harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
T^{\frac32 - 3b}\| u_0\|_{L^2}\lesssim 1, T^{\frac32 - 3b - \epsilon}\| u_0\|_{L^2}\lesssim 1, T^{\frac32 - 3b}(\|n_0\|_{H^{s_1}}+ \|n_1\|_{{H}^{s_1-1}}+ \|h\|_{H^{s_1}}) \lesssim 1, T^{\frac32 - 3b - \epsilon}\| u_0\|_{L^2}^2 \lesssim (\|n_0\|_{H^{s_1}}+ \|n_1\|_{{H}^{s_1-1}}).
f_{L}(x)=\sum_{j=2}^{k_1}\left\{\left\{p_{j}x+\xi_{j}\right\}+2L\hat{\theta}_{j}\right\}+\sum_{j=k_{1}+1}^{k}\left\{\left\{p_{j}x\right\}+2L\hat{\theta}_{j}\right\},\ \forall x\in [0,1],
\widetilde{\mathcal M}[{\mathcal O}_{Y'}]=\bigoplus_{\mu\in X^*(T)}\widetilde{M}_{\mu}\otimes_k{\mathcal L}_{Y'}(\sum_{\emptyset\ne\sigma\subsetneq\{0,\ldots,d\}}-\lceil\sum_{j\in\sigma}\overline{a}_j(\mu)\rceil (V_{\sigma}\cap Y')).
D_{\alpha\beta}I(p\|q)=T\;\left[\underbrace{\sum_{i}p^{\alpha+\beta-1}_{i}}_{A(p,q)}-\underbrace{\left(\sum_{i}p^{\alpha}_{i}q^{\beta-1}_{i}\right)^{\frac{\alpha+\beta-1}{\alpha}}}_{X(p,q)}\times\underbrace{\left(\sum_{i}q^{\alpha+\beta-1}_{i}\right)^{\frac{1-\beta}{\alpha}}}_{Y(p,q)}\right]
\frac{\partial^2 p}{\partial t^2}-c_1c_2\frac{\partial^2 p}{\partial x^2}+ (c_1-c_2)\frac{\partial^2 p}{\partial x\partial t}= -2\lambda \frac{\partial p}{\partial t}+ \lambda(c_2-c_1) \frac{\partial p}{\partial x}
\min\limits_{\sum\limits_{i=1}^n c_i^2=1}f(c_1,\cdots,c_n)=f(\boldsymbol{c^*}) =\mu^*\forall 1\leq i\leq n: (\lambda_i-\mu^{*})c_i^*=\frac{1}{N^{1-\delta}}\sum\limits_{j=1}^{n}c_j^{*}.
(\begin{matrix}I_{1}\\I_{2}\\I_{3}\\I_{4}\end{matrix})=\frac{2e\cdot i}{h}(\begin{matrix}1&0&0&-1\\ -1&1&0&0\\ 0&-1&1&0\\ 0&0&-1&1\end{matrix})(\begin{matrix}\mu_{1}\\ \mu_{2}\\ \mu_{3}\\ \mu_{4}\end{matrix}).
\int_{\mathcal{L}_{j}}{\left\vert{T_{n}\left( \pm{u,t}\right) dt}\right\vert}\leq \sum\limits_{s=0}^{n-2}{\frac{1}{\left\vert u\right\vert^{s}}\int_{\mathcal{L}_{j}}{\left\vert{F_{s+1}\left( t\right) dt}\right\vert.}}
\begin{array}{lll}b^{(r)}_i(X) =\Big(J^{(r)}\Big[(\hat{C}C\hat{C}^{-1}-I)\underline{v}+ \hat{C}\Big(Dp - U(J^{(r)}X) - \mathbf{1}\frac{\beta}{2}\Big)\Big]\Big)_i =j^{(r)}_{ii}\Big((\hat{C}C\hat{C}^{-1}-I)\underline{v}+ \hat{C}\Big(Dp - U(J^{(r)}X) - \mathbf{1}\frac{\beta}{2}\Big)\Big)_i \geq 0. \end{array}
\left\vert \frac{1}{N}\sum_{j=1}^{N}\sum_{m\in\mathbb{Z}^{2}}\chi_{I\left(s,x\right) \cap\Omega}\left( t\left( j\right) +m\right) -\left\vert I\left( s,x\right) \cap\Omega\right\vert \right\vert
m_{1,2}=m_{4,5}=m_{7,9}, m_{1,5}=m_{1,6}=0, m_{1,3}=m_{4,6}=-\frac{m_{8,9}}{m_{7,9}}, 1-m_{7,9}^{2}+\frac{m_{8,9}^{2}}{m_{7,9}}=0.
