harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
L_{\rm BRS}'' = \mbox{}- \frac{1}{2}\partial_{\mu}A_{\nu}\partial^{\mu}A^{\nu}- \frac{m^{2}}{2}A_{\mu}A^{\mu}- \frac{1}{2}\partial_{\mu}\theta\partial^{\mu}\theta - \frac{m^{2}}{2}\theta^{2}.
\begin{array}{r l}<0|T\left(\hat{\phi}(x_f)\hat{\phi}(x_i)\right)|0>= \theta(x^0_f-x^0_i)\,<0|\hat{\phi}(x_f)\hat{\phi}(x_i)|0> + \theta(x^0_i-x^0_f)\,<0|\hat{\phi}(x_i)\hat{\phi}(x_f)|0>\ ,\end{array}
L(\theta,\dot{\theta},\dot{\psi},\dot{\phi})=\frac{I_{1}}{2}(\dot{\theta}^{2}+\dot{\phi}^{2}sin^{2}\theta)+\frac{I_{3}}{2}(\dot{\psi}^{2}+\dot{\phi}^{2}cos^{2}\theta)+I_{3}\dot{\psi}\dot{\phi}cos\theta-Mglcos\theta
S = \int d^4x \; \mbox{Tr}\;\left\{-{1\over 8}\nabla^{\alpha\alpha'}\phi^{ab}\nabla_{\alpha\alpha'}\phi_{ab}+{1\over 2}\chi^a_{\alpha'}\nabla^{\alpha\alpha'}\chi_{\alpha a}+{1\over 6}G^{\alpha'\beta'}F_{\alpha'\beta'}- \phi^{ab}\chi^\alpha_a \chi_{\alpha b}\right\}\ .
0 = \sum_{s=1}^n (\frac{k+1}{n+1}s-1)\frac{P_f[X_1=s,X_1+\cdots+X_{k+1}=n+1]}{P_f[X_1+\ldots+X_{k+1}=n+1]}= \frac{1}{\binom{k+1}{n+1}_f}\sum_{s=1}^n (\frac{k+1}{n+1}s-1){P[X_1=s]\cdot P_f[X_2+\cdots+X_{k+1}=n+1-s]}= \frac{1}{\binom{k+1}{n+1}_f}\sum_{s=1}^n(\frac{k+1}{n+1}s-1)f(s)\binom{k}{n+1-s}_f,
\langle{\cal A}\rangle_{0}\lim_{\tau^{\prime}\rightarrow \infty}\langle{\cal U}_{\tau^{\prime}}({\scriptstyle{\cal A B}}){\cal B}\rangle_{0}\; - \; \langle{\cal A}\rangle_{0}\lim_{\tau^{\prime}\rightarrow \infty}\langle{\cal U}_{\tau^{\prime}}({\scriptstyle{\cal B}}){\cal B}\rangle_{0}= 0 \; .
T_{\mu\nu}^{matter}= \frac{1}{4}\left( H^2_{\mu\nu}- \frac{1}{6}\tilde{G}_{\mu\nu}H^2 \right) e^{-\frac{8}{\sqrt{2|D-2|}}\tilde{\phi}}+ \left( \partial_{\mu}\tilde{\phi}\partial_{\nu}\tilde{\phi}- \frac{1}{2}(\partial \tilde{\phi})^2 \tilde{G}_{\mu\nu}\right)~.
\Big|\int_{E_m+y_0}g(x)dx-\int_{E_m}g(x)dx\Big| &= \int_{E_m}g(x+y_0)-g(x)dx &=y_0\cdot \int_{E_m}\nabla g(x)dx+\frac{1}{2}\int_{E_m}\Big( \langle D^2 g(x)y_0,y_0\rangle +2o_x(|y_0|^2)\Big)dx &=\frac{1}{2}\int_{E_m}( \langle D^2 g(x)y_0, y_0\rangle +2o_x(|y_0|^2)) dx;
\begin{array}{l l}{{f_{1}= \frac{1}{2}( M_{4 5}+ M_{6 7}+ M_{8 9}) \, ,}}&{{f_{2}= \frac{1}{2}( M_{4 5}- M_{6 7}- M_{8 9}) \, ,}}\\{{{}}}&{{{}}}\\{{f_{3}= \frac{1}{2}( - M_{4 5}+ M_{6 7}- M_{8 9}) \, ,}}&{{f_{4}= \frac{1}{2}( - M_{4 5}- M_{6 7}+ M_{8 9}) \, .}}\\ \end{array}
D \! \! \! \! /^{~ 2}= \left[{\frac{1}{\sqrt g}}\left({\sqrt g}g^{\nu \lambda}\right)_{, \lambda}+ ( 1 -{\cal D}) \phi^{\nu}- 2 \sigma^{q p}e_{~ q}^{\mu}e_{~ p}^{\nu}\phi_{\mu}+ \left( g^{\mu \nu}+ 2 \sigma^{\mu \nu}\right) D_{\mu}\right] D_{\nu}.
N^{'^{\prime}}|_{x_{h}}= \frac{1 0 \alpha^{2}}{x_{h}^{4}}( w_{x_{h}}^{2}- 1 )^{2}- \frac{8 \alpha^{2}w_{x_{h}}w^{'}|_{x_{h}}}{x_{h}^{3}}- \frac{2 \alpha^{2}}{x_{h}}N^{'}|_{x_{h}}( w^{'}|_{x_{h}})^{2}\left( 1 + \frac{8}{x_{h}^{4}}( w_{x_{h}}^{2}- 1 )^{2}\right)
Tr(W[{\cal C}])=Tr(e^{-\oint_{\cal C}A_{\ 2}^0(x^2)T_0dx^2})=Tr(e^{\oint_{\cal C}\frac{4\pi}{k}J^0(x^2)T_0dx^2})=2\cos({\frac{2\pi J^0_0}{k}}).
