harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
a_F(\rho) =a_F(\rho_0\otimes\chi_F)=\rm{dim}(\rho_0)\cdot\left(1+\rm{max}\{j(\rho_0),j(\chi_F)\}\right) =\rm{dim}(\rho_0)\cdot\rm{max}\{1+j(\chi_F), 1+j(\rho_0)\}=\rm{dim}(\rho_0)\cdot\rm{max}\{a(\chi_F), 1+j(\rho_0)\}=\rm{dim}(\rho_0)\cdot a_F(\chi_F),
\boldsymbol{\lambda}_{\mathcal{E}}^{*}=-(\vec{G}_{\mathcal{E}}\vec{H}^{-1}\vec{G}_{\mathcal{E}}^{\top})^{-1}(\vec{W}_{\mathcal{E}}+\vec{S}_{\mathcal{E}}\vec{x}),\forall \vec{x}\in \mathcal{R}(\mathcal{E}),
\begin{aligned}\tilde{E}^{-}_{[i,j]}\coloneqq [\cdots[[E_{i},E_{i+1}]_{v^{-2}},E_{i+2}]_{v^{-2}},\cdots,E_{j}]_{v^{-2}}, \tilde{E}^{-}_{[i,n,j]}\coloneqq [\cdots[[[\cdots[E_{i},E_{i+1}]_{v^{-2}},\cdots, E_{n}]_{v^{-2}},E_{n}],E_{n-1}]_{v^{-2}},\cdots,E_{j}]_{v^{-2}}.\end{aligned}
\Lambda^{(h:h')}_{n,2}=\Lambda^{(h:h')}_{n,2,0}+ o(1) = \Lambda^{(h:h')}_{n,2,0}(\mathcal S_k) +\Lambda^{(h:h')}_{n,2,0}(\mathcal{W}_k(0)\setminus \mathcal{S}_k) + o(1)
\|Vx\|^2=\|\langle V{x},Vx\rangle\| =\|\langle 1\otimes U_1{x}+K,1\otimes U_1{x}+K\rangle\|=\|\langle U_1{x},\tau(1)U_1{x}\rangle\| &\leq\|\tau(1)\|\|x\|^2.
\begin{aligned}1 =-\int_t^{+\infty}f'(s)ds=\int_t^{+\infty}-f'(s)[A(s)]^{\frac{1}{p}}[A(s)]^{-\frac{1}{p}}ds &\leq(\int_t^{+\infty}[f'(s)]^pA(s)ds)^{\frac{1}{p}}(\int_t^{+\infty}[A(s)]^{\frac{1}{1-p}}ds)^{1-\frac{1}{p}}\end{aligned}
\mathbf{P}\left(\max_{j\le n}|X(j)|>ar,\tau_x>n\right) \le\sum_{j=1}^n\mathbf{P}\left(|X(j)|>ar,\tau_x>n\right) &\le\sum_{j=1}^n\mathbf{P}\left(|X(j)|>ar,\tau_x>j-1\right) &=\mathbf{E}[\tau_x\wedge n]\mathbf{P}(|X|>ar).
{\rm lim}_{L\rightarrow +\infty}\frac{k_{\gamma,\Sigma}^{L}}{\sqrt{L}}=\frac{|\frac{d}{dt}(\omega(\dot{\gamma}(t)))|}{\left(\overline{q}\frac{\dot{\gamma}_1}{\gamma_1}-\overline{p}\dot{\gamma}_3\right)^2},~~if ~~\omega(\dot{\gamma}(t))= 0~~and~~\frac{d}{dt}(\omega(\dot{\gamma}(t)))\neq 0.
\mathbb{P}_{N}\left[\left|\left\{i:\sup_{0\le t\le C_{\epsilon}\eta}\left|B_{i}(t)\right|>\frac{\gamma\epsilon}{2}\sqrt{N}\right\}\right|\ge N\alpha\right] \le\sum_{i=1}^{N}\mathbb{P}_{N}\left[\sup_{0\le t\le C_{\epsilon}\eta}\left|B_{i}(t)\right|>\frac{\gamma\epsilon}{2}\sqrt{N}\right]
P(t) = \mathrm{Id}+ \sum_{n=1}^\infty (-i)^n \int_0^t \int_0^{t_1}\dots \int_0^{t_{n-1}}B(t_1) B(t_2) \dots B(t_n) \, dt_1 \dots dt_n := \sum_{n=0}^\infty P_n(t) \, .
\varphi\left(Z_2^{m_1}Z_1^{k_1}Z_2^{m_2}\cdots Z_1^{m_n}Z_2^{k_n}Z_1^{m_{n+1}}\right) = \varphi\left(Z_1^{k_1 + \cdots + k_n}\right) \prod^{n+1}_{j=1}\varphi\left(Z_2^{m_j}\right).
F_{{\sf GI}\,ij}^{a}({\bf r})=\partial_jA_{{\sf GI}\,i}^{a}({\bf r})-\partial_iA_{{\sf GI}\,j}^{a}({\bf r})-gf^{abc}A_{{\sf GI}\,i}^{b}({\bf r})A_{{\sf GI}\,j}^{c}({\bf r})
\rho_\ell (y_\ell, y_\ell^\prime) = \min_{(x_\ell, x^\prime_\ell) \in \mathcal{X}_\ell \times \mathcal{X}_\ell}\left(s_{\ell}-s_{\ell}^{\prime}\right)^{\top}V_{\ell}^{-1}\left(s_{\ell}-s_{\ell}^{\prime}\right) \leq c_\ell(s_\ell, s_\ell^\prime), \forall s_\ell, s_\ell^\prime \in \mathcal{S}_\ell.
