harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\limsup_{n\rightarrow\infty}\widetilde{E}\big[\frac{E\prod_{i=0}^{n-1}F(S_i,\widehat{{S}}_i;S_{i+1},\widehat{S}_{i+1})}{(E\prod_{i=0}^{n-1}\frac{\lambda\rho(S_i)\rho(S_{i+1})}{1+\lambda\rho(S_i)\rho(S_{i+1})})^2}\big]<+\infty.
d^2h(y)(w|z) =\lim_{\nu\to \infty}\frac{2[h_{\overline{k}}(y+\tau_{\!\nu}w+\frac{1}{2}\tau_{\!\nu}^2 z^{\nu})-h_{\overline{k}}(y)-\tau_{\!\nu}dh_{\overline{k}}(y)(w)]}{\tau_{\nu}^2}=\langle\nabla h_{\overline{k}}(y),z\rangle+\langle w,\nabla^2h_{\overline{k}}(y)w\rangle.
\Psi_{b_{j}+t}(x)=e^{i\left\langle b_{j}+t,x\right\rangle}+\left(A(b_{j})\right) e^{i\left\langle b_{j}+t,x\right\rangle}+\left(A(b_{j})\right)^{2}e^{i\left\langle b_{j}+t,x\right\rangle}+...
\dim \delta H_3^g = \; 5 + 8 (k+2) +2(k+1)(k+2) + k(k+1)/2 &\;+{(k-1)k(k+1)}/{6}-{(k+4)(k+5)(k+6)}/{6}= \; 2\,k+5, \dim \delta E_3^g = \; I_M(V_3^g\times W_3^g)=\; 3,
\widetilde{\theta}_n(t) =\frac{4}{t^2}\left(\sin \frac{t\left(\ln n + \ln (n-1) \right)}{2}\right)^2 \left(\sin \frac{t\left(\ln n - \ln (n-1) \right)}{2}\right)^2 &\leq 2^{-2}\left(\ln n + \ln (n-1) \right)^2 \left(\ln n - \ln (n-1) \right)^2 t^2.
S S^{\star}= \frac{( 1 - \frac{1}{R^{2}}) ( 1 - \frac{1}{{R^{\star}}^{2}})}{( e^{i \pi \nu}- \frac{e^{- i \pi \nu}}{R^{2}}) ( e^{- i \pi \nu^{\star}}- \frac{e^{i \pi \nu^{\star}}}{{R^{\star}}^{2}})}
d \omega_{\phantom{i}i}^{a}\, = \, - \omega_{\phantom{c}c}^{a}\wedge \omega_{\phantom{i}i}^{c}- \omega_{\phantom{j}j}^{a}\wedge \omega_{\phantom{i}i}^{j}\longrightarrow R_{\phantom{i}i}^{a}= 0
\frac{1}{2}\frac{d}{d t}\sum_{i}\dot{x}^{i}\dot{x}^{i}= - \sum_{i}\dot{x}^{i}\dot{x}^{i}- \sum_{i}\dot{x}^{i}\left( \Gamma_{k l}^{i}( x ) \dot{x}^{k}\dot{x}^{l}+ X_{k}^{i}( x , \dot{x}) \dot{x}^{k}\right)
W=\frac{1}{\sqrt{4}}[\begin{matrix}\omega^{0}&\omega^{0}&\omega^{0}&\omega^{0}\\ \omega^{0}&\omega^{1}&\omega^{2}&\omega^{3}\\ \omega^{0}&\omega^{2}&\omega^{4}&\omega^{6}\\ \omega^{0}&\omega^{3}&\omega^{6}&\omega^{9}\\ \end{matrix}]=\frac{1}{\sqrt{4}}[\begin{matrix}1&1&1&1\\ 1&-i&-1&i\\ 1&-1&1&-1\\ 1&i&-1&-i\end{matrix}]
(Inf^{K\times G}_Ka) \times^d b =A(_{K\times H \times G^{\Delta}}K\times G\times H\times G_{K\times G\times H\times G})((Inf^{K\times G}_Ka) \times b) &=A((_{K\times H \times G^{\Delta}}K\times G\times H\times G_{K\times G\times H\times G})\circ (Inf^{K\times G}_K \times H\times G))(a\times b) & =A(K\times H\times G)(a\times b) & =a\times b.
R(\varphi_t)(x) = \frac{1}{2}\sum_{p=1}^{q_t}\bigg( e^{-i \langle x, \xi_p^t \rangle}\phi_t(\xi_p^t) |V_p^t| + e^{i \langle x, \xi_p^t \rangle}\overline{\phi_t(\xi_p^t)}|V_p^t| \bigg),
(I- G_{0,0}vM_0^{-1}v^*)(I+G_{0,0}V) &=I-G_{0,0}vM_0^{-1}v^*+G_{0,0}V-G_{0,0}vM_0^{-1}v^*G_{0,0}V &=I-G_{0,0}vM_0^{-1}(U+v^*G_{0,0}v)Uv^* +G_{0,0}V &=I.
\frac{\partial F}{\partial y}(x,y) = \frac{1}{y+1/2}- \frac{1}{x+y}- \frac{1}{2}\frac{1}{(x+y)^2}> \frac{1}{y+1/2}- \frac{1}{1+y}- \frac{1}{2}\frac{1}{(1+y)^2}
\rho< \begin{cases}\frac{1}{2}\left( \frac{\nu}{2}\right)^{\frac{1}{1-2 \sigma}}\ \ \sigma \in (0, \frac{1}{2}), & \frac{1}{2}\ \ \sigma=\frac{1}{2}, & \frac{1}{2}\left( \frac{2}{\nu}\right)^{\frac{1}{2 \sigma-1}}\ \ \sigma \in (\frac{1}{2},1].\end{cases}
\phi(z_1^0+\tilde z)= (x_1^0+\tilde x)(y_1^0+\tilde y)+\frac{1}{3}(y_1^0+\tilde y)^3 = \tilde x\tilde y+\frac{1}{3}{\tilde y}^3+y_1^0{\tilde y}^2+ = (\tilde x+y_1^0 \tilde y)\tilde y+\frac{1}{3}{\tilde y}^3+ .
