harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\prod_{j=1}^3y\big(\bigl\{X_{ni}^{(j)}\bigr\}\big)=\sum_{l=1}^3Z_l+\sum_{l=4}^k B_l^{(3)}\bigl(\bigl\{X_{ni}^{(1)}\bigr\},\bigl\{X_{ni}^{(2)}\bigr\},\bigl\{X_{ni}^{(3)}\bigr\}\bigr).
\frac{g(tx,ty)}{h(t)}\sim \frac{u([tx],[ty])}{h([t])}\leq \frac{u([[t]x],[[t]y])}{h([t])}\to \lambda(x,y), \intertext{and on the other, since $([t]+1)x\geq tx,$}\frac{g(tx,ty)}{h(t)}\geq \frac{g( ([t]+1)x,([t]+1)y)}{h(t)}= \frac{u( ([[t]+1)x],[([t]+1)y)]}{h(t)},
\frac{\partial\overline{G}}{\partial v_{k}}=\varepsilon\frac{\sqrt{R}}{\sqrt{\theta}}\sum^{3}_{j=1}\frac{\partial\bar{\theta}}{\partial x_{j}}\partial_{v_{k}}A_{j}(\frac{v-u}{\sqrt{R\theta}})\frac{1}{\sqrt{R\theta}}+\varepsilon\sum^{3}_{i,j=1}\frac{\partial\bar{u}_{j}}{\partial x_{i}}\partial_{v_{k}}B_{ij}(\frac{v-u}{\sqrt{R\theta}})\frac{1}{\sqrt{R\theta}},
\widetilde J((\mu^{\alpha_{n_k}},\rho^{\alpha_{n_k}}),u^{\alpha_{n_k}})\, =\,\widetilde J\left({\cal S}_{\alpha_{n_k}}(u^{\alpha_{n_k}}),u^{\alpha_{n_k}}\right)\,\le\,\widetilde J\left({\cal S}_{\alpha_{n_k}}(\bar u),\bar u\right)\,,
H=\{(\begin{matrix}e^{2\pi i\theta}&0\\ 0&e^{2\pi ia\theta}\end{matrix}):\theta\in\mathbb{R}\}\subset\mathbb{T}^{2}=\{(\begin{matrix}e^{2\pi i\theta}&0\\ 0&e^{2\pi i\phi}\end{matrix}):\theta,\phi\in\mathbb{R}\},
\frac{d}{dt}Q^R(t) = \int_{\mathbb{R}^n}\partial_t \widetilde{v}_t \cdot \chi_R\, dx &= \int_{\mathbb{R}^n}\big( \partial_t \widetilde{v}_t -\mathcal{L}_{\beta_{s,p}}^* \widetilde{v}_t\big)\chi_R\, dx+\int_{\mathbb{R}^n}\mathcal{L}_{\beta_{s,p}}^* \widetilde{v}_t\cdot \chi_R\, dx.
\int\big|\frac1K{}^{\vee}{\mathfrak{m}_{m,m',l,l'}}(\frac{z-y_1}{K}) \big|\, dw_{[0,K]}(z) = \int \big|{}^{\vee}{\mathfrak{m}_{m,m',l,l'}}(z - \frac{y_1}{K}) \big|\, dw_{[0,1]}(z) \le C_M \int \bigl(1+ \bigl| \frac{y_1}{K}-z \bigr| \bigr)^{-M}\, dw_{[0,1]}(z)\sim C_M \bigl( 1+ \bigl| \frac{y_1}{K}\bigr| \bigr)^{-M}
{\frac{\partial \overline{{\mathrm{\Large v}}}_{D}( x )}{\partial ( m / \Lambda )}}={\frac{2 m}{\Lambda}}\left[{\frac{( 4 \pi )^{D / 2}}{{\mathrm{t r}}{\bf 1}g^{2}\Lambda^{D - 2}}}+ f_{D}( x ) \right] = 0 \ ,
{W_{\nu}^{d}}( Z , Z^{\prime}) = \frac{1}{2 ( 2 \pi )^{\frac{d}{2}}}\Gamma \left( \frac{d - 1}{2}+ i \nu \right) \Gamma \left( \frac{d - 1}{2}- i \nu \right) \, \, ( ( Z \cdot Z^{\prime})^{2}- 1 )^{- \frac{d - 2}{4}}\, P_{- \frac{1}{2}+ i \nu}^{- \frac{d - 2}{2}}\left({Z \cdot Z^{\prime}}\right) ,
\|A^{}_{N}v\|_{\mathcal{H}}^{2}\leq \int N^{-2s'}\|\pi_{s}(v)\|_{V_s}^{2}dm_{L}(s) \leq N^{-2s'}\int\|\pi_{s}(v)\|_{V_s}^{2}dm_{L}(s) = N^{-2s'}\|v\|_{\mathcal{H}}^{2}.
\mathbf{u}^{(1)}:=\begin{cases}u \;\;\Omega, u_{0}\;\;\Omega'\setminus\overline{\Omega}\end{cases}\;\;\;\;\;\;\mathbf{u}^{(2)}:=\begin{cases}u \;\;\Omega, v_{0}\;\;\Omega'\setminus\overline{\Omega}.\end{cases}
\frac{d\theta}{ds}=\frac{1}{\rho}=y^{\prime\prime}(s)x^{\prime}(s)-y^{\prime}(s)x^{\prime\prime}(s)=\frac{y^{\prime\prime}(s)}{x^{\prime}(s)}=-\frac{x^{\prime\prime}(s)}{y^{\prime}(s)},
\sum_{x_i, x_j}\big\langle \psi^1(x_i) - \psi^1(x_j), \psi^2(x_i) - \psi^2(x_j) \big\rangle= \sum_{x_i, x_j}\big\langle \psi^1(x_i) - \psi^1(x_j), \psi^2(x_i) - \psi^2(x_j) \big\rangle.