\mathcal{L}_n(x)=\dfrac{i}{\omega}\left[\left(\dfrac{x-iw}{2}\right)^{n+1}-\left(\dfrac{x+iw}{2}\right)^{n+1}\right]- \dfrac{i \kappa}{\omega}\left[\left(\dfrac{x-iw}{2}\right)^{n}- \left(\dfrac{x+iw}{2}\right)^{n}\right],
\sum\limits_{k=0}^{n-1}\frac{x^{2k}}{(2k+1)^s}=\sum\limits_{k=1}^n(-1)^{k-1}\binom{n}{k}{_{s+1}F_s}\left(\left\{\frac{1}{2}\right\}^{s},1-k;\left\{\frac{3}{2}\right\}^{s};x^2\right).
{\cal L}_{2}\equiv{\cal L}_{3}\left( \varphi , \partial_{\mu}\varphi , \partial_{\mu \nu}\varphi ; B_{\rho}, F^{( 1 )}, F^{( 2 )}, C^{( 1 )}, C^{( 2 )}; B_{\rho \sigma}, D^{( 1 )}, D^{( 2 )}, G^{( 2 )}, G^{( 3 )}\right) .
\begin{aligned}\Re\left(1-\left(\frac{\rho}{\rho-\tau}\right)^n\right) &\quad>-\frac{n(n-1)\tau^2}{\Im(\rho)^2}\left(\frac{\tau^2-\Im(\rho)^2}{2(\frac{\tau^2}{|\Im(\rho)|}+|\Im(\rho)|)^2}+\frac{0.184(n-2)\tau}{|\Im(\rho)|}\right). \end{aligned}
\rho_{r}(S_{1}) = \dfrac{| S_{1}\cap B_r|}{|B_r|}= (2r+1)^{nm}\dfrac{| \pi_{1}(S_{1}\cap B_r) |}{(2r+1)^{n(k+m)}}= \rho_{r}(V_{n, k, n}(\mathbb{Z})).
a_{1}\sim N^{7}s \Phi \; , \quad a_{2}\sim N^{7}\Phi \; , \quad b_{1}\sim N^{7}( t - 1 ) \Phi \; , \quad b_{4}\sim N^{7}( t - s ) \Phi \; , \quad c_{1}\sim N^{7}t \Phi \; .
\sum (\lambda_j-\lambda(\alpha))c_j^2 =\sum \lambda_j c_j^2-\lambda(\alpha)\sum c_j^2\ge \sum \lambda_j c_j^2-\frac{\lambda_\alpha}{\lambda(2m+1)}\sum \lambda_jc_j^2 &\ge \sum \lambda_j c_j^2-\frac{\lambda(2m+\frac12)}{\lambda(2m+1)}\sum \lambda_jc_j^2\ge C_2 \sum \lambda_j c_j^2=C_2 \|\nabla_\theta \phi\|_{L^2(\partial B_1)}^{2},
g_{1}(U_1,U_2)g_{1}(\xi,Z)=g_1(\mathcal{T}_{U_1}\phi U_2,Z)+g_{1}(\mathcal{C}\mathcal{T}_{U_1}U_2,Z)=g_1(\mathcal{T}_{U_1}\phi U_2,Z)-g_{1}(\mathcal{T}_{U_1}U_2,\varphi Z)
\{{\cal M}{}_{,}^{\otimes}v ( y ) \}= \Bigl ({\cal M}T_{\rho}{\cal G}_{\rho \sigma}( \ell , y ) - T_{\rho}{\cal M}{\cal G}_{\rho \sigma}( 0 , y ) \Bigr ) \otimes T_{\sigma}v ( y ) \quad ,
\sum_{xy\in Q^{(2)}}\left(\tfrac{r-1}{r}\bigl(d(x)+d(y)\bigr)-\tfrac{r-1}{r+1}n\right)=(r-1)\sum_{x\in Q}d(x)-\tbinom{r}{2}n &=\sum_{k=0}^{r+1}\left(k(r-1)-\tbinom{r}{2}\right) |Q_k|\le \sum_{k=0}^{r+1}\tbinom{k}{2}|Q_k|=\sum_{xy\in Q^{(2)}}|N(x)\cap N(y)|\,,
\int_{-1}^1 h^n_{n-j}(t)\frac{t^{n+2+2r}}{(n+2+2r)!}dt &=\frac{2^{j+1}(-1)^{\frac{j+1}{2}}\sqrt{2n-2j+1}}{(2+2r+j)!(n+3+2r)_{n+1}}\prod_{m=0}^{\frac{j-1}{2}}(m+r+1)\prod_{m=0}^{\frac{j-3}{2}}(n+1+r-m).
\sum_{k\in T_{n+1}}2^{-c_k-1}p_k\le\sum_{k=1}^\infty 2^{-c_k-1}p_k\le\sum_{i=1}^\infty\sum_{k : c_k=i}2^{-i-1}p_k\le\sum_{i=1}^\infty2^{-i}\le1,
d \Theta_{\underline{{\mu}}}^{2}= v_{\underline{{\mu}}}^{( \underline{{\alpha}})}\pi_{( \underline{{\alpha}})}^{2}= v_{\underline{{\mu}}q}^{+}\pi_{q}^{2 -}+ v_{\underline{{\mu}}\dot{q}}^{-}\pi_{\dot{q}}^{2 +},
\P(A\mid G) - \P(A\mid H)= \frac{p_1 q_1 + p_2q_2}{q_1+q_2}- \frac{p_2 q_2 + p_3q_3}{q_2+q_3}\leq\frac{q_1 + p_2q_2}{q_1+q_2}- \frac{p_2 q_2}{q_2+q_3}.
\int_{W^d}\det(q_{n+i -j}(y_j - x_i))_{i, j = 1}^d \det(q_{m+i-j}(z_j - y_i))_{i, j =1}^d dy_1 \ldots dy_d & = \det( q_{n+m+i-j}(z_j - x_i))_{i, j = 1}^d.
