harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
A_{ik,jk}x^ix^j = \sigma_1(A)_{,ij}x^ix^j+R_{lijk}A_{lk}x^ix^j+R_{lj}A_{il}x^ix^j = (\sigma_1(A)_{,ij}(p)+O(r))x^ix^j+(W_{lijk}(p)+O(r))(A_{lk,m}(p)x^m+O(r^2))x^ix^j+O(r^4) = \sigma_1(A)_{,ij}(p)x^ix^j+O(r^3).
\Delta(P-P_{e})+\frac{\rho}{2}[\frac{f}{D}+\frac{f_{e}}{D_{e}}(\frac{F}{F_{e}})^{2}]W^{2}\Delta X+\frac{\rho}{2}[(2-\beta)-(2-\beta_{e})(\frac{F}{F_{e}})^{2}]\Delta W^{2}=0
{\rm cd}(A) ={\rm sup}\{n : \H^n(A, M) \not=0 \{\rm for}\{\rm some}\A-{\rm bimodule}\M\}\in \mathbb N \cup \{\infty\}& ={\rm min}\{n : \H^{n+1}(A, M) =0 \{\rm for}\{\rm any}\A-{\rm bimodule}\M\}&={\rm pd}_{_A\mathcal M_A}(A)
i^*_{\mu^{-1}(0)}\hat{\omega}_1 = p_0^*i\partial\bar\partial_{I_1}f_{x_1}+ x_1 i^*_{\mu^{-1}(0)}d\eta = p_0^*(i\partial\bar\partial_{I_1}f_{x_1}+ x_1\Omega_0).
\lim_{n\to\infty}\tau_{\Lambda^{(n)}\times\Delta^{(n)},N^{(n)}}\big( \tilde A(\Lambda^{(n)}\times\Delta^{(n)},N^{(n)};f_1)\dotsm \tilde A(\Lambda^{(n)}\times\Delta^{(n)},N^{(n)};f_k)\big) &\quad=\tau(\tilde A(f_1)\dotsm \tilde A(f_k)).
k_n(x,y) = \sum_{k=0}^{n-1}p_k(x) q_k(y) = \sum_{k=1}^n \widetilde{p}_k(x) \widetilde{q}_j(y) & = \left( \prod_{j=1}^n (x-a_j) \right) \sum_{k=1}^n \left( \prod_{j \neq k}\frac{1}{a_k-a_j}\right)\frac{\delta(y-a_k)}{x-a_k}
I_2 = \int_{A_2}\frac{|w_\rho(x)\, \varphi_\delta(x) - w_\rho(y)\, \varphi_\delta(y)|^p}{|x - y|^{N+sp}}\, dx\, dy \le [w_\rho \varphi_\delta]_{s,p}^p \le C \delta^{N-ps}.
\frac{1}{\lambda_1}e^{-2\lambda_1 t}+\frac{1}{\lambda_2}e^{-2\lambda_2 t}=\frac{(\lambda_1+\lambda_2)e^{-2\lambda_2 t}}{\lambda_1 \lambda_2}+\frac{(e^{-2\lambda_1 t}-e^{-2\lambda_2 t})}{\lambda_1}&=\frac{\gamma e^{-2\lambda_2 t}}{\omega^2}+\frac{(e^{-2\lambda_1 t}-e^{-2\lambda_2t})}{\lambda_1}
\small \textbf{R}_k = \mathbb{E}\left\lbrace \textit{\textbf{h}}_k(f) \textit{\textbf{h}}_k^{\rm H}(f) \right\rbrace=\sum\limits_{p=1}^{N_{\rm S}+ 1}|\alpha_{k,p}|^2\textbf{a}\left( \theta_{k, p}\right)\textbf{a}^{\rm H}\left( \theta_{k, p}\right). \normalsize
\theta_{\gamma_e}= \omega_{1,1}\cdot t_1 + \omega_{2,1}\cdot t_2 = \frac{2\pi \cdot t_1}{S_1 - \ln|c|}\theta_{\gamma_m}= \omega_{1,2}\cdot t_1 + \omega_{2,2}\cdot t_2 = t_2 -\frac{S_2+ \arg c}{S_1 - \ln |c|}\cdot t_1
A_e = \left( \begin{array}{ccccc}a 0 \cdots 0 0 0 a \cdots 0 0 \vdots \vdots \ddots \vdots \vdots 0 0 \cdots a 0 0 0 \cdots 0 \frac{n(n-1)}{r}a \end{array}\right).
\beta_{n+1}(t) = - \alpha_{n+1}(t) \frac{\sigma_{n-1,-1}(t)}{\sigma_{n,-1}(t)}\ \ \mbox{and}\ \ \alpha_{n+1}(t) = \frac{\sigma_{n,n}(t)}{\sigma_{n-1,n-1}(t)}, \ \n \geq 1.
b_{j}^{\left( \nu ,2\right)}=8\pi^2\left[\frac{\left( 1+\nu^{2}\right)^{j}}{j!}-\frac{\nu^2\left( 1+\nu^{2}\right)^{j-1}}{\left( j-1\right) !}+\sum\limits_{i=2}^{j}\frac{\left( 1+\nu^{2}\right)^{j-i}}{\left( j-i\right) !}c_{i}^{(\nu,2)}\right]
F_{0}( n , k ) ={\frac{6}{\pi^{2}}}{\frac{( n - 1 )}{( 1 2 - 5 n ) ( 1 3 - 5 n ) ( n - 4 ) !}}{\frac{Z_{k + 2}^{F_{0}}Z_{n - k}^{F_{0}}}{a_{k + 2}a_{n - k}}},
Q_{\lambda}^{\mu}(z)=\frac{\sqrt{\pi}\Gamma(\lambda+\mu+1)}{2^{\lambda+1}\Gamma(\lambda+3/2)}\frac{1}{z^{\lambda+\mu+1}}(1-z^{2})_{2}^{\mu/2}F_{1}(\frac{\lambda+\mu+1}{2},\frac{\lambda+\mu+2}{2};\lambda+\frac{3}{2};\frac{1}{z^{2}})
\begin{aligned}R_{O1}=\max_{\mathcal{P}}Kr^*+\sum_{i=1}^K \Delta r_i \mathrm{s.t.}\sum_{i=1}^K P_i\leq P-P_{O1}^*, \Delta r_i=\frac{1}{K}\ln \Big(1+\frac{KP_i|\bar{h_i}|^2}{N_0}\Big) .\end{aligned}
A_{k+1}f(x_{k+1}) \leq \min_x A_{k+1}\phi_{k+1}(x)= \min_x \sum_{i=0}^k \alpha_i f_i(x) + A_0\Big(\phi_0(x) - A_0^{-1}\tilde \varepsilon_0\Big) + \tilde \varepsilon_{k+1}.
