harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
^0(\mathcal{T}_h) := \{\boldsymbol{\tau}\in (L^2(T))^2:\forall \, T \in \mathcal{T}_h \exists \, a,b,c \in \mathbb{R}\forall \, \boldsymbol{x}\in T, &\boldsymbol{\tau}(\boldsymbol{x}) = (a,b)^T + c \, \boldsymbol{x}[\boldsymbol{\tau}]_E \cdot n_E = 0 \,\, \forall E \},
\delta_h u(x)\,\eta(x)^p-\delta_h u(y)\,\eta(y)^p =\frac{\Big(\delta_h u(x)-\delta_h u(y)\Big)}{2}\,\Big(\eta(x)^p+\eta(y)^p\Big) &+\frac{\Big(\delta_h u(x)+\delta_h u(y)\Big)}{2}\,(\eta(x)^p-\eta(y)^p).
\leq \; \Bigg [ \int_{\Lambda}\prod_{a = 1}^{N}\prod_{i = 1}^{n_{a}}d^{2}x_{i}^{( a )}d^{2}y_{i}^{( a )}\prod_{b = 1}^{N}\Big | A ( b , b ) \Big |^{\frac{\pi}{\pi + g N}p}\Bigg ]^{\frac{1}{p}}
\pi(X)-m \geqslant T_{p_m}\cdot(1-\frac{1}{p_{m+1}})\cdot(1-\frac{1}{p_{m+2}})\cdots(1-\frac{1}{p_{m+K}}) &=\prod_{i=1}^{m}(p_i-1)\cdot\frac{\prod_{i=1}^{m+K}(1-p_{i}^{-1})}{\prod_{i=1}^{m}(1-p_{i}^{-1})}&=\prod_{i=1}^{m}p_i\cdot\prod_{i=1}^{m+K}(1-p_{i}^{-1})
\langle x^{+ i}x^{- j}\rangle ={\frac{\delta^{i j}}{\omega^{2}+ r^{2}}}+{\frac{4 ( v^{i}v^{j}) + \mathrm{c o n s t}\ \delta^{i j}v^{2}}{( \omega^{2}+ r^{2})^{3}}}.
\frac{\nu_y\nu_z^2-q^{-2}\nu_z\nu_y\nu_z}{q-q^{-1}}= q^{-5}\nu_z \frac{q\nu_y \nu_z-q^{-1}\nu_z\nu_y}{q-q^{-1}}+ q^{-3}(q^2-q^{-2})\nu_z.
\frac{27i}{20}\int \!\! d^4x \,\,\, \eta_{{\mathbf m}{\mathbf i}}[D^{\mathbf m}{}_{{\mathbf j}}{\bar D}_{{\mathbf k}{\mathbf l}}+{\bar D}^{\mathbf m}{}_{{\mathbf j}}D_{{\mathbf k}{\mathbf l}}] T^{{\mathbf i}{\mathbf j}{\mathbf k}{\mathbf l}}|.
\left[ 1 \, + \,{\frac{M_{*}^{N - 2}}{c_{1}\, M_{*}^{N - 2}+ c_{2}\, \nabla^{N - 2}}}\,{\frac{1}{( r_{c}\, \nabla )^{2}}}\right] \, R \, = \, 8 \pi G_{N}\, T_{\mu}^{\mu}\, .
x^{u+z}e^{w-2z\over2}\Big({u+z\over uz}e^{v\over2}+{p-v\over z}s(v)\Big)+ \cr+ x^{u}\Big(ns(v)s(w)-{u+z\over uz}e^{v+w\over2}-{m-w\over u}s(w)e^{v\over2}-{p-v\over z}s(v) e^{w\over2}\Big)+ \cr- x^{z}{u+z\over uz}e^{v+w-2z\over2}+ \cr+{u+z\over uz}e^{v+w\over2}+{m-w\over u}s(w) e^{v\over2}=0 \ .
1 2 d s^{2}= \frac{\Delta + b^{2}\sin^{2}\theta}{\Sigma}\left( d t - \omega d \varphi \right)^{2}- \Sigma \left( \frac{d r^{2}}{\Delta}+ d \theta^{2}+ \frac{\Delta \sin^{2}\theta}{\Delta + b^{2}\sin^{2}\theta}d \varphi^{2}\right) .
a_{\beta , 2}={\frac{\pi}{3 \alpha}}( 1 - \alpha^{2}) \int_{\Sigma}\left[ \frac 1 6 R +{\frac{1 + \alpha^{2}}{6 0 \alpha^{2}}}( 2 R_{\mu \nu \lambda \rho}n_{i}^{\mu}n_{i}^{\lambda}n_{j}^{\nu}n_{j}^{\rho}- R_{\mu \nu}n_{i}^{\mu}n_{i}^{\nu}) \right] .
\widetilde{K}_n^{[s, \xi^{(n)}]}(z_1, z_2) = \frac{1}{2 n^2}K_n^{[s, \xi^{(n)}]}\Big( 1 - \frac{z_1}{2 n^2}, 1 - \frac{z_2}{2 n^2}\Big) = \frac{(z_1 z_2)^{\frac{s}{2}}}{\left| \prod_{i=1}^m (z_1 - w_i) (z_2 - w_i )\right|}\cdot S_n(z_1, z_2),
\sum_{\sigma\in\mathcal{N}}\sum_{j=1}^{m}\|P_{S_{\sigma}^{\alpha}}a_j^{\sigma}\|^2 \leq \sum_{\sigma\in\mathcal{N}}\sum_{j=1}^{m}\|P_{S_{\sigma}^{\beta}}a_j^{\sigma}\|^2 \quad\forall \; \alpha \in \mathcal{Q}.