\frac{{\rm d}}{{\rm d}t}\int_{x\in\mathbb{T}^3}\frac{\big|\widetilde{U}\big|^2}{2h}+\int_{x\in\mathbb{T}^3}\frac{\widetilde{W}^{\rm T}Q(w^*)\widetilde{W}}{2h}+\int_{x\in\mathbb{T}^3}\widetilde{W}\cdot\mathrm{L}(w^*)=0
( \alpha , \beta ) = \int_{\Sigma}^{}\mathrm{t r}( \alpha \wedge{}^{\ast}\beta ) = \int_{\Sigma}^{}\frac{1}{m !}\alpha_{\mu_{1}\cdots \mu_{m}}^{a}\beta_{a}^{\mu_{1}\cdots \mu_{m}}\sqrt{g}d^{3}\! x ,
\alpha C \xi_{ra_{1}}(n) =\displaystyle \sum_{r'a' \in_{k}\hat{T}_{1}}\alpha \xi_{r'a'}\pi_{r'a'}C\xi_{ra_{1}}(n) &=\displaystyle \sum_{r'a' \in_{k}\hat{T}_{1}}\phi^{-1}\left(r'a'C_{r'a',ra_{1}}\right)n = \phi^{-1}\phi(ra_{1})n &=ra_{1}n=\alpha_{1}\xi_{ra_{1}}(n)
\| p_\mathfrak{B}(\phi(a) - \widetilde{\rho_1}( \psi_1(p_\mathfrak{A}(a))) ) \| = \| [\phi]_{\mathsf U}(p_\mathfrak{A}(a)) - \rho_1 ( \psi_1 (p_\mathfrak{A}(a))) \| < \epsilon/6
\begin{cases}dx_i =( Ax_i +BR^{-1}B^Ty_i +G m_i+\gamma p_i) dt +D dW_i, \dot{m_i}= (A + G) m_i +B\bar u^* +\gamma p_i, \dot{p_i}= -(A+G)^T p_i -(I-\Gamma)^T Q [\mathbb{E}x_i- (\Gamma m_i+\eta)] , dy_i=\left\{-A^T y_i +Q[x_i-(\Gamma m_i+\eta)]\right\}dt +\zeta_i dW_i, \end{cases}
\int_{SO_d}f_j(n(P,\vartheta Q;F,\vartheta G)) [F, \vartheta G]\nu (d\vartheta ) &= \int_{SO_d}f_j((N(P,F) +\vartheta N(Q,G))\cap S^{d-1}) [L_1, \vartheta L_2]\nu (d\vartheta ),
( - \partial_{0}^{2}+ \nabla^{2}- 2 e^{2}| \phi |^{2}) \chi_{\uparrow}+ i \kappa \partial_{0}\chi_{\uparrow}- \sqrt{2}e ( D_{-}\phi )^{*}\psi_{\downarrow}+ i \kappa \partial_{+}\chi_{\downarrow}= 0 ,
\left. \mathrm{D i s c}\; \left( \int d^{4}x{\frac{e^{i p \cdot x}}{| x |^{2 m + 4}}}\right) \right|_{- p^{2}= \omega^{2}- i \epsilon}^{- p^{2}\omega^{2}+ i \epsilon}={\frac{2 \pi^{3}i \omega^{2 m}}{4^{m}m ! ( m + 1 ) !}}
\Delta^{(m)}(x| z^{(n)}| \cdots |z^{(2)}| z^{(1)})=\det \left( A_{\lambda}^{(m)}(x_{\mu}| z^{(n)}| \cdots |z^{(2)}| z^{(1)}) \right)_{1\leq \lambda , \mu \leq M}
\mu_{1}^{AA^{\prime}}=(L^{-1})_{B^{\prime}}^{A^{\prime}}(L^{-1})_{B}^{A}\mu^{BB^{\prime}}\nu_{1}^{AA^{\prime}}=(L^{-1})_{B^{\prime}}^{A^{\prime}}(L^{-1})_{B}^{A}\nu^{BB^{\prime}}+i(L^{-1})_{B^{\prime}}^{A^{\prime}}d[(L^{-1})_{B}^{A}]\mu^{BB^{\prime}}& -i[d(L^{-1})_{B^{\prime}}^{A^{\prime}}](L^{-1})_{B}^{A}\mu^{BB^{\prime}}.
-\Biggl.{\bf 1}_{\{i_1=i_2\ne 0\}}{\bf 1}_{\{j_1=j_2\}}\zeta_{j_3}^{(i_3)}-{\bf 1}_{\{i_2=i_3\ne 0\}}{\bf 1}_{\{j_2=j_3\}}\zeta_{j_1}^{(i_1)}-{\bf 1}_{\{i_1=i_3\ne 0\}}{\bf 1}_{\{j_1=j_3\}}\zeta_{j_2}^{(i_2)}\Biggr),
\Pi(p^2,\mu) \simeq \frac{b_0}{8\pi^2}\left[\log\frac{z_1}{z_0}+ \log m z_0 - \frac{1}{4}\log\mu z_0 - \frac{1}{4}\log\mu z_1 + \frac{m^2 z_0^2}{8}\left( \log\frac{z_1}{z_0}-\frac{1}{2}\right) \right] \;.
\sum_{\alpha = 1}^n z_{\alpha,i}\iota_{e_{\alpha, n+1}}(\varphi_1^{(j_s)}) = \sum_{\alpha = 1}^n z_{\alpha,i}\sum_{\beta=1}^n z_{\beta, j_s}\iota_{e_{\alpha, n+1}}(\omega_\beta) = \sum_{\alpha = 1}^n \sum_{\beta=1}^n z_{\alpha,i}z_{\beta, j_s}\delta_{\alpha, \beta}= r_{i,j_s}.
