harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\int_{\mathbb{S}^n}u\mathrm{d}A = -\dfrac{n}{2}\dfrac{\phi'(\rho)}{\phi(\rho)}\rho \int_{\mathbb{S}^n}u^2 \mathrm{d}A + O(\varepsilon) \|u\|_{L^2(\mathbb{S}^n)}^2,
\eta = \frac{C_0^UC_1^S + C_1^U - C_0^SC_1^U}{C_0^U-C_0^UC_0^S-(C_0^U)^2 + (1-C_0^U)\sigma_v^2/A^2}&= \frac{C_0^UC_1^S + C_1^U - C_0^SC_1^U}{C_0^UC_0^L + (1-C_0^S)\sigma_v^2/A^2},
V ( \phi , \chi ) = \frac{1}{2}\mu^{2}r^{2}( \phi^{2}- a^{2})^{2}+ \mu^{2}\phi^{2}\chi^{2}- \mu^{2}a^{2}s^{2}\chi^{2}+ \frac{1}{2}\mu^{2}\frac{s^{4}}{r^{2}}\chi^{4}.
\Omega_{k}^{\prime}:=\left\{P_{k-1}^{n,\zeta},\; TP_{k-1}^{n,\zeta},\ldots,\; T^{d_{k}-1}P_{k-1}^{n,\zeta},\;P_{k}^{n,\zeta},\; TP_{k}^{n,\zeta},\ldots,\; T^{m-d_{k}}P_{k}^{n,\zeta}\right\}
\begin{aligned}\sum_mh_{ijklm}^{p^{\ast}}\omega_m =dh_{ijkl}^{p^{\ast}}+\sum_mh_{mjkl}^{p^{\ast}}\omega_{mi}+\sum_mh_{imkl}^{p^{\ast}}\omega_{mj}+\sum_mh_{ijml}^{p^{\ast}}\omega_{mk}&\ \ +\sum_mh_{ijkm}^{p^{\ast}}\omega_{ml}+\sum_{m}h_{ijkl}^{m^{\ast}}\omega_{m^{\ast}p^{\ast}}\end{aligned}
\beta_1 \oplus \beta_2 = \begin{cases}\emptyset i_2 < j_1, \{(\alpha_{i_1, \overline{j}_1}, \alpha_{j_2, \overline{i}_2}), (\alpha_{i_1, \overline{j}_2}, \alpha_{j_1, \overline{i}_2})\}j_1 \leq j_2 < i_2, \{(\alpha_{i_1, \overline{j}_1}, \alpha_{i_2, \overline{j}_2}) \}j_1 \leq i_2 \leq j_2. \end{cases}
\begin{array}{l}{{{\displaystyle | F_{\nu}\left( \frac{2 \pi}{\sigma_{0}}| n | \tau , \frac{2 \pi}{\sigma_{0}}| n | \tau_{0}\right) |^{2}\sim}}}\\{{\sim{\displaystyle \frac{2^{2 \nu}}{\Gamma^{2}( 1 - \nu )}\left( \frac{2 \pi | n |}{\sigma_{0}}\right)^{2 - 2 \nu}( - \tau_{0}) ~ J_{\nu}^{2}\left( - \frac{2 \pi}{\sigma_{0}}| n | \tau_{0}\right) ( - \tau )^{1 - 2 \nu}}}}\\ \end{array}
\mathcal{H}'=\{(e_i,0,f_i,-\beta_{i}), (g_i,\theta_i\gamma_{i},h_i,0) : 1 \leq i \leq m \}\cup \{(e_{k},0,f_{k},0),(g_{\ell},0,h_{\ell},0) : m+1 \leq k \leq t+s m+1 \leq \ell \leq t \}.
ds_{11}^2 = \left( \frac{\pi^4}{15}Q \right)^{-1/3}l_p^{\prime 2}U^2 (-d \tau^2 + dx_6^2 + \cdots + dx_9^2 + dx_{11}^2 ) + \left( \frac{\pi^4}{15}Q \right)^{2/3}l_p^{\prime 2}( \frac{4 dU^2}{U^2}+ d \Omega_{(4)}^2 )
\bar{R}^{m_j}_j(\boldsymbol{x})=\sum_{k=1}^{m_j}( \sigma^{(j)}_k)^{-1}(f_{| \Omega_j}, \bar{u}^{(j)}_k)_{\ell_2(X_{N_j})}\bar{u}^{(j)}_k (\boldsymbol{x}), \boldsymbol{x}\in \Omega_j.
|L(f)|^{2^k}\leq L(f^2)^{2^{k-1}}L(1)^{2^{k-1}}\leq L(f^4)^{2^{k-2}}L(1)^{2^{k-2}+2^{k-1}}\leq \dots &\leq L(f^{2^k}) L(1)^{1+\dots+2^{k-1}}=0.
\prod_{d|i_j}\phi_{dr^{(j)}_m}(x) =\frac{\prod_{d|i_jr^{(j)}_m}\phi_{d}(x)}{\prod_{d|i_j}\phi_{d}(x)}&\equiv \frac{\left(x^{i_jr^{(j)}_m}-1\right)}{(x^{i_j}-1)}\mod x^{r+1}&\equiv \left(1-x^{i_j}\right)^{-1}\mod x^{r+1}.
\int_t^T\int_{\mathcal{O}}(u(s,x)-\hat{\xi}(s,x))\,\mu(dsdx) = \int_t^T\int_{\mathcal{O}}(u(s,x)-\xi(s,x)+\xi(s,x)-\hat{\xi}(s,x))\,\mu(dsdx) \leq 0.
