harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\begin{array}{rcl}\delta e^i_\mu = \varepsilon^{ij}e_{\mu j}\zeta+ e^i_\nu\partial_\mu\xi^\nu+(\partial_\nu e^i_\mu)\xi^\nu, \delta\omega_\mu = -\partial_\mu\zeta+ \omega_\nu\partial_\mu\xi^\nu+(\partial_\nu \omega_\mu)\xi^\nu, \delta\varphi = (\partial_\mu\varphi)\xi^\mu, \delta\varphi_i = \varepsilon_{ij}\varphi^j\zeta+ (\partial_\nu\varphi_i)\xi^\nu,\end{array}
|u_n(x)-u(x)| \le E_x\int_0^\zeta|f_n(\cdot,u)-f(\cdot,u)|(X_r)\,dr+E_x\int_0^\zeta dA^{|\mu_n-\mu|}_r &=R|f_n(\cdot,u)-f(\cdot,u)|(x)+R|\mu_n-\mu|(x)
\tilde{r}(v \cdot a) = h\left(S(a_{(2)})r(v_{(0)})_{(1)}S(v_{(1)})S^2(a_{(1)})\right)r(v_{(0)})_{(0)}\cdot a_{(3)}& = h\left(r(v_{(0)})_{(1)}S(v_{(1)})\right)r(v_{(0)})_{(0)}\cdot a &= \tilde{r}(v) \cdot a
\mathbf{d}\widetilde \alpha = - i_{\{\alpha_{m}}v_{\alpha_{2}\dots \alpha_{m}\}}F^{\alpha_{1}}\wedge \dots \wedge F^{\alpha_{m}}\equiv - \alpha_{m}( m ) k_{\alpha_{1}\dots \alpha_{m}}F^{\alpha_{1}}\wedge \dots \wedge F^{\alpha_{m}}\quad .
\begin{cases}C_p^T(\Psi''- \Delta \Psi+ \overline A_p^T\Psi)=0 \hbox{in}(0, +\infty)\times \Omega, C_p^T\Psi=0 \hbox{on}(0, +\infty)\times \Gamma_0 C_p^T(\partial_\nu\Psi + \overline B_p^T\Psi)=0 \hbox{on}(0, +\infty)\times \Gamma_1.\end{cases}
\int_0^1 c(t) \mathrm{d}t \geq \int_{0}^{t_0}\left(1-\frac12 t \right) \mathrm{d}t +\int_{t_0}^1 \left(\frac{1}{(e-1)^2}(1-t) +\frac{1}{e-1}\right)\mathrm{d}t &= -\frac{t_0^2}{4}+\frac{(t_0-1)^2}{2(e-1)^2}+\frac{t_0(e-2)+1}{e-1}.
\mathcal{M}^*_{1,s}\left(\tau\right) -\frac{1}{\sqrt{-i\tau}}\sqrt{\frac{2}{p}}\sum_{k=1}^{p-1}\sin\left(\frac{\pi k(p-s)}{p}\right)\mathcal{M}^*_{1,k}\left(-\frac{1}{\tau}\right)=i\sqrt{2p}\cdot r_{f_{p-s,p}}(\tau),
\tilde{\eta}:=\big(\|\nu-1\|_\infty + \| \bar{u}^h \|_{\dot{B}_{p,r}^{-1+\frac{d}{p}}}\big)\exp\big\{\frac{c_r}{2}\|\bar{u}^d\|^{4r}_{\dot{B}_{p,r}^{-1+\frac{d}{p}}}\big\}.
2(S\tau)(j)=\frac{1-\frac{1}{4}}{(1-j)^{2}}+\frac{1-\frac{1}{9}}{j^{2}}+\frac{1-\frac{1}{4}-\frac{1}{9}}{j(1-j)}=\frac{3}{4(1-j)^{2}}+\frac{8}{9j^{2}}+\frac{23}{36j(1-j)}
\rho_S(x) = \big( 1 - \frac{1}{q}\big)^+ \delta(x) + \frac{\sqrt{\Big(x - \big( \frac{1}{\sqrt{q}}-1 \big)^2\Big)\Big( \big(\frac{1}{\sqrt{q}}+ 1\big)^2 -x \Big)}}{2 \pi x}.
\begin{cases}||M_{v + \xi\delta_n}- M_l|| < \frac{\xi\Delta}{4}\delta_n, ||M_{v}- M_l|| < \frac{\xi\Delta}{4}\delta_n, \;\; \mbox{and}||M_{v - \xi\delta_n}- M_l|| < \frac{\xi\Delta}{4}\delta_n. \end{cases}
v=\hat{v}^{i}\hat{e}_{i}=((R^{-1})_{j}^{i}v^{j})(e_{k}R_{i}^{k})=((R^{-1})_{j}^{i}R_{i}^{k})v^{j}e_{k}=\delta_{j}^{k}v^{j}e_{k}=v^{k}e_{k}=v^{i}e_{i}
m\left(\Omega_{n}\cap \Omega_{0}\right) =m\left(\Omega_{0}\backslash \left(\Omega_{n}^{c}\cap \Omega_{0}\right)\right) &\geq m\left( \Omega_{0}\right)-m\left(\Omega_{n}^{c}\cap \Omega_{0}\right) &\geq \gamma_{0}-\frac{1}{n}>\frac{\gamma_{0}}{2}
\sum_{j=-\infty}^{nt}\operatorname E\Bigl\|b_n^{-1}\sum_{k=nu+1}^{nt}v_{k-j}\varepsilon_j\Bigr\|^p=\sum_{j=-\infty}^{nt}\operatorname E\Bigl[\int_\mathbb Sn^{-[3-2d(r)]}\bigl|\sum_{k=nu+1}^{nt}v_{k-j}(r)\bigr|^2\varepsilon_j^2(r)\mu(\mathrm dr)\Bigr]^{p/2}.
