harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
d s_{M 2}^{2}\approx{\frac{r^{4}}{R^{4}}}( - d t^{2}+ d x_{1}^{2}+ d x_{2}^{2}) \, + R^{2}\, \frac{d r^{2}}{r^{2}}+ R^{2}\, \frac{\Lambda}{6}\, d s_{X_{7}}^{2}.
m_{i,n-r}= \big(2-c(W^*;\beta)\big)\sum_{m\geq 0}\left(\frac{\beta}{2}\right)^m c_{\lambda_i+m}+ c(W^*;\beta)\,c'_{\lambda_i}-c(W^*;\beta) \sum_{\ell>0}\sum_{m\geq 0}\left(-\frac{1}{2}\right)^\ell \left( \binom{\ell+m-1}{m}+ \binom{\ell+m}{m}\right) \beta^{\ell+m}\, c'_{\lambda_i+\ell+m}.
Q(x)B(x_{1})\cdots B(x_{m})|\Omega\rangle = q_m(x)B(x_{1})\dots B(x_{m})W_{2m,0}(x)|\Omega\rangle +\sum_{k=1}^{m}q_{k-1}(x)B(x_{1})\dots B(x_{k-1})X_{2(k-1),0}(x,x_{k})B(x_{k+1})\dots B(x_{m})|\Omega\rangle \,.
u = \frac{x_{1 2}^{2}x_{3 4}^{2}}{x_{1 3}^{2}x_{2 4}^{2}}\mathrm{a n d}v = \frac{x_{1 2}^{2}x_{3 4}^{2}}{x_{1 4}^{2}x_{2 3}^{2}}.
\left\langle V^\square \psi^*_{01}, \psi^*_{01}\right\rangle = \left\langle V^\square \psi^*_{01}, \psi^*_{0\perp}\right\rangle = 0 \left\langle V^\square \psi^*_{0\perp}, \psi^*_{0\perp}\right\rangle \geq 0
M_{(\tau, \lambda_1, \lambda_2)}= \frac{\tau^{1/\tau}}{\Gamma(1-1/\tau)}\,\beta^{-1}_{22}(\tau, b_0=\tau,\,b_1=1+\tau \lambda_1, \,b_2=1+\tau \lambda_2) \times \times\beta_{1,0}^{-1}(\tau, b_0=\tau(\lambda_1+\lambda_2+1)+1),
\left\{\begin{array}{c}(m-1)\partial_A\partial_Bx^j+L^i{}_{jk}(x,y)\partial_Ax^i\partial_Bx^j=0 (n-1)\partial_A\partial_By^b+C^a{}_{bc}(x,y)\partial_Ay^a\partial_By^b=0 \end{array}\right\}
f\left(\boldsymbol{P}^*\right) =\ln\left(1+\frac{h_{1}P^*_1}{C+h_{2}(Q-g_1P^*_1)/g_2}\right)+\ln\left(1+\frac{h_{2}(Q-g_1P^*_1)/g_2}{C+h_{1}P^*_1}\right) &+\sum_{i=3}^K\ln\left(1+\frac{h_{i}P^*_i}{D_i+h_{1}P^*_1+h_{2}(Q-g_1P^*_1)/g_2}\right).
i\hbar\frac{\partial}{\partial t}\Pi_{k,\omega}=(\tilde{\epsilon}_{k}-\hbar\omega)\Pi_{k,\omega}+\Omega_{k,\omega}^{spont}-(1-f_{k}^{e}-f_{k}^{h})[\Omega_{\omega}^{stim}+\sum_{k^{\prime}}V_{k-k^{\prime}}\Pi_{k^{\prime},\omega}]+T[\Pi]
\int_U(-\det D^2u )u^2\psi\,dx + \int_U |Du|^4\psi \,dx &=-\int_U |Du|^2 (Du\cdot D\psi) u \,dx +\frac12\int_U u^2[ |Du|^2\Delta\psi -D^2\psi Du\cdot Du] \,dx.
\sum_{i=1}^\nu\bigl(\Phi_1\circ\Psi_1[e_i],\,\Phi''[e_i^\star]\bigr)^G=\sum_{i=1}^\nu{\vphantom{\bigm|}}^{\backprime}\bigl(\Phi_1\circ\Psi_1[e_i],\,\Phi''[e_i^\star]\bigr)^G.
- \int_{0}^{\sigma}\! \! \! d s \, \int_{- \infty}^{\infty}\frac{d \omega}{2 \pi}\frac{s K^{\frac{D - 3}{2}}}{( 4 \pi )^{\frac{D - 1}{2}}}\left| \frac{\Gamma ( \frac{D - 1}{2}+ i \beta )}{\Gamma ( 1 + i \beta )}\right|^{2}\Gamma ( - \frac{D - 3}{2}) \mathrm{t r}\mathrm{\boldmath ~ 1 ~}.
f{\rm ad}{x}(z)=f([x,z])=\sum f([[x_{i1},x_{i2}],z]) &=\sum[[f(x_{i1}),f(x_{i2})],f(z)]=[\sum[f(x_{i1}),f(x_{i2})],f(z)] =[\delta_{f}(x),f(z)]={\rm ad}{\delta_{f}(x)}f(z).
\sigma^{\mu \nu}=\frac{i}{2}\left[\gamma^{\mu}\;,\;\gamma^{\nu}\right]\;,\;\;\left(F(x)K(x)\right)^{\ast \mu \nu}=\frac{1}{2}\varepsilon^{\mu \nu\alpha \beta}\left(F(x)K(x)\right)_{\alpha \beta}\;,
\begin{array}{r l}{\displaystyle E ( L ) =}&{- \displaystyle \frac{\pi}{6 L}\left( c - 1 2 ( \Delta_{+}+ \Delta_{-}) \right) + \ldots \: ,}\\{P ( L ) =}&{\displaystyle \frac{2 \pi}{L}\left( \Delta_{+}- \Delta_{-}\right) + \ldots \: .}\\ \end{array}
H_1 =\int_0^{2\pi}\int_0^{\gamma}\frac{r dr d\theta}{|\gamma+r\sin(\theta+\varphi)|^\tau r^q|re^{i\theta}-1|^q}, H_2 =\int_0^{2\pi}\int_\gamma^R\frac{r dr d\theta}{|\gamma+r\sin(\theta+\varphi)|^\tau r^q|re^{i\theta}-1|^q}.
