harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\sum_{j=m_{1}}^{m_{2}}(m_{3}-j)e^{-jx}= \frac{(m_{3}-a_{1})e^{(2-m_{1})x}-(m_{3}-m_{2}-1)e^{(1-m_{2})x}-(m_{3}-m_{1}+1)e^{(1-m_{1})x}+(m_{3}-m_{2})e^{-m_{2}x}}{(e^x-1)^2}
\langle A^*(\Psi),\Psi\rangle_{\mathcal{H}}= -a_0\dfrac{(\partial_x^2v(L))^2}{2}+\dfrac{1}{2}\int_0^\infty g'(s)\|\partial_x^{k}\zeta^t\|^2ds\leq 0, \forall \Psi\in D(A^*)
\sum_{Q}m_{Q}^{2}K_{Q r}( p ) = k_{r}, \ \ \sum_{Q}m_{Q}^{2}K_{Q i}( p ) = k_{i}, \ \ \sum_{Q}m_{Q}^{2}H_{Q}( p ) = h , \ \ \sum_{Q}m_{Q}^{2}S_{Q}^{I}( p ) = s^{I},
d\mu^{(n)}_i(t) = 2 \sqrt{\mu^{(n)}_i(t)(1 - \mu^{(n)}_i(t))}dB^{(n)}_i(t) + 2 \Big((p - n + 1) + (p + q - 2n + 2)\mu^{(n)}_i(t)\Big) dt + \sum_{j \neq i}\frac{4 \mu^{(n)}_i(t)(1 - \mu^{(n)}_i(t))}{\mu^{(n)}_i(t) - \mu^{(n)}_j(t)}dt
\begin{array}{rcl}(i_{\hat k}{\hat \Omega})_{\mu_1\dots \mu_5 y}= -(i_k \Omega^{(6)})_{\mu_1\dots\mu_5}& (i_{\hat k}{\hat \Sigma})_{\mu_1\dots\mu_5 y}= -(i_k \Sigma^{(6)})_{\mu_1\dots\mu_5}\, . \end{array}
A^{(c/2\mu )}_{L,N',N^{\prime\prime}}=\left[2^{L+1}\Gamma (L+1)\right]^2\sqrt{\frac{n'!N'}{2\pi\Gamma (N'+L+1)}}\sqrt{\frac{n^{\prime\prime}!N^{\prime\prime}}{2\pi\Gamma (N^{\prime\prime}+L+1)}}B^{(c/2\mu )}_{L,N',N^{\prime\prime}}
\left|\frac{(-1)^j(W_\ell\mathbf{1}(Z=z^r)/P(Z=z^r)-z^{r\prime}\theta)}{\sigma_{P,\ell r}}-\frac{(-1)^j(W_\ell\mathbf{1}(Z=z^r)/P(Z=z^r)-z^{r\prime}\theta')}{\sigma_{P,\ell r}}\right| =\Vert z^r \Vert\frac{\Vert(\theta'-\theta)\Vert}{\sigma_{P,\ell r}(\theta)},~~\ell=0,1,
E_0=\Bigl({4 RT \lambda^2 \over \eta}\Bigr)^{{1 \over 2}}\sum_{n \neq 0,1}{1 \over |n|^{{1 \over 2}}}=\Bigl({4 RT \lambda^2\over \eta}\Bigr)^{{1 \over 2}}\Bigl[ 2\, \zeta(-1 / 2) -1\Bigr].
K \mathbf{c}_j + L(A_j) \sum_{i\in\mathcal{I}^-}\hat{\sigma}(\Psi_{i,T}\mathbf{c}_i)(F)\bar{\mathbf{n}}= K\mathbf{v}_j-L(A_j) \sum_{i\in\mathcal{I}^+}\hat{\sigma}(\Psi_{i,T}\mathbf{v}_i)(F)\bar{\mathbf{n}},~~j\in\mathcal{I}^-.
\|W_\lambda^\frac{4}{N-2}\|_{L^\frac{2N}{N+2}(|x|\leq \sqrt\lambda)}= \lambda^\frac{N-2}{2}\|W^\frac{4}{N-2}\|_{L^\frac{2N}{N+2}(|x| \leq \lambda^{-\frac 12})}\lesssim \lambda^\frac{N+2}{4},
b(h,y)= \sum\limits_{n=1}^\infty\sum\limits_{\mu=0}^\nu{{1-k_1+\nu}\choose \nu-\mu}{{k_2+\nu-1}\choose \mu}\qquad\qquad\qquad\times\frac{\Gamma(k_1-1)}{\Gamma(k_1-\mu-1)}n^{1-k_1+\mu}\overline{a_1(n)}\Gamma(k_1-\mu-1;4\pi ny)a_2(n+h)(n+h)^{\nu-\mu}.
in_{(0,\ldots,0,-e)}(x) = \sum w^N_{n+1}p'_i g_i + w^N_{n+1}p'_\ell \cdot (w_1 \cdot \ldots \cdot w_n) = \sum w^N_{n+1}p'_i in_{(0,\ldots,0,-e)}(g_i) + w^N_{n+1}p'_\ell in_{(0,\ldots,0,-e)}(g_\ell)
I_{1,\ast}(t)+\Vert\varphi(t)\Vert^2_{L^2(\mathbb R)}\le \frac{C_3}{\varepsilon}\int_0^t(I_{1,\ast}(s)+I_{1,n}(s)+\Vert\varphi(s)\Vert^2_{L^2(\mathbb R)})ds &+\varepsilon\left\{I(t)+I_{1,n}(t)\right\}+\frac{C_3}{\varepsilon}\Vert{\mathbf F}\Vert^2_{H^1_\ast(\Omega_t)}\,.
