harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\tan \varphi_{k+1}\leq \frac{\Vert L_{k}^{-1}\Vert \,\varepsilon_{k}}{1-\mathbf{c}^{T}L_{k}^{-1}V^{T}\mathbf{x}_{k}/\cos \varphi_{k}}\tan \varphi_{k}:=\beta_{k}\tan \varphi_{k}.
\left\|e^{t\mathcal{A}_b}\left( \begin{array}{c}u^0 v^0\end{array}\right)\right\|_{\bigcap_{j=0}^kC^{k-j}([0,\tau ];D(\mathcal{A}_b^j))}\leq C \left\|\left( \begin{array}{c}u^0 v^0\end{array}\right)\right\|_{D(\mathcal{A}_b^k)}.
a_{i}= \frac{i}{2}\Psi^{\dagger}\partial_{i}^{\raisebox{1 . 2 m m}{\leftrightarrow}}\Psi - \Psi^{\dagger}\frac{\sigma^{3}}{2}\Psi A_{i}= \frac{i}{2}\Psi^{\dagger}\nabla_{i}^{\raisebox{1 . 2 m m}{\leftrightarrow}}\Psi .
S_{n^1}^{k}u^{\alpha}=h^{\alpha}\left(x,n,[u]_{n^1},\left[S_{n^1}^{1}u\right]_{n^1},\left[S_{n^1}^{2}u\right]_{n^1},\ldots,\left[S_{n^1}^{k-1}u\right]_{n^1}\right).
W(x)-W(x_a)= p(m-p)\int_{x_a}^xA^{\tfrac{p}{2}-1}(s)\tanh s \,h'^2(s)\,ds \geq p(m-p)A^{\tfrac{p}{2}-1}(x_a)\tanh x_a \int_{x_a}^x h'^2(s)\,ds = p(m-p)(m-1)^{\tfrac{p}{2}-1}\cos^{p-2}h(x_a)\tanh x_a \int_{x_a}^x h'^2(s)\,ds
P(x + 1) = \sum_i \mathcal{C}(i, P)\binom{x + 1}{i}= \mathcal{C}(0, P) + \sum_{i = 1}^\infty \mathcal{C}(i, P)\left[\binom{x}{i - 1}+ \binom{x}{i}\right] = \sum_i \left[\mathcal{C}(i, P) + \mathcal{C}(i + 1, P)\right]\binom{x}{i}.
H_{ij}(\widehat{\eta}_1,\lambda\widehat{\eta}_1+dD\widehat{\eta}_3)= H_{ij}(\widehat{\eta}_1,\lambda\widehat{\eta}_1)+dD\widetilde{H}_{ij}(\widehat{\eta}_1,\widehat{\eta}_3 ,\lambda,d,D).
\psi_D(z_1,\ldots,z_n) = c^n \prod_{j=1}^n r(D,z_j)^{-1/8}\exp[ \mu_D(\gamma : N_{\gamma}(z_j) , | \overline\gamma \cap \{z_1,\ldots,z_n\}| \geq 2 ) ] & \exp\Big[ - \mu_D\big(\gamma: \textstyle \sum_{j=1}^n N_{\gamma}(z_i) ,| \overline\gamma \cap \{z_1,\ldots,z_n\}|\geq 2\big) \Big].
\frac{{u_{11}}}{{u_1}}= - (\frac{{\psi '}}{\psi}u_1 + \frac{{\rho_1}}{\rho}) = - \frac{{\psi '}}{{2\psi}}u_1 - \frac{{\psi 'u_1 \rho + 2\psi \rho_1}}{{2\psi \rho}}\le - \frac{{\psi '}}{{2\psi}}u_1.
d_{S(G,t)}(xw,xw')> d_{S(G,t-1)}(w,(v_1)^{t-1})+\sum_{i=0}^{\frac{r-5}{2}}d_{S(G,t-1)}((v_{2i+1})^{t-1},(v_{2i+3})^{t-1}) &+d_{S(G,t-1)}((v_{r-2})^{t-1},(v_0)^{t-1})\sum_{i=0}^{\frac{r-3}{2}}d_{S(G,t-1)}((v_{2i})^{t-1},(v_{2i+2})^{t-1}) &+d_{S(G,t-1)}((v_{r-1})^{t-1},w') \ge d_{S(G,t-1)}(w,w') ,
Z_T(s,f,\chi)=\nu_T(\overline{f},\chi)+\sigma_T(\overline{f},\chi)\frac{(1-q^{-1})q^{-s}}{(1-q^{-1-s})}+ \int\limits_{S_T(f)}\chi(ac\f(x,y))\ |f(x,y)|^s\ |dx dy|,
(\lambda_g\otimes I_{\mathcal{H}_0})\theta_A\theta_A^*=\theta_A\theta_A^*(\lambda_g\otimes I_{\mathcal{H}_0}), ~ (\lambda_g\otimes I_{\mathcal{H}_0})\theta_A\theta_\Psi^*=\theta_A\theta_\Psi^*(\lambda_g\otimes I_{\mathcal{H}_0}), (\lambda_g\otimes I_{\mathcal{H}_0})\theta_\Psi\theta_\Psi^*=\theta_\Psi\theta_\Psi^*(\lambda_g\otimes I_{\mathcal{H}_0}).
