harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\left[ \begin{array}{cc}x_{1,\tau}x_{2,\tau}t_\tau p_{1,\tau}p_{2,\tau}p_\tau \end{array}\right] = \left[ \begin{array}{cc}A_1(Q_1,T) A_2(Q_2,T) B(T) C_1(Q_1,T)P_1 + D_1(Q_1,T) C_2(Q_2,T)P_2 + D_2(Q_2,T) F(Q_1,Q_2,T,P_1,P_2,P_T) \end{array}\right]\;.
\partial_ta^{(m)}_l=(\varphi_l, u^{(m)}_t)_{L^2(S^1)}=-\big(\varphi_l,(\varepsilon+d\pi_N(v))u^{(m)}_r\big)_{L^2(S^1)}=-\sum_{j=0}^ma^{(m)}_j\lambda_j \big(\varphi_l,(\varepsilon+d\pi_N(v))\varphi_j\big)_{L^2(S^1)},\ 0\le l\le m.
k_{1}= \left( \begin{array}{c}{{e}}\\{{p}}\\ \end{array}\right) k_{2}= \left( \begin{array}{c}{{e}}\\{{- p}}\\ \end{array}\right) k_{3}= \left( \begin{array}{c}{{- e}}\\{{- p}}\\ \end{array}\right) k_{4}= \left( \begin{array}{c}{{- e}}\\{{p}}\\ \end{array}\right) .
[\hat{x}_{1}^{\prime},\hat{x}_{2}^{\prime}]=i\alpha^{2}, \hspace{0.4cm}[\hat{p}_{1}^{\prime},\hat{p}_{2}^{\prime}]=i\alpha^{-2}, \hspace{0.4cm}[\hat{x}_{i}^{\prime},\hat{p}_{j}^{\prime}]=i\delta_{ij}
\mathcal{T}^{2}_{S}(x,w) :=\big\{z\in\mathbb{X}\ |\ \exists\, \tau_k\downarrow 0\ \{\rm s.t.}\ \{\rm dist}(x+\tau_kw+\frac{1}{2}\tau_k^2 z,S)=o(\tau_k^2)\big\}, \mathcal{T}^{i,2}_{S}(x,w) :=\big\{z\in\mathbb{X}\ |\{\rm dist}(x+\tau w+\frac{1}{2}\tau^2 z,S)=o(\tau^2) \forall\tau\ge 0\big\}.
\sum_{k=0}^{\infty}\frac{\left( a\right)_{k}\left( b\right)_{k}}{k!\left(b+1\right)_{k}}\left[ \psi \left( a+k\right) -\psi \left( a\right) \right]z^{k}=b\,z^{-b}\frac{\partial}{\partial a}\mathrm{B}_{z}\left( b,1-a\right).
\sigma_i\sigma_j\sigma_i\alpha_j=\alpha_j, \sigma_j\sigma_i\sigma_j\alpha_i=\alpha_i, &\sigma_i\sigma_j\alpha_i=\sigma_j\alpha_i=\alpha_i+\alpha_j, \sigma_j\sigma_i\alpha_j=\sigma_i\alpha_j=2\alpha_i+\alpha_j.
M(w,\,x,\,y,\,z)\rhd M(t,\,a,\,b,\,c)= \begin{bmatrix}1 a+xe^{c}-ae^{z}w+ zc+ae^{-c}(b+y-be^{z})) 0 e^{z}(y-b)e^{-c}+be^{z-c}0 0 1\end{bmatrix}
p \cdot \left(\int_{\partial X}< v, \overrightarrow{z\xi}> < DF_p(v), \overrightarrow{F_p(z)f(\xi)}> - < DF_p(v), \overrightarrow{F_p(z)f(\xi)}>^2 d\mu^z_p(\xi)\right) & = \int_{\partial X}d^2 B^{f(\xi)}(DF_p(v), DF_p(v)) d\mu^z_p(\xi) & \geq 0
\begin{cases}\mathbb{L}'_e(\hat{{\mathbf U}},\hat{\Psi})\dot{{\mathbf U}}=\mathbf{f}\Omega_T, \mathbb{B}'_e(\hat{{\mathbf U}},\hat{\varphi})(\dot{{\mathbf U}},\varphi)=\mathbf{g}\Gamma_T, (\dot{{\mathbf U}},\varphi)=0 t<0, \end{cases}
x\ast[y,[z,u]]+y\ast[x,[z,u]] = x(y[z,u]-[z,u]y)+(y[z,u]-[z,u]y)x+y(x[z,u]-[z,u]x)+(x[z,u]-[z,u]x)y = (xy)[z,u]-x([z,u]y)+y([z,u]x)-[z,u](yx) &+(yx)[z,u]-y([z,u]x)+x([z,u]y)-[z,u](xy) = (xy+yx)[z,u]-[z,u](xy+yx) = [x\ast y, [z,u]].
2\hat g^{+,-}=-\sum_{j,k=0}^ng^{jk}\psi_{x_J}^+\psi_{x_k}^-, &2\hat g^{+,j}=\sum_{p,r=0}^n g^{pr}\psi_{x_p}^+\varphi_{jx_r},\ \ 1\leq j\leq n-1, &\hat g^{jk}=\sum_{p,r=0}^n g^{pr}\varphi_{j x_p}\varphi_{kx_r},\ \ 1\leq j,k\leq n-1,
\varphi(E_-(n_vw)) = \varphi(E_-(n_vw)E_{o_-}(\lambda_P^-))\sigma(E^P_{o_{-,P}}(\lambda_P^-)^{-1}) & \in C[q_s]\varphi(E_-(n_vw\lambda_P^-))\sigma(E^P_{o_{-,P}}(\lambda_P^-)^{-1}).
