harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\tilde N_{n}^2(k)=\sum\limits_{\substack{\mathbf{p}_{k,1},\mathbf{p}_{k,2}\in \mathcal{P}(d,k)}}\tilde N_{n, \mathbf{p}_{k,1}}(k) \tilde N_{n,\mathbf{p}_{k,2}}(k)= \sum_{\mu=0}^k \tilde N_{n,\mu}(k),
S_{C5}(q) =- \lim_{z\rightarrow 1}\frac{F(z)-F(1)}{(z-1)^2}\cdot \frac{1}{(-q;q)_\infty (q;q)_\infty^2}&=-\frac{F''(1)}{2(-q,q,q;q)_\infty}&=\frac{1}{(q^2;q^2)_\infty}\sum_{n=0}^\infty \frac{q^{2n+1}}{(1-q^{2n+1})^2}.
\epsilon = \tilde{c}_0 \times \frac{1}{\sqrt{\sum_{i \in{\mathcal{C}}_j}N_i}}+\tilde{c}_1\Delta_{\max}\sum_{i \in [M]}\sum_{t=0}^{T-1}\exp\left(-\tilde{c}_2 N_in_x \left(\frac{\rho^{(i)}\|\Sigma^{(i)}_t\|}{\rho^{(i)}\|\Sigma^{(i)}_t\| + \sqrt{n_x}}\right)^2\right),
Q_{1}=[\begin{matrix}1&0&0&0\\ 0&cos(\theta)&-sin(\theta)&0\\ 0&sin(\theta)&cos(\theta)&0\\ 0&0&0&1\end{matrix}]=exp(\theta[\begin{matrix}0&0&0&0\\ 0&0&-1&0\\ 0&1&0&0\\ 0&0&0&0\end{matrix}]).
\{\{\sigma_{1},\sigma_{2}\}\}\left( x\right) =\left\{\sigma_{1}\left(x\right) ,\sigma_{2}\left( x\right) \right\}+d\sigma_{1}\left(\xi_{\sigma_{2}\left( x\right)}\right) -d\sigma_{2}\left( \xi_{\sigma_{1}\left( x\right)}\right)
P_{k}^{(\alpha ,\beta )}(2t^{2}-1)=\frac{2\Gamma (\alpha +\beta +1)\Gamma(k+\beta +1)}{\Gamma (\beta +\frac{1}{2})\Gamma (k+\alpha +\beta +1)}\int_{0}^{1}(1-u^{2})^{\beta -\frac{1}{2}}C_{2k}^{\alpha +\beta +1}(tu)du,
\left( \frac{\delta \rho_{\phi}({\bf k})}{{\dot{\phi}}^{2}+ \frac{4}{3}\rho_{r}}\right)_{H C}\sim \left( \frac{\delta \rho_{\phi}({\bf k})}{\rho_{r}}\right)_{H C}\sim 1 0^{- 3}- 1 0^{- 5},
\begin{array}{l l}{D_{a}= \partial_{a}- i \sigma_{a \dot{a}}^{\mu}\bar{\theta}^{\dot{a}}\partial_{\mu},}&{}\\{\bar{D}_{\dot{a}}= - \partial_{\dot{a}}+ i \theta^{a}\sigma_{\dot{a}a}^{\mu}\partial_{\mu},}&{}\\ \end{array}
X_p^t = [ h_p^1 + q_p^0 | h_p^2 + q_p^1 |\cdots| h_p^t + q_p^{t-1}],Y_p^t = [ b_p^0 | b_p^1 + \lambda_p^1 m_p^{0}| \cdots | b_p^{t-1}+ \lambda_p^{t-1}m_p^{t-2}] M_p^t =[ m_p^0 | m_p^1 | \cdots |m_p^{t-1}],Q_p^t=[ q_p^0 | q_p^1 | \cdots| q_p^{t-1}].
N(t) =\sum_{j=1}^J\beta\int_0^{2-\sqrt{tD_+}\frac{\gamma(\theta_0^j)}{2}}\frac{e^{-z^2}}{\sqrt{\pi}}\mu(B_j,\sqrt{4D_+t}z+tD_+\gamma(\theta_0^j)) dz +O(t^\frac{3}{2}),
\rho(\gamma_1 m_t)R_j = \epsilon([g_\alpha \gamma_1 m_t g_j^{-1}]) &= \epsilon([g_\alpha \gamma_1 g_t^{-1}m g_t g_j^{-1}]) &= \epsilon([g_\alpha \gamma_1 g_t^{-1}g_{t}g_j^{-1}]) - \epsilon([g_\alpha \gamma_1 g_t^{-1}]) \epsilon([g_t g_j^{-1}]).
{^\mathbf{z}\Delta^{i}_x}:={^{z_i}\Delta^{-1}_x}\,{^{z_{i+1}}\Delta^{-1}_x}\dots{^{z_{-1}}\Delta^{-1}_x},{^\mathbf{z}\delta^{i}_x}:={^{z_i}\delta^{-1}_x}\,{^{z_{i+1}}\delta^{-1}_x}\dots{^{z_{-1}}\delta^{-1}_x}.
R_{2}^{\left( 3 \right)}= \sum_{i = 3}^{m + 1}\alpha_{i}( - 1 )^{i + 1}\sum_{k = 0}^{l_{2}- 1}g_{k + 1}^{\left( 2 \right)}( \Phi_{i , k}+ \Phi_{i - 1 , k}) .
\nabla_{z_{\alpha}}^{0}\left( \begin{array}{c}{\langle v |}\\{\hat{V}}\\ \end{array}\right) = \frac{1}{\theta_{\alpha}}\left( \begin{array}{c}{\langle v | \widehat{\bar{z}}_{\alpha}}\\{- \lbrack \widehat{\bar{z}}_{\alpha}, \hat{V}\rbrack}\\ \end{array}\right) .
