harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
(|D_{x^\prime}D^{m-1}u-(D_{x^\prime}D^{m-1}u)_{Q_r^+(X_0)}|)_{Q_r^+(X_0)}+\lambda^{\frac{1}{2}}(|u-(u)_{Q_r^+(X_0)}|)_{Q_r^+(X_0)}\le N\kappa^{-\gamma}\sum_{k=0}^m \lambda^{\frac{1}{2}-\frac{k}{2m}}(|D^k u|^2)_{Q_{\kappa r}^+(X_0)}^\frac{1}{2},
\int_{\Omega}\frac{1}{8}\vert \beta_{a}\vert^2_{1,}+\frac{M}{100 r^4}\vert Z_{ab}\vert^2_{2}d\mu_{\Omega}\leq{}\frac{19}{10}E_{\xi}(\Sigma_1) - \frac{1}{10}E_{\xi}(\Sigma_2)=\frac{9}{5}E_{\xi}(\Sigma_1).
\sigma_{2}:T^{*}M|_{Q,P}\rightarrow T^{*}M;(\sigma_{2}^{*}\Theta^{\alpha}:=\epsilon^{\alpha},\sigma_{2}^{*}\Theta_{\alpha}:=\epsilon_{\alpha},\sigma_{2}^{*}Q^{i},\sigma_{2}^{*}P_{i})\mapsto (\Theta^{\alpha},\Theta_{\alpha},Q^{i},P_{i}),
\sum_{r=0}^j\frac{1}{\beta_{j-r}}\cdot \sum_{\ell=1}^k\frac{c_{\ell+r}}{\beta_{k-\ell}}=-\sum_{r=0}^j\frac{1}{\beta_{j-r}}\cdot \sum_{i=0}^r\frac{c_i}{\beta_{k+r-i}}&=-\sum_{\ell=0}^j\frac{1}{\beta_{k+\ell}}\cdot\sum_{r=0}^{j-\ell}\frac{c_r}{\beta_{j-\ell-r}}=-\frac{1}{\beta_{k+j}}
\mathcal{F}^{S,\pi}_{\preceq_1}(F_1^\ast,\dots,F_d^\ast):= \bigg\{F : \begin{array}{l}F \mathbb{R}^d F_i\preceq_1 F_i^\ast i=1,\dots, d F(s)\leq \pi_s s\in S \end{array}\bigg\},
t_{ij}= \begin{cases}s_{ij}j \in K_i\backslash \{0,n\}a_{ij}j =0 s_{ik_i^*}j = n (1-\gamma_{ij})s_{il(i,j)}+ \gamma_{ij}s_{ir(i,j)}j \notin K_i \cup \{0\}\cup \{n\},\end{cases}
(q+q^{-1})\nu_x = q^2 + q^{-2}- \frac{q^4 y^2 z^2 - q^{-4}z^2y^2}{q^2-q^{-2}}, (q+q^{-1})\nu_y = q^2 + q^{-2}- \frac{q^4 z^2 x^2 - q^{-4}x^2z^2}{q^2-q^{-2}}, (q+q^{-1})\nu_z = q^2 + q^{-2}- \frac{q^4 x^2 y^2 - q^{-4}y^2x^2}{q^2-q^{-2}}.
\beta^{0}=\left( \begin{array}{ccccc}0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0\end{array}\right) ;\ \beta^{1}=\left( \begin{array}{ccccc}0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0\end{array}\right) ;
u = \Phi ( \chi_{x}, \chi_{x x}, \chi_{x x x}) \equiv \frac{\chi_{x x x}}{\chi_{x}}- \frac{\chi_{x x}^{\; 2}}{\chi_{x}^{2}}+ 2 \alpha \chi_{x x}+ \alpha^{2}\chi_{x}^{2}+ 2 \alpha \varepsilon + \frac{\varepsilon^{2}}{\chi_{x}^{2}}
0 =\sum\limits_{i\in I^g(x)}\lambda_i\nabla g_i( x)+\sum\limits_{j\in\mathcal P}\rho_j\nabla h_j(x)+\sum\limits_{l\in I^{\varphi^t_\textup{FB}}(x)}\xi_l\left(\alpha_l\nabla G_l(x)+\beta_l\nabla H_l(x)\right)
T_0 = 0; T_{i+1}= T_i + t_{i+1}; z_{i}= R^{k_i -k_{i+1}}. z(z_{i-1},t_i); \gamma(\kappa,z_0,t) = \gamma_{k_1}(z_0,T_0,T_1,t) \gamma_{k_2}(z_1,T_1,T_2,t)\cdots \gamma_{k_n}(z_{n-1},T_{n-1},T_n,t).
P = [2]P_0 =(x_1(t), y_1(t)) =\left(\frac{1}{144}t^2+\frac{1231}{72}t-\frac{5143775}{144}, -\frac{1}{1728}t^3-\frac{2335}{576}t^2+\frac{13493375}{576}t-\frac{29962489375}{1728}\right)
(G_{0}\sigma_{3}G_{0}^{-1})_{21}(G_{0}\sigma_{+}G_{0}^{-1})_{12}+(G_{0}\sigma_{3}G_{0}^{-1})_{12}(G_{0}\sigma_{+}G_{0}^{-1})_{21}+ 2(G_{0}\sigma_{3}G_{0}^{-1})_{11}(G_{0}\sigma_{+}G_{0}^{-1})_{11}= 0,
y=t^2(1-t)^2(2-t)^2sin(\pi x), &p=\nu(-t^2((2t-t)(t-2)^2-t^2(2t-4)(t-1)^2-2t(t-1)^2(t-2)^2-\pi^2t^2(1-t)^2(2-t)^2sin(\pi x).
