harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\lim_{n\longrightarrow +\infty}I(\psi_n,c,0) = \lim_{n\longrightarrow +\infty}\left( I(\psi_n,c,\gamma_n) -\gamma_n\int(\partial_x^{-1}\psi_n)^2dx\right) &\leq \lim_{n\longrightarrow +\infty}I(\psi_n,c,\gamma_n) &=\lim_{n\longrightarrow +\infty}m(c,\gamma_n)^3 &=m(c,0)^3.
w^{(1)}(\theta) = - 2b_\alpha S(\theta)^{-1}\boldsymbol{\Sigma}^{-1}\boldsymbol{\mu}, f^{(1)}({\bf{x}}) ={\bf{x}},w^{(2)}(\theta) = b_\alpha S(\theta)^{-1}\rm{vec}(\boldsymbol{\Sigma}^{-1}),f^{(2)}({\bf{x}}) =\rm{vec}({\bf{x}}{\bf{x}}^\top).
\Re\left(z^{-1}M(z)\right)=\Re\left(z^{-1}\left(\begin{array}{cc}1 0 0 0\end{array}\right)+\left(\begin{array}{cc}0 0 0 k^{-1}\end{array}\right)\right)\geq\left(\begin{array}{cc}\frac{1}{2r}0 0 \frac{1}{c}\end{array}\right).
G_{\lambda}(Q[A]) = \begin{bmatrix}A_0^{11}O U_1^{11}U_2^{11}A_0^{21}O U_1^{21}U_2^{21}V_1^{11}V_1^{12}W_1 + \lambda I_{n- r -\rho}W_2 M_1^{11}M_1^{12}M_2 W_3 + \lambda I_\rho \end{bmatrix},
{\vec{\lambda}}_{1 \, \pm}\equiv \pm ( 1 , 0 ; - 1 , 0 ) ,{\vec{\lambda}}_{2 \, \pm}\equiv \pm ( 0 , 1 ; 0 , - 1 ) ,{\vec{\lambda}}_{3 \, \pm}\equiv \pm ( 1 , 1 ; - 1 , 0 ) ,{\vec{\lambda}}_{4 \, \pm}\equiv \pm ( 1 , 0 ; - 1 , 1 ) .
\sum_{s\ge \nu+k/2}(2C\varkappa^2)^s s^{\nu+k/2}\le \sum_{s\ge 0}(2C\varkappa^2)^s s^{\nu+k/2}= \partial^{\nu+k/2}_\alpha|_{\alpha=0}\,(1-2e^\alpha C\varkappa^2)^{-1},
\frac{\partial \;\;}{\partial x_1}\left( \frac{H^f_{11}H^f_{22}-H^f_{12}{}^2+K(f_1{}^2G-2f_1f_2F+f_2{}^2E)}{K \Delta}\right)= -\frac{r_2}{f^2K^2 \Delta^2}, & \frac{\partial \;\;}{\partial x_2}\left( \frac{H^f_{11}H^f_{22}-H^f_{12}{}^2+K(f_1{}^2G-2f_1f_2F+f_2{}^2E)}{K \Delta}\right) = -\frac{r_3}{f^2K^2 \Delta^2}.
\sum_{\begin{subarray}{c}j_1, j_2, j_3, j_4 \in S j_1 + j_2 + j_3 + j_4 = 0 j_1^3 + j_2^3 + j_3^3 + j_4^3 = 0 \end{subarray}}u_{j_1}u_{j_2}u_{j_3}u_{j_4}= A_1 + A_2 + A_3
F_p^3(\gamma) = \frac{1}{2}\sum_{i,j=1}^n \Omega_{ij}^\sigma(0) \int_{\gamma}z_i dz_j + \frac{1}{3}\sum_{i,j,k=1}^n \left( (\partial_k \Omega_{ij}^\sigma)(0) + [ \omega_k^\sigma(0), \Omega_{ij}^\sigma(0) ]\right) \int_\gamma z_i z_k dz_j .
\frac{1}{\sqrt{n}}\Big(\sum_{k=2}^r\frac{\tilde\lambda_k}{\tilde\lambda_1-\tilde\lambda_k}+\frac{\tilde\lambda_1}{\tilde\lambda_1-\tilde\lambda_2}\Big)\leq \frac{2\tilde x}{z}\sum_{k=1}^r\frac{\lambda_k}{\tilde\lambda_1-\lambda_k}+\frac{1}{\sqrt{n}}=\frac{2}{z}+\frac{1}{\sqrt{n}}.
\begin{aligned}\mathbf{a}^t_\sigma=\boldsymbol{g}+\frac{q}{p}\left(\sum_{\beta=1}^{k}\sigma_\beta \boldsymbol{x}_{\pi_\beta(1)}+\sum_{\beta=1}^{k}t_\beta \boldsymbol{x}_{\pi_\beta(m_{\beta})}\right. & \left.+\sum_{\beta=1}^{\delta}t_{k+\beta}\boldsymbol{x}_{m-\delta+\beta}\right), \end{aligned}
p(M) =(1-u_1^2)(1-v_1^2)\cdots(1-u_n^2)(1-v_n^2) =(1-(u_1+v_1)^2)\cdots(1-(u_n+v_n)^2) =(1-(b_1 u_1)^2)\cdots(1-(b_n u_n)^2),
T(n_i) < \frac{1}{n_i}\cdot \left(1+\frac{2}{3}+\frac{4}{9}\right) =\frac{1}{n_i}\cdot \frac{19}{9}&< \frac{1}{\tfrac{32}{27}\cdot X_0}\cdot \frac{19}{9}= \frac{57}{32}\cdot \frac{1}{X_0}< \frac{97}{54}\cdot \frac{1}{X_0}.