\int dx \, \log \sin x = -x \log 2 - \frac{1}{2}\sum_{n=1}^{\infty}\frac{\sin (2nx)}{n^2},0 \leq x < \pi, \int dx \, \log \cos x = - x \log 2 + \frac{1}{2}\sum_{n=1}^{\infty}(-1)^{n-1}\frac{\sin(2 n x)}{n^2},|x| < \frac{\pi}{2}.
\begin{cases}(\alpha \mathrm{I}+ \mathcal{H}) \boldsymbol{z}^{(i+1/2)}= (\alpha \mathrm{I}- \mathcal{S}) \boldsymbol{z}^{(i)}+ \boldsymbol{v}_k, (\alpha \mathrm{I}+ \mathcal{S}) \boldsymbol{z}^{(i+1)}= (\alpha \mathrm{I}- \mathcal{H}) \boldsymbol{z}^{(i+1/2)}+ \boldsymbol{v}_k,\end{cases}i = 1, 2, \dots, \ell ~ \mbox{or}~ \ell_k
\frac{1}{2}\int_{\Omega}^{}|v_n|^2dx + \frac{1}{n}\int_{\Omega}^{}|\nabla v_n|^2dx +\frac{1}{2}\int_{\Gamma}^{}\bigl| (v_n)_{\Gamma}\bigr|^2 d\Gamma \le \frac{1}{2}\int_{\Omega}^{}|u|^2 dx+ \frac{1}{2}\int_{\Gamma}^{}|u_\Gamma |^2 d\Gamma =: M.
|\frac{{t_2}^{1-\gamma}}{\Gamma(\alpha)}\int_{t_1}^{t_2}(t_2-s)^{\alpha-1}f_{i}(s,x_{i}(s))ds| \leq\frac{M_i{(\frac{\tilde{{\delta}_{i}}}{2})}^{-\delta_i}}{\Gamma(\alpha)}\int_{t_1}^{t_2}{t_2}^{1-\gamma}(t_2-s)^{\alpha-1}ds &=\frac{M_i{(\frac{\tilde{{\delta}_{i}}}{2})}^{-\delta_i}}{\Gamma(\alpha+1)}{t_2}^{1-\gamma}(t_2-t_1)^{\alpha}.
L_{i_{1}\alpha_{1}}^{i_{2}\alpha_{2}}\hat{L}_{i_{2}\beta_{1}}^{i_{3}\beta_{2}}S_{\alpha_{2}\beta_{2}}^{\alpha_{3}\beta_{3}}=S_{\alpha_{1}\beta_{1}}^{\alpha_{2}\beta_{2}}\hat{L}_{i_{1}\beta_{2}}^{i_{2}\beta_{3}}L_{i_{2}\alpha_{2}}^{i_{3}\alpha_{3}},0<i_{i}\le1,0\le\alpha_{i},\beta_{i}\le N-1.
\sum_{\substack{i,j n\geq 1 m=1,\ldots,\min(n,\mathcal N)}}\frac{n! m!}{n (n-m)! m!}(x_iy_j)^{n-m}= \sum_{\substack{i,j n\geq 1 m=1,\ldots,\min(n,\mathcal N)}}\frac{(n-1)!}{(n-m)!}(x_iy_j)^{n-m}= \sum_{\substack{i,j n\geq 0 m=1,\ldots,\mathcal N}}\frac{(n+m-1)!}{(n)!}(x_iy_j)^{n}.
\Omega^{-}: \Gamma \subset W^u(\Lambda)\to \Omega^{-}(\Gamma)\subseteq \Lambda,\, \Omega^{-}(x)=x^-, \Omega^{+}: \Gamma \subset W^s(\Lambda)\to \Omega^{+}(\Gamma)\subseteq \Lambda,\, \Omega^{+}(x)=x^+,
\eta(\tau_{n-i+j}, \tau_{n-i+\ell+1})=(\tau_n \tau_1)^{i-j}\tau_{n-j+\ell+1}=(\tau_n \tau_1)^{i-j}\tau_n^{q-1}\tau_{n-j+\ell+1}=(\tau_n \tau_1)^{i-j}\tau_n^{q-1+D(n-j+\ell)}\tau_{n-j+\ell+1}.
2 \sqrt{\pi}\sum_{J = 2}^{N}U_{J b}\Phi_{0}^{( J )}= n_{b}2 \pi \; , \; n_{b}\; \in \; \mathrm{Z \hspace{- 1 . 3 5 m m}Z}\; \; , \; \forall \; \; b \; = \; 1 , 2 , \; . . . \; N \; .
U ( \prod_{i = 1}^{g}\left( g_{a_{i}}^{~}g_{b_{i}}^{~}g_{a_{i}}^{- 1}g_{b_{i}}^{- 1}\right) ) = e^{i \Theta_{s}^{\pm}}~ , ~ V ( \prod_{i = 1}^{g}\left( g_{a_{i}}^{~}g_{b_{i}}^{~}g_{a_{i}}^{- 1}g_{b_{i}}^{- 1}\right) ) = e^{i \Theta_{j}}~ ,
\begin{gathered}R_{0}+ \tilde{R}_{0}> H(Y_{0}|Y_{j}Z_{j}), R_{j}+ \tilde{R}_{j}> H(Y_{j}|Y_{0}Z_{j}), R_{0}+ \tilde{R}_{0}+ R_{j}+ \tilde{R}_{j}> H(Y_{0}Y_{j}|Z_{j}). \end{gathered}
J_{out}^{(V)}\!\left (s_{ch},J_{1},\ldots,J_{d_v\!-\!1}\right )\!=\!J\!\!\left (\!\sqrt{\sum_{i=1}^{d_v-1}\!\!\left (J^{-1}(J_{i})\right )^2\!+\!s_{ch}^2}\right )\!,
|\dfrac{dz}{dw}| = \dfrac{\Pi_{l=1,2}(e^{p_l\sum_{j=1}^{q_l}u^l_j}|W(f^l_0,f^l_1)|^{p_l})}{|h(\Phi)|\Pi_{l=1,2}\Pi_{j=1}^{q_l}|F^l_j|^{(1-\delta)p_l}}\ .