\sum_{s = 1 , 2 , 5}g ( s ) ( \varepsilon_{s})^{\mu}( \varepsilon_{s})^{\nu}= \sum_{s = 1 , 2 , 5}( \varepsilon_{s})^{\mu}( \varepsilon_{s})^{\nu}= g^{\mu \nu}+ \frac{1}{\kappa^{2}}k^{\mu}k^{\nu}= g^{\mu \nu}- \frac{1}{k^{2}}k^{\mu}k^{\nu}
\mathrm{d}_V u ( X_0 , \ldots, X_k ) = \sum_{i = 0}^k ( - 1 )^i X_i \bigl[ u ( X_0 , \ldots, \widehat{X}_i , \ldots, X_k ) \bigr] \hphantom{\mathrm{d}_V u ( X_0 , \ldots, X_k ) =}{}+ \sum_{0 \leq i < j \leq k}( - 1 )^{i + j}\big( [X_i , X_j ], X_0 , \ldots, \widehat{X}_i , \ldots, \widehat{X}_j , \ldots, X_k \big) ,
\frac{d\rho}{d t}=\frac{\partial}{\partial S}\mathcal H_0(\rho,S)+\frac{\partial}{\partial S}\mathcal H_1(\rho,S)dW_{\delta}(t), &\frac{d S}{dt}=-\frac{\partial}{\partial \rho}\mathcal H_0(\rho,S)-\frac{\partial}{\partial \rho}\mathcal H_1(\rho,S) dW_{\delta}(t).
n^{1/2}\widetilde \alpha_{0,0}(\widehat \theta) = \widetilde \xi_{-1,0}+ \widetilde \eta_{-1,0}+ \widetilde \upsilon_{-1,0}; n^{1/2}\widetilde \beta_{j,k}(\widehat \theta) = \widetilde \xi_{j,k}+ \widetilde \eta_{j,k}+ \widetilde \upsilon_{j,k};
{\bf E}\bigl[M_{(\tau=1,0,0)}^q\bigr] =\frac{\Gamma\bigl(1-q\bigr)\Gamma\bigl(3-2q\bigr)}{\Gamma\bigl(3-q\bigr)}\prod\limits_{m=1}^\infty m^{2q}\frac{\Gamma^3\bigl(1-q+m\bigr)}{\Gamma^3\bigl(1+m\bigr)}\frac{\Gamma\bigl(2-q+m\bigr)}{\Gamma\bigl(2-2q+m\bigr)}.
\frac{\partial f_{k}(A_{k}, w_{k})}{\partial A_{k}}= \frac{1 - A_{k}^{2}\frac{2 \log(2)}{\alpha}\left[\frac{(1 + \tau_{k})\rho(\tau_{k}, \alpha) + \tau_{k}}{\tau_{k}}(\frac{R_{k}\lambda_{u}}{\lambda_{k}Ww_{k}})\right]}{(1 + A_{k}\rho(\tau_{k}, \alpha))^{2}}.
H_{1}y_{i}= y_{i}(a q), B_{1}y_{i}= y_{i}(a q^{-1}), \allowdisplaybreaks H_{2}y_{i}= y_{i}(b q), B_{2}y_{i}= y_{i}(b q^{-1}), \allowdisplaybreaks H_{3}y_{i}= y_{i}(c q), B_{3}y_{i}= y_{i}(c q^{-1}).
\frac{1}{|\Omega|}\int_{\Omega}\exp \left( \frac{\alpha}{\ln N}\rho(x) \right){\rm d}x \geq \exp \left( \frac{\alpha}{|\Omega| \ln N}\int_{\Omega}\rho(x){\rm d}x \right) &= \exp \left( \frac{\alpha}{4\pi}\left( 1 + o(1)_{N\to \infty}\right) \right).
{\cal W}_{\Gamma{_0}}={\frac{1}{N}}\langle 0 | \mathrm{T r}\left[{\cal T}{\cal P}\mathrm{e x p}\left( i g \oint_{\Gamma{_0}}d x^{\mu}\A_{\mu}^{a}( x ) T^{a}\right) \right] | 0 \rangle \ \ ,
e_k(\lambda, \tau + \delta) \; =\; \sqrt{\tilde{\mu}_k(\lambda, \tau + \delta)}\, \tilde{e}_k(\lambda, \tau + \delta) \; =\; \sqrt{\tilde{\mu}_k(\lambda, \tau)}\, \tilde{e}_k(\lambda, \tau) \; =\; e_k(\lambda, \tau)
I = R_{k, \beta}(z)\frac{x^{1-z+\delta z}}{1-z+\delta z}+ \frac{1}{2\pi i}\left\{\int_{a-i T}^{c-i T}+\int_{c-i T}^{c+i T}+\int_{c+i T}^{a+i T}\right\}F_{k, \beta}(s,z)\frac{x^s}{s}\ds = R_{k, \beta}(z)\frac{x^{1-z+\delta z}}{1-z+\delta z}+J_1+J_2+J_3,
\sum\limits_{n = 0}^{\infty}\tau(n)q^{n}= \frac{(-q^{2};q^{2})_{\infty}}{(q;q^{\infty})_{\infty}}+ \sum\limits_{n = 1}^{\infty}q^{2n + 2n}(-q^{2n + 2};q^{2})_{\infty}(q^{2n + 1};q^{2})_{\infty}^{-1}= \sum_{n = 0}^{\infty}q^{2n + 2n}(-q^{2n + 2};q^{2})_{\infty}(q^{2n + 1};q^{2})_{\infty}^{-1}
\Psi_{2N+\nu}(a,b,c,d) = \sum_{i=0}^N \genfrac{[}{]}{0pt}{}{N}{i}_{Q}(-a;Q)_{N-i+\nu}(-c;Q)_{i}(ab)^{i}, \Phi_{2N+\nu}(a,b,c,d) =\frac{1}{(ac;Q)_{N+\nu}(Q;Q)_{N}}\sum_{i=0}^N \genfrac{[}{]}{0pt}{}{N}{i}_{Q}(-a;Q)_{N-i+\nu}(-c;Q)_{i}(ab)^{i},
\sum_{j=1}^2\int_{t_i}^{t_{i+1}}\{\mathcal{I}}_j(X(t)) \dt\leq C P^{\frac43\gamma}Q^{\frac13}\Delta+ C P^{{\frac53-\eta}}Q^{\frac13}\log P\Delta+\frac{CQ^{\frac23}}{P^{{\frac53-\eta}}}.
