harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
h\tau h^{-1}\tau^{-\frac{w}{z}}= \left( \begin{pmatrix}1 -1 0 1 \end{pmatrix}+ \begin{pmatrix}a -b -c d \end{pmatrix}p \right) \left( \begin{pmatrix}1 1 0 1 \end{pmatrix}+ \begin{pmatrix}a * c d \end{pmatrix}p \right) &= I + \begin{pmatrix}2a-c * 0 2d-c \end{pmatrix}p.
c h a n g e d \int d^{2 5 - p}q_{\bot}( 2 )^{q_{\bot}}\left( \frac{1 + W_{0}^{2}}{k_{0}W_{0}^{2}}\right)^{q_{\bot}^{2}}: c \partial X ( i ) \cdot \tilde{c}\bar{\partial}X ( i ) e^{i q_{\bot}\cdot X ( i )}: .
e^{q_{i}- q_{n + 1}}={\frac{X^{( n - i + 1 )}}{X^{( n - i )}}}, \quad p_{i}={\frac{Y^{( n - i + 1 )}}{X^{( n - i + 1 )}}}-{\frac{Y^{( n - i )}}{X^{( n - i )}}}
\|z\|^2\lambda\coth\lambda-i\lambda t ={\lambda\over \sinh\lambda}d_K^2(g)\bigg({\|z\|^2\over d_K^2(g)}\cosh\lambda-i{t\over d_K^2(g)}\sinh\lambda\bigg) &={\lambda\over \sinh\lambda}d_K^2(g)\cosh(\lambda-i\phi),
(u_{\mp}-v_{\pm}+hB_{ij})(u_{\pm}-v_{\mp}-hB_{ij})H_{i}^{\pm}(u)H_{j}^{\mp}(v) =(u_{\mp}-v_{\pm}-hB_{ij})(u_{\pm}-v_{\mp}+hB_{ij})H_{j}^{\mp}(v)H_{i}^{\pm}(u),
\Theta \left[^a_{b}\right] \left( z + n + \Omega \cdot m | \Omega \right) = e^{- \pi i m \cdot \Omega \cdot m - 2 \pi i m \cdot z + 2 \pi i ( a \cdot n - b \cdot m )}\Theta \left[^a_{b}\right] \left( z | \Omega \right) .
G(\omega ,\vec{k})=-\frac{\phi_{1}f_{2}\partial_{r}\phi_{2}^{(-)}-\phi_{2}^{(-)}f_{1}\partial_{r}\phi_{1}}{\phi_{1}f_{2}\partial_{r}\phi_{2}^{(+)}-\phi_{2}^{(+)}f_{1}\partial_{r}\phi_{1}}\ .
\mathbb{E}_{t_{n}}^{\sigma,X^{n}}\left[ \int_{t_{n}}^{t_{n+1}}Z_{t}^{t_{n},X^{n}}dB_{t}\Delta B_{n+1}\right] =\sigma^{2}\mathbb{E}_{t_{n}}^{\sigma,X^{n}}\left[ \int_{t_{n}}^{t_{n+1}}Z_{t}^{t_{n},X^{n}}dt\right] .
\det \left(z_j^{i-1}\right)_{1\le i,j\le n_2}=(-1)^{\frac{n_2(n_2+1)}2}\prod\limits_{j=1}^{n_2}z_j^{n_2}\int\limits_{(-\infty,a]^{n_2}}\prod\limits_{j=1}^{n_2}e^{(a-x_j)z_j}\det\left(\frac{(x_j-a)^{i-1}}{(i-1)!}\right)_{1\le i,j\le n_2}\,d^{n_2}x.
\left\langle P^{2}\left( a_{\hbar},a_{\hbar}^{\dag}\right) \right\rangle_{U_{\hbar}\left( \alpha\right) \psi}=\left( P_{0}^{2}\right) \left(\alpha\right) +\sum_{k=1}^{2d}\hbar^{k/2}\left\langle \left( P^{2}\right)_{k}\left( \alpha:a,a^{\dag}\right) \right\rangle_{\psi}
J(w)=\frac{h_d^2}{-\lambda_w \cdot \langle1,1\rangle}+ \frac{1}{4\pi i}\int_{(1/2)}\left|\frac{\zeta_k(s)}{\zeta(2s)}\right|^2 - \left|\frac{\zeta_k(w)}{\zeta(2w)}\right|^2 \frac{ds}{\lambda_s - \lambda_w},
a_m \equiv\begin{cases}(6r)^m \pmod{3^{m+1}}p = 3,~ 3^2 \mid n, 8^m \pmod{2^{3m+1}}p = 2,~ 2^4 \mid n,\end{cases}b_m,c_m \equiv \begin{cases}(3r)^{2m}\pmod{3^{2m+1}}p = 3,~ 3^2 \mid n, 2^{6m}\pmod{2^{6m+1}}p = 2,~ 2^4 \mid n.\end{cases}
II \ge (1+\eta) (1- \eta) - 2\eta \cdot \frac 23 \cdot \frac 1{1 +1}\cdot \frac{\frac 32 (1- \eta)\cdot 4}{\sqrt{3}+1}= (1-\eta)\Big( 1+ \eta - \frac 4{1+\sqrt{3}}\eta \Big) \ge 0,
S_{eff}=\int{d^5\sigma d\tau [\mu\dot{C}+\mu_\rho\dot{C}^{\rho}+P^{\mu\nu}\dot{B}_{\mu\nu}+\Pi_a\dot{X}^a+\hat{\mu}^i\dot{\hat{C}}_i+\widehat{\mu}\dot{\widehat{C}}+\frac{1}{\sqrt{W}}\hat{\delta}\mu]}
C(\hat{L}; n_1,n_2) \stackrel{\cong}{\rightarrow}(C(\Sigma^{2V_-}\hat{L},n_1, n_2), d_1, d_2), C(\hat{L}; n_1+1,n_2) \stackrel{\cong}{\rightarrow}( C(\Sigma^{2V_-'}\hat{L}; n_1+1, n_2), d_1, d_2 )
2d_{TV}(\mathbb{P}_{Gen(n)}, \mathbb{P}_{\mathcal{P}_n}) = \left \| \frac{d \mathbb{P}_{Gen(n)}}{d \mathbb{P}_{\mathcal{P}_n}}- 1 \right\|_1^{\mathbb{P}_{\mathcal{P}_n}}\le \left \| \frac{d \mathbb{P}_{Gen(n)}}{d \mathbb{P}_{\mathcal{P}_n}}- 1 \right\|_p^{\mathbb{P}_{\mathcal{P}_n}}= e^{-\Theta(n)}.
