harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
T_{l-1}=\frac{f^{(l)}(\tau)}{l!p^{l}}\times\left[E\{(\|X_{1}-X_{2}\|_{2}^{2}-p\tau)^{l}\}+E\{(\|Y_{1}-Y_{2}\|_{2}^{2}-p\tau)^{l}\}-2E\{(\|X_{1}-Y_{1}\|_{2}^{2}-p\tau)^{l}\}\right].
V_{\mathcal{P}}= V_{(\mathcal{M}, *)}\textrm{and}U_{\mathcal{P}}= U_{(\mathcal{M}, *)}\cup \left\{f_{\{\frac{1}{32}\}}(ad_{\hat{a}_{\pm i}})(\hat{a}_j - \hat{a}_{-j}) \right\}\cup \left\{f_{\{1, 0, \frac{1}{4}\}}(ad_{\hat{a}_{\pm i}})(\hat{a}_j + \hat{a}_{-j}) \right\}
A_1=106+20C_{\theta}^{\frac{1}{2-\theta}}, A_2=2^{q+4}(C_kq^k(2+\|f_q\|_\infty)^{k(q-1)})^{\frac{1}{1+k}}+16(C_{\theta}C_kq^k(2+\|f_q\|_\infty)^{k(q-1)})^{\frac{1}{2+k-\theta}}.
\delta_x^2Y-2\Bigl( \tfrac{t_1-t_2}{x}+\tfrac{t_2}{x-1}\Bigr)\delta_xY+ \Bigl(\tfrac{(t_1-2t_2)(t_2-1)+2(t_1-t_2)^2x}{x^2}+ \tfrac{t_1(2t_2-t_1+1)-2(t_1-t_2)^2(x-1)}{(x-1)^2}\Bigr)Y=0.
V^{q}( \psi \otimes \bar{w}, z ) ( \chi \otimes \bar{v}) : = \sum_{\bar{w}^{\prime}, \bar{v}^{\prime}}\left( \widetilde{F}^{- 1}\right)_{\bar{w}^{\prime}, \bar{v}^{\prime}}^{\bar{w}, \bar{v}}V ( \psi \otimes \bar{w}^{\prime}, z ) ( \chi \otimes \bar{v}^{\prime}) ,
\begin{array}{cl}\min\limits_{x\in\Re^{|V|}}\; \sum\limits_{i \in V}f_i(x_i) + \sum\limits_{(i,j) \in{\cal A}_{>}}\lambda_{i,j}(x_i - x_{j}) + \sum\limits_{(i,j) \in{\cal A}_{<}}\mu_{i,j}(x_{j}- x_{i}), begin{align*}5mm]{\rm s.t.}\; x_i = x_j, \forall \, (i,j) \in{\cal A}_{=},\end{array}
\hat{V}_{\mu \nu \rho}^{( 3 z ) a b}= \hat{V}_{\mu \nu \rho}^{( 3 x )}= \hat{V}_{\mu \nu \rho}^{( 3 y ) a b}= - Y_{L}^{( e ) 3}\mathrm{T r}\tau^{a}\tau^{b}V_{\mu \nu \rho}^{( 3 )}.
\mathbb E \left[ (\widehat{\mu}^{1,N}_t,\widetilde{\mu}^{j,N}_t) \right]\overset{(2.1)}{\leq}\mathbb E \left[ \frac{1}{N-1}\sum_{l=2}^N \boldsymbol{1}_{\widetilde{X}^{l,N}_t\neq \widehat{X}^{l,N}_t}\right]=\frac{1}{N-1}\sum_{l=2}^N \mathbb{P}\left(\widetilde{X}^{l,N}_t\neq \widehat{X}^{l,N}_t\right).
\sqrt{R}\varphi'_i(U)[\mathbf T_{N}]_{jk}= \partial_j(R\varphi'_i(U) U_k) - 2 \sqrt{R}U_k \partial_j \sqrt{R}+g_\varphi\,, \forall i,j,k\in \{1,\cdots,d\}, begin{align*}6pt] \mathbf S_K =\sqrt{R}\nabla^2 \sqrt{R}- \nabla \sqrt{R}\otimes \nabla \sqrt{R}\,.
g_{S_{n}}(s) =\left( e+\left( s-1\right) f\right)^{n}=\exp\left[n\log\left( e+\left( s-1\right) f\right) \right] & \leq\exp\left[ n\left( s-1\right) f\right] =\exp\left[ nsf\right]\exp\left[ -nf\right] & \leq\exp\left( ns\left\Vert f\right\Vert_{e}e\right) \exp\left(-nf\right) =\exp\left( t\right) \exp\left( -nf\right) .
\eta = \left[{m+1 \over 2n}+{3m^2+7m+6 \over 8n^2}+ O\left({1 \over n^3}\right)\right]\tilde \epsilon - \left[{m+1 \over 2n}+{3(m+1)^2 \over 4n^2}+O\left({1 \over n^3}\right)\right]\tilde \epsilon^2 +O(\tilde \epsilon^3).