\kappa(r,y) = \sum f_\lambda(r) \kappa_\lambda(y), \quad\quad\quad\quad\quad\quad\quad\;\tilde{d}^*\tilde{d}\kappa_\lambda = \lambda \kappa_\lambda , \eta(r,y) = \sum g_\lambda(r)\tilde{d}\kappa_\lambda(y) + h_\lambda(r) \eta_\lambda(y), \tilde{d}^*\tilde{d}\eta_\lambda = \lambda \eta_\lambda, \; \tilde{d}^*{\eta}_\lambda = 0.
d s^{2}= - 4 d x^{+}d x^{-}+ \sum_{i = 1}^{8}( d x^{i})^{2}- \left( \sum_{i = 1}^{4}( x^{i})^{2}+ 2 \sum_{i = 5}^{8}( x^{i})^{2}\right) ( d x^{+})^{2}
|\partial_{t}\left(-16 \int_{t}^{\infty}\lambda''(s) \left(K_{3}(s-t,\lambda(t))-K_{3,0}(s-t,\lambda(t))\right) ds\right)| &\leq C \sup_{x \geq t}|e'''(x)| + \frac{C}{t^{3}\log^{b+1}(t)}
\int_{t_0}^{T_{\max}}\frac{d}{dt}\left(\int_{\Omega}F_r(\overline U(x,t))^2dx\right)\phi(t)dt=-\int_{t_0}^{T_{\max}}\left(\int_{\Omega}F_r(\overline U(x,t))^2dx\right)\phi_t(t)dt.
= - \sum_{j = 0}^{\infty}\int d e^{\prime}d n^{\prime}\frac{i}{k}\mathrm{e x p}- [ 2 i \pi e^{\prime}( n - j - 1 / 2 ) / k ] \int_{C_{7}}\zeta_{e^{\prime}n^{\prime}}( z_{1}, z_{2}, w ; \tau ) d w
B=\begin{pmatrix}\tan \frac{\pi}{n}\tan \frac{\pi}{n}0 \cdots 0 0 \tan \frac{\pi}{n}\tan \frac{\pi}{n}\cdots 0 \vdots 0 \ddots \ddots \vdots 0 \cdots 0 \tan \frac{\pi}{n}\tan \frac{\pi}{n}\tan \frac{\pi}{n}0 \cdots 0 \tan \frac{\pi}{n}\end{pmatrix}_{n \times n},
f_x(z,x)-Tz^{10}=-\frac{25}{4}z^4x^6-\frac{17}{4}z^3x^6+4z^2x^6-4x^6z-12z^3x^4+6x^4z^2 -\frac{71}{4}z^6x^4+4z^5x^4+\frac{75}{4}z^4x^4-\frac{25}{2}z^8x^2+\frac{75}{4}z^7x^2+\frac{59}{4}z^6x^2-\frac{99}{4}z^5x^2 +12x^2z^4-4x^2z^3+z^4-4z^5+\frac{41}{4}z^6-\frac{25}{2}z^7+\frac{25}{4}z^9+x^8-Tz^{10}.
\vec{S}= \left[ \begin{array}{c}{\mathrm{R e}\left({\mathcal{W}}\right)}\\{\mathrm{I m}\left({\mathcal{W}}\right)}\\ \end{array}\right] , \qquad \vec{a}= - \left[ \begin{array}{c}{\mathrm{I m}( \phi \partial_{x}\bar{\phi})}\\{\mathrm{I m}( \phi \partial_{t}\bar{\phi})}\\ \end{array}\right] ,
\|u_x\|_{L^2}^2 + \alpha \|v_x\|_{L^2}^2 = E_4+ \int uv^2 \; \d x \leq E_4 + C\| v\|_{L^2}^2 \|u \|_{H^1}&\leq E_4 + C(\|u_x\|_{L^2}^2 + C)^{1/2}\leq (E_4 + C) + \tilde{C}\|u_x\|_2.
\left( \begin{array}{c c}{{\frac{\tilde{B}_{\lambda}}{( k^{2})^{3 / 2 - \beta_{\lambda}}}}}&{{i \vartheta \frac{\tilde{B}_{\lambda \sigma}}{( k^{2})^{\beta_{\lambda \sigma}}}}}\\{{i \vartheta \frac{\tilde{B}_{\lambda \sigma}}{( k^{2})^{\beta_{\lambda \sigma}}}}}&{{\frac{\tilde{B}_{\sigma}}{( k^{2})^{3 / 2 - \beta_{\sigma}}}}}\\ \end{array}\right)
\max_{0\leq j\leq n-1}|w_{j}^{(k)}|_p =\left\|W^{(k)}\right\|_p=\left\|q R_p^{(k)}B_p^{-1}\right\|_p &\leq\left\|R_p^{(k)}\right\|_p\left\|q B_p^{-1}\right\|_p<\infty\intertext{and analogously,}\max_{0\leq j\leq n-1}|w_{j}|_\infty <\infty.
{\rho}=\omega_{\beta,\mu,\Lambda,\lambda}(\frac{N_{\Lambda}}{V})= \frac{1}{V(\exp(-\beta \mu)-1)}+ \frac{|\lambda|^{2}}{\mu^{2}}+ \frac{1}{V}\sum_{{k}\ne{0}}\frac{1}{\exp(\beta(\epsilon_{{k}}-\mu))-1}\ .