\zeta_{\bar{{\cal V}}}( s ) = \frac{m L}{\Gamma ( s ) \sqrt{4 \pi}}\int_{0}^{\infty}\! \! d \beta \, \beta^{s - \frac{3}{2}}e^{- \beta}= \frac{m L}{\sqrt{4 \pi}}\frac{\Gamma ( s - \frac{1}{2})}{\Gamma ( s )}\, .
C_{n,A,L}[z; \psi_1, \psi_2] := \int_{(\Lambda_L^*)^{n+1}}\mathcal{P}_{A,L}(k) \overline{\widehat{\psi}}_{1,\#}(k_1) \widehat{\psi}_{2,\#}(k_{n+1}) \prod_{j=1}^{n+1}(\nu( k_j) - z )^{-1}d (k_1, \ldots, k_{n+1}).
\sigma_{1}= \left( \begin{array}{c c}{0}&{1}\\{1}&{0}\\ \end{array}\right) , \sigma_{2}= \left( \begin{array}{c c}{0}&{- i}\\{i}&{0}\\ \end{array}\right) , \sigma_{3}= \left( \begin{array}{c c}{1}&{0}\\{0}&{- 1}\\ \end{array}\right) , \,
\widehat{f_\delta}(a,z) = (2\pi)^{-p-q}|\lambda|^{p+q}\, \Big( \sum_{j=1}^{d(p,q)}P_{p,q}^j(z)\, \mathcal{G}_{n+p+q}(g_{\delta,j})(a(p,q))\Big) \, \varphi_{k-q,\lambda}^{n+p+q-1}(z).
\begin{array}{l}{{c_{1}^{[ \alpha ]_{k}| [ \beta ]_{l}}= 0 , \quad k \neq l , \quad k + l \neq n ,}}\\{{c_{1}^{[ \alpha ]_{k}| [ \beta ]_{n - k}}= c_{1}^{( k )}\varepsilon^{[ \alpha ]_{k}[ \beta ]_{n - k}}, \quad k , n - k \ge 4 , \quad 2 k \neq n ,}}\\{{\varepsilon ( c_{1}^{( k )}) = n \, ( m o d \, \, 2 ) , \quad c_{1}^{( k )}= - c_{1}^{( n - k )}.}}\\ \end{array}
\mathcal{L}_{\mathcal{I}_l}W_\epsilon(x_j,x_k) u_\epsilon=W_\epsilon(x_j,x_k)\mathcal{L}_{\mathcal{I}_l}u_\epsilon+(\nabla_jW_\epsilon(x_j,x_k))\mathcal{L}_{\mathcal{I}_l,j}u_\epsilon.
\lambda\int_{-a}^{a}f\left( x\right) u\left( x\right) dx =\int_{-a}^{a}L_{q}f\left( x\right) u\left( x\right) dx & =f(a)u^{\prime}(a)-f(-a)u^{\prime}(-a)+\lambda_{0}\int_{-a}^{a}f\left(x\right) u(x)dx
I_{2}=\frac{\delta \Gamma_{2}}{2\pi}\frac{\mbox{\mathversion{bold}$Y$}_{\xi}\times(\mbox{\mathversion{bold}$X$}-\mbox{\mathversion{bold}$Y$})}{|\mbox{\mathversion{bold}$X$}-\mbox{\mathversion{bold}$Y$}|^{3}}+O(\delta^{2}).
\sum q^{\frac{1}{2}p^{2}}= \sum_{n}q^{\frac{3}{2}\left( 2 n + 1 + \frac{m_{1}+ m_{2}}{3}\right)^{2}}= \sum_{n}q^{6 \left( n + \frac{m_{1}+ m_{2}+ 3}{6}\right)^{2}}= \theta_{2 m_{1}+ 2 m_{2}+ 6 , 6}\: ( \tau ) .
h_{\beta_1}:= + 2^{-1}\sigma_1 h_1 - 2^{-1}\sigma_2 h_2 - 2^{-1}\sigma_3 h_3, h_{\beta_2}:= - 2^{-1}\sigma_1 h_1 + 2^{-1}\sigma_2 h_2 - 2^{-1}\sigma_3 h_3, h_{\beta_3}:= - 2^{-1}\sigma_1 h_1 - 2^{-1}\sigma_2 h_2 + 2^{-1}\sigma_3 h_3, h_\theta:= + 2^{-1}\sigma_1 h_1 + 2^{-1}\sigma_2 h_2 + 2^{-1}\sigma_3 h_3.
u \in \mathrm{U S p}( 2 N ) : \qquad u_{i}^{I}u_{J}^{i}= \delta_{J}^{I}\; , \ \u_{i}^{I}\Omega^{i j}u_{j}^{J}= \Omega^{I J}\; , \ \u_{i}^{I}= ( u_{I}^{i})^{*}\; .
