harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
{\bar{Y}}^{n}(t) ={\bar{X}}_{0}^{n}(0)+\sum_{k=0}^{\infty}(k-2){\bar{B}}_{k}^{n}(t),{\bar{X}}_{0}^{n}(t) ={\bar{Y}}^{n}(t)+{\bar{\eta}}^{n}(t),{\bar{X}}_{k}^{n}(t) ={\bar{X}}_{k}^{n}(0)-{\bar{B}}_{k}^{n}(t),k\in \mathbb{N},
\pi_m(x) = \pi_m\left(v_1+B(T_1)\cdot v_2+\cdots +B(T_k)\cdot v_k\right) &= \pi_m(v_1)+\pi_m(B(T_1)\cdot v_2)+\cdots +\pi_m(B(T_k)\cdot v_k) &=\pi_m(v_1)+\lambda^{T_1}\pi_m(v_2)+\cdots \lambda^{T_k}\pi_m(v_k).
N \simeq \left( \frac{M_{s}}{M_{P}}\right)^{2}\frac{( M_{s}^{3}\, A_{T})^{3 / 2}}{( 2 \pi )^{7 / 2}g_{s}^{3 / 2}\, V_{6}^{1 / 6}\, F}\frac{\Psi_{*}}{M_{s}}\, .
\langle( \psi)'(u_{n})-\psi)'(u), u_{n}-u\rangle = B_{n}+\int_{M}(u_{n}-u) (|u_{n}(x)|^{p-2}u_{n}- |u(x)|^{p-2}u)d\mu_g(x) +\int_{M}(u_{n}-u)(f(x,u_{n})-f(x,u))d\mu_g(x).
T r_{( s + 1 )}( e^{- a_{s}^{\dagger}a_{s}/ M^{2}}) - T r_{( s )}( e^{- a_{s}a_{s}^{\dagger}/ M^{2}}) = T r_{( s + 1 )}( e^{- a_{s}a_{s}^{\dagger}/ M^{2}}) - T r_{( s )}( e^{- a_{s}a_{s}^{\dagger}/ M^{2}}) = 1
U(x)=\mathcal{C}_{N,p}\,\big(1+|x|^{\frac{p}{p-1}}\big)^{\frac{p-N}{p}},\qquad\mathcal{C}_{N,p}=\left(\int_{\mathbb{R}^N}\big(1+|x|^{\frac{p}{p-1}}\big)^{-N}\,dx\right)^\frac{N\,p}{p-N}.
W_\nu^{2,s}(\Omega):=\big\{\zeta\in W^{2,s}(\Omega)\,|\,\nabla\zeta\cdot \nu=0\partial\Omega\big\}, &W_+^{1,p}(\Omega):=\big\{\zeta\in W^{1,p}(\Omega)\,|\,\zeta\geq 0\Omega\big\}, &W_-^{1,p}(\Omega):=\big\{\zeta\in W^{1,p}(\Omega)\,|\,\zeta\leq 0\Omega\big\}
\tilde{a}_i=\left(\frac{d}{dv}+\tilde{a}\right)^i\tilde{a},\tilde{b}_i=\frac{d^i\tilde{b}}{dv^i}+\sum_{m=0}^{i-1}\frac{d^{i-1-m}(\tilde{a}_m\tilde{b})}{dv^{i-1-m}}.
\lim_{s\to 0}\int_{\left\lbrace u>\varepsilon\right\rbrace}\frac{1}{u^p}(P_s u)^p \Big(\sup_{t>0}P_tf\Big)^p\:d\mu= \int_{\left\lbrace u>\varepsilon\right\rbrace}\Big(\sup_{t>0}P_tf\Big)^p\:d\mu.
\frac{\partial E_{f}}{\partial t}+\overline{u_{j}}\frac{\partial E_{f}}{\partial x_{j}}+\frac{1}{\rho}\frac{\partial\overline{u_{i}}\overline{p}}{\partial x_{i}}+\frac{\partial\overline{u_{i}}\tau_{ij}^{r}}{\partial x_{j}}-2\nu\frac{\partial\overline{u_{i}}\overline{S_{ij}}}{\partial x_{j}}=-\epsilon_{f}-\Pi
v_j^+(r) = c_j^+\cdot r^{\gamma_j^+}; v_j^-(r) = \begin{cases}c_j^- \cdot r^{\gamma_j^-}, \ \mu_j > -\frac{(n-2)^2}{4}; c_j^- \cdot r^{\gamma_j^-}\log r, \ \mu_j = -\frac{(n-2)^2}{4}. \end{cases}
G_\mu(X_1,X_2,X_3,Y_1) \ , H_{\mu,2}(X_1,X_2,X_3,Y_1,Y_2) = H_{\mu,2}^{(1)}(X_1,X_2,X_3,Y_1)Y_2 + H_{\mu,2}^{(0)}(X_1,X_2,X_3,Y_1) \ , H_{\mu,3}(X_1,X_2,X_3,Y_1,Y_3) = H_{\mu,3}^{(1)}(X_1,X_2,X_3,Y_1)Y_3 + H_{\mu,3}^{(0)}(X_1,X_2,X_3,Y_1)
\hat{m}_{(A_r,B_r,C_r)}(\xi,\eta,\zeta) = J(A_r,B_r,C_r) + \langle{\rm grad}\,J(A_r,B_r,C_r), (\xi,\eta,\zeta) \rangle_{(A_r,B_r,C_r)}&+\frac{1}{2}\langle{\rm Hess}\, J(A_r,B_r,C_r)[(\xi,\eta,\zeta)], (\xi,\eta,\zeta) \rangle_{(A_r,B_r,C_r)}.
\begin{array}{rcl}\dot{V}(x) = \frac{d}{dt}\left( - \ln (W(x) + 1)\right) &= -\dfrac{1}{(W(x) + 1)}\dot{W}(x) &= -\dfrac{1}{(W(x) + 1)}(1 + W(x))\phi(x) &= - \phi(x) \end{array}
\bigl\langle(I\otimes J)W^*(I\otimes J)(v\otimes p),w\otimes q\bigr\rangle =\overline{\bigl\langle J(\omega_{v,w}\otimes\operatorname{id})(W)p,Jq\bigr\rangle}&=\bigl\langle (\omega_{v,w}\otimes\operatorname{id})(W)p,q\bigr\rangle=\bigl\langle W(v\otimes p),w\otimes q\bigr\rangle, \forall v,w\in{\mathcal H}, \forall p,q\in{\mathcal H}_{\varphi}.