\lim_{p\to 0}y_{1}(x_{1},x_{2},p,\Phi_{1}(2(1-\sqrt{1-p}))) = \frac{x_{1}+x_{2}}{2}\lim_{p\to 0}z(x_{1},x_{2},p,\Phi_{1}(2(1-\sqrt{1-p}))) = \frac{x_{1}+x_{2}}{2}\lim_{p\to 0}y_{2}(x_{1},x_{2},p,\Phi_{1}(2(1-\sqrt{1-p}))) = \frac{x_{1}+x_{2}}{2}
F_{\mu \mathrm{m}}={1\over 2}\biggl({\partial u^\mu\over \partial X^{\mathrm{m}}}+{\partial u^{\mathrm{m}}\over \partial X^\mu}+ \sum_{\mathrm{n}=1}^3{\partial u^\mu\over \partial X^{\mathrm{n}}}{\partial u^{\mathrm{m}}\over \partial X^{\mathrm{n}}}\biggr).
\begin{array}{l c}{{\mathrm{S e c t o r}}\quad \quad}&{S U ( 2 ) \otimes S U ( 2 ){\mathrm{r e p .}}}\\{{\mathrm{N S}}}&{2 ({\bf 1},{\bf 1}) ,}\\{{\mathrm{R}}}&{({\bf 1},{\bf 2}) .}\\ \end{array}
\tilde{w}_1(x) := \frac{u_2(2^{-k}x)- a_k u_1(2^{-k}x)}{b_k -a_k}, \ \tilde{w}_2(x):= \frac{b_k u_1(2^{-k}x) - u_2(2^{-k}x)}{b_k-a_k}.
\langle u , Z_{s}\rangle / N(u) = \sum_{i = 0}^{\ell_s}b_{i,\ell_s}\langle u , \zeta(V_{s,i}) \rangle / N(u) &\ge \sum_{i = 0}^{\ell_s}b_{i,\ell_s}+ b_{\ell_s,\ell_s}\lambda_M &\ge \sum_{i = 0}^{r}b_{i,r}+ b_{r,r}\lambda_M > 1.
c\delta(u)W_3(u,-\mu,z) = zI_{-\mu}(uz)\left(\sum_{s=0}^{N-1}\frac{\tilde A_s(z)}{u^{2s}}+g_3(u,z)\right) \hphantom{c\delta(u)W_3(u,-\mu,z) =}{}+\frac{z}{u}I_{1-\mu}(uz)\left(\sum_{s=0}^{N-1}\frac{B_s(z)}{u^{2s}}+zh_3(u,z)\right),
\sum\limits_{i=1}^{i=2}a_{i}x_{i}^{2}\pm \left( s^{2}+\frac{1}{2}s\left(\sum\limits_{i=1}^{i=2}x_{i}^{2}\right) +\frac{1}{16}\left(\sum\limits_{i=1}^{i=2}x_{i}^{2}\right)^{2}\right)
+i\hbar(V^{CE}_{AB}V^D_E(-1)^{\varepsilon_C}-(C\leftrightarrow D)(-1)^{\varepsilon_C\varepsilon_D})-[V^{CD}_{AB},\Omega](-1)^{\varepsilon_C+\varepsilon_D}-\frac{1}{2}(i\hbar)^2V^{EF}_{AB}V^{CD}_{FE}.
|E|n^{-H}S_{n}(s)|^2|\leq n^{2h(s)-3}\left( n|\gamma_{0}(s)|+\sum_{k=1}^{n}(n-k)\left| \gamma_{k}(s)\right| +\sum_{k=1}^{n}(n-k)\left| \overline{\gamma_{k}(s)}\right| \right) .
\mathbb E\int_0^T |f(s, Z_s)|\,ds \le N_R \|f\|_{L_{p+1}([0,T]\times \mathbb R^d)}\sum_{k=0}^{\infty}\mathbb{P}(T^k \le T) \le C N_R\|f\|_{L_{p+1}([0,T]\times D)},
( [ \hat{A}, \hat{B}] )_{\mu}^{\nu}= ( \hat{A}\hat{B}- \hat{B}\hat{A})_{\mu}^{\nu}= \sum_{\sigma = 0}^{3}( \hat{A}_{\mu}^{\sigma}\hat{B}_{\sigma}^{\nu}- \hat{B}_{\mu}^{\sigma}\hat{A}_{\sigma}^{\nu}) .
\lim_{t\to +\infty}\int_{vt -\lambda(t)}^{vt+ \lambda(t)}\left((\partial_{t}\Lambda)^{2}+(\partial_{x}\Lambda)^{2}+\sinh^{2}{\Lambda}((\partial_{t}\phi)^{2}+(\partial_{x}\phi)^{2}) \right)(t,x)dx=0.
f_{\varphi}(x_0, \cdots, x_k) :=\int_{G^{\times (k+1)}}c_X(g_0^{-1}x_0) \cdots c_X(g_k^{-1}x_k)\varphi(g_0, g_1, \cdots, g_k){\rm d}g_0\cdots{\rm d}g_k, \Phi(\varphi) :=(d_{x_1}d_{x_2}\cdots d_{x_k}f_{\varphi})(x, \cdots, x).
\begin{aligned}\sum_{j=1}^{n-N}(\frac jn)^{c'kn}=n\sum_{j=1}^{n-N}\frac1n(\frac jn)^{c'kn}\le n\int_0^{1-\frac{N-1}n}x^{c'kn}dx\le & \frac n{c'kn+1}(1-\frac{N-1}n)^{c'kn+1}\le \frac n{c'kn+1}e^{-\frac{N-1}n(c'kn+1)}.\end{aligned}
\frac{q}{16}\sum_{i=6}^n \frac{n!}{(n-i)!}(i-5) \leq\frac{q}{16}n! \sum_{i=6}^n \frac{n}{(n-i)!}&= \frac{q}{16}n(n!)\sum_{i=6}^n \frac{1}{(n-i)!}&\leq \frac{q}{16}n(n!) \sum_{m=1}^\infty \frac{1}{m!}&=\frac{q}{16}n(n!)e.
