harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\begin{aligned}\mathcal{L}_{\mathcal{I}_l}W_\epsilon(x_j,x_k) u_\epsilon = W_\epsilon(x_j,x_k)\mathcal{L}_{\mathcal{I}_l}u_\epsilon+(\nabla_jW_\epsilon(x_j,x_k))\mathcal{L}_{\mathcal{I}_l,j}u_\epsilon &+(\nabla_kW_\epsilon(x_j,x_k))\mathcal{L}_{\mathcal{I}_l,k}u_\epsilon+(\nabla_j\nabla_kW_\epsilon(x_j,x_k))\mathcal{L}_{\mathcal{I}_l,j,k}u_\epsilon.\end{aligned}
\frac{\partial}{\partial t}F= ~\Phi'\dot{F}^{kl}\nabla_k\nabla_lF+\Phi''\dot{F}^{kl}\nabla_kF\nabla_lF \quad+\Phi\left(\dot{F}^{ij}h_i^kh_{kj}+K_N\dot{F}^{ij}g_{ij}\right).
2(N-1)\Vert f\Vert_{\infty}\Vert g\Vert_{\infty}\delta^{-N(N-1) \vert \kappa\vert}\binom{N-1}{2}\left( 2\arcsin\frac{\delta^{1/2}}{2}\right) \leq c^{\prime}\delta^{1/2-N(N-1) \vert \kappa\vert},
B_{\vec\tau, \vec s}^{(j)}(r)=\frac{s_{j+1}-s_j}{2\pi i}\left(T^{-1}\partial_{s_m}T\right)_{21}(\lambda_j)=B_{\vec\tau, \vec s}^{(j,1)}(r)+B_{\vec\tau, \vec s}^{(j,2)}(r)+B_{\vec\tau, \vec s}^{(j,3)}(r),
{}[ \kern - 0 . 4 5 e m \llap [ J_{0}( x ) , J_{1}^{a b}( y ) ] \kern - 0 . 4 5 e m \llap ] = M^{a b}\delta^{\prime}( x - y ) = [ \kern - 0 . 4 5 e m \llap [ J_{1}( x ) , J_{0}^{a b}( y ) ] \kern - 0 . 4 5 e m \llap ~
: \because \frac{n + 2}{( n + 1 )^{2}}= \frac{n + 1 + 1}{( n + 1 )^{2}}= \frac{n + 1}{( n + 1 )^{2}}+ \frac{1}{( n + 1 )^{2}}= \frac{1}{n + 1}+ \frac{1}{( n + 1 )}
z(s,t)= \left(-4s^{1/2}\cos t+s\sin t+\frac 12\cos t,-4s^{1/2}\sin t-s\cos t+\frac 12\sin t,\right. &-4s^{1/2}\sin t-s\cos t-\frac 12\sin t,\left.-4s^{1/2}\cos t+s\sin t-\frac 12\cos t\right)
L^{(t)}(\mathbf{x}) =\frac{1}{t}\sum_{\tau=1}^{t}\sum_{n=1}^{N}(y_{n}^{(\tau)}-(\mathbf{g}_{n}^{(\tau)})^{T}\mathbf{x})^{2}\quad\quad+\mu^{(t)}\sum_{k=1}^{K}\mathcal{W}_{\mu^{(t)}}(|x_{\textrm{rls},k}^{(t)}|)\cdot|x_{k}|,
\dim_{\mathcal R}\biggl (\bigl ((f,\tilde a_{i_j}); j\in I_1\bigl )_{\mathcal R}\cap\bigl ((f,\tilde a_{i_j}); j\in A_2\bigl )_{\mathcal R}\biggl ) &\ge\dim_{\mathcal R}\left ((f,\tilde a_{i_j}); j\in I_1\right )_{\mathcal R}+\dim_{\mathcal R}\left ((f,\tilde a_{i_j}); j\in A_2\right )_{\mathcal R}-(k+1) &=k-n+\sharp I_1+k-n+\sharp A_2-(k+1)=1.
\{J_x,\Z_y\}=2ixJ_x\sum_{j=1}^{N+1}\frac{\Vert u_j\Vert^2}{\rho_j(1-x\rho_j^2)(1-y\rho_j^2)}{\mathrm{e}}^{i\varphi_j}=\frac{2ixJ_x}{x-y}(xZ_x-yZ_y)\ .
\langle f , 1 \rangle = \langle \theta_{i_1}\cdots \theta_{i_r}\overline{\theta}_{j_1}\cdots \overline{\theta}_{j_s}f^\prime, 1 \rangle &:= w (i_1, \dots, i_r) \delta_{r,s}\delta_{i_1, j_r}\cdots \delta_{i_r, j_1}\langle f^\prime , 1 \rangle &= w(i) \delta_{r,s}\delta_{i,j^T}\langle f^\prime , 1 \rangle
\left\Vert \bar{\lambda}(\hat{\gamma}_{\ell})-\bar{\lambda}(\gamma_{0})\right\Vert^{2}=\int[\int_{-\infty}^{\hat{\gamma}(x)-\gamma_{0}(x)}f(u|x)du-\int_{-\infty}^{0}f(u|x)du]^{2}F_{0}(dx) & \leq\int C|\hat{\gamma}(x)-\gamma_{0}(x)|^{2}F_{0}(dx)=C\left\Vert\hat{\gamma}-\gamma_{0}\right\Vert^{2}.
