harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\nabla\pi_L =\left[\begin{array}{cccc}1 0 0 0 * 0 \frac{2}{c_0}\pi_1\pi_{T\Psi}\hat{R}(u, v, s) \frac{2}{c_0}\pi_2\pi_{T\Psi}\hat{R}(u, v, s) * * \partial_u(\frac{2\tau}{c_0}\hat{R}(u, v, s)\cdot\dot{\gamma}(s)) \partial_v(\frac{2\tau}{c_0}\hat{R}(u, v, s)\cdot\dot{\gamma}(s)) 0 1 0 0\end{array}\right]
M=\begin{bmatrix}a_0 a_1 a_2 \dots a_m 0 0 \dots 0 a_0 a_1 a_2 \dots a_m 0 \dots 0 0 a_0 a_1 a_2 \dots a_m \dots \dots 0 \dots 0 a_0 a_1 a_2 \dots a_m b_0 b_1 b_2 \dots b_m 0 0 \dots 0 b_0 b_1 b_2 \dots b_m 0 \dots 0 0 b_0 b_1 b_2 \dots b_m \dots \dots 0 \dots 0 b_0 b_1 b_2 \dots b_m\end{bmatrix}.
\|v_h\|_{1,h,G}^2:=\sum\limits_{K\in G\cap\mathcal{T}_h}\|\nabla v_h\|_{L^2(K)}^2+\sum\limits_{E\in G\cap\mathcal{E}_h}h_E^{-1}\|[v_h]\|_{L^2(E)}^2.
\|M+N\|_{\widetilde{S}^{p}_q}= \lim_{n\to \infty}\|M^n+N^n\|_{\widetilde{S}^{p}_q}& = \lim_{n\to \infty}\|M^n+N^n\|_{\widetilde{S}^{p,\mathcal{T}}_q}& \leq \lim_{n\to \infty}\|M^n\|_{\widetilde{S}^{p,\mathcal{T}}_q}+\|N^n\|_{\widetilde{S}^{p,\mathcal{T}}_q}= \|M\|_{\widetilde{S}^{p}_q}+\|N\|_{\widetilde{S}^{p}_q}.
|J_{4,a}|+|J_{4,c}| \leq \frac{p'(\varrho)}{\varrho}\varepsilon^{-1}\|\rho-\varrho\|_{L^\infty(0,T;L^2(\Omega_R))}\left(\|v\|_{L^\infty(0,T;L^{q_1}(\Omega_R))}+\|\nabla\Psi\|_{L^\infty(0,T;L^{q_1}(\Omega_R))}\right)\|\nabla s\|_{L^1(0,T;L^{{q_2}}(\Omega_R))}&\leq c \varepsilon^{1-\frac{2}{q_2}}
U_{SW}(\theta,P_{BS},p_{s},\rho_{e}) = \mu T_{s}- \theta (AP_{BS}^{2}+ BP_{BS}) &~~~~~~ = \mu(1-\theta) \bigg[ \log\bigg( 1 + \frac{p_{s}|h_{s}|^{2}}{\sigma_{s}^{2}}\bigg) &~~~~~~ - \log (1 + \rho_{e})\bigg] - \theta (A P_{BS}^{2}+ B P_{BS}).
\hat{S}_{naive}(t)=\frac{1}{m(t)}\sum_{k:c_{k}\ge t}X_{k}=\frac{|\{1\le k\le n:\tilde{\tau}_{k}\ge t\}|}{|\{1\le k\le n:c_{k}\ge t\}|}=\frac{|\{1\le k\le n:\tilde{\tau}_{k}\ge t\}|}{m(t)},
U= \begin{pmatrix}0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 \end{pmatrix}\ \ \{\text and}\ \ \ |T|= \begin{pmatrix}0 0 0 0 0 1 0 0 0 0 2 0 0 0 0 1 \end{pmatrix}.
(\rho^{k}\otimes (^{k}\rho))\Delta(m_{1}rm_{2}) \Diamond \rho(z) = (\rho^{k}\otimes (^{k}\rho))(1 \otimes m_{1}rm_{2}+m_{1}\otimes rm_{2}) \Diamond \rho(z) &=(\bar{e_{k}}\otimes \rho(m_{1}rm_{2})) \Diamond \rho(z) &=cyc(\rho(m_{1}rm_{2}z)) &=\rho^{k}(\Delta(m_{1}rm_{2}) \Diamond z)
A=\frac12 \Big(([f(n)f(n+3k)-f(n+2k)f(n+k)] &+[f(n+2k)f(n+5k)-f(n+4k)f(n+3k)]\dots +[f(n+(2m-4)k)f(n+(2m-1)k- f(n+(2m-2)k)f(n+(2m-3)k)] +[f(n+(2p-2)k))f(n+k)-f(n)f(n+(2p-1)k)] \big) .
(T \bar{\otimes}T')(|e_k\rangle_{\mathcal{H}_A}\langle e_l|_{\mathcal{H}_A}\otimes|e_{k'}\rangle_{\mathcal{H}_C}\langle e_{l'}|_{\mathcal{H}_C}) &\quad:=T(|e_k\rangle_{\mathcal{H}_A}\langle e_l|_{\mathcal{H}_A})T'(|e_{k'}\rangle_{\mathcal{H}_C}\langle e_{l'}|_{\mathcal{H}_C}).
\left( J_{\varepsilon}^{\prime}\left( W_{\varepsilon,\xi(y)}+\phi_{\varepsilon,\xi(y)}\right) +\frac{\omega^{2}}{2}G_{\varepsilon}^{\prime}\left( W_{\varepsilon,\xi(y)}+\phi_{\varepsilon,\xi(y)}\right) \right)\left[ \frac{\partial}{\partial y_{h}}\phi_{\varepsilon,\xi(y)}\right]=o(1).