P_{11}(0)= \sum_{j,k\in S_{+}}p_{jk}(0)|x_j||x_k|, P_{12}(0) =\sum_{j\in S_{+},k\in S_{-}}p_{jk}(0)|x_j||x_k|, P_{22}(0) = \sum_{j,k\in S_{-}}p_{jk}(0) |x_j||x_k|.
J_\Gamma(x)=\bar\psi(x)\Gamma\psi(x)\equiv \frac{1}{8\pi}\int d\hat\epsilon e^{i\chi_\Gamma(\hat\epsilon)}\left\{\left[\psi^\dagger(x+\epsilon),\gamma^0\Gamma\psi(x-\epsilon)\right],e^{ie\int_{x-\epsilon}^{x+\epsilon}dx_iA_i^{T}}\right\}_{|\epsilon|,|\eta|\propto 1/\Lambda}
J_1(t,\delta) \leq 2\|f\| t \Pr(Y_1\wedge Y_2>b_0(t)\delta) =2\|f\|t\,\Pr(Y_1>b_0(t)\delta) \Pr( Y_2>b_0(t)\delta) \leq C_{1}t^{\frac{\alpha_0-\alpha-\alpha^*+\epsilon}{\alpha_0}}\stackrel{t\to\infty}{\to}0.
\|O_pu\|^2_{L^2(\mathbb{R}^2)}=\|O_p\Big(\frac{1}{L}\int_0^L\varphi_s(q,p)ds\Big)\|^2_{L^2(\mathbb{R}^2)}\le \frac{1}{L^2}\int_0^L\|O_p\varphi_s\|^2_{L^2(\mathbb{R}^2)}ds{~.}
\begin{aligned}\|(\tilde{\rho},\tilde{u})\|_{L^2(L^2)\times L^2(L^2)}=\|\Lambda_1^*(p)\|_{L^2(L^2)\times L^2(L^2)}&\leq \|\Lambda_1^*\| \ \|p\|_{L^2(0,T)}.\end{aligned}
\prod\limits_{j=1}^k |E_j| -\nu_k(0) \ge \prod\limits_{j=1}^k |E_j| -q^{-1}\prod_{j=1}^k |E_j|- q^{\frac{d}{2}}\left(\prod_{j=1}^k |E_j| \right)^{\frac{k-1}{k}}&\ge \frac{1}{3}\prod\limits_{j=1}^k |E_j| + \left( \frac{1}{3}\prod\limits_{j=1}^k |E_j|- q^{\frac{d}{2}}\left(\prod_{j=1}^k |E_j| \right)^{\frac{k-1}{k}}\right).
\sum_{k=0}^{n-1}[3k+1]\frac{(q;q^2)_k^3 q^{-{k+1\choose 2}}}{(q;q)_k^2 (q^2;q^2)_k}\equiv [n]q^{\frac{1-n}{2}}+\frac{(n^2-1)(1-q)^2}{24}[n]^3 q^{\frac{1-n}{2}}\pmod{[n]\Phi_n(q)^3}.
\tilde{\cal H}= \frac{1}{2}\,\pi^2_a + \frac{1}{2}\,{\sigma^\prime}^2_a - (\sigma_a\pi^a)\theta +\frac{1}{2}\lambda \bigl(\sigma^2_a - 1\bigr)+ \frac{1}{2}\,\sigma^2_a\theta^2 .
KL (P_{f_{0,n}}, P_{f_{1,n}}) = \frac{n}{2}\int_0^1 \big(2\sqrt{x+r_n}-2\sqrt{x}\big)^2 dx &\leq 4n \int_0^{r_n}(\sqrt{x+r_n})^2 +(\sqrt{x})^2 dx + 2n \int_{r_n}^1 \frac{(x+r_n-x)^2}{(\sqrt{x+r_n}+ \sqrt{x})^2}dx &\leq 8n r_n^2 + 2nr_n^2 \int_{r_n}^1 \frac 1x dx &\leq 10nr_n^2 \log(1/r_n),
\begin{array}{l}{{\mu ( R ) = p + \frac{1}{2}a_{i j}\, \alpha^{i}\wedge \alpha^{j}+ q \, \, \alpha^{1}\wedge \alpha^{2}\wedge \alpha^{3}\wedge \alpha^{4},}}\\{{\mu ( \hat{R}) = q - \frac{1}{4}\epsilon^{i j k l}a_{i j}\, \beta_{k}\wedge \beta_{l}+ p \, \, \beta_{1}\wedge \beta_{2}\wedge \beta_{3}\wedge \beta_{4}}}\\ \end{array}
\{\underline{1}, y, \widehat{y^2},\dots, \boxed{y^{2j+1}},\dots, \underline{y^l},\dots, \boxed{y^{2l-2j-1}},\dots, \underline{y^{2l}},\dots, \overline{y^{n+2r+1}},\dots ,\underline{y^{3l}},\dots, \overline{y^{2n-2r-1}},\dots, \widehat{y^{2n-2}},y^{2n-1}\}
\det{\left(U^{^{*}}TU\right)_{_{NE}}}=\left(\frac{-2\gamma^2\sigma^2}{\beta+6}\right)\left(\frac{1}{\frac{e^{i \theta}\sigma^2}{\gamma^2}}+\frac{e^{i \theta}\sigma^2}{\gamma^2}-\beta\right),
\frac{d s^{2}}{\alpha^{\prime}\sqrt{4 \pi g_{s}N}}= ( 1 + \delta_{1}) \frac{d \rho^{2}}{\rho^{2}- \frac{b^{2}}{\rho^{2}}}+ ( 1 + \delta_{2}) ( \rho^{2}- \frac{b^{4}}{\rho^{2}}) d t^{2}+ \rho^{2}d x_{i}^{2}+ d \Omega_{5}^{2},
g(u_1\otimes\cdots\otimes u_n) =f(u_1)\overline{u_2}\cdots \overline{u_n}+\overline{u_1}f(u_2)\overline{u_3}\cdots \overline{u_n}&+\cdots+\overline{u_1}\cdots \overline{u_{n-1}}f(u_n),
N_n=\sum_{k=0}^{n-1}\Delta N_k \lesssim \sum_{k=0}^{n-1}\|u-u_k\|^{-1/s}\le \|u-u_n\|^{-1/s}\sum_{k=0}^{n-1}\rho^{(n-k)/s}\lesssim \|u-u_n\|^{-1/s}.
