harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\Pr_{c\sim\mathcal{D}(n,d)}[|c|\leq \frac{1-2^{-j}}{2}N] \leq 2^{-(1-h(\frac{1-2^{-j}}{2}))\cdot \binom{n}{\leq d}}\leq 2^{-\frac{2^{-2j}}{2\ln2}\cdot \binom{n}{\leq d}}.
\begin{aligned}\left(x \frac{d}{dx}\right)^4 \log \left( \theta_3(x) \right) = x \frac{d}{dx}\left(x \, \psi(x) + 3 x^2 \, \psi'(x) + x^3 \, \psi''(x)\right) & = x \, \psi(x) + 7 x^2 \, \psi'(x) + 6 x^3 \, \psi''(x) + x^4 \, \psi'''(x).\end{aligned}
\|f\|_{\mathrm{H}_{\mathcal{H}_{\hbar,V}}^{s}}:=\left\|(I+\mathcal{H}_{\hbar,V})^{s / 2}f\right\|_{\ell^{2}(\hbar\mathbb{Z}^{n})}=\left(\sum\limits_{\xi\in\mathcal{I}_{\hbar}}\langle\xi\rangle^{2s}\widehat{f}(\xi)|^{2}\right)^{\frac{1}{2}}=\left(\sum\limits_{\xi\in\mathcal{I}_{\hbar}}\left(1+\lambda_{\xi}\right)^{s}\widehat{f}(\xi)|^{2}\right)^{\frac{1}{2}}.
[y_1, y_2]=y_4+\theta_1y_5, [y_2, y_1]=\frac{\alpha_5}{\alpha_3}y_4+\theta_2y_5, [y_2, y_2]=\beta_2y_5, [y_1, y_3]=\frac{\beta_4\gamma_1}{\alpha_3}y_5, [y_2, y_3]=y_4, [y_3, y_2]=\gamma_4y_5.
(-)^{{\bar\alpha}+{\bar\beta}+\bar\alpha\bar\beta}\frac{{\bar\vartheta}[^{\bar\alpha}_{\bar\beta}]}{{\bar\eta}}\rightarrow\frac{{\bar\vartheta}[^{\bar\alpha}_{\bar\beta}]^{5}}{{\bar\eta}^5}{1\over2}\sum_{\gamma,\delta}\frac{{\bar\vartheta[^{\gamma}_{\delta}]}^{8}}{{\bar\eta}^8}
= - \delta_{\epsilon , \bar{\epsilon}}\langle X \rangle = \frac{1}{2 \pi i}\oint_{C}d z \epsilon ( z ) \langle T ( z ) X \rangle - \frac{1}{2 \pi i}\oint_{C}d \bar{z}\bar{\epsilon}( \bar{z}) \langle \bar{T}( \bar{z}) X \rangle
A_{E,m, \xi}(t) = \frac{\kappa_{n-1}}{\kappa_m}\int_{G(\xi^\perp, m)}V_m ((E-t\xi) \cap H ) \, dH &=\frac{\kappa_{n-1}}{\kappa_m}\int_{G(\xi^\perp, m)}V_m (E\cap (H\vee\xi) \cap \{\xi^\perp +t\xi\}) \, dH,
[\begin{matrix}U_{n}\\U_{n+1}\end{matrix}]=[\begin{matrix}0&1\\ -Q&P\end{matrix}]\cdot[\begin{matrix}U_{n-1}\\U_{n}\end{matrix}]=[\begin{matrix}0&1\\ -Q&P\end{matrix}]^{n}\cdot[\begin{matrix}U_{0}\\U_{1}\end{matrix}]
\left|R_{n,s}(f)\right|\leq 2\pi\left(\max_{z\in\mathcal{E}_{\rho}}|f(z)|\right)\dfrac{\sum_{k=0}^s(-1)^{k}{2s+1 \choose s-k}\rho^{2n(s-k)}}{\rho^n(\rho^{2n}-1)^{2s}}\,.
\langle w_1(\det(D_\bullet\# 0)), v \rangle_{C^{\infty}_{X, L}}= \langle w_2(TL), v_*[S^1 \times \partial D] \rangle_L + I_{\mu_L}(A)\langle w_1(TL), v_*[S^1 \times \{1\}]\rangle_L.
[y_1, y_2]=y_3, [y_2, y_1]=-y_3+y_4, [y_2, y_2]=\beta_4y_5, [y_3, y_1]=\alpha_3\gamma_2y_5, [y_2, y_3]=\alpha_3\gamma_3y_5=-[y_3, y_2], [y_1, y_4]=(\alpha_4+\beta_1)\gamma_6y_5.
\partial_t R_\tau^{k+1}u|_{I_n}(t_{n,\mu}) = \partial_t I_\tau^{k+2}u(t_{n,\mu}) \,, \mu=0,\ldots, k\,, R_\tau^{k+1}u|_{I_n}(t_{n-1}) = I_\tau^{k+2}u(t_{n-1})\,.
\|y\partial_x y\|_{L^1(0,T;L^2(I))}\leq K\int_0^T\|y\|_{H^2(I)}\|\partial_x y\|dt\leq K\int_0^T\|y\|_{H^2(I)}^2dt=K\|y\|_{L^2(0,T;H^2(I))}^2,
\sum_{n \le N}R_{\textit{HL}}(n) \frac{(1 - n/N)^k}{\Gamma(k + 1)}= \frac{\pi^{1 / 2}}2 \frac{N^{3 / 2}}{\Gamma(k + 5 / 2)}- \frac 12 \frac{N}{\Gamma(k + 2)}- \frac{\pi^{1 / 2}}2 \sum_{\rho}\frac{\Gamma(\rho)}{\Gamma(k + 3 / 2 + \rho)}N^{1 / 2 + \rho}
\int\limits_0^\infty e^{-st}dK^{(f)}_{M,N}(t\,|\,b) =\int\limits_0^\infty e^{-st}e^{-b_0t}\prod\limits_{j=1}^N(1-e^{-b_j t}) f(t)\,dt/t^M<\infty, \;s>0.