\sum_{n=\ell}^\infty |A_k^{(1)}(m,n)|^2 e^{-2\pi \sqrt{3}n}\leq .242^2 \cdot 6.747^{2\ell}\cdot e^{-4\pi m \cdot 1.16}\sum_{n=\ell}^\infty e^{4\pi n \cdot .865-2\pi \sqrt{3}n}&\leq (4.5763)(45.523)^\ell \cdot e^{-4\pi m \cdot 1.16}e^{-.01288 \ell},
\frac{d}{dx}D_n(x|a_1,\dots,a_r) =\sum_{l=0}^{n-1}\binom{n}{l}(-1)^{n-l-1}(n-l-1)!D_l(x|a_1,\dots,a_r) &=n!\sum_{l=0}^{n-1}\frac{(-1)^{n-l-1}}{l!(n-l)}D_l(x|a_1,\dots,a_r)\,,
{\rm Vol}(\tilde A\setminus{\tilde \Sigma}^{r}) \geq{\rm Vol}(A \cap (\tilde A\setminus{\tilde\Sigma}^{r})\,) ={\rm Vol}(A) -{\rm Vol}(A\cap (\tilde B \cup{\tilde \Sigma}^{r})) & \geq{\rm Vol}(A) -{\rm Vol}({\tilde \Sigma}^{r}) - \sum_{i=s+1}^m{\rm Vol}(A\cap B_{p_i}(r)).
H^{d+2}_{}(Y,\mathbb{Q}/\mathbb{Z}(d+1)) = (H^{d+2}(Y,\mathbb{Q}/\mathbb{Z}(d+1))/(H^{d+2}(K,\mathbb{Q}/\mathbb{Z}(d+1))) \rightarrow \prod_{v \in X^{(1)}}H^{d+2}(Y_{K_v},\mathbb{Q}/\mathbb{Z}(d+1))/(H^{d+2}(K_v,\mathbb{Q}/\mathbb{Z}(d+1)))
\mathfrak{F}_{l+1}^{(k)}= \displaystyle\sum_{m=1}^{k}\left[\frac{\displaystyle\sum_{p=1}^{k}\left(\lambda_m^{k-p}-\displaystyle\sum_{i=1}^{k-p}\lambda_m^{(k-p)-i}\right)t_{p-1}}{\displaystyle \prod_{j=1, j\neq m}^{k}(\lambda_m - \lambda_j)}\right] \lambda_m^l +\mathfrak{F}_{l-(k-1)}^{(k)}+\mathfrak{F}_{l-(k-2)}^{(k)}+ \ldots +\mathfrak{F}_{l-1}^{(k)}
\rho \ddot{u}_i = \big(A_{iJKj}u_{j,K}- \beta_{Ji}\dot{\tau}-(C_{iJKLIj}u_{j,IL}+ M_{iJKL}\tau_{,L}, a \ddot{\tau}=-\beta_{Ki}\dot{u}_{i,K}+ m_{IJ}\dot{\tau}_{,IJ}+ M_{jLKI}u_{j,LKI}+ K_{IJ}\tau_{,IJ}.
\sum_{n= -\infty}^\infty \int d\tau\, d^{d-2}x^i \left( \epsilon^2 \partial_\tau \phi_n^* \partial_\tau \phi^{\vphantom x}_n+ \frac{2in}{R}\phi_n^* \partial_\tau \phi^{\vphantom x}_n- \partial_i \phi_n^* \partial_i \phi^{\vphantom x}_n -M^2 \phi^* \phi\right)\ .
D_{k}\Psi_{i , j}={\frac{1}{a}}( U_{i , j}^{k}\Psi_{( i , j ) + \hat{k}}- \Psi_{i , j}) \mathrm{w h e r e}U_{i , j}^{k}= \mathrm{e x p}( - i a A_{i , j}^{k})
\lambda W_i^\varepsilon (x) - \frac{b (x) \cdot DW_i^\varepsilon (x)}{\varepsilon}+ G(x, DW_i^\varepsilon (x)) = \lambda v_i \circ H(x) - \lambda a + \varepsilon \lambda \psi_i (x) + (-1)^i \lambda C (h_i - H(x)) - b(x) \cdot D\psi_i (x) + G \big( x, v_i' \circ H(x)DH(x) + \varepsilon D\psi_i (x) - (-1)^i CDH(x) \big),
\int_0^1\tau^{-1}\langle \nabla f(x+\tau(y-x)) -\nabla f(x),(x+\tau(y-x))-x\rangle d\tau &\leq \int_0^1\tau^{-1}\langle L\tau (x-y),\tau(x-y)\rangle d\tau &=\langle L (x-y),x-y\rangle\int_0^1\tau d\tau &=\tfrac{1}{2}\langle L (x-y),x-y\rangle.
\int_{\Omega_{bl}^+}\left( S(A+ D u_{bl}^1) - S(A + D u_{bl}^2) \right) : D (u_{bl}^1 - u_{bl}^2) \: + \: \int_{\Omega_{bl}^-}(S(D u_{bl}^1) - S(D u_{bl}^2) : D (u_{bl}^1 - u_{bl}^2) = 0 .
l_2(d_1,d_3) =\min(r_3+r_4, d_1+d_3) - \max (r_4-r_3, |d_1-d_3|) =\min (2r_3, r_3+r_4 -|d_1-d_3|, d_1 +d_3-r_4+r_3, 2 \min(d_1,d_3)), l_3(d_1,d_2) =\min(r_5+r_6, d_1+d_2) - \max (r_6-r_5, |d_1-d_2|) &=\min (2r_5, r_5+r_6 -|d_1-d_2|, d_1 +d_2-r_6+r_5, 2 \min(d_1,d_2)), c(d_1,d_2,d_3) = \min (l_2(d_1,d_3),l_3(d_1,d_2))
\begin{array}{c}{{I = 0 , \, \alpha_{i}, \, \alpha_{i_{1}}\alpha_{i_{2}}, \, \ldots , \, \alpha_{i_{1}}\! \cdots \alpha_{i_{k}}, \, \ldots \, , \quad i_{1}< i_{2}< \cdots < i_{k},}}\\{{e^{0}= 1 , \quad e^{\alpha_{i_{1}}\alpha_{i_{2}}\cdots \alpha_{i_{k}}}= \xi^{\alpha_{i_{1}}}\xi^{\alpha_{i_{2}}}\cdots \xi^{\alpha_{i_{k}}}, \quad k = 0 , 1 , \ldots , n .}}\\ \end{array}
G_2(z_1,z_2)^2= \left(\sqrt{1-\frac{|z_1|^2}{4}}\operatorname{Re}(z_1)+\sqrt{1-\frac{|z_2|^2}{4}}\operatorname{Re}(z_2) \right)^2 \, +\left(\sqrt{1-\frac{|z_1|^2}{4}}\operatorname{Im}(z_1)+\sqrt{1-\frac{|z_2|^2}{4}}\operatorname{Im}(z_2) \right)^2+\left(2-\frac{|z_1|^2+|z_2|^2}{2}\right)^2.