[ a_{1}, a_{1}^{\dagger}] = [ 2 \gamma \omega_{1}( \omega_{1}^{2}- \omega_{2}^{2}) ]^{- 1}, ~ ~ [ a_{2}, a_{2}^{\dagger}] = [ 2 \gamma \omega_{2}( \omega_{2}^{2}- \omega_{1}^{2}) ]^{- 1}, ~ ~ [ a_{1}, a_{2}^{\dagger}] = 0 , ~ ~ [ a_{1}, a_{2}] = 0 .
(1) \f \mapsto \Psi(f) \ \ \ \,, (2) \ \Psi(D f) = 0\,, (3) \ \Psi(f)^* = \Psi(\Gamma f) \,, (4) \ \Psi(f)^*\Psi(h) + \Psi(h)\Psi(f)^* = i\langle f ,\gamma_0 R h \rangle{\bf 1}\,.
r'(A)= -\frac{q-A}{q-A}-\ln(q-A)-\left[\frac{A}{A}+\ln(A)\right]-\left[-\frac{2(1-A)}{1-A}-2\ln(1-A)\right] &-\left[\frac{(q-2)+A}{(q-2)+A}+\ln((q-2)+A)\right]-\left[-\frac{(q-1)-A}{(q-1)-A}-\ln((q-1)-A)\right] = -\ln(q-A)-\ln(A)+2 \cdot \ln(1-A)-\ln((q-2)+A)+\ln((q-1)-A),
\int_{0}^{\infty}g(y \, | \, \sigma_{1}) \pi_{\gamma}(\sigma_{1}) \sigma_{1}= 2^{\tfrac{n-\gamma-3}{2}}\big ( \tfrac{1- \rho^{2}}{n s_{1}^{2}}\big )^{\tfrac{n - \gamma - 1}{2}}\Big [ \mathring{A}_{\gamma}+ \mathring{B}_{\gamma}\Big ] ,
\langle :X\otimes X:,\Psi_{\epsilon,\delta,\psi_\kappa}^f \rangle =\frac{1}{2\pi}\int_{\mathbb{R}^2}^{''}\widehat{\psi_\kappa}(x+y)\widehat{f_\epsilon}(y)\widehat{h_\delta}(y)Z_G(dx)Z_G(dy).
< \Phi ( Z_{1}){\bar{\Phi}}( Z_{2}) > = \mathrm{l n}( Z_{1 2}) -{\frac{\theta_{1 2}\bar{\theta}_{1 2}}{2 Z_{1 2}}}\; , \; \; <{\cal V}( Z_{1}){\bar{\cal V}}( Z_{2}) > = \mathrm{l n}( Z_{1 2}) -{\frac{\theta_{1 2}\bar{\theta}_{1 2}}{2 Z_{1 2}}}\; .
\begin{array}[c]{rl}X^{u^0}(t_i)-X^u(t_i)-X^{u^0}(t_{i-1})+X^u(t_{i-1}) = \displaystyle\int_{t_{i-1}}^{t_i}\left(b(X^{u^0}(t),u^0(t))-b(X^{u}(t),u^0(t))\right)dt &+\displaystyle\int_{t_{i-1}}^{t_i}\left(b(X^{u}(t),u^0(t))-b(X^{u}(t),u(t))\right)dt,\end{array}
\begin{array}{r c l}{C}&{=}&{C^{( 0 )}+ C^{( 1 )}+ C^{( 2 )}+ \ldots}\\{}&{}&{}\\{G}&{=}&{G^{( 0 )}+ G^{( 1 )}+ G^{( 2 )}+ \ldots}\\{}&{}&{}\\{\Lambda^{( \cdot )}}&{=}&{\Lambda^{( 0 )}+ \Lambda^{( 1 )}+ \ldots ,}\\ \end{array}
T_E=\frac{(4a+c^2)M+(6E+ac)T}{12{\rm disc}\left(E-V(\cdot;a,c)\right)},\{T,M\}_{a,E}=-\frac{T^2V'(M/T;a,c)}{12{\rm disc}\left(E-V(\cdot;a,c)\right)}&\qquad\qquad\qquad\{T,M,P\}_{a,E,c}=\frac{T^3(E-V(M/T;a,c))}{2{\rm disc}\left(E-V(\cdot;a,c)\right)}
\frac{1}{4 \beta}F_{\nu , \alpha \beta}^{\infty}( x , y ) = - \int_{1}^{\infty}\frac{d \alpha}{\alpha^{2}}j_{\nu , \alpha \beta}^{\infty}( \alpha x , y ) - \frac{1}{2}\frac{\partial}{\partial x^{\nu}}\int_{1}^{\infty}\frac{d \alpha}{\alpha}\left( 1 - \frac{1}{\alpha}\right) x^{\rho}j_{\rho , \alpha \beta}^{\infty}( \alpha x , y )
|B_{R^{(\ell-1)}}| \mu(R^{(\ell-1)}) \lesssim \Big| \int_{R^{(\ell-1)}}B_{R^{(\ell-1)}}b_{(R^{(\ell)})^a}\,d\mu\Big| = \Big| \int_{R^{(\ell-1)}}\Delta_{R^{(\ell)}}f\,d\mu\Big|.