\tilde \mu_1 |(y,p)|_{1,\frac{1}{2}}\leq \sup_{0 \not= (z,q) \in (H^{1,\frac{1}{2}}_{0,per}(Q_T))^2}\frac{\mathcal{B}((y,p),(z,q))}{|(z,q)|_{1,\frac{1}{2}}}\leq \tilde \mu_2 |(y,p)|_{1,\frac{1}{2}},
{\rm tr}({\bf A}_z{\bf A}_y^{-1})-n\frac{\langle{\bf A}_z{\bf A}_y^{-1/2}(z-y),{\bf A}_y^{-1/2}(z-y)\rangle}{|z-y|^2}+\langle{\bf B}_z{\bf A}_y^{-1/2},(z-y)\rangle,
\sum_{m=-\infty}^{\infty}f(m)e^{-im\pi}= \hat{f}(\pi)+\hat{f}(-\pi) + 2\sum_{k=1}^{\infty}\hat{f}(\pi +2k\pi) = \int_{-\infty}^{\infty}2\cos(\pi t)f(t)\, dt + 2\sum_{k=1}^{\infty}\hat{f}(\pi +2k\pi).
{\cal{Z}}^{F}( X_{1}, \cdots , X_{M}) ={\frac{\det ( X_{j}^{2 M + 2 - i}- X_{j}^{i})}{\det ( X_{j}^{2 M + 1 - i}- X_{j}^{i})}}~ ~ ; ~ ~ 1 \le i , j \le M \, \, \, \, .
R_{00}+ R_{01}+ R_{02}+ R_{03}= R_{0}, R_{11}+R_{12}+R_{13}= \log |\mathcal{D}_1(K_0)|, R_{21}+R_{22}+R_{23}= \log |\mathcal{D}_2(K_0)|, R_{31}+R_{32}+R_{33}= \log |\mathcal{D}_3(K_0)|.
L^{(1)}_{\ell}(x_i;f):= \frac{1}{x_i}\sum_{\substack{j=1 x_i \geqslant y_{j-1}}}^J \int_{y_{j-1}}^{y_j}X \sum_{t/(1+1/X) < p \leqslant t}\frac{1}{p}\big|\Psi_f^{\prime}(x_i/t,p)\big|^2{\rm d}t
\delta x_{i}( s ) \; = \; ( \partial_{s}x_{i}) ( s ) \delta \widetilde{\gamma}_{i}[ g ] ( s ) , \delta \varphi_{i}( s ) \; = \; ( \partial_{s}\varphi_{i}) ( s ) \delta \widetilde{\gamma}_{i}[ g ] ( s ) \; \; \;
-{\frac{1}{r^{2}}}\; \left( 1 -{\frac{r_{0}}{r}}\right){\frac{d}{d r}}\left( r ( r - r_{0}) \;{\frac{d R_{\ell}}{d r}}\right) +{\frac{\ell ( \ell + 1 )}{r^{2}}}\; \left( 1 -{\frac{r_{0}}{r}}\right) \; R_{\ell}= k^{2}\; R_{\ell}
= \frac{2}{( 1 - 1 3 x ) ( x - 3 )}+ \frac{1}{3 ( x - 1 3 )}= \frac{6 + x - 3}{3 ( x + 3 ) ( x - 3 )}= \frac{x + 5}{3 ( x - 5 ) ( x - 3 )}= \frac{1}{3 x - 9}
d_{\alpha}({\bf x},\Sigma) =\min\big\{\,|x_k|, \tfrac{1}{\sqrt{2}}|x_k-x_{j}|\,\big|\, k\in Q_{\alpha}, j\in\{1,\ldots,N\}, j\neq k\,\big\}\,, |{\bf x}| \ge d_{\alpha}({\bf x},\Sigma)\ge d({\bf x},\Sigma)\,.
L_n(w, \theta_1) - L_n(w, \theta_2)= \frac1{2n}\sum\limits_{i = 1}^n \log \frac{D_{\theta_1}(X_i)}{D_{\theta_2}(X_i)}+ \frac1{2n}\sum\limits_{i = 1}^n \log \frac{1 - D_{\theta_1}(g_w(Y_i))}{1 - D_{\theta_2}(g_w(Y_i))}
\sum_{i=1}^m\frac1{\eta_i}=\frac{r}{(m-1)(r-1)}\sum_{i=1}^{m-1}\frac1{\theta_i}+ \frac1{(m-1)(\frac{1-r}{r}\varrho)'}&=1-\frac{r}{(m-1)(r-1)}\sum_{i=1}^{m-1}\frac1{\delta_i}+\frac1{m-1}-\frac1{m-1}\frac{r}{(1-r)\varrho}&=1-\frac{r}{(m-1)(r-1)}\Big(\frac{1-r}{r}-\frac1{\varrho}\Big)+\frac1{m-1}-\frac1{m-1}\frac{r}{(1-r)\varrho}&=1.
S~=~-\frac{1}{2}\int e^{-{\bf \Phi}}\Bigl\{\epsilon_{abc}{\bf e}^a{\bf R}^{bc}+ d{\bf \Phi}~{^{*}d}{\bf \Phi}- 2{\bf H}~{^{*}{\bf H}}+{\alpha' \over 8}Tr{\bf F}~{^{*}{\bf F}}+ \frac{\Lambda(\Phi)}{3}\epsilon_{abc}{\bf e}^a{\bf e}^b{\bf e}^c~\Bigr\}
K\big(R(1+\cos\gamma_0)-1\big)-M\left(1+\frac{1}{R}\right) \geq K\Big\{\Big((1+\sin\gamma_0)\mathcal{M}\big(L^+_{\gamma_0}(t)\big)-1\Big)(1+\cos\gamma_0)-1\Big\}& -M\Big(1+\frac{1}{(1+\sin\gamma_0)\mathcal{M}\big(L^+_{\gamma_0}(t)\big)-1}\Big) &=: \Delta(t).