C \left( \begin{matrix}{{\lambda_{2}}}\\{{\lambda_{3}}}\\{{\vdots}}\\{{\lambda_{n_{f}}}}\\ \end{matrix}\right) - \mu \left( \begin{matrix}{{m_{2}}}\\{{m_{3}}}\\{{\vdots}}\\{{m_{n_{f}}}}\\ \end{matrix}\right) = 0 ,
|v_{1}^{\lambda}+v_{2}+v_{3}^{\lambda}| \leq \begin{cases}\frac{C r}{t^{2}\log^{b}(t)}, r \leq \frac{t}{2}\frac{C}{\sqrt{r}}, t-\sqrt{t}\leq r \leq t+\sqrt{t}\frac{C \log(r)}{|t-r|}, \frac{t}{2}\leq r \leq t-\sqrt{t}, r \geq t+\sqrt{t}\end{cases}
j(p_1, \ldots, p_m; t_1,\ldots, t_p):= & \min\left(1, \ \max\left(0, \sum_{i=1}^{p_1}t_i, \sum_{i=1}^{p_1+p_2}t_i, \ldots, \sum_{i=1}^{p_1+\ldots+p_{m-1}}t_i\right)+\max\left(0, \sum_{i=1}^{p_1}(-t_i), \ldots, \sum_{i=1}^{p_1+\ldots+p_{m-1}}(-t_i)\right)\right).
\frac{2 C_{1}C_{2}}{\pi}{\cal B}\left( \frac{1}{2}, - \frac{\lambda + m + 1}{2}\right){\cal B}\left( \frac{1}{2}, - \frac{\lambda + n + 1}{2}\right){\sigma}_{1}^{\frac{\lambda + m + 1}{2}}{\sigma}_{2}^{\frac{\lambda + n + 1}{2}}\leq
\kappa^{2}= - \frac{1}{4}\lim_{R \rightarrow R_{0}}\Big [ g^{t t}g^{R R}( \partial_{R}g_{t t})^{2}\Big ] = \frac{( d - 3 )^{2}}{4 R_{0}^{2}R_{T}^{2}\cosh^{2}\alpha \, A |_{R = R_{0}}}\ .
\begin{array}{rcl}\mathrm P(A_4,t) = t\mathrm P(A_3)\prec a_{44}+t\mathrm P(A_2)\prec a_{34}+t\mathrm P(A_1)\prec a_{24}+t\mathrm P(A_0)\prec a_{14}&= t^4((a_{11}\prec a_{22})\prec a_{33})\prec a_{44}+t^3(a_{11}\prec a_{23})\prec a_{44}& +t^3(a_{12}\prec a_{33})\prec a_{44}+t^3(a_{11}\prec a_{22})\prec a_{34}+t^2a_{12}\prec a_{34}+at^2_{11}\prec a_{24}& +ta_{14}.\end{array}
\sum_{{n=-\infty}\atop{n\ne \alpha,\beta}}^{\infty}\left(\int_{[0,2\pi]^{4s+1}}\frac{dx_1dx_2\cdots dx_{4s+1}}{\displaystyle\sum_{j=1}^{4s+1}\left|\cos\left(\left(\frac{\alpha-\beta}{2}\right)x_j\right)-\cos \left(\left(n-\frac{\alpha+\beta}{2}\right)x_j\right)\right|^{2s}}\right) |{\widehat f}(n)|^2 &\le M\sum_{n=-\infty}^{\infty}|{\widehat f}(n)|^2 &<\infty.
|I_k| = (4/5)^{K_{k-1}}, K_k \in [(5/4)^{K_{k-1}/(4\tau)}, 2(5/4)^{K_{k-1}/(4\tau)}], k = 1, 2, \dots, n; &M_k \in [(5/4)^{K_{k-1}/(2\tau)}, 2(5/4)^{K_{k-1}/(2\tau)}], k = 1, 2, \dots, n; &I_n \supseteq [\alpha - (4/5)^{K_n}, \alpha + (4/5)^{K_n}].
\left|\frac{1}{(\delta^2+|x|^2)^\frac{n}{2}}-\frac{1}{(\delta^2+|x'|^2)^\frac{n}{2}}\right|\leq \frac{n}{2}\frac{x_n^2}{(\delta^2+|x'|^2)^\frac{n+2}{2}}=O\left(\frac{|x'|^4}{(\delta^2+|x'|^2)^{n+2}}\right)
b_k(q) = \beta_k (M-1) (p_{\sf ave}T - q) \left[1- \frac{K M \sigma^2}{\beta_k^2 q}\ln \left( 1+ \frac{\beta_k^2 q}{K M \sigma^2}\right) \right], d_k(q) = \beta_k \left( p_{\sf ave}T - q \right).
