harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
U'(w')=\prod_{X'_{i'}\in{\it G}}e^{-\xi'\sum_{i_{\alpha_{1}}'\in J_{X'_{i'}}}t'_{i_{\alpha_{1}}'}-\lambda'\sum_{\stackrel{i_{\alpha_{1}}'< j_{\alpha_{1}}'}{\left\{i_{\alpha_{1}}',j_{\alpha_{1}}'\right\}\in J_{X'_{i'}}}}t'_{i_{\alpha_{1}}'}t'_{j_{\alpha_{1}}'}{\bf 1}_{(w(t'_{i'_{\alpha_{1}}})=w(t'_{j'_{\alpha_{1}}}))}}\times
d_{\alpha \beta}^{0}= \frac 1 2 \left( \begin{matrix}{{- 1}}&{{0}}\\{{0}}&{{- 1}}\\ \end{matrix}\right) , \qquad d_{\alpha \beta}^{1}= \frac 1 2 \left( \begin{matrix}{{1}}&{{0}}\\{{0}}&{{- 1}}\\ \end{matrix}\right) , \qquad d_{\alpha \beta}^{2}= \frac 1 2 \left( \begin{matrix}{{0}}&{{- 1}}\\{{- 1}}&{{0}}\\ \end{matrix}\right) .
\int ( x_{k})^{4}\mathcal{V}_{J}\, d^{2}x = 6 \int ( x_{k})^{2}\partial_{j}\partial_{j}h \, d^{2}x = 6 \int \left( \partial_{j}( ( x_{k})^{2}\partial_{j}h ) - 2 \partial_{k}( x_{k}h ) + 2 h \right) d^{2}x ,
( \exp t T_{3}) \circ [ g_{c}( z ) ]_{B}= [ g_{c}( z^{\prime}) ]_{B}= \left[ \left( \begin{array}{c c c}{1}&{0}&{0}\\{z_{1}^{\prime}}&{1}&{0}\\{z_{2}^{\prime}}&{z_{3}^{\prime}}&{1}\\ \end{array}\right) \right]_{B}.
\mathcal{C}_{T}:=\mathcal{C}\times_ST=\mathbf{Proj}_S(\bigoplus_{l\ge0}\pi_{\mathcal{X}*}\omega_{\mathcal{X}/S}^{[ler]})\times_ST\cong\mathbf{Proj}_T(\bigoplus_{l\ge0}\pi_{\mathcal{X}_{T}*}\omega_{\mathcal{X}_T/T}^{[ler]}).
b \left( \phi_{o}^{4}+ \phi^{4}\right){\bf M}^{2}= \frac{1}{6}\left[1 + \frac{2{\bf Q}^{2}}{9{\bf M}^{2}}\left( 1 + \frac{2{\bf Q}^{2}}{a^{2}}\right)^{2}\right]^{2}
\begin{aligned}0 =\gamma\big(1(m\,0)^km\big)-\gamma\big(z_{2k-1}z_{2k-2}\cdots z_0\big)\leq &\leq\underbrace{-\beta^{2k+1}+m\beta^{2k}+\cdots+(m-1)\beta^j+\cdots+m\beta+m}_{<m+1-\beta^j}-(m-z_1)\beta-z_0< &\leq m+1 - \beta^j - (m-z_1)\beta-z_0\,,\end{aligned}
\begin{aligned}\Vert P^a_Me^{itH_a}J_a^*P_a^{{\varepsilon}_a}(t)e^{-itH}f-P^a_Me^{isH_a}J_a^*P_a^{{\varepsilon}_a}(s)e^{-isH}f\Vert &\le\hskip-2pt M_1\hskip-2pt\left(\int_s^t\left\Vert\frac{1}{\sqrt{\tau}}Q_a^{{\varepsilon}_a}(\tau)e^{-i\tau H}f\right\Vert^2d\tau\hskip-2pt\right)^{1/2}\hskip-14pt+Cs^{-{\delta}}\Vert f\Vert\to 0,\end{aligned}
N_{m_{1},m_{2},m_{3}}^{\l_{1},\l_{2},\l_{3}}= &{\sum_{z}}\frac{(-1)^{l_{1}-l_{2}-m_{3}+z}}{z!(l_{1}+l_{2}-l_{3}-z)!(l_{1}-m_{1}-z)!(l_{2}+m_{2}-z)!(l_{3}-l_{2}+m_{1}+z)!(l_{3}-l_{1}-m_{2}+z)!}
c\langle \xi_\mu,\varphi\rangle + \langle \hat\rho_v (\xi_\mu),\varphi\rangle\geq2\pi c +\pi\sum_{\alpha\in R_0^+}\langle \alpha,\varphi^\vee\rangle \langle \varphi,\frac{\alpha^\vee}{m_\alpha}\rangle{=}2\pi (1+c+ \langle \hat\rho,\varphi\rangle ),
\begin{pmatrix}\dot{z_{1}}begin{align*}3mm]\dot{z_{2}}\end{pmatrix}=-\frac{i\alpha(\Gamma_{1}+\Gamma_{2})}{D^{3/2}}\begin{pmatrix}1 \ 0 begin{align*}3mm]0 \ 1 \end{pmatrix}\begin{pmatrix}z_{1}-C begin{align*}3mm]z_{2}-C\end{pmatrix}.
\sqrt{\ell!}\biggl( \int_{B_{u_p}^{\ell}}\lVert (H_{B_{u_p}^c}(\xi-x_1-\dots-x_\ell)+\omega^{(\ell)}(x_1,\dots.,x_\ell) &-\Sigma(\xi-k_0) -\omega^{(\ell)}(k_1,\dots,k_\ell) )\Gamma(P_{B_{u_p}^c})\psi_p \lVert^2 \lvert g_{u_p}(x)\lvert^2d\lambda_\nu(x)\biggl )^{1/2}:=\sqrt{\ell!}\gamma_p
= \frac{(-g)^{n}(gh^{2}-k^{2}+fhk)}{\left(f^{2}+4g\right)}\left| B^{-n}\right | \left |MB^n+(-g)^nB^{-n}\right | = \frac{gh^{2}-k^{2}+fhk}{\left(f^{2}+4g\right)}\left| B^{-n}\right | \left |MB^n+(-g)^nB^{-n}\right |.