\|P_{R_{j,k}}P_{N_1}e^{it\Delta}f_1 \cdot P_{N_2}e^{it\Delta}f_2\|_{L^2(M)}^2\lesssim \sum_{n\in\frac{2\pi}{\mathcal{T}}\mathbb{Z}}\#_{R_{j,k},N_2,n}\sum_{\substack{\lambda_1\in \Lambda^+\cap R_{j,k}, \lambda_2\in \Lambda^+, |\lambda_2|_\rho\sim N_2, |\lambda_1|^2_\rho+|\lambda_2|_\rho^2=n}}\| P_{\lambda_1}f_1 \cdot P_{\lambda_2}f_2\|_{L^2(M)}^2,
\lambda_{a|a|c|d}^{e|f|g|h}= \lambda_{a|a|c|c}^{e|y|g|z}\lambda_{a|a|c|d}^{y|f|z|h}+ \lambda_{a|a|c|d}^{e|y|g|z}\lambda_{a|a|d|d}^{y|f|z|h}.
Q_{\vec q}=\sum_{i,j}\varepsilon_i\varepsilon_j q_iq_j^{-1}E_i^j\otimes E_{-i}^{-j}, (Q_{\vec q})^{il}_{jm}=\varepsilon_i\varepsilon_jq_iq_j^{-1}\delta^{i,-l}\delta_{j,-m},
\hat{\Omega}^{( 1 )}= - \frac{\partial^{2}}{\partial \rho^{2}}+ \frac 1 2 \mu \left[ \sin^{2}g ( \rho ) V^{\prime \prime}( \cos g ( \rho ) ) - \cos g ( \rho ) V^{\prime}( \cos g ( \rho ) ) \right]
\sum_{n = 1}^{\infty}e^{- \beta k E_{n}}= \frac{1}{\sqrt{\pi}}\int_{0}^{\infty}d x \; e^{- \frac{x^{2}}{4}- \frac{\beta^{2}k^{2}m^{2}}{x^{2}}}\frac{1}{e^{\frac{2 \beta^{2}e B k^{2}}{x^{2}}}- 1}
\frac{1}{2\pi \textnormal{i}}\int_{L}\Gamma(\frac{1}{2}+z)\Gamma(1+z)\left(\frac{1}{4}s^{\alpha}|x|^{2}\right)^{-z}dz &= \textnormal{H}_{02}^{20}\left[\frac{1}{4}s^{\alpha}|x|^{2}\Big|\begin{array}{cc}(\frac{1}{2},1) (1,1) \end{array}\right]= 2\left(\frac{s^{\alpha/2}|x|}{2}\right)^{3/2}K_{1/2}(s^{\alpha/2}|x|),
\left( K\left( g_{1}g_{2}\right) q\right) \left( x\right) =-2\partial_{x}^{2}\log\tau_{g_{1}g_{2}W_{m}}\left( e_{x}\right) & =-2\partial_{x}^{2}\log\tau_{g_{1}\left( g_{2}W_{m}\right)}\left(e_{x}\right) =\left( K(g_{1})K\left( g_{2}\right) q\right) \left( x\right)
\Lambda(dd^cf)= \frac{1}{2}\sum_{i=1}^m\left(dd^cf\right)(e_i,Je_i) = \sum_{i=1}^m e_i\left(d^cf(Je_i)\right)-d^cf(JD^g_{e_i}e_i)-\left(d^cf\right)\left((D^g_{e_i}J)e_i\right) = -\Delta^g(f) - g(\theta,df).
f_{\mathrm{\footnotesize ~ 1 - l o o p}}= - \frac{i}{2}\Bigg ( \int \frac{d^{4}k}{( 2 \pi )^{4}}\frac{1}{k^{2}( k + p )^{2}}- \sum_{i}c_{i}\int \frac{d^{4}k}{( 2 \pi )^{4}}\frac{1}{( k^{2}- M_{i}^{2}) \Big ( ( k + p )^{2}- M_{i}^{2}\Big )}\Bigg )
\gamma_{\frac{q-2}{2}}(C(N,q),2)=\frac{q-1}{q-2}\cdot\frac{\frac{q^{(N+1)/2}-1}{q^{1/2}-1}-1}{\frac{q^{(N+1)/2}+1}{q^{1/2}+1}-1}=\frac{q-1}{q-2}\cdot \frac{\sqrt q+1}{\sqrt q-1}=\frac{(\sqrt q+1)^2}{q-2}.
A_0=S_0^{1/2}H_{e{\sf E}_0}S_0^{1/2}, H_{e{\sf E}_0}=\sum_{j\in\mathbb{J}_0}E_j \langle e_j, \cdot\rangle\,e_j,{\sf E}_0=\{E_j, j\in\mathbb{J}_0\}.
p_{11}(\zeta) + \eta_2 p_{12}(\zeta) = 25\zeta^2 (5\zeta^2 - 7), p_{21}(\zeta) + \eta_2 p_{22}(\zeta) = 625 \zeta^3 (5\zeta^2 - 7) (5\zeta^4 - 26\zeta^2 + 15).
\begin{aligned}|T'_1|+|T'_2| +|T'_3|-|A_{123}| &\ge |T_1|+|T_2|+|T_3|-|A_{123}|-|S'_{1}|-|S'_{2}|-|S'_{3}| &> 2(|A_{123}|+|A_1|+|A_2|+|A_3|+|A_{12}|) &\quad\quad-|U''_2|-|U''_3|-3N/2 &\ge N/2 -|U''_2|-|U''_3| > N/4.\end{aligned}
I_{2 p}= \left( \begin{array}{c c c}{1_{p}}&{0}&{0}\\{0}&{0}&{0}\\{0}&{0}&{1_{p}}\\ \end{array}\right) \! , I_{q}= \left( \begin{array}{c c c}{0}&{0}&{0}\\{0}&{1_{q}}&{0}\\{0}&{0}&{0}\\ \end{array}\right) . \vspace{2 m m}
\tilde{{\mathcal H}}^{^{\bullet}}\ : = d i s p l a y s t y l e \bigoplus_{k = 0}^{\infty}\ \tilde{{\mathcal H}}^{k}\quad \tilde{{\mathcal H}}^{k}\ : = \ \overline{{\tilde{\Omega}^{k}({\mathcal A})}}^{( \cdot , \cdot )}\ .
