harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\|h-h_{(r)}\|_{L^1(D(z,\frac{r}{2}))}\leq \|h-h_{z,r}\|_{L^1(D(z,\frac{r}{2}))}+\|h_{(r)}-h_{z,r}\|_{L^1(D(z,\frac{r}{2}))}= \|h-h_{z,r}\|_{L^1(D(z,\frac{r}{2}))}+\|\phi_{(r)}*(h-h_{z,r})\|_{L^1(D(z,\frac{r}{2}))}\leq \|h-h_{z,r}\|_{L^1(D(z,r))}+\|\phi_{(r)}\|_{L^1}\|(h-h_{z,r})\|_{L^1(D(z,r))}= 2A_0.
P(\hat{A}_n) = P(A_{t_n}) = P \Big( (t_n/3)^{-1/2}\, S_{t_n}^1 \ge x(t_n) \Big) & = (1+o(1)) \bar \Phi(x(t_n)) \ge c_1 x(t_n)^{-1}e^{-x(t_n)^2/2}\ge \frac{c_2}{n \sqrt{\log n}}\,,
P\Big(\widehat{\tau}_M<+\infty\Big|\widehat{\zeta}_0^M=n, \widehat{g}_0^M=m\Big)\geq &P\Big(\widehat{\zeta}_t^M+\widehat{g}_t^M>0t\geq 0\Big|\widehat{\zeta}_0^M=n, \widehat{g}_0^M=m\Big).
2p\hat{\beta}^TA\sum_{i}k_i\bar{q}_i-2p\hat{\beta}^Tq =2p\hat{\beta}^T(Aq-pA\hat{\beta})-2p\lambda_0\hat{\beta}^Tq &\equiv 2p\hat{\beta}^T(Aq-\lambda_0q) &\equiv 0~({\rm mod}~p^2).
\begin{array}{r c l}{{\tilde{f}_{1 , i}}}&{{=}}&{{\tilde{f}_{L , i}\cos \theta_{\tilde{f}_{i}}+ \tilde{f}_{R , i}\sin \theta_{\tilde{f}_{i}}}}\\{{\tilde{f}_{2 , i}}}&{{=}}&{{- \tilde{f}_{L , i}\sin \theta_{\tilde{f}_{i}}+ \tilde{f}_{R , i}\cos \theta_{\tilde{f}_{i}}}}\\ \end{array}\quad .
U_{[ 4 ] \times [ N ]}^{\prime}= - \left( \! \begin{array}{c c c c}{{U_{1 1}^{\prime}}}&{{U_{1 2}^{\prime}}}&{{\cdots}}&{{U_{1 N}^{\prime}}}\\{{U_{2 1}^{\prime}}}&{{U_{2 2}^{\prime}}}&{{\cdots}}&{{U_{2 N}^{\prime}}}\\{{U_{3 1}^{\prime}}}&{{U_{3 2}^{\prime}}}&{{\cdots}}&{{U_{3 N}^{\prime}}}\\{{U_{4 1}^{\prime}}}&{{U_{4 2}^{\prime}}}&{{\cdots}}&{{U_{4 N}^{\prime}}}\\ \end{array}\! \right) ,
{\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 \ ,
\begin{aligned}\int_{\mathcal{Z}}\left|\frac{D^k f^*(y|x)}{f^*(y|x)}\right|^{(2\beta+\varepsilon)/k}f^*(y|x) \, dy\, dx < \infty, \int_{\mathcal{Z}}\left|\frac{L(y,x)}{f^*(y|x)}\right|^{(2\beta+\varepsilon)/\beta}f^*(y|x) \, dy\, dx < \infty.\end{aligned}
\Delta_{2,2}\varphi(x_1, x_2; h_1, h_2): = \varphi (x_1 + h_1, x_2 + h_2) + \varphi (x_1 - h_1, x_2 + h_2) & + \varphi (x_1 + h_1, x_2 - h_2) + \varphi (x_1 - h_1, x_2 - h_2) - 2 \varphi (x_1 + h_1, x_2) - 2 \varphi (x_1-h_1, x_2 ) - 2 \varphi (x_1, x_2 + h_2) - 2 \varphi (x_1, x_2 - h_2) + 4 \varphi (x_1, x_2 ).
F_n\widehat{E}^{C_2}_* = E^{C_2}_{*+|n\sigma| -n\sigma}F_n\widehat{E}^{C_2}_*(\mathbf{CP}^\infty_{C_2}) = E^{C_2}_{*+|n\sigma|-n\sigma}(\mathbf{CP}^\infty_{C_2}).
P(x_{t}|o_{1:t},u_{1:t},m_{t})=\sum_{m_{t-1}}P(o_{t}|x_{t},m_{t},u_{1:t})\sum_{x_{t-1}}P(x_{t}|x_{t-1})P(x_{t-1}|m_{t},o_{1:t-1},u_{1:t})/Z
\nu_{m}= \begin{array}{r l}{{\frac{n m}{2},}}&{{\quad \mathrm{f o r ~ m ~ \ne ~ 0 ~ m o d ~ n + 1 ~}}}\\{{\frac{n m}{2}+ ( n + 1 ) b_{h_{m}+ 1},}}&{{\quad \mathrm{f o r ~ m ~ = ~ 0 ~ m o d ~ n + 1 ~}.}}\\ \end{array}
\dot{\eta}^{\prime}= \left\{\eta + \delta \eta , H \right\}^{( \eta , \Lambda )}+ \left\{\eta , \frac{\partial \delta W}{\partial t}\right\}^{( \eta , \Lambda )}= \left\{\eta + \delta \eta , H + \frac{\partial \delta W}{\partial t}\right\}^{( \eta , \Lambda )}.
S_n:=(X_n,Y_n,Z_n):= \Bigr(\frac{x_1}{\sqrt{n}},\frac{y_1}{\sqrt{n}},\frac{z_1}{\sqrt{n}}\Bigr) \cdots \Bigr(\frac{x_n}{\sqrt{n}},\frac{y_n}{\sqrt{n}},\frac{z_n}{\sqrt{n}}\Bigr).
