harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
R\left({\mathcal{H}}_{1, i}, m, k \right) =\left( 1-{P}_{\mathrm{d}}\right) \frac{{T}_{\mathrm{f}}-{T}_{\mathrm{s}}}{{T}_{\mathrm{f}}}P\left({\mathcal{H}}_{1, i}, m, k \right) &\times C\left({\mathcal{H}}_{1, i}, m, k \right).
\| \Psi_{t}^{n}(f_{0})\|_{C^{2}}\le C_{3}\sum_{m=1}^{n}t^{m\beta - m\beta/\alpha}K_{m}+ \sum_{m=1}^{n}t^{m(\beta - \beta/\alpha)}C_{m}\left(1 + \sup_{s \in [0,t]}\|f_{0}\|_{C^{2}}\right),
\displaystyle{\raisebox{1 . 1 e x}{\scriptsize{\diamond}}}\mathrm{}\tilde{S}^{( \epsilon_{j + 1})}( Q^{( j + 1 )}) \simeq \frac{1}{2}\sum_{\rho = \pm}\frac{{\cal N}^{( \rho )}( Q^{( j + 1 )})}{{\cal D}_{\rho}},
\omega_{i}\left( \lambda \right) :=W\left( \phi_{i\lambda},\chi_{i\lambda};x\right) =\phi_{i\lambda}\left( x\right) \chi_{i\lambda}^{\prime}\left( x\right) -\phi_{i\lambda}^{\prime}\left( x\right) \chi_{i\lambda}\left( x\right) ,\left( i=1,2,3\right)
\sum_{i=0}^{4}(\begin{matrix}12\\i\end{matrix})x^{i}z^{12-i}+\sum_{i=0}^{4}(\begin{matrix}12\\i+5\end{matrix})12x^{i}z^{12-5-i}+\sum_{i=0}^{2}(\begin{matrix}12\\i+10\end{matrix})12^{2}x^{i}z^{12-10-i}=5
Z = \int{\cal D}A_{\mu}{\cal D}\Pi^{\mu}{\cal D}\phi{\cal D}\Pi_{\phi}{\cal D}\theta{\cal D}\Pi_{\theta}\prod_{\alpha , \beta = 1}^{2}\delta ( \tilde{\Omega}_{\alpha}) \delta ( \Gamma_{\beta}) \det \mid \{\tilde{\Omega}_{\alpha}, \Gamma_{\beta}\}\mid e^{i S^{\prime}},
\mathcal{O}\Big(T^{1/2}\int_{1}^{T^{1/2}}(t^{1/3}+t^{1/6}|a|^{1/6})\frac{dt}{t}\Big)=\mathcal{O}\big(T^{1/2}(T^{1/6}+ T^{1/12}|a|^{1/6})\big).
\phi(2^\delta q_1^{e_1}\!\cdots \!q_k^{e_k})=2^{\delta-1}\phi( q_1^{e_1}\!\cdots\! q_k^{e_k})\ge \sum_{i=1}^k\!2^{\delta-1}\phi( q_i^{e_i})=\sum_{i=1}^k\!\phi(2^{\delta}q_i^{e_i})
\sup_{p_j(x) \leqslant 1}p_j(Ax) = \sup_{p_j(x) \leqslant 1}\left \{\inf_{p_j(z)=0}p_j(Ax) - p_j(z) \right \}\leqslant \sup_{p_j(x) \leqslant 1}\left \{\inf_{p_j(z)=0}p_j(Ax - z)\right \}= \sup_{p_j(x) \leqslant 1}\| [Ax]_j \|_j \leqslant \sup_{\| [x]_j \|_j \leqslant 1}\| A_j [x]_j \|_j < \infty
\rho_t +\Delta \rho = -\frac{\rho}{2(s-t)},\rho_t +\Delta \rho -\frac{\nabla \varphi^\varepsilon\otimes \nabla \varphi^\varepsilon}{|\nabla \varphi^\varepsilon |^2}:\nabla^2 \rho + \frac{(\nabla \rho \cdot \nabla \varphi^\varepsilon )^2}{\rho |\nabla \varphi^\varepsilon|^2}=0.
\sum_{n=1}^{\infty}\frac{(-1)^{n-1}z^n q^{n^2}}{(zq;q^2)_{n}(1-z q^{2n})}=\sum_{n=1}^{\infty}\frac{z^n q^{\frac{n(n+1)}{2}}(q;q)_{n-1}}{(zq; q)_n}.
\boldsymbol{y}_{N h}(\boldsymbol{x},t) = \sum_{k=0}^N [\boldsymbol{y}_{kh}^c(\boldsymbol{x}) \cos(k \omega t) + \boldsymbol{y}_{kh}^s(\boldsymbol{x}) \sin(k \omega t)], \boldsymbol{p}_{N h}(\boldsymbol{x},t) = \sum_{k=0}^N [\boldsymbol{p}_{kh}^c(\boldsymbol{x}) \cos(k \omega t) + \boldsymbol{p}_{kh}^s(\boldsymbol{x}) \sin(k \omega t)].
\prod_{i=1}^3\left(\frac{R(\mathcal{F}_i)}{[\mathrm{cok}(\delta(\mathcal{F}_i)_{tor})]}\right)^{(-1)^{i+1}}=\frac{\nu(\mathcal{H}om)_{\mathbb{R}}\nu(\mathcal{H}_S)_{\mathbb{R}}}{\nu(\mathcal{H}^0)_{\mathbb{R}}}.
\frac{d^n}{dx^n}\Big[(f\circ g)(x)\Big] = \sum_{k_1+2k_2+\cdots+nk_n=n}\frac{n!}{k_1!\cdots k_n!}\Bigg[\frac{d^{k_1+\cdots+k_n}f}{dx}\circ g\Bigg](x)\prod_{m=1}^n\Bigg(\frac{1}{m!}\frac{d^mg}{dx^m}\Bigg)^{k_m}.