L(e_{ii}) =\sum_{x<i}C_{xi}^{ii}e_{xi}+\sum_{x\in X}C_{xx}^{ii}e_{xx}+\sum_{y>i}C_{iy}^{ii}e_{iy}, L(e_{ij}) =\sum_{x<i}C_{xi}^{ii}e_{xj}+C_{ij}^{ij}e_{ij}+\sum_{y>j}C_{jy}^{jj}e_{iy},\ \mbox{if}\i<j.
\int_M (2qu_0v+ |\nabla v|^2+V_1v^2)\,dV_g =-\int_M |\nabla v|^2\,dV_g-\int_M V_1v^2\,dV_g, \int_M (qvr_t+V_1r_t^2+|\nabla r_t|^2+tqr_t^2)\,dV_g =0.
\rho(g_n^2) =\left(\begin{array}{cc}\cos(\frac{2\pi}{n}) -\sin(\frac{2\pi}{n}) \sin(\frac{2\pi}{n}) \cos(\frac{2\pi}{n})\end{array}\right), \sigma(g_n^2) =(1,3,\dots,2n-1)(2,4,\dots,2n) \tau(g_n^2) =1,
K(\theta_{1},\ldots,\theta_{p}):= 2\log 2+\sum_{k=1}^{p}\theta_{k}^2\, \log \left(\theta_{k}/4\right)+2\sum_{k=1}^{p-1}\left(\sum_{j=k+1}^{p}\theta_{j}\right) \theta_{k}\log \theta_{k},
\begin{minipage}[c]{35pc}For every $x\in M$ there is a{\em unique}$y\in M$satisfying $xyx=x$ and $yxy=y$ \cite[p.\,6, conditions~(1) and~(2)]{MVL},\cite[Definition~II.1.1]{MP},\cite[p.\,28, lines~9-11]{C+P1}.\end{minipage}
\frac{3^{9(2k-p_k)/5+1}}{10^{2k-p_k+1}}> \frac{3^{(9/10)\log_2(\pi k)+1}}{10^{(1/2)\log_2(\pi k)+41/40}}= \frac{3}{10^{41/40}}(\pi k)^{(9/10)\log_2 3 - (1/2)\log_2 10}& > \frac{1}{4}(\pi k)^{-1/4}> (2\pi k)^{-1/2},
\begin{array}{l l}{{\ell^{-}}}&{{= 0 , \frac{1}{2}, 1 , \dots , \frac{1}{2}( k^{-}- 1 )}}\\{{}}&{{}}\\{{\ell^{+}}}&{{= 0 , \frac{1}{2}, 1 , \ldots , \frac{1}{2}( k^{+}- 1 )}}\\ \end{array}
S_{T}^{\chi_{r}^{2}}=\frac{1}{a_{d,1}^{\chi_{r}^{2}}(\mathcal{D})\mathcal{L}(T)T^{d-\alpha}}\left[\int_{\mathcal{D}(T)}F(\chi_{r}^{2}(\mathbf{x}))d\mathbf{x}-C_{0}^{L}T^{d}|\mathcal{D}|\right],
\log\phi_s\left( \frac{x-\mu}{\sigma}\right) =\log\left[\frac{s}{2\Gamma(1/s)}\exp\left( -\left|\frac{x-\mu}{\sigma}\right|^s \right)\right] &=\log s-\log(2\Gamma(1/s))-\left| \frac{x-\mu}{\sigma}\right|^s.
T_{\underline{{A}}\ \ \underline{{B}}\dot{A}}^{\ \ \dot{A}\ \ \ \dot{D}}H_{\dot{D}\underline{{C}}}= - \frac{i}{1 2}\left( \nabla_{\underline{{A}}}G_{\underline{{B}}}^{\ \ \dot{D}}\right) \overline{{R}}\chi_{\underline{{C}}\dot{D}}
= \frac{\epsilon^{0 i j k 5}}{( 4 ! )^{2}}\left\{\partial_{0}\left[ \frac{\sigma^{\prime}}{( ( \sigma^{\prime}/ b )^{2}- ( \dot{\sigma}/ n )^{2})^{2}}\right] - \partial_{5}\left[ \frac{\dot{\sigma}}{( ( \sigma^{\prime}/ b )^{2}- ( \dot{\sigma}/ n )^{2})^{2}}\right] \right\}= 0 .
\widetilde{H}_\pm^{\prime\mspace{2mu}\textsc{cs}}= \frac{1}{2}\sum_{j=1}^N (z_j \, \partial_{z_j})^2 + \frac{\beta}{2}\sum_{i<j}^N \frac{z_i + z_j}{z_i - z_j}\, \bigl(z_i \, \partial_{z_i}- z_j \, \partial_{z_j}\bigr) + \beta \sum_{i<j}^N \frac{z_i \, z_j}{z_{ij}\, z_{ji}}\, (1 \mp P_{ij}) \, .
d(\mathcal{U}((\overline{w}_{\i})_{\i=1}^{\eta}),\mathcal{U}((\overline{v}_{\i})_{\i=1}^{\eta})\leq~\beta(\mathcal{M}'^{*}_{\eta}((\overline{w}_{\i})_{\i=1}^{\eta},(\overline{v}_{\i})_{\i=1}^{\eta}))\mathcal{M}'^{*}_{\eta}((\overline{w}_{\i})_{\i=1}^{\eta},(\overline{v}_{\i})_{\i=1}^{\eta})
(zI_d-J_d)^{-1}= \left[ \begin{array}{cccc}\displaystyle{\frac{1}{z-\lambda_1}}0 \cdots 0 0 \displaystyle{\frac{1}{z-\lambda_2}}\cdots 0 \vdots \vdots \ddots \vdots 0 0 \cdots \displaystyle{\frac{1}{z-\lambda_d}}\end{array}\right].