A^{(3,1)}_{(2,2)}=\{\begin{bmatrix}2 1 0 1 \end{bmatrix},\begin{bmatrix}1 2 1 0 \end{bmatrix}\}A^{(3,1)}_{(3,1)}=\{\begin{bmatrix}3 0 0 1 \end{bmatrix},\begin{bmatrix}2 1 1 0 \end{bmatrix}\}.
{\cal P}_3(y)\|=\frac{1}{r^n}\int\limits^u_{-\infty}d{\bar u}\,\log\Bigl(m\frac{u-{\bar u}}{2}\Bigr)\frac{d}{d{\bar u}}A({\bar u})+O\Bigl(\frac{1}{r^{n+1}}\Bigr)\quad .
H_NS_N(i,j) =\frac{1}{J_N}(2J_1J_{N-j}(J_{j-2}+J_{j-1})-J_{N-j}(J_{j-1}+J_j)+(-1)^{N+j-1}J_{j-1}J_0) =\frac{1}{J_N}(2J_{N-j}(J_{j-2}+J_{j-1})-J_{N-j}(J_{j-1}+J_j)) =\frac{J_{N-j}}{J_N}(2J_{j-2}+J_{j-1}-J_j) =0.
\sum_{(A,B,j,k)\in \mathcal{S}_1}\left(\frac{Cov(\Delta_j f(W)\Delta_jf(W^A), \Delta_k f(W) \Delta_kf(W^B))}{\binom{2n}{|A|}(2n-|A|)\binom{2n}{|B|}(2n-|B|)}\right) \leq 4n^2.
\boldsymbol{\Psi}_D^{(g)}\left[\boldsymbol{\Psi}_D^{(g)}\right]^H=\left(\mathbf{X}^{(g)}\otimes \left[\mathbf{S}_D^{(g)}\right]^H\right)\left(\mathbf{I}_{K_g}\otimes \mathbf{V}\mathbf{V}^H\right)\left(\left[\mathbf{X}^{(g)}\right]^H \otimes \mathbf{S}_D^{(g)}\right)
J_1g(\zeta_0)=\frac{1}{2\pi i}\int_{b\Omega_0}\frac{\Phi(\zeta,\zeta_0)-\rho_{\bar\zeta}^{-1}z_{\bar\zeta}\phi(\zeta,\zeta_0)}{(\zeta-\zeta_0)(z-z_0)}g(\zeta)\,d\zeta &J_2g(\zeta_0)=\frac{-\rho(\zeta_0)}{2\pi i}\int_{b\Omega_0}\frac{\rho_{\bar\zeta}^{-1}z_{\bar\zeta}g(\zeta)\,d\zeta}{(\zeta-\zeta_0)(z-z_0)}.
\mathcal{E}(\delta_a, \delta_b) = \int_M dG^{\delta_a}\wedge *dG^{\delta_b}= -\int_M G^{\delta_a}\wedge d*dG^{\delta_b}= \int_M G^{\delta_a}\wedge \left( \delta_b-\frac{1}{V}\,{\rm vol}\right) = G^{\delta_a}(b) = G(a,b).
\left\lVert \overset{1}{\mathbb{P}}\left(\tilde{W}\right)\right\rVert_{\mathbb{L}^2(B)}=\mathcal{O}(a^{1-h}),\left\lVert \overset{2}{\mathbb{P}}\left(\tilde{W}\right)\right\rVert_{\mathbb{L}^2(B)}=\mathcal{O}(a^{3}),\left\lVert\overset{3}{\mathbb{P}}\left(\tilde{W}\right)\right\rVert_{\mathbb{L}^2(B)}=\mathcal{O}(a^{3}).
P\left(x,\frac{2px-\ell}{2q}\right)= e^{-i\pi \frac{p}{q}x^2}\sum_{k=0}^{q-1}2(-1)^{kp}e^{i\pi \left( \frac{p}{q}k^2 - 2\frac{n k}{q}\right)}=2q c_{n,p,q}e^{-i\pi \frac{p}{q}x^2}
[..H_tH_s]_n=\begin{bmatrix}[..G_tG_s]_n * 0 0 q^n 0 0 * [..F_tF_s]_n\end{bmatrix}=\begin{bmatrix}[..H_t''H_s'']_n 0 * [..F_tF_s]_n\end{bmatrix}=\begin{bmatrix}[..G_tG_s]_n * 0 [..H_t'H_s']_n\end{bmatrix}
\omega_{a,b}(x, y)=-(-1)^{|x||y|}{\omega}_{a,b}(y,x)= -(-1)^{|x||y|+|b||y|}\omega(ay,bx)= (-1)^{|a||b|+|a||x|}\omega(bx,ay) = (-1)^{|a||b|}\omega_{b,a}(x,y).
\begin{array}{c c c}{V^{r}= r^{m +{\textstyle{\frac{1}{2}}}}\cos ( \delta + ( p - 1 ) \varphi ) ,}&{}&{V^{\varphi}= r^{m -{\textstyle{\frac{1}{2}}}}\sin ( \delta + ( p - 1 ) \varphi ) .}\\ \end{array}
\hat{T}{\bf f}=0\;,\;\;\hat{T}=\frac{\imath sm}{2\hat{\omega}}\hat{\zeta}\Sigma^3\left[\hat{\zeta}\hat{\omega}\Sigma^3+\imath\partial_1(\imath s\Sigma^2)+\imath\partial_2(-\imath s\Sigma^1)-sm\right]\;.