\| u - P_{k}[u] \|_{L^{2}(\mathcal{D}\times\omega)}^{2}= \left(\frac{1}{n}\sum_{\mu \in \Xi}\| u(\mu,\cdot) - \sum_{l=1}^{k}\int_{{\omega}}u(\mu,{z}) \kappa_{l}({z}) \,d{z}\,\kappa_{l}\|_{L^{2}({\omega})}^{2}\right) + \delta \leq \left(\underset{l = k + 1}{\overset{d(n)}{\sum}}\lambda_{l}^{n}\right) + \delta,
\frac1\tau(dp_j^{n+1}\wedge dq_j^{n+1}-dp_j^{n}\wedge dq_j^{n})-\frac1h(dp_j^{n+\frac12}\wedge dv_{j+1}^{n+\frac12}-dp_{j-1}^{n+\frac12}\wedge dv_j^{n+\frac12}) &-\frac1h(dq_j^{n+\frac12}\wedge dw_{j+1}^{n+\frac12}-dq_{j-1}^{n+\frac12}\wedge dw_j^{n+\frac12})=0,
c(\rho) = \left(2\rho \int_0^T \tilde{g}(s) g(s) ds+ \rho^2 \int_0^T g(s) \frac{d}{d\rho}\tilde{g}(s) ds + \rho^2 \int_0^T \tilde{g}(s) \frac{d}{d\rho}g(s) ds \right. & \left. + \int_0^T \tilde{g}dB(s)+\rho \int_0^T \frac{d}{d\rho}\tilde{g}(s) dB(s)-\rho \int_0^T \tilde{g}^2(s) ds -\rho^2 \int_0^T \tilde{g}(s) \frac{d}{d\rho}\tilde{g}(s) \right) .
\sum_{p = 0}^{M - k}( - 1 )^{p}\left( \begin{matrix}{{M - k}}\\{{p}}\\ \end{matrix}\right) \biggl ( \frac{1}{2}( \Delta - M ) + p \biggr )_{n}\, \biggl ( [ \frac{1}{2}( \Delta - M ) + k ] + [ ( M - k ) - p ] \biggr )_{n + m}\, .
d\big(y ,P^{y,y}_\gamma\big((nu^{-1}n^{-1})\cdot nx'\big)\big) \leq C d(nu^{-1}n^{-1},e)^\kappa\leq Ce^{C_1d(n,e)\varrho}d(u^{-1},e)^\kappa =Ce^{C_1d(y,x)}d(x,x')^\kappa;
\partial_1 f= \nabla_1 \tilde{h}_{11}= 2h^S_{22}H +(h^S_{\nu \nu}-3 h^S_{22}) h_{11}+ \nabla_\nu^S h^S_{22}\geq (h^S_{\nu \nu}+ h^S_{22}) h_{11}+\nabla_\nu^S h^S_{22}= \frac{1}{2}H^S \tilde{D}+\nabla_\nu^S h^S_{22}\geq 0
e^{-t\Box^+_{b,m}}(x,x)\sim\Bigr(\sum\limits^{p}_{s=1}e^{\frac{2\pi(s-1)}{p}mi}\Bigr)\sum^\infty_{j=0}t^{-n+j}\alpha^+_{n-j}(x)\mod O\Bigr(t^{-n}e^{-\frac{\varepsilon_0\hat d(x,X_{\mathrm{sing\,}})^2}{t}}\Bigr)
0= \{12- 2(\cot (2\vartheta) )^{2}- (\tan \vartheta )^{2}\}((\cot (2\vartheta) )(\cot \vartheta -1)^{2}-9 (\cot (2\vartheta) +\cot \vartheta )^{2}= -\frac{1}{8}(\tan \vartheta)^{2}\{(\cot \vartheta)^{8}-32(\cot \vartheta)^{6}+330(\cot \vartheta)^{4}-360(\cot \vartheta)^{2}+45\}.
\nu(\mathcal{E}_k) = \nu(A\mathcal{E}_{k_0}A^*) \leq ||A||\nu(\mathcal{E}_{k_0})||A^*|| \leq ||A||^2 \nu(\mathcal{E}_{k_0}) \leq ||\sigma||_p^{2-p}||\sigma||_p^p = ||\sigma||_p^2
\big\|\mathcal{R}_\varepsilon c\big\|_{L^2(\Omega^\prime)}^2 \leq K\int_{\Omega^\prime}\left|\int_{\Omega^c}J_\varepsilon(|x-y|)c(x)\:y\right|^2\:x + K\int_{\Omega^\prime}\left|\int_{\Omega^c}J_\varepsilon(|x-y|)\tilde{c}(y)\:y\right|^2\:x &=: K \big(I_\varepsilon^1 + I_\varepsilon^2\big).
\varphi^\sharp\left(\begin{array}{c c}a b 0 d \end{array}\right) =\left(\begin{array}{c c c}a^{2 \, p^{e_1}}a^{p^{e_1}}\, b^{p^{e_1}}\frac{1}{2}\, b^{2 \, p^{e_1}}0 1 b^{p^{e_1}}\, d^{p^{e_1}}0 0 d^{2 \, p^{e_1}}\end{array}\right) (\, e_1 \geq 0 \, ) .
