harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
H =\begin{pmatrix}\alpha_{00}\vec{v}_{0}+ \alpha_{01}\vec{v}_{1}+ \alpha_{02}\vec{v}_{2}\alpha_{10}\vec{w}_{0}+ \alpha_{11}\vec{w}_{1}+ \alpha_{12}\vec{w}_{2}\alpha_{20}\vec{v}_{0}+ \alpha_{21}\vec{v}_{1}+ \alpha_{22}\vec{v}_{2}\alpha_{30}\vec{w}_{0}+ \alpha_{31}\vec{w}_{1}+ \alpha_{32}\vec{w}_{2}\end{pmatrix}
[s, x] [d_1, y_i ] + [s, y_i] [d_1, x] = [s, x] [z_1 \dots z_{\ell}, y_i ] + [s, y_i] [z_1 \dots z_{\ell}, x] = \ \sum_{j=1}^{\ell}\bigl( [s, x] z_1 \dots z_{j - 1}[z_j , y_i ] z_{j+1}\dots z_{\ell}+ [s, y_i] z_1 \dots z_{j - 1}[z_j , x] z_{j + 1}\dots z_{\ell}\bigr)
\otimes^L_n(F_1,\dots,F_n)X \qquad{}= \sum_{X=\sum\limits_{\varnothing \ne S\subseteq \langle n \rangle}{A_S}}{L\left( \sum_{S}(\#S-1)\cdot A_S\right) \otimes{F_1\left( \sum_{1\in S}A_S\right) \otimes \dots \otimes F_n\left( \sum_{n\in S}A_S\right)}}.
\langle\underline t,\partial_{\underline x}\rangle^2g =\partial_{\theta}^2g, \Delta_{\underline y}g =-\partial_{\underline y}^2g=-\partial_{\underline y}(\underline\nu\partial_{\rho}g)=\partial_{\rho}^2g+\frac{q-1}{\rho}\,\partial_{\rho}g.
\epsilon \equiv \frac{M_{P}^{2}}{2}\left( \frac{V^{\prime}}{V}\right)^{2}\sim \frac{M_{P}^{2}}{f^{2}}, \quad \eta \equiv M_{P}^{2}\frac{V^{\prime \prime}}{V}\sim \frac{M_{P}^{2}}{f^{2}}
|\Gamma_1^h(u)|+|\Gamma_2^h(u)|= |\langle(I_h -I)z,I_h (z u)\rangle_{H_x^\mathfrak{s}}|+|\langle z,(I_h -I)(z u)\rangle_{H_x^\mathfrak{s}}| = |\langle I_h (I_h -I)z,z u\rangle_{H_x^\mathfrak{s}}|+|\langle (I_h -I)z,z u\rangle_{H_x^\mathfrak{s}}|,
\psi_{n,k}(z)=\int_{a_{k-1}}^{b_{k-1}}\frac{\psi_{n,k-1}(\tau)}{z-\tau}\,\tau\,d\sigma_{k-1}^{*}(\tau),\psi_{n+1,k}(z)=z\int_{a_{k-1}}^{b_{k-1}}\frac{\psi_{n+1,k-1}(\tau)}{z-\tau}\,d\sigma_{k-1}^{*}(\tau),
\partial_{2}f^{[ M 2 ]}= \partial_{2}f^{[ M 5 ]}\left( f^{[ M 5^{\prime}]}- D_{i}D_{i}Y \right) + \partial_{2}f^{[ M 5^{\prime}]}\left( f^{[ M 5 ]}- D_{p}D_{p}X \right) .
T_f(\partial^{\alpha}U_{\omega}) = \int_{\mathcal{D}}\partial^{\alpha}U_{\omega}(x)f(x)dx = \lim_{n\rightarrow \infty}\int_{\mathcal{D}}\partial^{\alpha}U_{\omega}(x)\phi_n(x)dx = \lim_{n\rightarrow \infty}(-1)^{|\alpha|}\int_{\mathcal{D}}U_{\omega}(x)\partial^{\alpha}\phi_n(x)dx
\int_{B_R(\xi)}V_{\delta,\xi}^q=\begin{cases}O(\delta^{q\frac{n-2}{2}}) 0<q<\frac{n}{n-3}, O(\delta^\frac{n(n-2)}{2(n-3)}|\log \delta|) q=\frac{n}{n-3}, O(\delta^{n-q\frac{n-4}{2}}) \frac{n}{n-3}<q<\infty.\end{cases}
\left( \sum_{j=1}^{\infty}\left( \sum_{i=1}^{\infty}|T(e_{i},e_{j})|^{\frac{p}{p-1}}\right)^{2\frac{p-1}{p}}\right)^{\frac{1}{2}}\leq\left( \sum_{i=1}^{\infty}\left( \sum_{j=1}^{\infty}|T(e_{i},e_{j})|^{2}\right)^{\frac{1}{2}\frac{p}{p-1}}\right)^{\frac{p-1}{p}}& \leq C_{p,\infty}\Vert T\Vert.
\left|\sum_{i=0}^{n_j-1}\sum_{r=0}^{j-1}\sum_{\ell \in A_r \cap [k_j,k_j+n_j-N]}\psi_\ell(z_i,k_j+i)\right| \leq\sum_{r=0}^{j-1}\sum_{\ell \in A_r}\left|\sum_{i=0}^{n_j-1}\psi_\ell(z_i,k_j+i)\right| &\leq \sum_{r=0}^{j-1}\sum_{\ell \in B_r}\left|\sum_{i=0}^{n_j-1}\psi_\ell(z_i,k_j+i)\right| &\leq N^2\sum_{r=0}^{j-1}n_r \leq \frac{N^2n_j}{2^{j+1}}
\widehat{H}_{\alpha t}(\hat{u},\hat{p}):=\,\frac{1}{2}\,\sum^{8N}_{k=1}\,\Big((1+\alpha){\beta}^{k}(t, \hat{u})\,\hat{p}_{k}-\,\alpha\,\hat{p}_{k}\,{\beta}^{k}(t, \hat{u})\Big)
u_{\theta,\vartheta_1\vartheta_2 \ldots \vartheta_n}(t) := \left(u_{\theta,\vartheta_1}\circ u_{\theta,\vartheta_2}\circ \ldots \circ u_{\theta,\vartheta_n}\right)(t) = \frac{1}{\displaystyle \vartheta_1 \theta +\frac{1}{\displaystyle \vartheta_2 \theta + \ddots +\frac{1}{\vartheta_n \theta+t}}}.