\partial_{\mu}\Omega (\Sigma ,x) =-\int_{\Sigma}d^{D-1}\tilde{\sigma}_{\mu}\delta^{D}(x-\bar{x}(\sigma ) ) -\int_{\Sigma}d^{D-1}\tilde{\sigma}_{\nu}(\delta_{\mu \nu}\partial^{2}- \partial_{\mu}\partial_{\nu})D(x-\bar{x}(\sigma ) ) \ .
| U_i / U_{i+1}|= l^{rd}, \,\,\,{\rm if}\,\,\, i \not\equiv 0 \pmod{d}\,\,\, |U_i / U_{i+1}|= l^{r(d-1)}, \,\,\,{\rm if}\,\,\, i \equiv 0 \pmod{d}.
\begin{array}{c}\displaystyle{\sum_{\nu=\nu_< +1}^{\nu_>}\left( f(\nu)-g(\nu) \right)= \int_{\nu_< +1}^{\nu_>}\left( f(\nu)-g(\nu) \right)\, d\nu +}\displaystyle{+\left( f(\nu_< +1)-g(\nu_< +1) \right) + \int_{\nu_< +1}^{\nu_>}\left( x-[x] \right)\left( f'(x)-g'(x) \right) \, dx}.\end{array}
F_0 \leq \delta^s F_s + \Theta G \sum_{i=0}^{s-1}\big(\delta 2^{\frac{n}{p}})^i \leq \delta^s \|f\|_{L^\infty(B_R)}+ \Theta G \sum_{i=0}^{s-1}\big(\delta 2^{\frac{n}{p}})^i
\|T_n(vu) - T_n(v) T_n(u)\|_F^2\;\leq\; 2\sum_{j=1}^n\sum_{k=1}^n \Big|\sum_{l=-\infty}^{j-n-1}v_l u_{j-k-l}\Big|^2\;+\;2\sum_{j=1}^n\sum_{k=1}^n\Big|\sum_{l=j}^\infty v_l u_{j-k-l}\Big|^2.
\|u\|_{C^{k,\alpha}_\nu(U)}=\sup_{x \in U}\left( r(x)^{k+\alpha-\nu}[\nabla^k u]_{C^{0,\alpha}(U \cap B(x,\frac{1}{2}r(x)))}\right) + \sum_{j = 0}^k \| r^{j-\nu}\nabla^j u\|_{L^\infty(U)}
d>2m, k\in [0,2m];\tag*{$\textup{(\ref*{11-1-24})}_1$}&m<d\le 2m, k\in [2m-d,d); \tag*{$\textup{(\ref*{11-1-24})}_2$}&d=m, k=d.\tag*{$\textup{(\ref*{11-1-24})}_3$}
\nabla^{2}f=\frac{1}{r^{2}}\frac{\partial}{\partial r}(r^{2}\frac{\partial f}{\partial r})+\frac{1}{r^{2}sin\theta}\frac{\partial}{\partial\theta}(sin\theta\frac{\partial f}{\partial\theta})+\frac{1}{r^{2}sin^{2}\theta}\frac{\partial^{2}f}{\partial\varphi^{2}}=0.
u_{b^{\beta}}= \begin{cases}u_b, \beta = - \nu_b, \beta = +,\end{cases}y_{b^{\beta}}= \begin{cases}(y_a, \eta_a, u_a, \nu_a, d_b,\delta_b), \beta = - (y_a, \eta_a, u_a, \nu_a, u_b,d_b,\delta_b), \beta = +.\end{cases}
I_2(n) \leq \frac 1{x^{\eta}}\left ( \int_{-Q(v)}^{Q(v)}\left \Vert \max_{2 \leq 2j \leq [n/q]}\left \vert \sum_{i=1}^j \tilde U_{2i}(t) \right \vert \right \Vert_{\eta}dt \right )^{\eta}\, .
{\cal H}^{(0)}= -\frac{2 \kappa}{\sqrt{-\gamma}}\frac{\delta I^{(0)}}{\delta \gamma_{\mu \nu}}\frac{\delta I^{(0)}}{\delta \gamma_{\lambda \rho}}\left( \gamma_{\mu \rho}\gamma_{\nu \lambda}-\frac{1}{d-1}\gamma_{\mu \nu}\gamma_{\lambda \rho}\right)+ \frac{\sqrt{-\gamma}}{\kappa}\Lambda=0,
\prod_{j=0}^{k-1}\frac{\Gamma(\frac{j}{4}+\frac{1}{2})}{\Gamma(\frac{j}{4}+\frac{5}{4})}= \frac{\Gamma(1/2)\Gamma(3/4)}{\Gamma(\frac{k+2}{4})\Gamma(\frac{k+3}{4})\Gamma(\frac{k+4}{4})}
\bar{C}(\mathcal G) =R_0(p(x_1), 0) &=-\min_{q(x_1)}\max_{y_2\in \mathcal{Y}_2}\log\big(\sum_{x_1:d(x_1,y_2)=0}q(x_1)\big) &=\max_{q(x_1)}-\max_{y_2\in \mathcal{Y}_2}\log\big(\sum_{x_1:d(x_1,y_2)=0}q(x_1)\big) &=\max_{q(x_1)}\min_{p(y_2|x_1)\in \mathcal{P}}I_0(X_1;Y_2),
\left\| \partial_{t}^{\ell}\nabla_{x}^{k}\tilde{J}_{1}(t)g\right\|_{2}+\left\| \partial_{t}^{\ell}\nabla_{x}^{k}\tilde{J}_{2}(t)g\right\|_{2}\le C(1+t)^{-\frac{n}{2}-(\ell+ k)}\| g \|_{1}+ C e^{-ct}\| \nabla_{x}^{\ell+k}g\|_{2},
H^{s_2}h^{s_1}:=\{\vec{\phi}=\{\phi_{p}\}: ||\vec{\phi}||_{H^{s_2}h^{s_1}}^2=\sum\limits_{p \in \mathbb{Z}^2}\langle p \rangle^{2s_1}||\phi_p||_{H^{s_2}}^{2}< +\infty \}.
