harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
S_{T\left( G\circledS \mathfrak{g}^{\ast}\right)}^{\prime}=\left\{\left( TR_{g}\left( \frac{\delta \bar{H}}{\delta \mu}\right) ,ad_{\frac{\delta\bar{H}}{\delta \mu}}^{\ast}\mu -TR_{g}^{\ast}\frac{\delta \bar{H}}{\delta g}\right) \in T\left( G\circledS \mathfrak{g}^{\ast}\right):\left( g,\mu \right) \in G\circledS \mathfrak{g}^{\ast}\right\}
E_{\mathrm{v a c}}= - \frac{1}{2}\sum_{j}( 2 G_{j}+ 1 ) | \epsilon_{j}| - \int_{0}^{\infty}\frac{d k}{2 \pi}\, \sqrt{k^{2}+ m_{f}^{2}}\, \frac{d}{d k}\, \overline{{{\delta}}}( k ) + E^{( 1 , 2 )}+ E^{( 3 , 4 )}\, ,
\frac{v^{2}}{\lambda}= \frac{( a \Lambda )^{2}}{8 \pi^{2}}+ \frac{1}{2}\hat{m}^{2}a^{2}\left( 1 + \frac{\lambda_{R}}{1 6 \pi^{2}}\bar{c}\right) \bar{I}_{2}+ Z_{\phi^{2}}\left( \frac{v_{\mathrm{R}}^{2}}{\lambda_{\mathrm{R}}}\right) ,
(\begin{matrix}n-1\\k\end{matrix})=\frac{(n-1)(n-2)\cdot\cdot\cdot(n-k)}{1\cdot2\cdot\cdot\cdot k}=\prod_{i=1}^{k}\frac{n-i}{i}\equiv\prod_{i=1}^{k}\frac{-i}{i}=(-1)^{k}modn.
\alpha_k=|E(uw)|^{2(n+k)}E(|w|^2) -(1+n)\mu^n|E(uw)|^{2(k-1)}\chi_{S_0}(E(|w|^2))^2E(|u|^2) &+\mu^{1+n}|E(uw)|^{2(k-1)}\chi_{S_0}E(|w|^2),
t^+_{k}(2) = \tfrac{1}{6}\left( 2^k - (-1)^{k}\left(1+ (-3)^{\left\lceil \frac{k}{2}\right\rceil}\right)\right) t^-_{k}(2) = \tfrac{1}{6}\left( 2^k - (-1)^{k}\left(1- (-3)^{\left\lceil \frac{k}{2}\right\rceil}\right)\right)
\sum \sum_{b < a}^{N}\lambda_{a}\lambda_{b}= \sum_{a = 1}^{N}r_{a}q_{a}^{2}- \sum_{a = 1}^{N}q_{a}^{2}\sum_{b = 1}^{N}r_{b}+ \sum \sum_{b < a}^{N}r_{b}r_{a}
||(u_n^1)_+||_{L^{p+1}}^{p+1}=||(u_n^1(\cdot+y_n^1)-v_1)_+||_{L^{p+1}}^{p+1}+||(v_1)_+||_{L^{p+1}}^{p+1}+o(1) &=||(u_n^1-v_1(\cdot-y_n^1))_+||_{L^{p+1}}^{p+1}+||(v_1)_+||_{L^{p+1}}^{p+1}+o(1) &=||(u_n^2)_+||_{L^{p+1}}^{p+1}+||(v_1)_+||_{L^{p+1}}^{p+1}+o(1),
T_{(A,j+n+i+n\ \left( \operatorname{mod}2n\right) )}=-T_{(A,j+n+i\ \left( \operatorname{mod}2n\right) )}, T_{(A,j+i\ \left( \operatorname{mod}2n\right) )}=-T_{(A,(j+i)+n\ \left( \operatorname{mod}2n\right) )},
\psi (a + b ) = \big( (a / \alpha) \oplus (b / \alpha), \, \psi_2 (a) +_o \psi_2 (b) \big) = \big( \psi_1 (a) \oplus \psi_1 (b), \, \psi_2 (a) +_o \psi_2 (b) \big) = \big( \psi_1 (a), \psi_2 (a) \big) \boxplus \big(\psi_2 (b), \psi_2 (b) \big) = \psi (a) \boxplus \psi (b)
\tilde a(x):= \begin{cases}a(2-x), x \in [1,2], a(x), x \in [0,1], a(-x), x \in [-1,0],\end{cases}\tilde b(x):= \begin{cases}b(2-x), x \in [1,2], b(x), x \in [0,1], b(-x), x \in [-1,0]\end{cases}
e^{\textstyle -J\alpha_-\Phi(\sigma, \tau )}={1\over \sqrt{\varpi}}\sum_{M=-J}^J\> (-1)^{J-M}\>e^{ih(J-M)}\>\xi_M^{(J)}(x_+)\,{\overline \xi_{-M}^{(J)}}(x_-) \sqrt{\varpi}
e_R(\mathcal I(1)_t^{[d_1]},\ldots,\mathcal I(r)_t^{[d_r]};R)=\frac{1}{s_t^d}e_R((I(1)_{t,s_t})^{[d_1]},\ldots,(I(r)_{t,s_t})^{[d_r]};R)
\mathbb{E}L=\sum_{i=1}^{n}p_{i}l_{i}\le\sum_{i=1}^{n}p_{i}(-log_{2}p_{i}+1)=-\sum_{i=1}^{n}p_{i}log_{2}p_{i}+\sum_{i=1}^{n}p_{i}=H(X)+1.
