harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
J(k,c)=[\begin{matrix}\frac{\partial\dot{k}}{\partial k}&\frac{\partial\dot{k}}{\partial c}\\ \frac{\partial\dot{c}}{\partial k}&\frac{\partial\dot{c}}{\partial c}\end{matrix}]=[\begin{matrix}f_{k}(k)-(n+\delta)&-1\\ \frac{1}{\sigma}f_{kk}(k)\cdot c&\frac{1}{\sigma}[f_{k}(k)-\delta-\rho]\end{matrix}]
H^{-(0)}_0=\mu[\frac{(\alpha-\imath)}{2{\alpha}}]\exp(-i\frac{\partial}{{\partial}{\alpha}}),\quad\widetilde{H}^{-(1/2)}_0=\mu[\frac{(\alpha-\frac{3i}{2})}{2({\alpha}-\frac{i}{2})}]\exp(-i\frac{\partial}{{\partial}{\alpha}}),
\sum_{n=0}^{\infty}\sum_{m=-\infty}^{\infty}C(m,n)z^{m}q^{n}=\dfrac{1}{(q;q^{2})_{\infty}(zq^{2};q^{2})_{\infty}(z^{-1}q^{2};q^{2})_{\infty}}.
dt^{\prime}=\sqrt{\frac{(1+2\frac{V(r^{\prime})}{c^{2}})}{1+2\frac{V(r^{\prime\prime})}{c^{2}}}}dt^{\prime\prime}\approx(1+\frac{V(r^{\prime})-V(r^{\prime\prime})}{c^{2}})dt^{\prime\prime}
m^{GP}=\Delta_\eta(k):=\frac{1}{2}\left( \begin{array}{cc}\chi_\eta(k)+\chi^{-1}_\eta(k) i\left(\chi_\eta(k)-\chi_\eta^{-1}(k)\right) -i\left(\chi_\eta(k)-\chi_{\eta}^{-1}(k)\right) \chi_\eta(k)+\chi_\eta^{-1}(k) \end{array}\right),
u(t) =V_0(t)+u(t)-V_0(t) &=V_0(t)+U_1'(t) &=V_0(t)+V_1(t)' + (V_1(t)-U_1(t))' &=V_0(t)+V_1'(t) + U_2''(t) &\,\,\,\vdots &=\sum_{\ell=0}^m \frac{d^\ell}{dt^\ell}V_{\ell}(t)+\frac{d^{m+1}}{dt^{m+1}}U_{m+1}(t).
\mathrm{p g h}\left( V_{\alpha ( \lambda )}\right) = \mathrm{p g h}\left( A_{\alpha}^{( \lambda )}\right) = 0 , \mathrm{p g h}\left( \eta_{( \lambda )}\right) = \mathrm{p g h}\left( C^{( \lambda )}\right) = 1 ,
\langle \lfloor p_{i}; h_{i}\rceil, \lfloor p_{j}; h_{j}\rceil \rangle_{B_{1}}= \langle p_{i}, p_{j}\rangle_{\mathcal{Q}_{W_{1}}}= \langle \phi(p_{i}), \phi(p_{j}) \rangle_{\mathcal{Q}_{W_{2}}}= \langle \lfloor \phi(p_{i}); h_{i}\rceil, \lfloor \phi(p_{j}); h_{j}\rceil \rangle_{B_{2}}.
\beta:= \inf\{\pi^{\top}x\, | \ \A^{i_0}x\succeq_{\mathcal{L}_{m_{i_0}}}b^{i_0}\}, &\sup\{(b^{i_0})^{\top}y^{i_0}\, | \ (y^{i_0})^{\top}A^{i_0}= \pi^{\top}, \y^{i_0}\in \mathcal{L}^*_{m_{i_0}}\}.
V_{G}( \phi_{0}) ={\frac{1}{V}}\sum_{n = 0}^{\infty}\frac{( - i )^{n}}{n !}\int_{- \infty}^{0}d t_{n}. . . \int_{- \infty}^{0}d t_{n}\langle 0 | T ( H_{I}( 0 ) H_{I}( t_{1}) . . . H_{I}( t_{n}) ) | 0{\rangle}_{c o n n}
\lim_{L \to +\infty}\big(q^{L+1}\xi\big)^{\mu}\frac{\big(q^{\lambda+L+2}\xi/x, q^{L+1}\xi\alpha_1, q^L\xi\alpha_2 ;q\big)_{\infty}}{\big(q^{L+1}\xi/x, q^{L+1}\xi\beta_1, q^{L+1}\xi\beta_2 ;q\big)_{\infty}}= 0 .
\widetilde{J}^{-1}(\mu) = \widetilde{J}_1^{-1}(c_1) \cap \widetilde{J}_2^{-1}(c_2) = \{(\theta,\phi,\psi,p_\theta,p_\phi,p_\psi) \in T^*\widetilde{Q}: p_\theta = \tfrac{c_1}{E(\phi)}+c_2 \ \p_\psi=c_2 \}.
Pr(X>\frac{n}{2})\le\frac{2}{n}E[X]\le\frac{1}{n}\sum_{i=3}^{g-1}p^{i}n^{i}=\frac{1}{n}\sum_{i=3}^{g-1}n^{\frac{i}{g}}\le\frac{g}{n}n^{\frac{g-1}{g}}=gn^{-\frac{1}{g}}=o(1).
\{a_{1,1}^{x_{1,1}},\ldots a_{1,b_{1}}^{x_{1,b_{1}}};\ldots;a_{i,1}^{x_{i,1}},a_{i,2}^{x_{i,2}}\ldots ,a_{i,b_{i}}^{x_{i,b_{i}}},\ldots\}:=\prod_{i=1}^{h-1}\prod_{j=1}^{b_{i}}\left( \frac{\sin \pi \tau(a_{i,j}+i)}{\sin \pi \tau i}\right)^{x_{i,j}}\,.
