harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\int g(\vec{x})e^{\frac{f(\vec{x})-\sum_{i=1}^nx_i\frac{\partial f(\vec{t})}{\partial t_i}}{\hbar}}d\vec{x}=(2\pi\hbar)^{\frac{n}{2}}g(\vec{t}) \det\Bigg(\frac{\partial^2f(\vec{t})}{\partial t_i\partial t_j}\Bigg)_{i,j}^{-\frac{1}{2}}\cdot e^{\frac{1}{\hbar}\big(f(\vec{t})-\sum_{i=1}^{\infty}t_i\frac{\partial f(\vec{t})}{\partial t_i}\big)}.
\operatorname{div}\mathbf{v}_A= \;\operatorname{div}(\zeta \circ d_\Gamma)(\mathbf{v}_A^{in}-\mathbf{v}_A^+\chi_+ -\mathbf{v}_A^-\chi_-)+(\zeta \circ d_\Gamma)\operatorname{div}\,\mathbf{v}_A^{in}& +(1-\zeta \circ d_\Gamma)(\operatorname{div}\,\mathbf{v}_A^+\chi_+ +\operatorname{div}\,\mathbf{v}_A^-\chi_-)\triangleq I_1+I_2+I_3.
\int_Q |\nabla u_{\delta_k}|^2 \,dxdt \,- \int_\Sigma \partial_{\boldsymbol{\nu}}u_{\delta_k}v_{\delta_k}\,d\Gamma dt &{}+\int_Q \xi_{\delta_k}u_{\delta_k}\,dxdt =\int_Q \bigl( \mu_{\delta_k}-\pi(u_{\delta_k})+f \bigr) u_{\delta_k}\,dxdt.
q_{n+2}=\frac{3}{2}q_{n+1}+6\sum_{j=0}^{n}(\begin{matrix}2n+2\\ 2j+2\end{matrix})q_{n-j}\sum_{k=0}^{j}(\begin{matrix}2j+1\\ 2k+1\end{matrix})q_{k}q_{j-k},q_{0}=1,q_{1}=\frac{3}{2}.
\frac{\alpha(x+h_n)-\alpha(x)}{\sigma \ell h_n\sqrt{-\log|h_n|}}= \frac{1}{\sigma \ell \sqrt{-\log|h_n|}}\sum_{i=1}^{N(x,h_n)}\phi\left( f^{i-1}(x)\right) + r(x,h_n),
\frac{\partial L}{\partial w_{ijkl}}(g_{ij},g_{ij,k},g_{ij,kl})=\frac{\partial x^i}{\partial \tilde{x}^a}\frac{\partial x^j}{\partial \tilde{x}^b}\frac{\partial x^k}{\partial \tilde{x}^c}\frac{\partial x^l}{\partial \tilde{x}^d}\frac{\partial L}{\partial w_{abcd}}(\tilde{g}_{ij},\tilde{g}_{ij,k},\tilde{g}_{ij,kl}).
\frac{\partial N_{i}}{\partial\alpha_{j}}=\frac{[N_{i},N_{j}]}{\alpha_{i}-\alpha_{j}}fori\ne j,\frac{\partial N_{i}}{\partial\alpha_{i}}=-\sum_{1\le j\le n,j\ne i}\frac{[N_{i},N_{j}]}{\alpha_{i}-\alpha_{j}}.
\begin{tabular}{c|ll|c|ll|c|ll|c}$1$ $it$ $1$ $0$ $x$ $0$ $K_C=G_C=$ $0$ $1$ $-ix_1$ $1$ $0$ $1/x$ \end{tabular}\begin{tabular}{c|ll|c|ll|c}$1$ $it$ $x$ $0$ $=$ $0$ $1$ $ip$ $1/x$ \end{tabular}
\small\boldsymbol{\Omega}= \left[{\begin{array}{*{20}{c}}{w_1^{in}}{w_{W}^{in}}\cdots{w_2^{in}}{w_2^{in}}{w_1^{in}}\cdots{w_3^{in}}\vdots \vdots \cdots \vdots{w_{W}^{in}}{w_{W-1}^{in}}\cdots{w_1^{in}}\end{array}}\right].
\sup_{f\in\mathcal{F}}\frac{\sqrt{n}}{v_{n}}\left\vert \mathbb{E}\left(\mathbb{P}_{n}^{(N)}(f)\right) -P(f)\right\vert =\sup_{f\in\mathcal{F}}\left\vert \frac{1}{v_{n}}\mathbb{E}\left( \mathbb{R}_{n}^{(N)}(f)\right)\right\vert \leqslant\mathbb{E}\left( \frac{r_{n}^{(N)}}{v_{n}}\right).
\left( \begin{array}{c c}{{0}}&{{A^{-}( \ell )}}\\{{A^{+}( \ell )}}&{{0}}\\ \end{array}\right) \left( \begin{array}{c}{{\chi_{E \ell}( r )}}\\{{\chi_{E \ell + 1}( r )}}\\ \end{array}\right) = \sqrt{E + \frac{1}{2 ( \ell + 1 )^{2}}}\left( \begin{array}{c}{{\chi_{E \ell}( r )}}\\{{\chi_{E \ell + 1}( r )}}\\ \end{array}\right) ,
\frac{u}{v}-\sqrt{\frac{b}{a}}\leqslant \frac{ba^{-1}-uv^{-1}}{uv^{-1}+\sqrt{ba^{-1}}}\varepsilon B^{-\frac{1}{r}}\leqslant \frac{ba^{-1}-\sqrt{ba^{-1}}}{2\sqrt{ba^{-1}}}\varepsilon B^{-\frac{1}{r}}=\frac{1}{2}\left(\sqrt{\frac{b}{a}}-1\right)\varepsilon B^{-\frac{1}{r}}.