\sum_{k=0}^{n-1}\binom{n-1}{k}S_{k+1}^{(a,b)}x^{k}(x-b)^{n-1-k}(x-a-b)^{n-1-k}= \sum_{k=0}^{n-1}\frac{1}{n}\binom{n}{k}\binom{n}{k+1}x^{2k}(a+b)^{n-k}b^{n-1-k},
\mathbb{E}\left[ \mathcal{I}_{\mathrm{u}}\right] = \mathbb{E}\left[ \sum_{x \in \Phi_{\mathrm{b}}\setminus i}\frac{g_{xik}}{\| x \|^{\alpha}}\right] & \stackrel{(a)}{=}\int_{t}^\infty r^{-\alpha}r dr = \frac{2 \pi \lambda t^{-(\alpha-2)}}{\alpha-2}
-2s_0 e^2+s_1he-eh = 2e -2s_0ef+s_1hf+s_0^2eh-s_0s_1h^2-s_1^2fh = -s_0 h-2s_1 f ef+s_0^2e^2-s_0s_1he-s_1^2fe = -s_0 e + \frac{s_1+1}{2}h.
\begin{cases}\pi_1(j_0)=\pi_0(a_1) \pi_1(a_k)=\pi_0(a_{k+1})\, 1\leq k\leq m-1 \pi_1(a_m)=\pi_0(j_0) \pi_1(r)=\pi_0(r), r\notin \{j_0,a_1,...,a_m\}=:U.\end{cases}
2\cdot(\frac{Vol(\partial X,\hat{g})}{\mathcal{A}(p)})^{\frac{1}{n}}\leq \delta e^{\coth^{-1}(1+a(\delta))}=\delta e^{\frac{1}{2}\ln\frac{a(\delta)+2}{a(\delta)}}=\delta(1+\frac{2}{a(\delta)})^{\frac{1}{2}}
F_N(z,u;\tau)=\sum_{\substack{n\geq0 r\geq\frac N2}}(-1)^{Nn}q^{\frac N2n^2+nr}\zeta^rw^{r+Nn}-\sum_{\substack{n\geq1 r<\frac N2}}(-1)^{Nn}q^{\frac N2n^2-nr}\zeta^rw^{r-Nn},
\sup\{s\leq t:X_s=0\}\,\mathbb{Q}^{\delta,\mu}_0 &~~\stackrel{\mathcal{L}}{=}\sup\{s\leq t:X_s+X_{(s-E_\mu)^+}'=0\}\,\mathbb{Q}_0^\delta(X)\times\mathbb{Q}_0^{4,\mu}(X') &~~\stackrel{\mathcal{L}}{=}\sup\big\{s\leq\min\{t,\,E_\mu\}:X_s=0\big\}\,\mathbb{Q}_0^\delta.
\int_{0}^{1}\frac{dy}{y}\frac{e^{-y}}{\log^{a+1}(t/y)}= \int_{0}^{1}\frac{dy}{y \log^{a+1}(t/y)}+\int_{0}^{1}dy \frac{\left(e^{-y}-1\right)}{y \log^{a+1}(\frac{t}{y})}
q^{-k}\gamma(\psi^{-1})^{-1}(\gamma(s,\pi,\eta,\psi)-\gamma(s,\pi',\eta,\psi)) = \int_{F^\times}( W_{v_k}(t(a)w)-W_{v_k'}(t(a)w))\psi^{-1}(2a) \chi(a)\eta^{-1}(a)|a|^{-s}da.
a_{\mathbf{m},j}= \sum_{j=1,2}(\mathbf{e}_{\mathbf{m},j})_i b_{\mathbf{m},i}a_{\mathbf{m},j}^+ = \sum_{j=1,2}(\mathbf{e}_{\mathbf{m},j})_i b_{\mathbf{m},i}^+.
\begin{gathered}u^\alpha J_{\nu+\alpha}(ux; q^2)= -\frac{(1-q^2)^{\alpha}q^{2\alpha+\nu-2}x^{\alpha+\nu-1}}{\Gamma_{q^2}(1-\alpha )}D_{q,x}\int_x^\infty \rho^{-2\alpha-\nu+1}({x^2}/{\rho^2}; q^2)_{-\alpha}J_\nu (u\rho; q^2) \,d_q \rho. \end{gathered}
\mathcal{I}_\kappa[f,g]=1/\kappa\int_0^1f(x/\kappa){\rm e}^{{\rm i}\kappa g(x/\kappa)}{\rm d}x+1/\kappa\int_1^\kappa f(x/\kappa){\rm e}^{{\rm i}\kappa g(x/\kappa)}{\rm d}x.
\widetilde{c}_{ji}(r-2)+\widetilde{c}_{ji}(r+2)-\sum_{|j-k|=1}\widetilde{c}_{ki}(r)=0\ \j\in I\setminus\{n\}, &\widetilde{c}_{ni}(r-1)+\widetilde{c}_{ni}(r+1)-\widetilde{c}_{n-1 i}(r-1)-\widetilde{c}_{n-1 i}(r+1)=0.
\gamma_{m}^{f}=\frac{\rho_{0,1}(1-\pi_{m-1}^{in})(1-\gamma_{m-1}^{f})+\rho_{1,1}\pi_{m-1}^{in}\gamma_{m-1}^{f}}{(1-\pi_{m-1}^{in})(1-\gamma_{m-1}^{f})+\pi_{m-1}^{in}\gamma_{m-1}^{f}},
\begin{pmatrix}1 1 0 1 & I_2 \end{pmatrix}, \begin{pmatrix}U I_2 \end{pmatrix}, \begin{pmatrix}1 U & 1 \end{pmatrix}, \begin{pmatrix}I_2 U \end{pmatrix}, \ \ \begin{pmatrix}I_2 1 1 0 1 \end{pmatrix},
\Psi^{(2)}(\eta_1,\eta_2):=\lim_{M\to\infty}N_1^{1/3}Q_2'(0) &=\sum\limits_{r=1,s,t=0}^\infty\frac 1{r!s!(r-1)!t!}\int\limits_{(-\infty,0]^r}d^rx\int\limits_{(-\infty,0]^s}d^sx'\int\limits_{[0,\infty)^{r-1}}d^{r-1}y\int\limits_{[0,\infty)^t}d^ty' W_{r,s,r-1,t}^{(2)}(\mathbf{x},\mathbf{x'},\mathbf{y},\mathbf{y'}),
\eta^{2}=\frac{{\sigma_{\overline{y}}}^{2}}{{\sigma_{y}}^{2}},where{\sigma_{\overline{y}}}^{2}=\frac{\sum_{x}n_{x}(\overline{y}_{x}-\overline{y})^{2}}{\sum_{x}n_{x}}and{\sigma_{y}}^{2}=\frac{\sum_{x,i}(y_{xi}-\overline{y})^{2}}{n},
f_{i}^{2}d V = \frac{f_{i}^{2}d \Omega}{- \tilde{u}}\rightarrow \frac{2 \pi^{3}\epsilon}{\xi_{0}^{2}}S ( t ) \delta ( t ) \delta ( r ) \left( \Sigma + 2 m_{i}r \right) \sin \theta d r d \theta
\quad\sum_{n=0}^N (-1)^nn!(n+3)\Gamma(\frac{1}{a}-n)=\frac{\Gamma(\frac{1}{a})(5a+3)}{(2a+1)(a+1)}&+\frac{(-1)^{N+1}(N+1)!\Gamma(\frac{1}{a}-(N+1))((N+6)a+(N+4))(a(N+1)-1)}{(2a+1)(a+1)}.