\begin{aligned}w_0^+\circ X^{-1}_\gamma(s,0)+\frac{\eta}{\lambda\mu_\lambda}\left(\partial_n w_0^+\circ X^{-1}_\gamma\right)(s,0) =-a_0\eta+b_0+2\log\mu_\lambda-\log h_\gamma, w_0^- \circ X^{-1}_\gamma(s,0)+\frac{\eta}{\lambda\mu_\lambda}\left(\partial_n w_0^-\circ X^{-1}_\gamma\right)(s,0) =a_0\eta+b_0+2\log\mu_\lambda-\log h_\gamma.\end{aligned}
\log \left [ \left ( \frac{f(m)-p}{f(m)-q}\right )^{h(m)}\left ( \frac{g(m)-q}{g(m)-p}\right )^{c\, g(m)}\right ] &=c\, g(m) \, \log \frac{(f(m)-p)\,(g(m)-q)}{(f(m)-q)\,(g(m)-p)}&+ (h(m) - c\, g(m)) \, \log \frac{f(m)-p}{f(m)-q}.
G_{\psi}^{i j}( \nu ) \equiv ( - 1 )^{i + j}\Sigma ( ( - 1 )^{i}\bar{R}_{i}) \Sigma ( ( - 1 )^{j}\bar{R}_{j}) \bar{\gamma}_{i}^{- 2}\bar{\gamma}_{j}^{- 2}\frac{\partial^{2}}{\partial r \partial r^{\prime}}G ( r , r^{\prime}, \nu ) \Bigg |_{{r^{\prime}= ( - 1 )^{i}\bar{R}_{i}}\atop{r = ( - 1 )^{j}\bar{R}_{j}}}\, .
\delta^{( 1 )}( A_{j})_{\overline{{m_{j}^{\prime}- 1}},{\bf m}_{\perp}, n}= i g{\frac{1}{2}}[ \delta \omega_{m_{j},{\bf m}_{\perp}, n}+ \delta \omega_{m_{j}- 1 ,{\bf m}_{\perp}, n}, ( A_{j})_{\overline{{m_{j}^{\prime}- 1}},{\bf m}_{\perp}, n}] ,
\frac{1}{g_a^2(p^2)}= \frac{k_a}{g_a^2(M^2_{X})}+ \frac{b_a}{16 \pi^2}\log \frac{M^2_{X}}{p^2}+\frac{\Delta_{a}}{16 \pi^2}+\frac{{\tilde \Delta}_a}{16 \pi^2},
\frac{v_{n+1}}{v_n}=\frac{(1-a_1q^n)(1-a_2q^n)\cdots(1-a_rq^n)}{(1-q^{n+1})(1-b_1q^n)\cdots(1-b_sq^n)}(-q^n)^{1+s-r}z =\frac{(a_1-q^{-n})(a_2-q^{-n})\cdots(a_r-q^{-n})}{(1-q^{-n-1})(b_1-q^{-n})\cdots(b_s-q^{-n})}\frac zq.
\mathbb{R}_{U_1^n}(d) = \frac{1}{n}\sum_{i = 1}^n \frac{1}{2}\log \max\left(\mu_{n, i},~\frac{\sigma^2}{\theta_n}\right), d = \frac{1}{n}\sum_{i = 1}^n \min\left(\theta_n, \frac{\sigma^2}{\mu_{n,i}}\right),
\sum_{n=1}^\infty e^{-h_\mu\psi(n)+\sqrt{2\rho_\mu \psi(n)}g(\psi(n))}=\sum_{n=1}^\infty \#\psi^{-1}(n)e^{-h_\mu n+\sqrt{2\rho_\mu n}g(n)}&\geq\sum_{k=1}^\infty \#\psi^{-1}(n_k)e^{-h_\mu n_k+\sqrt{2\rho_\mu n_k}g(n_k)}&\geq \sum_{k=1}^\infty1=\infty.
\nabla_{n}^{2}\phi_{2,m}=\nabla_{n}H_{2,m}+(\nabla_{n}M^{-1})(H_{1,m}\pm\nabla_{n}\phi_{2,(m-1)})+(M^{-1})(\nabla_{n}H_{1,m}\pm\nabla_{n}^{2}\phi_{2,(m-1)})
g=\frac{\pi^2s^2}{\sin^2(\frac{\pi s}{s_w})}\left(\frac{ds_w^2}{s_w^4}+d\theta_w^2\right) =\frac{\pi^2s^2}{\sin^2(\frac{\pi}{1+\rho_z})}\left(\frac{d\rho_z^2}{s^2(1+\rho_z)^4}+d\theta_z^2\right).
e' = (U_{j-1}^*U'''_jL) < \frac{e}{2q}\sum_{i=1}^{2q}\frac{k'_i}{k_i}+\sum_{i=1}^{2q}\sigma_i\left(k'-(k+1)\frac{k'_i}{k_i}\right) &=\frac{e}{2qk_{j-1}}+\sum_{i=1}^{2q}\sigma_i\left(1-(k+1)\frac{k'_i}{k_i}\right) &=\frac{e}{2qr_j}+ \sigma_{j-1}\left(1-\frac{r}{r_j}\right)+\sum_{i=1, i\neq j-1}^{2q}\sigma_i
w_n(x, y):=\frac{|u_n(x)-u_n(y)|^{p-2}(u_n(x)-u_n(y))}{|x-y|^{\frac{N+ps}{p'}}}w(x, y):= \frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))}{|x-y|^{\frac{N+ps}{p'}}},
E_{0}(\mathbf{z})B_{\mathbf{m}}^{(d)}\left(\mathbf{z}\right) = \sum_{i=1}^{r}B_{\mathbf{m}_{i}}^{(d)}\left(\mathbf{z}\right) \left(m_{i}+\frac{d}{2}(r-i)\right) h_{-,i}^{(d)}(\mathbf{m}).