\Phi_{z}(0,0)=a_{-1},\Phi_{\overline{z}}(0,0)=a_{1},\Phi_{w}(0,0)=-1,\Phi_{\overline{w}}(0,0)=\frac{1}{2\pi}\int_{0}^{2\pi}f(e^{i\theta})^{2}d\theta=b.
\big\|T_\lambda\|_{L^\infty}\leq 3\lambda\big(\|E\|_{L^\infty}+ \eta_t\big)\Big(\|C\|_{L^1}\|E\|_{L^\infty}+ \|C\|_{L^2}^2\Big) &\leq 3\lambda \big(\eta_t + \|E\|_{L^\infty}\big)\Big(\frac{\|E\|_{L^\infty}}{m^2_t}+ \frac{c}{m^2_t}{\bf 1}_{d=2}+ \frac{c}{m_t}{\bf 1}_{d=3}\Big),
{\cal O}_V (P) = V =V_{{\bar{i}}_1{\bar{i}}_2 \cdots{\bar{i}}_p}^{j_1 j_2 \cdots j_q}(P)\eta^{\bar{i}_1}\eta^{\bar{i}_2}\cdots \eta^{\bar{i}_p}\psi_{\bar{j}_1}\psi_{\bar{j}_2}\cdots \psi_{\bar{j}_q},
{\bf v}_G(\bf q,\bf p,\bf 0) =\left(A_{{\bf q p}}^T({\bf q, p})\frac{\partial G}{\partial \bf p}-A_{{\bf p p}}({\bf q, p})\frac{\partial G}{\partial \bf q}\right)\frac{\partial}{\partial \bf q}&+\left(-A_{{\bf q q}}({\bf q, p})\frac{\partial G}{\partial \bf p}+A_{{\bf q p}}(\bf q,\bf p)\frac{\partial G}{\partial \bf q}\right)\frac{\partial}{\partial \bf p},
\boxed{S_k(B(0),B_1)= \sum_{j=\frac k2}^{2N-\frac k2}\begin{pmatrix}j \frac k2 \end{pmatrix}\begin{pmatrix}2N-j-1 \frac k2 -1 \end{pmatrix}B(0)^{2j}\,,k=2,4,\dots, 2N\,.}
g_{n+1}=\cfrac{u_{m,n+1}\, u_{m,n}\, (\Delta x_m)^2\, (\delta y_n)^2}{u_{m+1,n}\, \Delta x_{m+1}\Delta\bigg[ \cfrac{1}{u_{m,n}\, \Delta x_m}\Delta\bigg(\cfrac{u_{m,n+1}}{u_{m,n}}\bigg)\bigg]}.
|g''(t)|=\frac{2}{2\pi}\left|\int_{|\zeta|=\rho}\frac{g(\zeta+t)}{\zeta^3}dt\right|\leq\frac{2}{\rho^2}\sup_{|\zeta|=\rho}|g(\zeta+t)|\leq\frac{2}{\rho^2}\sup_{|\lambda-1/2|\leq\rho+1/2}|g(\lambda)|.
\begin{aligned}D^2u_{-1}= D^2f_{-1}+ \frac{\delta^2}{h_0(h_{-1}+h_0)}(u''_{\delta}-v''_{\delta}) + \frac{\delta^3}{3h_0(h_{-1}+h_0)}(u'''_{\delta}-v'''_{\delta}) + \frac{\delta^4}{12h_0(h_{-1}+h_0)}(u''''_{\delta}-v''''_{\delta}) + \mathcal O(h^3),\end{aligned}
\sum_{i=1}^{n-1}\big(r_id_{i+1}-r_{i+1}d_i + r_ir_{i+1}t\big)+(1-g)\left(\sum_{i=1}^{n-1}r_ir_{i+1}-\sum_{i=1}^n r_i^2\right)+\min_{1\leq i \leq n}\{h^0(\emph{End}U_i)\}.
\sum_{y\in \mathcal{S}_i}(\mathbf{P}^\varepsilon-\mathbf{I}_N)_{x,y}=-\sum_{\substack{j=1 j\neq i}}^n\sum_{y\in\mathcal{S}_j}(\mathbf{P}^\varepsilon-\mathbf{I}_{N})_{x,y}.
\begin{aligned}g_{2}=(v^{6}+1)x_{2,1}x_{2,2}x_{2,3}+(v^{6}+1)x_{1,1}x_{1,2}(x_{2,1}+x_{2,2}+x_{2,3}) \ \ \ \ \ \ -v^{3}(x_{1,1}+x_{1,2})(x_{1,1}x_{1,2}+x_{2,1}x_{2,2}+x_{2,1}x_{2,3}+x_{2,2}x_{2,3}).\end{aligned}
\eta_p^{\epsilon,{\bf h}}(t)=e^{-\frac{t}{\epsilon^2}}h_p+\frac{1}{\epsilon}\int_0^t e^{-\frac{t-s}{\epsilon^2}}Df(q^\epsilon(s)).\eta_q^{\epsilon,{\bf h}}(s)ds+\frac{1}{\epsilon}\int_0^t e^{-\frac{t-s}{\epsilon^2}}D\sigma(q^\epsilon(s)).\eta_q^{\epsilon,{\bf h}}(s)d\beta(s).
