harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\quad\mathbb{P}[\bigcap_{m=1,l=1}^{M,L}\{(U_m,V_l)\notin \mathcal{F}\}] &\le\mathbb{P}[(U,V)\notin\mathcal{F}]+\mathbb{P}\left[\exp(\imath_{U;V}(U;V))>ML\exp(-\gamma)-\delta\right] &+\frac{\min\{M,L\}-1}{\delta}+e^{-\exp(\gamma)}.
\overline{F}=\frac{i\rho}{2}[2\pi i\frac{a_{0}\Gamma}{\pi i}]=i\rho a_{0}\Gamma=i\rho\Gamma(v_{x\infty}-iv_{y\infty})=\rho\Gamma v_{y\infty}+i\rho\Gamma v_{x\infty}=F_{x}-iF_{y}.
\frac{\lambda}{4\pi}\int d^2 z \left\{\sum_{\alpha_i>0}\left[E^{\alpha_i}(z)\overline{E}^{-\alpha_i}(\bar{z}) +E^{-\alpha_i}(z)\overline{E}^{\alpha_i}(\bar{z}) \right]+ \sum_{i=1}^{2}H^i(z)\overline{H}^i(\bar{z}) \right\}\ .
\hat{S}= \int d^{1 0}\hat{x}\sqrt{| \hat{g}|}\, \left\{e^{- 2 \hat{\phi}}\left[ \hat{R}- 4 ( \partial \hat{\phi})^{2}\right] +{\textstyle \frac{1}{2 \cdot 7 !}}e^{2 \hat{\phi}}\hat{\tilde{H}}{}^{2}\right\}\, ,
\sum_{t=1}^T \theta_t \| u_t - u^*\| \leq \frac{2}{T(T+1)}\O\!\left(\sum_{t=1}^T t \beta^t\right) &\leq\frac{2}{T(T+1)}\O\!\left( \frac{\beta(1 - (T+1)\beta^T + T \beta^{T+1})}{(1-\beta)^2}\right)= O\!\left( \frac{1}{T^2}\right) = o\!\left(\frac{1}{T}\right).
\left( g_{m n}\right)^{D}= \left( \begin{array}{c c}{g_{m n}+ g_{t s}{\Gamma^{t}}_{m}{\Gamma^{s}}_{n}}{{\Gamma^{t}}_{n}g_{t m}}{{\Gamma^{t}}_{m}g_{t n}}{g_{m n}}\end{array}\right) .
\tilde{\alpha}= \alpha \beta^{- m}= \left( \begin{array}{c c c}{\omega_{k}^{- 2 y}\omega_{2 d}^{2 m}}&{0}&{0}\\{0}&{\omega_{k}^{y}\omega_{2 d}^{- m}}&{0}\\{0}&{0}&{\omega_{k}^{y}\omega_{2 d}^{- m}}\\ \end{array}\right)
\begin{aligned}\| \nabla u_n^h \|_{L^{2r}_t L^{\frac{dr}{2r-1}}_x}+ \| u_n^h \|_{L^{2r}_t L^{\frac{dr}{r-1}}_x}\leq \bar{C}_1\tilde{\eta}&\|\nabla u_n^d\|_{L^{2r}_t L^{\frac{dr}{2r-1}}_x}+ \| u_n^d \|_{L^{2r}_t L^{\frac{dr}{r-1}}_x}\leq \bar{C}_2\| \bar{u}^d \|_{\dot{B}_{p,r}^{\frac{d}{p}-1}}+ \bar{C}_3\end{aligned}
\hat{x}_{i+2}\coloneqq x_{i-1}+ \frac{y_{i+2}-y_{i-1}}{a_i}= x_{i-1}+ \frac{y_i-y_{i-1}}{a_i}+ \frac{y_{i+1}-y_i}{a_i}+ \frac{y_{i+2}-y_{i+1}}{a_i}= x_{i-1}+ (x_i-x_{i-1}) + \frac{a_{i+1}}{a_i}(x_{i+1}-x_i) + \frac{a_{i+2}}{a_i}(x_{i+2}-x_{i+1}) = x_i + \frac{a_{i+1}}{a_i}(x_{i+1}-x_i) + \frac{a_{i+2}}{a_i}(x_{i+2}-x_{i+1}).
\mathcal{Q}\coloneqq \frac1{2\pi (1-e^{-2s})}\begin{pmatrix}\big( 1-(1-e^{-2s})\frac1p \big){\rm id}_{\mathbb{R}^n}-e^{-s}{\rm id}_{\mathbb{R}^n}-e^{-s}{\rm id}_{\mathbb{R}^n}\big( 1-(1-e^{-2s})\frac1{q'}\big){\rm id}_{\mathbb{R}^n}\end{pmatrix}, &c_1 \coloneqq \frac1p,\;\;\; c_2\coloneqq \frac1{q'},
S_{E}={\frac{4 \, \pi}{\lambda}}\int d \tau \int d r \; r^{2}\left[{\frac{1}{2}}{({\frac{\partial \varphi}{\partial \tau}})^{2}+ ( \nabla \varphi )^{2}}+{\frac{1}{2}}( \varphi^{2}- 1 )^{2}- f \varphi^{3}\right] .
SO_3(G) = \sum_{uv\in E(G)}\sqrt{2}\frac{d_u^2+d_v^2}{d_u+d_v}\pi &\leq \sqrt{2}\pi \left( \sum_{uv\in E(G)}\frac{|d_u^2-d_v^2|+2\Delta^2}{2\delta}\right) &\leq \sqrt{2}\pi \left(\frac{1}{\delta}\left( \frac{1}{2}\sum_{uv\in E(G)}|d_u^2-d_v^2| \right)+ \frac{m\Delta^2}{\delta}\right) &= \sqrt{2}\pi \left(\frac{SO_1(G)+m\Delta^2}{\delta}\right).
