harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\int |\tilde K_\ell (\mathbf x,\mathbf y)|(1+d(\mathbf x,\mathbf y))^{\delta'}\,dw(\mathbf x) =\int |\tau_{-\mathbf y}(\mathcal{F}^{-1}m_\ell )(\mathbf x)|(1+ d(\mathbf x,\mathbf y))^{\delta'}\, dw(\mathbf x) & \leq C_{\delta', s}\| m_{\ell, 1}\|_{W^s_2}\leq CM.
2(\mbox{\mathversion{bold}$v$}^{R}_{sss},\mbox{\mathversion{bold}$\varphi$}_{sss})=\frac{2}{R^{2}}(\mbox{\mathversion{bold}$v$}^{R}_{ss},\mbox{\mathversion{bold}$\varphi$}_{ss})=-\frac{1}{R^{6}}\| \mbox{\mathversion{bold}$\varphi$}\|^{2}.
N_{d}=\sum_{d_{A}+d_{B}=d}N_{d_{A}}N_{d_{B}}d_{A}^{2}d_{B}(d_{B}(\begin{matrix}3d-4\\ 3d_{A}-2\end{matrix})-d_{A}(\begin{matrix}3d-4\\ 3d_{A}-1\end{matrix}))
D_{2N}^{BS,(p\to p)}(n-\lambda) =(-2)^N(\lambda-\frac n2-2N)_N(2N-1)!!\times \times(\lambda-n+p-2N+1)_{2N-1}(\lambda-p-2N+1)_{2N}D^{(p\to p)}_{2N}(n-\lambda).
Y(\mathcal{I})_{\alpha\beta}^{ij}:=c_{1}c_{2}\cdot\frac{\Delta_{\mathbf{R}(\mathbf{t})}(\mathcal{I}_{\alpha}^{i})}{\Delta_{\mathbf{R}(\mathbf{t})}(\mathcal{I}_{\beta}^{i})}\cdot\frac{\Delta_{\mathbf{R}(\mathbf{t})}(\mathcal{I}_{\beta}^{j})}{\Delta_{\mathbf{R}(\mathbf{t})}(\mathcal{I}_{\alpha}^{j})}, i,j\in\mathcal{I},\,\alpha,\beta\in\mathcal{I}^{\mathtt{C}}
f_2f_3 / (x-a)^2 = s_2 s_3 x (x-b) (x-c) (x-d) f_1f_3 / x^2 = s_1 s_3 (x-1) (x-a) (x-c) (x-d) f_1f_2 / (x-b)^2 = s_1 s_2 x (x-1) (x-a) (x-c) f_1f_2f_3 / (x(x-a)(x-b))^2 = s_1 s_2 s_3 (x-1) (x-c) (x-d).
\frac{d}{dt}\sum_{i=1}^{N}\|v_i\|_p^p + C\sum_{i=1}^{N}\|v_i^{p/2}\|_{H^1(\Omega)}^2 \leq C\sum_{i=1}^{N}\|v_i\|_{\mu+p-1}^{\mu+p-1}+ C.
\phi\big(M_1(D_{i(1)})^\circ\cdots M_t(D_{i(t)})^\circ\big) &\qquad=\phi\left(\prod_{s=1}^t\Big(M_s(a_{1;i(s)})^\circ\otimes M_s(a_{2;i(s)})\otimes\cdots\otimes M_s(a_{K;i(s)})\Big)\right) &\qquad=\phi_1\big(M_1(a_{1;i(1)})^\circ\cdots M_t(a_{1;i(t)})^\circ\big)\prod_{k=2}^K\phi_k\big(M_1(a_{k;i(1)})\cdots M_t(a_{k;i(t)})\big).
\|u_{k}\|_{L^{q}}^{q}=\frac{\omega_{d}}{\nu(d,\alpha)^{q}}r_{k}^{d}\phi_{k}^{q}&=\frac{\omega_{d}}{\nu(d,\alpha)^{q}}(\varepsilon \phi_{k}^{-q/d}k^{-\alpha q/d})^{d}\phi_{k}^{q}&=\frac{\omega_{d}}{\nu(d,\alpha)^{q}}\varepsilon^{d}k^{-\alpha q},
\varphi_0(1_2)=\begin{cases}|m|^{-1}(1+2^{\frac{z+1}{2}}) (\tau \in \{\pm2, \pm10\}), |m|^{-1}(1+2^{-\frac{z+1}{2}}) (\tau \in \{-1,-5 \}), |m|^{-1}3^{-1}2(1+2^{-z}) (\tau =5).\end{cases}
\mathcal{F}= \int\big{(}\varepsilon_y^2\psi_B+\varepsilon^2\zeta_B-p\psi_B\mathcal{Q}_p^{p-1}\varepsilon^2\big{)}-p\int\psi_B(Q_b^{p-1}-\mathcal{Q}_p^{p-1})\varepsilon^2 &-\frac{2}{p+1}\int\psi_B\Big{[}|Q_b+\varepsilon|^{p+1}-Q_b^{p+1}-(p+1)\varepsilon Q_b^{p}-\frac{p(p+1)}{2}Q_b^{p-1}\varepsilon^2\Big{]}.
Z ( s ) = e^{- \Gamma ( s )}= \exp ( - \eta \int d^{3}x \frac i 2 \widehat{s}\varepsilon \partial \widehat{s}) = \int{\cal D}A_{\mu}\exp \int d^{3}x ( - \frac i{2 \eta}A \varepsilon \partial A - i A \varepsilon \partial \widehat{s}) \; .
\int_{(0,\,\rho/n)}\,\sup_{u\in[s/n,\,sn]}|l'(u)|\,ds = \int_{(0,\,\rho/n)}\,\sup_{u\in[s/n,\,sn]}\big(\alpha(-\log u)^{-\alpha-1}\,u^{-1}\big) \,ds &\le \alpha n \int_{(0,\,\rho/n)}(-\log(sn))^{-\alpha-1}\,s^{-1}\,ds<\infty.
