harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
Q^{0}_{P''}=\omega^{0}_{P''_{1}}\times\omega^{0}_{P''_{2}}\times\omega^{0}_{P''_{3}}:=\{(\xi^{2}_{1},\xi^{2}_{2},\xi^{2}_{3})\in\mathbb{R}^{3}: (2^{k''}\xi^{2}_{1},2^{k''}\xi^{2}_{2},2^{k''}\xi^{2}_{3})\in Q_{P''}\}
\tilde C_{\beta,\zeta,\gamma}\Phi\left(\frac{3}{2}\gamma\sqrt{\log p}\right) \le\P\left(|\hat\tau(Y_{1:n},Z_{1:n})-\tau(Y_1,Z_1)|\ge \gamma\sqrt{\frac{\log p}{n}}\right) &\le C_{\beta,\zeta,\gamma}\Phi\left(\frac{3}{2}\gamma\sqrt{\log p}\right),
j\theta = c_{k+1}q_k \theta + c_{k+2}q_{k+1}\theta + \dots + c_{k+\ell}q_{k+\ell-1}\theta &\le \tilde r_{k+1}q_k \theta + a_{k+3}q_{k+2}\theta + a_{k+5}q_{k+4}\theta + \dots < \tilde r_{k+1}\| q_k \theta \| + \|q_{k+1}\theta\|, j\theta \ge a_{k+2}q_{k+1}\theta + a_{k+4}q_{k+3}\theta + \dots > - \|q_{k}\theta\|.
{I_{1 0}( \mathrm{s t r i n g}) ={\frac{1}{2 \kappa^{2}}}\int d^{1 0}x \sqrt{- g}e^{- 2 \phi}\big ( R + 4 ( \partial \phi )^{2}-{\frac{1}{2 \cdot 3 !}}H^{2}\big ) ,}
\begin{aligned}0 \geq \eta \nabla_{\nu}w - w \nabla_{\nu}\eta & = 2\eta \nabla_1 v \, \nabla_{\nu}(\nabla_1 v) + 0.5 c_0 Bw & = - 2\eta g(e_1, \nabla_{e_1}\nu) |\nabla v|^2 + 0.5 c_0 Bw & > \big(0.5 c_0 B - 2\eta g(e_1, \nabla_{e_1}\nu) \big)|\nabla v|^2 \geq 0,\end{aligned}
\inf_{\psi\in CX,\psi(x)=1}(\varphi_1(y)\otimes\psi(y))=\varphi_1(y)\otimes\left(\inf_{\psi\in CX,\psi(x)=1}\psi(y)\right)=\varphi_1(y)\otimes a_0(y,x)\le\varphi_1(x)<u.
\phi^{-1}(h_1h_2)=(\sum^m_{k=1}(a^1_k +ib^1_k ) f_k+\sum^m_{k=1}(a^2_k +ib^2_k ) f_k, t_1+t_2+\tfrac{1}{2}\sum_{k=1}^m (a_k^1 b_k^2-a_k^2b_k^1));
\tilde{B}_2= \frac{(3+\alpha+(1-\alpha_0)\rho_0)\rho_1+3+(1-\alpha_0)\rho_0}{A^2(1+(2+\alpha)\rho_1+(1+\alpha)\rho_1^2)}\tilde{B}_3= \frac{(1+(1+\alpha)\rho_1)(1+(1-\alpha_0)\rho_0)}{2A^2(1+(2+\alpha)\rho_1+(1+\alpha)\rho_1^2)}.
\lambda_0(f_1,f_2) := \inf\{t \geq 0: (f_1(s_1) - f_2(s_1))(f_1(s_2) - f_2(s_2))> 0 s_1,s_2 \in (t,t + 1)\}\lambda_1(f_1,f_2) := \inf\{t > \lambda_0 : f_1(t) = f_2(t)\}.
2|V(G)|-2\geq r_2(E(G))=\sum_{i=1}^t r_2(E_i) &= \sum_{E_i: \textrm{balanced}}(2|V(E_i)|-3) +\sum_{E_i: \textrm{unbalanced}}(2|V(E_i)|-2) &=\sum_{i=1}^t (2|V(E_i)|-a_i)+\sum_{E_i: \textrm{balanced}}(a_i-3) +\sum_{E_i: \textrm{unbalanced}}(a_i-2).
\begin{aligned}\sum^\infty_{n=1}\int_{0}^{\infty}d_n^2(s,f)ds =\frac{2qe^{-2qt}m^2(T-t)}{b^2(p+2q)^2\{1 - \phi^{2}(t)\}}&=\frac{2 q m^2(T-t)}{b^2\{(p+2q)^2 e^{2qt}-p^2\}}.\end{aligned}
{d^2q_k \over dt^2}= 16\nu_0^2 \left[ \langle\vec{n}_{k+1}, \vec{n}_k\rangle e^{q_{k+1}-q_{k}}- \langle\vec{n}_{k},\vec{n}_{k-1}\rangle e^{q_{k}-q_{k-1}}\right],
\tilde{x}(t) =x_1(t)+\frac{1}{\Gamma(\alpha)}\int_{\nu}^{t}(t-s)^{\alpha-1}f(s,\tilde{x}(s))ds &=\frac{x_0}{\Gamma(\gamma)}t^{1-\gamma}+\frac{1}{\Gamma(\alpha)}\int_{0}^{t}(t-s)^{\alpha-1}f(s,\bar{x}(s))ds
\d{u}{\xi_r}=\d{v}{\xi_i}=\frac{\Re \left( \sqrt{g_\Omega (\xi)}\right)}{|g_\Omega (\xi)|}\d{u}{\xi_i}=-\d{v}{\xi_r}=\frac{\Im \left( \sqrt{g_\Omega (\xi)}\right)}{|g_\Omega (\xi)|}
K(p, \tilde{\pi}')= K(p,\pi)+\varepsilon Q(g,R)(\pi,\bar{\pi})+\varepsilon Q(g,R)(\pi,J\bar{\pi})+O(\varepsilon^2) = K(p,\pi)+\varepsilon Q^c(g,R)(\pi;\bar{\pi})+O(\varepsilon^2).