T^{U_m A U_m^*,B,B}_{\psi_m}(X_m,Y_m) =\sum_{i=1}^{m_0}\sum_{j,k=1}^{m_1}\psi_m(\lambda_i, \mu_k, \mu_j)E_i^{U_m A U_m^*}X E_k^{B}YE_j^{B}&= U_m\Big(\sum_{i=1}^{m_0}\sum_{j,k=1}^{m_1}\psi_m(\lambda_i,\mu_k, \mu_j)E_i^{A}(U_m^* X) E_k^{B}YE_j^{B}\Big) &=U_m T^{A,B,B}_{\psi_m}(U_m^* X,Y).
\mathbb{P}_x\bigl(X_n=o d(o,X_m) \leq r \bigr) \leq \mathbb{P}_x\bigl(\tau_\dagger > n d(o,X_m) \leq r \bigr) \leq \left(1-\frac{c_1 (\log r)^\gamma}{r^2}\right)^n \leq \exp\left[- \frac{c_1(\log r)^\gamma n}{r^2}\right],
\mu_p(C_1\cap C_2)-\mu_p(C_1)\mu_p(C_2) =\frac{q^{10}}{Z_p}-\left(\frac{q^4+q^{10}}{Z_p}\right)^2 =\frac{-q^8+q^{10}+q^{14}+3q^{16}}{Z_p^2}<0
{\cal Z}( G , \lambda A , N ) = \int_{\Sigma ({\cal M})}d \nu \; e^{- \frac{( n + \tilde{n}) \lambda A}{2}}\frac{( - 1 )^{i + \tilde{t}}N^{2 - 2 g}}{| S_{\nu}|}\prod_{j = 1}^{\max ( 0 , - I )}( - 1 )^{x_{j}},
\mathbb P[E_2^c] \leq \sum_{|\alpha|>CL}\mathbb P\left[|a_{\alpha}|^2>\frac{|\alpha|!\Gamma(n)}{\Gamma(n+|\alpha|)}|\alpha|^{2n}\right]=\sum_{m>CL}\mathbb P\left[|\xi|^2>\frac{m!\Gamma(n)}{\Gamma(n+m)}m^{2n}\right]\frac{\Gamma(n+m)}{\Gamma(n)m!}& \lesssim \sum_{m>CL}\mathbb P\left[|\xi|^2>c_nm^{n+1}\right]m^{n-1}=\sum_{m>CL}e^{-c_nm^{n+1}}m^{n-1}.
h_{2k_1,2k_2}=\frac{1}{2}\max\left(\frac{2k_1^2+n}{2k_1^2},\frac{2k_2^2+m}{2k_2^2}\right)^{\frac{1}{2}}\frac{k_1!k_2!}{(\frac{n}{2}+2k_1)^{k_1}(\frac{m}{2}+2k_2)^{k_2}}, j_{2k_1,2k_2}=\frac{1}{c_{2k_1,2k_2}}\left(\frac{k_1!k_2!}{(\frac{n}{2}+2k_1)^{k_1}(\frac{m}{2}+2k_2)^{k_2}}\right)^{\alpha_{2k_1,2k_2}},
(hg)^{z^2}=(hg)^{R(h)yR(c_{m-3})yR(h^{-1}c_{m-3})}&=(hgh)^{yR(c_{m-3})yR(h^{-1}c_{m-3})}&=\left((hgh)^\tau\right)^{R(c_{m-3})yR(h^{-1}c_{m-3})}&=\left((hgh)^\tau c_{m-3}\right)^{yR(h^{-1}c_{m-3})}&=\left((hgh)^\tau c_{m-3}\right)^\tau h^{-1}c_{m-3}&=hghc_{m-3}h^{-1}c_{m-3}&=hg.
|\Omega(x')| =\bigg[|\Omega(x')|-\frac1{\omega_{n-1}}\int_{\mathbf S^{n-1}}|\Omega(y')|d\sigma(y')\bigg]+\frac1{\omega_{n-1}}\int_{\mathbf S^{n-1}}|\Omega(y')|d\sigma(y') &:=\Omega_0(x')+C(\Omega,n),
\int_\mathbb C\log(|z-w|)d\mathrm{arg}(z)d\mathrm{arg}(z-1)=\int_{|z|< 1}\log(|z-w|)d\mathrm{arg}(z)d\mathrm{arg}(z-1)+\int_{|z|> 1}\log(|z-w|)d\mathrm{arg}(z)d\mathrm{arg}(z-1).
\|Tf\|_{L^q(d\nu)}\le C\|w^{-1}\|_{L^r(d\mu)}^{\frac{1}{s}}\|f\|_{L^p(d\mu)}^{\frac{p}{s}}=C\|f\|_{L^p(d\mu)}^{1-\frac{p}{s}}\|f\|_{L^p(d\mu)}^{\frac{p}{s}}=C\|f\|_{L^p(d\mu)}.
\varphi(X) = \varphi(XE^{P'}_{o_{-,P'}}(\lambda)^{-(n + 1)})j_{P'}^+(E^{P'}_{o_{-,P'}}(\lambda)^{n + 1}) & = \varphi(XE^{P'}_{o_{-,P'}}(\lambda)^{-(n + 1)})E_{o_-}(\lambda)^{n + 1}& = \varphi(XE^{P'}_{o_{-,P'}}(\lambda)^{-(n + 1)})z_\mathcal{O}^nE_{o_-}(\lambda) = 0.