\mathcal{A}_p u(t) = \big[A^{\frac{1}{2}}_{p'}\big]^* \Phi^{-1}A_p \int_0^t e^{- (t - s) A_p}\Phi \big[ A_{p^{\prime}}^{- \frac{1}{2}}\big]^* f(s) \; d s.
I_{\nu}\left({\nu z}\right) =\frac{\nu^{\nu}}{e^{\nu}\Gamma \left({\nu+1}\right) \left({1+z^{2}}\right)^{1/4}}\exp \left\{{\nu \xi+\sum\limits_{s=1}^{n-1}{\frac{\tilde{{E}}_{s}\left( p\right) -k_{s}}{\nu^{s}}}}\right\}\left\{{1+\eta_{n,1}\left({\nu ,z}\right)}\right\},
\frac{2f^2}{r^6}\left(1-\frac{3M}{r}\right)^2\lambda^2-\frac{2f^2}{r^6}\left( 4-\frac{18M}{r}+\frac{12M^2}{r^2}\right)^2\lambda = \frac{2f^2}{r^6}\left(1-\frac{3M}{r}\right)^2\left(\lambda-\left( 4-\frac{18M}{r}+\frac{12M^2}{r^2}\right)^2\bigg/ \left(1-\frac{3M}{r}\right)^2\right)\lambda,
J_T(x) ={}\begin{pmatrix}1 0 i(x^2-1)^{1/2}_- e^{-N\xi_-(x)}1 \end{pmatrix}\begin{pmatrix}0 \frac{1}{\sqrt{1-x^2}}- \sqrt{1-x^2}0 \end{pmatrix}&\times\begin{pmatrix}1 0 -i(x^2-1)^{1/2}_+ e^{-N\xi_+(x)}1 \end{pmatrix}.
\operatorname{DET}| \mathfrak{N}_{n+k-i-j}(\beta) |_{1 \le i,j \le k}=\beta^{{k \choose 2}}\sum_{(\mathfrak{p}_1,\ldots,\mathfrak{p}_k)}\beta^{\pi(\mathfrak{p}_1) + \cdots+\pi(\mathfrak{p}_k)},
g_{\mu \nu}= \left( \begin{array}{c c}{h_{i j}- B a_{i}a_{j}}&{- B a_{i}}\\{- B a_{j}}&{- B}\\ \end{array}\right) , g^{\mu \nu}= \left( \begin{array}{c c}{h^{i j}}&{- a^{i}}\\{- a^{j}}&{a^{2}- B^{- 1}}\\ \end{array}\right) ,
d_{r}^3 \gamma(x,y,z,t) &=B(\left[ [x,y],\alpha(z)\right] ,t)+B\left( \left[ [x,z],\alpha(y)\right] ,t\right) +B\left( \left[ \alpha(x),[y,z]\right] ,t\right) + B(x,\left[ y,[\alpha(z),t]\right] )+B(x, \alpha\left(\left[ [y,z],t\right] \right))+B(x,\left[ z,[\alpha(y),t]\right] )
\int{\cal D}v \delta^{n}\left( \int v d \tau \right) F_{q G}[ v , I ] = \int{\cal D}v \delta^{n}\left( \int \phi d \tau \right) F_{q G}[ \phi , I ] \mathrm{D e t}\frac{\delta \phi_{\tau}( v )}{\delta v ( \tau^{\prime})},
C_1 := \frac{1}{1-s_*\beta}, \lambda_1 = - \frac{\log (1-s_*\beta)}{2s_*}C_2 := \frac{1}{1 - \epsilon \delta (s_*-\delta) \beta}, \lambda_2 = - \frac{\log ( 1 - \epsilon \delta (s_*-\delta) \beta)}{2s_*}
= \int d t d \theta d \phi d t^{\prime}d \theta^{\prime}d \phi^{\prime}\frac{\sin \theta \sin \theta^{\prime}\Phi_{-}( t , \theta , \phi ) \Phi_{-}( t^{\prime}, \theta^{\prime}, \phi^{\prime})}{[ \cosh ( \frac{t - t^{\prime}}{a}) - ( \sin \theta \sin \theta^{\prime}\cos ( \phi - \phi^{\prime}) + \cos \theta \cos \theta^{\prime}) ]^{h}}.
V e f f ( \phi ) = V_{0}( \phi ) + \frac{N}{2}\int \frac{d^{D}p}{( 2 \pi )^{D}}\ln \left[ p^{2}+ M^{2}( \phi ) \right] - \frac{N}{4 !}( \lambda_{0}+ \frac{\eta_{0}\phi^{2}}{1 0 N}) F ( \phi )^{2}- \frac{2 N \eta F ( \phi )^{3}}{6 !}.
I_2 \leq \int_{[0,1]^n}\prod\limits_{k=1}^2\left(\frac1{\omega_k\left(B\right)}\int_{B}\left|f_k\left(s_k(t)x\right)\right|^{p_k}\omega_k(x)dx \right)^{1/p_k}\times &\qquad\qquad\times\prod_{k=1}^2\left(\frac1{\omega_k\left(B\right)}\int_{B}\left|b_k(s_k(t)x)-b_{k,\omega_k,s_k(t)B}\right|^{q_k}\omega(x)dx\right)^{1/q_k}\psi(t)dt
\frac{\partial^{\beta_1}u(x_i,y_j,t)}{\partial x^{\beta_1}}\cong\sum\limits_{m=0}^{M_x}{a_{im}^{(\beta_1)}u(x_m,y_j,t)}, &\frac{\partial^{\beta_2}u(x_i,y_j,t)}{\partial y^{\beta_2}}\cong\sum\limits_{m=0}^{M_y}{b_{jm}^{(\beta_2)}u(x_i,y_m,t)},
\sum_{j = 0}^k{\sum_{s = 0}^{k - j}{( - 1)^{n(j + s)}\binom kj\binom{k-j}sF_n^{j + s}G_{m - 2nk + (n - 1)j + (n + 1)s}}}= ( - 1)^{nk}G_m\,,
\mu_{g}^{h,d}(\Delta_{1},\cdots,\Delta_{k})= \sum_{\Delta^{\prime}}\mu_{g_1}^{h_1,d}(\Delta_{1},\cdots,\Delta_{l},\Delta^{\prime})\frac{d!}{|C_{\Delta^{\prime}}|}\mu_{g_2}^{h_2,d}(\Delta^{\prime},\Delta_{l+1},\cdots,\Delta_{k}).