\dim((X+V)/V) + \dim((Y+W)/W) = \dim(X+Y + V+ W) + \dim((X+V) \cap(Y+W)) - \dim(V+W) -\dim(V \cap W)= \dim((X+Y + V+ W)/(V+W))+ \dim((X+V) \cap(Y+W)) -\dim(V \cap W)\geq \dim((X+Y + V+ W)/(V+W))+\dim(((X\cap Y)+(V\cap W))/(V\cap W)),
|e_{J - i}(B)p_{i + 1}(B) - e_{J - i}(A)p_{i + 1}(A)| = |e_{J - i}(B)(p_{i + 1}(B) - p_{i + 1}(A)) + p_{i + 1}(A)(e_{J - i}(B) - e_{J - i}(A))| &\leq \max(|p_{i + 1}(B) - p_{i + 1}(A)|, |e_{J - i}(B) - e_{J - i}(A)|) \leq \delta
\lim_{t\rightarrow+\infty}\frac{y^{(t)}}{x_{2}^{(t)}}=\begin{cases}+\infty \mbox{if}\gamma_{2}=0,\delta_{1}\neq0 \frac{\gamma x_{1}^{(t_{0})}+\delta x_{2}^{(t_{0})}}{\delta_2x_{2}^{\left(t_{0}\right)}}\mbox{if}\gamma_{2}=\delta_{1}=0, \frac{\delta}{\delta_2}\mbox{if}\gamma_{2}=0,\delta_{1}\neq0.\end{cases}
V_2(R_2)= kR_2^2 \left[ 3(1+\sigma_0)^{1/2}(1+\sigma_1)-\frac{3}{2}\left( \frac{3-d}{12\pi}\right)^{2/3}(\sigma_0+2\sigma_1)k^{1/3}\rho^2+ \frac{(d-3)^2}{48\pi^2}\rho^6 \right],
\widetilde{\theta}_B \left( \widetilde{\tau}_B (y,t,s) \right) = \widetilde{\theta}_B \left( \tau (y,t), t - s \right) &= \left( \theta_{B,1}(\tau (y,t)), \theta_{B,2}(\tau (y,t)), \theta_{B,2}(\tau (y,t)) - (t-s) \right) &= (y,t,s) \; .
{\rm E}_0 = \langle{\rm ground};{\rm ex}| \hat{\rm H}_{\rm phys}|{\rm ground};{\rm ex}\rangle ={\hbar}^2 \frac{q^2}{\rm L}\sum_{p>0}\frac{1}{{\rm E}_p^2({\rm N})}j_p^{\rm ex}(j_p^{\rm ex})^{\star}.
\frak f_x := \frac 1{4\gamma}\left(q_{x+1}- q_x\right)\left(p_x + p_{x+1}\right) + \frac 14 \left(q_{x+1}- q_x\right)^2 ,x=0, \dots, n-1, \mathfrak F_x = p_x^2 + \left(q_{x+1}- q_x\right) \left(q_{x}- q_{x-1}\right) -\omega_0^2 q_x^2,x=0,\dots, n,
R^T = \left(\begin{array}{cc}R_{11}^T R_{12}^T R_{22}^T\end{array}\right) = \widehat{Q}\left(\begin{array}{cc}\widehat{R}_{11}\widehat{R}_{12}& \widehat{R}_{22}\end{array}\right) \approx \widehat{Q}_1 \left(\begin{array}{cc}\widehat{R}_{11}\widehat{R}_{12}\end{array}\right),
Q(g+f^{n}t) = Q(g)+f^{n}t Q'(g)+(f^{n}t)^2 \tilde{Q}(g,f^nt) \ \ \ \ \ \tilde{Q}&\equiv Q(g)+f^{n}t Q'(g) \pmod{f^{n+m}}\ \ \ \ \n \geqslant m.
\frac{\Gamma_2(z+1-k\,|\,1,\tau)}{\Gamma_2(z+1\,|\,1,\tau)}= \prod\limits_{j=0}^{k-1}\Gamma_1\bigl(z-j\,|\,1,\tau\bigr), = \bigl(\frac{1}{2\pi\tau}\bigr)^{k/2}\tau^{\sum\limits_{j=0}^{k-1}(z-j)/\tau}\;\prod\limits_{j=0}^{k-1}\Gamma\bigl(\frac{z}{\tau}-\frac{j}{\tau}\bigr).
\log x(1 + 2\log \log t) = \frac{\log^{5/3}t(\log \log x)^{1/3}(1 + 2\log\log t)}{B_0^{5/3}}= 3c\log^{5/3}t(\log\log t)^{4/3}\cdot \left(1 + \frac{1}{2\log\log t}\right) \cdot \left(\frac{\frac{3}{5}\log\log x}{\log\log t}\right)^{1/3}\geq 3c\log^{5/3}t(\log\log t)^{4/3}\cdot \left(\frac{\frac{3}{5}\log\log x}{\log\log t}\right)^{1/3},
\int d^{2 K N}w = 2^{N K - K ( K + 1 ) / 2}\int \big | \det_{K}W \big |^{N - K}d^{K^{2}}W \Big \{\prod_{a = 1}^{K}d^{2 N - 2 a + 1}\hat{\Omega}_{a}\Big \}\ .
h(\xi,\eta)=-\intop_{0}^{\infty}\left[P_{\xi}^{+}+P_{\xi}^{0}\right]\left[X^{s}(\xi)\right]^{-1}w(s,y_{\ast}(s,\xi,\eta),\xi)\mathrm{d}s\in\mathbb{L}_{\xi}^{+}\oplus\mathbb{L}_{\xi}^{0},
[N_i,{\cal P}_j] = i\, \delta_{ij}\left(\kappa\sinh \frac{{\cal P}_0}\kappa - \frac{\vec{{\cal P}}\,{}^2}{2\kappa}\, e^{{\cal P}_0/\kappa}\right),\quad [N_i,{\cal P}_0] = i{\cal P}_i\, e^{{\cal P}_0/\kappa}.