K_n(t) = \sum_{0\leq \tau^n_{j+1}\leq t}\left|\sum_i\int_{(\tau^n_j, \tau^n_{j+1})}v_i^{\ast}(s)\,dA(s) + \sum_iv_i^{\ast}(\tau^n_{j+1}) \Delta A(\tau^n_{j+1})\right|^2 & = \sum_{0\leq \tau^n_{j+1}\leq t}(J^{(1)}_j+J^{(2)}_j-J^{(3)}_j)
\displaystyle{\mbox{$_{q+1}$F$_q$}(a,b,\dots,b;\,b+1,\dots,b+1;\,1)}={\frac{\left( -1 \right)^{q-1}{b}^{q}\Gamma \left( 1-a \right) \mbox{}}{\left( q-1 \right) !}{\frac{\partial^{q-1}}{\partial{b}^{q-1}}}\left({\frac{\Gamma \left( b \right)}{\Gamma \left( 1+b-a \right)}}\right)}.
Z_{\mathrm{s c}}( \beta ) = \int_{- \infty}^{\infty}d x_{0}\sum_{j = 1}^{N ( x_{0}, \beta )}\mathrm{e}^{- S_{j}( x_{0}, \beta ) / \hbar}\Delta_{j}^{- 1 / 2}( x_{0}, \beta ) ,
\Phi_1 =\frac{a_2}{a_1}, \Phi_2 =\frac{1}{a_1^2}\left(\frac{1}{2}a_2^2 -a_1 a_3\right), \Phi_3 =\frac{1}{a_1^3}\left( \frac{1}{3}(a_2)^3-a_1a_2a_3+a_1^2 a_4 \right).
A_\mu^{(3)}=\mp{2i\over g}\,(H_2\,{v_\mu^1\over\rho_1}+H_3\,{v_\mu^2\over\rho_2})+\kappa\,(E_{21}^\pm\,R^1_\mu+E_{31}^\pm\,R^2_\mu)\,\delta(R^1\cdot R^2).
\sum_{(i,j) \in N}\left(1 - \cos\left(2 \pi x_{(i,j)}\right) \right) (x_{(i,j)})_{(i,j) \in N}\in \left[-\frac{1}{2}, \frac{1}{2}\right]^N, \forall\, (k,\ell) \in S, \; 4x_{(k,\ell)}- \sum_{\|(i,j)-(k,\ell)\|_1 = 1}x_{(i,j)}= v_{(k,\ell)},
G_{k}^{( 2 )}({\vec{x}},{\vec{y}}) = D_{k}({\vec{x}},{\vec{y}}) -{\frac{\lambda_{2}( \mu )}{1 - ( \lambda_{2}( \mu ) / 4 \pi \alpha ) \ln ( k_{4}^{2}/ \mu^{2})}}D_{k}({\vec{x}}, 0 ) D_{k}( 0 ,{\vec{y}})
B'' =q^{2i+2}\left[\mu_{n-i-1}(q^{2})(q^{2}-1)+2(q^{4}\mu_{n-i-3}(q^{2})+q^{2}-1)\right]\; , C'' =q^{2i+2}(q^{4}\mu_{n-i-3}(q^{2})+q^{2}-1)\; .
{\cal E_{C}}[ \phi_{0}] = \Gamma^{( 2 )}[ \phi_{0}] + \sum_{\ell = 0}^{\infty}( 2 \ell + 1 ) \left[ \int_{0}^{\infty}\frac{d k}{\pi}E \frac{d}{d k}\left( \delta_{\ell}( k ) - \delta_{\ell}^{( 1 )}( k ) - \delta_{\ell}^{( 2 )}( k ) \right) + \sum_{j}\mathrm{m a x}( \omega , E_{j , \ell}) \right] \, .
0 = \nabla_a \nabla_b D(x^{(c)}||m_\zeta) - \nabla_a \nabla_b D(x||m_\zeta) &= \partial_a \partial_b D(x^{(c)}||m_\zeta) -{\omega^d}_{ab}(\zeta) \partial_d D(x^{(c)}||m_\zeta) - g_{ab}(\zeta) &= \partial_a \partial_b D(x^{(c)}||m_\zeta) -{\omega^d}_{ab}(\zeta){A_d}^c(\zeta) - g_{ab}(\zeta).
(u_p^1 \gamma_1^{c_1(p)}\otimes u_p^2)(u_{q}^1 \gamma_1^{c_2(q)}\otimes u_{q}^2) = (-1)^{c_1(p)c_2(q)}\sigma_1(p,q)\sigma_2(p,q) (u_{pq}^1 \gamma^{c(pq)}_1 \otimes u_{pq}^2).
\left({D}^{\alpha}_{N}U^{n-\sigma}, v_h \right) \, - \, a\big(l(U^{n, \sigma})\big) \, (\Delta_h U^{n, \sigma}, v_h) \, = \, (f^{n-\sigma}, v_h), \; \, \forall v_h \in V_h.