{\frac{d}{d u}}W_{\gamma_{u}}^{R}= \oint d v \, \gamma_{u}^{' \mu}( v ) \dot{\gamma}_{u}^{\nu}( v ) U_{\gamma_{u}}^{R}( s , v ) F_{\mu \nu}( \gamma ( v ) ) U_{\gamma_{u}}^{R}( v , s ) ,
y_i = \frac{\sqrt{w_i(1+\gamma_i)|h_{i,i}|^2p_i}}{\sum_{j=1}^{L}|h_{i,j}|^2p_j+\sigma^2_j}, \tilde{y}_k = \frac{\sqrt{\sum_{j\ne k}|\tilde{h}_{k,j}|^2 p_j+\tilde{\sigma}^2_k}}{w_k(1-\tilde{\gamma}_k)|\tilde{h}_{k,k}|^2p_k+\varepsilon}.
M_{s}^2=\frac{\left(M_{s}^1\backslash \phi_2( (0,s^{1/4}]\times X\times [-\eta/2,\eta/2])\right) \bigsqcup\left( \{\mathbf{r}_s\leq 1\}\times I_{\eta}^{2}\right)}{\thicksim_2},
\widetilde\Omega:\Gamma (\mathcal{H}\times \mathbb{R})\to \mathbb{C}, |\Psi(0)\rangle\mapsto \langle \Psi(0)|\Phi_\mathcal{H}(t,|\Psi (0)\rangle )\rangle =\,\langle\Psi(0)|\mathcal{U}(t)|\Psi (0)\rangle .
\begin{aligned}\left|\frac{Df^{n}_{a}(\mu_{0,j}(a))}{Df_{a}^{n}(\xi_{0,j}(a))}- 1 \right| =\left|\prod_{k=0}^{n-1}\frac{Df_{a}(\mu_{k,j}(a))}{Df_{a}(\xi_{k,j}(a))}- 1 \right| &\leq \exp\left(\sum_{k=0}^{n-1}\left|\frac{Df_{a}(\mu_{k,j}(a))}{Df_{a}(\xi_{k,j}(a))}- 1\right| \right)-1.\end{aligned}
\int_{\tau^{1/2}+|z|<y}(\tau^{1/2}+|z|)^\alpha\,dz\,d\tau = 2\int_{\rho+|z|<y}\rho(\rho+|z|)^\alpha\,dz\,d\rho \le 2\int_{\rho+|z|<y}(\rho+|z|)^{\alpha+1}\,dz\,d\rho\le Cy^{\alpha+n+2}.
V_{e f f}= \left[ e^{2}a^{i}a_{i}- \frac{1}{2 \alpha^{\prime}}+ \left( e \phi_{a}- \frac{\delta^{a}}{2 \pi \alpha^{\prime}}\right)^{2}\right] T \overline{{T}}+ \frac{e^{2}}{2}\left( T \overline{{T}}\right)^{2}
D_1 \lesssim \sum_{k=-\infty}^{M-1}2^{-k\frac{q}{p-q}}\bigg( \int_{x_k}^{{x_{k+1}}}\bigg( \int_t^{{x_{k+1}}}u \bigg)^{\frac{q}{1-q}}u(t) V_r(0,t)^{\frac{q}{1-q}}\bigg)^{\frac{p(1-q)}{p-q}}+ \sum_{k=-\infty}^{M-1}2^{-k\frac{q}{p-q}}\bigg( \int_{x_{k}}^{{x_{k+1}}}u \bigg)^{\frac{p}{p-q}}V_r(0,x_k)^{\frac{pq}{p-q}}.
~ \inf_{x}(\max\{\psi_1(x),\psi_2(x)\}+(\phi(x)-X(x,x))) = ~ \phi\circ(\psi_1\wedge\psi_2) = ~ \max\{\phi\circ\psi_1,\phi\circ\psi_2\}= ~\max\Big\{\inf_{x}(\psi_1(x)+(\phi(x)-X(x,x))), \inf_{x}(\psi_2(x)+(\phi(x)-X(x,x)))\Big\}= ~\max\{\lambda_1+k+\phi(x_1),\lambda_2+k+\phi(x_2)\}< ~\delta_1+\delta_2+k.
H' (v(s),\mu (s))\left[\begin{array}{c}\frac{dv}{ds}\frac{d\mu}{ds}\end{array}\right]=0, |( \frac{dv}{ds},\frac{d\mu}{ds})\|=1, \v(0)=v^{(0)}, \ \mu(0)=1, \ \frac{d\mu}{ds}(0)<0,
\epsilon_{(1,2)}=\epsilon_{1}\epsilon_{2}^{\prime}-\epsilon_{1}^{\prime}\epsilon_{2},\quad (\epsilon \lambda )_{(1,2)}=\epsilon_{1}\lambda_{2}^{\prime}-2\epsilon_{1}^{\prime}\lambda_{2},\quad \lambda_{(1,2)}=\lambda_{1}\lambda_{2}^{\prime}-\lambda_{1}^{\prime}\lambda_{2}.
{\cal S}= \sum_{m = 1}^{i + j + i^{\prime}+ j^{\prime}}\sum_{l \mathrm{' \scriptsize{s}}}\sum_{\mathrm{\scriptsize{g r}}}{\cal S}_{c}( \cdots ){\cal S}_{c}( \cdots ) \, \cdots \,{\cal S}_{c}( \cdots ) \, .
= i \int_0^P \left(\begin{array}{cc}\psi_1 \bar{\psi}_2 -\bar{\psi}_2^2 \psi_1^2 -\psi_1\bar{\psi}_2 \end{array}\right)dz +\left(\begin{array}{cc}\bar{\psi}_1 \psi_2 \bar{\psi}_1^2 -\psi_2^2 -\bar{\psi}_1 \psi_2 \end{array}\right)d\bar{z}=
|\overline{\Delta_j \Gamma(Z^A)}|^q = | \sum_{\substack{A \subset [n] j \notin A}}K_{n,A}\Delta_j \Gamma(Z^A) |^q \leq \sum_{\substack{A \subset [n] j \notin A}}K_{n,A}|\Delta_j \Gamma(Z^A)|^q .