\left( j^{k}i,j^{N-k}i\right) \ast \left( j^{p}i^{\prime},j^{N-p}i^{\prime}\right) =\sum_{k}\left( j^{p+k}\left( i\ast i^{\prime}\right)_{k},j^{N-\left( p+k\right)}\left( i\ast i^{\prime}\right)_{k}\right) ,
r_1+r_2+\cdots +r_{k-d}\le d+r_{12}+r_3+\cdots +r_{k-d}\le 2d+r_{123}+r_4+\cdots +r_{k-d}\phantom{..}\vdots \le d(k-d-1)+r_{12\ldots (k-d)}&= d(k-d-1).
- \Delta v_k + V^\infty v_k = (V^\infty - V(x))\widetilde{u}_k + \frac{\mu_k}{2 \rho(u_k)}|u_k|^{p(x) - 2}u_k &-\frac{c}{\rho_0}\Big(|w^{(1)}|^{p(x) - 2}w^{(1)}+ \sum_{n=2}^m|w^{(n)}(x - y^{(n)}_k)|^{p^\infty - 2}w^{(n)}(c - y^{(n)}_k)\Big) + \eta_k,
[B,K] = \sum_{k=0}^M\lambda_k\{A^k,C\}+ \sum_{k=0}^{L-1}F_k\{A^k,C\}+ \sum_{k=0}^LG_k\{A^k,C\}+ \sum_{k=0}^{2L-1}Z_k\{A^k,C\}- \sum_{k=0}^MH_k\{A^k,C\},
2\widetilde{\nabla}_{[\mu}\widetilde{\nabla}_{\nu ]}\sigma_{\lambda}^{AA^{\prime}}=-\widetilde{R}_{\mu \nu \lambda}{}^{\rho}\sigma_{\rho}^{AA^{\prime}}+(\widetilde{C}_{\mu \nu B}{}^{A}\sigma_{\lambda}^{BA^{\prime}}+),
x_j^+(-1)=-(r_ks_k)^{-\frac{a_{kj}}{2}}[-a_{kj}]_k^{-1}{\gamma'}^{-\frac{1}{2}}\,\bigl[\,a_k(-1),\,x_j^+(0)\,\bigr]\in{\mathcal U}^k_{r,s}(\hat{\frak{g}}), x_j^-(1)=-(r_ks_k)^{\frac{a_{kj}}{2}}[a_{kj}]_k^{-1}{\gamma}^{-\frac{1}{2}}\,\bigl[\,a_k(1),\,x_j^-(0)\,\bigr]\in{\mathcal U}^k_{r,s}(\hat{\frak{g}}).
\Delta_n \leqslant\frac{A}{n \bar{B}_n^{3/2}}\sum\limits_{i,j=1}^n\left(|\mu_{ij}|^{3}+ E|\bar{X}_{ij}|^3 \right)+\frac{1}{n}\sum\limits_{i,j=1}^n P(|X_{ij}-\mu_{ij}| \geqslant b_n)+ \Theta_n + \Upsilon_n.
[\tilde{\alpha}_1, \ldots, \tilde{\alpha}_{n-1}, (\beta^*f) \tilde{\alpha}_n ] = (\beta^*f) [\tilde{\alpha}_1, \ldots, \tilde{\alpha}_{n-1}, \tilde{\alpha}_n ] + \Pi^{\sharp}(\tilde{\alpha}_1 \wedge \cdots \wedge \tilde{\alpha}_{n-1})(\beta^*f)\tilde{\alpha}_n.
\mathcal{L}w_{11}= \mathcal{L}u_{11}-\mathcal{L}A_{11}\geq -F^{ij,kl}D_1w_{ij}D_1w_{kl}- C[F^{ii}(1+w_{jj})+w_{ii}]-CF^{ii}w^2_{ii}-CF^{11}w^2_{11}.
\prod \limits^{s_l + k-1}_{i = 0}|c(\theta_i)p'(\xi_i)| < (3/5)^{(1/2 + 1/2^{n+1})s_l}\cdot (5/3)^{3k}\leq (3/5)^{(1/2 + 1/2^{n+1})s_l - 3k}.
\lim_{n \to \infty}\mathrm{P}[\tilde{g}_n(x) = \tilde{g}_n(y)] = \sum_{i=1}^3 \mathrm{P}[X \in B_i, Y \in B_i] >\sum_{i=1}^3 \mathrm{P}[X \in A_i, Y \in A_i]
\int_{D_0}|\tilde{F}-(1-b_{t_0}(\Psi_1))fF^{2}|^2_{\tilde{h}}e^{v_{t_0}(\Psi_1)- \tilde{M}}c(-v_{t_0}(\Psi_1)) \le \left(c(T_1)e^{-T_1}+\int_{T_1}^{t_0+1}c(s)e^{-s}ds\right) \int_{D_0}\mathbb{I}_{\{-t_0-1<\Psi_1<-t_0\}}|fF|^2_{\tilde{h}},
\sum_{k=1}^\infty\int_{D(z_k, 3r)}\frac{|f(z)(1+|z|)^m|^p}{e^{\frac{\alpha p}{2}|z|^2}}dV(z) \lesssim \sum_{k=1}^\infty\int_{D(z_k, 3r)}\frac{|f(z)|z|^m|^p}{e^{\frac{\alpha p}{2}|z|^2}}dV(z).
\big|\langle b, \Pi_{j}(G,\tilde G)\rangle\big|= \big|\langle G, [b, R_j](\tilde G)\rangle\big|\leq C \|b\|_{{\rm BMO}(\mathcal G)}\|\tilde G\|_{L^{p}(\mathcal G)}\|G\|_{L^{p'}(\mathcal G)}.