\frac{1}{2}\sum_{i=1}^{K}Q_{i}(t+1)^{2}\le\frac{1}{2}\sum_{i=1}^{K}Q_{i}(t)^{2}+\frac{1}{2}\sum_{i=1}^{K}y_{i}(t)^{2}+\sum_{i=1}^{K}Q_{i}(t)y_{i}(t)
\prod_{i,j\geq0}\, z^{2i+j+1}\, s_{(1^{2i+1})}(X) \, s_{(j)}(X) = \prod_{a,b\geq0}\, z^{a+b+1}\,s_{(a+1,1^b)}(X) \cr\mbox{and}\cr \prod_{i,j\geq0:(i,j)\neq(0,0)}\, z^{2i+j}\, s_{(1^{2i})}(X) \, s_{(j)}(X) = \prod_{a,b\geq0}\, z^{a+b+1}\, s_{(a+1,1^b)}(X)
C^{FB,A.1}=\sum_{y\in\{0,1\}}\bigg(\sum_{x\in\{0,1\},z\in\{0,1\}}\log\Big(\frac{q(z|y,x)}{\nu^{*,\infty}(z|y)}\Big)q(z|y,x)\pi^{*,\infty}(x|y)\bigg)\nu^{{\pi^{*,\infty}}}(y).
\begin{array}{lll}\lim_{m\rightarrow\infty}\frac{\ell_R(H^1(X,\mathcal O_X(-mA))}{m^d}= \lim_{m\rightarrow\infty}\frac{1}{[R/m_R:k_1]}\frac{\dim_{k_1}H^1(X,\mathcal O_X(-mA))}{m^d}&= \lim_{m\rightarrow\infty}\frac{1}{[R/m_R:k_1]}\frac{h^0(Y,R^1\Psi_*\mathcal O_Z(m(aL-A))}{m^d}&= 0\end{array}
2\mu{\sqrt{J_1 J_2 J_3}\over N}\big(\bar{O}^{\prime J_3}\Pi^{\prime J_1}\bar{\Pi}^{\prime J_2}\delta_{J_3+J_1,J_2}+ O^{J_3}\Pi^{\prime J_1}\bar{\Pi}^{\prime J_2}\delta_{J_3+J_2,J_1}\big).
<D^{\alpha \beta \,6}> = \sum_{a=1}^{5}q^6{}_a <( h^{* a \alpha}h^{a\beta}+ h^{* a \beta}h^{a \alpha})> = -2 \, ( v^{* \alpha}v^{\beta}+ v^{* \beta}v^{\alpha})) \neq 0 .
\,\, (a+b+c) F_1(\tilde d_m +1;c,b;a+b+c;u,v) = \,\, a F_1(\tilde d_m +1;c,b;a+b+c+1;u,v) + \,\, b F_1(\tilde d_m +1;c,b+1;a+b+c+1;u,v) + \,\, c F_1(\tilde d_m +1;c+1,b;a+b+c+1;u,v).
\int_0^t\frac{1}{\Delta J_0(s)}\int_0^{\Delta J_0(s)}f(W(s-)+x)dxdJ_0(s) &=c_0 \int_0^tf(W(s))ds+\sum_{0<s\le t}\int_0^{\Delta J_0(s)}f(W(s-)+x)dx\ ,
\mathbf{H}=\left[\begin{array}{cc}\frac{\partial^2{g(x,y)}}{\partial{x^2}}\frac{\partial^2{g(x,y)}}{\partial{x}\partial{y}}\frac{\partial^2{g(x,y)}}{\partial{y}\partial{x}}\frac{\partial^2{g(x,y)}}{\partial{y^2}}\end{array}\right].
\sum_{n=0}^\infty (p_{2,5}+p_{3,5})(n) q^n =\frac{1}{(q^2; q^5)_\infty (q^3 ; q^5)_\infty}= \Big( \sum_{n=0}^\infty p(n) q^n \Big) \Big( 1+ \sum_{n=1}^\infty (-1)^n (q^{P_{7,n}}+ q^{Q_{7,n}})\Big).
A \sharp_{\nu}B \leqslant A \sharp_{\nu}B + \sum_{k=0}^{n}r_{k}(A\sharp_{\frac{m_k}{2^k}}B -2 A\sharp_{\frac{2m_k+1}{2^{k+1}}}B+ A\sharp_{\frac{m_k+1}{2^k}}B) \leqslant A \nabla_{\nu}B.
\mathbf{\theta}[\diamond_i, q_1]= \mathbf{A}[\diamond_i ,q_1] + \sum_{*}{\alpha_{* \diamond_i}\mathbf{A}[* ,q_1]}\mathbf{\theta}[\diamond_i ,q_2]= \mathbf{A}[\diamond_i, q_2] + \sum_{*}{\alpha_{* \diamond_i}\mathbf{A}[*, q_2]}.
r=\epsilon c_1+2\gamma\gamma_1+c\epsilon_1\,\,,\,\,\,\,s=\gamma\epsilon_1-\epsilon\gamma_1\,\,,\,\,\,\,A=\epsilon\alpha_1-2\beta_1 v+\frac{\epsilon_1}{\epsilon}v^2+2\frac{s}{\epsilon}v'\,.
