harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\tilde{Z}_{3}= 1 - \frac{\tilde{e}^{2}}{8 \pi}\int_{2 m}^{\infty}d a \frac{( 1 / a^{2}) ( a^{2}- 2 m \tilde{\mu})^{2}+ ( 2 m - \tilde{\mu})^{2}}{( a^{2}- \tilde{\mu}^{2})^{2}}+ O ( \tilde{e}^{4}) .
\tau \Delta t+{\textstyle\sum_{l=1}^{r_1}}\langle \mathcal{G}(\Delta\widetilde{X}_{a_la_l}), \mathcal{G}(\widetilde{Z}_{a_la_l})\rangle\le 0 \quad\ \forall(\tau,Z)\in\mathcal{T}_K(\overline{t},\overline{X})\cap (\overline{\zeta},\overline{\Gamma})^{\perp},
\nabla_{\partial_x}\partial_x =\lambda_1 e^{-2\lambda_1 z}\partial_z \nabla_{\partial_y}\partial_y =\lambda_2 e^{-2\lambda_2 z}\partial_z \nabla_{\partial_x}\partial_z =\nabla_{\partial_z}\partial_x=-\lambda_1 \partial_x \nabla_{\partial_y}\partial_z =\nabla_{\partial_z}\partial_y=-\lambda_2 \partial_y.
{\cal L}_{S C M \lambda}= \frac 1 2 \left( J_{\mu}^{j}M_{j k}^{\mu \nu}J_{\nu}^{k}- 2 \varepsilon^{\mu \nu}~ J_{\nu}^{j}~ \partial_{\mu}\Phi^{j}~ + i \overline{{{\chi}}}^{j}/ \kern - . 5 e m \partial ~ \chi^{j}+ i \varepsilon^{j k l}\overline{{{\chi}}}^{j}/ \kern - . 5 e m J^{k}\chi^{l}+ \frac 1 4 ( \bar{\chi}\chi )^{2}\right) ,
p(h_{i},x_{i}^{m}|X_{i}^{o},\Theta)=\frac{p(h_{i},x_{i}^{m},X_{i}^{o}|\Theta)}{\sum_{h_{i}\in H_{b}}\int p(h_{i},x_{i}^{m},X_{i}^{o}|\Theta)dx_{i}^{m}}
\begin{matrix}H^{2}(Ri_{*}\mathbb{Q}_{U}|_{p=0})&=&\mathbb{Q}_{p=0}\\H^{1}(Ri_{*}\mathbb{Q}_{U}|_{p=0})&=&\mathbb{Q}_{p=0}^{\oplus2}\\H^{0}(Ri_{*}\mathbb{Q}_{U}|_{p=0})&=&\mathbb{Q}_{p=0}\end{matrix}
\left|g\circ \pi(\omega)-\int g\, dT_{\omega_{0}*}\ldots_\ast T_{\omega_{n}*}\mu_{n}\right| =\left|\int g\circ \pi(\omega)\,d\mu_{n}-\int g\circ T_{\omega_{0}}\circ\cdots\circ T_{\omega_{n}}(x)\,d \mu_{n}\right| &\leq \int |g\circ \pi(\omega)- g\circ T_{\omega_{0}}\circ\cdots\circ T_{\omega_{n}}(x)|\, d\mu_{n}\le \epsilon.
G_k\subset \bigcup_i Q(x_{i},t_{i};\,{5r}), &(x_{i},t_{i}) \in G_k, &Q(x_{m},t_{m};\,{r})\cap Q(x_{n},t_{n};\,{r})=\varnothing, \m\neq n, &r\leq \varepsilon^{-1}\iint_{Q(x_i,t_i;\,{r})}|\nabla u|^2 dxds.
S_{1}( q , P , t ) = - \frac{B}{2 A}q^{2}+ \frac{\sqrt{\kappa}}{2 A}\Bigl [ q \sqrt{\frac{A P}{\kappa}- q^{2}}+ \frac{A P}{\kappa}\sin^{- 1}\Bigl ( \sqrt{\frac{\kappa}{A P}}q \Bigr ) \Bigr ] + s_{1}( P , t ) ,
L^t = -\frac{1-\chi (s,\xi)}{|s|^2 +|\xi|^2}\sum_{1 \leq j \leq n}(\xi_j D_{s_j}+ s_j D_{\xi_j}) - \frac{4is\cdot\xi(1-\chi(s,\xi))}{(|s|^2+|\xi|^2)^2}+ \widetilde{\chi}(s,\xi) ,
E_i\,=\,H_{ii}\cup H_{i,\,i-1}\cup\ldots \cup H_{i1}, & H_{ii}\,=\,\overline{\pi_i^{-1}(C_{i-1})\,-\,\pi_i^*(E_{i-1})}, & \pi_i|_{H_{ij}}:H_{ij}\xrightarrow{\cong}H_{i-1,\, j},j<i.
\begin{array}{l}{x^{-}x^{0}= q x^{0}x^{-}, x^{0}x^{+}= q x^{+}x^{0},}\\{x^{+}x^{-}- x^{-}x^{+}= ( q^{1 / 2}- q^{- 1 / 2}) x^{0}x^{0}.}\\ \end{array}
A_{\nu}(x)=-\frac{2(\nu+1)}{3}\cdot\frac{\alpha^2_{\nu,1}-x^2}{x}\cdot\frac{\mathcal{D}'_{\nu}(x)}{\mathcal{D}_{\nu}(x)}=\alpha^2_{\nu,1}+\frac{4(\nu+1)}{3}\sum_{m\geq 1}A_mx^{2m},
\int_{\mathbb{R}^{2}}dx dy\, e^{-\frac{1}{2}x^{2}- \frac{1}{2}y^{2}}\frac{x-y}{x+y}\Big(e^{\alpha x^{2}+ \beta y^{2}}- e^{\alpha y^{2}+ \beta x^{2}}\Big) = \frac{4\pi(\alpha - \beta)}{(1 - \alpha - \beta) \sqrt{1 - 2\alpha}\sqrt{1 - 2\beta}}.