\left(\left[\prod_{i=1}^k\left(S_i^{m_i}(S_i^*)^{n_i}-\frac{1}{q_i}S_i^*S_i^{m_i}(S_i^*)^{n_i}S_i\right)\right] T\right) &\qquad= \left(\left[\prod_{i=1}^k S_i^{m_i}(S_i^*)^{n_i}\right] \left[\prod_{i=1}^k\left(I-\frac{\tau_i}{q_i}\right)\right] T\right).
dS_{c}\omega+S_{c}d\omega=dt\wedge\omega_{B}(t)+\int_{c}^{t}\left(\frac{\partial\omega_{A}}{\partial t}(\tau)\right)d\tau=dt\wedge\omega_{B}(t)+\omega_{A}(t)-\omega_{c}(t)=\omega-i_{c}^{*}\omega.
\begin{pmatrix}u, v, w\end{pmatrix}\begin{pmatrix}I \gamma 0 0 \varepsilon 0 \vartheta \psi \varepsilon^{t^{-1}}\end{pmatrix}= \begin{pmatrix}u+w\vartheta, u\gamma+v\varepsilon+w\psi, w\varepsilon^{t^{-1}}\end{pmatrix}
\frac{(2-\theta_0)}{k+(2-\theta_0)/\theta_0}\leq \theta_k \leq \frac{2}{k+2/\theta_0}&\frac{1-\theta_{k+1}}{\theta_{k+1}^2}=\frac{1}{\theta_k^2},\enspace \forall k=0,1,\dots
\langle x,m,q | x', m',q' \rangle\; =\; \delta(x-x')\, \delta_{m\, m'}\,\, \delta_{q\,q'},\sum_{m,q}\, \int_{-1}^{+1}|x,m,q\rangle\, dx\, \langle x,m,q|\; =\;{\cal I}.
\Theta_{i}^{- 1}\, \exp \left( 2 \pi i \mathbf{a}_{i}\right) \, \Theta_{i}\ = \ \exp \left( 2 \pi i \theta_{i}( \mathbf{a}_{i}) \right) \, L e f t r i g h t a r r o w \ \mathbf{a}_{i}= \theta_{i}( \mathbf{a}_{i}) .
\begin{aligned}M_{i,i}=\frac{1}{(b-a)^2}\Big(\int^{b}_{a}(x-a)^2\,d\mu\circ T_{J_i}+\int^{b}_{a}(b-x)^2\,d\mu\circ T_{J_{i+1}}\Big),1\le i\le N^m-1, &M_{i,i-1}=\frac{1}{(b-a)^2}\int^{b}_{a}(x-a)(b-x)\,d\mu\circ T_{J_{i}},2\le i\le N^m-1, &M_{i,i+1}=\frac{1}{(b-a)^2}\int^{b}_{a}(x-a)(b-x)\,d\mu\circ T_{J_{i+1}},1\le i\le N^m-2.\end{aligned}
{\hat{\cal H}}\psi ({\bf X}) = \int d^{3}x \left[ \frac{g^{2}}{2}{\hat{X}}^{a x}{\hat{X}}^{a x}+ \frac{1}{4 g^{2}}( \partial_{[ a}{\hat{P}}_{b ] x})^{2}\right] \psi ({\bf X})
4S \sum_{\substack{\beta\geq\frac{1}{2}-\frac{R}{\log q}0\leq\gamma\leq \frac{2S}{3\log q}L(\beta+i\gamma,f)=0}}\sinh\left(\frac{\pi \big(R+(\beta-1/2)\log q\big)}{2S}\right) \; \leq \; I_1(f)+I_2(f)+I_3(f).
\left[ E_{i}^{a}- \cos \theta ( D_{i}A_{4})^{a}- \sin \theta ( D_{i}A_{5})^{a}\right]^{2}+ \left[ B_{i}^{a}+ \sin \theta ( D_{i}A_{4})^{a}- \cos \theta ( D_{i}A_{5})^{a}\right]^{2}+
D_{\mu \mu}= - \frac{g_{\mu \mu}}{3 2 \pi^{2}}\left[ \frac{\alpha^{2}}{( 3 - d )}+ 3 2 \pi^{2}\alpha \left( \frac{\mu_{R}^{2}}{\lambda_{R}}+ \frac{1}{2}\varphi^{2}\right) - 1 6 \pi^{2}\frac{\alpha^{2}}{\lambda_{R}}\right] \; \; ,
W_{q}(\pi_{k}(x,y_{0:k},\cdot),\pi_{k}(x^{\prime},y_{0:k},\cdot))\leq\exp\left[-\sum_{j=1}^{k}\int_{0}^{\Delta}\lambda(j,y_{j},t)\mathrm{d}t\right]\|x-x^{\prime}\|,\qquad\forall x,x^{\prime}\in\mathbb{R}^{p},
\sup_{s \leq r_{t}}z^n_{s}= \max \Bigl( \sup_{s \leq \max(0,r_{t}-\delta)}z^n_{s},\sup_{\max(0,r_{t}- \delta) \leq s \leq r_{t}}z^n_{s}\Bigr) &\leq \max \Bigl( \sup_{s \leq \max(0,r_{t}-\delta)}z^n_{s},z_{r_{t}}\Bigr) + \varepsilon\leq \max \Bigl( \sup_{s < r_{t}^n}z^n_{s},z_{r_{t}}\Bigr) + \varepsilon.
\|w^{(k)}\|_{L^{\infty}([0,1])}\leq 2C_m \|w\|_{L^{\infty}([0,1])}^{1-k/m}\max\left\{\|w\|_{L^{\infty}([0,1])}^{k/m}, \|w^{(m)}\|_{L^{\infty}([0,1])}^{k/m}\right\}, k=1, \dots, m-1.
\begin{tabular}{|c||c|c|}\hline ${\bf 28}$ $\overline{\bf 28}$ \hline\hline${\bf 63}$ ${\bf 36}+{\bf 420}$ $\overline{\bf 36}+\overline{\bf 420}$ \hline${\bf 70}$ $\overline{\bf 420}$ ${\bf 420}$ \hline\end{tabular}
S = \int d x^{+}d x^{-}\, \mathrm{t r}\Bigg [ \partial_{+}\phi \partial_{-}\phi + i \psi \partial_{+}\psi + \frac{g^{2}}{2}J^{+}\frac{1}{\partial_{-}^{2}}J^{+}- \frac{1}{2}i g^{2}[ \phi , \psi ] \frac{1}{\partial_{-}}[ \phi , \psi ] \Bigg ] .
