harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\int_{\mathbb{R}^n}u(x,y)\,dx=C_n\sigma_{n-1}y^{1-\beta}\int_0^{\infty}\frac{r^{n-1}K_{\nu}\left(\lambda\sqrt{r^2+y^2}\right)}{\left(\sqrt{r^2+y^2}\right)^{\nu}}\,dr=
<\rho ,\tau, \kappa \; \mid \; \rho \tau \rho= \tau \rho \tau \mbox{,}\; (\rho \tau \rho)^4 =1 \mbox{,}\;\kappa \tau \kappa^{-1}= \tau^{-1}\mbox{,}\; \kappa \rho \kappa^{-1}= \rho^{-1}\mbox{,}\; \kappa^2=1> \mbox{,}
\begin{aligned}c_1 \left(t^{-d/2}\wedge t^{-d/\beta}\right) \wedge \left( p_0(t,c_2x, c_2y)+ \frac{t}{|x-y|^{d+\beta}}\right) \leq p_1(t, x, y) \leq c_3\left(t^{-d/2}\wedge t^{-d/\beta}\right) \wedge \left( p_0(t,c_4x, c_4y)+ \frac{t}{|x-y|^{d+\beta}}\right) \end{aligned}
f(A+B)-f(A)-\frac{d}{dt}\bigl(f(A+tB)\bigr)\Big|_{t=0}=T_{f^{[1]}}^{A+B,A}(B)-T_{f^{[1]}}^{A,A}(B) =T_{f^{[2]}}^{A+B,A,A}(B,B).
{\bf P}_{c_L}= \sum\limits_{q=1}^{R}\sum\limits_{k_1,k_2,k_3=0}^{L-1}{\cal W}_{({k_1})}{\bf p}^{(1)}_{q}\otimes{\cal W}_{({k_2})}{\bf p}^{(2)}_{q}\otimes{\cal W}_{({k_3})}{\bf p}^{(3)}_{q}.
\mathbb{E}[N_w^{\hat{x}}(H[+])\mid \hat{x}] = \binom{n^{++}+ n^{+-}}{2}\frac{a}{n}- \frac{(a-b)}{n}(n^{++}- n_C^{++}+ n_C^{+-}) (n^{+-}+ n_C^{++}- n_C{^+-}),
\tilde{\mathcal{M}}(\tilde{\gamma},u)=\left(\tilde{Q}_{\eta_1,\eta_2}(u)\right)^{-1/2}\left(H_{\tilde{\gamma}}(u)+\frac{1}{2}\nabla^{\tilde{\gamma}}_{\nu}\log Q_{\eta_1,\eta_2}(u)\right)
U_{n-1}U_{n}y =\left( A_{n-1}\partial_{x}-B_{n-1}\right) \left(A_{n}y^{\prime}-B_{n}y\right) & =A_{n-1}\left( \partial_{x}A_{n}y^{\prime}+A_{n}y^{\prime\prime}-\partial_{x}B_{n}y-B_{n}y^{\prime}\right) -B_{n-1}\left( A_{n}y^{\prime}-B_{n}y\right) ,
\begin{pmatrix}X \begin{pmatrix}0 B_{11}B_{12}\end{pmatrix}\begin{pmatrix}0 B_{11}^* B_{12}^* \end{pmatrix}\begin{pmatrix}Y B_{21}B_{22}B_{21}^* D_{11}D_{12}B_{22}^* D_{12}^* D_{22}\end{pmatrix}\end{pmatrix}\in \Gamma
\Gamma_{a}( p , q )_{\mu}= - i \int \frac{d^{3}k}{( 2 \pi^{3})}\frac{\bar{u}( p - q ) \gamma_{\nu}( p \! \! \! / - q \! \! \! / - k \! \! \! + m ) \gamma_{\mu}( p \! \! \! / - k \! \! \! / + m ) \gamma_{\lambda}u ( p )}{[ ( p - k )^{2}- m^{2}] [ ( p - k - q )^{2}- m^{2}]}G ( k )^{\nu \lambda}
X^{opt}=\frac{W}{\Delta}[(r^{T}V^{-1}r)V^{-1}1-(1^{T}V^{-1}r)V^{-1}r]+\frac{\mu}{\Delta}[(1^{T}V^{-1}1)V^{-1}r-(r^{T}V^{-1}1)V^{-1}1]
\begin{cases}y(t) = e^{tA}\phi_0 + \displaystyle\int^t_0 e^{(t-s)A}Fy_sds + \int^t_0 e^{(t-s)A}f(s)ds,\hskip 20pt t\ge 0, y(0)=\phi_0,\,\, y(\theta)=\phi_1(\theta),\,\,\theta\in [-r, 0],\,\,\,\Phi=(\phi_0, \phi_1)\in{\cal X},\end{cases}
\Omega ( q ) = \rho ( q , q ) = \frac 1{N !}\sum_{P , P^{\prime}}\int d x^{+}d x^{-}e^{i ( q_{P P^{\prime}}^{-}x^{+}+ q_{P P^{\prime}}^{+}x^{-})}\rho ( x^{+}, x^{-})
\deg_{X_f}\left(\psi_{\vec{i}_{(u,v)}}\circ \chi_{\vec{i}_{(u,v)}}\right)^*(X_g)=\deg_{X_f}\left(\tilde{p}_{\vec{i}_{(u,v)}}\circ \tilde{\psi}_{\vec{i}_{(u,v)}}\circ\tilde{\chi}_{\vec{i}_{(u,v)}}\right)^*(X_g)
\gamma_n:= C_1^{n+1}C_2^{(\beta_1-M)n}\prod_{j=1}^n B\left({M}/{2},{jM}/{2}\right)=C_1 \Gamma(M/2) \frac{\left(C_1 C_2^{\beta_1-M}\Gamma(M/2)\right)^n}{\Gamma \left((n+1)M/2\right)}.