F^V_{{\rm L}}\Big(\frac{t}{\tau},-\frac{1}{\tau}\Big) =2^{[(N+\dim V)/2]}\tau^{(\dim M +1)/2}e^{\pi\sqrt{-1}nt^2/\tau}F^V_{{\rm W}}(t,\tau),{F'}^V_{{\rm W}}\Big(\frac{t}{\tau},-\frac{1}{\tau}\Big) =\tau^{(\dim M +1)/2}e^{\pi\sqrt{-1}nt^2/\tau}{F'}^V_{{\rm W}}(t,\tau),
ds^2_{(\mu<0)}=l^2\left[- d\tau^2 +\frac{1+m}{4}\frac{dz^2}{z^2}+\frac{1+m}{4}\frac{d\bar{z}^2}{\bar{z}^2}+\left(e^{-2\tau}+\frac{(1+m)^2}{16}\frac{1}{z^2 \bar{z}^2}e^{2\tau}\right)dzd\bar{z}\right],
{\frac{r^{\frac n2}\alpha_n}3}\geq \begin{Vmatrix}1_{S^n}-{\frac 1{(2k)^l}}\sum_{s\in{\hat S}}H_t(x_0s,x)\end{Vmatrix}_{L^2}=\begin{Vmatrix}{\frac 1{(2k)^l}}\sum_{s\in{\hat S}}\begin{pmatrix}1_{S^n}-H_t(x_0s,x)\end{pmatrix}\end{Vmatrix}_{L^2}&\geq{\frac 23}\begin{pmatrix}\int_{B(x_0s,r)}d\sigma(x)\end{pmatrix}^{\frac 12}
K_{n}^{\ast}(f;x)=\frac{n}{e_{\mu}(n r_n(x))}\sum_{k=0}^{\infty}\frac{(n r_n(x))^{k}}{\gamma_{\mu}(k)}\int_{\frac{k+2\mu \theta_{k}}{n}}^{\frac{k+1+2\mu \theta_{k}}{n}}f\left(\frac{nt+\alpha}{n+\beta}\right)\mathrm{d}t,
\nabla_i = g_0^{-1}h_1^{-1}(\partial_i-A_i\partial_2) ~,~~ \nabla_2 = g_0^{-1}h_2^{-1}\partial_2 ~,~~ \nabla_p = g_0^{-1}h_3^{-1}(\partial_p-A_p\partial_2) .
\sum_{i = 1}^{4}\bigl [ [ X , M_{j}] , M_{j}\bigr ] ={\frac{1}{2}}\bigl ( \bigl [ [ X , S_{k l}] , S_{- k - l}\bigr ] + \bigl [ [ X , S_{- k l}] , S_{k - l}\bigr ] \bigr ) . \nonumber
{||C||}_{W^{2,1}_p(Q_T)}\le \mu(\omega)\left({||C ||}_{L^p(Q_T)}+{||\psi ||}_{W^{2,1}_p(Q_T)}+||e^{-\sum_{k=0}^mh_k(x)B_k}g ||_{L^p(Q_T)}\right),
_{1}\phi_{2}\left(\begin{array}{ccc}\begin{array}{c}a b\sqrt{q},-b\sqrt{q}\end{array}\vert q,zq\end{array}\right)=\frac{1}{\sqrt{\pi\log q^{-4}}}\int_{-\infty}^{\infty}\frac{\left(aze^{ix};q\right)_{\infty}\exp\left(\frac{x^{2}}{\log q^{4}}\right)dx}{\left(ze^{ix};q\right)_{\infty}\left(b^{2}q,-b^{2}e^{ix};q^{2}\right)_{\infty}}
\Phi(r)=m(r)\left(\begin{array}{ccc}1 0 0 0 -\frac{1}{2}0 0 0 -\frac{1}{2}\end{array}\right)+q(r)\left(\begin{array}{ccc}0 0 0 0 \frac{3}{2}0 0 0 -\frac{3}{2}\end{array}\right)
\int_0^\infty(AL_1^u(y)+BL_1^v(y))\mathrm{e}^{-\lambda y^2}\mathrm{d}y=\frac{-mx^{1+\sqrt{mn}+c}}{\sqrt{mn}(1+\frac{4t^\alpha}{\alpha}\lambda)^{\frac{3+\sqrt{mn}-c}{2}}}\mathrm{e}^{-\frac{\lambda x^2}{(1+\frac{4t^\alpha}{\alpha}\lambda)}},
\theta_1=\biggr(1-\sum_{\alpha=2}^N\theta_\alpha^2\biggr)^{1/2},\frac{\partial\theta_1}{\partial\theta_\alpha}=-\theta_1^{-1}\theta_\alpha,\frac{\partial^2\theta_1}{\partial\theta_\alpha\partial\theta_\beta}=-\delta_{\alpha\beta}\theta_1^{-1}-\theta_1^{-3}\theta_\alpha\theta_\beta,
\begin{array}{l}\delta_bx^\mu={b^\mu}_\nu n^\nu, \delta_bA^\mu_a=-{b^\mu}_\nu\displaystyle\left(\varepsilon_{ab}\frac{g^{bc}}{\sqrt{-g}}{\Pi_c}^\nu-\xi_an^\nu+i(\bar\theta\Gamma^\nu\partial_a\theta)\right).\end{array}
\delta_{i_1i_2\cdots i_mi_{m+1}\cdots i_k}^{j_1j_2\cdots j_mj_{m+1}\cdots j_k}\delta^{i_{m+1}}_{j_{m+1}}\cdots\delta^{i_k}_{j_k}= \delta_{i_1i_2\cdots i_m}^{j_1j_2\cdots j_m}{n-m \choose k-m}(k-m)!,
\langle B^{*}_{j}w, z \rangle_{V^{*}, V}= \langle B^{*}_{j}z, w \rangle_{V^{*}, V}\quad\mbox{for all}\w, z \in V, B^{*}_{j}w = B_{j}w \quad\mbox{for all}\w \in D(B_{j}) \cap V.