c(T^*)=e(T)+f_4(T)+\varepsilon^* =(2t+3\left\lfloor{k}/{2}\right\rfloor-16+\varepsilon)+(t-3\left\lfloor{k}/{2}\right\rfloor+10-\varepsilon)+\varepsilon^* &=(3n+9\left\lfloor{k}/{2}\right\rfloor+3\varepsilon-42-\varepsilon^*)/{2}&\ge (3n+9\left\lfloor{k}/{2}\right\rfloor+3\varepsilon-43)/{2},
\alpha+u = \alpha+\sum_{i=1}^u\nu_{p}(b_{\rho p^i})+\nu_{p}\left(\prod_{i=1}^u\prod_{\substack{e\mid m e>1}}b_{\rho p^ie}\right) & = \alpha+u+\nu_{p}\left(\prod_{i=1}^u\prod_{\substack{e\mid m e>1}}b_{\rho p^ie}\right) & = \alpha+u+\sum_{i=1}^u\sum_{\substack{e\mid m e>1}}\nu_{p}(b_{\rho p^ie})
1 \geq \left(1 - x^{\lambda}\right)^{\nu_2 M - 1}\geq \exp\left(- \frac{\nu_2}{M^{\lambda \varepsilon}}\right) \left[1 + O\left(\frac{1}{M^{1 + \lambda \varepsilon}}\right)\right]= 1 + O\left(\frac{1}{M^{\lambda \varepsilon}}\right).
\begin{cases}\nabla u_1(x_1) = \sum^N_{i=2}x_i \cdots \nabla u_j(x_j) = \sum^N_{i=1,i\neq j}x_i \cdots \nabla u_N(x_N) = \sum^{N-1}_{i=1}x_i \end{cases}\begin{cases}x_1 + \nabla u_1(x_1) = \sum^N_{i=1}x_i \cdots x_j + \nabla u_j(x_j) = \sum^N_{i=1}x_i \cdots x_N + \nabla u_N(x_N) = \sum^N_{i=1}x_i \end{cases}, \medskip
\int_{D\setminus U}M_D(y,\infty)\omega_U^x(dy) =\frac{1}{\mathbb{E}_{x_0}\tau_D}\mathbb{E}_x\big[\mathbb{E}_{X_{\tau_U}}\tau_D\big]=\frac{1}{\mathbb{E}_{x_0}\tau_D}\mathbb{E}_x\left[\int_{\tau_U}^{\tau_D}\mathbf 1dt\right] &<\frac{\mathbb{E}_{x}\tau_D}{\mathbb{E}_{x_0}\tau_D}=M_D(x,\infty),
(1-\frac{6h}{n})\frac{n^{3}}{6h}\sum_{s=1}^{h-1}(\begin{matrix}h\\s\end{matrix})\frac{1}{s}\le S(h,n)\le\frac{n^{3}}{6h}\sum_{s=1}^{h-1}(\begin{matrix}h\\s\end{matrix})\frac{1}{s}
\infty=\int_{B_R(0)}\frac{\lvert v(k)\lvert^2}{\omega(k)^2}d\lambda_\nu=\lambda_\nu(B_1(0))\int_{0}^\infty 1_{B_R(0)}(ke_1)\frac{\lvert v(ke_1)\lvert^2}{\omega(ke_1)^2}k^{\nu-1}d\lambda_1.
\|h\|_{W^{1,2}}\geq\frac{\int_{0}^{1}f(s)dh(s)}{\|f\|_{L^{2}([0,1])}}=\frac{\|h\|_{\bar{\mathcal{H}}}\|f\|_{\mathcal{H}}}{\|f\|_{L^{2}([0,1])}}\geq C_{H}\|h\|_{\bar{\mathcal{H}}},
E = -{\frac{2}{\pi a}}\sum_{l = 0}^{\infty}( l + 1 ) \int_{0}^{\infty}d x x \cos x \epsilon{\frac{d}{d x}}\ln [ ( e_{l}^{2}+ e_{l + 1}^{2}) ( s_{l}^{2}+ s_{l + 1}^{2}) ] .
| \{(\xi_1, \eta_1) \ | \ (\tau_1, \xi_1, \eta_1) \in E(\tau_2, \xi_2,\eta_2) \}|= \int_{\theta_1}\int_{r_1}{\chi}_{E(\tau_2,\xi_2,\eta_2)}(|\xi_1|, \theta_1) r_1 d r_1 d \theta_1 \lesssim A^{-1}M N_1^{-1}\max (L_1, L_2).
S_{G N}= \int d^{D}x \left[ \frac{i}2 \overline{\psi}\not \! \partial \psi - \frac{M}2 \overline{\psi}\psi - \frac 1 2 \sigma \star ( \overline{\psi}\star \psi ) - \frac{N}{4 g}\sigma^{2}- \frac{N}{2 g}M \sigma \right] .
\Vert v_\tau(t)\Vert^2=\frac 1{4\pi}\int_0^{t-\tau}|h(t-r)|^2dr+\frac{m^2}{4\pi}\int_0^{t-\tau}r^2|p_\tau(r,t)|^2dr+\frac{m}{2\pi}\int_0^{t-\tau}h(t-r)p_\tau(r,t)r\,dr,t\in [\tau,T].
