harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
M_t = Y_t -Y_0 -A^{W,W}_t(-\nabla u^*\, b - \tilde f) = u(t, W_t) - u(0, W_0) - u(t, W_t) + u(0, W_0) + \int_0^t \nabla u^* (r, W_r) \mathrm d W_r
\|u_\varepsilon - \mathfrak{U}_{M}^{(\varepsilon)}\|_{C(\overline{\Omega}_\varepsilon\times [0,T])}:= \max_{\overline{\Omega_\varepsilon}\times [0, T]}|u_\varepsilon - \mathfrak{U}_{M}^{(\varepsilon)}| \le C_M \, \varepsilon^{\gamma (M +\lfloor \alpha\rfloor -1)}.
I_{1}(h) \leq \sup_{t\in \lbrack t_{0},t_{0}+h]}\Vert f(t,\xi (t))\Vert\int_{t_{0}}^{t_{0}+h}(t_{0}+h-\tau )^{\alpha -1}\;d\tau &= \sup_{t\in \lbrack t_{0},t_{0}+h]}\Vert f(t,\xi (t))\Vert \times \frac{h^{\alpha}}{\alpha}.
F^{\mu\lambda_2\dots \lambda_{p+2}}F_{\nu\lambda_2\dots\lambda_{p+2}}=\bar{\delta}^\mu_\nu (p+1)!\F_{t y_1 \dots y_p r}F^{t y_1 \dots y_p r}= -\bar{\delta}^\mu_\nu(p+1)!\e^{-2a\phi}{Q^2\over (Gr)^{2(d-1)}},
Z_{\mathrm{e f f}}= \int \mathcal{D}\Phi^{\Delta}\mathcal{D}\Phi_{\Delta}^{* ( a )}\mathcal{D}\bar{\Phi}_{\Delta}^{( a )}\mathcal{D}\tilde{\Phi}_{\Delta}\mathcal{D}\mu_{a}^{\Delta}\mathcal{D}\nu_{a}^{\Delta}\mathcal{D}\omega^{\Delta}\exp \left( i S_{\mathrm{e f f}}\right) .
J(r) = \chi_l(r)^{-1}\sqrt{r}\, J_{l+\frac{1}{2}}(r), H(r) = \chi_l(r) \sqrt{r}\, H_{l+\frac{1}{2}}^{(1)}(r), \chi_l(r)=\left(\frac{r^2}{1+r^2}\right)^{(l+1)/2}.
\frac{\Gamma(n+\alpha+1)\Gamma(m+\alpha+1)}{n!m!\Gamma(\alpha+1)}\frac{(a-1)^{n-m+\alpha+1}}{a^{n+m+2\alpha+2}}{}_{2}F_{1}(n+\alpha+1;\frac{m+\alpha+1}{\alpha+1};\frac{1}{a^{2}})
P_{H(\mathbb{M}^{min}(\Sigma))}(t)=P_{\Sigma\otimes\mathbb{M}_0^{min}(\Sigma)}(t)-P_{\Lambda^3(\Sigma)}(t)-P_{\mathbb{M}_1^{min}(\Sigma)}(t) ={t^3\over 1-t^2}\cdot{t^8\over(1-t^2)(1-t^6)}-{t^{15}\over(1-t^2)(1-t^4)(1-t^6)}-(t^3+t^5)\cdot{t^8\over(1-t^2)(1-t^6)}={t^{17}\over(1-t^2)(1-t^4)(1-t^6)},
\lim_{m \to \infty}\| u_m(x) - u(x) \|_{L^2(P)}= \lim_{m \to \infty}\|\frac{m}{m+1}f_{\frac{1}{r(m)}}(x) - f(x) \|_{L^2(P)}\le \lim_{m \to \infty}\frac{m}{m+1}\| f_{\frac{1}{r(m)}}(x) - f(x) \|_{L^2(P)}+ \lim_{m \to \infty}\|\frac{m}{m+1}f(x) - f(x) \|_{L^2(P)}= 0.
\begin{array}{rcl}{\displaystyle \frac{\partial^{k-1}\lambda_{i}^{t}}{\partial t^{k-1}}}= u_{k-1,0}c^{\frac{k}{2}}\|T\|^{k}+u_{k-1,2}c^{\frac{k-2}{2}}\|T\|^{k-2}(\lambda_{i}^{t})^{2}+u_{k-1,4}c^{\frac{k-4}{2}}\|T\|^{k-4}(\lambda_{i}^{t})^{4}+\ldots+u_{k-1,k}(\lambda_{i}^{t})^{k}.\end{array}
\begin{array}{l}\left \vert u_{00}^{\left( i\right)}\left( x,t\right) \right \vert=l^{\left( i\right)}\left( x,t\right) \leq g\left( x,t\right) =\frac{1}{R_{i}^{2}}o\left( t^{-1}\right) .\end{array}
\nabla_{(X^iG^l_i+y^i\frac{\partial X^l}{\partial x^i})\frac{\partial}{\partial y^l}}\frac{\partial}{\partial x^k}=(X^iG^l_i+y^i\frac{\partial X^l}{\partial x^i})C^m_{~lk}\frac{\partial}{\partial x^m}&=(y^m\frac{\delta X^l}{\delta x^m}+y^mX^rF^l_{rm})C^m_{~lk}\frac{\partial}{\partial x^m}&=y^i\nabla_iX^lC^m_{~lk}\frac{\partial}{\partial x^m}.
\begin{aligned}\hat{y}^n_t=\zeta_t=\mathbb{E}[\zeta|\mathcal{F}_t],\hat{z}^n_t=\hat{\tilde{z}}^n_t=\hat{\gamma}^n_{(t,e)}=g(x,\hat{y}^n,\hat{z}^n,\hat{\tilde{z}}^n,\hat{\gamma}^n)=0.\end{aligned}
\Psi^{(6)}_0(x,t) = \Psi^{(3)}_0(x,t) S^{(3)}_0 S^{(4)}_0= \Psi^{(3)}_0(x,t)\begin{pmatrix}1 0 a 1\end{pmatrix}\begin{pmatrix}1 -a 0 1\end{pmatrix}\hphantom{\Psi^{(6)}_0(x,t)}{}= \Psi^{(3)}_0(x,t)\begin{pmatrix}1 -a a 1-a^2\end{pmatrix}.