[a_1, b_{2i+1}]=[a_2, b_{2i+2}]=1, \hskip5mm i=0,1,\ldots, n-1, & [b_1,b_2] = [b_3,b_4] = \cdots = [b_{2n-1}, b_{2n}]=1, & [b_i,b_j]=1 \mbox{if}i\equiv j \pmod 2.
x^{\gamma}y^{\alpha}z^{\beta}-x^{\alpha}y^{\gamma}z^{\beta}= -(x-y)x^{\gamma}y^{\gamma}z^{\beta}\sum_{i=0}^{\alpha-\gamma-1}x^i y^{\alpha-\gamma-1-i}= -(x-y)\bigg( x^{\gamma}z^{\beta}\sum_{i=0}^{\alpha-\beta-1}x^i y^{\alpha-1-i}+x^{\alpha-\beta+\gamma}z^{\beta}\sum_{i=0}^{\beta-\gamma-1}x^i y^{\beta-1-i}\bigg).
\lambda_{\rho}(K_{\rho})=\sum_{m=0}^{d}C^m_d W_m(K) \rho^m=\sum_{j=0}^{d}\omega_{d-j}V_j(K) \rho^{d-j}, \, \rho \geq 0, C^m_d=\left( \!\! \begin{array}{c}d m\end{array}\!\! \right)
=C_{m_{1},m_{2},m}^{l_{1},l_{2},l}C_{0,0,0}^{l_{1},l_{2},l}+\frac{1}{2\sqrt{j(j+1)}}C_{m_{1},m_{2},m}^{l_{1},l_{2},l}P(l_{1},l_{2},l)+O((j(j+1))^{-1})
\left\langle u, v \right\rangle_{H^2_{\frac{1}{a}}(0, 1)}=\int_{0}^{1}\frac{f v}{a}\,dx \Longleftrightarrow \int_{0}^{1}\biggl (\frac{u v}{a}+u''v''\biggr ) dx=\int_{0}^{1}\frac{f v}{a}\,dx,
\int_{\mathbb{R}}\frac{\varepsilon ( q^{\varepsilon,\delta}_r )^2}{2}+ \frac{F_\delta (q^{\varepsilon,\delta})}{\varepsilon}\,dr = \int_{\mathbb{R}}\sqrt{2F_\delta (q^{\varepsilon,\delta})}q^{\varepsilon,\delta}_r \,dr = \int_{-(1-\delta)^{-1}}^{(1-\delta)^{-1}}\sqrt{2F_\delta ( s )}\, ds=:\sigma_{\delta}.
M_{m+1}(u| x_0,\ldots,x_m,x_{m+1}) \, = \frac{(u-x_0)M_m(u | x_0,\ldots,x_m)}{x_{m+1}-x_0}&+ \frac{(x_{m+1}-u)M_m(u | x_1,\ldots,x_{m+1})}{x_{m+1}-x_0}.
Z_{l}^{S U}( g , \theta ) \sim \frac{\sqrt{2 \pi}}{g}\exp \frac{g^{2}}{8}( \frac{\theta}{\pi})^{2}\mathrm{d e t}_{j k}\exp [ - \frac{g^{2}}{2 N}\{( j - k )^{2}- \frac{1}{4}( \frac{\theta}{\pi})^{2}\}] .
M_{n}^{\left( 2\right)}\left( \upsilon\right) =2\gamma_{1}^{2}+2\gamma_{1}\mu_{2}^{\left( 2\right)}\left( \rho_{1}\right)\mathbf{A}_{0}\left( n/\upsilon\right) \left( 1+o_{\mathbf{P}}\left(1\right) \right) +o_{\mathbf{P}}\left( \upsilon^{-1/2}\right) .
\begin{aligned}\ss_1\circ \ss^{-1}_2(x) =\int_{\ss_2^{-1}(0)}^{\ss_2^{-1}(x)}1_{G_1}(y)d\ss(y)=\int_{\ss_2^{-1}(0)}^{\ss_2^{-1}(x)}1_{G_1}(y)\cdot 1_{G_2}(y)d\ss(y) &= \int_{\ss_2^{-1}(0)}^{\ss_2^{-1}(x)}1_{G_1}(y)d\ss_2(y)=\int_0^x 1_{G_1}(\ss_2^{-1}(z))dz &=\int_0^x 1_{\ss_2(G_1)}(z)dz. \end{aligned}
\| v \|^2_{L^2(0,\infty)}= \int_0^{\infty}[V(z)]^2 dz = \arctan\left( e^{-2a}\right)= \frac{\pi}{4}- 2 \pi \varepsilon^2 + \frac{100}{3}\pi^3 \varepsilon^6 + \mathcal{O}(\varepsilon^{10})
|\chi_{\mathbb{R}^+}(x)e^{i(\lambda+i\epsilon)x}*g_n| = \left|\int_{-\infty}^x e^{-\epsilon(x-y)}e^{i\lambda(x-y)}g_n(y)~dy\right| &\le e^{-\epsilon x}(\sup_{y\le x}e^{\epsilon y}[w(y)]^{2+s-s_1}) \|w^{-2-(s-s_1)}g_n\|_{L^1}
\begin{cases}v_t^\varepsilon+A_\varepsilon^+ v^\varepsilon = Q_\varepsilon(\bar{\lambda})f(v^\varepsilon+z^\varepsilon):=H_\varepsilon(v^\varepsilon,z^\varepsilon) z_t^\varepsilon+A_\varepsilon^- z^\varepsilon = (I-Q_\varepsilon(\bar{\lambda}))f(v^\varepsilon+z^\varepsilon):=G_\varepsilon(v^\varepsilon,z^\varepsilon).\end{cases}
f_{(s,r)}= \tilde{f}_{\tilde{\nu}(s,r)}\circ f_r: \Big(\dot r U^- P^+_K \Big) \bigcap f_r^{-1}\Big(\dot{\tilde{\nu}}(s,r) \tilde{U}^{-}\tilde{B}^+ \Big) \rightarrow \tilde{G}.