\begin{aligned}-\frac{d}{dt}(F(\Gamma_t)-F(\Gamma)) =-dF_{\Gamma_t}([u]_{\Gamma_t}\cdot\nu_t)=B_{\Gamma_t}(u,u) &=\|d F_{\Gamma_t}\|_{B'_{\Gamma_t}}^2\ge \frac1c \|dF_{\Gamma_t}\|_{X'_{\Gamma_t}}^2\ge \frac1c \|dF_{\Gamma_t}\|_{Y'_{\Gamma_t}}^2\end{aligned}
\left(\begin{array}{cccc}A_{11}A_{12}\cdots A_{1n}A_{21}A_{22}\cdots A_{2n}\vdots \vdots \ddots \vdots A_{n1}A_{n2}\cdots A_{nn}\end{array}\right)\left(\begin{array}{cccc}C_1 C_2 \vdots C_n\end{array}\right)=\left(\begin{array}{cccc}X_1 X_2 \vdots X_n\end{array}\right).
m m^{\dagger}= \rho^{2}\left( \begin{array}{c c c}{x^{2}}&{x y}&{x z}\\{y x}&{y^{2}}&{y z}\\{z x}&{z y}&{z^{2}}\\ \end{array}\right) = \rho^{2}\left( \begin{array}{c}{x}\\{y}\\{z}\\ \end{array}\right) ( x , y , z ) ,
\left( \begin{array}{c}{\chi^{( 1 )}}\\{\chi^{( 2 )}}\\ \end{array}\right) = \frac{1}{\sqrt{2}}\left( \begin{array}{c c}{1}&{1}\\{- 1}&{1}\\ \end{array}\right) \left( \begin{array}{c}{\chi^{L}}\\{\chi^{R}}\\ \end{array}\right)
Z_g(s, \chi, D)= q^{-e_2s}Z_{g_1}(s, \chi, D) = q^{-e_2s}\Big(v(\bar{g}_1, D, \chi)+\sigma(\bar{g}_1, D, \chi)\dfrac{(1-q^{-1})q^{-s}}{1-q^{-1-s}}+Z_{g_1}(s, D_{S(g_1, D)})\Big) = \dfrac{M_{2, 3}(q^{-s})}{1-q^{-1-s}}+q^{-e_2s}Z_{g_1}(s, \chi, D_{S(g_1, D)}),
|E_{v_{3},ip}| \leq \frac{C}{t^{2}\log^{b+2}(t)}\int_{t}^{\infty}\frac{dx}{(\log^{(\alpha-1)b}(t)+x-t)(1+x-t)^{3}}&\leq \frac{C \log(\log(t))}{t^{2}\log^{b+2}(t)}
L_{s e l f i n t}^{f e r m}= 3 \left[ \frac{1}{b}\left( \frac{m ( h - b )}{4 M ( k - b )}\overline{{\Psi}}^{\prime}\Psi^{\prime}\right)^{4}e^{2 \alpha \bar{\phi}/ M_{p}}\right]^{1 / 3}.
=i_{0}e^{-\frac{R}{L}t}+\frac{E}{2jL}e^{-\frac{R}{L}t}2jIm(\frac{1}{\frac{R}{L}-j\omega})+\frac{E}{2jL}2jIm(e^{j\omega t}\frac{1}{\frac{R}{L}+j\omega})
E_{M}= \frac{l}{4}\sum_{n = 1}^{\infty}\left[ \left( 1 + \frac{r^{2}k_{n 1}^{2}}{l^{2}}\right) \omega_{n 1}^{2}A_{n 1}^{2}+ \left( 1 + \frac{r^{2}k_{n 2}^{2}}{l^{2}}\right) \omega_{n 2}^{2}A_{n 2}^{2}\right]
d{\bf M}= \lim_{\delta \to 0}\lim_{N \to \infty}\{1 \over (2 \pi \delta )^{NM\over 2}}exp\Big [-{1\over 2 \delta}\sum_{n=1}^N \Delta{\mathrm w}_n^T \Delta{\mathrm w}_n \Big ]\ \prod_{n=2}^N \prod_{j=1}^M d{\mathrm w}_{jn}.
\Phi_j=\frac{\zeta(s_j)[\xi(s_{j-1})-\xi(s_{j+1})]+\zeta(s_{j-1})[\xi(s_{j+1})-\xi(s_{j})]+\zeta(s_{j+1})[\xi(s_{j})-\xi(s_{j-1})]}{[\xi(s_j)\zeta(s_{j-1})-\xi(s_{j-1})\zeta(s_j)][\xi(s_{j+1})\zeta(s_j)-\xi(s_j)\zeta(s_{j+1}]}.
\dim \delta H_3^g = \; 6 + 9 k + (k-1)k +{3(k-1)^2}+(k-2)(k-1)^2/2 \; -\left({(k+1)^2(k+2)/2+k+3}\right) = \; k+4, \dim \delta E_3^g = \; I_M(V_3^g\times W_3^g)=\; k+2,
\partial_{\hat \xi}H = (1+\beta) |z|^{\beta-1}\left[ \hat \xi \cos{((1+\beta)\theta)}- \hat \zeta \sin{((1+\beta)\theta)}\right] \partial_{\hat \zeta}H = (1+\beta)|z|^{\beta-1}\left[ \hat \zeta \cos{((1+\beta)\theta)}- \hat \xi \sin{((1+\beta)\theta)}\right],
f(y)-\sum_{\alpha\in\mathbb{Z}_+^n,\,|\alpha|\leq N}\frac{(y-z)^\alpha}{\alpha!}\partial^\alpha f(z) &\quad=\sum_{\alpha\in\mathbb{Z}_+^n,\,|\alpha|=N}\frac{(y-z)^\alpha}{\alpha!}\int_0^1[\partial^\alpha f(z+t(y-z))-\partial^\alpha f(z)]N(1-t)^{N-1}\,dt,
\phi(n) =\left[ \mathbb Q(w_n): \mathbb Q\right] begin{align*}10pt] =\left[ \mathbb Q(w_n): \mathbb Q(\sqrt{p})\right]\cdot \left[ \mathbb Q(\sqrt{p}): \mathbb Q\right] begin{align*}10pt] =2\left[ \mathbb Q(w_n): \mathbb Q(\sqrt{p})\right].