K ={\frac{( d - 2 + \sum_{i}{\frac{1}{a_{i}^{2}}})}{( \sum_{i}{\frac{z_{i}^{2}}{a_{i}^{4}}})^{1 / 2}}}-{\frac{( \sum_{i}{\frac{z_{i}^{2}}{a_{i}^{6}}})}{( \sum_{i}{\frac{z_{i}^{2}}{a_{i}^{4}}})^{3 / 2}}}~ ~ .
\begin{aligned}|\left|k\right|_g^2-\left| \ell \right|_g^2 | = \beta \Big| \sum_{ij}\overline g_{ij}\left(k_i k_j- \ell_i \ell_j\right) \Big| & \geq \beta_1 \dfrac{\gamma}{\Big( \sum_{i , j}|k_i k_j - \ell_i \ell_j| \Big)^\tau}\,.\end{aligned}
i \frac{\partial}{\partial t}\langle \vec{q}^{\prime}, t \mid \vec{q}, 0 \rangle = - \frac{1}{2}\sum_{\nu = 1}^{3}\frac{\partial^{2}}{\partial q_{\nu}^{2}}\langle \vec{q}^{\prime}, t \mid \vec{q}, 0 \rangle + V ( \vec{q}^{\prime}) \langle \vec{q}^{\prime}, t \mid \vec{q}, 0 \rangle
=\frac{-1}{8\pi^2}\ln\left(\frac{R_2}{R}\right)\left(\frac{-2\delta_{\mu\nu}\delta_{\alpha\beta}}{4}+\frac{4\left(\delta_{\mu\nu}\delta_{\alpha\beta}+\delta_{\mu\alpha}\delta_{\nu\beta}+\delta_{\mu\beta}\delta_{\nu\alpha}\right)}{24}\right)\frac{1}{2}A_{\mu ax}\hat{x}_\alpha\hat{x}_\beta A_{\nu ax}
w_1=\frac{y_1y_2-1}{y_1y_2+1},\w_2=\frac{y_1-y_3}{y_1+y_3}+\frac{y_1y_3-1}{y_1y_3+1},\w_3=\frac{y_1-y_3}{y_1+y_3}-\frac{y_1y_3-1}{y_1y_3+1}.
x\cdot\nabla u(x)=(n-\bar{\lambda})u+\lim_{d \to 0}\int_{R^n\setminus B_d(0)}\frac{z\cdot\nabla v^{q_2}(z)dz}{|x|^{\beta_1}|x-z|^{n-\alpha}|z|^{\beta_2}}.
\Phi(\zeta,\zeta_0)=z(\zeta)-z(\zeta_0)-z_\zeta(\zeta)(\zeta-\zeta_0)-z_{\bar\zeta}(\zeta)\bar{(\zeta-\zeta_0)}, &\phi(\zeta,\zeta_0)=\rho(\zeta)-\rho(\zeta_0)-\rho_\zeta(\zeta)(\zeta-\zeta_0)-\rho_{\bar\zeta}(\zeta)\bar{(\zeta-\zeta_0)}.
x = \sum_{i=1}^\ell x_i g_i - \sum_{i=1}^{\ell-1}w_{n+1}^N p_i g_i - w^N_{n+1}p_\ell \cdot g_\ell + w^{N+1}_{n+1}\left( \sum_{i=1}^{\ell-1}q_i g_i \right) = \sum_{i=1}^{\ell}x'_i g_i
\begin{dcases}Lu_\lambda - \lambda u_\lambda = -g \mbox{in}\mathbb{R}^2, & Lv_\mu - \mu v_\mu = -\psi \mbox{in}\mathbb{R}^2, & Lw_{\mu + \lambda}- (\mu + \lambda) w_{\lambda + \mu}= -\frac{\partial u_\lambda}{\partial y}\frac{\partial v_\mu}{\partial y}\mbox{in}\mathbb{R}^2.\end{dcases}
a^{1243}=a^{2143},a_{1}^{1243}=a_{1}^{2143},a_{2}^{1243}=a_{2}^{2143},a_{3}^{1243}=a_{3}^{2143},a_{4}^{1243}=a_{4}^{2143}, a_{12}^{1243}=a_{12}^{2143},a_{13}^{1243}=a_{13}^{2143},a_{14}^{1243}=a_{14}^{2143},a_{23}^{1243}=a_{23}^{2143},a_{24}^{1243}=a_{24}^{2143}, a_{123}^{1243}=a_{123}^{2143},a_{124}^{1243}=a_{124}^{2143}
f_k(\tau)=\begin{cases}\dfrac{1}{2^{2k}\cdot E_{2k}}\displaystyle\sum_{n\geq 1}\chi_{-4}(n)\dfrac{q^{n}P_{k}(q^{n})}{(1-q^{2n})^{2k+1}}, k\equiv 0\pmod{2}, -\dfrac{1}{E_{2k}}\displaystyle\sum_{n\geq 1}\chi_{-4}(n)\dfrac{q^{2n}A_{2k}(q^{2n})}{(1-q^{2n})^{2k+1}}, k\equiv 1\pmod{2}, \end{cases}+T_{2k+1}(\tau)
\frac{Z_{n}^{( N_{f}+ 1 )}( M , M_{N_{f}+ 1})}{C_{n}^{( N_{f}+ 1 )}( M , M_{N_{f}+ 1})}\ = \ \frac{Z_{n}^{( N_{f})}( M )}{C_{n}^{( N_{f})}( M )}\, \frac{( n + N_{f}) !}{( N \Sigma^{2})^{n + N_{f}}}\, \frac{1}{\prod_{f = 1}^{N_{f}}( M_{f}^{2}- M_{N_{f}+ 1}^{2})}
=\frac{\omega_{n}}{(4\pi t)^{n}}\left[ \sum\limits_{d=0}^{+\infty}c_{d}^{\left( \nu ,n\right)}\t^{d}\right] \left[ \sum\limits_{i=0}^{+\infty}\frac{1}{i!}\left( \frac{n^{2}}{4}+\nu^{2}\right)^{i}\right]
X = m \left( \begin{array}{c c}{{0}}&{{1}}\\{{0}}&{{0}}\\ \end{array}\right) , \hspace{1 c m}Q = m \left( \begin{array}{c c}{{0}}&{{\sqrt{2}}}\\{{0}}&{{0}}\\ \end{array}\right) , \hspace{1 c m}\tilde{Q}= 0 .
