harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\limsup_{n\rightarrow\infty}\nu_{n}(E,\mathrm{i}(y+2\epsilon))\leq \frac{1}{\epsilon}(L(E,\mathrm{i}(y+3\epsilon))-L(E,\mathrm{i}(y+2\epsilon)))=\omega^{+}(E,\mathrm{i}y).
g_1(\lambda,s) =\begin{cases}\overline{f_1}(2\lambda,s), 0\leq \lambda\leq\frac{1}{2}\overline{f_1}(1,s), \frac{1}{2}\leq \lambda\leq 1\end{cases}g_2(\lambda,s) =\begin{cases}\overline{f_2}(0,s), 0\leq \lambda\leq\frac{1}{2}\overline{f_2}(2\lambda-1,s), \frac{1}{2}\leq \lambda\leq 1\end{cases}.
\begin{cases}-\Delta v_\lambda (0,c_0) + \frac{1}{|\Omega|}\{\int_\Omega (mc_0 +a c_0^{p-1}) + \int_{\partial \Omega}b c_0^{q-1}\}= mc_0 + a c_0^{p-1}\mbox{in $\Omega$}, \dfrac{\partial v_\lambda (0,c_0)}{\partial \mathbf{n}}= b c_0^{q-1}\mbox{on $\partial \Omega$}, \end{cases}
{\frac{2 e^{\left({\frac{\cal H}{\hbar}}+{\frac{1}{2}}\right) \tau}}{e^{\tau}+ 1}}= \sum_{n = 0}^{\infty}{\frac{\tau^{n}}{n !}}E_{n}\left({\frac{\cal H}{\hbar}}+{\frac{1}{2}}\right) \quad ( | \tau | < \pi ) .
\frac{1}{n}\,\sum_{k=1}^n \big|X^{(n)}_{(k)}(s)-\bar{X}^{(n)}_{(k)}(s)\big|^p=W_p\big(\rho^{(n)}(s),\bar{\rho}^{(n)}(s)\big)^p\le \frac{1}{n}\,\sum_{i=1}^n \big|X^{(n)}_i(s)-\bar{X}^{(n)}_i(s)\big|^p
i\int_{[s, s+1] \times Y_+}K^2ds \wedge \pi_+^*dt \wedge (K^{-1}*_3 E_A) + i\int_{[s, s+1] \times Y_+}K^2ds \wedge \pi_+^*dt \wedge (F_{A(s)}+ \frac{1}{2}F_{A_{K^{-1}}}) = \int_{[s, s +1] \times Y_+}r(1-|\alpha|^2) + \mathfrak{e},
g(T):= \|v_{k}\|_{C_{T}L_{x}^{3}\cap L_{T}^{4}L_{x}^{6}}+\big\|\nabla|v_{k}|^{\frac{3}{2}}\big\|^{\frac{2}{3}}_{L_{T}^{2}L_{x}^{2}}, a:= \|v_{k}(\cdot,0)\|_{L^{3}},\;\, b(T):=e^{C_{2}\int_{0}^{T}\|w\|_{L^{6}}^{4}dt}.
\lim_{m\to+\infty}\int_{X_j}v''_{\epsilon}(\Psi_m)|fF^{1+\delta}|^2_{h_{m'}}e^{-\phi-\delta M_{\eta_m}}= \int_{X_j}v''_{\epsilon}(\Psi)|fF^{1+\delta}|^2_{h_{m'}}e^{-u(-v_{\epsilon}(\Psi))-\delta \max{\{\psi+T,2\log|F|\}}}.
Cl_{3,0}~[\sim M_2({\cal C})]~,~~~ Cl_{1,2}~[\sim M_2({\cal C})]~,~~~ Cl_{0,3}~[\sim{\cal H}\oplus{\cal H}]~,~~~ Cl_{2,1}~[\sim M_2({\cal R}) \oplus M_2({\cal R})]~.
\delta_tv^{n-\frac{1}{2}}+ \frac{1}{k_n}\int_{t_{n-1}}^{t_n}Av(t)dt + \frac{1}{k_n}\sum\limits_{j=1}^m\int_{t_{n-1}}^{t_n}(\omega_{\alpha_j}*B_jv)(t)dt - \frac{\kappa}{k_n}\int_{t_{n-1}}^{t_n}v(t)dt = \frac{1}{k_n}\int_{t_{n-1}}^{t_n}g(t)dt,
\left(wq^{4};q^{4}\right)_{\infty}\sum_{n=0}^{\infty}\frac{q^{12n^{2}}\left(-w^{2}\right)^{n}}{\left(q^{8},wq^{4},wq^{8};q^{8}\right)_{n}}=\left(-wq^{2};q^{4}\right)_{\infty}\sum_{n=0}^{\infty}\frac{q^{3n^{2}}\left(-q^{-1}w\right)^{n}}{\left(q^{2};q^{2}\right)_{n}\left(-wq^{2};q^{4}\right)_{n}}.
t_1 = \frac{-x_1+x_2+x_3-x_1x_2x_3}{1-x_1x_2+x_2x_3-x_3x_1}, t_2 = \frac{x_1-x_2+x_3-x_1x_2x_3}{1-x_1x_2-x_2x_3+x_3x_1}, t_3 = \frac{x_1+x_2-x_3-x_1x_2x_3}{1+x_1x_2-x_2x_3-x_3x_1}.