\mathtt{R}=\left[\mathbf{1}+\lambda\mathtt{G}+\lambda^2\mathtt{H}+\lambda^3\mathtt{I}\right]^{-1}\left(\left[\mathtt{G}+\lambda\mathtt{H}+\lambda^2\mathtt{I}\right]\left[\mathtt{H}-\mathtt{G}^2\right] + \left[\mathtt{H}+\lambda\mathtt{I}\right]\mathtt{G}- \mathtt{I}\right)\,.
\sum_{j=0}^{m-2}\sum_{k=0}^{m-1}\prod_{i_2=j+1}^{m-1}\rho_{i_2+k}+ \sum_{k=0}^{m-1}1=\sum_{j=0}^{m-2}\sum_{k=0}^{m-1}\prod_{i_2=j+1}^{m-1}\rho_{i_2+k}+ m.
{\cal L}= g^{\mu \nu}\frac{\partial f^{*}}{\partial x^{\mu}}\frac{\partial f}{\partial x^{\nu}}- \frac{1}{r^{2}}\frac{\partial f^{*}}{\partial \phi}\frac{\partial f}{\partial \phi}+ \mu^{2}f^{*}f - \lambda ( f^{*}f )^{2}.
\det(A_n)w_{1}= \big( \operatorname{adj}(A_n)A_n \vec w )_{1}= \big(\operatorname{adj}(A_n) (T_n, \ldots, T_{n+k+p-1})^T \big)_{1}= \sum_{j=0}^{k+p-1}\gamma_j T_{n+j}
\Psi_{0,1}(M)_1R_{21}\Psi_{0,1}(M)_2R^{-1}_{21}= L^{(+)}_1L^{(-)-1}_1 R_{21}L^{(+)}_2L^{(-)-1}_2R_{21}^{-1}& = L^{(+)}_1 L^{(+)}_2 R_{21}L^{(-)-1}_1 L^{(-)-1}_2R_{21}^{-1}= L^{(+)}_1 L^{(+)}_2 L^{(-)-1}_2 L^{(-)-1}_1 & = L^{(+)}_{12}L^{(-)-1}_{12}= \Psi_{0,1}(M)_{12}.
jO_{n}^{(3)}-JO_{n+2}^{(3)}=(j_{n}^{(3)}-J_{n+2}^{(3)})e_{0}+(j_{n+1}^{(3)}-JO_{n+3}^{(3)})e_{1}+\cdots +(j_{n+7}^{(3)}-J_{n+9}^{(3)})e_{7}&=1-e_{1}+e_{3}-e_{4}+e_{6}-e_{7}
\operatorname{Re}\psi_3(i\rho,i\tau,\xi+i\eta)=-\alpha\rho\tau-\beta\tau\eta\leqslant -\frac{\alpha}{2}(1+3\rho^2)\leqslant-\frac{\alpha}{2},\shortintertext{and}\operatorname{Re}\psi_4(i\rho,i\tau, \xi+i\eta)=-\alpha\rho\tau-\beta\rho\eta \leqslant-\frac{\alpha(1+3\rho^2)}{2}\leqslant-\frac{\alpha}{2},
\frac{2}{n}\sum_{i=1}^n \left(\boldsymbol{\hat{\tilde{H}}_{k2}}(t_p) - \boldsymbol{\hat{\tilde{H}}_{k2}}(t_o)\right)^T \! \hat{\mathbf{\Sigma}}_\mathbf{k}^{-1}\tilde{\mathbf{\Sigma}}_\mathbf{ki}\hat{\mathbf{\Sigma}}_\mathbf{k}^{-1}\left(\boldsymbol{\hat{\tilde{H}}_{k2}}(t_p) - \boldsymbol{\hat{\tilde{H}}_{k2}}(t_o)\right),
u_\alpha-1 =\sum_{j=1}^sa_j(\sum_{i=1}^k w_{ji}X_i)+\sum_{j=1}^tb_j(\sum_{i=1}^k v_{ji}X_i) \equiv\sum_{j=1}^sa_j(\prod_{i=1}^k g_i^{w_{ji}}-1)+\sum_{j=1}^tb_j(\prod_{i=1}^k g_i^{v_{ji}}-1)\mod J^2 =\sum_{j=1}^sa_j(f_j-1)+\sum_{j=1}^tb_j(h_j-1),
\begin{cases}\displaystyle{\partial_t m_k = - \frac{ik}2 \sum_{k_1 + k_2 = k}e^{-it(k^3 - \alpha k_1^3 - \alpha k_2^3)}n_{k_1}n_{k_2}}+ p_k \displaystyle{\partial_t n_k = - ik \sum_{k_1 + k_2 = k}e^{-it(\alpha k^3 - k_1^3 - \alpha k_2^3)}m_{k_1}n_{k_2}}+ q_k.\end{cases}
\begin{cases}1<p<\infty \mbox{when}\ \n=1, \displaystyle{1<p\leqslant \max\left\{1+\frac{2}{n},\frac{1}{2(n-1)}\left(1+\sqrt{4n^2-3}\,\right)\right\}}\mbox{when}\ \n\geqslant 2.\end{cases}
\mathrm{Td\,}(\nabla^{T^{1,0}U},T^{1,0}U)(z)=\mathrm{Td_b\,}(\nabla^{T^{1,0}X},T^{1,0}X)(z,\theta) &\mathrm{ch\,}(\nabla^{E\otimes L^m},E\otimes L^m)(z)=\Bigr(\mathrm{ch_b\,}(\nabla^{E},E)\wedge e^{-m\frac{d\omega_0}{2\pi}}\Bigr)(z,\theta)
\sum_{\substack{r=0}}^{2n-1}x_{r/2}y^*_{r/2+t\bmod n}+ \sum_{\substack{r=0}}^{2n-1}y_{(r-1)/2}x^*_{(r+1)/2+t\bmod n}&= R_{X,Y}(t) + R_{Y,X}(t+1) .