\sum_{i=0,1}\big| T_{k,j;n}^{\mu,i}( \mathfrak{m}, E)(t,x, \zeta) + \hat{\zeta}\times T_{k,j;n}^{\mu,i}( \mathfrak{m}, B)(t,x, \zeta)\big|+ \big|\widetilde{T}_{k,j;n}^{bil;\mu,0}( \mathfrak{m}, E)(t,x, \zeta )+ \hat{\zeta}\times \widetilde{T}_{k,j;n}^{bil;\mu,0}(\mathfrak{m}, B )(t,x, \zeta)\big|
\log \lambda_{g}^{\left( p,q\right)}\left( f\right) =\underset{\sigma\rightarrow +\infty}{\underline{\lim}}\left[ \log^{\left[ p+1\right]}M_{g}^{-1}\left( \sigma \right) -\log^{\left[ q+1\right]}M_{f}^{-1}(\sigma )\right] .
\widetilde{\mathfrak{D}_I}:=\mathfrak{D}_{I}\prod_{i \notin I,\, j \in I\atop i < j}\big(1+t \beta-t^2 q_{ij}\big) = \prod_{b \in I}^{\nearrow}\Bigg(\prod_{a \notin I \atop a < b}^{\searrow}h_{ba}\Bigg) \prod_{a \in I}^{\nearrow}\Bigg((1+z_a) \prod_{b \notin I \atop a < b}^{\searrow}h_{ab}\Bigg).
\frac{1}{16\pi}\int_{B_{e}}\left(2\Lambda - F^{2}\right) \sqrt{g}d^{4}x = \beta_{RN}\Lambda \left[ \frac{r_{s}^{3}- r_{h}^{3}}{12}\right] \,+\, \frac{q^{2}}{4}\beta_{RN}\left( \frac{1}{r_{h}}- \frac{1}{r_{s}}\right)
\mathcal{L}_{j+1}(x) =\mathcal{P}_{j+1}(x)-\alpha_{j+1}\mathcal{P}_{j}(x), j=n,n-1,n-2, \mathcal{P}_{j+1}(x) = \rho_j(x-c_j)\mathcal{P}_j(x)-d_j (x^2+\omega^2)\mathcal{P}_{j-1}(x), j=n,n-1.
\frac{d^\ell_x\lambda^c_{x^u}(\tau'(\ell)+k)}{d^\ell_y\lambda^c_{y^u}(\tau'(\ell))}=\frac{d^\ell_x}{d^\ell_y}\times\frac{\lambda^c_{x^u}(\tau'(\ell))}{\lambda^c_{y^u}(\tau'(\ell))}\times \lambda^c_{x^u_{\tau'(\ell)}}(k).
\hat{\Phi}= \hat{B}^{- 1 / 2}\hat{\varphi}\hat{B}^{- 1 / 2}= \hat{B}^{- 1 / 2}\left( \begin{matrix}{\hat{\varphi}^{R}}&{0}\\{0}&{\hat{\varphi}^{L}}\\ \end{matrix}\right) \hat{B}^{- 1 / 2}
\mathcal{A}=\begin{pmatrix}A_{11}A_{12}0_{d\times d}0_{d\times d}0_{d\times d}0_{d\times d}L_{12}^T L_{22}^T 0_{d\times d}0_{d\times d}L_{11}^T L_{21}^T A_{41}A_{42}0_{d\times d}0_{d\times d}\end{pmatrix}
N^{-1}\psi(R(2^{2n_1+1}))N=\exp A_x \equiv \exp \left( \begin{matrix}0 \mu_x 0 0 \end{matrix}\right) \equiv \left( \begin{matrix}1 \mu_x 0 1 \end{matrix}\right) \pmod{2^{U-n_1-2}}.
\sum_{j=0}^n\binom{j}{t}\binom{n-j}{r-t}W_{\mathcal{C}}^j=q^{k-r}\sum_{i=0}^t(-1)^i(q-1)^{t-i}\sum_{j=0}^r\binom{n-j}{r-j}\binom{j}{i}\binom{r-j}{t-i}W_{\mathcal{C}^{\perp}}^j,0\leq t\leq r\leq n
\bigl\{(\sigma_1,\sigma_2)\mid \sigma_1(I_1)=I_1,\,\sigma_1(K_1)=K_1,\,\sigma_2(I_2)=I_2,\, \sigma_2(K_2)=K_2,\, (\sigma_2\circ \mu)_{|\bar K_1}=(\mu\circ \sigma_1)_{|\bar K_1}\bigr\}\,.
A = \begin{pmatrix}\!0 \!0 \!\binom{3k + 1}{k}t^{6k + 3}\!\binom{3k + 1}{k}t^{3k + 1}\!0 \!0 \!0 \!\binom{3k + 1}{k}t^{5k + 2}\!0\end{pmatrix}=\binom{3k + 1}{k}\!\begin{pmatrix}\!0 \!0 \!t^{6k + 3}\!t^{3k + 1}\!0 \!0 \!0 \!t^{5k + 2}\!0\end{pmatrix}.
\frac{2p}{2-p}\log (\frac{\left \|f\right\|_{L^p(O_N^+)}}{\left \|\widehat{f}\right \|_{\ell^{p'}(\widehat{O_N^+})}}) \geq \frac{2p}{2-p}\frac{p-2}{p}\log(\sup_{k\geq t}\frac{C(1+k)}{n_k^{\frac{1}{2}}}) &=-2 \log (\sup_{k \geq t}\frac{C(1+k)}{n_{k}^{\frac{1}{2}}})=2 \inf_{k \geq t}\log (\frac{n_{k}^{\frac{1}{2}}}{C(1+ k)}).