\frac{\partial^2 u}{\partial t^2}= \frac{\partial^2 u}{\partial x^2}, \Omega \times (0,T], u = 0, \mathbf{x}\in \partial \Omega, \frac{\partial u}{\partial t}(0,\mathbf{x}) = 0, u(0,\mathbf{x}) = s(\mathbf{x}).
C_{N U T}= - \frac{( 4 \pi )^{( n - 2 ) / 2}\beta}{1 6 \pi^{3 / 2}\ell^{2}}\Gamma \left( \frac{3 - n}{2}\right) \Gamma \left( \frac{n}{2}\right) N^{n - 3}\left[ n ( n - 1 ) ( n - 2 ) N^{2}- ( n - 2 ) ( n - 3 ) \ell^{2}\right]
E_{0}(\mathbf{z}){_{0}\mathcal{F}_0}^{(d)}\left( \begin{matrix}\end{matrix};\mathbf{z},\mathbf{u}\right) ={_{0}\mathcal{F}_0}^{(d)}\left( \begin{matrix}\end{matrix};\mathbf{z},\mathbf{u}\right)|\mathbf{u}|.
\sigma_{\rm a}^2 (x[n])=\lambda_\mathrm{a}[n]T_{\rm s}\exp\left(-\frac{\lambda_\mathrm{a}[n]\tau_{\rm d}}{N_{\rm a}}\right)-\frac{\lambda_\mathrm{a}^2 [n]T_{\rm s}\tau_{\rm d}}{N_{\rm a}}\exp\left(-\frac{2\lambda_\mathrm{a}[n]\tau_{\rm d}}{N_{\rm a}}\right)\left(2-\frac{\tau_{\rm d}}{T_{\rm s}}\right)
\sum_{m=2}^N2^{3m}c_0^{\frac{m}{2}}\|J_{m}^{l+1}\|_{0,r}=2^6c_0\|J_2^{l+1}\|_{0,r}+\sum_{m=4}^N2^{3m}c_0^{\frac{m}{2}}\|J_m^{l+1}\|_{0,r}&\le c\alpha^{-2}+c\alpha^{-4}\le c\alpha^{-2}.
(M'(t))^2 = 4 \left( |{\rm Re}(u,u_t)|^2 + 2b{\rm Re}(u,u_t) \int_0^t{\rm Re}(u(s),u_s(s))ds + b^2 \left(\int_0^t{\rm Re}(u(s),u_s(s))ds\right)^2 \right) &\leq 4 \left( ||u||^2_{L^2(\mathbb{H}^n)}+ b \int_{0}^t ||u(s)||^2_{L^2(\mathbb{H}^n)}ds \right)\left( ||u_t||^2_{L^2(\mathbb{H}^n)}+ \int_{0}^t ||u_s(s)||^2_{L^2(\mathbb{H}^n)}ds \right),
\begin{array}{l}T_{e,j,1}= \{4kd+j, 4kd+e+j, \ldots, 4kd+(4k-1)e+j\}, T_{e,j,2}= \{4kN+j, 4kN+2e+j\}, U_{e,\ell,1}= \{4kd+4ke+\ell, 4kd+(4k+1)e+\ell, \ldots, 4kd+(8k-1)e+\ell\}, U_{e,\ell,2}= \{4kN+e+\ell, 4kN+3e+\ell\},\end{array}
\nu \left|r_\rho\right|_{H^1}^2+\left(\mathbf{curl}\,\sigma\left(\mathbf r_\rho\right)\times \mathbf y+ \mathbf{curl}\,\sigma\left(\mathbf y_\rho\right)\times \mathbf r_\rho+\mathbf{curl}\,\sigma\left(\mathbf y_\rho-\mathbf y\right)\times \mathbf z,\mathbf r_\rho\right)=0.
A^{4}=[\begin{matrix}1/2&1/2\\ -\frac{1}{\sqrt{6}}&\frac{1}{\sqrt{6}}\end{matrix}]\cdot[\begin{matrix}(1-\sqrt{6})^{4}&0\\ 0&(1+\sqrt{6})^{4}\end{matrix}]\cdot[\begin{matrix}1&-\sqrt{6}/2\\ 1&\sqrt{6}/2\end{matrix}]=[\begin{matrix}73&84\\ 56&73\end{matrix}].
K_p( x,b,c,d,\nu) :=\inf\left\{\int_{Q_{\nu}}f_p^{\infty}(x,w( y),\eta( y) ,\nabla w(y))dy:w\in\mathcal{A}( c,d,\nu), \eta\in L^{\infty}( Q_{\nu};\mathbb{R}^{m}), \int_{Q_{\nu}}\eta\,dy=b\right\},
b^{T,sym}_{h}(u_{h},v_{h}):=\frac{1}{2}\int_{T}(\nabla \cdot \vec{b}) \, \Pi_{k}(u_{h}) \, \Pi_{k}(v_{h}) \, dT+s^{T,sym}((I-\Pi_{k})u_{h},(I-\Pi_{k})v_{h}) \, dT
p ( x ) ={\frac{1}{x}}+{\frac{-{\frac{1}{2}}}{x + \tilde{Q}_{5}}}+{\frac{\frac{1}{2}}{x + \tilde{Q}_{5}- \ell^{2}}}+{\frac{-{\frac{1}{2}}}{x + \cos^{2}\gamma \tilde{Q}_{5}}}.