\displaystyle\max_{t\geq0}\left(\frac{a}{2}t^2-\frac{b}{2_\alpha^*}t^{2_\alpha^*}\right)=\frac{\alpha}{N}\left(\frac{a^2}{b^\frac{2}{2_\alpha^*}}\right)^\frac{2_\alpha^*}{2_\alpha^*-2}
\left\{\stackrel{n-1}{\underset{j =0}\oplus}H^0(E, \Omega^i_{E/k}(j))\right\}_n\to\left\{\frac{H^0(E, p_*(\Omega^i_{V/k}|_{E_n}))}{d(H^0(E, p_*(\Omega^{i-1}_{V/k}|_{E_n})))}\right\}_n
\widetilde{M}:=\bar{e_{k}}M\bar{e_{k}}\oplus Me_{k}M \oplus (e_{k}M)^{\ast}\oplus^{\ast}(Me_{k}) \widetilde{P}:=[P]+\displaystyle \sum_{sa \in_{k}\hat{T},bt \in \tilde{T}_{k}}[btsa]((sa)^{\ast})(^{\ast}(bt))
\begin{cases}\displaystyle\frac{dy(t)}{dt}=Ay(t) + \alpha A y(t-r)+ f \dot B(t),\,\,\,\,t\ge 0, y(0)=\phi_0,\,\,\,\,y_0=\phi_1,\,\,\,\Phi=(\phi_0, \phi_1)\in{\cal X},\end{cases}
P_{\min}^{\rm{N}}\left({\overline S,\overline \varepsilon ,\overline R}\right) ={p_K}\left({\overline S,\overline \varepsilon ,W}\right) + \frac{{{p_K}\left({\overline S,\overline \varepsilon ,W}\right){{\left|{{{\widetilde h}_b}}\right|}^2}+ W{N_0}}}{{{{\left|{{{\widetilde h}_b}}\right|}^2}}}\left({{2^{\frac{{\overline R}}{W}}}- 1}\right).
\left(\frac{|x|+\delta s}{\rho}\right)^{\frac 1\delta}= \left(\frac{|x|}{\rho}\right)^{\frac 1\delta}\, \left(1+\frac{\delta s}{|x|}\right)^{\frac 1\delta}\, \geq\, \frac{|x|}{\rho}\, \left(1+\frac{s}{|x|}\right) = \frac{|x|+s}{\rho}\, ,
\mbox{LHS of}\eqref{prop3.1}=\Pi^{h}_{x}\left[\int_{0}^{t}q(\xi^{h}_{s})e_{q}(s){\rm d}s\int_{E}f(s,\xi^{h}_{s},y)\pi^{h}(\xi^{h}_{s},{\rm d}y)\right].
\vert \lambda \rangle ={e^{\lambda{J_{+}}}}\left( \begin{matrix}{{0}}\\{{1}}\\ \end{matrix}\right) ,{J_{+}}= \left( \begin{matrix}{{0}}&{{1}}\\{{0}}&{{0}}\\ \end{matrix}\right) \equiv{e_{1 2}}, \langle \lambda \vert \lambda \rangle = ( 1 +{\lambda}{\bar{\lambda}})
\frac{1}{2 \pi i}\int_{C}d s x^{- s}\frac{\Gamma ( s )}{\Gamma ( s + \frac{1}{2})}\;{}_{2}F_{1}( \frac{s}{2}, \frac{s + 1}{2}; s + \frac{1}{2}; z^{2}) = \frac{1}{\sqrt{\pi}}\frac{\Theta ( 1 - x + \frac{x^{2}z^{2}}{4})}{\sqrt{1 - x + \frac{x^{2}z^{2}}{4}}}.
|[\partial_\xi^j O_{<k;x}, \partial_\xi^{n-j}V_{<k;\xi}]| \lesssim 2^{k(1+ \delta(\frac12-j))}|x-y| 2^{k(1 - \delta(\frac12+(n-j)))}= 2^{k}|x-y| 2^{k(1 - \delta n)}
W_{m ( 1 )}= - \bigr ( \bar{Q}_{i}R_{\alpha}^{i}( A + Q ) - \epsilon^{a b}E_{\alpha a b}^{*}\bigr ) B^{\alpha}- \bigr ( Q_{i a}^{*}R_{\alpha}^{i}( A + Q ) + H_{\alpha a}\bigr ) C^{\alpha a}.
T_{2 2}^{a b k}= \frac{\Theta}{4 \ell_{P}^{7}}< s_{K}^{a}s_{K}^{b}\epsilon^{i j k}\epsilon^{I J K}\langle W , \xi | \hat{w}_{i I \Delta}( v ) \hat{w}_{j J \Delta}( v ) | W , \xi \rangle >
| 0_{M}> = \frac{1}{Z}\, \exp \left[{\sum_{\sigma}\sum_{p}\; \tanh \epsilon ( p ) d_{p}^{( \sigma ) \dagger}( \epsilon ) \bar{d}_{\tilde{p}}^{( - \sigma ) \dagger}}( \epsilon ) \right] \, | 0 ( \epsilon ) > \,{,}
\lim_{n\rightarrow\infty}\frac{1}{n}\log\left[ \sum_{i=1}^m(1-d^t_{i})\right]=0,~\mbox{and}&\lim_{n\rightarrow\infty}\frac{1}{n}\log\left[\sum_{i=1}^m(1- \hat d_{in})\right] = 0.
\sum_{t=t_d}^T \ell(f_{d,t},y_t) - \inf_{f \in \mathcal{L}_L^d}\sum_{t=t_d}^T \ell(f(y_{t-d}^{t-1}),y_t) \leq M(3+L) \Bigl(\sqrt{T}+ 2(3d)^{\frac{d}{2(d+2)}}T^{\frac{d+1}{d+2}}\Bigr) \,.
\hat{W}^{\hat{a}\hat{b}\hat{c}}\equiv \hat{\cal H}^{\hat{a}\hat{d}\hat{e}}\,\hat{C}^{\hat{b}}_{\hat{d}}\,\hat{C}^{\hat{c}}_{\hat{e}}= \hat{P}^{\hat{a}\hat{b}\hat{d}}\,\hat{C}^{\hat{c}}_{\hat{d}}\, .
\mathcal G_0(X \xleftarrow f M \xrightarrow g Y) :=f_*g^*:\mathcal G_0(Y) \to \mathcal G_0(X), \mathcal H^{Todd}_*(X \xleftarrow f M \xrightarrow g Y) := f_* \bigl(td(T_g)\cap g^* \bigr):\mathcal H^{Todd}_*(Y) \to \mathcal H^{Todd}_*(X).