E_1(J_1)\cdots E_k(J_k)F(V)^{-1}=E_1(J_1)\cdots E_{k-1}(J_{k-1})F(V^{(1)})^{-1}E_k(J'_k) &=E_1(J_1)\cdots E_{k-2}(J_{k-2})F(V^{(2)})^{-1}E_{k-1}(J'_{k-1})E_k(J'_k) &=\cdots=F(V^{(k)})^{-1}E_1(J'_1)\cdots E_{k-1}(J'_{k-1})E_k(J'_k),
|S_6|^2 \ll M \log^3 u \sum_{h=1}^H \sum_{\frac{X_3}{M_1}< n, n+h \leq \frac{X_4}{M}}\Lambda(n) \Lambda(n+h) \frac{X}{qh \log^{13}X}+ M \log^3 u \, \cdot \frac{X_3}{M}\, \log \frac{X_3}{M}&\ll \frac{X X_3 \log^3 u}{\log^{11}X}+ X \log^3 u \, \log X \ll \frac{X^2}{\log^6 X}.
A ( x^{+}) = a{\frac{x^{+}+{\frac{\lambda ( 1 - c ) x_{0}^{+}}{2}}( x^{+}- x_{0}^{+})}{1 +{\frac{\lambda ( 1 - c )}{2}}( x^{+}- x_{0}^{+})}}, \x^{+}> x_{0}^{+}
\tilde{Y}_{w_{j}}\stackrel{\sigma_{j}}{\rightarrow}\tilde{Y}_{w_{j - 1}}\stackrel{\sigma_{j - 1}}{\rightarrow}\cdots \stackrel{\sigma_{2}}{\rightarrow}\tilde{Y}_{w_{1}}\stackrel{\sigma_{1}^{- 1}}{\rightarrow}Y_{i_{1}}\rightarrow F .
\frac{t_n-s_l}{t_n -s_m}\bigg( \frac{\overline{A(s_l)}}{\bar s_l-\bar t_n}- \overline{B(s_l)}\bigg) = d_{l,m}\bigg( \frac{\overline{A(s_m)}}{\bar s_m-\bar t_n}- \overline{B(s_m)}\bigg).
<\tilde \Sigma(k_1,\theta_1)\tilde \Sigma(k_2,\theta_2)>=-\frac{1}{2 N}I(k_{1}^{2})\frac{D^2-2m}{k^{2}_{1}-4m^2}\delta^5_{12}\simeq-\frac{16i}{N}\frac{D^2-2m}{\sqrt{-k^{2}_{1}+4m^2}}\delta^3(k_1-k_2)\bar \delta_{12}.
\begin{aligned}C_{3,4}(r) =\frac{V_3^{4/3}(r)}{(\pi^2/2)^{1/3}\,V_4(r)}, C_{2,3}(r) =\frac{V_2(r)}{(\pi^2/2)^{1/3}\,V_3^{2/3}(r)}, C_{1,2}(r) =\frac{V_1^{2/3}(r)}{(\pi^2/2)^{1/3}\,V_2^{1/3}(r)},\end{aligned}
T_i^{-1}(u)(v\otimes v_\lambda)= \sum_hu_h (k_h v\otimes k_h v_\lambda)=\sum_h q^{\langle\lambda,h\rangle}u_h(k_h v\otimes v_\lambda) = \sum_h q^{\langle\lambda,h\rangle}u_hk_h v\otimes v_\lambda, T_i^{-1}(u)v\otimes v_\lambda= \sum_h u_h k_h v\otimes v_\lambda,
d s^{2}= \left( 1 +{\frac{r_{0}^{2}}{r^{2}}}\right)^{- 1}[ - d u d v + K ( u , x , y ) d u^{2}] + \left( 1 +{\frac{r_{0}^{2}}{r^{2}}}\right) d x_{i}d x^{i}+ d y_{i}d y^{i}
\, c > 0 \, , \liminf_{r\to \infty}\frac{1}{r}\log \left( \int_{B_r \cap \Omega}w^2 \psi_+^2 \right) > 0 \, ; \, c = 0 \, , \liminf_{r\to \infty}\frac{1}{r^2}\int_{B_r\cap\Omega}w^2 \psi_+^2 > 0
s_{\mathrm{id}}:=s_1=\{0\}\ ,\s_{(1,2)}:=s_2=\tau_{(1,2),1}\ ,\s_{(1,2,3)}:=s_3=\tau_{(1,2,3),2}\ , &s_{(2,3)}:=s_4=\tau_{(2,3),2}\ ,\s_{(1,3,2)}:=s_5=\tau_{(1,3,2),1}\ ,\s_{(1,3)}:=s_6=\sigma_{(1,3)}
\mathcal{E}^D(f,g) :=\frac{1}{2}\int_D \left(\sum_{i=1}^d\dfrac{\partial f}{\partial x_i}\cdot\dfrac{\partial g}{\partial x_i}\right)dx_1...dx_d, \intertext{with domain}W_0^{1,2}(D) :=\overline{C_c(D)}^{W^{1,2}(D)},
\|f\|^2_2=\int_{\mathcal O}\sum_{k\in \mathbb N_0}\left|\widehat{f}\left(\xi+u(k)\right)\right|^2 d\xi=\int_{\mathcal O}|r(\xi)|^2\sum_{k\in \mathbb N_0}\left|\widehat{\varphi}\left(\xi+u(k)\right)\right|^2 d\xi=\|r\|^2_{L^2\left(\mathcal O, w\right)},
0<B_1(\epsilon) :=\frac{1}{\gamma}(1-\epsilon)^2 \leq\frac{\log \Gamma_1(x)}{\log x}&\qquad\qquad\leq \frac{1}{\gamma}(1+\epsilon)^2=:B_2(\epsilon),x\in(0,g(\delta(\epsilon)/2)], g(\delta(\epsilon)/2) <1, g'(x)>0 x\in(0,\delta(\epsilon)).
|M(t) - j| \leq \frac{1}{N}\sum_{\ell = 1}^{\infty}\ell X_{j + \ell}(t) + \frac{1}{N}\sum_{\ell = 1}^j \ell X_{j - \ell}(t) &= \frac{1}{N}\sum_{\ell = 1}^{\infty}\ell X_{j + \ell}(t) + \frac{1}{N}\sum_{\ell = 1}^{j - k^* - 1}\ell X_{j - \ell}(t) + \frac{1}{N}\sum_{i=0}^{k^*}(j - i) X_i(t).
