harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
H_NS_N(i,j) =\frac{1}{J_N}(-J_{i-2}J_{N-j}(J_{j-i+1}+J_{j-i+2})-J_{i-1}J_{N-j}(J_{j-i}+J_{j-i+1}) ~+2J_iJ_{N-j}(J_{j-i-1}+J_{j-i})) =\frac{J_{N-j}}{J_N}(-2J_{i-2}(J_{j-i}+J_{j-i+1})-J_{i-1}(J_{j-i}+J_{j-i+1}) ~+J_i(J_{j-i}+J_{j-i+1})) =\frac{J_{N-j}}{J_N}(-2J_{i-2}-J_{i-1}+J_i)(J_{j-i-1}+J_{j-i}) =0.
S_k\widetilde{u^{\alpha}_{J_1;J_2}}=S_k\left(S_{J_2}\left(\widetilde{D_{J_1}}\widetilde{u}^{\alpha}\right)\right)=S_{J_2+\bold{1}_k}\left(\widetilde{D_{J_1}}\widetilde{u}^{\alpha}\right)=\widetilde{u^{\alpha}_{J_1;J_2+\bold{1}_k}}.
{\vert O_k\rangle}_{\nu\bar\nu}=\epsilon^{l_1..l_{k-2}}\,\epsilon^{t_1 \cdots t_{k-2}}\, (\Gamma^z_{A_1B_1}\, \nu^{A_1}_{l_1}\, \nu^{B_1}_{t_1}) \cdots (\Gamma^z_{A_{k-2}B_{k-2}}\,\nu^{A_{k-2}}_{l_{k-2}}\,\nu^{B_{k-2}}_{t_{k-2}}) \, \vert\Phi\rangle
D_{23}= s_3 \int_{at_3}^\infty \frac{v-[v]-\frac{1}{2}}{v^{s_2}}\int_v^\infty \frac{1}{u^{s_1+s_3+1}(1+\frac{v}{u})^{s_3+1}}dudv &= \frac{s_3}{2 \pi i \Gamma(s_3+1)}\int_{at_3}^\infty \int_v^\infty \int_{(c)}\frac{(v-[v]-\frac{1}{2}) \Gamma(s_3+1+z) \Gamma(-z)}{v^{s_2-z}u^{s_1+s_3+1+z}}dzdudv
\mathcal{K}_{\partial \Omega}[\phi]\equiv \frac{1}{2}\bigg(\gamma_{0}\mathcal{D}_{\partial\Omega}[\phi]+\gamma_{0}^\complement\mathcal{D}_{\partial\Omega}[\phi]\bigg)\, , \mathcal{W}_{\partial \Omega}[\phi]\equiv-\gamma_{1}\mathcal{D}_{\partial\Omega}[\phi] =-\gamma_{1}^\complement\mathcal{D}_{\partial\Omega}[\phi]
\| u(t)-u(t_0) \|_{H^n(M )}\leqslant C \| u(t)-u(t_0) \|_{H^{2n}(M )}^{1/2}\| u(t)-u(t_0) \|_{L^2(M )}^{1/2}\leqslant CF_2(t_0)^{1/4}.
\begin{aligned}X_{{12}}(-j-1)\sp{b_{{12}}}X_{{11}}(-j-1)\sp{b_{{11}}}X_{{22}}(-j)\sp{a_{22}}\,,b_{{12}}+b_{{11}}+a_{{22}}= k+1, &X_{{22}}(-j-1)\sp{b_{22}}X_{{12}}(-j-1)\sp{b_{12}}X_{{22}}(-j)\sp{a_{22}}\,,b_{22}+b_{12}+a_{22}= k+1.\end{aligned}
\left( \begin{pmatrix}1 1 0 1 \end{pmatrix}+ \begin{pmatrix}a b c d \end{pmatrix}p \right) \left( I + \begin{pmatrix}\alpha 0 0 \delta \end{pmatrix}p \right) = \begin{pmatrix}1 1 0 1 \end{pmatrix}+ \begin{pmatrix}a + \alpha b + \delta c d + \delta \end{pmatrix}p.
\begin{array}{c}{{C^{- 1}\left( \begin{array}{c}{{\gamma_{4}}}\\{{\gamma_{5}}}\\ \end{array}\right) C = R_{C}\left( \begin{array}{c}{{\gamma_{4}^{T}}}\\{{\gamma_{5}^{T}}}\\ \end{array}\right) ,}}\\{{\Pi^{- 1}\left( \begin{array}{c}{{\gamma_{4}}}\\{{\gamma_{5}}}\\ \end{array}\right) \Pi = R_{P}\left( \begin{array}{c}{{\gamma_{4}}}\\{{\gamma_{5}}}\\ \end{array}\right) ,}}\\ \end{array}
R_{\mu \nu}= T r \{2 J_{( \mu}^{z}\bar{J}_{\nu )}^{z}+ i ( z - \bar{z}) ({\cal F}_{\mu \lambda}{{{\cal F}^{+}}_{\nu .}}^{\lambda}+ \bar{\cal F}_{\mu \lambda}{{{\cal F}^{T}}_{\nu .}}^{\lambda}) \},
q^{1/24}(-q^2;q^3)_\infty(-q;q^3)_\infty(q^3;q^3)_\infty = q^{1/24}\frac{(q^4;q^6)_\infty}{(q^2;q^3)_\infty}\frac{(q^2;q^6)_\infty}{(q;q^3)_\infty}(q^3;q^3)_\infty = q^{1/24}\frac{\varphi_2/\varphi_6}{\varphi_1/\varphi_3}\varphi_3 = q^{1/24}\frac{\varphi_2\varphi_3^2}{\varphi_1\varphi_6}= \frac{\eta_2\eta_3^2}{\eta_1\eta_6}.