F_{\mu\nu}= \mathrm{tr}\,\Psi^{\dagger}\left( \partial_{\left[ \mu\right.}\mathcal{D}_z^{\dagger}\frac{1}{\mathcal{D}_z\mathcal{D}_z^{\dagger}}\partial_{\left. \nu \right]}\mathcal{D}_z\right)\Psi\,.
g^{\epsilon}_{t}\ \L^{1}\left(0,T; W^{-m,1}(\Omega)\right) \ \m\ge 0, \left\|h^{\epsilon}-h^{\epsilon}(\cdot +y,t)\right\|_{L^{q_{1}}(0,T; L^{q_{2}}(\Omega))}\rightarrow 0 \ \ |y| \rightarrow 0.
\min\Bigl(X^{23/80}(Q_0E_0)^{27/77},\frac{Q_0^3E_0^2}{X^{9/8}}\Bigr) \ll \Bigl(X^{23/80}(Q_0E_0)^{27/77}\Bigr)^{77/100}\Bigl(\frac{Q_0^3E_0^2}{X^{9/8}}\Bigr)^{23/100}&=\frac{Q_0^{96/100}E_0^{73/100}}{X^{(90-77)\times 23/8000}}&\ll Q_0^{1-\epsilon}E_0^{1-\epsilon}.
\widehat{L^{(n)}_s}=(x^2+1)\frac{d^2}{dx^2}+\left[2x-\left(2-2n-2\Re(s)\right)x-2\Im(s)\right]\frac{d}{dx}&=(x^2+1)\frac{d^2}{dx^2}+\left[\left(2n+2\Re(s)\right)x-2\Im(s)\right]\frac{d}{dx},
[\sigma_{y,z}] = [D_{y^4z}\cap D_{y^3z^2}] + 2[\varphi_{y^3z^2}] - [\gamma_{vy^2z^2,y^3z^2}]- [\gamma_{wy^2z^2,y^3z^2}]- [\gamma_{xy^2z^2,y^3z^2}],
{\mathbb E}(\|{\mathbb E}(U_i|{\mathcal F}_{(i-2)q})\|_H) \leq\sum_{j=(i-1)q+1}^{iq}\|{\mathbb E}(Y_j|{\mathcal F}_{(i-2)q})\|_H \leq\sum_{j=(i-1)q+1}^{iq}\theta(j-(i-2)q) \leq q \theta(q) \, ,
W=\frac{1}{2}c_1a_1c_1a_1+\frac{1}{2}c_2a_2c_2a_2+\frac{1}{2}c_3a_3c_3a_3+c_1c_2a_2a_1-c_1a_1a_3c_3-c_2c_3a_3a_2.
u^{(m)}(y,s) \, =\,\lambda_m u(\lambda_m y_1,\lambda_m y_2,\lambda_m y_3+x_3^{(m)}, \lambda_m^2 s)\,, \phi^{(m)}(y,s) \, =\,\,\,\, \,\,\,\phi(\lambda_m y_1,\lambda_m y_2,\lambda_m y_3+x_3^{(m)},\lambda_m^2 s)\,.
\begin{aligned}\mu_{\eta^,i}= E_i\{\eta^\}&= \int_{\Omega}p_{\eta^,i}(x) x\;dx &= \int_{C_0}^{C_1}\left(A_{0,i}\delta(x-C_0) + A_{1,i}\delta(x-C_1)+A_{2,i}\right)x\;dx &= A_{0,i}C_0 + A_{1,i}C_1 + \frac{1}{2}\left[A_{2,i}x^2\right]_{C_0}^{C_1}&= A_{0,i}C_0 + A_{1,i}C_1+A_{2,i}\frac{C_1^2-C_0^2}{2},\end{aligned}
\|u_n-u\|_{L^1(\Omega)}\leq\|u_n-u\|_{W_0^{1,q}(\Omega)}&\leq \|(\bar{g}(x,v_n)+\mu)-(\bar{g}(x,v)+\mu)\|_{\mathcal{M}(\Omega)}&=\|\bar{g}(x,v_n)-\bar{g}(x,v)\|_{L^1(\Omega)}\rightarrow 0, ~~n\rightarrow\infty.
\sum_{m=0}^{\infty}\frac{y^mq^{m^2+m}}{(yq;q^2)_{m+1}}\cdot\frac{z^mq^{m^2+m}}{(zq;q^2)_{m+1}}=\sum_{n=0}^{\infty}\sum_{m=0}^{\infty}\sum_{\ell=0}^{\infty}b_\omega(\ell,m,n)z^\ell y^m q^n=\sum_{m=0}^{\infty}\frac{y^mq^m}{(zq;q^2)_{m+1}}.
\Delta:\Gamma\mathcal{E}\to \mathbb{K},\,E\mapsto \lim_{t\to 0}\,\left(\Omega\circ \Phi_\mathcal{E}(t,E)-\,\Omega\circ \Phi^c_\mathcal{E}(t,E)\right) & =\, \lim_{t\to 0}\,\left(\Omega\circ\Phi_\mathcal{E}(0,E)-\,\Omega\circ \Phi^c_\mathcal{E}(0,E)\right) & =\, \Omega(E)-\,\Omega(E)=\,0,
\|D(t\partial_tG(m,t))\|_{\mathbb{B}}= \|te^{-2t}(I_{m+2}(2t) - 3I_{m+1}(2t) + 3I_{m}(2t)-I_{m-1}(2t))\|_{\mathbb{B}}&\le C \,\frac{T_1^{1/2}+T_2^{1/2}+T_3^{1/2}}{\Gamma(m-1/2)},
\begin{aligned}\rho^n_t = \rho^R_{t-s}\left( \rho^n_s \right) + \int_s^t \rho^R_{t-u}\left( \left[ \left( \alpha_R \left( u \right) a^\dagger - \overline{\alpha_R \left( u \right)}a \right)+\left( \overline{\beta_R \left( u \right)}\sigma^{-}- \beta_R \left( u \right) \sigma^+ \right) ,\rho^n_u \right] \right) du \end{aligned}
\partial_t \langle*F_{h(t) kA_+}+ \mu(h(t) ku_+),\xi \rangle &=\left\langle - \left[ *F_{h(t)kA_+}+ \mu(h(t)ku_+), \alpha(t) \right] , \xi \right \rangle+ \left\langle \mathcal{L}_{h(t)k(A_+,u_+)}\beta(t), \mathcal{L}_{h(t)k(A_+,u_+)}\xi \right\rangle &= 0
x = a \cos \sigma \, , \hspace{0 . 5 c m}y = 0 \, , \hspace{0 . 5 c m}z = \int_{0}^{\sigma}\sqrt{1 - a^{2}\sin^{2}\sigma}\, d \sigma \, , \hspace{0 . 5 c m}| \sigma | \le \pi / 2 \, .