L_q = \frac{1}{2e}\left(\dot{x}_\mu - \frac{i}{2}v \varepsilon_{\mu\nu\lambda}\xi^\nu_a \xi^\lambda_a \right)^2 - \frac{1}{2}e m^2 - i q m v \xi_1^\mu\xi_{2\mu}+ \frac{i}{2}\xi_a^\mu \dot{\xi}_{a\mu},
f\left( k_{\mathrm{o}}\right) =\frac{e^{i\theta_{\mathrm{o}}}\sin\theta_{\mathrm{o}}}{k_{\mathrm{o}}}\overset{\theta\left( k_{\mathrm{o}}\right) \ll1}{\approx}\frac{\theta\left( k_{\mathrm{o}}\right)}{k_{\mathrm{o}}}+\mathcal{O}\left( \theta^{2}\right) ,
E[|Z_iW_i|] \le \|\sum_{r=0}^{2l-2}S_i^{r}S_{i-1}^{2l-2-r}\|_\frac{l}{l-1}\|X_i\|_{2l}\|W_i\|_{2l}&\le c_l A_{2l}^2 ((E S_i^{2l})^\frac{l-1}{l}+(ES_{i-1}^{2l})^\frac{l-1}{l}).
(WTW^*)(\xi)(x) = \sum_{z\in Z}\phi_z(x)\sum_{w\in Z}T_{z,w}\int_G \xi(y)\phi_w(y)\d\mu(y) =\int_G \sum_{z,w \in Z}\varphi_z(x) \varphi_w(xy) T_{z,w}\xi(xy) \, d\mu(y),
T_{n n}^{( 2 )}( K ) = - \frac{1}{2 \pi^{2}}F_{1}( n , k ; \Delta^{2}= \frac{( 2 n - 1 )}{R^{2}}) - \frac{1}{2 \pi^{2}}F_{2}( n , k ; \Delta^{2}= \frac{( 2 n + 1 )}{R^{2}}) ;
S_{G,A}=\frac 12\int_0^Td\tau \left[ \begin{array}{c}\left( D_\tau X_i^M\right) \varepsilon^{ij}X_j^N\,\,G_{MN}\left(X_1,X_2\right) +\left( D_\tau X_i^M\right) \varepsilon^{ij}A_{jN}\left( X_1,X_2\right)\,.\,\end{array}\right]
\tilde{g}(\tilde{\nabla}\Delta_{\tilde{g}}\Phi,\tilde{\nu})r^{n-1}=- \frac{n-4}{4(n-1)}\underbrace{u^{-2\frac{n-3}{n-2}}}_{\xrightarrow[r\rightarrow\infty]{}1}\partial_r R_{g}r^{n-1}+ \underbrace{O_1(r^{n-2\delta-4})}_{\xrightarrow[r\rightarrow\infty]{}0},
\int_{0}^{T}- \psi'(t) (\tilde{\textbf{u}}^{\varepsilon}, \phi)_{0}~ dt + \sum_{j=1}^{d}\int_{0}^{T}\psi(t) (A_{0}^{-1}J_{\varepsilon}A_{0}A_{j}(J_{\varepsilon}(\tilde{\textbf{u}}^{\varepsilon}+\bar{\textbf{u}}))\partial_{x_{j}}J_{\varepsilon}\tilde{\textbf{u}}^{\varepsilon}, \phi)_{0}~ dt = 0.
\hat{H_q}=\frac{\hbar^2(I-I_3)}{2II_3}J_3^{\prime 2}+\frac{\hbar^2}{2I}\left(\frac{\gamma}{\sinh\gamma}\hat{J}_{+}^\prime \hat{J}_{-}^\prime +[\hat{J}_{3}^\prime ][\hat{J}_{3}^\prime +1]\right)~.
\int_{{\cal M}_{g , N}}\! \! [ d m ] [ d z ] \; \langle \; \prod_{i = 1}^{N}\: \oint_{C_{z_{i}}}\! \! G^{-}\bar{G}^{-}\phi_{i}( z_{i}) \; \prod_{a ,{\bar{a}}= 1}^{3 g - 3}\: G^{-}( \chi_{a}) \bar{G}^{-}( \bar{\chi}_{\bar{a}}) \; \rangle_{\Sigma_{g , N}}\label 2
H_{\mu,N}= H_{\nu,N}+ \sum_{w \in L_{[1,N]^2}(Y_{1,n})}\nu(w) H( \mu( \cdot \mid \pi^{-1}(w) )) \leq H_{\nu,N}+ \sum_{w \in L_{[1,N]^2}(Y_{1,n})}\nu(w) \biggl( F_{N_0}(w) \cdot h' + F_{\partial}(w) \log m \biggr).
\bar{X}_{L,t_{k}}=\Big(\frac{1}{N_{0}}\sum^{N_{0}}_{i=1}\tilde{X}_{0,t_{k}}^{i}\Big)+\sum^{L}_{l=1}\Big(\frac{1}{N_{l}}\sum^{N_{l}}_{i=1}\big(\tilde{X}_{l,t_{k}}^{i}-\tilde{X}_{l-1,t_{k}}^{i}\big)\Big).
\frac{n+a_{p+1}+p+1}{n+1}\frac{c_{m-1}(n+1)}{(p-m+1)!}\frac{\prod_{i=1}^p(n+1+a_i+m-1)}{\prod_{i=1}^{m-1}{(n+1+i)}}-\frac{c_m(n)}{(p-m)!}\frac{\prod_{i=1}^p(n+a_i+m)}{\prod_{i=1}^m{(n+i)}}=\frac{\prod_{i=1}^{p+1}{(n+a_i+m)}}{(p+1-m)!\prod_{i=1}^m(n+i)}\frac{(n+a_{p+1}+p+1)c_{m-1}(n+1)-(p+1-m)c_m(n)}{n+a_{p+1}+m}.