\psi_{j}^{( n )}( z ) = C_{j}^{( n )}\Theta \left( I ( z ) +{\cal D}^{j ; n}| \Omega \right){\frac{\sigma ( z )^{2 n - 1}E ( z , P_{+})^{j - s ( n )}}{E ( z , P_{-})^{j + s ( n )}}},
w \asymp \int\limits_{b(v)}^1 \frac{(bD^r)^{(r-1)/r}}{r \sqrt{1-b^2}- D^{-1}O(b^{-1/r}\sqrt{1-b^2})}\, db &= D^{r-1}\int\limits_{b(v)}^1 \frac{b}{r b^{1/r}\sqrt{1-b^2}- D^{-1}O( \sqrt{1-b^2})}\, db.
Z [ \eta_{l}] \approx \int{\cal D}[ \chi ] \exp \left[ -{\frac{e^{2}}{4 \pi^{2}m_{W}}}\int d^{3}x \left({\frac{1}{2}}( \nabla ( \chi_{\mathrm{c l}}- \eta_{l}) )^{2}- \omega^{2}\cos \chi_{\mathrm{c l}}\right) \right] \; ,
\begin{aligned}\chi(CY_3; \tau, z) =\frac{e(CY_3)}2\,\varphi_{0,\frac32}=\frac{e(CY_3)}2\left(\zeta^{-\frac{1}2}+\zeta^{\frac{1}2}+q(\ldots)\right), \chi(CY_5; \tau, z) =\frac{e(CY_5)}{24}\,\varphi_{0,\frac32}\cdot \varphi_{0,1}=\frac{e(CY_5)}{24}\left(\zeta^{\pm \frac 32}+11\zeta^{\pm \frac 12}+q(\ldots)\right)\end{aligned}
\int_{\mathbb{R}}\frac{P_{m}(a,q;x)P_{n}(a,q;x)}{\gamma\big(a\psi_{a}(x)-\varphi_{a}(x)\big)^{2}+\gamma^{-1}(a-1)^{2}\varphi_{a}(x)^{2}}\,\mbox{d}x=\frac{\pi}{(a-1)^{2}}\,\delta_{mn}
{\mathbb E}( \| S_n-P_m(S_n) \|_H^2) =n{\mathbb E}(\| Y_0-P_m(Y_0) \|_H^2)+ 2 \sum_{k=1}^{n-1}(n-k){\mathbb E}(<Y_0-P_m(Y_0), Y_k-P_m(Y_k)>) &\leq n \Big({\mathbb E}(< Y_0-P_m(Y_0), Y_0> )+ 2 \sum_{k=1}^{n-1}|{\mathbb E}(<Y_0-P_m(Y_0), Y_k>)| \Big)\, ,
(\delta^o_{\frac14}B_o) \cap (\phi^{o,\epsilon}_t)^{-1}\delta^o_{\frac1\epsilon}E = (\phi^{o,\epsilon}_t)^{-1}\left( \phi^{o,\epsilon}_t(\delta^o_{\frac14}B_o) \cap \delta^o_{\frac1\epsilon}E\right) &\subset (\phi^{o,\epsilon}_t)^{-1}\left( \delta^o_{\frac12}B_o\cap \delta^o_{\frac1\epsilon}E\right)\subset B_o\cap \delta^o_{\frac1\epsilon}E
E[\psi(W,\gamma,\alpha,\theta_{0})] =E[m(W,\gamma)]-\theta_{0}+E[\alpha_{0}(X)\{Y-\gamma(Z)\}] & =E[m(W,\gamma)]-\theta_{0}+E[\alpha_{0}(X)\{Y-\gamma_{0}(Z)+\gamma_{0}(Z)-\gamma(Z)\}] & =E[\alpha_{0}(X)\{\gamma(Z)-\gamma_{0}(Z)\}]+E[\alpha_{0}(X)\{\gamma_{0}(Z)-\gamma(Z)\}]=0.Q.E.D.
\|\tilde{h}^{(m)}\|_{\bar{{\cal H}}([0,a_{m+1}])}^{2}\leq C_{H}(Q_{1}+Q_{2}+Q_{3})\leq C_{H,l_{0}}\sum_{k=1}^{m}k^{2H+1}|I_{k}|^{2(1-H)}.
(p-q)P_{q+1}^{n-2,p-q}(x)+2s\Big(P_{q+1}^{n-2,p-q}(x)\Big)' =(p-q)P_{q+1}^{n-2,p-q}(x)+\frac{1}{2}(x+1)(n+p)P_q^{n-1,p-q+1}.
a_\pm = \frac{1}{2}\biggl(1\pm\frac{\rho_2}{\rho_3}-\frac{i\kappa_3}{\kappa}\cdot\frac{\rho_1-\rho_2}{\rho_1+\rho_2}\biggr),b_\pm = 1 \pm \frac{\rho_2}{\rho_3}\,.
\beta([x_{\textup{f}}])=\beta^{\textup{cyc}}\left(\left[x_{\textup{f}}\right]\right)\cdot(\gamma-1)+\beta^{\textup{wt}}\left(\left[x_{\textup{f}}\right]\right)\cdot\varpi_\kappa=-\left(\partial_1(\frak{d}_{\textup{cyc}}(\frak{X}))\cdot(\gamma-1)+\partial_1(\frak{d}_{\textup{wt}}(\frak{X}))\cdot\varpi_{\kappa}\right)
(m+2) \left( \#(e_{\geq m+2})+\#(h_{\geq m+2})+\#(f_{\geq m+2})\right) &+ (m+1)\left( \#(e_{m+1})+\#(h_{m+1})+\#(f_{m+1})\right)+m\#(f_m) ={}C + (m+1) \left( \#(e_{\geq m+1})+\#(h_{\geq m+1})+\#(f_{\geq m}) \right) - \#(f_m) ={}C + (m+1)(k+1) - \#(f_m).