\epsilon([ m_t^{-1}\gamma]) = \epsilon([ g_t^{-1}m^{-1}g_t \gamma]) = \epsilon([ g_t^{-1}m m^{-1}g_t \gamma]) + \epsilon([ g_t^{-1}])\epsilon([m^{-1}g_t\gamma]) &= \epsilon([\gamma])+ \epsilon([g_t^{-1}])\epsilon([m^{-1}g_t\gamma]) = 0.
\int_C \frac{d\nu \: \nu}{i 4\sqrt{2}\sin \pi\nu}\left[ J_\nu(z_1) + J_{-\nu}(z_1) \right] \left[ J_\nu(z_2) + J_{-\nu}(z_2) \right] = \; \; \; \; \; \; \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;
\tilde{\rho}(U, x) = P_{\tilde{\pi}}(\theta_1 < U|x) = \frac{\int^{U}\ldots \int \tilde{L}(\theta|x) \pi(\theta) d\theta_d \ldots d\theta_1}{\int \ldots \int \tilde{L}(\theta|x) \pi(\theta) d\theta_d \ldots d\theta_1}= \alpha\,.
\begin{cases}(ce+1)x^2+(2cde+2d)xy+(c^2e+cd^2e+c+d^2)y^2=ux+cvy,\cr(de+1)x^2+(2ce+2d^2e+d-2)xy+(2cde+d^3e-c-d)y^2=vx+(u+dv)y.\end{cases}
\lim_{t \to \infty}\frac{1}{t}\sum_{m=0}^{t-1}\mathbf{1}_{\{X(m)=s\}}=\lim_{t \to \infty}\frac{1}{t}\sum_{m=0}^{t-1}\prod_{h=1}^r \mathbf{1}_{\{X_{C_h}(m)=s_{C_h}\}}=\prod_{h=1}^r \mu_{C_h}(s_{C_h})=\mu(s),
\bar{h}_n^*(x) = \begin{cases}h_n^*(x), x\in K_n h_n^*(\Delta_n), x\in K_n^c,\end{cases}\intertext{and}\bar{f}_n^*(x) = \begin{cases}f_n^*(x), x\in K_n f_n^*(\Delta_n), x\in K_n^c.\end{cases}
(a)=2\pi\int_{\lambda\varepsilon_{n}}^{\frac{T_{\varepsilon_{n}}}{20d_{k}}}p\frac{f_{0}^{2}}{r}dr= \underbrace{2\pi\int_{\lambda\varepsilon_{n}}^{\frac{T_{\varepsilon_{n}}}{20d_{k}}}(p-p_{0})\frac{f_{0}^{2}}{r}dr}_{(1)}+ \underbrace{2\pi p_{0}\int_{\lambda\varepsilon_{n}}^{\frac{T_{\varepsilon_{n}}}{20d_{k}}}\frac{f_{0}^{2}}{r}dr}_{(2)}.
1 =g(y_i)g(x_i)-qg(x_i)g(y_i) &=Y_iX_i+a_iY_i+d_iX_i+a_id_i-q(X_iY_i+a_iY_i+d_iX_i+a_id_i) &=Y_iX_i-qX_iY_i+(1-q)[a_i Y_i+d_i X_i+a_id_i].
\widehat{(\partial_{\Sigma_0, \epsilon}^{*})^{\otimes 2}\overline{u}_n}(\xi, \eta) = -4 \xi_u \eta_u \hat{\delta}_{\epsilon}(2 \xi_v) \hat{\delta}_{\epsilon}(2 \eta_v) \int_{\mathbb{R}^{2}}\overline{u}_n(s - s') e^{i(\xi_u - \xi_v)s}e^{i(\eta_u - \eta_v)s}\, \mathrm{d}s \, \mathrm{d}s'.
\ell_n =\lfloor\sqrt[3]{n-d+d}\rfloor=\left\lfloor \sqrt[3]{n-d}\,\sqrt[3]{1+\frac{d}{n-d}}\right\rfloor &=\left\lfloor s\,\sqrt[3]{1+\frac{d}{n-d}}\right\rfloor=\left\lfloor s\,\Big(1+\frac{1}{3}n^{-1/3}\,+\,{\rm o}(n^{-1/3})\Big)\right\rfloor &= s+\left\lfloor \frac{1}{3}\,+\,{\rm o}(1)\right\rfloor
{n\brack i}_r:= \begin{cases}\frac{2}{2n+r}{2n+r\choose 2i+r}B_{2(n-i)}, n\ge\operatorname{max}\{0,\lfloor-r/2\rfloor+1\}, \frac{2}{2i+r}{-2i-r\choose -2n-r}B_{2(n-i)}, 0\le n\le\lfloor-(r+1)/2\rfloor.\end{cases}
\nabla \Phi^g(x,y)=\begin{bmatrix}e^{y_1}\nabla g_1(x) \dots e^{y_m}\nabla g_m(x) e^{y_1}g_1(x) \dots 0 . \dots . 0 \dots e^{y_m}g_m(x) \end{bmatrix}=\varphi^g(x) \times \begin{bmatrix}e^{y_1}\dots 0 . \dots . 0 \dots e^{y_m}\end{bmatrix}\,.