4 \left( \begin{array}{c}{{\bf E}^{\prime}}\\{{\bf B}^{\prime}}\\ \end{array}\right) = R \left( \begin{array}{c}{{\bf E}}\\{{\bf B}}\\ \end{array}\right) \equiv \left( \begin{array}{l l}{\cos \theta}&{\sin \theta}\\{- \sin \theta}&{\cos \theta}\\ \end{array}\right) \left( \begin{array}{c}{{\bf E}}\\{{\bf B}}\\ \end{array}\right)
(\begin{matrix}n\\k_{1},k_{2},...,k_{m}\end{matrix})=(\begin{matrix}n-1\\k_{1}-1,k_{2},...,k_{m}\end{matrix})+(\begin{matrix}n\\k_{1},k_{2}-1,...,k_{m}\end{matrix})+\cdot\cdot\cdot+(\begin{matrix}n-1\\k_{1},k_{2},...,k_{m}-1\end{matrix})
\|\nabla h_1\|_{L^2}^2=-\langle\Delta h_1,h_1\rangle=-\langle(\partial_t \kappa+\nu\Delta \kappa)\partial_x\partial_yf,h_1\rangle \leq \|\partial_t \kappa+\nu\Delta \kappa\|_{L^{4}}\|\partial_x\partial_yf\|_{L^{2}}\|h_1\|_{L^{4}}\leq C\nu\|\partial_x\nabla f\|_{L^{2}}\|h_1\|_{H^{1}}\leq C\nu\|\nabla\partial_x^2 f\|_{L^{2}}\|\nabla h_1\|_{L^2}.
\mathbb{E}e^{i y_1 (Z(t_3)-Z(t_2))+i y_2 (Z(t_2)-Z(t_1))}= \mathbb{E}\bigl [ \mathbb{E}\bigl(e^{i y_1 (M(T_3)-M(T_2))+i y_2 (M(T_2)-M(T_1))}|T_1,T_2,T_3 \bigr) \bigr] &= \mathbb{E}\bigl [ \mathbb{E}\bigl(e^{i y_1 (M(T_3)-M(T_2))}|T_2,T_3\bigr) \mathbb{E}\bigr(e^{i y_2 (M(T_2)-M(T_1))}|T_1,T_2\bigr)\bigr],
\mathcal{L}^zw =\Delta w +\nabla\cdot( z w - w\nabla\phi_{n_\infty}-n_\infty\nabla\phi_w)=\nabla\cdot\Big(n_\infty\nabla\Big(\frac{w}{n_\infty}-\phi_w\Big)\Big) &=\nabla\cdot\Big(n_\infty\nabla(\mathcal{M}^z w)\Big),
dX_t = \left(-i\left[X_t,H_t\right] + \mathcal{L}_L (X_t)\right)dt + d\mathbf{A}^{\dagger}_t\mathbf{S}^{\dagger}_t\left[X_t,\mathbf{L}_t\right] + \left[\mathbf{L}^{\dagger}_t,X_t\right]\mathbf{S}_t d\mathbf{A}_t & + \left[\left(\mathbf{S}^{\dagger}_t X_t \mathbf{S}_t - X_t\right)d\mathbf{\Lambda}_t^{\top}\right],
q^{\mu}\left\{\tilde{\Gamma}_{\mu}( q ; p , p^{\prime}) - i e \mu^{\frac{\epsilon}{2}}( p^{\prime}+ p )_{\mu}\tilde{D}\right\}= - e \mu^{\frac{\epsilon}{2}}\left\{( \tilde{\Gamma_{2}}( p^{\prime}) + i p^{\prime 2}\tilde{D}) - ( \tilde{\Gamma_{2}}( p ) + i p^{2}\tilde{D}) \right\}\; .
\begin{array}{rcl}U_f = \displaystyle +[D_m(k_1)D_m(k_2)+D(k_1)D(k_2)]D(k_1+k_2-p)(1+K_{12}) & \displaystyle +2D_m(k_1)D(k_2)D(k_1+k_2-p)(1-K_{12}) & \displaystyle +4[(N-1)D(k_1)D(k_2) & \displaystyle +D(k_1+p)D(k_2+p)K_{12}]D(k_1+k_2+p), \end{array}
g(V_{\gamma(t)},\nabla_{\dot{\gamma}(t)}\dot{\gamma}(t))_{\gamma(t)}=\dot{\gamma}(t)g(V,\dot{\gamma})+g(\dot{\gamma}(t),[V,\dot{\gamma}]_{\gamma(t)})_{\gamma(t)}-\frac{1}{2}V_{\gamma(t)}g(\dot{\gamma},\dot{\gamma}).
\frac{d E}{d \tau}= \frac{d}{d \tau}\int_{V}d^{3}\sigma \: [ \frac{\vec{E}_{\perp}^{2}+ \vec{B}^{2}}{2}] ( \tau , \vec{\sigma}) = - \int_{S}d \Sigma \: \vec{n}\cdot ( \vec{E}_{\perp}\wedge \vec{B}) ( \tau , \vec{\sigma}) .