\aligned d_1(x)d_1(x)^{tr}=s^2 (x_3^2+x_4^2),d_2(x)d_2(x)^{tr}=s^2(x_1^2+x_2^2). &\tau d_1(x)Jg_1(x)^{tr}=0=\tau d_2(x)Jg_2(x)^{tr}.\endaligned
\mathcal{I}_{\mathrm{D}}\left(0, \delta_2\right)=\sup_{\pi \in \Pi\left(\mu_1, \mu_2\right)}\inf_{\lambda_2 \in \mathbb{R}_{+}}\left[\lambda_2 \delta_2+\int_{\mathcal{V}}g_{\lambda, 2}\, d \pi\right]= \sup_{\pi \in \Pi(\mu_1, \mu_2)}\inf_{\lambda \in \mathbb{R}^2_{+}}\left[ \langle \lambda , \left( 0, \delta_2 \right) \rangle + \int_{\mathcal{V}}g_{\lambda}\, d \pi\right] .
H^s(\mathbb{R})\!=\!\{\psi_i \! \in \! L^2(\mathbb{R})\! : \! \frac{|\psi_i(x_i)-\psi_i(y_i)|}{|x_i-y_i|^{\frac{1}{2}+s}}\! \in \! L^2(\mathbb{R}\times \mathbb{R})\}
\|u_x(t)\|_{L^2}^2 + \int_0^t\|u_{xxx}(s)\|_{L^2}^2\,ds \leq \|u_x(0)\|_{L^2}^2 + \int_0^t\|u_x\|_{L^2}^2\,ds &\leq \|u_x(0)\|_{L^2}^2 + \int_0^t e^s\|u_x(0)\|_{L^2}^2\,ds = e^t\|u_x(0)\|_{L^2}^2.
v_t-d_2\Delta v+\chi_2 \nabla v \cdot \nabla w & \geq v\left\{b_0-(b_2-\frac{\chi_2}{d_3}k)v-(b_1+|\Omega|(b_3)_+)\overline{u}-\left(|\Omega|(b_4)_++l\frac{\chi_2}{d_3}\right)\overline{v}+|\Omega|(b_4)_-\underline{v}\right\}&-v\left\{\left(b_1+|\Omega|((b_3)_-+|b_4|)+l\frac{\chi_2}{d_3}+2k\frac{\chi_{1}}{d_{3}}\right)\epsilon\right\},
\mathbf{f}_{1/(1+t),1}\boxtimes\mathbf{f}_{1/(1+s),\rho}=\mathbf{f}_{1/(1+s+t),\rho}, & \mathbf{b}_{1/(1+t),1}\boxtimes\mathbf{b}_{1/(1+s),\rho}=\mathbf{b}_{1/(1+s+t),\rho}
\sqrt{q}(r')^{-2/q}=(q \cdot \left(\frac{1}{(r')^2}\right)^{2/q})^{1/2}\leq (q \cdot \left(\frac{\log(1/\gamma)}{2}\right)^{2/q})^{1/2}=(x_0 \cdot a_0^{2/x_0})^{1/2},
\bar K_1^{(n)}(x,y) :=\frac{1}{4(n+1)^2}\sum_{j,j'=-n-1}^n \Phi\left(\frac{j}{n+1}, \frac{j'}{n+1}\right) \cos\left(\frac{\pi j(y-x)}{n+1}\right) \cos\left(\frac{\pi j'(y-x)}{n+1}\right),
{\cal L}_I = \frac{2}{9}\left[ \left\{\phantom{|}^{\mu\nu}_{ab}\right\}\Omega_\mu{}^{acde}\Omega_\nu{}^{bcde}- \frac{1}{48}\left\{\phantom{|}^{\mu\nu\alpha\beta\gamma}_{abcde}\right\}\Omega_\mu{}^{abcd}E_{\nu\alpha\beta\gamma}{}^e \right]
u(x)=\left(\int_{\partial\Omega}y^\perp\cdot\big(T(u,p)\nu\big)\,d\sigma_y+\int_\Omega y^\perp\cdot f\,dy\right)\frac{x^\perp}{4\pi|x|^2}+(1+|a|^{-1})\,o(|x|^{-1}),
\tilde{m}_{2}={\frac{2 \int d \tilde{p}( \Sigma ( \tilde{p}) )^{2}f ( \tilde{p}) -{\frac{\beta^{2}}{1 6 \pi}}( \tilde{\mu}^{2}( \varphi_{0}) - \tilde{m}_{0}^{2}) ( \int d \tilde{p}{\frac{\Sigma ( \tilde{p})}{f ( \tilde{p})}})^{2}}{\int d \tilde{p}( \Sigma ( \tilde{p}) )^{2}}},
\frac{\mbox{Vol}(\mathcal{E}_V)}{\mbox{Vol}(B_V)}= \prod_{v\in V}\eta_v^{1/2}= \Big(\prod_{j=0}^{m-1}\eta_j^{1/2}\Big)^{|I|\,m^{|I|-1}}\asymp a^{-\frac{1}{2}|I|\, m^{|I|}}e^{-\frac{|I|\,m^{|I|+2}}{8a^2}},
\|u\|_{H^3}:= \Big(\int_{M}(|\nabla \Delta u|_g^2 d\mu_g+|\nabla^2 u|_g^2+|\nabla u|_g^2+u^2)d\mu_g\Big)^{\frac{1}{2}}\approx \Big(\int_{M}(|\nabla^3 u|_g^2 d\mu_g+|\nabla^2 u|_g^2+|\nabla u|_g^2+u^2)d\mu_g\Big)^{\frac{1}{2}}, ~~u \in H^3(M,g).
\pi_{a,b}(x)=\left(\int_{a}^{b}m(x,\theta) \mathrm{d}x\right)^{-1}=\frac{e^{-\frac{2}{\sigma^{2}}\int_{a}^{x}f(y,\theta) \mathrm{d}y}}{\int_{a}^{b}e^{-\frac{2}{\sigma^{2}}\int_{a}^{x}f(y,\theta) \mathrm{d}y}\mathrm{d}x}.