\Gamma_{a - c}^{i}( \vec{p}, - \vec{p}) = \varepsilon_{\beta}^{*}( - \vec{p}, s ) \Gamma_{a + b + c}^{i \alpha \beta}\varepsilon_{\alpha}( \vec{p}, s ) = \frac{3 e^{3}}{8 \pi \theta}\epsilon^{i j}q_{j}.
(H(x,y,z)-h)\cdot [\nabla C(x,y,z) \times (\nabla C(x,y,z) \times \nabla H(x,y,z))] &=(H(x,y,z)-h)\cdot y^{\lambda-1}\cdot\left[ \begin{array}{c}y(\lambda x^2 -y^2 -z^2) x(-\lambda x^2 +y^2 -\lambda z^2) (\lambda +1)xyz \end{array}\right].
\frac{\theta_{\Gamma,\mathbf{P_1}}^2(t_\Gamma(P_m))^2}{\theta_{\Gamma,\mathbf{P_2}}^2(t_\Gamma(P_m))^2}\frac{\theta_{\Gamma,\mathbf{P_2}}^2(t_\Gamma(P_h))}{\theta_{\Gamma,\mathbf{P_1}}^2(t_\Gamma(P_h))^2}=\frac{\lambda_h-\lambda_m}{\lambda_h-\lambda_l}\frac{\lambda_k-\lambda_m}{\lambda_k-\lambda_l}
F ( l ) = \left( \frac{d}{d l / s}\right)^{( d + 3 ) / 2}{\frac{\left[ \sqrt{\left( \frac{l}{s}\right)^{2}+ 1}-{\frac{l}{s}}\right]^{( d - 1 ) / 2}}{\sqrt{\left( \frac{l}{s}\right)^{2}+ 1}}}.
\sum_{i , j , k = 0 , 1 . . .}\left( g^{2}\right)^{i}\left( g^{2}l n{\frac{\Lambda^{2}}{m_{0}^{2}}}\right)^{j}\left( g^{2}{\frac{\Lambda^{2}}{M^{2}}}\right)^{k}C_{i j k}\left( \Lambda^{2}, M^{2}, m_{0}, p^{2}\right)
(t^\vee)^{-1}= \gamma t^\vee \gamma^{-1}= \zeta n^\vee \cdot t^\vee \cdot (n^\vee)^{-1}\zeta^{-1}&= \zeta n^\vee \zeta^{-1}\cdot \zeta t^\vee \zeta^{-1}\cdot \zeta (n^\vee)^{-1}\zeta^{-1}&= \sigma({}^{w^\vee}t^\vee).
\sum_{n=1}^{N}\frac{(-1)^{n-1}\left(\frac{c}{d}\right)_{n}d^n q^{\frac{n(n+1)}{2}}}{(q)_n (q)_{N-n}(cq)_{n}}\left(n + \sum_{k=1}^n \frac{q^k}{1- q^k}\right) =:T_1+T_2,
u_s(z, t_0) := z + (x_s - x_{t_0}) + ( - \inf_{t_0 \leq u \leq s}[z + (x_u - x_{t_0})]) \vee 0, \tilde{u}_s(\tilde{z}, t_0) := \tilde{z}+ (\tilde{x}_s - \tilde{x}_{t_0}) + ( - \inf_{t_0 \leq u \leq s}[\tilde{z}+ (\tilde{x}_u - \tilde{x}_{t_0})]) \vee 0.
\it{II}\leq \frac{2^k}{C_0^k}\int_{|\beta|\geq \frac{2}{C_1}d_I(t)}|\beta|^{-k}d\beta=2\frac{2^k}{(k-1)C_0^k}\frac{C_1^{k-1}}{2^{k-1}}d_I(t)^{-k+1}=\frac{4C_1^{k-1}}{(k-1)C_0^{k-1}}d_I(t)^{-k+1}.
\sum_{|k|\le N}\int_{\eta_{k-1}}^{\eta_k}\frac{v_1^{-p'}(x)}{\bigl[V_1^-(x)\bigr]^{p}}\biggl(\int_{\eta_{k-1}}^xv_1^{-p'}(t)\bigl|G^{(\delta)}_{1,k}(t)\bigr|^{p'-1}\,dt\biggr)^p\,dx\lesssim\int_{\eta_{k-1}}^{\eta_k}v_1^{-p'}\bigl|G_{1,k}^{(\delta)}\bigr|^{p'}=\bigl[\mathbf{G}^{(\delta)}_{1,N}(g)\bigr]^{p'-1},
\Omega_{j}( x_{n}) \overline{{{\Omega}}}_{i}( x_{m}) = \delta_{i , j}\frac{\delta_{m , n}}{\Delta x_{n}}\pm q^{\pm 1}\hat{{\cal R}}_{i k j l}( q ) \overline{{{\Omega}}}_{l}( x_{m}) \Omega_{k}( x_{n})
\int_{0}^{T}e^{\left( T-s\right) A}G u(s) ds = \int_{0}^{T}e^{\left( T-s\right) A}\left (\begin{array}[c]{c}0 \psi_2(s) \end{array}\right) ds + \int_{0}^{T}e^{\left( T-s\right) A}G \psi_1'(s) ds.
\tau^l \frac{\Gamma\bigl(2-2l+\tau(1+\lambda_1+\lambda_2)\bigr)}{\Gamma\bigl(2-l+\tau(1+\lambda_1+\lambda_2)\bigr)}=\prod\limits_{j=0}^{l-1}\frac{1}{1+\lambda_1+\lambda_2-(l+j-1)/\tau}.
\sum_{\beta=1}^{\frac{n(n-1)}{2}}\omega_{\alpha\beta}^{(l)}\, K_\beta^{(2,l)}(x) = S^{(2,l)}(U^{(l)})^{-1}J_\alpha^{(1,l)}U^{(l)}(S^{(2,l)})^{-1}(x-q^{(2,l)}) = S^{(1,l)}\exp(\sigma^{(l)}) J_\alpha^{(1,l)}\exp(-\sigma^{(l)}) (S^{(1,l)})^{-1}(x-q^{(2,l)}).