(\begin{matrix}16\\ 8\end{matrix})\cdot(\begin{matrix}8\\ 2\end{matrix})\cdot(\begin{matrix}6\\ 2\end{matrix})\cdot(\begin{matrix}4\\ 2\end{matrix})\cdot(\begin{matrix}2\\ 1\end{matrix})\cdot(\begin{matrix}1\\ 1\end{matrix})=64,864,800
S_{P , \Sigma}( X , h ) \, \to \, S_{\mathrm{t o t . ,}\Sigma}( X , h ) \; : = \; S_{P , \Sigma}( X , h ) \, + \, S_{\Sigma}^{\prime}( X , h ) \, + \, S_{\Sigma}^{\prime \prime}( X , h ) \,
\int_{-\pi/2}^{\pi/2}(y_{2,2}')(y_{2,3}') = \frac{(n-1)^2(n-3)^2}{48^2}\frac{1}{160}\left(\frac{855\pi}{32}-\frac{75\pi^3}{4}+\frac{17\pi^5}{4}-\frac{83\pi^7}{168}+\frac{16\pi^9}{315}\right) &\frac{(n-1)^2(n-3)^2}{48^2}\approx 5.157
v_{j}\left( t,\beta\right) =w_{j}\left(\beta\right) e^{-\mathrm{i}\zeta_{j}\left( \beta\right) t},\text{\ \}1\leq j\leq N_{R}; v_{j}\left( t,\beta\right) =w_{j}\left(\beta\right) e^{-\mathrm{i}\zeta_{j}\left( \beta\right) t},\text{\ \}N_{R}+1\leq j\leq2N.
\begin{array}{c}{C_{2}= \frac 1 2 L_{A B}L^{A B}= E_{o}( E_{o}+ 3 ) + s ( s + 1 )}\\{C_{4}= W_{A}W^{A}= - s ( s + 1 ) ( E_{o}- 1 ) ( E_{o}- 2 )}\\{W_{A}= \frac 1 8 \epsilon_{A B C D E}L^{B C}L^{D E}}\\ \end{array}
I(X;Y_1|X_1,W_1) = I(X_1,W_1,X;Y_1) - I(X_1,W_1;Y_1) &= I(X;Y_1) + I(W_1;Y_1|X) - I(X_1;Y_1) - I(W_1;Y_1|X_1) &\stackrel{(d)}{=}I(X;Y_1),
{_2F}_{1}\left(\begin{array}{cc}\frac{1}{4}, \frac{3}{4}1+\frac{1}{2}\end{array}\mid-1\right)=\frac{\Gamma(1+\frac{1}{2})\Gamma(1+\frac{3}{8})}{\Gamma(1+\frac{3}{4}) \Gamma(1+\frac{1}{8})}.
q(s)=\frac{-s.\Gamma(s)}{\frac{1}{x_0}.\underset{k=0}{\overset{\infty}{\sum}}\beta_{2.k+1}.\Gamma(s+2.k+1)-\frac{\Gamma(s)}{x_0}.\sum_{n=1}^{n=N}\left(\frac{n}{x_0}\right)^{-s}}
\left. \frac{1}{g_{a}^{2}( \mu^{2})}\ \right|_{\mathrm{1 - l o o p}}\ = \ \frac{1}{l}\ + \ \sum_{k}s_{a k}m_{k}\ - \ \frac{b_{a}}{1 6 \pi^{2}}\: \ln \frac{\mu^{2}}{M_{I}^{2}}\ .
F_{m,N}= \kern-7pt \sum_{(\boldsymbol{j}_m|\boldsymbol{k}_m)\in\mathcal{I}_m^+}\frac{a_{N-k_1}}{|\lambda_N-b_{N-j_1}|}\frac{a_{N-k_2}}{|\lambda_N-b_{N-j_2}|}\cdots \frac{a_{N-k_m}}{|\lambda_N-b_{N-j_m}|},
\triangle_{i}^{\left( k \right)}= \frac{\frac 1 2 \left( 1 - \left( - 1 \right)^{i}\right) \left( k + 1 \right) + 2 \left( - 1 \right)^{i}\left[ \frac{i + 1}2 \right]}{\sqrt{\left( k - 1 \right) \left( k + 1 \right)}}
\frac{1}{n}\mathbb{E}\sum_{k=1}^{n}|I_{n,k}|=\frac{1}{n}\mathbb{E}\sum_{k=1}^{n}|(\mathbb{E}_{k-1}X_{n,k})^{2}-2X_{n,k}(\mathbb{E}_{k-1}X_{n,k})| \leq\mathbb{E}((\mathbb{E}_{-1}X_{n,0})^{2})+\frac{2}{n}\sum_{k=1}^{n}||X_{n,k}||_{2}||\mathbb{E}_{k-1}X_{n,k}||_{2}\leq\mathrm{cov}(X_{n,0},X_{n,2})+C\sqrt{\mathrm{cov}(X_{n,0},X_{n,2})}.
\begin{array}{l}L_1=(1+c\lambda)D_x-\displaystyle\frac{u_{xz}}{u_{zt}}D_t-c\lambda \displaystyle\frac{u_{xt}}{u_{zt}}D_z, begin{align*}5mm]L_2=(1-b\lambda)D_y-\displaystyle\frac{u_{yz}}{u_{zt}}D_t+b\lambda \displaystyle\frac{u_{yt}}{u_{zt}}D_z.\end{array}
\begin{bmatrix}1 0 0 0 0 \overline\alpha\alpha \alpha J(T^{-2r},T^{-3r}) \overline\alpha^2 0 \overline\alpha J(T^{-r},T^{-r}) 0 \alpha J(T^{-3r},T^{-3r}) 0 \alpha^2 \overline\alpha J(T^{-2r},T^{-r}) \overline\alpha\alpha\end{bmatrix}
I_{\times}= \{\lambda_1,\lambda_2 -1,\ldots,\lambda_{n-i}- (n-i) + 1, - n + i, \ldots, -n + 1 \}I_{\circ}= \{1-n,2-n,\ldots,i-n,i+1-n - \lambda_{n+1+i},\ldots, - \lambda_{2n}\}.