= e x p \mathrm{~}[ \mathrm{~}( \frac{1}{4 N_{0}}) \sum_{{\bf{q}}}f_{{\bf{q}}}( -{\bf{x}}, - t ) \mathrm{~}] \langle \psi ({\bf{x}}, t ) \psi^{\dagger}({\bf{0}}, 0 ) \rangle_{f r e e}
\begin{array}{c}{{( a_{v}+ \{\epsilon_{v}, . \}) R_{v}( \epsilon , a ) \equiv ( a_{v}+ \{\epsilon_{v}, . \}) ( a H_{v}+ \{\epsilon , H_{v}\})}}\\{{= R_{v}( \{\epsilon_{v}, \epsilon \}, \{\epsilon_{v}, a \}- \{\epsilon , a_{v}\}) ,}}\\ \end{array}
\int_X\left(n\mu_{\hat\omega}\omega_0-Ric(\omega_0)\right)^n =\lim_{t\to n\mu_{\hat\omega}}\int_X\left(t\omega_0-Ric(\omega_{0})+\sqrt{-1}\partial\bar\partial\Phi(t)\right)^n &=\lim_{t\to n\mu_{\hat\omega}}\int_Xe^{\Phi(t)}e^{-t\psi}\omega_{0}^n &\ge\liminf_{t\to n\mu_{\hat\omega}}\int_Xe^{\Phi(t)}\omega_{0}^n &\ge c_4,
p_2(s) = (1+e^{-hs})\zeta k_p + (1-e^{-hs})k_{p}\;, p_1(s) = (1+e^{-hs})(bk_p+a)\zeta k_p + (1-e^{-hs})(b\zeta^2k_p^2+ak_p+k_i) \;, p_0(s) = (1+e^{-hs})bk_i \zeta k_p + (1-e^{-hs}) a k_i \;.
(u_1,\ldots,u_m,s_1,\ldots,s_m)\longmapsto (u_1,\ldots,u_m,q_1,\ldots,q_m) &\qquad=(u_1,\ldots,u_m,u_1s_1,\ldots,u_ks_k,s_{k+1},\ldots,s_m).
W(\lambda_1,\lambda_2,\lambda_3) = \Phi(\lambda_1,\,\lambda_2,\,\lambda_3,\,(\lambda_2\lambda_3),\,(\lambda_1\lambda_3),\,(\lambda_1\lambda_2),\,(\lambda_1\lambda_2\lambda_3)) &= \Psi(\lambda_1+\lambda_2+\lambda_3,\,(\lambda_2\lambda_3)+(\lambda_1\lambda_3)+(\lambda_1\lambda_2),\,(\lambda_1\lambda_2\lambda_3))
\sum_{\alpha}''' w\left(\frac{N(\alpha)}{X}\right) \frac{L'\left(\chi_{\alpha},\frac{1}{2}+ \left(c - \frac{1}{2}+it\right)\right)}{L\left(\chi_{\alpha},\frac{1}{2}+ \left(c - \frac{1}{2}+ it\right)\right)}= \frac{\partial}{\partial\nu'}R_w(\nu';\nu) \Biggr|_{\nu' = \nu = c - 1/2 + it}.
L_m(R\eta_*\Omega_{Z_{\bullet}/Y}(\log E_{\bullet}))^n=\bigoplus_{p \ge -m}(R(\eta_p)_*\Omega_{Z_p/Y}(\log E_p))^{n-p}, F^r(R\eta_*\Omega_{Z_{\bullet}/Y}(\log E_{\bullet}))^n=\bigoplus_pF^r(R(\eta_p)_*\Omega_{Z_p/Y}(\log E_p))^{n-p}
Q = \int_{0}^{t}\bigg\{\int_{0}^{1}\chi_{\{x+z\in[0,1]\}}\bigg|\int_{0}^{1}\chi_{\{|x-y|> |z|\}}|G(t-s,x+z,y)-G(t-s,x,y)| & \times v^{2\delta+1}(s,y)\d y\bigg|^p\d x\bigg\}^{\frac{1}{p}}\d s,
\mathrm{Cov}\left(\mathbb{G}^{(N)}(f),\mathbb{G}^{(N)}(g)\right) =\mathrm{Cov}\left(\mathbb{G}(f),\mathbb{G}(g)\right)-\sum_{k=1}^{N}\Phi_{k}^{(N)}(f)^{t}\cdot\mathbb{V}\left( \mathbb{G}[\mathcal{A}^{(k)}]\right) \cdot\Phi_{k}^{(N)}(g).
\mathcal{L}^{lin}_{I}= - \frac{i}{4}\left( h^{a\mu}\bar\psi \gamma_{a}\frac{1\pm\gamma_{5}}{2}\stackrel{\leftrightarrow}{\partial^{\psi}_{\mu}}\psi + h^{\mu}_{\ \mu}\bar\psi \gamma^{a}\frac{1\pm\gamma_{5}}{2}\stackrel{\leftrightarrow}{\partial^{\psi}_{a}}\psi \right)
F (t) = \int_{0}^{t}\exp \left( - \frac{2r}{\sigma^{2}}(t-s) - \frac{(\sqrt{t}\eta (t) - \sqrt{s}\eta (s))^{2}}{t-s}\right) \Big\{1+ \frac{\sqrt{t}\eta (t) - \sqrt{s}\eta (s)}{t-s}\Big\}\frac{\mathrm{d}s}{\sqrt{t-s}}.
\delta_r^{-1}\widetilde{T}_r - \delta_r^{-1}\widetilde{T}_s = &= \mu[U]\left(x^{(1)}\otimes\ldots\otimes t^{(i_0)}_r\otimes\ldots\otimes x^{(m)}\right) - \mu[U]\left(x^{(1)}\otimes\ldots\otimes t^{(i_0)}_s\otimes\ldots\otimes x^{(m)}\right) = &= \mu[U]\left(x^{(1)}\otimes\ldots\otimes \left(t^{(i_0)}_r - t^{(i_0)}_s\right) \otimes\ldots\otimes x^{(m)}\right).
d l^{2}\sim \biggl ( 1 + \frac{\psi_{i}^{\prime}}{r_{i}^{\prime}}\biggr )^{2 + 2 a}\biggl ( 1 + \frac{\chi_{i}^{\prime}}{r_{i}^{\prime}}\biggr )^{2 - 2 a}( d r_{i}^{\prime 2}+ r_{i}^{\prime 2}d \Omega^{2}) \, ,
a \leqq_{\rm lex}b,\ \{\rm iff}\ \a_1<b_1\ \{\rm or}\ \a_1=b_1\ \{\rm and}\ \a_2\leq b_2, a <_{\rm lex}b,\ \{\rm iff}\ \a_1<b_1\ \{\rm or}\ \a_1=b_1\ \{\rm and}\ \a_2< b_2.