\vert u(x_k)\vert \leq \sum_{i=0}^{k-k_0-1}2^{m_0(s-s')-\frac{(k-i-1)p(s-s')}{Q}}\bigg(\sum_{j=-\infty}^{m_0-(k-i-1)p/Q}2^{js'p}\Big[g_j(x_{k-i})+g_j(x_{k-i-1})\Big]^p\bigg)^{1/p}&\qquad+\vert u(x_{k_0})\vert &\leq C2^{m_0(s-s')}\sum_{i=0}^{k-k_0-1}2^{-\frac{(k-i-1)p(s-s')}{Q}}2^{k-i}+\vert u(x_{k_0})\vert.
f_2(t) =(t^2+4t+6)e^{6 t}-4 t(2t^2+2 t+1) e^{5 t}-3(t^2+4 t+6)e^{4 t}&\quad-8t(t^2-t-1) e^{3 t}+3(t^2+4 t+6) e^{2 t}-4t e^t -t^2-4 t-6
\begin{aligned}J_1(x,x,y) = x, J_{2n+1}(x,y,y) = y, \begin{aligned}J_i(x,y,x) = x, 0\leq i\leq 2n+1, J_{2i+1}(x,y,y) = J_{2i+2}(x,y,y) 0\leq i\leq n-1, J_{2i}(x,x,y) = J_{2i+1}(x,x,y) 1\leq i\leq n, \end{aligned}\end{aligned}
\begin{cases}\nabla\times(\epsilon^{-1}\nabla\Pi_{\eta,\ell})-k^{2}\mu\Pi_{\eta,\ell}=(k^{2}\mu+\epsilon_{\max}^{-1})U\mbox{in}\Omega, \nu\times\Pi_{\eta,\ell}=0\mbox{on}\partial\Omega. \mbox{}\end{cases}
{\cal L}= \frac{1}{2}h_{i j}( y ) \dot{y}^{i}\dot{y}^{j}+ \frac{1}{2}L_{A B}( y ) ( \dot{\xi}^{A}+ w_{i}^{A}( y ) \dot{y}^{i}) ( \dot{\xi}^{B}+ w_{j}^{B}( y ) \dot{y}^{j}) \, ,
\lim_{N \to \infty}N R_v\left( \pmb{\rho}^{(N)}\right) = \frac{e^{-\gamma_u\left({\rho}^{(\infty)}\right)}\sum_{k=0}^N\sum_{n_1+\cdots+ n_V = k}\prod_{u=1}^V \frac{\gamma_u^{n_u}\left({\rho}^{(\infty)}\right)}{n_u!}\bar r_v(k)}{e^{- \gamma_u\left({\rho}^{(\infty)}\right)}+ \tau\left( 1 - e^{- \gamma_u\left({\rho}^{(\infty)}\right)}\right)}\, .
t\mathcal T^\phi_t-\omega\mathcal T^\phi_\omega-\phi\mathcal T^\phi_\phi-2T\mathcal T^\phi_T=-2a_{11}\mathcal T^\phi+a_{13}c_p(\mathcal T^p)^\kappa -\tilde{\boldsymbol\gamma}^6_{tt}\cdot\tilde{\mathbf x}+\tilde\alpha^6.
d s^{2}={\frac{1}{g_{s}}}\left( \mu_{6}V Z_{2}- \mu_{2}Z_{6}\right) d x_{\perp}^{2}={\frac{\alpha^{- 3 / 2}}{( 2 \pi )^{2}g}}\Biggl ( \frac{V}{V_{*}}- 1 - \frac{g_{s}N \alpha^{1 / 2}}{r}\Biggr ) ( d r^{2}+ r^{2}d \Omega_{2}^{2}) \ .
\begin{cases}(-\Delta)^s u(x)=v^p(x), x\in\Omega, (-\Delta)^t v(x)=u^q(x), x\in\Omega, u(x)=-\Delta u(x)=\cdots =(-\Delta)^{s-1}u(x)=0, x\in\partial\Omega, v(x)=-\Delta v(x)=\cdots =(-\Delta)^{t-1}v(x)=0, x\in\partial\Omega.\end{cases}
\left\lbrace \begin{aligned}p_{n+1}=p_n+R(p_n,\kappa_n) t_{n+1}=t_n+\dfrac{\ell_*}{\|p_{n+1}\|}q_{n+1}= q_n+\dfrac{\ell_*}{\|p_{n+1}\|}p_{n+1},\end{aligned}\right.
x_1^{d+1}+ x_2^{d+1}+ \cdots + x_i^{d+1}= (x_1 + x_2 + \cdots + x_i ) x_{i+1}^d = (x_1^2 + x_2^2 + \cdots + x_i^2 )x_{i+1}^{d-1}= \cdots = (x_1^d + x_2^d + \cdots + x_i^d)x_{i+1}
|L_1[h]| \lesssim\begin{cases}e^{-\alpha \lambda t/(d+\alpha)}(1+e^{-\lambda t}|x|^{d+\alpha})^{-1 -\alpha/(d+\alpha)}, \e^{-\lambda t}|x|^{d+\alpha}> 1 e^{-\alpha \lambda t/(d+\alpha)}, .\end{cases}
\left(D_{\gamma,\xi}\,u_\xi,u_\xi\right)=\left(|D_{\xi}|\,u_\xi,u_\xi\right)+\left((D_{\gamma,\xi}-D_\xi)\,u_\xi,u_\xi\right)\geq (1-\kappa)\left(|D_{\xi}|\,u_\xi,u_\xi\right).
q^*(t)= q^*(T) \exp \Bigl[ -\int_t^T \frac{\lambda^*(s)+\lambda_1^*(s)+c^*(s)}{s}ds \Bigr] - \int_t^T \exp \Bigl[ -\int_t^{\tau}\frac{\lambda^*(s)+\lambda_1^*(s)+c^*(s)}{s}ds \Bigr] \frac{\gamma^*(\tau)u^*(\tau)}{\tau}d\tau.