\int d^{4}x \left\{\sqrt{g_{1}^{( 4 )}}\lambda_{1}\left( \Phi^{2}( - l ) - v_{1}^{2}\right)^{2}+ \sqrt{g_{2}^{( 4 )}}\lambda_{2}\left( \Phi^{2}( l ) - v_{2}^{2}\right)^{2}\right\}.
M^{0 5}( n_{1}^{\prime}n_{2}^{\prime}n^{\prime}k^{\prime}n_{1}n_{2}n k ) = ( 2 + s + 2 n + 2 n_{1}+ 2 n_{2}) \delta_{n_{1}n_{1}^{\prime}}\delta_{n_{2}n_{2}^{\prime}}\delta_{n n^{\prime}}\delta_{k k^{\prime}}
S_{0}={\frac{1}{\pi}}\int d^{2}x [ e^{- 2 \phi}( 2 \partial_{+}\partial_{-}\rho - 4 \partial_{+}\phi \partial_{-}\phi + \lambda^{2}e^{2 \rho}) +{\frac{1}{2}}\sum_{i = 1}^{N}\partial_{+}f_{i}\partial_{-}f_{i}] \ \ .
\|\nabla^2 \log\, \frac{v_t}{\gamma}(y) \|_{\mathrm{HS}}^2= \sum_{i=1}^n \lambda_i(y)^2\le(1-\frac1\beta) \sum_{i=1}^n \lambda_i(y)=(1-\frac1\beta) \Delta \log\, \frac{v_t}{\gamma}(y).
\tilde{g}^{(q)}(b;h) =W^{(q+r)}(b)\Bigg[-\dfrac{\rho^{(q)}_{b}(b;h)+\tilde{g}^{{(q)}}(b;h)}{W^{(q)}(b)}r\int_b^{\infty}e^{-\Phi(q+r)y}W^{(q)}(y)dy+\int_b^{\infty}e^{-\Phi(q+r)u}h(u)du &+r\int_{b}^{\infty}e^{-\Phi(q+r)y}\rho_b^{(q)}(y;h)dy\Bigg]+\tilde{g}^{(q)}(b;h)+\dfrac{r}{\Phi(r+q)}W^{(q+r)}(b)e^{-\Phi(q+r)b}\tilde{g}^{(q)}(b;h).
\begin{array}{lcl}X^{V_\alpha}(f^V)=0, X^{V_\alpha}(f^{(\beta)})=\delta^\alpha_\beta (X(f))^V , \noalign{\medskip}X^C(f^V)=(X(f))^V, X^C(f^{(\alpha)})=(X(f))^{(\alpha)}\, , \end{array}
w \in \bar{g}_{v(t),1}^{-1}\circ \varphi_{v(t)+n,v(t)}\left( \mathbb{B}(0,s_v(t)) \right)=\Phi_{v(t)+n}\left( \bar{f}_{v(t),1}^{-1}\left( \mathbb{B}(0,s_v(t))\right) \right) & \subseteq \Phi_{v(t)+n}\left( \bar{f}_{v(t),1}^{-1}\left( \bar{\mathbb{B}}(0,s_v(t))\right) \right),
|\partial_{3}F_{3}(r,\rho,z)| \leq \frac{C r^{2}\lambda_{0,0}(s)^{2\alpha-3}}{(1+(-\rho^{2}+r^{2})\lambda_{0,0}(s)^{2\alpha-2})^{2}+4 \rho^{2}\lambda_{0,0}(s)^{2\alpha-2}}&\leq \frac{C r^{2}\lambda_{0,0}(s)^{2\alpha-3}}{1+2(\rho^{2}+r^{2})\lambda_{0,0}(t)^{2\alpha-2}+(\rho^{2}-r^{2})^{2}\lambda_{0,0}(t)^{4\alpha-4}}
H = \int \! d^{2}\!{\bf r}\left\{\sum_{p = 1}^{M}\frac{\hbar^{2}}{2 m_{p}}\mathrm{\boldmath \Pi}_{p}^{\dagger}({\bf r}, t ) \cdot \mathrm{\boldmath \Pi}_{p}({\bf r}, t ) + U ( \Psi^{\dagger}, \Psi ) \right\},
\nu_k^*\omega_{\log}(s_1,\ldots,s_k) =\tilde{\omega}_{\log}(s_1,\ldots,s_k)\cdot F, \nu_k^*\omega'_{\log}(s_1,\ldots,s_k) =\tilde{\omega}_{\log}(s_1,\ldots,s_k)\cdot F'.
\rho =\left[\Gamma^2 -\frac{1}{\kappa^2}\Big(\sum_{i\in I\smallsetminus J}\frac{1+\alpha_i}{1-\alpha_i}\Big)^{-1}\Big(\frac{1 +\varepsilon}{1 -\varepsilon}\Big)\right]_+^{\frac{1}{2m}}<1.
\int dx \, \sqrt{g}\, V [\sigma]= \int dx \, \sqrt{g}\, \frac{N}{2\lambda_B}\sigma^2 +\frac{N}{2}\mbox{Tr}\, \int_{1/\Lambda^2}^\infty \frac{dt}{t}\, e^{-t\sigma^2}e^{\not\nabla^2t}.
1-([\eta]_{m}-2^{-j})^2 =1-[\eta]_{m}^2+2.[\eta]_{m}.2^{-j}-2^{-2j}&=\frac{2m+2}{\eta^2+2m+2}+2.[\eta]_{m}.2^{-j}-2^{-2j}&=\frac{2m+2}{\eta^2+2m+2}\left(1-2^{-j}\frac{1}{m+1}\eta\sqrt{\eta^2+2m+2}+2^{-2j}\frac{\eta^2+2m+2}{2m+2}\right)
w_{21}(0)=\frac{f_{21}\langle \Psi_{1}+\Psi_{2}, \,\rho\rangle-2g_{11}\langle \widetilde{\rho}, w_{20}\rangle-(g_{20}+2\overline{g}_{11})\langle \widetilde{\rho},w_{11}\rangle -\overline{g}_{02}\langle \widetilde{\rho},w_{02}\rangle}{2r i +2}.