\begin{array}{lll}\tau(\varphi,g)=(\rho''+\frac{\sigma'}{\sigma}\rho'-\frac{k^2\lambda\lambda'(\rho)}{\sigma^2})\frac{\partial}{\partial\rho}, \tau(\varphi,\bar{g})=f\tau(\varphi,g)=f(\rho''+\frac{\sigma'}{\sigma}\rho'-\frac{k^2\lambda\lambda'(\rho)}{\sigma^2})\frac{\partial}{\partial\rho},\end{array}
\rho_i(x)\leq a\sin \sqrt{a}d(x)\frac{\dfrac{\sqrt{a}}{\kappa_+}\tan \sqrt{a}d_0+1}{\dfrac{\sqrt{a}}{\kappa_+}-\tan \sqrt{a}d_0}=\frac{a\sin \sqrt{a}d(x)}{\tan\sqrt{a}(\bar{h}-d_0(x))}\leq a\cos \sqrt{a}d(x).
\frac{d^k}{d\xi^k}\frac{d}{d\sigma}h(\sigma, \xi) = \frac{d}{d\sigma}\frac{d^k}{d\xi^k}h(\sigma, \xi) = i^k\frac{d}{d\sigma}\int{(x-m)^k\exp(i(x-m)\xi)\mu_{\sigma}(dx)}.
\langle m_{1}; m_{2}; . . . ; m_{n}; A ; \pm | \overline{{{m}}}_{1}; \overline{{{m}}}_{2}; . . . ; \overline{{{m}}}_{n}; A ; \pm \rangle ={\delta}_{m_{1}\overline{{{m}}}_{1}}{\delta}_{m_{2}\overline{{{m}}}_{2}}. . .{\delta}_{m_{n}\overline{{{m}}}_{n}}.
\frac{\partial^2}{\partial z_1 \partial z_1^\rho}= \frac{1}{4}\left( \frac{\partial^2}{\partial x_2^2}- \frac{1}{u}\frac{\partial^2}{\partial x_1^2}\right), \frac{\partial^2}{\partial z_2 \partial z_2^\rho}= \frac{1}{4}\left( \frac{\partial^2}{\partial x_4^2}- \frac{1}{u}\frac{\partial^2}{\partial x_3^2}\right),
{\bf R}^{(pq)}_{\bf i}:= \left(\mathbf{R}^{(pq)}_{{\mathbf i}, 1}, \dots, \mathbf{R}^{(pq)}_{{\bf i}, k}\right){\bf X}^{(pq)}_{\bf i}:= \left(\mathbf{X}^{(pq)}_{{\bf i}, 1}, \dots, \mathbf{X}^{(pq)}_{{\bf i}, k}\right).
[v,e_3]+J[Jv,e_3] + J[v,Je_3]-[Jv,Je_3] = 2v-u+2YJv-YJu+J[v,Ce_3]-[Yv,Ce_3] = 2v-u+2Y^2v-\lambda Yu+2CYv-\lambda Cu-2YCv+CYu = (2+Y^2+2CY-2CY)v+(-1-\lambda Y-\lambda C+CY)u
\operatorname*{ess\,sup}_{t \in [0, \tau]}\|u_{k-1}(s)\|_{X}\leq \max\{C_{1}, C_{3, k}\}+ C_{2}\sum_{j = 0}^{k-1}C_{3, k}e^{(k - j - 2) C_{2}}\leq C_{1}+ C_{2}e^{C_{2}k}\sum_{j = 0}^{k}C_{3, k}.
\sum_{k=N}^{\infty}|b_k||z|^k = M_{r}^{N}(g) \leq M_{r}^{N}(f) & = \int_{0}^{r}\frac{M_{t}^{N}(G) M_{t}^{N}(\psi\circ \omega)}{t}dt & \leq \int_{0}^{r}\frac{M_{t}^{N}(h_{\psi}) M_{t}^{N}(\psi)}{t}dt =: R^N(r),
\begin{array}{l l}{{\vec{L}_{0}( x ) = \vec{l}( x ) \; \;{;}\vec{L}_{1}( x ) = \gamma \partial_{x}\vec{n}\wedge \vec{n}}}\\{{\vec{J}_{0}( x ) = \vec{l}( x ) - \gamma \partial_{x}\vec{n}{;}\vec{J}_{1}( x ) = \gamma \partial_{x}\vec{n}\wedge \vec{n}- i \vec{\pi}= - i ( \vec{\pi}+ i \gamma \partial_{x}\vec{n}\wedge \vec{n})}}\\ \end{array}
F = i \frac{d g_{1}\wedge d \overline{{{g}}}_{1}}{( 1 - g_{1}\overline{{{g}}}_{1})^{2}}- i \frac{d g_{2}\wedge d \overline{{{g}}}_{2}}{( 1 + g_{2}\overline{{{g}}}_{2})^{2}}= - \frac{1}{2}( R_{2}^{1}+ R_{4}^{3}) ~ .
(u-\phi^h)(x_h,t_h) \leq u(x_0,t_0)-\phi^h(x_0,t_0)= \phi(x_0,t_0)-\phi(x_0+h\eta, t_0) = -\frac{1}{2}h^2 \eta^t \nabla^2\phi(x_0,t_0)\eta+o(h^2)<0,
L(t, U(t,s)x) + \int_s^t \|H(\tau) U(\tau,s) x\|^p d\tau =2^{p-1}\int_t^{+\infty}\|H(\tau) U(\tau,s) P(s) x\|^p d\tau - 2^{p-1}\int_0^t \|H(\tau) U_{Q}(\tau,t)Q(t)U(t,s)x\|^p d\tau +\int_s^t \|H(\tau) U(\tau,s) x\|^p d\tau.