\Bigg| \int_{\mathbb{R}^5}\left( \Delta^2 \varphi_R \right) |u|^2 dx\Bigg| =\Bigg| \frac{1}{R^2}\int \left( \Delta^2 \varphi \right) \left( \frac{x}{R}\right) u(t,x)^2 dx \Bigg| & \le \frac{C}{R^2}\| u \|_2^2 = \frac{C}{R^2}\| Q \|_2^2 \le 2s_c(p-1)\delta_1 \le 2s_c(p-1)\delta(t)
\begin{array}{r c l r c l}{F_{1}}&{=}&{\bar{U}d V - V d \bar{U},}&{F_{3}}&{=}&{\bar{U}H + V \bar{H},}\\{F_{5}}&{=}&{d A_{4}+ \mathrm{i}( B \bar{H}- \bar{B}H ) ,}&{H}&{=}&{d B ,}\\ \end{array}
\Omega ( N , T , V ) ={\frac{V^{N}}{N !}}\Bigl \{1 +{\frac{N^{2}}{2 V}}\int_{0}^{2 \pi}d \theta \int_{0}^{\infty}d r r \Bigl ( \mathrm{e}^{-{\frac{\phi ({\bf r}_{i j})}{k T}}}- 1 \Bigr ) + \cdots \Bigr \}\ .
\left|\begin{array}{cc}X Y x y\end{array}\right|^{k-2}=2^{2-k}\left|\begin{array}{cc}Y+iX Y-iX x-yi -iy-x\end{array}\right|^{k-2}=2^{2-k}\sum_{j=0}^{k-2}\binom{k-2}{j}P_{j}(X,Y)z^j\bar z^{k-2-j},
( X , Y ) = \int d^{n}x \left( \frac{\delta^{R}X}{\delta \Phi^{M}( x )}\frac{\delta^{L}Y}{\delta \Phi_{M}^{*}( x )}- \frac{\delta^{R}X}{\delta \Phi_{M}^{*}( x )}\frac{\delta^{L}Y}{\delta \Phi^{M}( x )}\right)
\begin{cases}dX_{x}^{\varepsilon}(t)=[AX_{x}^{\varepsilon}(t)+B(t,X_{x}^{\varepsilon}(t))]dt+\sqrt{{\varepsilon}}G(t,X_{x}^{\varepsilon}(t))dw(t), X_{x}^{\varepsilon}(0)=x\in E.\end{cases}
(\,\Lambda \,\vert\Lambda\,)=1\,,\,\,(\,\Phi \vert X\Psi\,)=(\,X^*\Phi\vert \Psi \,)\,\,{\rm for}\,\,\, \Phi\,,\Psi\,\in{\cal V}_\Lambda\,, \,\,X\in U_q^{(n)}\,.
\mathcal{L}(\mathcal{X}) =\lim_{t\to \infty}\frac{1}{t}\mathbb{E}\bigl(\log \Vert \mathbf{X}_{i_1}\cdots \mathbf{X}_{i_t}\Vert \bigr) &=\lim_{t \to \infty}\frac{1}{t}\log \Vert \mathbb{E}(\boldsymbol{\alpha}_t)\Vert=\lim_{t \to \infty}\frac{1}{t}\mathbb{E}(\log \Vert \boldsymbol{\alpha}_t\Vert) &=\mathcal{L}(\mathcal{S})\,.
\int_{- \infty}^{\infty}{\frac{d p}{2 \pi}}e^{- \alpha_{1}p^{2}}\, e^{- \alpha_{2}( p + k )^{2}}={\frac{1}{\sqrt{4 \pi ( \alpha_{1}+ \alpha_{2})}}}e x p \left[ - k^{2}\left({\frac{\alpha_{1}\alpha_{2}}{\alpha_{1}+ \alpha_{2}}}\right) \right]
u (t) =\hat{f}(\eta(t_{k}),\hat{\xi}(t_{k}),e(t_{k})) \dot{\eta}(t) =\hat{g}(\eta(t),\eta(t_{k}),\hat{\xi}(t_{k}),e(t_{k})) \dot{\hat{\xi}}(t) =\hat{l}(\hat{\xi}(t),\hat{\xi}(t_{k}),e(t_{k})),~t\in[t_{k},t_{k+1}),~k\in\mathbb{S}
\partial_1{\mathbf V}^+_n=\left[\begin{array}{c}\mathcal{K}_{1}\mathcal{K}_{2}-\partial_2(\dot{H}^+_2\partial_1\hat{\Phi}^+) \end{array}\right],\partial_1{\mathbf V}^-_n=\left[\begin{array}{c}\mathcal{K}_{7}\mathcal{K}_{8}-\partial_2(\dot{H}^-_2\partial_1\hat{\Phi}^-) \end{array}\right]\,.
\lim_{\lambda\to0^+}\lambda\left\|\left[\int_{\mathbb{R}^n}\mathbf{1}_{E_{\lambda,\frac{\gamma}{q}}[f]}(\cdot,y)\left|\cdot-y\right|^{\gamma-n}\,dy\right]^\frac{1}{q}\right\|_{L^{{r}(\cdot)}(\mathbb{R}^n)}=\left[-\frac{\kappa(q,n)}{\gamma}\right]^\frac{1}{q}\left\|\,\left|\nabla f\right|\,\right\|_{L^{{r}(\cdot)}(\mathbb{R}^n)}.
\frac{\partial \tau_1}{\partial \xi}\ddot{\omega}_{11*}^H +\frac{\partial \tau_2}{\partial \eta}\ddot{\omega}_{22*}^E +\frac{\partial^2 \alpha_1}{\partial \zeta^2}\ddot{\omega}_{33*}^0 +2\frac{\delta \omega}{c}\alpha_1=0.
\Big\|C\star({\bf 1}^\epsilon_0 \|C^3\|_{L^1}-{\bf 1}^\epsilon_0 \|C_\infty^3\|_{L^1})\star S\Big\|_{L^1\cap L^\infty}\leq \frac{\gamma_t\|E\|_{L^1\cap L^\infty}}{m^2_t}+ \gamma_t\Big( \frac{1}{m^4_t}+ \frac{c}{m^2_t}{\bf 1}_{d=2}+ \frac{c}{m_t}{\bf 1}_{d=3}\Big).