\Pi_{\mu_1\mu_2}^{{\rm UV}}=\left(\frac{13}{3}- \xi \right)\frac{g^2 C_G}{\left(4\pi\right)^2}\left(\frac{1}{2\epsilon}+{\rm ln}\left(M\right)\right)\left(\eta_{\mu_1\mu_2}k^2 - k_{\mu_1}k_{\mu_2}\right) ,
P_{\vec{\sigma}, a, \vec{1}}(\eta,\theta) = \frac{e^{\sigma_1 \eta}e^{\sigma_2 \theta}}{f_{\vec{\sigma}, a, \vec{1}}(\vec{0})}\left(\frac{H(a,e^\eta+e^\theta)}{\exp(e^\theta)-1}+ \frac{H(1,e^\theta)}{e^\eta + e^\theta}\right), \eta,\theta \in{\mathbb{R}},
\begin{array}{l}{{Y_{2}R_{1 2}K_{1}R_{2 1}= R_{1 2}K_{1}R_{1 2}^{- 1}Y_{2}+ q^{2}R_{1 2}{\cal P}\quad ,}}\\{{\tilde{Y}_{2}R_{2 1}^{- 1}K_{1}R_{2 1}= R_{1 2}K_{1}R_{2 1}\tilde{Y}_{2}+ q^{- 2}R_{1 2}{\cal P}\quad .}}\\ \end{array}
2\mu \left(a^{22}a^{11}- a^{21}a^{12}\right) e_{1||3}^1 + \rho \left( a^{22}a^{11}- a^{21}a^{12}\right)\dot{e}_{1||3}^1 = 2\mu ae_{1||3}^1 + \rho a\dot{e}_{1||3}^1 = 0,
{\rm Re}\, \chi^{\rm (TM)}_T(\omega) = \frac{\hbar A}{4\pi^2}\left(\frac{k_B T}{\hbar}\right)^5\int_0^{\Omega}dv\frac{\left(\frac{\Omega}{2}-v\right)^2}{\sqrt{v(\Omega-v)}}\int_{0}^{\infty}du\, \frac{u^2}{e^{\frac{u}{2}}-1}\left[1+{\cal O}(\Omega^2)\right].
p\approx(w^{-},w^{+})=\frac{1}{1+\frac{z^{2}}{n}}(\hat{p}+\frac{z^{2}}{2n})\pm\frac{z}{1+\frac{z^{2}}{n}}\sqrt{\frac{\hat{p}(1-\hat{p})}{n}+\frac{z^{2}}{4n^{2}}}
\tau_{g_1}(b)\tau_{g_2}(c) =Q\circ\pi(c(g_1)c(g_2),bc)=Q\circ\pi(\sigma(g_1,g_2)c(g_1g_2),bc) &=Q\circ\pi(c(g_1g_2),\gamma(\sigma(g_1,g_2))bc)=\tau_{g_1g_2}(b\cdot_{\gamma\circ\sigma}c).
I(\gamma) = I_0(\gamma(0)) + \sup_{k \geq 1}\sup_{\substack{0 = t_0 < t_1 < \dots, t_k t_i \in \Delta_\gamma^c}}\sum_{i=1}^{k}I_{t_i - t_{i-1}}(\gamma(t_i) \, | \, \gamma(t_{i-1})).
\xi-\zeta = \sum_{l=1}^{k}(\xi_{l}-\xi_{l-1})=\sum_{l=1}^{k}n_{l}z_{\xi_{l-1}}^{-1}\gamma_{l}^{\vee}, [\xi-\zeta]=\sum_{l=1}^{k}n_{l}[\gamma_{l}^{\vee}] \in Q_{J^{c}}^{\vee +}.
Z_{\pm}= \sum_{h = 0}^{\infty}\left( g_{s}^{- 1}e^{i \Theta_{\pm}}\right)^{2 - 2 h}\left( \lambda_{1}^{R}\right)^{{\frac{5}{2}}( 1 - h )}Z_{h}, \qquad \Theta_{+}= 0 , \; \Theta_{-}= \pi / 2 .
\frac{h^{-1}}{4}\int_K6\xi_1^2(a_5+a_7)h\frac{\partial^3v_h}{\partial x_1^3}\;\mathrm{d}x_1\mathrm{d}x_2=\frac{1}{2}\int_K(a_5+a_7)\frac{\partial^3v_h}{\partial x_1^3}\;\mathrm{d}x_1\mathrm{d}x_2.
{\cal Z}( a_{1}, a_{2}, \dots , a_{N}) ~ = ~ \int \prod_{\mu}{\cal D}P^{\mu}e^{\sum_{i}[ a_{i}\sum_{\mu}( p_{i}^{\mu})^{2}]}\prod_{\mu ; i , j}\delta ( ( 1 - D_{i j}) P_{i j}^{\mu}) \prod_{\mu ; i}\delta ( \sum_{j}P_{i j}^{\mu}) ~ ~ ,
\mathcal{E}_{q}(u)\mathcal{E}_{q}(-v)= \sum_{n=0}^{\infty}\frac{(-1)^{n}q^{n^{2}/4}}{(q;q)_{n}}v^{n}\sum_{k=0}^{n}{n \brack k}_{q}q^{k(k-1)/2}\left(-q^{\frac{1-n}{2}}\frac{u}{v}\right)^{k}.
F_i^{(p)}G^{\mathrm{low}}(b)=\sum_{\tilde{b}\in B(\infty)}c_{-pi, b}^{\tilde{b}}G^{\mathrm{low}}(\tilde{b}), &(_ie')^p(G^{\mathrm{low}}(b))=\sum_{\tilde{b}\in B(\infty)}d_{b, \tilde{b}}^{i, p}G^{\mathrm{low}}(\tilde{b}), &(e'_i)^p(G^{\mathrm{low}}(b))=\sum_{\tilde{b}\in B(\infty)}\widehat{d}_{b, \tilde{b}}^{i, p}G^{\mathrm{low}}(\tilde{b}).