\mathcal{K}= \begin{bmatrix}S_M^{1/2}0 0 S_N^{1/2}\end{bmatrix}\begin{bmatrix}I - V^\top V I \end{bmatrix}^{-1}\begin{bmatrix}S_M^{1/2}0 0 S_N^{1/2}\end{bmatrix}.
\sum_{\substack{m,p_1,\dots,p_j,p_1'\dots p_r' p_j<p_{j-1}}}^*a_m=\frac{1}{2\pi i}\int_{-T}^{T}\sum^*_{m,p_1,\dots,p_j,p_1'\dots, p_r'}a_m\Bigl(\frac{p_{j-1}-1/2}{p_j}\Bigr)^{c+it}\frac{dt}{c+it}+O\Bigl(\frac{x^{1+o(1)}}{T}\Bigr),
J_i(\tau_i)=\mathbb{E}\Bigg\{\bar{\theta}_1e^{-\beta\tau_i}\Big(x_i(\tau_i)-\bar{K}_1\Big)\Bigg\}=\bar{\theta}_1\mathbb{E}\Bigg\{e^{-\beta\tau_i}\Big(x_i(\tau_i)-\bar{K}_1\Big)\Bigg\}
\zeta_\alpha(s) = \sum_{n=1}^\infty [d(n)]^\alpha n^{-s}= \prod_{j=1}^\infty \left(1 + \sum_{k=1}^\infty (k+1)^\alpha p_j^{-sk}\right) = \left[\zeta(s)\right]^{2^\alpha}\phi_\alpha(s).
\partial^\mu\partial_\nu\partial_\lambda\sigma^2+\delta^\mu{}_\nu\partial_\lambda[\sigma\partial_\alpha\partial^\alpha\sigma-2(\partial_\alpha\sigma)(\partial^\alpha\sigma)]-[\delta^\mu{}_\nu\partial_\lambda+\delta^\mu{}_\lambda\partial_\nu+\delta_{\lambda\nu}\partial^\mu](\partial_\alpha\sigma)(\partial^\alpha\sigma)=0.
\tfrac{1}{4}\mu(dp)^b|V_i||Y'| = \tfrac{1}{8}\mu(dp)^b|V_i||Y'|+\tfrac{1}{8}\mu(dp)^b|V_i||Y'| > \frac{\mu d^b}{8\rho}p^b|Y'||W| +\frac{\mu d^b}{8 \kappa r_1}p^b n|Y'| > 7p^b|Y'||W|+\rho p^b n|Y'|/r_1\,,
\lambda_{k}= -2a \cos \left(\tfrac{n k \pi}{n+3}\right) - 2b \cos \left[\tfrac{(n + 1) k \pi}{n+3}\right] - 2c \cos \left[\tfrac{(n + 2) k \pi}{n+3}\right] - d \cos \left(k \pi \right), k = 1,2,\ldots,n+2.
emb^{*}\left( WZ\right) =\frac 1{\left( p+1\right) !}dx^{\mu_1}\cdots dx^{\mu_{p+1}}\cdot \Pi_{,\mu_1}^{A_1}\cdots \Pi_{,\mu_{p+1}}^{A_{p+1}}\cdot \left( WZ\right)_{A_{p+1}\cdots A_1}\left( \phi\right) \quad ;
\tau_{k,Y_{k+1,k+1}}(x) = \frac{1}{k+1}\left(\sqrt{x}\frac{d\tau_{k,Y_{k,k}}(x)}{dx}- \frac{k^2+k}{2\sqrt{x}}\tau_{k,Y_{k,k}}(x) \right) + \frac{k}{\sqrt{x}}\tau_{k,Y_{k,k}}(x)-\frac{1}{k+1}\tau_{k,Y_{k-1,k-1}}(x) .
\mathcal{S}_{1,k}(z_{1},z_{2}) = \frac{N-k}{N}\frac{s(z_{1})s(z_{2})}{1-\frac{N-k}{N}s(z_{1})s(z_{2})}+\frac{E_{N,k}}{1-\frac{N-k}{N}s(z_{1})s(z_{2})}
{}_2F_1\bigg[\begin{matrix}\alpha \frac12 &1\end{matrix}\bigg|\,z\bigg]=(1-z)^{-\frac12\alpha}{}_2F_1\bigg[\begin{matrix}\frac12\alpha \frac12-\frac12\alpha &1\end{matrix}\bigg|\,-\frac{z^2}{4z-4}\bigg].
L(h,V)=V^{\underline n}\left(D_{\underline m}h_{\underline l \underline n}F^{\underline l\underline m}-h_{\underline l}^{\underline m}D_{\underline m}F^{\underline l}_{\quad \underline n}\right)-\frac{1}{2}A' F^{r \underline m}V_{\underline m}h_i^i+\frac{D_1-2}{2}A' F^{\underline l \underline m}V_{\underline m}h_{r\underline l}~,
K^{a}_{j}\partial_{a}( -\gamma^{cd}\partial_{c}X^{\mu}\partial_{d}X^{\nu}g_{\mu \nu}+ \frac{\varepsilon^{a_{1}a_{2}...a_{d+1}}}{\sqrt{-\gamma}}\partial_{[a_{1}}A_{a_{2}...a_{d+1}]}) = 0
p^{\alpha_2}(2,t)=\sum_{k=2}^{\infty}(-\lambda)^k\underset{\Theta^{k}_{2}}{\sum}\frac{t^{k_0\alpha_0+k_1\alpha_1+k_2\alpha_2}}{\Gamma\left(k_0\alpha_0+k_1\alpha_1+k_2\alpha_2+1\right)},
\Omega_{\gamma}= t r \{d P \, i \, u^{- 1}\, d u - \frac{1}{2 \gamma}( e^{i \gamma P}u^{- 1}\, d u \, e^{- i \gamma P}\, u^{- 1}\, d u + L_{+}^{- 1}d L_{+}L_{-}^{- 1}d L_{-}) \}.