\alpha = \pi^{n'}_{X'_0}\circ \bigl(\sum a_i \cdot Z_i\bigr) \circ \pi^n_{X_0}\, ,\alpha^{-1}= \pi^n_{X_0}\circ \bigl(\sum b_i\cdot{}^{\mathsf{t}}Z_i\bigr) \circ \pi^{n'}_{X'_0}\, .
y_i = \left( L (i-k) + y_k \right)_+ = \left( L (i-n+m-1) + \frac{U}{m}-\frac{L(m+1)}{2}+L \right)_+ = \left( L (i-n+m) + \frac{U}{m}-\frac{L(m+1)}{2}\right)_+
a_{i}^{I}({\bf r}) = - \epsilon^{i j}\nabla_{j}U^{I}({\bf r}) \, , \hspace{6 m m}A_{i}^{\mathrm{e x}}({\bf r}) = - \epsilon^{i j}\nabla_{j}V^{\mathrm{e x}}({\bf r}) \, , \hspace{6 m m}
\begin{array}{l}\nabla f(x^*)+\left[\displaystyle \sum_{i=1}^p[y^I]^*_j\nabla^2 c_j(x^*)-A^TA-{\cal J}c_{\gamma}(x^*)^T{\cal J}c_{\gamma}(x^*)\right]\eta^*_1+{\cal J}c_{\beta}(x^*)^T[\xi_b]^*_{\beta}=0, begin{align*}8pt]{\cal J}c_{\beta}(x^*)\eta^*_1+[\xi_a]^*_{\beta}+[\xi_b]^*_{\beta}=0. \end{array}
\partial_{\xi}^{3}\left(\frac{\xi^{2}}{\log^{b-1}(\frac{1}{\xi})}\chi_{\leq \frac{1}{4}}(\xi)\right) = \left(\frac{(b-1)b(b+1)}{\xi\log^{b+2}(\frac{1}{\xi})}+ \frac{3 b(b-1)}{\xi\log^{b+1}(\frac{1}{\xi})}+ \frac{2(b-1)}{\xi\log^{b}(\frac{1}{\xi})}\right)\chi_{\leq \frac{1}{4}}(\xi)+\psi(\xi)
\frac{\partial^{k}\varphi_{0}^{-}\left( \rho,x\right)}{\partial x^{k}}=\frac{\left( 2i\right)^{k}}{2\pi i}\int_{0}^{\infty}\left(q(s)L_{k}^{\left( 1\right)}\left( s,\rho\right) +Q^{\prime}(s)L_{k}^{\left( 2\right)}\left( s,\rho\right) \right) ds
\begin{array}{ll}t_k^{\pm}(2) = t_{k-1}^{\mp}(0) + t_{k-1}^{\pm}(1) k \geq 1 begin{align*}3pt]t_k^{\pm}(2) = 2 t_{k-2}^{\mp}(2) + t_{k-2}^{\pm}(0) + t_{k-2}^{\pm}(1) k \geq 2.\end{array}
\sum_{i=0}^{m}a_{i}f_{s+m-i}^{(t)}=\sum_{i=0}^{m}a_{i}\left(\sum_{\ell=1}^{m}c_{\ell}^{(t)}\lambda_{\ell}^{s+m-i}\right) &=\sum_{\ell=1}^mc_{\ell}^{(t)}\lambda_\ell^{s}\left(\sum_{i=1}^{m}a_{i}\lambda_{\ell}^{m-i}\right).
\lim_{N \to \infty}\left(\frac{2}{N}\right)^k R_{N}^{(k)}\left(\frac{2u_1}{N}, \ldots, \frac{2u_k}{N}\right)= \mathrm{Pf}{\begin{bmatrix}DS^{(\mathrm{hard})}(u_i, u_j) S^{(\mathrm{hard})}(u_i, u_j) -S^{(\mathrm{hard})}(u_j, u_i) IS^{(\mathrm{hard})}(u_i, u_j) \end{bmatrix}}_{i,j = 1}^{k}
\frac{4n}{(1-z)^{N}(1+z)^{M}}\frac{d}{dz}B ( z ) = \big( 1-z^{2}\big) \frac{N+M+n+1}{2}\big( P_{n-1}^{( N+1,M+1 )}(z) \big)^{2}{}+\big( 1-z^{2}\big) P_{n}^{(N,M)}(z) \frac{d}{dz}P_{n-1}^{(N+1,M+1)}(z){}+ [ (M-N)-z ( N+M+2 ) ] P_{n}^{(N,M)}(z) P_{n-1}^{(N+1,M+1)}(z) .
\|x-x_J\|_2\leq\sum_{j\geq1}\|x_{T_j}\|_2\leq\frac{1}{\sqrt{J}}\sum_{j\geq0}\|x_{T_j}\|_1=\frac{1}{\sqrt{J}}\|x\|_1<\frac{1}{\rho\sqrt{J}}\|x\|_2,
D_{1}= \left(\left\langle\Psi_{j}^{0}, \frac{1}{ik}\mathcal{A}_{0}[\phi]\partial_{\tau}\Phi_{\ell}^{\tau}\bigg|_{\tau = 0}+ \mathcal{A}_{1}[\phi]\Phi_{\ell}^{0}\right\rangle + \left\langle\frac{1}{ik}\partial_{\tau}\Psi_{j}^{\tau}\bigg|_{\tau = 0},\mathcal{A}_{0}[\phi]\Phi_{j}^{0}\right\rangle\right)_{j,\ell = 1}^{3}.