\biggr|\frac{\sum_{l=1}^\eta c^{\bf a}_l\gamma_l^{k+2}}{\sum_{l=1}^\eta c^{\bf a}_l\gamma_l^{k+1}}-\frac{\sum_{l=1}^\eta c^{\bf a}_l\gamma_l^{k+1}}{\sum_{l=1}^\eta c^{\bf a}_l\gamma_l^{k}}\biggr|<\epsilon.
( u_{t_1}^*u_0,u_{t_2}^*u_{t_1},....,u_{t_{N}}^*u_{t_{N-1}},u_{t_N}1_{mN}) =(u_{t_1}^*,....,u_{t_N}^*,1_m,1_{mN})( u_0,u_{t_1},....,u_{t_N},1_{mN}).
h(E_1,E_1) = -\frac{\langle E_1,d\xi_qE_1\rangle}{\langle\eta,\xi\rangle}= -\frac{\lambda_1}{\langle\eta,\xi\rangle}\ \ \mathrm{and}h(E_2,E_2) = -\frac{\langle E_2,d\xi_qE_2\rangle}{\langle\eta,\xi\rangle}= -\frac{\lambda_2}{\langle\eta,\xi\rangle},
L_{D R}= -{\frac{a}{4}}( B + \gamma A )_{\mu \nu}( B + \gamma A )^{\mu \nu}-{\frac{m}{6 \pi}}B_{\mu}B^{\mu}+ \alpha \epsilon^{\mu \nu \lambda}( B + \gamma A )_{\mu}\partial_{\nu}( B + \gamma A )_{\lambda}.
\|u_h\|^2_{L^2\left ((0,1)\times \left (0, \frac{2}{\pi}\right )\right )}\geq \sum^{\infty}_{n=-\infty}n^2 \pi^2\frac{1}{|n|^{2\beta}}= \sum^{\infty}_{n=-\infty , \n\ne 0}\frac{\pi^2}{|n|^{2\beta -2}}\, .
S^{\left( \Sigma_{1 + 1}\right)}= - 2 \pi i \theta f \left( \overline{{t_{*}}}\right) g \left( t_{*}\right) \int_{\Sigma_{1 + 1}}\left[ i \overline{{\psi}}^{L}\sigma^{a}D_{a}\psi^{L}- \overline{{\psi}}^{L}m \psi^{L}\right] ,
\mathcal E_1 \subset \left\{\max_{|z|\leq r}\log |f_L(z)|^2 \leq L\log\frac{1}{1-r^2}-L \delta\right\}&=\left\{\max_{|z|=r}\log |f_L(z)|^2 \leq L\bigl(\log\frac{1}{1-r^2}- \delta\bigr) \right\}.
X^{r,\mathbf{w}}_{T_{k-1}^{(n)}}=f_{(T_{k-1}^{(n)}-T_{k}^{(n)})}{(X^{r,\mathbf{w}}_{T_{k}^{(n)}},U_{T_{k}^{(n)}})}, \,\,\,\,k=M_r^{(n)},\ldots,1
0 = \displaystyle\int_{\mathbb{R}^N}(- \Delta)^{\frac{\gamma}{2}}\tilde{v}^k(- \Delta)^{\frac{\gamma}{2}}\tilde{\delta}_{i}^kdx = \displaystyle\int_{\mathbb{R}^N}(- \Delta)^{\frac{\gamma}{2}}\tilde{v}^k(- \Delta)^{\frac{\gamma}{2}}\frac{\partial\tilde{\delta}_{i}^k}{\partial \lambda_{i}^k}dx i = 1, 2 k,
G_{R}(X)=\left[{\rm Diag}\left(\overline{\cal C}{\rm diag}(S(H_1))\right)0\right] +\left[\begin{array}{ccccc}\overline{\gamma}_{1}S(H_{a_1a_1}) 0 0 0 0 0 \ddots 0 0 0 0 0 \overline{\gamma}_{r}S(H_{a_ra_r}) 0 0 0 0 0 \overline{\gamma}_{r+1}H_{bb}\overline{\gamma}_{r+1}H_{bc}\end{array}\right]+o(\|H\|)\,.
h_j(t) =\log\Big(1+\sum_{l\in[n]\setminus\{b_j\}}\exp\big(a_j^\top (x_l+t(y_l-x_l))-a_j^\top (x_{b_j}+t(y_{b_j}-x_{b_j}))\big)\Big) &=\log\Big(\sum_{l\in[n]}\exp\big(a_j^\top (x_l+t(y_l-x_l))\big)\Big)-a_j^\top (x_{b_j}+t(y_{b_j}-x_{b_j})),
f_{9}^A=2\zeta(3) r_A e^{-2\phi}+ \frac{2\pi^2}{3 r_A}+ 0+~ 4\pi e^{-\phi}\sum_{m\ne 0}\sum_{n\ne 0}\left|\frac{m}{n}\right| K_1\left(2\pi r_A e^{-\phi}|mn|\right) e^{2\pi i m n{\cal A}}
\begin{array}{ll}\|w\|_{\mathcal{B}_{0,\ell}^{0,1}}+ \frac{1}{\sqrt{\beta + C_{\eta}}}\big\|w|_{y=0}\big\|_{A^0}begin{align*}12pt]\leq C(\lambda T)\|w|_{t=0}\|_{\mathcal{A}_{\ell}^{0}(\mathbb{R}_{+}^2,t=0)}+ \frac{C(\lambda T)}{\max\{\sqrt{\beta -C_{\eta}},\sqrt{\delta}\}}\big\|w|_{y=0,t=0}\big\|_{A^0(\mathbb{R},t=0).}\end{array}
F ( x ) = F ( \vartheta , 1 - \vartheta ; 1 ; x ) = \frac{1}{\pi}\sin ( \vartheta \pi ) \int_{0}^{1}d y \, y^{- \vartheta}( 1 - y )^{- ( 1 - \vartheta )}( 1 - x y )^{- \vartheta}
\displaystyle\sum_{k=1}^{\infty}\dfrac{k+1}{\big(x^{\sigma-\frac{1}{2}}\big)^k}\Bigg|\displaystyle\sum_{n\leq x}\Lambda(n) \Bigg(\dfrac{1}{n^{\sigma}(k\log x+\log n)^{2m+2}}-\dfrac{1}{x^{2\sigma-1}\,n^{1-\sigma}((k+2)\log x-\log n)^{2m+2}}\Bigg)\Bigg| & \ \ \ \ll_{c}\dfrac{x^{1-\sigma}}{(1-\sigma)^{2}(\log x)^{2m+3}}.