A = \begin{pmatrix}1- \frac{\sqrt2}{2}0 \sqrt2-1 1-\frac{\sqrt2}{2}\end{pmatrix},~\Gamma = \begin{pmatrix}0 0 2-\sqrt2 0 \end{pmatrix},~b = \begin{pmatrix}\frac{1}{2}\frac{1}{2}\end{pmatrix}^T .
X^{T}\Sigma X=\frac{1}{T}\sum_{t=1}^{T}e_{t}^{2}x_{t}x_{t}^{T}+\frac{1}{T}\sum_{l=1}^{L}\sum_{t=l+1}^{T}w_{l}e_{t}e_{t-l}(x_{t}x_{t-l}^{T}+x_{t-l}x_{t}^{T})
\frac{(3\gamma -2)\kappa_{4)}^{2}\rho_{0}}{6}a^{-3\gamma}+\frac{(3\gamma -1)\kappa_{(4)}^{2}\rho_{0}^{2}}a^{-6\gamma}+\frac{2}{3}\sigma_{0}a^{-6}-\frac{\Lambda_{(4)}}{3}a +\frac{2{\cal U}_{0}}{\lambda \kappa_{(4)}^{2}}a^{-4}< 0.
\delta_{( n )}| A \rangle ={\cal D}_{n}| \Lambda_{n}\rangle \, , \qquad{\cal D}_{n}\equiv \sum_{j = 0}^{n - 1}( - )^{j}\! \! \! \! \sum_{l_{1}, \ldots , l_{j + 1}= 1}^{n}\delta_{n l_{j + 1}}\prod_{i = 1}^{j}\frac{\varepsilon^{l_{i}l_{i + 1}}}{\lambda_{l_{i}n}}{\bf a}_{l_{i + 1}l_{i}}^{}D_{l_{1}}\, ,
\frac{1}{g_M(E_i,E_i)}Jd\varphi([E_i^\prime,E_i])=\frac{1}{g_M(E_i,E_i)}d\varphi(\nabla^M_{E_i}E_i)+\frac{1}{g_M(E_i^\prime,E_i^\prime)}d\varphi(\nabla^M_{E_i^\prime}E_i^\prime)
\left(\sum_{0\le k<n}\,\prod_{0\le j<m}y_{k}^{(j)}\right)=\sum_{0\le k<n}\mbox{Prod}_{\mathbf{P}_{k}}\left(\left(\mathbf{x}^{(m-1)}\right)^{\top^{\left(m-1\right)}},\cdots,\left(\mathbf{x}^{(j)}\right)^{\top^{j}},\cdots,\left(\mathbf{x}^{(0)}\right)^{\top^{0}}\right)=
=P_{n,k}^{2}\mathbb{E}\bigg[\bigg(\sum\limits_{i=1}^{Z_{n}}(\mathbf{\Lambda}_{ii}+\sigma_{k}^{2})^{-1}\bigg)^{2}+\sum\limits_{i=1}^{Z_{n}}(\mathbf{\Lambda}_{ii}+\sigma_{k}^{2})^{-2}\bigg].
C_{u x u x}={\frac{- 3 \,{{\left( \cos ({\frac{u}{2 \, a}}) + \sin ({\frac{u}{2 \, a}}) \right)}^{2}}}{{a^{2}}\,{{\left( \cos ({\frac{u}{2 \, a}}) - \sin ({\frac{u}{2 \, a}}) \right)}^{4}}}}
\|Sf\|_{L^p}\lesssim\int_0^{\infty}s^{p-1}(\gamma\otimes \mu)(x\colon |Sb_{s}(x)|>s/2)\,ds+\int_0^{\infty}s^{p-1}(\gamma\otimes \mu)(x\colon |Sg_{s}(x)|>s/2)\,ds &:= E_1+E_2.
(\mathcal{E}_k) = (\mathcal{E}_{\tilde{k}}) = \sum_{n=0}^{+\infty}\langle \phi_n,\mathcal{E}_{\tilde{k}}\phi_n\rangle_{L^2}= \sum_{n=0}^{+\infty}\int_{\mathcal{D}}\int_{\mathcal{D}}\phi_n(x)\phi_n(y)\tilde{k}(x,y)dydx
\begin{aligned}[t] d u\wedge\alpha\wedge \tilde \Omega = \frac 12\,\biggl(u_y^2+\Bigl(u_t+\frac 12\,u\Bigr)^2\biggr)(u_{xx}+1)\Omega^2 &+\frac 12\,u_x^2(u_{yy}+u_{tt}+u_t+1)\Omega^2 &-\biggl(u_xu_yu_{xy}+u_x\Bigl(u_t+\frac 12\,u\Bigr)u_{xt}\biggr)\Omega^2 &-\frac 18\,u^2(u_{xx}+1)\Omega^2.\end{aligned}
\begin{cases}\eta_t + w_x+w_{xxx}= 0, \,\, (0,L)\times (0,T), w_t +\eta_x +\eta_{xxx}=0, \,\, (0,L)\times (0,T), \eta(0,t)=\eta(L,t)=\eta_{x}(0,t)=0, \,\, (0,T), w(0,t)=w(L,t)=w_{x}(L,t)=0, dg \,\, (0,T), \eta(x,0)= \eta_0(x), w(x,0)= w_0(x), \,\, (0,L), \end{cases}
HT_{g^{-1}}f(x) = \sum_{y \in X}b(x,y)(f(gx) - f(gy)) + c(x)f(gx) = \sum_{y \in X}b(gx,gy)(f(gx) - f(gy)) + c(gx)f(gx) = \sum_{y \in X}b(gx, y)(f(gx) - f(y)) + c(gx)f(gx) = Hf(gx) = T_{g^{-1}}Hf(x)
\mathbf{A}\cdot\left(\mathbf{B}\times\mathbf{C}\right)=\begin{vmatrix}\begin{array}{ccc}A_{1}A_{2}A_{3}B_{1}B_{2}B_{3}C_{1}C_{2}C_{3}\end{array}\end{vmatrix}=\epsilon_{ijk}A_{i}B_{j}C_{k}
[ \delta_{1}, \delta_{2}] M ={\frac{t_{1}\delta ( \epsilon_{1 2}, t_{1}) - t_{2}\delta ( \epsilon_{1 2}, t_{2})}{t_{1}- t_{2}}}M +{\frac{t_{1}t_{2}}{1 - t_{1}t_{2}}}\left( \delta ( \epsilon_{1 2}^{\prime}, t_{2}) - \delta ( \epsilon_{2 1}^{\prime}, t_{1}) \right) M .