-t^{\frac{N+2A_1}{2}}(\partial_r^2u_{1,1})(x,t)\ge -C, -t^{\frac{N+2A_1}{2}}(\partial_{\theta_\alpha}\partial_{\theta_\beta}u_{1,1})(x,t)\ge\frac{q_NM_{1,1}}{2}U_1(r_*)\delta_{\alpha\beta}-\epsilon, -t^{\frac{N+2A_1}{2}}(\partial_r\partial_{\theta_j}u_{1,1})(x,t)\ge- C|\theta_\alpha|U_1'(r)+O(t^{-1})\ge-\epsilon,
\mu (\Delta ) =\sum_{i=1}^{k}\sum_{\substack{\sigma \in \mathcal{A}^{n}\varrho^{-n}(a-S_{\sigma}(0))=a_{i}}}p_{\sigma}\mu [a_i, a_i + \ell_n(\Delta)] =\sum_{i=1}^{k}\mu [a_i, a_i + \ell_n(\Delta)] \sum_{\substack{\sigma \in \mathcal{A}^{n}\varrho^{-n}(a-S_{\sigma}(0))=a_{i}}}p_{\sigma}
\zeta\left(\frac32\right) =\frac{2}{\sqrt{\pi}}\int_{0}^\infty \frac{t^{1/2}}{e^t-1}\mathrm{d}t =\frac{2}{\sqrt{\pi}}\left(\int_{0}^1+\int_{1}^\infty \frac{t^{1/2}}{e^t-1}\mathrm{d}t\right)=\frac{2}{\sqrt{\pi}}(\mathrm{J}_1+\mathrm{J}_2).
a_h(u_h,v_h) = \int_{Q}(\partial_{t}u_h v_h + \theta h \partial_{t}u_h \partial_{t}v_h + \nabla_x u_h \nabla_x v_h + \theta h \nabla_x u_h \cdot \nabla_x \partial_{t}v_h)\, dxdt, l_h(v_h) = \int_{Q}f \left[ v_h + \theta h \partial_{t}v_h \right]\, dxdt,
\frac{d}{dt}\big(\frac{p_x}{\sqrt{1+a(t)^2y^2}}+\frac{a(t)y p_z}{\sqrt{1+a(t)^2y^2}}\big)\big\vert_{p_z=a(t)y p_x}=mg\frac{k_x+a(t)y k_z}{\sqrt{1+a(t)^2 y^2}}+\epsilon\frac{\dot y}{\sqrt{1+a(t)^2 y^2}}
\frac{\partial\rho\langle v_{r}\rangle}{\partial t}+\frac{\partial\rho\langle v_{r}^{2}\rangle}{\partial r}+\frac{\rho}{r}[2\langle v_{r}^{2}\rangle-\langle v_{\theta}^{2}\rangle-\langle v_{\phi}^{2}\rangle]+\rho\frac{\partial\Phi}{\partial r}=0
\begin{array}{ccc}\Psi_I = (w_2)^{\frac{1}{2}}\Psi~~; F^{\mu\nu}_I = (w_3)^{\frac{1}{2}}F^{\mu\nu}~~; \end{array}\quad \begin{array}{ccc}m_I = \frac{w_0}{w_2}m e_I = \frac{w_1}{w_2w^{\frac{1}{2}}_3}e \end{array}
\begin{cases}V_{n,2}(y,x) \triangleq \delta \left (\frac{y^2}{2}+ U(x) \right ) + xy + C_V, V_{n,3}(\eta,y,x) \triangleq \Gamma_1(\eta) + V_{n,2}(y,x), V_{n,4}(\zeta,\eta,y,x) \triangleq \Gamma_2(\zeta,\eta) + V_{n,2}(y,x),\end{cases}
{\textbf{Y}}=\sum\nolimits_{k=1}^{K}\left(\sqrt{\alpha_k}{\textbf{h}}_k{\textbf{x}}_k^{\mathsf{H}}+\sqrt{\alpha_{k'}}{\textbf{h}}_{k'}{\textbf{x}}_{k'}^{\mathsf{H}}\right)+{\textbf{G}}^{\mathsf{H}}{\textbf{S}}+{\textbf{N}},
q_{sqL}= \frac{1}{k}\sum_{i=1}^n\sqrt{s_i^*}\leq \max_i\frac{n\sqrt{s_i}}{k}= q_U c_{sqL}= \max_i \frac{1}{k}\sqrt{s_i^*}\sum_{i=1}^n\sqrt{s_i^*}\leq \max_i \frac{ns_i^*}{k}= c_U
\partial_{+}\tilde{j}^{+}( x ) = - i [ \tilde{j}^{+}( x ) , P^{-}] = i m ( : \psi_{L}^{\dagger}\psi_{R}( x ) - \psi_{R}^{\dagger}\psi_{L}( x ) : )_{\sim}+{\frac{e}{2 \pi}}\partial_{-}\tilde{A^{-}}.
\mathbb{E}\big [(\xi_{i_1}-p)\cdots (\xi_{i_r}-p)\mid X=x\big ]= \begin{cases}0 \mbox{if~}i_r \geq x+2, q(-p)^{r-1}\mbox{if~}i_r = x+1, (-p)^r \mbox{if~}i_r \leq x.\end{cases}
a_h = \mathcal{L}^2\left( \left\{f_h \neq g_h \right\}\cap B_1 \right) \lesssim \left( \mathcal{H}^1\left( \left\{f_h \neq g_h \right\}\cap B_1 \right) \right)^2 ={\rm o}(\gamma_h) \mbox{as}h \to +\infty
Y^{(n)}(t_{n, i+1})=Y^{(n)}(t_{n, i})+ \int_{t_{n, i}}^{t_{n, i+1}}g(s, W_0^{t_{n, i}}, Y_0^{(n), t_{n, i}}) ds + B(t_{n, i+1})-B(t_{n, i}),
\int_{0}^{\infty}T(x\sqrt{t})\frac{\dot{R}(t)}{t}dt=\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty}\frac{\bar{T}(s)\Gamma(1-\frac{s}{2})}{\zeta(s)}x^{-s}ds.