\zeta_{p,p^{\prime}}(s,c)=\frac{2}{1+\frac{1}{\pi p^{\prime}}}\,\zeta_{p}(2s)+\frac{2\sqrt{\pi}\,c^{\frac{1}{2}-s}\Gamma\left(s-\frac{1}{2}\right)}{\Gamma(s)}\zeta_{p^{\prime}}(2s-1)+H_{p,p^{\prime}}(s,c),\,\,\,\,\,(s)>1,
R^*_{n,pt}\le C_3R^{SC}_{n-l,pt}R^*_{l,pt}\le 2\rho^{m_1(l)}3^{l-m_1(l)}C_2^{m_2(l)}\cdot C_2\rho^{n-l}&=2\rho^{m_1(n)}3^{n-m_1(n)}C_2^{m_2(n)}.
a_{g_0, g_{i_k}\cdots g_{i_1}}w_{i_1}\cdots w_{i_k}+a_{g_{i_1}, g_{i_k}\cdots g_{i_2}}z_{i_1}w_{i_2}\cdots w_{i_k}&+ a_{g_{i_1}g_{i_2}, g_{i_k}\cdots g_{i_3}}z_{i_1}z_{i_2}w_{i_3}\cdots w_{i_k}+\cdots +a_{g_{i_1}\cdots g_{i_k},g_0}z_{i_1}\cdots z_{i_k}=0
y_i = \frac{\sqrt{|h_{ii}|^2p_i}}{\sum^L_{j=1,j\ne i}|h_{ij}|^2p_j+\sigma^2_i}, \tilde y_k = \frac{\sqrt{\sum^L_{j=1}|\tilde h_{kj}|^2p_j+\tilde\sigma^2_k}}{|\tilde h_{kk}|^2p_k+\varepsilon}.
\frac{\partial^{4}F}{\partial x^{4}}+2\frac{\partial^{4}F}{\partial x^{2}\partial y^{2}}+\frac{\partial^{4}F}{\partial y^{4}}=E[(\frac{\partial^{2}w}{\partial x\partial y})^{2}-\frac{\partial^{2}w}{\partial x^{2}}\frac{\partial^{2}w}{\partial y^{2}}]
(u,v) = h \sum_{i=1}^{N-1}u_i v_i, \langle \delta_h u, \delta_h v\rangle = h \sum_{i=1}^{N-1}(\delta_h u_{i-1/2})(\delta_h v_{i-1/2}), \|u\| = \sqrt{(u,u)}, |u|_1 = \sqrt{\langle \delta_h u, \delta_h u\rangle}.
X_t \ = \e^{tA}\,x_0 + \int_0^t e^{(t-s)A}\,b_s(X,\alpha_s)\,ds + \int_0^t e^{(t-s)A}\,\sigma_s(X,\alpha_s)\,dW_s &\ + \int_0^t\int_{U \setminus \{0\}}e^{(t-s)A}\,\gamma_s(X, \alpha_s, z)\,\big(\pi(ds\,dz)-\lambda_\pi(dz)\,ds\big), \qquad0\leq t\leq T.
E_1(n, i) := 2^{n}(1+2i)+2^{n+s}\mathbb{Z}_{2}\ (n\geq 1, 0\leq i < 2^{s-1}), E_2(n,i) := 1+2^{n}(1+2i)+2^{n+s+1}\mathbb{Z}_{2}\ (n\geq 2, 0\leq i < 2^{s}), E_3(n,i) :=-1+2^{n}(1+2i)+2^{n+s+1}\mathbb{Z}_{2}\ (n\geq 2, 0\leq i < 2^{s})
\rho_{\mu_{1}\dots \mu_{p - 1}\alpha_{1}}^{( 1 )}= ( C \Gamma_{\nu \mu_{1}\dots \mu_{p - 1}})_{\beta \alpha_{1}}\Pi^{\nu}\Pi^{\beta}\quad , \quad \quad \rho_{\mu_{1}\dots \mu_{p - 1}\alpha_{1}}^{( 2 )}= ( C \Gamma^{\nu})_{\beta \alpha_{1}}\Pi_{\nu \mu_{1}\dots \mu_{p - 1}}\Pi^{\beta}\quad .
\epsilon_{a}^{i}(\theta )\approx r\,a\,m_{i}\cosh \theta-\sum_{b=1}^{k-1}\left( \delta_{ab}-2\left( K^{A_{k-1}}\right)_{\,\,\,ab}^{-1}\right) L_{b}^{i}(\theta )-\sum_{b=1}^{k-1}\left( K^{A_{k-1}}\right)_{\,\,\,ab}^{-1}L_{b}^{j}(\theta -\sigma )\;.
f^{*}(\overline{h\otimes \alpha})=f^{*}(\overline{h}\otimes\overline{\alpha})=\overline{h}\otimes f(\overline{\alpha})=\overline{h}\otimes \overline{f(\alpha)}=\overline{h\otimes f(\alpha)}=\overline{f^{*}(h\otimes \alpha)},
B_{n,k}= \frac{1}{k}\sum_{\alpha\ge 0}\binom{n}{\alpha}x_{\alpha}B_{n-\alpha,k-1}, B_{n,k}= \sum_{\alpha\ge 1}\binom{n-1}{\alpha-1}x_{\alpha}B_{n-\alpha,k-1}, B_{n,k}(x_1,\ldots,x_{n-k+1}) = \sum_{\alpha\ge 0}\binom{n}{\alpha}x_1^\alpha B_{n-\alpha,k-\alpha}(0,x_2,\ldots,x_{n-k+1}).
{\displaystyle{\frac{\delta^{2}\Gamma_{m}[ 0 , \Psi , \bar{\Psi}]}{\delta \Psi ( x^{\prime}) \ \delta \bar{\Psi}( x )}}}\, B i g g \vert_{\bar{\Psi}= \Psi = 0}\ = \S_{m}^{- 1}( x - x^{\prime}) \hspace{0 . 3 c m}. \hspace{0 . 3 c m}
R <\min_{Q: Q_Y = W_Y, Q_X = P_X, f(W)= f(Q)}I_Q(X;Y) & = \min_{\substack{Q: Q_Y = W_Y, Q_X = P_X, H_W(Y|X)= D(Q||W|P_X)+H_Q(Y|X)}}H_W(Y)-H_Q(Y|X) & = I_W(X;Y) &\ \ \ \ \ \ +\min_{\substack{Q: Q_Y = W_Y, Q_X = P_X, H_W(Y|X)= D(Q||W|P_X)+H_Q(Y|X)}}D(Q||W|P_X) & = I_W(X;Y),
\partial_tf''=\mathcal{L}_*f''=-\mathcal{Q}(\underline{\theta}\cdot\partial_v \mathcal{Q}f'')-\mathcal{Q}\big((\underline{h}+1)\nabla^{\perp}\underline{\phi}\cdot\nabla \mathcal{Q}f''\big)-\mathcal{Q}\big((\underline{h}+1)(\partial_v\underline{f}\cdot\partial_z-\partial_z\underline{f}\cdot\partial_v)\underline{\Delta_t}^{-1}\mathcal{Q}f''\big).