C_0^n := \frac{16 \bigl( 1 +| a^n|_{W^{1, \infty}(Q)}+|b^n|_{L^\infty(Q)}+|\mu^n|_{L^\infty(0, T; H)}^2 +|\omega^n|_{L^\infty(Q)}^2 \bigr)}{\min \{1, \nu^2, \frac{\delta_*(a)}{2}\}},
\Phi_{0 , 1}^{( 3 , 0 )}= a_{1 1}^{*^{2}}V^{( 3 , 0 )}\hspace{2 0 m m}\Phi_{1 , 1}^{( 3 , 0 )}= ( a_{1 1}^{*^{2}}a_{2 2}^{*}- a_{1 2}^{*}a_{1 1}^{*}) V^{( 3 , 0 )}
(M_{t})_{i,j}=\begin{cases}G(a_1,\dots,a_{J+1})^t, i-1=\sum_{\ell=1}^J a_{\ell+1}10^{\ell-1},\, j-1=\sum_{\ell=1}^J a_\ell 10^{\ell-1}&a_1,\dots,a_{J+1}\in\{0,\dots, 9\}, 0, \end{cases}
f_{m+1}\le f_m + d\,\epsilon^{2}n^2 + \epsilon f_m m \|\tilde{P}^{(1)}\| + \epsilon^2 m\sum_{i=1}^{m}i\|\tilde{P}^{(1)}\|^2 &\le f_m(1+\epsilon m \|\tilde{P}^{(1)}\|) + d\,\epsilon^{2}n^2 + \epsilon^2 n^3\|\tilde{P}^{(1)}\|^2 &\le f_m(1+\epsilon n\|\tilde{P}^{(1)}\|) + d''\epsilon^{2}n^3
B_t A = \sum_{j=j_0(t)}^\infty M_{\psi_{j,t}}B_{j,t}A M_{\varphi_{j,t}}M_{\psi_{j,t}}+ \sum_{j=j_0(t)}^\infty M_{\psi_{j,t}}B_{j,t}[M_{\varphi_{j,t}},A]M_{\psi_{j,t}}&+ \sum_{j = j_0(t)}^\infty M_{\psi_{j,t}}B_{j,t}M_{\varphi_{j,t}}AM_{1-\psi_{j,t}}
E_{0}(T) := \sum_{m = 0}^{s}\|z^{m}\|_{H^{s - m}(\Omega)}^{2}+ \sum_{k = 0}^{s - 2}\|\theta^{k}\|_{H^{s + 1 - k}(\Omega)}^{2}+ \|\theta^{s - 1}\|_{H^{1}(\Omega)}^{2}.
p_1(t) = \frac{1}{6}\left(1+e^{-at}- e^{-bt}- e^{-ct}\right) p_2(t) = \frac{1}{6}\left(1 - e^{-at}- e^{-bt}+ e^{-ct}\right) p_3(t) = \frac{1}{6}\left(1- 2 e^{-at}+ 2 e^{-bt}- e^{-ct}\right).
\frac 1 2 L_{t}^{0}(Y)+J_{t}^{+,p}+\int_{0}^{t}dA_{r}^{\nu^{+}}=-\int_{0}^{t}\mathbf{1}_{\{u=h_{1}\}}(X_{r})f_{h_{1}}(X_{r})\,dr &\quad+\int_{0}^{t}\mathbf{1}_{\{u=h_{1}\}}(X_{r})\,d(A_{r}^{\nu^{-}}-A_{r}^{\mu}+A_{r}^{\alpha}),
(1-\eta^{2})\frac{d^{2}Y_{mn}(c,\eta)}{d\eta^{2}}-2(m+1)\eta\frac{dY_{mn}(c,\eta)}{d\eta}-(c^{2}\eta^{2}+m(m+1)-\lambda_{mn}(c))Y_{mn}(c,\eta)=0,
\min_{\substack{V_{X\widetilde{X}Y}\,:\, V_{XY}= P_{XY}, P_{\widetilde{X}}= P, q(V_{\widetilde{X}Y}) \ge q(P_{XY}), d(P_{X\widetilde{X}}) \ge \Delta}}D\big( V_{X\widetilde{X}Y}\| P \times P_{XY}\big),
(n+2) \Delta_{P, n+1}^{( \alpha, \beta; \,\ell+1)}(x) + \ell \Delta_{P, n+2}^{( \alpha, \beta; \,\ell )}(x)+ \ell (1-x) \Delta_{P, n+2}^{( \alpha+1, \beta; \ell-1)}(x) + (n + \frac{\alpha + \beta}{2}+ 2) (1-x) \Delta_{P, n+1}^{( \alpha + 1, \beta; \ell)}(x)= \alpha \Delta_{P, n+1}^{( \alpha, \beta; \,\ell )}(x).
\frac{(k/n - \alpha_0)^{1/2}}{n^{1/2}(1-k/n)^{3/2}}e^{-n(k/n-\alpha_0)^3 g(k/n)}= \frac{c^{1/2}}{n^{2/3}(1-k/n)^{3/2}}e^{-c^3 g(\alpha_0+c/n^{1/3})}.
g(x)= - \min_{y\in D}\Big \{\langle \tilde{B_1}x - \alpha, y - x\rangle+ \beta\sum_{i=1}^N y^2_i +\sum_{i=1}^N h_i(y_i)\Big \}+\beta \sum_{i=1}^Nx_i^2 +\sum_{i=1}^N h_i(x_i)
Q(z',z'')\lesssim \begin{cases}r''^{-n}r'^{s}(r'/r'')^{1-\frac n2+\nu_0-s},r'<\frac{r''}2; d(z',z'')^{-(n-s)},r'\sim r''; r'^{-n+s}(r''/r')^{1-\frac n2+\nu_0},r'>2r'';\end{cases}
\begin{cases}\sum_{j,k=1}^3t_{1jk}x_jx_k=\lambda x_1^2, \sum_{j,k=1}^3t_{2jk}x_jx_k=\lambda x_2^2, \sum_{j,k=1}^3t_{3jk}x_jx_k=\lambda x_3^2,\end{cases}
\frac{(-248)^m\prod_{i = 0}^{m-1}(r+(2i+e-1)3\nu)}{(3\nu)^mm!}+((e-1)3\nu+r)\times \left(\frac{2^{8m}\prod_{i=2m+e}^{3m+e-2}(3i\nu+r)}{m!(3\nu)^m}\right. &\left. +\sum_{x=1}^{m-1}\binom{m}{x}\frac{2^{8(m-x)}(-248)^{x}\prod_{i = 0}^{x-1}(r+(2i+e-1)3\nu) \prod_{i=2(m-x)+e}^{3(m-x)+e-2}(3i\nu+r)}{(3\nu)^{m}m!}\right).