\begin{array}{ll}\displaystyle{\kappa_j(s)=\langle\frac{de_j(s)}{ds},e_{n+j}(s)\rangle, 1\leq j\leq n-1}, \displaystyle{\kappa_n(s)=\langle\frac{de_n(s)}{ds},e_{2n}(s)\rangle}, \displaystyle{\tau(s)=\langle\frac{d\gamma(s)}{ds},T\rangle}.\end{array}
\widetilde{S}_{0}^{( D_{0})}( k ) = - \frac{1 + i \left( 2 \omega_{_\mathrm{( g s )}}^{- 1}/ \pi - Q_{_\mathrm{( g s )}}\right)}{1 - i \left( 2 \omega_{_\mathrm{( g s )}}^{- 1}/ \pi - Q_{_\mathrm{( g s )}}\right)},
\mu(\{\lambda_{k}\})=\frac{(-1)^{k}\Gamma(c-a)\Gamma(c-b)}{\Gamma(c)\Gamma(c-a-b)}\left(1+\frac{2k}{a+b-c}\right)\frac{(a+b-c)_{k}(a)_{k}(b)_{k}}{(a-c+1)_{k}(b-c+1)_{k}\, k!}.
\frac{Q_1}{\alpha F^{-\alpha-2}}= ~ \ddot{F}^{kl,mn}\nabla_1h_{kl}\nabla_1h_{mn}-(\alpha+1)F^{-1}(\nabla_1F)^2 &+(\alpha+1)F^{-2}\dot{f}^{k}(\nabla_kF)^2\kappa_1 +2 \sum_{k=1}^n\sum_{l=2}^n\frac{\dot{f}^k}{\kappa_l-\kappa_1}(\nabla_kh_{1l})^2~\geq~0.
U(su_2):{}\Delta(X^{\pm}) = X^{\pm}\otimes 1 + 1 \otimes X^{\pm}\,, \Delta(H) = H \otimes 1 + 1 \otimes H \,, U_q(su_2):{}\Delta(X^{\pm}) = X^{\pm}\otimes K + K^{-1}\otimes X^{\pm}\,, \Delta(K^{\pm 1}) = K^{\pm 1}\otimes K^{\pm 1}.
\binom{k+\mu}{k}S_{a_{1},b_{1}}^{a_{2},b_{2},p_{2}}\left( p_{1},k+\mu\right) =\sum_{j_{2}=0}^{p_{2}}\sum_{j_{1}=0}^{p_{1}}\binom{p_{1}}{j_{1}}\binom{p_{2}}{j_{2}}S_{a_{1},0}^{a_{2},0,p_{2}-j_{2}}\left( p_{1}-j_{1},\mu\right) S_{a_{1},b_{1}}^{a_{2},b_{2},j_{2}}\left( j_{1},k\right) .
|N_0(t)-N(B(t),C(t))| \leq\frac{1}{V_F}\int_{\mathbb{R}}|y|p_{t,0}(y)dy=\frac{1}{V_F}e^{-t}\int_{\mathbb{R}}|y|p_{0,0}(y)dy &=\frac{1}{V_F}e^{-t}\int_{\mathbb{R}}\int_{0}^{V_F}|y|p_{}(v,y)dvdy\leq Ce^{-t}
{\cal P}_{0,n}^{B}(\kappa)\triangleq\Big\{\pi_t(x_t|y^{t-1}), ~t=0, \ldots, n: \frac{1}{n+1}{\bf E}^{\pi}\Big( c^{B}_{0,n}(X^n, Y^{n-1}) \Big)\leq \kappa\Big\},~ \kappa \in [0,\infty)
L_{\psi}^{\mathrm{c u t o f f}}= - \frac{1}{2}\int \frac{d \nu}{2 \pi}\sum_{i = 1}^{2}\frac{\omega^{3}}{\nu^{2}}\frac{v_{i}\bar{\gamma}_{i}^{2}}{\omega^{2}\Sigma ( ( - 1 )^{i}\bar{R}_{i})}\left( G_{\psi}^{i i}( - i \nu ) - G_{\psi}^{i i}( 0 ) \right) \, .
{\frac{2 i \pi ( \beta_{2 1}+ i \pi )}{( \beta_{2 1}+ 2 i \pi ) \beta_{2 1}}}[ \operatorname{e x p}( \beta_{1}) + \operatorname{e x p}( \beta_{2}) ] \operatorname{t a n h}^{2}{\frac{\beta_{2 1}}{2}},
\left( \begin{matrix}{{X^{I}}}\\{{F_{I}}}\\ \end{matrix}\right) \rightarrow \left( \begin{matrix}{{{\bf a}}}&{{{\bf b}}}\\{{{\bf c}}}&{{{\bf d}}}\\ \end{matrix}\right) \left( \begin{matrix}{{X^{I}}}\\{{F_{I}}}\\ \end{matrix}\right)
u^{\prime}( 4^{\prime}3^{\prime}2^{\prime}1^{\prime}) = s^{4 - D}u ( \frac{4^{\prime}}{s}\frac{3^{\prime}}{s}\frac{2^{\prime}}{s}\frac{1^{\prime}}{s}) = s^{4 - D}u ( 4 3 2 1 ) .