\int_{\alpha}^{\beta}\varphi(t)dt-(\beta-a)\varphi(\alpha) =\sum_{n\geq1}\left( \int_{\alpha}^{\beta}\varphi_{n}(t)dt-(\beta-a)\varphi_{n}(\alpha)\right) & =\sum_{n\geq1}\int_{\alpha}^{\beta}\varphi_{n}(t)dt-(\beta-a)\sum_{n\geq1}\varphi_{n}(\alpha),
G{}_{T}^{*}\left(\omega\right)=g_{T}\left(s_{0}\right)^{2}\geq\frac{1}{n}\left(g_{T}\left(s_{1}\right)^{2}+\sum_{s_{i}\in\left\{s_{1},\dots,s_{n}\right\}\setminus\left\{s_{1}\right\}}\frac{1}{2T^{2\left(\alpha_{i}-\alpha_{1}\right)}}g_{T}\left(s_{i}\right)^{2}\right),
\left| \sum_{k=0}^{N-1}\varphi (2 \pi x)^k \varphi (2 \pi y)^{N-k-1}\right| \le \frac{2}{\left| \varphi (2 \pi x) - \varphi (2 \pi y) \right|}\ll \frac{1}{\| x-y \|}.
\sum_{k = 1}^\infty{\frac{{\zeta (2k)}}{{k(2k + 1)}}}= -1 + \ln (2\pi ), \sum_{k = 1}^\infty{\frac{{\zeta (2k)}}{{k(2k + 3)}}}=- \frac{1}{9}+ \frac{\ln (2\pi )}{3}+ \frac{\zeta (3)}{2\pi^2}.
\int_{B}|f(y)-f_{B}|^{p}d\mu(y) =\int_{0}^{+\infty}p\lambda^{p-1}\mu\left(\left\{z\in B:\left\vert f(z)-f_{B}\right\vert >\lambda\right\}\right)d\lambda & \leq2\int_{0}^{+\infty}p\lambda^{p-1}e^{-b_{n}\lambda/[f]_{n}}\mu(B)d\lambda\, & =\mu(B)[f]_{n}^{p}2p\int_{0}^{+\infty}t^{p-1}e^{-b_{n}t}dt,
f_{3}( r ) = 2 \sqrt{\frac{\bar{s}^{2}- Q c}{Q c}}e^{- \tilde{c}\sqrt{\bar{s}^{2}- Q c}}\frac{\left( 1 - \frac{r_{0}^{2}}{r^{2}}\right)^{( 1 + \sqrt{( \bar{s}^{2}- Q c ) / D_{2}}\ ) / 2}}{1 - e^{- 2 \tilde{c}\sqrt{\bar{s}^{2}- Q c}}\left( 1 - \frac{r_{0}^{2}}{r^{2}}\right)^{\sqrt{( \bar{s}^{2}- Q c ) / D_{2}}}}.
\mathcal{C}_{1,j,k}^{J}=\frac{1+2x_{1}}{2x_{1}}\left[ \frac{1}{2+2x_{1}}\mathcal{A}_{1,j}^{J\setminus k}+\tilde{\mathcal{D}}_{1,j}^{J\setminus k}\right]{\mathcal{A}}_{1,k}^{J}+\frac{1+2x_{1}}{2x_{1}}\left[\frac{1}{2x_{1}}\mathcal{A}_{1,k}^{J\setminus j}+\tilde{\mathcal{D}}_{1,k}^{J\setminus j}\right]{\mathcal{A}}_{1,j}^{J}\,.
y(t)=\sqrt{jk}.e^{j2\pi.f_{0}t}.e^{-j\pi.t^{2}k}=\sqrt{\frac{jB}{T}}.e^{j2\pi.f_{0}t}.e^{-j\pi.t^{2}.k}=\sqrt{\frac{jB}{T}}.e^{j2\pi(f_{0}t-t^{2}k/2)}
\hspace{-1.5cm}\Psi_{int}\left( x^{m}\right) =\widetilde{U}_{int}\left( x^{\mu},x^{m}\right) \bar{\Psi}_{int}\left( x^{m}\right) =\widetilde{U}_{int}\left( x^{\mu},x^{m}\right) \bar{U}_{int}\,\left( x^{m}\right)n_{int}^{1/2}\left( x^{m}\right) \eta_{int}
T_{3}(x,y,z) =(z_{1}+z_{2})\left(x_{1}y_{1}+x_{1}y_{2}+x_{2}y_{1}-x_{2}y_{2}\right) & +(z_{1}-z_{2})\left( x_{3}y_{3}+x_{3}y_{4}+x_{4}y_{3}-x_{4}y_{4}\right) ,
{\bf E\,}\left( \sqrt{n}\int_{\overline{C}_{n}}\{|f_{n}(x)-{\bf E\,}f_{n}(x)|-{\bf \,E\,}|f_{n}(x)-{\bf E\,}f_{n}(x)|\}\,dx\right)^{2}\leq d_{n}\stackrel{\rm def}{=}d(n,\overline{C}_{n}),
M_x[v_{\delta}^3]-M_{\varphi, x}[v_{\delta}^3]=(M_x[\overline{v}^3]-M_{\varphi, x}[\overline{v}^3])+M_x[v_{\delta}^3-\overline{v}^3]-M_{\varphi, x}[v_{\delta}^3-\overline{v}^3],
\langle \varphi | \chi \rangle_{{\cal L}}^{l}= ( 2 l - 1 ) \int_{| z | < 1}\frac{d z d \bar{z}}{2 \pi i}( 1 - z \bar{z})^{2 l - 2}\overline{{\varphi ( \bar{z})}}\chi ( \bar{z})
\Big(\Big\{{\cal X}_t:~t=0,\ldots,n\Big\}, \Big\{{\cal Y}_t:~t=0,\ldots,n\Big\},{\cal C}_{0,n}\triangleq \Big\{{\bf P}_{Y_t|Y^{t-1},X^t}:~t=0,\ldots,n\Big\}, &\qquad\qquad{\cal P}_{0,n}\triangleq\Big\{{\bf P}_{X_t|X^{t-1}, Y^{t-1}}:~t=0, \ldots, n\Big\}\Big)
\mathbb{E}_0\left[ U_h^{(13)}U_h^{(23)}U_h^{(14)}U_h^{(24)}\right] ={n \choose k}^{-4}\sum_{\substack{\mathbf{i}^\omega \in\mathcal{P}(n, k) 1 \leq \omega \leq 4}}\mathbb{E}_0\left[B({\bf i}^1, \dots,{\bf i}^4) \right].