S_p := \inf_{u \in W^{1,p}_0(B)}\frac{\ \|\nabla u\|_{L^p(B)}\}{\|u\|_{L^{p^*}(B)}}= \sqrt{\pi}N^{\frac1p}\left(\frac{N-p}{p-1}\right)^{\frac{p-1}{p}}\left[ \frac{\Gamma\left(\frac{N}{p}\right)\Gamma\left(N+1-\frac{N}{p}\right)}{\Gamma(N)\Gamma\left(1+\frac{N}{2}\right)}\right]^{\frac1N}.
|\Psi_{(n,0)}\rangle = \int_0^{P^+}{dq_1 \dots dq_n \over \sqrt{q_1...q_n}}\delta(q_1 + \cdots + q_n - P^+)\hspace{1mm}f^{(n,0)}(q_1,\dots,q_n) \mbox{tr}[a^\dagger(q_1)\dots a^\dagger(q_n)]|0\rangle .
f^{(2)}(\tau_1)E(\widetilde{X}_{1,2}^2\mid X_1)-f^{(2)}(\tau_3)E(\widetilde{Z}_{1,1}^2\mid X_1) &=p^{-2}f^{(2)}(\tau)\left\{E(\|X_1-X_2\|_2^4\mid X_1)-E(\|X_1-Y_1\|_2^4\mid X_1)\right\}&=-p^{-2}f^{(2)}(\tau)\big\{4E(\|X_2\|_2^2X_2^\top-\|Y_1\|_2^2Y_1^\top)X_1+E(\|X_2\|_{2}^{4}-\|Y_1\|_{2}^{4})\big\}.
P(x,t)-P(x,0) = \frac{I_{2H}^{-\alpha, \alpha}}{2^{\alpha}}\left(t^{2H\alpha}\frac{\partial^2}{\partial x^2}P(x,t)\right) = \frac{2^{1-\alpha}H}{\Gamma(\alpha)}\int_0^t \tau^{2H-1}\left(t^{2H}-\tau^{2H}\right)^{\alpha-1}\frac{\partial^2}{\partial x^2}P(x,t)\,\mathrm{d}\tau,
\begin{aligned}{\bf B}_\alpha(t, T) v := \sum_{j=1}^\infty \frac{E_{\alpha ,1}(-\lambda_j t^\alpha )}{E_{\alpha ,1}(-\lambda_j T^\alpha )}\langle v,\varphi_j\rangle\varphi_j ,{\bf P}_\alpha(t) v:= \sum_{j=1}^\infty t^{\alpha -1}E_{\alpha ,\alpha}(-\lambda_j t^\alpha ) \langle v,\varphi_j \rangle\varphi_j. \end{aligned}
\mathcal{Q}_{1}(u,V)=\lim_{\tau\to \infty}\frac{1}{\tau}\int_{0}^{\tau}\mathcal{L}(-s)\left(\begin{array}{c}0 \frac{1}{\sqrt{\rho_{0}}}\mathbb{H}^{\perp}_{\rho_{0}}B_{1}(u,V)\end{array}\right)ds
c_{n-1,\xi_{\boldsymbol{s}k}}^{\ast}=\frac{\Gamma\left((n-1)/2\right)}{(2\pi)^{(n-1)/2}}\left[\int_{0}^{\infty}x^{(n-3)/2}\overline{G}_{n}\left(\frac{1}{2}\xi_{\boldsymbol{s}k}^{2}+x\right)\mathrm{d}x\right]^{-1},
\mathbb{E}\left[ \nabla \mathfrak{L}(\boldsymbol{\theta}) \right] = \frac{2}{n}\mathbb{E}\left[ \mathbf{X}^T (\mathbf{x}_1^n-\mathbf{X}\boldsymbol{\theta}) \right]=\frac{2}{n}\mathbb{E}\left[ \mathbf{X}^T \mathbf{w}_1^n \right]= \mathbf{0}.
|g_j(\lambda,u)-g_j(\lambda',u)|\leq |\frac{\partial}{\partial\lambda}g_j(\lambda,u)|_{\tilde\delta_{j+1}}|\lambda-\lambda'|<\frac{(2\tilde R_j)^{m^2}}{\tilde\delta_j/2}\tilde\delta_{j+1}=\frac{1}{2K_j},
p_{j+1}= \theta_K\left(\frac{\alpha}{(1-K)^k}\right) - 2k\frac{1+2K}{6-6K}\frac{\alpha}{(1-K)^k}&= \frac{\theta_K(\alpha)(1-K)^k + k\alpha(1-K)^{k-1}K}{(1-K)^{2k}}- k\frac{(1+2K)\alpha}{3(1-K)^{k+1}}&= \frac{3\theta_K(\alpha)(1-K) + 3k\alpha K-k(1+2K)\alpha}{3(1-K)^{k+1}}&= \frac{3\theta_K(\alpha) - k\alpha}{3(1-K)^{k}}.