\frac{\partial Z_{R}}{\partial t}=-K^{\prime}(\theta)[\frac{\partial\psi(\theta)}{\partial z}-1]-K(\theta)[\psi^{\prime\prime}(\theta)\frac{\partial\theta}{\partial z}+\psi^{\prime}(\theta)\frac{\partial^{2}\theta/\partial z^{2}}{\partial\theta/\partial z}]
+\frac{C^{2}\, e^{2\, \alpha +2\, \alpha^{*}}\, n^{4}\, \Theta (y_{0}-x_{0})}{\left( \left( x_{0}-y_{0}-i\, \epsilon \right)^{2}-\left(x_{s}-y_{s}\right)^{2}\right)^{2}}+\frac{C^{2}\, n^{4}\, \Theta (y_{0}-x_{0})}{\left( \left( y_{0}-x_{0}-i\, \epsilon \right)^{2}-\left(x_{s}-y_{s}\right)^{2}\right)^{2}}
\begin{aligned}\min_{\{\delta_{m,n}^{(k)}\}}\sum_{p=1}^{N_t}\left|\left|\mathbf{q}_p^{(k)}- \mathbf{\Delta}_p^{(k)}\mathbf{G}_p^{(k)}\right|\right|_2^2 = \sum_{p=1}^{N_t}\min_{\mathbf{\Delta}_p^{(k)}}\left|\left|\mathbf{q}_p^{(k)}- \mathbf{\Delta}_p^{(k)}\mathbf{G}_p^{(k)}\right|\right|_2^2.\end{aligned}
\mu(s\widetilde{\alpha},\sigma)=\gamma(G/P)^2q_F^{n(\sigma \times \Pi(\sigma))-n(\sigma)}\dfrac{(1-\chi^2\chi'^{-1}(\varpi_L)q_L^{-s})(1-\chi^{-2}\chi'(\varpi_L)q_L^s)}{(1-\chi^2\chi'^{-1}(\varpi_L)q_L^{-1-s})(1-\chi^{-2}\chi'(\varpi_L)q_L^{-1+s})}
\begin{tabular}{cc}$\hat{H}_1\sim \rho^{|n|}$, $1-\hat{H}_2\sim \rho^{|n|+1}$, $\hat{F}_3=[\frac{\rho}{r}\hat{H}_3 + \frac{z}{r}(1-\hat{H}_4)]\sim \rho^{|n|+1}$, $\hat{F}_4=[\frac{z}{r}\hat{H}_3 -\frac{\rho}{r}(1-\hat{H}_4)]\sim \rho$. \end{tabular}
\frac{1}{c^{2}(1-\frac{r_{s}}{r})}(\frac{\partial S}{\partial t})^{2}-(1-\frac{r_{s}}{r})(\frac{\partial S}{\partial r})^{2}-\frac{1}{r^{2}}(\frac{\partial S}{\partial\varphi})^{2}=c^{2}
{\mathcal{F}_{D}^{0}}[ \hat{g}] ={\frac{Q^{2}}{1 6 \pi}}\int{d^{2}}\xi{d^{2}}\xi^{\prime}\sqrt{\hat{g}( \xi )}\hat{R}( \xi ){\hat{G}_{D}}( \xi , \xi^{\prime}) \sqrt{\hat{g}( \xi^{\prime})}\hat{R}( \xi^{\prime}) .
y(qx) = (qx)^{\mu}\frac{(\alpha qx;q)_{\infty}}{(\beta qx;q)_{\infty}}= q^{\mu}\frac{1-\beta x}{1-\alpha x}y(x) = \bigg( q^{\mu}\frac{\beta}{\alpha}+\frac{q^{\mu}(1-\beta /\alpha)}{1-\alpha x}\bigg) y(x) .
\int_{x}^{y}m_{-2,\beta}[f](t)\overline{g}(t)w_{-2,\beta}(t)dt+(1-t)^{-1}(1+t)^{\beta+1}f^{\prime}(t)\overline{g}(t)\big|_{x}^{y}= \int_{x}^{y}f^{\prime}(t)\overline{g}^{\prime}(t)(1-t)^{-1}(1+t)^{\beta+1}dt+\int_{x}^{y}f(t)\overline{g}(t)(1-t)^{-2}(1+t)^{\beta}dt.
\times \; \Big\langle \prod_{b=1}^N \prod_{i=1}^{n^\prime_b}\overline{\psi}^{(b)}({x^\prime}^{(b)}_i) P_+ \psi^{(b)}({x^\prime}^{(b)}_i)\prod_{i=1}^{m^\prime_b}\overline{\psi}^{(b)}({y^\prime}^{(b)}_i) P_- \psi^{(b)}({y^\prime}^{(b)}_i) \Big\rangle_0 \; .