\mathfrak{n}_{T}:= \left\{x\in M\mid \| T(x^*x)\|_\infty<+\infty \right\}, \mathfrak{m}_{T}:=(\mathfrak{n}_{T})^*\mathfrak{n}_{T}= \left\{\sum_{i=1}^nx_i^*y_i \mid n \geq 1, x_i, y_i\in \mathfrak{n}_{T}1 \leq i \leq n \right\}.
{\bf T}exp\biggl(ie\sum_{a=1}^N\int_0^T dt\,{\dot{\bf x}}_a\cdot{\hat{\bf A}}({\bf x}_a)\biggr) =exp(i\omega)\,exp\biggl(ie\sum_{a=1}^N\int_0^T dt\,{\dot{\bf x}}_a\cdot{\hat{\bf A}}({\bf x}_a)\biggr) \,,
\frac{J^{w_1}(\textbf{X}_{RSS}^{(n)})}{J^{w_1}(\textbf{X}_{SRS}^{(n)})}\leq \frac{n^{2n}}{(n-1)^{2(n-1)(n-2)}}\prod_{i=2}^{n-1}\left(\binom{n-1}{i-1}^2(i-1)^{2i-2}(n-i)^{2n-2i}\right).
\lim_{N\rightarrow\infty}\frac1N\, K_N(\frac{x}{N^2}, \frac{y}{N^2})=\frac{u(1)}{2}\frac{\sqrt{x}J_{\alpha+1}( u(1)\sqrt{x}) J_\alpha ( u(1)\sqrt{y})-J_\alpha ( u(1)\sqrt{x}) \sqrt{y}J_{\alpha+1}( u(1)\sqrt{y})}{x-y}.
g_{\zeta_1(0)}\left( \tilde{\varphi}_i^{-1}\circ \tilde{\chi}_{\sigma_1}\right) = \delta\!\left(s_{0,1}\, e^{\pi i\left( \frac{[k_0-1]_3}{3}- \frac{k_1}{3}+ \frac{1}{2}\right)}\right) \delta(\tilde{\alpha}) \tilde{t_0}\delta(\tilde{\alpha})^{-1}
S_{\Psi}=\sum_{n=m}^\infty\frac{1}{nh(n)}\mathbb P \left(|S_n|>(1+\varepsilon)\sigma_n\sqrt{2\ln\Psi(n)}\right) &\propto \sum_{n=m}^\infty\frac{1}{nh(n)\sqrt{\ln\Psi(n)}\Psi(n)^{(1+\varepsilon)^2}}&=\sum_{n=m}^\infty\frac{\Psi'(n)}{\sqrt{\ln\Psi(n)}\Psi(n)^{(1+\varepsilon)^2}}.
\begin{cases}-\operatorname{div}_{x,y}(y^a\varphi(x)\nabla_{x,y}w)=0, \hbox{in}~\mathcal{C}_{\Omega}, w=0, \hbox{on}~\partial_{L}\mathcal{C}_{\Omega}, w(x,0)=u(x), \hbox{on}~\Omega,\end{cases}
\begin{bmatrix}\Phi_k U_k\end{bmatrix}d =\begin{bmatrix}M 0 0 T\otimes I_m\end{bmatrix}\begin{bmatrix}u_{[k-n,k]}u_{[k-n+1,k+L-1]}\end{bmatrix}&=Zu_{[k-n,k+L-1]}.
S_{S D}[ \vec{B}_{m}, V_{m n}] = \int d^{5}x \left\{-{\textstyle \frac{1}{4}}m^{2}\vec{B}_{m}^{T}\vec{B}_{m}-{\textstyle \frac{1}{4}}{\cal F}_{m n}{\cal F}_{m n}-{\textstyle \frac{m}{8}}\epsilon \vec{B}_{m}^{T}\eta \mathcal{D}\vec{B}_{m}\, \right\}\, ,
W^{[ a_{1}, a_{2}]}( \theta = 0 ) = C^{i_{1}\cdots i_{a_{1}+ a_{2}}j_{1}\cdots j_{a_{2}}}u_{i_{1}}^{1}\cdots u_{i_{a_{1}+ a_{2}}}^{1}u_{j_{1}}^{2}\cdots u_{j_{a_{2}}}^{2},
e + \delta \, r(S_n) = e + \delta \, r\left( \sum_{i=0}^{n-1}\alpha_r(\delta,i) r^{i}(e) \right) = e + \delta \sum_{i=0}^{n-1}\frac{\alpha_r(\delta,i+1)}{\delta}r^{i+1}(e) &= e + \sum_{i=1}^{n}\alpha_r(\delta,i) r^i(e)= \sum_{i=0}^{n}\alpha_r(\delta,i) r^i(e) = S_n + \alpha_r(\delta,n) r^n(e).
\begin{aligned}\lVert u^\tau(t) - u(t) \rVert \leq t_{p-1}(2\lVert y_{p-1}\rVert +\lVert r_{p-1}\rVert+ \lVert w_{p-1}\rVert) +t_{q}(2\lVert y_{q}\rVert +\lVert r_{q}\rVert+ \lVert w_{q}\rVert) + \left\lVert \sum_{\ell=p}^{q-1}t_\ell w_\ell\right\rVert + \left\lVert \sum_{\ell=p}^{q-1}t_\ell r_\ell\right\rVert.\end{aligned}
{\cal V}( x_{0}, \theta_{0}, \bar{\beta}, \bar{\zeta}, \bar{\omega}, \rho ) = e^{i P x_{0}}e^{- i Q \theta_{0}}e^{- i \bar{S}\bar{\beta}}e^{- i \bar{Q}\bar{\zeta}}e^{i \bar{M}\bar{\omega}}e^{i D \ln \rho}\, .