p_{\sigma,x_0}(\gamma\gamma',a)=I_{x_0}(\sigma(a))(\gamma\gamma'x_0)=I_{x_0}(\sigma(a^{(1)}))(\gamma x_0)I_{\gamma x_0}(\sigma(a^{(2)}))(\gamma\gamma'x_0) =I_{x_0}(\sigma(a^{(1)}))(\gamma x_0)I_{x_0}(\sigma(a^{(2)}))(\gamma'x_0)=p_{\sigma,x_0}(\gamma,a^{(1)})p_{\sigma,x_0}(\gamma',a^{(2)}),
\varphi\left[ f(x+h)\right] - \varphi\left[ f(x)\right] = \sum_{1\leq j \leq d}h_j \int_0^1 \varphi\left[\partial_{x_j}f(x+th)\right]dt & = \sum_{1\leq j \leq d}h_j \varphi\left( \int_0^1 \partial_{x_j}f(x+th)dt\right).
M_{U \ \scriptstyle \mathrm{m i n}}^{E_{8}- E_{6}}\approx 5 . 4 9 \times 1 0^{1 7}\times g_{U}^{E_{8}- E_{6}}\times \frac{1}{\sqrt{1 + 0 . 0 8 \times \left( g_{U}^{E_{8}- E_{6}}\right)^{2}}}\mathrm{G e V}.
\eta^{\prime\prime\prime}(x)-\lambda\eta^{\prime\prime}(x)-2\lambda\eta^{\prime}(x)+\lambda^2\eta(x)=0 , \ \ \forall x \in (0,1), &\eta(0)=0,\ \ \eta(1)=0,\ \ \eta^{\prime\prime}(0)=\eta^{\prime\prime}(1).
\mathrm{U}_{0}({\tau}^{( 1 )}) \mathrm{U}_{0}({\tau}^{( 2 )}) = \exp \{2 \pi i{\omega}_{2}({\tau}^{( 1 )},{\tau}^{( 2 )}) \}\mathrm{U}_{0}({\tau}^{( 1 )}+{\tau}^{( 2 )}) ,
T_{0}^{\mu \nu}= - \left( \frac{g_{s t r}\alpha^{\prime}}{2}\right)^{2}\prod_{u}\frac{\delta ( \sqrt{\alpha^{\prime}}k_{u})}{\sqrt{\tau_{2}}}\left( \frac{\eta^{\mu \nu}}{2 \pi \alpha^{\prime}}\prod_{o}\frac{\delta ( \sqrt{\alpha^{\prime}}k_{o})}{\sqrt{\tau_{2}}}X_{1 , 1}+ \frac{k_{\mu}k_{\nu}}{4}2^{- ( d + 1 )}X_{1 , 0}\right)
\biggl\{- 4z(1-z) \frac{d^2}{dz^2}+ 2 [-8b_2z^3- 4(b_1-2b_2)z^2 + (4a+4b_1+3)z - 4a + 2l + d -3] \frac{d}{dz}+ [B_2 + 4b_1^2 - 4b_2(4a+4b_1+5)]z^2 + [B_1 - 2b_1(4a+2b_1+3) + 4b_2(4a-2l-d+5)]z - A + 2a(2a+1) + 2b_1(4a-2l-d+3)\biggr\}\phi(z) = 0
o(\alpha_1,\alpha'_2)=\left((\widehat\psi'_{21}\circ \widehat\psi_{1}- \widehat\psi_{2}\circ \widehat\psi'_{21})-(\widehat d'_{2}\circ \widehat{\frak h}+ \widehat{\frak h}\circ \widehat d_{1})\right)_{\alpha'_2\alpha_1}.
\xi(t;\xi_f)=\left(\xi_f+\frac{b}{a}\right)e^{a(t-t_f)}-\frac{b}{a}\ \ ,\ \ \gamma(\xi_f)=\frac{1}{a}\left(\xi_f+\frac{b}{a}\right)\left(1-e^{-a(t_f-t_i)}\right)-\frac{b}{a}(t_f-t_i)\ \ ,
\Gamma_1={1\over \lambda_1}{1\over \lambda_2}\left({2\over\lambda_1+\lambda_2}\right)^2,\quad\Gamma_2={1\over 2}{1\over \lambda_1^2}\left({2\over \lambda_1+\lambda_2}\right)^2,\quad\Gamma_3={1\over \lambda_1^3}{2\over \lambda_1+\lambda_2}.
u^N_{t}(x) - u_{t}(x) = e^{t\Delta}(u^N_{0}- u_{0})(x) - \int_{0}^t \nabla \cdot e^{(t-s)\Delta}\left( \langle \mu_s^N, V^N (x-\cdot) F\big( K\ast u^N_s (\cdot)\big) \rangle - u_{s}F(K\ast u_s) (x) \right) \, ds &- \frac{1}{N}\sum_{i=1}^N \int_0^t e^{(t-s)\Delta}\nabla V^N (x-X_s^{i,N})\cdot dW^i_s .