\Gamma\Bigl({D\over 2}-1\Bigr){{\dot x_a\cdot\dot x_b\over{\Bigl[{(x_a - x_b)}^2\Bigr]}^{{D\over 2}-1}}\quad}-{1-\alpha\over 4}\Gamma\Bigl({{D\over 2}-2}\Bigr){\partial\over\partial\tau_a}{\partial\over\partial\tau_b}{\Bigl[{(x_a-x_b)}^2\Bigr]}^{2-{D\over2}}
\int_{\{-t'_3\le\Psi<-t'_4\}}|F_{t_0,\tilde{c}}|^2_h \le \int_{\{z\in M:-\Psi(z)\in (t'_4,t'_3]\backslash U_k\}}|\tilde F|^2_h+\int_{\{z\in M:-\Psi(z)\in (t'_4,t'_3]\backslash U_k\}}|F_{t_0,\tilde{c}}-\tilde F|^2_h &+\int_{\{z\in M: -\Psi(z)\in(t'_4,t'_3]\cap U_k\}}|F_{t_0,\tilde{c}}|^2_h.
\sum_{i=0}^{\mathrm{deg}(h)}h_i t^i = \sum_{i=0}^{d_0}h_i t^i = \sum_{i=0}^{pd+p-1}h_i t^i \, h_i=0 i>\mathrm{deg}(h) &=\sum_{\theta=0}^{d}\left( h_{p\theta}t^{p\theta}+h_{p\theta+1}t^{p\theta+1}+\cdots+h_{p\theta+p-1}t^{p\theta+p-1}\right) = \sum_{\ell = 0}^{p-1}\sum_{\theta=0}^{d}h_{p\theta+\ell}t^{p\theta+\ell}.
\sigma(\xi, \eta) = \sum_{|\alpha|\le s_1}\frac{(\partial^{\alpha}_\xi \sigma) (0,\eta)}{\alpha !}\xi^{\alpha}+ \sum_{|\beta|\le s_2}\frac{(\partial_{\eta}^{\beta}\sigma)(\xi,0)}{\beta!}\eta^{\beta}+\sigma_{\operatorname{err}},
|B_{R+|z|}|-|B_{R}| =\big((1+|z|R^{-1})^r-1\big)|B_R| \le r|z|R^{-1}(1+|z|R^{-1})^r|B_R| \le r|z|R^{-1}e^{r|z|R^{-1}}|B_R|.
\int \Big{|}\frac{1}{L}\sum \limits_{m=1}^{L}\chi_{B}\circ T^{-m t_n}- \mu(B)\Big{|}^{2}d\mu = \frac{1}{L^{2}}\sum \limits_{r,m=1}^{L}\mu(T^{(m-r)t_{n}}(B) \cap B) - \mu(B)\mu(B) &\leq \frac{1}{L}+ \frac{1}{L}\sum \limits_{\ell=1}^{L-1}\frac{L-\ell}{L}\mu(T^{\ell t_{n}}(B) \cap B) - \mu(B)\mu(B) < 2\epsilon^{2}/2 = \epsilon^{2}
\left( D_{i}\right)_{\;\;b}^{a}=\delta_{\;\;b}^{a}\partial_{i}+gf_{\;\;bc}^{a}A_{i}^{c},\;\left( D_{i}\right)_{b}^{\;\;a}=\delta_{b}^{\;\;a}\partial_{i}-gf_{\;\;bc}^{a}A_{i}^{c}.
\xi_{2-2k,z}\left(\mathbb{P}_{2-2k,n,N}(z,\mathfrak{z})\right) =(4\mathfrak{z}_2)^{2k-1}\Psi_{2k,-n-1,N}(z,\mathfrak{z}), \mathcal{D}_z^{2k-1}\left(\mathbb{P}_{2-2k,n,N}(z,\mathfrak{z})\right) =-(2k-2)!\left(\frac{\mathfrak{z}_2}{\pi}\right)^{2k-1}\Psi_{2k,n+1-2k,N}(z,\mathfrak{z}).
\hat{B}(C,t)|\vec{A},\vec{E};\phi_1,\phi_2{\rangle}_0 \propto |\vec{A}+L_3A^A(x_1,x_2),\vec{E}; \phi_1+L_3\phi^A_1(x_1,x_2),\phi_2{\rangle}_{e_3}=
\left\langle \psi_1^n T_a, T_j, \tau^{(k)}\right\rangle^\textnormal{coh}_{0,k+2,0}= \sum_{u+v=k}\sum_{x+y=u-1}\frac{k!}{v! x! y!}\tau_2(0)^x \left\langle \psi_1^{n-y+1}(T_a \cup \tau_2^{y-1}), T_j, \tau_2, \tau'^{(v)}\right\rangle^\textnormal{coh}_{0,v+3,0}
I_\mu = \Big\langle \mu_\xi \Big\rangle_{\xi \in \mathfrak{g}}^{}=\left\{\left.\sum_{i=1}^n f_i ~ \mu_{\xi_i}\right|~n\geq0,~ f_i \in C^\infty(M),~ \xi_i\in \mathfrak{g},~ 1 \leq i \leq n\right\}~.