\alpha^j := \lim_{k\to\infty}\alpha^j_k, \beta^j:=\lim_{k\to\infty}\beta^j_k, t:= \lim_{k\to\infty}t_k, f^j:= \lim_{k\to\infty}f^j_k g^j:= \lim_{k\to\infty}g^j_k.
dx_t^n = \beta_t^n(y_t^n, x_{\kappa(n,t)}^n) dt + \alpha_t(y_t^n, x_{\kappa(n,t)}^n) dw_t +\int_{Z}\lambda_t( y_t^n, x_{\kappa(n,t)}^n, z)\tilde N(dt,dz), \,\,\,t \in [0, T], x_t^n = \xi_t, \,\,\, t \in [-H, 0],
R(M_s,M_p)=\frac{\Big(\sum_{j=1}^{\mathcal{L}_1}\binom{\Lambda}{t_s+1}-\binom{\Lambda-|U_j|}{t_s+1}\Big) F_s}{\binom{\Lambda}{t_s}}+\frac{\binom{K}{t_p+1}F_p}{\binom{K}{t_p}}.
Lf = \partial^*\sum_{i=1}^m \phi^i\otimes \frac{\partial F}{\partial x^i}= \sum_{i,j=1}^m \frac{\partial^2 F}{\partial x^i\partial x^j}Z^{ij}+ \sum_{i=1}^m \frac{\partial F}{\partial x^i}L\phi^i
c_{n}= \frac{( - 1 )^{n ( n + 1 ) / 2}x^{n^{2}/ 2}}{( x^{2}; x^{2})_{\infty}^{n ( n - 3 ) / 2}( x^{2 r}; x^{2 r})_{\infty}^{3 n ( n - 1 ) / 2}}\frac{\{x^{2}\}_{\infty}^{n}\{x^{6}\}_{\infty}^{n}\{x^{2 r + 2}\}_{\infty}^{n}\{x^{2 r + 6}\}_{\infty}^{n}}{\{x^{4}\}_{\infty}^{n}\{x^{8}\}_{\infty}^{n}\{x^{2 r}\}_{\infty}^{n}\{x^{2 r + 4}\}_{\infty}^{n}},
\mathrm{Res}_{s_1=0}\Big( \mathcal{B}(s_1,s_2) \Phi(s_1) \zeta(s_1+1)^{k}X^{s_1}\Big) = \sum_{\substack{i_1+i_2+i_3+i_4=k i_1,i_2,i_3,i_4 \ge 0}}\frac{\mathcal{B}^{(i_1,0)}(0,s_2)\Psi^{(i_2)}(0) \alpha_{i_3, k}(\log X)^{i_4}}{i_1! i_2! i_4!}.
P_{| n_{1}, n_{2}, \dots , n_{k}\rangle}^{\pm}( \eta , \vartheta , u ) = P^{\pm}( \eta , \vartheta , u ) \prod_{i \textrm{o d d}}a_{n_{i}}( \eta , u ) \prod_{i \textrm{e v e n}}a_{n_{i}}( \bar{\eta}, u ) \, \, \, ,
V_{n s}({\tilde{x}}) = \left( \frac{{\tilde{m}}N^{2}}{1 6 \pi}\right) N g^{2 n s - 1}{\tilde{x}}^{2}\left\{{\tilde{x}}^{2}- \frac{2{\tilde{b}}}{3}{\tilde{x}}+ \frac{{\tilde{b}}^{2}}{3}- ( - 1 )^{n s}{\tilde{c}}\right\}\, .
\varepsilon_{u}= \eta_{P}[ 4 \sum_{a^{\prime}< b^{\prime}}({\bf P}_{a^{\prime}b^{\prime}})^{2}]^{- 1 / 2}{\bf P}_{a b}\gamma^{a b}\varepsilon_{\ell}\ \ ( \eta_{P}\equiv \pm 1 ) .
= - i \frac{3 \hbar c}{2}\int d^{3}\vec{r}[ \epsilon_{0}( \varphi^{\dagger}\varphi + \chi^{\dagger}\chi ) + \varphi^{\dagger}( \vec{\sigma}\cdot \vec{\epsilon}) \chi + \chi^{\dagger}( \vec{\sigma}\cdot \vec{\epsilon}) \varphi ] \,{.}
a_1(k)=\textnormal{Wr}\left(\psi_1^{(1)}(0,0,k),\psi_2^{(2)}(0,0,k)\right), k\in\overline{\mathbb{C}_+}\backslash\{0\}, a_2(k)=\textnormal{Wr}\left(\psi_2^{(1)}(0,0,k),\psi_1^{(2)}(0,0,k)\right), k\in\overline{\mathbb{C}_{-}}, b(k)=\textnormal{Wr}\left(\psi_2^{(1)}(0,0,k),\psi_1^{(1)}(0,0,k)\right), k\in\mathbb{R}.
\begin{array}{l l l}{}&{\}&{\left( \sigma_{i}, \sum_{j = 1}^{\infty}\sigma_{j}S a_{i n}\left( \overline{{\sigma}}_{j}\right) S^{- 1}+ \overline{{\sigma}}_{j}S a_{i n}^{\dagger}\left( \sigma_{j}\right) S^{- 1}\right)_{K . G .}}\\{}&{=}&{S a_{i n}\left( \overline{{\sigma}}_{i}\right) S^{- 1}.}\\ \end{array}
\Delta\nu_{c}=\frac{1}{2\pi\tau_{c}}\Rightarrow\tilde{\gamma}_{q}(\nu)=\frac{1}{\pi}\frac{\Delta\nu_{c}/2}{(\Delta\nu_{c}/2)^{2}+(\nu-\nu_{q})^{2}}=\frac{2}{\pi}\frac{\Delta\nu_{c}}{(\Delta\nu_{c})^{2}+4(\nu-\nu_{q})^{2}},
\inf_{\xi\in M}\inf_{v\in C(\xi)}\|D_\xi T^{-n}v\|> c_0\nu_0^{-n}\|v\| & \inf_{\xi\in M}\inf_{v\not\in C(\xi)}\|D_\xi T^{n}v\|> c_0\lambda_0^{n}\|v\|.