H^k(\mathbb{P}, \Omega_{\mathbb{P}}^{n-1}\otimes \mathcal{O}(-mY-mK_{\mathbb{P}})) \cong H^{n-k}(\mathbb{P}, \Omega_{\mathbb{P}}^1 \otimes \mathcal{O}(mY+mK_{\mathbb{P}})) = 0
\begin{aligned}\Delta = 4 X(0)^2 (1-\beta \gamma_g)^2 - 4 (1 - 2\beta \gamma_g) ( 1 + X(0)^2 ) &= 4 \Big( \beta^2 \gamma_g^2 X(0)^2 + 2 \beta \gamma_g -1 \Big) > 0.\end{aligned}
\begin{array}{cl}P_{n}^{c}\left( x\right) =P_{n}\left( x\right)-\sum\limits_{i=1}^{d}A_{i}\left( x\right) P_{n-k-i}^{\left( k+i\right)}\left( x\right) ,d\geq 1,n,k\geq 0, P_{n}^{c}\left( x\right) =P_{n}\left( x\right) ,n\leq k.\end{array}
\delta_{\tilde g^{(4)}}(\alpha^2) =-\frac{1}{u^2}\{-\nabla_{\partial_t}[\alpha^2(\partial_t)]+\alpha^2(\nabla_{\partial_t}\partial_t)\}&=\frac{1}{u^2}\nabla_{\partial_t}\alpha=-\frac{1}{u^2}\nabla_{\partial_t}(\frac{1}{u^2}\xi) &=-\frac{1}{u^4}\nabla_{\partial_t}\xi=-u^{-3}d u.
\nabla_{k}\nabla_{l}\Psi^{i}-\nabla_{l}\nabla_{k}\Psi^{i}=\Psi^{r}R^{i}_{rkl}-\dot{\nabla}_{r}\Psi^{i}R^{r}_{0kl}-\nabla_{r}\Psi^{i}S^{r}_{kl},
\tilde{\varepsilon}_T^{-1}\frac{J2^{Jd/2}(\log T)^\eta}{\sqrt{T}}E^{\Pi^{D_T}}\Big[ \max_{\lambda\leq J, k,j}\sqrt{T}|\langle b_j-b_{0,j},a_\lambda \Phi_{\lambda,k}\rangle_{L^2}| \big|X^T \Big] & =O_{P_{b_0}}\Big(\tilde{\varepsilon}_T^{-1}\frac{J^{3/2}2^{Jd/2}(\log T)^\eta}{\sqrt{T}}\Big) = O_{P_{b_0}}\Big((\log T)^{3/2+\eta-\delta}\Big).
\|w_nv_nw_n^* - [1-2q_{k_n}]v_n\|_2 = \|[1 - 2p_{k_n}q_{k_n}][1- 2(v_np_{k_n}v_n^*)q_{k_n}] - [1 - 2q_{k_n}]\|_2 \le \|2(1-p_{k_n})q_{k_n}- 2(v_np_{k_n}v_n^*)q_{k_n}\|_2 \le 2\cdot \frac{1}{2^{n+1}}= \frac{1}{2^n},
d_{n,n+1}(T) = C\Big(d_{n-1,n}(T) + 2d_{n,n}(T) + \frac{\beta_n T r_{n,n}(T)}{2}&+ \frac{k_{n,n}(T)}{T}\Big)\frac{r_{n+1,n+1}(T)}{\alpha_{n+1}}\le\frac{H_{6,n+1}}{T^3}\,.
\Lambda_{\alpha}\bar{\Lambda}_{\dot{\beta}}\bar{\mathrm{\raisebox{0 e x}[ 1 . 7 5 e x ] [ 0 e x ]{~ D ~}}}\bar{\Theta}^{\dot{\beta}}= 0 , \ \ \ \ \ \, L a m b d a_{\alpha}\bar{\Lambda}_{\dot{\beta}}D \Theta^{\alpha}= 0
A \tilde{g}_i = \tilde{c}_i \tilde{u}_i^A, A^T \tilde{u}_i^A = R^T \left(\tilde{c}_i R \tilde{g}_i + \alpha_{k+1}v_{k+1}e_{k+1}^T x_i\right), B \tilde{g}_i = \tilde{s}_i \tilde{u}_i^B, B^T \tilde{u}_i^B = R^T \left(\tilde{s}_i R \tilde{g}_i + \check{\beta}_{k}v_{k+1}e_{k}^T \hat{x}_i\right).
\overline{\mathbf{x}}_{k+1}= \mathbf{F}\;\mathbf{x}_{k+1}+ \mathbf{z}_{k+1}, \mathbf{z}_{k+1}= \mathbf{G}\; \overline{\mathbf{x}}_{k}= \left[ \mathbf{x}_{k-J+1}^T\:\mathbf{x}_{k-J+2}^T\:\ldots\:\mathbf{x}_{k}^T \:\mathbf{0}_{1 \times N_t}\right]^T.
\mathcal{P}:=\left\{\mathbf{p}:=\left[p_{n}\right]{}_{n\in\mathcal{N}}\left|p_{1}\in\mathcal{T}_{1},p_{n}\in\mathcal{T}_{n}\left(p_{n-1}\right),\forall n\in\mathcal{N}\backslash{1}\right.\right\}.
A_2(s, t) = \frac{1}{16\pi^2}\left( \frac{-t}{s}\right)\!\int\limits_{1}^{\infty}dx_1 \!\int\limits_{0}^{1}\!\! dx_2\frac{(x_1^2 - x_2^2)}{\sqrt{(x_1^2 - 1)(1 - x_2^2)}}&\times A_B[s, \sqrt{-t}(x_1 + x_2)/2] A_B[s, \sqrt{-t}(x_1 -x_2)/2] \;.