A_{\mu}= \Bigl ( \bigl ({\frac{Q_{e}r}{r^{2}+ A^{2}\cos^{2}\theta}}\bigr ) , 0 , 0 ,{}- \bigl ({\frac{Q_{e}r A \sin^{2}\theta}{r^{2}+ A^{2}\cos^{2}\theta}}\bigr ) \Bigr )
g^{i_{0}}\rightarrow g^{i_{0}}\; , \; g^{i_{1}}\rightarrow \lambda g^{i_{1}}\; , \; \ldots , g^{i_{n}}\rightarrow \lambda^{n}g^{i_{n}}\quad \quad ( g^{i_{p}}\rightarrow \lambda^{p}g^{i_{p}}, \; p = 0 , \ldots , n )
\begin{array}{l c r}{{\sum_{\nu = 1}^{8}X_{m_{1}n_{1}}^{\mu}( \sigma ) X_{m_{2}n_{2}}^{\nu}( \sigma ) X_{m_{3}n_{3}}^{\nu}( \sigma ) = - 8{\gamma}^{\mu}X_{m_{1}n_{1}}( \sigma ) X_{m_{2}n_{2}}( \sigma ) X_{m_{3}n_{3}}( \sigma ) .}}\\ \end{array}
\dfrac{1}{\sqrt{d_1}}\; \Phi \left.\left(\vec{\alpha}=\left\langle - \frac{1}{2}, - \frac{1}{2}\right\rangle, \vec{\beta}=\left\langle - 1, - \frac{1}{2}\right\rangle\; \right| \, \mathbf{v}\right) \;,
\frac{|\xi-z|^2}{|\xi-z^*|^2|z|^2}=1-\frac{|\xi z-z^*z|^2-|\xi-z|^2}{|\xi-z^*|^2|z|^2}=1-\frac{(|\xi|^2|z|^2-2{\rm Re}\, (\xi \bar z) +1)- (|\xi|^2- 2{\rm Re}\,(\xi \bar z) +|z|^2)}{|\xi-z^*|^2|z|^2}= 1-\frac{(1-|\xi|^2)(1-|z|^2)}{|\xi-z^*|^2|z|^2}
{\mathcal C}\approx (1-1) [- iV_4 (8\pi^2\alpha')^{-2}] \frac{1}{2N}\int^{\infty}_{0}\frac{dt}{t^2}\sum_{k=0}^{N-1}\prod^{5}_{r=3}\vert 2 \sin (\pi k v_r) \vert^{-1}\cdot \vert Tr\gamma_{k, 9}+\prod_{r=3}^{5}[2 \sin(\pi k v_r)] \sum_{i=3}^{5}{[ \frac{Tr\gamma_{k,5_{i}}}{2 \sin(\pi k v_i)}]}\vert^{2}
\begin{aligned}L^{(1)}_{(0,0)\to(t,n)}-(1-\rho)Z_n(t) =\sup_{\vec{\pi}\in\Pi(0,0;t,n)}w^{(1)}(\vec{\pi})-(1-\rho)Z_n(t) =w^{(1)}(\vec{\pi}^*)-(1-\rho)s_0^*=w^{(\rho)}(\vec{\pi}^*).\end{aligned}
\frac{1}{2}\delta \Lambda_I = - \gamma^\mu\epsilon_I \partial_\mu\phi + \frac{1}{\sqrt 2}\sigma^{\mu\nu}\left(e^{-\phi}F_{\mu\nu}\alpha_{IJ}-e^\phi \tilde G_{\mu\nu}\beta_{IJ}\right)^-\epsilon^J = 0\ .
\sum_{m = - \infty}^{\infty}q^{( p + m N + \theta N / 2 \pi )^{2}/ 2 N}= \sqrt{2 l / \beta N}\sum_{Q = - \infty}^{\infty}e^{- i Q ( 2 \pi p / N + \theta )}\tilde{q}^{Q^{2}/ 2 N}
u = \Phi(\chi_x,\chi_{xx},\chi_{xxx}) \equiv\frac{\chi_{xxx}}{\chi_x}- \frac{\chi_{xx}^{\;2}}{\chi_x^2}+ 2\alpha \chi_{xx}+ \alpha^2 \chi_{x}^2 + 2 \alpha \varepsilon +\frac{\varepsilon^{2}}{\chi_x^2}
Jq^k(\xi^J) = \xi_1^{j_1}Jq^k(\xi_2^{j_2}\cdots \xi_n^{j_n}) & + \binom{j_1}{1}\xi_1^{j_1+1}Jq^{k-1}(\xi_2^{j_2}\cdots \xi_n^{j_n}) & + \dots & + \binom{j_1}{k}\xi_1^{j_1+k}\xi_2^{j_2}\cdots \xi_n^{j_n}.
\mathrm{\underset{x\in\widetilde{\,T\,}(\epsilon,{\bf{W}})\cap\Pi^{-1}\left(\widetilde{y}_{n}\right)}{max}\vert \widetilde{f}_{n}\vert^{2}=\vert \widetilde{f}_{n}\vert^{2}\left(x_{n}\right)=1}
J_{0}= - 1 + \frac{V^{\prime}\left( - \left| \lambda \right|^{2}\right)}{\left| \lambda \right|^{2}}\varphi \star \varphi^{\dagger}+ \left( 1 - \frac{1}{2}J_{i j}\star J^{i j}\right)^{1 / 2}.