{\mathcal J}^\mu\;=\;\sum_{i=1 \atop s=\uparrow,\downarrow}^N{\mathcal J}^{(i,s)\mu}~,~~ \widehat{{\mathcal J}}_{\ell/r}^\mu \;=\;\sum_{i=1\atop s=\uparrow,\downarrow}^N\widehat{{\mathcal J}}_{\ell,r}^{(i,s)\mu}~.
G\cdot \left( \frac{\omega_{0}^{n}}{\omega_t^{n}}\right)^{\frac{\alpha}{2\beta+\alpha}}\ = \G\cdot \left(e^{-(1+\frac{\alpha}{2\beta})\varphi_t}\right)^{\frac{\alpha}{2\beta+\alpha}}\ = \G\cdot e^{-\frac{\alpha}{2\beta}\varphi_t}.
[F_{\overline{j}}, e_{\overline{i}}] = [[f_{\overline{j}},f_j],e_{\overline{i}}] &= [e_{\overline{i}},[f_{\overline{j}},f_j]] &= [f_{\overline{j}},[f_j, e_{\overline{i}}]] - [f_j, [e_{\overline{i}}, f_{\overline{j}}]].
\frac{3}{8\pi^{2}}\omega^{(4)}_{i}+ \frac{1}{8\pi^{2}}\sum \Omega^{(2)}_{ijk}~\omega^{(2)}_{ijk}+ \frac{1}{4\pi^{2}}\sum \Omega^{(4)}_{ijklm}=\frac{1}{2},
\sum_{k=1}^3\int_{I^2}\epsilon^{ijk}\frac{\kappa^2}{16\pi}e^{-\kappa^2|\hat{y}_{k,\bar{s}}^{\bar{u}}- \hat{y}_{k,s}^{u}|^2/16}[-\zeta_{\kappa}^k(\hat{s})]\eta_\kappa(\hat{s})\y_{i,s}^{u,\prime}y_{j,\bar{s}}^{\bar{u},\prime}\d\hat{s}.
e_j \;=\; -\frac12\,\Bigl(\sigma_j^x\sigma_{j+1}^x \;+\;\sigma^y_j\sigma_{j+1}^y \;+\;\frac{q+q^{-1}}2\,(\sigma^z_j\sigma^z_{j+1}-1) \;+\;\frac{q-q^{-1}}2\,(\sigma^z_j-\sigma^z_{j+1}) \Bigr)\,.
\frac{d^{2}}{d\xi^{2}}\left(e^{-\frac{\xi}{2}}\xi^{\frac{k}{2}}F(\xi)\right) =e^{-\frac{\xi}{2}}\xi^{\frac{k}{2}}\left[\frac{d^{2}F}{d\xi^{2}}+\left(\frac{k}{\xi}-1\right)\frac{dF}{d\xi}+\left(\frac{1}{4}-\frac{k}{2\xi}+ \frac{\frac{k}{2}\left(\frac{k}{2}-1\right)}{\xi^{2}}\right)\right].
\frac{d}{dt}[(u(t)-v(t))^{\top}(u(t)-v(t))]\left|_{(\ref{r2})}\right.=2(u-v)^{\top}[f(u)-f(v)+\theta-\vartheta] &\ge{2\sigma}(u-v)^{\top}(u-v)-\mu (u-v)^{\top}(u-v)-\frac{1}{\mu}(\theta-\vartheta)^{\top}(\theta-\vartheta)
q(x) = (x - x_1)(x - \overline{x}_1) (x + 1/x_1)(x + 1/\overline{x}_1) = x^4 - x^3 + \frac{1}{2}x^2 + x + 1, r(x) = \frac{2}{3}x^3 - 2 x^2 + x.
\sharp S_{d,\varepsilon,B}=\frac{6\varepsilon}{\pi^2 \alpha^2}B^{\frac{2}{3}+\delta}+O(B^{\frac{2}{3}+\frac{2}{3}\varepsilon+(1-\varepsilon)\frac{\delta}{2}})=\frac{6}{\pi^2 \alpha^2}B^{\frac{2}{3}+\delta_0}(\int\chi(\varepsilon)(x)\operatorname{d}x+o(1)).
f(u,v) := \prod\nolimits_{\substack{\nu<0 v\nu>0 uv\nu<0}}(-1)^{uv\nu}\in{}^2H^\vee; a(u,v) := f(u,v)\cdot \prod\nolimits_{\substack{\nu<0 v\nu>0 uv\nu<0}}\left(x^{-\nu}\right)^{uv\nu}:H \to H^\vee
\mathcal{X}_{\infty}=\bigcap_{s\ge0}\bigoplus_{i=0}^K W^s\left(\Phi_i(U_i),\mathbf{R}^{2n}\right)=\bigoplus_{i=0}^K C^{\infty}\left(\overline{\Phi_i}(\overline{U_i}),\mathbf{R}^{2n}\right).
\mathbb{V}(\overline{S}_n)=\frac{1}{n^2}\sum_{1\leq i\leq n}\mathbb{E}({X_i}^2)+\frac{2}{n^2}\sum_{1\leq i\textless j\leq n}\mathbb{E}(X_iX_j)-\frac{1}{n^2}\mathbb{E}^2\Big(\sum_{1\leq i\leq n}{X_i}\Big).
E_{\lambda \mu}^{(r)}= \dfrac{\alpha_{\lambda}}{\alpha_{\mu}}\tilde{E}_{\lambda \mu}^{(r)}&= \dfrac{\alpha_{\lambda}}{\alpha_{\mu}}\left( \varepsilon_1 (x + \dfrac{1}{2}i) + \varepsilon_2 (y - \dfrac{1}{2}i) \right)^{r}\tilde{E}_{\lambda \mu}.
I_5 = \frac{-16 \pi^{\frac{D}{2}}\Gamma (\frac{D}{2}-1)}{(4-D)^2}\left[(n_0^2 +i\eta^{\prime})^{2-\frac{D}{2}}+ (-n_0^2 +i\eta^{\prime})^{2-\frac{D}{2}}\right],\eta^{\prime}> 0.