H=\left( \begin{array}{ccccc}1 \alpha \alpha^{2}... \alpha^{n-1}1 \alpha^{8}\alpha^{16}... \alpha^{8(n-1)}1 \alpha^{15}\alpha^{30}... \alpha^{15(n-1)}1 \alpha^{22}\alpha^{44}... \alpha^{22\left( n-1\right)}\end{array}\right) .
\begin{aligned}{\xi_{i}^n}= \Pi{L^{i}u(t^n)- L_h^{i}u_h^n}= \; \Pi L^{i}u(t^n) - L_h\Pi L^{i-1}u(t^n) + L_h\Pi{L^{i-1}u(t^n)- L_h^{i}u_h^n}= \; \left(\Pi L - L_h\Pi\right) L^{i-1}u(t^n) + L_h\left(\Pi L^{i-1}u(t^n)- L_h^{i-1}u_h^n \right) = \; \zeta_{i}^n + L_h \xi_{i-1}^n.\end{aligned}
\langle N \rangle_{\mathrm{T}}= \frac{1}{2 \pi}\int_{- \infty}^{\infty}d x \, \theta^{\prime}- \sqrt{\frac{2 m T}{\pi}}\, e^{- m / T}\, \int_{- \infty}^{\infty}d x \, \mathrm{s i n h}\left( \frac{\theta^{\prime}}{2 T}\right) + \dots
\frac{1}{n}\Omega^{(\alpha,\lambda)}(\{Q_{i}\}_{i=1}^{n}) =-\frac{1}{n}\sum_{i=1}^{n}\log\Lambda_{i}^{(\alpha,\lambda)}(\{Q_{j}\}_{j=1}^{i}) * \geq\frac{\Omega^{(\alpha,\theta)}}{1+2\bar{\alpha}\theta}.
\Phi \left( \int_{0}^{\mu \left( A_{n+1}\right)}\varphi \left( \gamma\right) d\gamma \right) =\Phi \left( \int_{0}^{\mu \left( TA_{n}\right)}\varphi \left( \gamma \right) d\gamma \right) \leqslant \Psi \left(\int_{0}^{\mu \left( A_{n}\right)}\varphi \left( \gamma \right) d\gamma\right) .
{\cal L}= - \bf E_{\mathrm{T}}\cdot \dot{A}_{\mathrm{T}}\mathrm{+ p_{\it a}\dot{\t h e t a_{\it a}}- H_{C}^{( 2 )}- H_{C}^{( 3 )}+ O ( \t h e t a^{4}) . \vspace{1 e m}}
\frac{(p_{1}+q_{1}-a)(p_{2}+q_{2}-c)}{(p_{1}+q_{1}+a)(p_{2}+q_{2}+c)}=\frac{(p_{2}+q_{1}-b)(p_{1}+q_{2}-d)}{(p_{2}+q_{1}+b)(p_{1}+q_{2}+d)}
\psi( b^-_1(n) n z_0)-\psi(nz_0)) = \int_0^1 \frac{\partial}{\partial t}\tilde{\psi}_t( b^-_t(n) n z_0) \;dt =\int_0^1 L_{\tilde{w}_t}( \tilde{\psi}_t)( p^-_t(n) n z_0) \;dt.
W_{pq}(n)=(\frac{\mu_{p}}{\mu_{q}})^{n}\prod_{j=1}^{n}\frac{y_{q}-x_{p}\omega^{j}}{y_{p}-x_{q}\omega^{j}},\overline{W}_{pq}(n)=(\mu_{p}\mu_{q})^{n}\prod_{j=1}^{n}\frac{\omega x_{p}-x_{q}\omega^{j}}{y_{q}-y_{p}\omega^{j}},
||D_{k+\frac{1}{2}}\delta\Psi_k||_{s,\ast,T}\lesssim||\delta\Psi_k||_{4,\ast,T}||R_k||_{s,\ast,T}+||\delta\Psi_k||_{s,\ast,T}||R_k||_{4,\ast,T}&+||\delta\Psi_k||_{4,\ast,T}||R_k||_{4,\ast,T}||\Psi^a+\Psi_{k+\frac{1}{2}}||_{s+2,\ast,T}.
d\lambda=-p(\lambda)\left( \frac{dF}{d\lambda}\right)^{-1}dt,\frac{dF(\lambda_{n}(t))}{d\lambda}=-\varphi(1,\lambda_{n}(t))(\Phi_{+}(x,\lambda_{n}(t)),\overline{\Phi_{-}(x,\lambda_{n}(t))})
\frac{x}{y}+ \frac{y}{x}= \frac{\sqrt{2}- 1}{\sqrt{2}+ 1}+ \frac{\sqrt{2}+ 1}{\sqrt{2}- 1}= \frac{( \sqrt{2}- 1 )^{2}}{( \sqrt{2}+ 1 ) ( \sqrt{2}- 1 )}+ \frac{( \sqrt{2}+ 1 )^{2}}{( \sqrt{2}- 1 ) ( \sqrt{2}+ 1 )}
\prod_{k=1}^\infty (1+\alpha^{-1}X^{2k-1})(1-X^{2k})=\prod_{k=0}^\infty\frac{1}{1+\alpha X^{2k+1}}\sum_{k=-\infty}^\infty X^{k^2}\alpha^{k}=\prod_{k=1}^\infty\frac{1}{1+\alpha X^{2k-1}}\sum_{k=-\infty}^\infty X^{k^2}\alpha^{k}.
n_{2r-1,\ell}(c,d) =-c\left(\dfrac{\ell^{2r}-1}{24}\right)-\ell\cdot d\left(\dfrac{\ell^{2r-2}-1}{24}\right). n_{2r,\ell}(c,d) =-c\left(\dfrac{\ell^{2r}-1}{24}\right)-\ell\cdot d\left(\dfrac{\ell^{2r}-1}{24}\right) .