D_{\mu \nu}= \overline{{P}}_{\mu \nu}\frac{1}{p^{2}+ \Pi_{\mathrm{T}}}+ Q_{\mu \nu}\frac{1}{p^{2}+ \Pi_{\mathrm{L}}}+ R_{\mu \nu}\frac{1}{p^{2}+ \tilde{\Pi}_{\mathrm{T}}}+ \xi \frac{p_{\mu}p_{\nu}}{p^{2}}
\frac{1}{nh}\sum_{k=0}^{n-1}\partial_{\theta^{2}}f(X_{t_{k}},\theta_{0})\int_{t_{k}}^{t_{k+1}}\big(f(X_{t},\theta_{0})-f(X_{t_{k}},\theta_{0})\big)\mathrm{d}t\rightarrow0,
\frac{\sin\left((b-a)\pi\right)}{\pi}{}_2 \tilde{F}_1 (a,b;c;z) = \frac{(-z)^{-a}}{\Gamma(b)\Gamma(c-a)}{}_2 \tilde{F}_1(a,a-c+1;a-b+1;\frac{1}{z}) - \frac{(-z)^{-b}}{\Gamma(a)\Gamma(c-b)}{}_2 \tilde{F}_1(b,b-c+1;b-a+1;\frac{1}{z})
[\begin{matrix}A_{11}&A_{12}&A_{13}\\A_{21}&A_{22}&A_{23}\\A_{31}&A_{32}&A_{33}\end{matrix}]=\frac{2h}{1-\nu_{12}\nu_{21}}[\begin{matrix}E_{1}&\nu_{12}E_{2}&0\\ \nu_{21}E_{1}&E_{2}&0\\ 0&0&2G_{12}(1-\nu_{12}\nu_{21})\end{matrix}]
\begin{aligned}\frac{1}{\tau^k}[f_j(x_j)-f_j(\breve{x}_j^{k})]-\frac{1}{\tau^{k-1}}[f_j(x_j)-f_j(\breve{x}_j^{k-1})]+(x_j-\widetilde{x}^k_j)^T[-A_j^T\widetilde{\lambda}^k + \tau^k\beta^k\sum_{i=2}^{j}A_j^TA_i(\widetilde{x}_i^k-\bar{x}_i^{k})+(1-\gamma)(1-\tau^k)\beta^kA_j^T(A\breve{x}^{k-1}-b)]\geq 0,~\forall x_j.\end{aligned}
\sum_{i_{1}+i_{2}+\cdots+i_{r}=N}\frac{a_{1}^{i_{1}}a_{2}^{i_{2}}\cdots a_{r}^{i_{r}}q^{\binom{i_1}{2}+\binom{i_2}{2}+\cdots+\binom{i_r}{2}}}{(q;q)_{i_{1}}(q;q)_{i_{2}}\cdots(q;q)_{i_{r}}}&\quad\quad\times \left(1+\sum_{s=1}^{r}q^{i_1+i_2+\cdots+i_s}a_s q^N \prod_{k=1}^{s-1}\left(1+a_k q^N\right)\right).
d_k= \dim Span_{\mathbb{R}}Z_{k}\leq \dim Span_{\mathbb{R}}\left\{\prod\limits ({x_i})^{p_{i}}\cdot\prod(x_s)^{a_s}\prod(y_s)^{b_s}\prod(y_{i})^{f_{i}}\right. &\quad\quad\quad\quad\quad\left. |\sum p_{i}=\sum f_{i}+\sum a_s+\sum b_s=k \right\}\approx \alpha_{11}k^{2n-2}, \emph{~for some constant}\alpha_{11}.
\vec v(t,x) = \left ( \frac{1}{\lambda(t)^{(N-2)/2}}u_C \left ( t , \frac{x - x(t)}{\lambda(t)}\right ), \frac{1}{\lambda(t)^{N/2}}\partial_t u_C \left ( t , \frac{x - x(t)}{\lambda(t)}\right ) \right ).
\sum_{k=0}^{n}\left( -1\right)^{k}\left( \begin{array}{c}n k\end{array}\right) \frac{\left( z+n-k-1\right) ...\left( z-k\right)}{\left(z+k-1\right)}=n!\frac{\left( 2z+n-2\right) ...\left( 2z\right) \left(2z-1\right)}{\left( z+n-1\right) ...\left( z\right) \left( z-1\right)}.
\sum_{r=1}^{3}\Biggl( E\otimes e^{-2h_{t}(\zeta)}e_{t}^{r}(\zeta) &+H\otimes (e^{-2h_{t}(\zeta)}f_{t}(\zeta)e_{t}^{r}(\zeta)+h_{t}^{r}(\zeta)) &+F\otimes (f_{t}^{r}(\zeta)-e^{-2h_{t}(\zeta)}f_{t}(\zeta)^{2}e_{t}^{r}(\zeta)-2f_{t}(\zeta)h_{t}^{r}(\zeta))\Biggr) dB_{t}^{(r)}
I=\int_{0}^{\infty}rdr \frac{\left(\omega'-\omega\cosh 2\Omega r\right)}{\left(1+r\omega\sinh 2\Omega r\right)}J_{0}(qr)J_1(q'r) \left[\exp\left(2\pi i(1+r\omega\sinh 2\Omega r)\right)-1\right].