\begin{cases}\limsup_{|x|\rightarrow\infty}|x|^{\min\{\beta,n\}-2}\left|u(x)-\left(\frac{1}{2}x^TAx+b\cdot x+c\right)\right|<\infty, \ \beta\neq n, \limsup_{|x|\rightarrow\infty}|x|^{n-2}(\ln|x|)^{-1}\left|u(x)-\left(\frac{1}{2}x^TAx+b\cdot x+c\right)\right|<\infty, \ \beta=n.\end{cases}
\det\big(A(e^{\chi})A(e^{-\chi})\big) =\det\big(\big\{e^{-A(\chi_+)}e^{A(\chi_-)},A(e^{\chi_0})\big\}\big) \hphantom{\det\big(A(e^{\chi})A(e^{-\chi})\big) =}{}\times\det\big(\big\{e^{-A(\chi_+)},e^{A(\chi_-)}\big\}\big)\det\big(A(e^{\chi_0})A(e^{-\chi_0})\big)
\overline{{G}}_{\sigma}^{( n )}( \zeta^{\prime}, x^{2}\zeta_{2 n})^{\varepsilon^{\prime}\varepsilon_{2 n}}= \overline{{G}}_{\sigma}^{( n )}( \zeta_{2 n}, \zeta^{\prime})^{\varepsilon_{2 n}\varepsilon^{\prime}}\prod_{j = 1}^{2 n - 1}\frac{\zeta_{2 n}}{\zeta_{j}}.
\left( \begin{array}{r l}{{Q_{1 1}}}&{{Q_{1 2}}}\\{{Q_{2 1}}}&{{Q_{2 2}}}\\ \end{array}\right) = \frac{1}{6}N \omega_{1}R_{1}^{2}\left( \begin{array}{r l}{{- \sin 2 \omega_{1}t}}&{{\cos 2 \omega_{1}t}}\\{{\cos 2 \omega_{1}t}}&{{\sin 2 \omega_{1}t}}\\ \end{array}\right)
{\cal L}=\bar\psi\gamma^\mu(i\nabla_\mu + A \omega_{\mu}+a_{\mu})\psi+\frac{g^2}{4}\bar\psi\gamma^\mu\psi\bar\psi\gamma_\mu\psi-\epsilon^{\mu\nu}\rho(\partial_\mu a_\nu-\frac{B}{2}\epsilon_{ab}e^a_\mu e^b_\nu),
\begin{aligned}c_{F,1}(T_0)\le 2C_{F,1}(T_0), c_{F,2}\le 2C_{F,2}(T_0)+C_{F,1}(T_0)\log{2}, c_{F,3}(T_0)\le\frac{3C_{F,3}(T_0)}{2}. \end{aligned}
|E_{ij}(\mathcal{P})|=|E_{ji}(\mathcal{P}| < \min\left\{n_{i}\min\left\{n_{j},\tau_{r_{i}}(r_{j})\right\},n_{j}\min\left\{n_{i},\tau_{r_{j}}(r_{i})\right\}\right\}&=\min\left\{n_{i}\min\left\{n_{j},A^{}(3,\theta_{i}^{j})\right\},n_{j}\min\left\{n_{i},A^{}(3,\theta_{j}^{i})\right\}\right\}
\left(\mathcal{F}_M\right)_{j\,l}=e_j\delta_{j\,l}-\langle\varphi_j,\varphi_l\rangle=(\int_{-\infty}^\infty-\int_{-\infty}^x)\varphi_j(y)\varphi_l(y)dy=\int_x^\infty \!\!\varphi_j(y)\varphi_l(y)dy.