g_{z}(R,z)=-\partial_{z}\Phi(R,z)=-\frac{GM_{0}z}{2z_{0}^{2}}[\frac{1}{\sqrt{R_{0}^{2}+R^{2}}}-\frac{1}{\sqrt{(R_{0}+2z_{0})^{2}+R^{2}}}],z=\pm z_{0},
{\cal G}_{z z}\left(\rho,z\right)={\displaystyle\frac{1}{4\pi A{\cal C}_{66}}\sum_{l=1}^{4}{\frac{\Lambda_{\rm{c}}\left(a=-A_l\right)}{\sqrt{A_l\rho^2+z^2}\prod_{j=1,(j\neq l)}^{4}{\left(A_j-A_l\right)}}}}
\sum_{n=0}^3 \left( \frac{b_2 c_n}{a_2}- d_n \right) \frac{q_n}{\omega^n}=2 a_3 k_1 + a_4 k_2 + 2 a_5 k_3 &\quad+ 2 b_3 \ell_1 + b_4 \ell_2 + 2 b_5 \ell_3 \,,
\mathcal{E}=\frac{1}{\lambda(t)^{3+\frac{2}{p-1}}}\bigg{[}-\Phi_b+b_sP_b-\bigg{(}\frac{\lambda_s}{\lambda}+b\bigg{)}\Lambda Q_b-\bigg{(}\frac{x_s}{\lambda}-1\bigg{)}Q_b'\bigg{]}\bigg{(}t,\frac{x-x(t)}{\lambda(t)}\bigg{)}, &f(\tilde{u})=(Q_{S}+\tilde{u})|Q_{S}+\tilde{u}|^{p-1}-Q_{S}|Q_{S}|^{p-1},
\frac{Z_{t}}{\, 1 + \varepsilon Z_{t}\,}\, =\, \frac{1}{\, 1 + \varepsilon \,}+ \int^{t}_{0}\frac{Z_{s}b(s, X_{s}, F_{s})}{(1+\varepsilon Z_{s})^{2}}{\mathrm d}B_{s}- \int^{t}_{0}\frac{\varepsilon Z_{s}^{2}\lvert b(s, X_{s}, F_{s}) \rvert^{2}}{\, ( 1 + \varepsilon Z_{s})^{3}\,}{\mathrm d}s \, ; t \ge 0 \, ,
(A_{[B]}f,g)-(f,A_{[B]}g) =(Tf,g)-(f,Tg) =(\Gamma_1 f,\Gamma_0 g)-(\Gamma_0 f,\Gamma_1 g) =(\Gamma_1 f,B\Gamma_1 g)-(B\Gamma_1 f,\Gamma_1 g) = 0,
i \neq j \Rightarrow \langle(\mathcal H^A_\emph{CK})_{i}, (\mathcal H^A_\emph{GL})^{j}\rangle = 0,\langle x, (\mathcal H^A_\emph{GL})^{|x|}\rangle = 0 \Rightarrow x = 0,\langle (\mathcal H^A_\emph{CK})_{|y|}, y \rangle = 0 \Rightarrow y = 0
Q_1Q_2 e_{L|(x,x)}= \frac{1}{(2\pi)^n}\int_{K_L}\sigma_{Q_1}(i\xi) \overline{\sigma_{Q_2}(i\xi)}|d\xi| +O(L^{\frac{n+|\sigma_{Q_1}|+|\sigma_{Q_2}| - 1}{m}}),
H^{31,T}(\tau,t,x)=\begin{cases}H(t,x) H^{21}(\tau+T,t,x) =H(t,x) H^2(t,x) = H(t,x) H^{32}(\tau-T,t,x) = H_{\tau-T,t}(x) H^3(t,x) = H'(t,x) .\end{cases}
T_1=\left(\begin{array}{c|c|c|c|c|c}0 W_1 \hline-W_1' 0 W_2 \hline -W_2' 0 W_3 \hline -W_3' \ddots \ddots \hline \ddots 0 W_{l}\hline -W_{l}' 0\end{array}\right).
q(T) = \Phi_{22}(T) q(0)-\int_0^T \Phi_{22}(T-s) (I-\Gamma)^T Q\check z(s)ds = H\check z(T)+H \Phi_{12}(T)q(0) -H\int_0^T\Phi_{12}(T-s) (I-\Gamma)^T Q\check z(s)ds,
M^{(\alpha)}(\lambda,\rho) = \frac{(c'_\lambda)^{(\alpha)}}{n \alpha c_\rho^{(\alpha)}}\sum_{\tau \vdash n-1}\frac{\phi^{(\alpha)}(\lambda/\tau)\phi^{(\alpha)}(\rho/\tau) c_\tau^{(\alpha)}}{(c'_\tau)^{(\alpha)}}
\sup_{\substack{u\in\theta([0,1])}}J(u) =\sup_{\substack{u\in\eta\big(1,\gamma([0,1])\big)\cap \overline{\pm D^0_m}}}J_m(u) &\leq \sup_{\substack{u\in\eta\big(1,J_m^{-1}(]-\infty,c^\pm_{m}+\varepsilon])\cap \overline{\pm D^0_m}\big)}}J_m(u) &\leq c^\pm_{m}-\varepsilon.
D_{3,a}=\frac{1}{|z|}\int_0^1 \frac{d}{d\tau}\big[Q(X_\tau,z+t_0(X_\tau), D_t \tilde h(X_\tau, z+t_0(X_\tau) ) ) &\qquad - Q(X_\tau, z+t_0(X_\tau), D_t \tilde h(X_\tau, t_0(X_\tau) ) )\big]\, d\tau
\mu_{n}(c) =\mathcal{L}\big( x^{n}\big) =\sum\limits_{k=1}^{n}\genfrac{\{}{\}}{0pt}{}{n}{k}\mathcal{L}( \varphi_{k}) =\sum\limits_{k=1}^{n}\genfrac{\{}{\}}{0pt}{}{n}{k}\nu_{k}(c) ,
\int_\Omega |\widetilde{\mathbf{w}}|^2dx \leq C\sqrt{\mu}+\epsilon\iint_{Q_t}|\widetilde{\mathbf{b}}_x|^2 dxds + \frac{C}{\epsilon}\int_0^tD(s)\mathbb{H}(s)ds,\quad\forall \epsilon \in (0, 1).