k^{(1)}=\frac{EA}{L}[\begin{matrix}1&0&-1&0\\ 0&0&0&0\\ -1&0&1&0\\ 0&0&0&0\\ \end{matrix}]\rightarrow K^{(1)}=\frac{EA}{L}[\begin{matrix}1&0&-1&0&0&0\\ 0&0&0&0&0&0\\ -1&0&1&0&0&0\\ 0&0&0&0&0&0\\ 0&0&0&0&0&0\\ 0&0&0&0&0&0\\ \end{matrix}]
S = d + i \theta \cdot \sigma^{\mu}\cdot \bar{\theta}\partial_{\mu}d + \frac{1}{4}\theta^{2}\bar{\theta}^{2}\partial^{2}d + \sqrt{2}\theta \cdot \delta - \frac{i}{\sqrt{2}}\theta^{2}\partial_{\mu}\delta \cdot \sigma^{\mu}\cdot \bar{\theta}+ \theta^{2}Z
<{\frac{\delta W}{\delta \beta^{a}}}>_{_{\beta^{a}= 0}}= \, < ( D_{\mu}J_{R}^{\mu}\, )^{a}+{\frac{1}{4 \pi}}\epsilon^{\mu \nu}\partial_{\mu}A_{\nu}^{a}>_{_{\beta^{a}= 0}}\, = \, 0 \, \, .
\sup_{\substack{\pi, \pi=\lbrace \kappa_0=s,\kappa_{1},...\kappa_{m}=t \rbrace}}\big{\lbrace}\sum_{j}\big{[}|h_{\kappa_{j},\kappa_{j+1}}|^{\frac{1}{{\gamma}^{\prime}}}\big{]}\big{\rbrace}<\infty.
a_1 = \frac{3 \sigma_1^2}{\sigma_2^2 \sqrt{\sigma_2^2-\sigma_1^2}}, a_2 = \frac{\sqrt{2}\sigma_1^2}{\sqrt{\sigma_2^2}\sqrt{\sigma_2^2-\sigma_1^2}}+2, a_3 =\frac{\sigma_1^2}{\sqrt{\sigma_2^2-\sigma_1^2}}\left( \sqrt{|\log(2\pi\sigma^2_2)|^2 + \frac{24 (\sigma_2^2-\sigma_1^2)^2}{\sigma^4_2}+ \pi^2}\right).
\xi=\tau \dot q-g^{-1}\partial_{\dot q}N_2= \binom{\tau \dot x}{\tau \dot y-\dot x}, f=\tau L+N_2-\partial_{\dot q}g^{-1}\partial_{\dot q}N_2= \tau \bigl(\dot x\dot y-G(x)y\bigr)-\frac{1}{2}\dot x^2+\int G(x)dx
\frac{dT}{dx}= - \frac{4c\varepsilon^2HZ^4n}{\nu_0^2v^2}\int\limits_0^\infty \xi d\xi \left\{1 - e^{- \alpha \xi \int\limits_{\xi}^\infty \frac{du}{u^2}B( u )}\right\}\frac{\int\limits_\xi^\infty \frac{du}{u}B( u )}{\alpha \xi \int\limits_\xi^\infty \frac{du}{u^2}B( u )}~~.
\nu\biggl(\binom{16(x+8\ell)+p}k-\binom{16x+p}k\biggr)\ge\begin{cases}\nu\binom{16x+p}k+7-\lg(p) 1\le k\le p \nu\binom{16x+p}k+3-\nu(x) k>p,\ 1\le x\le7 7-\nu(p+1)-\nu\binom k{p+1}k>p,\x=0.\end{cases}
P\bigg(|\mathrm{Nbd}(B[0,n/3], n^{1/2-\delta}) | \le (1-\epsilon)^{2}\hat{\psi}(n)^3 \omega_3 \bigg) = P\bigg(| \mathrm{Nbd}(B[0, n^{2\delta}], 3^{-1/2}) | \le (1-\epsilon)^{2}\bigg(\frac{\pi}{\sqrt{2}}\bigg)^3 (n^{2\delta}(\log_2 n)^{-1})^{3/2}\omega_3 \bigg) \le C \exp(-(1-\epsilon)^{-2}(\log_2 n)).
\pi(z_{2n-1})h(\alpha_{1},\alpha_{2},\cdots ,\alpha_{2n-1})= q^{\alpha_{1}}\pi(z_{1})^{\alpha_{1}}\pi(z_{2n-1})\pi(z_{2})^{\alpha_{2}}\pi(z_{3})^{\alpha_{3}}\cdots \pi(z_{2n-1})^{\alpha_{2n-1}}h.
\log \bigg( k_N^2 \bigg( \frac{\mu}{s}\bigg)^{1/3k_N}\bigg) = 2 \log k_N - \frac{1}{3k_N}\log \bigg( \frac{s}{\mu}\bigg) = 2 \log k_N - \frac{[\log(s/\mu)]^2}{3 \log N},
S_n=\frac{2i(2\pi)^{(A-9)/2}}{n^{(A+9)/2}}\sum_{\ell=-2}^2 u_\ell J_{\ell,n}\textup{and}\widehat{S}_n=-\frac{2i(2\pi)^{(A-9)/2}}{n^{(A+9)/2}}\sum_{\ell=-2}^2 u_\ell \widehat{J}_{\ell,n}
b(\gamma^{-1}z) = \lim_{n \rightarrow \infty}B_\Omega(\gamma^{-1}z,y_n)-B_\Omega(y_n,z_0) = \lim_{n \rightarrow \infty}B_\Omega(z,\gamma y_n)-B_\Omega(y_n,z_0) =b(z)
\psi_n(t) =\sum^{n}_{k=0}{\binom{n}{k}^2\left(\frac{t}{t+1}\right)^{2k}\left(\frac{1}{t+1}\right)^{2n-2k}}&=\sum^{n}_{k=0}{\binom{n}{k}^2x^{2k}(1-x)^{2n-2k}}.