ch^{\pm}_{H(\Lambda)}(\tau, z):=tr^{\pm}_{H(\Lambda)}q^{L_0-\frac{c_k}{24}}e^{2\pi izJ_0}= (\widehat{R}^{\pm}ch^{\pm}_{\Lambda})(\tau,-\tau x+J_0z,\frac{\tau}{4})\overset{3}{R}^{\pm}(\tau,z)^{-1},
\begin{cases}u(x,t)=0,\,\,\,(x,t)\in\Gamma\times\mathbb{R}^+, &\frac{\partial \mu}{\partial \vec{n}}=0,\,\,\,(x,t)\in\Gamma\times\mathbb{R}^+, &\frac{\partial \phi}{\partial t}-\alpha\Delta_\Gamma\phi+\frac{\partial \phi}{\partial \vec{n}}+\beta\phi+g(\phi_1)-g(\phi_2)=0,\,\,\,(x,t)\in\Gamma\times\mathbb{R}^+\end{cases}
=-\sum_{j}\nabla_{e_{j}}\langle\bigtriangleup,e_{j}\cdot\bar{s}^{+}\otimes \bar{f}\rangle_{h})-\sum_{j}(\nabla_{e_{j}}e_{j})\cdot\langle\star,\bar{s}^{+}\otimes \bar{f}\rangle_{h}+\sum_{j}\langle\bigtriangleup , e_{j}\cdot\nabla_{e_{j}}(\bar{s}^{+}\otimes \bar{f})\rangle_{h},
\mathfrak{u}_{\mathcal{M}}(T, x) \,= \,\inf \left\{r >0 : \exists\{\it \Pi}\in \mathfrak{P}\ \ \ \ \mathbb{P}^{\mathcal{M}}\left[Z^{\,rX^{\mathcal{M}}(0),{\it \Pi}}(T) \ge X^{\mathcal{M}}(T) \right]= 1\right\}\in ( 0,1]\,.
\overline{H}_n^{\star}(\mathbf{s})=\overline{H}_n^{\star}(s_1,s_2,\ldots,s_r):=\sum\limits_{0\leq k_1\leq k_2\leq \cdots\leq k_r\leq n-1}\prod_{j=1}^{r}\frac{1}{(2k_j+1)^{s_j}}.
\int_{0}^{\pi}\int_{0}^{2\pi}\Psi_{n;N}^{(e_1,e_2,e_3)}(\theta,\phi)\Psi_{n';N'}^{(e_1',e_2',e_3')}(\theta,\phi)\;\sin \theta\,\mathrm{d}\phi\,\mathrm{d}\theta=\delta_{nn'}\delta_{NN'}\delta_{e_1e_1'}\delta_{e_2e_2'}\delta_{e_3e_3'}.
g_{2i+1}^{2j+1}=\frac{(-1)^{\beta+j+\frac{k}{2}}(2j+1)!(k+i)!}{4^{\beta}i!(\beta-i)!(\beta+k+i+1)!}\sum\limits_{b=0}^{j}\frac{\prod\limits_{d=0,\;d\neq i}^{\beta}((2b+1)^2-(k+2d+1)^2)}{\prod\limits_{d=0,\;d\neq b}^{j}((2b+1)^2-(2d+1)^2)},
M_{K}= \left( \begin{array}{c c c c}{f^{- 1}}&{0}&{- \chi f^{- 1}}&{0}\\{0}&{- 1}&{0}&{0}\\{- \chi f^{- 1}}&{0}&{f + \chi^{2}f^{- 1}}&{0}\\{0}&{0}&{0}&{- 1}\\ \end{array}\right) ,
\begin{array}{lcl}\Phi_{n+1,k}\big((\bar{r})\big) = \displaystyle(-1)^{n-j-1}\phi_{n}(\{\bar{r}',r_j\})\wedge\frac{dr_k}{r_k}, &= \displaystyle(-1)^{j}\log(1+i_j)\frac{d\bar{s}'}{\Pi_{\bar{s}'}}\wedge\frac{dr_k}{r_k}+(-1)^{n-j-1}\omega\wedge\frac{dr_j}{r_j}\wedge\frac{dr_k}{r_k}.\end{array}
\begin{array}{l}k_1=C^{(2)}\otimes 1\,,\ \k_2=1\otimes C^{(2)}\,,\ \k_3=\delta^{(2)}(C^{(2)})\,, begin{align*}0.5em]l_1=C^{(3)}\otimes 1\,,\ \l_2=1\otimes C^{(3)}\,,\ \l_3=\delta^{(2)}(C^{(3)})\ .\end{array}
{\cal L}_{\mathrm{b u l k}}+{\cal L}_{\mathrm{s o u r c e}}\sim \int d^{n}\xi \Bigl [{\frac{1}{g_{\mathrm{I}}^{2}}}( \partial_{\xi}m_{a})^{2}+{\frac{1}{g_{\mathrm{I}}}}\sum_{s}f_{s}( m_{a}) \delta ( \xi - \xi_{s}) \Bigr ] .
*(e_{i}) =e_{i}, *(f_{i}) =f_{i}, *\left(q^{h}\right) =q^{-h}, \varphi\left(e_{i}\right) =f_{i}, \varphi\left(f_{i}\right) =e_{i}, \varphi\left(q^{h}\right) =q^{h}\psi\left(e_{i}\right) =q_{i}^{-1}t_{i}^{-1}f_{i}, \psi\left(f_{i}\right) =q_{i}^{-1}t_{i}e_{i}, \psi\left(q^{h}\right) =q^{h}.