(p_2 - p_1 + s)(E_{1}^{n-1}F_{2}^{m}E_2 \otimes F_2) \cdot w_s = - (E_{1}^{n}F_{2}^{m}\otimes F_2) \cdot w_{s+1}& + ms(p_1 - p_2 - s) (E_{1}^{n-1}F_{2}^{m-1}\otimes F_2) \cdot w_s.
\theta_{1,1}\bigg(\frac{a\tau+b}{c\tau+d},\frac{z+\lambda\tau+\mu}{c\tau+d}\bigg)(c\tau+d)^{-1/2}\,\times begin{align*}2mm] \times\exp\bigg(\pi i\bigg(\lambda^{2}\tau+2\lambda z-\frac{c(z+\lambda\tau+\mu)^{2}}{c\tau+d}\bigg)\bigg)=\chi\bigg(\begin{matrix}a b c d\end{matrix}\bigg)\,\theta_{1,1}(\tau,z)
\partial_{\nu}S(\nu,\varepsilon) =\left( \partial_{\nu}\left[ \bar{K}(\nu,\varepsilon)-\bar{K}(\nu,0)\right] \left[ Z^{\perp}(\nu,\varepsilon)+\mathbf{I}\right] r_{0},r_{0}\right) & +\left( \left[ \bar{K}(\nu,\varepsilon)-\bar{K}(\nu,0)\right]\partial_{\nu}Z^{\perp}(\nu,\varepsilon)r_{0},r_{0}\right) .
\begin{cases}e^{i\theta_{k_l}}(w_{k_l},\phi_i)_{L^2(\Omega_{k_l})}+(\sigma_{ik_l}\nabla w_{ik_l},\nabla\phi_i)_{L^2(\Omega_{k_l})}=(t_{k_l},\phi_i)_{L^2(\Omega_{k_l})}, e^{i\theta_{k_l}}(w_{k_l},\phi_e)_{L^2(\Omega_{k_l})}-(\sigma_{ek_l}\nabla w_{ek_l},\nabla\phi_e)_{L^2(\Omega_{k_l})}=(t_{k_l},\phi_e)_{L^2(\Omega_{k_l})}, w_{k_l}=w_{k_l}-w_{k_l}. \end{cases}
S = -{\frac{1}{2 \pi}}\int d \tau d \sigma \left( ( U^{'})^{2}+ \frac{U^{4}L^{2}}{d_{3}e^{2}}\right)^{1 / 2}+{\frac{1}{2 \pi \alpha^{\prime}}}\int d \tau A_{0}\big \vert_{\sigma = 0}^{\pi}.
\phi_1^2 = -|\xi| \cos\theta \chi_{in}(\cos \theta - c \rho_{R}(r)) ( \rho_R \chi_{> 5R})'(r), \phi_2^2 = - \chi_{in}'(\cos \theta - c \rho_{R}(r)) \frac{|\xi|}{r}\sin^2 \theta ( \rho_R \chi_{> 5R})(r) , \phi_3^2 = c |\xi| \cos \theta \chi_{in}'(\cos \theta - c \rho_{R}(r)) (\rho_R \chi_{> 5R})(r) \rho_R'(r).
\Gamma\bigl(Y,\mathcal{C}(\mathcal{F})\bigr)=\varinjlim_{K\subset K_Y}\Bigl(\prod_{g\in G}\prod_{x\in X}\Gamma(Y,g^*i_{\overline{x}*}\mathcal{F}_{\overline{x}})\Bigr)^K =\varinjlim_{K\subset K_Y}\Bigl(\prod_{g\in \Omega_K}\prod_{x\in X}\Gamma(Y,g^*i_{\overline{x}*}\mathcal{F}_{\overline{x}})\Bigr),
\begin{array}{lll}g_0 := [C_0B_0,C_0A_0B_0,\dots,C_0A_0^{2T-2}B_0] \in \mathcal{M}_{p\times (2T-1)r}(\mathbb{R}) \bar{g}_0 := [C_0A_0^{2T-1}B_0,C_0A_0^{2T}B_0,\dots,C_0A_0^{N-1}B_0] \in \mathcal{M}_{p\times \bar{N}r}(\mathbb{R}) h := [C_0,C_0A_0,\dots,C_0A_0^{N-1}] \in \mathcal{M}_{p \times Nd_0}(\mathbb{R}).\end{array}
\int_{a_{k}}^{b_{k}}P_{n,k}(\tau)\,\tau^{s}\,d|\nu_{n,k}|(\tau) =0,s=0,\ldots,Z(n,k)-1, \int_{a_{k}}^{b_{k}}P_{n+1,k}(\tau)\,\tau^{s}\,d|\nu_{n+1,k}|(\tau) =0,s=0,\ldots,Z(n+1,k)-1,
I_1(x,t):=\dfrac{1}{(2\pi)^d\, t^{d/\ell}}\int\limits_{R_{\pi t^{1/L}}}(-1)^J \left(A(t^{-1/\ell}\xi)\right)^J e^{-tA(t^{-1/\ell}\xi)}\cdot e^{ix t^{-1/\ell}\xi}\, d\xi,
\langle u_T, \phi_p \rangle= \begin{cases}\dfrac{1}{n}\sum_{k=1}^n u_T(x_k) - \widetilde{G}_{n0}, p=0, \dfrac{\pi}{n}\sum_{k=1}^n u_T(x_k) \phi_p(x_k) - \widetilde{G}_{np}, 1 \le p \le n-1.\end{cases}
\sum_{y\neq 0,1}\chi_4(y)\chi_4(1-y)\sum_{x\neq 0,1,y}\overline{\chi_4}(x)\chi_4(x-y) =\sum_{y\neq 0,1}\chi_4(y)\chi_4(1-y)\left[-1-\chi_4(y-1) \right] &=-\rho-\sum_y \chi_4(y)\varphi(1-y)=-2\rho,
T= \left(\begin{tabular}{cc}$ k_{0}k$ $ \frac{k_{1}x}{k-x}$ $k_{0}$ $ \frac{k_{1}}{k-x}$\end{tabular}\right) ,\T= \left(\begin{tabular}{cc}$ k_{1}$ $ k_{0}x$ $0$ $ k_{0}$\end{tabular}\right)
L_f(\varphi_1\vee \varphi_2)+L_g(\varphi_1\vee \varphi_2)+\varepsilon =\frac12\left(L_{f+g}(\varphi_1\vee\varphi_2)+L_{f-g}(\varphi_1\vee\varphi_2)\right)+\varepsilon &\leq\frac12\left(\mathcal{E}_L(f+g)+\mathcal{E}_L(f-g)\right)+\varepsilon
\tilde{\mu}(\omega)=\frac{\omega^2 \left( C_1 - \eta_{1}C_2 / \mu_{1}\right)}{C_2^2+ \omega^2 C_1^2}+ i\ \frac{\omega \left( C_2 + \eta_{1}\omega^2 C_1 / \mu_{1}\right)}{C_2^2+ \omega^2 C_1^2}\ ,
\partial^{\nu}\!\left( \left(f_j-T^{\ell,x_0}[f_j]\right)-\left(f_k-T^{\ell,x_0}[f_k]\right)\right)\!(x) = \partial^{\nu}\!\left(g_{j,k}- T^{\ell,x_0}[g_{j,k}]\right)\!(x) &=\partial^{\nu}g_{j,k}(x) - T^{\ell-|\nu|,x_0}[\partial^{\nu}g_{j,k}](x).