\mathfrak{X}_j^+(v, \zeta) * \mathfrak{X}_i^+(u, \lambda)= (-1)^{a+1}g_{\zeta_j}(z_j-v) g_{\lambda_i}(z_i-u) \prod_{m\in S}\vartheta(z_{i}-z_j+m\frac{\hbar}{2}) = - g_{\zeta_j}(z_j-v) g_{\lambda_i}(z_i-u) \prod_{m\in S}\vartheta(z_j-z_i-m\frac{\hbar}{2})
\begin{aligned}\frac{\partial u(x_i,y_j,t)}{\partial t}-\varepsilon_x\sum\limits_{m=0}^{M_x}{a_{im}^{(\beta_1)}u(x_m,y_j,t)}&\ -\varepsilon_y\sum\limits_{m=0}^{M_y}{b_{jm}^{(\beta_2)}u(x_i,y_m,t)}=f(x_i,y_j,t),\end{aligned}
\texttt{p}^\dagger_t(x,y) = \texttt{p}_t(y-x) - \mathbf{P}_{0,x}[\mathbf{1}_{\{\texttt{k}^\Omega<t\}}\texttt{p}_{t-\texttt{k}^\Omega}(y-\vartheta_{\texttt{k}^\Omega})], x,y\in\Omega.
c(x,\xi)\widetilde{\Gamma}(\frac{x+p'(\xi)}{\sqrt{h}}) \sharp (x+p'(\xi)) = c(x,\xi)\widetilde{\Gamma}(\frac{x+p'(\xi)}{\sqrt{h}}) (x+p'(\xi)) & + \frac{h}{2i}\left\{c(x,\xi)\widetilde{\Gamma}(\frac{x+p'(\xi)}{\sqrt{h}}), (x+p'(\xi))\right\}+ h r_1\, ,
\xi_2'(y) + \frac{p}{p-1}\xi_2(y)^{(p-1)/p}\ge \psi_1'(y) + M +\frac{p}{p-1}\psi_1(y)^{(p-1)/p}& \ge M(1-y) + \frac{p a_1^{2-p}}{p-1}(1-y) = \frac{p a_2^{2-p}}{p-1}(1-y) & = \psi_2'(y) + \frac{p}{p-1}\psi_2(y)^{(p-1)/p}\ .
\frac{1}{\pi}\int_{1}^{\infty}\frac{| D ( \nu , \alpha ) |}{\alpha + \tau}d \alpha \leq \frac{\frac{1}{\pi}\int_{1}^{\infty}\omega ( \nu , \alpha ) / \alpha d \alpha}{1 + \frac{1}{\pi}\int_{1}^{\infty}\frac{\omega ( \nu , \beta )}{\beta}d \beta}< 1 .
\begin{array}{lllllllllll}0=\omega(C)-\omega(C) \le |E_{f=(0,0)}\cap E(C)|+\Omega|E(C)-E_{f=(0,0)}|-\omega(C) &\le \frac{\omega(C)}{4}+\Omega(\frac{\omega(C)}{4}+1-\Omega)-\omega(C) &\le \frac{1}{64}\left(\omega(C) -20\right)^2-6<0,\end{array}
\int_\mathbb{T}|u|^{p-1}(x,t)\,dx = 2\pi\sum_{0=k_1-k_2+\cdots -k_{p-1}}\widehat{u}_{k_1}\overline{\widehat{u}}_{k_2}\cdots\widehat{u}_{k_{p-2}}\overline{\widehat{u}}_{k_{p-1}}= \int_\mathbb{T}|v|^{p-1}(x,t)\,dx,
\big [ \big ( C_{1}^{[ r ]}\big )_{2 V , 2 V}\big ]_{\Lambda = 0}= \frac{1}{P^{2}}\Big \{\sum_{n = 1}^{r}B_{n}^{[ r ]}( \Lambda = 0 ) \frac{t ( P )}{u_{r , 2 n + 1}}B_{n}^{[ r ]}( \Lambda = 0 ) \Big \}.
\|e_{m+1}\|_{\mathcal{PG}^{\omega}}=\|e_{m+1}\|+\left(\sum_{n=1}^{m}\left(\omega(n)\|e_{m+1}\|\frac{1}{n}\right)^{q}\right)^{\frac{1}{q}}\ge \|e_{m+1}\|\omega(1)\left(\sum_{n=1}^{m}\frac{1}{n}\right)^{\frac{1}{q}}.
l(E_{T_J}) = \frac{1}{2}|L(E_{T_J}, t)| & = \frac{1}{2}\Big|\bigsqcup_{K \in [J]}c_{K,J}X(S,J) \Big(\bigsqcup_{u \in \lfloor t \rceil}c_{u,t}N_{W_J}(W_{\{s\}})^v \Big)\Big| & = |\lfloor t \rceil| \cdot |[J]| \cdot |X(S,J)| \cdot |X(J,\{s\})| \frac{1}{2}|N_{W_J}(W_{\{s\}})| = |X(J,\{s\})|,
\lim_{m \to \infty}\frac{1}{m}\sum_{j = -m/2+1}^{m/2}\frac{e^{2 \pi i \frac{j + \theta_1/2}{m}\lfloor sm \rfloor}}{1 - \exp \left\{- 2 \pi i \frac{j + \theta_1/2}{m}\right\}+ \frac{z}{m}}= \frac{e^{- z s}}{1 + (-1)^{\theta_1 - 1}e^{-z}}.
\vert \tilde y - y \vert = \left \vert \pi_{L(z_0,12r_j)}^\perp(y-z_0) \right \vert \leq \left \vert \pi_{L(z_0,12r_j)}^\perp (y-w') \right \vert + \left \vert \pi_{L(z_0,12r_j)}^\perp (w'-z_0) \right \vert \leq \vert y - w'\vert + 12 r_j \delta.
\begin{matrix}min&p\\subjectto&\Omega_{b}(A)\le pQ,\\ &A\in S_{n}(A_{1\cdot\cdot\cdot n},C_{1\cdot\cdot\cdot n}),\\ &Q\in co-S_{n}(B_{1\cdot\cdot\cdot n},D_{1\cdot\cdot\cdot n})\end{matrix}
\sup_{x,y\in[0,1]}\bigg|\frac{1}{T}\int_0^T\zeta_N\left(\frac{1}{2}+it+ix\right)\zeta_N\left(\frac{1}{2}-it-iy\right)dt \; -\prod_{k=1}^N \sum_{j_k=0}^{n_0}p_k^{-(1+i(x-y))j_k}\bigg|< \frac{\varepsilon}{2}.