\frac{1}{2}\sum_{x \in V_1}d^{J'}(x) \frac{p^2 n^2}{12}- \sum_{x \in V_1}\frac{p^2n^2}{100}d^{J'}(x) \ge \frac{1}{4}\sum_{x \in V_1}d^{J'}(x) \frac{p^2 n^2}{12}= \frac{1}{24}|J'| p^2n^2.
\left\vert \int_\Omega R\xi_E^2u^2dV_g\right\vert=\left\vert \int_{\Omega\cap E}R\xi_E^2u^2dV_g\right\vert\sim\left\vert \int_{\tilde M}\left(R\xi_E^2u^2\right)^{\sim}dV_{\tilde g}\right\vert.
\Pi^{in}=-\frac{i}{R}\left\{\partial_{x}+\frac{\sigma^{3}}{x}\left[l-l_{0}-\frac{1}{2}\left( 1-\sigma^{3}\right) +\vartheta \left( x\right)+\xi \rho_{R}x^{2}\right] \right\}\sigma^{1},\;\rho_{R}=\gamma R^{2}/2\;.
\varphi_{-1,\frac{1}{2}}(\tau,z)=\frac{\vartheta(\tau,z)}{\eta^3(\tau)}, \ \varphi_{-1,2}(\tau,z)=\frac{\vartheta(\tau,2z)}{\eta^3(\tau)},\ \varphi_{0,\frac{3}{2}}(\tau,z)=\frac{\vartheta(\tau,2z)}{\vartheta(\tau,z)}.
\begin{array}{r c l}{Q}&{=}&{\int_{0}^{L}d x \psi^{\dagger}\psi = - \lambda ( p_{+}- p_{-}) \ ,}\\{Q_{5}}&{=}&{\int_{0}^{L}d x \psi^{\dagger}\gamma_{5}\psi = - \frac{i \lambda \sqrt{2 \mu}}{2 \sqrt{\pi}\sqrt{L}}\left[ \varphi_{0}- \varphi_{0}^{\dagger}\right] \ .}\\ \end{array}
\left(I_{-}^{\alpha,\alpha',\beta,\beta',\gamma}\left(t^{-\frac{\rho}{k}}\sum_{n=0}^{\infty}\frac{\prod_{i=1}^{p}\Gamma_k(a_i+n\alpha_i)}{\prod_{j=1}^{q}\Gamma_k(b_j+n\beta_j)}\frac{(at^{-\frac{\mu}{k}})^n}{n!}\right)\right)(x).
\lVert \mu_\beta \rVert_{\ast}=\inf\left\{\sum\nolimits_{i=1}^{r}\lvert\lambda_{i}\rvert : \mu_\beta =\sum\nolimits_{i=1}^{r}\lambda_{i}u_{i}\otimes v_{i}\otimes w_{i}, \; r \in \mathbb{N}\right\}
\mathrm{t r}^{\prime}e^{- ( \Delta + m^{2}) t}= \sum_{\gamma \neq{\bf 1}}\int_{H / \Gamma_{b}}{\frac{d x d y}{y^{2}}}{\hat{G}}( z , \gamma [ z ] ; t ) + A ( H / \Gamma ) \cdot{\hat{G}}( z , z ; t ) .
\frac{\partial \bar{R}^\mu}{\partial x^i}= \left(\frac{\partial{R}^\mu}{\partial x^i}\circ\iota_\mu\right) + \left( \frac{\partial{R}^\mu}{\partial \hat{\rm v}^a}\circ\iota_\mu\right) \frac{\partial{\hat\iota}^a_\mu}{\partial x^i}.
f_1(\alpha;\vec y) =\frac{y_1(y_1-1)(y_1-\alpha)}{\alpha(\alpha-1)}\left(2y_2+\frac{v_1-v_2}{y_1}-\frac{(v_1+v_2)(\alpha-1)}{(y_1-1)(y_1-\alpha)}\right), f_2(\alpha;\vec y) =\frac{y_2}{\alpha(\alpha-1)}\big\{\big[-3y_1^2+2(\alpha+1)y_1-\alpha\big]y_2-2(v_1-v_2)y_1+2v_1\alpha-2v_2\big\}.
\rho_A(\tau) =\int_0^\tau t^{-1/2}\|B(t)\|_2^2 dt =\int_0^\tau t^{-1/2}(-\Delta e^{2t\Delta}A_0, A_0)dt = c_1 ((-\Delta)^{1/2}A_0, A_0) -\int_\tau^\infty t^{-1/2}(-\Delta e^{2t\Delta}A_0, A_0)dt = c_1 \|A_0\|_{H_{1/2}}^2+ O(\tau^{-1/2}\|e^{\tau \Delta}A_0\|_2^2)
- \int_{0}^{\sigma}\! \! \! d s \int_{- \infty}^{\infty}\frac{d \omega}{2 \pi}\frac{s K^{\frac{D - 3}{2}}}{( 4 \pi )^{\frac{D - 1}{2}}}\left| \frac{\Gamma ( \frac{D - 1}{2}+ i \beta )}{\Gamma ( 1 + i \beta )}\right|^{2}\Gamma ( - \frac{D - 3}{2}) \mathrm{t r}\mathrm{\boldmath 1}.
F_{N}(a, b; t) = \frac{1-tq^N}{1-t}+ \frac{(1-tq^N)(b-a)(1-q^N)tq}{(1-t)(1-bq)(1-tq)}\sum_{n=0}^{N-1}\frac{(q^{1-N})_{n}q^{Nn+n}(bq/a)_{n}(at)^{n}}{(bq^2)_{n}(tq^2)_{n}}&= \frac{1-tq^{N+1}}{1-t}+ \frac{(1-tq^{N+1})(b-a)(1-q^{N+1})tq}{(1-t)(1-bq)(1-tq)}\sum_{n=0}^{N}\frac{(q^{-N})_{n}q^{(N+1)n+n}(bq/a)_{n}(at)^{n}}{(bq^2)_{n}(tq^2)_{n}}.