-\sigma\int_{0}^{t}\int_{z=0}\nabla_y\cdot(\frac{\mathcal{Z}^\alpha \nabla_yh}{\sqrt{1+|\nabla_yh|^2}}-\frac{\nabla_yh\langle\nabla_yh,\mathcal{Z}^\alpha \nabla_yh\rangle}{(1+|\nabla_yh|^2)^\frac{3}{2}})\mathcal{C}^\alpha(h)dyd\tau,
{W}_{\alpha}= \frac{1}{8}\;\bar{D}^2\, \left({e}^{-V}\! D_{\alpha}{e}^V\right) = i\left( \lambda_{\alpha}+ i\theta_{\alpha}D - \theta^{\beta}\, G_{\alpha\beta}- i\theta^2{\cal D}_{\alpha\dot\alpha}\bar{\lambda}^{\dot\alpha}\right)\, .
\mathrm{I}_{t_0}(t)=\int_{\vert x-x(t)\vert<R_\varepsilon}\big(u^2+u_x^2\big)\Psi+\int_{\vert x-x(t)\vert>R_\varepsilon}\big(u^2+u_x^2\big)\Psi=\mathrm{I}_1+\mathrm{I}_2.
\tilde{w}={\hat{T}_{k}}w{\hat{T}_{k}}^{- 1}= - e^{- \lambda_{0}^{( k )}}+ 2 \lambda_{1}^{( k )}( \hat{L}_{0}- \lambda_{- 1}^{( k )}w ) - \lambda_{1}^{( k ) 2}e^{\lambda_{0}^{( k )}}( \hat{L}_{1}+ 2 \lambda_{- 1}^{( k )}\hat{L}_{0}-{\lambda_{- 1}^{( k ) 2}}w ) ,
\mathbf{u}_0 (\cdot ,t)=e^{-\mathcal{A}_D^0t}\boldsymbol{\varphi}(\cdot)+\int_0^t e^{-\mathcal{A}_D^0(t-\widetilde{t})}\mathbf{F}(\cdot ,\widetilde{t})\,d\widetilde{t}.
\Phi_{\{[\{1,3,5\},3],[\{2,4\},1]\}}\Phi_{\{[\{1,2,5\},4],[\{3\},1],[\{4\},2]\}}=\Phi_{\{[\{1,3,5\},3],[\{2,4\},1],[\{6,7,10\},4],[\{8\},1],[\{9\},2]\}}.
\dot{\xi}_{\underline{{c}}}-{h}_{\ \underline{{c}}}^{i}{h}_{\ \underline{{b}}}^{j}T_{i j}^{\underline{{a}}}\xi_{\underline{{a}}}\xi^{\underline{{b}}}-{h}_{\ \underline{{c}}}^{i}T_{i 0}^{\underline{{a}}}\xi_{{a}}= 0 ,
|\mathbb{E}\circ\mathrm{tr}(f_{1}^{(1)}\otimes f_{1}^{(2)})(f_{2}^{(1)}\otimes f_{2}^{(2)})\ldots(f_{p}^{(1)}\otimes f_{p}^{(2)})|\leq\frac{1}{n}\mathbb{E}\prod_{i=1}^{p}|\langle f_{i}^{(2)},f_{i+1}^{(1)}\rangle|.
\begin{cases}0 < \lambda < 1 0 < \lambda \le \frac{n-1}{2}\end{cases}\Longleftrightarrow \begin{cases}\beta +(j-2)< \alpha < \beta +(j-1) \beta +(j-1) - \frac{n-1}{2}\le \alpha < \beta +(j-1).\end{cases}
(\begin{pmatrix}A 0 0 \alpha\end{pmatrix},\,\begin{pmatrix}B tb^{\top}b \beta+t\chi\end{pmatrix},\,\begin{pmatrix}I X\end{pmatrix},\begin{pmatrix}1\!\!1_{k}0 0 t1\!\!1_{n-k}\end{pmatrix}).
AG \geq \,\, \frac{2}{n}\phi G^{2}-\bigg(2K+\frac{12\left\vert b\right\vert}{n}\bigg)\phi G-2\frac{C_{1}\phi^{\frac{1}{2}}G^{\frac{3}{2}}}{R}+\bigg( \frac{2(\lambda-1)aV^{p}}{n}+(2-\lambda)apV^{p-1}&+\lambda a p(p-1)V^{p-2}\bigg)\phi G.
\sum_{\substack{nm\leqslant N (n,m)=1}}\frac{r(n)^2r(m)\omega(n)\omega'(m)}{\sqrt{m}}=\big(1+o^{\star}(1)\big)\prod_p\left(1+r(p)^2\omega(p)+\frac{r(p)}{\sqrt{p}}\omega'(p)\right).