\frac{1}{x}- \frac{1}{x + 2}+ \frac{1}{x + 1}- \frac{1}{x + 3}+ \frac{1}{x + 2}- \frac{1}{x + 4}\dot{s}- \frac{1}{x + 9}+ \frac{1}{x + 8}- \frac{1}{x + 1 0}= 0
{\cal S}^{n}\equiv{\cal S}\left( \varphi_{n}^{\alpha}, \varphi_{n + 1}^{\alpha}, \nabla \varphi_{n}^{\alpha}, \nabla \varphi_{n + 1}^{\alpha}, T \right) \rightarrow{\cal S}^{n}\left( T_{n}\right) \equiv{\cal S}\left( \varphi_{n}^{\alpha}, \varphi_{n + 1}^{\alpha}, \nabla \varphi_{n}^{\alpha}, \nabla \varphi_{n + 1}^{\alpha}, T_{n}\right) ,
{\bf E}\,\Delta_{\eta}^{r}(x)=\frac{1}{\left( \sqrt{n}\,h_{n}\right)^{r}}\,{\bf E}\left|{\bf \,}\sum_{i{\bf \,}=1}^{\eta}K\left( \frac{x-X_{i}}{h_{n}}\right) -n\,{\bf E}\,K\left( \frac{x-X}{h_{n}}\right) \,\right|^{r}.
b_1(\tau(x),\tau(y)) = - Q_1(1_{\tau(x)},1_{\tau(y)}) = - \beta Q_2 (U1_{\tau(x)},U1_{\tau(y)}) = - \beta h(x)h(y) Q_2(1_x,1_y) = \beta h(x)h(y) b_2(x,y).
U^N(s,x) =U(0,x)-\alpha\int_0^{s}\int_0^1 A(x,y)U(u\wedge \tau_N,y) 1_{(0,\tau_N)}(u) \mathrm{d}y\mathrm{d}u +\alpha \beta\int_0^s 1_{(0,\tau_N)}(u)\mathrm{d}\xi_2(u\wedge \tau_N,x)+\alpha \zeta\int_0^s 1_{(0,\tau_N)}(u)\mathrm{d}\xi_3(u\wedge \tau_N,x).
0 = F_{l}^{\prime \prime}+ \frac{2}{r}F_{l}^{\prime}\left[ 1 + r L_{-}^{\prime}( k r ) \right] - \frac{s^{\prime}}{s - \frac{\omega}{m}}\left[ F_{l}^{\prime}+ F_{l}L_{-}^{\prime}( k r ) \right] + W ( r , m ) F_{l}- \left[ K , F_{l}\right]
\dot{x}=f(t,x,u):=F(t,x,H(t,u)x,u), &(t,x,u)\in{\mathbb R}_{\ge 0}\times{\mathbb R}^{n}\times\mathbb{R}^{p}, y =h(t,x,u):=H(t,u)x,\; y\in{\mathbb R}^{k},
\frac{\partial_t R_N(x,t)}{a'(t)}\ge \; c\Big\{(a^2N^{-1/4}+aN^{-6/4}+N^{-11/4})(x-N)^4 &\;\;\;\;+ N^{-7/4}(x-N)^3+N^{-3/4}(x-N)^2\Big\}.
\begin{array}[c]{ll}\operatorname*{curl}\operatorname{curl}\mathbf{E}-k^{2}\mathbf{E}=\operatorname*{i}k\mathbf{\tilde{\boldsymbol{\jmath}}}\Omega \gamma_{T}\operatorname{curl}\mathbf{E}-\operatorname*{i}kT_{k}\Pi_{T}\mathbf{E}=\mathbf{0}\Gamma.\end{array}
I_{12}\le c_9\begin{cases}y_d^{2\alpha-p+\beta_1+\beta_2}(\log (1/y_d))^{\beta_4}2\alpha-2p+\beta_1+\beta_2<0; y_d^p \left(\log (1/y_d)\right)^{\beta_4+1}2\alpha-2p+\beta_1+\beta_2=0. \end{cases}
\mathrm{tr}(Y^{2j-1}) = \mathrm{tr}(Y^{2(j-1)}Y)=\mathrm{tr}((xX)^{j-1}Y) &= \mathrm{tr}(x^{j-1}x^{j-2}X Y)= x^{2j-3}\mathrm{tr}(XY).
\begin{aligned}\begin{pmatrix}f_{b}e_{b}\end{pmatrix}= \begin{pmatrix}-1 0 0 E(\cdot) \end{pmatrix}\begin{pmatrix}\frac{\delta\bar{H}}{\delta\phi_{\partial}}\frac{\delta\bar{H}}{\delta\Sigma}\end{pmatrix}. \end{aligned}
S(e^{-i\pi}\theta_\pm z) = S(\omega^{1/2}\theta_\pm z) = \sum_k i^k c_k\left( \frac{3f}{2z^{3/2}\theta_\pm^{3/2}}\right)^k = 1 + i c_1\frac{3f}{2z^{3/2}\theta_{\pm}^{3/2}}+ O\left( f^{2}z^{-3}\right).
A_1 = \begin{pmatrix}1 1 \end{pmatrix}\begin{pmatrix}0 1 \end{pmatrix}, A_3 = \begin{pmatrix}1 1 \end{pmatrix}\begin{pmatrix}1 0 \end{pmatrix}, A_5 = \begin{pmatrix}1 1 \end{pmatrix}\begin{pmatrix}1 -1 \end{pmatrix}, O = \begin{pmatrix}1 1 \end{pmatrix}\begin{pmatrix}0 0 \end{pmatrix}.
G ( u_{1}, \cdots , u_{2 n}| u_{0}) = \sum_{k_{0}, k_{1}, \cdots k_{2 n}}t_{k_{1}}^{k_{0}}( u_{1}- u_{0}) \otimes \cdots \otimes t_{k_{2 n}}^{k_{2 n - 1}}( u_{2 n}- u_{0}) F ( u_{1}, \cdots , u_{2 n}| u_{0})^{k_{0}k_{1}\cdots k_{2 n}}.
S_{\omega_{\xi,\eta}}(\hat{T})(x) =\int_G\int_G\xi(g^{-1}h)\overline{\eta(h)}R_{h^{-1}}(\hat{T}_g)(x)\d\mu(g)d\mu(h) =\int_G\int_G k(g,h)\sum_{z,w\in Z}\phi_z(xh^{-1})\phi_w(xh^{-1}g)T_{z,w}\Delta(g)^{-1/2}\d\mu(g)d\mu(h).