\max \Big\{n^{-1}\sum_{i=1}^n K(P_{f_0,i}, P_{f,i}), n^{-1}\sum_{i=1}^n V(P_{f_0,i}, P_{f,i}) \Big\}\leq \|f_0 - f\|_{2,n}^2 /\sigma^2 \leq \|f_0 - f\|_{\infty}^2/\sigma^2.
B_0(v,w,t,q^{\frac{1}{2}})= B_0(v,w,t)=F^*(v\,|\,w,-t), D_0(v,w,t,q^{\frac{1}{2}})= G^*(v\,|\,w-t\tau/2,w+t\tau/2,-t), B_n(v,w,t,q^{\frac{1}{2}})= B_0(v+nw,w,t), D_n(v,w,t,q^{\frac{1}{2}})= D_0(v+nw-nt\tau/2,w,t,q^{\frac{1}{2}}) \cdot \prod_{k=0}^{n-1}B_0(v+nw+(1-n+2k)t\tau/2,w+t\tau/2,t) , q^{\frac{1}{2}}=(\pi i \tau).
\frac{1}{\tau}\big\|(e_n-e_{n-1})_{n=k}^N\big\|_{L^p(X)}+ \big\|(e_n )_{n=k}^N\big\|_{L^p(D)}\le C\delta , & \big\|(e_n )_{n=k}^N\big\|_{L^\infty(W)}\le C\delta,
\sigma_{lsbw}(T) :=\{\lambda \in \mathbb{C}: \lambda I-T \thinspace \mbox{is not lower semi B-Weyl}\}, \sigma_{usbw}(T) :=\{\lambda \in \mathbb{C}: \lambda I-T \thinspace \mbox{is not upper semi B-Weyl}\}, \sigma_{bw}(T) :=\{\lambda \in \mathbb{C}: \lambda I-T \thinspace \mbox{is not B-Weyl}\}, \thinspace \mbox{respectively.}
\int_{Z}D'D^{*}\phi(z)\overline{\psi(z)}\,dz =\int_{Z}\phi(z)\overline{DD'^{*}\psi(z)}\,dz=\int_{Z}\phi(z)\overline{D'^{*}D\psi(z)}\,dz &=\int_{Z}D^{*}D'\phi(z)\overline{\psi(z)}\,dz,
\psi_{A}= \rho_{E}^{1 / 2}\cdot [ C_{2}K_{i_{p}}( 2 \mu \rho_{E}) + C_{1}I_{i_{p}}( 2 \mu \rho_{E}) ] \longrightarrow_{\rho_{0}\gg 0}\sqrt{\frac{\pi}{4 \mu}}[ C_{2}e^{\omega \rho_{0}}+ C_{1}e^{- \omega \rho_{0}}]
{\cal Z}_2=\frac{(1+\xi\bar{\xi})^2}{2|\sigma|{\cal H}_{10}^2}\left[A\partial|\sigma|\bar{\partial}|\sigma|+{\textstyle{\frac{1}{2}}}B(e^{-i\phi}(\partial|\sigma|)^2+e^{i\phi}(\bar{\partial}|\sigma|)^2)\right],
-{\frac{1}{8}}f_{1}f_{2}{\frac{( - G )}{( - G_{5})^{2}}}\epsilon_{\rho^{\prime}\mu \nu \lambda \sigma}\partial_{\rho}\xi G_{5}^{\rho \sigma}\tilde{H}^{\rho^{\prime}\mu}\tilde{H}^{\nu \lambda}( G \tilde{H}G )_{\eta \zeta}\tilde{H}^{\zeta \eta}= - z_{1}f_{1}f_{2}W .
\mathcal{L}_{^{m,2}_{inter}}(s) ={\rm exp}\left(-\lambda_{c}\int_{\mathcal{B}(x_m,\mathcal{R}_s)}f_{y_{m,2}}(y) \int_{\mathbb{R}^2}\left(1 - \frac{1}{\frac{s}{{L\left(||y +x_m-x||\right)}}+1}\right) dx dy\right).
Ch_{\mathbb{C}[x_0^2,\ldots,x_{p-1}^2]}(z^k,t) =\frac{1}{(1-\omega^{2k}t^2)^p}Ch_{\mathbb{C}[x_0^2,\ldots,x_{p-1}^2]}(e_1^ke_2^l,t) =\frac{1}{1-t^{2p}}, (k,l)\neq (0,0).
\frac{p_{n}}{n}= \frac{p_{n}k_{n}}{nk_{n}}\leq \frac{2(h_{m+1}+c_{m+1})}{n(h_{m}+c_{m})}\leq \frac{4}{n}\frac{(r_{m}+1)(h_{m}+ c_{m}+ r_{m})}{(h_{m}+c_{m})}= \frac{4r_{m}}{n}\Big{(}1 + \frac{r_{m}}{h_{m}+c_{m}}\Big{)}\to 0
tr_{M_n(N)}(Q) =(1/n)\sum_i tr( q_{ii}) = (1/n)\sum_i tr(E_N(\lambda_i{\lambda_i}^*)) = (1/n)\sum_i tr(\lambda_i{\lambda_i}^*) = ({\tau}^{-1}/n ) \sum_i tr(e_1 \lambda_i{\lambda_i}^*) ~~\textrm{(Markov Property)}= ({\tau}^{-1}/n)\sum_i tr({\lambda_i}^*e_1 \lambda_i) = ({\tau}^{-1}/n).