\mathcal M_{A}^{m,main}\phi = 2i \sum_k \P_{\geq k+m}A^j \partial_j P_k \phi + P_{<k-m}(P_{< k-m}A^j \partial_j P_k \phi) + P_{\geq k-m}( P_{[k-m,k+m)}A^j \partial_j P_k \phi)
\hat{U}_{m}( F ) = \mathrm{e x p}\{( \hbar / i ) \hat{T}_{m}( F ) \}, \qquad \hat{T}_{m}( F ) = \mathrm{~ \frac{1}{2}~}\epsilon_{a b}\{\bar{\Delta}_{m}^{b}, [ \bar{\Delta}_{m}^{a}, F ] \}+ ( i / \hbar )^{2}m^{2}F ,
\mathcal T_*P^t=\varepsilon\tilde P^t,\mathcal T_*P^x=\varepsilon\varepsilon'a\tilde P^x,\mathcal T_*P^y=\varepsilon a\tilde P^y,\mathcal T_*G^x=\varepsilon'a\tilde G^x,\mathcal T_*G^y=a\tilde G^y, \mathcal T_*D =\tilde D,\mathcal T_*\Pi=\varepsilon\tilde\Pi,\mathcal T_*J =\varepsilon'\tilde J,
{[}L_{n}, \psi ( L_{- 1}\xi | z ){]}= \sum_{k = - 1}^{l ( n )}z^{n - k}\left({n + 1 \atop k + 1}\right) \left( \frac{n - k}{z}\psi ( L_{k}\xi | z ) + \psi ( L_{- 1}L_{k}\xi | z ) \right)
d(f,g) = \displaystyle\sup_{x\in M}|d_M(f(x),g(x))|+\sum_{i=1}^k\||f^{(i)}(x)-g^{(i)}(x)\|+\int_I|f^{(k+1)}-g^{(k+1)}| &= d_{C^k}(f,g)+\|f^{(k+1)}-g^{(k+1)}\|_{L^1}
y_{\textrm{NF}}=\sqrt{\beta_{\textrm{N}}\eta P_\textrm{B}d_{\textrm{BN}}^{-\alpha}d_{\textrm{NF}}^{-\alpha}|h_{\textrm{BN}}|^2}h_{\textrm{NF}}x_{\textrm{F}}+n_{\textrm{F}},
I_{2}^{q}=\sum_{k=-\infty}^{+\infty}\int_{2^{k}\leq|x|\leq 2^{k+1}}\left(\int_{B(0,2|x|)\setminus B\left(0,\frac{|x|}{2}\right)}|G_{\xi, \sigma}(y^{-1}x)u(y)| dy\right)^{q}\frac{dx}{|x|^{\beta q}}.
\mu^\sigma_t(\d{x}) = e^{-|\sigma|t}\sum_{k=0}^\infty \frac{t^k \sigma^{k*}(\d{x})}{k!}= e^{-|\sigma|t}\delta_0(\d{x}) + e^{-|\sigma|t}\sum_{k=1}^\infty \frac{t^k \sigma^{k*}(x)\d{x}}{k!}.
\sum_{n=0}^{\infty}\frac{nc^nq^{n^2}}{(q)_n(cq)_n}-\sum_{k=1}^{\infty}\frac{(-c)^kq^{\frac{k(k+1)}{2}}}{(q)_k(1-q^k)}\sum_{j=0}^{\infty}\frac{c^jq^{(j+k)^2}}{(cq)_{j+k}(q)_j}=\frac{1}{(cq)_{\infty}}-1-\frac{1}{(cq)_{\infty}}\sum_{k=1}^{\infty}\frac{(-c)^kq^{\frac{k(k+3)}{2}}}{(q)_k(1-q^k)}.
\mathbf{c}(\varepsilon,\kappa,\tau,\zeta)=\frac{\varepsilon^{-d+1}}{\rho^\alpha}\int_{\mathbb{R}^d}\varphi_{\zeta\rho}(z'-z)\frac{\langle\phi^{e,\varepsilon}((z'-z)/\rho),\eta^{\kappa}_{y_{\tau}}\rangle}{|z'-z|^{d-\alpha}}dz=\varepsilon^{-d+1}\int_{\mathbb{R}^d}\varphi_{\zeta}(z)\frac{\langle\phi^{e,\varepsilon}(z),\eta^{\kappa}_{y_{\tau}}\rangle}{|z|^{d-\alpha}}dz.
0= \left( \frac{\partial U}{\partial u_\nu}\right)_{u_\mu = e_\mu}\rightarrow \frac{1}{g}+ \sum_{\mu=1 ( \neq \nu)}^{N}\frac{2}{e_\mu - e_\nu}= \sum_{j=1}^\Omega \frac{1}{z_j - e_\nu}\; ,
\begin{matrix}p(\mu|\Sigma,X)&\sim&N(\frac{n\overline{x}+m\mu_{0}}{n+m},\frac{1}{n+m}\Sigma),\\p(\Sigma|X)&\sim&W^{-1}(\Psi+nS+\frac{nm}{n+m}(\overline{x}-\mu_{0})(\overline{x}-\mu_{0})^{\prime},n+n_{0}),\end{matrix}
\nabla^1_k=d+\sum_{1\leq i<j\leq k}\theta^0_{ij}\t_{ij}+\cdots,\ \theta^1_{ij}=\frac{1}{2\pi i}d\log(\overline z_i-\overline z_j)=\frac{1}{2\pi i}\frac{d\overline z_i-d\overline z_j}{\overline z_i-\overline z_j},
P_g\Big\{\big[v(t),w(t)\big];[\bar{v}(t),\bar{w}(t)\big]\Big\}=\frac{1}{Z}\exp\Bigg[-\frac{1}{2\sigma^2}\left(\begin{array}{c}v(t)-\bar{v}(t) w(t)-\bar{w}(t)\end{array}\right)^{\top}\Theta_{ij}^{-1}(t) &\times\left(\begin{array}{c}v(t)-\bar{v}(t) w(t)-\bar{w}(t)\end{array}\right)\Bigg].