(f_{h\cdot q}, \upsilon) = (f_{h\cdot q}, \mathfrak{J}_L \mathfrak{J}_L \upsilon) = (L_h W_q, \mathfrak{J}_L \mathfrak{J}_L \upsilon)= (\mathfrak{J}_L L_h W_q, \mathfrak{J}_L \upsilon) = (\mathfrak{J}_L W_q , L_{h^{-1}}\mathfrak{J}_L \upsilon)= \sum_{n \in \mathbb{N}}(\mathfrak{J}_L W_q ,e_n)(e_n , L_{h^{-1}}\mathfrak{J}_L \upsilon)
{\sf M}\left\{\left(I_{T,t}^{(i_1 i_2)}-I_{T,t}^{(i_1 i_2)q}\right)^{2n}\right\}\le C_{n,2}\left(\frac{3(T-t)^{2}}{2\pi^2}\left(\frac{\pi^2}{6}-\sum\limits_{r=1}^q\frac{1}{r^2}\right)\right)^n\ \to 0\ \ \ \hbox{if}\q\to\infty,
\hskip-5mm\left[S_{2,a}(u)(x')\right]^2=\iint_{\Gamma_a(x')}|\nabla u|^{2/p}| u|^{1-2/p}|\nabla u|^{2/p'}| u|^{1-2/p'}x_0\,dx'\,dx_0 &\le \left(\iint_{\Gamma_a(x')}|\nabla u|^2|\u|^{p-2}x_0\,dx'\,dx_0\right)^{1/p}\left(\iint_{\Gamma_a(x')}|\nabla u|^2| u|^{p'-2}x_0\,dx'\,dx_0\right)^{1/p'}\le S_{p,a}(u)(x')S_{p',a}(u)(x').
I=(1,\dots,r_1,i_{r_1+1},\dots,i_{l(I)}),\r_1<n<i_{l(I)}<\dots <i_{r_1+1}\le 2n, J=(1,\dots,r_2,j_{r_2+1},\dots,j_{l(J)}),\r_2<n<j_{l(J)}<\dots <j_{r_2+1}\le 2n
\mathcal{J}(y(v),v) = \frac{1}{2}\|y-\eta\|^2 + \int_\Omega\left(y-\eta\right)\left(\eta-y_d\right)d\boldsymbol{x}+ \frac{1}{2}\|\eta-y_d\|^2 + \frac{\lambda}{2}\|v-u_d\|^2
\mathcal{E}(\mathcal{B}_{d, k}) = \sum_{j = 1}^{k-1}(d-1)^{k-1-j}\Psi(E_{j+1}(x, d-1)) - \sum_{j = 1}^{k-1}(d-1)^{k-1-j}\Psi(E_j(x, d-1)) = \sum_{j = 1}^{k-1}(d-1)^{k-1-j}\left( \Psi(E_{j+1}(x, d-1)) - \Psi(E_j(x, d-1)) \right) .
p^{\mu}A_{\mu}=\eta_{\mu\nu}p^{\mu}A^{\nu}=\eta_{\mu\nu}p^{\mu}\frac{d}{d\tau}\frac{p^{\nu}}{m}=\frac{1}{2m}\frac{d}{d\tau}p\cdot p=\frac{1}{2m}\frac{d}{d\tau}(-m^{2}c^{2})=0.