K^\prime+\bar K^\prime=\frac{1}{2}\int\limits_0^Td\tau\Biggl[\frac{m_1^2}{\mu_1(\tau)}+\mu_1(\tau)\left(1+\dot{\bf z}^2(\tau)\right)+\frac{m_2^2}{\mu_2(\tau)}+\mu_2(\tau)\left(1+\dot{\bar{\bf z}}^2(\tau)\right)\Biggr],
(|x-y| +|x-z|)^{3\beta}+ \Phi(x)^\beta \Phi(y)^\beta \Phi(z)^\beta & \gtrsim (|x-y|+|x-z|)^{3\beta}+ \Phi(x)^{3\beta}+\Phi(y)^{3\beta}+\Phi(z)^{3\beta},
\tilde{W}_{1}^{V C}( \beta , \alpha , y , \epsilon ) ={\frac{1}{1 2}}\left( \alpha -{\frac{1}{\alpha}}\right) \left( \ln{\frac{\mu}{\epsilon}}- \ln{\frac{2 \pi \mu \alpha}{\beta}}- \frac 1 2 \ln y - \frac 1 2 +{\frac{1}{2 y}}\right)
=\frac{\rho}{(2\pi RT)^{D/2}}e^{-\frac{(\vec{e})^{2}}{2RT}}(1+\frac{\vec{e}\vec{u}}{RT}+\frac{(\vec{e}\vec{u})^{2}}{2(RT)^{2}}-\frac{\vec{u}^{2}}{2RT}+...)
x_{1}^{3}+6sx_{1}+\sum\limits_{i=2}^{i=n}a_{i}x_{i}^{2}+2\left(\sum\limits_{i=2}^{i=n}a_{i}\right) sa_{i}\neq 0\text{pourtout}i\sum\limits_{i=2}^{i=n}a_{i}\neq 0
C(n,\tau)=\left(\theta_3^4(0,\tau)-\theta_4^4(0,\tau)-e^{i\pi n}\theta_2^4(0,\tau)\right)\left(\bar\theta_3^4(0,\tau)-\bar\theta_4^4(0,\tau)-e^{i\pi n}\bar\theta_2^4(0,\tau)\right)
<T(T(X,Z),A),W>=<(\nabla_{A}T)(X,Z),W>=\frac{1}{2}\sigma_{X,Z,W}<R_{A,JX}Z.W> &=<\alpha(A,W),\alpha(JX,Z)>+<\alpha(A,Z),\alpha(JX,W)> &+\frac{1}{2}<\alpha(A,X),\alpha(JZ,W)-\alpha(JW,Z)>,
|K_\alpha(x,y)| \lesssim \int_0^\infty \frac1{V(x,y,\sqrt t)}\frac t{\|x-y\|^2}e^{-cd(x,y)^2/t}\frac{dt}{t^{1-\alpha/2}}&\leqslant \frac1{\|x-y\|^2}\bigg(\int_0^{d(x,y)^2}+ \int_{d(x,y)^2}^\infty\bigg) \frac1{V(x,y,\sqrt t)}e^{-cd(x,y)^2/t}t^\alpha dt &=: I_1 +I_2.
\sum_{k=1}^{M}\left(\frac{2^{n_{k}}}{2^{n_{1}}+2^{n_{2}}+\cdots+2^{n_{M}}}\right)^{1+s}\leq \sum_{k=1}^{M}\frac{2^{n_{k}}}{2^{n_{1}}+2^{n_{2}}+\cdots+2^{n_{M}}}=1, \left|\left(\frac{2^{n_{1}}+2^{n_{2}}+\cdots+2^{n_{M}}}{N}\right)^{1+s}-1\right|\leq 1-(1-\epsilon)^{1+s},
\textrm{(i)}~u\phi_0 (x,u,v)\in N_\mu \big((Q,\circ_0 )\big),~\textrm{(ii)}~u[u\backslash (xv)]\in N_\mu \big((Q,\circ_0 )\big),~\textrm{(iii)}~u\phi_0 (x,u,v)=u[u\backslash (xv)]\Leftrightarrow
\delta_{\eta}q_{i}^{a}= \{q_{i}^{a}, \eta \phi \}= \eta \epsilon^{a b}q_{i}^{b}\ \ , \ \ \delta_{\eta}p_{i}^{a}= \{p_{i}^{a}, \eta \phi \}= \eta \epsilon^{a b}p_{i}^{b}\ \ , \ \ \delta_{\eta}\lambda = \dot{\eta}\ \ ,
\int_K\frac{\partial^2 R_Ku}{\partial x_1^2}\frac{\partial^2 v_h}{\partial x_1^2}\;\mathrm{d}x_1\mathrm{d}x_2=\frac{1}{2}\int_K(a_5+a_7)\frac{\partial^3v_h}{\partial x_1^3}\;\mathrm{d}x_1\mathrm{d}x_2.
\begin{cases}\exists \T_{0}>0,\mbox{such that all the solutions}w \mbox{of}w'' + A w = 0 \mbox{satisfy}\int_0^T |C_{n+1 n}w^{\prime}|^2 dt \geq \gamma(T)e_1(W)(0)\,, \forall \T>T_{0}\,,\end{cases}
\frac{1}{|{\bf x}-{\bf x'}|}=\sum_{\ell =0}^{\infty}\sum_{m=-\ell}^{\ell}\frac{4\pi}{2\ell +1}\frac{R_<^{\ell}}{R_>^{\ell +1}}\ \bar Y^{(\ell)}_m (\theta' ,\phi') \Y^{(\ell)}_m (\theta ,\phi) \ .