\mathcal{A}_\chi:= \left( \left( \widehat{\Pi}_\chi^{\rm stiff}\right)^*\right)^{-1}\widehat{\Lambda}_\chi^{\rm stiff}\left( \widehat{\Pi}_\chi^{\rm stiff}\right)^{-1}, \mathcal{B}_\chi (z):= zI + \left( \left( \widehat{\Pi}_\chi^{\rm stiff}\right)^*\right)^{-1}\widehat{M}_\chi^{\rm soft}(z) \left( \widehat{\Pi}_\chi^{\rm stiff}\right)^{-1},
\tilde{\cal Z}_{\{n_{E}^{a}, n_{M}^{a}\}}= \sqrt{2}Q^{a}a_{a}^{*}, \qquad \left| \tilde{\cal Z}_{\{n_{E}^{a}, n_{M}^{a}\}}\right| = \left|{\cal Z}_{\{- n_{E}^{a}, n_{M}^{a}\}}\right| ,
df_n(u)[X_H(u)] = d\tilde{f}_n(\psi_n^{-1}(u))\circ d\psi_n^{-1}(u)[X_H(u)] = d\tilde{f}_n(t,v)\left[ \frac{\partial}{\partial t}\right] = \frac{\partial \tilde{f}_n}{\partial t}(t,v) = - \frac{\chi(\|v\|/r_n)}{1+t^2}< 0.
\langle \Psi_{k}\, | \, k \,^{\prime}\rangle_{\epsilon}= c_{k}\, \frac{2^{\epsilon + i k \,^{\prime}}}{4 \sqrt{2 \pi}}\, \Gamma \left( \frac{\epsilon + i k \,^{\prime}- i k}{2}\right) \Gamma \left( \frac{\epsilon + i k \,^{\prime}+ i k}{2}\right) ,
0 =\sum_{i=1}^{n}\frac{g_{-,b,C}(x_{i})}{\sigma^{2}(x_{i})}\left(x_{i},\dotsc,x_{i}^{p-1}\right)', 0 =\sum_{i=1}^{n}\frac{g_{+,b,C}(x_{i})}{\sigma^{2}(x_{i})}\left(x_{i},\dotsc,x_{i}^{p-1}\right)', 0 =\sum_{i=1}^{n}\frac{g_{+,b,C}(x_{i})}{\sigma^{2}(x_{i})}+\sum_{i=1}^{n}\frac{g_{-,b,C}(x_{i})}{\sigma^{2}(x_{i})}.
\omega_{D}^{3}=\frac{18\pi^{2}N}{V}\frac{(v_{s,1}v_{s,2}v_{s,3})^{3}}{(v_{s,2}v_{s,3})^{3}+(v_{s,1}v_{s,3})^{3}+(v_{s,1}v_{s,2})^{3}}.
\begin{pmatrix}S_1^T E_{2\lfloor \frac{1}{4}(n-2k-2l_2) \rfloor}S_3^T E_{2\lfloor \frac{1}{4}(n-2k-2l_2+2) \rfloor}\end{pmatrix}\begin{pmatrix}S_2 S_4 \end{pmatrix}=0.
v(x,t)= \frac{1}{\pi}\int^{\infty}_0 e^{-i\beta^2 t}e^{i\beta x}\beta \int^{\infty}_0 e^{i\beta^2 \tau}h(\tau) d\tau d\beta +\frac{1}{\pi}\int^{\infty}_{0}e^{i\beta^2t}e^{-\beta x}\beta \int^{\infty}_0 e^{-i\beta^2 \tau}h(\tau) d\tau d\beta.
d\Gamma^\lambda_{t}=\sum_{\alpha=1}^{d}\tilde{W}_{\alpha}(\Gamma^\lambda_{t})d(B_{t}^{\alpha,\lambda}) =\lambda\sum_{\alpha=1}^{d}\tilde{W}_{\alpha}(\Gamma^\lambda_{t})d\left(\lambda^{-1}B^{\alpha,\lambda}_{t}\right)_t.
A_{n}(x) =\left( l_{a,n}\right)^{-1}a_{n}(x) \, , a_{n}(x)= l_{a,n}x^n + \, (\textrm{terms\, of\,degree\,less\,than\,}\, n ) &B_{n}(x) =\left( l_{b,n}\right)^{-1}b_{n}(x) \, , b_{n}(x)= l_{b,n}x^n + \, (\textrm{terms\, of\,degree\,less\,than\,}\, n )
{\cal L}= - \frac 1 2 \varphi \partial^{2}\varphi + \frac{i}{2}\overline{{\psi}}\not \! \partial \psi + \frac 1 2 F^{2}+ \sigma \varphi_{j}F_{j}+ \frac 1 2 \lambda \varphi^{2}- \frac 1 2 \sigma \overline{{\psi}}\psi - \overline{{\xi}}\psi_{j}\varphi_{j}- \frac{N}{2 g}\lambda ,
A_{m}^{\alpha,\lambda}\Phi_{\lambda,\bar{\lambda}}(w,\bar{w})=\int_w{\frac{dz}{2\pi{i}}(z-w)^{m-\frac{\alpha\lambda}{2k}}\boldsymbol{A}_\alpha(z)\Phi_{\lambda,\bar{\lambda}}(w,\bar{w})}.
g = \frac{1}{2}u_{b}^{2}b_{\mathrm{e x t}}- L ,{\cal E}\simeq 2 \pi \frac{\Delta u}{u_{b}}\left[ f^{2}g^{2}+ \frac{1}{2}\kappa^{2}u_{b}^{2}f^{4}- \kappa^{2}u_{b}^{2}f^{2}\right] .