\displaystyle{\left. \Pi_{22}(p) \right|_{\rm G}}= \displaystyle{\frac{1}{4\pi}\left[ - \frac{p^2}{8}{\rm ln}\left( \frac{p^2}{\mu^2}\right) - \frac{53}{6}\frac{(p_1)^2 (p_2)^2}{p^2}\right] \, .}
S = - \frac{1}{2 \pi}\int d^{2}\sigma \left[ \partial_{\alpha}X^{\mu}\partial^{\alpha}X_{\mu}- i \bar{\psi}^{\mu , j}\rho^{\alpha}\partial_{\alpha}\psi_{\mu , j}+ i \bar{\phi}^{\mu , k}\rho^{\alpha}\partial_{\alpha}\phi_{\mu , k}\right]
\frac{\partial f_{i}}{\partial t}+\vec{e}_{i}\cdot\nabla f_{i}+(\frac{1}{2}\vec{e}_{i}\vec{e}_{i}:\nabla\nabla f_{i}+\vec{e}_{i}\cdot\nabla\frac{\partial f_{i}}{\partial t}+\frac{1}{2}\frac{\partial^{2}f_{i}}{\partial t^{2}})=\frac{1}{\tau}(f_{i}^{eq}-f_{i}).
0 = R_{S_{0}}( e_{\otimes}^{S_{\mathrm{i n t}}}, \tilde{s}_{1}( x ) S_{\mathrm{i n t}}+ \tilde{s}_{0}( x ) \sum_{n \ge 2}S_{n}\lambda^{n}- \sum_{n \ge 2}\d j^{( n )}( x ) \lambda^{n}) \ , \quad x \in \mathcal{O}_{1},
S ={\textstyle{\frac{g^{2}}{2}}}\int d t \int d^{3}{x}\int d^{3}{y}\left\{\, - \dot{X}^{a x}( \gamma ) \frac{1}{4 \pi \mid x - y \mid}\dot{X}^{a y}( \gamma ) - X^{a x}( \gamma ) \, \delta ( x - y ) \, X^{a y}( \gamma ) \, \right\}
H_1^1 =Y^3 H_1^2 =Z^3 H_2^1 =XH_1^1 H_2^2 =XH_1^2 H_3^1 =X^2H_1^1 H_3^2 =X^2H_1^2 H_4^1 =X^3H_1^1 H_4^2 =X^3H_1^2 H_5^1 =X^4H_1^1 H_5^2 =X^4H_1^2.
\begin{cases}(\bar{a}_s^j)_{j=0}^{m-1}= \frac{\sqrt{\hat{v}_s}}{\sqrt{\sum_{j=0}^{m-1}|{\hat{a}}_s^j|^2}}(\hat{a}_s^j)_{j=0}^{m-1}, (\hat{a}_s^j)_{j=0}^{m-1}\neq 0_m, (\bar{a}_s^j)_{j=0}^{m-1}\in \sqrt{\hat{v}_s}\mathbb{S}, (\hat{a}_s^j)_{j=0}^{m-1}= 0_m. \end{cases}
\Delta_{\varepsilon}Q_{12}Q_{11}-\Delta_{\varepsilon}Q_{11}Q_{12}=\varepsilon^2\partial_x^2Q_{12}Q_{11}-\varepsilon^2\partial_x^2Q_{11}Q_{12}+\partial_y^2Q_{12}Q_{11}-\partial_y^2Q_{11}Q_{12}
\Omega \cdot E^P_{s,\varphi}= \Omega \sum_{{\gamma\in P\cap \Gamma \backslash \Gamma}}\varphi^P_{s}(\gamma\cdot g) = \sum_{{\gamma\in P\cap \Gamma \backslash \Gamma}}(\Omega \cdot \varphi^P_{s})(\gamma\cdot g) &= \lambda_s \sum_{{\gamma\in P\cap \Gamma \backslash \Gamma}}\varphi^P_{s}(\gamma\cdot g)=\lambda_s\cdot E^P_{s,\varphi}.
\exp(-r2^{j}l)< \Pr(B_{i_1,k_1,\ldots,i_l,k_l}) = \prod_{s=1}^{\ell}\frac{\Pr(B_{i_1,k_1,\ldots,i_s,k_s})}{\Pr(B_{i_1,k_1,\ldots,i_{s-1},k_{s-1}})}&\leq \exp(-r2^{\rho-1}|\{s\colon\f(s)\geq \rho\}|).
Q^{(\Sigma_m,cF+dH),\delta}(y)=Q^{\delta}_{s(c,m,d)}(y)=\sum_{\Gamma}M(\Gamma)\sum_{k=1-\epsilon_o(\Gamma)}^{d-\ell(\Gamma)+\epsilon_1(\Gamma)}\Phi_{s(c,m,d)}(\Gamma_{(k)}),
{{\rm{Var}}}[f_{n, j}(t)] = \tbinom{n}{j + 1}\sum_{i = 0}^{j + 1}\tbinom{j + 1}{i}\tbinom{n - j - 1}{j + 1 - i}\tfrac{p^{2 \binom{j + 1}{2}}}{p^{\binom{i}{2}}}- \tbinom{n}{j + 1}^2 p^{2 \binom{j + 1}{2}}.
\bar J_i^b(v_i) = \bar J_i^a (\bar v_i) +{\mathbb E}\int_0^T \left[|z_i-{\mathbb E}z_i|_Q^2 +| v_i-{\mathbb E}v_i|_R^2\right] dt +{\mathbb E}|z_i(T)-{\mathbb E}z_i(T)|_H^2 \ge \bar J_i^a (\bar v_i)\ge 0.
\hat{\mathcal{O}}= \big\{\hat z\in \mathbb{R}^{n-1}\ \big | \ | (\hat z -\hat x) \cdot e_i |<\gamma mbox{for all}i=1, \ldots, n-1\big\}\mathcal{O}= \big\{z=(\hat z, z^n)\in \mathbb{R}^n \ \big | \ \hat z\in\hat{\mathcal{O}}\ \mbox{and}\ |z^n - x^n|<M\gamma\sqrt{n-1}\big\}.
\frac{H}{p}ww^{\prime}\Bigg|_{r_1}^{r_2}= \int_{r_1}^{r_2}dr \, \Bigg\{\frac{H}{p}(w^{\prime})^2 -\frac{p}{H}u^2 w^2 -\frac{1}{pr^2}w^2(1-w^2) \Bigg\}~~.