\Lambda=\frac{k}{1-\sigma}+\left( \frac{(k^3-k-(k^3-2k^2+2)\theta^{t-3})(1-\sigma)-(k-1+\sigma)(k^2+k-k\theta^{t-3})}{(k^2+k-k\theta^{t-3})(1-\sigma)}\right) \phi^u,
Z=\overline{\mathcal{O}_M}: \, \dim Z = \dim(\textbf{d})-\dim(M)<\dim(\textbf{d}) Z=\displaystyle{\overline{\bigcup_{M\in C}\mathcal{O}_M}}: \,\dim Z=\dim(\textbf{d})-\dim (M)+1\leq \dim(\textbf{d})
g(x) = (x-a^1_j)\ldots(x-a^{k_j}_j)h'_j(x) + e_1(\widehat{x-a^1_j})(x-a^2_j)\ldots(x-a^{k_j}_j)h_j(x) + e^2_j(x-a^1_j) (\widehat{x-a^2_j})\ldots(x-a^{k_j}_j)h_j(x) + e_k(x-a^1_j)\ldots (x-a^{k_j-1}_j) (\widehat{x-a^{k_j}_j})
\sqrt{\max\left\{_0\widehat{\tau}^{(n)}_{\widehat{M}_n(a)},_{n}\widehat{\tau}^{(n)}_{\widehat{M}_{n}(a)}\right\}}=o(t_n) =o\left(\mathbb{E}_0\widehat{\tau}^{(n)}_{\widehat{M}_n}\right).
2\sum_{i=1}^{n}\frac{x_{i}}{1+e^{\frac{2\mu x_{i}}{\sigma^{2}}}}-\sum_{i=1}^{n}\frac{x_{i}(1+e^{\frac{2\mu x_{i}}{\sigma^{2}}})}{1+e^{\frac{2\mu x_{i}}{\sigma^{2}}}}+n\mu=0
I= \frac{\bar{\beta}}{\sqrt{\pi}}e^{- \bar{m}^2 \pi}\sum_{k=1}^{\infty}\frac{\left( \bar{t}_f \right)^{2k-1}\left( -1\right)^{k+1}}{\left( 2k-1\right)\left(k -1\right)!}\int_{0}^{\infty}\frac{du u^{(2k-5)/2}}{u+ \pi}e^{- u \bar{m}^2 - \frac{\bar{\beta}^2}{4u}}
2\frac{\partial^2w_i}{\partial z\partial \bar z}+e^{2(w_i-w_{i-1})}-e^{2(w_{i+1}-w_i)}=0, \ \forall \i\in I, \footnote{Here we set $w_{i}=w_{n+1+i}$ for any $i\in\mathbb{N}\cup\{0\}$.}
{\mathfrak B}(E,F) = \sum_{x\in \mathbb F_q^{d*}}\sum_{s\in\mathbb F_q^*}E(sx) |\widehat{F}(x)|^2 &= \sum_{x\in \mathbb F_q^{d*}}|E\cap l_x| |\widehat{F}(x)|^2,
d\omega_1 =df\wedge dx_1\wedge \cdots \wedge dx_n=\left(\sum_{i=1}^n\frac{\partial f}{\partial x_i}dx_i+\frac{\partial f}{\partial z}dz+\sum_{i=1}^n\frac{\partial f}{\partial p_i}dp_i\right)\wedge dx_1\wedge \cdots \wedge dx_n \hphantom{d\omega_1}{}\equiv \sum_{i=1}^n\frac{\partial f}{\partial p_i}dp_i\wedge dx_1\wedge \cdots\wedge dx_n, \rm{mod}\ \theta,
\det[(z-\mu^{(t)})I_{k+1}-{\cal A}_{k+1}^{(s,t)}] =(z-Q_{k+1}^{(s,t)}-E_{k}^{(s,t)}-\mu^{(t)})\det[(z-\mu^{(t)})I_{k}-{\cal A}_{k}^{(s,t)}] &\quad\quad-Q_{k}^{(s,t)}E_{k}^{(s,t)}\det[(z-\mu^{(t)})I_{k-1}-{\cal A}_{k-1}^{(s,t)}].
g(x)p_{xy}=\frac{1}{d}\frac{g(x)g(y)}{\sum_{z\in\Theta:z\sim_{j}x}g(z)}=\frac{1}{d}\frac{g(y)g(x)}{\sum_{z\in\Theta:z\sim_{j}y}g(z)}=g(y)p_{yx}
\begin{array}{r@{\;=\;}l}R_A(c(R_Af^*\times_{A_0}c(g\times_{A_0}S_Af^*)))^* c(R_Ag^*\times_{A_0}S_Af^*)\quad S_A(c(R_Af^*\times_{A_0}c(g\times_{A_0}S_Af^*)))^* c(R_Af^*\times_{A_0}S_Ag^*).\end{array}
w(z)=(\frac{z-a}{z-b})^{\alpha}(\frac{z-c}{z-b})^{\gamma}{}_{2}F_{1}(\alpha+\beta+\gamma,\alpha+\beta^{\prime}+\gamma;1+\alpha-\alpha^{\prime};\frac{(z-a)(c-b)}{(z-b)(c-a)}),
\begin{bmatrix}S(\lambda) X(\lambda) & W(\lambda) Y(\lambda) Z(\lambda)\end{bmatrix}=\begin{bmatrix}I & I & I\end{bmatrix}\begin{bmatrix}S(\lambda) X(\lambda) Y(\lambda) Z(\lambda) & W(\lambda)\end{bmatrix}\begin{bmatrix}I & I & I\end{bmatrix},
{\mathbb E}\,e^{-\alpha \bar Y(T_{\beta+\omega})-\bar\alpha G(T_{\beta+\omega})}&=\exp\left(-\int_0^\infty \int_{(-\infty,0)}\frac{1}{t}e^{-(\beta+\omega) t}\,(1-e^{-\bar \alpha t-\alpha x})\,{\mathbb P}(Y(t)\in{\rm d}x)\,{\rm d}t\right).