\frac{{d^2}}{{dr^2}}{K_{p,q}(r)}= \left({\frac{{q\left({r'}\right)^q}}{p}+ qr^q - 2\left({r'}\right)^q}\right)E_{p,q}(r) + \left({2\left({r'}\right)^{2q}- \frac{q}{p}\left({r'}\right)^q}\right)K_{p,q}(r)
(\ell_\gamma\circ g)\circ h\ =\ \ (\ell_{\lambda_g+\gamma}- \lambda_{g;\gamma}+\epsilon) \circ h\ =\ (\ell_{\lambda_g+\gamma}\circ h) -\lambda_{g;\gamma}+ (\epsilon\circ h) =\ \big(\ell_{\lambda_h+\lambda_g+\gamma}- \lambda_{h;\lambda_g+\gamma}+\epsilon^*\big)- \lambda_{g;\gamma}+ (\epsilon\circ h).
\tilde{z}_{R^\prime}(t)-\tilde{z}_R(t)= &\int_0^t S(t-s)(i\nu (\tilde{z}_{R^\prime}(s)-\tilde{z}_R(s))+\epsilon(\gamma (\tilde{z}_{R^\prime}(s)-\tilde{z}_R(s)) -\mu (\overline{\tilde{z}_{R^\prime}}(s)-\overline{\tilde{z}_R}(s))))\d s+ I(t),
Q_N(\gamma_1,\gamma_2)=2 \int_{0<s_1<s_2<1}ds_1ds_2\int_\mathbb{R}[s_1-x_1]_N^{\gamma_1}[s_2-x_1]_N^{\gamma_2}dx_1 \int_{\mathbb{R}}[s_1-x_2]_N^{\gamma_2}[s_2-x_2]_N^{\gamma_1}dx_2.
-q_*^{-1}t^{\frac{N+2A}{2}+1}(\partial_r^2 u_0)(x,t) =-[M_{0,1}+o(1)]t(\partial_r^2 U)(|x|)+\left[\frac{N+2A}{2}M_{0,1}+o(1)\right]\partial_r^2[UF](|x|)+O(t^{-1}) \ge \left[\frac{N+2A}{2}M_{0,1}+o(1)\right]\partial_r^2[UF](|x|)+O(t^{-1})\ge\frac{N+2A}{4N}M_{0,1}U(r_*)>0
(W^{\dagger})_{q,k}=(-1)^q\sum_{w=q}^{2p-1}\frac{1}{w!}s(w+1,q+1)\sum_{t=w}^{2p-1}\frac{\binom{t+w}{w}\binom{r-w-1}{r-t-1}}{\binom{2t}{t}\binom{r+t}{2t+1}}\sum_{j=0}^{\min(t,k)}(-1)^{j+1}\binom{k}{j}\binom{j+t}{j}\binom{r-j-1}{r-t-1},
\Delta_i(\xi_{1}^{i})= \sum_{u_{i+1}^{k}\in\{0,1\}^{k-i}}\P(u_{i+1}^{k}|\xi_{1}^{i})g(\xi_1^iu_{i+1}^{k})-\sum_{u_{i}^{k}\in\{0,1\}^{k-i+1}}\P(u_{i}^{k}|\xi_{1}^{i-1})g(\xi_1^{i-1}u_{i}^{k}) \leq |\sum_{u_{i+1}^{k}}\P(u_{i+1}^{k}|\xi_{1}^{i-1}1)g(\xi_1^{i-1}1u_{i+1}^{k})-\sum_{u_{i+1}^{k}}\P(u_{i+1}^{k}|\xi_{1}^{i-1}0)g(\xi_1^{i-1}0u_{i+1}^{k})|
V_{\pm}=\frac{a(a\mp \frac{\mu}{\sqrt{2}})f^2+(a(b\mp \frac{\nu}{\sqrt{2}})+b(a\mp \frac{\mu}{\sqrt{2}}))f+b(b\mp\frac{\nu}{\sqrt{2}})}{(f^{\prime})^2}\pm \frac{a}{\sqrt{2}}.
H = 2 \prod_{\alpha < \beta}^{M}( z_{\alpha}- z_{\beta})^{\frac{\kappa^{2}}{\lambda}}\prod_{i < j}^{N}( x_{i}- x_{j})^{\lambda}{\cal L}_{0}\prod_{i < j}^{N}( x_{i}- x_{j})^{- \lambda}\prod_{\alpha < \beta}^{M}( z_{\alpha}- z_{\beta})^{- \frac{\kappa^{2}}{\lambda}}=
\begin{cases}\partial_tg_\varepsilon+\frac 1 \varepsilon v\cdot\nabla_{x}g_\varepsilon+\frac 1{\varepsilon^2}\mathcal{L}g_\varepsilon=\frac{1}{\varepsilon}\Gamma(g_\varepsilon, g_\varepsilon), g_\varepsilon|_{t=0}=g_{\varepsilon, 0}\,,\end{cases}
\int_{-\infty}^\infty x^m e^{-\alpha x^2/2 + \beta x}\, dx = e^{-\beta^2/2\alpha}\int_{-\infty}^\infty (x + \beta/\alpha)^m e^{-\alpha x^2/2}\, dx = e^{-\beta^2/2\alpha}\sum_{i = 0}^m \binom{m}{i}\frac{C_i \sqrt{2\pi}}{\alpha^{(i + 1)/2}}\frac{\beta^{m - i}}{\alpha^{m - i}}.
\Phi^L(g,\xi) =\widehat L_g(\xi) +\lambda_R\widehat R_{g\pi_G(\xi)}\bigl(\theta(g)\bigr)\,, \Phi^R(\xi,g) =\widehat R_g(\xi) +\lambda_L\widehat L_{\pi_G(\xi)g}\bigl(\theta(g^{-1})\bigr)
(1-zq^{2N+3})\sum_{n=0}^{N}\frac{(q;q^2)_{n}(q^2;q^2)_N(zq^2;q^2)_{N-n}z^n q^{2n}}{(zq;q^2)_{n+1}(q^2;q^2)_{N-n}(zq^2;q^2)_{N+1}}&=\sum_{n=0}^{N}\left(\frac{(q;q)_{2n}z^{2n}q^{2n^2+3n}}{(zq;q)_{2n+1}}+\frac{(q;q)_{2n+1}z^{2n+1}q^{(2n+1)(n+2)}}{(zq;q)_{2n+2}}\right)\frac{(q^2;q^2)_N(zq^5;q^2)_N}{(q^2;q^2)_{N-n}(zq^5;q^2)_{N+n}}.