{\varpi^\sigma}_{\sigma\sigma}(\mu,\sigma) = \frac{\partial_\sigma^2 D^\prime(p^{(\sigma)}||p_{\mu,\sigma}) - g_{\sigma\sigma}(\mu,\sigma)}{\partial_\sigma D^\prime(p^{(\sigma)}||p_{\mu,\sigma})}&= \frac{\mu^2 + 3\sigma^2 - [(\sigma + \delta \sigma)^2 + \mu^2] -2\sigma^2}{\sigma(\mu^2 + \sigma^2 - [(\sigma+\delta \sigma)^2 + \mu^2])}&= \frac{1}{\sigma}.
F=u_{tt}-u_{xx}+ b(u)u_t + c(u)u_x +m(u)=\frac{\delta L}{\delta v}= 0 , & F^*= v_{tt}-v_{xx}-b v_t -c v_x +(b' u_t+c' u_x+m')v=\frac{\delta L}{\delta u}= 0 ,
\left(\mathbf{P}\mathbf{A}_{f}\mathbf{A}_{g}\right)_{N,m}+\left(\mathbf{P}\mathbf{A}_{fg}\right)_{N,m}=\left(\mathbf{P}\mathbf{\Delta}\left(\mathbf{S}_{f}\mathbf{\Delta}\mathbf{S}_{g}+\mathbf{S}_{fg}\right)\right)_{N,m}.
\sum\limits_{n\geq 0}a(n)(f(n)-\sum\limits_{j=0}^{B-1}f(Bn+j))-2\sum\limits_{m\geq 0}a(B^{|w|}m+v_B(m))f(B^{|w|}m+v_B(m)) &= \begin{cases}0 \mbox{if}w \neq 0^j -2a(0)f(0) \mbox{if}w=0^j \end{cases}.
+ \, \frac{D - 2}{2 \pi}\frac{m}{M_{0}^{2}}\, \frac{1}{R^{2}}\, - \, ( D - 2 ) \, \sqrt{\frac{2}{\pi \alpha r_{s}}}\, \frac{1}{\sqrt{R}}\, \exp [ - 2 \sqrt{2}\, R / \sqrt{\alpha}r_{s}] \,{,}\quad R \to \infty \,{.}
ME(B_{n,3,3,3}^{(n-5)})=2\sqrt{\frac{n+1+\sqrt{(n+1)^{2}-4(3n-9)}}{2}}+2\sqrt{\frac{n+1-\sqrt{(n+1)^{2}-4(3n-9)}}{2}}; ME(B_{n,3,3}^{(n-5)})=2+2\sqrt{\frac{n+\sqrt{n^{2}-4(n-5)}}{2}}+2\sqrt{\frac{n-\sqrt{n^{2}-4(n-5)}}{2}}.
g_{2}=x_{1}x_{2}-x_{0}x_{3}=\sum_{\mu=1}^{n-1}\alpha_{\mu}\left[x_{2}^{(n+1)-\mu}x_{3}^{\mu}-x_{0}^{(n+1)-\mu}x_{1}^{\mu+1}\right] &+\sum_{t=0}^{n}\beta_{t}\left[x_{1}^{(n+1)-t}x_{3}^{t}-x_{0}^{n-t}x_{2}^{t+1}\right]+\gamma_{1}(x_{0}^{n+1}-x_{3}^{n}),
\sum_{r,s=1}^{\nu}\widetilde{\gamma}_{irs}\widetilde{\gamma}_{ipr}=\sum_{r=1}^{\nu}\widetilde{\gamma}_{ipr}\sum_{s=1}^{\nu}\widetilde{\gamma}_{irs}=\tfrac{1}{2}\sum_{r=1}^{\nu}\widetilde{\gamma}_{ipr}=\tfrac{1}{4}
\sum_{n = 1}^{\infty}\frac{1}{t^{2}+ n^{2}z^{2}/ 4}= \frac 1 2 \left[ \frac{2 \pi}{z t}\frac{1}{\mathrm{t h}( 2 \pi t / z )}- \frac{1}{t^{2}}\right] ,
\sum_{t\in T}\sum_{w_{1},\ldots,w_{n-1}\in W}(-1)^{|w_{1}|+\cdots+|w_{n-1}|}\Delta_{t(w_{1}(\Lambda_{1})+\cdots+w_{n-1}(\Lambda_{n-1}))}^{(n-2)}=\sum_{\Lambda_{n}\in\Delta_{\Lambda_{1},\ldots,\Lambda_{n-1}}}\Delta_{\tilde{W}(\Lambda_{n})},
x_{1}={\frac{1}{r}}-{\frac{r + \ln ( r - 1 )}{N r}}+{\frac{1}{2 N^{2}r}}\, \Bigl ( \left( \ln ( r - 1 ) \right)^{2}-{\frac{2 r \ln ( r - 1 )}{r - 1}}\Bigr ) +{\cal O}\bigl ( 1 / N^{3}\bigr ) .