(\#S^{g^n_Z}_Z)^*(\{(z, S) \in Z \times \#S^{Y^n}\mid S(\vec{y})\}) := \{(z, R) \in Z \times \#S^{X^n}\mid \bigvee_{\vec{x}\in X^n}((g^n_Z(z, \vec{x}) = (z, \vec{y})) \wedge R(\vec{x}))\}.
\Vert \left(M_{4R}f_j\right)^\lambda \Vert_q =\Vert M_{4R}f_j\Vert^{\lambda}_{p_3}\leq C\Vert(N_{4R}f_j)^\sigma \Vert^{\lambda}_{p_2/\sigma}\leq C\Vert f_j \Vert^{\lambda\sigma}_{p}
\begin{aligned}\mu_{0}^{2s}\partial_t\phi = -(-\Delta)^s_y\phi + pU^{p-1}(y)\phi + \Bigg\{p\mu_{0}^{\frac{n-2s}{2}}\frac{\mu_{0}^{2s}}{\mu^{2s}}U^{p-1}\left(\frac{\mu_{0}}{\mu}y\right)\psi(\xi + \mu_{0}y, t)+\cdots \Bigg\}\chi_{B_{2R}(0)}(y), y\in \mathbb{R}^n.\end{aligned}
\xi \frac{\partial V^{(1)}}{\partial \xi}=\frac12(m_2^2\varphi )\xi e^2\int \frac{d^dk}{(2\pi )^d}\frac1{D_N\,(k^2+m_{gh}^2)}\left\{(\varphi -2u)k^2-\xi ue^2\varphi^2)\right\}.
|(\eta_0 + s) - (\xi_{0}+ r)^{2}| = |(\eta_{0}- \xi_{0}^{2}) + (s - 2r\xi_{0}- r^2)| & \leq \max(|\eta_{0}- \xi_{0}^{2}|, |s|, |2r\xi_{0}|, |r^2|) \leq |J|^2
E_n\big( X_n; \{q_{ij}\}_{1 \le i < j \le n}\big) = (t+x_n)E_{n-1}\big( X_{n-1}; \{q_{ij}\}_{1 \le i < j \le n-1}\big) \hphantom{E_n\big( X_n; \{q_{ij}\}_{1 \le i < j \le n}\big) =}{}+\sum_{j=1}^{n-1}q_{jn}E_{n-2}\big( X_{[1,n-1]{\setminus}\{j \}};\{q_{a,b}\}_{1 \le a < b \le n-1 \atop a\not=j, b \not= j}\big).
\begin{array}{rcl}G = (\varphi(x)\varphi(y))\star i\Delta(z,m^2)+i\Delta(z,m^2)\star(\varphi(y)\varphi(x)) +\varphi(x)\star_x i\Delta(z,m^2)\star_y\varphi(y)+\varphi(y)\star_y i\Delta(z,m^2)\star_x\varphi(x), \end{array}
\left\vert \widetilde{\varphi}_{x,y}\left( k\right) \right\vert =\left\vert\varphi_{x,y}\left( k\right) \right\vert \prod\limits_{j=1}^{m_{1}}\frac{1}{\left\vert k-a_{j}\right\vert},\forall k\in \mathbb{R}.
\langle g_{x y}\rangle = \left( \begin{array}{c c c c}{{b}}&{{0}}&{{0}}&{{0}}\\{{0}}&{{b}}&{{0}}&{{0}}\\{{0}}&{{0}}&{{\langle g_{3 3}\rangle}}&{{\langle g_{3 4}\rangle}}\\{{0}}&{{0}}&{{\langle g_{4 3}\rangle}}&{{\langle g_{4 4}\rangle}}\\ \end{array}\right)
=\sqrt{\frac{(n+1)\cdot(n-l_{1})!(n-l_{2})!(n-l_{3})!}{(n+l_{1}+1)!(n+l_{2}+1)!(n+l_{3}+1)!}}\displaystyle{\sum_{k}}\frac{(-1)^{k}(n+1+k)!}{(n+k-l_{1}-l_{2}-l_{3})!R(l_{1},l_{2},l_{3};k)}\ ,
\int_{S_{y}}D\left(\alpha_{x,g}\parallel\alpha_{x,e}\right)d\eta^{y}(x)=\sup_{n}\left\{\int_{S_{y}}H_{q_{x,g}^{n}\parallel q_{x,e}^{n}}\left(\mathcal{P}_{x,n}\right)d\eta^{y}-\delta_{n}\right\}.