r_{i}^{-t_k}= \sum\limits_{j\in J^k}r_{j}^{t_k}\geq \sum\limits_{j\in J^k}r_{j}^{\dim_H(\mathcal{G}(F^{\alpha}))}= \sum\limits_{j\in J^k}r_{j}^{t_*}r_{j}^{\dim_H(\mathcal{G}(F^{\alpha}))-t_*}\geq \sum\limits_{j\in J^k}r_{j}^{t_*}r_{max}^{k(\dim_H(\mathcal{G}(F^{\alpha}))-t_*)}= r_{max}^{k(\dim_H(\mathcal{G}(F^{\alpha}))-t_*)},
\begin{multlined}[t][12.5cm]y_{b}=\frac{b+\nu_{G}}{2b+\nu_{G}}=\frac{1+\frac{\nu_{G}}{b}}{2\left(1+\frac{\nu_{G}}{2b}\right)}=\frac{1}{2}+\frac{\nu_{G}}{4b}+O\left(\frac{1}{b^{2}}\right),\end{multlined}
\int_{\mathcal{A}}l(x)^KR(x,n)^jdx=D^j\int_{\mathcal{A}}l(x)^{K-j}dx+\sum_{h=1}^j\binom{j}{h}D^{j-h}\int_{\mathcal{A}}l(x)^{K-j+h}Q(x)^hdx.
I_{D_{i_1}}^{+,i_1}(z)=\frac{1}{i_1!j_1!k_1!}\int_{0<t_1<z}L_1(t_1,z)^{i_1}L_{-1}(t_1,z)^{j_1}L_0(t_1,z)^{k_1}\frac{dt_1}{1-t_1}, I_{D_{i_1}}^{-,i_1}(z)=\frac{1}{i_1!j_1!k_1!}\int_{0<t_1<z}L_1(t_1,z)^{i_1}L_{-1}(t_1,z)^{j_1}L_0(t_1,z)^{k_1}\frac{-dt_1}{1+t_1}.
\left\langle \left[ \partial_{\mu y}A_{\nu}^{m}( y ) \right] \left[ g \epsilon^{3 m n}a^{\mu}( x ) A^{n \nu}( x ) \right] \right\rangle \Bigr |_{y = x}= - 8 i g a^{\mu}( x ) \partial_{\mu y}\mathcal P ( y , x ) \Bigr |_{y = x}.
\chi_{(\lambda,\mu)}= \frac{\sum_{w\in W}(-1)^{l(w)}e^{\left(w\left(\frac{\lambda+\mu}{2}+\rho\right) ,w\left(\frac{\lambda+\mu}{2}+\rho\right)\right)}}{\sum_{w\in W}(-1)^{l(w)}e^{(w(\rho),w(\rho))}}e^{\left(\frac{\lambda-\mu}{2},-\frac{\lambda-\mu}{2}\right)},
v^{k+1}(\ell) - [y^{k+1}]_\ell\mathbf{1}= \left(W(k)-\frac{1}{N}\mathbf{1}\mathbf{1}^TW(k)\right) v^k(\ell) + \left(\mathbb{I}- \frac{1}{N}\mathbf{1}\mathbf{1}^T \right)\zeta^{k+1}(\ell),
d\omega(f,g,y) = \omega([f,g],y) + \omega([g,y],f) + \omega([y,f],g) = \omega_0([f,g],\lambda \otimes y) + \omega_0([g,y], f) + \omega_0([y,f],g) = \omega_0([f,g],\lambda \otimes y) + \omega_0([g, \lambda \otimes y], f) + \omega_0([\lambda \otimes y, f] , g)= d\omega_0 (f,g,\lambda \otimes y) =0.
Q_n(z) =\sum_{\frac{h}{k}\in F_N}\int_{M_{h,k}}P(z,e^{-\tau})e^{n\tau}d\tau, =\sum_{\frac{h}{k}\in F_N}\int_{- \frac{1}{k(k+k')}}^{\frac{1}{k(k+k'')}}e^{-2\pi in \frac{h}{k}}e^{2\pi n (\alpha_n - iv)}P(z,e^{-2\pi (\alpha_n - iv) +2\pi i \frac{h}{k}})dv.
\frac{\partial}{\partial u}g(z,u)|_{u=1}=\frac{1}{1-z}\sum_{k\ge m}(\begin{matrix}k\\m\end{matrix})\frac{z^{k}}{k}=\frac{1}{1-z}\frac{1}{m}\frac{z^{m}}{(1-z)^{m}}=\frac{1}{m}\frac{z^{m}}{(1-z)^{m+1}}.
0=\left\langle \phi',\left( (5 h_{0}+1)\;h_{0}\;\Delta\,(Q_{-2})- 12 q_{0}\; \Delta\,(L_{-1})^{2}+ 6 q_{0}\,(h_{0}+1)\; \Delta\,(L_{-2})\right)\left(\phi\otimes \phi_{0}\right) \right\rangle.
{{A}_{i}^{\gamma}}({\bf{r}}) = \frac{1}{g r}\left\{\left({\delta}_{{i}\,{\gamma}}- \frac{r_{i}r_{\gamma}}{r^{2}}\right) \,{\cal N}+ \frac{r_{i}\, r_{\gamma}}{r}\frac{d \,{\cal N}}{d r}\right\}+{\delta}_{{i}\,{\gamma}}\,{\cal T}_{A}+ \frac{r_{i}\, r_{\gamma}}{r^{2}}\,{\cal T}_{B}+{\epsilon}_{i{\gamma}n}\frac{r_{n}}{r}\,{\cal T}_{C}\, ,
R_{\mu \nu \rho \sigma}R^{\mu \nu \rho \sigma}= \lambda_{s}^{2}( \gamma_{2}+ \frac{\gamma_{1}^{2}}{2})^{2}+ \frac{2 \gamma_{1}^{2}\lambda_{s}^{2}}{r_{s}^{2}}+ \frac{4 ( 1 - \lambda_{s})^{2}}{r_{s}^{4}}+{\cal{O}}( r - r_{s})
P'(X_1 ; \,Y_1, Y_2) = \sum_{k=0}^3 i^k P_k'(X_1 ; -i Y_1, -i Y_2) P''(X_1, X_2, X_3 ;\, Y_1, Y_2) = \sum_{k=0}^2 i^k P_k''(X_1, -X_2, -X_3; -i Y_1, -i Y_2)
( \partial^{*}F_{\ell})_{\bar{z}}=_{\mathcal{H}^{n-1}}B^{n-1}\left( 0 , r_{\ell}^{\vee}(\bar{z})\right) \backslash B^{n-1}\left( 0 , r_{\ell}^{\wedge}(\bar{z})\right).