P_{s}({\bf h^{i}_{sp,n^{*}}}(k),{\bf w_{n^{*}}}(k)) = \frac{Q_{p}}{\left|\frac{1}{\sqrt{M}}\sum_{i=1}^{M}e^{j\theta_{n^{*},i}(k)}\, h^{i}_{sp,n^{*}}(k)\right|^{2}}.
\frac{P_{\mathrm{1 C G}}}{m_{\mathrm{\scriptscriptstyle{M T}}}^{2}}\; \sim \; 1 - \frac{\pi T}{m_{\mathrm{\scriptscriptstyle{M T}}}g_{\mathrm{\scriptscriptstyle{M T}}}}\; \gg \; \frac{1}{2}.
J_\mu(U_*,a) = \lambda_1(U_*,a) +\mu m \leq \lambda_1(tU_1,a) + \mu m \leq t^{-2}[\lambda_1(U_1,a)+\sigma|U_1|^{-\frac{2}{d}}] + \mu t^{d}|U_1| =: f(t)
f_\#((\gamma, g)) = f_\# \big((\gamma, id_G) (id_\Gamma, g) \big) = f_\# \big((\gamma, id_G)\big) f_\# \big((id_\Gamma, g) \big) &= \left(\alpha(\gamma), \rho(\gamma) \right)(id_\Gamma, L(g)) = (\alpha(\gamma), \rho(\gamma)L(g)).
K_{2,r}\geq\frac{\left( \sum_{i,j=1}^{N}\left\vert \left(y_{ij}\right) \right\vert^{2}\right)^{\frac{1}{2}}}{\left({\displaystyle\int\limits_{I^{2}}}\left\vert \sum_{i,j=1}^{N}r_{i}\left( t\right) r_{j}\left( s\right)y_{ij}\right\vert^{r}dtds\right)^{\frac{1}{r}}}& =\frac{2}{2^{^{\frac{2r-2}{r}}}}=2^{\frac{2-r}{r}}.
< \varphi^{p}( \tau ) > = \int d \varphi \sum_{n , n^{\prime}}\bar{a}_{n}\chi_{n}( \varphi ) e^{E_{n}\tau}\varphi^{p}e^{- E_{n^{\prime}}\tau}a_{n^{\prime}}\chi_{n^{\prime}}( \varphi )
\frac{\rm d}{{\rm d}\tau}\left[\sum_{j=1}^dJ_{ij}(\tau,x) \bar{\bf b}_j(x)\right]= \frac{\rm d}{{\rm d}\tau}\left[\sum_{j=1}^d\frac{\partial \Phi^i_{-\tau}}{\partial x_j}(x)\bar{\bf b}_j(x)\right]=-\sum_{j=1}^d\frac{\partial}{\partial x_j}\Big(\bar{{\bf b}}_i\left(\Phi_{-\tau}(x)\right)\Big)\bar{\bf b}_j(x).
Q_i \big( \mu(\omega_{q_0}),\dots,\mu(\omega_{q_{\rho(X)}}) \big) = \frac{1}{2}\Lambda(q_i)\mu(\omega_{q_i}) = \frac{1}{2}\frac{\Lambda(q_i)}{\phi(q_i)}\log\log x + O\bigg(\frac{\Lambda(q_i) \log q_i}{\phi(q_i)}\bigg).
\begin{cases}\varphi^{(\infty)}_1(z) = \frac{1}{z}\,{}_2F_1(\alpha, 1\!-\!\alpha\,; 1 \,; \frac{1}{z}), \varphi^{(\infty)}_2(z) = \frac{1}{z}\left({}_2F_1(\alpha, 1\!-\!\alpha\,; 1 \,; \frac{1}{z}) \log(-z)+{{}_2F_1}^*(\alpha, 1\!-\!\alpha\,; 1 \,; \frac{1}{z})\right).\end{cases}
\frac 1{z-\alpha w}=-\int_0^\infty e^{\tau_1(z-\alpha w)}d\tau_1\,,\,\frac 1{z-\zeta}=\int_0^\infty e^{-\tau_2(z-\zeta)}d\tau_2\,,\,\frac 1{w-\omega}=\int_0^\infty e^{-\tau_3(w-\omega)}d\tau_3.
(-1)^{F}= \psi_{11}(-1)^{\sum\limits_{m=1}^{\infty}\psi_{-m}\cdot \psi_m}~~,~~{G}= - \gamma_0 \beta_0 - \sum\limits_{m=1}^{\infty}\left[ \gamma_{-m}\beta_m + \beta_{-m}\gamma_m \right]~~.
||u_j\cdot \nabla u_j||_{\tilde{L}^{q}}=\max\{||u_j\cdot \nabla u_j||_{L^{q}},||u_j\cdot \nabla u_j||_{L^{2}}\}\leq \frac{CK_j^{2}}{s^{\frac{3}{2}(1-\frac{1}{q})}}.
\frac{F^2(1+r)}{F^2(1 +\tilde{r})}= \exp \left( \frac{a}{n\log\frac{2k}{\tilde{r}}}+ O\left(\frac{1}{n\log^3 n}\right)\right) = 1 + \frac{a}{n\log\frac{2k}{\tilde{r}}}+ O\left(\frac{1}{n\log^3n}\right).