\sum_{1 \leq h_1,\dots,h_k \leq H}\left|\sum_{1 \leq n \leq X}\prod_{j=1}^k g_j(a_j n + b_j + h_j)\right| \ll A^2 k\left( \exp( - M / 80) + \frac{\log\log H}{\log H}+ \frac{1}{\log^{1/3000}X}\right) H^k X
L(\lambda) = \sum\sb{i=1}\sp N \left( p\sb i + \sum\sb{\alpha =1}\sp M{f\sb{ii}\sp\alpha\over\lambda-\epsilon\sb\alpha}\right) e\sb{ii}+\sum\sb{\stackrel{i,j =1}{i\neq j}}\sp N \left({f\sb{ij}\over q\sb i-q\sb j}+\sum\sb{\alpha =1}\sp M{f\sb{ij}\sp\alpha\over\lambda-\epsilon\sb\alpha}\right) e\sb{ij},
ds^2~=~\alpha^{\prime}\left (\frac{u^2h(u)}{R^2}~e^{\gamma A}dx_0^2+\frac{u^2}{R^2}~e^{\gamma C}dx_i^2+\frac{R^2}{u^2h(u)}~e^{\gamma B}du^2+R^2~e^{\gamma D}d\Omega_5^2\right )~,
G_{l+\nu}(r'',r';E) =\sum_{n=0}^{\infty}\,\prod_{\alpha=1}^{n}\left[ \int_{0}^{\infty}d r_{\alpha}V(r_{\alpha}) \right]\,\prod_{\beta=0}^{n}\left[ G^{(0)}_{l+\nu}(r_{\beta +1},r_{\beta};E) \right]\; .
\begin{aligned}\frac{d}{dt}\mathcal{E}^m_l(t)+C_5 \left(\|\nabla^{l+1}\varrho\|_{H^{m-l-1}}^2+\|\nabla^{l+1}u\|_{H^{m-l}}^2 +\|\nabla^{l+1}\nabla n\|_{H^{m-l}}^2\right)\le 0,\end{aligned}
\theta(T) = \underset{k\to+\infty}{\lim}\, \Big(\inf \big\{\big\| T^k - Q \big\|_{\mathcal{B}(E)},\Q \ \mathrm{compact}\big\}\Big)^{\frac{1}{k}}= \underset{k \geq 1}{\inf}\, \Big(\inf \big\{\big\| T^k - Q \big\|_{\mathcal{B}(E)},\Q \ \mathrm{compact}\big\}\Big)^{\frac{1}{k}}.
\frac{d}{dt}g(\rho+t\sigma_1)= -g(\rho+t\sigma_1)\frac{d}{dt}g(\rho+t\sigma_1)^{\dagger}g(\rho+t\sigma_1) = -g(\rho+t\sigma_1)\frac{d}{dt}(-\Delta_{\rho+t\sigma_1})g(\rho+t\sigma_1) = g(\rho+t\sigma_1)\Delta_{\sigma_1}g(\rho+t\sigma_1),
T_u(\mathcal{S}) = T_u(\mathcal{S}_p) + T_u(\mathcal{S}_s) &= \overline{W}_u + D_u(\mathcal{S}_p) + T_u(\mathcal{S}_s) &= \overline{W}_u + D_u(\mathcal{S}_p) + \sum_{t \in \mathcal{S}_s}M - W_u^1(t) +1.
\varphi_{0,1}(A(b,z)) =\sum_{0\leq i<g}\left(\int^z_b \omega_i \right)\kappa_i , \varphi_{1,2}(A(b,z)) =\sum_{0\leq j<g}\left(d_j +\sum_{0\leq k<g}a(Z)_{jk}\int^z_b \omega_k \right)\kappa_j^* (1).
[\begin{matrix}\dot{x}_{1}(t)\\ \dot{x}_{2}(t)\end{matrix}]=[\begin{matrix}0&1\\ -\frac{k}{m}&-\frac{b}{m}\end{matrix}][\begin{matrix}x_{1}(t)\\x_{2}(t)\end{matrix}]+[\begin{matrix}0\\ \frac{1}{m}\end{matrix}]u(t)
\inf\bigg\{k>0:~(L)\int_{\mathcal{J}}\theta\bigg(t, (\xi_n-\xi)(t)\bigg)d \overline{\mu}\leq 1\bigg\}\to 0 \implies ~(L)\int_{\mathcal{J}}\theta\big(t, (\xi_n-\xi)(t)\big) \to 0 & ~i.e.,~\theta(t, (\xi_n-\xi)(t)=0~\overline{\mu}-a.e.~as~n \to \infty &\xi_n-\xi=0~a.e.~as~n \to \infty
\left[\Phi^*(e^{\lambda\mathtt{P}})^*e^{\lambda\mathtt{P}}\Phi\right]^{-1}=\mathbf{1}-\Phi^*\big[\lambda\mathtt{U}(\mathtt{P})+\lambda^2\mathtt{V}(Q,\mathtt{P})\big]\Phi+\lambda^3\,\mathtt{R}^{(\lambda)}(\Phi,\mathtt{P})\,,
U=\left(\begin{array}{cc}\textrm{sign}(\lambda_1) 0 0 \textrm{sign}(\lambda_2)\end{array}\right),\,\,\,\,\, v=\left(\begin{array}{cc}a b c d\end{array}\right):=\left(\begin{array}{cc}\eta_1 0 0 \eta_2\end{array}\right) B.
\int\limits_{\pi/2}^{\pi-\psi_0}Cf(0,e_n,\psi)g(\psi_0,\psi)d\psi &=\frac{(\cos\psi_0)^{n-3}}{|S^{n-3}|}\int\limits_{S^{n-2}}\int\limits_{-\infty}^0 \int\limits_{\mathbb{R}^{n-1}}f(\bar{x},z) \delta(\bar{x}\cdot (\cos\psi_0)\omega+z\sin\psi_0)d\bar{x}dz d\omega.