Y_{4}^{\ast}\left( M,g\right) =\frac{n-4}{2}\inf_{\substack{\widetilde{g}\in \left[ g\right] \widetilde{R}>0}}\frac{\int_{M}\widetilde{Q}d\widetilde{\mu}}{\left( \widetilde{\mu}\left( M\right) \right)^{\frac{n-4}{n}}}.
x_{+}(t) = x_{\alpha_{7}}(t^{5^r})x_{\alpha_2}(t^{5^r})x_{\alpha_{5}}(t^{5^s})x_{-1122100}(t^{5^r})x_{\alpha_{1}}(t^{5^r}) x_{-1112100}(t^{5^{r+1}})x_{1010000}(t^{5^{r+1}}), x_{-}(t) = x_{-\alpha_{7}}(t^{5^r})x_{-\alpha_{2}}(t^{5^r})x_{-\alpha_{5}}(t^{5^s})x_{1122100}(t^{5^r})x_{-\alpha_{1}}(t^{5^r}) x_{1112100}(t^{5^{r+1}})x_{-1010000}(t^{5^{r+1}}),
\tilde{w}_{\mu_2}\psi_{w_{\mu_2}(\mu_1)}(x)\tilde{w}_{\mu_2}^{-1}=\tilde{w}_{\mu_2}\tilde{w}_\theta\tilde{x_1}\tilde{w}_\theta^{-1}\tilde{w}_{\mu_2}^{-1}=\tilde{w}_\theta\tilde{w}_{w_\theta(\mu_2)}(z)\tilde{x_1}\tilde{w}_{w_\theta(\mu_2)}(z)^{-1}\tilde{w}_\theta^{-1}
\begin{aligned}(\tilde{\Delta}_E\tilde{h}_1,\tilde{h}_2)_{L^2(\tilde{g})}= -12\lambda_i(\lambda_i-4)\int_0^{\infty}\varphi\psi\cosh \sinh^{2}dr, (\tilde{\Delta}_E\tilde{h}_2,\tilde{h}_3)_{L^2(\tilde{g})}= 20\lambda_i\int_0^{\infty}\psi\chi \cosh \sinh^{2}dr.\end{aligned}
\sum_{q \ge -1}2^{qs}\sum_{\vert j-q \vert \le 4}2^{-js}c_j =\sum_{q\ge -1}[(\mathbf{1}_{\vert j \vert\le 4}2^{js})*c_j](q) &\le \sum_j \mathbf{1}_{\vert j\vert\le 4}2^{js}\cdot\sum_j c_j <\infty.
\Vert(0, E_1(u))\Vert^2_{G}(t) =\sum_{i\in I_1}(\mathcal{E}_{i,1}(t))^2+(\mathcal{E}_{i,2}(t))^2+\sum_{i\in I_2}(\mathcal{E}_{i,3}(t))^2+(\mathcal{E}_{i,4}(t))^2+\sum_{i\in I_3\cup I_4}(\mathcal{E}_{i}^+(t))^2+(\mathcal{E}_{i}^-(t))^2 =O(e^{2(\gamma_*-\varepsilon_0)t}).
i \Pi_{( \beta , \mu )}^{\mu \nu}( 0 , \overline{{p}}) = - \frac{4 g^{2}}{\beta}\sum_{n}\int \frac{d^{2}k}{( 2 \pi )^{2}}\frac{{\mathit{M}}^{\mu \nu}. s i n^{2}( \frac{\tilde{p}k}{2})}{( ( k - p )^{2}+ m^{2}+ \omega_{n}^{2}) ( k^{2}+ m^{2}+ \omega_{n}^{2})},
\mathrm{DTV}_{(p,q)}(X) =-\mathrm{DTE}_{(p,q)}^{2}(X)+\mu^{2}+2\mu\sigma\mathrm{DTE}_{(p,q)}(Y)+\sigma^{2}\left[L_{1}+\frac{\Psi_{1}^{\ast}(-1,\frac{1}{2},1)}{\Psi_{2}^{\ast}(-1,\frac{1}{2},1)}L_{2}\right],
\mathcal L(\psi, \phi, x, t) = \mathcal L_{\mathrm{Dirac}}(\psi) + \mathcal L_{\mathrm{meson}}(\phi) + \mathcal L_{\mathrm{Yukawa}}(\psi,\phi) + \mathcal L_{\mathrm{noise}}(\psi, \phi, x, t)
R_q^{p+1,q}=\left|\begin{array}{ccccc}\hat{a}_{11}a_{12}\cdots a_{1p}b_1 a_{21}\hat{a}_{22}\cdots a_{2p}b_2 \vdots \vdots \ddots \vdots \ddots a_{p1}a_{p2}\cdots \hat{a}_{pp}b_p a_{q1}a_{q2}\cdots a_{qp}b_q \end{array}\right| ,
\Vert P_\frac{t}{2}(1_A) \Vert_2^2=\mu(A)^2 + \Vert P_\frac{t}{2}(1_A-\mu(A)) \Vert_2^2 \le \mu(A)^2 +e^{-\lambda_1 t}\| 1_A-\mu(A) \|_2^2
c_{p}\geq \frac{1}{2^{p^2}(p^2-1)}\cdot 2^{(p^2-1)\cdot(p^2-2)/(p^2-1)}\phi(p^2-1)=\frac{1}{4}\cdot\frac{\phi(p^2-1)}{p^2-1}\gg \frac{1}{\log\log p^2}, p\to \infty,
\int_{r_{1}}^{r_{2}}d r \sqrt{2 \mu \left( E - \tilde{V}( r ) - \frac{\eta + \frac{1}{4}}{2 \mu r^{2}}\right)}= \left( n + \frac{1}{2}\right) \pi . \; \; ( n = 0 , 1 , 2 , \cdots )
\mathrm{P}\Big( \sum_{i=1}^k Y_{[i]}\ge Ck \log(\tfrac{en}{k})+c\Big) \le A_k(t) \exp\{-(Ct-1)k \log(\tfrac{en}{k})-ct\}, &\mathrm{E}\sum_{i=1}^k Y_{[i]}\le t^{-1}\big[k \log(\tfrac{en}{k})+\log (A_k(t))\big].