\frac{1}{2^d}\sum_\Phi h^\mathrm{Falt}_{(E,\Phi)}= - \frac{1}{2}\cdot \frac{L'(0,\chi)}{L ( 0 , \chi )}- \frac{1}{4}\cdot \log\left| \frac{D_E}{D_F}\right| - \frac{d}{2}\log(2\pi ) + \sum_p b_E(p) \log(p)
[V_\lambda] = q^{-\frac{c}{24}}q^{\frac{(\lambda| \lambda + 2\rho)}{2(k+h^\vee)}-\lambda(x)}\ \prod_{n=1}^\infty \prod_{\alpha \in \Delta_{}^{0}}(1-q^{n-\epsilon(\alpha)})^{-1}\prod_{\alpha \in \Delta_{}^{+, \sharp}}(1-e^{\alpha(h)}q^{n-\epsilon(\alpha)})^{-1}(1-e^{-\alpha(h)}q^{n+\epsilon(\alpha)-1})^{-1},
s^{(+)}_m(y)=\big[y +2(m-1)(\lambda+m-1) +\tfrac{\beta}{2}(\lambda-\tfrac{\beta}{2}+1) \big]s^{(+)}_{m-1}(y) -(m-1)(\lambda+m-2)(\lambda-\tfrac{\beta}{2}+m-1)(\tfrac{\beta}{2}+m-2)s^{(+)}_{m-2}(y),
\{u ( \sigma , \tau ) , \dot{\bar{u}}( \sigma^{\prime}, \tau ) \}= \{\bar{u}( \sigma , \tau ) , \dot{u}( \sigma^{\prime}, \tau ) \}= 2 \gamma^{2}( 1 + u \bar{u}) \delta_{2 \pi}( \sigma - \sigma^{\prime}) .
\overline{g}((\nabla'_{X}\mathcal{A}_{\widehat{W}})Y,Z)=\overline{g}(Y,(\nabla'_{X}\mathcal{A}_{\widehat{W}})Z),\;\;\overline{g}((\nabla'_{X}T_{r})Y,Z)=\overline{g}(Y,(\nabla'_{X}T_{r})Z),
\mathbf h(D_x) \mathbf f = \begin{pmatrix}h_1(D_x) f_1 \vdots h_n(D_x) f_n \end{pmatrix}, \mathbf S(t) \mathbf f = \begin{pmatrix}S_{h_1(\xi)}(t)f_1 \vdots S_{h_n(\xi)}(t)f_n \end{pmatrix},
E_{A D M}= \left.{\Delta}\left[{\lambda}\sqrt{\kappa}({\psi}_{0}-{\psi}_{1}) + \sqrt{\kappa}({\psi}_{0}-{\psi}_{1}){\phi}^{\prime}- \frac{\sqrt{\kappa}}{2}({\psi}_{0}^{\prime}-{\psi}_{1}^{\prime}) \right] \right|_{- \infty}^{\infty}
L_{+}^{C}= \left( \begin{array}{l l}{{( q t^{- 1})^{S_{3}}}}&{{\Omega S^{-}}}\\{{0}}&{{( q t )^{- S_{3}}}}\\ \end{array}\right) \ \ \ , L_{-}^{C}= \left( \begin{array}{l l}{{( q t )^{- S_{3}}}}&{{0}}\\{{- \Omega S^{+}}}&{{( q t^{- 1})^{S_{3}}}}\\ \end{array}\right) \ ,
S_{| T_{l = a}|}^{W Z}( \eta ) = \frac{\kappa}{4 \pi \sqrt{- 1}}\int_{V_{M}}\widehat{h}^{\ast}\chi_{S U ( 2 )}- \frac{\kappa}{4 \pi \sqrt{- 1}}\sum_{j = 1}^{N_{0}}\int_{U_{\rho^{2}( j )}}\left. \widehat{h}\right|_{U_{\rho^{2}( j )}}^{\ast}\omega_{j},
\exp(\partial^\rho\otimes x_\rho)(x_\mu\otimes 1)\exp(-\partial^\sigma\otimes x_\sigma) = x_\mu\otimes 1 + 1\otimes x_\mu, \exp(\partial^\rho\otimes x_\rho)(1\otimes x_\mu)\exp(-\partial^\sigma\otimes x_\sigma) = 1\otimes x_\mu.
R_{iklm}=\frac{1}{2}(\frac{\partial^{2}g_{im}}{\partial x^{k}\partial x^{l}}+\frac{\partial^{2}g_{kl}}{\partial x^{i}\partial x^{m}}-\frac{\partial^{2}g_{il}}{\partial x^{k}\partial x^{m}}-\frac{\partial^{2}g_{km}}{\partial x^{i}\partial x^{l}})
J_{\nu}( z ) = \frac{( z / 2 )^{\nu}}{\Gamma ( \nu + 1 )}\prod_{m = 1}^{\infty}\left( 1 - \frac{z^{2}}{z_{\nu , m}^{2}}\right){,}\quad \nu \not = - 1 , \, - 2 , \, \ldots
|K(x,y)| \leq \frac{1}{|x-y|^d}, & |K(x+h,y)-K(x,y)|+|K(x,y+h)-K(x,y)| \leq \frac{\omega\left(\textstyle\frac{|h|}{|x-y|}\right)}{|x-y|^d}\forall |h|<\frac{|x-y|}{2}
[z^n] N_h = [z^{n-1/2}] \sqrt{B}(1-B)^{h-1}&= \frac{1}{n-\frac12}[t^{n-3/2}] \frac{1}{2\sqrt{t}}(1-(2h-1)t) (1-t)^{h-2}(1-t)^{-2k(n-1/2)}&= \frac{1}{2n-1}[t^{n-1}] (1-(2h-1)t) (1-t)^{-2kn+k+h-2}&= \frac{1}{2n-1}\binom{(2k+1)n-k-h}{n-1}- \frac{2h-1}{2n-1}\binom{(2k+1)n-k-h-1}{n-2}&= \frac{k+1-h}{2kn-k-h+1}\binom{(2k+1)n-k-h-1}{n-1}
\begin{array}{cc}\displaystyle{\langle \phi^{\alpha}(z_{1})\phi^{\beta}(z_{2})\rangle =C_{\phi}\frac{I^{\alpha\beta}(\hat{{\cal X}}_{12})}{(\det{\cal X}_{12})^{\frac{1}{2}\eta}}}~~~~ ~~~~I^{\alpha\beta}(\hat{{\cal X}}_{12})=\hat{{\cal X}}_{12}^{\alpha\beta}\end{array}
{\cal L}= \overline{{\psi}}_{a}({\ooalign{\hfil / \hfil \crcr \partial}}- m ) \psi_{a}+ \overline{{\psi}}_{b}({\ooalign{\hfil / \hfil \crcr \partial}}- m ) \psi_{b}+ \frac{1}{2}g A_{\mu}j_{a}^{\mu}+ \frac{1}{2}g B_{\mu}j_{b}^{\mu}- \frac{1}{2}g A_{\mu}B^{\mu}.