\frac{\lambda}{\alpha}\frac{d}{dt}\int_U u^{\alpha}dx= - (\alpha-\lambda)\int_U K(|\nabla u+Z(u)|)|\nabla u+Z(u)|^2u^{\alpha-\lambda-1}dx & +(\alpha-\lambda)\int_U K(|\nabla u+Z(u)|)(\nabla u+Z(u))\cdot Z(u) u^{\alpha-\lambda-1}dx+\int_\Gamma u^{\alpha-\lambda}B d\sigma +\int_U f u^{\alpha-\lambda}dx.
\int_{-\infty}^{\infty}L(\Delta f)d\Delta f=\frac{f_{\Delta}}{\pi}\int_{-\infty}^{\infty}\frac{d\Delta f}{f_{\Delta}^{2}+\Delta f^{2}}=\frac{1}{\pi}tan^{-1}(\frac{\Delta f}{f_{\Delta}})|_{-\infty}^{\infty}=1
\begin{array}{r l}{{\tau^{\prime}- \tau}}&{{= \delta \tau = 2 \Lambda ( \tau , \eta ) - \eta D \Lambda ,}}\\{{\eta^{\prime}- \eta}}&{{= \delta \eta = - i D \Lambda ,}}\\{{D^{\prime}- D}}&{{= \delta D = - \partial_{\tau}\Lambda D ,}}\\ \end{array}
\begin{cases}\eta_1(x) = \xi(x)(1-\omega(x)) \equiv{\bf 1}_{\{\eta(x) =1\}}\, , \eta_2(x) = (1-\xi(x))\omega(x) \equiv{\bf 1}_{\{\eta(x) =2\}}\, , \eta_3(x) = \xi(x)\omega(x) \equiv{\bf 1}_{\{\eta(x) =3\}}\, . \end{cases}
\|B f\|_{L^2(K)}= \bigg\|\sum_{j = k}^{\infty}q^{-(j+1)}~ g_j \ast f \bigg\|_{L^2(K)}\leq ~ \sum_{j = k}^{\infty}q^{-(j+1)}~ \|g_j \ast f\|_{L^2(K)}\leq ~ \sum_{j = k}^{\infty}q^{-(j+1)}~ \| f\|_{L^2(K)}= ~ q^{-k}(q-1)^{-1}\| f\|_{L^2(K)}.
\int_{0}^{1}F_{1}^{p\nu}(x,Q^{2})-F_{1}^{p\overline{\nu}}(x,Q^{2})dx=\int_{0}^{1}F_{1}^{p\nu}(x,Q^{2})-F_{1}^{n\nu}(x,Q^{2})dx=1-\frac{2}{3}\frac{\alpha_{s}(Q^{2})}{\pi},
\|(\partial_xG\ast fgh)\varphi_N\|_{\infty}= \frac{1}{2}\varphi_N \int_{\mathbb{R}}e^{-|x-y|}(fgh)(y)dy &\leq \frac{1}{2}(\varphi_N\int_{\mathbb{R}}e^{-|x-y|}\frac{1}{\varphi_N(y)}dy)\|f\|_{\infty}\|g\|_{\infty}\|h\varphi_N\|_{\infty}&\leq C_0\|f\|_{\infty}\|g\|_{\infty}\|h\varphi_N\|_{\infty}.
\begin{array}{l}{{\left. X_{1}^{+}\right.^{*}= X_{1}^{+}~ , ~ ~ \left. X_{1}^{-}\right.^{*}= X_{1}^{-}~ , ~ ~ H_{1}^{*}= - H_{1}~ ,}}\\{{\left. X_{2}^{+}\right.^{*}= - X_{2}^{+}~ , ~ ~ \left. X_{2}^{-}\right.^{*}= - X_{2}^{-}~ , ~ ~ H_{2}^{*}= - H_{2}~ .}}\\ \end{array}
\xi(\xi_{0},t)\approx\xi_{0}+\frac{\hat{u}}{\omega}cos(k\xi_{0}-\omega t)-\frac{1}{4}\frac{k\hat{u}^{2}}{\omega^{2}}sin2(k\xi_{0}-\omega t)+\frac{1}{2}\frac{k\hat{u}^{2}}{\omega}t.
\propto \prod_{k=1}^n x_k^{-n-\nu}\, \det(I_n - (X^{-1})^* Y^* Y X^{-1})^{m-2n-\nu}\theta(I_n-(X^{-1})^* Y^* Y X^{-1}) \, dY = \prod_{k=1}^n x_k^{-m+n}\, \det(X^* X - Y^* Y)^{m-2n-\nu}\theta(X^*X - Y^* Y) \, dY.
\sum_{k=0}^{n-1}\frac{(q;q^2)_k(-q^2;q^2)_k}{(q^2;q^2)_k}q^{n^2-(k+1)^2}begin{align*}7pt] =\sum_{\substack{0\le k \le n-1 begin{align*}3pt]k\not= (n-1)/2}}c_{n,k}+c_{n,(n-1)/2}begin{align*}7pt] \equiv \frac{1-(-1)^{\frac{n-1}{2}}}{2}(1-q^n)+(-1)^{\frac{n-1}{2}}\left(1+\frac{(1-n)(1-q^n)}{2}\right) \pmod{\Phi_n(q)^2}.
\nu^{2}{\frac{\theta_{i j}}{( z_{i j})^{2}}}\xi_{i}\epsilon_{j}K_{i j}^{\mathrm{t o t}}, \quad \nu^{2}{\frac{\theta_{i j}}{( z_{i j})^{2}}}\xi_{i}\epsilon_{j}K_{i j}^{\mathrm{t o t}}K_{j k}^{\mathrm{t o t}}\, .