\begin{array}{ll}\displaystyle \frac{\partial \bar{U}}{\partial t}\displaystyle = K_{\alpha}\,e^{-k_\ast\,t}\,\frac{\partial^{1-\alpha}}{\partial t^{1-\alpha}}\left(e^{k_\ast\,t}\,\int_{\Omega}\Delta U\,d\Omega\right) - k_\ast\,\bar{U}= - k_\ast\,\bar{U}.\end{array}
\Big( \sum_{k=B+1}^{N}\sigma(\vec{A})_k + 1 \Big)^{-1}\sum_{\lambda=1}^{N+1-B}\prod_{j=B+1}^{B+\lambda-1}\Bigg( \Big( \sum_{k=j+1}^{N}\sigma(\vec{A})_k + 1 \Big)^{-1}\Big( \sum_{k=j+1}^{N}\sigma(\vec{A})_k + \sigma(\vec{A})_j \Big) \Bigg).
\sideset{}{^*}\sum_{\mathbf{s},\mathbf{t}}\frac{\mu(s)}{\phi(s)\phi_{\omega}(st)}=\prod_{p}\sideset{}{^*}\sum_{\substack{\mathbf{s},\mathbf{t}s_i|p, t_i|p\forall i}}\frac{\mu(s)}{\phi(s)\phi_{\omega}(st)},
\frac{1}{\varphi_{\infty}(f;q)}= 1 + \frac{\sum_{(6)}}{\varphi_{\infty}(f;q)}&= 1 + \frac{\sum_{(6)}}{1-\varphi_{\infty}(f;q)\sum_{(5)}}&= 1+\cfrac{\sum_{(6)}}{1-\cfrac{\sum_{(5)}}{1/\varphi_{\infty}(f;q)}}.
I = \frac{1}{16\pi G_N}\int d^4x\sqrt{-g}\left[ R + \tilde{a}_1R^2 + \tilde{a}_2R^{\mu}_{\nu}R_{\mu}^{\nu}+ \tilde{a}_3R^{\mu\nu}_{\ \ \rho\sigma}R^{\rho\sigma}_{\ \ \mu\nu}\right],
x_{k}=\|r_{0}\|V_{k}T_{k}^{-1}e_{1},\quad\mbox{and}\|x_{k}\|^{2}=\|r_{0}\|^{2}e_{1}^{T}T_{k}^{-1}V_{k}^{T}V_{k}T_{k}^{-1}e_{1}.
\Delta^{( 2 )}{\cal V}= - \varepsilon^{2}\frac{3 m^{3}}{2{\sqrt{2}}\lambda}~ m^{2}t^{2}\int \frac{d z}{\cosh^{2}z}= - \varepsilon^{2}\frac{3 m^{5}}{\lambda{\sqrt 2}}t^{2}\equiv \frac{M V^{2}}{2}.
\int_{\Omega_{2}}\partial_{2}\phi_{n}(x)\partial_{2}\phi_{m}(x)\,dx = &{4n_{2}m_{2}\over \pi^2}\int_{0}^\pi\int_{\ell_{2}}^{\ell_{2}+\delta_{2}}\sin(n_{1}x_{1})\sin(m_{1}x_{1})\, dx_{1}\cos(n_{2}x_{2})\cos(m_{2}x_{2})\, dx_{2}= 0.
\left\vert \frac{f(z+c_{j})}{f(z+c_{l})}\right\vert \leq \exp_{p}\left[\left\{\log_{q-1}\left( r\right) \right\}^{\rho_{f}\left( p,q\right)+\varepsilon}\right] <\exp_{p}\left[ \left\{\log_{q-1}\left( r\right)\right\}^{\rho -2\delta}\right] ,\left( j\neq l,j=1,2,....,k\right)
\begin{pmatrix}A B -B A \end{pmatrix}=\begin{pmatrix}1 -AB^{-1}0 1 \end{pmatrix}\begin{pmatrix}{}^{\rm t}B^{-1}0 0 B \end{pmatrix}\begin{pmatrix}1 -1 \end{pmatrix}\begin{pmatrix}1 -B^{-1}A 0 1 \end{pmatrix},
\widehat{a}=\sum_{i=1}^n x_{i-1}g^{i}+ y g^{n+1}+ \sum_{i=n+2}^{2n}x_{i-2n-1}g^{i}\in \mathbb{C}[C_{2n}], \widehat{b}=\sum_{i=1}^{n}z_i g^{-i}\in \mathbb{C}[C_{2n}].
f_{r_m,r_t}(x,y) = f_{r_m|r_t}(x)f_{r_t}(y) = 4y(\lambda_c\pi)^{t}e^{-\lambda_c \pi y^2}\frac{x^{2m-1}(y^2-x^2)^{t-m-1}}{(t-m-1)!(m-1)!}.
\int_{0}^{r}s_1^{-(2N-2)}\Big(\int_{B(x,s_1)}u^{-(4N-1)}dy \Big)ds_1 &= - r^{-(2N-2)}\int_{\partial B(x,r)}(-\Delta)^{N-1}u d\sigma + \omega_{2N-1}(-\Delta)^{N-1}u(x).