\sum_{\gamma\in\mathcal{V}_{[n,\ell]}}w(\gamma)=\sum_{k=\ell}^{n-1}\sum_{\gamma\in\mathcal{V}_{[n,\ell,k]}}w(\gamma)=\sum_{k=\ell}^{n-1}a_{0}^{(\ell)}A^{(1)}_{[k-1,\ell-1]}W_{[n-k-1,0]}.
I_2=\int_0^r \left[-\frac{z}{2}(\bar\psi \bar\psi'\cdot \phi+\bar\psi \bar\phi'\cdot\psi)\right]{\rm{d}}z \leq\Vert (\phi,\psi)\Vert_{L^2}\cdot \Vert (\bar\phi',\bar\psi')\Vert_{L^2}\cdot\frac{1}{2}\Vert z\bar\psi\Vert_{L^\infty},
<{\lambda^{+}},\,Z_{1}-Ad_{p^{-1}}\bigr([u_{0},u_{\bf i}]\bigr)>=0, &<{\lambda^+},\, Z_{2}-Ad_{p^{-1}}\bigr(\alpha(v)[u_{0},u_{\bf j}]-[u_{\bf i},\,u_{\bf k}]\bigr)>=0, &<{\lambda^+},\, Z_{3}-Ad_{p^{-1}}\bigr(\alpha(v)[u_{0},u_{\bf k}]+[u_{\bf i},\,u_{\bf j}]\bigr)>=0,
\sum_{\lambda \in \sigma_{disc}(H)}\frac{\textup{dist}\big( \lambda,\cup_{j=0}^{\infty}\lbrace \Lambda_{j}\rbrace \big)^{p}}{\big( 1 + \vert \lambda \vert \big)^{2p}}\leq C_{3}\Vert F \Vert_{L^{p}}^{p},
S_{2 + 1}= - \frac{L m^{3}\lambda}{8 \pi}\sum_{j = 0}^{\infty}\frac{1}{j !}\left( \frac{1}{2 e B \lambda^{2}}\right)^{j}\sum_{k = 1}^{\infty}\frac{( 2 k + j - 1 ) !{\cal B}_{2 k + 2 j}}{( 2 k ) ! ( 2 k + j - \frac{1}{2}) ( 2 k + j - \frac{3}{2})}\left( \frac{2 e B}{m^{2}}\right)^{2 k + j}
u(x,t)\,\to u_{\ell}(x,t):=u\left(x-\frac{x\cdot \ell}{|\ell |^2}\ell +\frac{\frac{x\cdot\ell}{|\ell |}\frac{\ell}{|\ell |}-\ell t}{\sqrt{1-|\ell |^2}},\,\frac{t-x\cdot\ell}{\sqrt{1-|\ell|^2}}\right),
\int_{\Omega}\chi_A \det \nabla \phi~dx=\lim_{\nu\rightarrow \infty}\int_{\Omega}\chi_A \det \nabla \phi_{\nu}~dx \geq \lim_{\nu\rightarrow \infty}\int_{\Omega}(1-C\sqrt{|K_{\nu}|}\tau)\chi_A~dx= \int_{\Omega}(1-C\sqrt{|M|}\tau)\chi_A~dx,
\left(\frac{\dot a_0}{a_0}\right)^2+\frac{K}{a_0^2}= 2k^2\frac{(-1-\bar\rho_0\bar\rho_{\frac12})\sinh(kb)+(\bar\rho_0+\bar\rho_{\frac12})\cosh(kb)}{\sinh(kb)-\bar\rho_{\frac12}(\cosh(kb)-1)}.
p=\frac{(\begin{matrix}26\\ 13\end{matrix})(\begin{matrix}26\\ 13\end{matrix})+2(\begin{matrix}26\\ 12\end{matrix})(\begin{matrix}26\\ 14\end{matrix})+...+2(\begin{matrix}26\\ 13-n/2\end{matrix})(\begin{matrix}26\\ 13+n/2\end{matrix})}{(\begin{matrix}52\\ 26\end{matrix})}
f(p^{-1}\|X_{i_1}-X_{i_2}\|_{2}^2)=f(\tau_1)+\sum^{l-1}_{s=1}\frac{f^{(s)}(\tau_1)}{s!}\widetilde{X}_{i_1,i_2}^s+c_{l,\tau_1}(\widetilde{X}_{i_1,i_2})\widetilde{X}_{i_1,i_2}^{l},
g(t) := 1 - \frac{b(q - 2)(6 - q)}{a(p - 2)(6 - p)}t - \frac{c(r - 2)(6 - r)}{a(p - 2)(6 - p)}t^{\frac{r - p}{q - p}}h(t) := 1 - \frac{b(q - 2)}{a(p - 2)}t - \frac{c(r - 2)}{a(p - 2)}t^{\frac{r - p}{q - p}}.