\langle B\phi_k,\phi_k\rangle \geq \epsilon_{1}\int_{\ell_{1}}^{\ell_{1}+ \delta_{1}}|\phi_{k}(x)|^2dx ={2\epsilon_{1}\over \pi}\int_{\ell_{1}}^{\ell_{1}+ \delta_{1}}\!\!\! \sin^2(kx)\,dx \geq c.
= \; \mbox{Tr}\Big[\frac{- \not{\!\partial}+m}{-\triangle + m^2}\not{\!\!B}\frac{- \not{\!\partial}-m}{-\triangle + m^2}\not{\!\!B}\frac{- \not{\!\partial}+m}{-\triangle + m^2}\not{\!\!B}\frac{- \not{\!\partial}-m}{-\triangle + m^2}\not{\!\!B}\Big]\; .
\partial_{t}|\psi\rangle=-\alpha\sum_{i}(2a_{i}^{\dagger}a_{i}-a_{i-1}^{\dagger}a_{i}-a_{i+1}^{\dagger}a_{i})|\psi\rangle=-\alpha\sum_{i}(a_{i}^{\dagger}-a_{i-1}^{\dagger})(a_{i}-a_{i-1})|\psi\rangle.
S_{0}\!\!=\!\int\!\!dtd^{3}\!\vec{r}\,\Psi^{\!\dag}\!(\vec{r},t)i\!\left\{\partial_{t}\!+\!\alpha_{1}\!\left(\!\partial_{x}\!-\!\frac{i}{2}{\mathcal{B}}y\!\right)\!+\!\alpha_{2}\!\left(\!\partial_{y}\!+\!\frac{i}{2}{\mathcal{B}}x\!\right)\!+\!\alpha_{3}\partial_{z}\!+\!i\beta m\right\}\!\Psi(\vec{r},t),
\widetilde{R}^{\ast}(X,Y)V =-(\nabla^{\ast}_{X}A^{\ast})_{V}Y+(\nabla^{\ast}_{Y}A^{\ast})_{V}X+(R^{\ast})^{\perp}(X,Y)V-\sigma^{\ast}(X,A^{\ast}_{V}Y) &\quad+\sigma^{\ast}(Y,A^{\ast}_{V}X),
\lim_{n \to \infty}n^{s -2}T_1^{(n)}( 1 + \gamma^{(n)}) =\sqrt{2}s \cdot u^{\frac{s-2}{2}}\int_{-\infty}^\infty e^{-s \tau - \sqrt{u}\cosh \tau}d \tau = 2 \sqrt{2}s \cdot u^{\frac{s-2}{2}}K_s (\sqrt{u}).
D_{1}(v) = (dF)_{0}(v) = * (-(n+1)* (\rho_{\phi}[\iota] f) + d * (\iota^{*}g (\rho_{\phi}[\iota]Y, \cdot))) &=-(n+1) \rho_{\phi}[\iota] f - d^{*}(\iota^{*}g (\rho_{\phi}[\iota]Y, \cdot)).
(_{c}\mathbf{L})_{b}^{a}=[\delta_{c}^{a}+(dL_{d}^{a})(L^{-1})_{c}^{d}\mathbf{m}](L)_{b}^{c}, (_{c}\mathbf{L}^{-1})_{b}^{a}=[\delta_{c}^{a}-(L^{-1})_{d}^{a}dL_{c}^{d}\mathbf{m}](L^{-1})_{b}^{c}.
e_ig(\nabla_{e_j}e_j,e_i) + e_iT_{jij}- e_jg(\nabla_{e_i}e_j,e_i) - e_jT_{jii}= 2s(p) + g(\nabla_{e_i}e_i, \nabla_{e_j}e_j) - g(\nabla_{e_i}e_j, \nabla_{e_j}e_i) - 2e_jT_{jii}.
a_{2i}^{2t}= \frac{t}{i}\frac{t+k}{i+k}a_{2i-2}^{2t-2}&=\frac{t!}{(t-i)!i!}\frac{(t+k)!}{(t+k-i)!}\frac{k!}{(i+k)!}a^{2t-2i}_0\intertext{and}a_{2i+1}^{2t+1}= \frac{t+k+1}{i+k+1}\frac{t}{i}a_{2i-1}^{2t-1}&=\frac{(t+k+1)!}{(t+k-i+1)!}\frac{(k+1)!}{(i+k+1)!}\frac{t!}{(t-i)!i!}a^{2t-2i+1}_1.
r_1^{n-\frac{1}{2}}= \left(\frac{\partial u}{\partial t}\right)^{n-\frac{1}{2}}- \left(\frac{\partial u}{\partial t}\right)(t_{n-\frac{1}{2}}) + \left(\frac{\partial u}{\partial t}\right)(t_{n-\frac{1}{2}}) - \delta_tu^n.
J^{i^{\prime}j^{\prime}}= \widehat{J}^{i^{\prime}j^{\prime}}- \frac{i}{4}\int d \sigma ( \psi_{+}^{1}\gamma^{i^{\prime}j^{\prime}}\psi_{+}^{1}+ \psi_{-}^{2}\gamma^{i^{\prime}j^{\prime}}\psi_{-}^{2}) .
E_{s,3}= \frac{2}{3(2\pi)^{\frac12}}\int_{\xi_1+\xi_2+\xi_3=0}\frac{(1+\xi_1^2)^s \xi_1 +(1+\xi_2^2)^s\xi_2 +(1+\xi_3)^s\xi_3}{\xi_1\xi_2\xi_3}\hat u(\xi_1) \hat u(\xi_2) \hat u(\xi_3) d\xi_1 d\xi_2
= \{\chi^{\mu \nu}\}+ \left( \begin{array}{c c c c}{{( ( p^{2}- q^{2}) a + 2 p q b )}}&{{( p a + q b )}}&{{( p b - q a )}}&{{( ( p^{2}- q^{2}) a + 2 p q b )}}\\{{( p a + q b )}}&{{0}}&{{0}}&{{( p a + q b )}}\\{{( p b - q a )}}&{{0}}&{{0}}&{{( p b - q a )}}\\{{( ( p^{2}- q^{2}) a + 2 p q b )}}&{{( p a + q b )}}&{{( p b - q a )}}&{{( ( p^{2}- q^{2}) a + 2 p q b )}}\\ \end{array}\right) .