{\mathbf T}_1(E) ={\rm e}^{-\frac{\pi}{3}i}\frac{{\rm Ai}(E{\rm e}^{\frac{2\pi}{3}i})}{{\rm Ai}(E)}+{\rm e}^{\frac{\pi}{3}i}\frac{{\rm Ai}(E{\rm e}^{-\frac{2\pi}{3}i})}{{\rm Ai}(E)}=-{\rm e}^{\frac{2\pi}{3}i}\frac{{\rm Ai}(E{\rm e}^{\frac{2\pi}{3}i})}{{\rm Ai}(E)}-{\rm e}^{-\frac{2\pi}{3}i}\frac{{\rm Ai}(E{\rm e}^{-\frac{2\pi}{3}i})}{{\rm Ai}(E)}.
x_1(t) = \frac{\Gamma_2}{\Gamma_1+\Gamma_2}D\cos \omega t, x_2(t) = -\frac{\Gamma_1}{\Gamma_1+\Gamma_2}D\cos \omega t, y_1(t) = \frac{\Gamma_2}{\Gamma_1+\Gamma_2}D\sin \omega t, y_2(t) = -\frac{\Gamma_1}{\Gamma_1+\Gamma_2}D\sin \omega t,
M_1:=\left({\begin{array}{*{20}c}b^2 0 0 0 0 a^2 0 0 0 0 -b^2 0 0 0 0 -a^2 \end{array}}\right)\,\,\,\,\,\,\,\,M_2:=\left({\begin{array}{*{20}c}0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 \end{array}}\right).
\frac{ds^2}{\sqrt{g_s}l_s}= \sqrt{\frac{u}{\sqrt{N}l_s}}\left[ - \left(1-\frac{u_0^2}{u^{2}}\right) dt^2+ \sum_{i=1}^5 (dy^i)^2 + N l_s^2 \left( \frac{du^2}{u^2-u_0^2}+d\Omega_{3}^2 \right) \right] \ ,
ds_6^2 ={3{\tilde R}^2 \over 2^{2/3}}K(\tau)[{1 \over 3 K^3(\tau)}(d\tau^2+(g^5)^2)+\cosh^2(\tau/2)((g^3)^2+(g^4)^2)+\sinh^2(\tau/2)((g^1)^2+(g^2)^2)].
\delta_\mathrm{mRBA}(u) =~ (\delta_\mathrm{Hoch}(u), -u), ~ u \in M, \delta_\mathrm{mRBA}(\chi, \Phi) =~ \big( \delta_\mathrm{Hoch}(\chi), ~ - \widetilde{\delta}_\mathrm{Hoch}(\Phi) - \Psi^n (\chi) \big), ~ (\chi, \Phi) \in C^n_\mathrm{mRBA}((A, R), (M,S)).
M_{Y}(t) = 2p(1-p) \bigg[\frac{(p_{\alpha_{+}}- p_{\alpha_{-}})}{(p_{\alpha_{-}}-\sigma t)(p_{\alpha_{+}}-\sigma t)}\bigg]\exp\Big\{\mu t + \frac{1}{2}\alpha^2 \sigma^2 t^2 \Big\}\;\Phi\big(|\alpha| \sigma t \big).
R_{a b c d}= - \frac{1}{1 6}W^{2}( g_{a c}g_{b d}- g_{a d}g_{b c}) + \frac{1}{4 p^{2}}W^{, x}W_{, x}( p_{a}g_{b [ d}p_{c ]}- p_{b}g_{a [ d}p_{c ]})
\begin{dcases}\partial_t \rho + \nabla \cdot (\rho u) = 0 \partial_t (\rho u) + \nabla \cdot \big(\rho u\otimes u\big) + \nabla p_1(\rho) + \rho \nabla h_2^\prime(n) = 0 -\delta \Delta h_2^\prime(n) + n = \rho \ , \end{dcases}
\Vert u_{n+N}-u_n\Vert \le d\sum_{k=\phi_4(\varepsilon/2d)}^\infty\vert\lambda_{k+N}-\lambda_k\vert+d\prod_{k=\phi_4(\varepsilon/2d)}^n(1-\lambda_{k+N}(1-\tau)) \le\frac{\varepsilon}{2}+d\prod_{k=\phi_4(\varepsilon/2d)}^n(1-\lambda_{k+N}(1-\tau)).
\left\|\sum\limits_{i=1}^{j}f(s_i)\Delta_i\right\| \leq \left\|\sum\limits_{i=1}^{j}f(s_i)\Delta_i-f(t_1)(t-t_1)\right\|+\left\|f(t_1)(t-t_1)\right\| &<\frac{\epsilon}{4}+\frac{\epsilon}{4}=\frac{\epsilon}{2}
c(x,\lambda )= - \kappa x^4 \lim_{y\rightarrow\infty}\lim_{y'\rightarrow\infty}e^{4\phi (y')}\partial_{y'}\int\limits_{-\lambda}^{y}dy_1\int\limits_{y_1}^{\infty}dy_2\int\limits_{-\lambda}^{y_2}dy_3 F(x,y_1,y_2,y_3,y',\lambda )\,,
\log(d^2/y) = \log( (m y f)^{1/2}/ y ) = \log( (m/y)^{1/2}\log^{1/2}(m/y) ) \approx \log(m/y) = f \log(m/d) \geq \log(\frac{m}{\sqrt{m}}) \approx \log m \log((m/d)^2/y) \geq \log( \frac{m^2}{(\sqrt{m})^2 y}) = \log(\frac{m}{y}) \approx f
P_{\mathrm{gen},p}( z)= 4\pi\Im(\sigma_{p}^{-1}z)+\lim_{w\rightarrow z}\lim_{s\rightarrow 1}\bigg(\sum_{n\neq0}\frac{1}{|n|}W_{s}(n\sigma_{p}^{-1}z)\overline{V_{\overline{s}}(n\sigma_{p}^{-1}w)}-g_{\mathbb{H},s}(\sigma_{p}^{-1}z,\sigma_{p}^{-1}w)\bigg).
T =\frac{(3\rho T+\rho|U|^2)-|\rho U|^2\rho^{-1}}{3\rho}\cr =\frac{\int_{\mathbb{R}^3}f|v|^2dv-\Big|\int_{\mathbb{R}^3}fvdv\Big|^2\Big(\int_{\mathbb{R}^3}fdv\Big)^{-1}}{3\int_{\mathbb{R}^3}fdv}.