\Theta_{}=\sum_{i=1}^{k-1}\Theta_{h_i}^2+2\sum_{i=1}^{w}\Theta_{x_i}\Theta_{x^*_i}+\sum_{i=1}^{k-1}\sum_{j=1}^{w}\beta_{\bar0j}(h_i)\Theta_{h_i}-\sum_{i=1}^{k-1}\sum_{j=1}^{\ell}\beta_{\bar1j}(h_i)\Theta_{h_i}+2\sum_{i=1}^{\ell}\Theta_{y_i}\Theta_{y^*_i}.
\,\,\mathbb M\subseteq\mathbb N\,\,\,\,t_m\in K\,\,\,\,m\in\mathbb M &\,\,t_m\to t_0\,\,\,\,a,b\,\,\,\,0<a<b<2\pi &\,\,[a/m,b/m]\cap(K-t_m)\cap(t_m-K)\neq\emptyset &\,\,m\in\mathbb M.
\ll\frac{F_{max}^2\varphi(VP_f)^2X(\log R)^2}{\varphi(V)(VP_f)^2\log X}\biggl(\sum_{p>D_0}\frac{\varphi(p)^4}{g(p)^2p^4}\biggr)\biggl(\sum_{\substack{r<R (r,VP_f)=1}}\frac{\varphi(r)^2}{g(r)r^2}\biggr)^{k-1}&\ll\frac{F_{max}^2\varphi(VP_f)^{k+1}X(\log X)^k}{\varphi(V)(VP_f)^{k+1}D_0}.
g(\omega,\mathbf{n},i) = \i c(\omega) \sum_l e\frac{\sqrt{\omega_0}}{2\pi}\Pi(\mathbf{n})_{i,l}(\psi_{0}| \mathbf{x}|\psi_1)_l (Kf)(\omega,\mathbf{n},i) = \omega \sum_l \Pi(n)_{il}f(\omega,\mathbf{n},l).
{\cal F}= \frac{1}{2}\sum_{n_{1}\geq n_{2}\geq \cdots \geq n_{N}\geq 0}^{\infty}\det \left[ d_{n_{j}+ N - j , i}\right] \det \left[ t_{i}^{n_{j}+ N - j}+ t_{i}^{- ( n_{j}+ N - j )}- \delta_{j N}\delta_{n_{N}0}\right] .
X=\left\{\omega=\sum_{k\in\mathbf{Z},\k\neq0}e^{ik\alpha x}\omega_{k}\left( y\right) ,\ \Vert\omega\Vert_{X}^{2}=\Vert\frac{1}{\sqrt{K_{2}\left( y\right)}}\omega\Vert_{L^{2}}^{2}<\infty\right\}.
\left(\left(g_1'\right)^2,\left(g_2'\right)^2 \right)=({\lambda_1}^2 \cdot \operatorname{Id}+{l_1}^2 + 2 \lambda_1 l_1,{\lambda_2}^2 \cdot \operatorname{Id}+{l_2}^2 + 2 \lambda_2 l_2 ).
y'(0) =\frac{d}{dx}(b_1xE_{\alpha, 2}(\lambda x^{\alpha}))|_{x=0}+ \frac{d}{dx}\left(\int_0^x (x-t)^{\alpha-1}E_{\alpha, \alpha}[\lambda(x-t)^{\alpha}]f(t)dt\right)\Big|_{x=0}
S = -\frac{1}{2\kappa_P^2}\int d^4 x\sqrt{-g}\left[\frac{1}{2}R+\frac{1}{4}(\nabla \varphi)^2 +\frac{3}{4}(\nabla \beta)^2 +\frac{q_5}{2}e^{(\beta -\varphi)}(\nabla z)^2 \right] \; ,
d=\frac{1}{4K}\log\bigg[\frac{1}{3}\underline{R}\dot{R}(T)-\bigg(\frac{M^{2}}{6\underline{R}K}-\frac{M^{2}}{4K}\bigg)\bigg]-\frac{1}{4K}\log\bigg(\frac{1}{4}\frac{M^{2}}{K}+\frac{M^{2}}{2\underline{R}K}\bigg).
480\left\{\widehat{A}(TX){\rm det}^{\frac{1}{2}}{\rm cosh}(\frac{\sqrt{-1}}{4 \pi}R^V)\right\}^{(12)}=\left\{\widehat{A}(TX){\rm det}^{\frac{1}{2}}{\rm cosh}(\frac{\sqrt{-1}}{4 \pi}R^V)\right. &\left.\cdot{\rm ch}(\widetilde{TX}+2\wedge^2\widetilde{V_C}-\widetilde{V_C}\otimes \widetilde{V_C}+\widetilde{V_C})\right\}^{(12)}.
E_2(i,m)=\underbrace{\frac{1}{e\sqrt{i+1}}}_{\geq\frac{1}{e\sqrt{m+1}}}\underbrace{\Big(\frac{e^{1-\frac{1}{A(m+1)}}}{2A(m+1)}\Big)^{i+1}}_{\geq\Big(\frac{1}{2A(m+1)}\Big)^{i+1}}\geq\Big(\frac{1}{2A(m+1)}\Big)^{i+2}
\sum_{i=1}^{K}\sum_{j=1}^{S}I^{[i,j]}= \sum_{i=1}^{K}\sum_{j=1}^{S}\sum_{s=1}^{S}\eta^{[i,j]}\mathsf{SNR}=S\sum_{i=1}^{K}\sum_{j=1}^{S}\eta^{[i,j]}\mathsf{SNR}.