0 =\int_{\Omega\times Y\times Z}\widehat{u}\,\nabla_{x}\Psi dxdydz, 0 =\int_{\Omega}\operatorname{div}_{x}\left( \int_{Y\times Z}\widehat{u}\left( x,y,z\right) dydz\right) \Psi dx,\ \ \ \forall\Psi\in\mathcal{D}\left( \Omega\right) ,
\Delta(d^2\bar{\eta}) = 2|\bar{\nu}|^2\bar{\eta}+2d\{(\mathrm{div}\,\bar{\nu})\bar{\eta}+2\bar{\nu}\cdot\nabla\bar{\eta}\}+d^2\Delta\bar{\eta}= 2\bar{\eta}+2d(\mathrm{div}\,\bar{\nu})\bar{\eta}+d^2\Delta\bar{\eta}
\widehat{\boldsymbol{K}}_e = \sum_{k=1}^{N}\boldsymbol{A}_{\Omega^{(k)}_e}\boldsymbol{K}_e^{(k)}\boldsymbol{A}^T_{\Omega^{(k)}_e}\widehat{\boldsymbol{f}}_e=\sum_{k=1}^{N}\boldsymbol{A}_{\Omega^{(k)}_e}\boldsymbol{f}^{(k)}_e.
\delta_{\xi}\left<T_{z\overline z}\left(z,\overline z\right)\right>={1\over96}\left(2\,\partial_z^{\,2}\partial^z\xi^z\left(z,\overline z\right)+2\,\partial_{\overline z}^{\,2}\partial^{\overline z}\xi^{\overline z}\left(z,\overline z\right)\right)\;.
b_{1}^{2}=-c\displaystyle\frac{(\lambda_{2}-\lambda_{1})}{\lambda_{1}+\lambda_{2}}\left\{\lambda_{1}\lambda_{2}+2c\nu^{2}-\displaystyle\frac{c\lambda_{2}(1-\nu^{2})}{\lambda_{2}-\lambda_{1}}+\displaystyle\frac{2\nu^{2}(1-\nu^{2})}{(\lambda_{2}-\lambda_{1})^{2}}\right\}.
G_{\mu \nu}^{( p )}= \eta_{\mu \nu}+ \partial_{\mu}\phi^{i}\partial_{\nu}\phi^{i}- \bar{\lambda}( \Gamma_{\mu}+ \Gamma_{i}\partial_{\mu}\phi^{i}) \partial_{\nu}\lambda - \bar{\lambda}( \Gamma_{\nu}+ \Gamma_{i}\partial_{\nu}\phi^{i}) \partial_{\mu}\lambda + \bar{\lambda}\Gamma^{m}\partial_{\mu}\lambda \bar{\lambda}\Gamma_{m}\partial_{\nu}\lambda ,
{\rm{\tilde M}}= \frac{{{\rho_1}\exp \left({-{\lambda_2}\gamma}\right)}}{{{\Phi_2}}}{\left({\frac{1}{{{{\bar \gamma}_0}}}\gamma +{\beta_{j,k}}}\right)^{-{\rho_2}}}\frac{{1 - \exp \left({-{\Phi_2}\gamma}\right)}}{{\left({{\rho_3}+{\rho_4}}\right)\gamma +{\rho_5}}}
\begin{array}{ccc}(k_1)_t = \left(\frac1{k_0}(k_1r_1+k_2r_2+ r_3')\right)'-k_1r_3+\frac{k_2}{k_0}r_2'\hskip 6ex (k_2)_t = \left(\frac1{k_0}\left(\frac1{k_0}r_2'\right)'+k_2r_1-k_1r_2\right)'-k_2r_3-\frac{k_1}{k_0}r_2'\end{array}
u\left( x\right) =\frac{1}{\pi}\left( \nu \,u_{0}\left( x\right) +\mu\,v_{0}\left( x\right) \right) \ast_{x}\int_{0}^{\infty}\frac{1}{\rho^{1+\beta}\sin \frac{\beta \pi}{2}+\omega}\cos (\rho x)\mathrm{d}\rho,\;\;x\in\mathbb{R},
\begin{array}{rcl}tt^{-1}=1=t^{-1}t~, & tet^{-1}=q^2e~, & \Delta(e)=e\otimes 1+t\otimes e~,~~~~\Delta(t)=t\otimes t~, & \epsilon(e)=0~,~~~~\epsilon(t)=1~, & S(e)=-t^{-1}e~,~~~~S(t)=t^{-1}~.\end{array}
\Big\langle e^{-ie\sum_b(j^{(b)}_\mu,a^{(b)}_\mu)}\Big\rangle =e^{-\frac{e^2}{2\pi}\sum_{b,c}(\varepsilon_{\mu \nu}\partial_\mu a^{(b)}_\nu , M^{(b,c)}\varepsilon_{\rho \sigma}\partial_\rho a^{(c)}_\sigma )}\; ,
\begin{aligned}\sup_{B_{r_{k+1}}}v\leq\sup_{B_{\tau_2r_{k}}}v \leq (1-\mu)\left(4M r_{k}^{\beta}-M r_{k}^{\alpha}\right)+M r_{k}^{\alpha}&\leq 4M r_{k+1}^{\beta}\cdot \frac{1-\mu}{\tau_1^{\beta}}+\mu M r_{k}^{\beta}&\leq 4M r_{k+1}^{\beta}\left(\frac{1-\mu}{\tau_1^{\beta}}+\frac{\mu}{4\tau_1^{\beta}}\right).\end{aligned}
f'_{n,d}(n) = Lag\Big\{n, \eta G_1G_2 m_s h_{sr}h_{rr}h_{rd}, \eta G_1G_2 (m_{br_1}+ n_{sp}) & \times h_{rr}h_{rd}+ \eta G_2 (m_{br_2}+ n_{sp})h_{rd}+ \eta m_{bd}, D \Big\}.