S_{2}^{( 0 )}={\frac{1}{2}}x^{i}\Lambda_{i j}x^{j}+ x_{i a}^{*}D_{1 \alpha}^{i}c^{\alpha a}+ y_{\mu a}^{*}D_{2 \alpha}^{\mu}c^{\alpha a}+ \bar{x}_{i}D_{3 \alpha}^{i}B^{\alpha}+ \bar{y}_{\mu}D_{4 \alpha}^{\mu}B^{\alpha}- \varepsilon^{a b}c_{\alpha a | b}^{*}D_{5 \beta}^{\alpha}B^{\beta}.
\pi=\sqrt{12}\sum_{k=0}^{\infty}\frac{(-3)^{-k}}{2k+1}=\sqrt{12}\sum_{k=0}^{\infty}\frac{(-\frac{1}{3})^{k}}{2k+1}=\sqrt{12}(\frac{1}{1\cdot3^{0}}-\frac{1}{3\cdot3^{1}}+\frac{1}{5\cdot3^{2}}-\frac{1}{7\cdot3^{3}}+\cdot\cdot\cdot)
w_{n+m}^{\left( \alpha\right)}\left( x\right) =\sum_{k=0}^{m}\sum_{j=0}^{n}\genfrac{\{}{\}}{0pt}{}{m}{k}\binom{n}{j}\binom{\alpha+k-1}{k}k!k^{n-j}x^{k}w_{j}^{\left( \alpha+k\right)}\left( x\right) ,
\begin{aligned}Q_{i+1,j+1}^t=(\min[Q_{i+1,j}^t,W_{i+1,j}^t]-\min[Q_{i,j}^t,W_{i,j}^t])+Q_{i,j}^t, &W_{i,j}^{t+1}=(\min[Q_{i+1,j}^t,W_{i+1,j}^t]-\min[Q_{i,j}^t,W_{i,j}^t])+W_{i,j}^t.\end{aligned}
\delta \Psi_{\hat{a}}= \partial_{\hat{a}}\epsilon +{\frac{1}{8}}H^{- 1}( \partial_{\hat{a}}H ) \epsilon +{\frac{1}{8}}W \gamma^{\underline{{{u x y}}}}\gamma_{\underline{{{\hat{a}}}}}( i \sigma_{2}) \epsilon = 0 \,
\mathcal{H}=-\frac{d^2}{dx^2}+U(x), \mathcal{H}\psi(x) =\mathcal{E}\psi(x)\ \bigl(\mathcal{E},U(x)\in\mathbb{C}\bigr), \mathcal{H}\varphi_j(x) =\tilde{\mathcal{E}}_j\varphi_j(x) (\tilde{\mathcal{E}}_j\in\mathbb{C}\ ;\j=1,2,\ldots,M),
\mathcal G^{\leq N}_{\perp \psi_t}= \Big( \bigoplus_{k=0}^N \mathcal F_{b,\psi_t}^{(k)}\Big) \otimes \mathcal F_a = \bigoplus_{k=0}^N \Big( \mathcal F_{b,\psi_t}^{(k)}\otimes \mathcal F_a \Big) \; \subset \;\mathcal G_{\perp \psi_t}= \mathcal{F}_{b,\psi_t}\otimes \mathcal{F}_{a}\; \subset \;\mathcal{G}= \mathcal{F}_b \otimes \mathcal{F}_a
l_{\varphi^{*}}(\underline{a},k):=\min\{l\in\mathbb{N}:[F\in\widehat{\mathcal{Q}}_{b},\forall_{i=1\dots s}\,\,\,\nu_{M,a_{i}}(\widehat{\varphi}^{*}_{a_{i}}(F))>l]\Rightarrow F\in\bigcap_{i=1}^{s}\mbox{Ker}\,\widehat{\varphi}_{a_{i}}^{*}+\widehat{m}_{b}^{k+1}\}.
\Sigma ( p ) = - J + \frac{\alpha}{2 \pi^{2}}\int \frac{d^{2}k \Sigma ( k )}{k^{2}+ \Sigma^{2}( k )}\int_{0}^{\infty}\frac{d q \exp \left( - q l^{2}/ 2 \right)}{( k - p )^{2}+ q}.
I_i=\frac{h_1^i+h_2^i}{\alpha^{\prime}\pi}\left[ \sum\limits_{n=1}^\infty\left( \overline{b}_n^ib_n^i-\overline{a}_n^ia_n^i\right) -\frac{\left|\epsilon^i\right|}{\epsilon^i}\overline{a}_0^ia_0^i\right] ,
(2{\rm Re}z)^3 |Sh(z)|^2 \leq 2 (2{\rm Re}z)^3 |(Ph(z))'|^2 + 1/2 \left((2{\rm Re}z) |Ph(z)|\right)^2 (2{\rm Re}z) |Ph(z)|^2 & \leq 2 (2{\rm Re}z)^3 |(Ph(z))'|^2 + 18 (2{\rm Re}z) |Ph(z)|^2.
f_\delta'(0) =\int_{\partial E}\frac{h(x)-h(p)}{|x - p|^{N+s}}\eta_\delta(x-p)\, d x+h(p)\int_{\partial E}\frac{1-\langle \nu(x) ,\nu(p) \rangle}{|x-p|^{N+s}}\eta_\delta(x-p) \, d x.
\sum_{ij\in E^\prime}|g_i^p-g_j^p| =\sum_{ij\in E^\prime}|g_i-g_j|\frac{g_i^p-g_j^p}{g_i-g_j}&\le \left(\sum_{ij\in E^\prime}|g_i-g_j|^p\right )^{1/p}\left(\sum_{ij\in E^\prime}\left(\frac{g_i^p-g_j^p}{g_i-g_j}\right)^q\right )^{1/q}
Q_{0}= - \sum_{\kappa = - \infty}^{\infty}\vert \kappa \vert \left\{N_{\kappa}^{+}- N_{\kappa}^{--}n_{l}^{+}+ n_{\overline{l}}^{-}+ \frac{2}{\pi}\int_{0}^{\infty}\mathrm{d}r \varphi \left( r \right) \right\},
\begin{gathered}\Psi = r^{\lambda+1}P^{-m}_{\lambda+1}(\cos\theta) \cos(m\varphi), \Theta = r^{\lambda+1}P^{-m}_{\lambda+1}(\cos\theta) \sin(m\varphi), \mbox{and}\Lambda = r^{\lambda}P^{-m}_{\lambda}(\cos\theta) \sin(m\varphi),\end{gathered}
\mathcal{L}_{^{m,2}_{inter}}(s) = \mathcal{E}\left\{\prod_{x_j\in \Phi_c\backslash x_m}{\rm exp}\left( -s \frac{|h_{j,m2}|^2}{{L\left(||y_{m,2}+x_m-x_j||\right)}}\right)\right\}.