\P (G_0v_{||E||^{1/3}}(d, 0, E + 2 |E|)) \leq \P (G_0v_{||E||_1^{1/3}}(d, ||E||_1^{1/3}, E)) + \mathcal{O}_{d, \upsilon}\left( \sum_{k=2}^{\infty}N^{-3/2}(1+ k^3) e^{- \frac{(k-1)^2}{4 d^2 ||E||_1^{1/3}}}\right).
\frac{1}{2}\frac{d}{dt}\int_\mathbb{R}\big(|W_x|^2 + 2W_{x}W_{xt}\big) dx+ a^2\|W_{xx}\|^2 - \|W_{xt}\|^2\leqslant O(1) \delta^{-1/4}\big(\|W_x\|^2 + \|W_t\|^2\big) &\quad+O(1) \delta^{-1/4}\int_\mathbb{R}\big(|u^a_x W|^2 +|u^p_{xx}|^2+|u^p_{tt}|^2+ e^2 +|u^p_{xt}|^2\big)dx+ O(1)\delta^{1/4}\|W_{xx}\|^2.
\lim_{m \rightarrow 0}I m \, T_{1}( t ) = - \lim_{m \rightarrow 0}\frac{m^{2}}{4 t^{2}}\left( 1 - \frac{4 m^{2}}{t}\right)^{\frac{1}{2}}\theta ( t - 4 m^{2}) = - \frac{1}{2 4}\delta ( t ) \, .
\Psi_{i,g_1}^{[L][M]}(x_1) \Psi_{j,g_2}^{[M][N]}(x_2) \ =\ \sum_k (x_1-x_2)^{h_i+h_j-h_k}\Psi_{k,g_{12}}^{[L][N]}(x_2) C_{(i,g_1) (j,g_2) (k,g_{12})}^{[L][M][N]}+ \dots
J =\left(\cos(ax) \frac{\sin(bx)}{b}\right)_0^1 + a \int_0^1 \sin(ax) \frac{\sin(bx)}{b}dx =\frac{1}{b}\cos(a)\sin(b) +\frac{a}{b}\left[ \left(\sin(ax)\frac{\cos(bx)}{b}\right)_1^0+\frac{a}{b}\int_0^1 \cos(ax) \cos(bx) dx\right] =\frac{1}{b}\cos(a) \sin(b) -\frac{a}{b^2}\sin(a)\cos(b) + \left(\frac{a}{b}\right)^2 J
A=\frac{1}{2}\Big( f(n)f(n+3k)+f(n+2k)f(n+5k)+\dots +f(n+(2m-4)k)f(n+(2m-1)k) &+f(n+(2m-2)k))fn(n+k) &-f(n+2k)f(n+k)-f(n+4k)f(n+3k)-\dots &-f(n+(2m-2)k)f(n+(2m-3)k)-f(n)f(n+(m-1)k)\Big).
R ( x ) e^{2 k r_{0}}\left( 1 +{\cal O}( l^{2}) \right) \mathrm{T r}T_{\mu \nu}= \frac{r ( x )}{\sqrt{6}M_{P l}e^{- k r_{0}}}\mathrm{T r}T_{\mu \nu}\left( 1 +{\cal O}( l^{2}) \right) ,
\beta \circ G_{s,0}\circ \alpha^{-1}\circ \gamma(x) =\beta \circ F_{b^*}\circ \alpha^{-1}(x, s) &=(\beta_x(\phi_{b^*}(\alpha_x^{-1}(x),\alpha_y^{-1}(x,s)), \alpha_x^{-1}(x)), \beta_y (\alpha_x^{-1}(x)) )
\nu \biggl(\bigcap_{i=1}^rT^{\mathbf{n}_i}B_i\biggr) \longrightarrow \prod_{i=1}^r\nu (B_i)\enspace \textup{as}\enspace \|\mathbf{n}_i-\mathbf{n}_j\|\to\infty \enspace \textup{for}\enspace 1\le i < j \le r.
\lambda_{\ell}(k) := \begin{cases}4-4\cos(k/2) \ \k \in (0,\pi) 4+4\cos(k/2) \ \k \in (\pi,2\pi) \end{cases}\lambda_{r}(k) := \begin{cases}4+4\cos(k/2) \ \k \in (0,\pi) 4-4\cos(k/2) \ \k \in (\pi,2\pi). \end{cases}
-\theta Te^{-\theta T\mathcal{I}(\boldsymbol{\mathrm{x}}_t;\boldsymbol{\mathrm{y}}_t)}\frac{\partial\mathcal{I}(\boldsymbol{\mathrm{x}}_t;\boldsymbol{\mathrm{y}}_t)}{\partial\boldsymbol{\mathrm{K}}_t}+\lambda\boldsymbol{\mathrm{I}}-\Phi=0,
{\cal S}= \int d^{4}x \sqrt{- \bar{g}}\ \left[ \frac{1}{2}\bar{R}- \frac{1}{4}\left( \frac{\partial Y}{Y}\right)^{2}- \frac{3}{4}\left( 1 - \frac{c}{Y}\right) ( \partial \sigma )^{2}- V ( Y , \sigma ) \right] +{\cal S}_{m}\hspace{5 m m}\ ,
\hskip-36pt \begin{array}{lcp{10.7cm}}\hline\noalign{\smallskip}P_{an}= $(x-1)^2 \partial^2 -(x-1)(3a+3b+4c+2g+2-(a+c)/x) \partial$ $+ (2a+2b+2c+g+2)(a+b+2c+g+1-c/x)$, Q_{an}= $ (x-1)^2 \partial^2 -(x-1)(1+2g+3b+4c+3a-(1+a+c)/x)\partial$ & $+(2a+2b+2c+g+1)(a+b+2c+g+1)$ & $-(c+1)(2a+2b+2c+g+2)/x +(a+c+1)/x^2$. \noalign{\smallskip}\hline \end{array}
F_{\textrm{B}i}(x)=\bar{F}_i(x)\tilde{H}(x)+F_{\textrm{BR}i}(x), \deg(F_{\textrm{BR}i}) < d, i=1, \dots, n, F_{\textrm{S}i}(x)=\bar{F}_i(x)\tilde{H}(x)+F_{\textrm{SR}i}(x), \deg(F_{\textrm{SR}i}) < d, i=1, \dots, n.