( \vec{x}_{a}- \vec{x}_{b}) A_{0 a b}( A_{0 a a}- A_{0 b b}) = \displaystyle{\sum_{c \neq a , b}\left( ( \vec{x}_{a}- \vec{x}_{c}) + ( \vec{x}_{b}- \vec{x}_{c}) \right) A_{0 a c}A_{0 b c}}.
\| \delta_0^m(a)(D+i\lambda)^{-p-1}\|_1 \leq 2^{-m}\sum_{k=0}^m \binom{m}{k}\left(\frac{2}{\pi}\right)^k \|\Lambda^{m+k}(a)(D+i\lambda)^{-p-1}\|_1\int_{[0,\infty)^k}\prod_{l=1}^k \frac{d\lambda_l}{\lambda_l^{1/2}(1+\lambda_l)}.
\begin{aligned}(p',\, p'') = (\{p^{(j)}\}_{j \in J'},\, \{p^{(j)}\}_{j \in J''}),\, (p'_*,\, p''_*) = (\{p_*^{(j)}\}_{j \in J'},\, \{p_*^{(j)}\}_{j \in J''}).\end{aligned}
\mathcal{V}_e^{(k)}= \mathcal{V}^{(k)}\cup \{\bigcup_{l\in{\mathcal{I}}_{\mathcal{F}}^{(k)}}\partial F^{(lk)}\}, \mathcal{V}^{(k)}= \{\bigcup_{l\in{\mathcal{I}}_{\mathcal{F}}^{(k)}}\partial F^{(kl)}\},
\mathcal{K}_1(t) = 1+\|\varphi^-(t)\|_{L^\infty}^\frac{(2-a)\alpha}{(1-a)(\alpha-\lambda-1-\mu_0)}+ \|\varphi^-(t)\|_{L^\infty}^\frac\alpha{\alpha-\lambda-1}+ \int_0^t\int_\Gamma |\varphi_t|^{\frac{\alpha}{\alpha-\lambda-1}}d\sigma d\tau.
d^2(x_{k+1},\mathrm{Arg}\min f) = \|x_{k+1}-x_{k+1}^\prime\|^2\leq \|x_{k+1}-x_k^\prime\|^2 = \|x_k-h\cdot\nabla f(x_k)-x_k^\prime\|^2 = d^2(x_k,\mathrm{Arg}\min f) -2h\langle \nabla f(x_k),x_k-x_k^\prime\rangle+h^2\|\nabla f(x_k)\|^2,
\begin{array}{c c c c c}{\delta \sqrt{- g}}&{=}&{-{\frac{1}{2}}\sqrt{- g}g_{a b}\delta g^{a b}}&{\}&{}\\{\delta R}&{=}&{\delta \left( R_{a b}g^{a b}\right)}&{=}&{\delta R_{a b}g^{a b}+ R_{a b}\delta g^{a b},}\\ \end{array}
\begin{aligned}F_\rho(x^k) = \min_x\Big\{F(x) + \frac{\rho}{2}\big\|x - x^k\big\|^2 \Big\}\geq\min_x \Big\{\widetilde{F}^k(x) + \frac{\rho}{2}\big\|x - x^k\big\|^2 \Big\}= \eta^k.\end{aligned}
f(t,m,x) \ = \ 2m \Big(\Phi\Big(\frac{m-x}{\sqrt{T-t}}\Big) - \frac{1}{2}\Big) + 2x \Big(1 - \Phi\Big(\frac{m-x}{\sqrt{T-t}}\Big)\Big) + \sqrt{\frac{2(T-t)}{\pi}}e^{-\frac{(m-x)^2}{2(T-t)}},
\Bigl(\frac{1}{2}\Bigr)_{\frac{p-1}{2}+ p}= \Bigl(\frac{1}{2}\Bigr)_{\frac{p-1}{2}}\prod_{k=0}^{p-1}\Bigl( \frac{p}{2}+ k \Bigr), \frac{(\frac{1}{2})_{\frac{p-1}{2}}}{(1)_{\frac{p-1}{2}}^2}= \frac{\binom{p-1}{\frac{p-1}{2}}}{2^{p-2}(p-1) (1)_{\frac{p-3}{2}}}
\begin{array}{l}\Psi (z,\bar{z})=S_{-1/2}\phi (z,\bar{z}); \bar{\Psi}(z,\bar{z})=\bar{S}_{-1/2}\phi (z,\bar{z}); F(z,\bar{z})=S_{-1/2}\bar{S}_{-1/2}\phi (z,\bar{z})\end{array}
\int \frac{dw_2 dw_3 dw_4 dw}{{(2\pi)}^4}\frac{-(w_2+w_3+w_4) X_i^8[-(w_2+w_3+w_4)]w_2 X_j^8(w_2) w_3 X_k^8(w_3) w_4 X_l^8(w_4)}{[(w+w_2)^2-R^2][(w+w_2+w_3)^2-R^2][(w+w_2+w_3+w_4)^2-R^2][w^2-R^2]}
\sup_{\gamma\in \Gamma}\rho(\gamma x, \gamma 0_X) =\sup_{\gamma \in \Gamma}\sum_{\chi\in \widehat{X}}h(\chi)\Psi_{x, \chi}(\gamma) &\le \varepsilon_0/2+\sup_{\gamma \in \Gamma}\sum_{\chi\in W}h(\chi)\Psi_{x, \chi}(\gamma) &\le \varepsilon_0/2+(\max_{\chi\in W}h(\chi))\sum_{\chi\in W}\|\Psi_{x, \chi}\|_\infty & < \varepsilon_0,
h_{\mathbf{f}}^{\textup{c-wt}}=h^{\textup{wt}}_{V_{\psi_k(\mathbf{f})},\mathbb D_{\psi_k(\mathbf{f})}}\,:\,\widetilde H^1_{\textup{f}}(V_f)\times \widetilde H^1_{\textup{f}}(V_f^*(1)) \rightarrow \frak p/\frak p^2.