\mathcal{E}(\tilde{u},-\psi^{1+q}\tilde{u}^{-q}) = \int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}(\tilde{u}(s,x)-\tilde{u}(s,y))(\psi^{1+q}(y)\tilde{u}^{-q}(s,y)-\psi^{1+q}(x)\tilde{u}^{-q}(s,x))\frac{k(x,y)}{2}dxdy \geq \frac{1}{2(q-1)}- \frac{\vartheta(q)}{2},
I(\frac{r}{2}) = \big(\frac{r}{2}\big)^{1- n}\int_{b= \frac{r}{2}}u^2|\nabla b|\leq \frac{C(n)}{V_M}D(r)\cdot \big(\frac{r}{2}\big)^{1- n}\int_{b= \frac{r}{2}}|\nabla b| &= \frac{C(n)}{V_M}D(r) I_1(r) = C(n) D(r)
\sum_{\lambda\in \mathcal{P}\atop |\lambda|_a=j}\prod_{i\geq 1}{\lambda_i-\lambda_{i+2}+1\choose \lambda_{i+1}-\lambda_{i+2}}q^{{\lambda_1\choose 2}+|\lambda|}=\frac{q^{2j\choose 2}}{\prod_{i\geq 1}(1-q^{2i})^2}.
a(\eta) = -\rho_r^6\mu^2\vartheta^2 b(\eta) =\rho_r^2\mu^2[3\rho_r^2\mu^2-1-3(|\zeta_s(0,s)|^2+|\chi_{i0}^\prime|^2)]+\frac{1}{4}\rho_r^4\vartheta^2 &c(\eta) =- \frac{1}{4}[3\rho_r^2\mu^2-1+|\zeta_s(0,s)|^2+|\chi_{i0}^\prime|^2] \,,
L = \frac{1}{4 \pi \alpha^{\prime}}\sum_{n}\left[ \epsilon ( \dot{X}_{n}^{i})^{2}- \frac{1}{\epsilon}\left( X_{n + 1}^{i}- X_{n}^{i}\right)^{2}+ 4 \pi \alpha^{\prime}B_{i j}\dot{X}_{n}^{i}\left( X_{n + 1}^{j}- X_{n}^{j}\right) \right]
\begin{cases}w_{tt}-\mathrm{e}^{2t}\Delta w + M^2 w= g(t,x), x\in \mathbb{R}^n, t>0, w(0,x)= w_0(x), x\in \mathbb{R}^n, w_t(0,x)= w_1(x), x\in \mathbb{R}^n, \end{cases}
V_1=\exp (\nu (x))\cos \bigl(\psi (x)\bigr)|x|^m,\tag*{$\textup{(\ref*{24-A-34})}_{1}$}&V_2=\exp (\nu (x))\sin \bigl(\psi (x)\bigr)|x|^m, V_3=0\tag*{$\textup{(\ref*{24-A-34})}_{2,3}$}
\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}}}}_{\mbox{one $\sum$ for each link}}.
{\cal L}={\frac{k}{2}}\varepsilon^{\alpha \beta \gamma}A_{\alpha}\partial_{\beta}A_{\gamma}+ i\Psi^{\ast}(\partial_{t}+iA_{0})\Psi -{\frac{1}{2}}\left |(\nabla - i{\bf A})\Psi \right|^{2}+{\frac{g}{2}}\left|\Psi \right|^{4}\, .
\left[\frac{1}{2\hat{j}+ (2 k + 1) \hat{I}}, \, \tau^\dagger_\theta\right] = \tau^\dagger_\theta \left(\frac{1}{2\hat{j}+ (2 k + 1) \hat{I}}- \frac{1}{2\hat{j}+ (2 (k-\theta)+1) \hat{I}}\right).
H_{l,r}^{-}=\begin{cases}H_l^{-1}H_{l+1}^{-1}\cdots H_{r}^{-1}\hbox{if}l\leq r, H_l^{-1}H_{l-1}^{-1}\cdots H_{r}^{-1}\hbox{if}l>r,\end{cases}
\langle \Upsilon^{\varepsilon_1}_{l_1 m_1}| \Upsilon^{\varepsilon_2}_{l_2 m_2}\rangle = \int_0^{2\pi}d\phi \int_0^\pi (\Upsilon^{\varepsilon_1}_{l_1 m_1})^\dagger \Upsilon^{\varepsilon_2}_{l_2 m_2}\, \sin \theta\, d\theta = \delta^{\varepsilon_1 \varepsilon_2}\, \delta_{l_1 l_2}\, \delta_{m_1 m_2}.
\begin{array}{rcl}p_0(u,v) = \sum_{i=0}^{d-1}{\binom{d-1}{i}}a_iu^{d-1-i}v^{i}, p_1(u,v) = \sum_{i=0}^{d-1}{\binom{d-1}i}b_iu^{d-1-i}v^{i}, p_2(u,v) = \sum_{i=0}^{d-1}{\binom{d-1}i}c_iu^{d-1-i}v^{i}, g(u,v) = \sum_{i=0}^{d}{\binom{d}i}g_iu^{d-i}v^{i}. \end{array}
[\begin{matrix}c_{3}c_{1}-c_{2}s_{1}s_{3}&-c_{3}s_{1}-c_{1}c_{2}s_{3}&s_{2}s_{3}\\c_{2}c_{3}s_{1}+c_{1}s_{3}&c_{3}c_{2}c_{1}-s_{3}s_{1}&-c_{3}s_{2}\\s_{1}s_{2}&c_{1}s_{2}&c_{2}\end{matrix}]
B_2(x)=-\int_{0}^{x}\df_{t+h}^\star\left( \frac{D^2f_{t+h}}{|Df_{t+h}|^3}\nu_2 \right) + \int_{0}^{x}df_{t}^\star\left( \frac{D^2f_{t}}{|Df_{t}|^3}\nu_2 \right) ,
t^3_{3\overline{2}}\eta^{\overline{2}}-\overline{t}^3_{3\overline{2}}\eta^{2}=\left(-\overline{f}^1_{12}+\overline{f}^2_{22}+\overline{f}^1_{21}-\epsilon t^2_{1\overline{1}}\right)\eta^{\overline{2}}-\left(-f^1_{12}+f^2_{22}+f^1_{21}-\epsilon\overline{t}^2_{1\overline{1}}\right)\eta^{2}.