\int_\Omega |v|^p\mu \ = \ \int_\Omega v^{p-1}(v-\langle v\rangle_{\mu,\Omega})\mu + \left(\int_\Omega v^{p-1}\mu\right) \frac{1}{\mu(\Omega)}\left(\int_\Omega v\mu\right) & = \ \int_\Omega v^{p-1}(v-\langle v\rangle_{\mu,\Omega})\mu - \frac{1}{\mu(\Omega)}\left(\int_\Omega v^{p-1}\mu\right) \left(\int_{\Omega^c}v\mu\right),
{\cal S}[\psi]=\int\, d^{5}x\sqrt{g}\left\{\frac{i}{2}\,\left[\overline{\psi}\tilde\gamma^{\hat\mu}\tilde\nabla_{\hat\mu}\psi-(\overline{\tilde\nabla_{\hat\mu}\psi})\tilde\gamma^{\hat\mu}\psi\right] - M\overline{\psi}\psi\right\}
\overline{f_{\phi_i, t}}= \nu_i^{2t}\phi_i, \overline{f_{\phi_i, t}f_{\phi_j, t}}= (\nu_i \nu_j)^{2t}\left(\sum_{s=0}^{t}\frac{\Phi^s}{(\nu_i \nu_j d)^{2s}}\right) (\phi_i \circ \phi_j)
\begin{aligned}S_f^{\sharp}(x) = \sum_{d\le x/N_f}f(d) \sum_{x/(d+1)<n\le x/d}1 & = \sum_{d\le x/N_f}f(d) \Big(\frac{x}{d}-\psi\Big(\frac{x}{d}\Big)-\frac{x}{d+1}+\psi\Big(\frac{x}{d+1}\Big) \Big) & = x \sum_{d\ge 1}\frac{f(d)}{d(d+1)}+ \mathcal{R}_1^{f}(x, N_f) - \mathcal{R}_0^{f}(x, N_f) + O(N_f),\end{aligned}
\widetilde{W}_{i}^{T}= -{\frac{\beta \beta_{H}}{( 4 \pi )^{D / 2 + 1}}}\Gamma \left( 1 - \frac D 2 \right) n_{i}m_{i}^{D - 2}\left[ \int_{0}^{\infty}{\frac{d t}{t^{2}}}\sum_{k = 1}^{\infty}\eta_{i}^{k + 1}e^{-{\frac{\beta^{2}k^{2}}{4 t}}}\right]{\cal A}^{H}+ P_{i}( \epsilon , D ) .
\! \! D_{2}\eta ({\xi}_{1}, . . ,{\xi}_{q - 1}) ({\phi}_{1}, . . ,{\phi}_{p + 1}) ={\displaystyle \sum_{i}}\eta \left({\phi}_{i},{\xi}_{1}, . . ,{\xi}_{q - 1}\right) \left({\phi}_{1}, . . , \widehat{{\phi}_{i}}, . . ,{\phi}_{p + 1}\right) ,
\left[\int_0^1B(t)\mathrm{d}t\right]^2 =\left[\sum_{k=1}^{\infty}\sqrt{\lambda_k}Z_k\int_{0}^{1}f_k(t)\sqrt{t(1-t)}\mathrm{d}t\right]^2 &=\frac{1}{6}\left[\sum_{k=1}^{\infty}\sqrt{\lambda_k}Z_k\int_{0}^{1}f_k(t)f_1(t)\mathrm{d}t\right]^2 &=\frac{1}{6}\lambda_1 Z_1^2\ .