A_1=\sum^m_{\stackrel{i=0}{i\even}}(M^{0,0}_{\frac{i}{2},\frac{2m-i}{2}}+M^{0,0}_{\frac{2m-i}{2},\frac{i}{2}})+\sum^m_{\stackrel{i=0}{i\odd}}(M^{0,0}_{\frac{2m-i-1}{2},\frac{i-1}{2}}+M^{0,0}_{\frac{i-1}{2},\frac{2m-i-1}{2}})+M^{0,0}_{\lfloor\frac{m}{2}\rfloor,\lfloor\frac{m}{2}\rfloor}
f_\Theta(a) = \underbrace{\sum_{i \in S(\eta)}[\Theta^{-1}y]_i U(a)_i}_{\eqqcolon f_1(\eta,a)}+ \underbrace{\sum_{i \notin S(\eta)}[\Theta^{-1}y]_i U(a)_i}_{\eqqcolon f_2(\eta,a)}.
P\lbrace N(s)\ \ |\ \mathcal{T}(s) = x\rbrace = \frac{(a_1-a_2)s\,I_1\bigl(\frac{2\lambda}{a_1-a_2}A(s,x) \bigr)}{2A(s,x)\,I_0\bigl(\frac{2\lambda}{a_1-a_2}A(s,x)\bigr) +(a_1-a_2)s\,I_1\bigl(\frac{2\lambda}{a_1-a_2}A(s,x) \bigr)}
\Gamma^{-1}_M(z\,|\,a) = e^{P(z\,|\,a)}\,w\prod\limits_{n_1,\cdots , n_M=0}^\infty{}' \Bigl(1+\frac{z}{\Omega}\Bigr)\exp\Bigl(\sum_{k=1}^M \frac{(-1)^k}{k}\frac{z^k}{\Omega^k}\Bigr),
= \int D h_{\mu \nu}\exp \left\{- \int d^{3}x \left( g^{2}h_{\mu \nu}^{2}+ V_{\mathrm{t o t a l}}\left[ \varepsilon_{\mu \nu \lambda}\partial_{\mu}h_{\nu \lambda}\right] \right) + 2 \pi i \int_{\Sigma}^{}d \sigma_{\mu \nu}h_{\mu \nu}\right\},
H_{R}( N N ) = N_{F}^{R}( N N ) + \frac{1}{1 2}= \sum_{n = 1}^{\infty}n ({d_{- n}}^{- j}{d_{n}}_{j}+{d_{- n}}^{j}{d_{n}}_{- j}) + \frac{1}{1 2}\; ( j = 2 )
\Phi \ = \ \int d \bar{\theta}d \theta \ \theta ({\cal A}_{\theta}^{a}{\cal D}_{i}^{a b}{\cal E}_{i}^{b}+ \frac{1}{2}f^{a b c}{\cal A}_{\theta}^{a}{\cal A}_{\theta}^{b}{\cal E}_{\bar{\theta}}^{c})
F_X(x)=\frac{1}{2}+\frac{\alpha(x-\mu)}{2}\bigg[K_{\nu}(\alpha|x-\mu|)\mathbf{L}_{\nu-1}(\alpha|x-\mu|)+\mathbf{L}_{\nu}(\alpha|x-\mu|)K_{\nu-1}(\alpha|x-\mu|)\bigg],
D^{\mu \nu}( x , y ) = i \int \frac{d^{4}k}{( 2 \pi )^{4}}\frac{e^{- i k_{\lambda}( x - y )^{\lambda}}}{k_{\lambda}k^{\lambda}+ i \epsilon}\left[ g^{\mu \nu}- \frac{\gamma k^{\mu}k^{\nu}}{k_{\lambda}k^{\lambda}+ i \epsilon}\right] ;
\rho^* \nu_G = \rho^*\big(\sum_{i=1}^N (-1)^i x_i dx_1 \wedge \ldots \wedge \widehat{dx_i}\wedge \ldots \wedge dx_N\big) = \prod_{i=1}^m (y_i^*)^{|E(G_i)|-1}\nu_G.
\begin{array}{l l}c_{n-1}(\alpha_1)=(n-2)!\Sigma^2v_{2n-4}c_n(\alpha_1)=\frac{(n-1)!}{2}\Sigma^2v_{2n-2}c_{n-1}(\alpha_2)=0 c_n(\alpha_2)=(n-1)!\Sigma^2v_{2n-2}.\end{array}
\|W_N^k\|_{H_2^j}\leq C \|W_N\|_{H_2^2}\; \; j \leq 2, \; \; \|W_N^k\|_{H_2^j}\leq C k^{\frac{j-2}{2}}\| W_N\|_{H_2^2}\; \; j=3,\dots, 6.
\epsilon^{-1}\left(f^n(u_{h,1}^n)-f^n(u_{h,2}^n),U_h^n\right)_{\mathcal T_h}=\epsilon^{-1}\left((u_{h,1}^n)^3-(u_{h,2}^n)^3,U_h^n\right)_{\mathcal T_h}&=\epsilon^{-1}\left((u_{h,1}^n)^2+u_{h,1}^n u_{h,2}^n+(u_{h,2}^n)^2,(U_h^n)^2\right)_{\mathcal T_h}&\geq 0.