\displaystyle g_{j , 0}( x , w ) ={\cal F}_{\xi}^{- 1}\left[ P . V . \left[ \int_{0}^{\infty}t^{\nu - j - 2}\ \tilde{d}_{- 1 - j}( x , t ; \xi / \vert \xi \vert , t ; 0 ) \d t - M_{j}( x ) \right] \vert \xi \vert^{- ( \nu - 1 )}\right] ( w ) .
\frac{{\partial [\frac{{\sigma_k (\lambda )}}{{\sigma_l (\lambda )}}]}}{{\partial \lambda_1}}\geqslant \frac{n}{k}\frac{{k - l}}{{n - l}}\frac{1}{{n - k + 1}}\sum\limits_{i = 1}^n{\frac{{\partial [\frac{{\sigma_k (\lambda )}}{{\sigma_l (\lambda )}}]}}{{\partial \lambda_i}}}, \forall 0 \leq l < k.
S_{\mathrm{r e p l i c a}}={\frac{\mathrm{l n}({\frac{L}{\epsilon}})}{\pi}}\int_{0}^{\infty}d \omega \{{\frac{2 \pi \omega}{( e^{2 \pi \omega}+ 1 )}}+ \mathrm{l n}( 1 + e^{- 2 \pi \omega}) \}.
c_4(\rho) t K \frac{K^{\alpha}}{n^\alpha}n^{2\alpha-3}\sum_{x}f'\big( \tfrac{x}{n}\big)^2 \sum_{y=1}^{K-1}y^2 a(y)^2 \leq C(f,\rho) \frac{K^{1+\alpha}t}{n^{2-\alpha}}.
|II_{r,2}| \leq C \int_{0}^{\infty}|\partial_{\xi}^{2}\left(\left(1-\chi_{\leq 1}(r \xi)\right) \xi (r\xi)^{2}\Phi_{2}(r \xi) \frac{\chi_{\leq \frac{1}{4}}(\xi)}{\log^{b-1}(\frac{1}{\xi})}\right)|\frac{d\xi}{(t-r)^{2}}&\leq \frac{C}{(t-r)^{2}\log^{b-1}(r)}, r \geq \frac{t}{2}
\sum_{k = 0}^n{\binom nk \frac{F_{jk + m}}{L_j^{k}}B_{n - k}}= \begin{cases}F_m B_n \Big(\frac{\alpha^j}{L_j}\Big), begin{align*}6pt]\frac{{L_m}}{{\sqrt 5}}B_n\Big(\frac{\alpha^j}{L_j}\Big), \end{cases}
\mathfrak{\widehat D}_k= -ka^{k-2}X^d+\frac{k(k-3)}{2}a^{k-4}X^{2d}-\frac{k(k-4)(k-5)}{6}a^{k-6}X^{3d}+\cdots +(-1)^{\frac{k}{2}-1}\,\frac{k^2}{4}a^2X^{d(\frac{k}{2}-1)}+(-1)^{\frac{k}{2}}\cdot 2\cdot X^{\frac{dk}{2}}.
\phi^{\rm rob}_0\ge \frac{7r^\diamond}{n}\stackrel{\eqref{eq:rs}}{\ge}\frac{3r_1}{n}=\frac{3\gamma_1m_1}{n}\ge \frac{\gamma_1}{K_1}\ge 2\phi_0\ \ \ \ \ \ 2 \phi^{\rm rob}_0 n \stackrel{\eqref{D4eq}}{=}D_1 - D_4.
\mu(A) = \lim_{n\to\infty}m^{-\alpha (n+1)}\nu(p_{n+1}^{-1}A) = m^{-\alpha}\lim_{n\to\infty}m^{-\alpha n}\nu(p_n^{-1}(p_n/p_{n+1})A) = m^{-\alpha}\mu(m^{-1}A),
E_P[\sum_{i=1}^n\hat{\mathbb{E}}_{t_i}[\xi_i]I_{A_i}] \leq\lim_{m\rightarrow\infty}\lim_{l\rightarrow\infty}\hat{\mathbb{E}}[\sum_{i=1}^{n}\xi_i{\varphi^{i,m}_{l}}\prod_{j=1}^{i-1}{\widetilde{\varphi}^{j,m}_l}].
\mathcal{G}_{p}^{2,\ell}(H;\mathbb{R})=\Big\{\Phi: \, H \to \mathbb{R}, \; | \Phi |_{\mathcal{G}_{p}^{2,\ell}(H;\mathbb{R})}=\sup_{u \in H}\frac{\| \Phi^{(2)}(u) \|_{\mathcal{B}(H;\mathbb{R})}}{(1+ \| u \|_{H}^{\ell-2})}<\infty\Big\},
\delta^d\,\sum_{k\in \mathbb{Z}^d}\,\sup_{\gamma \in Q_\delta}\,|\psi(\xi-\delta k-\gamma)| \le \sum_{k\in \mathbb{Z}^d}\,C_1\, \prod_{i=1}^d\,\delta\, \sup_{|\gamma_i|\le \delta /2}\,g(\xi_i-\delta k_i-\gamma_i) &=C_1\, \prod_{i=1}^d\,\delta\, \sum_{k_i\in \mathbb{Z}}\,\sup_{|\gamma_i|\le \delta/2}\,g(\xi_i-\delta k_i-\gamma_i)\le C_1\,c^d=C<\infty.