\Gamma^{a_{1}\cdots a_{N}}[ k_{1}, \varepsilon_{1}; \ldots ; k_{N}, \varepsilon_{N}] = ( - i g )^{N}\mathrm{t r}( T^{a_{1}}\cdots T^{a_{N}}){\frac{{( 4 \pi \mu^{2})}^{-{\frac{\epsilon}{2}}}}{3 2{\pi}^{2}}}\sum_{\mathrm{d i a g r a m s}}D_{i}
\partial_{j}u_{i(k)}(x,t)= \int\limits_{A-1}^{0}\eta_{\frac{1}{k}}(t-\tau)\int\limits_{\mathbb{R}^{3}_{+}}\omega_{\frac{1}{k}}(x-y)\frac{\partial}{\partial y_{j}}\phi_{k}(y)u^{(\frac{2}{k})}_{i}(y,\tau)dyd\tau+
[ \delta_{1}( \Lambda_{1}) , \delta_{2}( \Lambda_{2}) ] = ( 1 - 2 ) \delta_{2}( \Lambda_{2}^{\prime}= \sum_{n = 0}^{\infty}\frac{k^{n + 1}}{n !}[ \Lambda_{1}, \Lambda_{2}, \Psi^{n}] ) ,
d s^{2}= - \left( 1 - \frac{1 6 \pi{\cal G}M}{( D - 2 ) \Omega_{D - 2}}\frac{1}{r^{D - 3}}\right) d t^{2}+ \left( 1 - \frac{1 6 \pi{\cal G}M}{( D - 2 ) \Omega_{D - 2}}\frac{1}{r^{D - 3}}\right)^{- 1}d r^{2}+ r^{D - 2}d \Omega_{D - 2}^{2}.
0 = \lim_{y\to \bar x^+}((y-\bar x) \mathcal D(\alpha,\bar x)) & \leq \lim_{y\to \bar x^+}(\log M_y(\alpha) - M_{\bar x}(\alpha)) & = \lim_{y\to \bar x^+}\log M_y(\alpha) - \lim_{y\to \bar x^+}\log M_{\bar x}(\alpha) & = \lim_{y\to \bar x^+}\log M_y(\alpha) - \log M_{\bar x}(\alpha),
\P^U \left( \frac{m-1}{2\sqrt{b}}\underset{i=1}{\overset{b}{\sum}}\left( (p_{f,U})_i - \frac{1}{2}\right)^2 \leq N_\alpha \right) \leq \P^U \left( \frac{m-1}{24\sqrt{b}}\underset{i=1}{\overset{b}{\sum}}\min \left\{\frac{n}{m}(Uf)_i^2, 1 \right\}\leq N_\alpha \right).
\Lambda \geq{\frac{3 ( 3 \gamma + 4 n - 2 ) k}{3 \gamma + 4 n}}\left[ -{\frac{3 ( 3 \gamma + 4 n - 2 ) k c_{0}^{2}}{4 \pi ( 3 \gamma + 2 n ) ( 3 \gamma + 2 n - 2 ) G B}}\right]^{\frac{2}{3 \gamma + 2 n - 2}}.
g=( 1,n+2,2,n+3,5,n+6,7,n+8,\dots,2i-1,n+2i,\dots,n-2,2n-1,n, &n+1,3,n+4,4,n+5,6,n+7,\dots,2j,n+2j+1,\dots,n-1,2n).
L(t)=\frac{n-1}{t},W(t)=\frac{C}{t^{(2-\gamma)p}}, w=t^{\gamma p},G(t)=\frac{a}{t^{2p-2}}\quad\mbox{and}H(t)=\frac{b}{t},\qquad\forall t\in(0,\sup_\Omega\rho),
d s_{Q^{1 , 1 , 1}}^{2}= \frac{3}{8 \Lambda}( d \psi -{\cal A})^{2}+ \frac{3}{4 \Lambda}\sum_{i = 1}^{3}\left( d \theta_{i}^{2}+ \sin^{2}\theta_{i}\, d \phi_{i}^{2}\right) \, ,
\lim_{z \to 0}\left| \ln |\ln z^2 |\right|^{-1}\mathfrak{n}(T^\pm (z), a ) = \lim_{z \to 0}\left| \ln |\ln z^2 |\right|^{-1}\mathfrak{n}\left (\frac 3{2\pi c_0^2}T^\pm (z), \frac{3a}{2\pi c_0^2}\right) = \frac 4{3\pi a}
A_j \leq \sum_{i=-\infty}^{1}(2^{j}\rho)^d\int_{S^{d-1}}\sup_{r\in [2^{j}\rho,2^{j+1}\rho]}|\mathbf{K}_{i,n}(r\theta)| + (2^{j}\rho)^d\int_{S^{d-1}}\sup_{r\in [2^{j}\rho,2^{j+1}\rho]}|\varphi_{\rho}(r\theta)\mathbf{K}_{n}(r\theta)| &\lesssim 2^{jd}+ \sum_{i=-\infty}^{1}(2^{j}\rho)^d\int_{S^{d-1}}\sup_{r\in [2^{j}\rho,2^{j+1}\rho]}|\mathbf{K}_{i,n}(r\theta)| ,
\left(\begin{array}{c}\theta^{\mathrm{descr}}_n \gamma^{\mathrm{descr}}_n\end{array}\right)=\left(\begin{array}{cc}W^{XX}_n W^{XZ}_n W^{ZX}_n W^{ZZ}_n\end{array}\right)^{-1}\left(\begin{array}{c}W^{XY}_n W^{ZY}_n\end{array}\right),
\displaystyle \lim\limits_{\|\bigtriangleup_{n}\| \longrightarrow 0}\mathbb{E}| \sum_{\substack{j\in J_{K}}}\Phi(\frac{B_{t_{j}}+B_{t_{j+1}}}{2},t_{j}) (B_{t_{j+1}}-B_{t_{j}}) |^2 =0
\begin{aligned}\sup_{t_a \in \Omega_a}||B\exp(t+t_a) \sum_{\zeta}\alpha_{\zeta}e_{J_0}\wedge e_{\zeta}|| \geq \kappa_2 \sup_{\xi}|(\sum_{\zeta}\alpha_{\zeta}\beta^{\xi}_{\zeta}) | ||B A_0^{-1}\exp(t) e_{J_0}\wedge e_{\xi}|| \end{aligned}
\psi^{\alpha_1^1\alpha_1^2,\alpha_2^1\alpha_2^2}_{R_1 f_1,R_2 f_2}(x_1,x_2)=-\psi^{\alpha_2^1\alpha_2^2,\alpha_1^1\alpha_1^2}_{R_2 f_2,R_1 f_1}(x_2,x_1).