\begin{array}{ll}\int_{_{B(x_{i_{l_{k}}}, R_{i_{l_k}}) / \bigcup_{k^{'}=k+N}^{K}B (x_{i_{l_{k^{'}}}},R_{i_{l_{k'}}})}}\ |u(\bar{t})|^{2}\, dx \geq \frac{1}{2}\int_{B(x_{i_{l_{k}}},R_{i_{l_k}})}|u(\bar{t})|^{2}\, dx & \geq \frac{(c')^{2}}{8}|J_{i_{l_k}}|,\end{array}
t\left( 2n,2l\right) =\left( 2n\right) !~2^{2l}\sum_{m=1}^{j+1-n}\left(V^{-1}\left[ j\right] \right)_{2l+1,m}\frac{\left( j+1-m\right)}{n}\binom{j+n-m}{2n-1}\ ,
\begin{cases}y''+\cot ry'-\frac{4\sin^{2}\frac{r}{2}}{\sin^2r}y=0, y=f x, x=-\frac{6+3\tan^4\frac{r}{2}+9\tan^2\frac{r}{2}+\cot^2\frac{r}{2}}{2(1+2\tan^2\frac{r}{2})^{\frac{3}{2}}}.\end{cases}
0\,\longrightarrow\,{\mathcal K}\, \stackrel{h}{\longrightarrow}\,{\mathcal K}(H^{m-1}) \, \longrightarrow\,\bigoplus_{j=2}^{d-m+1}{\mathcal K}(H^{m-1})_{x_j}/\widetilde{H}_j\,\longrightarrow\, 0\, .
\sum_{\lambda_1\geq n\geq \max\{\lambda_2, \mu_1\}}{B(n + N - 1, \mu_1 + N - 1)}\leq const\cdot(1 + \sum_{n = \lambda_2}^{\lambda_1}{n^{-1-2a}}) \leq const\cdot(\lambda_1 - \lambda_2 + 1).
(-1)^{l}(-x + \mathbf{a}\cdot \mathcal{B})^{m}(-x + A + \mathbf{a}\cdot \mathcal{B})^{l}= (-1)^{m}(x - \mathbf{a}\cdot \mathcal{B})^{m}( x- A - \mathbf{a}\cdot \mathcal{B})^{l}
\beta(\mathbf{v}(a),\mathbf{g}(b)) =-(-1)^{|a|(1+|b|)}\beta(\mathbf{g}(b),\mathbf{v}(a))=\boldsymbol{\pi}_1(a_+b), \beta(\mathbf{f}(a),\mathbf{w}(b)) =-(-1)^{(1+|a|)|b|}\beta(\mathbf{w}(b),\mathbf{f}(a))=\boldsymbol{\pi}_2(ab_+),
\begin{cases}a = p_1+p_2-u-v-2s \ , \ \b = q_1+q_2-u-v+s \ , \ \ \mbox{for (\ref{psi-1case})}\ , a = p_1+p_2-u-v+s \ , \ \b = q_1+q_2-u-v-2s \ , \ \ \mbox{for (\ref{psi-2case})}\ , \end{cases}
\sqrt{2\pi}a_n \int_0^{\infty}e^{- y}I_1(y) dy = \sqrt{2\pi}\pi_{s,f}(\varphi r_{s,f}^{-1}) \phi(0) \int_0^{\infty}e^{- y}y dy + \|\varphi\|_{\infty}O \left( \frac{1}{\sqrt{n}}\right) & = \pi_{s,f}(\varphi r_{s,f}^{-1}) + \|\varphi\|_{\infty}O \left( \frac{1}{\sqrt{n}}\right)
I(X;Y_2|V_0,V_1) \geq I(V_2;Y_2|V_0,V_1) &=I(V_2;Y_2,V_1|V_0)-I(V_1;V_2|V_0) &\geq I(V_2;Y_2|V_0)-I(V_1;V_2|V_0)
u_{1}={\frac{1}{v_{1}}}, \quad u_{2}={\frac{v_{2}}{v_{1}}}, \quad v_{1}={\frac{w_{1}}{w_{2}}}, \quad v_{2}={\frac{1}{w_{2}}}, \quad w_{1}={\frac{1}{u_{2}}}, \quad w_{2}={\frac{u_{1}}{u_{2}}},
\left(\sum_{i=1}^n\tilde{\theta}_0^i(\tilde{\delta}^i)^t\right)^\gamma\left(\hat{\phi}+\frac{\eta}{\gamma}\,\hat{x}_t\right)^{-\gamma}=\left(\sum_{i=1}^n\tilde{\theta}_0^i(\tilde{\delta}^i)^{t+1}\right)^\gamma\left(\hat{\phi}+\frac{\eta}{\gamma}\,\hat{x}_{t+1}\right)^{-\gamma}f'(k_{t+1}).
\overline{y}= \bigg \{\frac{v}{c}-\kappa \, g -\left[ \frac{\delta \left (\delta+ c \, g'\right)}{c \, \kappa}\right]^{'}\bigg \}\, \overline{e}+\frac{\delta \left (\delta+ cg'\right)}{c\kappa}\,\overline{n}- g \, \overline{z}.
(H_{ij}f)(x) =\int_{\Omega}K_{ij}(x,y)f(y)\,dy+\int_{\Omega}G_{ij}(x,y)f(y)\,dy &+f(x)\int_\Omega\frac{(x-y)_i(x-y)_j}{|x-y|^2}\omega(y)\,dy\,\, \hbox{for $x \in \Omega$,}
[ a_{m}, \tilde{a}_{n}^{\dagger}] = [ \tilde{a}_{m}, a_{n}^{\dagger}] = \delta_{m n}\mid 0 \rangle \langle 0 \mid , [ a_{m}, \tilde{a}_{n}] = [ a_{m}^{\dagger}, \tilde{a}_{n}^{\dagger}] = 0
d^{-1}=\left( \left( x^{\ast}\right)^{-1}y^{\ast}yx^{-1}\right)^{-1}=xy^{-1}\left( y^{-1}\right)^{\ast}x^{\ast}=\left\vert \left(y^{-1}\right)^{\ast}x^{\ast}\right\vert^{2}.