[e_1, e_1]=\alpha_1e_4+ \alpha_2e_5, [e_1, e_2]=\alpha_3e_3+\alpha_4e_4+ \alpha_5e_5, [e_2, e_1]=-\alpha_3e_3+ \beta_1e_4+ \beta_2e_5, [e_2, e_2]=\beta_3e_4+ \beta_4e_5, [e_1, e_3]=\gamma_1e_5, [e_3, e_1]=\gamma_2e_5, [e_2, e_3]=\gamma_3e_5, [e_3, e_2]=\gamma_4e_5, [e_3, e_3]=\gamma_5e_5, [e_1, e_4]=\gamma_6e_5, [e_2, e_4]=\gamma_7e_5, [e_3, e_4]=\gamma_8e_5.
\prod_{i=1}^k r_i(x_i) \le \frac{1}{\Bigl( \sum_{j=1}^n \bigl( |(x_1)_j|^{\frac{1}{k}}\cdots |(x_k)_j|^{\frac{1}{k}}\bigr)^2 \Bigr)^{\frac{k}{2}}}.
( 3 , 1 ) : \quad \left( \begin{array}{c}{u_{L}^{i}}\\{d_{L}^{i}}\\{b_{L}^{i}}\\ \end{array}\right) , \quad ( 1 , \bar{3}) : \quad \left( \begin{array}{c}{\hat{u}_{R}^{i}}\\{\hat{d}_{R}^{i}}\\{\hat{b}_{R}^{i}}\\ \end{array}\right)
m^{k,k_1,k_2}(\xi,\eta) := m(\xi,\eta)\cdot\varphi_k(\xi) \varphi_{k_1}(\xi-\eta) \varphi_{k_2}(\eta), &m^{k_1,k_2,k_3,k_4}(\xi,\eta,\rho) := m(\xi,\eta,\rho) \cdot\varphi_{k_1}(\xi) \varphi_{k_2}(\eta)\varphi_{k_3}(\rho-\xi)\varphi_{k_4}(-\rho-\eta).
\Psi_{( 1 )}= \sqrt{\frac{V}{2}}\left( \begin{matrix}{0}&{1}\\{1}&{0}\\ \end{matrix}\right) , \Psi_{( 2 )}= \sqrt{\frac{V}{2}}\left( \begin{matrix}{\frac{1}{\tau}}&{0}\\{0}&{- \tau}\\ \end{matrix}\right) .
C_{\theta}( \tau ) : = \lim_{\tau \rightarrow \infty}\left[ \langle{\cal A}( \tau ){\cal B}( 0 ) \rangle_{\theta}\; - \; \langle{\cal A}( 0 ) \rangle_{\theta}\; \langle{\cal B}( 0 ) \rangle_{\theta}\right] \; .
\frac{H_{\mu}^{( 1 )}( \lambda z )}{H_{\nu}^{( 1 )}( z )}F ( z ) = \frac{H_{\mu}^{( 2 )}( \lambda z_{1})}{H_{\nu}^{( 2 )}( z_{1})}F ( z_{1}) , \, z_{1}= e^{- \pi i},
S = -\frac{1}{4 \pi \alpha^\prime}\int\!{{d^2}\xi}[{R^2}{\partial_\alpha}{y}{\partial^\alpha}{y}+ 2kR{A_\mu}{\partial_\alpha}{y}{\partial^\alpha}{x^\mu}+ (g_{\mu \nu}+{k^2}{A_\mu}{A_\nu}){\partial_\alpha}{x^\mu}{\partial^\alpha}{x^\nu}] .
\hat{R}_{\mu\nu\alpha\beta}=R_{\mu\nu\alpha\beta}= \partial_\mu\Omega_{\nu,\alpha\beta}-\partial_\nu\Omega_{\mu,\alpha\beta}+\Omega_{\mu,\alpha}^{\ \ \ \gamma}\Omega_{\nu,\gamma\beta}-\Omega_{\nu,\alpha}^{\ \ \ \gamma}\Omega_{\mu,\gamma\beta}\hskip1.5cm
\lim_n[v_n]_{s,p}^p=\lim_n\mu\int_\Omega\frac{|v_n|^q}{|x|^\alpha}\, dx+\lambda\int_\Omega|v_n|^r\, dx=\mu\int_\Omega\frac{|v|^q}{|x|^\alpha}\, dx+\lambda\int_\Omega|v|^r\, dx=[v]_{s,p}^p.
\begin{cases}\frac{\partial^2 u}{\partial\nu^2}=\lambda\frac{\partial u}{\partial\nu},{\rm on\}\partial B, -{\rm div}_{\partial\Omega}(D^2u\cdot\nu)_{\partial\Omega}-\frac{\partial\Delta u}{\partial\nu}=\mu u,{\rm on\}\partial B,\end{cases}
_{1}\phi_{1}\left(\begin{array}{ccc}\begin{array}{c}-a b\end{array}\vert q,-bq^{\alpha}\end{array}\right)q^{\alpha^{2}/2}=\int_{-\infty}^{\infty}\frac{\left(-abq^{-1/2}e^{ix};q\right)_{\infty}\exp\left(\frac{x^{2}}{\log q^{2}}+i\alpha x\right)dx}{\sqrt{\pi\log q^{-2}}\left(b;q\right)_{\infty}\left(be^{2ix}q^{-1};q^{2}\right)_{\infty}}
\begin{aligned}M_{i,r}^{n^{-1}}=\frac{\Phi_{i}}{(1+\lambda_{i})^{2}}\binom{n}{r}\left[\frac{2^{2i}(i!)^{2}}{(2i)!}\Psi_{k,i}^{n,r}+\left(\frac{2^{2i}(i!)^{2}}{(2i)!}\right)^{2}\sum_{d=0}^{i}\frac{(2d)!\lambda_{d}}{2^{2d}(d!)^{2}}\Psi_{j,d}^{n,r}\right],\end{aligned}
\mathbb{P}_\eta\left(\left. \mathcal{T}_t(x) < \frac{1-p}{4}t\right|\mathcal{F}_{k}\right) \leq \frac{\int_{t/2}^{t}p \mathrm{d}s}{\left(1-\frac{1-p}{2}\right)\frac{t}{2}}= \frac{p}{1-\frac{1-p}{2}}.