{\rm Re}\, \Gamma (\varphi , \Lambda) \;=\; \frac{1}{2}\,\sum_n \, \log \left( 1 \,+\, \frac{\Lambda^2}{(2 \pi)^2 \, (n + \varphi)^2}\right) \;-\; \frac{1}{2}\,\sum_n \, \log \left( 1 \,+\, \frac{\Lambda^2}{(2 \pi)^2 n^2}\right) \; .
\Lambda ( \beta ) = \frac{\alpha^{\prime}}{2}\lim_{\mu^{2}\rightarrow 0}\mathrm{T r}\left[ \int_{\infty}^{\mu^{2}}d m^{2}\left( \Delta_{B}^{\beta}( p , P ; m^{2}) + \Delta_{F}^{\beta}( p , P ; m^{2}) \right) \right]
V_{0}=\left[\begin{array}{cccc|cccc|ccc|cccc}1 1 \cdots 1\, \,2 2 \cdots 2\, \,\bullet \bullet \bullet\, \,n n \cdots n 1 2 \cdots n\, \,1 2 \cdots n\, \,\bullet \bullet \bullet\, \,1 2 \cdots n\end{array}\right].
\mathcal{B}_3(s_1,s_2) = \prod_{p \mid h}(1+O_{k,\ell}(p^{-\sigma_1-\sigma_2 -2})) \ll_{k,\ell}\prod_{p}(1 + O_{k,\ell}(p^{-1.8})) \ll_{k,\ell}1 (s_1,s_2) \in D_1 \times D_2.
| \lambda_{i}> = \flat_{i}^{+}\flat_{i - 1}^{+}\cdots \flat_{1}^{+}| 0 > , \mathrm{~ a n d , r e s p e c t i v e l y ,}< \lambda_{i}| = < 0 | \flat_{1}\cdots \flat_{i - 1}\flat_{i}.
\xi_{2}( \mathrm{z}) = \vartheta [ \begin{array}{c}{\frac{1}{3}}{0}\end{array}] ( \mathrm{z}, \tau ) \ \ \ \mathrm{o r \}\ \ \vartheta [ \begin{array}{c}{0}{\frac{1}{3}}\end{array}] ( \mathrm{z}, \tau ) \ .
\widetilde{\Delta}_{\ell}^{(2)}=\frac{1}{\alpha_n-f_{2\ell-1}}\Delta_{2\ell-1}^{(2)}+\frac{1}{\alpha_n-f_{2\ell}}\Delta_{2\ell}^{(2)}+\dddot{Z}_{\ell,1}^{(2)}.
\Phi ( x , y ) = \sum_{\alpha}\int \frac{d^{3}{\bf k}}{( 2 \pi )^{3}}\frac{1}{2 k_{0}}\biggl ( a_{\alpha}({\bf k}) e^{i k \cdot x}\psi_{\alpha}( y ) + \mathrm{h . c .}\biggr ) \ .
\lim_{r\to0}\lim_{k\to+\infty}\frac{1}{2\pi}\int_{|x|<r}|x|^{a \beta_k -4}e^{a \hat{v}_k}= \frac{a \beta_\infty (2 \beta - 4(N+1) - \beta_\infty)}{4(1-a(N+1))}
\hat{V}( r ; \lambda ) = V ( r ) - 2 [ \ln ( I_{0}+ \lambda ) ]^{\prime \prime}= V ( r ) - \frac{4 u_{0}u_{0}^{\prime}}{I_{0}+ \lambda}+ \frac{2 u_{0}^{4}}{( I_{0}+ \lambda )^{2}},
\frac{1}{2}W^2_2(\rho_0 \cdot \gamma, \rho_1 \cdot \gamma) = 2 W^2_2(\rho_0 \cdot \gamma, \rho_{1/2}\cdot \gamma )= W^2_2(\rho_0 \cdot \gamma, \rho_{1/2}\cdot \gamma )+ W^2_2(\rho_1 \cdot \gamma, \rho_{1/2}\cdot \gamma ) & \le \int \rho_0 \log \rho_0 d \gamma + \int \rho_1 \log \rho_1 d \gamma.
\begin{multlined}[t][12.5cm]\frac{(3c+\nu_{1}+2\nu_{2}+2)!}{\pi^2(c+\nu_{2}+\nu_{1})!(c+\nu_{2})!c!}\sim \frac{9\sqrt{3}c}{2\pi^3}\left[\frac{3c+\nu_{1}+2\nu_{2}}{c+\nu_{2}+\nu_{1}}\right]^{c+\nu_{2}+\nu_{1}}\left[\frac{3c+\nu_{1}+2\nu_{2}}{c+\nu_{2}}\right]^{c+\nu_{2}}\left[\frac{3c+\nu_{1}+2\nu_{2}}{c}\right]^c\end{multlined}
A_{w,1}f(v_{\ell,m,m})= q^3f(v_{\ell-1,m-1,m-1})+(q^2+q)f(v_{\ell,m+1,m})+f(v_{\ell+1,m,m}) A_{w,2}f(v_{\ell,m,m})= (q^4+q^3)f(v_{\ell-1,m,m-1})+q^2f(v_{\ell,m-1,m-1}) &+q^2f(v_{\ell,m+1,m+1})+(q+1)f(v_{\ell+1,m+1,m}) A_{w,3}f(v_{\ell,m,m})= q^3f(v_{\ell-1,m,m})+(q^2+q)f(v_{\ell,m,m-1})+f(v_{\ell+1,m+1,m+1})
c_n =\frac{(-1)^{n(n+1)/2}}{x^{n(n-1)/2}(x^2 ; x^2 )_\infty^{n(n-1)/2}}\frac{(x^2 ; x^4 , x^4 )_\infty^{2n}}{(x^8 ; x^4 , x^4 )_\infty^{2n}}.