{{\bf{H}}_{k,{\rm{R}}}}= \sum\nolimits_{l \in{\mathcal L}_{k,{\rm{R}}}}{{\alpha_{{k,{\rm{R}}},l}}{{\bf{a}}_{\rm{R}}}\left({\Theta_{{k,{\rm{R}}},l}^{\rm{A}}}\right){\bf{a}}_{{\rm{S,}}k}^H\left({\Theta_{{k,{\rm{R}}},l}^{\rm{D}}}\right)},
V_{\pm}( r_{\pm}) \approx \Big \{\mp \gamma_{1}r_{\pm}+ \gamma_{2}r_{\pm}^{2}\mp \gamma_{3}r_{\pm}^{3}+ \left( \gamma_{4}- d_{4}\ln{r_{\pm}^{2}}\right) r_{\pm}^{4}\mp \beta_{5}r_{\pm}^{5}\ln{r_{\pm}^{2}}- \alpha_{5}^{\pm}r_{\pm}^{5}+ \dots \Big \}.
K(\Psi^{i};\Psi^{i};\gamma) = K(\Psi^i_0 ; \Psi^i_0; \gamma) = K(\Psi^i_1 ;\Psi^i_1;\gamma) = \cdots = K(\Psi^i_{i-1};\Psi^i_{i-1};\gamma) = K(\Psi^{i+1};\Psi^{i+1};\gamma)\,,
0 = \sum_Y \int_{\tilde{\Delta}_Y^+}-2q(b)\big{(}((n+1)\Lambda_Y-\bar{S}_{\Theta})P_{DH}(q)+d_qP_{DH}(\chi^{ac}-\chi) \big{)}dq +\int_{\Delta^+}2\sum_{\alpha\in\Phi_{Q^u}}\alpha(b)P_{DH}(q)dq
\nabla^G_{\xi}v(t,x) =\int_{t}^{T}\nabla^G_{\xi}R_{s-t}\Big [ e^{-(s-t){A}}GB(s,\cdot)+e^{-(s-t){A}}L(s,\cdot)B\left( s,\cdot\right) \big] ( x) ds,\;\;\; (t,x) \in [0,T] \times H,
A(x,y)=\left(\begin{matrix}a_{1}c_1 x 0 0 \hdots 0 b_1 y a_2 c_2x^2 0 \hdots 0 0 b_2 y^2 a_3 c_3 x^3 \hdots 0 \vdots \hdots \ddots \ddots \ddots \vdots 0 \hdots 0 b_{n-2}y^{n-2}a_{n-1}c_{n-1}x^{n-1}0 0 \hdots 0 b_{n-1}y^{n-1}a_n \end{matrix}\right)
a_{ij}(g)a_{ik}(h) = \delta_{ik}a_{ij}(gh) a_{ji}(g)a_{ki}(h) = \delta_{jk}a_{ji}(gh) \sum_{k=1}^{N}a_{ik}(e) = 1 = \sum_{k=1}^{N}a_{kj}(e)
(1+|u|)^{\lambda/2}|u| \Big(\int_1^2 (1+|u-t2^{-k}| )^{-\lambda-2}dt \Big)^{1/2}& \lesssim_\lambda (1+|u|)^{\lambda/2+1}(1+|u|)^{-\lambda/2-1}= 1 \leq 2^{k(-1-\lambda)/2}.
{\cal D}\Psi_{D}= \left( \begin{array}{c}{- B ( \bar{Z}e^{2}+ \bar{Y},_{3}e^{3})}\\{d B + B [ h ,_{4}+ ( Z - \bar{Z}) h / 2 ] e^{3}}\\{d C + C [ h ,_{4}- ( Z - \bar{Z}) h / 2 ] e^{3}}\\{C ( Z e^{1}+ Y ,_{3}e^{3})}\\ \end{array}\right) .
\lim_{n \to \infty}n^{s -2}\left[ \frac{d}{d x}\widehat{Q}_n^{(s)}\right] ( 1 + \gamma^{(n)}) = 2 \sqrt{2}u^{\frac{s - 2}{2}}\left( s K_s(\sqrt{u}) - \frac{\sqrt{u}K_{s + 1}(\sqrt{u}) + \sqrt{u}K_{s-1}(\sqrt{u})}{2}\right) = 2 \sqrt{2}u^{\frac{s-2}{2}}L_{s}(\sqrt{u}).
\int_{0}^{T}\bigl{(}v(t)f'(u(t))u''(t) + v'(t)f'(u(t))u'(t) \bigr{)}~\!dt = \Bigl{[}v(t)f'(u(t)) u'(t)\Bigr{]}_{t=0}^{t=T}- \int_{0}^{T}v(t) f''(u(t))u'(t)^{2}~\!dt = -\int_{0}^{T}v(t) f''(u(t))u'(t)^{2}~\!dt.
\int_{v_0 \in \mathbb{S}^{d-1}}v_{0,i}^2dS = \frac{1}{d}\int_{v_0 \in \mathbb{S}^{d-1}}\left(\sum_{i=1}^{d}v_{0,i}^2 \right)dS = \frac{1}{d}\int_{v_0 \in \mathbb{S}^{d-1}}dS.
A(t)u(t) =A(t)e^{-tA(t)}u_{0}+ A(t)\int_{0}^{t}{e^{-(t-l)A(t)}(\mathcal{A}(t)-\mathcal{A}(l))u(l)dl}+A(t)\int_{0}^{t}{e^{-(t-l)A(t)}[f(l)]}dl
E ={1 \over n!}\int d\Omega \left[ E_1 \left| Y^{(l)}_{a_1 \cdots a_n}\right|^2 + E_2 \left| Y^{(l)}_{a_2 \cdots a_n}\right|^2 + E_3 \left| Y^{(l)}_{a_3 \cdots a_n}\right|^2 \right].