S = \int d^{4}x \sqrt{| g_{E}|}\left[ \hat{R}( \hat{g}_{E}) -{\textstyle \frac{1}{2}}\frac{\partial_{\hat{\mu}}\hat{\lambda}\partial^{\hat{\mu}}\bar{\hat{\lambda}}}{\left( \Im \mathrm{m}\hat{\lambda}\right)^{2}}+{\textstyle \frac{1}{4}}\hat{F}^{I}\{}^{\star}\tilde{\hat{F}^{I}}\right] .
\langle V_{i_{1}}(k_{1}^{\mu})\cdot\cdot\cdot V_{i_{p}}(k_{p}^{\mu})\rangle=\sum_{h=0}^{\infty}\int\frac{Dg_{mn}DX^{\mu}}{N}exp(-I[g,X])V_{i_{1}}(k_{1}^{\mu})\cdot\cdot\cdot V_{i_{p}}(k_{p}^{\mu})
\operatorname{Cap}_{\mathbf{G}_{\frac{2-q}{q}},q'}(O)=\inf\left\{\int_{\mathbb{R}^{N}}|f|^{q'}dx: f\in L^{q'}_+(\mathbb{R}^{N}), \mathbf{G}_{\frac{2-q}{q}}*f\geq \chi_O\right\},
u^{n+2}_j = u^n_j - a \Delta t (u_x)^n_j + a^2 \theta_o \Delta t^2 (u_{xx})^n_j - a \Delta t (u_x)^{n+1}_j + a^2 \theta_e \Delta t^2 (u_{xx})^{n+1}_j + O(\Delta x^3).
-{\frac{t_{1}}{t_{3}}}= ( E_{1}^{( B = 0 )})^{- 2}={\frac{1}{\xi_{1}}}, \qquad -{\frac{t_{2}}{t_{4}}}= ( E_{2}^{( B = 0 )})^{- 2}={\frac{1}{\xi_{2}}}
\partial_t\left[L\partial_{t}\left(\frac 1N\right)-M\partial_{s}\left(\frac1 N\right)\right]+\partial_s\left[N\partial_{s}\left(\frac1N\right)-M\partial_{t}\left(\frac1 N\right)\right] +\frac{\widetilde A_{2}}{N}\partial_t\left(\frac 1N\right)+\frac{\widetilde A_{1}}{N}\partial_s\left(\frac 1N\right) -\frac{A_0}{N^2}=0,
P_{j, k}^{(u)}= \prod_{n=1}^\infty \frac{(1-q^{u(n-1)+j+k})(1-q^{un-j-k})(1-q^{un})^2}{(1-q^{un-j}y)(1-q^{u(n-1)+j}y^{-1})(1-q^{un-k}y^{-1})(1-q^{u(n-1)+k}y)}.
\frac{(-248)^m\prod_{i = 0}^{m-1}(3r+(2i+e-1)n)}{n^mm!}+((e-1)n+3r)\times \left(\frac{2^{8m}\prod_{i=2m+e}^{3m+e-2}(ni+3r)}{n^mm!}+\right. &\left.\sum_{s =1}^{m-1}\binom{m}{s}\frac{(-248)^{s}2^{8(m-s)}\prod_{i = 0}^{s-1}(3r+(2i+e-1)n)\prod_{i=2(m-s)+e}^{3(m-s)+e-2}(ni+3r)}{n^{m}m!}\right)
\partial_\tau w_g^n(y,\tau) - \frac{1}{2}\partial_{yy}w_g^n(y,\tau) + (y + a_n'(y)) \partial_y w_g^n(y,\tau) = g(y,\tau), \: \forall (y,\tau) \in \mathbb{R}\times [0,\infty),
{\left( \rho_{a b}^{( 2 )}\right)}_{i j}^{- 1}= \left( \begin{array}{c c c c}{{0}}&{{- \delta_{a b}D_{i j}}}&{{0}}&{{\partial_{i}^{x}\nabla^{- 2}}}\\{{\delta_{a b}D_{i j}}}&{{0}}&{{\partial_{i}^{x}\nabla^{- 2}}}&{{0}}\\{{0}}&{{- \partial_{j}^{x}\nabla^{- 2}}}&{{0}}&{{\nabla^{- 2}}}\\{{- \partial_{j}^{x}\nabla^{- 2}}}&{{0}}&{{- \nabla^{- 2}}}&{{0}}\\ \end{array}\right) \delta ( \vec{x}- \vec{y}) ,
\lambda (n(r+1)-1) A_{r+1}(X) = \left( r + \frac{1}{2}\right) Y_{1}(X) A_{r}(X) - (nr+1) P^{2}(X) A_{r-1}(X), \lambda(n(r+1)-1)B_{r+1}(X) = \left( r + \frac{1}{2}\right) Y_{1}(X) B_{r}(X) - (nr+1)P^{2}(X)B_{r-1}(X).