\triangle|\mathring{Ric}|^2=2|\nabla \mathring{Ric}|^2+2W_{kijl}\mathring{R}_{ij}\mathring{R}_{kl}+2\frac{n}{n-2}\mathring{R}_{ij}\mathring{R}_{jl}\mathring{R}_{li}+2\frac{R}{n-1}|\mathring{Ric}|^2.
\frac{(-2\pi i y)^k}{\Gamma(k)}\sum_{n=1}^{\infty}n^{k-1}e(nz) = \frac{(-2\pi i y)^k}{\Gamma(k)}\sum_{|n-\frac{k}{2 \pi y}| \leq C \frac{k^{1/2}}{2\pi y}}n^{k-1}e(nz) + O(e^{-C/4}).
\sum_{m_2}\lambda_{2}(m_2)e\left(\frac{dm_2}{c}\right)V_{-\eta H'}\left(\frac{n-m_2}{H'}\right)=\frac{H'}{c}\sum_{m_2}\lambda_{2}(m_2)e\left(-\frac{\bar{d}m_2}{c}\right)V^{\star}_{-\eta H'}\left(\frac{m_2n}{c^2},\frac{m_2H'}{c^2}\right),
\sum_{i=1}^{\nu}\sum_{n=0}^{\infty}\int_{A^n}\lvert \psi^{(n)}(k) \partial_i K(\xi-m^{(n)}(k))\lvert^2d\lambda_{n\nu}&\leq \sum_{n=0}^{\infty}\int_{A^n}C_K\lvert \psi^{(n)}(k) K(\xi-m^{(n)}(k))\lvert^2d\lambda_{n\nu}+C_K\lVert \psi\lVert^2 &=C_K\lVert K (\xi-d\Gamma_A(m))\psi\lVert^2+C_K\lVert \psi\lVert^2.
\left|\left\langle F\left(\cdot,s;e^{\rho_{\varepsilon}\left(T-s\right)}v_{n}^{\varepsilon}\right)-F\left(\cdot,s;e^{\rho_{\varepsilon}\left(T-s\right)}w_{n}^{\varepsilon}\right),\phi_{j}\right\rangle\right| \le CL_{F}e^{\rho_{\varepsilon}\left(T-s\right)}\sum_{k=1}^{n}\left|V_{k}^{\varepsilon}-W_{k}^{\varepsilon}\right|,
I_{4}(\epsilon) =\epsilon^{N-4}R^{2}(4-N)^{2}\int_{\mathbb{R}^{N}}\frac{4|\varphi|^{2}}{(\epsilon^2+|x|^{2})^{N-2}}+(2-N)^{2}\frac{|x|^{4}|\varphi|^{2}}{(\epsilon^2+|x|^{2})^{N}}+4(2-N)\frac{|x|^{2}|\varphi|^{2}}{(\epsilon^2+|x|^{2})^{N-1}}dx &=K_{1}(\epsilon)+K_{2}(\epsilon)+K_{3}(\epsilon).
\int_{D_{\rho}}|\mathrm{Ric}_{\tilde{g}}|^2_{\tilde{g}}\Phi dV_{\tilde{g}}=-\int_{S_{\rho}}\tilde{g}(\tilde{\nabla}\Delta_{\tilde{g}}\Phi,\tilde{\nu})d\tilde{\omega}_{\rho}- 4\int_{S_{\rho}}\mathrm{Ric}_{\tilde{g}}(\nabla\Phi,\tilde{\nu})d\tilde{\omega}_{\rho}.
\delta A_d[q_d] = \sum_{m=1}^{N-1}D_1 L_d (q_m,q_{m+1}).\delta{q_m}+\sum_{m=0}^{N-2}D_2 L_d (q_m,q_{m+1}).\delta{q_{m+1}}. = \sum_{m=1}^{N-1}[ D_1 L_d (q_m,q_{m+1}).\delta{q_m}+D_2 L_d (q_{m-1},q_{m})].\delta{q_{m}}.
R_h^0(z)-\tilde{R}_h(z)=-i \tilde{R}_h(z)WR_h^0(z) =ih^2\left(\int_0^\infty e^{-ish^{-1}(-ihX -i(Q_\infty+q_1+W)-z)}ds\right) W \left(\int_0^\infty e^{-ith^{-1}(-ihX -i(Q_\infty+q_1)-z)}dt\right).
\chi_{\Lambda}( C^{\Lambda_{0}}) = \sum_{( \lambda_{0}^{\prime}, \kappa_{0}, - s ) \in D^{+}}q^{2 ( \lambda_{0}^{\prime}, \lambda + \rho_{0})}\sum_{s = 0}^{\infty}n_{\lambda_{0}^{\prime}, s}q^{- 2 s ( \kappa + g )}
\Pi_i^a \left({\bf x}\right)\left| \Omega \right\rangle =\overline \psi \left({\bf y}\right)U({\bf y},{\bf 0})\psi\left({\bf 0}\right)\Pi_i^a \left({\bf x}\right)\left| 0\right\rangle + gT^a \int\limits_{\bf 0}^{\bf y}{dz_i}\delta\left({{\bf x}-{\bf z}}\right)\left| \Omega \right\rangle.