\alpha_{-2}:= \alpha_0 \cdot \alpha_1 - \frac{1}{2^2}(\alpha_0 + \alpha_1) - \frac{8}{3}\alpha_{-1}&= -\frac{2t}{3}a_1 - \frac{1}{3}v_{(2,3)}+ \frac{4}{3}(a_2 + a_{-2}) \cdot v_{(1,3)}+ \frac{4}{3}(a_3 + a_{-3}) \cdot v_{(1,2)}\in M_0^{(a_1)}.
H_V(t) = \frac{\sum_i t^{|a_i|}(1-t)^{n+1-|Z'_i|}}{(1-t)^{n+1}}= \frac{\sum_i t^{|a_i|}(1-t)^{n-|Z_i|}}{(1-t)^{n+1}}= \frac{1}{1-t}H_{I/J}(t) = H_{I^1/J^1}(t).
F_{k}(\vec{q})=[\int d^{3}\vec{v}\rho_{k}(\vec{v})e^{-i\vec{q}\cdot\vec{v}}][\int d^{3}\vec{u}p_{k}(\vec{u})e^{-i\vec{q}\cdot\vec{u}}]e^{-i\vec{q}\cdot\vec{x}_{k0}}
\sum_{i=1}^{N}d_i\frac{\|\nabla v_i\|_2^2}{u_{i,\infty}}\geq P(\Omega)\min_{i=1,\ldots, N}\{d_i\}\sum_{i=1}^{N}\frac{\|v_i - \overline{v}_i\|_2^2}{u_{i,\infty}}.
e^{-\frac{\lambda}{\xi}z}z^{\frac{\mu}{\xi}-1}P_{1}(z) = \int_{0}^{z}e^{-\frac{\lambda}{\xi}s}s^{\frac{\mu}{\xi}-1}\left[ -\frac{\gamma}{\xi(1-s)}P_{0}(s) + \frac{\gamma}{\xi s (1-s)}p_{0,0}+ \frac{\mu - \xi}{\xi s(1-s)}p_{1,1}\right]ds.
E\big[|I_2(t)|^2\big] = E\big[|\hat{\textbf{h}}~\!\!^{H}_{1,k}\textbf{h}_{1,k'}|^2\big],~~ k' \neq k, E\big[|I_3(t)|^2\big] = E\big[|\hat{\textbf{h}}~\!\!^{H}_{1,k}\textbf{h}_{j,k}|^2\big],~~ j \neq 1.
\|g(A) (-A)^{\frac{1}{2}}e^{A\varepsilon}e^{A \cdot}x\|_{L^2}\leq \ \|g(A) (-A)^{\frac{1}{2}}e^{A \cdot}x\|_{L^2}= \ \|M_g \left((-A)^{\frac{1}{2}}e^{A \cdot}x\right)\|_{L^2}\leq \ \|g\|_{\infty}\|(-A)^{\frac{1}{2}}e^{A \cdot}x \|_{L^2}\leq \ \|g\|_{\infty}m \|x\|,
=\frac{p}{2\left( p-1\right)}\inf \left\{\ \frac{\left\Vert \left\vert\Delta u\right\vert^{\frac{p}{2}}\right\Vert_{2}}{\left\Vert \nabla \left(\left\vert u\right\vert^{\frac{p-2}{2}}\left\vert u\right\vert \right)\right\Vert_{2}}\left\vert \u\in W^{2,p}\cap W_{0}^{1,p}\right. \right\}
\begin{gathered}I-\int_{\Omega}f(x)\big(g_k(\phi_{h,t})-g_k(z)\big)(\phi_{h,t}-z)\,dx =\int_{\Omega}g_v(x)\,dx-\int_{\Omega}f(x)g_k(\phi_{h,t})\big(\phi_{h,t}-z-t\phi_h\big)\,dx+t\int_{\Omega}\mathcal{A}(x,\nabla\phi_{h,t})\cdot\nabla\phi_h\,dx- t\int_{\Omega}f(x)g_k(\phi_{h,t})\phi_h\,dx.\end{gathered}
S_{\mu \nu}: = \left( \begin{array}{c}{G_{\mu \nu}}\\{E_{\mu \nu}}\\ \end{array}\right) = \left( \begin{array}{c}{g_{\mu \nu}}\\{H_{\mu \nu}}\\ \end{array}\right) + i \left( \begin{array}{c}{B_{\mu \nu}}\\{M_{\mu \nu}}\\ \end{array}\right) ,
{C}_{X^n \rightarrow Y^n}^{FB,A.B}(\kappa) = \sup_{{\cal P}_{0,n}^{B}(\kappa)}\sum_{t=0}^n{\bf E}^{\pi}\left\{\log\Big(\frac{q_t(\cdot|Y_{t-M}^{t-1},X_t)}{\nu_t^{{\pi}}(\cdot|Y^{t-1})}(Y_t)\Big)\right\},
F(X,Y,Z,0) = F(0,Y,Z,0) + X \int_{0}^1 \partial_X F(\alpha X,Y,Z,0){\rm d \alpha}&= X \int_{0}^1 \int_{0}^1 \partial_X F(\alpha X,0,Z,0) + Y \partial_X \partial_Y F(\alpha X,\beta Y,Z,0){\rm d \beta}{\rm d \alpha}&= XY\int_{0}^1 \int_{0}^1 \partial_X \partial_Y F(\alpha X,\beta Y,Z,0){\rm d \beta}{\rm d \alpha}.