\begin{cases}\Phi''-{\Delta}\Phi+A^T\Phi=0 \hbox{in}(0, +\infty) \times \Omega, \Phi = 0 \hbox{on}(0, +\infty) \times \Gamma_0,\cr\partial_\nu \Phi + B^T\Phi= 0 \hbox{on}(0, +\infty) \times \Gamma_1 \end{cases}
P_{N}^{( N_{f})}( W , M ) \ = \ \frac{1}{\widetilde{Z}_{N}^{( N_{f})}( M )}\, \prod_{f = 1}^{N_{f}}\det ( W W^{\dagger}+ M_{f}^{2}) \exp ( - N \Sigma^{2}{\mathrm{t r}\,}W W^{\dagger})
a(\lambda):=(24 c_1^2-12 c_4) \lambda^6+(\frac{14}{3}c_2^2-4 c_6) \lambda^4+(12 c_2 c_3 -12 c_7)\,\lambda^2-6\,c_3^2, b(\lambda):= (-48 c_1^2+24 c_4) \lambda^6+(-\frac{32}{3}c_2^2+8 c_6) \lambda^4+(-16 c_2 c_3 +24 c_7)\,\lambda^2.
\mu_{l,n}\bigl(\Omega_{+}(\mathbf{A})\bigr) =\mu_{k,n}\bigl(\Omega_{-}(\mathbf{B})\bigr) =\frac{l!(n-k)!\prod_{j=l-k+1}^{l}{\omega_j}}{n!(l-k)!\prod_{j=n-k+1}^{n}\omega_j}.
\sum_{n=0}^{\infty}\frac{z^nq^{n^2+n}}{(q;q^2)_{n+1}}=\sum_{n}^{\infty}\sum_{m}^{\infty}b^1_\nu(m,n)z^m q^n=\sum_{n=0}^{\infty}(-zq;q^2)_n q^n,
C_{J}= \left( \frac{\partial E}{\partial T}\right)_{J}= 1 2 \pi^{1 1 / 4}\kappa^{- 3 / 2}N^{1 / 2}m^{3 / 2}r_{+}^{- 1}\frac{2 r_{+}^{2}+ l^{2}}{2 r_{+}^{4}+ 5 l^{2}r_{+}^{2}- l^{4}},
{\cal E}\equiv M = \int_{- \infty}^{\infty}d x \left[ \frac{1}{2}{\dot{\phi}}^{2}+ \frac{1}{2}{{\phi}^{\prime}}^{2}+ \frac{\lambda}{4}\left( \phi^{2}- \frac{m^{2}}{\lambda}\right)^{2}+ \varepsilon \frac{m^{3}}{\sqrt \lambda}\phi \right] = \frac{2{\sqrt 2}m^{3}}{3 \lambda}
\Tilde{c_2}(T_2)\Tilde{\gamma_2}(T_2)\dfrac{\partial T_2}{\partial t}=\dfrac{1}{r^{\nu}}\dfrac{\partial}{\partial r}\bigg(\Tilde{\lambda_1}(T_2)r^{\nu}\dfrac{\partial T_2}{\partial r}\bigg),\;\;\; \beta(t)<r<\infty,\;\;t>0,
(\widetilde W n+ A )^{it_N}= (\widetilde W n'+ A )^{it_N}\exp\bigg( O\Big( \frac{\log N}{M(\log \log N)^{1/(2R)}}\Big)\bigg) = (\widetilde W n'+ A )^{it_N}(1 + O((\log \log N)^{-1/(2R)})) = (\widetilde W n'+ A )^{it_N}+ O((\log\log N)^{-1/(2R)})
\frac{1}{2\pi i}\int_{1/2-i T_0}^{1/2+i T_0}\Bigl(\sum_{n\asymp x}\frac{c_j^{(II)}(n)}{n^s}\Bigr)x^{s-1}ds=\frac{1}{\widehat{g}_1(0)}\sum_{n\in [x,x+h_1]}c_j^{(II)}(n)g_1(n)+O\Bigl(\frac{1}{\log^{B}{x}}\Bigr).
S_3 = \begin{bmatrix}42 -48 27 -8 1 -48 69 -56 28 -8 1 27 -56 70 -56 28 -8 1 -8 28 -56 70 -56 28 -8 1 1 -8 28 -56 70 -56 28 -8 1 \ddots \ddots \ddots \ddots \ddots \ddots \ddots \end{bmatrix}_{(N-1)\times (N-1)},
T= (1,0,1),\ (1,1,1){\rm if}\ 10\le k \le 18, T= (1, 0, 1),\ (1, 0, 2),\ (1, 1, 1),\ (1, 1, 2),\ (2, 0, 2), &(2, 1, 2),\ (2, 2, 2){\rm if}\ 20 \le k\le 28.
\mathcal{X}(t)+\int^t_{0}\mathcal{X}(\tau)d\tau\leqslant C_0\sum_{j=1}^{m}\int^t_{0}\mathcal{X}(\tau)^{\alpha_j}d\tau+C_0\sum_{k=1}^{n}\mathcal{X}(t)^{\beta_k}+C_0\sum_{k=1}^{n}\mathcal{X}(0)^{\beta_k}+C_0\mathcal{X}(0),
-\frac{b^{1 - s}}{(q^{1 - s})_{s - 1}}\sum_{i = 0}^{\min\left\{m, s \right\}}\frac{(q^{-s})_{i}}{(q)_{i}}q^{i}\sum_{i_{1}= 0}^{n_{1}- i}\dotsm \sum_{i_{r}= 0}^{n_{r}- i}\sum_{j = 0}^{\infty}H (i, i_{1}, \dotsc, i_{r}, j).