I=\langle ((x_2)_1^m-m!(x_2)_m)j,\;((x_6)_1^m-m!(x_6)_m)k, & (x_2)_a(x_6)_b(\gamma_1^2+\gamma_2^2),\;(x_2)_a(x_6)_b\gamma_1^4,\;2(x_2)_a(x_6)_b(y_1\gamma_1+y_2\gamma_2),\;(x_6)_b4y_1\gamma_1^3)\rangle
\frac{dQ_c}{dP}= \frac{1}{D}\left[ (M(P) - c) - \frac{f(P)}{Q}\right] > \frac{1}{D}\left[ (M(P) - \widehat{c}\, ) - \frac{f(P)}{Q}\right] = \frac{dQ_{\widehat{c}}}{dP}\, ,
\Psi:SU(2)\rightarrow S^{3},g=\left(\begin{array}[c]{cc}z_{1}-\bar{z}_{2}z_{2}\bar{z}_{1}\end{array}\right) \mapsto \Psi(g)= \left(\begin{array}[c]{c}z_1 z_2 \end{array}\right) \sim (z_{1},z_{2})=\mathbf{z}
- ({\frac{e}{c}}) \int d \vec{y}\psi^{( 2 ) +}( y ) ) K_{i}( y , x ) ( i \hbar \vec{\partial}_{y^{i}}- ({\frac{e}{c}}) A_{i}^{[ 1 ]}( y ) ) \psi^{( 2 )}( y ) \left. \right] \psi^{( 2 )}( x ) ,
R = ( 1 +{\frac{1}{2 \beta_{B I}^{2}}}F_{\mu \nu}^{a}F^{a \mu \nu}-{\frac{1}{1 6 \beta_{B I}^{4}}}( F_{\mu \nu}^{a}\tilde{F}^{a \mu \nu})^{2})^{\frac{1}{2}}
L_I(s) =e^{-\rho_I\sum\limits_{l=0}^{\infty}w_{Il}\delta_0 (-1 +{}_2F_1(-\frac{1}{\alpha_I}, l+\mu, 1-\frac{1}{\alpha_I}, -\frac{s \gamma_0}{c d_0^{\alpha_I}}))}
\mathcal{I}:= \left\{0 \leq A\leq 1, B\geq \frac{2}{K}\right\}\cup \left\{0\leq B\leq \frac{2}{K}\right\}; \mathcal{F}:= \left\{A>1,\,\,\,A\leq \frac{K}{2}B\right\}.
H(X_i|\tilde{X}_i,\{\tilde{Z}_j\},\{X_w\}_{w \neq i}) = H\left(X_i|\tilde{X}_i,\{\hat{X}_i(u)\}_{u \in T_k(i)}, \{\tilde{Z}_j\}_{j \notin T_k(i)},\{X_w\}_{w \neq i}\right).
\left\Vert u_{n}\right\Vert_{1,2}^{2}=\int_{\Omega}f(x,\mathrm{T}_{1}(z_{1,n}),\mathrm{T}_{2}(z_{2,n}))u_{n}\,\mathrm{d}x\leq M\int_{\Omega}u_{n}\,\mathrm{d}x
V=M^2-\gamma^k M_{,k}+\frac 14 \bar{R}+{1 \over 16}F^{jk}F_{jk}-\left(eA_0+\frac i8F\right)^2+{e \over 2}({\cal F}-2\breve{\gamma}^0\gamma^k\nabla_k A_0)~.
\max\{\lim_{k \to \infty}-\mathcal{E}^{\mathbf{x}_n}(W,J_k),\lim_{k \to \infty}-\mathcal{E}^{\mathbf{x'}_n}(W,J_k)\}=\alpha^2\beta_1 + (1-\alpha)^2\beta_2 + \max\{\alpha^2\beta_1, (1-\alpha)^2\beta_2\},
{\cal D}^{- 1}= \left( \begin{array}{c c}{2 \beta p^{2}\theta_{\mu \nu}+ 2 \left( 1 + \frac{m \xi^{2}}{2 \pi}\right) p^{2}\omega_{\mu \nu}}&{- \frac{i m \xi \alpha}{2 \pi}p_{\mu}}\\{\frac{i m \xi \alpha}{2 \pi}p_{\nu}}&{p^{2}- \mu^{2}- \frac{m \alpha^{2}}{\pi}}\\ \end{array}\right) .