\frac{\xi_1}{u_1}\frac{\partial}{\partial \xi_1}\wedge\frac{\partial}{\partial u_1},\,\,\,\,\,\frac{1}{u_1^{n+1}}\frac{\partial}{\partial \xi_1}\wedge \frac{\partial}{\partial u_1},\,\,\,\,\,\frac{1}{u_1^k}\frac{\partial}{\partial \xi_1}\wedge \frac{\partial}{\partial u_1}\in C^1(\mathcal{U},\wedge^2\Theta_S),\,\,\,\,\,k=1,...,n-1,
\| a-w \|_W \leq \| a - v \|_W + \| v - w \|_W < \| a - v \|_W + \frac{\varepsilon}{2}\leq c \, \| a - v \|_V + \frac{\varepsilon}{2}< c \cdot \frac{\varepsilon}{2c}+ \frac{\varepsilon}{2}= \varepsilon .
\lambda(R)=\frac{\nu+\frac{1}{2}+(\nu-\frac{1}{2})\frac{\Gamma(1-\nu)}{\Gamma(1+\nu)}\left( \frac{kR}{2}\right)^{2\nu}}{1-\frac{\Gamma(1-\nu)}{\Gamma(1+\nu)}\left( \frac{kR}{2}\right)^{2\nu}}.
\begin{array}{ll}\frac{1}{2}(L_Z g)(X,Y) + g(X, Y) div(H')+ \frac{1}{\lambda^2}Ric^N (\tilde{X}, \tilde{Y}) - \frac{1}{4}\lambda^4 g(X, Y) |\nabla_{\nu}\frac{1}{\lambda^2}|^2 + \mu g(X,Y) + \frac{n \lambda^2}{2}g(X, Y) \left(H' \frac{1}{\lambda^2}\right) = 0.\end{array}
\overset{2}{R}(X,Y)Z + \overset{2}{R}ic(AX,Z) A Y - \overset{2}{R}ic(AY,Z) AX = \overset{g}{R}(X,Y)Z + \overset{g}{R}ic (AX,Z) A Y - \overset{g}{R}ic (AY,Z) AX.
2(1-\delta) k' \cdot \frac{2}{3}(n-1) \le d(v_1)+d(v_2) &= e(\{v_1,v_2\},S) + e\bigl(\{v_1,v_2\},V-(S\cup\{v_1,v_2\})\bigr) + 2w(v_1v_2) &\le \left(k'+h_5+\frac{1}{2}\right)|S|+\biggl(k'+\frac{h-h_5}{2}\biggr)(n-|S|-2) + 2k.
f_{jk}\,^{p}f_{pi}\,^{m}=f_{ji}\,^{p}f_{pk}\,^{m}+f_{ki}\,^{p}f_{jp}\,^{m}, &f_{jk}\,^{p}f_{ip}\,^{m}=f_{ij}\,^{p}f_{pk}\,^{m}+f_{ik}\,^{p}f_{jp}\,^{m}.
{\cal A}_{99}(\phi^i,{\phi^{\bar{\imath}}})=-\;\frac{1}{4N}\int_0^1 \frac{dq}{q}\int_0^q \frac{dz}{z}\int \frac{d^{4}p}{{(2\pi)}^{4}}\sum_{k=0}^{N-1}{\rm Tr}\Bigl[\theta^k V(\xi_1,p_1;z) V(\xi_2,p_2;q) \, q^{L_0}\Bigr]
\xi_1^2+\frac{\xi_2^2}{a(t)^2}+\frac{\xi_3^2 \csc^2(\eta )}{a(t)^2}+\frac{\xi_4^2 \sec^2(\eta )}{a(t)^2}+\xi_5^2+\xi_6^2= u_1^2 + \frac{1}{a(t)^2}( u_2^2 + u_3^2+u_4^2 ) + u_5^2 + u_6^2,
g_{1}(\nabla^{^{M_1}}_{X}Y,W) =-g_{1}(\nabla^{^{M_1}}_{X}Y,\phi^{2}W)-g_{1}(\nabla^{^{M_1}}_{X}Y,\omega\phi W)+g_{1}(\nabla^{^{M_1}}_{X}\mathcal{B}Y,\omega W) &+g_{1}(\nabla^{^{M_1}}_{X}\mathcal{C}Y,\omega W)-\eta (Y)g_{1}(X,\omega W).
\mathcal{J}_i(\tau_i,\tau_{-i})=\mathbb{E}\left\{\theta e^{-\beta\tau_i}x_i(\tau_i)+(1-\theta)\frac{e^{-\beta\tau_i}x_i(\tau_i)}{l_1+l_2\left(\frac{1}{N}\sum\limits_{j=1}^Ne^{-\beta\tau_j}x_j(\tau_j)\right)}-e^{-\beta\tau_i}K\right\}.
\partial_{\hat{\imath}}\hat{\epsilon}( u , x , x^{\prime}) +{\frac{1}{8}}W H^{- \frac{1}{2}}\gamma^{\underline{{{u x y}}}}\gamma_{\underline{{{\hat{\imath}}}}}( i \sigma_{2}) \hat{\epsilon}( u , x , x^{\prime}) = 0 \, .
\begin{aligned}u(y)-\sum\limits_{j=1}^4 b_j(y)u(\Omega_j(y))-\int\limits_{G_1}c(y,\eta)u(\eta)d\eta-\int\limits_{\Gamma}d(y,\eta)u(\eta)d\Gamma_\eta =0, y\in\Gamma_1, u(y) =0, y\in\overline{\Gamma_2}.\end{aligned}
\left( \begin{array}{c c}{{a}}&{{b}}\\{{c}}&{{d}}\\ \end{array}\right) = \left( \begin{array}{c c}{{0}}&{{1}}\\{{- 1}}&{{0}}\\ \end{array}\right) \left( \begin{array}{c c}{{- c}}&{{- d}}\\{{a}}&{{b}}\\ \end{array}\right)
\Psi^*g_a = V(dx_1)^2 + \frac{1}{4x_1^2}[\rho^2 -2\rho^2(m) + g_{x_1,0,0}(\xi_3(m),\xi_3(m))]((dx_2)^2+(dx_3)^2)+ + a^2\sum_{j = 1}^3(dx_j)^2 + \beta_3dx_2 - \beta_2dx_3 + g_{x_1,0,0}+ \frac{1}{V+a^2}\eta^2.