\int_M\Delta\log|\psi|dM=\lim_{\epsilon\to 0}\int_{M\setminus\cup_{j=1}^k D_\epsilon(p_j)}\Delta\log|\psi|dM=-\lim_{\epsilon\to 0}\sum_{j=1}^k\int_{\partial D_\epsilon(p_j)}\frac{\partial}{\partial r}\log|\psi|d\theta,
0=D_{t}D_{t}X^{I}=\left(\textstyle{\frac{d^{2}\,}{dt^{2}}}X^{I}-[A_{0},[A_{0},X^{I}]]\right)+i\left([\textstyle{\frac{d\,}{dt}}A_{0},X^{I}]+2[A_{0},\textstyle{\frac{d\,}{dt}}X^{I}]\right)\,.
\overline{{{\delta}}}( k ) = \sum_{\sigma \in \{+ , - \}}\sum_{G = 0}^{\infty}( 2 G + 1 ) \left( \delta_{G , \sigma}( k ) - \delta_{G , \sigma}^{( 1 )}( k ) - \delta_{G , \sigma}^{( 2 )}( k ) \right) + \delta_{\mathrm{l i m}}( k ) \, .
R_0\leq\min\{I(U^m_{0};\hat{S}^m_{1}),I(U^m_{0};\hat{S}^m_{2})\}, &R_0+R_i\leq I(U^m_{0},U^m_{i};\hat{S}^m_{i}),i=1,2, &R_0+R_1+R_2\leq\min\{I(U^m_{0};\hat{S}^m_{1}),I(U^m_{0};\hat{S}^m_{2})\}+I(U^m_{1};\hat{S}^m_{1}|U^m_{0})+I(U^m_{2};\hat{S}^m_{2}|U^m_{0})-I(U^m_{1};U^m_{2}|U^m_{0})
d(\underline{S},f) =(b({\mathbf{0}})\Gamma_{\mathbf{0},0}(\underline{S}), \ldots, b({\mathbf{i}})\Gamma_{\mathbf{i},\vert \mathbf{i}\vert}(\underline{S}), \ldots, b({\mathbf{k}})\Gamma_{\mathbf{k},\vert \mathbf{k}\vert}(\underline{S})).
a^\alpha=\hat{g}_{+,\alpha}\hat{g}_{+,\alpha+1}\cdots \hat{g}_{+,d}\hat{A}^\alpha\,, b^\alpha=\hat{B}^\alpha \hat{g}_{-,d}^{-1}\cdots \hat{g}_{-,\alpha+1}^{-1}\hat{g}_{-,\alpha}^{-1}\,.
\epsilon^{i}( \theta ) + \varphi * L^{i}( \theta ) - \varphi * L^{j}( \theta - \sigma_{j i}) = r M^{i}\cosh \theta , \quad \varphi ( \theta ) = - \frac{4 \sqrt{3}\cosh \theta}{1 + 2 \cosh 2 \theta}\; .
\mathbf{T_{n,s}}= \{\langle \sigma, \tau, n \rangle \mid \langle \sigma,\tau,n \rangle \in \mathbf{J}\wedge \exists t \geq s (\langle \sigma, \tau, n \rangle \in I_t) \}, \mathbf{U_{n,s}}= \{\tau \mid \exists \sigma (\langle \sigma, \tau, n \rangle \in T_{n,s}) \}.
\frac{\rho_i}{\rho_i-\rho_{i+1}}=\frac{\tau-(\tau-\sigma)\frac{1-2^{-i}}{1-2^{-k}}}{(\tau-\sigma)\frac{2^{-(i+1)}}{1-2^{-k}}}\leq \frac{2^{i+1}\tau}{\tau-\sigma}
I_1 := \sum_{j=1}^N \beta( a_j^{1/2}[ r_{j,k}, a_j^{1/2}] );\;\;\;\;\;\;\;\; I_2 := \sum_{j=1}^N [\beta(a_j)-a_j]\beta(r_{j,k}) ; I_3 := \sum_{j=1}^N a_j[ \beta(r_{j,k}) - r_{j,k+1}];\;\;\;\;\;\;\;\;I_4 := \sum_{j=1}^N a_j^{1/2}[ a_j^{1/2}, r_{j,k+1}].
f_0= 1+2q+2q^4+2q^9+2q^{16}+O(q^{25}) f_3= q^{-3}-248q+26752q^4-85995q^5+1707264q^8-4096248q^9+O(q^{12}) f_4= q^{-4}+492q+143376q^4+565760q^5+18473000q^8+51180012q^9+O(q^{12}) \vdots
\widetilde{\Delta}_{\ell,i}^{(t)}=\sum_{j\in[(\ell-1)R_w^{(t)}+1:\ell R_w^{(t)}]}\frac{1}{\alpha_n-f_{j,i}}\Delta_{j,i}^{(t)}+\sum_{x\in[X_{\Delta}]}\alpha_n^{x-1}\dddot{Z}_{\ell,i,x}^{(t)}.
T_i f_{\ldots,\lambda_i,\lambda_{i+1},\ldots}= t^{1/2}f_{\ldots,\lambda_i,\lambda_{i+1},\ldots}\lambda_i = \lambda_{i+1}, T_i f_{\ldots,\lambda_i,\lambda_{i+1},\ldots}= t^{-1/2}f_{\ldots,\lambda_{i+1},\lambda_i,\ldots}\lambda_i > \lambda_{i+1}, \omega f_{\lambda_n,\lambda_1,\ldots,\lambda_{n-1}}= q^{\lambda_n}f_{\lambda_1,\ldots,\lambda_n}.