\int_{\mathbb{R}^n}{}\frac{v(y)^q}{|x-y|^{n-\alpha}|y|^{\sigma_1}}\,dy ={}\left( \int_{B_{R}(0)}+ \int_{B_{|x|/2}(x)}+ \int_{B_{R}(0)^{C}\backslash B_{|x|/2}(x)}\right) \frac{v(y)^q}{|x-y|^{n-\alpha}|y|^{\sigma_1}}\,dy :={}I_1 + I_2 + I_3,
\Gamma_{l , s}^{\alpha , \alpha_{0}}( \vec{p}) = \Gamma_{l , s}^{\alpha , \alpha_{0}}( 0 ) + \sum_{j = 1}^{3}\sum_{\mu = 0}^{3}p_{\mu , j}\int_{0}^{1}d \lambda \left( \partial_{\mu , j}\Gamma_{l , s}^{\alpha , \alpha_{0}}\right) ( \lambda \vec{p}) \quad ,
\Phi=\,{-1\over2i\pi}\,G^0_1G^0_2G^0_3\,\beta_1\beta_2\beta_3\,\left[\,V^0_{12}\,\psi_{12}\,+\,V^0_{23}\,\psi_{23}\,+\,V^0_{31}\,\psi_{31}\,\right]
\int L^{- 1}( \tau , \tau^{\prime}) j ( \tau^{\prime}) d \tau d \tau^{\prime}= \frac{e^{\frac{g e_{0}F}{2}}}{\cosh \frac{g e_{0}F}{2}}\int_{0}^{1}e^{- g e_{0}F \tau^{\prime}}j ( \tau^{\prime}) d \tau^{\prime}.
P ( q , q^{\prime}) = \sum_{k = 0 , k^{\prime}= 0}^{\infty}\frac{1}{k ! k^{\prime}!}q^{n_{1}}. . . q^{n_{k^{\prime}}}q^{\prime}{}^{m_{1}}. . . q^{\prime}{}^{m_{k}}P_{\{m_{1}. . . m_{k}| n_{1}. . . n_{k^{\prime}}\}}( \partial_{l}) .
T_{t_{n+1}}T_{\sigma_{n}}T_{t_{n+1}}T_{\sigma_{n}}= - T_{t_{n+1}}T_{\sigma_{n}}T_{t_{n+1}}-T_{\sigma_{n}}T_{t_{n+1}}T_{\sigma_{n}}- T_{t_{n+1}}T_{\sigma_{n}}-T_{\sigma_{n}}T_{t_{n+1}}- T_{t_{n+1}}- T_{\sigma_{n}}-1
\partial_{+}( \Gamma^{( 1 ) - 1}\partial_{-}\Gamma^{( 1 )}) = - \Gamma^{( 1 ) - 1}\Gamma^{( 2 )}, \qquad \partial_{+}( \Gamma^{( 2 ) - 1}\partial_{-}\Gamma^{( 2 )}) = \Gamma^{( 1 ) - 1}\Gamma^{( 2 )},
(P_{1}) =\frac{4(n-2)}{n(n-1)}Q_{1}+\frac{2}{n(n-1)}Q_{2}, (P_{2}) =\frac{4(m-2)}{m(m-1)}Q_{1}+\frac{2}{m(m-1)}Q_{2}, (P_{3}) =\frac{4(n+m-2)}{nm}Q_{1}+\frac{4}{nm}Q_{2}.
\begin{aligned}x = \frac{2}{3}y + \frac 13 \cdot 0, y = \frac{\rho v}{\rho v + 3}x + \frac{1}{\rho v + 3}z + \frac{2}{\rho v + 3}\cdot 0, z = \frac{2\rho v}{2\rho v + 1}y + \frac{1}{2\rho v +1}\cdot 1. \end{aligned}
\mu ( k \eta , q ) \rightarrow \sqrt{\frac{\eta_{c}}{q ( \eta + 2 \eta_{c})}}( c_{-}e^{\frac{i}{2}q \frac{( \eta + 2 \eta_{c})^{2}}{\eta_{c}}}+ c_{+}e^{- \frac{i}{2}q \frac{( \eta + 2 \eta_{c})^{2}}{\eta_{c}}})
\frac{1}{n!}\frac{d^n}{dx^n}(G\circ F)(0) = C_{f,g}(n)= \sum_{k_1+2k_2+\cdots+nk_n=n}\binom{k_1+\cdots+k_n}{k_1,\ldots,k_n}g(k_1+\cdots+ k_n)\prod_{i=1}^n f(i)^{k_i},
\Phi^{a}( X ) = \Phi^{a}( \tilde{X}) + \partial_{\mu}\Phi^{a}( \tilde{X}) \xi^{\mu}+ \frac{1}{2}\partial_{\mu}\partial_{\nu}\Phi^{a}( \tilde{X}) \xi^{\mu}\xi^{\nu}+ o ( \xi^{3}) .