S_{2}=c \big\langle (\nabla_r\mathsf{f}) \cdot w, (\mathsf{f}\cdot \nabla w_r)|w|^{q-2}\big\rangle + c(q-2)\big\langle (\nabla_r\mathsf{f}) \cdot w, w_r|w|^{q-3}\mathsf{f}\cdot \nabla|w| \big\rangle.
\langle \, \phi ( x_{1}) \dots \phi ( x_{n}) \, \rangle = \frac{1}{{\cal Z}}\, \int \prod_{x}\mathrm{d}\phi ( x ) \, p h i ( x_{1}) \dots \phi ( x_{n}) \ \mathrm{e}^{- S_{\mathrm{E}}[ \phi ( x ) ]}\, ,
||n\left( Y_{l_{1}}^{m_{1}}\star_{1}^{n}Y_{l_{2}}^{m_{2}}-Y_{l_{2}}^{m_{2}}\star_{1}^{n}Y_{l_{1}}^{m_{1}}\right) - 2i\{Y_{l_{1}}^{m_{1}},Y_{l_{2}}^{m_{2}}\}|| = \frac{1}{n}||R^{m_{1},m_{2}}_{l_{1},l_{2}}(n)|| \ ,
\Vert f_{t}\Vert_{L_{(\frac{T}{2},T)\times\mathbb{R}^{n}}^{\infty}}\leq\left(\prod_{k=1}^{\infty}C^{\frac{1}{2p_{k-1}}}(\vert\alpha\vert^{2}p_{k-1}^{2}+p_{k-1}^{\frac{2}{2-\gamma}}\Vert b\Vert_{L_{t}^{l}L_{x}^{q}}^{\frac{2}{2-\gamma}}\vert\alpha\vert^{\frac{2}{2-\gamma}}+\sigma\frac{4^{k}}{T})^{\frac{1}{2p_{k-1}}}\right)\Vert f_{t}\Vert_{L_{I_{0}\times\mathbb{R}^{n}}^{2}}.
\qquad\qquad\qquad\qquad\qquad\frac{dv}{dx}= - \frac{c\varepsilon^2HZ^4n}{m\nu_0^2 v^3}\left({0.96 + \log \frac{h\nu_0 v^4}{8\pi c\varepsilon^2HZ^4}}\right) ~~.\qquad\qquad\qquad\qquad (17^\prime)
\langle\pi_{[N]}(U^m(U^*)^n)\xi^{[N]},\eta^{[N]}\rangle = \sum_{\lambda\in\Lambda}\left\langle S_{[N]}^m\left(S_{[N]}^*\right)^nf_\lambda,g_\lambda\right\rangle= \left(S_{[N]}^m\left(S_{[N]}^*\right)^n T\right),
B = (X_{- \alpha_i}f)^N (v_1 \otimes v_2)\ & ={\displaystyle{\sum_{k_1 + k_2 = N}}}C_i(X_{-\alpha_i}f)^{k_1}v_1 \otimes (X_{-\alpha_i}f)^{k_2}v_2\in K,
f^{\#}(\omega)(x)\left(v_{1}\wedge\dots\wedge v_{r}\right) =\omega\left(f(x)\right)\left(Df(x)v_{1}\wedge\dots\wedge Df(x)v_{r}\right), =\omega\left(f(x)\right)\left(\bigwedge_{r}Df(x)\left(v_{1}\wedge\dots\wedge v_{r}\right)\right),
{\cal S}_{W Z W}\, = \, - \frac{k}{4 \pi}\int_{\Sigma}T r ( g^{- 1}d g \, g^{- 1}d g ) + \frac{k}{6 \pi}\int_{B}T r ( g^{- 1}d g \wedge g^{- 1}d g \wedge g^{- 1}d g ) \, ,
P(t)=a_{d_1}t^{d_1}+a_{d_1-1}t^{d_1-1}+\dots+a_{e_1}t^{e_1}, 1\le e_1\le d_1, a_{d_1}a_{e_1}\neq 0 Q(t)=b_{d_2}t^{d_2}+b_{d_2-1}t^{d_2-1}+\dots+b_{e_2}t^{e_2}, 1\le e_2\le d_2, b_{d_2}b_{e_2}\neq 0.
- ({\frac{e}{c}}) \int d \vec{y}\psi^{( 3 ) +}( y ) ) K_{i}( y , x ) ( i \hbar \vec{\partial}_{y^{i}}- ({\frac{e}{c}}) A_{i}^{[ 3 ]}( y ) ) \psi^{( 3 )}( y ) \left. \right] \psi^{( 3 )}( x ) ,
\pi(X) \leqslant T_{p_m}\cdot 2\cdot(1-\frac{1}{p_{m+1}})\cdot(1-\frac{1}{p_{m+2}})\cdots(1-\frac{1}{p_{m+K}}) &=2\prod_{i=1}^{m}p_i\cdot\prod_{i=1}^{m+K}(1-p_{i}^{-1})+m &=2(X-1)\cdot\frac{e^{-\gamma}}{\log X}\biggl(1+O(\frac{1}{\log X})\biggr)+m.
W_{\nu}^M(z,z') = e^{(v+v')(\frac{d}{2}+\nu)}\widetilde{W}_{\nu}\left( \eta\left(v,z \right) ,\eta'\left(v',z'\right)\right)= \frac{1}{2\pi^{\frac{d}2}}\frac{\Gamma(\nu+ \frac{d}2 )}{\Gamma(\nu+1)}\frac1{{[-(z-z')^2 ]}^{\frac{d}2+\nu}}.