\int_1^T \Delta^2(a_1,a_2;x)dx=c_2 T^{1+\frac{1}{a_1+a_2}}+O\left(T^{1+\frac{1}{a_1+a_2}-\frac{a_1}{a_2(a_1+a_2)(a_1+a_2-1)}+\varepsilon}\right),
\sum_{n=1/2,\cdots}{\Delta p^+ \over p_n^+}= \sum_{n=n_{\rm IR}+1}^{n_{\rm UV}-1}\frac{1}{n}\simeq{\rm ln}\frac{n_{\rm UV}}{n_{\rm IR}}\simeq{\rm ln}\frac{2\Lambda^2}{M^2 +{\bf p}_\bot^2},
e^{- 2 \phi}= C + a x^{+}+ b x^{-}- \lambda^{2}x^{+}x^{-}- \frac{\mu^{2}}{3 \beta}\left[ \log [{\cosh ({\Delta - \Delta_{0}})}] + \frac{1}{4}\tanh^{2}({\Delta - \Delta_{0}}) \right] ,
0 =(y'(x)z''(x)-z'(x)y''(x))[1+y_{x}^{2}+z_{x}^{2}]^{-3/2}z =z''(x)[1+y_{x}^{2}+z_{x}^{2}]^{-3/2}y = y''(x)[1+y_{x}^{2}+z_{x}^{2}]^{-3/2}.
\prod_{\rho=0}^{\infty}g(\rho)^{n_\rho}\leq \left(\frac{C_8C_5r 2^{j}\sigma_{r,X}(rl)}{t}\right)^{rl}2^{rl(2j+4)}\left(1+\frac{d^r}{\sigma_{r,X}(rl)^r}\right)^m\exp\left(-\frac{rl}{2}(2^{j-1}-1)\right),
\left|\mathbb{E}\left(\overline{\alpha}_{t,\varepsilon}\right)^n\right| \leq c\,\int_{[0,t]^n}\prod_{j=1}^n|s_j-s_{j-1}|^{-\frac{2H_1H_2}{H_1+H_2}}ds &\leq c\int_{\{0<s_1<\cdots<s_n<t\}}\prod_{j=1}^n|s_j-s_{j-1}|^{-\frac{2H_1H_2}{H_1+H_2}}ds,
(i)\,\,\{j_2\neq -j_1, j_3=-j_1, j_4=-j_2\}(ii)\,\,\{j_{2}\neq -j_1, j_3\neq -j_1, j_3=-j_2, j_4=-j_1\}begin{align*}2mm] (iii)\,\, \{j_1+j_2=0\}.
F(x,y) : = \left\{\frac{u(x)}{1-\varepsilon}- \frac{v(y)}{1+\varepsilon}- \alpha_2\Psi_2(x,y) - \frac{\varepsilon}{1-\varepsilon}\Upsilon(x) - \frac{\varepsilon}{1+\varepsilon}\Upsilon(y)\right\}, G(x,y) := \Psi_1(x,y).
\zeta(2)=\sum_{k=1}^{\infty}\frac{1}{k^{2}}=lim_{m\rightarrow\infty}(\frac{1}{1^{2}}+\frac{1}{2^{2}}+\cdot\cdot\cdot+\frac{1}{m^{2}})=\frac{\pi^{2}}{6}
\hat{J}_{m_{0},m_{1},n}^{\,(k,N)}(\mathbf{s})=J_{\Delta_{1},n-m_{0}}^{\,(k,N)}(\mathbf{s})=1-\frac{1-H_{\Delta_{1}}^{(k,N)}\left(\mathbf{1}-(\mathbf{1}-\mathbf{s})\otimes \mathbf{Q}_{n-m_{1}}\right)}{Q_{n-m_{0}}^{(k,N)}}.
\sum_{n\ge 1}\frac{\max_{1\le i\le n}V_n^q}{n^p}\le \sum_{r\ge 0}\sum_{n=2^r}^{2^{r+1}-1}\frac{\max_{1\le i\le n}V_n^q}{n^p}\le \sum_{r\ge 0}\frac1{2^{(p-1)r}}(\sum_{k=0}^rV_{2^k})^q
\varphi^{a}= \Bigl ( \delta_{b}^{a}- \frac{1}{2}\xi^{a e}\S_{e c}\delta_{j}^{c}G^{j k}\delta_{k}^{d}\xi_{d b}\Bigr ) \delta_{i}^{b}\phi^{i}- \Bigl ( \xi^{a c}\S_{c b}\delta_{j}^{b}G^{j i}\Bigr ){\bar{\phi}}_{i},
\frac{d}{dt}\int_{P_t}\sigma_u\,d{\rm vol}_{P_t}=\frac{d}{dt}\int_{L_t}\int_{P_{(p,t)}}\sigma_u\,d{\rm vol}_{P_{(p,t)}}\,d{\rm vol}_{L_t}&=\int_L\int_{P_{(p,0)}}\left(\frac{d}{dt}\right)^h\sigma_u\,d{\rm vol}_{P_{(p,0)}}\,d{\rm vol}_L &+\int_L\int_{P_{(p,0)}}\sigma_u\,d{\rm vol}_{P_{(p,0)}}\,d\frac{d}{dt}{\rm vol}_{L_t}.
\lambda_{2n}^\pm= 2n\pi-i \sum_{k=1}^{N+1}\frac{1}{(2i\lambda_{2n}^\pm)^k}\int_0^1\varphi_1r_kdt + \sigma_n^\pm\sqrt[\circ ]{e_2e_3+h_{2n}^\pm}+\frac{\ell^2_n}{n^{N+1}}.