\lim_{t \rightarrow \infty}I_n(t) = \lim_{t \rightarrow \infty}\left|\mathfrak{Re}(\alpha_n(t))\right|\exp\left[\mp\frac{j}{2}e^{t}\mathfrak{Re}(\alpha_n(t)) \mathfrak{Im}(\alpha_n(t)) \right] = \lim_{t \rightarrow \infty}\left|\mathfrak{Re}(\alpha_n(t))\right|.
f^{3,0}(Y,w) = f^{0,3}(Y,w) = 1, f^{1,1}(Y,w) = f^{2,2}(Y,2) = k(Y,w), f^{2,1}(Y,w) = f^{1,2}(Y,w) = ph(Y,w) -2 + h^{2,1}(Z),
U (c_1,c_2) = \begin{pmatrix}1 c_1 0 -c_1^2/2 0 1 0 -c_1 0 0 1 0 0 0 0 1\end{pmatrix}\begin{pmatrix}1 0 c_2 c_2^2/(2 \epsilon) 0 1 0 0 0 0 1 c_2/ \epsilon 0 0 0 1\end{pmatrix},
{\cal A}_{\mathrm ZMS}=\int{\mathrm d}^2x\left[\frac{1}{2}\partial_\nu\Phi\partial^\nu\Phi+\frac{m_0^2}{\gamma}\Big(\exp\left(2i\sqrt{\gamma}\Phi\right)+2\exp\left(-i\sqrt{\gamma}\Phi\right)\Big)\right]\,.
E^3 = (E|_E)^2= - \deg N_{C/X}, (\mu^*L)^2.\,E = 0\quad({}\eqref{zero1}), (\mu^*L).\,E^2 = (\mu^*L|_E, E|_E)_E = (\mu^*(L|_C), E|_E)_E=-L.C.
\infty =\int_E\delta\Phi_2(\frac{f(x)}{\parallel C_Tf\parallel_{\Phi_2}})d\mu &\leq\int_Ef_0(x)\Phi_2(\frac{f(x)}{\parallel C_Tf\parallel_{\Phi_2}})d\mu &\leq\int_X\Phi_2(\frac{C_Tf(x)}{\parallel C_Tf\parallel_{\Phi_2}})d\mu &\leq1,
\psi_{( m_{1}, n_{1})}\otimes \psi_{( m_{2}, n_{2})}= \sum_{k = | m_{1}- m_{2}| + 1}^{m_{1}+ m_{2}- 1}\sum_{l = | n_{1}- n_{2}| + 1}^{n_{1}+ n_{2}- 1}\left[ \psi_{( k , l )}\right] ,
\sqrt{C_{s}^{n}C_{n}^{2}}C_{n - 2}^{2}\Phi^{n}+ 2 \sqrt{C_{s}^{n - 2}}C_{n - 2}^{2}C_{s - n + 2}^{2}\Phi^{n - 2}+ \sqrt{C_{s}^{n - 4}}C_{s - n + 4}^{2}C_{s - n + 2}^{2}\Phi^{n - 4}= 0 ,
\max_{-N \leq j \leq N}\Vert M_{H_j}: S_{1}^{2N+1}\rightarrow S_{1}^{2N+1}\Vert\leq \Vert M_{\widetilde{H}\vert_{\mathbb{Z}_N \times \mathbb{Z}_N}}^{(N)}: S_{p_1}^{2N+1}\times S_{p_2}^{2N+1}\rightarrow S_{1}^{2N+1}\Vert.