E(\zeta_{n}(r,t)\overline{\zeta_{n}(s,u)})= E(S_{\lfloor nt \rfloor}(r)\overline{S_{\lfloor nu \rfloor}(s)})+\{nu\}E(S_{\lfloor nt \rfloor}(r)\overline{X_{\lfloor nu \rfloor+1}(s)})+ &\{nt\}E(S_{\lfloor nu \rfloor}(s)\overline{X_{\lfloor nt \rfloor+1}(r)})+\{nu\}\{nt\}E(X_{\lfloor nu \rfloor+1}(r)\overline{X_{\lfloor nt \rfloor+1}(s)}).
\begin{aligned}g = \Theta +Y Y:= -\frac 1 2\sum_{(\alpha , \beta )\in M_{min}}{{z}}^\alpha \overline{{z}}^\beta R_{-\sigma_3 B}^{+}( \widetilde{{\omega}}\cdot (\beta-\alpha )) \sigma_3{G}_{\alpha \beta}.\end{aligned}
\lambda_{j}:=\min\left\{\lambda_{j}\left( p_{j}^{\operatorname*{ext}}\right) ,\lambda_{j}\left( \mathbb{A}_{j}^{\operatorname*{ext}}\right)\right\}\quad\Lambda_{j}:=\max\left\{\Lambda_{j}\left(p_{j}^{\operatorname*{ext}}\right) ,\Lambda_{j}\left( \mathbb{A}_{j}^{\operatorname*{ext}}\right) \right\}.
{\rm e}^{\theta y}\mathbb{P}_i(T_y^+<\infty,H^+_{T_y^+}=y)=v_i(\theta)\sum_{j=\pm1}\frac{1}{v_j(\theta)}\delta_j u^{\theta, +}_{i,j}(y)\to v_i(\theta)\sum_{j = \pm1}\delta_j \frac{\pi^{\theta}_j}{v_j(\theta)\mu^+_\theta},
M_{\mu\nu}^{(p)}= \eta_{\mu\nu}+ F_{\mu\nu}+ \partial_\mu \phi^i\partial_\nu \phi^i - 2 \bar\lambda (\Gamma_\mu + \Gamma_i \partial_\mu \phi^i)\partial_\nu \lambda+ \bar\lambda \Gamma^m \partial_\mu \lambda \bar\lambda \Gamma_m \partial_\nu\lambda,
H(X_{[1,n]}) = H(X_{[j+1,n]}) + \sum_{i \leq j}H(X_i | X_{[i+1,n]}), H(X_{[1,n] \backslash j}) = H(X_{[j+1,n]}) + \sum_{i < j}H(X_i | X_{[i+1,n] \backslash j}).
{\bf\Xi}_{p,t}=\mathrm{blkdiag}({\bf\Xi}_{p,t,1},\cdots,{\bf\Xi}_{p,t,k}),\;{\bf\Xi}_{p,c}=\mathrm{blkdiag}({\bf\Xi}_{p,c,1},\cdots,{\bf\Xi}_{p,c,k}),{\bf R}_{u}=\mathrm{blkdiag}({\bf R}_{1,u},\cdots,{\bf R}_{u,k}),\; \widetilde{\bf R}_{u}=\mathrm{blkdiag}(\widetilde{\bf R}_{1,u},\cdots,\widetilde{\bf R}_{u,k}).
\|\|2^{(\alpha-1)m}\Delta_m(\partial_{x'}(U_{m+r}V_{m+r}))\|_{\ell^q(m)}\|_{L^p}\lesssim 2^{-(\alpha-1-a)r}\sum_{|\gamma|=a}\|\|2^{(\alpha-1-a)m}(\partial^{\gamma}\partial_{x'}(U_{m}V_{m}))\|_{\ell^q(m)}\|_{L^p}.
d_{b,c,j}= \begin{cases}(-1)^j\left(c_{n-j}- 2\sum_{i=0}^{i=j-1}(-1)^i b_i\right) j \neq n b_j j =n \end{cases}C_{b,c,h}= \prod_{j=1}^{j=h}C_{d_{b,c,j}}.
H^D_{\ell ,\ell +1}=g^2_{\ell +1}H^A_{\ell ,\ell +1}g^{-2}_{\ell +1}=(t^{-1}S^+_{\ell}S^-_{\ell +1}+tS^-_{\ell}S^+_{\ell +1}+2\cos\gamma S^3_{\ell}S^3_{\ell +1})\ ,
\hat{\theta}_{1,n}- \hat{\theta}_{n}= J_n(\hat{\theta}_{0,n})^{-1}\left( J_n(\hat{\theta}_{0,n}) - \int_0^1 J_n(\hat{\theta}_{n}+ t(\hat{\theta}_{0,n}- \hat{\theta}_{n}))dt \right) (\hat{\theta}_{0,n}- \hat{\theta}_{n}),
r_{b}(j)=\big[\underbrace{B\log_{2}\left(1+\zeta_{m,1}(j)\right)}_{r_{bm}(j)=C_{m1}(j)}- \underbrace{B\log_{2}\left(1+\zeta_{e,3}(j)\right)}_{r_{be}(j)=C_{e3}(j)}\big]^+.
A_S(t_1:t_0) =A_S(t_1:\tau)A_S(\tau:t_0)+A_{S\bar S}(t_1:\tau)A_{\bar S S}(\tau:t_0)\cr A_{\bar S S}(t_1:t_0) =A_{\bar S S}(t_1:\tau)A_{S}(\tau:t_0)+A_{\bar S}(t_1:\tau)A_{\bar S S}(\tau:t_0).
\int_{r,\theta}(r + 1)^{-2s}(|u|^2 + |hu'|^2) \lesssim \frac{1}{h^2}\int_{r,\theta}(r + 1)^{2s}|P^{\pm}_\varphi(h)u|^2 + \frac{\varepsilon}{h}\int_{r, \theta}|u|^2, h \in (0, h_0].