\frac{\partial\,r}{\partial\,x}= 1-xy^{-1}x^{-1}+xy^{-1}x^{-1}y+xy^{-1}x^{-1}yxy^{-1}-xy^{-1}x^{-1}yxy^{-1}xyx^{-1}, \frac{\partial\,r}{\partial\,y}= -xy^{-1}+xy^{-1}x^{-1}-xy^{-1}x^{-1}yxy^{-1}+xy^{-1}x^{-1}yxy^{-1}x -xy^{-1}x^{-1}yxy^{-1}xyx^{-1}y^{-1}.
\int_{\mathbb{R}}\lvert f(x)\rvert^p\,\mathrm{d}x = \int_{\mathbb{R}}\big\lvert F\big(\Phi(x)\big)\big\rvert^p \cdot \lvert \Phi'(x)\rvert\,\mathrm{d}x = \int_{\Gamma}\lvert F(\zeta)\rvert^p \lvert\mathrm{d}\zeta\rvert \leqslant \lVert F\rVert_{H^p(\Omega_+)}^p < +\infty,
\frac{\prod_{(i,j) \in \Delta_r}(1- \bar t_i/\bar t_j)}{\prod_{(i,j) \in C(\lambda)}(1- t_i/\bar t_j)}=\sum_{I\subset D(\lambda)_r}\prod_{(i,j)\in I}(-t_i/\bar t_j) \prod_{(i,j)\in \Delta_r}\frac{1- \bar t_i/\bar t_j}{1- t_i/\bar t_j}.
\sum_{n,k,\ell\ge 0}L_1(n,k,\ell)x^ky^{\ell}q^n = \sum_{m\ge 0}\frac{(-yq/x)_m x^m q^{\binom{m+1}{2}}}{(q)_m}, \sum_{n,k,\ell\ge 0}A(n,k,\ell)x^ky^{\ell}q^n = \sum_{k,\ell\ge 0}\frac{x^k y^{\ell}q^{\binom{k+\ell+1}{2}+\binom{\ell+1}{2}}}{(q)_k(q)_{\ell}}.
\begin{aligned}|\partial_j \Phi_k(t,x) \partial_i \Phi_h(t,x)-\partial_j\tilde\Phi_k(t,x)\partial_i\tilde\Phi_h(t,x)|\le \frac{2}{c_1^2}\Vert D\Lambda\Vert_\infty \Vert D^2\Lambda\Vert_\infty |x-r(t)|\,,\end{aligned}
\begin{array}{l}L_{\bar 0}= \mathfrak{sl}(V_1)\oplus \mathfrak{sl}(V_2), L_{\bar 1}= V_1 \otimes \bigwedge^2V_2 , L_{\bar 2}= \bigwedge^2V_1 \otimes \bigwedge^4V_2, L_{\bar 3}= \bigwedge^3V_1 \otimes V_2 , L_{\bar 4}= \bigwedge^4V_1 \otimes \bigwedge^3V_2.\end{array}
\frac{\partial G}{\partial t}= \big(1+\frac{\alpha'}{2}R\big)\Delta G-2\big(1+\frac{\alpha'}{2}R\big)t|\nabla^2R|^2+\alpha't\nabla|\nabla R|^2\nabla R &+|\nabla R|^2\big(-1+t(4R-3r+\frac{5}{4}\alpha'R^2+\alpha'\Delta R)\big)+2R^2(R+\frac{\alpha'}{4}R^2-r).
0 \le\overbrace{b_1=\dots =b_1}^{d_1}<\dots <\overbrace{b_{k_2}=\dots=b_{k_2}}^{d_{k_2}}&<r_1\le\dots\le r_l &<\underbrace{p+a_1=\dots=p+a_1}_{c_1}<\dots<\underbrace{p+a_{k_1}=\dots=p+a_{k_1}}_{c_{k_1}}<p,
\int_{Y}D_{\pi}(\varphi(y)) \times \delta_{y}~d\eta(y) = \int_{Z}\int_{Y}D_{\pi}(z) \times \delta_{y}~dD_{\varphi}(z)(y)~d\eta(y) &= \int_{Z}D_{\pi}(z) \times D_{\varphi}(z)~d\zeta(z) = \rho
\ddot{\nabla}_X\left(\begin{array}{ccc}\phi_{11}\cdots \phi_{1r}\vdots \ddots \vdots \phi_{r1}\cdots \phi_{rr}\end{array}\right)=\left(\begin{array}{ccc}X\phi_{11}\cdots X\phi_{1r}\vdots \ddots \vdots X\phi_{r1}\cdots X\phi_{rr}\end{array}\right)
E_{\rm gap}\approx \frac{(2n+1)\pi}{3^{(1/4)}8 \sqrt{\eta}}\left(1-\frac{\eta}{2\sqrt{3}}\right) \ , \qquad \Gamma_{\rm gap}\approx \frac{64 \sqrt{3}\eta}{\pi^3 (2n+1)^3}\ , \qquad n=0,1,2...