S_{\mathrm{e f f}}[ \vec{a}^{\alpha}] \equiv - \ln \int \prod_{\beta \neq \alpha}[ d a_{0}^{\alpha}] [ d A_{\mu}^{\alpha \beta}] [ d \theta ] [ d \mathrm{g h o s t}] e^{- S_{\mathrm{Y M}}- S_{\mathrm{g f}}- S_{\mathrm{g h}}}.
\mu_{\hat{g}}(\alpha) = \mu \{n\in \mathbb{Z}: |\hat{g}(n)| >\alpha \}& \ge \sum_{j=0}^{k-1}\mu \{n\in (jM-M/2, jM+M/2) : |\hat{g}(n)| >\alpha \}& \ge \sum_{j=0}^{k-1}\mu \{n\in (jM-M/2, jM+M/2) : |\hat{f}(n-jM)| - \varepsilon >\alpha \}& = k \mu \{n\in (-M/2, M/2) : |\hat{f}(n)| >\alpha + \varepsilon \}
\begin{array}{c}V(x)=-(\eta x-a)^2-F^2\mu^4x^4-2 F \mu^2 (\eta x-a)x^2+ +48 \exp \left(-2 \sqrt{3}x-2 \sqrt{3}\mu^2Ex^2+ \frac{\sqrt{3}\theta a}{\mu \lambda}\right),\end{array}
\int_0^t \psi^T(t-s) \int_0^s \phi^T(s-r) M^T_r dr ds = \int_0^t \int_0^t \mathrm{1}_{r\leq s}\psi^T(t-s) \phi^T(s-r)dsM^T_r dr = \int_{0}^t \int_0^{t-r}\psi^T(t-r-s) \phi^T(s)ds M^T_rdr &= \int_{0}^t \psi^T \ast \phi^T(t-r) M^T_rdr &= \int_{0}^t \psi^T (t-r) M^T_rdr-\int_{0}^t \phi^T (t-r) M^T_rdr.
\frac{1}{2\sqrt{\pi t}}\int\limits_{-\infty}^{+\infty}\Lambda_n(s_2) \exp\left\{- \frac{(s_2-x_2)^2}{4t}\right\}\,ds_2= = \exp(-\eta_2^2) \sum\limits_{m=1}^{N}t^{-m/2}Q_{n,m-1}(\eta_2) + O((x_2^2+t)^{-\rho_3 N}), \rho_3>0,
\left\langle\sum_{j\in \mathbb{S}}F_j^*A_jx, \sum_{k\in \mathbb{S}}F_k^*A_kx\right\rangle= \sum_{j\in \mathbb{S}}\left\langle A_jx, F_j\left(\sum_{k\in \mathbb{S}}F_k^*A_kx\right) \right\rangle=\sum_{j\in \mathbb{S}}\langle A_jx,A_jx \rangle
\max_{\hat{\mathbf{c}}^{\mbox{\tiny \upshape (\itshape k\upshape)}}\in \mathcal{S}^{(bf)}_k}\hat{\mathbf{c}}^{\mbox{\tiny \upshape (\itshape k\upshape)}\top}\mathbf{x}= \min\Big\{S^{\mbox{\tiny \upshape (\itshape k\upshape)}}+ \sum_{a \in \mathcal{A}}\max\{0, x_a - x^{\mbox{\tiny \upshape (\itshape k\upshape)}}_a\}; h\Big\}
E\left\vert \sum\nolimits_{i=1}^{n-1}s_{i}P^{i}(X)\right\vert^{2}=\sum\nolimits_{i=1}^{n-1}s_{i}^{2}E|P^{i}(X)|^{2}=\sum\nolimits_{i=1}^{n-1}s_{i}^{2}\left( E|E^{i}X|^{2}-E|E^{i+1}X|^{2}\right) \leq\sum\nolimits_{j=1}^{n-1}\left( s_{j}^{2}-s_{j-1}^{2}\right) E|E^{j}X|^{2}.
D F(t,p^{\pm})=\left(\begin{array}{cccc}f_{1,u_1}(t,p^{\pm}) f_{1,u_2}(t,p^{\pm}) \cdots f_{1,u_m}(t,p^{\pm}) f_{2,u_1}(t,p^{\pm}) f_{2,u_2}(t,p^{\pm}) \cdots f_{2,u_m}(t,p^{\pm}) \vdots \vdots \ddots \vdots f_{m,u_1}(t,p^{\pm}) f_{m,u_2}(t,p^{\pm}) \cdots f_{m,u_m}(t,p^{\pm}).\end{array}\right).
\vert C_a\rangle= \left[{2^{d(J_a)}V(J_a)\over{V(RJ_a)}}\right]^{1\over 2}\sum_{{\widetilde p}\in 2\Lambda(J_a)}\sum_{{\widetilde w}\in{\widetilde\Lambda}(RJ_a)}\sum_{\bf n}\zeta_{\bf n}V_L({\bf n}) V_R({\bf n}) \vert{\widetilde w},{\widetilde p}\rangle~.
\left.\begin{array}{ll}\displaystyle\frac{1}{2}\Lambda\left(\dfrac{a_2}{m_1}\int_{\mathbb{R}^N}|\nabla \xi|^2dx+\dfrac{a_1}{2m_2}\int_{\mathbb{R}^N}|\nabla \eta|^2dx\right)=0,\end{array}\right.