\frac{1}{n}\sum_{w^n = (-1)^{\theta_2}}\frac{w^{h-h'}e^{- (Tw+\theta_1 \pi i+ \lambda)[t' - t]}}{1 - (-1)^{\theta_1}e^{-(Tw+\lambda)}}= \mathrm{1}_{(t,h) = (t',h')}+ \frac{1}{n}\sum_{w^n = (-1)^{\theta_2}}\frac{w^{h-h'}e^{- (\lambda+\theta_1 \pi i+Tw)([t'-t]+\mathrm{1}_{t'=t})}}{1 - e^{- (\lambda+\theta_1 \pi i +Tw)}}.
(-h^{2}+ 2h+\mu)(h-4)^{n}=-h^{n+2}+2(2n+1)h^{n+1}&+\sum_{k=0}^{n-2}\bigg(\mu{n\choose k}(-4)^{k}+2{n\choose k+1}(-4)^{k+1}&-{n\choose k+2}(-4)^{k+2}\bigg)h^{n-k}&+\big(\mu n(-4)^{n-1}+2(-4)^{n}\big)h+\mu(-4)^{n},
Z_m'= \{(x_{i+1}, w(x_i)) \mid i \in [1,k-1]\}\sqcup\{(x_1, S(y))\}\sqcup\{(y, w(x_k))\}, Z_m'' = \{(x_i, w(x_{i})) \mid i \in [1,k]\}, Z_m''' = \{(y, S(y))\}.
H(\mathbb{Z}_p) \cong B([1]) \hookrightarrow B([1],[0]) & \hookrightarrow B([1 | 0],[0 | 1]) & \hookrightarrow B\left(\begin{bmatrix}1 0 \hline 0 -1 \end{bmatrix},\begin{bmatrix}0 1 \hline 1 0 \end{bmatrix}\right)\cong H(\mathbb{F}_{p^2}).
R_l(\beta)={\Gamma({2l\over\xi}+{i\beta\over\pi\xi})\Gamma(1+{2l\over\xi}+{i\beta\over\pi\xi})\over \Gamma({2l+1\over\xi}+{i\beta\over\pi\xi})\Gamma(1+{2l-1\over\xi}+{i\beta\over\pi\xi})}\ .
\begin{array}{l}{{\partial_{i}x^{j}= \delta_{i}^{j}+ q \hat{\cal R}_{i l}^{j k}x^{l}\partial_{k}~ ,}}\\{{\tilde{\partial}_{k}x^{v}= \delta_{k}^{v}+ q^{- 1}\left. \hat{\cal R}^{- 1}\right._{k j}^{v i}x^{j}\tilde{\partial}_{i}~ .}}\\ \end{array}
{\bf \Psi}( x ) = \left( \begin{array}{c}{{\varphi ( x )}}\\{{\Psi ( x )}}\\ \end{array}\right) , \; \; \hat{x}^{\mu}= x^{\mu}\; , \; \; \hat{\pi}_{\mu}= - i \partial_{\mu}\; , \; \; \hat{\psi}^{n}= \frac{i}{2}\Gamma^{n},
\widehat\Theta\left[{\rm d}w^{\varepsilon}-w_{xx}^{\varepsilon}{\rm d}t-\left(x+\varepsilon\right)^{2\gamma}w_{yy}^{\varepsilon}{\rm d}t-\frac{\sigma}{(x+\varepsilon)^2}w^{\varepsilon}{\rm d}t\right]=Q_1+(Q_2+Q){\rm d}t.
\ln t -\ln s -\frac{t-s}{st}<\sum_{r=1}^{t}\frac{1}{r}< \ln t -\ln s , \frac{t-s}{st}-\frac{t^2-s^2}{(st)^2}<\sum_{r=1}^{t}\frac{1}{r^2}< \frac{t-s}{st},
d_{A B C}h^{A}h^{B}h^{C}= 3 \Big \{h^{1}\big ( h^{2}\big )^{2}- h^{1}\big ( h^{\mu}\big )^{2}- h^{2}\big ( h^{i}\big )^{2}+ \gamma_{\mu i j}h^{\mu}h^{i}h^{j}\Big \}\ .
\frac{\mu_k(v_{1})}{\pi(v_{1})^p}> \frac{f_{1}(w_{1})}{\pi(w_{1})^p}= C^2 \frac{f_{1}(w_{1}')}{\pi(w_{1}')^p}> C^2 \frac{f_0(w_{1}')}{\pi(w_{1}')^p}\ge C^2 \frac{\mu_k(v_{1}')}{\pi(v_{1}')^p},
\varepsilon_{p}^{D}\left( L_{1}, L_{2}, \cdots , L_{p}\right) = - \frac{\pi^{( D - p ) / 2}}{2^{D - p + 1}}\Gamma \left( - \frac{D - p}{2}\right) E_{p}\left( \frac{1}{L_{1}^{2}}, \frac{1}{L_{2}^{2}}, \cdots , \frac{1}{L_{p}^{2}}; - \frac{D - p}{2}\right) \hspace{3 c m}
\sum_{n=1}^N p_n \sum_{\ell=0}^{L-1}\left(1-p_n\right)^\ell\left(1-r_n\right)^{\ell+1}, 0 \le r_n \le 1, \forall n\in[1:N], & \sum_{n=1}^N r_n = R_{\sf c}.
\tilde{\varphi}_{m n}( \vec{k}) = \frac{1}{\sqrt{m ! n !}}e^{\frac{k_{c}k_{\bar{c}}}{2}}\left( i \frac{\partial}{\partial k_{c}}\right)^{m}\left( i \frac{\partial}{\partial k_{\bar{c}}}\right)^{n}\tilde{\chi}_{0 0}( \vec{k}) ,
\sum_{a=2}^{n-6}\sum_{b=a+4}^{n-2}\sum_{q=a+2}^{b-2}(2^{q-1-a}-1)(2^{b-1-q}-1) &=\sum_{a=2}^{n-6}\sum_{b=a+4}^{n-2}\left[(b-3-a)2^{b-2-a}-2(2^{b-2-a}-2)+(b-3-a)\right] &=\sum_{a=2}^{n-6}\sum_{b=a+4}^{n-2}(b-5-a)2^{b-2-a}+\sum_{a=2}^{n-6}\left[4(n-5-a)+\binom{n-4-a}{2}\right] &=\sum_{a=2}^{n-6}\sum_{b=a+4}^{n-2}(b-5-a)2^{b-2-a}+4\binom{n-6}{2}+\binom{n-5}{3}
= \mathrm{T r}\Big [ \frac{- \not{\! \partial}+ m}{- \triangle + m^{2}}\not{\! \! B}\frac{- \not{\! \partial}- m}{- \triangle + m^{2}}\not{\! \! B}\frac{- \not{\! \partial}+ m}{- \triangle + m^{2}}\not{\! \! B}\frac{- \not{\! \partial}- m}{- \triangle + m^{2}}\not{\! \! B}\Big ] .