u_i(X_{t})=\int_{t}^{\zeta}f_{i}(X_r,u_i(X_r))\,dr+\int_{t}^{\zeta}dA_{r}^{\mu_i}+\int_{t}^{\zeta}dA_{r}^{\nu_i}+\int_{t}^{\zeta}dM^i_{r},0\le t\le\zeta,
k_{2}^{\nu}T_{\rho \sigma \mu \nu}^{a b c d}( p_{1}, p_{2}; k_{1}, k_{2}) = - i g^{2}e^{\rho}( p_{1}) e^{\sigma}( p_{2}) [ k_{1 \mu}S_{\rho \sigma}^{a b c d}+ \widetilde{G}_{\rho \sigma \mu}^{a b c d}]
N^{''}|_{x_h}=\frac{10\alpha^2}{x_h^4}(w_{x_h}^2-1)^2-\frac{8\alpha^2 w_{x_h}w^{'}|_{x_h}}{x_h^3}-\frac{2\alpha^2}{x_h}N^{'}|_{x_h}(w^{'}|_{x_h})^2\left(1+\frac{8}{x_h^4}(w_{x_h}^2-1)^2\right)
\sigma ={\frac{g^{2}M}{3 2 \pi^{2}}}\int_{- \infty}^{+ \infty}d y_{3}\left\{\left({\frac{d \bar{\chi}}{d y_{3}}}\right)^{2}+ 2 \left( 1 - \cos \bar{\chi}\right) \right\}={\frac{3 g^{2}M}{8 \pi^{2}}}\ldots
\lim_{t \to \infty}\frac{S_j(t)}{M_j(t)}= \lim_{N \to \infty}\frac{S_j(\tau_N)}{M_j(\tau_N)}= \frac{\int_0^\infty e^{-\lambda s}p_j(s) ds}{\int_0^\infty e^{-\lambda s}\big(\sum_{k=j}^\infty p_k(s)\big) ds}, j \geq 1,
Q(r_1,\theta)\approx \frac{\pi^2\lambda R_S(R_{max}-r_1)}{NR_E}\sum_{n=1}^N\sqrt{1-\phi_n^2}c_n \times\left(1-\prod_{m=1}^{M}{\left(1+\frac{m\eta\theta r_1^{\alpha}}{Mc_n^{\alpha}}\right)^{-Mb_m}}\right)
S_{a_{1},b_{1}}^{a_{2},b_{2},p_{2}}\left( p_{1},k\right)=\sum_{j_{1}=0}^{p_{1}}\sum_{j_{2}=0}^{p_{2}}\binom{p_{1}}{j_{1}}\binom{p_{2}}{j_{2}}a_{1}^{j_{1}}a_{2}^{j_{2}}b_{1}^{p_{1}-j_{1}}b_{2}^{p_{2}-j_{2}}S\left( j_{1}+j_{2},k\right) .
\mathcal A(\widetilde \rho,\widetilde m)=\frac 12\int_0^1\frac{((\Sigma_1-\Sigma_2)\theta_{12}(\widetilde \rho)\dot W^{\delta})^2}{\theta_{12}(\widetilde \rho)}+(m_{23}+\frac 12\rho_1(\Sigma_2-\Sigma_3)\dot W^{\delta})^2 dt.
{\cal{F}}_{V}^{( \tilde{\eta}_{o})}( Y , W ) = C_{\phi}^{2}W^{\frac{1}{3}\eta}\Bigl [ ( 1 - Y )^{- \frac{1}{2}\eta}- ( 1 - Y )^{\frac{1}{2}\eta}\Bigl ] + \frac{g_{*}^{2}}{C_{\tilde{O}}}{\cal{F}}_{V , 1}^{( \tilde{\eta}_{o})}( Y , W ) + O ( \frac{1}{N^{2}}) ,
\mbox{\textbf{P1-2:}}\;\; \max_{\{x_{k_j}\}}\sum_{k=1}^K\sum_{j=1}^J{x_{k_j}c_{k_j}}\mbox{s.t.}\;\; \sum_k x_{k_j}\leq L_j,~j=1,2,\cdots,J \;\; \sum_j x_{k_j}\leq 1,~k=1,2,\cdots,K \;\; \mbox{Constraints (\ref{eqn:mmhn95}),~(\ref{eqn:mmhn80})}.
\hat{\theta}_{0 4}={\frac{x_{0 4}}{( 1 2 )}}\xi_{1 2 4}= \theta_{0 4}+{\frac{( 2 4 )}{( 1 2 )}}x_{0 4}x_{0 1}^{- 1}\theta_{0 1}+{\frac{( 4 1 )}{( 1 2 )}}x_{0 4}x_{0 2}^{- 1}\theta_{0 2}
\widetilde{A}^{AVF}_\alpha:=\left(\begin{array}{c}\alpha\delta_n^+u_{m,n}+\delta_m^{(2)}\mu_nv_{m,n}+\mu_n(v_{m,n}^2+\frac{2}3u_{m,n}^2)\mu_nv_{m,n}+\frac{1}3\mu_n(u_{m,n}^2v_{m,n}) -\alpha\delta_n^+v_{m,n}+\delta_m^{(2)}\mu_nu_{m,n}+\mu_n(u_{m,n}^2+\frac{2}3v_{m,n}^2)\mu_nu_{m,n}+\frac{1}3\mu_n(v_{m,n}^2u_{m,n})\end{array}\right)=\mathbf{0},
A^{( 1 )}= \left( \begin{array}{c c c}{{A^{( 1 ) i j , k l}}}&{{A^{( 1 ) i j , 0}}}&{{A^{( 1 ) i j , b}}}\\{{A^{( 1 ) 0 , k l}}}&{{A^{( 1 ) 0 0}}}&{{A^{( 1 ) 0 b}}}\\{{A^{( 1 ) a , k l}}}&{{A^{( 1 ) a 0}}}&{{A^{( 1 ) a b}}}\\ \end{array}\right)
\begin{aligned}I_4 =P_1(\alpha_2)P_1(\alpha_1)\Phi(\alpha_1,\alpha_{2}) -P_1(\alpha_2)P_1(\beta_1)\Phi(\beta_1,\alpha_{2}) &\quad-P_1(\beta_2)P_1(\alpha_1)\Phi(\alpha_1,\beta_{2}) +P_1(\beta_2)P_1(\beta_1)\Phi(\beta_1,\beta_{2}).\end{aligned}
\sum_{m=0}^{[\frac{1}{2}n]}\sum_{s=0}^{[\frac{1}{2}n]}(-1)^{m+s}\alpha_{2m-1,n-1}\alpha_{2s-1,n-1}\frac{\mu_{2n-2m-2s+2}}{(b_{n-1}^2)!}= b_{n-1}^2 +b_{n}^2;n=0,1,2,...,
\begin{aligned}\tilde{b}_{\boldsymbol{\rho},f,t}^i = W \Delta \mu_{\boldsymbol{\rho},f,t}^{i}\log_{2}\Big(1 + \frac{p_{f,t}^{i}h_{\boldsymbol{\rho},f,t}^{i}}{N_0 W}\Big) \ge 0, \; i \in \{S,R\}. \end{aligned}
\begin{array}{rclrcl}e^{0}= \lambda^{1/2}/R\left( dt +\omega_{\aleph}d\varphi \right)\, ,\hspace{.5cm}e^{1}= \lambda^{-1/2}Rdr \, , & e^{2}= R\, d\theta\, , e^{3}= R\, \Omega_{\aleph}\, d\varphi \, , \end{array}
\left. \frac{d}{dt}\mbox{Cap}_{\mathcal{A}}( E_1 + t E_2 ) \, \right|_{t=t_2}= (p-1)\int_{\mathbb{S}^{n-1}}h_2 ( \mathbf{g}(z, E_1 + t_2 E_2 ) ) \, d \mu_{E_1 + t_2 E_2}(z).