\Bigl ( \prod_{( j , \kappa ) \in \Lambda^{1}}\sum_{a_{j \kappa}, b_{j \kappa}}\Bigr ) = \underbrace{\sum_{a_{j \kappa}, b_{j \kappa}= 1}^{\dim V_{\tau_{j \kappa}}}\cdots \sum_{a_{j \kappa}, b_{j \kappa}= 1}^{\dim V_{\tau_{j \kappa}}}}_{\mathrm{o n e \sum f o r e a c h l i n k}}.
S_{1,A}^m(x , t; B,T)=\int_0^{\Psi_m(T)(t) \wedge \sigma_k(B)}+ \int_{\Psi_m(T)(t) \wedge \sigma_k(B)}^{\Psi_m(T)(t)}, &S_{1,A}^0(x , t; B, T)=\int_0^{T(t) \wedge \sigma_k(B)}+ \int_{T(t) \wedge \sigma_k(B)}^{T(t)},
\begin{aligned}\int_{A}V_{tt}(x,0)dx = \int_{A}\int_{\partial A}\mathrm{div}_{y}(G(x,y)X(y))\langle X(y),N(y)\rangle dydx &=\int_{\partial A}\mathrm{div}_{y}\left[\left(\int_{A}G(x,y)dx\right) X(y)\right]\langle X(y), N(y)\rangle dy &\stackrel{\eqref{Btwo5.1}}{=}\int_{\partial A}\mathrm{div}(V(x,0) X(x))\langle X(x),N(x)\rangle dx\end{aligned}
A(n,p,\ell,\phi )= 2^{p'-1}p'^{-\frac{1}{p}}p^{\frac{1-p'}{p}}\| \Phi^{p'}\phi^{\ell-2p'}\|_{L^{1}(\mathbb R^n)}^{\frac{1}{p}}\| \phi^{\ell}\|_{L^1(\mathbb R^n)}^{\frac{1}{p'}},
\sum\limits_{\gamma}\left\|H_{\overline{f}}\left(\frac{z^{\gamma}}{c_{\gamma}}\right)\right\|^2=\sum\limits_{\alpha}\left| f_{\alpha}\right|^{2}\sum\limits_{\gamma}\left(\frac{c_{\alpha+\gamma}^{2}}{c_{\gamma}^{2}}-\frac{c_{\gamma}^{2}}{c_{\gamma-\alpha}^{2}}\right)
P_{i+1}=A_{i}(P_{i}-P_{i}C_{i}^{\prime}(C_{i}P_{i}C_{i}^{\prime}+W_{i})^{-1}C_{i}P_{i})A_{i}^{\prime}+V_{i}+\tau_{\perp i+1}\Psi_{i}^{1}\tau_{\perp i+1}^{\prime},P_{0}=E(x_{0}{x^{\prime}}_{0})
\mathrm{supp}((\mu^{\boxtimes t_0})^{\mathrm{ac}})\cap [\epsilon, \infty)\subset \overline{\{1/x: x=h_{t_0}(r), r\in V_{t_0}^+\cap [a, b] \}}\subset B_\epsilon(\mathrm{supp}(\mu^{\boxtimes s})^{\mathrm{ac}}),
F_E(g_r,0,T) =e^{-1+\frac{m}{k_B}\frac{g_{r}}{T}-\left(1+\frac{\mathcal{I}}{mc^2}\right)\frac{cp^0}{k_B T}}\cr =\frac{n}{\int_{\mathbb{R}^3}\int_0^\infty e^{-\left(1+ \frac{\mathcal{I}}{mc^2}\right)\frac{cp^0}{k_B T}}\phi(\mathcal{I})\,d \mathcal{I}dp}e^{-\left(1+\frac{\mathcal{I}}{mc^2}\right)\frac{cp^0}{k_B T}}.
\left(\begin{matrix}x=\infty\,\left( \frac12 \right) \overbrace{\begin{matrix}1 0 0 0 \frac32 t_1 0 -\frac{t_2}{2}0 -1 0 0 0 -\frac32 t_1 0 \frac{t_2}{2}0\end{matrix}}\end{matrix}\right).
\mathbb{P}\left( J_0\neq \hat{J}_0 \right)\leq\mathbb{P}\left(\mathcal{E}_{\delta,q^*}^c\right)+\exp\left(-\frac{3}{16}\frac{n}{d_{q^*}}\right) &+\sum_{l=1}^{s}\sum_{m=0}^{l}\binom{s}{l}\binom{q-s}{m}4\exp\left( -c_1\frac{n^2(1-\rho_{s}^2)^2\kappa_l^2}{\sigma^4d_{l}+\sigma^2n(1-\rho_{s}^2)\kappa_l}\right).
\bigl[ (ce+1)x^2+(2cde+2d)xy+(c^2e+cd^2e+c+d^2)y^2\bigr]\bigl[vx+(u+dv)y\bigr]\cr -\bigl[(de+1)x^2+(2ce+2d^2e+d-2)xy+(2cde+d^3e-c-d)y^2\bigr](ux+cvy)=0.
\begin{array}{cc}R_P = \left( \begin{array}{cc}-\cos 2\phi_P -\sin 2\phi_P -\sin 2\phi_P \cos 2\phi_P\end{array}\right), R_C = \left( \begin{array}{cc}-\cos 2\phi_C \sin 2\phi_C \sin 2\phi_C \cos 2\phi_C\end{array}\right).\end{array}
\omega^{ij}_{g_i}(g_jh_j) =\tau_j(g_jh_j)^{-1}\tau_i(g_i)^{-1}\tau_j(g_jh_j)\tau_i(g_i) & =\tau_j(h_j)^{-1}(\tau_j(g_j)^{-1}\tau_i(g_i)^{-1}\tau_j(g_j)\tau_i(g_i))\tau_i(g_i)^{-1}\tau_j(h_j)\tau_i(g_i) & = \omega^{ij}_{g_i}(g_j)\omega^{ij}_{g_i}(h_j).