g_{2}((\nabla\pi_*)(U_{2},V_{2}),\pi_*Z) =\cos^{2}\theta g_{2}((\nabla\pi_*)(U_{2},V_{2}),\pi_*Z)+g_{2}(\nabla\pi_*(U_{2},\omega\phi V_{2}),\pi_*Z)-g_{1}(\mathcal{T}_{U_{2}}\omega V_{2},\mathcal{B}Z) &+g_{2}(\nabla\pi_*(U_{2},\omega V_{2}),\pi_*(\mathcal{C}Z))-g_{1}(V_{2},\phi U_{2})\eta(Z),
\left\langle i_{1},i_{1}+1,\dots,\widehat{i_{j}+\epsilon},\dots,i_{r},i_{r}+1\right\rangle =\det\left(\begin{matrix}C I_{i_{1},i_{1}+1,\ldots,\widehat{i_{j}+\epsilon},\ldots,i_{\frac{m+1}{2}},i_{\frac{m+1}{2}}+1}\end{matrix}\right)\det(\mathcal{Z}),
J_n (\theta) = \begin{pmatrix}\nabla_\theta G_n^1 (\theta)^T \nabla_\theta G_n^2 (\theta)^T \nabla_\theta G_n^3 (\theta)^T \nabla_\theta G_n^4 (\theta)^T\end{pmatrix}= \begin{pmatrix}J_n^{11}(\theta) J_n^{21}(\theta) J_n^{22}(\theta)\end{pmatrix}
e^{-\pi (H + 1/2 H Q_0^{-1}H + o(\lVert H\rVert^2))[m]t}= 1 - \pi (H + 1/2 H Q_0^{-1}H + o(\lVert H\rVert^2))[m]t + o(\lVert H\rVert^2) = 1 - \pi H[m] t + o(\lVert H\rVert)
G(f)^{(n)}_{\beta}=\Sigma_{m=0}^n\{\tilde\Theta^{(n-m)}_\beta ,\tilde f^{(m)}\}_{(q)}+\Sigma_{m=0}^{(n-2)}\{\tilde\Theta^{(n-m)}_\beta ,\tilde f^{(m+2)}\}_{(\phi )}+\{\tilde\Theta^{(n+1)}_\beta ,\tilde f^{(1)}\}_{(\phi )}
\int_{0}^{L}\int_{0}^{\infty}(-1)^{k}g(s) \partial_{x}^{2 k}\eta^{t}\partial_{t}\eta^{t}d s d x +\int_{0}^{L}\int_{0}^{\infty}(-1)^{k}g(s) \partial_{x}^{2 k}\eta^{t}\partial_{s}\eta^{t}d s d x &=\int_{0}^{L}\int_{0}^{\infty}(-1)^{k}u g(s) \partial_{x}^{2 k}\eta^{t}d s d x
m_{\phi}^{2 \ ( - )}= \lambda_{o b s}^{- 2}\m_{\phi}^{2 \ ( + )}= \lambda_{o b s}^{- 2}\{\frac{d^{2}V_{\mathrm{\footnotesize \it e f f}}}{d \phi^{2}}}\approx \mp \lambda_{o b s}\left({\frac{5 \pi^{3}\zeta_{R}^{\prime}( - 4 )}{6 M^{3}l^{5}}}\right) .
\|P_m(U_n)\|_m\leq\begin{cases}0, m=1,\dots,j-1, \|X_{n1}\|_1^j, m=j, \sum_{(i_1,\dots,i_m)\in\mathbb{I}_j^m}|\gamma_{(i_1,\dots,i_m)}|\|X_{n1}\|_1^m, m=j+1,\dots,k.\end{cases}
J^{w}(\textbf{X}_{minRSSU}^{(n)}) =-\frac{(n!)^2}{2}\prod_{i=1}^{n}\left(\int_{0}^{1}(1-u)^{2i-2}w(F^{-1}(u)) f(F^{-1}(u))du\right) &=-\frac{(n!)^2}{2}\prod_{i=1}^{n}\left(\int_{0}^{1}(1-u)^{2i-2}\alpha (1-u)^{\frac{\alpha - m+1}{\alpha}}du\right) &=-\frac{(n!)^2 \alpha^n}{2}\prod_{i=1}^{n}\frac{1}{2i\alpha -m+1}.
D=\left(\begin{array}{c|cc|cc}1 1 0 0 1 \hline2 0 1 1 0 2 1 0 1/2 1/2 \hline1 \sqrt{2}1-\sqrt{2}1 -1 1 0 1 0 0\end{array}\right)\ \d=\left(\begin{array}{c}1 \hline 0 0 \hline 2 2\end{array}\right).
T_i =\{z;\nu_{(f,H_i),\leqslant k_i}(z)>0\}, S_i=\bigcup_{u=1}^3\{z;\nu_{(f_u,H_i),> k_i}(z)>0\}, R_i =\bigcap_{u=1}^3\{z;\nu_{(f_u,H_i),> k_i}(z)>0\}, \nu_i =\{z;k_i\geqslant\nu_{(f^u,H_i)}(z)\geqslant \nu_{(f^v,H_i)}(z)=\nu_{(f^t,H_i)}(z) (u,v,t)(1,2,3)\}.
I_{r e p}\equiv \mu_{0}\int_{\Sigma}d^{2}\sigma \Phi^{- 1}\mathrm{d e t}\left[ \gamma_{a b}( \sigma ) \right] \ , \qquad \Phi^{- 1}\equiv \epsilon_{i j}\epsilon^{m n}\partial_{m}\phi^{i}\partial_{n}\phi^{j}
A_{\bf r}({\bf v}) =\frac{1}{1+n+k}\frac{(1+n+k)_{k}}{0!k_1!k_2!\cdots k_n!}, &=\frac{1}{n+k+1}\frac{(n+k+1)!}{k_1!k_2!\cdots k_n!(n+1)!}, &=\frac{1}{n+1}\frac{(n+k)!}{k_1!k_2!\cdots k_n!n!}, &=\frac{1}{n+1}\frac{(n+k)!}{k!\;n!}\frac{k!}{k_1!k_2!\cdots k_n}, &=\frac{1}{n+1}\binom{n+k}{k}\binom{k}{k_1,k_2,...,k_n}.
\vartheta_{\mathcal L}(z) = \frac{1}{(1-z^q)^n}\sum_{\ell=0}^n \sum_{t=0}^n z^{tq}\sum_{s=0}^{n}\Phi_{\mathcal L}^{(\ell+s)}(z)\, 2^s\binom{\ell+s}{s}\binom{\ell}{t-s}(-1)^{t-s}= \sum_{t=0}^n \frac{z^{tq}}{(1-z^q)^n}\sum_{s=0}^{n}2^s (-1)^{t-s}\sum_{\ell=s}^{n}\Phi_{\mathcal L}^{(\ell)}(z)\binom{\ell}{s}\binom{\ell-s}{t-s}.