A^{0}(\partial_{x}^{\alpha}u)_{t}-B^{ij}\partial_{i}\partial_{j}(\partial_{x}^{\alpha}u) =A^{0}\partial_{x}^{\alpha}\left((A^{0})^{-1}[f-A^{i}\partial_{i}u-Du]\right) &+A^{0}G_{\alpha}((A^{0})^{-1}B^{ij},\partial_{i}\partial_{j}u).
l''(t_0) =-\frac{1}{2}\int^1_0\bigg(\frac{9}{2}U_{\xi^4}^2+3(\varphi^2-\epsilon)U_{\xi^3}^2+\frac{(\varphi^2-\epsilon)^2}{2}U_{\xi^2}^2 \qquad\qquad\qquad\qquad\qquad\qquad\qquad+\frac{(\varphi^2-4\epsilon)(2\varphi^2+\epsilon)^2}{162}U_{\xi}^2\bigg)gdp.
a_{i,u}^k = a_i^k + u q_i^k,0\le u \le L_i^k,L_i^k = [1/q_i^k],q_i^k = \theta_i^k (y_i^k)^{-\frac{2}{\alpha}},\theta_i^k = \left(y_i^k \right)^{-\frac{8}{\alpha}},y_i^k = f_p(a_i^k).
\theta_{m,p}(f)\bullet g-\theta_{m+n-1,p+q}(f\bullet g) =(-1)^{m(q-n-1)+p+q}d(f\otimes g\otimes 1) g\bullet \theta_{m,p}(f)-\theta_{m+n-1, p+q}(g\bullet f) =(-1)^{(p+1)(q-n-1)+p+q}d(f\otimes g\otimes 1).
\phi \left(t_{n+1}, \left[A^{a,m}_{i,\sqrt{T}}\right]\right)-\phi(t_n, [m]) &= \sum_{j: i\in j}\frac{a(j)}{\hat a(i)}\int \left( \phi\left(t_{n+1}, x-\frac{e_{j^c}}{\sqrt T}\right)-\phi(t_n,x) \right) m(dx).
S_{GB}^{volume}= 2 (e^{2U})'|_{r=r_h})^{-1}\left\{\left[(e^{2U})'\left(1 - (e^{U}R')^2\right)\right]_{r=r_h}- \left[(e^{2U})'\left(1 - (e^{U}R')^2\right)\right]_{r = r_0}\right\}\ .
n\left[ \frac{\alpha_{2}+\beta_{2}\left( -1\right)^{n}}{\alpha_{1}+\beta_{1}\left( -1\right)^{n}}\right] -k\left[ \frac{\alpha_{2}+\beta_{2}\left( -1\right)^{k}}{\alpha_{1}+\beta_{1}\left( -1\right)^{k}}\right] =0n\neq k.
\exp_{\widetilde{U}}({\rm Ad}(\exp_{\widetilde{U}}( \sum_{i=1}^p \lambda_i (t) \widetilde{X}_i)) \sum_{j=1}^q \mu_j (s) \widetilde{Y}_j)= \exp_{\widetilde{U}}(\sum_{j=1}^q \mu_j (s) \widetilde{Y}_j + [ \sum_{i=1}^p \lambda_i (t) \widetilde{X}_i, \, \sum_{j=1}^q \mu_j (s) \widetilde{Y}_j]).
e_i :=\begin{cases}Sw_{D_{1,i}}(f) - Sw_{pr_i^{-1}(x_i)}(f) \mbox{if $x_i$ is of type (I),}Sw_{D_{1,i}}(f) + Sw_{D_{2,i}}(f) - Sw_{pr_i^{-1}(x_i)}(f) \mbox{if $x_i$ is of type (II) ,}\end{cases}
{\boldsymbol\Sigma}^{a}{}_{b}:=\frac{\hbar}{i}{\mathbf L}^{a}{}_{b}=\frac{\hbar}{i}\varphi^{a}{}_{K}\frac{\partial}{\partial \varphi^{b}{}_{K}}, \widehat{{\boldsymbol \Sigma}}^{A}{}_{B}:=\frac{\hbar}{i}{\mathbf R}^{A}{}_{B}=\frac{\hbar}{i}\varphi^{m}{}_{B}\frac{\partial}{\partial \varphi^{m}{}_{A}}.
1 \le \lim_{k\to \infty}\frac{\gamma_{\parallel}'(h_k) + |\gamma_\perp'(h_k)| \ (|\gamma_\perp(h_k)| + |v_\perp|)}{|y_k-x|}= \lim_{k\to \infty}\frac{h_k}{|y_k-x|}\left( 1 + \frac{|v_\perp|}{r_0}\right)
\int_{D_j}|F_{m,\epsilon,j}-(1-v'_{\epsilon}(\Psi_m)){fF^{1+\delta}}|^{2}e^{v_{\epsilon}(\Psi_m)-\varphi_{m}-\Psi_m}&\leq\int_{D_j}(v''_{\epsilon}(\Psi_m))|fF^{1+\delta}|^2e^{-\phi-\varphi_{m}-\Psi_m}.
V(x) -V(\theta(x))+\frac{1}{2\rho}(H\mu(\theta(x))-V'(\theta(x)))^{2}\ge \frac{2\rho-1}{4}(\theta(x)-x)^{2}+(x-\theta(x))V'(\theta(x)) +\frac{1}{2\rho}(H\mu(\theta(x))-V'(\theta(x)))^{2}=- (1/4)(\theta(x)-x)^{2}+ (x-\theta(x))H\mu(\theta(x))+ \frac{1}{2\rho}\left[\rho(x-\theta(x)) + V'(\theta(x))-H\mu(\theta(x)) \right]^{2}
G(p)_{\mu \nu}=-i \left[ \int_0^{\infty}ds \frac{\rho_1(s)}{p^2-s^2+i \epsilon}{\cal P}_{\mu \nu}(p)-iM\epsilon_{\mu \nu \rho}\frac{p^{\rho}}{p^2}\int_0^{\infty}ds \frac{\rho_2(s)}{p^2-s^2+i\epsilon}\right]
\mathbf{z}_{\phi,slow}= \mathrm{sign}\left(\left(\mathbf{p}_{min}\times \mathbf{\dot{p}}_d\right) \cdot \mathbf{z}_I\right) \mathbf{z}_I \mathbf{z}_{\phi,fast}= \mathrm{sign}\left((\mathbf{\dot{p}}_o \times \mathbf{\dot{p}}_d) \cdot \mathbf{z}_I\right) \mathbf{z}_I
\underset{\begin{array}{c}s=s_{m,n}\end{array}}{\mathrm{Res}}\zeta_{U}\left( s,z\right)= \frac{(a-1)^{\tfrac{2n\pi i}{\log a}}}{\log a}\left(z+\frac{1}{1-a}\right)^m\frac{1}{m!}\prod_{j=1}^{m}\left(j-\frac{2n\pi i}{\log a}\right).