M_1(t)=\sum_{k}\int_{\mathbb{R}}A_k(t,\xi)\big|\widehat{g_1'}(t,k,\xi)\big|^2\,\mathrm{d}\xi+\sum_{j=2}^3\int_{\mathbb{R}}A_*(t,\xi)\big|\widehat{g_j'}(t,0,\xi)\big|^2\,\mathrm{d}\xi ,
| n \rangle = |{\vec{p}}_{1}\nu_{1};{\vec{p}}_{2}\nu_{2}. . . \rangle = \prod_{i}{\frac{{( 2 \pi )}^{\frac{\nu}{2}}}{V^{\frac{1}{2}}}}{\frac{1}{( \nu_{i}! )^{\frac{1}{2}}}}a^{\dag \nu_{i}}({\vec{p}}_{i}) | 0 \rangle
\big(\rho_z(z,b)\xi\,|\,\rho_z(z,b)\xi\big) =\big(\rho_{z^{-1}}(z^{-1},b)^*\rho_z(z,b)\xi\,|\,\xi\big) &=\big(\rho_e(e,b^*b)\xi\,|\,\xi\big) &=\big(\rho_e(e,b)\xi\,|\,\rho_e(e,b)\xi\big),
\sum\nolimits_{i,j=1}^{k,l}(a_i^\mathsf{T}b_j)^{2}= \lVert A^\mathsf{T}B \rVert_F^2 = \sum\nolimits_{i=1}^{\min(k,l)}\cos^2 \theta_i = \min(k,l) - \sum\nolimits_{i=1}^{\min(k,l)}\sin^2 \theta_i .
0 = k \cdot \varepsilon_{1 , 2}= \vec{k}\cdot \vec{\varepsilon}_{1 , 2}- ( k^{0}) \varepsilon_{1 , 2}^{0}= \left[ \vec{k}- \frac{k^{0}}{\sqrt{( \vec{n})^{2}+ 1}}\vec{n}\right] \cdot \vec{\varepsilon}_{1 , 2}.
\sum_{0<s\leq t}\langle\epsilon_{J_1},\mathbb{X}_{s^-}\rangle \Delta X_s^{S(J_2)}=\int_0^t\int_{\mathbb R}\langle\epsilon_{J_1},\mathbb{X}_{s^-}\rangle x^{S(J_2)}(\mu-\nu)(ds,dx) &\qquad+\int_0^t\langle\epsilon_{J_1},\mathbb{X}_{s^-}\rangle ds\int_{\mathbb R}x^{S(J_2)}F(dx).
\sum\limits_{n=1}^{\infty}e^{-cn}\|\rho^n\|^2 \leq \frac{\mathcal{C}(1+e^{\frac{c}{2}})^4}{(e^{c}-1)^2}k^2 \left[\sum\limits_{n=1}^{\infty}e^{-\frac{c}{2}n}\left( \left\|r_1^{n-\frac{1}{2}}\right\| + \left\|r_2^{n-\frac{1}{2}}\right\| \right) \right]^2.
R_{+}( q ) = \left( \begin{matrix}{{q^{1 / 2}}}&{{0}}&{{0}}&{{0}}\\{{0}}&{{q^{- 1 / 2}}}&{{q^{1 / 2}- q^{- 3 / 2}}}&{{0}}\\{{0}}&{{0}}&{{q^{- 1 / 2}}}&{{0}}\\{{0}}&{{0}}&{{0}}&{{q^{1 / 2}}}\\ \end{matrix}\right) \ .
\tilde U(X) = -X^{11}- X^{10}- X^9 + X^8 - X^7 + X^6 - X^5 - X^4 + X^3 + X^2 + X - 1, \tilde V(X) = -51 X^{11}+ 39X^{10}+ 63X^9 + 60X^8 + 45X^7 + 62X^6 \qquad\qquad- 18X^5 - 47X^4 + 52X^3 - 24X^2 - 32X - 19.
\sum^n_{\ell=0}t^{-\ell}\mathrm{STr\,}\alpha_\ell(x)dv_X(x) &=\frac{1}{2\pi}\left[\mathrm{Td_b\,}(\nabla^{T^{1,0}X},T^{1,0}X)\wedge\mathrm{ch_b\,}(\nabla^{E},E)\wedge e^{-m\frac{d\omega_0}{2\pi}}\wedge\omega_0\right]_{2n+1}(x)
\sum_{j=1}^{\infty}(D^k \Delta^{-1}\nabla^{\perp}\tilde \omega^j)(t,x-x_j) = \tilde \eta_k (x)+ \sum_{j=1}^{\infty}\chi_{<1}(\frac{x-x_j}{r_j}) \cdot (D^k \Delta^{-1}\nabla^{\perp}\tilde \omega^j)(t,x-x_j),
\lim_{t\to\infty}\,t^{\frac{N+A}{2}}u_m\left(t^{\frac{1}{2}}y,t\right)=0\quad\mbox{in}L^2({\bf R}^N,e^{|y|^2/4}\,dy)\,\cap\,L^\infty(K), \lim_{t\to\infty}\,t^{\frac{N+2A}{2}}\frac{u_m(x,t)}{U(|x|)}=0 \qquad\,\mbox{in}L^\infty(B(0,R)),
\|{h}^{n,i}-h_{0,2r}^{n,i}\|_{L^1(D_{2r})}+r^{\frac{2p-2}{p}}\|\tau(w_n^i)\|_{L^p(D_1)}= C_{m}(8r)^{\frac{2p-2}{p}3^{1-m}}+r^{\frac{2p-2}{p}}\leq CC_{m}r^{\frac{2p-2}{p}3^{1-m}},
\langle\psi^{\dagger}P_+\psi\rangle_{\cal{R}}=\langle\psi^{\dagger}P_+\psi\rangle_{{\cal{R}}=0}\cdot\big(\frac{{\cal{R}}}{2m_\gamma^2}\big)^{\frac{\pi}{2\pi+g_2^2}}\exp\Big[-\frac{\pi}{4e^2}{\cal{R}}- \frac{\pi m_\gamma^2}{4e^2}\gamma\Big]\hspace{.5cm}\hbox{for}\;\frac{{\cal{R}}}{m_\gamma}\rightarrow\infty.