T_{n_{w_Gw_P}}T_{n_w}^* = E_{o_+\cdot (w_Gw_P)^{-1}}(n_{w_Gw_P})E_{o_+}(n_w) & = q_{w_Gw_P}^{1/2}q_{w}^{1/2}q_{w_Gw_Pw}^{-1/2}E_{o_+\cdot (w_Gw_P)^{-1}}(n_{w_Gw_P}n_w) & \in \sum_{v\in W_0,v\le w_Gw_Pw}C[Z_\kappa]T_{n_v}.
\|D_{\bf p}a\|_{L^1}\ \lesssim \ \begin{cases}|c_{j_1, k_1}2^{p j_1 + q k_1}|^{-\delta}\mbox{\rm if}\p\le 0 |c_{j_1, k_1}2^{p (j_1 - 1) + q k_1}|^{-\delta}\mbox{\rm if}\p\ge 0 \end{cases}.
dX_{k}^{\epsilon,u_k}(t) = f_k\bigl(X_{k}^{\epsilon, u_k}(t), u_{k}^{\epsilon}(t)\bigr) dt + \sqrt{\epsilon}\sigma_k\bigl(X_{k}^{\epsilon, u_k}(t)\bigr) d W(t), X_{k}^{\epsilon,u_k}(0)=x_{0}, k = 1, 2, \ldots, n,
\nu_{\beta,h}^+(\eta_0 =+ \mid +^{-(n+1)}_{-\infty}\omega^{-1}_{-n}) - \nu_{\beta,h}^+(\eta_0=+ \mid -^{-(n+1)}_{-\infty}\omega^{-1}_{-n})| \leq \lim_{n \rightarrow \infty}|\mu_{\beta,h}^{\Lambda_n,\omega^+}(\eta_{(0,0)}=+1) - \mu_{\beta,h}^{\Lambda_n,\omega^-}(\eta_{(0,0)}=+1)| \leq \lim_{n \rightarrow \infty}10n Ce^{-cn},
\kappa^* = \arg \min_{\kappa \in \mathcal{P}(\mathcal{M})}\sum_{u \in \mathcal{U}}T_u(\kappa) &= \arg \min_{\kappa \in \mathcal{P}(\mathcal{M})}\sum_{u \in \mathcal{U}}\cfrac{\overline{W}_u + D_u(t,\kappa) + D_u(t-1)}{1-p_u}&= \arg \min_{\kappa \in \mathcal{P}(\mathcal{M})}\sum_{u \in \mathcal{U}}\cfrac{D_u(t,\kappa)}{1-p_u}
\det(I+M_{\sqrt{\chi_A}}VM_{\sqrt{\chi_A}})= 1+\sum\limits_{B\subset F_A,B\neq\phi}t^{|B|}(\prod\limits_{x\in B}(\sum\limits_{i\in A}1_{\{x\in F_i\}}))\det(V_B) \sim t^{|F_A|}(\prod\limits_{x\in F_A}(\sum\limits_{i\in A}1_{\{x\in F_i\}}))\det(V_{F_A})t\rightarrow\infty.
\mathbb{E}\left[|\Delta V^{i,n}_\tau|^p\right]=\mathbb{E}\left[|\Delta V^{i,n}_\tau-\Delta V^{(m),i,n}_\tau|^p\right] \leq 2^p \sup_{\tau \in \mathcal{T}_0}\mathbb{E}\left[|V^{i,n}_\tau-V^{(m),i,n}_\tau|^p\right],
e_{M_i}(\ell(\xi_1)\cdots\ell(\xi_a)\ell(\eta_b)^\ast\cdots\ell(\eta_1)^\ast r(\gamma_c)\cdots r(\gamma_1)r(\rho_1)^\ast\cdots r(\rho_d)^\ast U_tp_{\geq b+d}(\mu)) &\qquad= \ell(q_i \xi_1)\cdots\ell(q_i \xi_a)\ell(\eta_b)^\ast\cdots\ell(\eta_1)^\ast r(q_i \gamma_c)\cdots r(q_i \gamma_1)r(\rho_1)^\ast\cdots r(\rho_d)^\ast Z_i(\mu)
\overline{\textrm{OPT}}_{\eta}(T) \leq 3 + 3 \times T \times \bigg( \sum_{k = t_{\eta}(T)}^T P(\tau = k) \times c^2_k \bigg)^{\frac{1}{2}}+ 15 \times T^2 \times \sum_{k = t_{\eta}(T)}^T P(\tau = k) \times c^2_k.
\sum_{n \geq 0}|p_n^* - e q_n^*| = \sum_{n \geq 0}\frac{2^{n+1}}{1 \times 3 \times 5 \cdots \times (2n+1)}- e= \sum_{n \geq 0}\frac{2^{2n+1}\, n!}{(2n+1)!}- e
\|x^{AB}- (\bar x + h_0 v_0)\| = \|x^{AB}- (\bar x + h_0 v_0^A) -h_0(v_0- v_0^A)\| &\leq \|x^{AB}-x^A\| + h_0\|v_0- v_0^A\| \le ( K+1) \left\| x^A -x^B\right\|\ +h_0 \eta &\leq ( K+1) 2\eta h_0 +h_0 \eta = h_0\eta \left( 2K+3\right) = \varepsilon h_0 \, .
\|{\bf F}\|_{k,\varphi}^2 : =\sum_{i=1}^3 \|{\bf F}_i\|_{k,\varphi}^2, \ \ \|{\bf T}\|_{k,\varphi}^2 : =\sum_{i,j=1}^3 \|{\bf T}_{ij}\|_{k,\varphi}^2.