\partial_\xi^\alpha f(\xi)\sim\sum_{j\geq 0}\partial_\xi^\alpha f_{m-j}(\xi) g^{B,\alpha}(\xi) := \partial_x^\alpha\big|_{x=0}\Big( \alpha_{-x}\big( g(\xi+Bx) \big) \Big) \sim\sum_{j\geq 0}g_{m'-j}^{B,\alpha}(\xi) ,
\Delta_0 \, z_j = \Delta \, z_j - \sum_{i=k+1}^n \langle \Pi (\partial_{z_j}) , \nabla_{E_i}E_i \rangle+ \sum_{i=1}^k \langle \Pi_0 (\partial_{z_j}^T) , \nabla_{E_i}E_i \rangle\, .
\lim_{k\to\infty}\int_{B_1}e^{n(v_k+c_{v_k})}dx\geq N\lim_{k\to\infty}\int_{B_{\mu_k}(\xi_{1,k})}e^{n(v_k+c_{v_k})}dx=N\frac{1}{t_\infty}\int_{B_1}e^{n\eta}dx.
\lim_{t\to\infty}K^{\rm resc}_t(s_1,s_2) \frac{e^{t^{1/2}f_1(\alpha^{-1},s_2)}}{e^{t^{1/2}f_1(\alpha^{-1},s_1)}}= K_{\rm GUE(M)}(s_1+\xi_c,s_2+\xi_c).
A_{3}=\sum_{j=1}^{n}{B_{j}}\leq s\left\vert{q}\right\vert (p+1)(1-\left\vert{z}\right\vert^{2})^{p-1/2}\left\vert{R(sz)}\right\vert^{2}\log^{-}\left\vert{f(sz)}\right\vert .
n |\rho_{k, m_n}| \leq \frac{m_n}{\max\big( \sqrt{g(m_n)}, 1 / \sqrt{\log (m_n)}\big) \log^{k-1}(m_n)}\cdot \frac{\log^{k-1}(m_n)}{m_n}g(m_n) &= \min\big( \sqrt{g(m_n)}, g(m_n) \sqrt{\log (m_n)}\big) \underset{n\to\infty}{\longrightarrow}0.
\big[ f( e^{-s}\big[ A(\Lambda z) + c \big] + \sqrt{1-e^{-2s}}u ) \gamma(u)^\frac1\alpha \big]^\alpha=C\big[ e^{\frac{1}{\sqrt{e^{2s}-1}}\langle u, A^*z - A(\Lambda z) - c \rangle}\gamma(u)^{1-\frac1\alpha}\big]^{\alpha'}
\frac{1}{N^{\frac{n}{2}+ 1}}\sum^N_{\substack{j_1, \ldots, j_s = 1 j_k (\ref{part:first-j-relation}), (\ref{part:second-j-relation}) \pi \in \Theta}}\varphi_q\left(z_1\left(j_{s^{-1}_\chi(1)}, j_{s^{-1}_\chi(1) + 1}; k_1\right) \cdots z_n\left(j_{s^{-1}_\chi(n)}, j_{s^{-1}_\chi(n) + 1}; k_n\right) \right).
\textrm{IV}=\frac{1}{T_n}\bigg(P_t\int_0^{T_n}\nu_s\,\d s-\int_0^{T_n}\nu_s\,\d s\bigg)(f)=\frac{1}{T_n}\bigg(\int_0^{T_n}\nu_{s+t}\,\d s-\int_0^{T_n}\nu_s\,\d s\bigg)(f) &=\frac{1}{T_n}\bigg(\int_{T_n}^{T_n+t}\nu_s\,\d s-\int_0^{t}\nu_s\,\d s\bigg)(f)\le \frac{2t}{T_n}\max |f|\xrightarrow[n\to\infty]{}0.
= \; m^{( 1 )}c^{( 1 )}: \cos \Bigg ( 2 \sqrt{\pi}\sqrt{\frac{\pi}{\pi \! + \! g N}}U_{1 b}\Phi^{( 1 )}( x ) + 2 \sqrt{\pi}\sum_{I = 2}^{N - 1}U_{I b}\Phi^{( I )}( x )
W_{\bf a}= -{\frac{1}{2}}\int_{\frac{4 \pi \epsilon}{\bf a}}^{+ \infty}{\frac{d \tau}{\tau}}e^{\tau^{2}}( 1 - \Phi ( \tau ) ) \simeq{\frac{1}{2}}\ln ({\frac{4 \pi \epsilon}{\bf a}}) ~ ~ .
\langle \mathcal{L}_g f, \psi \rangle_{Q_t}+ \langle \mathcal{L}_g^{*}\psi, f \rangle_{Q_t}begin{align*}2pt] =\; \int_{\Omega\times\mathbb{R}^3}\! \big[(f\psi)(t,x,v) - (f\psi)(0,x,v)\big]dxdv + \int_{\Sigma^{t}}\!\gamma f \gamma\psi (v\!\cdot\! n_x) dS_xdvd\tau ,
\frac{1}{h}\left(p_{ij}(s,t,u,d) - p_{ij}(s,t,u,d-h)\right) =\frac{1}{h}p_{ij}(s,t,u,(d-h,d]) =\frac{1}{h}P(Z_t = j,d-h<U_t\le d | Z_s = i,U_s = u).
x y^{2}= \left[ x \prod_{a=1}^{n_{c}}(x-\phi_{a}^{2})^{2}+ 2 \Lambda^{2n_{c}+2-n_{f}}m_{1}\cdots m_{n_{f}}\right]^{2}- 4 \Lambda^{2(2n_{c}+2-n_{f})}\prod_{i=1}^{n_{f}}(x+m_{i}^{2}), \quad USp(2 n_c).