\aligned \widetilde{\mathcal W}{}^i_{jmn}=R^i_{jmn}-\delta^i_{[m}\zeta_{jn]}-\frac12 \mathcal F^i_{jm|n}+\frac12\mathcal F^i_{jn|m}, \endaligned \aligned \widetilde{W}{}^i_{jmn}=R^i_{jmn}+\frac1{N-1}\big(\delta^i_mR_{jn}-\delta^i_nR_{jm}\big) -\frac12\mathcal F^i_{jm|n}+ \frac12\mathcal F^i_{jn|m}. \endaligned
a^{p^*}= (2\Lambda)^{p^*}= 2^{\frac{p}{p-1}}\Big(\frac{p-2}{p\delta}\Big)^{\frac{p-2}{2}\cdot\frac{p}{p-1}}L^{\frac{p}{2}\cdot\frac{p}{p-1}}= O_p\bigg(\Big(\frac{1}{\epsilon}\Big)^{\frac{p(p-2)}{p-1}}L^p\bigg).
\frac{\partial F}{\partial x}(w,p, \tilde{\alpha}) = 2m \Big( 1- \tilde{t}^2 \frac{\vert \hat{w}\vert^2 p_1^2}{\vert w_1 \vert^2}\Big) \vert w_1 \vert^{2m}x^{2m-1}+ 2 \vert \hat{w}\vert^2 \tilde{t}^2 x = 0.
\limsup_{x \rightarrow\infty}\frac{\overline{F_{X_1}*F_{X_2}}(x-1)}{\overline{F_{X_1}*F_{X_2}}(x)}\leqslant\limsup_{x \rightarrow\infty}\sup_{z \geqslant x-D}\frac{\overline{F}_{X_1}(z-1)}{\overline{F}_{X_1}(z)}&= \limsup_{y \rightarrow\infty}\frac{\overline{F}_{X_1}(y-1)}{\overline{F}_{X_1}(y)}&<\infty
\partial_{\mu}S^{\mu}_0 = \frac{1}{3}\sum_i \left( \gamma^{\ast}_i + \gamma_i \right) \Phi^{\alpha}_i \frac{\partial W}{\partial \Phi^{\alpha}_i}\Bigg| + \frac{1}{48 \pi^2}\left[ 3T (G) - \sum_i T(R_i) (1-(\gamma^{\ast}_i + \gamma_i)) \right] F^a_{\mu v}\widetilde{F}^{\mu va}\, ,
\mathcal{D}_{G, \kappa}^{\mathrm{ch}}\simeq \bigoplus_{\lambda\in P^+}\mathbb{V}^{\kappa}_\lambda\otimes \mathbb{V}^{\kappa^*}_{\lambda^*},\frac{1}{\kappa+h^\vee}+\frac{1}{\kappa^*+h^\vee}=0.
\eta \in C_c^{\infty}(\mathbb{R}),0\leq \eta \leq 1,\eta(s)=\begin{cases}1\mbox{if}|s|\leq 1, 0 \mbox{if}|s|\geq 2,\end{cases}\zeta \in C_c^{\infty}(\mathbb{R}^2),0\leq \zeta \leq 1,\zeta(s)=\begin{cases}1\mbox{if}|x'|\leq 1, 0 \mbox{if}|x'|\geq 2.\end{cases}
\max_{j<\beta N}\left\{\Pr_{v\sim \mu_t}[|v H|=j]\cdot 2^{2\epsilon N\log(1-\frac{2j}{N})}\right\}\leq\max_{\alpha> 1}\left\{2^{-\frac{2\alpha\tilde{\epsilon}(1-\tilde{\epsilon})}{\ln2}(1-\eta) N}\cdot 2^{\epsilon N\log(4\alpha\tilde{\epsilon}(1-\tilde{\epsilon}))}\right\}.
\omega\left( T\left( a_{1}x_{1},a_{2}x_{2}\right) \right) =\omega\left(T\left( a_{1},a_{2}\right) \right) .\omega\left( T\left( x_{1},x_{2}\right) \right) =\omega\left( T\left( a_{1},a_{2}\right) T\left(x_{1},x_{2}\right) \right)
x_0\,a_1-x_1\,a_0=\begin{vmatrix}x_0 x_1 a_0 a_1\end{vmatrix}\ , y_0\,(x_0\,h_{01}+x_1h_{11})-y_1\,(x_0\,h_{00}+x_1h_{10})=\begin{vmatrix}y_0 y_1 x_0\,h_{00}+x_1h_{10}x_0\,h_{01}+x_1h_{11}\end{vmatrix}\ ,
\int_\Omega \frac{|\nabla u(z)|}{|x-z|^{d-1}}\,\d z = \int_{0}^{\infty}\int_{\{|\nabla u|>s\}}|x-z|^{1-d}\,\d z\, \d s\cr \le \int_{0}^{\infty}\bigl[\mu_{|\nabla u|}(s)\bigr]^{\frac{1}{d}}\, \d s= \frac{1}{d}\|\nabla u \|_{d,1(\Omega)}
\lim_{a\uparrow a^*}(a^*-a)^{-p_{j_0}/p}K(x_{j_0}+(a^*-a)^{1/p}x) |v_a({x+(a^*-a)^{-1/p}x_{j_0}})|^{2+4/d}&= \lambda_{j_0}|x|^{p_{j_0}}\Big(b^{d/2}Q_0(b(x+x_0)) \Big)^{2+4/d}.
\frac{1}{\sqrt{- g}}j_{\phi}^{\mu}= f_{\phi}g^{\mu \nu}\partial_{\nu}\phi + a \exp ( - 2 \frac{\phi}{f_{\phi}}) g^{\mu \nu}\partial_{\nu}a - \frac{1}{2}\exp ( \frac{\phi}{f_{\phi}}) F^{\mu \nu}A_{\nu}+ \frac{q^{2}}{1 6 \pi^{2}}\frac{a}{f_{a}}\tilde{F}^{\mu \nu}A_{\nu}.