i\frac{d}{dt}U^{M}\left( t,s\right) \varphi =Q_{M}\left( t\right)U^{M}\left( t,s\right) \varphi i\frac{d}{ds}U^{M}\left( t,s\right) \varphi =-U^{M}\left( t,s\right)Q_{M}\left( s\right) \varphi.
\left|\partial_x^\alpha\partial_y^\beta K(x,y)-\partial_x^\alpha\partial_y^\beta K(x,y+v)\right| &\quad\lesssim|v|^{\rho}|x-y|^{-n-|\alpha|-|\beta|-\rho}\quad\begin{cases}|\alpha|\leq\lfloor\widetilde s\rfloor, |\beta|=\lfloor \widetilde J\rfloor-n-|\alpha|.\end{cases}
\sum_{i=0}^n e_i(\boldsymbol{x}) (t_n x_n)^{n-i}s_{\lambda-\rho}(\boldsymbol{x}) = \sum_{i=0}^n (t_n x_n)^{n-i}\sum_{\mu \in (\lambda-\rho) \otimes 1^i}s_{\mu}(\boldsymbol{x}),
L_{T}:= \tilde{\sigma}^{*}_{3}(t)\left[P_{1}\dot{Q}^{1}+\Psi_{a}\dot{\Xi}^{a}+\Theta_{1}\dot{\Theta}^{1}-H_{T}\right] = \frac{d}{dt}\left[\Psi_{2}\Xi^{2}\right]+P_{1}\dot{Q}^{1}-H_{T}+{constant},
\gamma_\xi^{(0)}= - \frac{1}{32 \pi^2}{\cal M}_\xi^4 (X) \, \ln \frac{-{\cal M}_\xi^2 (X)}{e^{3 / 2}\mu_d^2}+ \frac{i}{32 \pi}{\cal M}_\xi^4 (X) \;\;\;\;\;\; ({\cal M}_\xi^2 (X) < 0) .
\xi^TA\xi =e^{\frac{|\Lambda+\nabla_y\widetilde{w}|^2}{2}+V}\xi^T\big( (\Lambda+\nabla_y\widetilde{w}) (\Lambda+\nabla_y\widetilde{w})^T+I\big )\xi &=e^{\frac{|\Lambda+\nabla_y\widetilde{w}|^2}{2}+V}\big| (\Lambda+\nabla_y\widetilde{w})^T\xi\big |^2+e^{\frac{|\Lambda+\nabla_y\widetilde{w}|^2}{2}+V}|\xi |^2\geq e^{\inf V}|\xi |^2.
\int_\Omega p_\varepsilon \nabla u^\varepsilon \nabla \varphi\,dx+\int_\Omega (\lambda+V_\varepsilon)u^\varepsilon \varphi\,dx=\int_\Omega g\varphi\,dx,\quad\forall\,\varphi\in X_\varepsilon^{\frac{1}{2}},\,\,\varepsilon\in(0,\varepsilon_0];
u_M(x):=\mathbb{E}\left\{g(X_M^{x})+\frac{1}{d}\sum\limits_{k= 1}^M \widetilde{r}^2(X^{x}_{k-1})f(X^{x}_{k-1}+\widetilde{r}(X^{x}_{k-1})Y)\right\}x\in D,
A=\frac{1}{2}\int_{\varphi_{1}}^{\varphi_{2}}a^{2}\varphi^{2n}d\varphi=\frac{a^{2}}{2(2n+1)}(\varphi_{2}^{2n+1}-\varphi_{1}^{2n+1}),ifn\ne-\frac{1}{2},
\mu_{k}^{\prime \prime}+ \biggl \{k^{2}\biggl [ \frac{\epsilon_{1}}{1 + \mathrm{e}^{x}}+ \frac{\epsilon_{3}\mathrm{e}^{x}}{( 1 + \mathrm{e}^{x})^{2}}\biggr ] - ( 1 - 6 \xi ) \frac{\beta ( \beta + 1 )}{\eta^{2}}\biggr \}\mu_{k}= 0 ,
\begin{cases}(d_0, d_1, d_2, d_3) \in I_0 \times I_1 \times I_2 \times I_3, \min_{i \in \{1, 2, 3 \}}\rho_i^{d_i}(0) \geq d_0, \nu_i^{d_0}(h_i) \geq d_i \ \ \i \in \{1, 2, 3 \}.\end{cases}
\begin{cases}l'_{k\,1}z_1 + l'_{k\,2}z_3+ \cdots + l'_{k\,n}z_{2n-1}=0 l'_{k\,1}z_2+ l'_{k\,2}z_4 +\cdots +l'_{k\,n}z_{2n}= b_{\mathbf{l'_k}}(P) z_{2n+1}\end{cases}\mbox{for}k=1, \ldots, n-r.