\begin{aligned}\sum\limits_{i_1 = 1}^n \sum\limits_{i_2 = 1}^n \dotsb \sum\limits_{i_{k-1}= 1}^n d_{i}X_{i_1,i}\cdot d_{i_1}X_{i_2,i_1}\dotsb d_{i_{k-1}}X_{j,i_{k-1}}.\end{aligned}
\gamma_{j_{k}}=\frac{w_{j_{k'},i}}{w_{j_k,i}}\Bigl(\frac{y_{\sigma(\iota_i)}}{w_{j_{k'},i}(1+\lambda_{\ell_{\iota_i}}/\mu_{i})}-\frac{y_{\sigma(\iota_{i'})}}{w_{j_{k'},i'}(1+\lambda_{\ell_{\iota_{i'}}}/\mu_{i'})}\Bigr),
F_3 :=E(2C_R+\frac{3}{2})\int_{t_0}^{u \wedge \tau_{nR}}|x^n_s-x^n_{\kappa(n,s)}|^2I_{B(R)}ds & \leq (f(R)+1)K\sup_{t_0 \leq t \leq t_1}E|x^n_t-x^n_{\kappa(n,t)}|^2
G^n_{ij}\!= f^n_{ij}\!-\!\kappa_x\big(a_{i0}^{(1)}g^n_{0j}+a_{iM_x}^{(1)}g^n_{M_xj}\big) \!-\!\kappa_y\big(b_{j0}^{(1)}g^n_{i0}+b_{jM_y}^{(1)}g^n_{iM_y}\big) +\varepsilon_x\big(a_{i0}^{(2)}g^n_{0j}+a_{iM_x}^{(2)}g^n_{M_xj}\big) +\varepsilon_y\big(b_{j0}^{(2)}g^n_{i0}+b_{jM_y}^{(2)}g^n_{iM_y}\big),
\langle A(x,y;\xi,\zeta) (x-y), \xi-\zeta\rangle\geq 0,\forall x,y\in (T), \xi\in T(x),\zeta\in T(y)\footnote{When $h(x)=|x|^2$, this is the well-known notion of monotone map that has been the subject of large amount of research and applications, see for example the classical and polished book by H. Br\'ezis \cite{brezis-book-monotone-maps}.}
\prod_{m=1}^n q^{-2N_{\ell_m +1}(\eta)}\;\mathbb E_{\xi^{(\ell_1, \ldots, \ell_n)}}\left[D(\eta,\xi(t))\right]:=\lim_{L \to \infty}\prod_{m=1}^n q^{-2N_{\ell_m +1}(\eta^{(L)})}\;\mathbb E_{\xi^{(\ell_1, \ldots, \ell_n)}}\left[D(\eta^{(L)},\xi(t))\right]
|\alpha k^3 - k_1^3 - (n-k_1)^3 - \alpha(k-n)^3| = |k^3[\alpha - (d_1 + \mu')^3 - (1 - d_1 + \mu - \mu')^3 + \alpha \mu^3] | &= |k^3[(1-\alpha) + 3(d_1^2 - d_1) + \mathcal{O}(\epsilon)] | \gtrsim |k|^3.
\begin{array}{rcl}J_1 = \{0, 1, \cdots , m \}\ni j_1, J_2 (j_1) = \{j_1 , m+1, \cdots , 2m \}\ni j_2, & \vdots J_{n-2}(j_1 , \cdots , j_{n-3}) = \{j_{n-3}, (n-3)m+1, \cdots , (n-2)m \}\ni j_{n-2}J_{n-1}(j_1 , \cdots , j_{n-2}) = \{j_{n-2}, (n-2)m+1, \cdots , (n-1)m \}\ni j_{n-1}\end{array}
(f \cdot_\sigma b^\tau)\ast_g a^\rho = (f \ast_g (\alpha_{\rho(i_1)}\otimes \cdots \otimes \alpha_{\rho(i_n)})) \cdot_\sigma ((\beta_{\tau(1)}\otimes \cdots \otimes \beta_{\tau(m)}) \ast_g (\alpha_{\rho(i_1)}\otimes \cdots \otimes \alpha_{\rho(i_m)})),
C_{I}:=\max\{C_{m_{i}}\mid 1\leq i\leq r\}; c_{j,p}:=\sum_{i=1}^{r}a_{i,p}m_{j}^{\beta_{i}}p^{(\sigma-\lambda_{i})m_{j}}, 1\leq j\leq r; D_{p}:=\det(B_{p}); D_{k,l,p}:=(-1)^{k+l}\det\big((m_{j}^{\beta_{i}}p^{(\sigma-\lambda_{i})m_{j}})_{j\neq k, i\neq l}\big),1\leq k,l \leq r.
P_{\min}^{\rm{O}}\left({\overline S,\overline \varepsilon ,\overline R}\right) = \mathop{\arg \min}\limits_{{W_s}\in \left[{{W^{low}},{W^{up}}}\right]}P_{\min}^{\rm{O}}\left({\overline S,\overline \varepsilon ,\overline R |{W_s}}\right).