\big(x^{-}_{\alpha}\otimes t^{(\lambda|\alpha)}\big)^{\langle (k^\prime-i^\prime)\theta,\, \alpha^{\vee}\rangle +1}\big(x^{-}_{\theta}\otimes t^{(\lambda|\theta)+1}\big)^{i^\prime+2k-\ell+1}\,v=0, \forall \,\,\alpha\in R^+,\,\,\, 0\leq i^\prime\leq k^\prime, \big(x_{\theta}^-\otimes t^{(\lambda|\theta)+1}\big)^{(2k^\prime-m^\prime+1)+(2k-\ell+1)}\,v=0.
\mathbb P\Big( \sup_{s \in [0,t]}\{X_k(s)\}\ge \Gamma \Big) \le 2 \mathcal G\Big(\frac{\ell \, \Gamma^{(k)}- t - \sum_{j=1}^{\ell}X_j(0)}{\sqrt{\ell t}}\Big) + 4 k \mathcal G \Big(\frac{\Gamma^{(k)}}{\sqrt t}\Big) \,.
\sum_{l=1}^{n_{k}}\int_{E_{k}\cap\{\varphi_{E_{k}}>n_{k}-l\}}\widehat{T}^{l}u\,d\mu\longrightarrow\alpha t\int_{0}^{1}[1-\widetilde{G}\left(t\,(1-r)^{\alpha}\right) ]\,r^{\alpha-1}\,dr
\pi \left( a,b,c\right) +J\pi \left( a,b,c\right) J^{\dagger}=diag\left( \begin{array}{c}2b a+b1_2 c+\left( ub-b\right) 1_3 c+\left( ub+b\right) 1_3 a\otimes 1_3+1_2\otimes c+ub\otimes 1_6\end{array}\right) \otimes 1_N
Y=Y(b_2,c_2,t_3)=\left (\begin{array}{ccc}-\textstyle\frac{1}{2}- \textstyle\frac{3b_2c_2}{2}\textstyle\frac{3b_2}{2}t_3 -\textstyle\frac{3c_2 (1+b_2c_2)}{2}1+ \textstyle\frac{3b_2c_2}{2}c_2 t_3 \textstyle\frac{3(1+b_2c_2)}{4 t_3}-\textstyle\frac{3b_2}{4t_3}\textstyle\frac{1}{2}\end{array}\right), t_3 \neq 0.
\operatorname{Re}\psi_3(i\rho,i\tau,\xi+i\eta)=\frac{\alpha\eta}{\tau}-\beta\tau^2\leqslant 0,\shortintertext{and}\operatorname{Re}\psi_4(i\rho,i\tau, \xi+i\eta)=\frac{\alpha\eta}{\rho}-\beta\tau^2 \leqslant 0,
Z_{\rm SS}={{1}\over{4}}{{1}\over{\eta^4{\bar{\eta}}^4}}\left [ (\Theta_3^4 - \Theta_4^4 - \Theta_2^4) ({\bar{\Theta}}_3^4 -{\bar{\Theta}}_4^4 -{\bar{\Theta}}_2^4 )\right ] \quad ,
\begin{aligned}\overline H(f) = \sup_{\phi}\, \inf_{g}\bigg[\int \d\P_0(\omega)\,\phi(\omega) \log\bigg(\sum_{e\in\mathcal U_d}\pi_\omega(0,e)\exp\big\{f(\omega,e)+ g(\omega)-g(\tau_e\omega)\big\}\bigg)\bigg] &=\sup_{\phi}\, \inf_{g}\,\,\int \d\P_0(\omega)\,\phi(\omega) L(g,\omega),\end{aligned}
\Phi'' = \mathcal{X}(\Phi), \mathcal{X}(\Phi) = \mathcal{F}^{-1}(\Phi) - e^{-\Phi}, \mathcal{F}(N) = \frac{u^0_3(0)^2}{2 N^2}\: + \: T^i \ln(N) \: - \: \frac{u^0_3(0)^2}{2}.
\sum_{i=0}^{p-1}\sum_{j=0}^{p-1}\frac{p^2(-1)^{i+j}}{(2i-1)(i+j+1)}=\sum_{i=0}^{p-1}\sum_{j=0}^{p-1}\left(\frac{2p^2(-1)^{i+j}}{(2i-1)(2j+3)}-\frac{p^2(-1)^{i+j}}{(i+j+1)(2j+3)}\right),
c_{\mathcal{N}}\leq\max_{t\geq0}\left\{\frac{t^{2}}{2}S_{q}(\psi,\phi)^{2}\|(\psi,\phi)\|_{L^{q}}^{2}-\vartheta t^{q}\|(\psi,\phi)\|_{L^{q}}^{q}\right\}=\left(\frac{1}{2}-\frac{1}{q}\right)\frac{S_{q}(\psi,\phi)^{2q/(q-2)}}{(q\vartheta)^{2/(q-2)}}.
f(f(\prod_{i=1}^{r}p_{i}^{k_{i}}))=f(\prod_{i=1}^{r}f(p_{i}^{k_{i}}))\le\prod_{i=1}^{r}f(f(p_{i}^{k_{i}}))\le\prod_{i=1}^{r}p_{i}^{k_{i}}
\Upsilon(x,t)= \sum_{j=0}^{k_2/2}c_{k_1,k_2,j}\frac{\omega^{k_2-2j}}{|x|^{2j}}+ O\left(\sum_{j=0}^{k_2/2}\frac{\omega^{k_2-2j+1}}{|x|^{2j}}+\frac{1}{|x|^{k_2+2}}\right);
G_1 = \left( \frac{1}{1-x}+ \frac{x}{1-x}G_1 \right) x \left( \frac{x^2}{1-x}(1+xG_1) + \frac{x^5}{(1-x)^2}(1+xG_1)^2\right) &=\frac{x^3}{(1-x)^2}(1+xG_1)^2 \left( 1+ \frac{x^3}{1-x}(1+xG_1)\right).