\frac{d^a}{dX^a}\sum_{j=0}^{n'-1}T_{\o_j}(1,R(X))X^j\Big|_{X=\eta}=\frac{d^a}{dX^a}\sum_{j=0}^{n'-1}T_{\o'_j}(1,R(X))X^j\Big|_{X=\eta}
(\sigma^{n})^{2}\delta Z_{n}\Delta t=\mathbb{E}_{t_{n}}^{TR,\sigma^{n},X^{n}}\left[ \delta Y_{n+1}\Delta B_{n+1}\right] +\tilde{R}_{z}^{n,\sigma^{n}}+R_{z}^{T,n,\sigma^{n}}.
\wp^{- 1}= \sigma{_{2}F_{1}}\left({\frac{1}{2}}, \,{\frac{1}{3}}; \,{\frac{4}{3}}; \, - 4 \sigma^{3}\right) -{\frac{4^{1 / 3}}{3}}B \left({\frac{1}{3}}, \,{\frac{1}{6}}\right)
x_{n+1}=x_{n}-\frac{f(x_{n})}{f^{\prime}(x_{n})}=x_{n}-\frac{x_{n}^{2}-a}{2x_{n}}=\frac{1}{2}(2x_{n}-(x_{n}-\frac{a}{x_{n}}))=\frac{1}{2}(x_{n}+\frac{a}{x_{n}})
S_{a,b,r}^{\alpha_{s},\beta_{s},r_{s},p_{s}}\left( p,1\right) =\left( r!\binom{a+b}{r}\right)^{p}\prod_{s=2}^{L}\left( r_{s}!\binom{\alpha_{s}+\beta_{s}}{r_{s}}\right)^{p_{s}}-S_{a,b,r}^{\alpha_{s},\beta_{s},r_{s},p_{s}}\left( p,0\right) ,
S^{\mathrm{h e}}= \frac{\mu^{4}}{2 \kappa_{1 0}^{2}}\int d^{6}x \sqrt{- g^{\mathrm{h e}}}\, e^{- 2 \varphi^{\mathrm{h e}}}\, \Bigg [ R ( g^{\mathrm{h e}}) + 4 \partial_{\mu}\varphi^{\mathrm{h e}}\partial^{\mu}\varphi^{\mathrm{h e}}
\left\Vert u(t) - e^{\gamma_*t}\sum_{i\in I_1}\textup{Re}\left(w_ie^{\mathbf{i}\beta_i t}\right)\varphi_i- e^{\gamma_*t}\sum_{i\in I_2}(tc_{i,3}+c_{i,4}) \varphi_i\right\Vert_{H^1}= O(e^{(\gamma_*-\varepsilon ) t}).
\dot{\iota}_t(x)=X_{E_t}(\iota_t(x)) &\dot{\phi}_t=\left(\vartheta(X_{E_t})-E_t\right)(\iota_t(p)) &\frac{\rm{d}}{\rm{d}t}\left(\iota_t^*\left(\frac{\partial D}{\partial U}\right)\mu_t\right)=0,
\begin{array}{cl}T_{--}^f ={1\over2}\,\partial_- f_i\partial_- f_i=-\sqrt{\kappa}\,\partial_-^{\,2}\Omega={\kappa\over4}\,\frac{\Delta}{x^-_s\left(x^-_s+\Delta\right)}\,\delta\left(x^--x^-_s\right) \ =-{{\kappa\lambda}\over4}\left(1-e^{-\frac{4M}{\kappa\lambda}}\right){\left(-\lambda\left(x^-_s+\Delta\right)\right)}^{-1}\delta\left(x^--x^-_s\right)\;.\end{array}
{\cal R}={\frac{p}{L^{2}}}\left[ ( p + 1 ) ( \bar{P}_{\phi}^{2}- \bar{E}^{2}) \left({\frac{L}{a}}\right)^{2 ( p + 1 )}+ ( p - 1 ) \bar{E}^{2}\left({\frac{L}{a}}\right)^{2 p}\right] .
{\rm\mbox{(MOG)}}_{\lambda}\begin{cases}\dot u_{\lambda}(t) + \bigg(N_K(u_{\lambda}(t)) + \mbox{{\rm Conv}}\left\{\nabla f_{i,\lambda}(u_{\lambda}(t)) \right\}\bigg)^{0}= 0, u_{\lambda}(0)= u_0.\end{cases}
\lim_{n \rightarrow \infty}W_{n+1}(x) =\lim_{n \rightarrow \infty}\lim_{m \rightarrow \infty}\sum_{j=1}^{l_m}\sum_{k=0}^n G^x_k (w_{m,j-1})[W_{n-k}(w_{m,j}) - W_{n-k}(w_{m,j-1})].
\begin{array}{c}{\Bigl ( a^{+}( P^{0}- P_{2}) + a^{-}\left( i P_{1}- P_{3}\right) \Bigr ) | \psi \rangle = 0 ,}\\{\Bigl ( a^{-}( P^{0}+ P_{2}) - a^{+}\left( i P_{1}+ P_{3}\right) \Bigr ) | \psi \rangle = 0 .}\\ \end{array}
I_2= \sum_{1\leq i,j\leq n}a^{ij}\bigg\{h' u_{ij}+\big[h''-(1+\log|Dw|^2)h'^2\big]u_iu_j+\big[g''-(1+\log|Dw|^2)g'^2\big]\gamma^i\gamma^j &\qquad\qquad\quad-2(1+\log|Dw|^2)h'g'\gamma^iu_j+g'(\gamma^i)_j\bigg\}.