\sum\limits_{\mathsf{w}=1}^{\mathsf{q}}\gamma_{\mathsf{w}}\;=\;\lim\limits_{N\rightarrow\infty}\frac{1}{N}\,\mathbb{E}\,\log\,\left\|\Lambda^{\mathsf{q}}(\mathcal{T}_N\cdots\mathcal{T}_1)\right\|_{\Lambda^{\mathsf{q}}\mathbb{C}^{\mathsf{L}}}\,,\qquad\mathsf{q}=1,\dots,\mathsf{L}\;.
\widetilde{V}^{( 0^{2})}(x;N,M,\lambda_{1}) =V( x;N,M) +2\frac{(1-z)^{2N+1}(1+z)^{2M+1}}{\big( \lambda_{1}+Q_{0}^{(N,M)}(z) \big)^{2}}\hphantom{\widetilde{V}^{( 0^{2})}(x;N,M,\lambda_{1}) =}{}+4(1-z)^{N}(1+z)^{M}\frac{M-N-(N+M+1)z}{\lambda_{1}+Q_{0}^{(N,M)}(z)}.
\kappa=\sum_{i<j}(\nu_i-\nu_j)+\sum_{i\neq j}(\nu_i-2\nu_j) \hphantom{\kappa}{}= \sum_{m=1}^n(n-2m+1)\nu_m-(n-1)\sum_{m=1}^n\nu_m=2\sum_{m=1}^n(1-m)\nu_m,
S = - i \left[ \int d^{2}\sigma d \theta_{0}^{+}d \theta_{0}^{-}g_{i j}D_{0 +}\varphi^{i}D_{0 -}\varphi^{j}+ \int d^{2}\sigma d t d \theta_{0}^{+}d \theta_{0}^{-}H_{i j k}\partial_{t}D_{0 +}\varphi^{j}D_{0 -}\varphi^{k}\right] ,
M \le{}C (\sup_{\Omega}d)^{\frac{1}{n}(1-2q\theta +q)}M^{1-\frac{q}{n}}\left( \int_{\Omega}(f^+)^q \right)^{1/n}\le C' M^{1- \frac{(q\theta-1)}{n(1-\theta)}}.
\mu_{1 , k_{c}}^{( a )}\, \longrightarrow \,{\frac{\tilde{g}^{2}T^{2}}{2 4}}-{\frac{\tilde{g}^{2}k_{c}T}{4 \pi}}\qquad \mathrm{f o r ~ \tilde{~}m = 0 ~}\; .
X ( p_{0}) = \frac{i \Gamma}{2 \pi}\, \left[ \left( p_{0}^{2}- \frac{3 m^{2}}{2}+ \frac{m^{4}}{2 p_{0}^{2}}\right) \, \log \left| \frac{p_{0}^{2}- m^{2}}{m^{2}}\right| + \frac{m^{2}}{2}- \frac{5 p_{0}^{2}}{4}\right] \, .
\vec{\phi}_{\mathrm{T K 2 L}}( x ) = \frac{( - 1 )^{\alpha}}{4}\left( 1 - ( - 1 )^{\beta}\tanh \frac{x + a}{2}\right) \vec{e}_{1}+ \frac{( - 1 )^{\gamma}}{\sqrt{2}}\sqrt{1 + ( - 1 )^{\beta}\tanh \frac{x + a}{2}}\vec{e}_{2}
{\gamma_k}= \frac{{D_{ak}^{-{\alpha_{ak}}}{P_{ak}}h_{ak}^2}}{{{\kappa^2}+ D_{jk}^{-{\alpha_{jk}}}{P_{jk}}h_{jk}^2}}\triangleq \frac{{{C_{1k}}{X_k}}}{{{\kappa^2}+{C_{2k}}{Z_k}}},
\sum_{\gamma\in\mathcal{S}_{1}(n+1,r)}w(\gamma) =\sum_{\gamma\in\mathcal{S}_{1}(n,r-1)}w(\gamma)+\sum_{k=0}^{p}a_{r+q}^{(k)}\Big(\sum_{\gamma\in\mathcal{S}_{1}(n,r+k)}w(\gamma)\Big) & =\langle\mathcal{H}_{q}^{n}e_{r-1},e_{0}\rangle+\sum_{k=0}^{p}a_{r+q}^{(k)}\langle\mathcal{H}_{q}^{n}e_{r+k},e_{0}\rangle & =\langle\mathcal{H}_{q}^{n+1}e_{r},e_{0}\rangle
\left\{\prod_{i=0}^n (\gamma_{\mu_i})_{21}\frac{1}{\tilde{x}_0 - \tilde{x}_{\pi(1)}}\frac{1}{\tilde{x}_{\pi(1)}- \tilde{x}_{\pi(2)}}.....\frac{1}{\tilde{x}_{\pi(n)}- \tilde{x}_0}\; + \; c.c. \right\}\; ,
t_{i}^{~ m}t_{k}^{~ n}\hat{\Phi}_{m n ~ b d}^{~ p r ~ a c}S ( t_{p}^{~ j}) S ( t_{r}^{~ l}) = S ( t_{b}^{~ s}) S ( t_{d}^{~ p}) \hat{\Phi}_{i k ~ s p}^{~ j l ~ m n}t_{m}^{~ a}t_{n}^{~ c}
\frac{k_1}{k}= \frac{n - m}{2}+ \nu \; , \; \; \; \frac{k_2}{k}= \frac{n - m}{2}- \nu \; , \; \; \; \nu \equiv \sqrt{\frac{(n - m)^2}{4}+ \frac{\Omega^2}{k^2}}\; .