\sum_{n=1}^{m-1}[3n]\frac{(1-q)^2(q;q^2)_{n}(q^{m+2};q^2)_{n-1}^2 q^{m-{n+1\choose 2}-\frac{(2n+1)(m-1)}{2}}}{(q;q)_{n}^2 (q^2;q^2)_{n}}\equiv 0\pmod{\Phi_m(q)},
\partial_t \partial^2 \omega + \Delta^{-1}\nabla^{\perp}\partial^2 \omega \cdot \nabla \omega+ \Delta^{-1}\nabla^{\perp}\partial \omega \cdot \nabla \partial \omega & \qquad+ (\Delta^{-1}\nabla^{\perp}\omega\cdot \nabla) (\partial^2 \omega) =0.
ds^{2}=-\frac{\Delta}{\rho^{2}}(dt-asin^{2}\theta d\phi)^{2}+\frac{sin^{2}\theta}{\rho^{2}}((r^{2}+a^{2})d\phi-adt)^{2}+\frac{\rho^{2}}{\Delta}dr^{2}+\rho^{2}d\theta^{2}
d_{n}H_{n - 1 +{\frac{D}{2}}}^{( 1 )}( | \omega | r^{\prime}) = e_{n}H_{n - 1 +{\frac{D}{2}}}^{( 1 )}( | \omega | r^{\prime}) + f_{n}H_{n - 1 +{\frac{D}{2}}}^{( 2 )}( | \omega | r^{\prime}) ,
[\begin{matrix}E_{Y}^{\prime}\\E_{I}^{\prime}\\E_{Q}^{\prime}\end{matrix}]=[\begin{matrix}0.30&0.59&0.11\\ 0.599&-0.2773&-0.3217\\ 0.213&-0.5251&0.3121\end{matrix}][\begin{matrix}E_{R}^{\prime}\\E_{G}^{\prime}\\E_{B}^{\prime}\end{matrix}]
\sum_{n}\frac{1}{\omega_{n}}\partial_{i}\phi_{n}\partial_{j}\phi_{n}^{*}= \lim_{\vec{y}\rightarrow \vec{x}}\frac{\partial}{\partial x^{i}}\frac{\partial}{\partial y^{i}}\zeta ( s = \frac{1}{2}\mid \vec{x}, \vec{y}) .
\left|\int_{\mathbb R}g'(x) V(x)\,\mathrm dx\right| ={}\left|\int_{\mathbb R}g(x) \left(|f'(x)|^2+q(x)|f(x)|^2\right)\,\mathrm dx\right| &\leq{}\|g\|_\infty \left(\|f'\|_{L^2}^2 + \|qf^2\|_{L^1}\right) \leq 12 \|q_-\|_{L^1}^2\|f\|_{L^2}^2.
G^\prime=\begin{pmatrix}1 1 \cdots 1 g_{21}^\prime g_{22}^\prime \cdots g_{2n}^\prime \vdots \vdots \vdots g_{k1}^\prime g_{k2}^\prime \cdots g_{kn}^\prime\end{pmatrix}=\begin{pmatrix}G_1^\prime G_2^\prime \cdots G_n^\prime\end{pmatrix}.
C'\sum_{i=1}^{4m}A_i(T-T_{1})^{1-2i}=C'\sum_{i=1}^{i=4m}\int_{T_{1}}^{2T-T_1}\frac{|\phi^{(i)}(t)|^{2}}{\phi(t)}dt \geq\int_{T_{1}}^{2T-T_1}w^{2}(t)\phi(t)dt - C'', & \geq \int_{T_{1}}^{T}w^{2}(t)dt - C'',
\lefteqn{P\lambda_\infty(f)-\lambda_\infty(f)=}\qquad &(P\lambda_\infty(f)-P^{L}\lambda_\infty(f))+(P^L\lambda_\infty(f)-P^L\lambda_n(f))+(P^L\lambda_n(f)-P\lambda_n(f)) &+(P\lambda_n(f)-\lambda_n(f))+(\lambda_n(f)-\lambda_\infty(f))=\textrm{I}+\textrm{II}+\dotsb+\textrm{V}.
-4 \int_{t}^{\infty}\frac{e''''(s) ds}{\log(\lambda_{0,0}(s))(1+s-t)}+ 4\alpha e''''(t) - 4 \int_{t}^{\infty}\frac{e''''(s) ds}{\log(\lambda_{0,0}(s))(\lambda_{0,0}(t)^{1-\alpha}+s-t)(1+s-t)^{3}}=RHS_{4}(t)
\overline{\partial}u =\overline{\partial}\sum_ku_k\otimes e_k=\sum_k\overline{\partial}u_k\otimes e_k=\sum_k f_k\otimes e_k=f, ||u||^2_{I_F\otimes h_E\otimes\mathrm{det}\,h_E,\omega}=\int_U|u|^2_{I_F\otimes h_E\otimes\mathrm{det}\,h_E,\omega}dV_\omega=\sum_k\int_U|u_k|^2_{h_E\otimes\mathrm{det}\,h_E,\omega}dV_\omega<+\infty.
t_{\infty}((s_{i,-1},1,\lambda_{i,2}^{2})) = \lim_{b\to\infty}\frac{b^{2}s_{i,-1}^{2}-bs_{i,-1}+s_{i,-1}+s_{i,-1}\lambda_{i,2}^{2}}{b^{2}-b\lambda_{i,2}^{2}}&= s_{i,-1}^{2}> 0, i\in\mathbb{N}\cap[1,\eta].
\left\{v: \exists \, t \in I_j^{(k)}\textrm{with}\textrm{sys}(\phi_t(v)) = T_k^{-\xi}\right\}\subset \left\{v: \textrm{sys}(\phi_{a_j^{(k)}}(v))\in \left[\frac{1}{2}T_k^{-\xi}, 2T_k^{-\xi}\right]\right\}.