\phi_{*}\left(J_{1}\frac{\partial}{\partial x^{i}}\right)_{p}= \phi_{*}\left(\frac{\partial}{\partial y^{i}}\right)_{p}= \displaystyle\sum_{j=1}^{m}\frac{\partial a_{j}}{\partial y^{i}}(p)\frac{\partial}{\partial u^{j}}|_{\phi(p)}+ \displaystyle\sum_{j=1}^{m}\frac{\partial b_{j}}{\partial y^{i}}(p)\frac{\partial}{\partial v^{j}}|_{\phi(p)}
\frac{\partial \bar{V}_{1}(\zeta_{1})}{\partial \zeta_{1}}\varphi_{1}(\zeta_{1},\zeta_{2},\psi,\mu)\!\leq -\!\Delta_{1}(\zeta_{1})\|\zeta_{1}\|^{2}\!+\!\bar{\phi}_{1}(\zeta_{2})|\zeta_{2}|^{2}+\bar{\pi}_{1}(\psi)\|\psi\|^{2}
\int d^{2}k \frac{- ( q_{2}+ q_{3}+ q_{4}) \cdot q_{2}\; q_{3}^{2}\; q_{4}^{2}}{( k^{2}\! + \! m^{2}) ( ( k \! + \! q_{2})^{2}\! + \! m^{2}) ( ( k \! + \! q_{2}\! + \! q_{3})^{2}\! + \! m^{2}) ( ( k \! + \! q_{2}\! + \! q_{3}\! + \! q_{4})^{2}\! + \! m^{2})}\; .
\Pr\left\{\frac{1}{t}\sum_{\tau=0}^{t-1}p_0(\tau) - p^{(opt)}> \epsilon_0 \right\}\leq u_t \exp\left\{\frac{-2 \epsilon^2 t^2}{u_{,0}^2 u_t^2}\right\}+ t \beta_{\texttt{ADPP},k}(u_t).
\{F ( \bar{\psi}, \psi ) , G ( \bar{\psi}, \psi ) \}= ( 2 i N )^{- 1}\int d x ~ g^{i j}\left( \frac{\delta F}{\delta \bar{\psi}_{i}( x )}\frac{\delta G}{\delta \psi_{j}( x )}- \frac{\delta G}{\delta \bar{\psi}_{i}( x )}\frac{\delta F}{\delta \psi_{j}( x )}\right)
\sigma_{C}=Z\frac{8}{3}\pi r_{e}^{2}\frac{1}{(1+2k)^{2}}(1+2k+\frac{6}{5}k^{2}-\frac{1}{2}k^{3}+\frac{2}{7}k^{4}-\frac{6}{35}k^{5}+\frac{8}{105}k^{6}+\frac{4}{105}k^{7})
\mathcal{C}_{ad}:=\Big\{v\in C(\overline{\Omega}\times [0, T])~~a.s.: ~v~\textup{is}~\mathfrak{F}(t)-\textup{adapted},~ l^{*}_1(x,y,t)\leq v(x,y,t)\leq l^{**}_1(x,y,t)~~a.s.\Big\},
\delta^2 H(U_0)[\Phi,\Phi] = \int\left[ |\Phi'|^2 + \Phi^t D^2 W(U_0)\Phi \right] dx &= \int \left[ (\varphi'_1)^2 + W_{p_1 p_1}(u_0,0) \varphi_1^2\right] dx + \int \left[ (\varphi'_2)^2 + W_{p_2 p_2}(u_0,0) \varphi_2^2\right] dx =: (\varphi_1, L_1\varphi_1) + (\varphi_2, L_2\varphi_2),
\zeta_{\infty,p^{\prime}}(s,c):=\sum_{m,n\neq0}\frac{p^{\prime2}+\lambda_{n}^{\prime2}}{p^{\prime}(p^{\prime}+\frac{1}{\pi})+\lambda_{n}^{\prime2}}\,\frac{1}{\left(m^{2}+c\lambda_{n}^{\prime2}\right)^{s}},\,\,\,\,(s)>1.
A = \left( \begin{array}{c c}{{\alpha}}&{{- \overline{{{\beta}}}}}\\{{\beta}}&{{\overline{{{\alpha}}}}}\\ \end{array}\right) , \quad \alpha = \cos{\frac{\theta}{2}}e^{i ( \psi + \phi ) / 2}, \quad \beta = \sin{\frac{\theta}{2}}e^{i ( \psi - \phi ) / 2},
\kappa_{\bar\epsilon_j}^{-1}h_L(a_{\epsilon_j}) = \kappa_{\bar\epsilon_j}^{-1}h_R(p_{\epsilon_j}F^{-2}a) = \kappa_{\bar\epsilon_j}^{-1}h_R(p_{\epsilon_j}E^R_{\epsilon_j}(F^{-2})a) = \kappa_{\bar\alpha}^{-1}h_R(p_\alpha E^R_{\epsilon_j}(F^{-2})a) .
||\dot{{\mathbf U}}||_{s,\ast,T}+||\varphi||_{H^s(\Gamma_T)}&\quad\leq C(K_0)\Big(||\mathbf{f}||_{s+2,\ast,T}+||\mathbf{g}||_{H^{s+2}(\Gamma_T)}+(||\mathbf{f}||_{8,\ast,T}+||\mathbf{g}||_{H^{8}(\Gamma_T)})||\hat{W}||_{s+4,\ast,T}\Big),
d s^{2}={\frac{\tau_{3}}{2}}{\frac{\cosh^{2}\chi}{\rho^{2}}}e^{2 A}d r^{2}+{\frac{\tau_{3}}{2}}{L^{2}}{e^{2 A}}\rho^{2}\left({\cosh^{2}\! \chi}\, \sigma_{3}^{2}+ \sigma_{1}^{2}+ \sigma_{2}^{2}\right) \ ,
[\alpha_1, \ldots, \alpha_n] = \sum_{k=1}^n (-1)^{n-k}\mathcal{L}_{\Pi^{\sharp}(\alpha_1 \wedge\cdots \wedge \hat{\alpha}_k \wedge\cdots \wedge \alpha_n)}\alpha_k - (n-1) d (\Pi(\alpha_1, \ldots, \alpha_n))
U_{e}= \frac{1}{\beta}\left[ \frac{e^{2}}{8}\left( | \sigma_{1}|^{2}- | \sigma_{2}|^{2}\right)^{2}+ \left( \beta \tilde{m}^{2}- \frac{1}{2}\alpha e \tilde{k}\right) | \sigma_{1}|^{2}+ \left( \beta \tilde{m}^{2}+ \frac{1}{2}\alpha e \tilde{k}\right) | \sigma_{2}|^{2}\right] ,
R_{l}^{m}(r+a)=\sum_{\lambda=0}^{l}(\begin{matrix}2l\\ 2\lambda\end{matrix})^{1/2}\sum_{\mu=-\lambda}^{\lambda}R_{\lambda}^{\mu}(r)R_{l-\lambda}^{m-\mu}(a)\langle\lambda,\mu;l-\lambda,m-\mu|lm\rangle,
\mathcal{L}_1 : \{x^d = -a_1/a_0 \}\times \{y^d = - a_2/a_3 \}, \mathcal{L}_2 : \{x^d = -a_2/a_0 \}\times \{y^d = - a_3/a_1 \}, \mathcal{L}_3 : \{x^d = -a_3/a_0 \}\times \{y^d = - a_1/a_2 \}.