(P_k)_j=\begin{bmatrix}0 0 U_0(\alpha_r) \overbrace{0 \ldots 0}^{(k-1)}\mu^{\frac{1}{2}}U_1(\alpha_r) \ldots \overbrace{0 \ldots 0}^{(k-1)}\mu^{\frac{n-k}{2k}}U_{\frac{n-k}{k}}(\alpha_r) \overbrace{0 \ldots 0}^{(k-3)}\end{bmatrix}^{T}
\mathrm{P}_{m,1}= \mathrm{P}\left( z_m<\frac{\epsilon_1}{\rho\xi_1}\right) =\mathrm{P}\left( \alpha_1=1, z_m<\frac{\epsilon_1}{\rho\xi_1}\right)+\mathrm{P}\left( \alpha_1<1, z_m<\frac{\epsilon_1}{\rho\xi_1}\right).
\begin{bmatrix}a_1 a_2 a_3\end{bmatrix}\in V_{-4k,k}\to \begin{bmatrix}ka_1/2\sqrt{k}a_2/2\sqrt{k}a_3/2\sqrt{k}\end{bmatrix}\in V_{-1,1}\to z_a=\frac{-a_3+i2\sqrt{k}}{2ka_1}\in H.
\int_{D(z,\textsf{d})}\frac{d\xi d\eta}{|\eta|^{\tau}|\zeta-z|^{q}}=\int_{0}^{2\pi}\int_{0}^{\textsf{d}}\frac{drd\theta}{|r\sin\theta+\Im(z)|^{\tau}r^{q-1}}\leq M(q,\tau)\,\textsf{d}^{2-\tau-q}.
\displaystyle\int_X\Phi_3\circ\Phi_2^{-1}(f_0(x))d\mu =\sum^{\infty}_{j=1}\displaystyle\int_{F_j}\Phi_3\circ\Phi_2^{-1}(f_0(x))d\mu &\leq\sum^{\infty}_{j=1}\Phi_3\circ\Phi_2^{-1}(ess.sup_{x\in F_j}f_0(x))\mu(F_j) &=\sum^{\infty}_{j=1}\phi_3\circ\phi_2^{-1}(Q_T(F_j))\mu(F_j).
-2d_{j-1}+ \frac{d}{r}+d_{j+1}+3t = \frac{e}{2q}\left(\frac{1}{r_j}+\frac{1}{r_{j+1}}\right)+ \sigma_{j-1}\left(2-\frac{r}{r_j}\right) + \sigma_{j}\left(2-\frac{r}{r_{j+1}}\right)+\sum_{i=1, i\neq j-1,j}^{2q}2\sigma_i
\varkappa(m,p_1,p_2):=\sum_{s=0}^{n}\sum_{\nu\in{I}_s}k^{\frac{sr}{2}(\frac{1}{4}-\frac{1}{p_1})}(\ln (k))^{sr}\prod_{\nu_j>k}[(\ln(\nu_j))^r\nu_j^{\frac{r}{2}(\frac{1}{4}-\frac{1}{p_1})}]|m(\nu)|^r<\infty,
S_{A}=I+V^{\ast}A^{\dagger}U, H=-V^{\ast}A^{\dagger}, K=A^{\dagger}U, \Phi=(I+H^{\ast}E_{S_{A}}H)^{-1},\Psi =(I+KF_{S_{A}}K^{\ast})^{-1}, \Sigma=\Psi(A^{\dagger}+KS_{A}^{\dagger}H)\Phi.
\eta_{m}(z)=1+\frac{e^{i2m\pi/3}}{\sqrt[3]{\left|c_{3}\right|}e^{i\theta/3}}(z-1)^{1/3}+e^{i2m\pi/3}\sum_{n=2}^{\infty}d_{n,m}(z-1)^{n/3},m=0,1,2,
\begin{array}{l}\int dy_{1}dy_{2}dy_{3}dy_{4}\langle\phi^{2}(y_{1})h^{n}(y_{1})\partial h (y_{1})\partial h(y_{1})\phi^{2}(y_{2})h^{m}(y_{2})\partial h (y_{2})\partial h(y_{2}) \cr\phi^{2}(y_{3})h^{r}(y_{3})\partial h (y_{3})\partial h(y_{3}) h(x_{1})h(x_{2})\rangle \end{array}
\langle D^2_1 J_1(f;v^1,v^2),(w^1,w^1) \rangle = \iint_{\mathcal{O}_1 \times (0,T)}\eta w^1 dxdt + \mu_1 \iint_{\mathcal{O}_1 \times (0,T)}|w^1|^2 dxdt.
C^{-1}=\frac{1}{2}\begin{pmatrix}2 1 2 2\end{pmatrix}C(z) =\begin{pmatrix}z^2+z^{-2}-1 -z-z^{-1}z+z^{-1}\end{pmatrix}\widetilde{C}(z) =\frac{1}{z^{3}+z^{-3}}\begin{pmatrix}z+z^{-1}1 z+z^{-1}z^{2}+z^{-2}\end{pmatrix}.
\left[{\cal D}^{\mu}\cdot \Gamma \right]_{\mathrm{f l a t}}= x^{\nu}\left[{\Theta}_{\nu}^{~ \mu}\left( x \right) \cdot \Gamma \right]_{\mathrm{f l a t}}- x^{\nu}\left[ w_{\nu}^{~ \mu}\left( x \right) \cdot \Gamma \right]_{\mathrm{f l a t}}- \left[ \Lambda^{\mu}\left( x \right) \cdot \Gamma \right]_{\mathrm{f l a t}}\, \, ,
Q_{I}( \theta ) \rightarrow \mathrm{e}^{i \beta}Q_{I}( \mathrm{e}^{- i \beta}\theta ) , \qquad{\overline{{Q}}}^{I}( \theta ) \rightarrow \mathrm{e}^{i \beta}{\overline{{Q}}}^{I}( \mathrm{e}^{- i \beta}\theta ) .