w(t)= T_A(t)w+\int_0^tT_A(t-s)[L_\Lambda x(s)+\bar{L}u_1(s)]ds+\Phi_{A,B}(t)u_2 = T_A(t)w+\int_0^tT_A(t-s)[L_\Lambda w_s+\bar{L}w(s)]ds+\Phi_{A,B}(t)v.
b(x_0\otimes x_1\otimes \dots\otimes x_n) = x_0x_1\otimes x_2\otimes \dots x_n &+\sum_{0}^{n-1}(-1)^i x_0\otimes \dots\otimes x_ix_{i+1}\otimes \dots\otimes x_n &+(-1)^n x_nx_0\otimes x_2\otimes \dots \otimes x_{n-1}.
F_d^0(r,t) =\int_0^r s^{1-d}[\nu_d(s)]^{-1}\left(\int_0^s \tau^{d-1}\nu_d(\tau)(\partial_t u_*)(\tau,t)\,d\tau\right)\,ds =\int_0^r s^{1-d}[\nu_d(s)]^{-1}\left(\int_0^s \tau^{d-1}\nu_d(\tau) \biggr[(\partial_tu_*)(0,t)+F_d^1(\tau,t)\biggr]\,d\tau\right)\,ds =(\partial_tu_*)(0,t)F_d(r)+G_d(r,t)
\left\langle \prod_{i=1}^{k}\prod_{a=1}^{n_i}(x_a^{(i)}+t)-p_{n_1,\ldots,n_k}({\boldsymbol{y}},t) \right\rangle,\big\langle \Delta_{n_1,\ldots,n_k}({\boldsymbol{x}},t) -t^n \big\rangle
\eqref{eq3_9}= \sum_{k=h}^N \sum_{\substack{w_0,w_1,\ldots,w_h\in \{e_0,e_1,e_0'\}^* \deg_{e_0'}(w_0w_1\cdots w_h) = k-h \deg (w_0)= j_1-1 \deg (e_1'w_m)=j_{m+1}-j_m \ (1\le m\le h) j_{h+1}=N+1}}\varphi \Big( w_0 \prod_{m=1}^h \big(e_1' w_m \big) \Big).
R^{n+1/2}a(z^{-n-1})^* T_R = T_R(a(s^{1/2}z^{-1}(sz^{-1}+ \overline{w})^n)^* \otimes 1) + T_R(\Gamma \otimes a(\tilde W_{\phi_t}z^{-n-1})^*),
\overline{\Pi}(U)\cong C\left\langle \alpha_1,\cdots,\alpha_g,\beta_1,\cdots,\beta_g,\sigma_1,\cdots,\sigma_r\middle|\prod_{i=1}^{g}[\alpha_i,\beta_i]\prod_{j=1}^{r}\sigma_j=1\right\rangle .
\varphi_t(u)=E[e^{iuX(t)}] =\sum_{n=0}^{\infty}P[N(t)=n]E[e^{iuS_n}]=\sum_{n=0}^{\infty}\frac{e^{-\Lambda_t}\Lambda_t^n}{n!}(E[e^{iuJ}])^n &=e^{\Lambda_t(E[e^{iuJ}]-1)}=e^{\Lambda_t(\varphi_J(u)-1)}.
U_{m , k}= \Delta^{2}Z_{m , k + 1}+ \Delta^{1}Z_{m + 1 , k}+ u^{i}C_{i ( n - m ) ( n - k )}({\frac{\partial}{\partial u_{1}^{*}}})^{n - m}Q_{1}({\frac{\partial}{\partial u_{2}^{*}}})^{n - k}Q_{2},
c^{\pm\nu}_{2r}(h^2) M^{(1)}_{\pm\nu}(z,h)=\sum^{+\infty}_{l=-\infty}(-1)^lc^{\pm\nu}_{2l}(h^2)J_{l-r}(he^{-z})J_{\pm\nu+l+r}(he^{z})
G(z+\tau\,|\,\tau) = (2\pi)^{\frac{(\tau-1)}{2}}\tau^{-\frac{1}{2}}e^{\gamma\frac{(z-z^2)}{2\tau}}\prod\limits_{m=1}^\infty(m\tau)^{z-1}e^{\frac{(z^2-z)}{2m\tau}}\frac{\Gamma(1+m\tau)}{\Gamma(z+m\tau)},
\sum_{\delta\in\{0,1\}^{n}}u_{\delta}\prod_{i=1}^{n}g_{i}^{\delta_{i}}(x_{i})=\sum_{\delta\in\{0,1\}^{n}}v_{\delta}f_{n}^{\delta_{n}}(x_{n})\prod_{i=1}^{n-1}f_{i}^{\delta_{i}}(x_{i}-x_{n}).
\iota_{{\sf j}W}(\iota_{{\sf j}X}\Omega_L)\vert_{{\cal E}_L}=\iota_{{\sf j}W}{\sf d_h}\rho^X\vert_{{\cal E}_L},\rho^X=\sum_{i,j=1}^n\rho^{ij}\vartheta^j_i\wedge \nu^{ji}.
\sum\limits_{i=0}^{m-1}\Big[\big(b_n(v_i(k))_{\bar t}-b_n(\tilde v_i(k))_{\bar t}\big)\big(v_i(k)-\tilde v_i(k)\big)+\big(v_{ix}(k) -\tilde v_{ix}(k)\big)^2\Big]=0.