\partial_{\omega}^n\widehat{g}^{t}(k,\omega) = (-1)^n\frac{L^{n}}{|k|^{n}}\int_{B_1(0)}\left(\hat{k}\cdot \nabla_v\right)^n \left(\frac{f(p(v),k)}{(1-|v|^2)^{5/2}}\right) \;\;\;\;\cdot\delta\left(\omega + \frac{k}{L}\cdot v \right)d^3v,
s= \begin{cases}\lfloor \alpha_{m,t}(-1)\rfloor , \ \ \{\alpha_{m,t}(-1)\}< 1/2; \lfloor \alpha_{m,t}(-1)\rfloor \ \ \lfloor \alpha_{m,t}(-1)\rfloor +1, \ \ \{\alpha_{m,t}(-1)\}= 1/2; \lfloor \alpha_{m,t}(-1)\rfloor +1, \ \ \{\alpha_{m,t}(-1)\}> 1/2.\end{cases}
J_z = \sum\limits_{\mu = -s}^{s}\mu a_{\mu}^{\dagger}a_{\mu},J_+ = \sum\limits_{\mu = -s}^{\mu = s - 1}\sqrt{(s + \mu + 1)(s - \mu)}a_{\mu + 1}^{\dagger}a_{\mu}= (J_-)^{\dagger}.
K_{y}=K_{2}=i\frac{\partial\hat{B}(\varphi,\hat{e}_{y})}{\partial\varphi}|_{\varphi=0}=i(\begin{matrix}0&0&1&0\\ 0&0&0&0\\ 1&0&0&0\\ 0&0&0&0\\ \end{matrix}),
{S_{P}^{4}}^{\star}= \int d^{4}x \Bigl [{\frac{1}{2}}T^{\mu \nu}F_{\mu \nu}+{\frac{1}{4}}T^{\mu \nu}T_{\mu \nu}-{\frac{{\mu}^{2}}{2}}A^{\mu}A_{\mu}\Bigl ] ,
\mathcal{L}_{F}^{\alpha,-1}\left[\frac{1}{(S_{F}^{\alpha}(s)^{\zeta}+S_{F}^{\alpha}(a)S_{F}^{\alpha}(s)^{\mu})^{n+1}}\right] &= S_{F}^{\alpha}(x)^{\zeta(n+1)-1}\sum_{k=0}^{\infty}\frac{-(S_{F}^{\alpha}(a))^k}{\Gamma_{F}^{\alpha}(k(\zeta-\mu)+(n+1)\zeta)}& \binom{n+k}{k}S_{F}^{\alpha}(x)^{k(\zeta-\mu)}.
\hat{\mathcal{U}}_s^\eta = \left\{Z_s = (X_s,V_s) \in\mathcal{U}_s^\eta \left| \forall \tau,\tau^\prime > 0,(\psi_s^{-\tau}Z_s,\psi_s^{-\tau^\prime}Z_s) \in\mathcal{V}_s^\eta \right. \right\}
\begin{aligned}\uppercase\expandafter{\romannumeral2}=\sum_{i=-\infty}^{+\infty}\sum_{\widetilde{Q}\in B_{i,1}}\sum_{Q\subset\widetilde{Q},Q\in B_{i,1}}\vert Q\vert(\psi_{Q}\ast h)(u_Q)\psi_{Q}(x-u_Q) &=:\sum_{i=-\infty}^{+\infty}\sum_{\widetilde{Q}\in B_{i,1}}\lambda_{\widetilde{Q}}^{i}a_{\widetilde{Q}}^{i}(x),\end{aligned}
P_{}=P^T_r = \frac{4}{\pi^2}M^2\cdot (\mu^T_h|g|\beta^T)^2 + M\cdot\big\{\sigma^R_h|g|^2(\beta^T)^2 + 2\mu^T_h|g|\beta^T \mathbb{E}[|h^T_d|] \big\}+\mathbb{E}[|h^T_d|^2].
\inf_{|x|<r_\lambda}\big(\sigma_\lambda(x)-\tilde u_\lambda(x)\big)= \inf_{|x|<\lambda^{-\frac 12}r}\lambda\,\big(\sigma(\sqrt{\lambda}x)-\tilde u_\lambda(\sqrt{\lambda}x)\big) \qquad= \inf_{|y|<r}\lambda\,\big(\sigma(y)-\tilde u_\lambda(y)\big)\ge c_0\lambda.
h_{g_{u}}(\theta) =I(g_{u}(\theta))-\theta^{2}I(u) &=(\theta^{2-2\beta}-\theta^{2})A(u)+(\theta^{(1-\frac{3}{2}\beta)p+3\beta}-\theta^{2})C(u)-\theta^{2}B(u) &\ \ \ +\frac{1}{4}\theta^{4-\beta}\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}\frac{1-e^{-\theta^{\beta}|x-y|}}{|x-y|}u(x)^{2}u(y)^{2}dxdy,
7 . 1 2 \Gamma_{\hat{h}}( \phi ) \ = \ \frac{1}{9 6 \pi}\int_{\Sigma}d^{2}\xi \sqrt{| \hat{h}( \xi ) |}\ \left( \hat{h}^{\alpha \beta}( \xi ) \, \partial_{\alpha}\phi ( \xi ) \, \partial_{\beta}\phi ( \xi ) \; + \; 4 r_{\hat{h}}( \xi ) \, \phi ( \xi ) \right) \ .