\widetilde{I}_{t_{1},\cdots, t_{k}}(x_{1},\cdots, x_{k})=\sup_{\delta>0}\liminf_{m\to\infty}\inf_{|y-x| \le \delta}I_{m, t_{1}, \cdots, t_{k}}(y_{1},\cdots, y_{k}), ~~x=(x_{1}, \cdots,x_{k}).
z_{0}^{2}\varepsilon^{2}<{\frac{1}{3 2}}\left( - 3 2 \omega z_{0}^{2}+ 9 l^{2}- 3 \sqrt{- 6 4 \omega z_{0}^{2}l^{2}+ 9 l^{4}- 6 4 l^{2}z_{0}^{2}\omega^{2}}\right) / ( \omega^{2}l^{2}) .
f(x;\mu,\Sigma,k)=\frac{\Gamma(\frac{1+k}{2})}{\Gamma(\frac{1}{2})\pi^{\frac{k}{2}}|\Sigma|^{\frac{1}{2}}[1+(x-\mu)^{T}\Sigma^{-1}(x-\mu)]^{\frac{1+k}{2}}}.
S_{F}( x - x^{\prime}) = \int \frac{d \omega}{2 \pi}\frac{d k}{2 \pi}\; \frac{i e^{- i \omega ( t - t^{\prime})}}{\omega{}^{2}- k{}^{2}- m^{2}+ i \epsilon}\left( \begin{array}{c c}{{m}}&{{- i ( \omega + k )}}\\{{i ( \omega - k )}}&{{m}}\\ \end{array}\right) e^{i k ( x - x^{\prime})}.
\inf \ \Big\{\sum_{i=1}^d g_{k}(x_i,y_i) : (x_i)_{i=1}^d \in R, (y_i)_{i=1}^d \in \{0,1\}^d \Big\}\geqslant \inf \ \Big\{\sum_{i=1}^d g_{k; \ast}(x_i) : (x_i)_{i=1}^d \in R \Big\}& \geqslant \inf \ \{g_{k;\ast}(x) + g_{k;\ast}(\lambda-x) : \lambda \in \mathcal{I}, x \in [0,\lambda] \}& \geqslant E+\epsilon/2.
\eta_(A) = \sum_{1 \leq i < i' \leq n,\, 1 \leq j' \leq j \leq n}a_{ij}a_{i'j'}&=\sum_{1 \leq i' \leq n}\sum_{1 \leq j \leq n}\sum_{1 \leq i < i'}\sum_{1 \leq j' \leq j}a_{ij}a_{i'j'}&=\sum_{1 \leq i' \leq n}\sum_{1 \leq j \leq n}\left(\sum_{1 \leq i < i'}a_{ij}\right) \left(\sum_{1 \leq j' \leq j}a_{i'j'}\right).
{j^{W}}^{( L )}( x ) = \left( \begin{array}{l}{\theta ( x_{1}+ L ) \theta ( L - x_{1}) [ \delta ( x_{2}- L ) - \delta ( x_{2}+ L ) ]}\\{\theta ( x_{2}+ L ) \theta ( L - x_{2}) [ \delta ( x_{1}+ L ) - \delta ( x_{1}- L ) ]}\\ \end{array}\right) .
\frac{d}{dr}\left[ \frac{p-1}{p}|g|^{p/(p-1)}+ \frac{1}{2}f^2 \right] = |g|^{(2-p)/(p-1)}g (f-|g|) - |g|^{(2-p)/(p-1)}g f & = - |g|^{1/(p-1)}g & \le \frac{p}{p-1}\left[ \frac{p-1}{p}|g|^{p/(p-1)}+ \frac{1}{2}f^2 \right]\ ,
tH_{\mathrm{FS}}^{A_3}\left({\alpha_,\beta \atop \gamma,\delta}; t;{q_1, p_1 \atop q_2, p_2}\right) =tH_{\mathrm{III}(D_6)}(\alpha, \beta; t; q_1,p_1 )+tH_{\mathrm{III}(D_6)}(\gamma, \delta; t; q_2,p_2 )-p_1p_2(q_1q_2+t).
\begin{array}{| c | c | c |}\hline{{{\bf D}}}&{{\lambda}}&{{\eta}}\\ \hline{{3}}&{{- 1}}&{{- 1}}\\{{5}}&{{- 1}}&{{+ 1}}\\{{7}}&{{+ 1}}&{{- 1}}\\{{9}}&{{+ 1}}&{{+ 1}}\\{{1 1}}&{{- 1}}&{{- 1}}\\ \hline{{}}\\ \end{array}
\|\Psi_\lambda \ast (Qf)\|_s \leq c_1\|\, |\cdot|^{2-N}\|_{\frac{N}{N-2},w}\|Qf\|_{t_1}+c_2\|\, |\cdot|^{\frac{1-N}{2}}\|_{\frac{2N}{N-1},w}\|Qf\|_{t_2}&\leq \Bigl\{c_1\|\, |\cdot|^{2-N}\|_{\frac{N}{N-2},w}|\Omega|^{\frac1{t_1}}+c_2\|\, |\cdot|^{\frac{1-N}{2}}\|_{\frac{2N}{N-1},w}|\Omega|^{\frac1{t_2}}\Bigr\}\|Q\|_\infty \|f\|_\infty=:C(\lambda)\|f\|_\infty,
J_\nu(z) = \left(\frac{2}{\pi z}\right)^{\!\!\frac{1}{2}}\left(\cos(\mu)\! \!\sum_{m = 0}^{M-1}(-1)^m\frac{a_{2m}(\nu)}{z^{2m}}- \sin(\mu)\!\!\sum_{m=0}^{M-1}(-1)^m\frac{a_{2m+1}(\nu)}{z^{2m+1}}\right) + R_{\nu,M}(z),
\begin{array}{r c l}{\delta_{\omega}V^{a}}{=}{\omega^{b}\partial_{b}V^{a}- V^{b}\partial_{b}\omega^{a}+{\textstyle{\frac{1}{2}}}\partial_{b}\omega^{b}V^{a},}{\delta_{\omega}X}{=}{\omega^{a}\partial_{a}X .}\end{array}
\frac{4n}{(1-z)^{N}(1+z)^{M}}\frac{d}{dz}B(z) \qquad{}=-2n\big( P_{n}^{(N,M)}(z) \big)^{2}+\big( 1-z^{2}\big) \frac{N+M+n+1}{2}\big( P_{n-1}^{( N+1,M+1)}(z) \big)^{2},
\mathbf{0}=X_0 \xrightarrow{\alpha_1}X_1\cdots \xrightarrow{\alpha_{{\bar\imath}}}X_{{\bar\imath}}\xrightarrow{\alpha_{{\bar\imath}+1}}X_{{\bar\imath}+1}\xrightarrow{}\cdots \xrightarrow{\alpha_n}X_n=X
\sum_{d|n}s_{d,k}^{(-1)}= \sum_{m=0}^n \sum_{j=1}^{n-m}\left( s_o(n-m, j) - s_e(n-m, j)\right) s_{j,k}^{(-1)}\cdot p(m) = \sum_{m=0}^n \delta_{n-k,m}\cdot p(m) = p(n-k),
{\rm det}\, ( \mathbb{D}^2_\lambda i.[(\nabla_\lambda i)^{-T}\nabla_\lambda g])=k^2 e^{k \log^2\frac{\lambda_1}{\lambda_2}}\left(\log^2 \frac{\lambda_1}{\lambda_2}\right)\frac{1}{(\lambda_1-\lambda_2)^2}\left(\frac{1}{\lambda_1}+\frac{1}{\lambda_2}\right)^2\geq 0.