\int_{[0,1] \times [-m,m]^{|SP(\pi)|}\times[-\lambda,1]^{|DP(\pi)|}}\prod_{s=1}^{r}\chi_{[0,\lambda]}\left(x_0+ \sum_{i=1}^{2s-1}\epsilon_\pi(i) x_{\pi(i)}\right)\chi_{[0,1]}\left( x_0+ \sum_{i=1}^{2s}\epsilon_\pi(i) x_{\pi(i)}\right) \prod_{l=0}^{r}\mathrm{~d}x_l,
\frac{8\pi^2}{e^2}\left(\sum_A \left(\frac{1}{y_A}d\vec y_A^2 + y_A\,(D\psi_A)^2\right)+\frac{\left(\,\sum_A d\vec y_A\,\right)^2}{|\vec y_N+2\pi \vec \zeta|}-\frac{\left(\,\sum_A y_A\,D\psi_A\,\right)^2}{|\vec y_N+2\pi \vec \zeta|+ \sum_A |\vec y_A|}\right),
\left(\delta g\right)_{\alpha\alpha'\beta_1\cdots\beta_{p+1}}= g_{\alpha'\beta_1\cdots\beta_{p+1}}g_{\alpha\beta_1\cdots\beta_{p+1}}^{-1}g_{\alpha\alpha'\beta_2\cdots\beta_{p+1}}\cdots g_{\alpha\alpha'\beta_1\cdots\beta_p}^{(-1)^p}= 1
(z(x+z)^{n+l-1},(x+z)^{n+l})+ c(z(x+z)^{l-1},(x+z)^l)) + ((x+z)^{l-1}, (x+z)^{l-1}) = (z(x+z)^{n+l-1},(x+z)^{n+l})+((cz+1)(x+z)^{l-1},(1+x+z)(x+z)^{l-1})) &= (z(x+z)^{n+l-1},(x+z)^{n+l})+((x+z)^{l-1},(x+z)^{l-1}))\in \textrm{Diag}(A)
\frac{1}{2}\frac{d}{dt}\|\nabla u\|_{L^{2}}^{2}+\nu\|\mathbb{P}\Delta u\|_{L^{2}}^{2}=\int_{\Omega}u\cdot\nabla u\mathbb{P}\Delta udx-\int_{\Omega}\theta e_{2}\mathbb{P}\Delta udx.
\widehat{c}_e(k,l;s)=c_e(k,l;s|\sqrt{q/t}a,\sqrt{q/t}c|q), &\widehat{c}_o(i,j; s)=c_o(i,j;s|\sqrt{q/t}a,\sqrt{q/t}b,\sqrt{q/t}c,\sqrt{q/t}d|q).
[{\hat M}^{\mu\nu},{\hat J}^{\rho}(x)]= -\imath\{(x^{\mu}\partial^{\nu}-x^{\nu}\partial^{\mu}){\hat J}^{\rho}(x)+g^{\mu\rho}{\hat J}^{\nu}(x)-g^{\nu\rho}{\hat J}^{\mu}(x)\}
v|S|+\circ(1) = \int_{S}v_j^+= (a_j)^\lambda \int_{S}\Big( \big(1+ \frac{\eta_j^+}{a_j}\big)^\lambda -1\Big) \le C (a_j)^\lambda \int_{S}\Big( \big( \frac{\eta_j^+}{a_j}\big)^\lambda + \frac{\eta_j^+}{a_j}\Big)= C \int_{S}\Big( (\eta_j^+)^\lambda +(a_j)^{\lambda-1}\eta_j^+\Big).
L(\xi_1, \cdots, \xi_K, \lambda_1, \cdots, \lambda_K, \nu)= -\sum \limits_{k=1}^K \ln (b_k \xi_k + d_k) - \sum \limits_{k=1}^K \lambda_k \xi_k + \nu \left( \sum \limits_{k=1}^K\xi_k - 1 \right).
v(-\infty) =\int^{w_0}_{-\infty}\int^{w_0}_y\frac{2}{\mathfrak{s}^2(z)}\exp\left(2\int^z_y\frac{\mathfrak{b}(x)}{\mathfrak{s}^2(x)}dx\right)dzdy\geq \int^{a}_{-\infty}\int^{a}_y\frac{2}{\mathfrak{s}^2(z)}dzdy &=\int^{a}_{-\infty}\int^{a}_y\frac{4R}{(c-rz)^2}dzdy=\infty.
\int\limits_{\lbrack0,2\pi)}{\displaystyle\sum\limits_{k\in\mathbb{S}}}a_{k}(t)\Psi_{k,t}dt=\sum_{k\in\mathbb{S}}\int\limits_{I(\delta)}a_{k}(t)\Psi_{k,t}dt+{\displaystyle\sum\nolimits_{j=1}^{m}}\int\limits_{(t_{j}-\delta,t_{j}+\delta)}{\displaystyle\sum\limits_{k\in\mathbb{S}}}a_{k}(t)\Psi_{k,t}dt.
\sum_{i_0=0}^\infty \sum_{i_1\geq i_0}^\infty \cdots \sum_{i_{d}\geq i_{d-1}}^\infty 2^{-(i_0+i_1+\cdots +i_{d})}\sum_{s=0}^k |\Delta(s,i_0,\dots,i_{d})| q^{-s(d-k)}\lesssim 1.
\widetilde Y_{\tau}^{t,x}=\int_\tau^Te^{-s{A}}G B(s,\Xi^{t,x}_s)\,ds+\int_\tau^Te^{-s{A}}Z_s^{t,x}\, B(s,\Xi^{t,x}_s)\,ds - \int_\tau^Te^{-s{A}}Z^{t,x}_{s}\;d W_s = \widetilde Y_{0}^{t,x}- \int_0^\tau e^{-s{A}}G B(s,\Xi^{t,x}_s)\,ds - \int_0^\tau e^{-s{A}}Z_s^{t,x}\, B(s,\Xi^{t,x}_s)\,ds + \int_0^\tau e^{-s{A}}Z^{t,x}_{s}\;d W_s
H_{d}(z)=\frac{(z-\xi_{1}^{\prime})(z-\xi_{2}^{\prime})\cdot\cdot\cdot(z-\xi_{N}^{\prime})}{(z-p_{1}^{\prime})(z-p_{2}^{\prime})\cdot\cdot\cdot(z-p_{N}^{\prime})}
\int_0^{2\pi}\mathrm e^{-\mathrm ipr\cos(\omega- \theta)}\mathrm e^{\mathrm im\theta}\mathrm d\theta= 2\pi\mathrm i^m J_m(-pr)\mathrm e^{\mathrm im\omega}= 2\pi(-\mathrm i)^mJ_m(pr)\mathrm e^{\mathrm im\omega}.