1 \underline{{{3 4 5 6}}}+ 2 \underline{{{3 4 5 6}}}\rightarrow \underline{{{1 2}}}3 4 5 + \underline{{{1 2}}}3 4 6 + \underline{{{1 2}}}4 5 6 + \underline{{{1 2}}}3 5 6 ,
\zeta_0 = -\sqrt\tau +{3\over 2}\sqrt\tau \coth^2\left (\sqrt{3\over 2}\tau^{1/4}D-{1\over 2}\log{\sqrt{\zeta+\sqrt\tau}-\sqrt{3\over 2}\tau^{1/4}\over\sqrt{\zeta+\sqrt\tau}+\sqrt{3\over 2}\tau^{1/4}}\right )
\lambda^{-m}\, \int_{S^{n-1}}\widehat{f}(\lambda, \omega)\, Y_m(\omega)\, d\omega = c_{n,m}\int_0^\infty r^{-m}f_m(r) \, \frac{J_{n/2+m-1}(\lambda r)}{(\lambda r)^{n/2+m-1}}r^{n+2m-1}dr
R_{ijkl}= -\left(g_{ik}g_{jl}-g_{il}g_{jk}\right)+h_{ik}h_{jl}-h_{il}h_{jk}= (\frac{H^2}4-1)\left(g_{ik}g_{jl}-g_{il}g_{jk}\right)+\mathring{h}_{ik}\mathring{h}_{jl}-\mathring{h}_{il}\mathring{h}_{jk}+\frac H2\left(\mathring{h}_{ik}g_{jl}+\mathring{h}_{jl}g_{ik}-\mathring{h}_{il}g_{jk}-\mathring{h}_{jk}g_{il}\right)
\frac{\lambda_2^{K_{\ell}}}{\lambda_2^L}=\Bigl\{2^{(\dim M_2^{K_{\ell}}-m+1)/2}\cdot|M_2^{K_{\ell}}|^{-1}\cdot\prod_{i=1}^{m-1}(2^{2i}-1)\Bigr\}\Bigl\{2^{(\dim M_2^L-m)/2}\cdot|M_2^L|^{-1}\cdot\prod_{i=1}^{m}(2^{2i}-1)\Bigr\}^{-1}\leq 2^{5/2}\cdot 3\cdot\frac{2^{n_{2,\nu_2}}-1}{2^{n+1}-1}.
\xi_k^{\frac{q^3-1}{p}}=(g^k\xi_0)^{\frac{q^3-1}{p}}=g^{k\frac{q^2+q+1}{p}(q-1)}\xi_0^{\frac{q^3-1}{p}}=\xi_0^{\frac{q^3-1}{p}}\ne1,
\Omega_\lambda=\lambda!\omega_\lambda =\lambda![B_1^+,\dots,B_n^+]^{\lambda_n-\lambda_{n+1}}\cdots [B_1^+,B_2^+]^{\lambda_2-\lambda_3}(B_1^+)^{\lambda_1-\lambda_2}v_0 &=\frac{\lambda!}{\mu!}[B_1^+,\dots,B_n^+]^{\lambda_n-\lambda_{n+1}}\cdots[B_1^+,\dots,B_j^+]^{\lambda_j-\lambda_{j+1}}\Omega_\mu.
\psi(x)=\min\left\{\frac{2e_G\left(S_{x}^t\right)+2e_G\left(T_{x}^t\right)+\textnormal{cut}_G\left(S_{x}^t\cup T_{x}^t\right)}{\left|S_{x}^t\cup T_{x}^t\right|}\colon 0\le t< \max_{i\in V}\{|x_i|\}\right\}.
\sigma_{a,b}^{*}\pi^{*}\widetilde{\tau}_{g}={\frak c}_{g}\,\|\theta_{a,b}\|^{-4},{\frak c}_{g}:=(4\pi)^{-g}e^{6(1-g)(2\zeta_{\bf Q}'(-1)+\zeta_{\bf Q}(-1))},
e^{\gamma_{b}}= \frac{( r_{b}- \mu_{b})^{2}+ \sigma_{b}^{2}- a_{b}^{2}\operatorname{s i n}^{2}\theta_{b}}{( r_{b}- \mu_{b})^{2}+ \sigma_{b}^{2}\operatorname{c o s}^{2}\theta_{b}},
\|P^{\#}_{t-s}\pi_0 \Phi_s Q^{\frac{1}{2}}\|_{L_2(L^2,L^2(\rho))}^2 = \sum_k \|P^{\#}_{t-s}\pi_0 \Phi_s Q^{\frac{1}{2}}e_k\|_{\rho}^2 \leq e^{-\kappa(t-s)}\sum_k \| Q^{\frac{1}{2}}e_k\|_{\rho_s}^2 & \leq L_{\rho}e^{-2\kappa(t-s)}\sum_k \| Q^{\frac{1}{2}}e_k\|_{\rho}^2 \leq L_{\rho}e^{-\kappa(t-s)}tr(Q)
\varphi(v) = \varphi(u)\varphi^2(u)\cdots\varphi^{n-1}(u) \varphi^{n}(u)= \varphi(u)\varphi^2(u)\cdots\varphi^{n-1}(u) u &=u\varphi^{-n}\big(\varphi(u)\varphi^2(u)\cdots\varphi^{n-1}(u) \big)=u\varphi\big(\varphi^{-n}(u)\big)\varphi^2\big(\varphi^{-n}(u)\big)\cdots\varphi^{n-1}(\varphi^{-n}(u)\big) &=u\varphi(u)\cdots\varphi^{n-1}(u)=v,
\left( a+\frac{b}{2}\, \vartheta_x \right) \left\{\sum_{k=0}^{\infty}\frac{\left(\frac12\right)_k^2}{(1)_k^2}x^{2k}\right\}=\left( a+b \, \frac{1+t}{1-t}\, \vartheta_t \right) \left\{(1+t) \sum_{k=0}^{\infty}\frac{\left(\frac12\right)_k^2}{(1)_k^2}t^{2k}\right\}.