\begin{multlined}[t][12.5cm]\frac{(2b+\nu_{G}+1)!}{\pi(b+\nu_{G})!b!}\sim \frac{2}{\sqrt{\pi}}\sqrt{b}\left(\frac{1}{y_{b}}\right)^{b+\nu_{G}}\left(\frac{1}{1-y_{b}}\right)^{b}\end{multlined}
\alpha-\chi(\alpha) = (\underbrace{0,\ldots,0}_n,r_n,r_{n+1}) - (\underbrace{0,\ldots,0}_{n+1},-1,0,0,\ldots) = (\underbrace{0,\ldots,0}_n,r_n,r_{n+1}+1,\ldots)\in S
{\cal A}=h^{(\alpha)}dy^{0}\wedge dy^{1}\wedge ...\wedge dy^{q-1}\wedge[dz^{1}_{1}\wedge ...\wedge dz^{r}_{1}H^{-1}_{1}+...+dz^{1}_{n}\wedge ...\wedge dz^{r}_{n}H^{-1}_{n}]
Q(\varphi;U, \mathcal{F}):= \int_{\mathcal{F}}\left(|\varphi'|^2 + (\lambda-(p-1)\rho|U|^{p-2}) \varphi^2\right)\,dx, \forall \varphi \in H^1(\mathcal{F}) \cap C_c(\mathcal{F}).
\mathrm{K}_0(a,b) :=\lim_{p \to 0}\mathrm{K}_p(a,b)=\frac{ab}{\mathrm{L}(a,b)}, \mathrm{K}_1(a,b) :=\lim_{p \to 1}\mathrm{K}_p(a,b)=\mathrm{L}(a,b).
\begin{pmatrix}\mathbf{f}(t) \mathbf{f}(t+\varepsilon) \vdots \mathbf{f}(t+(N-1)\varepsilon)\end{pmatrix}=\mathbf{D}\begin{pmatrix}\mathbf{u}(t+(N-1)\varepsilon) \vdots \mathbf{u}(t-N\varepsilon)\end{pmatrix}
\big(e_{i_1}\otimes \cdots \otimes e_{i_d}\big) \stackrel{r}{\smallfrown}\big(e_{j_1}\otimes \cdots \otimes e_{j_p}\big) = \prod_{l=0}^{r-1}\langle e_{i_{d-l}}, e_{j_{l+1}}\rangle e_{i_1}\otimes \cdots \otimes e_{i_{d-r}}\otimes e_{j_{r+1}}\otimes \cdots \otimes e_{j_{p}},
\Phi(X)\cdot\varphi_\pm =\tfrac{1}{2}(X^\flat\cdot\pi^*\omega-\pi^*\omega\cdot X^\flat)\cdot\varphi_\pm =\tfrac{1}{2}i\cdot(\pm 2)X^\flat\cdot\varphi_\pm =\pm i X^\flat\cdot\varphi_\pm
M_s(t)(\omega) = \sigma(s,\omega, X(s)(\omega)) + \int_s^t U(s, r, \omega) dr + \int_s^t V(s, r, \omega) dW(r) &+\int_s^t \nabla_x b(r,\omega, X(r)(\omega)) M_s(r)(\omega) dr + \int_s^t \nabla_x \sigma(r,\omega, X(r)(\omega)) M_s (r)(\omega) dW(r),
\frac{\partial\eta}{\partial t}+(u\frac{\partial\eta}{\partial x}+v\frac{\partial\eta}{\partial y})+\eta(\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y})+(H_{0}+\alpha_{0}y)(\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y})+\alpha_{0}v=0.
\| K(z)-e^{-z\tau}K(\chi(z)) \| \le |e^{-z\tau}|\| K(z)-K(\chi(z)) \| + |1-e^{-z\tau}|\|K(z)\| \le c\| K(z)-K(\chi(z)) \| + c\tau |z|\|K(z)\| \le c\|K(z)-K(\chi(z))\| + c\tau,
|\nabla H(\pi_{p_0}^{-1}(x))|=|\nabla H(\pi_{p_0}^{-1}(x))-\nabla H(p_0)|\leq C \max_{S^m}|\nabla^2 H|\cdot |\pi_{p_0}^{-1}(x)-p_0|\leq C (1+|x|^2)^{-\frac12}.
\int_a^b f(v) \, dv =\int_a^{v_0}\cdots + \int_{v_0}^b\cdots\ge f(a)(v_0- a) + f (v_0) (b-v_0) & =\mbox{$ \frac12 $}f(a)(b- a) +\mbox{$ \frac12 $}f(v_0)(b- a) + (f(v_0)- f(a)) (\mbox{$ \frac12 $}(a+b) - v_0),
[w^{\ell}]\frac{1-x(1+w)}{1-\xi x(1+w)}= \,\frac{\xi-1}{\xi}[w^{\ell}] \frac{1}{1-\xi x(1+w)}= \frac{1-\xi}{\xi^2x}\,[w^{\ell}]\left(w-\frac{1-\xi x}{\xi x}\right)^{-1}= \frac{1-\xi}{\xi^2 x}\binom{-1}{\ell}(-1)^{-1-\ell}\left(\frac{1-\xi x}{\xi x}\right)^{-1-\ell}= -\frac{1-\xi}{\xi^2 x}\left(\frac{\xi x}{1-\xi x}\right)^{1+\ell}.
\left(F^{-}(x)\right)^{-1}= \frac{1}{\Delta_{-}}\left( \begin{array}{lr}(p(x) - q(x))^{-}(p(x) - q(x))^{-}(p(x) + q(x))^{-}- (p(x) + q(x))^{-}\end{array}\right),
\begin{aligned}X_1 =a-(2m_1-1)d_1, X_2 = a+(2m_1+1)d_1+2d_2, X_3 =-(2m_2-1)d_2, X_4 = (2m_2+1)d_2+2d_1, X_5 = b+(2n+1)d_1, X_6 = b-(2n-1)d_1+2d_2, Y_1 =a+(2m_1+1)d_1, Y_2 = a-(2m_1-1)d_1+2d_2, Y_3 = (2m_2+1)d_2, Y_4 = -(2m_2-1)d_2+2d_1, Y_5 =b-(2n-1)d_1, Y_6 =b+(2n+1)d_1+2d_2,\end{aligned}
|\alpha\rangle=\sum_{n=0}^{\infty}|n\rangle\langle n|\alpha\rangle=e^{-\frac{1}{2}|\alpha|^{2}}\sum_{n=0}^{\infty}\frac{\alpha^{n}}{\sqrt{n!}}|n\rangle=e^{-\frac{1}{2}|\alpha|^{2}}e^{\alpha a^{\dagger}}e^{-\alpha^{*}a}|0\rangle.