\left( \sum_{i \in I}x_i \right)^d = \left( \sum_{i \in I}x_i \right)^{p^{s_1}+ \cdots + p^{s_w}}= \prod_{j=1}^{w}\left( \sum_{i \in I}x_i^{p^{s_j}}\right) = \sum_{t_1, \dots, t_w \in I}x_{t_1}^{p^{s_1}}\cdots x_{t_w}^{p^{s_w}}.
\varepsilon_0 = \max_{k\geq 1}|\lambda^{(1)}_{k + E}- \lambda^{(2)}_{k + E}|, \varepsilon_1 = \sum_{k\geq 1}k^{-4m/n}\|\partial_\nu\phi^{(1)}_{k + E}- \partial_\nu\phi^{(2)}_{k + E}\|_{L^2(\partial B_R)}, \varepsilon_2 = \sum_{k\geq 1}k^{-4m/n}\|\partial_\nu(\Delta\phi^{(1)}_{k + E}) - \partial_\nu(\Delta\phi^{(2)}_{k + E})\|_{L^2(\partial B_R)}.
\liminf_{N \to \infty}\;\displaystyle\sum\limits_{i = 1}^L \displaystyle\sum\limits_{j = 0}^{T_i-1}\frac{1}{2}\log\left( 1 + \tilde{g}(j)\right) \bigg / \displaystyle\sum\limits_{i = 1}^{L+1}T_i &\;\; = \;\; \mathbb{E}\left[ \displaystyle\sum\limits_{j = 0}^{T_1-1}\frac{1}{2}\log\left( 1 + \tilde{g}(j)\right)\right] \bigg / \mathbb{E}[T_1].
\int [ d \delta \phi ]_{e^{\phi}{\hat{g}}}~ e^{- S_{L}[ \phi ,{\hat{g}}] + S_{\mathrm{b o u n d a r y}}[ \phi ,{\hat{e}}]}= \int [ d \delta \phi ]_{{\hat{g}}}^{\prime}\int_{- \infty}^{\infty}d \phi_{0}~ e^{- S [ \phi_{R},{\hat{g}}]}\quad ,
\left(\beta^\ell-1\right)\beta^{k}\left(1-\frac{d_1}{\beta}-\cdots-\frac{d_k}{\beta^{k}}\right)-\beta^{\ell}\left(\frac{d_{k+1}}{\beta}+\cdots+\frac{d_{k+\ell}}{\beta^{\ell}}\right)=0.
L= \left(\begin{array}{cc|c}-2 \phantom{-}1 \phantom{-}2 \phantom{-}1 -2 \phantom{-}2 \hline \phantom{-}1 \phantom{-}1 -4 \end{array}\right), \widehat L= \left(\begin{array}{cc|c}-1 \phantom{-}2 -2 \phantom{-}2 -1 -2 \hline -1 -1 \phantom{-}4 \end{array}\right).
\begin{cases}\phi(x) \begin{displaystyle}\frac{\partial p}{\partial t}\end{displaystyle}= \nabla \cdot (K(x,|\nabla p|)\nabla p) U\times(0, \infty), p=\psi \Gamma \times (0, \infty), p(x,0)=p_0(x) U,\end{cases}
F(G) = \left\lfloor \frac{6k-1}{2}\right\rfloor a + 2 (6k) = \left\lfloor \frac{4t-1+2 \epsilon}{2}\right\rfloor a + 2 (6k) & = (2t+ \epsilon-1) a + 2 (6k) = (2 t + \epsilon + 1) a -2.
\tilde{\Psi}_{K_{Y}}( Y , T ) = \frac{1}{L}\sum_{n = - \infty}^{\infty}e^{\frac{\pi^{2}n}{H_{5}L}}H_{2 i \pi n / H_{5}L}^{( 2 )}( m t ) e^{i{\frac{2 \pi n}{L}}( y - H_{5}^{- 1}\sinh^{- 1}( \frac{K_{Y}}{m}) )}.
h_{1}^{-1}(U) = h_{1}^{-1}((\mbox{int}(K(h_{2})))\setminus K(h_{1})) = (\mbox{int}(h_{1}^{-1}(K(h_{2}))))\setminus h_{1}^{-1}(K(h_{1})) = \mbox{int}(h_{1}^{-1}(K(h_{2}))))\setminus K(h_{1}) \subset (\mbox{int}(K(h_{2})))\setminus K(h_{1})=U.
\mathcal{Y}:= \left [ L^{2}(0,T;H^{2}) \cap H^{1}(0,T;(H^{2})^{*}) \cap C^{0}([0,T];L^{2}) \right ] & \times L^{2}(Q) \times \left [ L^{\infty}(0,T;H^{1}) \cap H^{1}(0,T;L^{2}) \right ].
c = c_0 + \varepsilon^2, \psi^c(x,y) = \varepsilon^2 A(\varepsilon x)\varphi_0(y) + O(\varepsilon^4), \rho^c(x,y) = \bar{\rho}(y) - \frac{1}{c_0}\varepsilon^2 A(\varepsilon x) \bar{\rho}'(y) \varphi_0(y) + O(\varepsilon^4),
(\gamma^{(1)})'(s^{(1)}(t))=\frac{A^{-1/2}(r(t))\gamma'(s(t))}{|A^{-1/2}(r(t))\gamma'(s(t))|}\,,n^{(1)}(s^{(1)}(t))=\frac{A^{1/2}(r(t))n(s(t))}{|A^{1/2}(r(t))n(s(t))|}\, &\dot s^{(1)}(t)=|A^{-1/2}(r(t))\gamma'(s(t))|\dot s(t)\,,D\chi(r(t))=A^{-1/2}(r(t))\,,
\displaystyle{E^{T E}( a ) = \left. \frac{\hbar c}{a}\sum_{\nu = 3 / 2}^{\nu_{0}}\nu \left[ \zeta_{\nu}^{T E}( s , 0^{+}) - \zeta_{\nu}^{T E}( s , x ) \right] \right|_{s \rightarrow - 1}}.