\begin{aligned}x \partial_t w^{(3)}+ p(D-3) w^{(3)}& = r^{(3)}+ x \tilde q_3(D) \partial_t v^{(3)}+ x q_2(D) p(D-2) \left(\prod_{i \in I_3}(D-i+1)\right) \partial_t v^{(2)}\mbox{for}\, t, x > 0,\end{aligned}
k ={u_i \over 2}\,\Big[ \,{{\cal E}+\sqrt{{\cal E}^2 + \mu_i^2}\over K + \sqrt{K^2 + \mu_i^2}}-{K + \sqrt{K^2 + \mu_i^2}\over{\cal E}+\sqrt{{\cal E}^2 + \mu_i^2}}\,\Big]\,,
V_{\alpha}= \frac{\mu\Omega_{\alpha}^2}{2}\sum_{p=0}^{N-1}u_p^*(2-D-D^{\dagger})^{\frac{\alpha}{2}}u_p =:\frac{\mu}{2}\sum_{p=0}^{N-1}\sum_{q=0}^{N-1}u_q^*f^{(\alpha)}(|p-q|)u_p
{\cal A}( r , v ) = V_{2 6}\int_{0}^{\infty}{\frac{d l}{l}}( 8 \pi^{2}\alpha^{\prime}l )^{- p / 2}e^{- r^{2}l / 2 \pi \alpha^{\prime}}[ l^{- 1 / 2}\eta ( i / l ) ]^{- 2 2}[{\frac{\eta ( i / l )}{- \Theta_{1 1}( - i u / \pi , i / l )}}] ,
\nu(n,r)\sim\frac{1}{\sqrt{2\pi nr}}\exp\left\{-\frac{n\log n}{r}+\frac{n}{r}+\sum_{k=1}^{r-1}d_{rk}n^{(r-k)/r}-\frac{1}{r}\sum_{j=2}^r\frac{1}{j}\right\}
T_{band}=\sqrt{n}\max_{|j-k|\geq\iota}\left|\frac{\widehat{\gamma}_{jk}(0)}{\sqrt{\widehat{\gamma}_{jj}(0)\widehat{\gamma}_{kk}(0)}}\right|=\max_{|j-k|\geq\iota}\frac{1}{\sqrt{n}}\left|\sum^{n}_{i=1}\frac{x_{ij}x_{ik}}{\sqrt{\widehat{\gamma}_{jj}(0)\widehat{\gamma}_{kk}(0)}}\right|.
A_{T o t a l}^{A b e l i a n}= 1 6 \frac{G_{0}^{2}}{2 \alpha^{'}}[ \frac{K}{s t}c o s ( \frac{p_{1}\theta p_{2}- p_{3}\theta p_{4}}{2}) + \frac{K}{s u}c o s ( \frac{p_{1}\theta p_{2}+ p_{3}\theta p_{4}}{2}) + \frac{K}{u t}c o s ( \frac{p_{1}\theta p_{4}+ p_{2}\theta p_{3}}{2}) ]
\sigma \cdot c^{\psi}=\dfrac{1}{h}\sum_{i =1}^h \ \psi^{-1}(\mathfrak{a}_i) \ \sigma \cdot (\mathfrak{a}_i \cdot c ) =\dfrac{1}{h}\sum_{i =1}^h \ \psi^{-1}(\mathfrak{a}_i) \ \sigma(\mathfrak{a}_i \cdot c ) \sigma^{-1}= \dfrac{1}{h}\sum_{i =1}^h \ \psi^{-1}(\mathfrak{a}_i) \ \mathfrak{a}_i \cdot \sigma c \sigma^{-1}= (\sigma c)^{\psi}.
s_{n-2}s_{n-1}(x) = \sum_{\substack{1\leq l<k\leq n 1\leq i_1<\cdots<i_k\leq n}}(-1)^k d_{i_1}^{(n-i_1)}\cdots d_{i_l}^{(n-i_l)}d_{i_l+1}^{-(n-i_{l+1})}\cdots d_{i_k}^{-(n-i_k)}(x).
{\mathrm d}X_{t, n}^{(u)}\, =\, b \Big( t, X_{t, n}^{(u)}, u \cdot \delta_{X_{t, 1}^{(u)}}+ (1-u) \cdot \frac{1}{n}\sum_{j=1}^{n}\delta_{X_{t, j}^{(u)}}\Big){\mathrm d}t +{\mathrm d}W_{t, n}\, .