\vert \tau_{j+k}(y)- \tau_j(y)\vert \leq \sum_{i=1}^k \vert \tau_{j+i}(y)-\tau_{j+i-1}(y) \vert \leq \frac 5 4 \left( 36C_1(m)+24 \right) \delta \sum_{i=1}^k r_{j+i-1}= \frac 5 4 \left( 36C_1(m)+24 \right) \delta r_j \sum_{i=0}^{k-1}4^{-i}\xrightarrow [j \to \infty ]{}0.
\sum_{(d,r)=1}\int_{0}^{Y/d^2}\psi(v)dv =\sum_{\substack{d\leq D (d,r)=1}}\int_{0}^{Y/d^2}\psi(v)dv+\sum_{\substack{d> D (d,r)=1}}\int_{0}^{Y/d^2}\psi(v)dv &=\sum_{\substack{d> D (d,r)=1}}\int_{0}^{Y/d^2}\psi(v)dv+O(D).
\sum_Pe_{HK}(J, T/P)e(\underline{x}, T_P)=\sum_Pe_{HK}(J, T/P)e_{HK}(\underline{x}, T_P)=\lim_{e\to\infty}\frac{1}{p^{en}}e_{HK}(J^{[p^e]}+\underline{x}, T).
\sum_{l=0}^{n}q^{l}C^\prime_lC^\prime_{n-l}=\sum_{l=0}^{n}q^{l}\sum_{\tau_0+\tau_1+\cdots\tau_n=l}q^{\sum_{i= 0}^{n}i\tau_i}P_1(\tau_0,\tau_1,\cdots,\tau_n)\sum_{\mu_0+\mu_1+\cdots\mu_n=n-l}q^{-\sum_{i= 0}^{n}i\mu_i}P_1(\mu_0,\mu_1,\cdots,\mu_n).
\frac{x_{b}\left( t\right)}{x\left( t\right)}=\frac{1+\int_{0}^{t}ds\lambda_{b}\left( s\right) e^{\Lambda \left( s\right)}}{1+\int_{0}^{t}ds\left( \lambda_{b}\left( s\right) -\lambda_{d}\left(s\right) \right) e^{\Lambda \left( s\right)}}\rightarrow \frac{\lambda_{b}^{-}}{\lambda_{b}^{-}-\lambda_{d}^{-}}
(\eta_1,\dots, \eta_n)=(\xi'_1,\dots, \xi'_n) \begin{pmatrix}X^{-1}\cdots 0 0 \cdots 0 \vdots \ddots \vdots \vdots \ddots \vdots 0 \cdots X 0 \cdots 0 0 \cdots X 1 \cdots 0 \vdots \ddots \vdots \vdots \ddots \vdots 0 \cdots 0 0 \cdots 1 \end{pmatrix}= (\xi'_1,\dots, \xi'_n) e^{-i}h^{i}(X).
V^{\alpha}= \int d \xi^{a}\; \partial_{a}X^{\alpha}\; e^{i p \cdot X}\; , \ \ \V^{m}={\frac{1}{2 \pi \alpha^{\prime}}}\int d \xi_{a}\; \epsilon^{a b}\partial_{b}X^{m}\; e^{i p \cdot X}\; ,
\mathrm{E}_{0}= \langle \mathrm{g r o u n d}; \mathrm{e x}| \hat{\mathrm{H}}_{\mathrm{p h y s}}| \mathrm{g r o u n d}; \mathrm{e x}\rangle ={\hbar}^{2}\frac{q^{2}}{q^{2}}{L}\sum_{p > 0}\frac{1}{\mathrm{E}_{p}^{2}( \mathrm{N})}j_{p}^{\mathrm{e x}}( j_{p}^{\mathrm{e x}})^{\star}.
\delta H_B^{(m)}=\frac{mc_m\kappa}{2^{m-1}}\;\oint_Bd^{D-2}\!x\left(\sqrt{\tilde{h}}\tilde{\delta}_{d_1\cdots c_md_m}^{b_1\cdots a_mb_m}\;\delta\tilde{h}_{b_1}{}^{d_1}\cdots \tilde{R}_{a_mb_m}{}^{c_md_m}\right)\ .
\int_{0}^{\infty}x^{\lambda}K_{\mu}(ax) \sin(bx) dx = \frac{{2^{\lambda}b\, \Gamma \big(\frac{2+\mu+\lambda}{2}\big) \Gamma\big(\frac{2+\lambda-\mu}{2}\big)}}{a^{2+\lambda}}& \;\;\;\; \times{}_{2}F_{1}\Big(\frac{2+\mu+\lambda}{2}, \frac{2+\lambda-\mu}{2}; \frac{3}{2};-\frac{b^2}{a^2}\Big).
F+G \leq (\Omega_3+\Omega_2 + \langle M_{\Omega_1}\rangle)^{-1}[(\theta + \vert 1-\theta \vert) \vert \Omega_3+N_{\Omega_1}\vert + \vert\Omega_2 - M_{\Omega_2}\vert+\vert N_{\Omega_2}\vert]:=\tilde{\mathcal{L}}.
\begin{array}{c}{{\theta = B^{- 1}, \quad G = - ( 2 \pi l_{s}^{2})^{2}B g^{- 1}B , \quad G_{s}= g_{s}\det ( 2 \pi l_{s}^{2}B g^{- 1})^{\frac{1}{2}}.}}\\ \end{array}
P_Y(z,w) = \sum_{P \in \mathfrak{P}}w^{Y(P)}\,z^{\deg P}= \prod_{P \in \mathfrak{I}}\frac{1+(w-1)z^{\deg P}}{1-z^{\deg P}}&=\exp\left(\sum_{k=1}^\infty \sum_{P \in \mathfrak{I}}\frac{z^{k\deg P}}{k}- \frac{((1-w)z^{\deg P})^k}{k}\right) = \exp\left(\sum_{k=1}^\infty \frac{I(z^k)\,(1-(1-w)^k)}{k}\right)