\frac{\delta{\cal W}}{\delta \psi^{\alpha}}= \frac{\delta \Gamma}{\delta \psi^{\alpha}}- \frac{\delta \Gamma}{\delta \Sigma^{m}}\sigma_{, \alpha}^{m}( \varphi , \psi ) , \; \; \; \frac{\delta{\cal W}}{\delta \phi_{A}^{*}}= \frac{\delta \Gamma}{\delta \phi_{A}^{*}}\, ,
\rho_{(1),\infty}^{(\widetilde{\rm cJ})}( x ;\beta,p,q) := \lim_{N \to \infty}{2 \pi \over N}\rho_{(1),N}^{(\widetilde{\rm cJ})}( 2\pi \chi_{x < 0}+ 2 \pi x/ N ;\beta,p,q).
\varphi^{(8)}_1(\tau,z) :=\varphi_1^{(3)}(\tau,z)\varphi_1^{(6)}(\tau,z)-5\varphi_1^{(4)}(\tau,z)\varphi_1^{(5)}(\tau,z), \varphi^{(8)}_2(\tau,z) :=\varphi_1^{(4)}(\tau,z)\varphi_1^{(5)}(\tau,z)-\varphi_1^{(8)}(\tau,z).
\mu_0(\varepsilon, \Omega):= \max\, \Big\{\,\dim S:\alpha(S):= \sup_{u\in S\setminus\{0\}}\dfrac{\Vert u| E^m_{p,\sigma}(\Omega)\Vert}{\Vert u| L_p(\Omega)\Vert}\leq \frac{1}{\varepsilon}\,\Big\}
\mathrm{}[ \hat{\Phi}( x^{0}, x^{1}) \, , \, \hat{\Phi}( x^{0}, y^{1}) ] \, = \, - \frac{i}{2}\left\{\frac{1}{i \pi}{\sum_{n = - \infty}^{+ \infty}}^{\prime}\left( \frac{\pi}{R}\right) \frac{e^{\frac{i n \pi}{R}( x^{1}- y^{1})}}{\frac{n \pi}{R}}\right\}.
\bigoplus^{d}_{i=0}H^{2i}({X})(i) \cong \bigoplus^{d}_{i=0}H^{2i}({Y})(i), \bigoplus^{d}_{i=1}H^{2i-1}({X})(i) \cong \bigoplus^{d}_{i=1}H^{2i-1}({Y})(i).
H_NS_N(i,j) =\frac{1}{J_N}(-J_1J_{N-j}(J_{j-2}+J_{j-1})+4J_2J_{N-j}(J_{j-4}+J_{j-3})) =\frac{1}{J_N}(-2J_{N-j}(J_{j-3}+J_{j-2})+4J_{N-j}(J_{j-4}+J_{j-3})) =\frac{2J_{N-j}}{J_N}(2J_{j-4}+J_{j-3}-J_{j-2}) =0.
\Pi^{\alpha}{}_{\beta}\equiv ( \gamma^{1}\bar{\gamma}^{2}\gamma^{3}\bar{\gamma}^{4})^{\alpha}{}_{\beta}\, , \qquad ( \Pi^{\prime})^{\alpha}{}_{\beta}\equiv ( \gamma^{5}\bar{\gamma}^{6}\gamma^{7}\bar{\gamma}^{8})^{\alpha}{}_{\beta}\, .
\tilde{\tau}_m(\xi\times 1)\tilde{\tau}_m^{-1}=\left[\begin{array}{cccccc}-h -l -d+1 d \frac{l\zeta}{2}0 -\zeta^{-1}g -\zeta^{-1}(e-1) -\zeta^{-1}f \zeta^{-1}f \frac{e+1}{2}0 a-1 b c -c \frac{-b\zeta}{2}1\end{array}\right].
\left( -\frac{d^{2}}{d\varphi_2^{2}}+\frac{9\lambda_2}{2\sin^{2}(3\varphi_2 )}\right) \Phi_{n_2}^{(2)}(\varphi_2 )=B_{n_2}^{(2)}\Phi_{n_2}^{(2)}(\varphi_2 ),
v_{\alpha}(\sigma,\xi) \leq M \sum_{k=1}^{\infty}b_k^2 \int_0^{\infty}e^{-z}\dfrac{z^{-2\alpha}}{2\mu_k^{1-2\alpha}}c_k^2 dz \leq M \sum_{k=1}^{\infty}b_k^2 \mu_k^{2\alpha -1}c_k^2 < \infty.
{}_2F_1\bigg[\begin{matrix}-a-p p-a &1\end{matrix}\bigg|\,z\bigg]\equiv\sum_{k=0}^a\frac{(-a)_k^2}{(1)_k^2}\cdot z^k-z^p\sum_{k=0}^a\frac{(-a)_k^2}{(1)_k^2}\cdot\big(1+2p(H_{a-k}-H_k)\big)\cdot z^k\pmod{p^2}.
\frac{\underline{q}_i}2|x|^\rho<q_i(x)<2 \bar{q}_i|x|^\rho,\ \ \frac12\underline{p}<p(x)<2\bar{p},\ \ \frac{\underline{q}_i}{2\bar{p}}{|x|^\rho}<{u_i(x)}<\frac{2\bar{q}_i}{\underline{p}}{|x|^\rho},\ \i=1,2.