2h_{00}^{\frac{m+1}{2}}W_{0}^m \Phi^{i}_{\parallel l}= h_{00}^{m+1}B^{i}_{(1) \parallel l}+ mh_{00}^{m+1}B^i_{(21) l}+h_{00}^{\frac{m+1}{2}}W_{0}^m B^{i}_{(22)l}+ W_{0}^{2m}B^i_{(3)l}+ m W_0^{2m-1}B^i_{(4)l},
8\pi i \frac{d}{d\theta}F(\theta ) =\int_C \frac{dz}{\sinh^2{\frac{1}{2}(z - \theta)}}\ln F(z)=\int^\infty_\infty \frac{dz}{\sinh^2{\frac{1}{2}(z - \theta)}}\ln{\frac{F(z)}{F(z+2\pi i)}}\ .
T_{\alpha \beta}={\frac{1}{4}}\left( \nabla_{\alpha}\tilde{\psi}\gamma_{\beta}\psi + \nabla_{\beta}\tilde{\psi}\gamma_{\alpha}\psi - \tilde{\psi}\gamma_{\alpha}\nabla_{\beta}\psi - \tilde{\psi}\gamma_{\beta}\nabla_{\alpha}\psi \right) .
\frac{\partial}{\partial t}\begin{pmatrix}u'_k v'_k\end{pmatrix}=-i\begin{pmatrix}T_{\vert u\vert^2}\alpha(u\vert 1) -(1\vert u) T_{\vert u\vert^2}-\sigma_k^2\end{pmatrix}\begin{pmatrix}u'_k v'_k\end{pmatrix}\ ,
R_{\mathcal{BS}(\alpha)}(\mathcal{S^{*}}[A,B]) =\begin{dcases}\min \left\{1,\dfrac{2\sqrt{\alpha}}{(1+\alpha)\sqrt{4\alpha (A-B)^2 +B^2}}\right\}, 4A \alpha \geq (3\alpha+1)B, \min \left\{1,\dfrac{1}{(1-\alpha)(A-B)+B}\right\}, 4A \alpha\leq (3\alpha+1)B .\end{dcases}
\exp{\left( - i \xi F_{1}- i \eta F_{2}\right)}= \left( \begin{matrix}{{1}}&{{0}}&{{0}}&{{\xi}}\\{{0}}&{{1}}&{{0}}&{{\eta}}\\{{\xi}}&{{\eta}}&{{1}}&{{( \xi^{2}+ \eta^{2}) / 2}}\\{{0}}&{{0}}&{{0}}&{{1}}\\ \end{matrix}\right) .
\frac{f_{\ell_1,\ell_2,\ell_3,\ell_4}}{f_{t_1,t_2,t_3,t_4}}= x_1^{t_4-\ell_4}x_2^{t_3-\ell_3}x_3^{t_2-\ell_2}x_4^{t_1-\ell_1}.
X'_{\Pi_{\imath}}X'_{\Pi_{\jmath}}=\begin{cases}v^{-\delta(\imath\in{{}^{\jmath}B})}X'_{\Pi_{\jmath}}X'_{\Pi_{\imath}}\ \Xi'_{\imath}> \Xi'_{\jmath}X'_{\Pi_{\jmath}}X'_{\Pi_{\imath}}\ \Xi'_{\imath}=\Xi'_{\jmath}. \end{cases}
d s^{2}= - \left( 1 - \left({\frac{r_{0}}{r}}\right)^{2}\right)^{2}d t^{2}+ \left( 1 - \left({\frac{r_{0}}{r}}\right)^{2}\right)^{- 2}d r^{2}+ r^{2}d \Omega_{3}^{2},
A_{p^e}= \left(\begin{matrix}A_{p^{e-1}}A_{p^{e-1}}\dots A_{p^{e-1}}A_{p^{e-1}}I_{p^{e-1}}0 \dots 0 -I_{p^{e-1}}0 I_{p^{e-1}}\dots 0 -I_{p^{e-1}}\vdots \ddots \vdots 0 0 \dots I_{p^{e-1}}-I_{p^{e-1}}\end{matrix}\right)
\mathsf{h}_{B_{n+1}}^\vee=\mathsf{h}_{A_{2n}}^\vee=2n+1, \ \mathsf{h}_{C_n}^\vee=\dfrac{1}{2}\mathsf{h}_{D_{n+1}}^\vee=\mathsf{h}^\vee_{A_n}=n+1.