P_{s_i}^{mwtp}=\sqrt{\frac{A_{s_i,r_j}\left( 2+C_{s_i,r_j}+\sqrt{C_{s_i,r_j}^2+8C_{s_i,r_j}}\right)}{2p_{th}^{s_i}}}, P_{r_j}^{mwtp}=f\left( P_{s_i}^{mwtp}\right).
\begin{array}{c}\dot{t}_0+[T_0,t_0]+[T_1,t_1]+[T_2,t_2]+[T_3,t_3]=0, \dot{t}_1+[T_0,t_1]-[T_1,t_0]+[T_2,t_3]-[T_3,t_2]=0, \dot{t}_2+[T_0,t_2]-[T_1,t_3]-[T_2,t_0]+[T_3,t_1]=0, \dot{t}_3+[T_0,t_3]+[T_1,t_2]-[T_2,t_1]-[T_3,t_0]=0.\end{array}
|D_4(S)| \geq 1 + |D_{\{2,4 \}}(S)| = 1+ |D_{\{4\}}(S) \cup [ D_{\{2 \}}(S) \setminus D_{\{4\}}(S)]| \geq 1 + n-4 +\frac{n}{3}+ \frac{5}{3}= \frac{4}{3}n - \frac{4}{3}.
f(z)= C(\pi,m)\int_0^1\int_{B^n}\frac{(1-|\zeta |^2)^m Df(\zeta)}{(1-r<z,\zeta> )^{n+1+m}}d\nu(\zeta)dr &=C(\pi,m)\int_{B^n}(1-|\zeta |^2)^m Df(\zeta)\int_0^1\frac{dr}{(1-r<z,\zeta> )^{n+1+m}}d\nu(\zeta) &=\widetilde C(\pi,m)\int_{B^n}\frac{(1-|\zeta |^2)^m((1-<z,\zeta>)^{n+\alpha}-1)}{<z,\zeta>(1-<z,\zeta>)^{n+m}}Df(\zeta)d\nu(\zeta).
\sum_{t=1}^k \left(\frac{(d+1)(d-\ell_t+t)}{d(k+1)}-\ell_{t+1}+\ell_t \right) = -\frac{(d+1)}{k+1}+\ell_1 + \frac{d+1}{d(k+1)}~\sum_{t=1}^k (t-\ell_t) &\le -\frac{(d+1)}{k+1}+\ell_1.
\alpha_0 := -\frac{1}{2}a_1 + 2a_2 + 2a_{-2}+ v_{(1,2)}\in M_0^{(a_1)}\beta_0 := -\frac{1}{3}(a_1 + a_{-1}+ 2a_2 + 2a_{-2}) + v_{(1,2)}\in M_{\frac{1}{2^2}}^{(a_1)}\alpha_1 := -\frac{1}{2}a_1 + 2a_3 + 2a_{-3}+ v_{(1,3)}\in M_0^{(a_1)}\beta_1 := -\frac{1}{3}(a_1 + a_{-1}+ 2a_{-2}+ 2a_{-3}) + v_{(1,3)}\in M_{\frac{1}{2^2}}^{(a_1)}.
\P\left(\bigcap_{Q\in{\cal Q}:\, Q_{Y}=\tilde{Q}_{Y}}\left\{\hat{N}(Q|\mathbf{y})<e^{nJ(Q)}\right\}\right)\:\begin{cases}=1-o(n), \mathbf{S}(\tilde{Q}_{Y};J,{\cal Q})>R \doteq e^{-n\infty}, \mbox{otherwise}\end{cases},
\bigl|[v_n,w_n] - [v_m,w_m]\bigr| \ =\ \bigl|[v_n,w_n-w_m] + [v_n - v_m,w_m]\bigr| \ \leq\ \|v_n\| \|w_n - w_m\| + \|v_n-v_m\|\|w_m\|.
{\bf E}\bigl[\beta_{2,1}(a, b)^q\bigr] =\frac{\Gamma_2(q+b_0\,|\,a)}{\Gamma_2(b_0\,|\,a)}\frac{\Gamma_2(b_0+b_1\,|\,a)}{\Gamma_2(q+b_0+b_1\,|\,a)}.
\begin{aligned}\mu(B_i) = \sum_{I\in\mathcal S}\lambda_I^s (g_I)_*(\mu)(B_i) = \sum_{I\in\mathcal S}\lambda_I^s (g_I)_*(\mu)(B_i\cap \bar W_I) \le C(\delta)\max_{I\in\mathcal S, \bar W_I\cap B_i\neq \phi}\lambda_I^s. \end{aligned}
{L}_{n}(\xi) = \left( \begin{array}{c}{1 +{N(n)\over \Delta}\xi}\qquad{- i \kappa^{1 \over 2}\phi(n)\xi}{i \kappa^{1 \over 2}\psi(n)\xi}\qquad{{1\over\Delta}\xi}\end{array}\right).\ll{nl3l}
\ddot{\chi}_{m}+ \frac{3 \Delta+14}{3 \Delta+8}t^{-1}\dot{\chi}_{m}+ \left[ p^{2}\left( H_{0}t \right)^{- \frac{4}{3 \Delta+8}}+ \frac{m^{2}}{H_{0}^{2}}t^{-2}\right] \chi_{m}= 0.
\left\langle \prod_{b = 1}^{N}\; e^{- 2 \sum_{b = 1}^{N}\beta^{( b )}\epsilon^{( b )}\int_{\Lambda}d^{2}x \; t ( x ) \; : \cos \big [ 2 \sqrt{\pi}\big ( \theta_{1}^{( b )}( x ) + \theta_{2}^{( b )}( x ) \big ) \big ] :_{C^{1}- C^{2}, C^{2}}}\right\rangle_{C^{1}- C^{2}, C^{2}}