x_N(0) - a_E b_E x_E(0) (e^{k_E \tau_1}- 1) - a_M b_M k_M (\tau_s - \tau_1) = 0 \frac{1}{2}b_M k_M k_E (\tau_s - \tau_1)^2 + (b_E k_E - b_M k_M) (\tau_s - \tau_1) + (1 - \frac{a_E}{a_M}) b_E k_E (T - \tau_s) = 0,
|v(1-a_n z)| \sim |2a_n z|^{-\gamma}L(1/a_n), v(|1-a_n z|) = v\left(1-a_n \frac{1- |1-a_n z|}{a_n}\right)\sim 2^{-\gamma}(1- |1-a_n z|)^{-\gamma}L(1/a_n)
H \left( \begin{array}{c}{\psi_{+}}{\psi_{-}}\end{array}\right) = \left( \begin{array}{c}{P_{-}P_{+}\qquad 0}{0 \qquad P_{+}P_{-}}\end{array}\right) \left( \begin{array}{c}{\psi_{+}}{\psi_{-}}\end{array}\right) .
\overline{\omega}_x([Z,\widehat{W}]_x) = \overline{\omega}_x([Z,\widehat{W}^\top]_x) = \overline{\omega}_x([Z, W]_x) = \overline{\omega}_x(\lambda_x(Z)(W_x)) = \lambda_x(Z)(\overline{\omega}_x(W_x)) = 0,
\frac{1}{n^{k_1+k_2+1}}\sum_{J \in \Pi(\pi)}\sum_{i=1 \atop i^\prime=1}^{n}|\Pi(\pi)| V(\pi) =(\kappa-1)\frac{1}{n^{k_1+k_2+1}}\sum_{i=1 \atop i^\prime=1}^{n}\sum_{J \in \Pi(\pi)}g(i,i^\prime,J_1,J_2,p,n),
D_{i \alpha}\tilde{\mathrm{A}}^{\beta \gamma}( \theta ) = \textstyle{\frac{1}{3}}\left( \delta_{\alpha}^{~ \beta}D_{i \delta}\tilde{\mathrm{A}}^{\delta \gamma}( \theta ) - \delta_{\alpha}^{~ \gamma}D_{i \delta}\tilde{\mathrm{A}}^{\delta \beta}( \theta ) \right)
{\sl L}= -\frac{1}{4}(F_{\mu \nu}-G_{\mu \nu})^2 +\frac{1}{2 (3!)}H_{\mu \nu \lambda}^2 +\frac{1}{2}m B_{\mu \nu}\epsilon^{\mu \nu \lambda \rho}(F_{\lambda \rho}-G_{\lambda \rho})+ \Phi_\mu \epsilon^{\mu \nu \lambda \rho}\partial_\nu G_{\lambda \rho}
\int_{\mathbb T}\varepsilon_\eta^{-1}\bigl(\tfrac{\cdot}{\eta}\bigr)\,{\rm curl}\,u\cdot{\rm curl}\,v,\ \ \ \ \ \u,v\in [H^1_{\#}({\mathbb T})]^3\cap L^2_{\rm{\#}sol}(\mathbb T)=:{\mathcal H}.
D_{i\alpha}\tilde{{\rm A}}^{\beta\gamma}(\theta)=\textstyle{\frac{1}{3}}\left(\delta_{\alpha}^{~\beta}D_{i\delta}\tilde{{\rm A}}^{\delta\gamma}(\theta)-\delta_{\alpha}^{~\gamma}D_{i\delta}\tilde{{\rm A}}^{\delta\beta}(\theta)\right)
C_2:= - \max_{\partial \mathcal{D}_{\delta}}\, \big({\widehat{\Phi}}^* - \varphi \big)(t,x){\mathrm{and}}C^{\star}_3:=\frac{C_2(||x_0||_1 - n\delta)}{2(||x_0||_1 + n\delta)}\left(\frac{1}{\,2\,}-16\, \delta C^2-2\, \delta^2 C^4\right),
S' \ll \widehat{C}_1\widehat{C}_2 |d| \widehat{V}^{n(1-\kappa)}\widehat{Y}^{n/2+1+\varepsilon}\sum_{\substack{|r|}=\widehat{Y}}|b_2|^{1/2}|r_3''|^{(1-\kappa)/2}|r_3|^{\kappa(n-1)/2}\ll \widehat{C}_1\widehat{C}_2 |d| \widehat{V}^{n(1-\kappa)}\widehat{Y}^{n/2+2+\varepsilon}\sum_{|b_2r_3'r_3''|\leq \widehat{Y}}|b_2|^{-1/2}|r_3'|^{(n-1)/(2n-4)-1}.
\mathsf{ch}_a\mathsf{ch}_b(\mathsf{H}) = \mathsf{ch}_a(\mathsf{p})\mathsf{ch}_b(\mathsf{H})+\mathsf{ch}_a(\mathsf{L})\mathsf{ch}_b(\mathsf{L})+ \mathsf{ch}_a(\mathsf{H})\mathsf{ch}_b(\mathsf{p})
\phi^{*}f =f\circ \phi, \phi^{*}(\alpha,\beta) =(\alpha\circ \phi,\beta), \phi^{*}T = T, \psi_{*}f = \psi\circ f, \psi_{*}(\alpha,\beta) = (\alpha,\beta\circ \hat{\psi}), \psi_{*}T =T.