G_{U\times\overline{V}}(z,\overline{w})=J_U^{\widetilde{U}}(z)\cdot\widetilde{G}_{\widetilde{U}\times \overline{\widetilde{V}}}(\widetilde{z},\overline{\widetilde{w}})\cdot\overline{J_V^{\widetilde{V}}(w)}^t
a_2 = \frac{3281}{13824}- \frac{a_1}{64}, a_3 = \frac{203}{4608}- \frac{9 a_1}{256},a_4 = \frac{1763}{13824}+ \frac{3 a_1}{128}, &a_5 = -\frac{277}{13824}- \frac{a_1}{256},a_6 = \frac{1133}{1728}+ \frac{3 a_1}{64}, a_7 = -\frac{77}{1728}- \frac{a_1}{64}.
\begin{cases}u_h^{l}(z) =\max \left(\cfrac{-f_{l}(z,u_h^{l}(z))h^{2}}{4}+\overline{u}_h^l(z) - \sum\limits_{p \neq l}\overline{u}_h^p(z) , \, 0\right),\;\;z\in\Omega_h^o, u_h^{l}(z) =\phi_h^{l}(z)= \phi_{l}(z),\;\;z\in \partial\Omega_h.\end{cases}
i\sum_{n=1}^\infty{1\over n}\Bigl[e^{i{n\over L}(r+\ell)}- e^{i{n\over L}(r-\ell)}\Bigr]={1\over L}-i\ln{\sin{r+\ell\over 2L}\over \sin{r-\ell\over 2L}},\quad0<r-\ell\le r+\ell<2\pi L.
\left(I_{0+}^{\alpha,0,\beta,\beta',\gamma}f\right)(x)=\left(I_{0+}^{\gamma,\alpha-\gamma,-\beta}f\right)(x),\ \ \ \ \left(I_{-}^{\alpha,0,\beta,\beta',\gamma}f\right)(x)=\left(I_{-}^{\gamma,\alpha-\gamma,-\beta}f\right)(x)
u(x) =O\left(|x|^{-(n-2)}\log\frac{1}{a_b(x)}\right) |D^iu(x)| =O\left(|x|^{-(n-2+i)}a_b(x)^{-i}\right) u(x) =o\left(|x|^{-(n-2)}\log\frac{1}{a_b(x)}\right) |D^iu(x)| =o\left(|x|^{-(n-2+i)}a_b(x)^{-i}\right)
E^a_s \;=\; \Pi^a\:E_s \:+\: s\: \frac{\varepsilon}{2 \mu_k}\left( \Pi^a\:E_s\:{\cal{B}}_0\: E_{\bar{s}}+ E_{\bar{s}}\:{\cal{B}}_0\: \Pi^a\:E_s \right) \:+\:{\cal{O}}(\varepsilon^2)
\alpha \ = \ \left( \frac{c}{4 \kappa_{( 1 )}}+ \left( \left( \frac{c}{4 \kappa_{( 1 )}}\right)^{2}+ \left( \frac{\kappa_{( 2 )}}{2 \kappa_{( 1 )}}\right)^{3}\right)^{\frac{1}{2}}\right)^{\frac{1}{3}}\ ,
K_{m}^{\nu}(z,w)=\frac{\nu}{\pi}\left( \sum\limits_{n=0}^{m-1}\frac{\nu^n(z \overline{w})^{n}}{n!}+\frac{(z \overline{w})^{m}}{(m!)^2}\,{_2F_2}\left(\begin{array}{c}1, 1 m+1,m+1 \end{array}\bigg | \nu z \overline{w}\right)\right) .
4S \sum_{\substack{\beta\geq\frac{1}{2}-\frac{R}{\log q}0\leq\gamma\leq \frac{2S}{3\log q}L(\beta+i\gamma,f)=0}}\cos\left(\frac{\pi \,\gamma\, \log q}{2S}\right) \sinh\left(\frac{\pi \big(R+(\beta-1/2)\log q\big)}{2S}\right) \; \leq \; I_1(f)+I_2(f)+I_3(f),
\|f\|_{\Gamma_q(\sigma)}^q=\int_0^1\bigg(\frac1t\int_0^tf^*(s)\,ds\bigg)^q\sigma(t)\,dt+\bigg(\int_0^1 f^*(s)\,ds\bigg)^q\int_1^\infty\sigma(t)\,\frac{dt}{t^q}.
x_{d}^{( 1 / 2 )}( u , \xi ) = \biggl [{\frac{\sigma ( 2 \xi - u )}{\sigma ( 2 \xi ) \sigma ( u )}}+{\frac{\sigma ( 2 \xi - u - \omega_{1})}{\sigma ( 2 \xi ) \sigma ( u + \omega_{1})}}\exp [ 2 \eta_{1}\, \xi ] \biggr ] .