{\frac{1}{1 6 \pi G}}\left( - 2 \Lambda \right) +{\frac{1}{( 4 \pi )^{2}\epsilon}}\left\{c_{N / 2 + 2}\Lambda^{N / 2 + 2}r^{N}\right\}={\frac{1}{1 6 \pi G_{0}}}\left( - 2 \Lambda_{0}\right) ,
r^{(n)}< r^{(0)}e^{-\frac \eta\beta\min \{n,N_0-n\}}< (\frac{\alpha_2}{8K\ell^2})^{\frac 1\beta}e^{-\frac \eta\beta\min\{n,N_0-n\}}= (\frac{\alpha_2}{8K\ell^2 e^{\eta \min \{n,N_0-n\}}})^{\frac 1\beta}.
\pi_s(0)|_{B_s=K+1}-\pi_s(0)|_{B_s=K}= \frac{\lambda_s^{}\tau^{K}(\mu_s-\lambda_s^{})(\tau-1)}{(\mu_s-\lambda_s^{}\tau^{K+1})(\mu_s-\lambda_s^{}\tau^{K})}<0,
F[a]=\int d^{9}x~\left\{\frac{1}{(2\pi)^7(\alpha')^{5}g_s}\sum_{b}\sqrt{1+\left(2\pi \alpha'e_b\vec\nabla a(\vec x)\right)^2}+\sum_{b,c}V_{\rm eff}\Bigl[(e_b-e_c)a(\vec x)\Bigr]+\dots\right\}
\begin{cases}\widehat{U}''-{\Delta}\widehat{U}+A\widehat{U}=0 \hbox{in}(0,+\infty) \times \Omega, \partial_\nu \widehat{U}+ B\widehat{U}= D \widehat H \hbox{on}(0,+\infty) \times \Gamma t=0: \widehat{U}= \widehat{U}'=0 \hbox{in}\Omega.\end{cases}
E_{\ell,a}=(-1)^{\frac{\ell(\ell-1)}{2}}(1-q^{-2})^{1-\ell}[x^{a},[x^{a+2},\cdots,[x^{a+2(\ell-2)},x^{a+2(\ell-1)}]_{q^{-4}}\cdots]_{q^{-2(\ell-1)}}]_{q^{-2\ell}}\,.
\rho_{1}\otimes\rho_{2}(1,1)=\rho_{1}(1)\otimes\rho_{2}(1)=(\begin{matrix}0&0&0&-i&0&-i\omega\\ 0&0&0&0&-i\omega&0\\ 0&0&0&0&0&-i\omega^{2}\\i&0&i\omega&0&0&0\\ 0&i\omega&0&0&0&0\\ 0&0&i\omega^{2}&0&0&0\end{matrix})
\liminf_{z\rightarrow\alpha}\int_\mathbb{R}\frac{d\nu(s)}{|z-s|^2}=\int_\mathbb{R}\frac{d\nu(s)}{(\alpha-s)^2}=\lim_{z\rightarrow_{\sphericalangle}\alpha}\int_\mathbb{R}\frac{d\nu(s)}{|z-s|^2},
{\mathcal H}_{{\mathrm{\boldmath ~ \scriptstyle{\xi}~}}}= \mu^{- 2}H_{\mathrm{\boldmath ~ \scriptstyle{~ \xi}~}/ \mu}= - \nabla_{\mathrm{~ \boldmath ~ \scriptstyle{~ \xi}~}}^{2}- \lambda \,{\mathcal W}( \mathrm{\boldmath ~ \xi ~}) \; ,
v_1(v_i,\sigma v_i)=(v_i,\sigma v_i)\left(\begin{array}{rr}1 0 0 0\end{array}\right),\sigma v_1(v_i,\sigma v_i)=(v_i,\sigma v_i)\left(\begin{array}{rr}0 0 1 0\end{array}\right), v_2(v_i,\sigma v_i)=(v_i,\sigma v_i)\left(\begin{array}{rr}0 1 0 0\end{array}\right),\sigma v_2(v_i,\sigma v_i)=(v_i,\sigma v_i)\left(\begin{array}{rr}0 0 0 1\end{array}\right).
\sum_{j_{w}\le j\le j_c}w(x_j)\left(\mathcal{F}_{j+1/2}^n-\mathcal{F}_{j-1/2}^n\right)=w(x_{j_c})\mathcal{F}_{j_c+1/2}^n-w(x_{j_w})\mathcal{F}_{j_w-1/2}- \!\!\!\sum_{j_w\le j\le j_c-1}\left(w(x_{j+1})-w(x_j)\right)\mathcal{F}_{j+1/2}^n.
_nF_{n-1}(a_0,\ldots, a_n; b_1,\ldots, b_{n};z) = 1 + \sum_{r\geq 1}\frac{(a_0)_r(a_1)_r\cdots (a_n)_r}{(b_1)_r(b_2)_r\cdots (b_{n})_r}\frac{z^r}{r!},
\tau(x,y,t) \,\,=\,\, \sum_{I\in\binom{[n]}{k}}p_I\prod_{\substack{i,j\in I i<j}}(\kappa_i-\kappa_j) \cdot\exp\biggl[ x\cdot\sum_{i\in I}\kappa_i + y\cdot\sum_{i\in I}\kappa_i^2 + t\cdot\sum_{i\in I}\kappa_i^3\biggr],
{\cal L}= \sqrt{- \hat{G}}[{\cal R}_{\hat{G}}+ \hat{G}^{M N}\partial_{M}\Phi \partial_{N}\Phi -{\textstyle{\frac{1}{1 2}}}\hat{H}_{M N P}\hat{H}^{M N P}-{\textstyle{\frac{1}{4}}}\hat{F}_{M N}^{I}\hat{F}^{I M N}] ,
A=\begin{pmatrix}1 0 0 \cdots 0 -{\binom{p}{1}}1 1 0 \cdots 0 -{\binom{p}{2}}0 1 1 \cdots 0 -{\binom{p}{3}}\vdots \vdots \vdots \ddots \vdots \vdots 0 0 0 \cdots 1 -{\binom{p}{p-2}}0 0 0 \cdots 1 1-{\binom{p}{p-1}}\end{pmatrix}.