\begin{aligned}P_{\infty}\circ\widetilde{\phi}_{2}\circ (P_{\infty})^{-1}:H^{\bullet}(\overline{Z_{2}}, \overline{F})^{-}\longrightarrow H^{\bullet}(Z_{2}, Y, F)\otimes 1_{\mathbb{C}^{-}}.\end{aligned}
(q^*\alpha_0 +z(\theta)) \! \wedge \! (q^*d\alpha_0 + d(z(\theta) \! ))^{n}=(-1)^{\tfrac{s(s-1)}{2}}\! \tfrac{n!}{(n-s)!}q^*\alpha_0 \wedge \! (q^*d\alpha_0 + z(d\theta))^{n-s}\! \! \wedge \! \Omega \! \wedge [dz].
D_i H(x,\xi,t) = -(2 t A)^{-1}\left\langle e^{tS}x-\xi,e^{tS}e_i\right\rangle H(x,\xi,t), D_j D_i H(x,\xi,t) = \left(-\left(2 t A\right)^{-1}\delta_{ij}+\left(2 t A\right)^{-2}\left\langle e^{tS}x-\xi,e^{tS}e_i\right\rangle\left\langle e^{tS}x-\xi,e^{tS}e_j\right\rangle\right) \cdot H(x,\xi,t)
s_2 s_4 s_5 s_3 (s_{\alpha_7+\alpha_8}s_{\alpha_1 + \alpha_2 + \alpha_3 + 2\alpha_4 + \alpha_5 + \alpha_6 + \alpha_7}) s_3 s_5 s_4 s_2 = s_{\alpha_7+\alpha_8}s_{\alpha_1 + 2\alpha_2 + 2\alpha_3 + 3\alpha_4 + 2\alpha_5 + \alpha_6 + \alpha_7}.
\hat{\hat{S}}= \int d^{1 1}x \, s q r t{| \hat{\hat{g}}|}\ \left[ \hat{\hat{R}}-{\textstyle \frac{1}{2 \cdot 4 !}}\hat{\hat{G}}{}^{2}-{\textstyle \frac{1}{6^{4}}}\frac{1}{\sqrt{| \hat{\hat{g}}}|}\hat{\hat{\epsilon}}\partial \hat{\hat{C}}\partial \hat{\hat{C}}\hat{\hat{C}}\right] \, .
\Upsilon(\lambda;\mu,a_2) = \begin{pmatrix}\lambda_4 -\big(\mu_4 - 3 a_2^2\big) \lambda_6 -\big(\mu_6 - 2 a_2 \mu_4 + 2a_2^3 \big) \lambda_8 -\big({-}2a_2 \mu_6 + a_2^2 \mu_4\big) \lambda_{10}- a_2^2 \mu_6 \end{pmatrix}.
\left(\frac{x+1}{x+2}-\frac{x}{x+1}\right)\sum_{i=0}^k x_i\ln(x_i) \geq \left(\frac{x+1}{x+2}-\frac{x}{x+1}\right) (x+1)\ln\left(\frac{x+1}{x-1}\right) &= \frac{1}{x+2}\ln\left(\frac{x+1}{x-1}\right)
0 =d_{Y,0}^{i-1,j-1}s_{0}^{ij}+s_{0}^{i+1,j}d_{X,0}^{ij}, f_{0}=d_{Y,1}^{i-1,j-1}s_{0}^{ij}+ d_{Y,0}^{i-1,j}s_{1}^{i j}+s_{0}^{i+1,j+1}d_{X,1}^{ij}+s_{1}^{i+1,j}d_{X,0}^{ij}.
u := \ell(\gamma_2 \gamma_1 \gamma_2 \gamma_1 \gamma_1 \gamma_1 \gamma_2{\gamma_1}^k \gamma_2) w_1 := \ell(\gamma_2 \gamma_1 \gamma_1 \gamma_2 \gamma_1 \gamma_1 \gamma_2{\gamma_1}^k \gamma_2) w_2 := \ell(\gamma_2 \gamma_1 \gamma_1 \gamma_1 \gamma_2 \gamma_1 \gamma_2{\gamma_1}^k \gamma_2)
L^*\varphi = \sum\limits_{i=1}^u \lambda_i \mathbf{x_i^*}+ \sum\limits_{j=1}^{d-u}\mu_j \Xi(\mathbf{w_j})^* \omega = \sum\limits_{i=1}^u \lambda_i^\prime \mathbf{x_i^*}+ \sum\limits_{j=1}^{d-u}\mu_j^\prime \Xi(\mathbf{w_j})^*.
\sum_{m\in \mathbb{Z}}\mu(T_m(\phi_1)) x^m = \left(\frac{1+x^{-1}}{2}\right) \left(\sum_{U \in \mathbb{Q}_2^\times/(\mathbb{Q}_2^\times)^2}x^{\mathrm{ord}_3 c_2(\phi_{1,s})}\mu_2(U) \right) \left(\sum_{U \in \mathbb{Q}_7^\times/(\mathbb{Q}_7^\times)^2}x^{\mathrm{ord}_3 c_7(\phi_{1,s})}\mu_7(U)\right),
R_f = \int\limits_{0}^{R}\underset{t>0}\int\left(\underset{j \in \psi}\prod\frac{1}{1+\max\{e^t-1,T\}r^\alpha d_j^{-\alpha}}+\frac{1}{3}\underset{i \in \phi}\prod\frac{P[\eta(r)<T]}{1+\max\{e^t-1,T\}r^\alpha d_i^{-\alpha}}\right)\mathrm{d}tf_R(r)\mathrm{d}r.