\frac{\partial U^{-1}}{\partial z_\alpha}+{1\over 2\pi\kappa}\sum_{\beta\neq \alpha}\frac{\frac{1}{2}(\hat Q^+(\xi_\alpha) \hat Q^-(\xi_\beta)+\hat Q^-(\xi_\alpha)\hat Q^+(\xi_\beta))+\hat Q^3(\xi_\alpha)\hat Q^3(\xi_\beta)}{z_\alpha-z_\beta}U^{-1}=0
\frac{1}{g}=\inf_{\tilde{n}\geq 1}\left(\!\sum_{n=1}^{\tilde{n}}\frac{1}{\pi_n\lambda_{n}\varphi_{n}\varphi_{n+1}}\sum_{q=1}^{n}\pi_{q}\varphi_{q}^{2}+\!\sum_{n=\tilde{n}+1}^{\infty}\frac{1}{\pi_{n}\lambda_{n}\varphi_{n}\varphi_{n+1}}\;\sum_{q=n+1}^{\infty}\pi_{q}\varphi_{q}^{2}\;\right).
n^{2}=1-\frac{X}{1-iZ-\frac{\frac{1}{2}Y^{2}sin^{2}\theta}{1-X-iZ}\pm\frac{1}{1-X-iZ}(\frac{1}{4}Y^{4}sin^{4}\theta+Y^{2}cos^{2}\theta(1-X-iZ)^{2})^{1/2}}
\left( \begin{array}{c c}{{{\varphi}_{o u t}^{\prime}}}\\{{{{\varphi}^{\prime}}_{o u t}^{*}}}\\ \end{array}\right) = \left( \begin{array}{c c}{{- 1}}&{{0}}\\{{0}}&{{- 1}}\\ \end{array}\right) \left( \begin{array}{c c}{{\varphi_{i n}}}\\{{\varphi_{i n}^{*}}}\\ \end{array}\right) ,
-\Delta z_1=(m_+(x_0)-m_-(x_0))\delta_0\ \mbox{in $B_{r_0}$},z_1=b_0\ \mbox{on $\partial B_{r_0}$}, &-\Delta z_2=-s_-\leq 0\ \mbox{in $B_{r_0}$},z_2=0\ \mbox{on $\partial B_{r_0}$}.
P_{2r-2}(m_1,n_2,\dots,m_{r-1},n_r) =1+\sum_{k\ge i_1> i_2\ge 1}n_{i_1}m_{i_2}+\sum_{k\ge i_1> i_2\ge i_3 >i_4\ge 1}n_{i_1}m_{i_2}n_{i_3}m_{i_4}&\quad+\dots+n_{r}m_{r-1}n_{r-1}m_{r-2}\cdots n_2m_1.
M(x+ \i y) = \mu(x)\delta(y) \mu(x) = \frac{1}{2\pi \i}(R(x-\i 0)-R(x + \i 0))=|\alpha_x\rangle\langle\alpha_x| |\alpha_x\rangle = (1+\pi^2)^{-1/2}\left(\frac{\mathcal{P}}{x-\Omega}|E\rangle+|\delta_x\rangle\right)
2\pi i\omega_{sp}^{(0)}(\{q\})= \delta_{sp}\ln k_p+{\sum_\Gamma}^{''}\ln\frac{[u_s-g_\Gamma(u_p)][v_s-g_\Gamma(v_p)]}{[u_s-g_\Gamma(v_p)][v_s-g_\Gamma(u_p)]}
\sum_{I\in\mathcal{I}}\big(\tfrac{ne}{|I|}\big)^{-c_1 |I|}=\sum_{k=0}^n \tbinom{n}{k}\big(\tfrac{en}{k}\big)^{-c_1k}\le \sum_{k=0}^n\big(\tfrac{en}{k}\big)^{-k(c_1-1)}\le (1-e^{1-c_1})^{-1}\triangleq C_0.
\sup_{\gamma \in \bar{\mathcal{P}}}I_{\lambda}[\gamma]=\sup_{\gamma \in \bar{\mathcal{P}}}\sup_{\pi \in \Pi\left(\mu_{13}, \mu_{23}, \gamma\right)}\int_{\mathcal{V}\times \mathcal{S}}\phi_\lambda (v, s^\prime ) \, d \pi (v, s^\prime) .
\begin{aligned}\L\Psi(X) =-\frac{\gamma}{m}v^2-\sum_{k\geq 1}\lambda_kk^{-2s}z_k^2 & -\frac{1}{m}\sum_{k\geq 1}\sqrt{c_k}z_kv+\sum_{k\geq 1}\sqrt{c_k}k^{-2s}z_kv+\frac{\gamma}{m^2}+\sum_{k\geq 1}k^{-2s}\lambda_k . \end{aligned}
((L,\varphi)^{d+1}) \geqslant ((L,\varphi_0)^{d+1})-\frac 12\nu(\Omega_\infty)(d+1)\ln\binom{r+\delta-1}{\delta}&\qquad-\frac 12\nu(\Omega_\infty)\delta(d+1)\ln(r)-\nu(\Omega_\infty)\ln(\delta!) &\geqslant((L,\varphi_0)^{d+1})-\nu(\Omega_\infty)\delta(d+1)\ln(r)-\nu(\Omega_\infty)\ln(\delta!),
\tilde{D}_a f (x) = \left( a^{p^{i}+ p-2}x - a^{p-1}x^{p^i}\right) - \left( a^{p^{n-i}+ p-2}x - a^{p-1}x^{p^{n-i}}\right) &= a^{p-1}x \left( - x^{p^i - 1}+ x^{p^{n-i}-1}+ a^{p^i - 1}- a^{p^{n-i}-1}\right).