\frac{W_{\xi',\eta}(i)}{W_{\xi',\eta}(i-1)}= \frac{M_{\xi'}}{M_{\eta}}\frac{Y_{\eta}-X_{\xi'}\omega^{i}}{Y_{\xi'}- X_{\eta}\omega^{i}}\qquad \frac{W_{\xi,\eta'}(i)}{W_{\xi,\eta'}(i-1)}= \frac{M_{\xi}}{M_{\eta'}}\frac{Y_{\eta'}-X_{\xi}\omega^{i}}{Y_{\xi}- X_{\eta'}\omega^{i}}
\mathbf{h}= \lambda_{1}\mathbf{e}_1+\lambda_{2}\mathbf{e}_2+\lambda_{123'}\mathbf{e}_{123'}= [\lambda_1+\lambda_{123'},\lambda_2+\lambda_{123'},\lambda_{123'},\lambda_1+\lambda_2+2\lambda_{123'}, &\lambda_1+2\lambda_{123'},\lambda_2+2\lambda_{123'},\lambda_1+\lambda_2+2\lambda_{123'}]^\intercal
\tilde{D}=\left( \begin{array}{c}D^{(1)}D^{(2)}\end{array}\right),~\tilde{E}=\left( \begin{array}{c}E^{(1)}E^{(2)}\end{array}\right),~\tilde{B_\mu}=\left( \begin{array}{c}B^{(1)}_\mu B^{(2)}_\mu\end{array}\right),~\tilde{\psi_a}=\left( \begin{array}{c}\psi^{(1)}_a \psi^{(2)}_a\end{array}\right)
a ( \tau ) = - \sinh^{2}\tau \mathrm{a n d}b ( \sigma , \tau ) = - \frac{\left( \sinh \tau - 1 \right)^{2}\left( \cosh \tau - \cos \sigma \right)^{2}}{\rho_{g}^{2}\left( \sinh \tau + 1 \right)^{6}},
A_{\alpha}^{\, \, \, \, i}= \xi^{\beta}f_{\beta \alpha}^{\, \, \, \, \, \, \, i}, \, \, \, \, \, \, \, \, \, \, \, \, A_{i}^{\, \, \alpha}= \xi^{\beta}f_{\beta i}^{\, \, \, \, \, \alpha}.
\lambda_{l}^{2}=\left\Vert (x+iy)^{l}\right\Vert^{2}=\frac{1}{4\pi}\int_{S^{2}}(1-z^{2})^{l}dS=\sum_{k=0}^{l}\binom{l}{k}\frac{(-1)^{k}}{2k+1}=\frac{(l!)^{2}2^{2l}}{(2l+1)!}
\begin{aligned}\{x_{11},x_{12}\}= x_{11}x_{12}, \{x_{12},x_{22}\}= x_{12}x_{22}, \{x_{11},x_{21}\}= x_{11}x_{21}, \{x_{21},x_{22}\}= x_{21}x_{22}, \{x_{12},x_{21}\}= 0, \{x_{11},x_{22}\}= 2x_{12}x_{21}.\end{aligned}
1 \ge \mathbb E [ Z_{t}] \ge \mathbb E \Big[ \frac{Z_{t}}{\, 1 + \varepsilon Z_{t}\,}\Big] \, =\, \frac{1}{\, 1 + \varepsilon\,}- \mathbb E \Big[ \int^{t}_{0}\frac{\varepsilon Z_{s}^{2}\lvert b(s, X_{s}, F_{s}) \rvert^{2}}{\, ( 1 + \varepsilon Z_{s})^{3}\,}{\mathrm d}s \Big] \,
S_{N,M,3}+S_{N,M,4}=\sum_{k=1}^{M}\left(\frac{2^{n_{k}}}{2^{n_{1}}+\cdots+2^{n_{M}}}\right)^2\left(\widetilde{r}(2^{n_{k}})+2^{-n_{k}+1}\,(2\,\widetilde{r}(2^{n_{k}+1})-\widetilde{r}(2^{n_{k}})) \sum_{j=k+1}^{M}2^{n_{j}}\right)+O(\epsilon),
\phi \left( \gamma \right) =\tau_{0}\,\delta \left( \gamma -\alpha_{0}\right) +\sum_{\nu =1}^{N}\tau_{\nu}\,\delta \left( \gamma -\alpha_{\nu}\right) ,\;\;\begin{array}{c}0\leq \alpha_{0}<\ldots <\alpha_{N}<1, \tau_{0},\tau_{1},\ldots ,\tau_{N}>0,\end{array}
{\rm Hilb}\big(3T_3^{\rm red},t\big)=(1,3,4,1), \dim \big(3T_3^{\rm red}\big) =9,{\rm Hilb}\big(3T_4^{\rm red},t\big)=(1,6,19,32,19,6,1),\dim \big(3T_4^{\rm red}\big)=84,{\rm Hilb}\big(3T_5^{\rm red},t\big)=(1,10,55,190,383,370,227,102,34,8,1), \dim\big(3T_5^{\rm red}\big)=1374.
H^{s,m}(\mathbb{R}^n) = \{f \in L^2(\mathbb{R}^n) ; \| f \|_{H^{s,m}}< \infty \}, \| f \|_{H^{s,m}}= \left( \sum_{|\alpha| \le s}\int_{\mathbb{R}^n}(1+|x|^2)^m |\partial_x^{\alpha}f (x) |^2\,dx \right)^{1/2}.
\tan \theta = \frac{\tilde{b}}{\alpha^{\prime}}= \frac{( 2 \pi )^{2}\tilde{b}^{2}}{\alpha^{\prime}\tilde{V}_{2}}\frac{N_{p - 2}}{N_{p}}\Longrightarrow \frac{N_{p - 2}}{N_{p}}= \frac{\tilde{V}_{2}}{( 2 \pi )^{2}\tilde{b}}.
\underset{i = p_1}{\overset{r_1}\sum}{_{r_1}}C_{i}\ 2^{i - p_1}\S(i, p_1) \left[ \underset{j = 0}{\overset{r_1 - i}\sum}{_{r_1 - i}}C_j \ (2s_1 + s_2)^{r_1-i-j}\left[ \sum_{\substack{p_2 - j \leq l \leq r_2 p_2-j \geq 0}}{_{r_2}}C_l \s_2^{r_2 - l}\S(l+j, p_2) \right] \right]
s_{n}( u ) = \frac{( u_{n}e_{\bar{z}}, e_{\bar{z}})}{( e_{\bar{z}}, e_{\bar{z}})}= \frac{( u_{n}e_{\bar{z}}) ( z )}{e_{\bar{z}}( z )}= \frac{u_{n}( 1 + z \bar{z})^{n - 1}}{( 1 + z \bar{z})^{n - 1}}.
p_{C}^{\alpha}(x) :=\alpha r_{C}^*(x)+\tfrac{1-\alpha}{2}\|x\|^2, P_{C}^{\alpha}(x) :=\nabla p_{C}^{\alpha}(x)=\alpha\Pi_{C}(x)+(1-\alpha)x R_C(x) :=2\Pi_C(x)-x.