\tilde{\gamma}^{a}p_{0a}= 0 = \tilde{\gamma}^{a}f^{\mu}{}_ap_{0\mu}= (\tilde{\gamma}^{m}f^{\alpha}{}_m +\tilde{\gamma}^{h}f^{\alpha}{}_h )p_{0\alpha}+ (\tilde{\gamma}^{m}f^{\sigma}{}_m +\tilde{\gamma}^{h}f^{\sigma}{}_h ) p_{0\sigma},
\mathbb{P}_H \left( \sup_{1\leq k\leq n}\left| \sum_{i=1}^{k}Y_{i,n}\right| > \eta \right) \leq c \mathbb{P}_H \left( \left|\sum_{i=1}^{\lfloor n/2 \rfloor}Y_{i,n}\right| > \eta/c \right)
{[}\hat{D}_{\lambda}^{( + )}( \vec{k}) , \hat{Q}^{a}] = - k_{0}\varepsilon_{\lambda \lambda^{\prime}}\hat{C}_{\lambda^{\prime}}^{a ( + )}( \vec{k}) , \varepsilon_{\lambda \lambda^{\prime}}= - \varepsilon_{\lambda^{\prime}\lambda}, \varepsilon_{1 2}= - 1 ,
T=\frac{\sqrt{x^{2}+a^{2}}}{v_{1}}+\frac{\sqrt{b^{2}+(l-x)^{2}}}{v_{2}}=\frac{\sqrt{x^{2}+a^{2}}}{v_{1}}+\frac{\sqrt{b^{2}+l^{2}-2lx+x^{2}}}{v_{2}}
\big\langle e^{- t \mathcal{A}_p}u , v \big\rangle_{W^{-1 , p}_{\sigma}, W^{1 , p^{\prime}}_{0 , \sigma}}= \frac{1}{2 \pi i}\int_{\gamma}e^{t \lambda}\big\langle (\lambda + \mathcal{A}_p)^{-1}u , v \big\rangle_{W^{-1 , p}_{\sigma}, W^{1 , p^{\prime}}_{0 , \sigma}}\; d \lambda.
(\epsilon_{0n},\,\epsilon_{1n})(x)=\left((\lambda^j_n)^{-\frac{d}{2}+1}\widetilde{\epsilon}^{\,j}_{0n}\left(\frac{x-c^j_n}{\lambda^j_n}\right),\,(\lambda^j_n)^{-\frac{d}{2}}\widetilde{\epsilon}^{\,j}_{1n}\left(\frac{x-c^j_n}{\lambda^j_n}\right)\right),
f_2(\tau)=\sum_{n=1}^\infty a_2(n)q^n=\eta(\tau)^2\eta(2\tau)\eta(4\tau)\eta(8\tau)^2,f_3(\tau)=\sum_{n=1}^\infty a_3(n)q^n=\eta(2\tau)^3\eta(6\tau)^3
\exp\Big{\{}-\frac{1}{\sqrt{M}}\sum_{j}\tilde{\mathfrak{s}}_{kj}Tr\boldsymbol{\omega}_k^{[1]}(\mathbf{w}_k^{[1]})^* JQ_1^{-1}\Xi_j P_1 \Big{\}}\in \mathfrak{Q}_{}(\mathfrak{S}_{2,k}; \kappa_1,\kappa_2,\kappa_3), \kappa_1=W^{O(1)},\kappa_2, \kappa_3=O(1), &~
T^{-2}_{+2-2}=4i\psi^{1-}_{+2\dot q}\psi^{1-}_{-2\dot q},\T^{-2}_{\pm 2-\dot q}=4i\chi \psi^{1-}_{\pm2\dot q},\T^{-2}_{-\dot q-\dot p}= -4i\delta_{\dot q\dot p}(1+\chi^2).
M_2:=\begin{bmatrix}0 \beta A_1^\top A_2 \beta A_1^\top A_3 \dots \beta A_1^\top A_s 0 0 0 \beta A_2^\top A_3 \dots \beta A_2^\top A_s 0 0 0 0 \dots \beta A_3^\top A_s 0 \dots \dots \dots \ddots \dots \dots 0 0 0 \dots 0 0 0 0 0 \dots 0 0 \end{bmatrix}.
\sigma_{0}^{2}= \left( \frac{R}{\xi_{1}h^{2}}- \frac{1}{\lambda_{1}h^{2}}+ \frac{2 R}{3 ( 4 \pi )^{2}}\right) \left( \frac{1}{6 \lambda_{2}h^{4}}- \frac{1 1}{1 2 ( 4 \pi )^{2}}\right)^{- 1}.
A_1(s, t) = 4\pi s \int\limits_0^{\infty}\! db \, b \, J_0(b\sqrt{-t}) \chi(s, b) \;, A_2(s, t) = 2\pi s \int\limits_0^{\infty}\! db \, b \, J_0(b\sqrt{-t}) \chi^2(s, b) \;, A_3(s, t) = \frac{2}{3}\pi s \int\limits_0^{\infty}\! db \, b \,J_0(b \sqrt{-t}) \chi^3(s, b) \;.
\begin{aligned}{\dot \phi}= \frac{1}{R}\int_{\mathbb R}\omega g(\omega) \int_{\mathbb T}\varrho(\theta, \omega, t) \cos(\theta -\phi) ~ d\theta d\omega -\frac{1}{2}\int_{\mathbb R}Kg(\omega) \int_{\mathbb T}\varrho(\theta, \omega, t) \sin 2(\theta - \phi) ~ d\theta d\omega \cr =:{\mathcal I}_{21}+{\mathcal I}_{22}.\end{aligned}
b = e^{\frac{i}{2}\ln ( \frac{2 \hbar}{\theta}) D}\left( \frac{1}{\sqrt{2}}\left( \frac{\tilde{x}_{1}+ i \tilde{x}_{2}}{\sqrt{2 \hbar}}\right) + \frac{i}{\sqrt{2}}\left( \frac{\tilde{p}_{1}+ i \tilde{p}_{2}}{\sqrt{2 \hbar}}\right) \right) e^{- \frac{i}{2}\ln ( \frac{2 \hbar}{\theta}) D}
P^\omega\left(A_t^c \mid S \right) =\frac{P^\omega(A_t^c\cap S)}{P^\omega(S)}\leq c(\omega) P^\omega(A_t^c\cap S) &\leq c(\omega) P^\omega(A_t^c\cap S_{\alpha t}) \to 0,\:\:\: t\to\infty.