\sum_{i=1}^{\infty}\tau_k(p^i) \Big( \frac{1}{p^i}- \frac{1}{p^{i+1}}\Big) = \Big( 1-\frac{1}{p}\Big) \sum_{i=1}^{\infty}\frac{\tau_k(p^i)}{p^i}= \Big( 1-\frac{1}{p}\Big) \Big( \Big(1 -\frac{1}{p}\Big)^{-k}-1 \Big) = \Big( 1-\frac{1}{p}\Big)^{-(k-1)}-1 + \frac{1}{p}
\mathbf{J}_0(\, \underline{\smash{j}}_0\,\underline{\smash{j}}_0^\intercal/j_{0,N+1}) = \sum_{s=-\infty}^\infty \sum_{t=0}^\infty \frac{(-1)^t2^{-2t-s}}{t!(t+s)!}D_{\underline{e}_N}^{2t+s}\mathbf{J}_{-s}(\, \tilde{\underline{r}}\,\underline{\smash{j}}_0^\intercal ) D_{\underline{\smash{j}}_0}^{2t+s}.
\Psi_{k,\ell,j,m,n}(r,\alpha,\theta ,\beta,\varphi )=\frac{F_{k,\ell,j,m,n}(r)}{r^2}\frac{G_{\ell,j,m,n}(\alpha)}{\sin^{3/2}\alpha}\frac{\Theta_{j,m,n}(\theta )}{\sin\theta}\frac{H_{m,n}(\beta)}{\sqrt{\sin \beta}}\Phi_n (\varphi ).
\underset{{\tilde \tau}_1 \rightarrow \tau_1}\lim \nu_{\theta_1}({\tilde \tau}_1) = \underset{{\tilde \tau}_1 \rightarrow \tau_1}\lim \int \limits_{\Theta_2}\nu_{\theta}({\tilde \tau}_1, \tau_2) \, d \tau_2 = \int \limits_{\Theta_2}\nu_{\theta}(\tau_1, \tau_2) \, d \tau_2 = \nu_{\theta_1}(\tau_1)
\kappa_\mathrm{v}= \mathrm{rank}\left\lbrace \mathrm{span}\left[ \mathbf{a}_\mathrm{v}\left(\theta \right) \right] \right\rbrace \approx \lceil M_\mathrm{v}\frac{d_\mathrm{v}}{\lambda}\left(\cos\theta_\mathrm{min}-\cos \theta_\mathrm{max}\right) \rceil,
\begin{array}{c}\mathbf t^1=(t_1^1, \cdots, t^1_{3d+1})\in sym^{3d+1}(\mathbf P^1) \mathbf t^2=(t_1^2, \cdots, t^2_{3d+1})\in sym^{3d+1}(\mathbf P^1).\end{array}
\begin{array}{l}\emph{Ker}_{Z_{k+1}}\bar\Omega(Z_1,\ldots,Z_k)_{l+1}=\bar\Omega(Z_1,\ldots,Z_k)_{l}(l>0), \emph{Ker}_{Z_{k+1}}\bar\Omega(Z_1,\ldots,Z_k)_0=\mathcal R^k,\ \emph{Ker}_{Z_{k+1}}\mathcal R^k=\mathcal R^k.\end{array}
E \, = \, \frac{\rho F}{2}\int_{0}^{l}d x \, \left( \dot{y}^{2}\, + \, r^{2}\dot{\psi}^{2}\right) \, + \, \frac{E I}{2}\int_{0}^{l}d x \, \psi^{2}\, + \, \frac{k F G}{2}\int_{0}^{l}d x \, \left( y^{\prime}\, - \, \psi \right)^{2}.
C^{( 2 )}= \{- \frac{1}{\omega_{A}}( \frac{1}{\omega_{C}}f^{C A B}\varphi_{C}+ f^{\lambda A B}\varphi_{\lambda}) + \frac{1}{2 \omega_{A}\omega_{B}}f_{C}^{\; A B}\varphi^{C}\}( P_{A}^{a}P_{B}^{b}- P_{A}^{b}P_{B}^{a}) .
\mathcal{A}= \Big\{T=\begin{pmatrix}t_{0}bt_{1}\ldots bt_{n-1}at_{n-1}t_{0}\ldots bt_{n-2}\vdots \vdots \ddots \vdots at_{1}at_{2}\ldots t_{0}\end{pmatrix}: t_0,\dots, t_{n-1}\in \mathbb{C}\Big\}.
(e_{uh}^0, e_{ph}^0, e_{\theta h}^0) = (u_h^0, p_h^0, \theta_h^0) - (\hat{u}_h^0, \hat{p}_h^0, \hat{\theta}_h^0) = \Pi_h^{\rm SP}(u^0, 0, \theta^0) - \Pi_h^{\rm SP}(u^0, p^0, \theta^0).
\exp\left\{-\sum_{i=j}^{n}\frac 1{\left(i+e_{[k]}^1\right)\ln \left(i+e_{[k]}^1\right)\dots \ln_k \left(i+e_{[k]}^1\right)}\right\}\le \frac{\ln_{k}(j+e_{[k]}^1)}{\ln_{k}(n+1+e_{[k]}^1)},
\frac{1}{n-1}\sum_{r=0}^{n-\ell}\binom{(4h-4h+3)(n-1)}{r}\binom{n+r+\ell-2}{\ell}\binom{(4h-2k-3)(n-1)-r-\ell}{n-r-\ell}&-\frac{1}{2n-1}\sum_{r=0}^{n-\ell}\binom{2(k+1-h)(2n-1)-n}{r}\binom{n+r+\ell-1}{\ell}\binom{(2h-k-2)(2n-1)+n-r-\ell}{n-r-\ell}
\psi(X_1,X_2) =-\psi(X_2,X_1), \psi(x,y)+\psi(y,z)+\psi(z,x) =0, x+y+z=0, \psi(t_{12},t_{23}+t_{34})+\psi(t_{13}+t_{23},t_{34}) =\psi(t_{23},t_{34})+\psi(t_{12}+t_{13},t_{24}+t_{34})+\psi(t_{12},t_{23}).
e^{i(t-a_{j+1})\Delta}w(a_{j+1})= e^{it\Delta}w(0)+i\int_{0}^{a_{j+1}}e^{i(t-s)\Delta}(F(\tilde{u}+w)-F(u))ds -i\int_{0}^{a_{j+1}}e^{i(t-s)\Delta}eds.