F(u) =1+z\sum\limits_{k\geq 1}u^k\biggl(1+\sum\limits_{\ell=0}^{k-1}g_\ell +\sum\limits_{\ell=0}^{k-1}h_\ell\biggr) &=1+\frac{zu}{1-u}+z\sum\limits_{k\geq 0}\frac{u^{k+1}}{1-u}g_k+z\sum\limits_{k\geq 0}\frac{u^{k+1}}{1-u}h_k &=1+\frac{zu}{1-u}(1+G(u)+H(u)), G(u) =\frac{z}{u}(F(u)+H(u)-1-H(0)), H(u) =z(F(u)+G(u)).
{\cal S}^{U ( q )}= \frac{1}{g_{S Y M}^{2}}\int d t \int d^{3}\sigma^{\prime}\sqrt{\det ( G^{i j})}\mathrm{t r}_{q}\left( \frac{1}{2}G_{i j}^{\prime}{\cal F^{\prime}}^{0 i}{\cal F^{\prime}}^{0 j}- \right.
Y(X_{j_1}, \dots, X_{j_k})= \begin{cases}X 1 \notin \{j_1\ldots, j_k\},\, 2 \in \{j_1\ldots, j_k\}, X^* 1 \in \{j_1\ldots, j_k\},\, 2 \not\in \{j_1\ldots, j_k\}, \mathbb C \{1,2\}\subset \{j_1\ldots, j_k\}\; \textrm{or}\; \{1,2\}\cap \{j_1\ldots, j_k\}=\varnothing.\end{cases}
\left\{\begin{array}{c}{{\partial_{\mu}( L A_{\nu}^{a}) - \partial_{\nu}( L A_{\mu}^{a}) + \epsilon^{a b c}A_{\mu}^{b}A_{\nu}^{c}L^{2}\left( \begin{array}{c c}{{0}}&{{1}}\\{{- 1}}&{{0}}\\ \end{array}\right)}}\\ \end{array}\right\}\left( \begin{array}{c}{{e^{a}}}\\{{f^{a}}}\\ \end{array}\right) = 0 ,
\begin{cases}x_1\mapsto \frac{4a^2}{s^2}(x_1-\tilde{\nu_1}'(ls)x_2-\tilde{\nu_1}(ls)x_3+\tilde{\nu_1}'(ls)lsx_3) x_2\mapsto \frac{2a}{s}(x_2-lsx_3) x_3\mapsto x_3\end{cases}
=\sum_{n_{1}=0}^{N_{1}-1}(\sum_{n_{2}=0}^{N_{2}-1}x_{N_{1}n_{2}+n_{1}}e^{-\frac{2\pi i}{N_{2}}n_{2}k_{2}})e^{-\frac{2\pi i}{N_{1}N_{2}}n_{1}(N_{2}k_{1}+k_{2})}
Z^{( 1 )}= \frac{1}{2 !}\int_{1 \cdots 6}V_{1 2 3}V_{4 5 6}( - 2 ) \, \frac{\delta^{3}}{\delta D_{3 6}^{- 1}\delta S_{1 2}^{- 1}\delta S_{4 5}^{- 1}}\, Z^{( 0 )}.
\zeta(s,x+y)=\sum_{k=0}^{\infty}\frac{y^{k}}{k!}\frac{\partial^{k}}{\partial x^{k}}\zeta(s,x)=\sum_{k=0}^{\infty}(\begin{matrix}s+k-1\\s-1\end{matrix})(-y)^{k}\zeta(s+k,x).
\int_{\omega(a+2b/3)}^\pi \frac{\sin^{n-1}\chi\,d\chi}{(\chi-\omega\,b/3)^{3n+2\epsilon}}\leq\int_{\omega(a+b/3)}^\infty\frac{(\chi+\omega\,b/3)^{n-1}\,d\chi}{\chi^{3n+2\epsilon}}&=\sum_{k=0}^{n-1}\binom{n-1}{k}\frac{(\omega\,b/3)^{n-1-k}}{(3n+2\epsilon-k-1)\left(\omega(a+b/3)\right)^{3n+2\epsilon-k-1}}&\leq\frac{\mathfrak c}{(a+b/3)^{2n+2\epsilon}},
\Big\{\{T_{j,q}\}_{j\geq 2,q=0,\dotsc,k},\ \{\mathsf{T}_{q}\}_{q=0,\dotsc,k},\ \{T_{\beta}\}_{\beta\neq \mathbf{0},\mathbf{1},\dotsc,k\mathbf{1}_{1}}\Big\},
{\bar{{\cal G}}}_{\mathrm{( A )}}^{a}\left[ \Delta ( x ) \cdot \Gamma \right] = 0 \, \, , \, \, \, \, \,{\cal G}_{\mathrm{( A )}}^{a}\left[ \Delta ( x ) \cdot \Gamma \right] = 0 \, \, , \mathrm{A}= \mathrm{I}, \, \mathrm{I I}\, \, ,
\frac{d}{dt}\Phi_t^1(x_1^0,x_2)=-\Phi_t^1(x_1^0,x_2)J(t)+ x_2\frac{\rho_{0}(D(t)\Phi_t^1(x_1^0,x_2),0)}{D(t)\Phi_t^1(x_1^0,x_2)}\int_0^tD(s)ds.
|\widehat{W}_{i,k+t+1}| \leq \left(\frac{2}{3}\right)^{t+1}\cdot \left ( \binom{k+t}{i-1}+ \binom{k+t}{i}\right) &=\left(\frac{2}{3}\right)^{t+1}\cdot \binom{k+t+1}{i},
1-\frac{\log b_B(N,2t)}{\log N!}\leq R_{B,opt}(N,t) \leq 1-\frac{\log b_B(N,t)}{\log N!}, 1-\frac{\log b_G(N,2t)}{\log N!}\leq R_{G,opt}(N,t) \leq 1-\frac{\log b_G(N,t)}{\log N!}.