\left(^{\rho}D_{a^+}^{\alpha}g\right)(x)=\frac{\rho^{\alpha-n-1}}{\Gamma (n-\alpha )}\left( x^{1-\rho}\frac{d}{dx}\right)^{n}\int_{a}^{x}\ (x^{\rho}-\tau^{\rho})^{n-\alpha+1}\tau^{\rho-1}g(\tau)d\tau,
A_1 =\max(r_4-r_3,2r_1-r_5+r_6-d_1,d_1), A_2 =\min(r_4+r_3,r_5+r_6 +d_1-2r_1), B_1 =\max(r_6-r_5,2r_1-r_3+r_4 -d_1, d_1), B_2 =\min(r_6+r_5,r_3+r_4 +d_1- 2r_1).
\partial_t u(t) =\partial_t \int_0^t w(t-y)\left(\int_0^\infty e^{-\lambda s}p^\beta_s(y)\,ds\right)dy &= w(0)\int_0^\infty e^{-\lambda s}p^\beta_s(t)\,ds+\int_0^t w'(t-y)\int_0^\infty e^{-\lambda s}p^\beta_s(y)\,ds\,dy, t>0,
U_t^{ph}(h,h')=\left(\frac{l}{i\hbar t}\right)^{r/2}\sum\limits_{\hat{s}\in W_A}(\kappa (h)\kappa (\hat{s}h'))^{-1}\exp\left(\frac{i\pi l(h-\hat{s}h')^2}{\hbar t}+itE_0\right)\ ,
\lim_{R\to\infty}\frac{\alpha^2}{N}K_{N}\left(\frac{x\alpha^2}{N}, \frac{y\alpha^2}{N}; (1-t^2)^{1/2}e^{-N V_{\alpha, \widetilde{\varepsilon}_R}(t)}\right) = \frac{\sin\pi c_{\alpha}^- (x-y)}{\pi (x-y)}
\Im \mathcal{P}[\varrho ]=\frac{1}{2}a(b^2-4) |X|^2 \Re(X) + \frac{1}{2}b(a^2-4) |Y|^2 \Re(Y) + \frac{5}{2}a^2 b \Re(X^2 \overline{Y}) + \frac{5}{2}a b^2 \Re(Y^2 \overline{X}) - 4a (b^2 + 1) |XY| \Re(X) - 4b (a^2+1) |XY| \Re(Y) + \frac{1}{2}b(a^2 + 5b^2 -4) |X|^2 \Re(Y) + \frac{1}{2}a(b^2 + 5a^2 -4) |Y|^2 \Re(X) .
y^{\prime}=\frac{1}{2}[(y^{\prime}(0)+\sqrt{{y^{\prime}(0)}^{2}+1})(1-\frac{x}{A_{x}})^{-\frac{V_{t}}{V_{d}}}+(y^{\prime}(0)-\sqrt{{y^{\prime}(0)}^{2}+1})(1-\frac{x}{A_{x}})^{\frac{V_{t}}{V_{d}}}]
\int_0^T \zeta \int_{\Omega}v \cdot \nu \, d\|\nabla \varphi (\cdot, t) \| dt=-\int_0^T \zeta_t \int_{\Omega}\varphi \, dxdt = -(2\mathcal{L}^d (U_0) -1) \int_0^T \zeta_t \, dt=0
\ell_0(s) = \frac{s}{2}, &\ell_1(s) = -4 s r'\left(s^2\right), &\ell_2(s) = \frac{8}{s}\left(r\left(s^2\right){}^2 r'\left(s^2\right)+2s^2 r'\left(s^2\right){}^2-2 r\left(s^2\right) \left(r'\left(s^2\right)+s^2 r''\left(s^2\right)\right)-2 r_2'\left(s^2\right)\right).
\psi = \left( \begin{array}{c}{{\psi_{1}}}\\{{\psi_{2}}}\\ \end{array}\right) \, , \qquad \psi_{c}= \left( \begin{array}{r}{{\psi_{1}^{\dagger}}}\\{{\ - \psi_{2}^{\dagger}}}\\ \end{array}\right) \, .
\mathcal{F}(V(s)) =s\mathcal{F}_1(V_1)+\frac{s^2}{2}\big(\mathcal{F}_1(V_2)+\mathcal{F}_2(V_1,V_1)\big)+\frac{s^3}{3!}\big(\mathcal{F}_1(V_3)+3\mathcal{F}_2(V_1,V_2)+\mathcal{F}_3(V_1,V_1,V_1)\big) &+\ldots+\frac{s^n}{n!}\big(\mathcal{F}_1(V_n)+n\mathcal{F}_2(V_1,V_{n-1})+\ldots+\mathcal{F}_n(V_1,\ldots,V_1)\big)+\ldots
\frac{1}{S}\| P_B - P \|_1 = \frac{1}{S}\sum_{i = 1}^S \frac{Sa}{n+Sa}|p_i - 1/S| & = \frac{1}{S}\sum_{i = 1}^S \frac{|1-p_i S|a}{n+Sa}& \leq \frac{2a}{n+Sa},
\sup_{f\in \Lambda U_{p}(\mathrm{P}^{d}(\mathbb{R}))}\,\|f-S_{2n}(f)\|_{p}=\mathcal{K}(\mathrm{P}^{d}(\mathbb{R}))|\lambda_{n+1}|n^{(d-1)/2}(1+o(1)),\,\,\,p=1,\infty .
f(x,t) = \frac{1}{2}(f(x-1,t-1)+ f(x+1,t-1)) &+ \phi(f(x+1,t-1)-f(x-1,t-1)) &+ N^{-1/4}y(x,t) - \frac{1}{\beta}\log m(N^{-1/4}\beta).
\begin{aligned}\frac{da}{dr}=*_{g_r}\left(d\hat{\mathfrak{A}}(r)-d\mathfrak{A}(r)+\frac{1}{2}[\hat{\mathfrak{A}}(r),\hat{\mathfrak{A}}(r)]-\frac{1}{2}[\mathfrak{A}(r),\mathfrak{A}(r)]\right) &=*_{g_r}\left(da+\frac{1}{2}[\mathfrak{A},a]+\frac{1}{2}[a,\hat{\mathfrak{A}}]\right).\end{aligned}
\log\frac{(K-i)x_{K-i}+i}{K}= \sum_{j=0}^{i-1}(1-N)^j \log\frac{(K-1)x_{K-1}+1}{K}&\,\,\quad\qquad\qquad\qquad\qquad-\sum_{j=1}^{i-1}(1-N)^j \log x_{K-i+j}.