zn(u;k)=\sum_{n=1}^{\infty}\frac{2\pi K(k)^{-1}sin[\pi K(k)^{-1}u]q(k)^{2n-1}}{1-2cos[\pi K(k)^{-1}u]q(k)^{2n-1}+q(k)^{4n-2}}
\tau_{s_i (\nu) + \alpha, \mu}^{2}= g(\langle \alpha,\nu-\rho \rangle - 1) \dfrac{1-\mathbf{z}^{-n\alpha}}{1-q^{-1}\mathbf{z}^{-n\alpha}}= g(c_i-c_j) \dfrac{1 - \mathbf{z}^{-n\alpha}}{1 - v \mathbf{z}^{-n\alpha}}\, ,
\omega_m-\omega_l = j c \; + \; \frac{1}{2}\left(\frac{v_m}{m}-\frac{v_l}{l}\right) \; + \; \frac{c^3}{2\lambda_l^{1/2}}- \frac{c^3}{2\lambda_m^{1/2}}+ O\left(\frac{1}{m^3}\right) + O\left(\frac{1}{l^3}\right),
\tilde{\mathcal{R}}_{k,\ell}(\alpha+1)-\tilde{\mathcal{R}}_{k,\ell}(\alpha) = X^{\alpha+1}\Big( \tau_{k-1}(p^{\alpha+1}) \sum_{i=0}^{\ell-2}\binom{i+\alpha}{i}(1-X)^{-\ell+i+1}+\tau_{\ell-1}(p^{\alpha+1}) \sum_{i=0}^{k-2}\binom{i+\alpha}{i}(1-X)^{-k+i+1}-\tau_{k-1}(p^{\alpha+1}) \tau_{\ell-1}(p^{\alpha+1}) \Big).
\lim_{s\to 1}\int_0^{s}\frac{1}{\sqrt{1-u^2}}\, e^{-\frac{(b-au)^2}{2 (1-u^2)}}du=\int_0^{1}\frac{1}{\sqrt{1-u^2}}\, e^{-\frac{(b-au)^2}{2 (1-u^2)}}du
\sum_{j=1}^m \rho_j^{3}=\sum_{j= j_{1n}+1}^{j_{2n}}\frac 1{(nF_j)^{3/2}}\asymp n^{\frac{1-3\beta}{2\beta+1}}\int_{x_{j_{1n}}}^{x_{j_{2n}}}f_0(x)^{-\frac{3\beta+4}{2\beta+1}}dx .
\mathbb{E}_{\underline{Z}}\sup_{g\in\mathcal{G}}\left\{\frac{1}{n}\sum_{i=1}^nZ_i(g^2(X_i)-g^2(X^{\prime}_i)) - \frac{\gamma}{n}L(g)-\frac{1}{2\gamma}s^2(g)\right\}\leq\frac{\gamma}{n}\log\mathbb{E}_{\underline{Z}}\sup_{g\in\mathcal{G}}\exp\left\{\frac{1}{\gamma}\sum_{i=1}^nZ_i(g^2(X_i)-g^2(X^{\prime}_i)) - L(g)-\frac{n}{2\gamma^2}s^2(g)\right\}.
\left \| \frac{d \mathbb{P}_{Gen(n)}}{d \mathbb{P}_{\mathcal{P}_n}}- \frac{d \mathbb{P}_{\mathcal{GS}_n}}{d \mathbb{P}_{\mathcal{P}_n}}\right\|_p^{\mathbb{P}_{\mathcal{P}_n}}=\left \| \frac{d \mathbb{P}_{Gen(n)}}{d \mathbb{P}_{\mathcal{GS}_n}}- 1\right\|_p^{\mathbb{P}_{\mathcal{GS}_n}}{\left( \frac{|\mathcal{P}_n|}{|\mathcal{GS}_n|}\right)^{1- 1/p}}.
\Phi_{d,\ell}^{(\alpha)}(\sigma) := \frac{2^{\alpha}\Gamma\left(\frac{\ell+\sigma+\alpha}{2}\right) \Gamma \left(\frac{d+\ell-\sigma}{2}\right)}{\Gamma \left(\frac{\ell+\sigma}{2}\right) \Gamma \left(\frac{d+\ell-\sigma-\alpha}{2}\right)}, \sigma \in (-\ell-\alpha,d+\ell),
\limsup_{\rho\to0^{+}}\|\Psi_{\rho}\|_{L^{q(\cdot)}_{\nu}(B_{2\rho}(x_i))}\geqslant \nu_{i}^{\frac{1}{q(x_i)}}=\nu_{i}^{\frac{1}{\overline{p}^{\star}_{s}}}.