\int_{B_R}(1-c^{-2}(y))\cdot \Delta^{-1}(c^{-2}f)(y)\cdot\Phi_0(x-y)\dy = \int_{B_R}(1-\widetilde{c}^{-2}(y))\cdot \Delta^{-1}(\widetilde{c}^{-2}\widetilde{f})(y)\cdot\Phi_0(x-y)\dy
M = \frac{V_{p}\Omega_{d - 2}2 \pi R_{T}}{1 6 \pi G}r_{0}^{d - 3}\frac{d - 1}{2}+ \sqrt{\frac{1}{4}\left( \frac{V_{p}\Omega_{d - 2}2 \pi R_{T}}{1 6 \pi G}( d - 3 ) r_{0}^{d - 3}\right)^{2}+ Q^{2}}
\begin{aligned}\bigg| \left(\nu\partial_y D^\alpha u\cdot D^\alpha u\right)|_{y=0}\bigg| \leq \delta_1\Vert \partial_y (u,h)\Vert_{H^m_0}^2+\frac{3\nu}{7}\Vert \partial_y D^\alpha u\Vert_{L^2_y}^2 & +C\delta_1^{-1}E_{u,h}^2(1+E_{u,h}^2)+C(u_b).\end{aligned}
\begin{array}{c c c}{{F_{i j}}}&{{=}}&{{\partial_{i}a_{j}- \partial_{j}a_{i}+ i ( c_{j}^{*}c_{i}- c_{i}^{*}c_{j}) ,}}\\{{}}&{{=}}&{{\partial_{i}b_{j}- \partial_{j}b_{i}- i ( c_{j}^{*}c_{i}- c_{i}^{*}c_{j}) .}}\\ \end{array}
\mathbb{E}\left((2m-1)^{-\frac12|W_l|_*}\right) \leqslant \left(l \frac{2m}{2m-1}2^{l-1}\right)^{\frac12}(2m)^{-(1-{\eta(m)})l}= \left(\frac{lm}{2m-1}\right)^{\frac12}\left(\frac{\sqrt2}{(2m)^{1-{\eta(m)}}}\right)^l.
R_2(g,\rho)(f) = \frac{L}{8\pi_N(x)}\frac{1}{L^2 \pi_N(x)}\sum_{(p,q) \atop{p \neq q \leq x \atop{(p,N) = (q,N) = 1}}}\left[ \sum_{l \geq 0}U(l) a_f(p^{2l}) \right] \left[ \sum_{l' \geq 0}U(l') a_f(q^{2l'}) \right] &\left[4 G(0) + \sum_{n \geq 1}2G(n)(a_f(p^{2n}) - a_f(p^{2n-2}))(a_f(q^{2n}) - a_f(q^{2n-2}))\right]
{\mathbb Y}_a{}^{A}{}_{B}{\mathbb X}^b{}^B{}_{A}=\delta_a{}^{b},{\mathbb Z}_a{}^{b}{}^A{}_{B}{\mathbb Z}_c{}^{d}{}^B{}_{A}=\delta_a{}^{d}\delta_c{}^{b},{\mathbb W}^{A}{}_{B}{\mathbb W}^{B}{}_{A}=1,
\left\Vert \tilde{\eta}_{\psi}(t)-\tilde{\eta}_{\psi}(s)\right\Vert_{-p}=\left\Vert \int_{\mathbb{X}\times \lbrack s,t]}G(X^{0}(r),y)\psi (y,r)\nu(dy)dr\right\Vert_{-p}& \leq (t-s)^{1/2}M(1+m_{T})\left( \int_{\mathbb{X}}M_{G}^{2}(y)\nu(dy)\right)^{1/2}& =(t-s)^{1/2}m_{T}^{1},
\mathcal{B}:=\{(j_1, j_2, j_3, j_4)\in S^4 \,\,:\,\,j_1+j_2+j_3+j_4=0,\,\, j_1^3+j_2^3+j_3^3+j_4^3=0,\,\,j_1+j_2\neq 0\}
\|\sum_{j=1}^{\ell-1}\binom{2\ell-1}{2(\ell-j)-1}(-\partial(m_{s_1},...,m_{s_{2^{\ell-1}}}))^{2(\ell-j)-1}(\partial(m_{t_1},...,m_{t_{2^{\ell-1}}}))^{2j}\alpha-\beta_2\|<\frac{\epsilon}{10}.
S_{\mathrm{g h}}= \int \sum_{\alpha \beta}\left\{\bar{c}^{\alpha \beta}\left( - D_{\mu}^{\alpha \beta}\right)^{2}c^{\beta \alpha}+ i g_{\mathrm{Y M}}\bar{c}^{\beta \alpha}D_{\mu}^{\alpha \beta}\left[ \bar{A}_{\mu}, c \right] \right\}
\sum_{n=2^{k-1}}^{2^k-1}1_{[-B,B]}\left(t + \sum_{i=0}^{n-1}h(T^i x) \right)\ge\beta \tau \sum_{n=0}^{2^k - 1}1_{[-B,B]}\left(t + \sum_{i=0}^{n-1}h(T^i x) \right)
W=-ic(\omega-\omega^{-1})\,{\eta_1+\eta_2^{1/N_1}+\eta_3^{1/N_1}+ \eta_1^{-1}+\eta_2^{-1/N_1}+\eta_3^{-1/N_1}\over (1-2\xi) \left(\eta_1-\eta_1^{-1}\right)+(1+\xi) \left(\eta_2^{1/N_1}+\eta_3^{1/N_1}- \eta_2^{-1/N_1}-\eta_3^{-1/N_1}\right)}\ .
\langle \mathbf{N},\delta\mathbf{r}_{ij}\rangle = \langle \mathbf{N}, (u\mathbf{N})_{ij}\rangle = u_{ij}+ u \langle \mathbf{N}, \mathbf{N}_{ij}\rangle = u_{ij}-u\langle h_i^l \mathbf{r}_l, h_j^k \mathbf{r}_k \rangle &= u_{ij}- uh_{il}h^l_j.