- \theta \Delta t \mathcal{L}(\tilde{u}^{[n]}) + \tilde{u}^{[n]}(x) = \tilde{u}^{[n-1]}(x) + \theta \Delta t f(t_n,x) + (1-\theta) \Delta t \left[ \mathcal{L}(\tilde{u}^{[n-1]}) + f(t_{n-1},x) \right].
T(\vec{k'},n';\vec{k},n)=\frac{-1}{8\pi^2}\int d\vec{x'}\int_{o}^{2\pi}d\phi'e^{-i \vec{k'}. \vec{x'}}e^{-in'\phi'}V(x',\phi') \Psi_{\vec{k},\vec{n}}(x',\phi'),
a_{2j+1}(Q^{2j+1}f)(\xi_{0})=a_{2j+1}E(f(\xi_{2j+1})|\xi_{0})= a_{2j+1}E_{0}E(f(\xi_{2j+1})|\xi_{j+1})=a_{2j+1}E_{0}E(f(\xi_{1})|\xi_{j+1})=a_{2j+1}E_{0}E^{j+1}f(\xi_{1}),
2\partial_{\bar z}(\mathcal P^*_2\mathcal P_1)=2(\partial_{\bar z}\mathcal P^*_2\mathcal P_1+\mathcal P^*_2\partial_{\bar z}\mathcal P_1)=-\mathcal P^*_2A_2\mathcal P_1+\mathcal P^*_2A_2\mathcal P_1=0.
f_{b}( \hbar k_{0}/ T ) + \frac{1}{2}= \frac{T}{\hbar k_{0}}+ \frac{1}{1 2}\frac{\hbar k_{0}}{T}- \frac{1}{9 0}\left( \frac{\hbar k_{0}}{T}\right)^{2}+ \, \ldots \, .
\rho\left\vert x_{k}\right\vert^{r}=\left\vert \sum_{i_{2},\ldots,i_{r}}a_{k,i_{2},\ldots,i_{r}}x_{k}x_{i_{2}}\cdots x_{i_{r}}\right\vert \leq\sum_{i_{2},\ldots,i_{r}}a_{k,i_{2},\ldots,i_{r}}\left\vert x_{k}\right\vert|x_{i_{2}}|\cdots\left\vert x_{i_{r}}\right\vert .
r\ =\sum\lambda_{i_1 i_2 \cdots i_{t-1}}h_{i_1}h_{i_2}\cdots h_{i_{t-1}}\ \in\ (u_1^n,u_2^n,\dots,u_{t-1}^n,v_1^n,v_2^n,\dots,v_{t-1}^n)S,
\rho(\bar{T^2}) = \left(\begin{matrix}\lambda^2 0 0 0 \bar{\lambda}0 0 0 \bar{\lambda}\end{matrix}\right), \rho(\bar{U}) = \rho(\bar{R}\bar{T}^{-2}\bar{R}^{-1})=\left(\begin{matrix}\lambda 0 0 0 \lambda 0 0 0 \bar{\lambda}^2 \end{matrix}\right).
h={\mbox e}^{a_0 L_0}\;{\mbox e}^{a_1 L_1}\;{\mbox e}^{b_{-1}J_{-1}}\;{\mbox e}^{b_{0}J_{0}}\;{\mbox e}^{b_{1}J_{1}}\;{\mbox e}^{b_{2}J_{2}}\; \tilde{h}\; ,
\Gamma_2^{\mathrm{hori}}(f,f) = (X^2f)^2 + (Y^2f)^2 + (XY f)^2 + (YX f)^2 - 2 (Xf) (YZf) + 2 (Yf) (XZf) \Gamma_2^{\mathrm{vert}}(f,f) = (XZf)^2 + (YZf)^2.
S_{M C S}= \int d^{3}x [ - \frac{1}{2}( \partial_{\alpha}A_{\beta}- \partial_{\beta}A_{\alpha}) \partial^{\alpha}A^{\beta}+ \frac{2 \pi}{g^{2}}\epsilon^{\alpha \beta \gamma}A_{\alpha}\partial_{\beta}A_{\gamma}] .
\left[ X_{\left( \eta ,\upsilon ,\zeta \right)}^{\mathfrak{z}_{d}},X_{\left( \bar{\eta},\bar{\upsilon},\bar{\zeta}\right)}^{\mathfrak{z}_{d}}\right] =X_{\left( \left[ \eta ,\bar{\eta}\right] ,ad_{\bar{\eta}}^{\ast}\upsilon -ad_{\eta}^{\ast}\bar{\upsilon},\left[ \eta ,\bar{\zeta}\right] -\left[ \bar{\eta},\zeta \right] \right)}^{\mathfrak{z}_{d}}.
\int_{\pi/2}^{3\pi/2}d \theta_3 \int_{-\pi/2}^{\pi/2}d \theta_2 \int_{-3\pi/2}^{-\pi/2}d \theta_1 \left| 3 \sin \frac{\theta_2-\theta_1}{3}\right|^{-1}\left| 3 \sin \frac{\theta_3-\theta_1}{3}\right|^{-1}\left| 3 \sin \frac{\theta_3-\theta_2}{3}\right|^{-1}= \frac{\pi^2}{3}.