J = \frac{f^p}{1-\alpha p}- \left(\frac{p}{p-1}\right)^q(1-\alpha)^q\frac{f^p}{1-\alpha p}= &=-\frac{f^p}{1-\alpha p}\left[\left(\frac{p}{p-1}\right)^q(1-\alpha)^q-1\right] = -f^p \,G(\alpha),
\begin{aligned}\big(\int_x^a dy\phi_3^{-2}(y)\int_y^a\phi_3(z)dz\big)\big(\int_{-\infty}^a\phi_3^{-2}(t)dt\big)= &\big(\int_x^a dy\phi_3^{-2}(y)\int_y^a\phi_3(z)dz\big)\big(\int_{-\infty}^x\phi_3^{-2}(t)dt\big)+ &\big(\int_x^a dy\phi_3^{-2}(y)\int_y^a\phi_3(z)dz\big)\big(\int_x^a\phi_3^{-2}(t)dt\big).\end{aligned}
c_{l , m}= q^{m - l}i \tau_{l}^{2}\tau_{l}^{1}+ q^{l - m}i \tau_{m}^{2}\tau_{m}^{1}- \eta^{m - l}i \tau_{l}^{2}\tau_{m}^{1}- \eta^{l - m}i \tau_{m}^{2}\tau_{l}^{1}\,
\beta^\varepsilon_{0,p,\boldsymbol{\pi}}(\lambda)=\prod_{\nu=1}^{f_p(\boldsymbol{\pi})}(\lambda-\tilde{a}^\varepsilon_{x_\nu,x_\nu})\cdot \prod_{\nu=f_p(\boldsymbol{\pi})+1}^{N-n-p}(-\tilde{a}^\varepsilon_{x_\nu,x_\nu})
A\cap Z(L) = A\cap (\cap_{x\in L}\ker ad_{x})\cr =\cap_{x\in L}(A \cap \ker ad_{x}) \cr =\cap_{x\in L}\ker ad_{x}|_{A}\cr =(\cap_{x\in C_{L}(A)}\ker ad_{x}|_{A})\cap (\cap_{x\in L\setminus C_{L}(A)}\ker ad_{x}|_{A})\cr =A\cap \ker ad_{x}|_{A}x\in T_{A}\cr = \ker ad_{x}|_{A}x\in T_{A}.
\log|f(z)| = \frac{1}{2\pi}\int_{0}^{2\pi}\Re\bigg(\frac{Re^{i\theta}+z}{Re^{i\theta}-z}\bigg)\log|f(Re^{i\theta})|d\theta - \sum_{m=1}^{k}\log\bigg|\frac{R^{2}-\overline{\alpha}_jz}{R(z-\alpha_j)}\bigg| + \sum_{m=1}^{l}\log\bigg|\frac{R^{2}-\overline{\beta}_jz}{R(z-\beta_j)}\bigg|
S_{0}(T) \coloneqq \sum_{n=1}^\infty \sum_{e^{n\ell(\gamma)}\leq T}\ell(\gamma) e^{n\ell_\psi(\gamma)},\ \ \hat{S}_{0}(T) \coloneqq \sum_{e^{\ell(\gamma)}\leq T}\ell(\gamma) e^{\ell_\psi(\gamma)},
R =\left( \begin{array}{cccccccc}a 0 0 0 0 0 0 p 0 a 0 0 0 0 q 0 0 0 \pm b 0 0 q 0 0 0 0 0 a r 0 0 p 0 0 0 q b 0 0 0 0 0 r 0 0 \pm a 0 0 0 r 0 0 0 0 a 0 qrp^{-1}0 0 0 0 0 0 b\end{array}\right),
F_{s}\left( \begin{matrix}{{\bf a}_{1}{\bf a}_{+}}\\{{\bf a}_{3}{\bf a}_{4}}\\ \end{matrix}\right) ( z , \bar{z}) = z^{- 2 \Delta_{+}}\sum_{p = 0 , \pm 1}M_{p s}F_{p}\left( \begin{matrix}{{\bf a}_{4}{\bf a}_{+}}\\{{\bf a}_{3}{\bf a}_{1}}\\ \end{matrix}\right) ( 1 / z , 1 / \bar{z}) ,
E_{1 , n}^{c}= - \frac{1}{2 \mathrm{L}}\int_{0}^{2 \mathrm{L}}d z V ( 0 , z ) + \frac{2{\pi}n}{\mathrm{L}}- \frac{1}{2}( e_{1}+ e_{2}) b , \hspace{5 m m}n \in \cal Z ,