\mathcal{B}_{\mathcal{I}, \lambda_1, \ldots, \lambda_r}&= \left\{\tilde{f}_{i_{1, 1}}^{x_1}\cdots \tilde{f}_{i_{1, N_1}}^{x_{N_1}}(b_{\lambda_1}\otimes b)~\Big|~ x_1, \ldots, x_{N_1}\in \mathbb{Z}_{\ge 0},\b \in \mathcal{B}_{(I_2, \ldots, I_r), \lambda_2, \ldots, \lambda_r}\right\}\setminus \{0\}.
\langle v_{I,A\tilde{A}}(\tau) v_{J,B\tilde{B}}^\dagger(\tau') \rangle=\delta^{IJ}\delta^{AB}\delta^{\tilde{A}\tilde{B}}\int\frac{dk}{2\pi}\frac{e^{ik(\tau-\tau')}}{k^2+r^2/\lambda^2}\equiv \delta^{IJ}\delta^{AB}\delta^{\tilde{A}\tilde{B}}\Delta(\tau-\tau').
0= (\cot (2\vartheta) -\tan \vartheta ) (\cot \vartheta \cot (2\vartheta) -1)-3(\cot \vartheta+\cot (2\vartheta)) = \frac{(\cot \vartheta )^{4}-24 (\cot \vartheta )^{2}+15}{\cot \vartheta}.
{\rm AS}_g(D):=\frac{\widehat{A}\big(\frac{R_{X^g}}{2\pi i}\big) \operatorname{tr}\big(g \exp(\frac{R^F}{2\pi i})\big) \exp(\operatorname{tr}(\frac{R^L}{2\pi i}))}{\operatorname{det}\big(1-g \exp(-\frac{R^{\mathcal{N}}}{2\pi i})\big)^{\frac{1}{2}}}.
\begin{aligned}\zeta_{S(F),\chi,\chi'}:S(F) \rightarrow \mathbb{C}^{\times}t \mapsto \prod_{\alpha\in \Phi_{\mathrm{sym}}/\Gamma_F}\zeta_{\alpha}(\iota_{\alpha}\alpha(t))\prod_{\alpha\in \Phi_{\mathrm{asy}}/\Gamma_F\times\{\pm1\}}\zeta_{\alpha}(\alpha(t)),\end{aligned}
J_{i}\boldsymbol{I}_{jk}^{\pm}=0,\;\ \ \;J_{i}\boldsymbol{I}_{ki}^{\pm}=\pm \frac{1}{2}w^{k}\boldsymbol{a}_{ik},\;\ \J_{i}\boldsymbol{I}_{ij}^{\pm}=\pm \frac{1}{2}w^{j}\boldsymbol{a}_{ij},
g_j(\xi) =r_j^n(-\Delta v(x_j+r_j\xi)) \le Cr_j^n|x_j|^{-\beta}\left(u(x_j+r_j\xi)+|x_j|^{-(n-2)}\right)^\sigma \le C|x_j|^{n-\beta-(n-2)\sigma}\left(1+((N_2f_j)(\xi))^\sigma\right).
\min(|x-y|,2\pi-|x-y|)\geq \min(|y_i-y_j|,2\pi-|y_i-y_j|)-\max(F_n(x_i),F_n(x_j)) \geq \min\{z_{i+1}-z_i|1\leq i\leq k\}-\min\{z_{i+1}-z_i|1\leq i\leq k\}/2 &=\min\{z_{i+1}-z_i|1\leq i\leq k\}/2:=a_0\in(0,2\pi),
P_{\hat{{\bf S}}_i |{\bf X}^i, Y^i}({\bf s}|{\bf x}^i, y^i) = P_{\hat{{\bf S}}_i |{\bf X}_i, Y_i, \Omega_{i-1}}({\bf s}|{\bf x}_i, y_i, \omega_{i-1}({\bf x}^i, y^i )) & = \begin{cases}\frac{1}{| \omega_{i}({\bf x}^i, y^i ) |}{\bf s}\in \Omega_{i}0 \end{cases},
W := \Bigg\{\bigg(x_{\bar{a}}: \bar{a}\in \bigcup_{I \subseteq [k]}\prod_{i \in I}Q_i \bigg) \in [0,1]^{K}: \bigwedge_{\bar{a}\in \prod_{i \in [k]}Q_i \cap R_k}\left( x_{\bar{a}_I}: I \subseteq [k] \right) \in S_{+}\land \bigwedge_{\bar{a}\in \prod_{i \in [k]}Q_i \setminus R_k}\left( x_{\bar{a}_I}: I \subseteq [k] \right) \notin S_{-}\Bigg\}.
I_{( c )}+ I_{( d )}\sim - G_{- +}^{( 0 )}( k + p_{2}) \left[ G_{+ +}^{( 0 )}( k - p_{1}) - G_{+ -}^{( 0 )}( k - p_{1}) \right] = - G_{- +}^{( 0 )}( k + p_{2}) G_{R}( k - p_{1}) .
\lim_{t\rightarrow{1}}(1-t)F_N(a,b;t) =\frac{(1-b)(1-tq^N)}{(1-bq^N)}F_N(a/b,1;b) &=\frac{(1-b)(1-tq^N)}{(1-bq^N)}\cdot\frac{(aq)_N}{(b)_N}&=\frac{(1-q^N)(aq)_N}{(bq)_N},
\left\{\tau_R = \tau \le t \right\}= \bigcap_P \left( \bigcup_{\substack{t_k \in P t_k < t}}\bigcap_{0 \le s < t_k}f_{s}^{-1}\left( \left[0,R\right] \right) \cap \{t_k \le \tau \le t_{k+1}\}\right),