a_n = \sum_{j=0}^n \frac{n!}{j!}h(n)^j = \sum_{j=0}^n \frac{n!}{(n-j)!}h(n)^{n-j}= \sum_{j=0}^n h(n)^{n-j}\prod_{i=0}^{j-1}(n-i) \equiv \equiv \sum_{j=0}^{kp-1}h(n)^{n-j}\prod_{i=0}^{j-1}(n-i) \pmod{p^k},
\big[\mathrm{PSL}_{2}(\mathbb{Z}):\Gamma_{0}(N)\big]=O\big(g_{Y_{0}(N)}\big),|\mathcal{P}_{Y_{0}(N)}|=O\big(N\log N\big), |\mathcal{E}_{Y_{0}(N)}|=O_{\epsilon}\big( N^{\epsilon}\big),
b_{0}\geq A_{N,\varepsilon}\left( \frac{\varphi (r)}{r}\right)^{\frac{1}{N-1-\varepsilon}}+\ln (e^8\gamma)=A_{N,\varepsilon}\left( \frac{\varphi (r)}{r}\right)^{\frac{1}{N-1-\varepsilon}}+\ln \frac{4e^8}{c^{2}\tau^{2}\left\Vert \phi \right\Vert_{X}^{2}\varepsilon^{2}}
{\cal L}_{SCM}=\frac 12 \!\left( J_\mu^j J^{\mu ~j}+i \overline{\chi}^j /\kern-.5em\partial ~\chi^j+i\varepsilon^{jkl}\overline{\chi}^j /\kern-.5em J^k \chi^l+\frac 14 (\bar \chi^j \chi^j )^2 \right).
\frac{1}{n}e^{n-1}_{i,g+1}+\left(-1+\frac{1}{n}\right)e_{i,g}^{n-1}+\left(\frac{1}{n}-\frac{1}{n^2}-1\right)e_{i,g+1}^{n-2}&+\left(-1+\frac{1}{n}\right)\sum_{\tau=2}^{n-2}e_{i,g+\tau}^{n-1-\tau}-\frac{1-n}{n}\sum_{\tau=0}^{n-2}e_{i,g+\tau}^{n-\tau-1},
\tilde{C}_{pq}= \left( \frac{q}{2}\left( \frac{p}{2}\right)^{\frac{2-q}{p-2}}\right)^{-1}\frac{q(p-2)}{2(p-q)}\left( \frac{p(2-q)}{2(p-q)}\right)^{\frac{2-q}{p-2}}= \frac{p-2}{p-q}\left( \frac{2-q}{p-q}\right)^{\frac{2-q}{p-2}}.
\begin{cases}-\Delta v_i=f(v_i)-\mu_iv_i+\lambda v_i \ \ \ \ \mathbb R^N, \lim_{|x|\to\infty}v_i(x)=0, i=1,2,\cdots, \ell, \sum_{i=1}^\ell |v_i|_2^2= \alpha. \end{cases}
S= - \frac{1}{\lambda^2_{s}}\int d^4 x ~\sqrt{-g}~e^{-\phi}\biggl[ R + g^{\alpha\beta}\partial_{\alpha}\phi\partial_{\beta}\phi - \frac{1}{2}e^{2 \phi}g^{\alpha\beta}\partial_{\alpha}\psi\partial_{\beta}\psi \biggr].
\mathbf{X}= \left( \begin{array}{c c c c}{{1}}&{{- 1}}&{{i}}&{{i}}\\{{- i}}&{{- i}}&{{1}}&{{- 1}}\\{{1}}&{{1}}&{{- i}}&{{i}}\\{{- i}}&{{i}}&{{- 1}}&{{- 1}}\\ \end{array}\right) , \quad \mathbf{Y}= \left( \begin{array}{c c c c}{{- 1}}&{{1}}&{{i}}&{{i}}\\{{- i}}&{{- i}}&{{- 1}}&{{1}}\\{{- 1}}&{{- 1}}&{{- i}}&{{i}}\\{{- i}}&{{i}}&{{1}}&{{1}}\\ \end{array}\right) .
\|u(t)\|^2_{\widetilde{L}^{\infty}([0,t];\,{B}^{s-1}_{2,2})}+ \|u\|^2_{\widetilde{L}^2([0,t];\,{B}^s_{2,2})}\leq C_1(\nu)M_0+C_1(\nu)\int^t_0\|u\|^{4+\frac4{r-2}}_{\widetilde{L}^{\infty}([0,t^{\prime}];\,B^{s-1}_{2,2})}dt^{\prime}+C_2\int^t_0\|b\|^4_{\widetilde{L}^{\infty}([0,t^{\prime}];\,B^{s}_{2,2})}dt^{\prime}.
(j^{-1}\cdot \theta)((j^{-1}|_\theta k)\cdot\beta) =(j^{-1}\cdot \theta)((j^{-1}|_\theta\cdot (k\cdot\beta)) &=j^{-1}\cdot(\theta(k\cdot \beta)) &=j^{-1}\cdot(j\cdot \alpha) &=(j^{-1}j)\cdot\alpha &=\alpha.
\pi_{\mathbb{W}}\left((b,0,c)\cdot (as,s,0)\cdot z \right) & =\left(s+z_2,c +\tfrac{b}{2}s+ z_3+\tfrac{1}{2}[b+as]z_2-\tfrac{1}{2}s z_1+\tfrac{1}{2}[b+as+z_1][s+z_2]\right) &= \left(s+z_2,\tfrac{a}{2}s^2 + bs +c+[b+as]z_2+z_3+\tfrac{1}{2}z_1z_2\right).