Z_m \leq C T^{3/2}\sum_{|(j,k)|\leq m}{m\choose j~k}Z_j Z_k V_{m-j-k}+ C T \sum_{j=0}^m{m \choose j}Z_j V_{m-j}+ C T^{1/2}V_m &\leq C T^{1/2}(T Z_0^2 V_m + T Z_m Z_0 V_0 + T^{1/2}Z_0 V_m + T^{1/2}Z_m V_0 + V_m) &+ C T^{3/2}\sum_{0 < |(j,k)| < m}{m\choose j~k}Z_j Z_k V_{m-j-k}+ C T \sum_{j=1}^{m-1}{m \choose j}Z_j V_{m-j}
w\partial_\eta^2 w\leq w^{-1}(\eta\partial_\xi w+\partial_\tau w) \leq w^{-1}[\eta(\partial_\xi w+\partial_\tau w)+(1-\eta)\partial_\tau w] \leq \frac{b\delta(1-\eta)^{\alpha_0+1}}{c_0(1-\eta)}+w^{-1}\eta\delta b(1-\eta) \leq \frac{5\delta}{4},on \,\,\{\xi=0\}\cup \{\tau=0\},
\overline{{{\psi}}}^{( 1 )}( x ) P_{+}\psi^{( 1 )}( x ) \; \longleftrightarrow \; \frac{1}{2 \pi}c^{( 1 )}: e^{- i a \Phi^{( 1 )}( x )}:_{M^{( 1 )}}: e^{- i b \Phi^{( 2 )}( x )}:_{M^{( 2 )}}\; e^{i \frac{\theta}{2}}\; ,
\begin{aligned}I_{0,1}(x_0,x_1)=\lambda^2\int_0^\tau du \int_{0}^{1}ds \R(s+\tau-u,x_1(s)+x_0(\tau)-x_0(u)), &I_{N,N+1}(x_N,x_{N+1})=\lambda^2\int_0^1 du\int_0^{T-\tau-N}ds \R(s+1-u,x_{N+1}(s)+x_N(1)-x_N(u)).\end{aligned}
\begin{array}{l}\mathbb{P}(log(R_{j}^{k})=r_{j}^{k}| Y_{ij}^{(k(}< a_1~ or ~ Y_{ij}^{(k(}> a_2, b_k^{(2)}) = \mathbb{E}[ ~ \mathbb{I}(log(R_{j}^{k})\le r_{j}^{*k}) ~ \vert ~ \mathbb{I}_{(Y_{ij}^{(k)}< a_1)}=1 ~ or ~ \mathbb{I}_{(Y_{ij}^{(k)}> a_2)}= 1 ~,~ b_{k}^{2}] \end{array}
\frac{\mathrm{e}^{- x^{2}/ 2 t}}{\left( 2 \pi t \right)^{D / 2}}= \int \frac{\mathrm{d}^{D}p}{\left( 2 \pi \right)^{D}}\mathrm{e}^{- i p \cdot x - \frac 1 2 p^{2}t},
(-\Delta)_p^s u(x,t) :=\mathrm{P.V.}\int_{\mathbb{R}^n}\frac{|u(x,t)-u(y,t)|^{p-2}(u(x,t)-u(y,t))}{|x-y|^{n+sp}}\,dy &=\lim_{\varepsilon \searrow 0}\int_{\mathbb{R}^n \setminus B_\varepsilon(x)}\frac{|u(x,t)-u(y,t)|^{p-2}(u(x,t)-u(y,t))}{|x-y|^{n+sp}}\,dy,
C'_w=q^{-\frac{\ell(w)}{2}}\left(T_w+T_{rs_j}+T_{s_ir}+\cdots\right)=q^{-\frac{\ell(w)}{2}}\left(T_w+T_{rs_j}+q^{\frac{\ell(s_i r)}{2}}C'_{s_ir}\right)
h_F(c) =\frac{\Gamma K}{\mu K_1}\frac{1}{1-b}\frac{(1+c)^2}{(K+c)^2}, \theta_F(c) =\frac{\Gamma}{\lambda}\left(\frac{c}{K+c}-\frac{K}{1-b}\frac{(1+c)(b+c)}{(K+c)^2}\right).