C_k(a_1,\ldots,a_n)=\begin{pmatrix}a_1 a_2 \cdots a_n a_{n-k+1}a_{n-k+2}\cdots a_{n-k}a_{n-2k+1}a_{n-2k+2}\cdots a_{n-2k}\vdots \vdots \vdots \vdots a_{k+1}a_{k+2}\cdots a_k\end{pmatrix}.
\psi_1\circ \varphi_1(z_1, 0)=(\psi_1+u)\circ \varphi_1(z_1, 0)\leq \frac{1}{\mu(B(z_1, 2))}\int_{B(z_1, 2)}(\psi_1+u)\circ \varphi_1(z, 0) d\mu_z
x_{1}x_{-1}^{k^{'}_1-1}\left\{\prod_{i=2}^{q}x_{c^{'}_{i-1}}x_{1}^{l_{i-1}}x_{-1}^{k^{'}_{i}-1}\right\}=z_{1}(k^{'}_1)\left\{\prod_{i=2}^{q}z_{c^{'}_{i-1}}(1)z_{1}(1)^{l_{i-1}-1}z_{1}(k^{'}_{i})\right\}
B = \int \partial_xu(\partial_x^2u)^2 \chi_n(x+\nu t) \; dx + 3 \int u(\partial_x^2u)^2 \chi_n'(x+\nu t) \; dx &+ \frac43 \int (\partial_xu)^3 \chi_n''(x+\nu t) \; dx- \int u(\partial_xu)^2\chi_n'''(x+\nu t) \; dx.
2it \int\mathbf{1}_{\{k\geq0\}}\phi(k)\mathcal{Q}(x,k)e^{ik^{2}t}h(k)\,dk =\int\mathbf{1}_{\{k\geq0\}}\phi(k) k^{-1}\mathcal{Q}(x,k)h(k)\,\partial_k e^{ik^{2}t}\, dk = \phi'(0) h(0) \mathcal{Q}(x,0)-\int\mathbf{1}_{\{k\geq0\}}e^{ik^{2}t}\partial_{k}\big( \phi(k) k^{-1}\mathcal{Q}(x,k)h(k) \big)\,dk.
\chi R_0(z_m) \chi = \sum_{j=1}^{\frac{d-1}{2}}\tilde{B}_j(r, -z_0 ) + 2 i\Big( \frac{z_0}{2 \pi}\Big)^{d-1}\mathcal{E}^*_{\chi}( i\bar{z_0}) \mathcal{E}_{\chi}( -i\bar{z_0}) + i m \Big( \frac{z_0}{2 \pi}\Big)^{d-1}\mathcal{E}^*_{\chi}(z_0 ) \mathcal{E}_{\chi}( \bar{z_0})
\bar{P}^{in}_{\mathbb{Z}^d,\vec{H}}(T^{\vec{H}}(x)=1)=\bar{E}^{in}_{\mathbb{Z}^d,\vec{H}}[p_x(\beta,H,H_x)]\leq Prob(|H_x|<\delta)+1-a^2(\beta,H,\delta).
E_n(t) := \inf\{r: D_n(r) > t\}= n^{-1}(N_{t / b(n)}+ 1) G_n(t) := D_n^-(E_n^-(t+)) = b(n) T(N_{t/b(n)}) H_n(t) := D_n(E_n(t)) = b(n) T(N_{t / b(n)}+ 1) X_n(t) := A_n^-(E_n^-(t+)) = B(n) S(N_{t / b(n)}) Y_n(t) := A_n(E_n(t)) = B(n) S(N_{t / b(n)}+ 1).
\partial_x^{2M+1}f(x,y,1, z/n)^{-\alpha}= \sum_{i=0}^{2M}A_i(x,y)f(x,y,1,z/n)^{-\alpha-i-1}, \partial_x^{2M+1}\partial_y^{2M+1}f(x,y,1, z/n)^{-\alpha}= \sum_{i=0}^{4M}B_i(x,y)f(x,y,1,z/n)^{-\alpha-i-2},
B_{M N}= ( - )^{M ( N + \underline{{N}})}\partial_{N}z^{\underline{{N}}}\partial_{M}z^{\underline{{M}}}B_{\underline{{M N}}}, E_{\pm}^{\underline{{A}}}= E_{\pm}^{M}\partial_{M}z^{\underline{{N}}}E_{\underline{{N}}}^{\underline{{A}}},
S_{e f f}( A_{\mu}) ={\cal N}\int{\cal D}q ~ e^{- S ( q , A_{\mu})}= \prod_{k = 1}^{\infty}d e t \left( \delta_{\mu \nu}- \frac{1}{\eta}F_{\mu \nu}\right)^{- 1}= \sqrt{d e t \left( \delta_{\mu \nu}- \frac{1}{\eta}F_{\mu \nu}\right)}.
h_{\beta,\gamma}(r) := \int_0^\infty \frac{dt}{t}\, t^\beta \int_0^\infty ds\, s^\gamma p_\zeta^{(\alpha)}(t,r,s) = C^{(\alpha)}\left(\beta,\gamma,\zeta\right) r^{\alpha\beta+\gamma-2\zeta}, r>0,
\mathrm{P}(\mathcal{O}_{2}) = \mathrm{P}\left(\log(1+\rho|h_{n^*}|^2\alpha_2^2)<2R_2,|\mathcal{S}_r|>0\right) + \mathrm{P}\left(\log(1+\rho|g_{n^*,2}|^2\alpha_2^2)<2R_2, \right. \left. \log(1+\rho|h_{n^*}|^2\alpha_2^2)>2R_2,|\mathcal{S}_r|>0\right).