V_4{}^{ij}\equiv \Phi^{ij}= -{1\over 2}e^{2U}\epsilon_{ijm}\partial_m\left ( \ln \chi^{(1)}\chi^{(2)}\right ) \qquad V_4{}^{i4}\equiv \Phi^{i4}={1\over 2}e^{2U}\partial_i \left ( \ln{\chi^{(1)}\over\chi^{(2)}}\right )
v_k=\dfrac{\lambda_k}{\mu_k}=\dfrac{(b+u)_k(c+u)_k}{(a+u)_k(d+u)_k}\dfrac{(s_1)_k\cdots(s_r)_k}{(t_1)_k\cdots(t_r)_k}\left(\dfrac{z}{w}\right)^k \qquad(k=0,1,2,\ldots,N),
h'(\bar{\alpha}_A(a,x)) = T(\bar{\alpha}_A(a,x))^{-1}\beta(a,x)\alpha_A(a,x) = T(\bar{\alpha}_A(a,x))^{-1}\left( T(\bar{\alpha}_A(a,x))i(a)\psi_a(T(x))^{-1}\right) a\cdot x = i(a)aT(x)^{-1}x = \alpha_{\mathcal{A}}(a, h'(x)),
|k r_2(z)|^2 = |z r_2^2(z)|= \bigg | \int_0^z\left(r_2(s)^2+2 sr_2(s) r_2'(s)\right) d s\bigg| &\leq \|r_2(z)\|_{L^2}+2 \|r_2'(z) \|_{L^2}\|z r_2 (z) \|_{L^2}\leq \|r_2(z)\|_{H^1\cap L^{2,1}},
A_{3}={\frac{q Q ( r - m ) \sin^{2}\theta}{T [ ( r - m )^{2}- m^{2}\cos^{2}\theta ]}}, \A_{4}= -{\frac{q s ( 1 - 8{\frac{1}{I_{0}^{3}}}e^{- \tau_{0}\tau}( 1 -{\frac{2 m}{r}})^{2}g_{2 2})}{2 \sqrt{2}T}}
g_{iu}^{(j)}={Y}_{i,(u-1)(p+q)+j}-{\mathbb{E}}(Y_{i,(u-1)(p+q)+j}|{\mathcal{F}}_{i,u-1}^{Y}):=Y_{iu}^{(j)}-{\mathbb{E}}(Y_{iu}^{(j)}|{\mathcal{F}}_{i,u-1}^{Y})\,.
J_{\alpha \beta}^{i(1)}=x^\gamma X_\gamma(J_{\alpha\beta}^{i})(q), J_{\alpha \beta}^{i(2)}=\frac{1}{2}\left(x^\gamma x^\delta X_\gamma X_\delta(J_{\alpha\beta}^{i})(q)+z^j Z_j(J_{\alpha\beta}^{i})(q)\right).
P^+=\int^\infty_{-\infty}dx^- \mbox{Tr}\biggl[2 \partial_- \phi_- \cdot \partial_- \phi_++{{\rm i}\over 2}\sum_h \left ( u^\dagger_h \cdot\partial_- u_h -\partial_-u^\dagger_h \cdot u_h \right )\biggr ]
D_{eff}=\frac{n\sum_{i=1}^{n}w_{i}^{2}}{(\sum_{i=1}^{n}w_{i})^{2}}=\frac{\frac{1}{n}\sum_{i=1}^{n}w_{i}^{2}}{(\frac{1}{n}\sum_{i=1}^{n}w_{i})^{2}}=\frac{\overline{w^{2}}}{\overline{w}^{2}}
|z-\zeta_t|\leq |G'(F(\zeta_t))| \frac{4\pi/t}{(1-4\pi/t)^2}\leq |G'(F(\zeta_t))| \frac{16\pi}{t}= \frac{16\pi}{|F'(\zeta_t)|t}\leq \frac{128\pi^2}{h(x)t}.
\sinh^{2}\big((d_{\mathbb{H}}(z,\gamma w)+\varepsilon)\slash 2\big)= 2\sinh^{2}\big(d_{\mathbb{H}}(z,\gamma w)\slash 2\big)\cosh^{2}(\varepsilon\slash 2)+ &\sinh^{2}(\varepsilon\slash 2)+\sinh\big(d_{\mathbb{H}}(z,\gamma w)\slash 2\big)\cosh\big(d_{\mathbb{H}}(z,\gamma w)\slash 2\big)\sinh(\varepsilon).
N(\ell^0_{\S_0})=\{4,5,6\},~N(\ell^1_{\S_0})=\{4,5,6\};~ N(\ell^0_{O_1})=\{5,6,6\};~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ N(\ell^0_{O_2})=\{4,5,6\},~N(\ell^1_{O_2})=\{5,6,6\}; N(\ell^0_{O_3})=\{4,5,6\},~N(\ell^1_{O_3})=\{5,6,6\}; N(\ell^0_{O_4})=\{5,6,6\}.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
S(R) = \sum_{i=1}^{N_{max}}\frac{R}{i\cdot\pi}\cdot &\int_{0}^{\infty}\bigg[ \mathcal{I}m \Big\{\frac{e^{-jt(2^R-1)}}{t}\cdot\left( \phi_{\gamma(i)}(t) - \phi_{\gamma(i-1)}(t) \right)\Big\}\bigg] dt.