\begin{array}{| c | | c | c | c | c |}\hline{{\mathrm{M a t r i x ~ o p e r a t o r s}}}&{{J_{\ell}^{+ i j}}}&{{J_{\ell}^{+ - i}}}&{{J_{\ell}^{i j k}}}&{{J_{\ell}^{- i j}}}\\ \hline{{\hline \mathrm{S U G R A ~ m o d e s}}}&{{v_{2}^{\ell}}}&{{u^{\ell}}}&{{\alpha^{\ell}}}&{{v_{1}^{\ell}}}\\ \hline{{}}\\ \end{array}
\mathcal{L}V_0(q,p) = \mathcal{L}(H + \psi)(q,p) = \mathcal{L}H(q,p) + \mathcal{A}\psi(q,p)+ (\mathcal{L}-\mathcal{A})\psi(q,p) &= -\gamma p^2 -\gamma T - 2\gamma T \frac{p^2 U''(q)}{(U'(q))^2}- 2\gamma^2 T \frac{p}{U'(q)}& \leq -\frac{3}{4}\gamma p^2 -\gamma T -2\gamma T\frac{p^2 U''(q)}{(U'(q))^2}+ \frac{4\gamma^3 T^2}{(U'(q))^2}
\sup_{I \in \mathcal{I}}\lim_{n \rightarrow \infty}\max_{\lambda \in I}n \mathbb{E}_{f_{n^{-1/2},g_\lambda}}\bigl[\bigl\{\hat{H}_n^w - H(f_{n^{-1/2},g_\lambda})\bigr\}^2\bigr] = V(f).
\left \{\phi_n\left(b_1 b_2 \ldots b_d\{\rm M}_{k,l}\right) \ \vert \b_1 \in \mathcal{B}_{{\rm TL}_{\mu_1}(u)}, b_i \in \mathcal{B}_{{\rm H}_{\mu_i}(u)}\mbox{for all}i= 2,...,d,1\leq k,l \leq m_\mu, \mu \in{\rm Comp}_d(n) \right\},
\begin{array}{r c l r c l}{e^{0}}&{=}&{\lambda^{1 / 2}/ R \left( d t + \omega_{\aleph}d \varphi \right) ,}&{e^{1}}&{=}&{\lambda^{- 1 / 2}R d r ,}\\{}&{}&{}&{}&{}&{}\\{e^{2}}&{=}&{R d \theta ,}&{e^{3}}&{=}&{R \Omega_{\aleph}d \varphi ,}\\ \end{array}
\left( \begin{array}{c}{{\partial_{\mu}\theta}}\\{{\sin \theta \hspace{1 m m}\partial_{\mu}\varphi}}\\ \end{array}\right) = \left( \begin{array}{c c}{{G_{\mu}}}&{{- H_{\mu}}}\\{{H_{\mu}}}&{{G_{\mu}}}\\ \end{array}\right) \left( \begin{array}{c}{{\beta}}\\{{- \gamma}}\\ \end{array}\right) ,
\Delta v+k_2^2v=0 \ \mbox{in}\ \mathbb{R}^2\setminus \overline{\Omega_2}, \Delta v+k_1^2v=0 \ \mbox{in}\{\Omega_2}, v^{-}=v^{+}, S_2\frac{\partial v^-}{\partial n}=S_1\frac{\partial v^+}{\partial n}\mbox{on}\partial \Omega \mbox{outgoing radiation condition with}k_2 \mbox{in}\mathbb{R}^2\setminus\overline{\Omega_2}.
N'=n_2^2 + \cdots + n_{k-1}^2 +n_i + \cdots + n_{k-1}+\sum_{j=2}^{k-2}\sum_{p=1}^{n_j -n_{j+1}}a^{(j)}_{p}+2 \sum_{p=1}^{n_{k-1}}b_p ,
\langle \Psi | \phi^{2}(x) \cdots \phi^{2}(x) \pi^{2}(x) ({\rm or}\;{\phi^{\prime}}^{2}(x))| \Psi \rangle =\langle \Psi | \phi^{2}(x)| \Psi \rangle \cdots \langle \Psi | \phi^{2}(x) | \Psi \rangle \langle \Psi |\pi^{2}(x) ({\rm or}\;{\phi^{\prime}}^{2}(x) )| \Psi\rangle.
\begin{aligned}\sum_{\epsilon_a-\epsilon_b\in\Phi^+_0}A_{b,a}(\epsilon_b-\epsilon_a)= \sum_{j=k+1}^{mn}{\sigma_j}{\beta_j}- \sum_{i=1}^{k}{\sigma_i}{\beta_i}+{\beta_k}=:(\nu^{(0)},\nu^{(1)})\in\mathbb Z^{m+n}.\end{aligned}
(J_{x_{0}}^{k}f)(z)=\sum_{i=0}^{k}\frac{f^{(i)}(x_{0})}{i!}z^{i}=f(x_{0})+f^{\prime}(x_{0})z+\cdot\cdot\cdot+\frac{f^{(k)}(x_{0})}{k!}z^{k}.
\tilde{A}^0 (x) = A^0(x) + \frac{1}{(2 \pi)^3}\int \frac{d^3 k}{2 k^0}[ q_0(\vec{k}) e^{-ikx}+ q_0^*(\vec{k}) e^{ikx}] = A^0(x) + q^0(x) = \tilde{A}^0 (x)^K \, ,
\mathcal E_0(\mathrm{id}, r_0, y_0) := \inf_{m_{\kappa_n}}\liminf_{n \to \infty}{E_{\kappa_n}(m_{\kappa_n}) - 8 \pi \over \kappa_n^2}\leq \lim_{n \to \infty}{E_{\kappa_n}(\phi_n) - 8 \pi \over \kappa_n^2}= -{2 \pi r_0^2 \over y_0^2}- 8 \pi r_0,
A_{\varepsilon}(t):=\left(\begin{array}{cc}0 1 a_{\varepsilon}(t) 0\end{array}\right), Q_{\varepsilon}(t):=\left(\begin{array}{cc}0 0 q_{\varepsilon}(t)-a_{\varepsilon}(t) 0 \end{array}\right), F_{\varepsilon}(t,\xi):=\left(\begin{array}{c}0 \widehat{f}_{\varepsilon}(t,\xi)\end{array}\right).