\sum_{k=0}^{m}p_{r, r- (m- k)}a_{k}= q_{r, r- 1 - m}\ , \ \ \ \ \ \ \ \ \ \ \ \ \ 0 \leq m \leq r-1 \ , D_{r-1}a_{m+r}+ \sum\nolimits_{k=0}^{r-1}\p_{r, k}a_{m+k}= 0 \ , \ \ \ \ \ \m \geq 0 \ .
\nu\Big(a^*_{p_1}(t) a_{p_1 - k}(t) a^*_{h_2}(s) a_{h_2 - k}(s)\Big) = |\Lambda |\delta( k )f_0 (p_1) f_0(h_2 ) & + |\Lambda |\delta(k)\delta(p_1 - h_2)e^{i(t-s) (E_{p_1}- E_{h_2})}f_0(p_1) \widetilde f_0 (h_2 ) .
L_{n}= \frac{1}{n}\, \left( \sqrt{w}\right)^{n}\, l_{n}\left( \left( \sqrt{w}\right)^{- 1}\, \pi_{n}\right) + u = \frac{1}{n}\, \pi_{n}^{n}\, g_{n}\left( w \, \pi_{n}^{- 2}\right) + u
P_1(f,g){}= \left(f-\frac{1}{2}((f-g) \vee 0),g+\frac{1}{2}((f-g)\vee 0)\right), P_{2,\alpha}(f,g){}= \left(g+\frac{1}{2}\varphi_\alpha\circ(f-g),f-\frac{1}{2}\varphi_\alpha\circ(f-g)\right),
P =\bigoplus_{i\in I_{13}}SP_{13}\oplus\bigoplus_{i\in I_{14}}SP_{14}\oplus\bigoplus_{i\in I_{1234}}SP_{1234}\oplus\bigoplus_{i\in I_{23}}SP_{23}\oplus\bigoplus_{i\in I_{24}}SP_{24}&\quad\oplus\bigoplus_{i\in I_{123}}SP_{123}\oplus\bigoplus_{i\in I_{124}}SP_{124}\oplus\bigoplus_{i\in I_{1}}SP_{1}\oplus\bigoplus_{i\in I_{2}}SP_{2}
T_{n}(k)=\{\sum_{\nu_{1},...,\nu_{n}\ge0}a_{\nu_{1},...,\nu_{n}}X_{1}^{\nu_{1}}\cdot\cdot\cdot X_{n}^{\nu_{n}},|a_{\nu_{1},...,\nu_{n}}|\rightarrow0for\nu_{1}+...+\nu_{n}\rightarrow\infty\}
0= \langle \tau_{H}, 2\alpha_{2}\rangle\{(2m_{1}+m_{2})(1-(\cot \langle 2\alpha_{2}, H\rangle )^{2}) &+n_{2}(1-(\tan \langle 2\alpha_{2}, H\rangle )^{2})-2m_{1}\}(\alpha_{1}+2\alpha_{2}).
{\cal Z}_{N}( J ) = \int{\cal D}\theta \exp [ - \frac{t}{2}\partial_{\mu}\theta \partial_{\mu}\theta - \frac{t}{2 g^{2}}J^{2}+ \theta ( - \frac{t}{g}\epsilon_{\mu \nu}\partial_{\mu}J_{\nu}+ i g \rho_{N}) ] .
{\widetilde J}= \langle x_{22}, x_{21}, x_{12}, x_{11}\rangle \cap\langle x_{33}, x_{21}, x_{12}, x_{11}\rangle \cap \langle x_{33}, x_{32}, x_{12}, x_{11}\rangle \ \cap \langle x_{33}, x_{23}, x_{21}, x_{11}\rangle \cap \langle x_{33}, x_{32}, x_{23}, x_{11}\rangle \cap \langle x_{33}, x_{32}, x_{23}, x_{22}\rangle
\begin{array}{c l l}{{E_{a}{\circ_{1 , Z}}E_{b}}}&{{= ( Z e_{a}) ( Z^{\dagger}( Z e_{b}) )}}&{{}}\\{{}}&{{= ( Z e_{a}) e_{b}}}&{{}}\\{{}}&{{= - ( Z e_{b}) e_{a}}}&{{\mathrm{S e e ~{\bf ~ [ 1 ]}}}}\\{{}}&{{= - ( Z e_{b}) ( Z^{\dagger}( Z e_{a}) )}}&{{}}\\{{}}&{{= - E_{b}{\circ_{1 , Z}}E_{a}.}}&{{}}\\ \end{array}
\tilde{t}=T(t), \tilde{x}=\sqrt{\dot{T}}(I+\omega(t)), &\tilde{u}=\nu(t)\exp\left[-\frac{1}{2}\int\frac{J}{\sqrt{a}}dx-\frac{1}{8}\frac{\ddot{T}}{\dot{T}}I^2-\frac{1}{2}(\dot{\omega}+\frac{\ddot{T}}{2\dot{T}}\omega)I\right]u,
D_{N}(q):=\sum_{k=1}^{\infty}\frac{1}{\iota_{k}(q){}^{N}}=\frac{Nq^{N}}{1-q^{N}}\sum_{m\in\mathcal{M}(N)}\alpha(m)\, q^{\epsilon_{1}(m)},\N\in\mathbb{N},
\Re(f(\zeta))-\Re(\zeta) \leq \Re(\beta) +\frac{1}{\Re\left(\zeta\right)\cdot\log\left(\Re\left(\zeta\right)\right)\cdots(\log^{\circ k}(\Re(\zeta))){}^{1+\varepsilon}}& =\rho_{\beta ,\varepsilon,k}^{+}\left(\Re\left(\zeta\right)\right) ,
{\cal B}_{N_{1}, \ldots , N_{1 0}}: ={S}_{\alpha_{1}\alpha_{2}\alpha_{3}\alpha_{4}}^{N_{1}N_{2}N_{3}N_{4}}\{S}_{\alpha_{4}\alpha_{5}\alpha_{6}\alpha_{7}}^{N_{4}N_{5}N_{6}N_{7}}\{S}_{\alpha_{7}\alpha_{3}\alpha_{8}\alpha_{9}}^{N_{7}N_{3}N_{8}N_{9}}\{S}_{\alpha_{9}\alpha_{6}\alpha_{2}\alpha_{1 0}}^{N_{9}N_{6}N_{2}N_{1 0}}\{S}_{\alpha_{1 0}\alpha_{8}\alpha_{5}\alpha_{1}}^{N_{1 0}N_{8}N_{5}N_{1}}.