\frac 12\Delta_f|\nabla u|^2 =|\nabla^2u|^2+\langle\nabla\Delta_f u, \nabla u\rangle+Ric_f(\nabla u, \nabla u) &\geq\frac{(\Delta_f u)^2}{m+n}+\langle\nabla\Delta_f u, \nabla u\rangle+Ric_f^m(\nabla u, \nabla u)
f^{i}_{r}(q) =\| (\Sigma^{i})^{r}_{[0,q]}-(\Sigma^{i})^{r}_{[q,1-q]}\|_{\textrm{F}}, f^{i}_{\tau}(q) =\| (\Sigma^{i})^{\tau}_{[0,q]}-(\Sigma^{i})^{\tau}_{[q,1-q]}\|_{\textrm{F}}, f^{i}_{\rho}(q) =\| (\Sigma^{i})^{\rho}_{[0,q]}-(\Sigma^{i})^{\rho}_{[q,1-q]}\|_{\textrm{F}}.
S={1\over{2\kappa^2_{10}}}\int dx^{10}\sqrt{-\hat{G}}\,e^{-\hat{\Phi}}\left[{\cal R}_{\hat{G}}+\partial_M\hat{\Phi}\partial^M\hat{\Phi}-{1\over{2\cdot 3!}}\hat{H}_{MNP}\hat{H}^{MNP}\right],
\dfrac{\beta Z^{(\theta)}(b^*) - 1 + \lambda \tilde{C}^{(\theta,r)}(b^*;w'_{+})}{W^{(\theta)}(b^*)}= \beta \theta \dfrac{Z^{(\theta)}(b^*;\Phi(\theta+r))}{Z^{(\theta)\prime}(b^*;\Phi(\theta+r))}.
\sup_{t \leq s \leq T}|Y_s(\omega)|^p \ = \ |Y_{\tau(\omega)}(\omega)|^p \ = \ \lim_{n\rightarrow\infty}|Y_{\tau(\omega)}^n(\omega)|^p \ \leq \ \liminf_{n\rightarrow\infty}\sup_{t\leq s\leq T}|Y_s^n(\omega)|^p.
\iota_+ \;=\; \iota_+(\varepsilon , r_0, T)\;=\; \frac 1{\log T}\log \frac{T^{r_0}+ T^{\varepsilon}}{T^{r_0}}\; , & \iota_- \;=\; \iota_-(\varepsilon , r_0, T)\;=\; \frac 1{\log T}\log \frac{T^{r_0}}{T^{r_0}- T^{\varepsilon}}\;\cdot
v_i(\tau, \xi) = \int_t^\tau \sum_{j=1}^n \Big( C_{ij}(x_i(s, \tau, \xi)) v_j(s, x_i(s, \tau, \xi)) + D_{ij}(x_i(s, \tau, \xi)) v_j(s, 0) begin{align*}6pt] + f_i\big(s, x_i(s, \tau, \xi) \big) \Big)\, ds + \sum_{j=1}^{m}B_{ij}v_{j+k}(t, 0) + g_i(t),
\left( \begin{matrix}{{{Q_{E}}^{U}}}\\{{{Q_{M}}^{U}}}\\ \end{matrix}\right) ={\frac{e}{4 \pi}}\int d S_{k}\, \sqrt{- g}\, U ( x ) \left( \begin{matrix}{{F^{k 0}}}\\{{\tilde{F}^{k 0}}}\\ \end{matrix}\right) U^{- 1}( x )
a_0 =y-\sum_{i=1}^{(k-5)/2-t}a_i-\sum_{i=1}^{(k-5)/2-t}b_i \ge y-A-\frac{1}{2}\sum_{i=2}^{(k-5)/2-t}\binom{n}{k-2-i}-\frac{1}{2}\left(\binom{n}{k-3}-\left(A-\sum_{i=2}^{(k-5)/2-t}i\right)\right) = y-\frac{1}{2}A-\frac{1}{2}\sum_{i=1}^{(k-5)/2-t}\binom{n}{k-2-i}-\frac{\left((k-1)/2-t\right)\left((k-7)/2-t\right)}{4}.
\begin{aligned}\sqrt{a \cdot b}\le \frac{a + b}{2}\, , \operatorname{for}a, b > 0 \operatorname{with}\, `` ="\, \operatorname{holds}\, \operatorname{if}\, \operatorname{and}\, \operatorname{only}\, \operatorname{if}\, a=b\, .\end{aligned}
\left[\begin{pmatrix}t \frac{t u}{1-t^2}0 \frac{1}{t}\end{pmatrix}, \begin{pmatrix}1 0 -\frac{(1-t^2)^2}{t^2 u}1 \end{pmatrix}\right] =\begin{pmatrix}t^2 u 0 \frac{1}{t^2}\end{pmatrix}
L^{\frak{g}}\Bigl(\Bigl(n-\frac{1}{2}\Bigr)\omega_1\Bigr)\Bigl|_{\frak{g}^\sigma}\simeq\bigoplus_{k=0}^{\infty}A_{\frak{q}'(1)}\Bigl(\Bigl(n-2l+k-\frac{3}{2}\Bigr)\mu_1\Bigr)\boxtimes F^{\frak{so}(2n-2l)}(k\nu_1)
\sum_{n=0}^{p-1}\omega_{q}^{f_{0,i^{(n)}}-f_{0,j^{(n)}}}=\omega_{q}^{f_{0,i}-f_{0,j}}\sum_{n=0}^{p-1}\omega_{q}^{\frac{q}{p}n a_{v-1}(i_{\pi(v)}-j_{\pi(v)})}=0.