K_{\mu}\left( \begin{array}{l}{r_{k}}{s_{k}}\end{array}\right) = \left( \begin{array}{l}{r_{k}}{s_{k}}\end{array}\right) + ( \, r_{k}\, q_{\mu}\, \, - s_{k}\, p_{\mu}\, ) \left( \begin{array}{l}{p_{\mu}}{q_{\mu}}\end{array}\right) \, .
P\left(\sum_{y\in F_{k+i}}\zeta_y<2\alpha^{(d+1)(i+k)}\right) &=P\left(\sum_{y\in F_{k+i}}\bigl(\zeta_y-E[\zeta_y]\bigr)<2\alpha^{(d+1)(i+k)}-\sum_{y\in F_{k+i}}E[\zeta_y]\right) &\leq P\left(\sum_{y\in F_{k+i}}\bigl(\zeta_y-E[\zeta_y]\bigr)<-\alpha^{(d+1)(i+k)}\Bigl(\frac{5}{2}2^{d-1}(1-\exp(-2))-2\Bigr)\right)
\mbox{minimize}\sum_{l =1}^{10}\left( c_{il}(g_{il}) - p_l(\bar s_l)s_{il}\right)\cr\mbox{subject to}\sum_{l=1}^{10}g_{il}= \sum_{l=1}^{10}s_{il}, & g_{il}, s_{il}\geq 0,g_{il}\leq \mathrm{cap}_{il}, l = 1,\hdots,{10},
\gamma^{\mu}{\stackrel{\leftrightarrow}{\nabla}}_{\mu}= e_{m}^{\mu}[ \gamma^{m}( \stackrel{\leftrightarrow}{\partial}_{\mu}- i g A_{\mu}^{a}\tau_{a}) \, +{\frac{i}{2}}\Gamma_{n \mu p}\{\gamma^{m}; \Sigma^{n p}\}] ,
= g_{2}^{2}g^{4}{\frac{\Lambda^{2}}{M^{4}}}\left( \lambda_{1}^{2}- \lambda_{2}^{2}\right) - g^{6}{\frac{m_{0}^{2}}{M^{4}}}\left( \lambda_{1}^{2}- \lambda_{2}^{2}\right) - g^{2}l n{\frac{m_{0}^{2}+ \lambda_{1}^{2}}{m_{0}^{2}+ \lambda_{2}^{2}}}+ . . .
f(y_1,\ldots, y_v) = y_1 \cdots y_{m-\mu}\cdot (y_{m-\mu+1}\cdots y_m \oplus y_{m+1}\cdots y_{m+\mu}), f(y_1,\ldots, y_v) = y_1 \cdots y_{m-2}\cdot (y_{m-1}\cdot y_m \oplus y_{m+1}\cdot y_{m+2}\oplus \cdots \oplus y_{m+2\nu-3}\cdot y_{m+2\nu-2}),
\left( z\gamma_{n}^{\prime}\right)^{\prime}=\frac{z\left( \gamma_{n}^{\prime}\right)^{2}}{\gamma_{n}}+\frac{4}{z}\gamma_{n}\left[\gamma_{n-1}\left( \gamma_{n-2}-\gamma_{n}+1\right) -\gamma_{n+1}\left(\gamma_{n}-\gamma_{n+2}+1\right) \right] .
\omega_{\mu}^{\a b}= \frac{1}{2}e^{a \nu}e^{b \rho}\left[ e_{\mu c}( \partial_{\nu}e_{\rho}^{\c}- \partial_{\rho}e_{\nu}^{\c}) - e_{\nu c}( \partial_{\rho}e_{\mu}^{\c}- \partial_{\mu}e_{\rho}^{\c}) - e_{\rho c}( \partial_{\mu}e_{\nu}^{\c}- \partial_{\nu}e_{\mu}^{\c}) \right] .
2^n \int_{U^n_j}W_r(t)\,d\mu(t) = 2^n \int^{(j+1)/2^n}_{j/2^n}\Big(\frac{r}{1-e^{-r}}\Big) e^{-rt}\,d\mu(t) = 2^n \Big[\Big(\frac{-1}{1-e^{-r}}\Big) e^{-rt}\Big]^{(j+1)/2^n}_{j/2^n}= \Big(\frac{-2^n}{1-e^{-r}}\Big)\Big(e^{-(j+1)r/2^n}- e^{-jr/2^n}\Big) = 2^n\Big(\frac{1 - e^{-r/2^n}}{1 - e^{-r}}\Big)e^{-jr/2^n}= 2^n(v^n_0)_j,
\frac{1}{n}\mathbb{E}[\exp(n \nu_n(f))] = B_{n,1}+B_{n,2}+B_{n,3}+ B_{n,4}+ R(2n - 2j + 2, x_{n, j}, x_{n, j}) + R(2n - 2j + 1, x_{n, j}, y_{n, j}) ~,
\left(R^{ac}_e\right)^{\pm 1}\big[ F^{acb}_d \big]_{eg}\left(R^{bc}_g\right)^{\pm 1}= \sum_f \big[ F^{cab}_d \big]_{ef}\, \left(R^{fc}_d\right)^{\pm 1}\, \big[ F^{abc}_d \big]_{fg}.