\left|\int_{\S}\phi(t, x)- \phi(0, x)\, dx\right|= \left|\int_{\S}\int_0^t \phi_t(s, x) ds dx\right| \lesssim \int_0^t\|\phi_t(s, x)\|_{L^2_x}ds, \left|\int_{\S}\phi(0, x)- p\, dx\right|\lesssim \|\phi(0, x)- p\|_{L^2_x},
\mathrm{R e s}\left[ I_{C}( k , q ) , q_{0}= | \vec{q}| \right] = \frac{1}{2 | \vec{k}| | \vec{q}|}\left. \left( 4 q \cdot u - 4 k \cdot u \frac{k \cdot q}{k^{2}}- k \cdot u \right) \right|_{q_{0}= | \vec{q}|},
Q_1(\widetilde{\Sigma})+Q_1(\Sigma) \leq\left(\frac{1/r}{1/R}\right)^{n-1}\left(\frac{1}{r}\right)^{n-1}\cdot\frac{2n|\mathbb{S}^n|}{|\widetilde{\Sigma}|^{\frac{n-1}{n}}}=\left(\frac{R}{r}\right)^{n-1}\left(\frac{1}{r}\right)^{n-1}\cdot\frac{2n|\mathbb{S}^n|}{|\widetilde{\Sigma}|^{\frac{n-1}{n}}}
\Phi_{\pmb{\phi}_p}(\alpha_n) = \langle \omega_{\overline{\psi}_p}(\alpha_n)\pmb{\phi}_p, \pmb{\phi}_p \rangle \Phi_{\pmb{\phi}_p}(\beta_m) = \langle \omega_{\overline{\psi}_p}(\beta_m)\pmb{\phi}_p, \pmb{\phi}_p \rangle
\frac{A'_{n}(\tau,1)(\varphi)}{A_n(\tau,1)(\varphi)}\ = \ \frac{W_{n,\infty}'(\tau,1)}{W_{n,\infty}(\tau,1)}\ +{\sum_{\ell \in \Sigma}}\frac{W_{n,\ell}'(1, \varphi )}{W_{n,\ell}(1,\varphi)}\ - \ \frac{L^{\Sigma,'}(2, \chi_F)}{L^{\Sigma}(2,\chi_F)}
\gamma ( s_{p j}, s_{q j}^{\prime}, t ) = \exp \left( \frac{t}{2}\left( \frac{\partial^{2}}{\partial s_{p j}^{2}}+ \frac{1}{s_{p j}}\frac{\partial}{\partial s_{p j}}\right) \right) \, \frac{\delta ( s_{p j}- s_{q j}^{\prime})}{\sqrt{s_{p j}s_{q j}^{\prime}}}
\frac{E_{0}\left( \ell , \mu , d \right)}{L^{d - 1}}\approx - \frac{C \left( d \right) \, \left( 2^{d}- 1 \right) \Gamma \left( d \right) \zeta_{R}\left( d + 1 \right)}{2^{3 d - 1}\pi^{d / 2}\Gamma \left( \frac{d}{2}\right) \, \ell^{d}}.
\mathcal{R}(x_0,t) \leq \int_M G(x_0,t;y,0)\, \mathcal{R}(y,0)\,d\mathrm{vol}_{g_0}(y) \leq \int^\infty_0 \frac{r^n}{t^{n/2}}\cdot \exp\left(-\frac{r^2}{C_1t}\right)\frac{C_1}{rt}k(x_0,r) dr.
S=\frac{1}{16\pi G}\int_M \sqrt{-^{(3)}g}(^{(3)}R+\frac{2}{l^2})d^3x+\frac{1}{8\pi G}\int_{\partial M}\sqrt{-^{(3)}g}(2D_{\mu}n^{\mu}+\frac{1}{l})d^2x.
r_2 \; = \; r_1, l_2\; = \; \lambda^* l_1 \, + \, r_1 \, n \, p_2^2 \, h, \chi_2 \; = \; \chi_1 \, + \, n \, p_2^2 \, I \, + \,r_1 \, n^2 \, p_2^4 \, \frac{h^2}{2}.
\phi^{\alpha}_{J_1;J_2+\bold{1}_k}=D_{J_1}\left(\phi^{\alpha}_{\bold{0};J_2+\bold{1}_k}-\xi^iu^{\alpha}_{\bold{1}_i;J_2+\bold{1}_k}\right)+\xi^iu^{\alpha}_{J_1+\bold{1}_i;J_2+\bold{1}_k}.
\begin{aligned}\overline{\mathrm{mdim}}_M(Y^\mathbb{N}\times X, F,D)=\overline{\mathrm{mdim}}_{I_\psi(F)}(F,D) \le \overline{\mathrm{mdim}}_{Y^\mathbb{N}\times I_\psi(F)}(F,D) &\le \overline{\mathrm{umdim}}_{Y^\mathbb{N}\times I_\psi(F)}(F,D).\end{aligned}
1+\frac{gg^{\prime \prime \prime}}{g^{\prime}g^{\prime \prime}}-2\frac{\left( f^{\prime}\right)^{2}}{ff^{\prime \prime}}+\left( \frac{m_{0}}{f}\right) \frac{1}{g^{\prime \prime}}+\left( \frac{m_{0}}{f^{\prime \prime}}\right) \frac{g^{\prime \prime \prime}}{g^{\prime}g^{\prime \prime}}=0.
\vartheta(\rho,Q) \approx - 2(1-\lambda)\log \cos(2\lambda\alpha)- 2\lambda\log \cos(2(1-\lambda)\alpha) & \approx (1-\lambda)(4\lambda^2\alpha^2)+\lambda(4(1-\lambda)^2\alpha^2) & = 4\lambda(1-\lambda)\alpha^2.
\phi(\tilde{\tau})=\frac{1}{2i}\left\{\frac{\pi\tilde{\tau}}{\omega_2}\;\frac{\theta_1'}{\theta_1}(\frac{\pi a}{2\omega_2})+\log\mid\frac{\theta_1(\frac{\pi(\tilde{\tau}-a)}{2\omega_2})}{\theta_1(\frac{\pi(\tilde{\tau}+a)}{2\omega_2})}\mid\right\},
D_{3}= \frac{\alpha^{2}}{3 2 \pi^{2}}\left( - \sigma^{2}+ A^{2}{\beta^{x}}^{2}+ A^{2}{\beta^{z}}^{2}\right) \left\{\frac{1}{( 3 - d )}- \frac{1}{2}\left[ \ln \left( \frac{\alpha}{\mu_{R}^{2}}\right) - \frac{1}{2}\right] \right\}.