S = \sum_{C}\sum_{x ( \ni C )}\bigg | \left( \frac{\delta}{\delta \sigma^{t \mu}( x )}- i A_{t \mu}( x ) \right) \Psi [ C ] \bigg |^{2}+ \sum_{x}- \frac{\varepsilon}{2 \cdot 3 !}F_{\mu \nu \lambda}F^{\mu \nu \lambda}- V ( | \Psi [ C ] |^{2}) ,
\left({\dot{a}_0\over a_0}\right)^2={{32\pi^2 G^2_5}\over{9c^6}}(\varrho^2c^4+2\sigma\varrho c^2)+{{c^2a^{\prime\prime}_{R\,0}}\over{a_0}}+{{8\pi G_5}\over{3c^2}}\left(\Lambda+{{4\pi G_5}\over{3c^4}}\sigma^2\right)-{{kc^2}\over a^2_0},
S_{0}[ B_{\mu \nu}( x ) , A_{\mu}( x ) ] = \int d^{4}x \Bigg \{\frac{1}{4}\epsilon^{\mu \nu \rho \sigma}B_{\mu \nu}( x ) F_{\rho \sigma}( x ) + \frac{1}{2}A_{\mu}( x ) A^{\mu}( x ) \Bigg \},
\frac{\rho_{a}^{r}( \theta )}{\rho_{a}( \theta )}= \frac{\sum_{k = 1}^{G_{a}}k \operatorname{e x p}\{( G_{a}- k ) \epsilon_{a}( \theta ) \}}{\sum_{k = 0}^{G_{a}}\operatorname{e x p}\{k \epsilon_{a}( \theta ) \}}\, .
Z(\ell)=\prod_p Z(p,\ell)>Z(2,\ell) \prod_{p\ge 3}\left( 1-C_p^{-1}p^{-(\ell+1)^2}\right)\ge 1 - (1-Z(2,\ell)) - \sum_{p\ge 3}C_p^{-1}p^{-(\ell+1)^2}.
S^{\prime}= \int d^{2}\sigma \Big ( - e^{- \phi}\sqrt{- \mathrm{d e t}\, ( G_{\mu \nu}+{\cal F}_{\mu \nu})}+ \frac{1}{2}\tilde{H}^{\mu \nu}( F_{\mu \nu}- 2 \partial_{\mu}A_{\nu}) + \frac{1}{2}e^{- \phi}\epsilon^{\mu \nu}C_{\mu \nu}- \frac{1}{2}C_{0}\epsilon^{\mu \nu}F_{\mu \nu}\Big )
\frac{\binom{2n-2}{m-2}}{\binom{n-2}{m-2}}\leq \frac{1}{4}\frac{\binom{2n}{m}}{\binom{n}{m}}\leq \frac14 \frac{\frac{(2n)^m}{m!}}{\frac{n^m}{4.m!}}= 2^m \leq 2^M\;.
\sum_{\substack{{n\geq1},{n\neq{k}}}}\frac{1}{j_{\nu,n}^4-{j_{\nu,k}^4}}=-\frac{1}{2j_{\nu,k}^2}\sum_{n\geq 1}\frac{1}{j_{\nu,n}^2+j_{\nu,k}^2}+\frac{\nu+2}{4j_{\nu,k}^4}.
\varphi (j_{P'}^+(E_{o_{+,P'}}^{P'}(w))T_n) = \varphi(T_nj_P^-(E^P_{o_{+,P}}(n^{-1}wn))) & = \varphi(T_n)\sigma(E^P_{o_{+,P}}(n^{-1}wn)) & = \varphi(T_n)(n\sigma)(E^{P'}_{o_{+,P'}}(w)).
\frac{t}{C_1}\geq \sum_{k=0}^{N(x,t)-1}\frac{1}{l(r_p(x)-k+1)}\geq \sum_{j=\lfloor r_p(x)\rfloor-N(x,t)+3}^{\lfloor r_p(x)\rfloor +2}\frac{1}{l(j)}\geq \int_{\lfloor r_p(x)\rfloor -N(x,t)+3}^{\lfloor r_p(x)\rfloor +3}\frac{\d r}{l(r)}
\operatorname{Det}_{x_{0},x_{1},x_{2}}\left( f\right) :=\det\left( \begin{array}{ccc}f_{0}\left( x_{0}\right) f_{0}\left( x_{1}\right) f_{0}\left(x_{2}\right) f_{1}\left( x_{0}\right) f_{1}\left( x_{1}\right) f_{1}\left(x_{2}\right) f\left( x_{0}\right) f\left( x_{1}\right) f\left( x_{2}\right)\end{array}\right)
L =\begin{pmatrix}\ddots \ddots \ddots \ddots \ddots \ddots \ddots \ddots m_1 m_0 m_{-1}m_{-2}m_{-3}\ddots \ddots m_2 m_1 m_0 m_{-1}m_{-2}\ddots \ddots m_3 m_2 m_1 m_0 m_{-1}\ddots \ddots \ddots \ddots \ddots \ddots \ddots \ddots\end{pmatrix}
(u,v)_{X_{\beta}}:=\sum_{n=1}^{\infty}{\mu_{n}^{2\beta}(u,\phi_{n})\overline{(v,\phi_{n})}},\quad\Vert u\Vert_{X_{\beta}}:=(u,u)_{X_{\beta}}^{1/2},\quad\forall u,v\in\operatorname*{span}\left\{\phi_{n}:n\geq1\right\}.
(\widetilde{m_\mu})^*(\pi_{ij}) = \int_G \pi_{ij}(xzy)d\mu(z) & = \left[\int_G \pi(xzy)d\mu(z)\right]_{ij}& = \left[\pi(x)\left(\int_G \pi(z)d\mu(z)\right)\pi(y)\right]_{ij},
C_{1,0}= C_{1,0}(\chi,\sigma,r,\omega_0, \mathcal{U}_1) = C_{\ref{LemmaCentralEstimateInNormalDirection},1,0}(\eta(\chi),R_1(\chi,\sigma,r,\omega_0,\mathcal{U}_1)), C_1 = C_1(\chi,\sigma,r,\omega_0,\mathcal{U}_1,T^*) = C_{\ref{LemmaCentralEstimateInNormalDirection},1}(\eta(\chi),R_1(\chi,\sigma,r,\omega_0,\mathcal{U}_1),T^*).