[J_{a},\hat{T}_{q}^{(2)}]=\sum_{q^{\prime}}{D(J_{a})}_{qq^{\prime}}^{(2)}\hat{T}_{q^{\prime}}^{(2)}=\sum_{q^{\prime}}\langle j=2,m=q|J_{a}|j=2,m=q^{\prime}\rangle\hat{T}_{q^{\prime}}^{(2)}
R=(I+A)(I-A)^{-1}=I+\frac{2\theta_{1}}{1+{\theta_{1}}^{2}}A_{1}+\frac{2{\theta_{1}}^{2}}{1+{\theta_{1}}^{2}}{A_{1}}^{2}+\frac{2\theta_{2}}{1+{\theta_{2}}^{2}}A_{2}+\frac{2{\theta_{2}}^{2}}{1+{\theta_{2}}^{2}}{A_{2}}^{2}
\tilde{A}(p) = \frac{3}{2}g^{2}\delta_{ab}\int \frac{d^{4}k}{(2\pi)^{4}}d^{2}\theta d^{2}{\overline\theta}\, \frac{m^{2}}{(k^{2}+m^{2})[(p-k)^{2}+m^{2}]}\left\{V^{a}(p,\theta,{\overline\theta}) V^{b}(-p,\theta,{\overline\theta}) \right\},
\int\limits_{\mathbb{R}^{N}}\left\vert V_{0}\right\vert^{r}\frac{dx}{\left\Vert x\right\Vert^{s}}=d^{1+\frac{p-1}{p}r}\int\limits_{\mathbb{R}^{N}}\frac{\left\vert U_{0}\right\vert^{r}}{\left\Vert x\right\Vert^{N+sd-Nd}}dx
\prod_{j=0}^{k-1}\left(1 + \frac{1}{x_j}\right) \leq \left(1 + \frac{1}{X_3(1)}\right)^{k/2}\left(1 + \frac{1}{X_3(2)}\right)^{k/2}= \frac{2^k}{3^{k/2}}.
(a,b)(1,c)(a,b)^t = (a,b)(1,c)(\bar{a},\bar{b})= (1, b c\bar{b}) = (1, c)\forall c \in \operatorname{Sp}(1), \bar{c}= -c \Leftrightarrow cb = bc \forall c \in \operatorname{Sp}(1), \bar{c}= -c.
\frac{d_{1}\cdot d_{2}}{c^{2}}+\frac{(d_{1}\cdot w)(d_{2}\cdot w)}{c^{4}}\le[\frac{\vec{d}_{1}^{2}}{c^{2}}+\frac{(d_{1}\cdot w)^{2}}{c^{4}}]^{1/2}[\frac{\vec{d}_{2}^{2}}{c^{2}}+\frac{(d_{2}\cdot w)^{2}}{c^{4}}]^{1/2},
\lceil 2b \rceil = \lceil b+c+b-c \rceil \leq 2B \leq \lceil b+c + (b-c) \rceil + 1 = \lceil 2b \rceil + 1 \leq 2\lceil b \rceil + 1; \lceil 2c \rceil - 1 = \lceil b+c - (b-c) \rceil - 1 \leq 2C \leq \lceil b+c-(b-c) \rceil = \lceil 2c \rceil \leq 2\lceil c \rceil.
\int d \vec{x}\biggl ( v^{k}\frac{\delta}{\delta A^{0 k}}+ \frac{1}{2}u^{k l}\frac{\delta}{\delta A^{k l}}+ \frac{1}{2}\omega^{k l}\frac{\delta}{\delta \pi^{k l}}\biggr ) \int d \vec{y}V^{( 0 )}= 0
\frac{\displaystyle \prod_{i=0}^{t-1}(q^m - q^{a+i}) \prod_{i=0}^{rd-t-1}(q^{a}- q^i)}{\displaystyle \prod_{i= 0}^{rd-1}(q^m - q^i)}\begin{bmatrix}rd t \end{bmatrix}_q.
\left[ ( h r^{2}{\frac{d}{d r}})^{2}+ \omega^{2}r^{4}( 1 +{\frac{r_{K}}{r}}) ( 1 +{\frac{R}{r}})^{3}- 2 h R^{2}( 1 +{\frac{R}{r}})^{- 2}\right] \nu = 0 \ .
u_{\varepsilon, 2}(z,t) = \left(\begin{array}{l}\ \ \ \bar{u}_{1}^{\ast}(x,t) - (1/\varepsilon)\left[\big(P_{20}^{*}\big)^{T}x+P_{30}^{*}y+h_{2 0}(t)\right]\end{array}\right),
\vec{\phi}( \tau , x ) = \frac{1}{\sqrt{1 + \left( f^{( \perp )}( \tau , x ) \right)^{2}}}\left( \begin{array}{c}{\sin f^{( \parallel )}( \tau , x )}\\{f^{( \perp )}( \tau , x )}\\{\cos f^{( \parallel )}( \tau , x )}\\ \end{array}\right)
\frac{d q_{2}( t )}{d t}= \frac{p_{2}( t )}{m_{2}}, \qquad \frac{d p_{2}( t )}{d t}= - m_{2}\omega_{2}^{2}q_{2}( t ) - g ( t ) \langle q_{1}( t ) \rangle_{\psi_{1}}.
\bar{\pi}(\vartheta|X) =\bigotimes_{i=1}^n \big[p_i\mathrm{N}\big(X_i, \tfrac{K \sigma^2}{K+1}\big) +(1-p_i) \delta_0\big], p_i= 1/\big[1+h\exp\{-\tfrac{X_i^2}{2\sigma^2}\}\big],
R_h^j:=- \sum_{\ell=1}^{k}\alpha_{k-\ell}U_h^{j-\ell}+ h \sum_{\ell=1}^{k}\beta_{k-\ell}f(U_h^{j-\ell}) + \sum_{\ell=1}^k\gamma_{k-\ell}g( U_h^{j-\ell})\Delta_h W^{j-\ell+1},
\psi(S(r+s)) =\int^{T_2}_{T_1}G(T_2,\tau)f_\downarrow\left(r(\tau)+s(\tau)\right)d\tau &\ge f_\downarrow\left(b+d\right)\int^{T_2}_{T_1}G(T_2,\tau)\;d\tau &>d.
|g(\xi\gamma)-g(\xi)|= \left| \int_\Omega (g(\xi\eta^{-1}))^{q'-1}(|\eta\gamma|^{\alpha-Q}-|\eta|^{\alpha-Q}) d\eta\right| \leq \|g\|_{L^\infty(\Omega)}^{q'-1}\int_\Omega \left| |\eta\gamma|^{\alpha-Q}-|\eta|^{\alpha-Q}\right| d\eta.