\frac{d}{dt}\frac{W_2^2(\mu_t,\nu)}{2}= \lim_{s\downarrow 0}\frac{W_2^2(\mu_{t},\nu) - W_2^2 ( \mu_{t-s}, \nu )}{2s}\le- \int_X \langle \nabla u , \nabla \varphi_t \rangle \,d\mu_t .
\varepsilon(X)= \begin{pmatrix}c b' a d \end{pmatrix},\ \varepsilon(Y)= \begin{pmatrix}a' d c b \end{pmatrix},\ \varepsilon(W_1)= \begin{pmatrix}a b b' a' \end{pmatrix},\ \varepsilon(W_2)= \begin{pmatrix}b a b' a' \end{pmatrix}.
|\mathcal{B}_{i,i-l}|=|\{(i-1,i-l,t,p):(i-1,i-l,t,p)\in \mathcal{I}'_{k-1}\}|,\ \ \ \ \ begin{align*}.2cm] |\mathcal{B}_{i-l,i}|=|\{(i-l,i-1,t,p):(i-l,i-1,t,p)\in \mathcal{I}'_{k-1}\}|.
Z = \int_{0}^{\infty}\frac{d s}{s}T r_{N S - R}\left[ \left( \frac{1 + ( - 1 )^{F}}{2}\right) \left( \frac{e + h_{1}+ h_{2}+ h_{3}}{4}\right) \mathrm{e}^{- 2 \pi s ( L_{0}- a )}\right]
\frac{\partial I_{s}}{\partial s}=\left| \nabla I_{s}\right| \left( \frac{\nabla I_{s}}{\left| \nabla I_{s}\right|}\right) =\Delta I_{s}-\frac{H\left( \nabla I_{s},\nabla I_{s}\right)}{\left| \nabla I_{s}\right|^{2}}
n P_{\mathcal{I}}E P_{\mathcal{J}}= \sum_{l = 1}^n \sum_{i \in \mathcal{I}}\sum_{j \in \mathcal{J}}(\langle X_l, u_i \rangle \langle X_l, u_j \rangle - \delta_{ij}\sqrt{\lambda_i \lambda_j}) u_i \otimes u_j=: \sum_{l = 1}^n Z_l.
|e(\mathcal{C}(G))| = \dfrac{(2^{k}+1)(2^{k}-1)(2^{k}-2)}{2}+\dfrac{2^{k-1}(2^{k}+1)(2^{k}-2)(2^{k}-3)}{2}+\dfrac{2^{k-1}(2^{k}-1)2^{k}(2^{k}-1)}{2}= \dfrac{2^{4k}-2\cdot2^{3k}-2\cdot2^{2k}+3\cdot2^{k}+2}{2}.
\chi_{\max}(f):=\lim_{t\to\infty}\frac{1}{t}\max_{z\in X_{f}}\log\|DT_{f}^{t}(z)\|,\chi_{\min}(f):=\lim_{t\to\infty}\frac{1}{t}\min_{z\in X_{f}}\log\|DT_{f}^{t}(z)\|.
\Pi^{(\mathfrak{c},(\tau,Z))}[t_0,x_0]=\mathbb{E}\left[\int_{t_0}^{\tau_S}e^{-\delta r}R(X^{t_0,x_0,(\tau,Z)}_r)dr+\sum_{j\geq 1}e^{- \delta \tau_{j}}c (\tau_{j}^-, z_j ) \cdot 1_{\{\tau_{j}\leq \tau_S \}}\right] ,\; \forall (t_0,x_0)\in [0,T]\times S,\forall (\tau,Z)\in U,
G_{EG-APC}={{\sum\limits_{l = 1}^N{{{\bigg{|}{\frac{{{{\bar \lambda}^{(0)}}}}{{\lambda_l^{(0)}}}{\lambda_l}}\bigg{|}^2}}}}}\big{/}\left({{\left({{\bf{w}}_t^H{{\bf{w}}_t}}\right)\left({{\bf{w}}_r^H{{\bf{w}}_r}}\right)}}\right),
\chi\left(\left(\bigsqcup_{a \in \mathcal{A}}^{P_A}G_a \right)[\mathcal{S}]\right) = \chi\left(\bigsqcup_{a \in \mathcal{A}}^{P_A}G_a[\mathcal{S}_a] \right) = \max_{a \in \mathcal{A}}\chi\left(G_a[\mathcal{S}_a]\right) = \max_{a \in \mathcal{A}}\omega\left(G_a[\mathcal{S}_a]\right),
F [ a ] = \int d^{9}x ~ \left\{\frac{1}{( 2 \pi )^{7}( \alpha^{\prime})^{5}g_{s}}\sum_{b}\sqrt{1 + \left( 2 \pi \alpha^{\prime}e_{b}\vec{\nabla}a ( \vec{x}) \right)^{2}}+ \sum_{b , c}V_{\mathrm{e f f}}\Bigl [ ( e_{b}- e_{c}) a ( \vec{x}) \Bigr ] + \dots \right\}
H_{g_0}(u)=\left(\frac{n-2k}{4k}\right)^ku^{\frac{2kn}{n-2k}}\sigma_k(A_{g})-\left(\begin{array}{c}n k\end{array}\right)\left(\frac{n-2k}{4k}\right)^k u^{\frac{2kn}{n-2k}}.