\sum_{j=0}^{a-1}\sum_{i=0}^j\sum_{r=0}^{b-1-a}\left[\binom{b-1-a}{r}-1\right] =(2^{b-1-a}-b+a)\sum_{j=0}^{a-1}\sum_{i=0}^j1=\binom{a+1}{2}(2^{b-1-a}-b+a)
D_1 \ll ( t_3^{-\sigma_3}+1 )\begin{cases}t_3^{-\sigma_1-\sigma_3}(\sigma_2>\frac{3}{2}) t_3^{-\sigma_1-\sigma_3}\log t_3 (\sigma_2=\frac{3}{2}) t_3^{\frac{3}{2}-\sigma_1-\sigma_2-\sigma_3}(\sigma_2<\frac{3}{2}). \end{cases}
\frac{2 \pi}{L}\nu \sin \theta ( \nu ) - m \cos \theta ( \nu ) + \frac{N g^{2}L}{1 6 \pi^{2}}\int \! \! \! \! \! \! - \mathrm{d}\nu^{\prime}\frac{\sin ( \theta ( \nu ) - \theta ( \nu^{\prime}) )}{( \nu - \nu^{\prime})^{2}}= 0 \ .
\tau(U^*TP) = \tau(PV|T|P) &= \tau\big((PV|T|^{1/2})(|T|^{1/2}P)\big) &\leq \frac{1}{2}\big(\tau(PV|T|V^*P) + \tau(P|T|P)\big) &= \frac{1}{2}\big(\tau(|T|V^*PV) + \tau(|T|P)\big) &\leq \max \{\tau(|T|Q),\ \tau(|T|P) \},
W^{t r \; ( 2 )}= - \frac{e^{- 1}}{3 2 \pi^{2}}K ( S ) \frac{1}{\eta ( T )^{6}}\frac{1}{j^{a}( T ) ( j ( T ) - 1 7 2 8 )^{b}}, \; a , b \; \in \; Z^{\dagger}.
Q^{(N)}\stackrel{\eqref{symbolic power partition containment 001}}{\subseteq}\sum_{\mathcal{A}\in S(N)}\prod_{i=1}^n (P_i R)^{(A_i)}\stackrel{\mathbf{(A)}}{\subseteq}\sum_{\mathcal{A}\in S(N)}\prod_{i=1}^n (P_i R)^{\lceil A_i / D_i \rceil}\stackrel{\mathbf{(B)}}{\subseteq}Q^{\lceil N / D \rceil}.
\begin{cases}u^3(x-1)^{n-r_1+k_5}p_5(x)-u^3(x-1)^{n-r_1-r_2+k_4+k_6}p_4(x)p_6(x) \mbox{if~}n-r_1+k_4 > r_2, u^3(x-1)^{r_2-k_4+k_5}p_5(x)-u^3(x-1)^{k_6}p_4(x)p_6(x) \mbox{if~}n-r_1+k_4 \le r_2, \end{cases}
V_{w_{1}}^{*}V_{w_{2}}=V_{i_1}^{*}V_{i_{2}}^{*}\cdots V_{i_{n-k}}^{*}V_{j_{m-k}}V_{j_{m-k-1}}\cdots V_{j_{1}}=0 \textrm{~~\big( Since $V_{i}^{*}V_{j}=0$ if $i \neq j$ and $i_{n-k}\neq j_{m-k}$ \big)}.
\lambda^{a/2 + 1}\left( (n-2-a)\int_{B_{\rho}}r^{-a}|\nabla u|^2 \d x + \frac{a(8-a)}{4}\int_{B_{\rho}}r^{-a}u_r^2 \d x\right) &\leq (n-2-a) \int_{B_1}|x|_{A^{-1}(0)}^{-a}|\nabla u|_{A(0)}^2 \zeta^2 \d x + \dfrac{a(8-a)}{4}\int_{B_1}|x|_{A^{-1}(0)}^{-a-2}|x \cdot \nabla u |^2 \zeta^2 \d x.
\begin{array}{rl}Z^{IP}= \max \ -x_1 + 2x_2 + x_3 \mbox{s.t.}\ -x_1 + x_2 \leq 0.5, & x_1 + x_2 \leq 1.5, & x_3 \leq 0.5, & x \in \{0,1\}^3.\end{array}
\begin{aligned}\frac{1}{\sqrt{2}}= \lambda_1 \phi^1 + \sum_{k>1}\lambda_k \phi^k \leq \lambda_1\phi^1 + (\lambda_2 + \cdots + \lambda_k)\sqrt{\frac{\lambda_2y^2+ \cdots + \lambda_k y^k}{\lambda_2 + \cdots +\lambda_k}}&< \sqrt{\lambda_1 y^1 + \cdots + \lambda_k y^k}= \frac{1}{\sqrt{2}},\end{aligned}
A_{i, a}= Y_{i, aq_{i}}Y_{i, aq_{i}^{-1}}\prod_{C_{ji}=-1}Y_{j, a}^{-1}\prod_{C_{ji}=-2}Y_{j, aq}^{-1}Y_{j, aq^{-1}}^{-1}\prod_{C_{ji}=-3}Y_{j, aq^{2}}^{-1}Y_{j, a}^{-1}Y_{j, aq^{-2}}^{-1},
\lambda_{i}= - \frac{1}{m}D_{i}\left( m \varphi_{0}+ i e A_{0}\varphi \right) ; l a m b d a_{i}^{\ast}= - \frac{1}{m}D_{i}^{\ast}\left( m \varphi_{0}^{\ast}+ i e A_{0}\varphi^{\ast}\right) .