\max_{v\in \Omega}\|P_v\| = \max_{v\in \Omega}\sum_{j = 1}^d|P_j(v)| + |a_0| \ge \max_{v\in \Omega}|P_{j_0}(v)| \ge \frac{\mu(\Omega)^d}{c_{3}}\|P_{j_0}\| &\ge \frac{\mu(\Omega)^d}{c_{7}c_{2}(d+1)}=: c_{1}\mu(\Omega)^d,
R=\left[\begin{array}{llll}q \, \, \, \, 1 0 \, \, \lambda 1 \, \, \, \, q\end{array}\right]\;,\quad \hat{R}=\left[\begin{array}{llll}q \, \, \, \, \lambda 1 \, \, 1 0 \, \, \, \, q\end{array}\right]\;, \quad{\cal P}R{\cal P}=R^t \;,
\begin{cases}d\tilde{p}(t,z) =- \frac{a_0(t,z)}{b_0(t,z)}\tilde{p}(t,z) dB(t) \tilde{p}(T,z) = \int_D U'(y(T,x,z))dx\mathbb{E}[\delta_Z(z)|\mathcal{F}_T]\end{cases}
\sum_{\substack{[i,j]\in\sigma C_{ij}(\sigma)\ni c}}r_{ij}= \sum_{[i,j]\in \sigma}r_{ij}- \sum_{\substack{[i,j]\in\sigma i\le x ; j> y}}r_{ij}- \sum_{\substack{[i,j]\in\sigma i > x ; j\le y}}r_{ij}&=ab-(a-x)b -(b-y)a &= ab\left(\frac{x}{a}+\frac{y}{b}-1\right).
\tilde{m}_{0}^{\ast}: = K^{\ast - 1}m_{0}K^{\ast}= \left( \begin{array}{c c}{0}&{- \kappa_{e}^{\frac{1}{2}}T \kappa_{o}^{-{\frac{1}{2}}}}\\{- \kappa_{o}^{-{\frac{1}{2}}}\bar{T}\kappa_{e}^{\frac{1}{2}}}&{0}\\ \end{array}\right) = - \tilde{m}_{0}.
(z)_N = \sum_{n=0}^{N}\left[\begin{matrix}N n \end{matrix}\right](-1)^{n}z^n q^{n(n-1)/2}, \frac{1}{(z)_{N}}=\sum_{j=0}^{\infty}\left[\begin{matrix}N+j-1 j\end{matrix}\right]z^j.
L^{2}( G , d g ) \cong \bigoplus_{\lambda_{G}, \lambda_{H}, \lambda_{H}^{\prime}}\overline{{M}}_{\lambda_{H}}^{\lambda_{G}}\otimes \overline{{V}}_{{\lambda_{H}}}\otimes M_{\lambda_{H}^{\prime}}^{\lambda_{G}}\otimes V_{\lambda_{H}^{\prime}}.
Z [ \zeta^{\mu}, \xi^{\mu}, \eta^{\mu}] ={\cal N}\, \int \prod_{x}\{\mathrm{d}v_{\mu}\, \mathrm{d}j^{\mu}\, \mathrm{d}| h |^{2}\, \mathrm{d}\alpha \}\, e^{\mathrm{i}\int \mathrm{d}^{4}x \,{\cal L}[ \zeta^{\mu}, \xi^{\mu}, \eta^{\mu}; v_{\mu}, j^{\mu}, | h | , \alpha ]}\ ,
\nabla \bar{J_3}(U) = 2 A U{\rm sym}(\nabla_{A_r}\bar{J}(U^TAU, U^TB, CU)) &+ B(\nabla_{B_r}\bar{J}(U^TAU, U^TB, CU))^T &+ C^T\nabla_{C_r}\bar{J}(U^TAU, U^TB, CU).
I_{\pm}(e_{\pm}) =\frac{1}{2\pi}\int_{-\min\beta}^{e_{\pm}}T_{\pm}(e_{\pm}^{\prime})de_{\pm}^{\prime}, \theta_{\pm}=\frac{2\pi}{T_{\pm}(e_{\pm})}\int_{0}^{x}\frac{dx^{\prime}}{\sqrt{2(e_{\pm}\mp\beta(x^{\prime}))}},
\prod_{\substack{h=1 h\neq\frac{d}{2}}}^{d-1}\frac{\Gamma_p(\langle(\frac{h}{d}+\frac{m}{q-1})p^i\rangle)}{\Gamma_p(\langle\frac{hp^i}{d}\rangle)}=\prod_{\substack{h=1 h\neq\frac{d}{2}}}^{d-1}\frac{\Gamma_p(\langle(\frac{-h}{d}+\frac{m}{q-1})p^i\rangle)}{\Gamma_p(\langle\frac{-hp^i}{d}\rangle)}.
z\,W_{0}(z)-1=a_{0}^{(0)}\, W_{0}(z)+\sum_{j=1}^{p}\sum_{s=0}^{j}a_{-s}^{(j)}\, A_{j-s-1}^{(1)}(z)\, B_{s-1}^{(1)}(z)\, W_{0}(z)
2(1+1/p)G + H_x + 2xG_x - (2/p)(vG_v +\bar{v}G_{\bar v})+ GH_v- HG_v + \bar G H_{\bar v}- \bar H G_{\bar v}=0 &\i G + (2-n+2/p)H + 2xH_x -(2/p)(vH_v +\bar{v}H_{\bar v})- HH_v - \bar H H_{\bar v}=kv^{1+p/2}\bar{v}^{p/2}
S_{\mathrm{e f f}}= \int \! d^{2}x \, \Bigl ({\frac{1}{2}}\, \partial_{\alpha}\chi \partial_{\alpha}\chi +{\frac{1}{2}}\, \partial_{\alpha}\chi^{\prime}\partial_{\alpha}\chi^{\prime}+{\frac{v_{N - 1}}{2}}\, \chi^{2}-{\frac{\tilde{\delta}v}{2}}\, \left( \chi^{2}+ \chi^{2}\right) +{\frac{\pi \tilde{V}}{N}}\, \left( \chi^{2}+ \chi^{2}\right)^{2}\Bigr ) .