\delta\left(R\mathrm{dvol}_g-\frac{1}{2}|G|^2\mathrm{dvol}_g\right) =\biggl(R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R &-\frac{1}{2}\langle i_{\partial_\mu}G,i_{\partial_\nu}G\rangle+\frac{1}{4}g_{\mu\nu}|G|^2\biggr)\mathrm{dvol}_g\delta g^{\mu\nu}+\mathrm{td}.
W_1^n(t) = X_1^n(0) + Y^{n}_1(t)+ \int_0^t(-\phi_0(W_1^n(s))+\phi_{B^n}(W_2^n(s)))ds, W_2^n(t) = X_2^n(0) + Y^{n}_2(t) + \int_0^t(-\phi_{B^n}(W_2^n(s)))ds +\psi_0\left(W_1^n(t)\right) .
{\cal{F}}_{V}^{(\tilde{\eta}_{o})}(Y,W) =C_{\phi}^{2}W^{\frac{1}{3}\eta}\Bigl[(1-Y)^{-\frac{1}{2}\eta}-(1-Y)^{\frac{1}{2}\eta}\Bigl]+\frac{g_{*}^{2}}{C_{\tilde{O}}}{\cal{F}}_{V,1}^{(\tilde{\eta}_{o})}(Y,W)+O(\frac{1}{N^{2}}),
I = T_{( 1 )}\int d^{2}\zeta \; \Bigl [ \sqrt{1 - ( \partial_{0}Y^{m})^{2}- ( 2 \pi \alpha^{\prime}\; F_{0 1})^{2}}\; + \;{\widehat C}_{0 1}+ 2 \pi \alpha^{\prime}F_{0 1}\;{\widehat C}\Bigr ] \ .
r^* \mu_s = \lim_{t \to +\infty}r^*\left(\frac{\pi}{2}t 1_{\{|\tilde{Q}| > t\}}d\lambda\right) = \lim_{t \to +\infty}\left(\frac{\pi}{2}t 1_{\{|\tilde{Q}\circ r| > t\}}d(r^* \lambda)\right) = \lim_{t \to +\infty}\left(\frac{\pi}{2}t 1_{\{|Q| > t\}}d\lambda \right)
\psi_0(v_1,v_2,v_0) =\psi_0(v_1,0,v_0)+_1\psi_0(v_1,v_2,0)=(\psi_0(v_1,0,0)+_2\psi_0(0,0,v_0))+_1\psi_0(v_1,v_2,0) &=a_0(v_0)+\mu(v_1,v_2).
\frac{H_{k}}{RT^{2}}=Q_{k}(\frac{\sum_{m}\theta\psi_{mk}^{\prime}}{\sum_{m}\theta\psi_{mk}}-(\sum_{m}\frac{\theta_{m}\psi_{k}^{\prime}m}{\sum_{n}\theta_{n}\psi_{nm}}-\frac{\theta_{m}\psi_{km}(\sum_{n}\theta_{n}\psi_{nm}^{\prime})}{(\sum_{n}\theta_{n}\psi_{nm})^{2}}))
\begin{aligned}\mathbb{E}T^{\rm Reac}_j=\tau^{\rm Reac}_{m}\mathbb{E}R_j+\mathbb{E}\tau^{\rm Reac}_{V_j}+\tau_{\rm f}, &\mathbb{E}\left(T^{\rm Reac}_j\right)^2=\mathbb{E}\left(\tau^{\rm Reac}_{m}R_j+\tau^{\rm Reac}_{V_j}+\tau_{\rm f}\right)^2. \end{aligned}
\|L_Gf\|_1 = \sum_{v \in V}|Lf(v)| \leq \sum_{v \in V}|f(v)| + \sum_{v \in V}\sum_{v\sim v'}|f(v')| \leq \|f\|_1 + D\|f\|_1 = (D+1)\|f\|_1.
M(g) \le |c_0|^{l-1}\prod_{1 \le i<j \le l}\max\{1,|\gamma_i \gamma_j|\}& \le|c_0|^{l-1}\prod_{1 \le i<j \le l}\max\{1,|\gamma_i|\}\max\{1,|\gamma_j|\}&= |c_0|^{l-1}\prod_{k=1}^l \max\{1, |\gamma_k|\}^{l-1}= M(f_1)^{l-1}& \le M(f)^{n-1}.
\|\eta\|^\ell_{\dot{B}_{2,1}^{\frac n2-2}}\le C \big(\|\eta\|^\ell_{\dot{B}_{2,\infty}^{\sigma-1}}\big)^{\theta_{1}}\big(\|\eta\|^\ell_{\dot{B}_{2,1}^{\frac n2}}\big)^{1-\theta_{1}},\theta_1=\frac{4}{n-2\sigma+2}\in(0,1),
A_1 \! := \! \sum_{1 \leq k < j}S_{N}\Big( \frac{t_{j}(x)}{2\pi \lambda^{j}}\Big) f_k(x) A_2 \! := \! \sum_{k > j}S_{N}\Big( \frac{t_{j}(x)}{2\pi \lambda^{j}}\Big) f_k(x) \, .
\tau_n := \begin{cases}n^k a(R_n) R_n^{d-1}f(R_ne_1)^k nf(R_ne_1) \to 0\,, a(R_n)R_n^{d-1}nf(R_ne_1) \to \xi \in (0,\infty)\,, n^{2k-1}a(R_n) R_n^{d-1}f(R_ne_1)^{2k-1}nf(R_ne_1) \to \infty\,.\end{cases}
\alpha = \frac{K \cdot (1-\xi)}{2\cdot \pi \cdot \Theta}\cdot e^{\Phi_U(\xi) - \Phi_L(\xi)},\beta = \frac{(2 \pi)^2\cdot \Theta}{f_L(\xi)},\mbox{and}\gamma = \pi^2 \cdot e^{\Phi_U(\xi)}.