\int_{\Omega}|u(x)v(x)|\,d\gamma(x)\leq\int_{\Omega^{\displaystyle\star}}u^{\displaystyle\star}(x)v^{\displaystyle\star}(x)\,d\gamma(x)=\int_{0}^{\gamma(\Omega)}u^{\circledast}(r)v^{\circledast}(r)\,dr.
\frac{h}{N}\Bigg(|F_0^l|+\sum_{n=2}^{\infty}|T_0^{l,(n)}|\Bigg)\le \alpha^{-4}, c_0\alpha^2\Bigg(\|F_2^l\|_{0,r}+\sum_{n=2}^{\infty}\|T_2^{l,(n)}\|_{0,r}\Bigg)\le 1, &M^{-2l}\sum_{m=2}^Nc_0^{\frac{m}{2}}M^{\frac{l}{2}m}\alpha^m \Bigg(\|F_m^l\|_{0,r}+\sum_{n=2}^{\infty}\|T_m^{l,(n)}\|_{0,r}\Bigg)\le 1.
|\langle v(t)\rangle_{K,\sigma}|^2 =\frac{\sigma^4\varepsilon^2}{(\sigma^2+h^2)^2}|\eta(K)|+\frac{2\sigma^2\varepsilon^4}{\sigma^2+h^2}\Im\big(\overline{\eta(K)}V_K^1(t)e^{-it\theta_K}\big) &\quad+\varepsilon^6\Big(|V_K^1(t)|^2+\frac{2\sigma^2}{\sigma^2+h^2}\Im\big(\overline{\eta(K)}V_K^2(t)e^{-it\theta_K}\big)\Big)+\mathcal{O}(\varepsilon^8).
\iint_{K \times K}\dfrac{1}{|x-y|^{\frac{p}{p-1}\left(n+\bar{s}-(n+sp)\frac{1}{p}\right)}}\,dxdy \leq \int_K \left(\,\int_{B_R(x)}\dfrac{dy}{|x-y|^{n+\frac{p}{p-1}(\bar{s}-s)}}\right)\,dx begin{align*}3mm] =C(n)|K|\int_0^R\dfrac{d \rho}{\rho^{1+\frac{p}{p-1}(\bar{s}-s)}}begin{align*}3mm] =C(n)|K|\dfrac{p-1}{p(s-\bar{s})}R^{\frac{p}{p-1}(s-\bar{s})}
\alpha^2\langle z^j,z^j\rangle_{1}= \alpha^2\beta^2_1(j) = \beta_2^2(j) + \Big(\dfrac{\alpha^2\beta_1(j)^2}{\beta_2(j)^2}-1\Big)\beta_2^2(j) & = \langle z^j,z^j\rangle_2 + \langle Kz^j,z^j\rangle_2.
\| g^\epsilon_m\|_{L^2_{\ell'}(\mathbb{R}^2_+)}^2 + \| \tilde{w}_\epsilon\|_{H^{m, m-1}_{k+\ell}(\mathbb{R}^2_+)}^2 & \le C_8 e^{C_2 t}\int^t_0e^{-C_2 \tau}\| \tilde{w}_\epsilon(\tau)\|_{H^m_{k+\ell}(\mathbb{R}^2_+)}^2 d\tau+C_9e^{C_2 t}\| \tilde{u}_0\|_{H^{m+1}_{k + \ell'-1}(\mathbb{R}^2_+)}^2 .
\int_{\{z^i \in W^d, \forall i = 1, \ldots, n-1\}}\prod_{j=1}^d \lambda_j^n e^{-\sum_{j= 1}^d \lambda_j(z_j^i - z_j^{i-1})}1_{\{z_j^i > z_j^{i-1}\}}& = \prod_{j=1}^d \lambda_j^n e^{- \sum_{j= 1}^d \lambda_j(z_j - x_j)}\int_{\{z^i \in W^d, \forall i = 1, \ldots, n-1\}}1_{\{z_j^i > z_j^{i-1}\}}.
\Big(\frac{1}{i}\partial_t-\Delta_{\mathbb{R}^6}\Big)\psi_{1,a}^{(2)}=-\frac{1}{2N}v_N(y_1-y_2)\big\{\overbrace{u(t,y_1,y_2)+(\bar p\circ u)(t,y_1,y_2)}^{\bar c\circ u=u\circ c=\frac{1}{2}s_2}\big\}.
\partial_{t}\left(\frac{\rho(u,t)^{2}}{2}\right)-\Theta\sigma^{ij}_{k}\nabla_{ij}\left(\frac{\rho(u,t)^{2}}{2}\right) &=(k+1)h\Theta\sigma_{k}-\rho^{2}+\sigma_{k}\nabla_{i}h\nabla_{j}\Theta-\Theta\sigma^{ij}_{k}w_{mi}w_{mj}.
T_2^1 =(\chi_\lambda(x/t))^3e^{it\phi(x/t)}\int_{|y|\leq t^a}a^2\overline{a}+a^2\overline{b}+2|a|^2b\,dy &+(\chi_\lambda(x/t))^3e^{it\phi(x/t)}\int_{|y|\leq t^a}2a|b|^2+b^2\overline{a}\,dy:=T_{2a}+T_{2b},
\int_{B_{1}}\phi \psi^{1+q}w^{2}dx +\frac{1}{2}g_{\alpha}*(\,\phi)\leq q g_{\alpha}*\int_{B_{1}}\psi^{1+q}w^{2}g_{1-\alpha}\phi dx & + \frac{\vartheta(q)(q-1)}{2}g_{\alpha}*(\,\phi)+ \int_{0}^{s}g_{\alpha}(s-\sigma)\dot{\phi}(\sigma)\left( g_{1-\alpha}*\int_{B_{1}}\psi^{1+q}w^{2}dx \right)(\sigma)d\sigma.