\delta U=-\int_{0}^{T}\{\int_{\Omega^{0}}[N_{\alpha\beta,\alpha}\delta u_{\beta}^{0}+M_{\alpha\beta,\beta\alpha}\delta w^{0}]dA-\int_{\Gamma^{0}}[n_{\alpha}N_{\alpha\beta}\delta u_{\beta}^{0}+n_{\alpha}M_{\alpha\beta,\beta}\delta w^{0}-n_{\beta}M_{\alpha\beta}\delta w_{,\alpha}^{0}]ds\}dt
\begin{array}{r c l}{}&{}&{t t^{- 1}= 1 = t^{- 1}t ,}\\{}&{}&{t e t^{- 1}= q^{2}e , t f t^{- 1}= q^{- 2}f ,}\\{}&{}&{[ e , f ] = \displaystyle \frac{t - t^{- 1}}{q - q^{- 1}}.}\\ \end{array}
\dfrac{M_{1}(\mathcal{C}(G))}{|v(\mathcal{C}(G))|}= \dfrac{(4n-4)(4n-5)^{2}+36n}{8n-4}\dfrac{M_{2}(\mathcal{C}(G))}{|e(\mathcal{C}(G))|}= \dfrac{(2n-2)(4n-5)^{3}+54n}{(2n-2)(4n-5)+6n}.
(\mathbf 1_{A_{\alpha^*}}\bar{m}_{\tilde\xi_i}',L u)_{L^2(d\nu)}=(u,L\bar{m}_{\tilde\xi_i}')_{L^2(d\nu)},\,\,\,\,{\rm with}\,\,\, |L\bar{m}_{\tilde\xi_i}'|\leq R_{\max}.
v(x)= c_1f^{-1/4}\exp\left(i\int_{x}^{\infty}f^{1/2}dx\right)\bigl(1+\textit{o}(1)\bigr) +c_2f^{-1/4}\exp\left(-i\int_{x}^{\infty}f^{1/2}dx\right)\bigl(1+\textit{o}(1)\bigr),
\delta G^{( 2 , 2 , 2 , 2 )}( 1 \vert 2 \vert 3 \vert 4 ) = - \lambda \cdot \partial G^{( 2 , 2 , 2 , 2 )}( 1 \vert 2 \vert 3 \vert 4 ) + 2 ( \Lambda ( 1 ) + \Lambda ( 2 ) + \Lambda ( 3 ) + \Lambda ( 4 ) ) G^{( 2 , 2 , 2 , 2 )}( 1 \vert 2 \vert 3 \vert 4 ) \; .
\left[ ( \lambda z - \mu j)(1-z) + c\xi (1-z) \right]P_{c}(z) - \xi z (1-z)P^{'}_{c}(z) = \gamma z P_{c-1}(z) - \mu z \sum_{n=1}^{c}nz^{n}p_{c,n}+ c(1-z)(\xi - \mu)\sum_{n=0}^{c}z^{n}p_{c,n}- \xi (1-z)\sum_{n=1}^{c}n z^{n}p_{c,n}.
M^{u t}_{\mu}g(x) := \sup_{\{r > 0, y\in X : \d(x,y) < t r \mbox{and}\mu B^{cl}(y, r) > 0\}}\frac{1}{\mu(B^{cl}(y, r))}\int_{B^{cl}(y, t r)}|g (u)| d\mu (u).
\left[1-P_0^{(\Lambda,N)}(\tilde{\boldsymbol{\phi}})\right]\frac{\partial}{\partial \phi_j}\hat{\Phi}_{0,m}^{(N)}(\tilde{\boldsymbol{\phi}})=P_{0,j}^{(\Lambda,N)}(\tilde{\boldsymbol{\phi}})\hat{\Phi}_{0,m}^{(N)}(\tilde{\boldsymbol{\phi}})\mbox{for\}j=1,2.
\P_x(S(n)\in dz,\tau>n) =\P_x(\widehat S_i(n+d-i)\in dz_i,i=1,\ldots,d,\widehat \tau=\infty) =\sum_{\pi\in\mathcal S_d}(\pi) \P_x(\widehat S_i(n+d-\pi(i))\in dz_{\pi(i)}, i=1,\ldots, d, \widehat \tau=\infty).
B_1(k) := \int_{0}^{t}iske^{-isk^2}\iiint u^{\#}(\ell)\overline{u^{\#}}(n)u^{\#}(m) \, \mu_{R,1}^{\#,(2)}\left(k,\ell,n,m\right)\,d\ell dmdn\,ds, B_2(k) := \int_{0}^{t}e^{-isk^2}\iiint u^{\#}(\ell)\overline{u^{\#}}(n)u^{\#}(m) \, \partial_{k}\mu_{R,1}^{\#,(2)}\left(k,\ell,n,m\right)\,d\ell dmdn\,ds.
\Omega_{j^*,\kappa}= \left\{u \in \mathbb{R}^N ~\middle|~a = e_1^{\sf T}u ,\,a_{-j^*}< a < a_{-j^*+1},\,\| u - a e_1 \| < \kappa \left( a - a_{-j^*}\right) \left( a_{-j^*+1}- a \right) \right\},
\delta\left(w_1,w_2\right)= \frac{\frac{1}{w_{1}}+\frac{1}{w_{2}}-\frac{1}{2}\coth\left(\frac{w_{1}}{2}\right)-\frac{1}{2}\coth\left(\frac{w_{2}}{2}\right)}{w_{1}+w_{2}}.