(\nu_1+i\nu_2) P=\overline{\Phi'(\mathcal P_1{\mbox{\bf a}},\mathcal P_2{\mbox{\bf b}})-\widehat{\tilde\Psi}}\quad\mbox{on}\,\widetilde \Gamma,(\nu_1-i\nu_2) \widetilde P=\overline{\bar\Phi'(\mathcal C_1\bar{\mbox{\bf a}},\mathcal C_2\bar{\mbox{\bf b}})-\widehat{\Psi}}\quad\mbox{on}\,\widetilde \Gamma
\P (G_0v_{2||E||^{1/3}_1}(d,2||E||_1^{1/3},0)) = \P (G_0v_{2||E||^{1/3}_1}(d,0,0)) + \mathcal{O}_{d}(||E||_1^{1/3}N^{-3/2}(1+ \upsilon - 2||E||^{1/3})^4) = \P (G_0v_{2||E||^{1/3}_1}(d,0,0))+ \mathcal{O}_{d,\upsilon}(||E||_1^{1/3}N^{-3/2}).
\left[ E^{a}_{i}- \cos \theta (D_{i}A_{4})^a - \sin \theta (D_{i}A_5 )^a\right]^2 + \left[ B^{a}_{i}+ \sin \theta (D_{i}A_{4})^a - \cos \theta (D_{i}A_5 )^a \right]^2 +
\int_{u^{-1}(\Sigma\times(0,e^{-4\nu}))}u^*d(m\epsilon\alpha_0) \,\,\,\leq\int_{u^{-1}(\Sigma\times(0,e^{-4\nu}))}u^*d(\epsilon\alpha_0) &\overset{}{=}e^{4\nu}\int_{u^{-1}(\Sigma\times(0,e^{-4\nu}))}u^* d\lambda
\begin{array}{l l}{\pi^{\kappa}( \lambda ) | 0 > = 0 ,}&{\pi^{\nu}( \lambda ) | 0 > = 0}\\{< 0 | q^{\kappa}( \lambda ) = \mathrm{l n}a ( \lambda ) < 0 | ,}&{< 0 | q^{\nu}( \lambda ) = \mathrm{l n}d ( \lambda ) < 0 |}\\ \end{array}< 0 | 0 > = 1
\tilde{F}^{-}(x) =\left(\begin{array}{cc}1 - 1 & - 1 2\end{array}\right),\; \tilde{F}^{+}(x) =\left(\begin{array}{cc}2 1 & 1 1\end{array}\right),\; \tilde{\Lambda}(x) =\left(\begin{array}{cc}1 0 & 0 \frac{x + i}{x - i}\end{array}\right).
\frac 12\frac{d}{dt}\int_{\mathbb{R}^2}(|V^{(\alpha,a)}|^2+|H^{(\alpha,a)}|^2)dx &+\frac 12\mu\int_{\mathbb{R}^2}|\nabla V^{(\alpha,a)}|^2 dx-\sum\limits_{l=0}^{\alpha-1}\mu C\int_{\mathbb{R}^2}|\nabla V^{(l,a)}|^2dx\lesssim \langle t\rangle^{-\frac 32}E_{\kappa-3}E_{\kappa-1}^{\frac 12}.
\frac{T}{4}\int_{\frac{1}{2}-\epsilon}^{\frac{1}{2}+\epsilon}\log \left(\frac{\sigma^2+T^2}{4}\right)\d\sigma=\frac{T}{4}\left[\sigma\log \left(\frac{\sigma^2+T^2}{4}\right)-2\sigma+2T\arctan\left(\frac{\sigma}{T}\right)\right]\Biggr|_{\frac{1}{2}-\epsilon}^{\frac{1}{2}+\epsilon}
\begin{array}{c}{{\displaystyle E ( L ) = - \displaystyle \frac{\pi \tilde{c}( l )}{6 L}= - \displaystyle \frac{\pi}{6 L}\left( c - 1 2 ( \Delta + \bar{\Delta}) \right) + \ldots}}\\{{P ( L ) = \displaystyle \frac{2 \pi}{L}( \Delta - \bar{\Delta}) + \ldots}}\\ \end{array}
{\frac{1}{\pi \Delta_{\alpha \beta}}}\left[ 1 -{\frac{1}{3}}( \pi \Delta_{\alpha \beta})^{2}\right] ={\frac{1 + ( 2 \pi \alpha^{\prime}B )^{2}q_{\alpha}q_{\beta}}{( 2 \pi \alpha^{\prime}B ) ( q_{\alpha}- q_{\beta})}},
|\left(e_{1}(t)-e_{2}(t)\right)\int_{0}^{\infty}d\xi \chi_{\leq \frac{1}{4}}(\xi) \frac{\sin(t\xi)}{t^{2}}F_{v_{2}}(\xi,\lambda_{1}(t))|\leq C |e_{1}(t)-e_{2}(t)| \frac{\lambda_{1}(t) |\log(\lambda_{1}(t))|}{t^{3}}
\varrho_{*}\Big ( \sum_{k = 1}^{n}f_{k}d^{\prime}f_{k}^{\prime}\Big ) \: = \: \sum_{k = 1}^{n}\sum_{\iota \in I}\big ( r_{U_{\iota}\cap U}^{U}( f_{k}^{\prime}) d^{\prime}r_{U_{\iota}\cap U}^{U}( f_{k}) \big ) \otimes 1 + 0 ,
h(1,\theta) = \sum_{k=a}^{\infty}\frac{\sin(k \theta)}{k}= \frac{\sin(a \theta)- \sin((a-1)\theta)}{a(2-2 \cos(\theta))}+O \left(\frac{1}{a^2 \sin^2(\theta/2)}\right)= \frac{\cos( (a-\frac{1}{2}) \theta)}{2 a\sin( \theta/2)}+ O \left(\frac{1}{a^2 \sin^2(\theta/2)}\right).