\forall t\leq t_0\|u(t)-\mathrm{id}\|_{L^2}^2 = 2\int_0^t \int_{\mathbb{R}^3}\langle u -\mathrm{id}, u_t\rangle ~\mbox{d}x + \|u_0-\mathrm{id}\|_{L^2}^2 & \leq 2T\big(\sup_{s\leq t}\|u_t\|_{L^2}\big) \Big(\sup_{s\leq t}\|u-\mathrm{id}|_{L^2}\Big) + \|u_0-\mathrm{id}\|_{L^2}^2,
\mathbb{P}(t_n=t)\ = \ \mathbb{P}(t_n=t, |w_n|\leq n/4)+\mathbb{P}(t_n=t, |w_n|\geq n/4) \leq \e^{-c_1n}+\mathbb{P}(t_n=t, |w_n|\geq n/4)
\varphi^1\varphi^1_\omega-\varphi^1_{\omega\omega}-\frac{(\varphi^2)^2}{\omega}-\frac1{\omega^3}=0, \varphi^1\varphi^2_\omega-\varphi^2_{\omega\omega}+\frac{\varphi^1\varphi^2}{\omega}+2\frac{\varphi^2}{\omega^2}=0.
\int_{S_2}\left|\frac{1}{|x-z|^2}-\frac{1}{|y-z|^2}\right| \rho^N(z,t) \, dz\leq C |x-y|\int_{S_2}\frac{\rho^N(z,t)}{|\xi_z-z|^3}\,dz &\leq CN^{3\alpha}|x-y|\int_{S_2}\frac{1}{|\xi_z-z|^3}\,dz
\left[\begin{array}{c}\hat{\tau}\hat{\gamma}^1 \hat{\gamma}^2\end{array}\right]:= \left[\begin{array}{c}\tau \gamma^1 \gamma^2\end{array}\right]+ \left[\begin{array}{ccc}y 0 0 0 y 0 0 0 y\end{array}\right] \left[\begin{array}{c}\eta^0 \eta^1 \eta^2\end{array}\right]; \y\in C^\infty(B_4,\mathbb{R}).
(kD_1+H)\cdot \Sigma_k \leq 2n\frac{(kD_1+H)\cdot m_2H_2 \cdot...\cdot m_nH_n}{-K_{\cal F}\cdot m_2H_2 \cdot ... \cdot m_nH_n}= 2n\frac{(kD_1+H)\cdot H_2 \cdot...\cdot H_n}{-K_{\cal F}\cdot H_2 \cdot ... \cdot H_n}
F(x) - \tilde F_{n,l}(x) = \F(x) -\frac 1 l \frac 1{(n-l+1)}\sum_{j=1}^n a_{n,j}1_{\{Y_j\leq x \}}= \ \frac 1 l \frac 1{(n-l+1)}\sum_{j=1}^n a_{n,j}(F(x) - 1_{\{Y_j\leq x \}}),
\frac{H_1'(\rho_b)}{H_1(\rho_b)}+ \frac{2(d-1)(d-2) \rho_b^{2d-6}-2(d^2-d-2) \omega M \rho_b^{d -3}+ (d-1)^2 \omega^2 M^2}{\rho_b(\rho_b^{d-3}- \omega M) [2(d-2) \rho_b^{d-3}- (d-1) \omega M]}= 0.
\exp \left[{\frac{F_{\pm}}{2}}\sigma_{\mu \nu}N_{\mu \nu}^{\pm}T_{3}\right] = \cosh{\frac{F_{\pm}}{2}}{\bf I}+ \sigma_{\mu \nu}N_{\mu \nu}^{\pm}\sinh{\frac{F_{\pm}}{2}}T_{3}\ .
\big|\mathcal{Z}_\iota\big|\leq N^{\frac{\varepsilon}{3}}\hat{\Pi}\leq N^{\frac{\varepsilon}{2}}\frac{|\mathcal{S}|+ \hat{\Lambda}}{(N\eta)^{\frac13}}, \iota=1,2 \Theta\Big(z,{\frac{N^{3\varepsilon}}{(N\eta)^{\frac13}}},{\frac{N^{3\varepsilon}}{\sqrt{N\eta}}}\Big)\cap\Omega_3(z),
\begin{array}{r c l}{{\hat{S}}_{t}}&{=}&{{\hat{P}}_{x},}\\{{\hat{P}}_{t}}&{=}&{{\hat{P}}{\hat{S}}_{x}+{\hat{Q}}_{x},}\\{{\hat{Q}}_{t}}&{=}&{{2}{\hat{Q}}{\hat{S}}_{x}.}\\ \end{array}
\sum_{j=1}^k (R-\rho)^j \|\nabla^j\sigma\|_{L^\infty(B^{g_P}(p,\rho))}\leq C_k ((R-\rho)^{k+\alpha}[\nabla^k\sigma]_{C^{\alpha}(B^{g_P}(p,R))}+ \|\sigma\|_{L^\infty(B^{g_P}(p,R))})
W_i=\mathrm{d}\zeta_{\widetilde{\gamma}_i}+\tau_{ij}\mathrm{d}\zeta_{\gamma^j}, W_i^{}=-\sum_{\gamma}\Omega(\gamma)q_i(\gamma)\left(A_{\gamma}^{}-\mathrm{i}V_{\gamma}^{}\mathrm{d}\zeta_{\gamma}\right)\,.
\begin{array}{ccccccc}1 0 0 0 h^{11}0 1 h^{12}h^{21}1 0 h^{22}0 0 0 1 \end{array}\ ;\qquad\mbox{with}\qquad \begin{array}{l}h^{11}=h^{22}h^{12}=h^{21}\ .\end{array}
\frac{1}{k!}\sum_{l=0}^k{k \choose l}(-1)^{k-l}l^k \frac{1}{1-lt}= \sum_{m=0}^\infty \frac{1}{k!}\sum_{l=0}^k{k \choose l}(-1)^{k-l}l^{m+k}t^m &= \sum_{m=0}^\infty \left( \frac{1}{k!}\Delta^k 0^{m+k}\right) t^m = \sum_{m=0}^\infty S_2(m+k,k) t^m,
c^{X^{-1}}_{w_0}= c^{X^{-1}}_{v_\lambda w_\lambda}= (w^{-1}_\lambda c^{X^{-1}}_{v_\lambda}) c^{X^{-1}}_{w_\lambda}= c^{X^{-1}}_{v_\lambda}c^{X^{-1}}_{w_\lambda}\qquad\hbox{and}p_\lambda^X c^{X^{-1}}_{v_\lambda}= c^{X^{-1}}_{v_\lambda}p^X_\lambda.