\sigma\big((\Phi^e_{vw})^*\Phi^e_{vw}\big) = \sigma\Big(\big((\Phi^e_{vw})^*\Phi^e_{vw}\big)^{-1}\Big)^{-1}= \sigma\Big( \big(|\Phi^e_{vw}|\big)^{-1}\big(|\Phi^e_{vw}|^* \big)^{-1}\Big)^{-1}= \sigma\big( (\Phi^{\overline{e}}_{wv})^* \Phi^{\overline{e}}_{wv}\big)^{-1}
W = \sum_{k}M_{k}q \bar{q}( a \bar{a})^{N - 1 - k}+ \sum_{m}\Big ( H_{m}q^{2}a ( a \bar{a})^{N - 2 - m}+ \bar{H}_{m}\bar{q}^{2}\bar{a}( a \bar{a})^{N - 2 - m}\Big )
\begin{aligned}\mathcal{F}_{i}=\rho\xi*f_{i}+\rho v_{i}^{0}t+\rho u_{i}^{0}, \mathcal{H}=\rho T_{0}\eta^{0}+l*\rho h, \mathcal{L}_{ij}=a_{ij}^{(14)}\theta^{0}e^{-t/\beta}, \mathcal{M}_{ij}=a_{ij}^{(47)}\theta^{0}e^{-t/\beta}, \mathcal{R}=c\theta^{0}e^{-t/\beta'}.\end{aligned}
H_t = \{r : 0 < r_2 \leq c_2-a_2 \textrm{and}\phi(c) \leq \phi(r)-t < \phi(d)\}&\cup \{r: c_2-a_2 < r_2 \leq m \textrm{and}\phi(d)-(\phi(a)-\phi(c)) \leq \phi(r)-t < \phi(d)\}.
\{Y_{l_{1},m_{1}}^{g},Y_{l_{2},m_{2}}^{g}\}={\displaystyle\sum\limits_{\mu_{1},\mu_{2},l,l^{\prime}}}C_{\mu_{1},\mu_{2},\mu^{\prime}}^{l_{1},l_{2},l^{\prime}}C_{m_{1},m_{2},m^{\prime}}^{l_{1},l_{2},l^{\prime}}D_{\mu^{\prime},m^{\prime}}^{l^{\prime}}(g)\K_{\mu_{1},\mu_{2},\mu}^{l_{1},\l_{2},\l}\Y_{l,\mu}\ .
Z:= \mathfrak{Op}\big((h_\circ\,\sharp\,\mathfrak{z}+\mathfrak{z}\,\sharp\, h_\circ)+\mathfrak{z}\,\sharp\, h_\circ\,\sharp\,\mathfrak{z}+\mathfrak{y}\,\sharp\, h_\bullet\,\sharp\, r\,\sharp\,h_\bullet^*\,\sharp\,\mathfrak{y}\big)\,.
\nabla u(t,x) =L_1\phi'(\delta)\hat a+L_2x \nabla u(t,y) =L_1\phi'(\delta)\hat a-L_2y u_t(t,x)-u_t(t,y) = L_2t \begin{bmatrix}\nabla^2u(t,x) 0 0 -\nabla^2u(t,y) \end{bmatrix}\le L_1 \begin{bmatrix}Z -Z -Z Z \end{bmatrix}+ L_2 I,
(u_i + a_i f_{q(i)}) (w_{n,i}^*) = (u_i + a_i f_{q(i)}) (z_n^*) = (u_i + a_i f_{q(i)}) (y_n^*) - a_i f_{q(i)}(y_n^*) = y_n^*(u_i) = 0,
\Big| \frac{1}{2 m \ell n}\sum_{x}\Phi_{n}(s,\tfrac{x}{n}) \sum_{j=1}^{m-1}\int_{\Omega_2^j(x)}\sum_{z = x-\ell}^{x-1}[ \eta(z) - \eta(z - j \ell) ] \overrightarrow{\eta}^{\varepsilon n}(x+1) [f( \eta_{1,x,j,z}) - f \big( \eta_{2,x,j,z}) ] d\nu_{b}\Big|.
A =(\alpha_1,\dots, \alpha_m),f=(x-\alpha_1)\cdots (x-\alpha_m)=\sum_{i=0}^m f_ix^i,\ \mbox{where}\f_m=1, B =(\beta_1,\dots, \beta_n),g=(x-\beta_1)\cdots (x-\beta_n)=\sum_{i=0}^n g_ix^i ,\ \mbox{where}\g_n=1.