\frac{Z_{3}}{Z_{\lambda}\lambda}\langle \partial A_{x}^{a}\partial A_{y}^{b}\rangle = \widetilde Z_{3}\Big ( ( 1 - \frac \alpha 2 ) \langle i ( \partial D_{r}c )_{x}^{a}\bar{c}_{y}^{b}\rangle + \frac \alpha 2 \langle c_{y}^{b}i ( \partial D_{r}\bar{c})_{x}^{a}\rangle \Big ) .
\lambda\left\{x \in \mathbb{I}:\frac{\log q_n(x)}{n}\geq b+ \varepsilon\right\}\geq \lambda \big\{x \in \mathbb{I}:a_1(x) = b_\varepsilon, \cdots, a_n(x)= b_\varepsilon\big\}&= |I(\underbrace{b_\varepsilon, \cdots,b_\varepsilon}_{n})| & \geq \frac{1}{3b_\varepsilon^2}\cdot |I(\underbrace{b_\varepsilon, \cdots,b_\varepsilon}_{n-1})|,
\begin{aligned}\omega \colon M_f \otimes^L_{A_Y}M_g^{A_W}\to M_p^{A_Z}\otimes^L_{A_Z}M_p\otimes^L_{A_X}M_f \otimes^L_{A_Y}M_g^{A_W}\to \to M_p^{A_Z}\otimes^L_{A_Z}M_q\otimes^L_{A_W}M_g\otimes^L_{A_Y}M_g^{A_W}\to M_p^{A_Z}\otimes^L_{A_Z}M_q\end{aligned}
\pi_\alpha^{(0)}* H^{(1)}* \pi_\alpha^{(0)}=\frac{i \hbar}{2}\lambda_\alpha <\Pi, \pi_\alpha^0 (F^B- F^0) \pi_\alpha^0>+ \frac{i \hbar}{2}\sum_{\gamma \neq \alpha}\lambda_\gamma< \Pi, S^{\gamma \alpha}S^{\alpha \gamma}>.
=y^{-\upsilon}\sum_{k=0}^{\infty}\bigg[~\frac{(\lambda)_{k}}{k!}\sum_{\ell=0}^{\infty}\frac{(-1)^{\ell}(\lambda b+b k)^{\ell}\Gamma\left(\upsilon+\frac{\ell}{2}\right)}{y^{\frac{\ell}{2}}~\ell!}\cos\left(\frac{\upsilon \pi}{2}+\frac{\ell\pi}{4}\right)\bigg],~~~~~~~~~
\big[Q^{(23)}, R\big]=-2\big(Q^{(23)}\big)^2-2\big\{Q^{(12)},Q^{(23)}\big\}\hphantom{\big[Q^{(23)}, R\big]=}{}+2\big(Q+Q^{(1)}+Q^{(2)}+Q^{(3)}\big)Q^{(23)}+2\big(Q^{(2)}-Q^{(3)}\big)\big(Q^{(1)}-Q\big).
Z^{(c)}_{\rm 1-loop}(M)={(2 \pi x)^{{1\over2}({\rm dim}\, H^0_{A^{(c)}}-{\rm dim}\, H^1_{A^{(c)}})}\over{\rm vol}(H_c)}{\rm e}^{-{1 \over x}S_{\rm CS}(A^{(c)}) -{i \pi\over 4}\varphi}{\sqrt{| \tau^{(c)}_R|}},
S_i=-\frac{a_2 \left(a_1 d \gcd \left(a_i-a_1,a_1-a_p,a_p-a_i\right)+(p-2) \left(a_1-a_i\right) \left(a_1-a_p\right)\right)}{\left(a_1-a_2\right) \gcd \left(a_i-a_1,a_1-a_p,a_p-a_i\right)}.
P_{i+1}(\nu,R) = P_i(\nu,R)^{n_i(\nu,R)}+\sum_{k=1}^{\lambda_i}c_kP_0(\nu,R)^{\sigma_{i,0}(k)}P_1(\nu,R)^{\sigma_{i,1}(k)}\cdots P_{i}(\nu,R)^{\sigma_{i,i}(k)}
\frac{d^{n-2}z}{dv^{n-2}}=\bar{P}_2+\frac{n}{2}v^2\int_0^v\frac{d^ny}{dv^n}(v')dv'-(n-1)v\int_0^vv'\frac{d^ny}{dv^n}(v')dv'+\frac{n-2}{2}\int_0^vv'^2\frac{d^ny}{dv^n}(v')dv'.