Y_{t}Y_{t+10}= Y_{t+3}Y_{t+7}Z_{t}Z_{t+1}Z_{t+2}+ Y_{t+2}^2Y_{t+5}^2Y_{t+8}^2 \quad Z_{t}Z_{t+1}Z_{t+2}Z_{t+3}= Y_{t+2}Y_{t+3}Y_{t+5}Y_{t+6}Y_{t+8}Y_{t+9}+ Y_{t+1}Y_{t+4}Y_{t+7}Y_{t+10}
P(|X|\ge t)=P(exp(\frac{X^{2}}{K_{2}^{2}})\ge exp(\frac{t^{2}}{K_{2}^{2}}))\le\frac{E[exp(X^{2}/K_{2}^{2})]}{exp(\frac{t^{2}}{K_{2}^{2}})}\le2exp(-\frac{t^{2}}{K_{2}^{2}}).
_{2}F_{1}\left({\frac{1}{3}},{\frac{5}{6}}; 1 ;{- \frac{2}{2 5}}\right) = \frac{\sqrt [ [object Object] ] ]{5}}{\sqrt{3}}{\,}_{2}F_{1}\left({\frac{1}{6}},{\frac{2}{3}}; 1 ;{- \frac{2}{2 5}}\right) ,
|u^{(n)}(1)|^2 \lesssim \|u \|^2_{H^{m_d}(\delta, 1)}\lesssim \| u\|^2_{L^2(0,1)}+ \sum_{j=1}^{m_d}\| (\cdot)^{j-1}u^{(j)}\|^2_{L^2(0,1)}
* R *^{\mu \nu}{}{}_{\rho \sigma}= - R^{\mu \nu}{}{}_{\rho \sigma}+ \delta_{\sigma}^{\mu}R_{\rho}^{\nu}+ \delta_{\rho}^{\mu}R_{\sigma}^{\nu}- \delta_{\sigma}^{\mu}R_{\rho}^{\nu}- \delta_{\rho}^{\nu}R_{\sigma}^{\mu}+ \frac{1}{2}R ( \delta_{\rho}^{\nu}\delta_{\sigma}^{\mu}- \delta_{\rho}^{\mu}\delta_{\sigma}^{\nu}) .
\sum_{n=0}^{\infty}\frac{\left(bc;q\right)_{n}S_{n}\left(x;q\right)z^{n}}{\left(c;q\right)_{n}}=\frac{\left(b^{-1},bcz;q\right)_{\infty}}{\left(c,z;q\right)_{\infty}}\sum_{n=0}^{\infty}\frac{\left(bc;q\right)_{n}S_{n}\left(xbz\right),}{\left(bcz;q\right)_{n}b^{n}}.
\sum_{k=0}^\infty r^k \Phi_k(x,x') =\pi^{-d/2}(1-r^2)^{-d/2}e^{-\frac{1}{2}\frac{1+r^2}{1-r^2}(|x|^2+|x'|^2)+\frac{2r x \cdot x'}{1-r^2}},
L(\vert y-\zeta_{i,n}^{\prime}\vert^{-\nu}) =-\Bigl(\nu\,(1-\nu)-\bigl(\varepsilon_n^2\,\lambda+5(V_1+V_2)^4\bigr)\,\vert y-\zeta_{i,n}^{\prime}\vert^2\Bigr)\,\vert y-\zeta_{i,n}^{\prime}\vert^{-(2+\nu)}&\leq -\frac{\nu\,(1-\nu)}{2}\,\vert y-\zeta_{i,n}^{\prime}\vert^{-(2+\nu)}
\frac{\partial v_{i}}{\partial t}+\sum_{j=1}^{3}\frac{\partial v_{i}}{\partial x_{j}}v_{j}=-\frac{1}{\rho}\frac{\partial p}{\partial x_{i}}+\nu\sum_{j=1}^{3}\frac{\partial^{2}v_{i}}{\partial x_{j}^{2}}+f_{i}(x,t).
d_{u}(e_{wu^{k+1}}) = e_{wu^{k+2}}+ e_{wu^{k}}- 2e_{wu^{k+1}}& = e_{wu^{k+2}}+ (\epsilon_{k-1}+ a_{k-1}e_{w}) - 2(\epsilon_{k}+ a_{k}e_{w}) & = e_{wu^{k+2}}- (2a_{k}- a_{k-1})e_{w}+ \epsilon_{k-1}- 2\epsilon_{k}& = \epsilon_{k+1}+ \epsilon_{k-1}- 2\epsilon_{k}
(\widehat{w},\widehat{z},\widehat{\zeta})=\Psi(w,z,\zeta):=\left(w+\widehat{h}^2,z_1+\widehat{f}_{1}^1,\ldots, z_{n-1}+\widehat{f}_{n-1}^1, \zeta+\widehat{g}^00\right)
A_{p,q,k}^{m,n}(z) := \frac{z^{b_k}\prod_{j=1, j\neq k}^{m}\Gamma(b_j - b_k ) \prod_{j=1}^n \Gamma( 1 + b_k -a_j )}{\prod_{j=m+1}^{q}\Gamma(1 + b_k - b_{j}) \prod_{j=n+1}^{p}\Gamma(a_{j}- b_k)}.
F_t = \big\{\left(k+1, (t+1)k+1\right),(k+2, (t+1)k+2), \ldots, (2k, (t+2)k) \big\}\ \ , \ \ \forall t\in[n-2]\,, H_s = \big\{\left(1, (s+1)k+1\right),\left(2, (s+1)k+2\right),\ldots, \left(k, (s+2)k\right) \big\}\ \ , \ \ \forall s\in[n-2]\,.
\int_{S^n}d(y,x)~H_t(y,x)~d\sigma(x) \leq \Omega_n^{\frac 12}\begin{pmatrix}\int_{S^n}d(y,x)^2~H_t(y,x)~d\sigma(x)\end{pmatrix}^{\frac 12}&=\Omega_n^{\frac 12}\sqrt t
P_{+}= \int_{- L}^{L}d x^{-}{\cal H}( x ) = \sum_{n}p_{n +}\left( b_{n}^{a \dagger}b_{n}^{a}- d_{n}^{a}d_{n}^{a \dagger}\right) + 2 L \left( -{\frac{N m_{0}}{g_{0}}}\sigma +{\frac{N}{2 g_{0}}}\sigma^{2}\right) .