\inf_{\pi\in \mathrm{Irr}(G)}H(\left |\chi_{\pi}\right |^2,\varphi_G)>-\infty~\mathrm{and~}H(\left |\widehat{\chi_{\pi}}\right |^2,\widehat{\varphi}_{\widehat{G}})= \log(n_{\pi}^2),
\gamma_{n,m}\left(\tfrac{m+1}{2}\right)=(-1)^{[\frac{m+3}{2}]}\cdot\frac{\Gamma_{\frac{n-m}{2}}\left(\tfrac{n-m}{2}\right)}{\Gamma_{\frac{n-m}{2}}\left(\tfrac{n+1}{2}\right)}\prod_{j=0}^{\frac{n-m-4}{4}}\frac{\Gamma(\tfrac{1}{2}-j)}{\Gamma(-[\tfrac{m}{2}]-\tfrac{1}{2}-j)}\cdot \frac{([\frac{m+1}{2}]+j)!}{j!}.
(q^{\mu -2}T_{N+2}T_{N+3}\dots T_{N+\mu-1}) (q^{\mu-1}T_{N+1}T_{N+2}\dots T_{N+\mu-1}) &= (q^{\mu-1}T_{N+1}T_{N+2}\dots T_{N+\mu-1}) (q^{\mu -2}T_{N+1}T_{N+2}\dots T_{N+\mu-2}).
\left\{x^{\alpha}, p_{\beta}\right\}_{D ( \Phi_{3 n}^{( 1 )})}= \delta_{\beta}^{\alpha}, \left\{\psi^{n}, \psi^{m}\right\}_{D ( \Phi_{3 n}^{( 1 )})}= \frac{i}{2}\eta^{n m},
\mathrm{d}X_{t}^{x}=(a-bX_{t}^{x})\mathrm{d}t+\sigma\sqrt{X_{t}^{x}}\mathrm{d}B_{t}+\mathrm{d}J_{t},t\geqslant0,X_{0}^{x}=x\in\mathbb{R}_{\geqslant0}.
|\begin{matrix}\frac{\partial x_{1}}{\partial x}&\frac{\partial x_{1}}{\partial y}&\frac{\partial x_{1}}{\partial z}\\ \frac{\partial x_{2}}{\partial x}&\frac{\partial x_{2}}{\partial y}&\frac{\partial x_{2}}{\partial z}\\ \frac{\partial x_{3}}{\partial x}&\frac{\partial x_{3}}{\partial y}&\frac{\partial x_{3}}{\partial z}\end{matrix}|
\tilde T_n (k) :=\sum\limits_{\substack{A \subset \{1, \dots, d\}|A|=k}}\tilde S_{n,A}^{\rm M}, \tilde S_{n,A}^{\rm M}:=\frac1n \sum_{i, j=1}^n \prod_{p \in A}\tilde I_{i,j}^{(p)},
S =\frac{\Gamma(M+1/2)}{\Gamma(M+1)}\frac{2^{2M-n-1}\Gamma(M-\frac n2) \Gamma(M-\frac{n-1}2)}{2^{2M}\Gamma(M+1/2)\Gamma(M+1)}\bigl(2M+n(n-1)\bigr) =\frac 1{2^{n}}\frac{\Gamma(M-\frac n2) \Gamma(M-\frac{n-1}2)}{(\Gamma(M+1))^2}\bigl(M+\frac{n(n-1)}2\bigr)\ .
I_{q,x_{l,n,i}^{(\alpha )}}^{(x)}(f(x)) = \frac{1}{{(q - 1)!}}I_{1,x_{l,n,i}^{(\alpha )}}^{(x)}\left({{{\left({x_{l,n,i}^{(\alpha )}- x}\right)}^{q - 1}}f(x)}\right).
\sum_{\mu=1}^{n}\omega(k_\mu)^s \Omega(\hat{K}_{n+1})^{-u(s)}\leq\omega(k_{j})^{s-u(s)}+ n^{\max(0,1-s)}\Omega(\hat{K}_{n+1,j})^s \Omega(\hat{K}_{n+1})^{-u(s)}&\leq \omega(k_{j})^{s-u(s)}+ n^{\max(0,1-s)}\Omega(\hat{K}_{n+1,j})^{s-u(s)}&\leq \omega(k_{j})^{s-u(s)}+ n^{\max(0,1-s)}\Omega(\hat{K}_{j})^{s-u(s)}\,,
\frac{\partial}{\partial \eta^{\prime \prime}{}^{\mu}}\frac{\partial}{\partial \overline{{\psi}}^{\prime \prime}}\Omega ( \zeta^{\prime \prime}) = \frac{\partial}{\partial \overline{{\psi}}^{\prime \prime}}\frac{\partial}{\partial \eta^{\prime \prime}{}^{\mu}}F ( \zeta^{\prime \prime}) ,
\frac{\partial\psi}{\partial x_N}(e_1) =\left.\frac{d}{d\theta}\psi(\cos\theta,0,\ldots,0,\sin\theta)\right|_{\theta=0}=\left.\frac{d}{d\theta}\phi(\cos\theta,0,\ldots,0,\sin\theta)\right|_{\theta=0}&=\frac{\partial\phi}{\partial x_N}(e_1)>0.