\sum_{L^{2B}<u\le y}\left|{\frak P}_{\ell}(u,L^B)-{\frak S}_{\ell}(u)\right|^2 =\sum_{L^{2B}<u\le y}\left|{\frak P}_{\ell}^{\prime}(u,L^B)f_{\ell}(u,L^B)-{\frak S}_{\ell}^{\prime}(u)f_{\ell}(u)\right|^2 &\ll L^2\sum_{L^{2B}<u\le y}\left|{\frak P}_{\ell}^{\prime}(u,L^B)-{\frak S}_{\ell}^{\prime}(u)\right|^2+L^{10\ell}R_{f}.
\sum_{b = 1}^{N}\Big ( a_{\mu}^{( b )}, j_{\mu}^{( b )}\Big ) = \sum_{I = 1}^{N}\sum_{b = 1}^{N}\Big ( U_{I b}A_{\mu}^{( I )}, j_{\mu}^{( b )}\Big ) : = \sum_{I = 1}^{N}\Big ( A_{\mu}^{( I )}, J_{\mu}^{( I )}\Big ) \; ,
\mathbf r^1 =d\mathbf e^1\times\mathbf v=(0,-dv_3,dv_2), \mathbf s^1 =(\mathbf r^1\times\mathbf v)/d=(\mathbf e^1\times\mathbf v)\times\mathbf v=\left(-v_2^2-v_3^2,v_1v_2,v_1v_3\right).
J_{-}= J_{+}^{T}=\left[\begin{array}[c]{cccccc}0 0 0 0 0 0 \sqrt{n^{.}1}0 0 0 0 0 0 \sqrt{(n-1)^{.}2}0 0 0 0 : : : : : : 0 0 0 \sqrt{2^{.}(n-1)}0 0 0 0 0 0 \sqrt{1^{.}n}0\end{array}\right]
U(t):= \{x^{\ast}\in X^{\ast}:|\langle x^{\ast},e_{i}\rangle |\leq \gamma(t),\;i=1,...,p\}, V(t):= \{x^{\ast}\in L^{\ast}:|\langle x^{\ast},e_{i}\rangle |\leq \gamma (t),\;i=1,...,p\}.
X^{-1}Y=\left(\begin{array}{cccc}7/5 -9/10 -4/5 -7/5 -2/5 2/5 -1/5 2/5 -1/5 1/5 2/5 1/5 7/5 -9/10 -4/5 -9/10 \end{array}\right) \left( \begin{array}{c}3e_1-8g_1 3e_1-8g_1 0 0 \end{array}\right).
\begin{array}{l l}{{Z_{1 2}=}}&{{\frac{1}{s}( \sum_{l = 1}^{g ( s )}e^{- 4 \pi i ( l - 1 ) / n s}+ e^{4 \pi i / n}\sum_{l = g ( s ) + 1}^{s}e^{- 4 \pi i ( l - 1 ) / n s})}}\\{{}}&{{\times ( \sum_{p = 0}^{p_{m a x}}e^{2 \pi i p / n}\sum_{u ( p ) > j \geq l ( p )}e^{- 2 \pi i ( j - 1 ) / M}) ,}}\\ \end{array}
= \frac{1}{1 6 R}\ \Big [ \ \frac{3}{r^{5}}+ \sum_{m = 1}^{\infty}e^{- m r / R}( \frac{m^{2}}{R^{2}}\frac{1}{r^{3}}+ \frac{m}{R}\frac{3}{r^{4}}+ \frac{3}{r^{5}}) 2 \cos ( m x_{1 1}/ R ) ~ \Big ]
\mathbf{x}^{AA^{\prime}}=-\frac{1}{2}\theta^{AA^{\prime}}\mathbf{m}-\frac{1}{2}\theta^{AA^{\prime}}\overline{\mathbf{m}}+\frac{1}{2}[\omega_{B^{\prime}}^{A^{\prime}}\theta^{AB^{\prime}}-\omega_{B}^{A}\theta^{BA^{\prime}}]\mathbf{m}\overline{\mathbf{m}}.
\lim_{b \rightarrow 0}I ( \lambda , b ) = \int \exp \{- \int_{- \frac{L}{2}}^{\frac{L}{2}}\sum_{\alpha}\frac{\chi^{* \alpha}\chi^{\alpha}}{B^{\alpha}L}[ 1 - \exp \{\frac{B^{\alpha}}{\lambda}( t - \frac{L}{2}) \}] d t \}d \chi^{*}d \chi =
\left({}^{c}D_{-}^{\alpha,\alpha',\beta,\beta',\gamma}t^{-\rho}\right)(x)=\frac{\Gamma\left(\beta'+\rho+m\right)\Gamma\left(-\alpha-\alpha'+\gamma+\rho\right)\Gamma\left(-\alpha'-\beta+\gamma+\rho+m\right)}{\Gamma\left(\rho\right)\Gamma\left(-\alpha'+\beta'+\rho+m\right)\Gamma\left(-\alpha-\alpha'-\beta+\gamma+\rho+m\right)}x^{\alpha+\alpha'-\gamma-\rho}.
\left\{\begin{array}{c}{\left( \begin{array}{c l c}{\frac{\delta - \sqrt{\delta^{2}+ \omega^{\ast}\omega}}{\omega}}{,}{1}\end{array}\right)}{\left( \begin{array}{c l c}{\frac{\delta + \sqrt{\delta^{2}+ \omega^{\ast}\omega}}{\omega}}{,}{1}\end{array}\right)}\end{array}\right\}.