\lim_{n \rightarrow \infty}{\sqrt{n}~\mathbb{E}X_n}= \int_{0}^{\infty}{x \frac{d}{dx}\left(-2\sum_{k=1}^{\infty}{(-1)^{k-1}e^{-2 k^2 x^2}}\right) dx}&= \sum_{k=1}^{\infty}{(-1)^{k-1}\int_{\mathbb{R}}{e^{-2k^2 x^2}dx}}&= \sqrt{\frac{\pi}{2}}\sum_{k=1}^{\infty}{\frac{(-1)^{k-1}}{k}}&= \sqrt{\frac{\pi}{2}}\log{2}\sim 0.86873\dots
\,=\, \left<\,f_{\,1},\, f_{\,2}\,|\, f_{\,3},\, \,\cdots,\, f_{\,n}\,\right>_{1}\,\left<\,g_{\,1},\, g_{\,2}\,|\, g_{\,3},\, \,\cdots,\, g_{\,n}\,\right>_{2},
\begin{aligned}\int\Big(|\nabla v_\lambda|^2-(\partial_r (rV))|u|^2-2 \varepsilon \mbox{Im}[(r\partial_ru)\overline{u}]+2 \varepsilon \lambda^{\frac12}r|u|^2\Big)dx =\int\Big(\mbox{Re}[f(2r\partial_r\overline u+(n-1)\overline u)]-2\lambda^{\frac12}\mbox{Im}(rf\overline u)\Big)dx.\end{aligned}
{\bf y}_r={\bf B}_1({\bf s}_{1,2}^p+{\bf s}_{2,1}^p)+{\bf B}_2({\bf s}_{1,3}^p+{\bf s}_{3,1}^p)+{\bf B}_3({\bf s}_{2,3}^p+{\bf s}_{3,2}^p) &+{\bf B}_4({\bf s}_{1,2}^c+{\bf s}_{3,1}^c)+{\bf B}_5({\bf s}_{2,3}^c+{\bf s}_{3,1}^c) &+{\bf B}_6{\bf s}_{2,3}^r+{\bf B}_7{\bf s}_{3,1}^r+{\bf n}_r,
\sum_{n=0}^{\infty}\frac{n!}{\Gamma(1+\alpha+n)}L_{n}^{(\alpha)}(x)L_{n}^{(\alpha)}(y)t^{n}=\frac{1}{(xyt)^{\alpha/2}(1-t)}e^{-(x+y)t/(1-t)}I_{\alpha}(\frac{2\sqrt{xyt}}{1-t}).
X_t( x_{t-1}, \xi_t)= \{x_t \in \mathbb{R}^n : x_t \in \mathcal{X}_t,\;g_t(x_t, x_{t-1}, \xi_t) \leq 0,\;\;\displaystyle A_{t}x_{t}+ B_{t}x_{t-1}= b_t \},
\Delta^{( r o t )}( \tau ,{\vec{\sigma}}_{o}) = \sqrt{[ \mu \, n_{o}({\vec{\sigma}}_{o}) ]^{2}+ \sum_{r s}\, A^{r s}({\vec{\sigma}}_{o}, q^{\mu}( \tau ) , \overline{{{\cal S}}}^{u}) \, \overline{{{\cal S}}}^{r}\, \overline{{{\cal S}}}^{s}}.
\begin{array}{l}{{v_{k + 1}= W \left[ \left( 2 F ( u_{k}, v_{k}) - F ( u_{k}, v_{k - 1}) \right) \, , u_{k}\right] ,}}\\{{u_{k + 1}= \bar{W}\left[ \left( 2 \bar{F}( u_{k}, v_{k}) - \bar{F}( u_{k - 1}, v_{k}) \right) \, , v_{k}\right] .}}\\ \end{array}
\Psi^{\alpha_{1}^{1}. . . . \alpha_{1}^{m}, \alpha_{2}^{1}. . . . \alpha_{2}^{m}}( X_{1}, X_{2}) = \sum_{R_{1}f_{1}, R_{2}f_{2}}\psi_{R_{1}f_{1}, R_{2}f_{2}}^{\alpha_{1}^{1}\alpha_{1}^{2}, \alpha_{2}^{1}\alpha_{2}^{2}}( x_{1}, x_{2}) \phi_{R_{1}f_{1}}^{\alpha_{1}^{3}. . . . \alpha_{1}^{m}}( y_{1}) \phi_{R_{2}f_{2}}^{\alpha_{2}^{3}. . . . \alpha_{2}^{m}}( y_{2}) .
\left\vert a_{\varepsilon}(t;\boldsymbol{u},\boldsymbol{\Phi})\right\vert\leq N^{3}\left\Vert A_{0}\right\Vert_{L^{\infty}(Q\times \mathbb{R}^{N+1})^{N^{2}}}\left\Vert \boldsymbol{u}\right\Vert_{V}\left\Vert \boldsymbol{\Phi}\right\Vert_{V}:=M\left\Vert \boldsymbol{u}\right\Vert_{V}\left\Vert \boldsymbol{\Phi}\right\Vert_{V}.
\left\{\tilde{T}^{n}(x,\omega)\in A\right\}=\left\{L_{\omega_{n}^{-1}.x}\in S\left(\omega_{n}^{-1}.x,\left(\omega_{n}^{-1}\omega_{m+n}\right)_{m\in\mathbb{Z}}\right)\right\}=\left\{L_{x}\omega_{n}\in S\left(x,\omega\right)\right\}\ \ \mbox{by compatibility (i)}.
\tfrac{1}{L_1}\left(E(L_1,\ell_\nu,\ell_\mu)+E(L_1,\ell_\nu,\ell_{\mu'})\right) =\tfrac{1}{L_1}\left(\hat{R}(L_1^A,L_1^B,\ell_\nu)L_1^A+\hat{R}(L_1^B,L_1^A,\ell_\nu)L_1^B\right) &<\tfrac{1}{L_1}(6e^{-\frac{1}{2}\ell_\nu}L_1^A+6e^{-\frac{1}{2}\ell_\nu}L_1^B)=6e^{-\frac{1}{2}\ell_\nu}.
h(t\cdot x) = \prod_{v\leq \infty}h_v(t\cdot x)=\prod_{v\leq \infty}|t|_v\cdot h_v(x)=\prod_{v\leq \infty}|t|_v \cdot \prod_{v\leq \infty}h_v(x)=1\cdot h(x)
\begin{aligned}|f(x)-f(x')| \leqslant C \Big(\frac{\|x-x'\|}{r}\Big)^\beta \Big\{\frac{1}{V(x,x_0,r+d(x,x_0))}\Big(\frac{r}{r+{d(x,x_0)}}\Big)^\gamma \qquad+ \frac{1}{V(x',x_0,r+d(x',x_0))}\Big(\frac{r}{r+{d(x',x_0)}}\Big)^\gamma\Big\}; \end{aligned}
\Psi_{1,0}(z_{v^{-1}AB^{-1}}) = \sum_I \mathrm{tr}\big( \Lambda_I \overset{I}{S^2(a_i)}S(\overset{I}{T}) \overset{I}{S(a_j)}b_i b_j \big) = \mathcal{D}^{-1}(z)\big( S^2(a_i) S(?) S(a_j) \big)b_ib_j &= S\big(\mathcal{D}^{-1}(z)\big) (S(a_i) a_j ? )b_ib_j = S\big(\mathcal{D}^{-1}(z)\big) \in \mathcal{O}(H) \otimes 1.