R(f)=(r_{j}^{i}(f))=\begin{pmatrix}r_{1}^{1}(f) r_{1}^{2}(f) \cdots r_{1}^{n}(f) \vdots \vdots \vdots \vdots r_{n}^{1}(f) r_{n}^{2}(f) \cdots r_{n}^{n}(f)\end{pmatrix}
(\omega_1,\{\omega_1,\omega_1\})+\{\omega_1,(\omega_1,\omega_1)\}=(-1)^{\varepsilon_A(1+\varepsilon_B)}\left(\omega_1,\left.{\partial_r^2 \Gamma\over \partial\Phi^A\partial\Phi^B}\right|_K\right)\left.{\partial\omega_1\over\partial J_B}\right|_K\left.{\partial\omega_1\over\partial J_A}\right|_K.
CT_{\vec{t}}\,\prod_{i=0}^{c-1}(1+t_i)^{i+a}\left(1+\frac1{t_i}\right)^b\cdot \prod_{i\neq j}^{0,c-1}\left(1-\frac{t_j}{t_i}\right)=c!\cdot \prod_{i=0}^{a-1}\prod_{j=0}^{b-1}\prod_{k=0}^{c-1}\frac{i+j+k+2}{i+j+k+1}.
H_\phi (x,r) = - \int |u (x+z)|^2 |z|^{-1}\phi' \left({\textstyle\frac{|z|}{r}}\right)\, dz= - \frac{1}{r^{m-1}}\int |u (x+r\zeta)|^2 |\zeta|^{-1}\phi' (|\zeta|)\, d\zeta
x_{1}= \varphi_{1},\;\; x_{2}= \sin \varphi_{2}\sin \varphi_{3},\;\; x_{3}=\varphi_{1},\;\; x_{4}=\cos \varphi_{2}\sin \varphi_{3}, x_{5}=\cos \varphi_{3},\;\;\mbox{where}\;\; \varphi_{2}\in[0,2\pi]\;\; \mbox{and}\;\; \varphi_{3}\in(0,\pi/2).
V_{N B P S}( z ) = 2 m^{2}\left( \begin{array}{c c}{\frac{2 \alpha}{\lambda}- ( 1 + \frac{2 \alpha}{\lambda}) s e c h^{2}( M z )}&{0}\\{0}&{\frac{\alpha}{2 \lambda}- \frac{3 \alpha}{\lambda}s e c h^{2}( M z )}\\ \end{array}\right) .
(\rho s)_t + (\rho u s)_x-\left(\frac{\kappa \theta_x}{\theta}\right)_x=\frac{\kappa \theta_x^2}{\theta^2}+ \frac{\varepsilon u_x^2}{\theta}+\frac{\mu |\mathbf{w}_x|^2}{\theta}+\frac{\nu |\mathbf{h}|_x^2}{\theta}
\begin{aligned}\Theta_{\alpha,\alpha}(\rho) =\rho^{2-2\alpha}\Delta_{\alpha,\alpha}-\rho\sin\rho+W_{\alpha,\alpha}(a,\rho), \Theta_{\alpha,\beta}(\rho) =\rho\Delta_{\alpha,\beta}-(-1)^{\alpha}(\cos\rho-W_{\alpha,\beta}(a,\rho)).\end{aligned}
\zeta^{(\delta,\ell,E,\tau)}_{n}(m,k+\delta\tau^{-1})=\zeta^{(\delta,\ell,E,\tau)}_{n-1}(m,k)e^{-\delta\varphi\Big(D^{(\delta,\ell,E,\tau)}_{n-1,k},i_m\Big)}.
J \leq (\log N_{1}\log N_{2})^{2}\sum\limits_{1\leq m_{1},\ldots m_{4}\leq L}\prod\limits_{i=1}^{2}r_{i}(2^{m_{1}}+2^{m_{2}}-2^{m_{3}}-2^{m_{4}}) &\leq 305.8869N_{1}N_{2}L^{4}.
\partial_{t}\omega_{s}^{\nu}=\nu\partial_{yy}\omega_{s}^{\nu}+P_{0}\left(U_{n}^{\nu}\cdot\nabla\omega_{n}^{\nu}\right) & =\nu\partial_{yy}\omega_{s}^{\nu}+P_{0}\left( U_{n1}^{\nu}\cdot\nabla\left( \omega_{n2}^{\nu}-\psi_{n2}^{\nu}\right) +U_{n2}^{\nu}\cdot\nabla\omega_{n2}^{\nu}\right) .
|\det(Y)|^2 = \det (Y^* Y) &= y_1 y_2 &= \frac{1}{2}\left((y_1 + y_2)^2 - (y_1^2 + y_2^2)\right) = \frac{1}{2}\left((x_1 + x_2)^2 - (x_1^2 + x_2^2)\right) = x_1 x_2 = \det (X^* X) = |\det(X)|^2,
U^{n}(t)=\|u^{n}\|_{L^{\infty}(0,t;D_{A_{q}}^{1-\frac{1}{p},p})}+\|\partial_{t}u^{n}\|_{L^{p}(0,t;L^{q})}+\|u^{n}\|_{L^{p}(0,t;W^{2,q})},
\mathrm{P}\left( z_m < \frac{\epsilon_{2,m}}{\frac{\left(z_m-\frac{\epsilon_{1,m}}{\rho}\right)}{z_m(1+\epsilon_{1,m})}\rho}\right) &=\mathrm{P}\left( z_m <\frac{\epsilon_{1,m}+\epsilon_{2,m}+\epsilon_{1,m}\epsilon_{2,m}}{\rho}\right).