F_{N}(a,b;t) = 1+\frac{(1-aq)(1-q^{N+1})(1-tq^{N+1})}{(1-tq^N)(1-bq^{N+2})(1-t)}\large\left(t-\frac{(b-aqt)}{(b-aq)}+\frac{(b-aqt)(1-b)}{(b-aq)(1-bq^{N+1})}\large\right) &\quad+ \frac{(1-aq)(1-q^{N+1})^2(b-aqt)(b-atq^2)}{(1-tq^N)(b-aq)(1-bq^{N+2})(1-bq^{N+1})(1-t)}F_{N}(aq,b;tq).
\rho P = \bigl(G_1(f)F_1(f)+G_2(f)F_2(f)\bigr)P = G_1(f)F_1(f)F_2(f)P_2 + G_2(f)F_2(f)F_1(f)P_1 = G_1(f)F(f)P_2 + G_2(f)F(f)P_1 \quad = 0\quad
\Big|\frac{z_j(n+1)}{1-|z_j(n+1)|^2}\Big|^2 = \frac{1}{4}\Big(\frac{|\Theta[n](e_j,e_j)|^2}{H[n]^2(e_j,\overline{e_j}) - |\Theta[n](e_j,e_j)|^2}\Big) \le \frac{1}{4c^2}|\Theta[n](e_j,e_j)|^2,
x_{\epsilon_1+\epsilon_2}^2=0,x_{\epsilon_1-\epsilon_2}^2=0,x_{\epsilon_1}^2-x_{\epsilon_1+\epsilon_2}x_{\epsilon_1-\epsilon_2}=0,x_{\epsilon_2}x_{\epsilon_1+\epsilon_2}=0, x_{\epsilon_1}x_{\epsilon_1-\epsilon_2}=0,x_{\epsilon_1}x_{\epsilon_1+\epsilon_2}=0,x_{\epsilon_2}^3=0, x_{\epsilon_2}^2 x_{\epsilon_1}=0.
{C_z}(z,t|{z_{\rm tx}},{t_0}) = \frac{1}{{{\sqrt{(4\pi D(t -{t_0}))}}}}{e^{\frac{{-{{(z -{z_{\rm tx}}- v(t -{t_0}))}^2}}}{{4D(t -{t_0})}}-{k_d}(t -{t_0})}}u(t-t_0).
\frac{1}{z^3}\left( -q^2 - \partial_z^2 + \frac{3}{z}\partial_z + \frac{m^2}{z^2}+ \frac{\lambda_{v2}}{z_v}\delta(z-z_v) +\frac{\lambda_{h2}}{z_h}\delta(z-z_h) \right) \Delta_q(z,z')= \delta(z-z').
\begin{array}{lccl}(0~0~0~0~0~1) \mid \mbox{ground}> (0~0~0~0~0~1) = ({\bf 1,2,-5})_F \Downarrow \beta^{5-}_6 \mid \mbox{1st}> (0~0~0~1~-1~0) = ({\bf \overline{5},1,-2})_B. \end{array}
S \cong ((C_{16}\rtimes C_4) \times C_2) \rtimes C_2 = ((\langle x \rangle \rtimes \langle y \rangle) \times \langle z \rangle) \rtimes \langle w \rangle, \x^y = x^{13}, \x^w = xy^2, \y^w = x^8yz, \z^w = z.
\eta_x=\left(1-\frac{p_-(x)+p_+(x)}{\mu(\{x\})}\right)\delta_x+\frac{1}{\mu(\{x\})}\int_{F_\nu(x-)-p_-(x)}^{F_\nu(x-)}\delta_{F_\nu^{-1}(v)}dv+\frac{1}{\mu(\{x\})}\int_{F_\nu(x)}^{F_\nu(x)+p_+(x)}\delta_{F_\nu^{-1}(v)}dv.
\Big\|\int_{t_1}^{t_2}\int_0^{\infty}\sigma_s(y)^2 \int_y^{\infty}\psi(s,z) G_{t_2-s}(x-z)dz dy ds \Big\|_p \leq C \int_{t_1}^{t_2}\int_{\mathbb{R}}G_{t_2-s}(x-z) dz ds &= C (t_2-t_1).
f^{+}(\tau) =\sum\limits_{r(2N)}\sum\limits_{\substack{D\gg \infty D \equiv -r^{2}(4N)}}c_{f}^{+}(D,r)q^{D/4N}\mathfrak{e}_r, f^{-}(\tau) =\sum\limits_{r(2N)}\bigg(c_{f}^{-}(0,r)v^{1-k}+ \sum\limits_{\substack{D < 0 D \equiv -r^{2}(4N)}}c_{f}^{-}(D,r)\Gamma(1-k,\pi |D|v/N)q^{D/4N}\bigg)\mathfrak{e}_r,
\int_M f_0(x) \,(f_1(\phi(x)) \, \frac{\phi''(x)}{\phi'(x)}\, dx = \int_M f_1(x) \,(f_0(\phi^{-1}(x))\frac{\phi''(\phi^{-1}(x)}{\phi'(\phi^{-1}(x)}\, (\phi^{-1})'(x) dx &= \int_M f_1(x) \,(f_0(\phi^{-1}(x)) \phi''(\phi^{-1}(x)\, (\phi^{-1})'(x)^2 dx &= - \int_M f_1(x) \,(f_0(\phi^{-1}(x)) \, \frac{(\phi^{-1})''(x)}{(\phi^{-1})'(x))}dx ;
\delta A_d[q_d] =D_1 L_d (q_0,q_1) \cdot \delta{q_0}+ \sum_{m=1}^{N-1}D_1 L_d (q_m,q_{m+1})\cdot \delta{q_m}+\sum_{m=0}^{N-2}D_2 L_d (q_m,q_{m+1}) \cdot \delta{q_{m+1}}+D_2 L_d (q_{N-1},q_{N}) \cdot \delta{q_{N-1}}.