\Psi^h_1(s) = i\Delta I_h z_R(s), \Psi^h_2(s) =i\kappa \theta_R(|z_R|_{X_s})I_h(|z_R(s)|^2z_R(s)), \Psi^h_3(s) =-i \nu I_h z_R(s)-\epsilon I_h (\gamma z_R(s) -\mu \overline{z_R}(s)), \Psi^h_4(s) = -\tfrac{1}{2}I_h (z_R(s)F_\Phi).
\Pi_{\mu \nu}^{a b}( 4 ) = \frac i{1 4 4 \pi^{2}}g^{2}\frac{N_{f}}{2}\delta^{a b}( p_{\mu}p_{\nu}- p^{2}g_{\mu \nu}) [ 1 2 \mathrm{l n}( \bar{\lambda}^{2}/ p^{2}) - 4 ] \, .
\delta ( t ) = \frac{\delta ( 0 )}{1 - \frac{\delta ( 0 )}{E_{d}^{( 0 )}}\int_{0}^{t}\left( \partial^{0}\tilde{\phi}_{d ( \rightarrow )}^{( 1 )}\partial^{\xi}\tilde{\phi}_{d ( \rightarrow )}^{( 1 )}\right) \mid_{\xi = 1 + \delta}d t^{\prime}}.
{\hat{\cal T}}_{J}^{R}= \sum_{J^{1}, J^{2}}\left\langle \begin{array}{l l}{R^{1}}&{R^{2}}\\{J^{1}}&{J^{2}}\\ \end{array}\right. \left| \begin{array}{l}{R}\\{J}\\ \end{array}\right\rangle{\hat{\cal T}}_{J^{1}}^{R^{1}}( 1 ){\hat{\cal T}}_{J^{2}}^{R^{2}}( 2 ) ,
D_{k}=\nu(\lambda-\mu)\theta_{kl}c_{l}-\mu(\nu-\lambda)\eta_{kl}b_{l}\,, E_{k}= \lambda(\mu-\nu)\omega_{kl}a_{l}-\nu(\lambda-\mu)\theta_{kl}c_{l}\,, F_{k}= \mu(\nu-\lambda)\eta_{kl}b_{l}-\lambda(\mu-\nu)\omega_{kl}a_{l}\,.
y(x)= \frac{y_a}{\Gamma(\gamma)}(x-a)^{\gamma-1}+\frac{1}{\Gamma(\alpha)}\int_{x_1}^{x}(x-t)^{\alpha-1}f(t,y(t))dt +\frac{1}{\Gamma(\alpha)}\int_{a}^{x_1}(x-t)^{\alpha-1}f(t,y(t))dt, x\in[x_1,x_2].
\sum_{(n,vw)=1}G \left( \frac{n}{N}\right) e \left(\frac{bn \overline{c \ell m v}}{w}\right) = \mu(w)\frac{N}{w}\frac{\varphi(v)}{v}+ \sum_{d \mid v}\mu(d) \frac{N}{dw}\sum_{|h| \leq W^{1+\epsilon}d/N}\widehat{G}\left(\frac{hN}{dw}\right) \sum_{(a,w)=1}e \left(\frac{abd \overline{c \ell m v}}{w}+ \frac{ah}{w}\right) + O(Q^{-100}).
H^i({{{{\mathfrak N}}}}^{\lambda},V^{{{\bf N}_{-\lambda}({\mathbb Z}_p)}})\longrightarrow\prod_{w\in W^{\lambda}\atop \ell(w)=i}H^i({\mathfrak N}_w,(V^{{{\bf N}_{-\lambda}({\mathbb Z}_p)}})_{{\mathfrak N}(w)}).
\underbar r_2=\frac{b_{0,\inf}-b_{1,\sup}\bar r_1-l\frac{\chi_2}{d_3}\bar r_2}{b_{2,\sup}-l\frac{\chi_2}{d_3}},\bar r_2= \frac{b_{0,\sup}-b_{1,\inf}\underbar r_1-l\frac{\chi_2}{d_3}\underbar r_2}{b_{2,\inf}-l\frac{\chi_2}{d_3}}.
C^2_1\leq \frac{\widetilde{C}T^{p-1}_*}{p L^{2(p-1)}_*}+\frac{{C^\prime}T^{p-2}_*}{(p-1) L^{2(p-1)}_*}, C^2_2=\frac{2^{p-1}\Gamma(\frac{2p-1}{2})}{p\sqrt{\pi}}\frac{T^{p-1}_*}{L^{2(p-1)}_*}\,,
u(x,t;u_0)= T(t)u_0+\int_0^t T(t-s)\nabla\cdot((\chi_2u\nabla v_2-\chi_1 u\nabla v_1)(\cdot,s;u_0))ds &\qquad +\int_0^t T(t-s)(((a+1)u-bu^2)(\cdot,s;u_0))ds.
\left[C_{\gamma}(\mathcal{Z}_{m,n}^{\gamma}) \right](z)= \overline{\left(1-|z|^2\right)^{\gamma+1}\mathcal{Z}_{n,m-1}^{\gamma+1}(\overline{z},z)}= \left(1-|z|^2\right)^{\gamma+1}\mathcal{Z}_{n,m-1}^{\gamma+1}(z,\bar{z}),
k_{\lambda w}^{\lambda v}|_{\sigma}=k_{\lambda w}^{\lambda v}|_{t_{0}\sigma}=k_{\lambda w}^{\lambda v}|_{t_{0}\sigma+0\nu}\leq k_{\lambda w}^{\lambda v}|_{t_{0}\sigma+1\nu}=k_{\lambda w}^{\lambda v}|_{\rho}.
{\cal M}_{\pi}^{2}( X ) = M_{\pi}^{2}( X ) + \frac{\lambda}{6}\, \vec{\varphi}^{\, 2}( X ) ,{\cal M}_{\sigma}^{2}( X ) = M_{\sigma}^{2}( X ) + \frac{\lambda}{2}\, \vec{\varphi}^{\, 2}( X ) .