0=\frac{\partial}{\partial a_{i,j}}\int_{x_{0}}^{x_{m}}(\sum_{p}\sum_{q}f_{p,q}(x)-g(x))^{2}dx=\int_{x_{i}}^{x_{i+1}}(\sum_{q}f_{i,q}(x)-g(x))(x-x_{i})^{j}dx
\begin{aligned}\varepsilon^3\int_0^t{|(\Delta^h_k(v\partial_yw^2)_\Theta, \varepsilon^2\Delta^h_kw^2_\Theta)_{L^2}|dt'}\lesssim \varepsilon d_k^22^{-k}||(\varepsilon^2 w^2)_\Theta||_{\tilde L^2_{t, \zeta'(t)}(B^1)}||u_\Theta||^\frac 12_{\tilde L^\infty_t(B^\frac 32)}||\partial_y(\varepsilon^2 w^2)_\Theta||_{\tilde L^2_{t}(B^{\frac 12})}\end{aligned}
\tilde{A}^n_i(t)= \frac{1}{b_n\sqrt{n}}(A^n_i(t)-\lambda^n_i t), \tilde{S}^n_i(t)= \frac{1}{b_n\sqrt{n}}(S^n_i(t)-\mu^n_i t), &\tilde{Q}^n_i(t)= \frac{1}{b_n\sqrt{n}}Q^n_i(t)\,.
S_{0}= \int d^{3}x ~ \left( \kappa \epsilon^{\mu \nu \rho}t r ( A_{\mu}\partial_{\nu}A_{\rho}- \frac{2}{3}i A_{\mu}A_{\nu}A_{\rho}) + ( D_{\mu}\phi )^{\dagger}( D^{\mu}\phi ) \right) ,
\int_{\|y\| \geq 4\|x\|}\sum_{2^j \|x\| \geq \|r\|}|k_j(y-x)-k_j(y)| dy \leq \sum_{2^j \|x\| \geq \|r\|}2 \int_{\|y\| \geq 2\|x\|}k_j(y) dy &= \sum_{2^j \|x\| \geq \|r\|}2 \int_{\|y\| \geq 2 \cdot 2^j \|x\|}\varphi(y+r) dy &\lesssim \sum_{2^j \|x\| \geq \|r\|}\frac{1}{2^j \|x\|}&\lesssim \frac{1}{\|r\|}\lesssim 1.
{\mathcal L}_i(f):=L(z^{q+m+k}f_{i}) (z) - Cz^{m}CG\left(z,\frac{d^lz^{m+k}f_{j}}{dz^l}(z),\;1\leq j\leq r,\; 0\leq l\leq m\right)=0.
\vec{v}(\vec{x})\equiv \epsilon^{i j}\,\partial_{i}\hat{n}\times\partial_{j}\hat{n}\quad , \quad q(\vec{x})\equiv\epsilon^{i j}\, \epsilon^{abc}\,\hat{n}^{a}\,\partial_{i}\hat{n}^{b}\, \partial_{j}\hat{n}^{c}
\{\beta_1^*F|_{\Gamma_1}: F \in \O(\Sigma_1;L_1)\}= \{\beta_1^*B^* F|_{\Gamma_2}: F \in \O(\Sigma_2;L_2)\}&= \{\beta_2^* F|_{\Gamma_2}: F \in \O(\Sigma_2;L_2)\}.
\int \frac{d^{2\sigma}Q}{(2\pi)^{2\sigma}}\frac{Q_\mu}{(Q^2+\vec{Q}^2)}\ =0\,, \qquad \int \frac{d^{2\sigma}K}{(2\pi)^{2\sigma}}\frac{K_\mu}{(K^2+\vec{K}^2)}\ =0\,,
T_1 := \Theta(p+\iota(O-P)) \, \Theta(p+\iota(A_1-A_2)), T_2 := \mathrm{e}^{4 \pi \sqrt{-1}\beta \cdot \iota(A_2-A_1)}\, \Theta(p+\iota(A_2-P)) \, \Theta(p+\iota(O-A_1)), T_3 := \Theta(p+\iota(A_1-P)) \, \Theta(p+\iota(O-A_2)).
\frac{\tilde{c}}{2}\leq \int_{|x| \geq R}|u_n|^2 \, dx = \int_{|x| \geq R}\frac{V(x)|u_n|^2}{V(x)}\, dx \leq \frac{1}{\inf_{|x| \geq R}V(x)}\int_{|x| \geq R}V(x)|u_n|^2 \, dx &\leq \frac{\|u_n\|^2}{\inf_{|x| \geq R}V(x)}\leq \frac{\sup \|u_n\|^2}{\inf_{|x| \geq R}V(x)}.
\widehat{\delta}(\eta_X(x), \eta_X(A)) = \sup\{\widehat{\psi}(\delta(-,\{x\}))\mid \psi\in \mathcal{R}X, \forall a\in A, \widehat{\psi}(\delta(-,\{a\}))=0\}= \sup\{\psi(x)\mid \psi\in \mathcal{R}X, \forall a\in A, \psi(a)=0\}=\delta(x,A).
P\left(\frac{n-A_n^*}{f_2(n)}\in O\right) \geq P\left(\frac{n-A_n^*}{f_2(n)}\in (\varepsilon_1,\varepsilon_2]\right) &=P\left(\frac{n-A_n^*}{f_2(n)}\leq\varepsilon_2\right)-P\left(\frac{n-A_n^*}{f_2(n)}\leq\varepsilon_1\right).