\frac{d}{d t}\biggl |_{t = 0}Z ( R ( S ) , \langle \cdot , \cdot \rangle^{A ( t )}) = \biggl ( \sum_{k = 0}^{N}\frac{1}{2}( - 1 )^{k}T r ( B_{k}) \biggr ) Z ( R ( S ) , \langle \cdot , \cdot \rangle ) .
\sigma_z^2 = \frac{\theta_1}{\theta_1 + \theta_2}\frac{P_1}{h_d^2 + h_c^2}, P_{p1}= \frac{\theta_2}{\theta_1 + \theta_2}P_1, & P_{p2}= \frac{\eta_1}{\eta_1 + \eta_2}P', P_{d2}= \frac{\eta_2}{\eta_1 + \eta_2}P', & P' = (P_2 - (h_d^2 + h_c^2)\sigma_z^2)^{+}.
\begin{aligned}0\le u\le \min\{p_1,q_2\}\ , \ \ 0\le v\le \min\{p_2,q_1\}\ , \begin{cases}0\le s \le \min\{p_1-u,p_2-v\}\ , \ \ \mbox{for (\ref{psi-1case})}\ , 0\le s \le \min\{q_1-v, q_2-u\}\ , \ \ \mbox{for (\ref{psi-2case})}\ . \end{cases}\end{aligned}
\mu (A) = \lim_{\omega}\frac{\dim ((a^{-1}W_i+W_i)\cap A)}{\dim (W_i)}\ge \lim_{\omega}\frac{\dim (a^{-1}W_i\cap A)}{\dim (W_i)}& = \lim_{\omega}\frac{\dim (a(a^{-1}W_i)\cap aA)}{\dim (W_i)}= \mu (aA).
H_{g-Q}=-\int d{\bf r}d{\bf x}\left[{\sf K}_g^b({\bf r})\left(4{\pi}|({\bf r}-{\bf x})|\right)^{-1}{\cal K}_0^b({\bf x})+{\cal K}_0^b({\bf x})\left(4{\pi}|({\bf r}-{\bf x})|\right)^{-1}{\sf K}_g^b({\bf r})\right]\,.
\frac{r_{n}^{3}}{h_{n}}\leq \frac{2^{3n^{\alpha}}}{2^{d(n-1)^{1+\alpha}}}\to 0 \quad\quad\quad\frac{d(n-1)^{1+\alpha}-3n^{\alpha}}{n^{\alpha}}= d\Big{(}1-\frac{1}{n}\Big{)}^{\alpha}(n-1) - 3 \to \infty.
\mathbb{E}I_j(1)\xi_j^2 =\mathbb{E}I_j(1)\bigg(S_j^{(1)}+\frac{\theta_{j}}{\sqrt{np_j}}I_j(1)\bigg)^2 &\leq \mathbb{E}I_j(1)\Big\{\mathbb{E}(S_j^{(1)})^2+2\mathbb{E}|S_j^{(1)}|+1\Big\}<4p_j,
{\cal L}_{\mathrm{{U ( 1 ) F P}}}= i \overline{{c}}\raisebox{- 0 . 5 m m}{\hspace{0 . 6 m m}ule{0.1 mm}{3.1 mm}ule{3.1 mm}{0.1 mm}ule{0.1 mm}{3.2 mm}\hspace{- 3 . 3 m m}ule{3.2 mm}{0.1 mm}\hspace{0 . 7 m m}}c - \kappa \widetilde{h}^{\mu \rho}i \partial_{\mu}\overline{{c}}\partial_{\rho}c + \mathrm{O}( \kappa^{2}) .
\displaystyle{\left( \delta^{I J}- \frac{( x_{a}^{I}- x_{b}^{I}) ( x_{a}^{J}- x_{b}^{J})}{( x_{a}- x_{b})^{2}}\right)}( \textstyle{\frac{d}{d t}x_{a}^{J}- \frac{d}{d t}x_{b}^{J}}) = 0 ,
\pi^{ij}=-2\gamma^{1/2}e^{-\frac{3}{2}(\phi-\tilde{\phi})}\left[ \frac{1}{2}\gamma^{ij}H+\gamma^{-1/3}\left( \frac{\partial H}{\partial h_{ij}}-\frac{1}{3}\frac{\partial H}{\partial h_{kl}}h_{kl}h_{ij}\right) \right] ~~~,
k=\frac{1}{\sqrt{1-|\bar{\nabla}f|^2}}\left[f_{rr}dr^2+2(f_{ra}-\bar{\Gamma}_{ar}^b f_b)drdu^a+(f_{ab}-\bar{\Gamma}_{ab}^r f_r-\bar{\Gamma}_{ab}^c f_c) du^a du^b\right].
\sum_{4 \leq 2^n \leq \lambda^{1/2}}\sum_{\substack{|\mathcal I| = |\mathcal J| = 2^{-n}\mathcal I \sim \mathcal J}}\|\widehat{g_{\mathcal I}}\widehat{g_{\mathcal J}}\|_{L^{\frac{q}{2}}(d\mu)}\lesssim \langle\mu\rangle_\alpha^{\,\,\frac2q}\lambda^{\max (\frac{1}{2}+\frac{(d-1)-\alpha}{q}, 1-\frac{2\alpha}{q}+\frac{2(d-2)}{q}) + \epsilon}\| g \|_2^2.