\mathcal{D}_{v_{p}}\Psi_{p}\left(v_{p},\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\right]\right)=\left[\mathcal{D}_{v_{p'}}\Psi_{p}\left(v_{p},\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\right]\right)^{\top},\mathcal{D}_{v_{p''}}\Psi_{p}\left(v_{p},\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\right]\right)^{\top}\right]^{\top}.
{\frac{d}{d \bar{t}}}\left( \bar{H}_{\mathrm{e l}}+ \bar{H}_{\mathrm{m a g}}\right) = -{\frac{d}{d \bar{t}}}\left( \bar{H}_{\mathrm{k i n}}+ \bar{H}_{\mathrm{p o t}}\right) = \sum_{x , i}\bar{E}_{x , i}^{a}\,{\bar{j}}_{x , i}^{a}
\begin{cases}\Delta(E)=2^8 \cdot 3^2 \cdot 5^2 \cdot 11^2 \cdot 13^2 \cdot 17^2 \cdot 19^2 \cdot 23^2 \cdot 29^2 \cdot 31^2 \cdot 37^3 \cdot p^\prime; j(E) = \frac{2^4 \cdot 7^3 \cdot 61^3 \cdot 347^3 \cdot (p'')^3}{3^2 \cdot 5^2 \cdot 11^2 \cdot 13^2 \cdot 17^2 \cdot 19^2 \cdot 23^2 \cdot 29^2 \cdot 31^2 \cdot p^\prime}, \end{cases}
\lim_{r_1, r_2\to \infty}\limsup_{N\to \infty}\P(|\tilde M_{N,r_1, r_2, \sigma, \delta}- \max_{B\in \mathcal B}\tilde\varphi_{N, r_1, r_2, \sigma, z_B}| \geq 1/\log\log N) = \limsup_{N\to \infty}\P( |M^*_{N, \sigma, \delta}- \max_{B\in \mathcal B}\varphi^*_{N, \sigma, z_B}| \geq 1/\log\log N) = 0\,.
\begin{aligned}u(x+h)+u(x-h)-2u(x) \leq \int_0^T e^{-s}L\left( \xi(s)+\left(1-\frac{s}{T}\right)h ,-\left(\dot{\xi}(s)-\frac{h}{T}\right) \right) &+ e^{-s}L\left( \xi(s)-\left(1-\frac{s}{T}\right)h ,-\left(\dot{\xi}(s)+\frac{h}{T}\right) \right) -2 e^{-s}L \left(\xi(s) ,-\dot{\xi}(s) \right) ds.\end{aligned}
T(X,Y) :=\nabla_XY-s\nabla_Y X-[X,Y] &=-s(\nabla_Y X - s\nabla_X Y + s[X,Y]) &=-s(\nabla_Y X - s\nabla_X Y + s(-s[Y,X]) &=-s(\nabla_Y X - s\nabla_X Y - [Y,X]) &=-sT(Y,X).
u_\delta(t_2,x) - u_\delta(t_1,x) = [u_\delta(t_2,x) - u_\delta(t_2,x_0)] + [u_\delta(t_2,x_0) - u_\delta(t_1,x_0)] + [u_\delta(t_1,x_0) - u_\delta(t_1,x)] & \geqslant -2K\mid x - x_0 \mid^\gamma + M (t_2 - t_1)^\mu & \geqslant \frac{M}{2}(t_2 - t_1)^\mu.
{\bf f}(x)=\left(\begin{array}{c}u_-(x) u_+(x)\end{array}\right),\;\;\hat{x}^\mu=x^\mu{\bf I}\;,\;\; \hat{\pi}_\mu=-\imath\partial_\mu{\bf I}\;,\;\; \hat{\psi}^a=\frac{\imath}{2}\gamma^a,
P(E)=\sum_{i}P(E|S_{i})P(S_{i})\le\sum_{i}P(X_{1}^{n}(i))(\frac{1}{M}\sum_{i^{\prime}}(\frac{P(X_{1}^{n}(i^{\prime}))}{P(X_{1}^{n}(i))})^{s})^{\rho}.
\int_{0}^{\infty}r^{2}dr\int_{0}^{\pi}sin\theta d\theta\int_{0}^{2\pi}d\varphi\psi_{nlm}^{*}(r,\theta,\varphi)\psi_{n^{\prime}l^{\prime}m^{\prime}}(r,\theta,\varphi)=\langle n,l,m|n^{\prime},l^{\prime},m^{\prime}\rangle=\delta_{nn^{\prime}}\delta_{ll^{\prime}}\delta_{mm^{\prime}},
g_a^{m}(y) =\frac{1}{\Sigma_n}\,\sum_{l=0}^\infty\frac{l+\lambda}{\lambda}\,(al)^m e^{-al}\,C_l^\lambda(\cos\theta) &=\frac{1}{\Sigma_n}\,\sum_{l=0}^\infty(al)^m e^{-al}\,\mathcal K_l^\lambda(\cos\theta).
\begin{aligned}\bar{R}_o(X,Y)Z = R(X,Y)Z-g(A(Y,Z),J)A(X,J)+g(A(X,Z),J)A(Y,J) +2g(X,\varphi Y)A(J,Z)+[\tilde{T}(X,Y),Z] = R(X,Y)Z-g(A(Y,Z),J)(\varphi X+\varphi hX) +g(A(X,Z),J)(\varphi Y+\varphi hY)+2g(X,\varphi Y)A(J,Z) +2g(X,\varphi Y)[J,Z], \end{aligned}
f_{\scriptscriptstyle{0,L}}(a)=\frac{1}{2 \pi}\int_0^a dr\int_0^{2 \pi}d\theta\, e^{i \theta}\,e^{2 \phi(r,\theta)}\,\sum_{l=0}^{\infty}b_{\scriptscriptstyle{l,L}}\,r^l \,e^{i l \theta}\,.