(i+i_1+\ldots i_n)\ldots(i+i_1+i_2)(i+i_1)[a^{i_1}q_1a^{i_2}q_2\ldots a^{i_n}q_n, a^i]=(i+i_1+\ldots i_{n-1})\ldots(i+i_1)i[q_1q_2\ldots q_n, a^{i+\sum_k i_k}].
\underline{hs(2,2)}:f(a,\bar{a})=\alpha+\alpha_{i\bar{j}}a^i\bar{a}^{\bar{j}}+\alpha_{i_1i_2\bar{j}_1\bar{j}_2}a^{i_1}a^{i_2}\bar{a}^{\bar{j}_1}\bar{a}^{\bar{j}_2}+\ldots
\zeta(0)=\frac{1}{-1}\sum_{n=0}^{\infty}\frac{1}{2^{n+1}}\sum_{k=0}^{n}(\begin{matrix}n\\k\end{matrix})(-1)^{k}=-\sum_{n=0}^{\infty}\frac{1}{2^{n+1}}(1-1)^{n}=-\frac{1}{2}.
I = ( i / 2 )^{2}\left\{C_{\tau}^{1 2}[ \alpha , \beta ] P_{\tau}^{1 2}[ \bar{\alpha}, \bar{\beta}] + C_{- \tau}^{1 2}[ \alpha , \beta ] P_{- \tau}^{1 2}[ \bar{\alpha}, \bar{\beta}] \right\}.
J_T(x) = \begin{pmatrix}e^{-2\pi i N \int_x^1 \psi_{\alpha,\varepsilon}(s) ds}\frac{1}{\sqrt{1-x^2}}0 e^{2\pi i N \int_x^1 \psi_{\alpha, \varepsilon}(s) ds}\end{pmatrix};
\bar{\rm P}_{j,j+i}= \sum^{M-j}_{m=i+1}{M-j \choose m}\mathbb{P}_{\rm TX}^m \left(1- \mathbb{P}_{\rm TX}\right)^{M-j-m}\times m \cdots (m-i+1)\mathbb{P}_K^{m}\gamma_i\gamma_{m,i}+{M-j \choose i}\mathbb{P}_{\rm TX}^i \left(1- \mathbb{P}_{\rm TX}\right)^{M-j-i}i!\mathbb{P}_K^{i}\gamma_i
\varphi_\ell (x, \lambda_\ell) = \frac{Q_{\ell, YY}^{-1}}{4\lambda_{\ell}}+ a_\ell x^{\top}V_{\ell, XX}^{-1}V_{\ell, XY}- \lambda_{\ell}x^{\top}V_{\ell, XX}^{-1}x.
\tau_{k, \ell - 1}^{( \alpha,\beta)}\tau_{k, \ell}^{( \alpha + 1,\beta)}= \tau_{k, \ell - 1}^{(\alpha + 1,\beta)}\tau_{k, \ell}^{(\alpha,\beta)}+\tau_{k + 1, \ell}^{(\alpha,\beta)}\tau_{k - 1, \ell - 1}^{(\alpha + 1,\beta)},
\mathbb{P}(s(u, W, v)>2\mu_{uv})<\exp\left(-2\frac{\mu_{uv}^2}{3s(u, I, v)}\right)=\exp\left(-2\frac{\mu_{uv}}{3p^2}\right)<\exp\left(-\frac{16\sqrt{n}}{3\log^4 n}\right).
G_{M N}^{( 1 1 )}= \left( \begin{array}{l l}{e^{-{\frac{\hat{\Phi}}{3}}}G_{\hat{\mu}\hat{\nu}}^{( 1 0 )}}{e^{{\frac{2}{3}}\hat{\Phi}}B_{M}}{e^{{\frac{2}{3}}\hat{\Phi}}B_{N}}{e^{{\frac{2}{3}}\hat{\Phi}}}\end{array}\right) ,
S = \{1,2,3,4,5, 6, 7, 8, 9, 10, 12, 13, 15, 16, 18, 21, 22, 24, 25, 28, 30, 33, 37, 40, 42, 45, 48, 57, 58, 60, 70, 72, 78, 85, 88, 93, 102, 105, 112, 120, 130, 133, 165, 168, 177, 190, 210, 232, 240, 253, 273, 280, 312, 330, 345, 357, 385, 408, 462, 520, 760, 840, 1320, 1365, 1848.\}
\sum\nolimits^{n}_{j=1}|\Theta_j(V)|_{\cal H}= \tfrac{1}{2}\sum\nolimits^n_{j=1}|\Gamma_j|_{\cal H}+ \tfrac{1}{2}\big|\partial\big( \bigcup\nolimits^n_{j=1}\overline{X_j}\big)\big|_{\cal H}.
K_{3}(w,\lambda(t))-K_{3,0}(w,\lambda(t)) = \left(\frac{w}{1+w^{2}}-\frac{w}{\lambda(t)^{2-2\alpha}+w^{2}}\right) \frac{w^{4}}{4(w^{2}+36\lambda(t)^{2})^{2}}+ \frac{1}{4(\lambda(t)^{1-\alpha}+w)(1+w)^{3}}
\int_{0}^{2 \pi}d \sigma \left( - \frac{1}{2 \pi}p^{+}\partial_{\sigma}X^{-}+{\cal P}^{I}\partial_{\sigma}X^{I}+ \frac{i}{2}\psi_{+}^{A}\partial_{\sigma}\psi_{+}^{A}+ \frac{i}{2}\psi_{-}^{A}\partial_{\sigma}\psi_{-}^{A}\right) = 0 .