\lim_{\beta\to\infty}\frac{J_{\{m\}}}{J_0}=\lim_{\beta\to\infty}\frac{\int_G d\mu_H\, \exp(-\beta (1-1/N\Re\mbox{tr}(g)))\,\, \frac{\mbox{tr}(\pi(g))}{\mbox{dim}(\pi)}}{\int_G d\mu_H \,\exp(-\beta(1-1/N\Re\mbox{tr}(g)))}=1
\begin{aligned}r_j^{(k)}=\beta_j r_j^{(k-1)}-d_k &=q^{-1}\left(\beta_j\sum_{i=1}^n w^{(k-1)}_i \beta_j^{-i}-q d_k\right) &=q^{-1}\left(w^{(k-1)}_1-q d_k+\sum_{i=1}^{n-1}w^{(k-1)}_{i+1}\beta_j^{-i}\right) &=q^{-1}\left(\sum_{i=1}^n w^{(k)}_i \beta_j^{-i}\right).\end{aligned}
\begin{array}{c c c c c c c}{\nabla_{i}\Phi}&{=}&{( \partial_{i}+{\frac{1}{2}}p \partial_{i}{\cal K}) \Phi}&{;}&{\nabla_{i^{*}}\Phi}&{=}&{( \partial_{i^{*}}-{\frac{1}{2}}p \partial_{i^{*}}{\cal K}) \Phi}\\ \end{array}
\sigma_{2}^{--}\sigma_{3}^{-}={\frac{\Gamma ( 1 + x )}{\Gamma ( 1 - x )}}{\frac{\Gamma ( 1 / 2 - x )}{\Gamma ( 1 / 2 + x )}}{\frac{\Gamma ( 1 / 2 - \Delta + x )}{\Gamma ( 1 / 2 - \Delta - x )}}{\frac{\Gamma ( - \Delta - x )}{\Gamma ( - \Delta + x )}}
\dot{A}=\hat{r}(\dot{A}_{r}-A_{\theta}\dot{\theta}-A_{\phi}\dot{\phi}sin\theta)+\hat{\theta}(\dot{A}_{\theta}+A_{r}\dot{\theta}-A_{\phi}\dot{\phi}cos\theta)+\hat{\phi}(\dot{A}_{\phi}+A_{r}\dot{\phi}sin\theta+A_{\theta}\dot{\phi}cos\theta)
N_k =\binom{\Delta}{\frac{\Delta+k-x}{2}}-\binom{\Delta}{\frac{\Delta-x-k}{2}}&=\binom{\Delta}{\frac{\Delta+k-x}{2}}\left(1-\frac{\Big(\frac{\Delta+k-x}{2}\Big)!\Big(\frac{\Delta-k+x}{2}\Big)!}{\Big(\frac{\Delta+k+x}{2}\Big)!\Big(\frac{\Delta-k-x}{2}\Big)!}\right).
\left| \frac{n^\gamma P_{\vec{n}-\vec{e}_j}(n^\gamma x)}{P_{\vec{n}}(n^\gamma x)}\right| \leq \sum_{k=1}^{|\vec{n}|}\frac{A_{\vec{n},k}}{|x-x_{\vec{n},k}/n^\gamma|}\leq \frac{1}{\delta},
|\int_{t}^{t+2(r+1)}ds \frac{\lambda''(s)}{r}\int_{0}^{s-t}d\rho \frac{\rho}{(s-t)}\left(1+\frac{r^{2}-1-\rho^{2}}{\sqrt{(r^{2}-1-\rho^{2})^{2}+4r^{2}}}\right)| &\leq C r \sup_{x \geq t}\left(|\lambda''(x)|\right) \int_{t}^{t+2(r+1)}\frac{(s-t)}{1+r^{2}}ds &\leq C r \sup_{x \geq t}\left(|\lambda''(x)|\right)
\delta^{( 2 )}S_{\phi}= - \frac{3}{8 \kappa_{5}^{2}}\int \sqrt{- g_{0}}\left( \frac{{\cal L}_{\phi}}{3}h_{a b}h^{a b}+ ( \partial \delta \phi )^{2}+ \frac{1}{2}\frac{\partial^{2}U}{\partial \phi^{2}}\delta \phi^{2}+ 2 h^{a b}\partial_{a}\delta \phi \partial_{b}\phi \right)
r = \Big[ d(P)^2 \Big(38 |P|^{12}+220 |P|^{11}+123 |P|^{10}+305 |P|^9+89 |P|^8-98 |P|^7+34 |P|^6+20 |P|^5-89 |P|^4 +98 |P|^3-43 |P|^2+7|P| \Big) \Big] \Big/ \Big[ (|P|-1)^2 (|P|+1)^2 \left(|P|^5+7 |P|^4-3 |P|^3+6 |P|^2-4 |P|+1\right)^2 \Big],
( - \partial_{0}^{2}+ D_{-}D_{+}- 2 e^{2}| \phi |^{2}) \psi_{\downarrow}- 2 i e N \psi_{\downarrow}- \sqrt{2}e ( D_{-}\phi ) \chi_{\uparrow}+ \sqrt{2}i e \kappa \phi \chi_{\downarrow}= 0 ,
T = \left( \begin{array}{c c c c c}{T_{( 1 )}}&{}&{}&{0}&{}\\{}&{T_{( 2 )}}&{}&{}&{}\\{0}&{}&{\cdots}&{}&{}\\{}&{}&{}&{T_{( \tilde{N}_{c})}}&{}\\{}&{}&{}&{}&{0}\\ \end{array}\right) ,
\pi(x^i \frac{\partial}{\partial x^j}-x^j\frac{\partial}{\partial x^i},\frac{\partial}{\partial r})=(\tilde{x}^i \frac{\partial \tilde{x}^j}{\partial u^a}-\tilde{x}^j\frac{\partial \tilde{x}^i}{\partial u^a})\tilde{\sigma}^{ab}\pi_{br}.