\langle \, p_{f}= 0 \, | e^{- i H T / \hbar}| \, q_{i}= 0 \, \rangle = \frac{1}{\sqrt{2 \pi \hbar}}\exp \left\{- \frac{i}{\hbar}\frac{1}{6}m \lambda^{2}T^{3}\right\}\; ,
\big{\langle}\mathsf{v}^* (g\alpha_{v_q}), (gw_q) \big{\rangle}= \Big{\langle}g\alpha_{v_q}, (gw_q)^\mathsf{v}_{gv_q}\Big{\rangle}=\Big{\langle}g\alpha_{v_q}, g(w_q)^\mathsf{v}_{v_q}\Big{\rangle}= \Big{\langle}\alpha_{v_q}, (w_q)^\mathsf{v}_{v_q}\Big{\rangle}= \big{\langle}\mathsf{v}^* (\alpha_{v_q}), (w_q) \big{\rangle},
\tilde{H}= ( 2 \pi \alpha^{\prime}) d \tilde{\cal F}- ( 2 \pi \alpha^{\prime}) G^{( 6 )}{\cal G}^{( 1 )}+ 3 ( 2 \pi \alpha^{\prime})^{2}{\cal H}^{( 3 )}d{\cal H}^{( 3 )}+{\textstyle \frac{m}{2}}{\cal G}^{( 7 )}.
\prod_{j=1}^\nu\phi_{n_j}(x)=-C_{n_1}\ldots C_{n_\nu}\cdot{\rm sign\,}\left(\int_0^1\phi_{n_1}\phi_{n_2}\ldots \phi_{n_\nu}\right),x\in I, \int_I\left(\prod_{j\in U}\phi_{j}\right)=0,U\subsetneq\{n_1,n_2,\ldots,n_\nu\}.
S_{\phi , V , \Lambda}= \int d^{d}\xi \left( G_{0 0}V^{a}V_{a}+ 2 G_{0 i}V^{a}\partial_{a}\phi^{i}+ G_{i j}\partial^{a}\phi^{i}\partial_{a}\phi^{j}+ \Lambda^{a b}\partial_{a}V_{b}\right) .
x_i^{\boldsymbol{\omega},-}\geq \tilde{z}_{-2i_{-1}N+i}-1 \geq \tilde{z}_{2N+i}-1 \geq \tilde{z}_{-i}-1 \geq \tilde{z}_{|i|-1}-1, x_i^{\boldsymbol{\omega},+}\leq \tilde{z}_{i+1}=\tilde{z}_{-(|i|-1)},
V(x\otimes1)\bigl(\Lambda_{\psi}(p)\otimes\Lambda(a)\bigr) =V\bigl(\Lambda_{\psi}(xp)\otimes\Lambda(a)\bigr) =(\Lambda_{\psi}\otimes\Lambda)\bigl(\Delta (xp)(1\otimes a)\bigr) &=(\Delta x)(\Lambda_{\psi}\otimes\Lambda)\bigl((\Delta p)(1\otimes a)\bigr) &=(\Delta x)V\bigl(\Lambda_{\psi}(p)\otimes\Lambda(a)\bigr),
\times \exp \left( -\frac{1}{2}\sum_{b, b^\prime =1}^N \sum_{j=1}^{n_b}\sum_{j^\prime=1}^{n_{b^\prime}}\Big(1 - \delta_{b b^\prime}\delta_{j j^\prime}\Big)\tilde{\tilde{V}}(x^{(b)}_j - x^{(b^\prime)}_{j^\prime}) \right)
Q_{4^*,q}(\psi(t)) =2N(q-2)E_{4^*,q}(\psi_{0})-(N(q-2)-8)\|\Delta \psi(t)\|_{2}^{2}-\frac{2N(4^*-q)}{4^*}\|\psi(t)\|_{4^*}^{4^*}&\leq\rho^*\frac{N(q-2)-8}{2}-1-\frac{N(q-2)-8}{2}\rho^*-\frac{N(q-2)-8}{2}\|\Delta \psi(t)\|_{2}^{2}.
P_0 (l)=\frac{4\pi^2}{l!^2(l+1)!^2}\left( \frac{\lambda \omega r_0}{4}\right)^{2l+2}=\frac{\pi^2 \omega^4 r_1^2 r_5^2}{4}\frac{(\omega^2 r_1 r_5)^{2l}}{2^{4l}l!^2(l+1)!^2}\,,
\hat{P}=-\frac{2\lambda^2}{4!}\left(\begin{array}{cc}3{\varphi_1}^2+{\varphi_2}^2 2{\varphi_1}{\varphi_2}2{\varphi_1}{\varphi_2}3{\varphi_2}^2+{\varphi_1}^2\end{array}\right),
\tau(a_1) = \tau(a_{-1}): (a_2, \, a_{-2})(a_3, \, a_{-3}); \tau(a_2) = \tau(a_{-2}): (a_1, \, a_{-1})(a_3, \, a_{-3}); \tau(a_3) = \tau(a_{-3}): (a_1, \, a_{-1})(a_2, \, a_{-2}).
\left( \begin{array}{l}1 \dot{0}0 \dot{0}\end{array}\left| \begin{array}{l}[\mu]^t{[\mu]}^{t-1}\end{array}\right. \right|\left. \begin{array}{l}[\mu]^t_{+k}{[\mu]}^{t-1}\end{array}\right)= \left( \frac{\prod_{i=1}^{t-1}(l_{i,t-1}-l_{kt}-1 )}{\prod_{i=1(\neq k)}^{t}(l_{it}-l_{kt})}\right)^{1/2}
{\cal E}=\frac{\sqrt{1-\lambda_9^2}}{\lambda_9}\left(u-iv\right)=\frac{(1-\lambda_9^2){\cal E}_0}{1-\lambda_9^2 Re{\cal E}_0},\;\;\;z=\frac{i}{\left(1-\lambda_9^2{\cal E}_0\right)}.
i \partial_t u(x,t) = -\frac{1}{2}\Delta u(x,t) +\frac{1}{2}x^\top \! \textbf{A}x\, u(x,t), x \in \mathbb{R}^2, t>0, & u(x,0) = \pi^{-\frac{1}{2}}\big(\frac{1}{2}x_1^2 + \frac{1}{2}(x_2-1)^2 \big), & \textbf{A}= \begin{pmatrix}2 -1 -1 3 \end{pmatrix}.