E^I_{R, l}\cap E^J_{R, l}\subseteq C^{i_{l - 1}, i_{l}}_{R^{s_l}}\cap B^{i_{l - 1}, i_l + 1}_{R^{s_l}}\cap C^{i_{l - 1}, j_{l}}_{R^{s_l}}\cap B^{i_{l - 1}, j_l + 1}_{R^{s_l}}= \emptyset.
f'(t)\leq \frac{f(dt)-f(t)}{dt-t}\leq \frac{\frac{1+\eta}{2}f''(0)d^2t^2-\frac{1-\eta}{2}f''(0)t^2}{dt-t}=\left(\frac{1+d}{2}+\eta\frac{1+d^2}{2(d-1)}\right)f''(0)t = \left(1+\sqrt{\eta}+\frac{\sqrt{\eta}}{2}+\eta+\eta^{3/2}\right)f''(0)t\leq \left(1+\frac{7}{2}\sqrt{\eta}\right)f''(0)t\quad.
\begin{aligned}\mathcal{F}_1= \{d_1=d_2=m_1=m_2= 0,l_1=-13/4,l_2=-3/2,n_2=-1/2\}, \mathcal{F}_2= \{d_1=d_2=m_1=m_2= 0,l_1=9/4,l_2=1/2,n_2=3/2\}, \mathcal{F}_3^\pm= \{d_1=d_2=m_1=m_2= 0, &l_1=(\pm 1+6 l_2+f(l_2))/4,n_2= 1/2\pm f(l_2)/5,l_2\not\in \mathcal{L}\},\end{aligned}
\beta_{i}^2 =\max\left\{0, \min\left\{\frac{y_{ii}\left(\frac{1}{y_{ii}}-\frac{\epsilon_{1,i}}{\rho}\right)}{(1+\epsilon_{1,i})},\frac{x_i-\frac{\epsilon_{1,i}}{\rho}}{x_i(1+\epsilon_{1,i})}, \right\}\right\}.
B(L_1,\dots,L_t) =\left\{\left[\begin{array}{c|ccc|c}1 a c \hline I_r \begin{array}{ccc}L_1b^t \cdots L_t b^t\end{array}\hline I_t \end{array}\right]: \begin{array}{l}a\in \mathbb{Z}_p^r b\in\mathbb{Z}_p^s c\in\mathbb{Z}_p^t\end{array}\right\}.
\hat{\boldsymbol{\theta}}=\arg\underset{\boldsymbol{\theta}}{\min}\{-\log(p(\mathbf{y}\vert\mathbf{U},\boldsymbol{\theta})\}=\arg\underset{\boldsymbol{\theta}}{\min}\{-\log(\mathcal{N}(\mathbf{y};\mathbf{\mathbb{\boldsymbol{\mu}}}(\mathbf{U}),\mathbf{K}_{\mathrm{u}})\}.
|\widetilde S_{ki}| \leq \xi\, ||z||_{L^{\infty}}\langle |\psi_k|, |\psi_i|\rangle_{L_z^2}\leq \xi\, C_z\, ||\psi_k||_{L^2_z}\, ||\psi_i||_{L^2_z}=C_2(\xi), \qquad k\neq i,
{\cal L}= -{\kappa \over 2}\epsilon_{ij}A^i \dot{A}^j + i \phi^* \dot{\phi}-A^0(\phi^* \phi - \kappa F_{12}) -{1 \over 2}{|{\vec{D}}\phi |}^2 +{1 \over{2 \kappa}}(\phi^* \phi )^2 ,
\rho(\eta_{i_j}| \eta_{[i_{j+1}, i_k]}\eta_{\Lambda \setminus \Delta}\omega_{\Lambda^c\cup [i_1, i_{j-1}]}) = \gamma_{i_j}(\eta_{i_j}| \eta_{[i_{j+1}, i_k]}\eta_{\Lambda \setminus \Delta}\omega_{\Lambda^c\cup [i_1, i_{j-1}]}).
\nabla_{\lambda}\mathcal{L}= \prod_{j=1}^n x_j- \prod_{j=1}^n y_j=0 \Leftrightarrow\prod_{j\not = i}y_j = \frac{\prod_{k=1}^nx_k}{y_i}, \nabla_{y_i}\mathcal{L}=\ln y_i -\lambda \prod_{j\not = i}y_j=0, \ \i \in N
E_{\gamma^1,\gamma^2}\biggl[ E_{\gamma^1,\gamma^2}\biggl[ \bigl( u^1 - u^2 \bigr)^2 \biggl| u^1 \biggr] \biggr] = E_{\gamma^1,\gamma^2}\biggl[ \bigl( u^1 - u^2 \bigr)^2 \biggr]
\|R_{\alpha}\tilde{v}\|_{H^s_{\overline{W}}}\leq \frac{1}{\alpha}\|\overline{v}\|_{L^2(\Omega)}, & \|A R_{\alpha}(\tilde{v})+ \sum\limits_{l=1}^{m}\tilde{\beta}_l z_l -\overline{v}\|_{L^2(\Omega)}= \|A R_{\alpha}(\tilde{v})-\tilde{v}\|_{L^2(\Omega)}\leq \|\overline{v}\|_{H^{s}_{\overline{\Omega}}}\left(\log\left(1/\alpha\right)\right)^{-\mu}.
\mathcal{G}^{\varepsilon}\nabla\theta^0 = \varepsilon\nabla\cdot(\mathbf{V}^{\varepsilon}\nabla\theta^0) - \varepsilon \mathbf{V}^{\varepsilon}\Delta\theta^0 - \varepsilon (\nabla_{x}\mathbf{V})^{\varepsilon}\nabla\theta^0.
\mathrm{d}Z^{v,\pi}(t)= Z^{v,\pi}(t) \pi'(t,\mathfrak{X})\sigma(t,\mathfrak{X}) \left[\vartheta(t,\mathfrak{X})\, \mathrm{d}t+ \mathrm{d}W(t) \right] \mathrm{with}Z^{v,\pi}(0)=v\,.