\renewcommand{\arraystretch}{1 . 5}\begin{array}{l c l}{{{[}L_{m}, L_{n}{]}}}&{{=}}&{{( m - n ) L_{m + n}+ \delta_{m + n , 0}\frac{m^{3}- m}{1 2}c}}\\{{{[}L_{m}, G_{r}{]}}}&{{=}}&{{( \frac{1}{2}m - r ) G_{m + r}}}\\{{{[}G_{r}, G_{s}{]}_{+}}}&{{=}}&{{2 L_{r + s}+ \frac{1}{3}( r^{2}- \frac{1}{4}) \delta_{r + s , 0}c}}\\ \end{array}
L^{(0)}_{eff}=\int\;d^{2}x\; [(\partial_{t}\mid\phi\mid)^{2}+(\partial_{t}\mid\chi\mid)^{2}+ \frac{\kappa}{q_{1}}B_{2}\partial_{t}\omega_{1}+ \frac{\kappa}{q_{2}}B_{1}\partial_{t}\omega_{2}]\;\;.
\mathcal{E}_{s,\Lambda'}^{\rm cp}(N|\det B|) \leq \inf_{\omega_N\in(\Omega_{\Lambda})^N}E_{s,\Lambda'}^{\rm cp}(S(\omega_N))=|\det B| \cdot \mathcal{E}_{s,\Lambda}^{\rm cp}(N)+N|\det B|(\zeta_\Lambda(s)-\zeta_{\Lambda'}(s)).
\tilde{f}(\omega) = \mathcal{F}[f (x); \omega] = \int_{-\infty}^{\infty}e^{i \omega x}f (x) dx, f (x) = \mathcal{F}^{-1}[\tilde{f}(\omega); x] = \frac{1}{2\pi}\int_{-\infty}^{\infty}e^{-i \omega x}\tilde{f}(\omega) d\omega.
\begin{aligned}\int_I |h_\epsilon-u_2|^2dm &=\sum_{n= 1}^M \int_{[e_n, b_n+\varepsilon_n)}\left(h^+_n-u_2\right)^2dm+\sum_{n= 1}^M \int_{(a_n-\varepsilon'_n, e_n]}\left(h^-_n-u_2\right)^2dm &+\int_G \left(h-u_2\right)^2dm.\end{aligned}
[\begin{matrix}a&b\\c&d\end{matrix}]=[\begin{matrix}1&0\\ \frac{-1}{\lambda R_{B}}&1\end{matrix}][\begin{matrix}1&\lambda D\\ 0&1\end{matrix}][\begin{matrix}1&0\\ \frac{-1}{\lambda R_{A}}&1\end{matrix}]=[\begin{matrix}1-\frac{D}{R_{A}}&-\lambda D\\ \frac{1}{\lambda}(R_{A}^{-1}+R_{B}^{-1}-R_{A}^{-1}R_{B}^{-1}D)&1-\frac{D}{R_{B}}\end{matrix}].
Q_{Y , i_{1}i_{2}. . . i_{2 n}}^{T , I_{1}I_{2}. . . I_{n}}= P_{Y}^{i_{1}i_{2}. . . i_{2 l}, j_{1}j_{2}. . . j_{2 l}}( \prod_{l = 1}^{n}C_{j_{2 l - 1}j_{2 l}}^{I_{l}})
\omega^{(m)}(y,s) \,=\, \frac{\Gamma}{\nu s}f\biggl( \frac{\sqrt{(\bar r + \epsilon_m y_1)^2 + \epsilon_m^2 y_2^2}-\bar r}{\epsilon_m \sqrt{\nu s}}\,,\frac{y_3}{\sqrt{\nu s}}\,,\epsilon_m^2 s \biggr)\,e_\theta(\bar x + \epsilon_m y)\,.
-\frac{v_t-2}{e_t}= -\frac{(r-2) \frac{\tau^t-1}{\tau-1}}{\tau^t}= -\frac{r-2}{\binom{r}{2}-2}\frac{\tau^t-1}{\tau^t}= - \frac{1}{\lambda}+ \frac{1}{\lambda \tau^t}.
\begin{aligned}k^{-1}\psi^{\pm}_{21}(x ; k) =-\int_{\pm \infty}^{x}e^{2iz(x-y)}\bar{u}_y e^{-ic_\pm}d y &-\int_{\pm \infty}^{x}e^{2 i z(x-y)}\bar{u}_y\left(\psi^{\pm}_{11}(y ;k)-e^{-ic_\pm}\right) d y.\end{aligned}
\langle C^{main}w, w \rangle = \| \Phi_1(x,D) w\|_{L^2}^2 + \| \Phi_2(x,D) w\|_{L^2}^2 + \| \Phi_3(x,D) w\|_{L^2}^2 + O(\| w\|_{L^2}^2).
|\nu_{p^{k+1}}| = 1+\sum\limits_{j=1}^{p^{k+1}-1}\phi(j) + |\nu_{p^k}| = 1+\sum\limits_{j=1}^{p^{k+1}-1}\phi(j) + k+\sum\limits_{i=1}^k\sum\limits_{j=1}^{p^i-1}\phi(j) = (k+1) + \sum\limits_{i=1}^{k+1}\sum\limits_{j=1}^{p^i-1}\phi(j).
{\displaystyle \lambda^N\int_Q f_\lambda(u_0(y), v_0(y), 0)dy = \int_{a_0+\lambda Q}f(u(x),v(x),0)dx,\hbox{and}(K_p)_{\lambda}(0,c,d, e_N)=\frac{1}{\lambda}K_p(0,c,d,e_N).}
\mathbb{E}[(X(x+1,t) - X(x-1,t))^{2k}] = \sum_{(x_1,s_1),\ldots,(x_{2k},s_{2k})}\mathbb{E}\left[ \prod_{i}\Delta(x_i,s_i) \xi(x_i,s_i) \prod_i \Gamma(x_i,s_i)\right].
