde{B}^{\frac{N}{2}+1-\alpha,\frac{N}{2}}}+\Vert u(t)\Vert_{\dot{B}^{\frac{N}{2}+1-\alpha}_{2,1}}\big) = 0.
\lim_{n\rightarrow\infty}H(X^n|Y^n, X^n\in\mathcal{A}, Y^n\in \mathcal{B})^{\frac{1}{n}}=\lim_{n\rightarrow\infty}H(Y^n|X^n, X^n\in\mathcal{A}, Y^n\in \mathcal{B})^{\frac{1}{n}}\leq \left(\frac{p_{12}p_{21}}{p_{11}p_{22}}\right)^{\frac 12}.
\sum_P (-1)^P{\cal J}^{-2}(\theta){\exp\{N \beta \sum_k \cos(\theta_k - \theta_{P(k)}+{2 \pi \over L})\}\over \prod_{i,j}\cos \bigl({\theta_i - \theta_{P(i)}+ \theta_j - \theta_{P(j)}\over 2}\bigr)}~~,
\sqrt{\eta}K(z,w)(\eta) =\int_{|\mu|\le |\eta|^{3/2}/2}e^{3i\mu^2/4}z\left(\eta+\frac{\mu}{\sqrt{\eta}}\right)w\left(\eta-\frac{\mu}{\sqrt{\eta}}\right)d\mu &+ \int_{|\mu|\le |\eta|^{3/2}/2}e^{3i\mu^2/4}z\left(\eta+\frac{\mu}{\sqrt{\eta}}\right)w\left(\eta-\frac{\mu}{\sqrt{\eta}}\right)d\mu.
T_{s}\cdot (v_w, v_{sw}) = \begingroup\renewcommand*{\arraystretch}{1.5}\begin{pmatrix}D^{(n)}_{\alpha}(w \mathbf{z}) \cdot I_{k}(b_{i,j}^s(sw\mathbf{z})) (b_{i,j}^s(w\mathbf{z})) D^{(n)}_{\alpha}(s w \mathbf{z}) \cdot I_{k}\end{pmatrix}\endgroup \cdot \begin{pmatrix}v_w v_{sw}\end{pmatrix},
z^{k+1}= (1-\alpha)z^k+\alpha R_{\gamma A}R_{\gamma B}z^k &=(1-\alpha)z^k+\alpha \delta \begin{bmatrix}\cos{\xi}\sin{\xi}-\sin{\xi}\cos{\xi}\end{bmatrix}\begin{bmatrix}\cos{(\pi-\xi)}\sin{(\pi-\xi)}-\sin{(\pi-\xi)}\cos{(\pi-\xi)}\end{bmatrix}z^k &=(1-\alpha)z^k-\alpha \delta z^k &=-(|1-\alpha|z^k+\alpha \delta )z^k.
\aligned (i\partial_t+\Delta_{x})u_j=\sum_{(j_1,j_2,j_3)\in R(j)}u_{j_{1}}\bar{u}_{j_{2}}u_{j_{3}}, &R(j)=\{(j_1,j_2,j_3) \in (\mathbb{Z}^2)^3:j_1-j_2+j_3=j and |j_1|^2-|j_2|^2+|j_3|^2=|j|^2 \}\endaligned
\int_0^u(\cos x-\cos u)^{\nu-1}\cos [(\nu+\beta)]x\,dx=\frac{\sqrt{\pi}\Gamma(\beta+1)\Gamma(\nu)\Gamma(2\nu)(\sin u)^{2\nu-1}}{2^{\nu}\Gamma(\beta+2\nu)\Gamma \big(\nu+\frac12\big)}C_{\beta}^{\nu}(\cos u),
\begin{aligned}\tau_1^{\beta}=(t_2+2t_3)/3, \tau_2^{\beta}=(t_2-t_3)/3, \tau_3^{\beta}=-(2t_2+t_3)/3, \end{aligned}\begin{aligned}\tau_1^{\gamma}=(t_4+2t_5)/3, \tau_2^{\gamma}=(t_4-t_5)/3, \tau_3^{\gamma}=-(2t_4+t_5)/3,\end{aligned}\begin{aligned}\tau_1^{\delta}=t_1/2, \tau_2^{\delta}=-t_1/2. \end{aligned}
\langle c | \mathrm{e}^{- \pi \ell H}| c \rangle_{\mathrm{N S - N S}}\sim \sum_{m , k}\left[ 1 + ( - 1 )^{m}\right] \left[ 1 + ( - 1 )^{k}\right] \mathrm{e}^{-{\frac{\pi \alpha^{\prime}\ell}{2 R^{2}}}( k - \gamma m )^{2}}
\left[ -\omega^2 \gamma^{tt}(r) f(r) -{l(l+n)\over r^2}{f(r)\over h(r)}+{\sqrt{-\gamma^{tt}f(r)}\over U(r)}{d\over dr}{U(r)\over \sqrt{-\gamma^{tt}f(r)}}{d\over dr}\right]\phi^l_\omega(r) =0.
\frac{d^a}{dX^a}\sum_{j=0}^{M-1}T_{\omega_{1,j}}(1,R(X))X^j\Big|_{X=\l}= \frac{d^a}{dX^a}\sum_{j=0}^{M-1}T_{\omega_{2,j}}(1,R(X))X^j\Big|_{X=\l}\frac{d^a}{dX^a}\sum_{j=0}^{M'-1}T_{\omega'_{1,j}}(1,R(X))X^j\Big|_{X=\l}= \frac{d^a}{dX^a}\sum_{j=0}^{M'-1}T_{\omega'_{2,j}}(1,R(X))X^j\Big|_{X=\l}
F'(y)= \frac{\cos(y/2)(-2 \sin(y/2) + 1 -\sqrt{4 \sin^{2}(y/2) - 12 \sin(y/2)+1}}{4 \sin(y/2) \sqrt{4 \sin^{2}(y/2) - 12 \sin(y/2)+1})}\, > 0
A'(e_1)= \begin{bmatrix}1 e_1 \cdot w_2 \cdots e_1 \cdot w_n 0 \vdots \bigg[w_{t_1}\cdot w_{t_2}- (e_1 \cdot w_{t_1})(e_1 \cdot w_{t_2})\bigg]_{( m-1) \times (m-1)}0 \end{bmatrix}.
\left|\begin{array}{cccc}f_1(x_1) f_2(x_1) \cdots f_n(x_1) f_1(x_2) f_2(x_2) \cdots f_n(x_2) \vdots \vdots \ddots \vdots f_1(x_n) f_2(x_n) \cdots f_n(x_n) \end{array}\right|\not = 0.
\gamma_{\psi}(xy)=\gamma_{\psi}(x)\gamma_{\psi}(y)(x,y)_2 ,\gamma_{\psi}(x^2)=1,\gamma_{\psi}^{-1}=\gamma_{\psi^{-1}},\quad\gamma_{\psi,a}(x)=(a,x)_2\gamma_{\psi}(x).