\mathcal{F}_c = \sum_{i_1,\dots,i_n = 0}^\infty \frac{x_1^{i_1}\cdots x_n^{i_n}\otimes 1}{i_1!\cdots i_n!}\prod_{\nu = 1}^n\big[\big(\Delta_{\hat{S}(\gg)}- \Delta_0\big)(\partial^\nu)\big]^{i_\nu}
\tau_{2 , \mathrm{e f f}}= \tau_{2}+ \frac{\theta_{2}}{2 \pi}= \frac{1}{2 g}\ , \quad \frac{\theta_{\mathrm{B}}}{2 \pi}= \frac{\pi - 2 g N \ln r}{2 \pi - 2 g N \ln r}\ .
\mathbb{X}^{N}_t=1+\int_0^t\mathbb{X}^{N}_{s^-}\otimes d\mathbf{X}_s+\sum_{0<s\leq t}\mathbb{X}^{N}_{s^-}\otimes (\exp^{(N)}(\Delta X_s)-1-\Delta X_s-\frac 1 2 (\Delta X_s)^{\otimes 2}),
\begin{aligned}\overline H(f) = \sup_{\phi}\, \inf_{g}\, \sup_{\widetilde\pi}\, \bigg[\int \d\P_0(\omega)\,\phi(\omega) \bigg\{\sum_{e\in \mathcal U_d}\tilde\pi(\omega,e)\big[F(\omega,e)-\log \widetilde\pi(\omega,e)\big]\bigg\}\bigg]\end{aligned}
{\alpha+j-1 \choose j}\Big(\sum_{k=0}^\infty d_k\frac{(x^{2k+4}+y^{2k+4})}{(x^2+y^2+z^2)}n^{-2k-2}\Big)^{j}= \sum_{k=0}^\infty \frac{F'_{k,j}(x,y)}{(x^2+y^2+z^2)^j}n^{-2(k+j)},
\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)}}
W_s-W_{t_{i-1}}=\tfrac{s-t_{i-1}}{t_i-t_{i-1}}\,(W_{t_i}-W_{t_{i-1}})+B_s, \widetilde W_s-\widetilde W_{t_{i-1}}=\tfrac{s-t_{i-1}}{t_i-t_{i-1}}\,(W_{t_i}-W_{t_{i-1}})+\widetilde B_s.
Z \left[ \eta \right] = Z \left[ 0 \right] \exp \left\{- i T^{2}\sum_{n , m = - \infty}^{\infty}\int d \mathbf{x \,}d \mathbf{y}\eta_{n}^{+}\left( \mathbf{x}\right) \gamma^{0}S_{F}^{n - m}\left( \mathbf{x - y}\right) \eta_{m}\left( \mathbf{x}\right) \right\}.
\liminf_{n \to \infty}\mathbb{P}_n\left( \theta^{*(j)}_{\hat{\mathbb{M}}_n, n}\in \mathrm{CI}_{1-\alpha, \hat{\mathbb{M}}_n}^{(j), \mathrm{est}}j = 1, \hdots, m(\hat{\mathbb{M}}_n) \right) \geq 1-\alpha.
(1\,2\,3\,4) =(1\,2)(3\,4)\circ(1)(2\,4)(3) & =(1\,3)(2)(4)\circ(1\,2)(3\,4) & =(1\,4)(2\,3)\circ(1\,3)(2)(4) & =(1)(2\,4)(3)\circ(1\,4)(2\,3).
\int\limits_a^b \left(E(H_2(r);[\lambda',\lambda''])f(r),g(r)\right)dr=\frac{1}{2\pi i}\int\limits_{\lambda'}^{\lambda''}\,d\mu\oint\limits_{\tilde\Gamma'}\left((H_2(z)-\mu)^{-1}f(z),g(\bar z)\right)dz.
\lim_{h\to 0}\mu \left\{x \in \mathbb{S}^1 :\frac{\alpha(x+h)-\alpha(x)}{\sigma(\phi) \ell h\sqrt{-\log |h|}}\le y\right\}= \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{y}e^{-\frac{t^2}{2}}dt.
\P\big({S_n}\geq x,\,{T_n}\leq y \big) =\P\left(sS_n\geq sx,\,-tT_n\geq -ty\right) &\leq\P\big(sS_n-tT_n\geq sx-ty\big) &\leq\P\Big(\exp\big(sS_n-tT_n\big)\geq \exp\big(sx-ty\big)\Big)\,.
tE =t\|\partial_xh\|_2^2=\int_{\mathbb{R}}|t^{1/3}k|^{3}\exp(-2|t^{1/3}k|^3)\frac{1}{|k|}|\hat h_0|^2dk, t^2D =t^2\||\partial_x|^{5/2}h\|_2^2=\int_{\mathbb{R}}|t^{1/3}k|^{6}\exp(-2|t^{1/3}k|^3)\frac{1}{|k|}|\hat h_0|^2dk.
- \int_{B_{1}}\partial_{s}(g_{1-\alpha,m}*(\phi\psi^{2}\tilde{u}^{1-q}))dx + (1-q)\phi \mathcal{E}(\tilde{u},-\psi^{2}\tilde{u}^{-q}) & \quad\quad\quad\leq \int_{0}^{s}\dot{g}_{1-\alpha,m}(s-\tau)(\phi(s)-\phi(\tau))\left( \int_{B_{1}}\psi^{2}\tilde{u}^{1-q}dx \right)(\tau)d\tau + R_{m}(s),
\sup_{t\leq t_0}\|u(t)-\mathrm{id}\|_{L^2}^2 \leq 4t_0^2\sup_{t\leq t_0}\|u_t(t)\|_{L^2}^2 + 2\|u_0-\mathrm{id}\|_{L^2}^2 \leq 4 t_0^2 \mathcal{E}_0(0) + 2\|u_0-\mathrm{id}\|_{L^2}^2.