\begin{aligned}V_{1,i}= L^2(J)\cdot f^2h_i, V_{2,i}= L^2(J)\cdot \tilde{g}, V_{3,i}= L^2(J)\cdot f^2\delta^*\omega_i\oplus L^2(J)\cdot dr\odot f\omega_i, V_{4,i}= L^2(J)\cdot f^2(n\nabla^2v_i+\Delta v_i\cdot g)\oplus L^2(J)\cdot dr\odot f\nabla v_i\oplus L^2(J)\cdot v_i(f^2g-ndr\otimes dr).\end{aligned}
[L^2,E]_{\frac{2}{3}}=(L^2,E)_{\frac{2}{3},2}={\lbrace u\in L^2,\;|D_{q_i}|^{\frac{2}{3}}u\in L^2,\;|\partial_{q_i}V(q_i)|^{\frac{2}{3}}u\in L^2\;\; \;1\le i\le d\rbrace~,}
V_{7}( \vec{x}) = -{\frac{\alpha}{\sqrt{x^{2}+ y^{2}+ z^{2}}}}+{\frac{\hbar^{2}}{2 M ( x^{2}+ y^{2})}}\Bigg ({\frac{k_{1}^{2}z}{\sqrt{x^{2}+ y^{2}+ y^{2}}}}+ F \bigg ({\frac{y}{x}}\bigg ) \Bigg ) \enspace .
\pi_s(k)\xi_j^s = q^{-j}\xi_j^s, \pi_s(e)\xi_j^s = ([s+j]_q[s-j+1]_q)^{1/2}\xi_{j-1}^s, \pi_s(f)\xi_j^s = ([s-j]_q[s+j+1]_q)^{1/2}\xi_{j+1}^s.
\lambda^{-1-2\alpha+\epsilon}\sum\limits_{i,j=1}^{n+1}[Y_i Y_j \tilde{v}]_{C^{0,\epsilon}_\ast(\mathcal{B}_{\lambda/4}(y))}+ \lambda^{-1-2\alpha}\sum\limits_{i,j=1}^{n+1}\|Y_i Y_j \tilde{v}\|_{L^{\infty}(\mathcal{B}_{\lambda/4}(y))}\leq C\|f\|_{Y_{\alpha,\epsilon}}.
q_{k,\ell}(t,h) = \frac{1}{(2 \pi i)^2}\int_{C_2}\int_{C_1}\zeta^k(s_1+1) \zeta^{\ell}(s_2+1) \mathcal{D}_{k,\ell}(s_1,s_2) t^{s_1}(t+h)^{s_2}ds_1 ds_2,
\psi_{1}^{\dagger}( x ) \psi_{1}( x ) = \frac{1}{8 \pi}\int d \hat{\epsilon}\left\{\left[ \psi^{\dagger}- 1 ( x + \epsilon ) , \psi - 1 ( x - \epsilon ) \right] , e^{i e \int_{x - \epsilon}^{x + \epsilon}d x_{i}A_{i}^{T}}\right\}_{| \eta | , | \xi | , | \epsilon | \propto 1 / \Lambda}
\vec{f}_{F}^{( 1 )}( x ; \omega ) = \frac{1 + i \gamma^{1}}{2}\vec{f}_{F}( x ; \omega ) = \frac{1}{2}\left( \begin{array}{c}{{\vec{f}_{F}^{+}( x ; \omega ) - \vec{f}_{F}^{-}( x ; \omega )}}\\{{- \vec{f}_{F}^{+}( x ; \omega ) + \vec{f}_{F}^{-}( x ; \omega )}}\\ \end{array}\right)
[(x_{12}\partial_1x_{12}\partial_2+x_{23}\partial_3x_{23}\partial_2+x_{31}\partial_1\partial_3x_{31});(2\log x_{12}x_{23}x_{31}+2\log\partial_1\partial_2\partial_3+\partial_2^{-1}(x_{12}^{-1}-x_{23}^{-1})]=0,
D =\sum_{x,y,z\in\mathbb{F}_{q}^{\times}}\theta(x^dz)\theta(ax^{d-1}z)\theta(bz)\theta(-zy^2) &=\frac{1}{(q-1)^4}\sum_{l,m,n,k=0}^{q-2}G_{-m}G_{-l}G_{-n}G_{-k}T^l(a)T^n(b)T^k(-1) &\times\sum_{x\in\mathbb{F}_{q}^{\times}}T^{l(d-1)+md}(x)\sum_{y\in\mathbb{F}_{q}^{\times}}T^{2k}(y)\sum_{z\in\mathbb{F}_{q}^{\times}}T^{l+m+n+k}(z).
n_1= \biggl\lceil \frac{(\Phi^{-1}( \alpha_1 ) - \Phi^{-1}(1-\beta_1) )^2}{(G_m^{-1}(AQL) - G_m^{-1}(RQL) )^2}\biggr\rceil, c_1 = - \frac{\sqrt{n_1}}{2}( G_m^{-1}(AQL) + G_m^{-1}(RQL) ).
D_{0}^{c}=D_{0}^{c}( x_{out},x_{in}) =\frac{1}{2( 2\pi )^{2}}\int_{0}^{\infty}\frac{de_0}{e_0^2}\exp \left[-\frac{i}{2}\left( e_0m^2+\frac{\Delta x^2}{e_0}\right) \right]\;.
\begin{cases}(-\Delta)^{\frac{n}{2}}u(x)=v^p(x), x\in\Omega, (-\Delta)^{\frac{n}{2}}v(x)=u^q(x), x\in\Omega, u(x)=-\Delta u(x)=\cdots =(-\Delta)^{\frac{n}{2}}u(x)=0, x\in\partial \Omega, v(x)=-\Delta v(x)=\cdots =(-\Delta)^{\frac{n}{2}}v(x)=0, x\in\partial \Omega.\end{cases}