\{x,y\}={\bf k}((x,y,z),[\nabla x,\nabla y])={\bf k}(x{\bf e}_x+y{\bf e}_y+z{\bf e_z},[{\bf e}_x,{\bf e}_y]) &={\bf k}\left(x{\bf e}_x+y{\bf e}_y+z{\bf e_z},-\frac{1}{\sqrt{2}}{\bf e}_z\right)=\frac{1}{\sqrt{2}}z.
\int(\hat{f}_{g,\delta}\overset{(k-1)-}{*\dots*}\hat{f}_{g,\delta})(-\omega) \hat{\nu}_{g}(\omega)d\omega=\sum_{j_1,j_2,\dots,j_k}\int(\hat{f}_{g,\delta,j_1}\overset{(k-1)-}{*\dots*}\hat{f}_{g,\delta,j_{k-1}})(-\omega) \hat{\nu}_{g,j_k}(\omega)d\omega.
\frac{1}{2\pi i}\int_{-T_2}^{+T_2}\frac{e^{iEt}e^{-\alpha t^2}}{t-z}dt =\frac{1}{2\sqrt{\pi \alpha}}\int_{-\infty}^{+\infty}dw\; \exp[-(w-E)^2/(4\alpha)] \times\frac{1}{2\pi i}\int_{-T_2}^{+T_2}dt\; \frac{e^{iwt}}{t-z}
\mathbf{1}_{2^{s}}^{\mathrm{T}}[P_1, P_2, \cdots, P_{2^{n-s}}]=\mathbf{1}_{2^{n}}^{\mathrm{T}}W_{[2^{n-s},2^s]}=\mathbf{1}_{2^{n}}^{\mathrm{T}},
\{a_n \mathbf Z_n \cdot \mathbf j ; P(\cdot | \mathbf Z_n \neq \mathbf 0, \mathbf Z_0 = \mathbf i)\}\xrightarrow[n\to \infty]{\operatorname{law}}(\mathbf v\cdot \mathbf j) \mathbf z^{(\alpha)},
\nabla_{\theta}\mathbb{E}_{z\sim q_{\phi}(\cdot|x)}[ln\frac{p_{\theta}(x,z)}{q_{\phi}(z|x)}]=\mathbb{E}_{z\sim q_{\phi}(\cdot|x)}[\nabla_{\theta}ln\frac{p_{\theta}(x,z)}{q_{\phi}(z|x)}]
A_{11}=-(L_{21}-L_{22}L_{12}^{-1}L_{11})^{-1}L_{22}L_{12}^{-1}, A_{12}=(L_{21}-L_{22}L_{12}^{-1}L_{11})^{-1}, A_{41}=-L_{12}^{-1}-L_{12}^{-1}L_{11}(L_{21}-L_{22}L_{12}^{-1}L_{11})^{-1}L_{22}L_{12}^{-1}A_{42}=-L_{12}^{-1}L_{11}(L_{21}-L_{22}L_{12}^{-1}L_{11})^{-1}L_{22}L_{12}^{-1}.
\begin{aligned}C_{2,4}(r) =C_{3,4}^{1/3}(r)\cdot C_{2,3}^{2/3}(r), C_{1,3}(r) =C_{2,3}^{1/4}(r)\cdot C_{1,2}^{3/4}(r), C_{1,4}(r) =C_{3,4}^{1/9}(r)\cdot C_{2,3}^{2/9}(r)\cdot C_{1,2}^{2/3}(r). \end{aligned}
t^{-\frac12}\tau^\vee_i E_\mu(0,t) = t^{-\frac12}T_i^{-1}E_\mu(0,t),\quad\hbox{when $q=0$ and $\mu_i>\mu_{i+1}$, and}t^{\frac12}\tau^\vee_i E_\nu(\infty,t) = t^{-\frac12}T_i E_\nu(\infty,t),\quad\hbox{when $q=\infty$ and $\nu_i<\nu_{i+1}$.}
[n]_q:= 1+q+q^2+\dots+q^{n-1}=\frac{1-q^n}{1-q}, [n]_q!:= [1]_q[2]_q\dotsm[n]_q=\frac{(1-q)(1-q^2)\dotsm(1-q^n)}{(1-q)^n},[0]_q!:=1,{n\choose i}_q:= \frac{[n]_q!}{[i]_q![n-i]_q!}\,,i=0,1,\dots,n.
(A^{(V)})^{\hat{b}}_{\mbox{}\hat{c}}\equiv(\omega^{a}J^{(V)}_{a}+E^{a}P^{(V)}_{a})^{\hat{b}}_{\mbox{}\hat{c}}= \left( \begin{array}{cccc}0 \omega^{2}-\omega^{1}-E^{0}\omega^{2}0 -\omega^{0}-E^{1}-\omega^{1}\omega^{0}0 -E^{2}-E^{0}E^{1}E^{2}0 \end{array}\right).
\mathrm{P}_{t,i}=e^{-\lambda_c\pi \left( \frac{\epsilon_1}{\rho}+\frac{(1+\epsilon_1)}{\rho \phi_i}\right)^{-\frac{2}{\alpha}}}\sum^{t-1}_{k=0}\frac{(\lambda_c\pi)^{k}\left( \frac{\epsilon_1}{\rho}+\frac{(1+\epsilon_1)}{\rho \phi_i}\right)^{-\frac{2k}{\alpha}}}{k!},
\{G_Q(\eta), G_Q(\rho) \}^* = G_Q([\eta,\rho]) + \int_{\partial \Sigma}\omega \mbox{\tiny $\wedge\,$}\omega \mbox{\tiny $\wedge\,$}\dots \mbox{\tiny $\wedge\,$}\omega \mbox{\tiny $\wedge\,$}\eta_a d\rho^a,
\mathbf{b}=[\underbrace{0,\cdots,0}_{k_{A,j+1}},\underbrace{\tilde{\mathbf{\mathbf{b}}}^{T}}_{k_{A,j}-k_{A,j+1}},\underbrace{0,\cdots,0}_{k-k_{A,j}}]^{T}\in\mathbb{F}_{\gamma}^{k}.