\frac{1}{2}y_{sB}\frac{\partial}{\partial y_{sB}}\psi_0=-\frac{1}{2}y_{sB}y_{sB}\psi_0=-\frac{1}{2}\sum_{sB}(y_{sB}^2-\frac{1}{2})\psi_0-\frac{1}{4}\cdot 2(d-1)\psi_0\;,
I = \int_{0}^{\infty}r d r \frac{\left( \omega^{\prime}- \omega \cosh 2 \Omega r \right)}{\left( 1 + r \omega \sinh 2 \Omega r \right)}J_{0}( q r ) J_{1}( q^{\prime}r ) \left[ \exp \left( 2 \pi i ( 1 + r \omega \sinh 2 \Omega r ) \right) - 1 \right] .
J =\frac{(2-\lambda)p-\sqrt{(2-\lambda)^{2}p^{2}+\frac{8\lambda(\lambda-1)}{n}p(p-1)}}{2\lambda p(p-1)}, V = \frac{pn+\sqrt{n^{2}p^{2}+8np(p-1)}}{4}\frac{\lambda}{\lambda-1},
p_{i,s}:= \pi_{u^{i,s}}\pi_{\theta_i^{s+1}}+ \Big( \sum_{\substack{t\geq1 j}}\frac{f^j}{f^i}\theta_j^{t+1}\frac{\partial}{\partial \theta_i^{s+1}}\frac{\partial}{\partial u^{j,t}}\int d u^{i,s}\Big) \pi_{\lambda -u^i}.
\Lambda:= \left\{\sum_{i=1}^{s}\beta_{i}B_{i}\mid0\leq\beta_{i}<1\right\}B_{i}:= (\log\left\vert \sigma_{1}(\varepsilon_{i})\right\vert,\ldots,\log\left\vert \sigma_{s}(\varepsilon_{i})\right\vert )^{t}
K_{X_{m},\chi_{\beta}}(x+\ell m,y;t) = \sum_{k\in \mathbb{Z}}e^{-2\pi i \beta k}e^{-t}I_{x-y+(k+\ell)m}(t) = \sum_{j\in \mathbb{Z}}e^{-2\pi i \beta (j-\ell)}e^{-t}I_{x-y+jm}(t) &= e^{2\pi i \beta\ell}K_{X_{m},\chi_{\beta}}(x,y;t).
\sum_{q\leq x^\theta}\max_{(l,q)=1}\left| \#\mathcal{P}_{L_i,\mathcal{A}}(x;q,l)-\frac{1}{\phi(q)}\int_{x}^{2x}\frac{g'(t)}{\log t}dt\right| \ll_A \frac{g(2x)+g(x)}{(\log N)^A}\,,
\begin{cases}D^{\alpha_1}_0y(t) = w(t), y(0) = y_0 = \frac{\gamma_1-b_1s}{a_1}D^{1-\alpha_1}_0w(t) = z(t), w(0) = 0 D^{\alpha_2-1}_0z(t) = f(t,y(t),w(t)), z(0) = s.\end{cases}
\tilde{\sigma_{5}}= \int{\frac{d^{d}k}{( 2 \pi )^{d}}}\gamma_{\alpha \beta}^{5}\sigma_{\alpha \beta}( k ) = - \tilde{G}_{5}= - \int{\frac{d^{d}k}{( 2 \pi )^{d}}}T r [ \gamma^{5}G^{( 2 )}( k ) ]
\tilde{R}_{L}^{( n )}( t , t^{\prime}; \{q \}) \approx \frac{\tilde{a}_{L}^{( n )}( \{q \}) + \tilde{\alpha}_{L}^{( n )}( \{q \}) ( \vartheta + \vartheta^{\prime})}{( z - z_{0}) ( z^{\prime}- z_{0})}- \frac{\tilde{b}_{L}^{( n )}( \{q \}) \vartheta \vartheta^{\prime}}{( z - z_{0}) ( z^{\prime}- z_{0})}\Biggl [ \frac{1}{z - z_{0}}- \frac{1}{z^{\prime}- z_{0}}\Biggl ] \, .
\mathbf{x}=\sum_{\mu =1}^8 \tilde{\sigma}_{\mu}x^{\mu}=\left( \begin{array}{@{\,}cc@{\,}}z_2 z_1 -\bar{z}_1 \bar{z}_2 \end{array}\right)\,,\quad \mathbf{x}'=\sum_{\mu =1}^8 \tilde{\sigma}'_{\mu}x^{\mu}=\left( \begin{array}{@{\,}cc@{\,}}z_4 z_3 -\bar{z}_3 \bar{z}_4 \end{array}\right)\,.
\begin{aligned}\,{\rm Area}(C,\omega)2^{-\zeta_{{\mathcal O}_{C}}(0)}\tau(C,\omega)2^{-\zeta_{\Sigma}(0)}\tau(C,\kappa\otimes\chi_{a,b};\omega)^{2}&=(4\pi e^{6(2\zeta_{\bf Q}'(-1)+\zeta_{\bf Q}(-1))})^{1-g}\|\theta_{a,b}(\varOmega(C))\|^{-4},\end{aligned}
w(b,\bar b)=C\oint \prod_{j=1}^{N+1}{dz_j\over z_j-z_{j+1}}\delta\left(b^\mu(\tau)-\sum_{j=1}^{N}k_j^\mu\delta(\tau-\tau_j)\right)\delta\left(\bar b^\mu(\tau)-\sum_{j=1}^{N}k_j^\mu\delta(\tau-\tau_j)\right)
\frac{Y}{2}= g_{s t r i n g}^{- 2}= S_{R}+ \frac{1}{8 \pi^{2}}\delta^{G S}\log ( 2 T_{R}) \simeq \frac{2}{3 ( k_{1}\tilde{\beta}_{1}^{- 1}- k_{2}\tilde{\beta}_{2}^{- 1})}\log \frac{\tilde{d}_{1}k_{1}\tilde{\beta}_{1}^{- 1}}{\tilde{d}_{2}k_{2}\tilde{\beta}_{2}^{- 1}}\ - \ \frac{0 . 1 9 5}{4 \pi^{2}}\ \delta^{G S}.