\prod_{m = 0}^{( p - 1 ) / 2}\prod_{n_{1}= - \infty}^{\infty \prime}\left[{\frac{\omega_{n_{1}0}^{( m )}}{2 \pi \alpha^{\prime}}}\right]^{- 1}= \prod_{m = 0}^{( p - 1 ) / 2}\prod_{n_{1}= 1}^{\infty}\left[{\frac{\omega_{n_{1}0}^{( m )}}{2 \pi \alpha^{\prime}}}\right]^{- 2}= ( 2 \pi \alpha^{\prime}l^{2})^{- ( p + 1 ) / 2}\mathrm{d e t}({\bf 1}+{\bf B}) \quad ,
=\frac{e^{4}}{(k-k^{\prime})^{4}}((\overline{v}_{k}\gamma^{\mu}v_{k^{\prime}})(\overline{u}_{p^{\prime}}\gamma_{\mu}u_{p}))^{*}((\overline{v}_{k}\gamma^{\nu}v_{k^{\prime}})(\overline{u}_{p^{\prime}}\gamma_{\nu}u_{p}))
\partial_t \begin{pmatrix}\varepsilon_0 \varepsilon_1 \end{pmatrix}= \begin{pmatrix}\varepsilon_1 - \psi_0 \Delta \varepsilon_0 + (f(\varphi_0 + \varepsilon_0) - f(\varphi_0)) - \psi_1. \end{pmatrix},
\int_{BC}e^{sx}\overline{G}(s,t)ds = \int_{r}^{-\lambda_2+R}e^{-x\lambda_2}e^{-wx}e^{t(c_1\lambda^{\alpha_1}+c_2\lambda_2^{\alpha_2})}e^{-t[c_1(\lambda_1-\lambda_2+we^{i\pi})^{\alpha_1}+c_2(w^{\alpha_2}e^{i\pi \alpha_2})]}dw.
{\tilde{z}}= \frac{\big ( \cos \frac{\theta}{2}+ i \omega_{3}\sin \frac{\theta}{2}\big ) z + \big ( \omega_{2}- i \omega_{1}\big ) \sin \frac{\theta}{2}}{\big ( - \omega_{2}- i \omega_{1}\big ) \sin \frac{\theta}{2}z + \big ( \cos \frac{\theta}{2}- i \omega_{3}\sin \frac{\theta}{2}\big )}.
\mathcal{A}=\frac{h^2}{2}\Delta_{\mathbb{S}^2}-\left[\lambda_1v_r\times g'(v_r)-\lambda_2v_r\times (v_r\times g'(v_r))\right]\cdot \nabla &\quad+\left[\lambda_1v_r\times \Delta v_r-\lambda_2v_r\times (v_r\times \Delta v_r)\right]\cdot \nabla\, .
2\pi{\rho}_{a,j}(\beta) = \sigma_j^{(\infty)}* (\rho_{a,L}(\beta)+\rho_{a,R}(\beta)) - \sum_{b=1}^{N-1}\sum_{l=1}^\infty A_{jl}^{(\infty)}* K_{ab}^{(N)}* \widetilde\rho_{b,l}(\beta)
P_G(x) = (-1)^{\sum a_i}x^{m_0(G)}(x+1)^{m_{-1}(G)}\{x^2 (x+1)^2 -x^2(x+1)(a_2+a_4) &-(a_1a_2 + a_1 a_4+a_3a_4)x(x+1) +(a_2 a_3 a_4)x -a_1a_2a_3 a_4 \}.
\int_0^\infty x^{\lambda-1}(1+\alpha x^2)^{-\mu}(1+\beta x^2)^{-\nu}\,dx = \frac{1}{2}\alpha^{-\frac{\lambda}{2}}B\left(\frac{\lambda}{2},\mu+\nu-\frac{\lambda}{2}\right){_2F_1}\left(\nu,\frac{\lambda}{2};\mu+\nu;1-\frac{\beta}{\alpha}\right).
\begin{aligned}|X| =|S\cap X|+|X\setminus S| &=|S\cap A_1\cap \cdots\cap A_i\setminus (A_{i+1}\cup \cdots\cup A_{k-1})|+|A_1\cap \cdots\cap A_i\setminus (A_{i+1}\cup \cdots\cup A_{k-1}\cup S)| &=b_{i+1}+b_{i},\end{aligned}
\left|\left(\frac{zf_1'(z)}{f_1(z)}\right)^2-1\right| =\left|\left(1-\frac{\rho_1}{(1-\rho_1^2)}\left( \frac{u\rho_1^2+4\rho_1+u}{\rho_1^2+u\rho_1+1}+\frac{v\rho_1^2+4\rho_1+v}{\rho_1^2+v\rho_1+1}+\frac{q \rho_1^2+4\rho_1+q}{\rho_1^2+q \rho_1+1}\right)\right)^2-1\right| =|1-(2-\sqrt2)^2-1|=2(1-\sqrt2)=2\left|\frac{zf_1'(z)}{f_1(z)}\right|.
\begin{array}{r c l}\psi'(t,x) = e^{-ie\epsilon_0(t,x)}\psi(t,x)\ ,{A^1}'(t,x) = A^1(t,x)-\partial_1\epsilon_0(t,x)\ , \pi'_1(t,x) = \pi_1(t,x)\ ,\{A^0}'(t,x) = A^0(t,x)+\partial_0\epsilon_0(t,x)\ ,\end{array}
T(y)(t,\omega) = -\int_{t}^{\infty}\frac{\sin((t-x)\sqrt{\omega})}{\sqrt{\omega}}\left(F_{2}(y)(x,\omega) - \mathcal{F}(\sqrt{\cdot}F(x,\cdot\lambda(x)))(\omega \lambda(x)^{2}) - \mathcal{F}(\sqrt{\cdot}F_{3}(u(y))(x,\cdot\lambda(x)))(\omega \lambda(x)^{2})\right)dx
\theta(E_{0,0})=d^{-1}[F_{n-1,0},\ldots,[F^-_{2,0},F^-_{1,-1}]_q\ldots]_qK_1\cdots K_{n-1}\,, \theta(F_{0,0})=d(K_1\cdots K_{n-1})^{-1}[\ldots [E_{1,1},E_{2,0}]_{q^{-1}},\ldots,E_{n-1,0}]_{q^{-1}}\,.