\delta^{(a_1\cdots a_r)}_{b_1\cdots b_r}+ \sum_{t=1}^s \sum_{r-s+1 \leqslant i_1 < \cdots < i_t \leqslant r}\varepsilon^t \frac{(k+r-1)!}{(k+r-t-1)!}\delta^{(a_1 \cdots \hat{a}_{i_1}\hat{a}_{i_2}\cdots \hat{a}_{i_t}\cdots a_r)}_{b_1 \cdots \hat{b}_{i_1}\hat{b}_{i_2}\cdots \hat{b}_{i_t}\cdots b_r}\delta^{a_{i_1}}_{b_{i_1}}\cdots \delta^{a_{i_t}}_{b_{i_t}}.
f(x)=\frac{1}{2\pi}\sum_{k\in\mathbb{T}}\int\nolimits_{l(\varepsilon)}a_{k}(t)\Psi_{k,t}(x)dt+\frac{1}{2\pi}\sum_{k\in\mathbb{Z}\backslash\mathbb{T}}\int_{[0,2\pi)}a_{k}(t)\Psi_{k,t}(x)dt,
\Theta(u|\lambda):={}\prod_{n=1}^\infty\Bigg\{\Bigg[ 1-e^{-2n{\pi\mathbf K(\sqrt{1-\lambda})}/{\mathbf K(\sqrt{\lambda})}}\Bigg]\times{}\times\Bigg[1-2e^{-(2n-1){\pi\mathbf K(\sqrt{1-\lambda})}/{\mathbf K(\sqrt{\lambda})}}\cos\frac{\pi u}{{\mathbf K(\sqrt{\lambda})}}+e^{-2(2n-1){\pi\mathbf K(\sqrt{1-\lambda})}/{\mathbf K(\sqrt{\lambda})}}\Bigg]\Bigg\}
\omega_{1,3}=(jkx_1)(\bar{k}x_3)+(jx_2)(x_3)+(kx_3)(\overline{jk}x_1)+(x_3)(\bar{j}x_2) =jk\bar{k}x_1x_3+jx_2x_3+k\overline{jk}x_3x_1+\bar{j}x_3x_2 &= j x_1x_3+jx_2x_3-jx_1x_3-jx_2x_3 \ \ \ \ \ \{\rm commuting \variables}&= j(x_1x_3-x_1x_3)+j(x_2x_3-x_2x_3) &=0.
\frac{\rho_1(knh)}{\rho_2(knh)}=\frac{\frac{\Delta_H(h)\Delta_K(k)}{\Delta_G(h)}\cdot \rho_1(n)}{\frac{\Delta_H(h)\Delta_K(k)}{\Delta_G(h)}\cdot \rho_2(n)}=\frac{\rho_1(n)}{\rho_2(n)}
a_{2\ell}=\sqrt{\frac{2}{\pi}}\frac{(-1)^{\ell+1}}{2^{\ell}\ell!(2\ell-1)},b_{k}=\begin{cases}(-1)^{k/2}(k-1)!!, \textrm{if}k\textrm{is even}0, \textrm{if}k\textrm{is odd}\end{cases}
f^{T \to 0}\approx -{\frac{( d + 1 ) 2^{- d - 2}\pi^{- d / 2 - 1}}{a^{d + 2}}}\Gamma ( d / 2 + 1 ) \zeta ( d + 2 ) \left[ 1 +{\frac{1}{d + 1}}\left( \frac{2 a}{\beta}\right)^{d + 2}\right] ,
\sum\limits_{i=l}^kI(U^m_{\mathcal{A}_i};Y^{\rho m}_{a(i)}|U^m_{\cup_{j=1}^{i-1}\mathcal{A}_j}) &\leq \sum\limits_{q=1}^{\rho m}I(Y^{\rho m}_{2,q+1};Y_1(q)|U^m_{\cup_{j=1}^{l-1}\mathcal{A}_j},Y^{q-1}_{1})+\sum\limits_{i=l}^k\sum\limits_{q=1}^{\rho m}I(U^m_{\mathcal{A}_i};Y_{a(i)}(q)|U^m_{\cup_{j=1}^{i-1}\mathcal{A}_j},Y^{q-1}_{1},Y^{\rho m}_{2,q+1}),
\int\limits_{\mathcal{K}}\nabla \tilde\phi_j (y)\nabla\varphi(y) dy-\int\limits_{\mathcal{K}}2\alpha_j^2{|y|^{\alpha_j-2}\over (1+|y|^{\alpha_j})^2}\tilde\phi_j (y)\varphi(y) dy=\int\limits_{\mathcal{K}}\rho_j(y)\varphi(y)dy.
\bigl(\eta(\lambda_{k+1}),\eta(\rho_{k+1})\bigr) =\Bigg( \begin{pmatrix}e_{2^{n-1}+\cdots+2^{n-(k-1)}+1}f_{2^{n-(k-1)}}\end{pmatrix}, \begin{pmatrix}g_{2^{n-1}+\cdots+2^{n-(k-1)}+1}h_{2^{n-(k-1)}}\end{pmatrix}\Bigg),
(f\ |\g)_{k\phi}=\int_D\!\langle\,f\,|\,g\,\rangle e^{-2k\phi}dv_X\,, (f\ |\g)_{k\phi_0}=\int_D\!\langle\,f\,|\, g\,\rangle e^{-2k\phi_0}dv_X\,,