\mathcal{S}_{jk}(A):=\sum\limits_{\ell = p_1 + p_2 + \ldots + p_{k-1}+ 1}^{p_1 + p_2 + \ldots + p_k}a_{p_1 + p_2 +\ldots + p_j, \ell}\, ,j,k=1,\ldots,m.
\sum_{i=1}^3 |T_i| = O \biggl( \max\biggl\{\frac{k^{-\frac{1}{2}+\frac{2\alpha}{\alpha+d}- \epsilon}}{n^{\frac{2\alpha}{\alpha+d}- \epsilon}}\, , \, \frac{k^{\frac{1}{2}+\frac{\beta}{d}}}{n^{1+\frac{\beta}{d}}}\log^{2+\beta/d}n\biggr\}\biggr)
\vec{f}^{n}=\frac{1}{1-\epsilon_{b}^{n}\epsilon_{f}^{n}}[\begin{matrix}\vec{f}^{n-1}\\ 0\end{matrix}]-\frac{\epsilon_{f}^{n}}{1-\epsilon_{b}^{n}\epsilon_{f}^{n}}[\begin{matrix}0\\ \vec{b}^{n-1}\end{matrix}]
\Sigma_{|j|\geq\ell}=\sum\limits_{|j|\geq\ell}i^{-\beta(\lambda_j)}j_{-\alpha(\lambda_j)}(rr')\Big(e^{i(\widetilde A+j)\theta}+R_j(\theta)\Big)\Big(e^{-i(\widetilde A+j)\theta'}+\overline{R_j(\theta')}\Big)
\delta \int d^{4}x \mathcal{L}= \int d^{4}x \partial_{\mu}\left( \frac 1 2 \{\partial^{\mu}\Phi , \delta_{0}\Phi \}_{\star}+ \delta x^{\mu}\mathcal{L}+ \frac{\lambda}{1 2}\frac{\sinh ( \Delta )}{\Delta}( \Phi \star \Phi ~ , ~ \tilde{\partial}^{\mu}[ \Phi , \delta_{0}\Phi ]_{\star}) \right) .
\vec{X}( \tau , \theta , \phi ) = \left( \begin{array}{c}{( R ( \tau ) + r ( \tau , \theta ) c o s \theta ) c o s \phi}\\{( R ( \tau ) + r ( \tau , \theta ) c o s \theta ) s i n \phi}\\{r ( \tau , \theta ) s i n \theta}\\ \end{array}\right) ,
\widehat Z_t^k:= Z_{\tau^k}+ \int_{\tau^k}^t \mathbf{1}(s<T^{k+1}) b_s(Z_s) ds + \int_{\tau^k}^t (\mathbf{1}(s<T^{k+1})\sigma_t(Z_s) + \mathbf{1}(s\ge T^{k+1}))dW_s,
\delta S [ \theta , \bar{A}_{+}, \bar{\Pi}^{\perp -}] = \frac{k}{8 \pi}\int_{\partial \cal M}\partial_{+}\theta \, \partial_{-}\theta + \partial_{-}\theta \left( \bar{A}_{+}- \mathrm{~ \frac{8 \p i}{k}~}\bar{\Pi}^{\perp -}\right) \, .
\int_{D}|Df|^2 \; =\; \int_{D\setminus \bigcup_{n=1}^{\infty}Q_n}|Df|^2 + \int_{\bigcup_{n=1}^{\infty}Q_n}|Df|^2 &\leq\;{\mathrm{Area}(D)}+ C\sum_{n=1}^{\infty}n^{2p}{\mathrm{Area}(Q_n)}=\;{\mathrm{Area}(D)}+ C\sum_{n=1}^{\infty}n^{2p}2^{-2n}<\infty\ .
F^{\mu \nu}( x ) \, = \, \sum_{\sigma = 0 , \pm 1}\int \frac{d^{3}{\bf p}}{( 2 \pi )^{3}2 E_{p}}\, \left[ F_{( + )}^{\mu \nu}({\bf p}, \sigma ) \, a ({\bf p}, \sigma ) \, e^{- i p x}+ F_{( - )}^{\mu \nu}({\bf p}, \sigma ) \, b^{\dagger}({\bf p}, \sigma ) \, e^{+ i p x}\right]
\left|K^{\beta}(\psi,t)\right|=\left|Y\tilde{K}^{\beta}(\psi,t)Y^{-1}\right|\leqslant |Y||Y^{-1}|\left|\tilde{K}^{\beta}(\psi,t)\right| =\kappa\max_{k=1,\ldots,N}\left|\tilde{K}^{\beta}_{kk}(\psi,t)\right|,
\frac{\partial u}{\partial s}= \frac{1}{\delta}\frac{\partial u}{\partial z}, &= c\left[1-\left(u^2/2i_0\right)\right]^{-6/25}, &= c \left[1+\frac{6}{25}\cdot \frac{u^2}{2i_0}+\frac{6}{25}\cdot \frac{u^4}{2(2i_0)^2}+\frac{1}{2}\cdot\frac{6}{25}\cdot \frac{u^4}{(2i_0)^2}+o(u^6)\right].