V_{(a,b)}-t\rho_G = (n-3-2\ell_k-t)\rho_G + \rho_G - 1 - \bigoplus_{i=1}^L \lambda(i) = (n-3-2\ell_k-t)\rho_G - \bigoplus_{i=\frac{p^k+1}{2}}^L \lambda(i).
0=\lim_{y\to\infty}\frac{{B}({t}_k,y)}{y^{n-1}}=\lim_{y\to\infty}\frac{{S}(y)\prod_{l=1}^n({t}_k-{x}_l)-{S}({t}_k)\prod_{l=1}^n(y-{x}_l)}{({t}_k-y)y^{n-1}}={S}({t}_k)
\mathcal{L}_{I_J}(s) = \mathbb{E}\Big[\exp(-s(\sum_{\mathbf{z}_i \in \Phi_J^s}P_J h_{\mathbf{z}_i, \mathbf{u}_0}D_{\mathbf{z}_i, \mathbf{u}_0}^{-\alpha})) \Big] &\overset{(a)}{=}\mathbb{E}_{\Phi_J^s}\Big[\prod_{\mathbf{z}_i \in \Phi_J^s}\mathcal{L}_h(s P_J D_{\mathbf{z}_i, \mathbf{u}_0}^{-\alpha}) \Big],
0= \sum_l A(e^{i_1}_{j_1},...,Ce^{i_l}_{j_l},...,e^{i_k}_{j_k}) = \left(\sum_l \lambda_{i_l}\right)A(e^{i_1}_{j_1},...,e^{i_k}_{j_k})+\sum_lA(e^{i_1}_{j_1},...,e^{i_l}_{j_l-1},...,e^{i_k}_{j_k}).
\int_{(0,+\infty)}f(t)\,d\|E_{\rho,\sigma}(t)\xi_\sigma\|^2 =\int_{(0,+\infty)}f(t)t^{-1}\,d\|E_{\sigma,\rho}(t^{-1})\xi_\rho\|^2 &=\int_{(0,+\infty)}tf(t^{-1})\,d\|E_{\sigma,\rho}(t)\xi_\rho\|^2,
\begin{aligned}\mathbb{P}_{*-\delta_0}\left\{\inf_{t\in[0,T_n]}I^\sigma(t)<2\rho_n,I^\sigma(t)\in(\frac{\rho_n}{2},*-\frac{\delta_0}{2})\right\}&\leq\mathbb{P}_{*-\delta_0}\left\{\sigma\sup\limits_{t\in[0,T_n]}\left|\int^t_0(N-I^\sigma(s))dB(s)\right|>\frac{\ln\rho^{-1}_n}{2}\right\}.\end{aligned}
I_1+I_3 = - u_0(x)+ \frac{1}{2}(1+t)^{-\frac{\mu}{2}}\big[ u_0(x+t)+ u_0(x-t) \big]+ \frac{1}{2^{\sqrt{\delta}}}\int_{x-t}^{x+t}u_0(y) \bigg[-\frac{\partial E}{\partial b}(t,x;b,y;\mu,\nu^2)\bigg]_{b=0}\mathrm{d}y & + \frac{1}{2^{\sqrt{\delta}}}\int_{x-t}^{x+t}\mu\, u_0(y) E(t,x;0,y;\mu,\nu^2) \, \mathrm{d}y .
0=\lim_{i\rightarrow\infty}\frac{dA}{dt}\left( t_{i}\right) =\frac{2A\left(\infty\right)}{L\left( \infty\right)}\int_{0}^{2\pi}w\left(\theta\right) d\theta-\int_{0}^{2\pi}w^{1-\frac{1}{\alpha}}\left(\theta\right) d\theta.
\begin{vmatrix}F_1^1 F_1^2 F_1^3 F_1^4 F_1^5 F_2^1 F_2^2 F_2^3 F_2^4 F_2^5 F_3^1 F_3^2 F_3^3 F_3^4 F_3^5 \end{vmatrix}= 0 \, ,
|Z|=\!\!\!\sum_{1\le i_1<\cdots<i_n \le m}|\det(u_{i_1},\dots,u_{i_n})| &|P_{e^\bot}Z|=\!\!\!\sum_{1\le i_2<\cdots<i_n\le m}|\det(P_{e^\bot}u_{i_2},\dots,P_{e^\bot}u_{i_n})|.
\mathbb{E}\Vert Y_n^M (t+\delta) - Y_n^M (t) \Vert_{H}^2 = \mathbb{E}\int_{t\wedge\tau_n^M}^{(t+\delta)\wedge\tau_n^M}2\langle A(r,Y_n(r)), Y_n(r) - Y_n(t\wedge\tau_n^M) \rangle dr & + \mathbb{E}\int_{t\wedge\tau_n^M}^{(t+\delta)\wedge\tau_n^M}\Vert P_n B(r,Y_n(r))Q_n \Vert_{L_2}^2 dr .
x_1 =w_1w_m^{-1}, x_2 =x_1^{-1}w_2w_m^{-1}x_1, x_3 =(x_1x_2)^{-1}w_3w_m^{-1}x_1x_2, \ldots \hbox to 3.5cm{\dotfill}x_{m-1}=(x_1x_2\cdots x_{m-2})^{-1}w_{m-1}w_m^{-1}x_1x_2\cdots x_{m-2}, x_m =(x_1x_2\cdots x_{m-1})^{-1}w_mx_1^{-1}.
\left\{\mu \frac{\partial}{\partial \mu}+\left( \beta_{p}-d_{p}p\right)\,\frac{\partial}{\partial p}+\gamma_{i}\int d^{4}x\sqrt{-g}\;\Phi_{i}\,\frac{\delta}{\delta \Phi_{i}}\right\}\, \Gamma \left[ g_{\alpha\beta},p,\Psi_{i},\mu \right] =0\,.