I_n(\lambda, 0, 0) = \frac{1}{\bigl(n-1+(2^{n+1}-2n-2)\lambda\bigr)\bigl(n+(2^{n+1}-2n-2)\lambda\bigr)}I_{n-2}(\lambda, 2\lambda, 2\lambda) , J_n(\lambda, 0, 0) = \frac{1}{\bigl(n-1+((n-2)2^{n}+2)\lambda\bigr)\bigl(n+\bigl((n-2)2^{n}+2)\lambda\bigr)}J_{n-2}(4\lambda, 2\lambda, 2\lambda).
T : \Upsilon_{l , \, m}= - i \sigma_{y}\Upsilon_{l , \, m}^{*}= \left( \begin{array}{r}{{- \beta_{l , \, m}^{*}}}\\{{\alpha_{l , \, m}^{*}}}\\ \end{array}\right) = ( - 1 )^{l - m}\, \left( \begin{array}{c}{{\alpha_{l , \, - m}}}\\{{\beta_{l , \, - m}}}\\ \end{array}\right) .
P_N\begin{pmatrix}1 0 0 w K u q v^{{}}p\end{pmatrix}:= \begin{cases}\begin{pmatrix}1 0 w + \frac{q}{1-p}u K + \frac{1}{1-p}uv^{}\end{pmatrix}, p < 1, & \begin{pmatrix}1 0 w + u K \end{pmatrix}, p = 1,\end{cases}
\eta(\tau_{n-i+j+1}, \tau_{n-i+\ell+2})=(\tau_n^q \tau_1)^{i-j-1}\tau_{n-j+\ell+1}=(\tau_n^q \tau_1)^{i-j-1}\tau_n^{q-1+D(n+\ell-j)}\tau_{n-j+\ell+1}.
\begin{array}{l}<x>_f= \langle \widehat{x}f, f \rangle_{L^2 (\mathbb{R})}= \int_{\mathbb{R}}x | f (x)|^2 dx <\xi>_f= \langle \widehat{\xi}f, f \rangle_{L^2 (\mathbb{R})}= \int_{\mathbb{R}}\xi | \widetilde{f}(\xi)|^2 d \xi .\end{array}
\left\{\widehat{A}(TX){\rm exp}(\frac{c}{2})({\rm ch}(\overline{W_j})+{\rm ch}(\overline{W_i})+{\rm ch}({W_i}){\rm ch}({W_j})-16{\rm ch}({W_j})-16{\rm ch}({W_i})+104\right. &\left.+B_1\wedge[{\rm ch}({W_j})+{\rm ch}({W_i})-16]+B_2)\right\}^{(14)}&=-196632\left\{\widehat{A}(TX){\rm exp}(\frac{c}{2})\right\}^{(14)}.
\phi^{(\mathrm{per})}_{l,k}(t) := \sum_{m\in\mathbb{Z}}2^{l/2}\phi ( 2^{l}t + 2^{l}m - k) \ \psi^{(\mathrm{per})}_{l,k}(t) := \sum_{m\in\mathbb{Z}}2^{l/2}\psi ( 2^{l}t + 2^{l}m - k)
\sum_{m|n}mA(m^2D,d)\sum_{k=1}^{D-1}\left(\frac Dk\right)\zeta_D^{kn/m}\equiv g(n)\pmod \ell nA(n^2D,d) \equiv \sum_{m|n}\nu(m)g\left(\frac nm\right)\pmod \ell A(n^2D,d) \equiv\frac{1}{n}\sum\limits_{m|n}\nu(m)g\left(\frac nm\right)\pmod \ell.
\begin{array}{l l l l}{{{\bf G}^{( N )}}}&{{=}}&{{({\bf 1}+{\bf K}) [ ({\bf 1}+{\bf \tilde{K}}) ({\bf 1}+{\bf K}) ]^{\frac{N - 1}{2}}-{\bf 1}}}&{{\; N \in \mathrm{o d d}\; ,}}\\{{\tilde{{\bf G}}^{( N )}}}&{{=}}&{{[ ({\bf 1}+{\bf K}) ({\bf 1}+{\bf \tilde{K}}) ]^{\frac{N}{2}}-{\bf 1}}}&{{\; N \in \mathrm{e v e n}\; .}}\\ \end{array}
\rho_{u}\rho_{u}\Phi^{v}= \rho_{u}( \Phi^{u}R_{u}\Phi^{v}) = \Phi^{v}\lambda_{u}^{2}= \rho_{u}\rho_{w}\Phi^{u}= \rho_{u}( \Phi^{w}) R_{v u}( \Phi^{u}) = \Phi^{v}R_{v}( \lambda_{u}^{2})
p|_i = \rho_i g \left(\sum_{j=i}^N D_j + b - z + \sum_{j=1}^{i-1}\frac{\rho_j}{\rho_i}D_j\right) = \rho_i g \left(\sum_{j=1}^N D_j + b - z + \sum_{j=1}^{i-1}\frac{\rho_j-\rho_i}{\rho_i}D_j\right),
\begin{matrix}(\overline{10}_{H}10)(\overline{10}_{H}10)&\overline{10}_{H\alpha\beta}10_{i}^{\alpha\beta}\overline{10}_{H\gamma\delta}10_{j}^{\gamma\delta}\\ \overline{10}_{H}10\overline{10}_{H}10&\overline{10}_{H\alpha\beta}10_{i}^{\beta\gamma}\overline{10}_{H\gamma\delta}10_{j}^{\delta\alpha}\end{matrix}
\sum_{m_1+\ldots+n_N=2k}\frac{1}{i^{\sum_{i=1}^Mm_i}}\frac{1}{(-i)^{\sum_{j=1}^N n_j}}\frac{(2k)!}{\prod_{i=1}^M m_i!\prod_{j=1}^N n_j!}=(\frac{M}{i}-\frac{N}{i})^{2k}=(M-N)^{2k}(-1)^k,
\int_{\Omega}\left| \frac{\nabla u}{K}\right|^{p(x)-2}\langle \frac{\nabla u}{K}, \nabla \eta \rangle = \lambda_1 S \int_{\Omega}\left| \frac{u}{k}\right|^{p(x)-2}\frac{u}{k}\eta,\,\,\forall\eta \in C^{\infty}_0 (\Omega),
R_{b}=x_0u_m\sum_{n\neq a_b^*}\delta_{n,b}\mathbb{E}[k_{n,b}] &\leq x_0u_m \left[\sum_{n\neq a_b^*}\frac{8\ln (t'_b-t_n)}{\delta_{n,b}}+ \left(1+ \frac{\pi^2}{3}\right)\sum_{n\neq a_b^*}\delta_{n,b}\right].