\gamma_F(\frac{1}{2}\psi \circ q_6) = \gamma_F(\frac{1}{2}\psi)^2 \cdot \gamma_F(-J_1, \frac{1}{2}\psi) \cdot (\frac{ab-cdJ}{b^2 - d^2J}- t, J_1)_F = \gamma_F(J_1, \frac{1}{2}\psi)^{-1}\cdot (\frac{ab-cdJ}{b^2 - d^2J}- t, J_1)_F.
R_1= \mu^{-1}h^{1-d}\int_{X'_+}\rho^d\rho_1^{-1}\gamma^{-2}\,dx , R_2= \mu h^{s-d}\int_{X'_-}\rho_1 \rho^{d-s-1}\gamma^{1-s}\,dx\shortintertext{provided}\mu \rho_1 \gamma \ge \rho
c_{n-1}-q^{-1}c_{n-3}= \Big[(n-2)+(n-4)q+(n-6)q^2+\ldots+5q^{\frac{n-7}{2}}+3q^{\frac{n-5}{2}}+q^{\frac{n-3}{2}}\Big]h^2 = \sum\limits_{k=1}^{n-2}(\sum\limits_{l=1}^{(k,n-1-k)}q^{-1+l})h^2.
K(t,{\bf x}, s\,\vert\, t,{\bf x}, 0) =\int{d\omega \over 2\pi}\int{dp_{y}\over 2\pi}\int{dp_{z}\over 2\pi}\; \langle x\vert\, \exp -i(-\omega^2-d_{x}^2+(p_{y}-qA)^2+{p_{z}}^2)s\, \vert x \rangle.
T\left(-k\right)=T(k),\ \ \R_{\pm}\left(-k\right)=\overline{R_{\pm}(k),}& \left|R_{\pm}(k)\right|^{2}+\left|T(k)\right|^{2}=1,\ \T(k)\overline{R_{-}(k)}+\overline{T(k)}R_{+}(k)=0.
\Delta_H (V) = H(V\cup \{0\})-H(V)= a_1H_1(V\cup \{0\})+a_2 H_2(V \cup \{0\}) - (a_1 H_1(V)+a_2 H_2(V) ) &= a_1\Delta_{H_1}(V)+a_2\Delta_{H_2}(V)
\mathrm{Hess}(V_t)(X,X)(x)= \left(\int_{B_R}\nabla_X V(x-z)d\mu_x(z)\right)^2-\int_{B_R}\left(|\nabla_XV(x-z)|^2-\mathrm{Hess}(V)(X,X)(x-z)\right)d\mu_x(z) = \frac{1}{t}-\left(\int_{B_R}|\nabla_XV(x-z)|^2d\mu_x(z)-\left(\int_{B_R}\nabla_X V(x-z)d\mu_x(z)\right)^2\right) &\mbox{since}\mathrm{Hess}(V)=\frac{1}{t}I.
\begin{cases}\varphi_t +\varphi_{xxx}+\frac{ab}{c}\psi_{xxx}=0, \,\, (0,L)\times (0,T), \psi_t +\frac{r}{c}\psi_x+a\varphi_{xxx}+\frac{1}{c}\psi_{xxx}=0, \,\, (0,L)\times (0,T), \varphi(x,0)= \varphi^0(x), \,\,\psi(x,0)= \psi^0(x), \,\, (0,L),\end{cases}
{\cal K}(z)=\sum_{\mbox{\footnotesize\boldmath $\alpha $}\in G_m}\omega^{2\alpha_1 \alpha_2}u^{(n)}_{2\mbox{\footnotesize\boldmath $\alpha $}}(z,v) I_{2\mbox{\footnotesize\boldmath $\alpha $}}=\sum_{\mbox{\footnotesize\boldmath $\alpha $}\in G_m}u^{(n)}_{2\mbox{\footnotesize\boldmath $\alpha $}}(z,v) J_{\mbox{\footnotesize\boldmath $\alpha $}},
\big\{f\in \Theta : \big| \widehat{f}_{2n_*}\big(\tfrac in \big) - f\big(\tfrac in\big)\big| \leq (C+1) L_{n_*}^{\beta/(\beta+1)}+C (L_{n_*}\widehat f_{2n_*}(x) )^{\beta/(2\beta+1)}, \ \i=1,\ldots,n\big\}.
\mathcal{F}\,=\, (1-\varphi ) F - \nabla \mathbb{B}[\nabla \varphi \cdot u]\,, g \,=\, F \cdot \nabla \varphi + 2 \nabla \varphi \cdot \nabla u + (\Delta \varphi + \alpha x^\bot \cdot \nabla \varphi ) u & - \alpha \big ( x^\bot \nabla \mathbb{B}[\nabla \varphi \cdot u] - \mathbb{B}[\nabla \varphi \cdot u]^\bot \big ) - ( \nabla \varphi ) p \,.
-(T^*-t_n)\|\nabla u(t_n)\|^2_{L^2}=g'(t_n) = \frac{g(T^*)-g(T_n)}{T^*-T_n}=-\frac{\displaystyle{\int}_{T_n}^{T^*}(T^*-\tau)\|\nabla u(\tau)\|^2_{L^2}d\tau}{T^*-T_n}.