\begin{aligned}\||\nabla v-O|^2\|_{L^{p(\cdot)}(\Omega)}\le (M+\sqrt{n})^{2-\mu}\|\nabla v-O\|_{L^{q(\cdot)}(\Omega)}^\mu &\le 2^{\mu-1}C^\mu(M+\sqrt{n})\|f\|_{L^{p(\cdot)}(\Omega)}+2^{\mu-1}C^\mu(M+\sqrt{n})^{2-\mu}\|g\|^\mu_{L^{q(\cdot)}(\Omega)}&\le (2^{\mu-1}C^\mu(M+\sqrt{n})+2^{\mu-1}C^\mu(M+\sqrt{n})^{2-\mu})\|f\|_{L^{p(\cdot)}(\Omega)}.\end{aligned}
\begin{array}{ll}ds^{2}= -\frac{(\zeta+\frac{1}{4}-\epsilon^{+}\cos t^{-})(\zeta+\frac{1}{4}-\epsilon^{-}\cos t^{+})}{\zeta}dt^{2}+l^{2}\frac{(\zeta-\frac{1}{4}+\epsilon^{+}\cos t^{-})(\zeta-\frac{1}{4}+\epsilon^{-}\cos t^{+})}{\zeta}d\phi^{2}&-2l(\epsilon^{+}\cos t^{-}-\epsilon^{+}\cos t^{+})dtd\phi+l^{2}\frac{d\zeta^{2}}{4\zeta^{2}}.\end{array}
- 1 6 \left[ 9 \hat{a}_{e}^{\dagger}\hat{a}_{c}^{\dagger}\hat{a}_{d}^{\dagger}W_{\; \; f g b}^{e}V^{a b c d}\hat{a}^{f}\hat{a}^{g}+ 1 8 \hat{a}_{e}^{\dagger}\hat{a}_{f}^{\dagger}W_{\; \; f b c}^{e}V^{a b c d}\hat{a}^{f}+ 6 \hat{a}_{e}^{\dagger}W_{\; \; b c d}^{e}V^{a b c d}\right]
\vec{a}\,^{\prime}( 0 ) = \left( \begin{array}{c}{{c o s \, \alpha}}\\{{s i n \, \alpha}}\\{{0}}\\ \end{array}\right) \; \; ; \; \; \vec{a}\,^{\prime}( 2 \pi ) = \left( \begin{array}{c}{{c o s \, \alpha}}\\{{- s i n \, \alpha}}\\{{0}}\\ \end{array}\right) ,
x(t)= C(t,0)x_0+S(t,0)y_0+\int_0^t S(t,s)f(s)ds+\int_0^t S(t,s)Bu(s,x)ds,f\in S_{F,x}, u(t,x) =B^*S(b,t)R(a,\Upsilon_0^b)p(x(\cdot)),
(-1)^{m+1}\frac{(a v)^m}{m}= \frac{(-1)^{m+1}}{s}a^m \cdot \frac{(v^2)^{2^{\ell-1}s}}{2^{\ell}}= \frac{(-1)^{m+1}}{s}a^m \cdot \frac{(\pi + 2\alpha)^{2^{\ell-1}s}}{2^{\ell}}.
\left|\mu(\tau_A>k+1)-(1-\mu(A'))\mu(\tau_A>k)\right| =\left| \mu(A')\mu(\tau_A>k)-\mu(\tau_A=k+1)\right| & =\left|\mu(A')\mu(\tau_A>k)-\mu(A\cap\{\tau_A>k\})\right| &\le \mu(A)\delta(A, A'),
\mathbf{x}\thinspace =\thinspace \left( x_{1},\thinspace x_{2},\mathellipsis \thinspace ,\thinspace x_{m}\right),\thinspace \mathbf{y}\thinspace =\thinspace \left( y_{1},y_{2},\mathellipsis \thinspace ,\thinspace y_{n}\right),\mathbf{z}\thinspace =\thinspace \left( z_{1},\thinspace z_{2},\mathellipsis \thinspace ,\thinspace z_{p}\right),
\lim_{\theta\rightarrow 0}M(\theta)\,=\,\lim_{\theta\rightarrow \pi}M(\theta) \,=\,\frac{2}{3}+\frac{1}{\pi^2},\;\lim_{\theta\rightarrow 0^{+}}\psi(t)\,=\,\lim_{\theta\rightarrow \pi^{-}}\psi(t)\,=\,+\infty
\dfrac{d}{dt}\mathtt{I}_{t_0}^R(t) =2\int \big(uu_t+u_xu_{xt}\big)\Psi-\dot z(t)\int (u^2(t)+u_x^2(t)\big)\Psi' =:\mathrm{J}-\dot z(t)\int (u^2(t)+u_x^2(t)\big)\Psi'.
\tilde{E}_{0}\sp{\underline{{a}}}( \vec{x}, t ) \rightarrow -{\frac{\lambda}{4 \pi}}\sum_{\alpha}\dot{\xi}_{\alpha}\sp{\underline{{a}}}\Phi ( \vec{x}) + O ( r \sp{- 1}) , \quad \mathrm{a s}\quad r \rightarrow \infty \
\mathfrak{s}(A_1,\ldots,A_N) =\inf\left\{s>0 \colon \mathbf{P}((A_1,\ldots,A_N),s)<0\right\}&=\sup\left\{s>0 \colon \mathbf{P}((A_1,\ldots,A_N),s)>0\right\}.
A\left(\sum_{i=0}^m x^1_ir_i(t), \sum_{j=0}^m x^2_jr_j(t)\right) =A\left(\sum_{k=1}^{2^m}X^1_{k,m}\chi_{I_{k,m}}(t),\sum_{k'=1}^{2^m}X^2_{k',m}\chi_{I_{k',m}}(t)\right) &=\sum_{k, k'=1}^{2^m}A\left(X^1_{k,m}\chi_{I_{k,m}}(t),X^2_{k',m}\chi_{I_{k',m}}(t)\right) &=\sum_{k=1}^{2^m}A\left(X^1_{k,m},X^2_{k,m}\right)\chi_{I_{k,m}}(t).
H_{n m}^{\{{\bf R}\}}( 0 , 0 ) \approx \frac{( - 1 )^{\frac{n + m}2}\epsilon^{- \frac{| n - m | + 1}2}}{\sqrt{2 \pi}}\Gamma \left( \frac{n + m + 1}2 \right) R_{1 1}^{n / 2}R_{2 2}^{m / 2}\left( r + \epsilon \right)^{\frac{n + m + 1}2}.
\exp(\gamma\Phi(\tau)/2)\left(\frac{\overline{C}}{(\Phi(\tau)/2)^{p-1}}\right)^{1-\frac{\gamma}{q^+}}=\left(\frac{\overline{C}}{(\Phi(\tau)/2)^{p-1}}\right)^{\frac{2}{3}-\frac{\gamma}{q^+}}\leq 1,
\sup_{j\ge 0}\ \|P_{n}T^{\,j}P_{l}\,x\| \le C_{l}\,\cdot\,(b_{l+1}-b_{l})^{1-\frac1{p}}\,\cdot\,\Bigl\|\sum_{k=b_{l}}^{b_{l+1}-1}\Bigl(\prod_{s=k+1}^{b_{l+1}-1}w_{s}\Bigr)\,x_{k}e_{k}\Bigr\|,