S_{\mathrm{P 5 h}}= - T_{\mathrm{P 5 h}}\int d^{6}\xi \e^{- 2{\phi}}k^{- 1}\sqrt{| \mathrm{d e t}\ ( \partial_{i}X^{\mu}\partial_{j}{X}^{\nu}g_{\mu \nu}- k^{2}F_{i}F_{j}) |}+ \mathrm{W Z}\, ,
\hat{R}d K^{\dagger}_{1}\hat{R}K_{1}= K_{1}\hat{R}dK^{\dagger}_{1}\hat{R}^{-1}\;, \quad\hat{R}d K^{\dagger}_1 \hat{R}d K^{\dagger}_1 = - d K^{\dagger}_{1}\hat{R}dK^{\dagger}_{1}\hat{R}^{-1}\;.
\zeta^1 = \phantom{-}\frac{\xi}{2}+ i \tau \rho_2 + \tau \sqrt{1 - \frac{|\xi|^2}{4\tau^2}+ \frac{i\omega\sigma_0\mu_0}{\tau^2}}\rho_1, \zeta^2 = -\frac{\xi}{2}+ i \tau \rho_2 + \tau \sqrt{1 - \frac{|\xi|^2}{4\tau^2}+ \frac{i\omega\sigma_0\mu_0}{\tau^2}}\rho_1.
\sum_lh^{p^{\ast}}_{ijkl}\omega_l=dh^{p^{\ast}}_{ijk}+\sum_lh^{p^{\ast}}_{ljk}\omega_{li}+\sum_lh^{p^{\ast}}_{ilk}\omega_{lj}+\sum_l h^{p^{\ast}}_{ijl}\omega_{lk}+\sum_{q}h^{q^{\ast}}_{ijk}\omega_{{q^{\ast}p^{\ast}}},
\begin{dcases}\partial_t \rho + \nabla \cdot (\rho u) = 0 \partial_t (\rho u) + \nabla \cdot (\rho u \otimes u) + \nabla p_1(\rho) = - \rho \nabla \phi \partial_t n + \nabla \cdot (n v) = 0 \varepsilon \big(\partial_t (nv) + \nabla \cdot (nv \otimes v)\big) + \nabla p_2(n) = n \nabla \phi -\delta \Delta \phi = \rho - n \ \end{dcases}
\|V_{12}f\|^2 \!\leq\! a_{12,p}^{2}(b_{12,p})\|f\|^2\!\!+\!b_{12,p}^2\|H_S f\|^2\!, \ \a_{12,p}(b_{12,p})\!:=\!C_p^{\frac{2p}{2p-d}}b_{12,p}^{-\frac{d}{2p-d}}\!,\ \b_{12,p}\!>\!0,
H^{\prime}( t ) = U_{A}^{\dagger}( t ) V ( t ) U_{A}( t ) = - \sum_{n , m , n \neq m}e^{- i ( \alpha_{m}( t ) - \alpha_{n}( t ) )}\langle m , t | i \hbar \partial_{t}| n , t \rangle | m , 0 \rangle \langle n , 0 |
M_{K}= \left( \begin{array}{rrrrrrrrrrrrrrrrrrrr}1 0 0 -1 0 -1 0 0 1 0 0 1 -1 0 0 0 0 0 0 0 0 -1 1 0 0 \end{array}\right),M_{L, M}= \left( \begin{array}{rrrrrrrrrrrr}1 0 0 0 -1 0 0 0 0 0 0 0 -1 0 0 0 1 0 0 1 0 -1 0 0 0 0 0 0 0 0 0 -1 0 1 0 0 \end{array}\right),
\frac{d^2}{dt^2}\Big|_{t=0}\lambda=\int_{M}\langle \mathcal{N}(h,K,v_{h,K}),(h,K,v_{h,K}) \rangle e^{-f}dV_g=\Big(\mathcal{N}(h,K,v_{h,K}),(h,K,v_{h,K})\Big)_{f}
\Delta(X)=X\otimes 1+ g\otimes X,\ \ \ \ \ \Delta(Y)=Y\otimes 1+ h\otimes Y, gX=\zeta_m^a\lambda_1Xg,\hX=\lambda_2Xh,\gY=\zeta_n^b\eta_1Yg,\hY=\zeta_n^c\eta_2Yh, X^{\vec{N}_1}=0,\Y^{\vec{N}_2}=0, \XY=\lambda_2^{-1}YX, \where \N_1=|\zeta_m^{a}\lambda_1|, \N_2=|\zeta_n^c\eta_2|,
\hat{\rho}_{AM}(t) = \left(\mathcal{N}_A\left(\frac{\lambda\, t}{\eta}\right)\otimes\mathbb{I}_M\right)(\hat{\rho}_{AM})\;, \hat{\rho}_{BM}(t) = \left(\mathcal{N}_B\left(\frac{(1-\lambda)\,t}{|1-\eta|}\right)\otimes\mathbb{I}_M\right)(\hat{\rho}_{BM})\;, \hat{\rho}_{CM}(t) = \left(\mathcal{N}_C(t)\otimes\mathbb{I}_M\right)(\hat{\rho}_{CM})\;,
R_{10}\le I(S_1;Y_1) + I(S_2;Y_1|S_1,T_1) + I(S_3;Y_1|S_1,T_1,S_2), R_{20}\le I(U;Y_1|S_1,T_1,S_2,S_3), R_{11}\le I(T_1;Y_1|S_1) + I(T_2;Y_1|S_1,T_1,S_2,S_3,U),
N_{(2,5)}(n) = ~ 8\sigma(\frac{n}{2}) - 32\sigma(\frac{n}{8}) + 8\sigma(\frac{n}{5}) -32\sigma(\frac{n}{20}) + 64\, W_{(2,5)}(n) + 1024\, W_{(2,5)}(\frac{n}{4}) - 256\, \biggl( W_{(5,8)}(n) + W_{(1,10)}(\frac{n}{2}) \biggr),