\rho ( \eta ) \sim \biggl ( \frac{\eta_{1}}{\eta_{2}}\biggr )^{3}\int_{\cal K}\mathrm{d}k_{\mathrm{p h y s}}\, k_{\mathrm{p h y s}}^{2}\omega_{\mathrm{p h y s}}( k ) = \biggl ( \frac{\eta_{1}}{\eta_{2}}\biggr )^{3}{\cal O}( k_{\mathrm{c}}^{4}) ,
\Delta_q\left( \sum_j \sum_{\vert j-j'\vert \le 1}(\Delta_{j'}f_M, \Delta_j g) \right) = \Delta_q \left(\sum_{\max\{j,j'\}\ge q-2}\sum_{\vert j-j' \vert \le 1}(\Delta_{j'}f_M, \Delta_j g)\right) &=0 \enskip \mbox{if $q \geq M+3$},
( \Delta_{0}\vec{n}( \vec{x}, x_{0}) )^{2}\equiv ( \vec{n}( \vec{x}, x_{0}+ 1 ) - \vec{n}( \vec{x}, x_{0}) )^{2}\to ( R_{q}^{- 1}\vec{n}( \vec{x}, x_{0}+ 1 ) - \vec{n}( \vec{x}, x_{0}) )^{2}
\mu_{j,l,l^{\prime}}=\frac{2\left( -i\right)^{-l+l^{\prime}+j}\left(-1\right)^{j}}{\pi}\left( 2j+1\right) \left[ \left( 2l+1\right) \left(2l^{\prime}+1\right) \right]^{1/2}\left(\begin{array}[c]{ccc}l l^{\prime}j 0 0 0\end{array}\right) .
\varphi_t(t) =-\left(\alpha\,\mathrm{tr}\left(\overline{A}(t)\right)+\delta\right)\varphi(t) t>0, N_t(t) =-2\overline{\alpha}N(t)\tilde{P}N(t)+\tilde{S}^TN(t)+N(t)\tilde{S}t>0.
J_N(z,u, k) = \mathbf{E}^u_{t_{k},z}\!\left[\sum_{j=k}^{K-1}\Big(r(z^{u,N}(t_j), \nu_j) + \int_{t_j}^{t_{j+1}}\phi(z^{u,N}(s), \nu_j) \,ds\Big) + \psi\big(z^{u,N}(T)\big)\right], U_N(z,k) = \inf_{u \in \mathcal{U}_{f,k}}J_N(z,u, k)\,,
m \geq\frac{1+\varepsilon}{(\sqrt{\delta+1}-1)^2}\left( \Big( \sum_{i =1}^{d_1}\sum_{\stackrel{j=1}{j\neq i}}^{d_1}\frac{\lambda_i \lambda_j}{(\lambda_i-\lambda_j)^2}+\sum_{i =1}^{d_1}(n-d_1)\frac{\lambda_i\sigma^2}{(\sigma^2-\lambda_i)^2}\Big)\right),
|\mathbf{e}_2\cdot \mathbf{w}_1|=\frac{|\mathbf{w}_1\cdot (\mathbf{w}_2\times(\mathbf{w}_1\times\mathbf{w}_2))|}{\|\mathbf{w}_2\|_2\|\mathbf{w}_1\times\mathbf{w}_2\|_2}=\frac{\|\mathbf{w}_1\times\mathbf{w}_2\|_2}{\|\mathbf{w}_2\|_2}.
[\partial_nS_n(\theta)](h_1, \ldots, h_{n-1}) &=[\partial_n\theta'](h_1, \ldots, h_{n-1}) &=\sum_{h_0\in H}h_0^{-1}\theta'(h_0,\ldots, h_{n-1})+\sum_{i=1}^{n-1}(-1)^i\sum_{\substack{h, \bar{h}\in H h\bar{h}=h_i}}\theta'(h_1,\ldots, h_{i-1},h,\bar{h}, h_{i+1},\ldots, h_{n-1}) &\quad{}+(-1)^n\sum_{h_n\in H}\theta'(h_1,\ldots, h_n).
I_t = \frac{e^{2\rho t}-1}{c_t}\int_{1+\frac{1}{c_t}}^{e^{2\rho t}+\frac{1}{c_t}}\frac{dv}{v^2-d_t^2}=\frac{e^{2\rho t}-1}{c_t}\left[ \frac{1}{2d_t}\log \left( \frac{v-d_t}{v+d_t}\right) \right]_{1+\frac{1}{c_t}}^{e^{2\rho t}+\frac{1}{c_t}}.
\left \|\sigma_1\sharp_0 \sigma_2(\xi)\right\| = \left\| J\left( \sigma_1\natural_0\sigma_2(\xi;\cdot,\cdot)\right)\right\| \leq C_{00}p_{N_0}^{(m_1)}(\sigma_1)p_{N_0}^{(m_2)}(\sigma_2)(1+|\xi|)^{m_1+m_2}.
{\mathcal H}\equiv{\mathcal H}( m_{1}, 0 ) \otimes{\mathcal H}( m_{2}, 0 ) = \int_{( m_{1}+ m_{2})^{2}}^{\infty}d{\mathsf{s}}\sum_{j = 0}^{\infty}\oplus{\mathcal H}({\mathsf{s}}, j ) ,
\Gamma = - \frac{1}{2}\lim_{L^{2}\rightarrow \infty}\frac{1}{( 4 \pi )^{w}}\int_{1 / L^{2}}^{\infty}\frac{d s}{s^{w + 1}}\exp ( - s m^{2}) \sum_{n = 0}^{\infty}s^{n}\int d^{\scriptscriptstyle{2 w}}x \sqrt{g}\hat{a}_{n}( x )