\textrm{Tr}L(\mu)\rho(\mu,r) =\mu^N e^{\frac{\mu}{2}\eta\frac{\partial}{\partial\mu}}\rho(\mu,r)+\frac{\Delta}{\mu^N}e^{-\frac{\mu}{2}\eta\frac{\partial}{\partial\mu}}\rho(\mu,r)= &=\mu^N\rho(\mu(1+\frac{\eta}{2}))+\frac{\Delta}{\mu^N}\rho(\mu(1-\frac{\eta}{2})).
\mathbb{P}(\#{\cal E}_i \geq M\log{n}) \leq \sum_{r = M\log{n}}^{n}\frac{T_re^{-r}}{C(r-1)!}e^{-\delta_0r}\leq e^{-\delta_0M\log{n}}\sum_{r=M\log{n}}^{n}\frac{T_re^{-r}}{C(r-1)!}.
D_{-1}\mapsto\left( \begin{array}{rr}0 1 0 0 \end{array}\right),\qquad D_0 \mapsto\left( \begin{array}{rr}{\textstyle \frac12}0 0 -{\textstyle \frac12}\end{array}\right),\qquad D_1 \mapsto\left( \begin{array}{rr}0 0 -1 0 \end{array}\right)
W_2(u,z_1)-\lambda_-W_2(u,z)=\pi i zI_\mu(uz)\left(\sum_{s=0}^{N-1}\frac{A_s(z)}{u^{2s}}+g_2(u,z_1)\right) \qquad{}+\pi i\frac{z}{u}I_{\mu+1}(uz) \left(\sum_{s=0}^{N-1}\frac{B_s(z)}{u^{2s}}+zh_2(u,z_1)\right) \qquad{}+\lambda_- zK_\mu(uz) (g_2(u,z_1)-g_2(u,z))-\lambda_-\frac{z^2}{u}K_{\mu+1}(uz)(h_2(u,z_1)-h_2(u,z)).
\frac{1}{(q;q)_\infty}\sum_{n=-k}^{k}(-1)^{n}\, q^{n(3n-1)/2}= 1+(-1)^k\frac{q^{k(k+1)/2}}{(q;q)_k}\sum_{n=0}^\infty \frac{q^{(n+k+1)(k+1)}}{(q^{n+k+1};q)_\infty}.
\mathrm{P}(\tau_j<T) =\mathrm{P}(\sup_{t\in [0,T]}|Z^{\l,j}_t|>j) & \leq \frac{1}{j}\mathbb{E}[ \sup_{t\in [0,T]}|Z^{\l,j}_t| ] \leq \frac{1}{j}\exp{KT}(M_T +\mathbb{E}[|Z_0|])
\pi_g \tau_e=\theta_\tau \lambda_g \theta_f \tau_e =\theta_\tau \lambda_g \theta_f \theta_\tau \delta_e=\theta_\tau \theta_f \theta_\tau \lambda_g\delta_e=I_\mathcal{X}\theta_\tau \lambda_g\delta_e=\theta_\tau\delta_g=\tau_g, \forall g \in G
L_i\,{}_{{}_\lambda}v_m=\begin{cases}(\partial+\b)v_{i+m}\ \mbox{if}\ \ \ (a_m,a_{i+m})=(0,0), begin{align*}4pt](\partial+\b+\lambda)v_{i+m}\ \mbox{if}\ \ \ (a_m,a_{i+m})=(1,1), begin{align*}4pt]v_{i+m}\ \mbox{if}\ \ \ (a_m,a_{i+m})=(0,1), begin{align*}4pt](\partial+\b)(\partial+\b+\lambda)v_{i+m}\ \mbox{if}\ \ \ (a_m,a_{i+m})=(1,0).\end{cases}
\rho_{+}= \frac \ell 2 ( M + \sqrt{M^{2}- \frac{J^{2}}{\ell^{2}}}) \qquad \mathrm{a n d}\qquad \rho_{-}= \frac \ell 2 ( M - \sqrt{M^{2}- \frac{J^{2}}{\ell^{2}}}) \qquad
\mathcal{D}^{(k)}\mathcal{X}_t = \left(\mathcal{D}^{j_1,j_2,\dots,j_k}_{r_1,r_2,\dots,r_k}\mathcal{X}_t\right)_{\substack{j_1,j_2,\dots,j_k\in\{1,2,\dots, d\}r_1,r_2,\dots,r_k \in [0,T]}}.
v_X(\nu t) \le \nu^{d_n}v_X(t) \le \frac{1/s+1/\mu_X}{1/s+1}\cdot v_X(t) \le \frac{v_X(t)/s+u_X(t)}{1/s+1}= v_X(t) + \frac{u_X(t)-v_X(t)}{1/s+1}.
\widetilde\gamma(s)+\frac{\zeta(s)\cdot\zeta(s)}{4z(s)}=-2h^2s^2(|K_1|^2+|K_2|^2+|K_3|^2)+4s^2h^2(K_1+K_2+K_3)\cdot K-6s^2h^2|K|^2 &\quad+\mathcal{O}(s^2\frac{h^4}{\sigma^2}+s^3\sigma^4) &=-2h^2s^2|K_1-K|^2-2h^2s^2|K_2-K|^2-2h^2s^2|K_3-K|^2+\mathcal{O}(s^2\frac{h^4}{\sigma^2}+s^3\sigma^4)