\psi_0 (\xi) = \big((G^{-1}K)^\ast \big)^N \psi_0 (\xi) \ge (g_2^{-1}\alpha_1)^N \gamma_0 \int\limits_{\mathbb{T}^d}\psi_0(\eta) d \eta = (g_2^{-1}\alpha_1)^N \gamma_0 \|\psi_0\|_{L^1(\mathbb{T}^d)}\forall \; \xi \in \mathbb{T}^d,
r_{0}\surd \sigma = 1 . 1 9 2 ( 1 0 ) - ( 0 . 3 2 \pm 0 . 0 8 ) \biggl ({\frac{a_{s}}{r_{0}}}\biggr )^{2}+ ( 0 . 6 6 \pm 0 . 1 1 ) \biggl ({\frac{a_{s}}{r_{0}}}\biggr )^{4}
{\rm Exp}_{(A_r,B_r,C_r)}(\xi,\eta,\zeta) := (A_r^{\frac{1}{2}}\exp (A_r^{-\frac{1}{2}}\xi A_r^{-\frac{1}{2}}) A_r^{\frac{1}{2}}, B_r+\eta,C_r+\zeta)
f \star g = \sum_{s = 0}^{\infty}\frac{\kappa^{s}}{s !}\sum_{t = 0}^{s}( - 1 )^{t}\left( \begin{matrix}{{s}}\\{{t}}\\ \end{matrix}\right) [ \partial_{x}^{s - t}\partial_{p}^{t}f ] [ \partial_{x}^{t}\partial_{p}^{s - t}g ] \, .
\frac{\mid \alpha_{k}\mid^{2}}{\mid \beta_{k}\mid^{2}}= \frac{\mid \sinh \pi ( \mid \mu \mid - e E_{0}/ H^{2}) \mid e^{2 \pi ( \mid \mu \mid - e E_{0}/ H^{2})}}{\mid \sinh \pi ( \mid \mu \mid + e E_{0}/ H^{2}) \mid}
0 = \displaystyle\int_{\mathbb{R}^N}(- \Delta)^{\frac{\gamma}{2}}v^k(- \Delta)^{\frac{\gamma}{2}}\delta_{i}^kdx = \displaystyle\int_{\mathbb{R}^N}(- \Delta)^{\frac{\gamma}{2}}v^k(- \Delta)^{\frac{\gamma}{2}}\frac{\partial\delta_{i}^k}{\partial \lambda_{i}^k}dx i = 1, 2 k,
\hat{\tilde{E}}_{i}^{a}( x ) \Psi_{\gamma}( A ) ={\frac{\delta}{\delta A_{a}^{i}( x )}}\Psi_{\gamma}( A ) = \oint d s \ \delta^{3}( x - \gamma ( s ) ) \dot{\gamma}^{a}( s ) \mathrm{T r}( U ( 0 , s ) \tau^{i}U ( s , 1 ) )
\langle V ( x ) \rangle = N^{- 1}\lim_{T \rightarrow \infty}\int d A_{0}\langle 0 | e^{i T H}e^{{\frac{2 \pi i}{g}}\int_{C}d y^{i}\epsilon_{i j}E_{i}( y )}e^{i \int A_{0}[ \partial_{i}E_{i}- J_{0}]}e^{i T H}| 0 \rangle .
\left( f \star g \right) \star h - f \star \left( g \star h \right) = \frac{1}{6}\left( \alpha^{i\ell}\partial_{\ell}\alpha^{jk}+ \alpha^{j\ell}\partial_{\ell}\alpha^{ki}+ \alpha^{k\ell}\partial_{\ell}\alpha^{ij}\right) \left( \partial_{i}f \partial_{j}g \partial_{k}h \right) + \mathcal{O}(\alpha^{3}).
E[\frac{1}{2}\frac{\partial^{2}z}{\partial x_{1}^{2}}(x_{1}-\overline{x}_{1})^{2}]=\frac{1}{2}\frac{\partial^{2}z}{\partial x_{1}^{2}}E[(x_{1}-\overline{x}_{1})^{2}]=\frac{1}{2}\frac{\partial^{2}z}{\partial x_{1}^{2}}\sigma_{1}^{2}
\gamma_{n}=A\,\left( \Psi_{n}^{3/2}\,\,\max_{i\in \Upsilon_{3}}\,p_{i,n}^{1/2}+\max_{1\leq i\leq s_{n}}\,q_{i,n}^{1/2}\right)\rightarrow 0{\it ,\as\}n\rightarrow \infty ,
\prod\limits_{l=0}^{m-1}\left( x^{2}-l^{2}\right) =\sum_{k=1}^{m}t\left(2m,2k\right) x^{2k}\ , t\left( 2m,2k\right) =\left( -1\right)^{m-k}\left\vert t\left(2m,2k\right) \right\vert =\left( -1\right)^{m-k}\frac{1}{k!}\left.\frac{d^{k}}{dz^{k}}\prod\limits_{l=0}^{m-1}\left( z+l^{2}\right)\right\vert_{z=0}\ ,
G -{\frac{1}{2}}= J_{1}+ J_{2}- 1 = \frac{1}{\pi^{2}\sin \pi \nu}\frac{\partial}{\partial \nu}\left( \sin \pi \nu \int_{0}^{\infty}d t \, \frac{\sinh ( 1 - \nu ) t}{\sinh t \cosh \nu t}\right) .
D(\frac{\partial^{4}w^{0}}{\partial x_{1}^{4}}+2\frac{\partial^{4}w^{0}}{\partial x_{1}^{2}\partial x_{2}^{2}}+\frac{\partial^{4}w^{0}}{\partial x_{2}^{4}})=-q(x_{1},x_{2},t)-2\rho h\frac{\partial^{2}w^{0}}{\partial t^{2}}.
\widetilde{\boldsymbol{\zeta}}_u^n =\widetilde{\boldsymbol{\eta}}_{u,t}^n+\widetilde{\boldsymbol{\xi}}_{u,h}^n=\boldsymbol{u}(t_n)-\boldsymbol{u}_h^n, \boldsymbol{\zeta}_u^n =\boldsymbol{\eta}_{u,t}^n+\boldsymbol{\xi}_{u,h}^n=\boldsymbol{u}(t_n)-\boldsymbol{u}_h^n, \zeta_p^n =\eta_{p,t}^n+\xi_{p,h}^n=p(t_n)-p_h^n