[d, z_1 z_2, z_3] = - [z_1 z_2, d, z_3] = - z_1 [z_2, d, z_3] - [z_1, d] [z_2, z_3] - [z_1, z_3] [z_2, d] - [z_1, d, z_3] z_2 & = z_1 [d, z_2, z_3] + [d, z_1] [z_2, z_3] + [d, z_2] [z_1, z_3] - \bigl[ [d, z_2], [z_1, z_3] \bigr] + [d, z_1, z_3] z_2 .
{\cal L}_{J}= \frac{N}{G}\Biggl [ ( \partial_{\mu}z )^{\dagger}( \partial^{\mu}z ) - J_{\mu}J^{\mu}+ \theta G \, \epsilon^{\mu \nu \rho}J_{\mu}\partial_{\nu}J_{\rho}+ \frac{1}{4}\theta^{2}G^{2}J_{\mu}( \partial^{2}g^{\mu \nu}- \partial^{\mu}\partial^{\nu}) J_{\nu}\Biggr ] ,
\left \{{\partial^2 \over{\partial r^2}}+{5 \over r}{\partial \over{\partial r}}+ \tilde \omega^2 \left ( 1 +{{\tilde R_{\pi\over 2}^4}\over r^4}\right) -{{2 \tilde R_{\pi\over 2}^8}\over{r^{10}\tilde f^2}}\right \}\hat \phi \simeq 0
R_1,~R_2,~R_3,~R_4,~R_5,~R_7,~R_9, A=\frac{1}{2}(R_2+R_3+R_5+R_6),~B=\frac{1}{2}(R_2+R_3+R_{11}+R_{12}), C=\frac{1}{2}(R_1+R_2+R_4+R_5+R_7+R_8+R_{10}+R_{11}).
\|y^k-\hat v_i^k\| \leq \theta \beta^{k}\sum_{\ell=1}^{N}\|v_\ell^0\|+ \theta N\sum_{s=1}^{k}\beta^{k-s}\alpha_{s-1}C \le \theta \beta^{k}M +\theta NC\sum_{s=1}^{k}\beta^{k-s}\alpha_{s-1},
\frac{1}{N}\sum_{i=1}^N |\langle \Theta_i, \xi \rangle| \geq \frac{c_2}{\sqrt{n}}\textrm{and}\frac{1}{N}\sum_{i=1}^N \exp \left \{\left( \frac{\sqrt{n}|\langle \Theta_i, \xi \rangle| + 1}{2}\right)^{\alpha}\right \}\leq 3 C_3.
\bar{\rho}_{\frac 1 2}= \frac{\sinh ( k b ) + \frac 1 2 ( \bar{\rho}_{0}^{2}- \bar{C}- 1 ) \sinh ( k b ) - \bar{\rho}_{0}\cosh ( k b )}{\cosh ( k b ) + \frac 1 2 ( \bar{\rho}_{0}^{2}- \bar{C}- 1 ) ( \cosh ( k b ) - 1 ) - \bar{\rho}_{0}\sinh ( k b )},
\sum_{r \geq y}\frac{1}{r^2}= \int_y^{\infty}\frac{d\pi(t)}{t^2}= \frac{\pi(t)}{t^2}\Bigg|_y^{\infty}+ 2 \int_y^{\infty}\frac{\pi(t)\,dt}{t^3}\ll \frac{\pi(y)}{y^2}\ll \frac{1}{y \log y},
\frac{\lambda^{\prime}}{6}\left( \vec{\varphi}^{\, ( 0 )}( X_{0}) \right)^{2}\varphi_{i}^{( 0 )}( X_{0}) \to \frac{\lambda^{\prime}}{6}\langle \left( \vec{\varphi}^{\, ( 0 )}( X_{0}) \right)^{2}\rangle_{\zeta}\varphi_{i}^{( 0 )}( X_{0}) + \frac{\lambda^{\prime}}{3}\langle \varphi_{j}^{\, ( 0 )}( X_{0}) \varphi_{i}^{\, ( 0 )}( X_{0}) \rangle_{\zeta}\varphi_{j}^{( 0 )}( X_{0})
D \big|u_0^{(2)}\big\rangle = \delta \big|u_0^{(2)}\big\rangle ,P_{\ell}\big|u_0^{(2)}\big\rangle = p \big|u_0^{(2)}\big\rangle , X \big|u_0^{(2)}\big\rangle = 0, X \in{\mathfrak g}_{\ell}^-,P_{\ell-2}\big|u_0^{(2)}\big\rangle = \frac{\ell+2}{2p (\ell+1)}P_{\ell-1}^2 \big|u_0^{(2)}\big\rangle .
\mathcal E_{1,\boldsymbol{\alpha}}(\tau):=-\frac{\sqrt{3}}{4}\int_{-\overline{\tau}}^{i\infty}\int_{w_1}^{i\infty}\frac{\theta_1(\boldsymbol{\alpha};\boldsymbol{w})+\theta_2(\boldsymbol{\alpha};\boldsymbol{w})}{\sqrt{-i(w_1+\tau)}\sqrt{-i(w_2+\tau)}}dw_2 dw_1
{f_{13}}={f_{14}}- \frac{1}{2}\left({{f_1}-{f_2}}\right) - \frac{1}{2}\left({{f_7}-{f_8}}\right) - \frac{1}{2}\left({{f_9}-{f_{10}}}\right) - \frac{1}{8}\left({2{F_x}-{F_z}}\right) - \frac{1}{6}\rho{u_z},
\phi^1=du_{yy}+(a_2u_{zz}+a_3u_{yz}+a_6u_{yy})dx-a_2dy-a_3dz, \phi^2=du_{yz}+(a_3u_{zz}+a_6u_{yz)}dx-a_3dy-a_6dz,~~\phi^3=du_{zz}+a_6u_{zz}dx-a_6dy.