\big[E^{(m)}\tilde{\circ}F^{(m)}-F^{(m)}\tilde{\circ}E^{(m)}\big]=\big[F^{(m)}\circ E^{(m)}\big] =s_{p,m}([\omega]) \prod\limits_{i=2p}^{mp+2p-1}([\omega]- h_{i,1}) \prod\limits_{1\leq i\leq m^2p, m\nmid i}\bigg([\omega]-h_{p+\tfrac{i}{m},1}\bigg)* [H],
V ( M_{1}, M_{2}) = \frac{1}{N}\mathrm{T r}\, M_{1}^{2}+ \frac{1}{N}\mathrm{T r}\, M_{2}^{2}+ \frac{1}{N^{2}}\mathrm{T r}\, M_{1}^{2}\, \mathrm{T r}\, M_{2}^{2}\; .
\widehat{\mu}_{\{b_{k}\},\{D_{k}\}}(x)=\int e^{2\pi i x \xi}d\mu_{\{b_{k}\},\{D_{k}\}}(\xi)=\prod\limits_{k=1}^{\infty}m_{D_{k}}\big(\frac{x}{b_1b_2\cdots b_k}\big) ,x \in\mathbb{R},
\Omega = \sum_{k}x_kx^*_k= \sum_{1\leq i\leq r}h_i^2 + \sum_{i<j}x_{ij}y_{ji}+ \sum_{i>j}y_{ij}x_{ji}= \sum_{1\leq i\leq r}h_i^2 + \sum_{i<j}x_{ij}y_{ji}+ y_{ji}x_{ij}&= \sum_{1\leq i\leq r}h_i^2 + \sum_{i<j}2x_{ij}y_{ji}+ [x_{ij}, y_{ji}]=\sum_{1\leq i\leq r}h_i^2 + \sum_{i<j}2x_{ij}y_{ji}- h_i + h_j
\int_{\mathbb{R}^{n+1}_+\setminus B_\rho^+}{y^a|\nabla U_k|^2\,dX}+\int_{\mathbb{R}^n\setminus\{B_\rho\cap\{y=0\}\}}{(U_k)_+^{2^*_s}(x,0)\,dx}\geq \eta_0.
F_4(a,b,c,d) \Big|_{d=a+b+c}\! = \int\limits_0^{x_0}\!\! dx xJ_0(ax) J_0(bx) J_0(cx) J_0[(a+b+c)x] &+ \int\limits_{x_0}^{\infty}\!\! dx x \Big\{J_0(ax) J_0(bx)J_0(cx) J_0[(a+b+c)x] - J_{asym}(x) \Big\}&+ \int\limits_{x_0}^{\infty}\!\! dx x J_{asym}(x)\;,
\sum_{\{k_{i}\}}\prod_{i = 1}^{4}\frac{1}{k_{i}!}\left( \partial_{i}^{\mu_{i}}\right)^{k_{i}}u ( 0 ) \int \prod_{i = 1}^{k}\mathrm{d}^{4}x_{i}\left( x_{i}^{\mu_{i}}\right)^{k_{i}}\Gamma_{x_{1}\cdots x_{k}0}.
dz_i(s) = -\frac{\partial V_0}{\partial z_i}\big(z(s), y(s)\big)\,ds + \sqrt{2\beta^{-1}}dw_i(s)\,,1 \le i \le m\,, dy_j(s) = -\frac{\partial V_0}{\partial y_j}\big(z(s), y(s)\big)\,ds -\frac{1}{\epsilon}\frac{\partial V_1}{\partial y_j}\big(y(s)\big)\,ds + \sqrt{2\beta^{-1}}\,dw_j(s)\,,m+1 \le j \le n \,.
\begin{array}{r c c c c c}\mbox{\underline{sector}}{}\mbox{\underline{state}}{}{}\mbox{\underline{${\cal R}$}}\mbox{NS}{}\psi^i_{-{1\over 2}}|0\rangle i=2,\cdots,9{}- \mbox{R}{}|+,2s_1,\cdots,2s_4\rangle 8s_1s_2s_3s_4=+1{}+ \end{array}
\Psi^{(p,g)}(S)\Psi^{(q,h)}(T) = \psi^{(p)}(S_0) \psi^{(q)}(T_0) = \psi^{(r)}(i_p^r(S_0) i_q^r(T_0)) = \Psi^{(r,k)}(\iota_{r,k}(i_p^r(S_0) i_q^r(T_0))) = \Psi^{(r,k)}(i_{(p,g)}^{(r,k)}(\iota_{p,g}(S_0)) i_{(q,h)}^{(r,k)}(\iota_{q,h}(T_0))) = \Psi^{(r,k)}(i_{(p,g)}^{(r,k)}(S) i_{(q,h)}^{(r,k)}(T)).
\Big(\frac{d}{dT}\Big)^k \big(a_1(T){\mathrm{e}}^{\omega_1T}+\cdots+a_n(T){\mathrm{e}}^{\omega_nT}\big) = a_{k,1}(T){\mathrm{e}}^{\omega_1T}+\cdots+a_{k,n}(T){\mathrm{e}}^{\omega_nT}
\Omega_A =\sum_{\tau\in S_\lambda}\omega_{A^\tau}&=\sum_{\tau\in S_{\lambda}}[B_{A^\tau(1,1)}^+,\dots,B_{A^\tau(\lambda_1',1)}^+]\cdots [B_{A^\tau(1,\ell(\lambda'))}^+,\dots,B_{A^\tau(\lambda_{\ell(\lambda')}',\ell(\lambda'))}^+]v_0.
u_{\delta^{i_{1},\dots,i_{k}}}=A^{k}-\sum_{j=k+1}^{n}A^{k}(i_{j})+\sum_{\substack{j,l=k+1 j<l}}^{n}A^{k}(i_{j},i_{l})+\cdots+\left(-1\right)^{n-k}A^{k}(i_{k+1},\dots,i_{n})=0