\begin{aligned}k=\frac{\lambda_2}{\lambda_4}\left(\lambda_2\lambda_4-\lambda_3\lambda_5\right)=\lambda_2^2-\frac{\lambda_2\lambda_3\lambda_5}{\lambda_4}<0 \ell=\frac{\lambda_4}{\lambda_2}\left(\lambda_2\lambda_4-\lambda_3\lambda_5\right)=\lambda_4^2-\frac{\lambda_3\lambda_4\lambda_5}{\lambda_2}<0.\end{aligned}
{d\rho(\omega)\over d \ln \omega}= \omega^4 \left[{S_{ex}(\omega)\over S_{re}(\omega)}+{S_{re}(\omega)\over S_{ex}(\omega)}\right]\simeq \omega^4 ~{\rm Max}\Biggl\{{S_{ex}\over S_{re}},{S_{re}\over S_{ex}}\Biggr\}.
\Phi^+_i(\alpha, \beta) ={n}_i(\zeta = \infty) + a_i \alpha + b_i \beta + \int_0^\infty\int_0^\infty (1-e^{-\alpha x -\beta y})n_i(\zeta \in dx, \epsilon_\zeta \in dy, J_\zeta = i),\alpha,\beta\geq 0,
C^{(3)}_{12}={}q^{-\frac{1}{3}}\frac{(1+y)^2}{1 + y + y^2}A^{(3)}_{\underline{\beta^1\beta^2}}(y) ={}q^{-\frac{1}{3}}\frac{(1+y)^2}{1 + y + y^2}\left( 1 + \frac{y^4 + 6y^3 + 6y^2 + 6y + 1}{(y + 1)^2y}\, q + \ldots \right)\,.
|\partial_{r}v_{5}(t,r)| \leq \frac{C}{\sqrt{r}}\int_{t}^{\infty}ds \int_{0}^{1}d\xi \sqrt{\xi}|\widehat{N_{2}(f_{v_{5}})}(s,\xi)| + \frac{C}{\sqrt{r}}\int_{t}^{\infty}ds ||\xi \widehat{N_{2}(f_{v_{5}})}(s)||_{L^{2}(\xi d\xi)}&\leq \frac{C \log^{3}(t)}{\sqrt{r}t^{3/2}}
\langle 0 | \phi_{>}( x ) \phi_{>}( y ) | 0 \rangle = \int_{\mathrm{s h e l l}}\!{\frac{d^{d - 1}k}{( 2 \pi )^{d - 1}}}\,{\frac{1}{2 \omega_{k}}}\exp [ i k \cdot ( x - y ) ] \quad .
r_{1}^{2}= \sqrt{Q_{2}^{2}+ \frac{r_{0}^{4}}{4}}- \frac{r_{0}^{2}}{2}\ , \r_{2}^{2}= \sqrt{Q^{2}+ \frac{r_{0}^{4}}{4}}- \frac{r_{0}^{2}}{2}.
-\log_{q}\frac{|x|}{R}=-\frac{\left(\frac{|x|}{R}\right)^{1-q}-1}{1-q}=\left(\frac{|x|}{R}\right)^{1-q}\frac{\left(\frac{R}{|x|}\right)^{1-q}-1}{1-q}=\left(\frac{|x|}{R}\right)^{1-q}\left(\log_{q}\frac{R}{|x|}\right)
\begin{multlined}[t][12.5cm]\left|\widetilde{\Pi}^{P}_{\nu_{G},6c+\nu_{1}+3\nu_{2}}\left((z_{0},z_{1},z_{2}),(z_{0},z_{1},z_{2})\right)\right| \leq \frac{9\sqrt{3}c}{2\pi^3}x^{\nu_{1}+\nu_{2}}y^{\nu_{2}}e^{-c\delta(z)}=O(c^{-\infty})\end{multlined}
H_{d}(z)=\frac{\frac{b_{0}K+b_{1}}{a_{0}K+a_{1}}+\frac{-b_{0}K+b_{1}}{a_{0}K+a_{1}}z^{-1}}{1+\frac{-a_{0}K+a_{1}}{a_{0}K+a_{1}}z^{-1}}.
\|P(\rho_j * P^\perp f)\|_{H^2}= \| P(\rho_j * P^\perp g) \|_2 \leq \int_{|y| \leq 1/4}\rho(y) \| P^\perp g (\cdot - \tfrac{1}{j}y) - P^\perp g\|_2 \, \mathrm{d}y \to 0.
{u_z}= \frac{1}{\rho}\left[{{f_0}+{f_1}+{f_2}+{f_3}+{f_4}+{f_7}+{f_8}+{f_9}+{f_{10}}+ 2\left({{f_5}+{f_{11}}+{f_{14}}+{f_{15}}+{f_{18}}}\right) + \frac{1}{2}{F_z}}\right] - 1,
\tilde{P}_{2q_n-1}(\theta)=\sum_{x_1\in I_0\cup I_{\ell}}\tilde{P}_{2q_n-1}(\theta_{x_1})\prod_{\substack{j\in I_0\cup I_{\ell}j\neq x_1}}\frac{\sin\pi(\theta-\theta_j)}{\sin\pi(\theta_{x_1}-\theta_j)}.