I_{ini}(W_0)= \int_{\mathbb{T}}\Bigg(\sum_{k=1}^3w_{0,k}(u)\log\left(\frac{w_{0,k}(u)}{\rho_{k-1}(u)}\right) &-\left(1-\sum_{k=1}^3w_{0,k}(u)\right)\log\left(\frac{1-\sum_{k=1}^3\rho_{k-1}(u)}{1-\sum_{k=1}^3w_{0,k}(u)}\right)\Bigg)du
m\mathbf{\ddot{p}}= \mathbf{f}+ m\mathbf{g}+ \mathbf{d}_{p}\mathbf{J}\mathbf{\dot{\omega}}= \mathbf{\omega}\times \mathbf{J}\mathbf{\omega}+ \mathbf{u}+ \mathbf{R}_{IB}\mathbf{d}_{\omega}
\begin{array}{*{20}{l}}{{G_{v,j}^q}= \rho_{v,j}^q\frac{{{A_{{\rm{PD}}}}\left({m_1 + 1}\right)}}{{2\pi d_{v,j}^2}}{\cos^{m_1}}\left({{\phi_{v,j}}}\right)T\left({{\psi_{v,j}}}\right)}{G\left({{\psi_{v,j}}}\right){\cos}\left({{\psi_{v,j}}}\right),}\end{array}
\|\partial_n (u - I^k_h u)\|_{L^2(\Gamma)}\lesssim \begin{cases}h^k \|u\|_{B^{k+3/2}_{2,1}(\Omega)}h^s \|u\|_{B^{s+3/2}_{2,\infty}(\Omega)}, s \in (0,k), \|u\|_{B^{3/2}_{2,1}(\Omega)}.\end{cases}
\delta_m\left(\sum_{j=1}^{\infty}p^{(i)}_jt^{-j}\right)=\left\{\sum_{j=1}^{\infty}p^{(i)}_j t^{m-1-j}\right\}=\sum_{j=1}^{\infty}p^{(i)}_{m-1+j}t^{-j}.
\mathcal{L}^{\circ}(\hat{Y}[y]) \cup \varphi^{{\mathbb{P}}}_{n_{{\textsc{g}}}*}(\mathcal{L}^{\circ}(Y_{n_{{\textsc{g}}}}))={\mathbb{P}}(H_1(\partial^{\textsc{g}}_{n_{{\textsc{g}}}}\mkern-2mu \hat{M})).
\pi_{a,i}\!\left(\boldsymbol{\tau}_i; \boldsymbol{d}_i\right) \!= \sum_{t\in\mathcal{T}}\left\{\tau^{[t]}_i d_i^{[t]}- p^{[t]}\!\!\left(\boldsymbol{d}^{[t]}\right) \cdot d_i^{[t]}\right\}
-\frac{\dot{\lambda}}{\lambda}= 80\gamma_{2}\frac{\Delta K(a)}{\lambda^{2}}(1+o_{\varepsilon}(1))+O(\vert \delta J(u)\vert^{2}) = -c\frac{\Vert a \Vert^{2}}{\lambda^{2}}+O(\vert \delta J(u)\vert^{2}) \leq -c\frac{\varepsilon^{-1}}{\lambda^{3}}+O(\vert \delta J(u)\vert^{2}),
{\cal R}_{M}{}^{N}(z;g)=\left(\begin{array}{cc}\Omega(z;g)\hat{R}_{\mu}{}^{\nu}(z;g) i\Omega(z;g)^{\frac{1}{2}}(L^{-1}(z;g)\gamma_{\mu}\Sigma^{b}(z;g))^{\beta}0 \Omega(z;g)^{\frac{1}{2}}L^{-1}{}^{\beta}{}_{\alpha}(z;g)U_{a}{}^{b}(z;g)\end{array}\right)\,.
T_{_{\alpha}}(\vec{f})(z) = T_{\alpha}(f_{1}^{0},\dots, f_{m}^{0})(z) + \sum_{(\rho_{1},\dots,\rho_{m})\in \rho}T_{\alpha}(f_{1}^{\rho_{1}},\dots, f_{m}^{\rho_{m}})(z) .
( h^{- 1}\partial_{n^{\alpha}}A^{0 1}h )^{\bot}= \sum_{\gamma , \beta}( h^{- 1}\phi_{\gamma}h )^{\dagger}\ ( \Delta^{- 1})_{\gamma \beta}\ \big \langle ( h^{- 1}\phi_{\beta}h )^{\dagger}\big | h^{- 1}\partial_{n^{\alpha}}A^{0 1}h \big \rangle
|J_1 |\le C \sum_{j=1}^k U_j \le \frac{C}{\lambda^{\frac{N-2}2}|y-\xi_1|^{N-2}}+ \frac{C}{\lambda^{\frac{N-2}2}|y-\xi_1|^{N-2-\tau}}\sum_{j=2}^k\frac1{|\xi_j-\xi_1|^\tau}\le \frac{C}{\lambda^{\frac{N-2}2-\tau}|y-\xi_1|^{N-2-\tau}}.
\L^{(de)}_{(X,\Delta)}:=\iota_*\L^{(de)}_{(U,\Delta|_U)}\quad\textup{and}\phi^{(de)}_{(X,\Delta)}:=\iota_*(\phi^{(de)}_{(U,\Delta|_U)}):{F_X^{de}}_*\L^{(de)}_{(X,\Delta)}\to\O_X
\frac{\partial^{2}\tilde{A}}{\partial p_{i}\partial \tilde{p}_{j}}\, d p_{i}\, d \tilde{p}_{j}= \frac{\partial^{2}\tilde{A}}{\partial p_{k}^{\prime}\partial \tilde{p}_{l}^{\prime}}\left( \frac{\partial p_{k}^{\prime}}{\partial p_{i}}\right) \left( \frac{\partial \tilde{p}_{l}^{\prime}}{\partial \tilde{p}_{j}}\right) \, d p_{i}\, d \tilde{p}_{j}