\underline{\underline{S}}=[\begin{matrix}S_{11}&S_{12}&S_{13}&0&0&0\\S_{12}&S_{22}&S_{23}&0&0&0\\S_{13}&S_{23}&S_{33}&0&0&0\\ 0&0&0&S_{44}&0&0\\ 0&0&0&0&S_{55}&0\\ 0&0&0&0&0&S_{66}\end{matrix}]
\frac{1}{N_p}\sum_{i=1}^{N_p}\phi_h(h_{p,i}^1,...,h_{p,i}^{t+1},x_{p,i}) \doteq \mathbb{E}[\phi_h(\tau_p^0\tilde{Z}_p^0,...,\tau_p^t\tilde{Z}_p^t,X)].
\frac{\delta S}{\delta A_{0}}= - \frac{\sqrt{3}}{8 \pi}\partial_{i}\mathrm{I m}\left[ \sigma_{\alpha \beta}^{0 i}\left( D^{\alpha \beta}{\cal F}^{\prime}+{\cal F}^{\prime \prime}D^{\alpha \beta}{\cal A}\right) \right] | .
\dfrac{M_{2}(\mathcal{NC}(D_{2m}))}{|e( \mathcal{NC}(D_{2m}) )|}- \dfrac{M_{1}(\mathcal{NC}(D_{2m}))}{|v( \mathcal{NC}(D_{2m}) )|}= \dfrac{m^{3}(m^{2}-12m+28)+m^{2}(24m-96)+64m}{3m(m-2)(2m-2)}:= \frac{f(m)}{g(m)}.
\begin{array}{l}{{\, s \; \omega_{D}^{g}\; + \; d \; \omega_{D - 1}^{g + 1}\; = \; 0 \; ,}}\\{{s \; \omega_{D - 1}^{g + 1}\; + \; d \; \omega_{D - 2}^{g + 2}\; = \; 0 \; ,}}\\{{\; . . . . .}}\\{{\; . . . . .}}\\{{s \; \omega_{1}^{g + D - 1}\; + \; d \; \omega_{0}^{g + D}= \; 0 \; ,}}\\{{s \; \omega_{0}^{g + D}= \; 0 \; ,}}\\ \end{array}
\int_{-\infty}^{\infty}ke^{-fx^{2}+gx+h}dx=\int_{-\infty}^{\infty}ke^{-f(x-g/(2f))^{2}+g^{2}/(4f)+h}dx=k\sqrt{\frac{\pi}{f}}exp(\frac{g^{2}}{4f}+h),
\prod_{i=1}^{k}\frac{z_{i}^{M'-1}}{(1+z_{i})^{M}}\langle \prod_{1\le i \le k}^{\curvearrowright}C_{\mu_{i}}^{[M, M']}(z_{i})\prod_{1\le i \le k}\beta_{\nu_{i}, x_{i}}^{*}\rangle_{[M', M]}.
|B_{c}| \geq \delta^{N^{2}\tau_{n + 1}}|\bar{B}_{c}| \stackrel{\eqref{defTau}}{\geq}\delta^{N^{2}N^{4N - 3(n + 1)}\epsilon + \epsilon}|B_{n}| & \,\,\, \geq \delta^{(N^{4N - 3n - 1}+ 1)\epsilon}|B_{n}| & \,\,\, \geq \delta^{N^{4N - 3n}\epsilon}|B_{n}| = \delta^{\tau_{n}}|B_{n}|.
\vartheta:= \vartheta_0 \big ( a\, \vartheta_0\big )^{\frac{-1}4}\,, \pi:= \pi_0 \big ( a\, \vartheta_0\big )^{\frac{-1}4}\,, \phi_0^{a,b}:= \phi^{a,b}\Big (\pi, \vartheta \frac{b\, \pi_0}{a\, \vartheta_0}\Big )\,,
\bigg| \iint_{\max(|x-y|, |x-z|) > 2\Phi(x)}K_{\Phi}(x,y,z) f(y) g(z)\,d\mu(y)\,d\mu(z) \bigg| &\le \sup_{\varepsilon > \Phi(x)}\bigg| \iint_{\max(|x-y|, |x-z|) > \varepsilon}K_{\Phi}(x,y,z) f(y) g(z)\,d\mu(y)\,d\mu(z) \bigg| = T_{\mu, \Phi, \sharp, \Phi(x)}(f,g)(x).
B_{2}^{f}= - \frac{V}{1 6 \pi^{2}}\frac{2^{1 - 2 \nu}}{\nu \Gamma ( \nu )^{2}}\int_{0}^{\infty}d t \, t^{2 \nu + 3}\, \frac{K_{\nu}^{\prime}( t )}{I_{\nu}^{\prime}( t )}> 0 ,
W_{p,N_r}^{(o)}(t)=(-1)^{e(N_r)}\sum_{N_o,N_o'}\left[E_{p,N_o}^{(-2)}\left(Y_{b,N_r}(t)\right)+\sum_N [\partial q_NY_{p,N_r}(t)]B_{NN_o}\right]A_{N_oN_o'}^{-1}\chi_{N_o'}(t)
\left( \begin{array}{c c}{V V^{\dag}}&{V U^{\dag}}\\{U^{\prime}V^{\dag}}&{U^{\prime}U^{\dag}}\\ \end{array}\right) = \left( \begin{array}{c c}{1 - K f K^{\dag}}&{- K f \Delta^{\dag}}\\{- \Delta^{\prime}f K^{\dag}}&{1 - \Delta^{\prime}f \Delta^{\dag}}\\ \end{array}\right) .
d \left( \mathrm{e}^{- \phi}{}^{*}H_{3}+ \mathrm{e}^{\phi}C_{0}{\tilde{F}}_{3}\right) -{\tilde{F}}_{5}\wedge F_{3}+ 2 \kappa^{2}\frac{\delta{\cal{L}}_{b}}{\delta B_{2}}= 0