\mu_{P(\ell)}^{(k)}(\{I\})\le \frac{(q+1)^{n^2}\beta(\ell)}{K^{n^2}}\le \frac{(q+1)^{n^2}\beta(\ell)}{\left[ qP(\ell)\right]^{n^2}}=\left(\frac{q+1}{q}\right)^{n^2}\frac{\beta(\ell)}{P(\ell)^{n^2}}\le 2^{n^2}\prod_{j=j^*}^{\ell}[p_j,n]\,,
x_{i,j}+ x_{i,j+1}= a_j, x_{i,n+1-j}+ x_{i,n-j}= S- a_j, x_{m+1-i,j}+ x_{m+1-i,j+1}= a_j, \mbox{and}x_{m+1-i,n+1-j}+ x_{m+1-i,n-j}= S- a_j, \mbox{\ \and}
\langle f , g \rangle = \langle \theta_{i_1}\cdots \theta_{i_r}\overline{\theta}_{j_1}\cdots \overline{\theta}_{j_s}f^\prime, \, \theta_{k_1}\cdots \theta_{k_t}\overline{\theta}_{l_1}\cdots \overline{\theta}_{l_u}g^\prime \rangle &:= w(i,l^T) \delta_{r+u, s+t}\delta_{(i,l^T), (k,j^T)}\, \langle f^\prime , g^\prime \rangle,
\frac{1}{2}y_{s B}\frac{\partial}{\partial y_{s B}}\psi_{0}= - \frac{1}{2}y_{s B}y_{s B}\psi_{0}= - \frac{1}{2}\sum_{s B}( y_{s B}^{2}- \frac{1}{2}) \psi_{0}- \frac{1}{4}\cdot 2 ( d - 1 ) \psi_{0},
-\overline{\pi}_{ii}(u_1,u_2) =\tilde{\mu}i+\lambda i+h_1(i,u_1)+h_2(i,u_2) &\leq i(\tilde{\mu}+\lambda)+2L &\leq \frac{1}{\theta}(\tilde{\mu}+\lambda) V(i)+2L V(i) &=(2L+\tilde{\mu}+\lambda)\frac{1}{\theta}V(i) &=C_3 V(i).
\left|\times_{j=1}^{d}{K_j}\right|\le \prod_{j=1}^{d}m_j\le\prod_{j=1}^{d}\max\Big\{O\Big(q_j^{-1}\,\rho_n^{-1}\,\big(n'_j\,p(\tau_n/\varkappa_n)\big)^{-1/2}\Big),1\Big\}.
\begin{aligned}\int_0^T \int_{B_{2R}^d}\int_{\mathbb{S}^{d-1}}\mathbf{1}_{\left(\tau,v_{s+k+1},\omega_{k+1}\right) \in \mathcal{B}^+_{IV}}d\omega_{k+1}dv_{s+k+1}d\tau \leq & \leq C_{d,\alpha}(s+k-1) T R \eta^{d-1}\end{aligned}
r_i(P^{-(r-1)}\mathcal{A}(H)P) =\sum_{i_2,\ldots,i_r=1}^np_i^{-(r-1)}a_{ii_2\cdots i_r}p_{i_2}\cdots p_{i_r}& =\sum_{\{i,i_2,\ldots,i_r\}\in E(H)}p_i^{-(r-1)}p_{i_2}\cdots p_{i_r}.
\int_{\mathbb{R}^{n-1}}\int_{\mathbb{R}^{n-1}}a(x-y)\:b((1-t)y+tx)\:\textup{d}x\textup{d}y = \left(\int_{\mathbb{R}^{n-1}}a(x)\:x\right)\left(\int_{\mathbb{R}^{n-1}}b(y)\:y\right) &=\|a\|_{L^1(\mathbb{R}^{n-1})}\|b\|_{L^1(\mathbb{R}^{n-1})}
E_2(\tau) = 1 - 24 \sum_{d=1}^\infty \sigma_1(d) q^d, E_4(\tau) = 1 + 240 \sum_{d=1}^\infty \sigma_3(d) q^d, E_6(\tau) = 1 - 504 \sum_{d=1}^\infty \sigma_5(d) q^d
U(L_{1},L_{2})=\frac{\pi}{48}\biggl(\frac{1}{L_{1}}+\frac{1}{L_{2}}\biggr)- \frac{L_{1}L_{2}}{32\pi}\sum^{\infty}_{p,q=-\infty}\hspace{-.15in}~'\hspace{.1in}(p^{2}L^{2}_{1}+q^{2}L^{2}_{2})^{-\frac{3}{2}}.
H_{1}= 1 + \sum_{a = 1}^{l_{1}}\frac{Q_{a}^{( 1 )}}{| x - x_{a}^{( 1 )}|}; ~ ~ H_{2}= 1 + \sum_{b = 1}^{l_{2}}\frac{Q_{b}^{( 2 )}}{| x - x_{b}^{( 2 )}|}, ~ ~ | x - x_{a}| = ( \sum_{i = 1}^{3}| x_{_{i}}- x_{a_{i}}|^{2})^{1 / 2}.
\left( \beta x^{1-\alpha/\beta}D\right)^{n}\left[ x^{r/\beta}f\left(x\right) \right] =x^{\left( r-n\alpha\right) /\beta}\sum_{k=0}^{n}S\left(n,k;\alpha,\beta,r\right) \beta^{k}x^{k}f^{\left( k\right)}\left(x\right) ,
\int_{t_{0}}^{\infty}r_{1}^{-\frac{1}{\alpha}}(v)\left(\int_{t_{0}}^{v}r_{2}^{-\frac{1}{\beta}}(u)\left(\int_{t_{0}}^{u}\pi^{\gamma}\left(\sigma(s)\right)q(s)s\right)^{\frac{1}{\beta}}u\right)^{\frac{1}{\alpha}}v = \infty ,
L(v(t_n)) := \Big[ \int_0^{\tau}S^{-1}(t_n+r) f \left( S(t_n+r)v(t_n)\right) d r - \tau S^{-1}(t_n+\tau\xi_n) f\left( S(t_n+\tau\xi_n)v(t_n)\right)\Big]