\Pi_Q= \int_{\mathbb T^2}(u\theta)\cdot \nabla(\theta_{\leq Q})_{\leq Q}\,dx = \int_{\mathbb T^2}(u\theta)_{\leq Q}\cdot\nabla\theta_{\leq Q}\,dx = \int_{\mathbb T^2}r_Q(u,\theta)\cdot \nabla\theta_{\leq Q}dx-\int_{\mathbb T^2}u_{>Q}\theta_{>Q}\cdot \nabla\theta_{\leq Q}\,dx.
(zxe_{1}+x^{2}e_{2})[X+1][k(1-\alpha)]=\Big(zx(p_{1}+1)+\frac{x^{2}}{2}(p_{1}^{2}+2p_{1}-p_{2})\Big)[k(1-\alpha)] \qquad\qquad\qquad\qquad\qquad=zx(k(1-\alpha)+1)+\frac{x^{2}}{2}(k^{2}(1-\alpha)^{2}+2k(1-\alpha)-k(1-\alpha^{2})) \qquad\qquad\qquad\qquad\qquad=(k+1)zx-kzx\alpha+{k+1 \choose 2}x^{2}+{k+1 \choose 2}x^{2}\alpha^{2}-(k^{2}+k)x^{2}\alpha
(g, \alpha \wedge \beta)(x) =~ (dl_{g^{-1}})^* ~(\alpha \wedge \beta)(g^{-1}x) =~ (dl_{g^{-1}})^* ~ ( \alpha (g^{-1}x) \wedge \beta (g^{-1}x)) =~ (dl_{g^{-1}})^* \alpha (g^{-1}x) \wedge (dl_{g^{-1}})^* \beta (g^{-1}x) = \big((g, \alpha) \wedge (g, \beta) \big)(x).
g \, t_{i_{1}\ldots i_{n}\Phi^{2}}+ U \, t_{i_{1}\ldots i_{n}\Phi}+ c \, ( t_{( i_{1}+ 1 ) i_{2}\ldots ( i_{n}- 1 ) \Phi}+ t_{( i_{1}- 1 ) i_{2}\ldots ( i_{n}+ 1 ) \Phi}) - \sum_{j = 1}^{n - 1}t_{i_{1}\ldots i_{j}}\, t_{i_{j + 1}\ldots i_{n}}= 0
\hat{r}_{j}(t_{j},t_{j+1})=\left( \begin{array}{cccc}1 0 0 0 0 1+\beta t_{j+1}\ominus t_{j}0 0 0 q^{-\delta_{j,N}}t_{j+1}\ominus t_{j}1 0 0 0 0 1\end{array}\right)_{j,j+1}
{\frac{d \omega_{n}^{F}}{d m}}={\frac{1}{\sqrt{k^{2}+ m^{2}}}}\left( m + k_{n}{\frac{d k_{n}}{d m}}\right) ={\frac{1}{\sqrt{k^{2}+ m^{2}}}}\left( m +{\frac{1}{L}}{\frac{k_{n}^{2}}{m}}\left( \delta^{\prime}( k ) +{\frac{\theta^{\prime}( k )}{2}}\right) \right)
\log \hat{\rho}_s(A)=\max_{\mu \in \mathcal{M}_f}\left\lbrace \inf_n \frac{1}{n}\int \log V_s(A^n(x))d\mu \right\rbrace =\max_{\mu \in \mathcal{M}_f}\left\lbrace \lim_n \frac{1}{n}\int \log \varphi^s_c(A^n(x))d\mu \right\rbrace.
\frac{\Gamma(d + 1)}{\Gamma(d+\beta)}= \frac{\Gamma(d+1)}{\Gamma\big(d+\underbrace{(\beta-\ell)}_{\in (0, 1]}\big)\prod_{i = 1}^{\ell}\big(d+\beta-i\big)}\leq \frac{(d + \beta - \ell)^{1 - (\beta -\ell)}}{\prod_{i = 1}^{\ell}\big(d+\beta-i\big)}\leq \frac{1}{d^{\beta-1}}\enspace,
\lim_{x\to-\infty}\frac{h_{}(x)}{\phi(\frac{x-\mu}{\sigma})}=\frac{1}{\sigma}\lim_{x\to-\infty}\frac{\phi(\frac{x-\mu}{\sigma})}{1-\Phi(\frac{x-\mu}{\sigma})}\frac{1}{\phi(\frac{x-\mu}{\sigma})}=\frac{1}{\sigma}\lim_{x\to-\infty}\frac{1}{1-\Phi(\frac{x-\mu}{\sigma})}=\frac{1}{\sigma}.
\Delta^{( r o t )}( \tau ,{\vec{\sigma}}_{o}) = \sqrt{[ \mu n_{o}({\vec{\sigma}}_{o}) ]^{2}+ \sum_{r s}A^{r s}({\vec{\sigma}}_{o}, q^{\mu}( \tau ) , \overline{{\cal S}}^{u}) \overline{{\cal S}}^{r}\overline{{\cal S}}^{s}}.