I_{0}(\lambda)\approx f(x^{0})e^{\lambda S(x^{0})}\int_{R^{n}}exp(\lambda\sum_{j=1}^{n}\frac{\mu_{j}}{2}w_{j}^{2})dw=f(x^{0})e^{\lambda S(x^{0})}\prod_{j=1}^{n}\int_{-\infty}^{\infty}e^{\frac{1}{2}\lambda\mu_{j}y^{2}}dy.
G ( z_{2}, \cdots , z_{n}, z_{1}\tau^{2 n})^{n \cdots \hat{i}\cdots 1 i}= \delta_{i}G ( z_{1}, \cdots , z_{n})^{i n \cdots \hat{i}\cdots 1}, \delta_{i}= \delta_{1}\tau^{2 - 2 i}.
\int_0^1 \sum_{j\neq i}\partial_j\left (\frac{y_j}{t^d}\eta \left (\frac{y}{t}\right) \right) + \frac{y_i}{t^{d+1}}\partial_i \eta \left ( \frac{y}{t}\right) dt= - \int_0^1 \frac{1}{t^{d-1}}\frac{d}{dt}\left (\eta \left (\frac{y}{t}\right ) \right)dt=-\eta (y)
\sum_i \lambda_i^4 = \sum_i\mu_i^4 +12H^4 -6H^2\vert A\vert^2 +4H\sum_i\lambda_i^3 &= \sum_i\mu_i^4 -12H^4 -6H^2\vert \mathring{A}\vert^2 +4H\sum_i\lambda_i^3,
\prod_i (1+x_i)^{e_i}= e^{\sum e_i\ln(1+x_i)}= 1 + \Bigl( \sum e_i\ln(1+x_i) \Bigr) + \frac{1}{2!}\Bigl( \sum e_i\ln(1+x_i) \Bigr)^2 + \cdots
H\left(p,t\right) =\left\Vert Q_{p}\left(T-t\right)\hat{B}^{\dagger}e^{-\left(T-t\right)\hat{A}^{\dagger}}p\right\Vert_{1}-\left\Vert Q_{e}\hat{D}^{\dagger}e^{-\left(T-t\right)\hat{A}^{\dagger}}p\right\Vert_{1},
S = \frac{1}{1 6 \pi}\int d^{4}\! x \, ( - g^{\mathrm{I V}})^{1 / 2}\left[ R^{\mathrm{I V}}- 2 \Lambda - \frac{1}{2}\sum_{i = 1}^{N}( \nabla^{\mathrm{I V}}f_{i})^{2}\right] ,
\sum_{i=1}^{n}E_{\theta_{i}}\big(T_{\tau}(X_{i})-\theta_{i}\big)^2 \lesssim \tilde{q}_n\log\big(\frac{1}{\tau^{2a}}\big)+ (n-\tilde{q}_n)\tau^{2a}\sqrt{\log\big(\frac{1}{\tau^{2a}}\big)}\mbox{as}\tau \rightarrow 0.
\left|\frac{d}{dt}\|\Delta v(t)\|_{L^2}^2\right| =\left|2\langle\Delta v(t),\Delta v_t(t)\rangle\right| &=\left|-2\langle\Delta^2\exp(-tA_o)v_0,A_o v(t)\rangle\right| \leq C\|v_0\|_{H^4}^2,
\phi'(t) = J(C|M)_{\hat{\rho}_{CM}(t)}- \frac{\lambda^2}{\eta}J(A|M)_{\hat{\rho}_{AM}(t)}- \frac{(1-\lambda)^2}{|1-\eta|}J(B|M)_{\hat{\rho}_{BM}(t)}\le 0\;.
[B_{i}, B_{j}] = i \mu r \epsilon_{ijk}B_{k}, \hspace{12mm}B_{1}^{2}+ B_{2}^{2}+ B_{3}^{2}= \mu^{2}r^{2}\frac{N^{2}- 1}{4}{\bf 1}_{N \times N}\textrm{for}(i,j,k=1,2,3)
V_{e f f}( z ) = \frac{3 ( 5 \alpha + 4 ) k^{2}}{6 4 \left({\frac{k z}{2}}+ 1 \right)^{2}}+{\frac{l^{2}}{2 R^{2}}}\left({\frac{k z}{2}}+ 1 \right)^{{\frac{5}{2}}( \alpha - 2 )}-{\frac{3 k}{4}}\delta ( z )
\begin{aligned}K(e_0,e_1) =g^{00}R^1_{010}=\mu_1^2, K(e_1,e_2) =g^{11}R^2_{121}=\mu_1\mu_2, K(e_0,e_3) =g^{00}R^3_{030}=\mu_2^2,\end{aligned}