[\mathbb{P}_\pm^{(k)}(z) f ] (p_k , q_k) = \hat \varphi_\pm (\sqrt{z^2 +q_k^2}; p_k) \int \hat \varphi_\pm^* (\sqrt{z^2 + q_k^2}; p'_k) f (p'_k , q_k)dp'_k ,
\sum_{j = 1}^{N}x_{j}= 0 ~ , \hspace{1 c m}\sum_{j = 1}^{N}x_{j}m_{j}^{2}= 0 ~ , \hspace{1 c m}\sum_{j = 1}^{N}x_{j}m_{j}^{2}\ln m_{j}^{2}={\frac{\pi}{G}}~ ,
\mathcal{K}\left( U^{n+1}\right) =\sigma \left( 1-2\alpha \right) \mathcal{L}_{A}\Gamma \left( U^{n}\right) +\sigma \alpha \mathcal{L}_{A}\Gamma \left(U^{n-1}\right) +U^{n-1}+\sigma \lambda F^{n}.
E(t) =\begin{bmatrix}C(t) \Lambda^{-1/2}S(t) -\Lambda^{1/2}S(t) C(t) \end{bmatrix}, E_h(t) =\begin{bmatrix}C_h(t) \Lambda_h^{-1/2}S_h(t) -\Lambda_h^{1/2}S_h(t) C_h(t) \end{bmatrix},
\frac{\partial}{\partial \phi}\Re{f(4;R e^{i\phi})}= - \Big(\frac{4\tau}{1 - \tau^{2}}+ \frac{\tau}{1 - 2\tau R \cos\phi + \tau^2 R^{2}}- \frac{\tau}{\tau^{2}- 2\tau R \cos\phi + R^{2}}\Big) R \sin\phi \begin{cases}<0, \phi > 0; >0, \phi < 0.\end{cases}
\begin{aligned}\sum_{i\geq0}E^{(i)}_v(b_{+1}) ({-D_z})^{i+1}f(b_{+1}) = \sum_{i\geq0}E^{(i+1)}_v(b) ({-D_z})^{i+1}((D_z f'(b{+1}))/a) &-\sum_{i\geq0}\sum_{j\geq 0}{\textstyle\binom{i+j}{j}}E^{(i+j)}_v(a) (({-D_z})^j b_{+1})({-D_z})^{i}((D_z f'(b_{+1}))/a) \end{aligned}
|{\cal X}_{1}|^{2}+ |{\cal X}_{2}|^{2}+ 2 |{\cal X}_{3}|^{3}+ 8 |{\cal X}_{4}|^{2}+ 1 2 |{\cal X}_{5}|^{2}- 2 4 |{\cal X}_{6}|^{2}= r ,
\|u_s^\star(\varphi) - u_s^\star(\psi)\|_{\eta,s}= \|\mathcal{G}_s(u_s^\star(\varphi),\varphi) - \mathcal{G}_s(u_s^\star(\psi),\psi)\|_{\eta,s}\leq K_\varepsilon \|\varphi - \psi\| + \frac{1}{2}\|u_s^\star(\varphi) - u_s^\star(\psi)\|_{\eta,s}.
A_{p,N}\rightarrow \sum_{A=0}^{2n-1}\int_S \left[v_p(d_1v_{p}\wedge\alpha)_A \tau( \mathbf{{n}})_{A0}+v_{p}(d_0 v_p\wedge\alpha)_A \tau( \mathbf{{n}})_{A1}\right]dS &= \int_B (d_0 v_p \wedge d_1v_{p}+ v_p d_0 d_1v_{p}) \wedge\alpha+\int_B ( d_1v_{p}\wedge d_0 v_p+ v_p d_1 d_0v_{p})\wedge\alpha=0,
{\cal L}_{bulk}= M^3 \left\{- \frac{1}{2}K(\chi) (\partial_y \chi)^2 - V(\chi) + \sum_{\vec{n}}\frac{a_{\vec{n}}(\chi)}{M^{|n|}}(\partial \chi)^{n_1}... (\partial^N \chi)^{n_N}\right\},
\int_{\eta_{0}}^{\eta}\frac{\varphi_{+}(\eta)\varphi_{-}(s)-\varphi_{-}(\eta)\varphi_{+}(s)}{W(\varphi_{+},\varphi_{-})}\tilde{g}(s,t)\tilde{\phi}(s)ds=o(1)(\varphi_{+}(\eta)+\varphi_{-}(\eta)),~~t\rightarrow+\infty,
C\left(\mu, \Psi\right) =\inf_{L\in A_\phi}c_\mu 2^{n_\mu}\phi\left[c\left(1+ \frac{L[\phi^{-1}]'(L)}{\phi^{-1}(L)}\right)\right] \left( \frac{\phi^{-1}(L)}{L[\phi^{-1}]'(L)}+1\right) \leq \inf_{\phi(s)\in A_\phi}c_\mu 2^{n_\mu}\phi\left[c\left(1+\frac{\phi(s)}{s\phi'(s)}\right)\right]\left(\frac{s\phi'(s)}{\phi(s)}+1\right),
+ \frac{1}{2 \pi}m c \int_{\Lambda}d^{2}x \left[ e^{+ 2 \sqrt{\pi}\sqrt{\frac{\pi}{\pi + g}}\big ( \delta ( x ) , C_{m_{d}}\alpha \big ) + i \theta}+ e^{- 2 \sqrt{\pi}\sqrt{\frac{\pi}{\pi + g}}\big ( \delta ( x ) , C_{m_{d}}\alpha \big ) - i \theta}\right]
- \delta{\cal I}= \int_{0}^{\infty}d \beta \; \mathrm{T r}\, ( - 1 )^{F}H e^{- \beta H}= \int_{{\cal E}> 0}d{\cal E}\: \left\{\, \rho_{+}({\cal E}) - \rho_{-}({\cal E}) \, \right\},
R^2 \sin^2\theta - 2 R t \sin\theta\cos\theta +t^2 = \frac{R^2}{2}+t^2 - \left(\frac{R^2}{2}\cos (2\theta) +R t \sin(2\theta)\right) &\leq M^2+ M^2 + \sqrt{\frac{R^4}{4}+R^2 t^2}&\leq 2 M^2 + \sqrt{3}M^2 < 4 M^2,
\langle \Phi_n^{(N)},\tilde{a}_0\Phi_{0,m}^{(N)}\rangle =\frac{1}{\sqrt{2\pi K}}\int_{-\infty}^{+\infty}dt\>\langle \Phi_n^{(N)},a_0\Phi_{0,m}^{(N)}\rangle e^{i\mathcal{E}_{n,m}^\kappa}e^{-t^2/(2K)}&=\langle \Phi_n^{(N)},a_0\Phi_{0,m}^{(N)}\rangle\exp\bigl[-K(\mathcal{E}_{n,m}^\kappa)^2/2\bigr],
\begin{array}{ll}(UU_{x})^{n+1}=U^{n+1}U_{x}^{n}+U^{n}U_{x}^{n+1}-U^{n}U_{x}^{n}(VV_{x})^{n+1}=V^{n+1}V_{x}^{n}+V^{n}V_{x}^{n+1}-V^{n}V_{x}^{n}(UV)_{x}^{n+1}=(U_{x}V)^{n+1}+(UV_{x})^{n+1}=U_{x}^{n+1}V^{n}+U_{x}^{n}V^{n+1}-U_{x}^{n}V^{n}+U^{n+1}V_{x}^{n}+U^{n}V_{x}^{n+1}-U^{n}V_{x}^{n}\end{array}
D_{1 2}^{( l )}\stackrel{1}{F^{( l )}}\stackrel{2}{F^{( 1 )}}D_{1 2}^{( l ) \; - 1}= \left( \begin{array}{c c}{F^{( l + 1 )}}{0}{0}{- F^{( l - 1 )}}\end{array}\right) .