\bar{\Gamma}_t = \frac{\mathbb{P}^{0}(a<\tau_{x}<b)}{\mathbb{P}^{0}(\tau_{x}>t)}+ \int_{0}^{t}\sigma^{1}(\Phi)(t,u) dQ_{u}-\int_{0}^{t}\bar{\Gamma}_u \sigma(\Phi)(t,u))^{2}du +\int_{0}^{t}\bar{\Gamma}_u\sigma(\Phi)(t,u)dQ_u -\int_{0}^{t}\sigma^{1}(\Phi)(t,u) \sigma(\Phi)(t,u) du .
\mathbf{u}^s(A_i)=\mathbf{u}^s(X)+ \left( (A_i-X)^T\otimes I_2 \right) \emph{Vec}( \nabla\mathbf{u}^s(X) )+ \mathbf{R}^s_{i}(X), ~ i\in\mathcal{I}^s, ~ \forall X\in T_{\ast}^s,
d_1(\lambda_{i_1})= (1\otimes \lceil y_{i_1}\rceil-\lceil y_{i_1}\rceil\otimes 1), \forall \lambda_{i_1}\in \Lambda_1 &d_k(\lambda_{i_1\cdots i_k})=\sum\limits_{j=1}^k(-1)^{k-j}\lambda_{i_1\cdots \hat{i_j}\cdots i_k}(1\otimes \lceil y_{i_j}\rceil-\lceil y_{i_j}\rceil\otimes 1), \forall \lambda_{i_1\cdots i_k}\in \Lambda_k,
\begin{array}{l}\Omega_1=arg\inf\limits_{\Omega \subseteq \mathbb{R}^n, \Omega~}\int\limits_{\Omega}d\mathbf{x}(k) s.t. \int\limits_{\Omega}dP(\mathbf{x}(k))=1, \ \ \forall P \in \mathcal{P}_{\mathcal{X}_k}(\mathbf{X}(k))\end{array}
\begin{pmatrix}\lambda_1{a_{11}}\lambda_2{a_{21}}\lambda_3{a_{31}}\lambda_1{a_{12}}\lambda_2{a_{22}}\lambda_3{a_{32}}\lambda_1{a_{13}}\lambda_2{a_{23}}\lambda_3{a_{33}}\end{pmatrix}\begin{pmatrix}k_1 k_2 k_3 \end{pmatrix}= \begin{pmatrix}-1 -1 -1 \end{pmatrix}
P^\omega(E_t, S_t)\leq P^\omega(E_{1,t})\prod_{j=2}^{t/h(t)}P^\omega\left(E_{j,t}\:\bigg\vert\: S_{j-1,t}\, ,\:\bigcap_{k=1}^{j-1}(E_{k,t}, F_{k,t})\right) + \sum_{j=1}^{t/h(t)}P^\omega(F_{j,t}^c).
2 \kappa^{2}S_{\Phi}={\frac{\kappa^{2}N}{2 \pi g_{s}}}\,{\frac{1}{\sqrt{G_{\Omega \Omega}}}}\left({\frac{3}{4}}{e^{3 \Phi ( r ) / 4}}\mu_{6}+{\frac{1}{4}}{\frac{e^{- \Phi ( r ) / 4}}{V_{\mathrm{E}}( r )}}\mu_{2}\right) \, .
V_{f 1}( \hat{\phi}) = - 2 \frac{1}{( 4 \pi )^{2}}\sum_{i}\{\frac{1}{2}M_{i}^{4}\ln \frac{M_{i}^{2}}{\mu^{2}}- \frac{3}{4}M_{i}^{4}+ C_{2}M_{i}^{2}+ C_{3}\}
g(x_1,x_2)= -\frac{1}{8\pi}\frac{1}{\beta}\sum_{p_0}\frac{1}{(x_1+p_0)^2-(x_2+p_0)^2}\log\left(\frac{m^2-(x_1+p_0)^2}{m^2-(x_2+p_0)^2}\right) \,.
\dim T/(F^\perp + (l_1 + l_2)) > \dim T/J_1\cap J_2 = \dim T/J_1 + \dim T/ J_2 - \dim T/(J_1+J_2) = \dim T_1/(F_1^\perp + (l_1)) + T_2/(F_2^\perp + (l_2)) - 1
\sum_{k=1}^{\nu}(Y_{k}X_{k}^{j+\nu}+\Lambda_{1}Y_{k}X_{k}^{j+\nu-1}+\Lambda_{2}Y_{k}X_{k}^{j+\nu-2}+\cdot\cdot\cdot+\Lambda_{\nu}Y_{k}X_{k}^{j})=0
\left(a_{h-1},a_h\right)\in\left\{\begin{aligned}\left(n-5,n-4\right), \left(n-5,n-3\right), \left(n-5,n-2\right), \left(n-5,n-1\right), \left(n-4,n-3\right), \left(n-4,n-2\right), \left(n-4,n\right), \left(n-3,n-1\right), (n-3,n),(n-2,n-1), (n-2,n),(n-1,n) \end{aligned}\right\}.