\sigma_{sc}=\sigma_{sc}^{(e)}+\sigma_{sc}^{(m)}=\frac{1}{(4\pi\epsilon\epsilon_{0})^{2}}\cdot\frac{8\pi}{3}k^{4}|\alpha^{e}|^{2}+\frac{1}{(4\pi\mu\mu_{0})^{2}}\cdot\frac{8\pi}{3}k^{4}|\alpha^{m}|^{2}
\begin{array}{rcl}h^1_{1}+h^2_{2}= \frac1{EG-F^2}[-G g(\nabla_{\tau_1}\tau_1,\nu)+Fg(\nabla_{\tau_2}\tau_1,\nu)+F g(\nabla_{\tau_1}\tau_2,\nu)-Eg(\nabla_{\tau_2}\tau_2,\nu)], \end{array}
\| (h_0(u,X_u) -h_0(u,X_u^{\epsilon}))\lambda+h_0(u,X_u^{\epsilon})\lambda_{\varepsilon}\|^p_{L^p(\Omega, L^p(\nu))}&\leq 2^{p-1}\Big(L\| X_u-X_u^{\epsilon}\|^p_{S^p(\Omega;\mathcal{D})}\|\lambda\|^p_{L^p(\nu)}+K(1+\| X_u^{\epsilon}\|^p_{S^p(\Omega;\mathcal{D})})\|_{L^p(\Omega)}\| \lambda_{\varepsilon}\|^p_{L^p(\nu)}\Big).
\partial_{z}\tilde{\varphi}( z , \bar{z}) = \tilde{\Phi}= \partial_{\varphi}\bar{S}( z , \bar{z}, \varphi = \tilde{\varphi}) \, , \quad \partial_{\bar{z}}\tilde{\varphi}( z , \bar{z}) = \tilde{\bar{\Phi}}= \partial_{\varphi}S ( z , \bar{z}, \varphi = \tilde{\varphi}) .
\int_{-\infty}^{\infty}x^{2n}e^{-\frac{1}{2}ax^{2}}dx=(\frac{2\pi}{a})^{\frac{1}{2}}\frac{1}{a^{n}}(2n-1)(2n-3)\cdot\cdot\cdot5\cdot3\cdot1=(\frac{2\pi}{a})^{\frac{1}{2}}\frac{1}{a^{n}}(2n-1)!!
\left\{x ( 1 + x )^{2}\frac{d^{2}}{d x^{2}}+ ( 1 + x ) ( 3 x + 1 ) \frac{d}{d x}- \frac{m^{2}( 1 + x )^{2}}{4 x}+ m ( 1 + x ) + 2 + \lambda^{2}\rho^{2}\right\}\xi^{( m )}= 0
S_{n}^{(2m)}=(S_{n}^{(2)})^{m}+\sum_{k=1}^{\lfloor\frac{m}{2}\rfloor}\frac{1}{2^{k}}(\begin{matrix}m\\ 2k\end{matrix})(\begin{matrix}2k\\k\end{matrix})(S_{n}^{(4)}-(S_{n}^{(2)})^{2})^{k}(S_{n}^{(2)})^{m-2k}
J_n(\lambda)=\left[\prod_{i<j}\frac{1-q^{-(\lambda_i+\lambda_j+1)}}{1-q^{-(\lambda_i+\lambda_j)}}\right]\left[\prod_{i=1}^n\frac{1+q^{-\lambda_i-\frac12}}{1-q^{-\lambda_i-\frac12}}\right].
\mathbf{B}(x)=\left(-\operatorname{sign}(x_2)\left[\frac{x_1}{|x_2|^2}\mathbf{1}_{|x_1|\leq |x_2|}+\mathbf{1}_{|x_1|>|x_2|}\right],-\left[\frac{1}{|x_2|}\mathbf{1}_{|x_1|\leq |x_2|}+\mathbf{1}_{|x_1|>|x_2|}\right]\right)
(M_{k,r}^*)^{m\ell}(\xi_1, \xi_2) = \frac{\big( (1-r) \xi_2 - r \xi_1 - k \big)^m}{| (1-r) \xi_2 - r \xi_1 -k |}(\xi_2^\ell - k^\ell) \frac{k \cdot (\xi_1 + k )}{|\xi_1 + k|}\frac{k \cdot (\xi_2 - k )}{|\xi_2 - k|}\hat K_{\approx 1}\left( \xi_1 \right) \hat K_{\approx 1}\left(\xi_2 \right)
Z ( x_{T}, z ) = \sum_{i = 1}^{n_{1}}{\frac{b^{( 0 )}}{( x_{T}^{2}+ | z - z_{i}^{( 1 )}|^{2})^{2}}}+ \sum_{i = 1}^{n_{2}}{\frac{1 - b^{( 0 )}}{( x_{T}^{2}+ | z - z_{i}^{( 2 )}|^{2})^{2}}}+ \mathrm{c o n s t}\int d^{2}w{\frac{| \partial \gamma ( w ) |^{2}}{( x_{T}^{2}+ | z - w |^{2})^{2}}}.