X^2 = b \left(\begin{array}{ccccc}\ddots \ddots \ddots \ddots 0 e^{i \omega t}\ddots \ddots e^{-i \omega t}0 e^{i \omega t}\ddots \ddots \ddots e^{-i \omega t}0 \ddots & \ddots \ddots \ddots\end{array}\right)
\sum_{e_S \otimes e_T \in \mathbb{B}(\mathcal{A},\mathcal{B})}\langle(\psi_{\mathcal{A},\mathcal{B}})_* (e_S \otimes e_T) ,e_S \otimes e_T \rangle_{\mathcal{A},\mathcal{B}}= \sum_{f \in \mathcal{F}(\mathcal{A},\mathcal{B})}m(f),
\int_{\mathbb{R}^N}(I_\mu*|u|^{p})|v|^{q}\leq D_0(|\nabla u|^2_2+|\nabla v|^2_2)^{\frac{\gamma_p+\gamma_q}{2}},\D_0:=\frac{1}{\gamma_p+\gamma_q}\Bigl(\frac{\gamma_p+\gamma_q-2}{2(\gamma_p+\gamma_q)l(a,b)}\Bigr)^{\frac{\gamma_p+\gamma_q-2}{2}},
\frac{z^{N-1}}{1-wx_1}\left(Q_N\left(\frac{1}{z}; y\right) - wP_N\left(\frac{1}{z}; x\right)\right)\,=\,z^{N-1}+ \sum\noindent_{j=1}^{N-1}(-1)^j \frac{y_j - wx_{j+1}}{1 - wx_1}z^{N-(j+1)}
\tilde{e}_1( \tilde{m}) = \frac{1}{\sqrt{1-\tilde{m}_1^2}}{e}_1 \times \tilde{m},\ \ \ \ \ \ \ \tilde{e}_2(\tilde{m})= \frac{1}{\sqrt{1-\tilde{m}_1^2}}({e}_1 \times \tilde{m}) \times \tilde{m}\ \ \ \ \ \ \ \{\rm if}\ \ |\tilde{m}_1|<1.
\frac{(q)_{N}}{(zq)_{N}(z^{-1}q)_{N}}=\frac{1}{(q)_N}+(1-z)\sum_{n=1}^{N}\left[\begin{matrix}N n \end{matrix}\right]\frac{(-1)^n(q)_nq^{n(n+1)/2}}{(q)_{n+N}}\left(\frac{1}{1-zq^n}-\frac{1}{z-q^n}\right),
W^{n}_{0,1,\ldots, \frac{\l}{2}-1}=\begin{cases}W^{n}_{0,1}\circ W^n_{2,3}\circ \dots \circ W^n_{\frac{\l}{2}-2,\frac{\l}{2}-1}\mbox{if}\l \mbox{is even}W^n_{0,1}\circ W^n_{2,3}\circ \dots \circ W^n_{\frac{\l}{2}-5,\frac{\l}{2}-4}\circ W^n_{\frac{\l}{2}-3,\frac{\l}{2}-2,\frac{\l}{2}-1}\mbox{if}\l \mbox{is odd}\end{cases}
+2z_\alpha(\xi)\frac{\partial}{\partial y_\alpha(\xi)}D_m^{(4)}(y(\xi)-y(\xi'))+z_\alpha(\xi)z_\beta(\xi')\frac{\partial^2}{\partial y_\alpha(\xi)\partial y_\beta(\xi')}D_m^{(4)}(y(\xi)-y(\xi'))\Biggr)\Biggr].
\sup_{X\in\Omega,Y\in\Omega}|\langle AX,Y\rangle| =\sup_{X\in\Omega,Y\in\Omega}\langle BX,BY\rangle &\leq\sup_{X\in\Omega,Y\in\Omega}\|BX\|_2\|BY\|_2 &=\sup_{X\in\Omega}\|BX\|_2^2.
(\theta_{11}\theta^{11}- 1) s_1^2 + 2\theta_{11}(\theta^{12}s_2 + \theta^{13}s_3) s_1 + \theta_{11}(\theta^{22}s_2^2+2\theta^{23}s_2 s_3+\theta^{33}s_3^2)= \theta_{11}\mathfrak{D}.
\{Q_{( \alpha )}, Q_{( \beta )}\}= g [ \mathcal{C}\Gamma_{s}( \hat{M}^{\hat{a}\hat{b}}) ]_{\alpha \beta}\hat{M}_{\hat{a}\hat{b}}= - i ( \mathcal{C}\gamma^{a})_{\alpha \beta}P_{a}-{\textstyle \frac{g}{2}}( \mathcal{C}\gamma^{a b})_{\alpha \beta}M_{a b},
I_1=\int_2^{599}\frac{|\theta(t) - t|}{t\log^2t}t, I_2=\int_{599}^{\exp(58)}\frac{|\theta(t) - t|}{t\log^2t}t, I_3=\int_{\exp(58)}^x\frac{|\theta(t) - t|}{t\log^2t}t.
U(t) = \hat{U}(t) = 1 (-r \le t \le 0), & U(t) = 1 + \int_0^t A_1 \big( X(s-r) \big) \, U(s) \, \mbox{d}W(s) (0 \le t \le T), & \hat{U}(t) = 1 - \int_0^t \hat{U}(s) \, A_1 \big( X(s-r) \big) \, \mbox{d}W(s) + \int_0^t \hat{U}(s) \, A_1 \big( X(s-r) \big)^2 \, \mbox{d}s (0 \le t \le T).
q[\psi;p] = (p-p_0)^2 \|\psi\|^2 + 2(\psi,(p-p_0)(p_0+Bx)\psi) begin{align*}.3em] \le (p-p_0)^2 (1+\delta^{-1}) \|\psi\|^2 + \delta \|(p_0+Bx)\psi\|^2 begin{align*}.3em] \le (p-p_0)^2 (1+\delta^{-1}) \|\psi\|^2 + \delta h_a[\psi;p_0].