\int_{I'}\mathcal{E}((h_{m}*u-u)(t,\cdot),\phi(t,\cdot))dt = \int_{I'}\int_{B}\int_{B}((h_{m}*V)(t,x,y) - V(t,x,y))\Phi(t,x,y)k(x,y)dxdydt & + 2 \int_{I'}\int_{B}\phi(t,x)\int_{B^{c}}((h_{m}*V)(t,x,y) - V(t,x,y))k(x,y)dydxdt =:_{m}+_{m}.
\|u_i\|_2 \leq\alpha\sqrt{k/(t-1)}&\leq\frac{\|D^*_{S_0}h\|_2}{\sqrt{t-1}}+\frac{2\sigma_k(D^*x_0)_1+\rho}{\sqrt{k(t-1)}}&\leq\frac{\|D^*_{S_0}h+h^{(1)}\|_2}{\sqrt{t-1}}+\frac{2\sigma_k(D^*x_0)_1+\rho}{\sqrt{k(t-1)}}&=\frac{z+R}{\sqrt{t-1}},
\pi_{k}^{a}( x^{0},{\bf x}) \equiv \frac{\delta L^{\prime}}{\delta \dot{A}_{a}^{k}( x^{0},{\bf x})}= \left( \nabla_{k}( A ) A_{0}\left\{{\bf A}\right\}\right)^{a}- \dot{A}_{k}^{a}- \left( \nabla_{k}( A ) C \right)^{a},
\tilde{c}={\frac{3 ( p - 1 ) V_{p}}{2 \pi G}}\left[{\frac{( p - 3 )^{2}+ 1 6}{( p - 3 )^{2}}}\right]^{\frac{p - 3}{4}}\bar{\cal E}^{-{\frac{p - 3}{2}}}\bar{\cal P}_{\phi}^{\frac{p - 1}{2}},
\begin{aligned}a_nG_{m,n}^+ \sim a_n \left( \frac{a_{n-1}}{b_n}\frac{a_{n-1}}{b_{n-1}}\frac{a_{n-1}}{b_n}\frac{a_{n-1}}{b_{n-1}}\cdots \right) n, a_nG_{m,n}^+ \sim a_n \frac{a_{n-1}}{b_n}\left( \frac{a_{n-2}}{b_{n-1}}\frac{a_{n-2}}{b_{n-2}}\frac{a_{n-2}}{b_{n-1}}\frac{a_{n-2}}{b_{n-2}}\cdots \right) n. \end{aligned}
| \Omega_{i_{1}\ldots i_{n}}^{( n ) \alpha_{1}\ldots \alpha_{n}}\rangle = \frac{1}{\sqrt{n !}}\left( a_{\alpha_{n}}^{i_{n}}\right)^{\dagger}\ldots \left( a_{\alpha_{1}}^{i_{1}}\right)^{\dagger}| \Omega \rangle .
L = ( \dot{x}n ) ( m + e \gamma ( F n ) ) + i s \frac{\dot{z}\bar{z}- \dot{\bar{z}}z}{1 - z \bar{z}}- e \dot{x}^{a}( A_{a}+ \mu F_{a}+ \delta F_{a b}n^{b}) +{\cal O}( e^{2})
\phi_{n}( \omega ) = P_{n + 1}( \omega ) - \int_{\omega}^{+ \infty}d z \frac{( \omega - z )^{n}}{n !}\left( \tanh ( z ) - 1 \right) \, , \quad n = 0 , 1 , 2 , \dots \, , \quad \mathrm{R e}( \omega ) > 0 ,
z(t)= S(t)z_0-\int_0^t S(t-s)(i\nu z(s) +\epsilon(\gamma z(s) -\mu \overline{z}(s)))\d s-\tfrac{1}{2}\int_0^t S(t-s)(z(s)F_\Phi )\d s +i \kappa \int_0^t S(t-s)(\theta_R(|z|_{X^{\mathfrak{s}}_s})|z(s)|^2z(s))\d s-i\int_0^t S(t-s)(z(s)\d W(s))
11628=(\begin{matrix}11628\\ 1\end{matrix})=(\begin{matrix}11628\\ 11627\end{matrix})=(\begin{matrix}153\\ 2\end{matrix})=(\begin{matrix}153\\ 151\end{matrix})=(\begin{matrix}19\\ 5\end{matrix})=(\begin{matrix}19\\ 14\end{matrix})
\sup_B\big|P \big(\big(X_{n:n},\dots,X_{n-k+1:n}\big)\in B\big)-H_k(B)\big| &\quad\leq\bigg(\sum_{j=1}^{k}\int_{x_0}^{r(H)}\left(\frac{nf(x)}{w(x)}-1-\log \left(\frac{nf(x)}{w(x)}\right) \right)dH_j(x) +H_k(x_0)+kH_{k-1}(x_0) \qquad+\sum_{j=1}^{k-1}\int_{x_j>x_0,x_k<x_0}\log \left(\frac{nf(x_j)}{w(x_j)}\right)dH_k(x) \bigg)^{1/2}+\frac{ck}{n}.
\big( \int_0^t e^{-i (t-s)H_0}(\tau_{x^n}F(s) \star\rho_q)(x)ds\big) \cdot 1_{x\geq 0}(x) &= \tau_{x^n}\big( \int_0^t (e^{-i (t-s)H_0}(F(s) \star\rho_q))(x)ds\big) \cdot 1_{x\geq 0}(x)
- i f_{\hat{\alpha}\hat{\beta}}^{Y}\hat{\gamma}^{\hat{\alpha}}\hat{\nabla}^{\hat{\beta}}= - \frac{x^{i}}{r}Q_{i}+ \frac{2 i}{\mu \sqrt{V}}\left( \begin{array}{c c}{0}&{\lambda - V}\\{1 - \lambda}&{0}\\ \end{array}\right) .