X V I_{c}^{0}\, , \, \, X V I I_{a}^{0}\, , \, \, X V I I_{b}^{0}\, , \, \, X V I I I_{a}^{0}\, , \, \, X V I I I_{b}^{0}\, , \, \, X V I I I_{c}^{0}\, , \, \, X X I^{0}\, , \, \, X X I I^{0}- X X I X^{0}\, \,
H_p(t;w,W) = \left \langle R_{\mathcal{S}_p(1)}(e_W\circ D_t), R_{\mathcal{S}_p(1)}(e_w\circ D_t)\right\rangle_{\mathcal{S}_p(1)}= \left \langle t^{1/2\mu}R_{\mathcal{S}_p(1)}(e_{\widehat{\rho}_t(W)}), t^{1/2\mu}R_{\mathcal{S}_p(1)}(e_{\widehat{\rho}_t(w)}) \right\rangle_{\mathcal{S}_p(1)}&= t^{1/\mu}H_p\left(1;\widehat{\rho}_t(w),\widehat{\rho}_t(W)\right),
\alpha_2(e_1)=e_1,\alpha_2(e_2)=\frac{b(1-a)}{a}e_1+a e_2, \beta_2(e_1)=e_1,\beta_2(e_2)=be_1+(1-a) e_2, \mu_2(e_1,e_1)=e_1, \mu_2(e_1,e_2)= be_1+(1-a) e_2, \mu_2(e_2,e_1)=\frac{b(1-a)}{a}e_1+ae_2, \mu_2(e_2,e_2)=\frac{b}{a}e_2,
W^{\prime}( r )^{2}= \frac{1}{V^{2}( r )}\left( \Delta_{L}E^{2}- V \left( m^{2}+ \frac{4 j^{2}}{r^{2}}\right) - \frac{4 \Delta}{r^{4}}a j_{R}E - \frac{4 g^{2}a^{2}}{r^{6}}j_{R}^{2}\right) \, ,
{\widetilde a}= a -{\frac{b c}{2 \pi^{2}}}\int_{k^{2}= \mu^{2}}^{k^{2}= \Lambda^{2}}{\frac{d^{2}k}{k^{2}}}= a -{\frac{b c}{2 \pi}}\ln \left({\frac{\Lambda^{2}}{\mu^{2}}}\right) ~ .
\Psi'(\varphi^{w}) - \Psi'(\overline{\varphi}) - \Psi''(\overline{\varphi}) \Phi^{w}= \Psi''(\overline{\varphi}) \theta^{w}+ (\varphi^{w}- \overline{\varphi})^{2}R_{1}^{w}, h(\varphi^{w}) - h(\overline{\varphi}) - h'(\overline{\varphi}) \Phi^{w}= h'(\overline{\varphi}) \theta^{w}+ (\varphi^{w}- \overline{\varphi})^{2}R_{2}^{w},
\psi_0(x)=0 x\le v, &0<\psi_0(x)<\infty v < x\le 1, &\psi_0(x)=\infty 1< x, & \psi_0 x\in[0,1], & \psi_0 x\in[0,1], & \psi_0 x\in[v,1].
\mathbf{F}_{1\bar{2}}\mathbf{F}_{\bar{2}1}=\mathbf{F}_{1\bar{2}\,\bar{4}3}+\mathbf{F}_{1\bar{4}\,\bar{2}3}+\mathbf{F}_{1\bar{4}3\bar{2}}+\mathbf{F}_{\bar{4}1\bar{2}3}+\mathbf{F}_{\bar{4}13\bar{2}}+\mathbf{F}_{\bar{4}31\bar{2}}
\langle \, : a_n^i \, (a_m^j)^{\dagger}: \, \rangle_{\beta , \gamma}=\langle \, : (a_m^j)^{\dagger}\, a_n^i : \, \rangle_{\beta , \gamma}=\frac{\delta^{ij}\, \delta_{nm}}{e^{\epsilon_n}- 1}\, ,
u(X) =\sum_{j=1}^{m}\int_{\partial D}\mathcal{K}_{j}(X,Q)\widetilde{g}_{j-1}(Q)d\sigma(Q), &=\sum_{j=1}^{m}\mathcal{M}_{j}\widetilde{g}_{j-1}(X),\,\, X\in D,
\frac{P_{{E}'/X}(n)}{\alpha_{\dim({E}')}({E}')}\le \frac{P_{{E}/X}(n)}{\alpha_{\dim({E})}({E})}\quad \quad \left(\frac{P_{{E}'/X}(n)}{\alpha_{\dim({E}')}({E}')}< \frac{P_{{E}/X}(n)}{\alpha_{\dim({E})}({E})}\right)\,, \qquad(n\gg 0)\,,
0 \leq \Big( \ln A_{n,m}-\frac{n-2m}2 \ln (\varepsilon + \frac{|x|^2}{\varepsilon}) \Big) |x|^\alpha &\leq \Big( \ln A_{n,m}-\frac{n-2m}2 \ln \varepsilon \Big) a_\varepsilon^\alpha &= O(\varepsilon^{\alpha /2}(-\ln \varepsilon)) & = o(1)_{\varepsilon \searrow 0}.
\begin{cases}\ h(R,\omega) = (c_1+c_2\log R)R^{\gamma_1^+}w_1(\omega) + O_2(R^{\gamma_1^+ -\epsilon}) \ h(R,\omega) = c_1 R^{\gamma_1^-}w_1(\omega) + O_2(R^{\gamma_1^- -\epsilon}) \end{cases}
\bar{B_1}=\frac{a_{0,\sup}(b_{2,\inf}-\frac{l\chi_2}{d_3})+\frac{l\chi_1}{d_3}b_{0,\sup}}{(a_{1,\inf}-\frac{k\chi_1}{d_3})(b_{2,\inf}-\frac{l\chi_2}{d_3}) -\frac{lk\chi_1\chi_2}{d_3^2}}
\alpha\xi^{m}-\beta \eta^{m}+q_{m}(\xi,\eta)= \eta^{m}\left(\frac{\alpha\xi^{m}}{\eta^{m}}-\beta +\frac{q_{m}(\xi,\eta)}{\eta^{m}}\right) \geq \frac{-\beta}{2}\eta^{m}>c |\eta|.