<O_{k_{0}}O_{k_{1}}... O_{k_{p-1}}P>_{0}= - \mid \ln \mu \mid \left( \begin{array}{c}2\Sigma \Sigma \end{array}\right)\frac{1}{\pi \sqrt{2|t_{2}|}}\frac{1}{(-4t_{2})^{\Sigma}}\frac{\partial^{p-1}}{\partial t_{0}^{p-1}}t_{0}^{\Sigma},
1 \ge 1-g(\theta_1,x_0,t,c_t)\ge 1-g(\theta_1,x_0,t,c)=2-\Phi\left(\frac{e^{t^c}-x_0e^{\theta_1 t}}{\sqrt{v(\theta_1,t)}}\right)-\Phi\left(\frac{e^{t^c}+x_0e^{\theta_1 t}}{\sqrt{v(\theta_1,t)}}\right) &\to2-\Phi\left(-\frac{x_0\theta_1^H}{\sqrt{H\Gamma(2H)}}\right)-\Phi\left(\frac{x_0\theta_1^H}{\sqrt{H\Gamma(2H)}}\right)=1.
\pi_{TM}({\bf x},{\bf y})=\sum_{i,j=1}^{N}\pi^{ij}({\bf x})\frac{\partial}{\partial x^i}\wedge \frac{\partial}{\partial y^j}+\dfrac{1}{2}\sum_{i,j,s=1}^{N}\dfrac{\partial\pi^{ij}}{\partial x^s}({\bf x})y^s\frac{\partial}{\partial y^i}\wedge \frac{\partial}{\partial y^j}
\tau_{me}= \frac{1}{1 + b \chi + \chi^2}\left( \begin{array}{ccc}(b+ \chi )^2 -2 b + \chi - \frac{1}{2}\chi^2 \frac{\chi}{2}-2(b + \chi ) -4 \chi -1 + b \chi + \chi^2 \end{array}\right)
Df^{q_n}(x)=Df^{k_1}(x)\! \left[\prod_{j=1}^{m-1}Df^{k_{j+1}-k_j-1}\big(f^{k_j+1}(x)\big)Df\big(f^{k_j}(x)\big)\right] \times Df^{q_n-k_m-1}\big(f^{k_m+1}(x)\big)Df\big(f^{k_m}(x)\big).
[ B_{i}, B_{j}] = i \mu r \epsilon_{i j k}B_{k}, \hspace{1 2 m m}B_{1}^{2}+ B_{2}^{2}+ B_{3}^{2}= \mu^{2}r^{2}\frac{N^{2}- 1}{4}{\bf 1}_{N \times N}\textrm{f o r}( i , j , k = 1 , 2 , 3 )
q([nx],[ny])=\Biggl(\frac{\Gamma(1+1/c_1+\lambda+\mu)}{c_1\Gamma(\lambda+1)\Gamma(\mu)}\Biggr)\times \frac{\Gamma([nx]+\lambda+1)/\Gamma([nx]+1)\cdot \Gamma([ny]+\mu)/\Gamma([ny]+1)}{\Gamma([nx]+[ny]+1/c_1+\lambda+\mu+2)/\Gamma([nx]+[ny]+1)}.
V_{G}^{( b )}( \Phi_{0}) = V_{G E P}^{2 + 1}( \Phi_{0}) + \frac{3{\hat{\lambda}}^{2}}{4 \pi^{3}}( \sqrt{x}- 1 ) ( 1 - \ln{\frac{4}{3}}) -{\frac{3}{4 \pi^{3}}}{\hat{\lambda}}^{2}\sqrt{x}\ln \sqrt{x}.
e^{iH_a t}\{i[H_a,J_a]+\frac{\partial J_a}{\partial t}\}U(t,0)\psi(0)-e^{iH_a t}iJ_a(H(t)-H_a) U(t,0)\psi(0)= &e^{iH_a t}D_{H_a}(J_a) U(t,0)\psi(0)+(-i)e^{iH_a t}J_a N(x,t,|\psi|) U(t,0)\psi(0). &H(t)=-\Delta+N(x,t,|\psi|), &H_a\equiv -\Delta, .
{\cal L}^{pzm}_{eff}= \frac{1}{2L}\pi \partial_{+}a_{-}-{\cal H}_D^{pzm}= \frac{1}{2L}\pi \partial_{+}a_{-}+\frac{1}{2L}\pi \frac{1}{\Delta_{\perp}}\ast \partial_i J_{pzm}^i +a_{-}J_{pzm}^{-}- \frac 1 2 J_{pzm}^i\frac{1}{\Delta_{\perp}}\ast J_{pzm}^i \;,
0 \leq \langle M', X' \rangle = \langle M,X\rangle -y_{\omega}X_{\omega}&= \langle F_{\emptyset},X\rangle - \sum_{\omega \in \Omega_k^d}y_{\omega}b_{\omega}+ \sum_{\omega \in \Omega_k^d}y_{\omega}\epsilon_{\omega}-X_{\omega}y_{\omega},
cos(j\theta)=2^{j-1}cos^{j}\theta-\frac{j}{1}2^{j-3}cos^{j-2}\theta+\frac{j}{2}(\begin{matrix}j-3\\ 1\end{matrix})2^{j-5}cos^{j-4}\theta-\frac{j}{3}(\begin{matrix}j-4\\ 2\end{matrix})2^{j-7}cos^{j-6}\theta+\cdot\cdot\cdot
F_{\lambda}^{(0)}[W] := \dfrac{1}{m}\left( \int_{\partial B_0}\partial_n v^{(0)}_{\lambda}[W]{\rm d}\sigma \right), \tau_{\lambda}^{(0)}[W] := \mathcal{J}^{-1}\left( \int_{\partial B_0}z^{\bot}\cdot \partial_n v^{(0)}_{\lambda}[W]{\rm d}\sigma \right),
\varphi^1\varphi^1_\omega-\varphi^1_{\omega\omega}-\frac{(\varphi^2)^2}{\omega}-\frac1{\omega^3}+\omega=0, \varphi^1\varphi^2_\omega-\varphi^2_{\omega\omega}+\frac{\varphi^1\varphi^2}{\omega}+2\frac{\varphi^2}{\omega^2}=0.