\begin{array}{l l}{{D_{\mu}^{i}= \nabla_{\mu}+ i \frac{q}{2 \sqrt{\beta}}Q_{i}\tilde{H}_{\mu},}}\\{{D_{\mu}^{I}= \nabla_{\mu}+ i \frac{e}{2}E_{I}\tilde{A}_{\mu}- i \frac{\alpha e}{2 \sqrt{\beta}}E_{I}\tilde{H}_{\mu},}}\\ \end{array}
\tilde{b}_{s(X), s(g)}=\tilde{b}_{X\otimes g^i\boxtimes g^{-i}, g^{ik+1}\boxtimes g^{-ik}}&=b_{X\otimes g^i, g^{ik+1}}b_{g^{-i}, g^{-ik}}^{-1}&=b_{X, g^{ik+1}}b_{g^i, g^{ik+1}}b_{g^{-i}, g^{-ik}}^{-1}&=t^{ik+1}t^{ik(ik+1)}t^{-(-ik)(-ik)}&=t^{2ik+1}
P_v=\exp\left\{-\mu \tilde{n}\int d{\bf x}\right\},\;\; \mu=\sum_{l=0}^{\infty}\frac{(-1)^{l}\beta (l+1)}{(l+1)^{3/2}\beta (1)}\exp \left\{-l\pi\frac{(aM)^2}{\rho}\right\}\;.
c^k_{\mathrm{s2u},2}\left( N_{\mathrm{sct}}\right) = \mathsf{E}\left\lbrace B \sum^{S'_{\mathrm{s2u}}\left( k \right)}_{s=1}\log_2\left[ \frac{\gamma'_{\mathrm{s2u},s}\left( k \right)}{\gamma'_{\mathrm{s2u},0}\left( k \right)}\right] \right\rbrace.
-1 + \prod_{j=1}^s \left(1 +A\,\omega^{\,a_j\left(\frac{n_j+1}{2}\right)^{b_j}}\right) \le -1+\exp(\gamma)\le (\mathrm{e}-1)\gamma\le\frac{C_{\beta,\delta}\,(\mathrm{e}-1)}{M^{1/2}}.
\frac{(q;q)_{\infty}}{(-q;q)_{\infty}}=\sum_{k=-\infty}^{\infty}(-1)^{k}q^{k^2}, \frac{(q^2;q^2)_{\infty}}{(-q;q^2)_{\infty}}=\sum_{k=-\infty}^{\infty}(-1)^{k}q^{2k^{2}-k}.
i(t)=i_{0}e^{-\frac{R}{L}t}+\frac{E\omega}{L[(\frac{R}{L})^{2}+\omega^{2}]}e^{-\frac{R}{L}t}+\frac{E}{L\sqrt{(\frac{R}{L})^{2}+\omega^{2}}}sin[\omega t-tan^{-1}(\frac{\omega L}{R})]
\|U^m-u(t_m)\| \leq \|e_{u}(0)\| + C_1{\mathcal E}_1^0(\Omega) + C_2{\mathcal E}_2^m(\Omega) + C_{3}\sum_{n=1}^{m}k_n{\mathcal E}_3^n+\sum_{i=1}^4 c_i{\mathcal E}^n_{2,i}(\Omega),
\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}) ,
\tilde{u}_{k}(K)=\frac{\frac{2m}{\hbar^{2}}\frac{A}{a}f(k)}{\frac{2mE_{k}}{\hbar^{2}}-(k+K)^{2}}=\frac{\frac{2m}{\hbar^{2}}\frac{A}{a}}{\frac{2mE_{k}}{\hbar^{2}}-(k+K)^{2}}u_{k}(0)
\alpha_{r,m,n,s}= \frac{(m-1)!(n-1)!}{(r-1)!(s-1)!}; \ \alpha_{r,m,n,0}= \frac{(m-1)!(n-1)!}{(r-1)!}; beta_{r,m,n,s}= \frac{m!n!}{r!s!}.
\partial_t m_t(x) = - a(m_t,x) m_t(x)\,,\,\,\, a(m,x):=\frac{\delta F}{\delta m}(m,x) + \frac{\sigma^2}{2}\log \left( \frac{m(x)}{\pi(x)}\right) - \frac{\sigma^2}{2}\operatorname{KL}(m|\pi).
\begin{aligned}\Delta_\gamma= \mathrm{Tr}(A_\gamma)^2-4\det(A_\gamma) &=\gamma^{2m}\Bigl [ (2G_{m}(\gamma)+G_{m-1}(\gamma))^2-4 \Bigl (G^2_m(\gamma)+G_{m-1}(\gamma)G_{m}(\gamma)+\frac{G_{m-1}^2(\gamma)}{\gamma}\Bigr ) \Bigr] &=\gamma^{2m-1}(\gamma-4)G^2_{m-1}(\gamma)\end{aligned}
C_{13}=-C_{23}+C_{123}-C_{12}+C_1+C_2+C_3 &C_{24}= C_{234}- C_{23}- C_{34}+ C_2 + C_3 + C_4\,, &C_{14}= C_{1234}- C_{123}- C_{234}+ C_{23}+ C_1 + C_4\,, &C_{124}=C_{1234}- C_{123}- C_{34}+ C_{12}+ C_{3}+ C_{4}\,, &C_{134}= C_{1234}- C_{234}- C_{12}+ C_{34}+ C_1 + C_2\,.
\dot{r}^{2}+ \tilde{V}( r ) = 0 , \dot{t}= \frac{- E}{M - \frac{J^{2}}{4 r^{2}}- \frac{r^{2}}{l^{2}}}, \dot{f}= \frac{E}{\frac{l}{r^{2}}( M - \frac{J^{2}}{4 r^{2}}) - \frac{1}{l}}.
S_{g r}= -{\frac{1}{4 \pi G l^{2}}}\left[ V_{R}-{\frac{A_{R}l}{2}}\right] = -{\frac{l}{4 G}}\left[{\frac{4 R}{( 1 + R )^{2}}}- 2 \ln{{\frac{1 + R}{1 - R}}}\right] .
\begin{aligned}\psi(F_{L})-\psi(F_{R}) =\int_{F_{L}}^{F_{R}}\eta''(y)(A(t,y)-A(t,F_{L}))dy-\eta'(F_{R})(A(t,F_{R})-A(t,F_{L})) &\geq\dot{s}(t)\int_{F_{L}}^{F_{R}}\eta''(y)(y-F_{L})dy-\eta'(F_{R})(A(t,F_{R})-A(t,F_{L}).\end{aligned}