R_{ab}(\lambda )=\left( \begin{array}{cccc}q\lambda -q^{-1}\lambda^{-1}begin{align*}-1mm] \lambda -\lambda^{-1}q-q^{-1}begin{align*}-1mm] q-q^{-1}\lambda -\lambda^{-1}begin{align*}-1mm] q\lambda -q^{-1}\lambda^{-1}\end{array}\right) \,,
X_\tau^{t,x}=e^{(\tau-t)A}x+\int_t^\tau e^{(\tau-s)A}GB(s,X_s)ds+\int_t^\tau e^{(\tau-s)A}GdW_s ,\tau\in\left[t,T\right], \;\; X_\tau^{t,x}=x, \; \;\tau \le t.
\vec{\chi}(x) = \left( \begin{array}{c}\vec{\chi}_{\rm vec}(x) \vec{\chi}_{\rm gr}(x) \end{array}\right) = \left(\begin{array}{c}A(x) \, \vec{n}B(x) \, \vec{p}\end{array}\right) \ , \qquad \partial_i \partial_i \vec{\chi}(x) =0 \ ,
\implies \begin{cases}a\left(r^k+\dfrac{1}{r^k}\right)=a\left(r^k+\dfrac{-1_(-1)^{k+1}}{r^k}\right)=P_k, \left(r^k-\dfrac{1}{r^k}\right)=\left(r^k+\dfrac{-1(-1)^{k+1}}{r^k}\right)=Q_k\end{cases}
\mathsf{a}(n,\mathbf{d})\partial_{\tau^1}\partial_{\tau^I}F^{(1)}(0)=- \sum_{i=0}^{n}\sum_{j=0}^n\sum_{k=0}^n(1-i)W_j^i M_{i}^k p_j \partial_{\tau^{I+\mathbf{e}_k- \mathbf{e}_j}}F^{(1)}(0)+\partial_{\tau^I}F^{(1)}(0).
\begin{array}{rclrcl}Z_1 = \partial_{\mu}Z_{\nu}-\partial_{\nu}Z_{\mu}, Z_2 = [Z_{\mu},Z_{\nu}], A_1 = \partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}, A_2 = [A_{\mu},A_{\nu}], M = [Z_{\mu},A_{\nu}]-[Z_{\nu},A_{\mu}]. \end{array}
\left\Vert f\right\Vert_{\boldsymbol{B}_{p(\cdot ),q(\cdot )}^{\alpha(\cdot )}}^{\blacktriangledown}:=\left\Vert \Phi^{\ast ,a}f\right\Vert_{p(\cdot )}+\left\Vert \left\Vert \varphi_{t}^{\ast ,a}t^{-\alpha (\cdot)}f\right\Vert_{p(\cdot )}\right\Vert_{L^{q(\cdot )}((0,1],\frac{dt}{t})}
|u(\hbox{e}^{t\Omega}z)|^{2}\le \mathfrak C(R)\left(|\hbox{e}^{t\Omega}z|\,s_\tau(\hbox{e}^{t\Omega}z)\right)^{-2}=\mathfrak C(R)\left(| z|\,s_\tau( z)\right)^{-2}
(\mathrm{curl}\ \boldsymbol{u}, \mathrm{curl}\ \boldsymbol{v}) + (\boldsymbol{u}, \boldsymbol{v})=(\boldsymbol{f}, \boldsymbol{v}),\forall \boldsymbol{v}\in \boldsymbol{H}_0(\mathrm{curl}; \Omega).
\pi ( d \alpha ) = \left( \begin{matrix}{{\gamma^{\mu \nu}\partial_{\mu}\alpha_{1 \nu}}}&{{- g^{\mu \nu}\gamma^{5}V_{\nu}( \partial_{\mu}\alpha_{5}+ \alpha_{1 \mu}- \alpha_{2 \mu})}}\\{{g^{\mu \nu}\gamma^{5}V_{\nu}( \partial_{\mu}\tilde{\alpha}_{5}+ \alpha_{1 \mu}- \alpha_{2 \mu})}}&{{\gamma^{\mu \nu}\partial_{\mu}\alpha_{2 \nu}}}\\ \end{matrix}\right)
V_{\chi_{1,1}}(z)=\sum_{j=1}^n q^{2j-1}s_1 M_{i,1}(z)+\frac{s_1-s_1^{-1}}{q-q^{-1}}\bigl(M_{0,1}(z)+M_{\bar{0},1}(z)\bigr)+\sum_{j=1}^n q^{-2j+1}s_1^{-1}M_{\bar j,1}(z)\,.
u_{n+1}^{p+1}- u_{n+1,r}^{p+1}= \sum\limits_{j=0}^{p}(u_{n+1}- u_{n+1,r}) u_{n+1}^j u_{n+1,r}^{p-j}, & \approx (p+1) (u_{n+1}- u_{n+1,r}) u_n^p.
\frac{\partial f(t)}{\partial t}=e_{\lambda}^{1-\lambda}(t)|z|^{2}f(t)=e_{\lambda}^{1-\lambda}(t)|z|^{2}\sum_{k=0}^{\infty}\frac{t^{k}}{k!}\phi_{k,\lambda}(|z|^{2}) &=|z|^{2}\sum_{k=0}^{\infty}\bigg(\sum_{l=0}^{k}\binom{k}{l}(1-\lambda)_{k-l,\lambda}\phi_{l,\lambda}(|z|^{2})\bigg)\frac{t^{k}}{k!}.