\lim_{s \to \infty}\left(\exp\left(i\left(kX_s^{(\Theta)}+t\right)\left(\frac{\theta(\gamma_j) + 2\pi\ell}{k}\right)\right)\right)_{j=1}^g = \left(\exp\left(i(\theta_j + 2\pi [\ell t]/k)\right)\right)_{j=1}^g,
P(\theta^2,\bar\theta^2)=1-{i\over{2R}}(\theta^2-\bar\theta^2)+{1\over{3!R^2}}\theta^2\bar\theta^2, \qquad \theta^2\equiv \theta^A\theta_A,\quad \bar\theta^2\equiv\bar\theta_{\dot A}\bar\theta^{\dot A},
\mu_0 = (-1)^{n+1}\det \left( H_1, \dots, H_n, H_{n+1}\right) \mu_1 = (-1)^{n}\sum_{1 \leq i \leq n}\det \left( H_1, \dots, V_i, \dots, H_n, H_{n+1}\right) \mu_2 = (-1)^{n-1}\sum_{1 \leq i < j \leq n}\det \left( H_1, \dots, V_i, \dots, V_j, \dots, H_n, H_{n+1}\right) &\vdots \mu_n = - \det \left( V_1, \dots, V_n, H_{n+1}\right).
\begin{aligned}d_Y(G(y,x),G(v,u))\leq a\d_Y(y,G(v,u))+ b\d_Y(v,G(y,x))+c\d_Y(y,v); & \forall x\leq_{P_1}u,\y\geq_{P_2}v;\ 2a+c<1\end{aligned}
S_{I,J,K}(x,y; f) =\left.\chi_I(x) \chi_K(y)\frac{F(z)}{(z-x)(z-y)}\right|_{z=s}^{t}begin{align*}1ex] \ \ \ + \chi_I(x)\chi_K(y)\int\limits_J \Biggl[\frac{F(z)}{(z-x)^2(z-y)}+\frac{F(z)}{(z-x)(z-y)^2}\Biggr]\, dz.
\sqrt{\gamma_{1}}\sqrt{\gamma_{2}}\C_{\mathrm{t o t}}( X_{1}, X_{2}) = 3 2 G_{N}T^{2}( \partial_{+}X_{1}^{\mu}\ \partial_{+}X_{2 \mu}) ( \partial_{-}X_{1}^{\nu}\ \partial_{-}X_{2 \nu}) .
dX_1(t) = k_1(X(t)) dt + \frac{1}{\sqrt{2}}dB_1(t)+ \frac{1 - \zeta_1(X(t))}{2}dL_+^{X_1-X_2}(t)+ \frac{1 - \eta_1(X(t))}{2}dL_+^{X_2-X_1}(t) dX_2(t) = k_2(X(t)) dt + \frac{1}{\sqrt{2}}dB_2(t)+ \frac{1 - \zeta_2(X(t))}{2}dL_+^{X_1-X_2}(t)+ \frac{1 - \eta_2(X(t))}{2}dL_+^{X_2-X_1}(t).
\sqrt{a_{1}a_{k}}\leq\left( \sqrt{n_{1}n_{k}}\right)^{-1/p}\left(\frac{\sqrt{l\left( l+1\right)}}{2l+1}\right)^{1/p}c^{1/p}=\left(\frac{l\left( l+1\right)}{n_{1}n_{k}}\right)^{1/2p}b.
\begin{cases}\frac{d}{dt}\int_{\Omega}u=\int_{\Omega}u\left(a_0-a_1u-a_2v-a_3\int_{\Omega}u-a_4\int_{\Omega}v\right),x\in \Omega \quad\cr\frac{d}{dt}\int_{\Omega}v=\int_{\Omega}v\left(b_0-b_1u-b_2v-b_3\int_{\Omega}u-b_4\int_{\Omega}v\right),x\in \Omega. \quad\cr\end{cases}
\psi_{{\bf k}s}^{\pm}= \left[ - \gamma^{0}\left({\frac{d}{d \tau}}+{\frac{1}{2 \tau}}\right) - i \gamma_{\bf{{\perp}}}\cdot{\bf{k_{\perp}}}- i \gamma^{3}\pi_{\eta}+ m \right] \chi_{s}{\frac{f_{{\bf k}s}^{\pm}}{\sqrt \tau}}.
\frac{\partial \psi_{t_1}(s)}{\partial t_1}= \Phi (\psi_{t_1}(s)) &\frac{\partial \psi_{t_1}^{'}(s)}{\partial t_1}= \Phi^{'}(\psi_{t_1}(s)) \psi_{t_1}^{'}(s) &\frac{\partial \psi_{t_1}^{''}(s)}{\partial t_1}= \Phi^{''}(\psi_{t_1}(s)) \psi_{t_1}^{'}(s)^2+\Phi^{'}(\psi_{t_1}(s)) \psi_{t_1}^{''}(s),
{\cal L}={\frac{1}{2}}D_{\mu}\phi{\overline{{{D^{\mu}\phi}}}}-{\frac{1}{4}}F_{\mu \nu}F^{\mu \nu}-{\frac{1}{8}}\left( | \phi |^{2}- 1 \right)^{2}, \qquad \phi = \phi_{1}+ i \phi_{2},
Q_{a}= \left[ \begin{matrix}{{Q_{a}^{( 1 )}}}&{{}}&{{}}&{{}}\\{{}}&{{Q_{a}^{( 2 )}}}&{{}}&{{}}\\{{}}&{{}}&{{\ddots}}&{{}}\\{{}}&{{}}&{{}}&{{Q_{a}^{( Q_{6})}}}\\ \end{matrix}\right] , \ \P_{a}= \left[ \begin{matrix}{{P_{a}^{( 1 )}}}&{{}}&{{}}&{{}}\\{{}}&{{P_{a}^{( 2 )}}}&{{}}&{{}}\\{{}}&{{}}&{{\ddots}}&{{}}\\{{}}&{{}}&{{}}&{{P_{a}^{( Q_{6})}}}\\ \end{matrix}\right] ,
P_{\Lambda,\eta,\beta,H}(\sigma)=\frac{1}{Z_{\Lambda,\eta,\beta,H}}\exp{\left[2\beta\sum_{\{u,v\}}\delta_{\sigma_u,\sigma_v}+2\beta\sum_{e=\{u,v\}:u\in\Lambda, v\in\partial_{ex}\Lambda}\delta_{\sigma_u,\eta_v}+2H\sum_{u\in \Lambda}\delta_{\sigma_u,1}\right]},