[{\cal Z}^I_A, \bar{\cal Z}^J_B]_{P.B.}= i \delta^{IJ}\delta_{AB}\,,\quad[{\cal Z}^I_A,{\cal Z}^J_B ]_{P.B.}= [\bar{\cal Z}^I_A, \bar{\cal Z}^J_B ]_{P.B.}= 0\,,
\gamma_j := \left\{q' = (q^{(k)})_{k \in J'}\in L;\,\begin{array}{ll}q^{(k)}=0 (k \in J' k \in J_{\prec j}\sqcup J_{\nparallel\, j}) \operatorname{Re}\langle q^{(k)}, p^{(k)}_* \rangle > 0 (k=j)\end{array}\right\}.
\Phi(n_{-}(c)) = \frac{\psi( c^{-1}s_\lambda)}{\mathrm{Vol}(U^{d(k,\lambda)}_{F,\mathfrak{p}})}\cdot \int_{U^{d(k,\lambda)}_{F,\mathfrak{p}}}\chi( y) \psi(- c^{-1}s_\lambda y)dy.
\mathrm{\Delta}H_{\Lambda}(\widehat{\gamma}_{\Lambda}|\widehat{\eta}):=\sum_{x\in \gamma_{\Lambda}}\sum_{y\in \eta_{\Lambda^{\mathrm{c}}}}\Phi (x,y)+\sum_{x\in \gamma_{\Lambda}}\sum_{y\in \eta_{\Lambda^{\mathrm{c}}}}W_{xy}(\sigma_{x},\xi_{y}).
\mu(s,\pi\times \tau , \psi) = \mu(s,\Pi_{v_0}\times \tau, \psi) = \gamma(s, \rho_{\Pi, v_0, l}\times \phi_{\tau}, \psi) \cdot \gamma(-s,\rho_{\Pi, v_0, l}^{\vee}\times \phi^{\vee}_{\tau}, \psi) \cdot \gamma(2s, r_2 \circ \phi_{\tau}, \psi) \cdot \gamma(-2s, r_2^{\vee}\circ \phi_{\tau}, \psi).
h_{\Lambda^{(1\,\,3)}_{k_1,k_2}}=(k+\frac{1}{2}) ((k_1+\frac{1}{2}) (k_1+2k_2+\frac{1}{2}) -\frac{1}{4})+ \frac{k_1}{2}= h_{\Lambda^{(4\,\,2)}_{k_1+1,k_2}},
\phi_{1}( x + y ) ={\frac{\biggl | \begin{matrix}{\phi_{2}( x )}&{\phi_{2}( y )}\\{\phi_{3}( x )}&{\phi_{3}( y )}\\ \end{matrix}\biggr |}{\biggl | \begin{matrix}{\phi_{4}( x )}&{\phi_{4}( y )}\\{\phi_{5}( x )}&{\phi_{5}( y )}\\ \end{matrix}\biggr |}}
\begin{gathered}X\begin{pmatrix}G_0 G_1\end{pmatrix}\begin{pmatrix}G_0^- G_1^-\end{pmatrix}=X\;, \begin{pmatrix}H_0^- H_1^-\end{pmatrix}\begin{pmatrix}H_0 H_1\end{pmatrix}Y=Y\;, \end{gathered}
(P(\Psi_t)P^T(\Psi_t)(x))^{-1}=\begin{bmatrix}I_n+O(t) O(t^2) O(t^2) 2t\cdot \begin{pmatrix}\begin{bmatrix}I_{\frac{n(n-1)}{2}}0 0 (3\cdot \Xi(\frac{1}{3}))^{-1}\end{bmatrix}+O(t)\end{pmatrix}\end{bmatrix}_.
\begin{array}{lcr}\sum\limits_{j}q^j_a(|X_j|^2-|Y_j|^2)\eta_j^+\overline{\eta}_j^+ =r^2_a\eta_a^+\overline{\eta}_a^+ \qquad (a) \sum\limits_{j}q^j_a(X_j\overline{Y}_j)\eta_j^+{\eta}_j^+ =0 \qquad (b) \sum\limits_{j}q^j_a(\overline{X}_j{Y_j})\overline{\eta}_j^+\overline{\eta}_j^+ =0. \qquad (c)\end{array}
\times \, \sum_{\ell = - \infty}^{+ \infty}\, e^{- i \frac{\pi}{2}| \ell |}\, e^{i \ell ( \varphi_{f}- \varphi_{i})}\, e^{- \frac{i}{2}\hbar \Delta t g^{2}\ell^{2}}\, J_{| \ell |}\left( \frac{\omega r_{f}\, r_{i}}{\hbar \sin \omega \Delta t}\right) \ \ \ ,
\Pi ( k_{0}) = e^{2}\,{\frac{\mathrm{s e c h}^{2}{\frac{\beta m}{2}}}{1 + \tanh^{2}{\frac{\beta m}{2}}\tan^{2}{\frac{\beta k_{0}}{2}}}}\,{\frac{\tan{\frac{\beta k_{0}}{2}}}{k_{0}}}
t_{s_h}^{2m+1}= \prod_{i=1}^m{}_{t_x^{2i-1}}(t_y t_a^{-1}t_{t_x(y)}t_a^{-1}) \cdot t_{t_x^{2m+1}(y)}t_a^{-1}\cdot t_x^{2m+1}t_b^{-2m-1}t_z^{2m+1}t_c^{-2m-1}.
\begin{aligned}\frac{1}{4C}\int |\nabla u|^2dx+\frac{1}{2}\int_0^t\int \rho |u_t|^2 dxd\tau & \le C\left(1+\|f\|_{L^2}^2+\int_0^t \|f_t\|_{L^2}^2 d\tau\right)+C\int_0^t \|\nabla u\|_{L^2}^2 d\tau +\varepsilon \int_0^t \|\nabla u\|_{H^1}^2 d\tau.\end{aligned}