\left[\begin{array}{ll}H_{j,e}G_{j,e}H_{j,o}G_{j,o}\end{array}\right]^{-1}=\left[\begin{array}{ll}I_{n_j}U_j 0 I_{n_j'}\end{array}\right]\left[\begin{array}{ll}H_{j,e}G_{j,e}^{[0]}H_{j,o}G_{j,o}^{[0]}\end{array}\right]^{-1}
r_1=\sqrt{4\nu-1}~;\;\;\;Q_t(z)=\;^t\bar{z}\Big[(e^{tM})^*(e^{tM})-\Big]z\;\;\;\;\;\;S_t(z_1,z_2)=\;^t\bar{z_1}\Big[(e^{tM})^*(e^{tM})-\Big]z_2
\delta \mathrm{A}= \mathrm{A}^{\prime}\xi_{\mathrm{B}}+ \xi_{\mathrm{B}}^{\prime}\mathrm{A}- [ \Lambda_{\mathrm{B}}\mathrm{A}- \mathrm{A}\Lambda_{\mathrm{B}}] + k \mu \Lambda_{\mathrm{B}}^{\prime}= 0 ,
b_{2n+1-k}\sigma_{k-1}^\ell+c_{2n+1-k}\sigma_{k+1}^\ell =- b_{2n+1-k}\sigma_{2n+2-k}^\ell - c_{2n+1-k}\sigma_{2n-k}^\ell &=-s_\ell( \sigma_{2n+1-k})=s_\ell(-\sigma_{2n+1-k})=s_\ell(\sigma_{k}),
\begin{aligned}\int_0^T \int_{B_{2R}^d}\int_{\mathbb{S}^{d-1}}\mathbf{1}_{(\tau,v_{s+k+1},\omega_{k+1}) \in \mathcal{B}_{VI}^+}d\omega_{k+1}dv_{s+k+1}d\tau \leq C_{d,s,k}C_{d,\alpha}T R^d \theta^{(d-1)/2}\end{aligned}
\big\| \tilde f(u+h) - \sum_{i = 0}^k \frac{\tilde{f}^{(i)}(u)}{i !}[h^i] \big\|_s\leq C \| f \|_{C^{s+ 2}}\, \| h \|_{L^\infty}^k ( \| h \|_{s}+ \| h \|_{L^\infty}\| u \|_{s}).
\begin{array}[c]{cl}-\Delta Z=0 \Omega, T_{\Delta}Z-\frac{\operatorname*{i}}{k}\operatorname*{div}_{\Gamma}T_{k}\nabla_{\Gamma}Z=\left\langle L_{\Omega}\mathbf{r},\mathbf{n}\right\rangle-\frac{\operatorname*{i}}{k}\operatorname*{div}_{\Gamma}T_{k}^{\operatorname*{low}}\mathbf{r}_{T}\Gamma.\end{array}
\max_{\sum_{k=1}^{r-1}N_{k}+1 \le j \le \sum_{k=1}^{r}N_{j}}M_j \lesssim \sigma^{-\lfloor d/2 \rfloor}\exp \left ( \frac{(p-1)r^2 \delta^2}{2\sigma^2 (1-\eta)}\right )
{\cal L}_{SCM}=\frac 12 \biggl( J_\mu^j J^{\mu ~j}+ i \overline{\chi}^j /\kern-.5em\partial ~\chi^j+i\varepsilon^{jkl}\overline{\chi}^j /\kern-.5em J^k \chi^l+\frac 14 (\bar \chi \chi )^2 \biggr).
\widehat\psi_r(\beta-\varepsilon) = \inf_{x\in[0,1]}J_r(x,\beta-\varepsilon) \geq\, \inf_{x\in[0,1]}[J_r(x,\beta)-\theta(x,\beta)\varepsilon] \geq\, \widehat\psi_r(\beta)-\sup_{x\in[0,1]}\theta(x,\beta)\varepsilon\rightarrow\widehat\psi_r(\beta), \varepsilon\downarrow 0.
\lambda_p|R|^{p-2}r^2|\Theta|^{p-2}=-\frac{1}{R\,r^{N-3}}\frac{\partial}{\partial r}\bigl(R_rr^{N-1}\bigr)-\frac{1}{\Theta\sin^{N-2}\theta}\frac{\partial}{\partial\theta}\bigl(\Theta_\theta \sin^{N-2}\theta \bigr)=:a(r)+b(\theta)
\chi_{\Lambda}( C_{m}^{\Lambda_{0}}) = \sum_{( \lambda + \lambda_{0}^{\prime}, \kappa + \kappa_{0}, - s ) \in D^{+}}\sum_{s = 0}^{\infty}\, m_{\lambda_{0}^{\prime}, s}\, \alpha_{\lambda_{0}^{\prime}, s}^{m}( \Lambda ) \frac{D_{q}[ ( \lambda + \lambda_{0}^{\prime}, \kappa + \kappa_{0}, - s ) ]}{D_{q}[ ( \lambda , \kappa , 0 ) ]}\, , m \in{\bf Z}^{+}
\langle A ( z_{1}) B ( z_{2}) A ( z_{3}) B ( z_{4}) \rangle = \frac{1}{( z_{1}- z_{3})^{2 \Delta_{A}}( z_{2}- z_{4})^{2 \Delta_{B}}}[ x ( 1 - x ) ]^{\Delta_{C}- \Delta_{A}- \Delta_{B}}F ( x )
\sum_{d_1,d_2<D}\frac{1}{\sqrt{d_1d_2}}\sum_{Id_1, Jd_2<D}\alpha_{Id_1}\bar{\alpha}_{Jd_2}\sum_{L, M\geq 1}\frac{G_k(LMd_1d_2\slash p)}{\sqrt{LM}}\sum_{f\in \mathcal{B}_{2k}(p)}\omega_f\lambda_f(IL)\lambda_f(JM)=0.
\mathbb{E}_n \bigg[ \sup_{0 \le t \le T}\bigg| \frac{1}{\sqrt{n}}\int_0^t \sum_{j \in \mathbb{Z}}\sum_{i=0}^{\ell-1}(\nabla_{j+i-1, j+i}(\overrightarrow{W}_{j+1}^\ell(s))^2) W_{j+i-1}(s) \psi_i \nabla^n \varphi^n_j (s) ds \bigg|^2 \bigg] .