\prod_{U,V \in I(\mathcal{C})}\left(\prod_{i,j}\frac{1}{1 - x_i^{(U)}y_j^{(V)}}\right)^{N_{U,V}^{(\mathbf{1})}}= \prod_{U,V \in I(\mathcal{C})}\left( \sum_{\rho \in \mathcal{P}}s_\rho(\mathbf{x}^{(U)}) s_\rho(\mathbf{y}^{(V)})\right)^{N_{U,V}^{(\mathbf{1})}}
\frac{d}{dt}\tilde{A}_{\alpha}(t)_{t=0}Y =\frac{d}{dt}\left(\tau_t^{-1}A^{e_{\alpha}(\gamma^h(t))}\tau_t Y\right)_{t=0}=\nabla^{\top}_X A_{\alpha}Y-A^{\nabla^{\bot}_Xe_{\alpha}}Y-A_{\alpha}(\nabla^{\top}_XY) &=(\nabla^{\top}_XA)_{\alpha}Y.
\int_{1}^{\infty}d l \, F ( l ) = -{\frac{1}{\epsilon}}\left( \frac{d}{d \epsilon}\right)^{( d + 1 ) / 2}{\frac{\left[ \sqrt{\epsilon^{2}+ 1}- \epsilon \right]^{( d - 1 ) / 2}}{\sqrt{\epsilon^{2}+ 1}}}= - F ( 1 ) .
y^2=x^3+\sum_{k=-4}^{4}\sum_{\ell=0}^{8-nk}f_{k\ell}\,z'{}^\ell z^{4-k}u^{\ell-k}x+\sum_{k=-6}^{6}\sum_{\ell=0}^{12-nk}g_{k\ell}\,z'{}^\ell z^{6-k}u^{\ell-k}.
\sum_{t=1}^{N_T}\frac{\binom{N_T}{t}}{d_{0,t}}a_{0,t}\ge\frac{1}{\min\{1,\frac{N_T}{N_R}\}}\sum_{t=1}^{N_T}\binom{N_T}{t}a_{0,t}=\frac{N_R}{\min\{N_T,N_R\}}.
I=[\begin{matrix}\frac{1}{10}m(y_{m}^{2}+z_{m}^{2})&0&0\\ 0&\frac{1}{10}m(x_{m}^{2}+z_{m}^{2})&0\\ 0&0&\frac{1}{10}m(x_{m}^{2}-y_{m}^{2})\end{matrix}].
l i m_{Q \to q}[{{\frac{Q^{\frac{k N_{B}}{2}}{( B^{-})}^{k}}{( [ k ] ! )^{\frac{1}{2}}}},{{\frac{{( B^{+})}^{k}Q^{{\frac{k N_{B}}{2}}}}{( [ k ] ! )^{\frac{1}{2}}}}}}] ~ = ~ 1
J_3 = \int_\Omega\Big[f_0(V)-f_\epsilon(V) \Big] \psi^h_{\mu_j,\xi_j}dx = \epsilon\int_\Omega V^p\ln\ln(e+V)\psi^h_{\mu_j,\xi_j}dx -\epsilon^2\int_\Omega V^p \Big(\ln\ln(e+V)\Big)^2 \psi^h_{\mu_j,\xi_j}dx.
\sigma_{\mathrm{a b s}}= A_{b h}{\frac{\omega}{2 ( T_{R}+ T_{L})}}{\frac{e^{\frac{\omega}{T_{H}}}- 1}{( e^{\frac{\omega}{2 T_{R}}}- 1 ) ( e^{\frac{\omega}{2 T_{L}}}- 1 )}}[ 1 + O ( \omega^{2}r_{1}^{2}) ] .
\frac{\partial{K_p}}{\partial{x_j}}(z)=\frac{p}{q}K_p(z)^{1-q/p}\frac{\partial{K_p^{q/p}}}{\partial{x_j}}(z)=\frac{p}{q}m_p(z)^{q-p}\frac{\partial{K_p^{q/p}}}{\partial{x_j}}(z).
\left\| \lambda_q^se^{-\mu\lambda_q^\alpha t}\|B_q(0)\|_{L^2}\right\|_{L^{\frac{\alpha}{s+\alpha-\frac52}}(0,T; l^2)}\leq \ \left\| e^{-\mu\lambda_q^\alpha t}B_q(0)\right\|_{L^{\frac{\alpha}{s+\alpha-\frac52}}(0,T; H^s)}\leq \C(T) \|B(0)\|_{H^{\frac{5}{2}-\alpha}}
f^{2}_{n}=\frac{\sqrt{2\alpha'}}{R_{0}\cos^2\tau}\sum_{m=-\infty}^{\infty}\left\{C^{1}_{m}\left(\ddot{C}^{2}_{n-m}-n(n-m)C^{2}_{n-m}+\frac{2\sin\tau}{\cos\tau}\dot{C}^{2}_{n-m}\right)+\dot{C}^{1}_{m}\dot{C}^{2}_{n-m}\right\}
P(x^mP(x^{n+k\theta(n)})) =\beta(m+\theta(n+k\theta(n)))\beta(n+k\theta(n))x^{\theta(m+\theta(n+k\theta(n)))}&=\frac{1}{k+1}\beta(m+(k+1)\theta(n))\beta(n)x^{\theta(m+(k+1)\theta(n))}, (\hbox{by Eq.~\eqref{thetam+kthetam}and \eqref{betam+kthetam}})
\bar{u}^{1}=\eta^{1}(\zeta_{[-1]}^{1},x^{1},x^{2},x^{3},y_{1,[0,3]}^{1,d},y_{2,[0,2]}^{1,d}) \bar{u}^{2}=\eta^{2}(\zeta_{[-1]}^{1},x^{1},x^{2},x^{3},y_{1,[0,3]}^{1,d},y_{2,[0,2]}^{1,d})\,.
x(t_{1}, t_{2}) = ab ( 1- e^{(b-a) B^{\infty}_{21}t_{2}})( b - a e^{(b-a)B^{\infty}_{21}t_{2}})^{-1}& y(t_{1}, t_{2}) =\frac{c_{0}}{(b-a)^{2}}( b-{a}e^{(b-a)B^{\infty}_{21}t_{2}})^{2}e^{(B^{\infty}_{21}(a-b)-1)t_{2}}.