R_{M N}^{a b}= E_{M}^{\ \ \Lambda}E_{N}^{\ \ \Pi}\left\{\partial_{\Lambda}\Omega_{\Pi}^{\a b}+ \Omega_{\Lambda}^{\a c}\Omega_{\Pi c}^{\ \, b}- \left( - \right)^{\Lambda \Pi}\left( \Lambda \leftrightarrow \Pi \right) \right\}
(z_n^+)^{\lambda_n}\cdots(z_1^+)^{\lambda_1}v_0 =\left(\prod_{j=1}^n\frac{(-1)^{\frac{\lambda_j(\lambda_j+1)}{2}}}{\lambda_j!j^{\lambda_j}}\prod_{k=0}^{\lambda_j-1}\prod_{\ell=1}^{j-1}(k-\lambda_\ell-j+\ell+1-[k-\lambda_\ell ]_2)\right)\Omega_\lambda.
|v(t')|^2 - |v(t)|^2 = 2\sum_{i=1}^m \int_{(t,t']}\langle v_i^{\ast}(s), v(t') \rangle dA(s) + 2(v(t), h(t')-h(t)) & + |h(t')-h(t)|^2 - |v(t')-v(t) - (h(t')-h(t))|^2.
\overline{\lambda_{\epsilon}^{E(\frac{1}{\epsilon^{\alpha}})}}-\lambda^{E(\frac{1}{\epsilon^{\alpha}})}=\epsilon^{2}\sum_{j=1}^{m}\sum_{l=1}^{N}\nabla_{x}u^{(\frac{1}{\epsilon^{\alpha}})j}(z_{l}).{M}^{l}\nabla_{x}u^{(\frac{1}{\epsilon^{\alpha}})j}(z_{l})+\Theta(\epsilon^{(\frac{5}{2}-\alpha)}),
\begin{cases}Y(t)=\psi(t) + \int_t^T \overline{g}(t,s) ds - \int_t^T Z(t,s) dW(s), \ \t\in [0,T]; Y(t)=\psi(t), \ \t\in [T,T+K]; Z(t,s)=\eta(t,s), \ \ (t,s)\in [0,T+K]^{2}\setminus [0,T]^{2}, \end{cases}
\liminf_{s \nearrow s_0}\int_0^1 |\phi_{s_0}'|^2 \frac{P_s-P_{s_0}}{s-s_0}- \lambda_{1,2}(\sigma_{s_0}) |\phi_{s_0}|^2 \frac{Q_s-Q_{s_0}}{s-s_0}\,dt &\ge \int_0^1 |\phi_{s_0}'|^2 \dot P_{s_0}- \lambda_{1,2}(\sigma_{s_0}) |\phi_{s_0}|^2 \dot Q_{s_0}\,dt
S^{2}( p , q ) \, = \, f ( p , q ) \, = \, - ( p^{0}q_{0}- p^{1}q_{1}+ p^{2}q_{2}+ p^{3}q_{3})^{2}+ 4 ( p^{2}p^{3}- p^{0}q_{1}) ( p^{1}q_{0}+ q_{2}q_{3})
\left( \omega^{g_1 g_2, g_3}_{\alpha}\right) \,\left( g_3^* \omega^{g_1, g_2}_{\alpha}\right) \: = \:\left( \omega^{g_1, g_2 g_3}_{\alpha}\right) \,\left( \omega^{g_2, g_3}_{\alpha}\right)
ds^2=a^2(d\psi +p \cos \theta_1 d\phi_1 +q \cos \theta_2 d\phi_2)^2+ b^2 (d\theta_1^2 +\sin^2 \theta_1 d\phi_1^2) +c^2(d\theta_2^2 +\sin^2 \theta_2 d\phi_2^2),
\partial^{\mu}( \sqrt{- g}R_{\mu}) \, = \, \frac{1}{1 9 2 \pi^{2}}\left[ ( N_{c}^{2}- 1 ) - \frac{1}{3}\sum_{i}( - 1 + c_{i}^{( 0 )}) \right] \epsilon_{\mu \nu \lambda \delta}R^{\mu \nu \sigma \rho}R_{\sigma \rho}^{\lambda \delta}\; ,
H^i_{P}(R) \cong H^i_P(W) \otimes_W R \cong H^i_P(W) \otimes_W \left(W \otimes_A A[X_{t+1}, \ldots, X_n]\right) \cong H^i_P(W) \otimes_A A[X_{t+1}, \ldots, X_n] &= \begin{cases}A[X_1^{-1}, \ldots, X_t^{-1}, X_{t+1}, \ldots, X_n](-1,-1,\cdots,-1,0,\cdots,0) \mbox{if}i = t 0 \mbox{otherwise}.\end{cases}
\langle\alpha_x|\beta_y\rangle =\delta(x-y) \langle\alpha_x|\beta_{\pm \i \pi}\rangle =0 (\alpha_{\pm \i \pi}|\beta_x\rangle =0 \langle\alpha_{\pm \i \pi}|\beta_{\pm \i \pi}\rangle =1 \alpha_{\pm \i \pi}|\beta_{\mp \i \pi}\rangle =0
n(4(k+1)s)^{2s+3}\left(1+4p\right)^{16s^2}2^{6s}K^{22s}\theta^{4sk}\left(1\vee \frac{d}{K}\right)^{5s}&\cdot\sum_{g=0}^{\infty}\left( \frac{2(d\vee K)^3K^{9}\left(1+4p\right)^{8s}(4(k+1)s)^{6s}}{n}\right)^g.