\mathcal{P}=\lim_{\textsf{SNR}\rightarrow \infty}\textrm{Pr}\left\{\sum_{i=1}^{K}\sum_{j=1}^{S}\eta^{[i,j]}\mathsf{SNR}\le \epsilon \right\}&\ge \lim_{\textsf{SNR}\rightarrow \infty}\textrm{Pr}\left\{\eta^{[i,j]}\le \frac{\textsf{SNR}^{-1}\epsilon}{KS^2},\forall i\in \mathcal{K}, \forall j\in\mathcal{S}\right\}.
{Y}^{\quad\quad\quad\quad[{j}_3, \psi_{j_3}]}_{[{j_1}, \psi_{j_1}][{j_2}, \psi_{j_2}]}= \sum_{[m]}\sum_{\psi_m}{{S}_{[m, \psi_m][{j_1}, \psi_{j_1}]}{P}_{[m, \psi_m][{j_2}, \psi_{j_2}]}{P}^*_{[m,\psi_m] [{j_3}, \psi_{j_3}]}\over{S}_{[m,\psi_m][0]}}\;\;\;\; .
H(Z_t) =H(Z_0)+\int_0^t(H'(Z_s)\cdot \widetilde\mu(Z_s)+\tfrac{1}{2}H''(Z_s)\cdot\widetilde\sigma^2(Z_s))\,ds+\int_0^t H'(Z_s)\cdot\widetilde\sigma(Z_s)\,dW_s &= x_0+\int_0^t \mu(H(Z_s))\,ds +\int_0^t \sigma(H(Z_s))\,dW_s.
\left( \prod_{i=0}^{k-1}Y_{1, aq^{6i}}\right) \left( \prod_{i=0}^{l-1}Y_{2, aq^{6k+2i+1}}\right), \left( \prod_{i=0}^{l-1}Y_{2, aq^{2i}}\right) \left( \prod_{i=0}^{k-1}Y_{1, aq^{2l+6i+5}}\right),
z_{1}=x_{1}^{\left( 1\right)},z_{2}=x_{0}^{\left( 1\right)},z_{3}=x_{2}^{\left( 2\right)}, z_{4}=x_{2}^{\left( 1\right)},z_{5}=x_{0}^{\left( 2\right)},z_{6}=x_{1}^{\left( 2\right)}.
g_c:=-2[\bar{f}, \mathfrak{H}\frac{1}{z_{\alpha}}+\bar{\mathfrak{H}}\frac{1}{\bar{z}_{\alpha}}]\bar{f}_{\alpha}+\frac{1}{\pi i}\int \Big(\frac{z_t(\alpha,t)-z_t(\beta,t)}{z(\alpha,t)-z(\beta,t)}\Big)^2(z-\bar{z})_{\beta}d\beta.
\dim \operatorname{Gr}(r', r) + \dim N = (r' + 1)(r - r') + [(r' + 1)d - (r' - 3)(g - 1) + n] &= (d - g + 1)(r - d + g) + (d - g + 1)d - (d - g - 3)(g - 1) + n &= (r + 1) d - (r - 3)(g - 1) + n,
\phi_{0}(\mathbf{x})=\frac{\phi^{\mathrm{TF}}(\mathbf{x})}{\|\phi^{\mathrm{TF}}\|}\phi^{\mathrm{TF}}(\mathbf{x})=\begin{cases}\sqrt{\left(\mu^{\mathrm{TF}}-V(\mathbf{x})\right)/\beta}, \mbox{if}V(\mathbf{x}) \leq \mu^{\mathrm{TF}}, 0, \mbox{otherwise},\end{cases}
G_s(\gamma_2) = \frac{\sqrt{J_2}}{E_q(J_1,J_2)}\left(F_q(J_1,J_2,\gamma_2)-F_q(J_1,J_2,-\gamma_2) \right) = \frac{2i\sqrt{J_1}}{E_q(J_1,J_2)}\sum_{n_1,n_2 =0}^\infty \frac{J_1^{n_1}J_2^{n_2}}{[n_1]_q![n_2]_q!}\sin(\gamma_2q^{2[n_2]_q}),