\begin{array}{c c c}{{\phi_{1}= 3 D A_{1}= 3 D A_{2}= 3 D 0}}&{{,}}&{{\phi_{2}( x ) = 3 D \phi_{2}( x_{3}, x_{4}= )}}\\{{A_{3}( x ) = 3 D A_{3}( x_{3}, x_{4})}}&{{;}}&{{A_{4}( x ) = 3 D A_{4}( x_{3}, x_{4})}}\\ \end{array}
\begin{array}{lll}Z = (\frac{\lambda}{2\pi})^n\ \displaystyle\int{\cal D}u{\cal D}\psi{\cal D}\chi \exp (-L) ~ = \displaystyle\int{\cal D}u{\cal D}\psi{\cal D}\exp(\frac{\lambda_{ijk}\psi^i \psi^j \chi^{2k-1}\chi^{2k}}{\lambda}) ~ = \displaystyle\int{\cal D}u (\frac{1}{2\pi})^n \prod\limits_{k=1}^{n}\lambda_{ijk}.\end{array}
\Big(\int_{0}^1\int_0^1 \Big\| \Big(\sum_{2^k\in A}2^{ksq}|\psi_k*( T_{t_1}f_{t_2})|^q \Big)^{1/q}\Big\|_q^q dt_1 dt_2\Big)^{1/q}&=\Big(\sum_{2^k\in A}2^{ksq}\Big\|\Big(\int_{0}^1\int_0^1 \big|\psi_k* T_{t_1}f_{t_2}(x)\big|^q dt_1 dt_2\Big)^{1/q}\Big\|_q^q \Big)^{1/q}.
h = - \frac{1}{k}\int_{\Sigma_{\tau}}d \bar{\eta}d \eta ~ ~{\cal E}( - \eta ) \sum_{i}q_{i}E_{i i}+ \frac{1}{2 k}\sum_{i \neq j}\int_{\Sigma_{\tau}}d \bar{\eta}d \eta ~ ~ \left( \frac{u_{j}}{u_{i}}w_{i j}v_{i j}+ \frac{u_{i}}{u_{j}}w_{j i}v_{j i}\right) \left( E_{i i}- \frac{1}{n}I \right) .
H=\frac{1}{2}e^{-2A_0(y)}(\dot{\phi}^2+(\vec{\nabla}\phi)^2)+\frac{1}{2b_0^2}(\phi^\prime)^2e^{-4A_0(y)}+\frac{1}{2}m^2 \phi^2 e^{-4A_0(y)}.
d_\mu(q) =(1 - \delta_{q_3,0})\,\frac{1}{2q^2\,q_3^2}\,q_\mu\left(1 - \delta_{\mu,3}\right)+ \delta_{q_3,0}\,\frac{q_\mu}{(q^2 + i\epsilon)^2}\,(1 - \xi) \ .
\psi_{0}( \vec{r}) \, = \, \sum_{n = - \infty}^{\infty}T_{n}^{0}( r , \phi ) \, = \, \sum_{n = - \infty}^{\infty}\int_{- \infty}^{\infty}d \lambda ( - i )^{| \lambda |}J_{| \lambda |}( k r ) e^{i \lambda ( \phi + \pi - \theta + 2 n \pi )}.
M = N \prod_{\ell_{1}\in \mathcal{R}_1}\ell_{1}^{\alpha(\ell_1)}\prod_{\ell_{2}\in \mathcal{R}_2}\ell_{2}^{\alpha(\ell_2)}0\leq \alpha(\ell_1)\leq 1 0\leq \alpha(\ell_2)\leq 2,
N^{\ast}d\widetilde{\mu_i}=N^{\ast}d\mu_i+N^{\ast}df_i =\lambda_id\mu_i+\sum_{j=1}^n\frac{\partial f_i}{\partial\lambda_j}N^{\ast}d\lambda_j =\lambda_id\mu_i+\sum_{j=1}^n\frac{\partial f_i}{\partial\lambda_j}\lambda_jd\lambda_j =\lambda_id\mu_i+\sum_{j=1}^n\beta_{ji}d\lambda_j.
D_{t t}- D_{t b}( Q^{- 1})_{b c}D_{c t}= g_{t t}+ \lambda^{2}D_{t}\Phi^{a}( Q^{- 1})_{a b}D_{t}\Phi^{b}= P [ g_{t t}+ g_{t a}( X^{a b}- \delta^{a b}) g_{b t}] \ ,
\phi(G) =\phi(S)T &=\left[\begin{array}{cc}* * 0 f_{22}(I_1)+g_{22}(M-Y)+h_{22}(B) \end{array}\right] \left[ \begin{array}{cc}A Y 0 I_2 \end{array}\right] &=\left[\begin{array}{cc}* * 0 f_{22}(I_1)+g_{22}(M-Y)+h_{22}(B) \end{array}\right].
\varphi_C=\frac{1}{gr}\left[{\cal{N}}{\sin}(\overline{\cal{N}}+{\cal{N}})+{\cos}(\overline{\cal{N}}+{\cal{N}})-1\right]+{\cal T}_C\left[{\cos}(\overline{\cal{N}}+{\cal{N}})-1\right]+{\cal T}_A\,{\sin}(\overline{\cal{N}}+{\cal{N}}).
D_\alpha\rho_t = -D_\alpha (\nabla\cdot(\rho Q\ast \mathbf{u})) & = -[D_\alpha (\nabla\cdot(\rho Q\ast\mathbf{u})) - Q\ast\mathbf{u}\cdot \nabla(D_\alpha\rho)]- Q\ast\mathbf{u}\cdot \nabla(D_\alpha\rho).
\Omega_{k,n,\varepsilon_n}:=\left\{\omega \in \Omega \bigg| \sup_{t_k^{(n)}\leq s \leq t_{k+1}^{(n)}}|X_s^{(n)}(\omega)-X_{t_k^{(n)}}^{(n)}(\omega)| \geq \varepsilon_n \right\}.
P^*(-t \frac{1}{d}\{\log |\det \left(d_x f^n|_E \right)|\}) = P(-t \log |\det \left(d_x f|_E \right)|) = h_\kappa(f) - t \int_{\Lambda}\frac{1}{d}\log |\det \left(d_x f^n|_E \right)| d\kappa.
\partial_t y_1 = \partial_{xx}y_1 + y_2f_2(y_1) - \sigma y_3f_3(y_1), \partial_t y_2 = d_2\partial_{xx}y_2 - y_2f_2(y_1), \partial_t y_3 = d_3 \partial_{xx}y_3 - \tau y_3f_3(y_1).