\left(\begin{array}{@{\,}c@{\,}}y_1 y_2 y_3 Z \varepsilon \end{array}\right) = \left(\begin{array}{@{\,}c@{\,}}u_{1}w^{2}u_{2}w^{5}u_{3}w^{8}\zeta w^{6}w^{7}\end{array}\right).
\mathbf{E}( \mathbf{x}) = \frac{1}{4 \pi}\int \rho_{e}( \mathbf{x}^{\prime}) \frac{\mathbf{r}}{r^{3}}d^{3}x^{\prime}- \frac{1}{4 \pi}\int \mathbf{J}_{m}( \mathbf{x}^{\prime}){\bf \times}\frac{\mathbf{r}}{r^{3}}d^{3}x^{\prime}
\begin{cases}u(0,t)=h_0(t),\,\,u(L,t)=h_1(t),\,\,u_{x}(L,t)=h_2(t), v(0,t)=g_0(t),\,\,v(L,t)=g_1(t),\,\,v_{x}(L,t)=g_2(t).\end{cases}
\mathcal M_{\alpha, \beta, m}\left(\frac{1}{|x|^{2\alpha}|y|^{2\beta}}\right)= 2^{2j}(\alpha)_m (\beta)_m\left( \frac{\langle x, y\rangle}{|x|^{2}|y|^{2}}\right)^m\left(\frac{1}{|x|^{2\alpha}|y|^{2\beta}}\right).
\begin{aligned}u-h \geq (u_0-h_0) &-2Cx\left(F_0+\int_0^x F\right)^{1/2}\left\{1-2C\vartheta_0^{-8}\left(F_0+\int_0^x F\right)^2x\right\}^{-1/4}.\end{aligned}
T^2f\cdot d_0\cdot k\cdot T{L}f\cdot q_f \xrightarrow{\theta\cdot k\cdot T{L}f\cdot q_f}Ti_B\cdot d_1\cdot k\cdot T{L}f\cdot q_f = Ti_B\cdot Tf \cdot q_f \xrightarrow{Ti_B\cdot\bar\alpha_2}Ti_B\cdot i_B\cdot{R}f
ds^2= \frac{sin^2{\theta}}{\rho^2}\left[\left( r^2 +a^2 \right) d\phi - a \, dt \right]^2-\frac{D}{\rho^2}\left[ dt - a \sin^2{\theta}\, d\phi \right]^2 +\frac{\rho^2}{D}\, dr^2 +{\rho^2}d\theta^2 \;,
\|s(p)\|_{v}=\begin{cases}\displaystyle{\frac{|\rho_{s}(p_{0},\dots, p_{s})|_{v}}{\sum_{j}|p_{j}|_{v}}}v \text{is Archimedean}, begin{align*}4mm]\displaystyle{\frac{|\rho_{s}(p_{0},\dots, p_{s})|_{v}}{\max_{j}|p_{j}|_{v}}}v , \end{cases}
\|h^{(2)}\|_1 =\|D^*_{S_0^c}h\|_1-\|h^{(1)}\|_1\leq k\alpha-\ell \cdot \frac{\alpha}{t-1}=(k(t-1)-\ell)\frac{\alpha}{t-1}, \|h^{(2)}\|_\infty \leq \frac{\alpha}{t-1}.
F(a,p,k;b,t,s;x)=\prod_{i=1}^{r}\prod_{j=1}^{q}\frac{\Gamma(\frac{b_{j}}{s_{j}})}{\Gamma(\frac{a_{i}}{k_{i}})\Gamma(\frac{b_{j}}{s_{j}}-\frac{a_{i}}{k_{i}})}\int_{0}^{1}t^{\frac{a_{i}}{k_{i}}-1}(1-t)^{\frac{b_{j}}{s_{j}}-\frac{a_{i}}{k_{i}}-1}e^{\frac{p_{i}}{t_{j}}xt}dt.
\left\|p/q \right\| = \sum_{j=1}^{m}w_{j}\Delta \left(p/q, b_j/c_j\right)= \sum_{j=1}^{N}w_{j}(b_{j}q-pc_{j}) + \sum_{j=N+1}^{m}w_{j}(-b_{j}q+pc_{j}).
\lim_{m\to\infty}\frac{\sum_{k=1}^m (N+\tau^{N,1}_k)(c)_k}{\sum_{k=1}^m(N+\tau^{N,1}_k)(c)_k+\tau^{N,0}_k}=\frac{c(N+E\tau^{N,1})}{c(N+E\tau^{N,1})+E\tau^{N,0}}
h_{s}^{(j)}(t)=\frac{w_{s}^{(j)}p_{s}(x^{(t)};\mu_{s}^{(j)},\Sigma_{s}^{(j)})}{\sum_{i=1}^{n}w_{i}^{(j)}p_{i}(x^{(t)};\mu_{i}^{(j)},\Sigma_{i}^{(j)})}.
\nabla \cdot \nabla u+k^2n(x)u=0, D, begin{align*}1mm] \Delta u+k^2u=0, \mathbb R^2 \setminus D, begin{align*}1mm] u(x)=e^{ikx\cdot d}+u^s({x}), \mathbb R^2 , begin{align*}1mm] \lim_{r\to\infty}\sqrt{r}\left(\frac{\partial u^s}{\partial r}-iku^s \right)=0,
T_{n}(2x-1):=\frac{2^{2n}(n!)^{2}}{(2n)!}\sum_{k=0}^{n}(-1)^{n+1}\frac{\binom{n-\frac{1}{2}}{k}\binom{n-\frac{1}{2}}{n-k}}{\binom{n}{k}}B_{k}^{n}(x),
W^{( 1 )}= - i g^{2}t^{a}t^{a}\int \frac{d^{d}q}{( 2 \pi )^{d}}\frac{v_{\mu}{\mit \Pi}_{R}^{\mu \nu}( q ) v_{\nu}^{\prime}}{q^{4}( q \cdot v ) ( q \cdot v^{\prime})}.