S^{(\mathrm{soft})}(\kappa;u,v) =\frac{1}{u-v}\Big( -2 \phi_{1}''(\kappa;u) \phi_{2}(\kappa;v)-2\phi_{1}(\kappa;u) \phi_{2}''(\kappa;v)+2\phi_{1}'(\kappa;u) \phi_{2}'(\kappa;v) +2\kappa \big( \phi_{1}'(\kappa;u) \phi_{2}(\kappa;v)-\phi_{1}(\kappa;u) \phi_{2}'(\kappa;v)\big) +(u+v) \phi_{1}(\kappa;u) \phi_{2}(\kappa;v)\Big).
\mathcal{B}_{1}=\underset{i\neq j}{\bigcup_{1\leq i,j\leq n}}\mathcal{B}_{i,j},\textrm{where}\mathcal{B}_{i,j}=\left\{\omega\in\mathbb{R}^{n-1}:\left|\omega_{i}-\omega_{i-1}\right|>T_{n}^{-1},\left|\omega_{j}-\omega_{j-1}\right|>T_{n}^{-1}\right\}.
\omega^n\leq 2\Psi_n+2\Gamma(2-\alpha)\Big(\lambda_1\tau^{\alpha}\sum_{j=1}^{n-1}p_{n-j}\omega^j+\lambda_2\tau^{\alpha}\sum_{j=1}^{n}p_{n-j}\omega^{j-1}\Big)~~,n\geq1,
S = a_{0}\int d^{D}x \, \sqrt{| G |}e^{- 2 \Phi}\, \left[ \Lambda - \frac{\alpha^{\prime}}{2}\left( \hat{R}+ 4 D^{2}\Phi - 4 ( D \Phi )^{2}\right) + \frac{\rho}{2}e^{2 \Phi}\right] . \,
\mathbf{\Sigma}^3(x_1)=\frac{(-x_1)^{-\frac{3}{2}}}{\sqrt{2\pi}}\begin{pmatrix}1 0\end{pmatrix}, \mathbf{\Sigma}^4(x_1)=\frac{(-x_1)^{-\frac{3}{2}}}{\sqrt{2\pi}}\begin{pmatrix}0 1\end{pmatrix}.
4\pi\epsilon_{0}V(r)=\frac{q}{|r_{1}|}+\frac{(-qR/p)}{|r_{2}|}=\frac{q}{\sqrt{r^{2}+p^{2}-2r\cdot p}}+\frac{(-qR/p)}{\sqrt{r^{2}+\frac{R^{4}}{p^{2}}-\frac{2R^{2}}{p^{2}}r\cdot p}}
\vert f_{s}(w)-f_{s-1}(w)\vert = x\vert w \vert \vert w-1\vert \vert w+1\vert^s &\leq \vert w^{r+1}-x(w-1)(w+1)^{s-1}-xw(w-1)(w+1)^{s-1}\vert \quad\quad+\vert w^{r+1}-x(w-1)(w+1)^{s-1}\vert &=\vert f_s(w)\vert +\vert f_{s-1}(w)\vert.
f(v) =\sum_{i=-d}^{d}(\left\vert v-i\right\vert +1)+\sum_{i=-(k-1)}^{k-1}(\left\vert v-i\right\vert +2)+ & +x(\left\vert v+(d-1)\right\vert +2)+x(\left\vert v-(d-1)\right\vert +2)+ & +r(\left\vert v+d\right\vert +2)+r(\left\vert v-d\right\vert +2).
\oint_{C}\frac{d \zeta}{\zeta^{j}}\hat{f}= \sum_{\alpha}\oint_{\Gamma_{\alpha}}\frac{d \zeta}{\zeta^{j}}\ 2 \log \left( \eta ( \zeta ) - P_{\alpha}( \zeta ) \right) + 4 \mu \oint_{0}\frac{d \zeta}{\zeta^{j + 1}}\eta .
\Xi^{2 p}= \left[ \oplus_{j = 0 , ( e v e n )}^{2 p}\Lambda^{j}( M ) \right] \oplus \left[ \oplus_{j = 0 , ( e v e n )}^{2 p}\Lambda^{j}( M ) \right] \oplus \left[ \oplus_{j = 1 , ( o d d )}^{2 p - 1}\Lambda^{j}( M ) \right] \oplus \left[ \oplus_{j = 1 , ( o d d )}^{2 p - 1}\Lambda^{j}( M ) \right]
\cos(\omega_{N-1}-\omega_{N-1}'+\xi)\sin^2(\psi_{N-1})+\cos((N-1)(\omega_{N-1}-\omega_{N-1}'))\cos^2(\psi_{N-1}) =1, \sin(\omega_{N-1}-\omega_{N-1}'+\xi)\sin^2(\psi_{N-1})-\sin((N-1)(\omega_{N-1}-\omega_{N-1}'))\cos^2(\psi_{N-1}) =0.
\int_{0}^{\infty}d k \sqrt{k^{2}+ 4}\left( \frac{4 \cos \rho}{k^{2}+ 4 \cos^{2}\rho}- \frac{4 \cos \rho}{k^{2}+ 4}+ \frac{4 c_{+}}{k^{2}+ 4 c_{+}^{2}}- \frac{4 c_{+}}{k^{2}+ 4}\right) ,