\frac{d}{dt}G(v(t)) + \frac{b_0}{16}Z_0(v(t)) \leq \frac{1}{2}\sum_{\mu=0}^{L-1}\int_{\partial \Omega_\lambda}h \cdot \sigma | \sigma \cdot \nabla \partial_t^\mu v|^2 dS + \bar{C}_\lambda \sum_{\mu=0}^{L-1}\langle \partial_t^\mu F_\lambda, \partial_t^{\mu+1}v + \partial_t^\mu v + [h ; \nabla \partial_t^\mu v] \rangle ,
\sum_{j=0}^J d_j^{-1+N/2}\|\tilde g - g_h\|_{L^2(\Omega_h \cap A_j)}\le\; C(hd_0^{-1}) \Big( \sum_{j=0}^J d_j^{N/2}\|\tilde g - g_h\|_{H^1(\Omega_h\cap A_j)}+ \|\tilde g - g_h\|_{W^{1,1}(\Omega_h)}\Big) + Ch^{3/2 - m},
k_{1}^{4}\, k_{2}^{2}\, = \,{\operatorname*{d e t}}_{G}\left( \frac{d x_{\mu}}{d s \,}, \frac{d^{2}x_{\mu}}{d s^{2}\,}, \frac{d^{3}x_{\mu}}{d s^{3}\,}\right) ,
= \frac{M_{i}+ \eta_{i}}{\sqrt{2}}\left( \begin{array}{c}{\psi_{l R}}\\{\psi_{l}+ \psi_{u}}\\ \end{array}\right) = \frac{M_{i}+ \eta_{i}}{\sqrt{2}}\left( \begin{array}{c}{\psi_{1 R}}\\{\sqrt{2}\psi_{1 L}+ \psi_{1 R}}\\ \end{array}\right) .
\Pr[X_1=1|\mathcal{B}]=\Pr[X_2=1|\mathcal{B}]=3p^2(1-p)+p(1-p)^2=\frac{1}{2}, &\Pr[X_1=X_2|\mathcal{B}]=\Pr[X_1=1|\mathcal{B}]\Pr[X_2=1|\mathcal{B}]+\Pr[X_1=0|\mathcal{B}]\Pr[X_2=0|\mathcal{B}]=\frac{1}{2}\times\frac{1}{2}+\frac{1}{2}\times\frac{1}{2}=\frac{1}{2}.
\mu_{R S}\hspace *{2 m m}^{A}= \eta^{A}\; \frac{A_{i j}}{2}n_{R}^{i}n_{S}^{j}= \eta^{A}\; \frac{1}{2 F^{2}}\xi_{[ 0 , i}\xi_{j ]}n_{R}^{i}n_{S}^{j},
T(r)\geqslant\dfrac{1}{2n+1}\sum_{i=1}^{2n+1}(3N(r,\min\{\nu_{(f^u,H_i),\leqslant k_i};1\leqslant u\leqslant 3\}) &+(4n-1)N^{(1)}_{(f,H_i),\leqslant k_i}(r))+o(T(r)).
\begin{array}{l}Q_a = \int dx^- d^2x_{\bot}(f^{abc}A_b^i \partial^+A_c^i + 2 \psi_+^{\dagger}T^a \psi_+) ~~~~~= \int dx^- d^2x_{\bot}(\rho_a^g(x^-,x_{\bot})+ \rho_a^q(x^-,x_{\bot})) \end{array}
\widehat{\eta}^2(T) := |T|^{2{m}/3}\Vert \widehat{\lambda}_h \widehat{u}_{\mathrm{nc}}\Vert^2_{L^2(T)}+|T|^{1/3}\sum_{F\in\widehat{\mathcal{F}}(T)}\Vert [{D}^{m}_{\mathrm{pw}}\widehat{u}_{\mathrm{nc}}]_F\times \nu_F\Vert^2_{L^2(F)}.
\hat{\varphi}(\xi) = \left( 1 + \frac{a^2b(i\xi)^6}{(1-a^2b)(i\xi)^6+r(i\xi)^4+(c+1)\lambda(i\xi)^3+r\lambda(i\xi)+c\lambda^2}\right)\frac{\left(\alpha+\beta e^{-iL\xi}\right)}{(i\xi)^3+\lambda},
[ \hat{A}^{i}, \hat{Q}^{a}] = 0 , \; [ \hat{\pi}_{( A ) i}, \hat{Q}^{a}] = i \{\varepsilon_{0 i j k}( \partial^{j}\hat{C}^{k a}+ \hat{A}^{j}\wedge \hat{C}^{k a}) + \hat{\pi}_{( C ) i b}\wedge \hat{C}^{a b}- \varepsilon^{a b}\hat{\pi}_{( C ) o b}\wedge \hat{B}_{o i}\},
q_k:= \int_{I_k}e^{-\gamma_k (t_k + \sigma_k t)}|\varphi'(t)|^2 + \sigma_k^2 \Bigl(\beta_k^2 \lambda e^{-\gamma_k (t_k + \sigma_k t)}|\varphi(t)|^2 - \bigl \langle U_k(t_k + \sigma_k t) \varphi (t),\varphi (t) \bigr \rangle\Bigr)\,dt <0
\begin{aligned}\mbox{Case A}:~\exists~ t^* \in [t_2, t_2 + \tau + 1] \mbox{such that}~v^k_{j}(t^*)\leq \Delta(t_1). & \mbox{Case B}:~\forall~t \in [t_2,t_2+\tau+1], v^k_{j}(t) > \Delta(t_1).\end{aligned}