\mathbf{E}[\|\mathcal{X}^{t+1}-\mathcal{X}^{\star}\|_{F(\mathcal{Q})}^{2}|\mathcal{X}^{t}] =\|\mathcal{X}^{t}-\mathcal{X}^{\star}\|_{F(\mathcal{Q})}^{2}-\mathbf{E}_{i^{t}\sim \mathbf{p}}[f_{i^{t}}(\mathcal{X}^{t})]\leq(1-\delta_{\mathbf{p}}^{2}(\mathcal{Q},\boldsymbol{\mathcal{S}}))\|\mathcal{X}^{t}-\mathcal{X}^{\star}\|_{F(\mathcal{Q})}^{2}.
c_S(F_1)=(-1, 0), c_S(F_2)=(-1, 1), c_S(F_3)=(0, 1), c_S(F_4)=(1, 1), c_S(F_5)=(1, 0), c_S(F_6)=(1, -1), c_S(F_7)=(-1, 0), c_S(F_8)=(-1, -1),
\mathcal{C}= \left\{\begin{aligned}(r_1,r_2): (r_1,r_2)\in\mathbb{R}_2^+, r_1\leq I(X_1,Y|X_2,Q), r_2\leq I(X_2,Y|X_1,Q) r_1+r_2\leq I(X_1,X_2;Y|Q), \textrm{for some}p(q)p(x_1|q)p(x_2|q)p(y|x_1,x_2) \end{aligned}\right\},
R_\mu :=(2\mu -1)|c_2|^4+\sum_{k=3}^{+\infty}(\mu k-1)|c_k|^4+\sum_{k=3}^{+\infty}|c_k|^2((\mu k-2)|c_0|^2+(\mu (k+1)-2)|c_1|^2+(\mu (k+2)-2)|c_2|^2)\ .
\varepsilon_{\pm}\left( \theta \right) = \mathrm{e}^{\pm \theta}+ k_{0}- \int_{- \infty}^{\infty}\frac{d \theta^{\prime}}{2 \pi}\varphi \left( \theta - \theta^{\prime}\right) \log \left( 1 + \mathrm{e}^{- \varepsilon_{\pm}\left( \theta^{\prime}\right)}\right)
\Delta \rightarrow \left( \begin{array}{c c}{{1_{2 N \times 2 N}}}&{{0_{2 N \times 2 k}}}\\{{0_{2 k \times 2 N}}}&{{g \times 1_{2 \times 2}}}\\ \end{array}\right) \Delta ~ g^{t}\times 1_{2 \times 2}
\nabla u_k\to\beta_0\frac{x}{\vert x \vert^2}+\frac{4}{\max\left\{1,a\right\}}\displaystyle{\sum_{j=1}^{n}\frac{x-z_j}{\vert x-z_j\vert^2}}\quad\mbox{uniformly in}C^0_{\rm{loc}}(\mathbb{R}^2\setminus S),
e_{n,P,0}^{\epsilon,\Delta t}=\bigl(\frac{1}{(1+\frac{\Delta t}{\epsilon^2})^n}-e^{-\frac{t_n}{\epsilon^2}}\bigr)P_0^{\epsilon,\Delta t}=\epsilon\bigl(\frac{1}{(1+\frac{\Delta t}{\epsilon^2})^n}-e^{-\frac{t_n}{\epsilon^2}}\bigr)p_0^{\epsilon},
\mathrm{t r}( \gamma^{[ A}\tilde{\gamma}^{B}\gamma^{C ]}\tilde{\gamma}_{[ D}\gamma_{E}\tilde{\gamma}_{F ]}) = 4 \epsilon^{A B C}{}_{D E F}- 2 4 \delta_{D}^{[ A}\delta_{E}^{B}\delta_{F}^{C ]}
\frac{\sum\limits_{t=1}^{n-1}\left[(-1)^t\cdot t\sum\limits_{1\leq j_1\leq\cdots\leq j_{t+1}\leq n}z^{(d_{j_1}+\cdots+d_{j_{t+1}}-1)}\right]+z^{-1}}{(1-z)^n}.
\frac{1}{q}\int_\Omega u_t^q(t,x)dx =-(q-1)\int_\Omega u^{q-2}(t,x)|\nabla u(t,x)|^2dx -\chi(q-1)\int_\Omega \frac{u^{q-1}(t,x)}{v(t,x)}\nabla u(t,x)\cdot\nabla v(t,x)dx &\,\,\,\, +\int_\Omega a(t,x)u^q(t,x)-\int_\Omega b(t,x)u^{q+1}(t,x) dx\forall\, t>s,
\lim_{m \to \infty}\alpha^m_t :=\lim_{m \to \infty}\phi^m(t,\xi,W_{t \wedge},B_{t \wedge \cdot})=\phi(t,\xi,W_{t \wedge},B_{t \wedge \cdot})=\alpha_t,\;\mathrm{d}\P_\nu \otimes \mathrm{d}t\;\mbox{--a.e.}
\Psi_0\colon \mathfrak{g}^*\times \mathfrak{g}\to \; \mathfrak{g}^*\times \mathfrak{g}(\pi, \omega) \mapsto \; \Psi_0(\pi, \omega)= (\pi, \omega), \Xi\colon \mathfrak{g}^*\times \mathfrak{g}\to \; \mathfrak{g}^*\times \mathfrak{g}^* (\pi, \omega) \mapsto \; \Xi(\pi, \omega)= ( \pi, ad^{*}_\omega \pi).
W_{i_{1}j}^{( 1 )}( x , y ) = \left( \begin{array}{c c}{\widetilde{W}_{i j}^{( 0 )}}\\ \hline{}\\{\begin{array}{c c c c c}{0}&{\partial_{1}^{x}}&{- e}&{- ( a - 1 ) e}\\ \end{array}}\\ \end{array}\right) \delta ( x - y ) ,
d_{\alpha}({\bf x},\Sigma) =|x_j| \ \j\in\{1,\ldots,N\}\intertext{or}d_{\alpha}({\bf x},\Sigma) =\tfrac{1}{\sqrt{2}}|x_j-x_k| \ \ \text{for some}j,k\in\{1,\ldots,N\}\,.
q_k^{(s)}=\frac{H_{k-1}^{(s,0)}H_{k}^{(s+1,0)}}{H_{k}^{(s,0)}H_{k-1}^{(s+1,0)}},k=1,2,\dots,m, &e_k^{(s)}=\frac{H_{k+1}^{(s,0)}H_{k-1}^{(s+1,0)}}{H_{k}^{(s,0)}H_{k}^{(s+1,0)}},k=1,2,\dots,m-1,