\int |\Delta W^{+}|^{2}= -\int \langle \nabla W^{+}, \nabla \Delta W \rangle = -\frac{R}{2}\int |\nabla W^{+}|^{2}+ 6 \int W^{+}_{ijkl}W^{+}_{ijpq,t}W^{+}_{klpq,t}
C=\left( \begin{array}{ccccc}\textsf{a}\textsf{b}0 0 0 \textsf{b}2\textsf{b}-\textsf{a}0 0 0 0 0 \textsf{b}-\textsf{a}0 0 0 0 0 \textsf{b}-\textsf{a}0 0 0 0 0 \textsf{b}-\textsf{a}\end{array}\right)
{C}_{n}(\kappa,{\bf P}_{Y_1})= \sup_{(\Lambda_t, K_{\overline{Z}_t}), t=1, \ldots, n, \frac{1}{n}{\bf E}\big\{\sum_{t=1}^{n}||X_t||_{{\mathbb R}^{n_x}}^2\big\}\leq \kappa}\sum_{t=1}^n \Big\{H(I_t)-H(\hat{I}_t)\Big\}
\begin{array}{l}\displaystyle [e_1,e_{n+1}]=e_1+(a-2)e_3,[e_3,e_{n+1}]=(a-1)e_3, [e_4,e_{n+1}]=(1-a)e_2+ae_4, \displaystyle[e_{j},e_{n+1}]=\left(a+j-4\right)e_{j},(5\leq j\leq n),[e_{n+1},e_1]=-e_1+(2-a)e_3, \displaystyle[e_{n+1},e_2]=-ae_2,[e_{n+1},e_i]=\left(4-i-a\right)e_i,(3\leq i\leq n,n\geq4).\end{array}
0= \langle\nabla\nabla S\vert k \vert m_{3}m_{2}km_{2}\rangle_{3}=-\nabla_{3}\nabla_{2}S_{23}+\nabla_{1}\nabla_{2}S_{02}=-\nabla_{2}S_{12}\nabla_{3}k_3, 0= \langle\nabla\nabla S\vert k \vert m_{2}m_{2}km_{2}\rangle_{3}=-\nabla_{2}\nabla_{2}S_{23}=-\nabla_{2}S_{12}\nabla_{2}k_{3}, 0= \nabla_{0}\nabla_{2}S_{23}=\nabla_{2}S_{12}\nabla_{0}k_{3},
|{\bf x}_i-{\bf x}_j|^2=\frac{2k(1-u^2)}{k-1},\ |{\bf y}_i-{\bf y}_j|^2=\frac{2l(1-v^2)}{l-1}, \ |{\bf z}_i-{\bf z}_j|^2=\frac{2m(1-w^2)}{m-1},
z=\sqrt{2\pi}e^{\frac{\epsilon_{0}}{4}}\sum_{j=0}^{\infty}\sum_{k=0}^{\infty}\frac{(j+\frac{1}{2})I_{\frac{1}{2}-k}(1)}{k!2^{k}e^{\epsilon_{0}(j+\frac{1}{2})^{2}}}(\tau^{2}-1)^{k},\tau\ge0
K_X^2=\frac{1}{4}c^2_1(\omega_X^{[2]})=\frac{1}{4}2c_1^2(\omega_Y^{[2]}\otimes \mathcal{O}_Y(10))=\frac{1}{2}c_1^2(\mathcal{O}_Y(2))=\frac{1}{2}4c_1^2(\mathcal{O}_Y(1))=\frac{1}{2}\cdot 4 \cdot \frac{1}{2}=1.
P_k(m, x) = \sum_{j=0}^{k}(-1)^{j-k}x^{j}\binom{j+m-k}{m-k}{m+1 \brack j+m-k+1}&= \sum_{j=m-k}^{m}(-1)^{j-m}x^{j+k-m}\binom{j}{m-k}\frac{m!}{j!}&\times B_{j}\left(H_m^{(1)},-H_m^{(2)},\ldots, (-1)^{j-1}(j-1)! H_m^{(j)}\right).
\int_{|i-x|\leq1}dx \int_{S_i}dvf^N(x,v,t)\leq C\int_{|i-x|\leq1}dx\int_{S_i}e^{-\bar{\lambda}|v|^2}dv\leq &e^{-\frac{\bar{\lambda}}{2}a_i^2}\int_{|i-x|\leq1}dx\int_{}e^{-\frac{\bar{\lambda}}{2}|v|^2}dv\leq \frac{C}{|i|^{2+\epsilon}}.
az^2+bz^{q+1}+z^{2q}\cr=\, e(c+db)(x+yb)^2+b(x+yb)(x+yb^q)+(x+yb^q)^2\cr=\, e(c+db)(x+yb)^2+b(x^2+dxy-cy^2)+\bigl[x+y(d-b)\bigr]^2\cr=\, \cdots\quad\cr=\, (ce+1)x^2+(2cde+2d)xy+(c^2e+cd^2e+c+d^2)y^2\cr +\bigl[(de+1)x^2+(2ce+2d^2e+d-2)xy+(2cde+d^3e-c-d)y^2\bigr]b.
\frac{\partial}{\partial x_i}U^p(x) = p\left( |x|_1 - \frac{d\lambda'}{\delta'}\right)^{p-1}; \frac{\partial^2}{\partial x_i \partial x_j}U^p(x) = 2p(2p-1)\left( |x|_1 - \frac{d\lambda'}{\delta'}\right)^{2p-2}.
\begin{array}{l}{{A_{1}^{( n )}= A^{( 1 )}\otimes I \otimes \cdots \otimes I ,}}\\{{\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots}}\\{{A_{n}^{( n )}= I \otimes I \otimes \cdots \otimes A^{( 1 )}.}}\\ \end{array}
{\bf g}_1 ({\bf X}_i) :={\bf h}_{\tau, 1}({\bf X}_i)- \Theta ,{\bf g}_2 ({\bf X}_i ,{\bf X}_j) :={\bf h}_\tau ({\bf X}_i,{\bf X}_j)-{\bf h}_{\tau, 1}({\bf X}_i) -{\bf h}_{\tau, 1}({\bf X}_j)+ \Theta.