\begin{aligned}\int_{G_m}S_{L,\pi \begin{pmatrix}I_m x I_m 1 \end{pmatrix}v}(diag(g,I_{m+1}))|g|^{s_0}dg =\int_{G_m}\theta(Tr(gx))S_{L,v}(diag(g,I_{m+1}))|g|^{s_0}dg &=\int_{\mathcal{M}_m}\theta(Tr(xy))\Phi(y)dy=\widehat{\Phi}(x).\end{aligned}
H\backslash\left(y^{-1}z\right)^*/H=H\backslash\left(\left[\overline{u}^{-1}x\right]_-^{-1}\overline{u}^{-1}x\overline{v^{-1}}\left[x\overline{v^{-1}}\right]_+^{-1}\right)^t/H,
\tau_\gamma^i(z, f)(g, s) = \, \begin{cases}f((g_1,\ldots, g_{i-1}, \gamma g_i, g_{i+1},\ldots, g_m), s) \textrm{if}\ \rho_z(\gamma, \rho_z(g_i\cdots g_m, s))=\rho_z(g_i\cdots g_m, s) f(g, s) \textrm{otherwise}.\end{cases}
\sum_c\sum_{Q''}\Gamma_S^{Q'bQ''c}(p)\left[\delta_{Q''Q}\delta^{ca}-N_{Q''}^{ca}(p)g_{Q''Q}{(-1)}^{a+1}\right]=\frac{i}{2}g_{Q'Q}\delta^{ba}{(-1)}^{a+1}, \ \Q',Q=U,D, \ \b,a=1,2,
V_{( j , m )}^{\omega}( z , \bar{z}) = \gamma^{j - m}( z ) \bar{\gamma}^{j - m}( \bar{z}) e^{\frac 2{\alpha_{+}}j \phi ( z , \bar{z})}e^{i \sqrt{\frac{2}{k}}m X ( z , \bar{z})}e^{i \sqrt{\frac{2}{k}}( m +{\frac{k}{2}}\omega ) Y ( z , \bar{z})}
\lim_{k\to+\infty}\left\vert I_u(x, r_k) - I_u(x, 0^+)\right\vert \leq \lim_{k\to+\infty}\left\vert I_u(x, r_k) - I_u(0,r_k)\right\vert+\left\vert I_u(x, 0^+) - I_u(0,0^+)\right\vert &\stackrel{\textup{(iii)}}{=}\lim_{k\to+\infty}\left\vert I_u(x, r_k) - I_u(0,r_k)\right\vert =0.
\beta(\xi)=\frac{\sqrt{2\pi}e^{\frac{i\pi}{4}}e^{-\frac{\pi \nu(-k_0)}{2}}}{{q}_{1}(-k_0)\Gamma(-i\nu(-k_0))}, q_1(-k_0)=e^{-2\chi(\xi,-k_0)}r_1(-k_0)e^{2i\nu\log 4}, \gamma(\xi)=\frac{\sqrt{2\pi}e^{-\frac{i\pi}{4}}e^{-\frac{\pi \nu(-k_0)}{2}}}{{q}_{2}(-k_0)\Gamma(i\nu(-k_0))},q_2(-k_0)=e^{2\chi(\xi,-k_0)}r_2(-k_0)e^{-2i\nu\log 4}.
\frac{\d V_i}{\d z_j}=-\frac{p_je^{z_j}}{p_\mu e^{z_\mu}}\,V_i+\frac{C_{ij}e^{z_j}}{p_\mu e^{z_\mu}}&=(-p_jV_i+C_{ij})\frac{e^{z_j}}{p_\mu e^{z_\mu}}
(v_h - u_h, \partial^m\eta)_{\Omega_h}= a_h(v_h - u_h, g_h) = a_h(v_h - \tilde u, g_h) + a_h(\tilde u - u_h, g_h) &= a_h(\tilde u - v_h, \tilde g - g_h) - a_h(\tilde u - v_h, \tilde g) + \operatorname{Res}_u(g_h).
R_a =R_{a'}\star M_{a^{(k)}}begin{align*}5pt] =(M_{a'}+\sum_{a'\prec_{{\rm lex}}^{*}b'}c_{b'}M_{b'})\star M_{a^{(k)}}begin{align*}5pt] =\sum_{\Theta\in R(a',a^{(k)})}M_{a'\Theta a^{(k)}}+\sum_{a'\prec_{{\rm lex}}^{*}b'}\ \sum_{\Theta\in R(b', a^{(k)})}c_{b'}M_{b'\Theta a^{(k)}}.
J(u):=\frac{\|u\|_{\lambda}^{2}}{\left(\int_{\mathbb{B}^{N}}a(x)|u(x)|^{p+1}\mathrm{~d}V_{\mathbb{B}^{N}}(x)\right)^{\frac{2}{p+1}}}, J_{\infty}(u):=\frac{\|u\|_{\lambda}^{2}}{\left(\int_{\mathbb{B}^{N}}|u(x)|^{p+1}\mathrm{~d}V_{\mathbb{B}^{N}}(x)\right)^{\frac{2}{p+1}}}
\sup_{\mathcal{F}}\sup_{\mathcal{X}}\left\vert \phi_{(N)}f\right\vert=\sup_{\mathcal{F}}\max_{1\leqslant j\leqslant m_{N}}\sup_{\mathcal{X}}\left\vert \phi_{(j,N)}f\right\vert \leqslant M\left( 1+\frac{1}{p_{N}}\right) \leqslant\frac{2M}{p_{N}},
G_\varphi'(s) = \int_X AT_s \varphi (T_{t-s}f)^2\,d\mu -2\int_X T_s \varphi T_{t-s}f AT_{t-s}f\,d\mu = 2\int_X \varphi T_s|DT_{t-s}f|^2\,d\mu & \geq2e^{2Ks}\int_X \varphi |DT_sT_{t-s}f|^2 \,d\mu =2e^{2Ks}\int_X \varphi |DT_{t}f|^2 \,d\mu.
\|f_1\partial_xf_2\|_{L^2}= \big\|\sum\limits_{m\in\mathbb{Z}}e^{2\pi imx}f_{1,m}(y,z)\partial_xf_2\big\|_{L^2}\leq \sum\limits_{m\in\mathbb{Z}}\|e^{2\pi imx}f_{1,m}(y,z)\partial_jf_2\|_{L^2}\leq \sum\limits_{l\in\mathbb{Z}}C\|f_{1,m}\|_{L^2}\|\Delta f_2\|_{L^2}\leq C\big(\|\partial_xf_1\|_{L^2}+\|f_1\|_{L^2}\big)\|\Delta f_2\|_{L^2}.