\lim_{N \to \infty}\frac{4}{N^{2}}\Big(I_{2}\big(\frac{2u}{N}, \frac{2v}{N}\big) + I_{3}\big(\frac{2u}{N}, \frac{2v}{N}\big)\Big) = \frac{1}{4 \pi \kappa^{2}}\int_{0}^{\delta}\frac{dt}{\pi}\Im\left\{\frac{1}{1+it}e^{-\frac{u}{\kappa}(2+it) - \frac{v}{\kappa}\frac{2+it}{1+it}}\right\}.
{(a_{31}t^3 + a_{32}t^2 + p^rt)}^2 = 2p^ra_{32}t^3 + p^{2r}t^2 = c_1 p^kt^3 + c_2(a_{21}t^3 + p^lt^2) + c_3(a_{31}t^3 + a_{32}t^2 + p^rt)
\mathcal{S}(\tau_1,\tau_2;h) := \sum_{g \mid h}\frac{1}{g^{\tau_1}}\prod_{(p,g)=1}\Big ( 1 - \frac{f_1(p) f_2(p)}{p^{\tau_2}}\Big ) \prod_{p^{\alpha}\mid \mid g}\Big( f_1(p^{\alpha})f_2(p^{\alpha}) - \frac{f_1(p^{\alpha+1})f_2(p^{\alpha+1})}{p^{\tau_2}}\Big)
\|\hat{f}_{l}\|_{s-1}\leq C\left(\|\hat{u}^{2}-\hat{u}^{1}\|_{s-1}+\|\hat{v}^{2}-\hat{v}^{1}\|_{s}+\|\hat{w}^{2}-\hat{w}^{1}\|_{s-1}\right),l=1,3, \|\hat{f}_{2}\|_{s-2}\leq C\|(\hat{u}^{2}-\hat{u}^{1},\hat{v}^{2}-\hat{v}^{1},\hat{w}^{2}-\hat{w}^{1})\|_{s-1},
\tilde{\Gamma}_{s}= \left\{z = z^{( s )}+ \vartheta^{( s )}\varepsilon_{s}( z^{( s )}) \, , \quad \vartheta = \vartheta^{( s )}\left( 1 + \frac{\varepsilon_{s}\varepsilon_{s}^{\prime}}2 \right) + \varepsilon_{s}( z^{( s )}) \right\}\, ,
0\leq s_{k,l}:= \sum_{m=0}^k \sum_{n=0}^l d_{m,n}^{q-2}\leq\lim_{r\to1^-}\sum_{m,n\geq0}d_{m,n}^{q-2}S_{m,n}^{q}(r) = g(1), k,l\in\mathbb{Z}_+.
P= \begin{bmatrix}F_0 F_1 F_0\otimes I_d F_2 F_1\otimes I_d F_0\otimes I_d^{\otimes 2}\vdots \vdots \vdots \ddots F_{m-1}F_{m-2}\otimes I_d F_{m-3}\otimes I_d^{\otimes 2}\cdots F_0\otimes I_d^{\otimes (m-1)}\end{bmatrix},\, R=F_0\otimes I_d^{\otimes m}
+{\frac{{\frac{( p^{2}\alpha + q a^{2})}{4}}\partial^{2}+ q \alpha ( \alpha + q )}{({\frac{p + a}{2}})^{2}\partial^{2}+ ( q + \alpha )^{2}}}\epsilon^{\mu \nu \lambda}B_{\nu \lambda}-{\frac{1}{2 g}}B^{\mu}.
\begin{aligned}-2 \leq\frac{-\sup_{Q_{k-1}}u+3\inf_{Q_{k-1}}u}{2}\leq \omega_k s+\mu_k \leq \frac{2\sup_{Q_{k-1}}u-2\inf_{Q_{k-1}}u+\sup_{Q_{k-1}}u+\inf_{Q_{k-1}}u}{2}\leq 2. \end{aligned}
\lim_{\sigma \rightarrow \infty}F_{n}^{\mathcal{O}| \mu_{1}\ldots \mu_{n}}( \theta_{1}+ \eta_{1}\sigma , \ldots , \theta_{n}+ \eta_{n}\sigma ) = F_{\nu_{1}\cdots \nu_{n}}^{\mathcal{O}| \mu_{1}\ldots \mu_{n}}( \theta_{1}, \ldots , \theta_{n}) ,
\psi^-(x) +\psi^-(y) = \int_{B_2}g(x-z)+g(y-z)\, d\mu(z) + \int_{B_3}g(x-w)+g(y-w)\, d\mu(w) &\geq -2 \mu(B_2\cup B_3) + \frac{C''}4\, |y-x|^2 (k_2+k_3)\geq -2 \mu(B_2\cup B_3) + \frac{C''}{10}\, |y-x|^2 \,.
\begin{cases}\Phi'(x-\underline h(t))<-\epsilon_1\mathbf{1}\mbox{for}x\in [\underline h(t)-K_0,\underline h(t)], \Phi'(-x-\underline h(t))<-\epsilon_1\mathbf{1}\mbox{for}x\in [-\underline h(t),-\underline h(t)+K_0].\end{cases}
B_1 = f^{-(n_0+n_1+m_0)}(V_1) = f^{-n_0-k^+}\circ f^{-n_1-m_0+k^+}(V_1) &\subset f^{-n_0-k^+}(U_1) \subset f^{-n_0}(V_0) \subset B_0.