\lim_{n\to\infty}\frac{1}{(c_V n)^{1+1/\theta}}K^{\alpha,\theta}_{V,n}\left(\frac{x}{(c_V n)^{1+1/\theta}},\frac{y}{(c_V n)^{1+1/\theta}}\right) =\mathbb K^{(\alpha, \theta)}(x,y).
{i_{c}}_{2}={i_{c}}_{2}^{\prime}, \quad{i_{c}}_{3}={i_{c}}_{2}^{\prime \prime}, \quad{i_{c}}_{3}^{\prime}={i_{c}}_{3}^{\prime \prime}, \quad{i_{b}}_{3}^{\prime}={i_{b}}_{3}^{\prime \prime}, \quad{i_{b}}_{1}^{\prime}={i_{b}}_{3}^{\prime \prime \prime}, \quad{i_{b}}_{1}^{\prime \prime}={i_{b}}_{1}^{\prime \prime \prime},
A_\mu^{(I)}:= \sum_{b=1}^N U_{I b}a_\mu^{(b)}\; \;\stackrel{U^T = U^{-1}}{\Longleftarrow \!\!=\!\!=\!\!=\!\!=\!\!=\!\!\! \Longrightarrow}\; \;a_\mu^{(b)}= \sum_{I=1}^N U_{I b}A_\mu^{(I)}\; .
\widetilde a_k^{11}=a^{11}_k, \widetilde a_k^{12}=\frac{a_k^{12}}{\sqrt{t_k}},\widetilde a^{22}_k=\frac{a_k^{22}}{t_k},\widetilde c_k=t_k\theta_k c_k, &\widetilde b^1_k=\sqrt{t_k}b^1_k, \widetilde b^2_k=b^2_k.
\Vert T_{H\vert_{\mathbb{Z}\times \mathbb{Z}}}^{(2N+1)}: L_{p_1}(\mathbb{T}, S_{p_1}^{2N+1}) \times L_{p_2}(\mathbb{T}, S_{p_2}^{2N+1}) \rightarrow L_{1}(\mathbb{T}, S_{1}^{2N+1}) \Vert \leq A_{p_1, p_2,2N+1}.
\nabla (\gamma(\tilde{r})(i_x(\tilde{x}-a))) = \left(\dfrac{\partial}{\partial x_i}(\gamma(\tilde{r}))(i_x(\tilde{x}-a))_j\right) + \left( \gamma(\tilde{r}) \dfrac{\partial (i_x(\tilde{x}-a))_j}{\partial x_i}\right) =: (M_{ij}) + (N_{ij}).
I_2^p:=\sum_{|k|\le N}\int_{\eta_{k}}^{\eta_{k+1}}v_1^{p}(x)\frac{\Bigl|\bigl[V_1^-(x)\bigr]'\Bigr|^p}{\bigl[V_1^-(x)\bigr]^{2p}}\biggl(\int_{a(x)}^{\eta_{k}}v_1^{-p'}(t)\biggl(\int_{a(x)}^t v_1^{-p'}\biggr)^{1-\delta}[V_1(t)]^{\delta}\bigl|G^{(\delta)}_{2,k}(t)\bigr|^{p'-1}\,dt\biggr)^p\,dx
\aligned \Bigl(\int_{\mathbb{R}^N}(I_\mu\ast|\eta_\epsilon|^{2^*_\mu})|\eta_\epsilon|^{2^*_{\mu}}\Bigr)^{\frac1{2^*_\mu}}\leq C(N,\mu)^{\frac1{2^*_{\mu}}}|\eta_\epsilon|^2_{2^*}=C(N,\mu)^{\frac{N}{2}\cdot\frac{1}{2^*_\mu}}S^{\frac{N}{2}}_{H,L}+O(\epsilon^{N-2}).\endaligned
\mathrm{var}\bigg(\sum_{i=1}^{d_{v}-1}\mu_{i}^{(l-1)}\bigg)=\sum_{i=1}^{d_{v}-1}\mathrm{var}\bigg(\mu_{i}^{(l-1)}\bigg)=(d_{v}-1){\sigma^{2}_{\mu}}^{(l-1)}.
|\Phi (s,a,z) \Gamma (s)| \le \int_0^1 \frac{x^{\sigma-1}e^{(1-a)x}}{|e^x-z|}dx + \int_1^\infty \frac{x^{\sigma-1}e^{(1-a)x}}{|e^x-z|}dx \le \int_0^1 \frac{x^{\sigma-1}e^{(1-a)x}}{|1-z|}dx + \int_1^\infty \frac{x^{\sigma-1}e^{(1-a)x}}{e^x-1}dx< \infty .
\Gamma ( s , s^{\prime}, t ) = \frac{1}{k_{1}! k_{2}!}\, \frac{\det \left[ \gamma ( s_{p 1}, s_{p^{\prime}1}^{\prime}, t ) \right]_{p , p^{\prime}= 1 , \ldots , k_{1}}\det \left[ \gamma ( s_{q 2}, s_{q^{\prime}2}^{\prime}, t ) \right]_{q , q^{\prime}= 1 , \ldots , k_{2}}}{B_{k_{1}k_{2}}( s^{2}) B_{k_{1}k_{2}}( s^{\prime 2})}\ ,
N_3={}f-L\tau -\partial_{\dot q}L\cdot\bigl(\xi-\tau \dot q\bigr)= ={}y\dot h-\partial_{\dot q}L\cdot\xi= ={}(c+x^2)G'(x)\dot xy-(c+x^2) G(x)\dot y-x \dot x^2 \dot y+\dot x^3 y.
0 =-v_t - (\mathbf{L}^r +\mathbf{L}^Y +\mathbf{L}^{S})v - r x v_x -\sup_{\theta}\left\{\theta (\mu\!-\!r)v_x \!+\!\theta \sigma^2 y s^{2 \beta-1}v_{xs}\!+\!\frac12 \theta^2\sigma^2 y s^{2\beta-2}\right\}