{\mathbf O}{\mathbf R}{\mathbf O}^{- 1}\: = \: - 8 m +{\cal M}_{\nu}^{\mu}{\mathbf a}_{\nu}^{\dagger}{\mathbf a}^{\mu}+{\cal N}_{\nu}^{\mu}{\mathbf b}^{\dagger \nu}{\mathbf b}_{\mu}.
A_{0}(\rho_0,\rho_1) =\ \tau_1(1+\alpha\frac{\rho_0}{2})\rho_1\lambda_1^{-1}\langle\sin^2\theta h_0\rangle_{M_{\lambda_1}}+ \tau_1\alpha \frac{c_0\rho_0}{2}\rho_1\lambda_1^{-1}\langle\sin^2\theta\cos\theta h_0\rangle_{M_{\lambda_1}}- \tau_0\alpha \frac{c_1\rho_1}{2}\rho_0\langle\sin^2\theta h_0\rangle_{M_{\lambda_0}},
b_{d}( x , \beta ) = -{\frac{7 \pi^{2}r_{d}}{7 2 0 \beta_{l}^{4}}}+{\frac{r_{d}}{4 8 \beta_{l}^{2}}}\left[{\frac{1}{1 2}}R + \frac 1 2 ( \nabla w - w^{2}) + m^{2}\right] +{\frac{a_{d , 2}( x )}{1 6 \pi^{2}}}\ln{\frac{\mu \beta_{l}}{2 \pi}},
[ \bar{\epsilon}Q \ , \bar{Q}\epsilon^{\prime}]^{*}= \bar{\epsilon}( \sl + \Gamma^{1 1}Z ){\frac{\sl}{2{\bf P}^{2}}}( \sl + \Gamma^{1 1}Z ) \epsilon^{\prime}= \bar{\epsilon}\Gamma^{\hat{m}}{\bf P}_{\hat{m}}\epsilon^{\prime}= \bar{\epsilon}( \sl + \Gamma^{1 1}Z ) \epsilon^{\prime}\ .
\textrm{Cov}_B[\gamma X,Y] =\gamma \textrm{Cov}_B[X,Y] = \textrm{Var}_B[Y] \gamma \beta(X|Y)= \frac{\textrm{Var}_B[Y]}{\textrm{Var}[Y]}\gamma \textrm{Cov}[X,Y] &= \frac{\textrm{Var}_B[Y]}{\textrm{Var}[Y]}\textrm{Cov}[\gamma X,Y]=0.
\left[1-\frac{1}{K+1}\left(1-(1-\gamma)^{K+1}\right)\right]^M\leq\left[1-\frac{1}{1/\gamma + K}\right]^M \Longleftrightarrow \frac{1}{K+1}\left(1-(1-\gamma)^{K+1}\right)\geq \frac{1}{1/\gamma + K}\Longleftrightarrow h(\gamma, K) = 1/\gamma-(1-\gamma)^{K+1}/\gamma - K(1-\gamma)^{K+1}- 1\geq 0
\sum_{m=\ell_2}^{\infty}\sum_{k=\max\{0,\ell_1+\ell_2-m\}}^{\ell_1}A_{m,k}=\sum_{m=\ell_2}^{\ell_1+\ell_2}\sum_{k=\ell_1+\ell_2-m}^{\ell_1}A_{m,k}+\sum_{m=\ell_1+\ell_2+1}^{\infty}\sum_{k=0}^{\ell_1}A_{m,k}.
K_0 = \mbox{span}\{1, x^1, x^2, x^3\}&K_1 = \mbox{span}\left\{\frac{\partial}{\partial x^1}, \frac{\partial}{\partial x^2}, \frac{\partial}{\partial x^3}, x\times \frac{\partial}{\partial x^1}, x\times \frac{\partial}{\partial x^2}, x\times \frac{\partial}{\partial x^3}\right \}.
r_{0,n}:= \frac{1}{n}\log |{\cal M}_0^{(n)}| + \frac{|{\cal X}||{\cal Y}| \log (n+1)}{n}+ (\alpha_n + \beta_n), r_{1,n}:= \frac{1}{n}\log |{\cal M}_1^{(n)}| + \frac{1}{n}+ 2^{- n \beta_n}\log |{\cal X}|, r_{2,n}:= \frac{1}{n}\log |{\cal M}_2^{(n)}| + \frac{1}{n}+ 2^{- n \beta_n}\log |{\cal Y}|
-\tfrac{1}{2}\alpha^{2}\operatorname{arccosh}\left({\frac{\zeta}{\alpha}}\right) +\tfrac{1}{2}\zeta\left({\zeta^{2}-\alpha^{2}}\right)^{1/2}=\left\vert \operatorname{Im}{\left\{{\sigma E\left({x;\sigma^{-1}}\right)}\right\}}\right\vert .
\frac{6 \ddot{a}}{a}+ \dot{\phi}^{2}= \frac{2 \ddot{a}}{a}+ \frac{4 ( \dot{a}^{2}+ k )}{a^{2}}= - \frac{1}{\alpha}\left( \ddot{\phi}+ \frac{3 \dot{a}\dot{\phi}}{a}\right) = \Lambda e^{\alpha \phi}\; .
\lambda \left\{x \in \mathbb{I}: q_n^{-2}(x) \geq \frac{\delta}{\beta^{n}}\right\}= \lambda \left\{x \in \mathbb{I}: q_n^{-2t}(x)\geq \left(\frac{\delta}{\beta^n}\right)^t\right\}\leq\frac{\beta^{tn}\mathrm{E}(q_n^{-2t})}{\delta^t}.
\mathrm{Ric}_{\tilde{g}}(\tilde{\nabla}\Phi,\tilde{\nu}) =o(\rho^{-3}), \tilde{g}(\tilde{\nabla}\Delta_{\tilde{g}}\Phi,\tilde{\nu}) =\frac{8(1-\alpha)}{\rho^3}+ O_1(\rho^{-4}),