a_{2}^{1234}=a_{2}^{1243}=a_{2}^{1324}=a_{2}^{1342}=a_{2}^{1423}=a_{2}^{1432}= a_{2}^{2134}=a_{2}^{2143}=a_{2}^{2314}=a_{2}^{2341}=a_{2}^{2413}=a_{2}^{2431}= a_{2}^{3124}=a_{2}^{3142}=a_{2}^{3214}=a_{2}^{3241}=a_{2}^{3412}=a_{2}^{3421}= a_{2}^{4123}=a_{2}^{4132}=a_{2}^{4213}=a_{2}^{4231}=a_{2}^{4312}=a_{2}^{4321}
\int_{F_\nu^n}f_\nu(x_\nu)f_\nu(\delta_\nu x_\nu) d\alpha_\nu^n = \alpha_\nu^n(A_\nu \cap \delta_\nu^{-1}A_\nu) & \leq \min(\alpha^n_\nu(A_\nu), \|\delta_\nu\|^{-n}_\nu \alpha^n_\nu(A_\nu)).
\nabla(w)-\nabla(ws_i) =\partial(w,s_i){}=\partial(w\overline{u}^{d-i}){}=\partial(w\overline{u},s_{i+1}){}=\nabla(w\overline{u})-\nabla(w\overline{u}s_{i+1}){}=\nabla(w\overline{u})-\nabla(ws_i\overline{u})
\kappa^{2}g^{2}\alpha^{\prime}\left( - \frac{1}{2}( \frac{2 \cdot 2^{8}\sqrt{2}}{3^{5}})^{2}\right) \left( A_{i}A_{j}\right)^{2}= - \kappa^{2}g^{2}\alpha^{\prime}\left( \frac{6 5 5 3 6}{5 9 0 4 9}\right) \left( 2 A_{i}A_{i}A_{j}A_{j}\right) .
b^{( n )}( x^{2}) = \int_{- \infty}^{x^{2}}d \lambda \int_{- \infty}^{\lambda}d \lambda^{\prime}H^{\prime \prime}( \lambda^{\prime}) b^{( n - 1 )}( \lambda^{\prime}) \quad \mathrm{f o r n = 1 , 2 , 3 , \dots .}
\frac{df_{m+1}}{dv}(v)-\frac{\lambda}{3\kappa^2}v=\frac{\lambda}{18\kappa^2}\left\{2v(y_{m}(v)+1)+\frac{1}{v^2}\int_0^vv'^3\frac{dy_{m}}{dv}(v')dv'\right\}+\mathcal{O}_{\delta_2}(v^2).
\widehat{\Sigma}_{\mathrm{D}}(\delta_1, \delta_2) = \left\{\gamma \in \mathcal{P}(\mathcal{S}_1 \times \mathcal{S}_2): \boldsymbol{K}_1( \gamma_1, \mu_1) \leq \delta_1, \widehat{\boldsymbol{K}}_2( \gamma_2, \mu_2) \leq \delta_2 \right\}.
\int_{x_i\ge0,\sum x_i\le1}e(\sum_{i=1}^m(\alpha_i-\alpha_{m+1}))dx_1\dots dx_m = \lim_{\epsilon\to0}\int_{x_i\ge0,\sum x_i\le1}e(\sum_{i=1}^m(\beta_i-\beta_{m+1}))dx_1\dots dx_m &=\frac{(-1)^m}{(2\pi i)^m}\lim_{\epsilon\to0}\frac{N(\epsilon)}{D(\epsilon)},
\sum_{r,s < 0}(-1)^{r+s}q^{4\binom{r}{2}+4\binom{s}{2}+12rs}q^{14r}q^{14s}q^9 &+\sum_{r,s < 0}(-1)^{r+s}q^{4\binom{r}{2}+4\binom{s}{2}+12rs}q^{10r}q^{10s}q^4 +\sum_{r,s < 0}(-1)^{r+s}q^{4\binom{r}{2}+4\binom{s}{2}+12rs}q^{6r}q^{6s}q &+\sum_{r,s < 0}(-1)^{r+s}q^{4\binom{r}{2}+4\binom{s}{2}+12rs}q^{2r}q^{2s}.
\mathbb{E}\left[ \frac{1}{t}\int_0^t \left( 1 - | a(\theta_s \omega) |^2 \right) \, ds \right] = \frac{1}{t}\int_0^t \mathbb{E}\left[ 1 - |a(\theta_s \omega) |^2 \right] \,ds &= \frac{1}{Z_\sigma}\int_{\mathbb{R}}(1 - y^2) \, e^{\frac{2}{\sigma^2}(\frac{1}{2}y^2 - \frac{1}{4}y^4)}\, dy
{\cal L}_{g}= -{\frac{1}{2}}\sqrt{- g}g^{\alpha \beta}G_{\alpha \beta}+{\frac{\tilde{Y}M \tilde{Y}}{2 \sqrt{- g}( \partial a )^{2}}}-{\frac{\tilde{Y}L \partial Y \cdot \partial a}{2 ( \partial a )^{2}}}.
\gamma = (W^\rho_u)^{-1}T^{-1}_1 \Big(\sum_i W^\rho_u(\gamma)(x_i) \psi_i \# W^\rho_u(u)\Big) \gamma = \sum_i \lambda_i\big(T^{-1}_2 W^\rho_u(\gamma)\big)\rho^*(x_i)u \gamma = \sum_i c_i\,\,(T^{-1}_3 W^\rho_u(\gamma))\,\rho^*(x_i)u
\widehat{D}_{0}\pi_{a}+ i \varsigma{\frac{\partial E_{a}}{\partial \phi_{i}}}\tau_{i}- i \overline{{p}}^{\prime}{\frac{\partial \overline{{J}}^{a}}{\partial \overline{{\phi}}_{i}}}\overline{{\tau}}_{i}+ i E_{a}\beta - i \overline{{J}}^{a}\overline{{\gamma}}= 0