_{1}\phi_{1}\left(\begin{array}{ccc}\begin{array}{c}a b\end{array}\vert q,-bq^{\alpha}\end{array}\right)q^{\alpha^{2}/2}=\frac{1}{\sqrt{\pi\log q^{-2}}}\int_{-\infty}^{\infty}\frac{\left(abq^{-1/2}e^{ix};q\right)\exp\left(\frac{x^{2}}{\log q^{2}}+i\alpha x\right)dx}{\left(b;q\right)_{\infty}\left(be^{2ix}q^{-1};q^{2}\right)_{\infty}}
\sum_{(r,s) \ne(0,0)}\delta_{1}^{r}\delta_{2}^{s}= \frac{1}{(1-\delta_{1})(1-\delta_{2})}-1 = \frac{\delta_{1}+\delta_{2}- \delta_{1}\delta_{2}}{(1-\delta_{1})(1-\delta_{2})}\leq\frac{\delta_{1}+\delta_{2}}{(1-\delta_{1})(1-\delta_{2})}.
\frac{d_n}{c_n}= \frac{s_n^2}{2n s_n + n^2}> \frac{1}{\ell}+ 1 \mbox{and}\frac{c_{n+1}}{d_n}= \frac{2(n+1) s_{n+1}+ (n+1)^2}{s_n^2}> \frac{1}{\ell}+ 1
b_1 B a_2 B^{-1}a_2^{-1}\stackrel{}{=}B a_2 B^{-1}b_1 a_2^{-1}= B (a_2 B^{-1}a_2^{-1}) ( a_2 b_1 a_2^{-1}), \; a_2 b_1 a_2^{-1}= a_2 B a_2^{-1}B^{-1}a_2 B^{-1}a_2^{-1},
0= \epsilon_{i_1 i_2 \dots i_n i_{n+1}}\epsilon_{j_1 j_2 \dots j_n j_{n+1}}= \sum_{P \in S_{n+1}}\ (-1)^{\pi(P)}\ \delta_{i_1 P(j_1)}\dots\delta_{i_{n+1}P(j_{n+1})},
\partial_{\mu}{\cal F}_{\alpha \psi}^{\left( \mu \right)}=\frac{\partial{\cal F}_{\alpha \psi}^{\left( \mu \right)}}{\partial \psi_{\alpha}^{\left( \mu \right)}}\otimes \frac{\partial \psi_{\alpha}^{\left( \mu\right)}}{\partial x^{\mu}}.
\frac{\partial}{\partial t}\left(\big(d\varphi^{0}_{t}(x_{0})\big)^{-1}\cdot\frac{\partial}{\partial \tau}\varphi^{\tau}_{t}(x_{0})_{|\tau=0}\right)=\big(d\varphi^{0}_{t}(x_{0})\big)^{-1}\cdot\left(\frac{\partial X_\tau}{\partial \tau}\big(\varphi^{0}_{t}(x_{0})\big)\right)_{|\tau=0}.
\frac{1}{2}\frac{d}{d t}\sum_i \dot{x}^i \dot{x}^i = -\sum_i \dot{x}^i \dot{x}^i - \sum_i \dot{x}^i \left( \Gamma^i_{k l}(x) \dot{x}^k \dot{x}^l + X^i_k(x,\dot{x}) \dot{x}^k \right)
\int_0^T\int \big|\partial_t^h u^h\big|^2\,dx\,dt \lesssim \int_0^T\int \big| G_h\ast \partial_t^h u^h\big|^2 \,dx\,dt +2 \sum_{n=0}^N \left( E_h(u^n)- E_{2h}(u^n) \right),
Pr \biggl( \lVert \sum_{l=1}^{L_{n}}\frac{1}{L_{n}}Q_{t^n}(X_{j,l}) -\Theta_{t^n}\rVert_1 \leq 2^{-\sqrt{n}\frac{1}{16}\hat{\beta}(\alpha)}\forall t^n ~ \forall j \biggr)\allowdisplaybreaks &\geq 1- 2^{n\vartheta}
{\cal L}= \frac{1}{2 g l_{s}}\mathrm{T r}\; \left[ \dot{X}^{a}\dot{X}^{a}+ \frac{1}{2}[ X^{a}, X^{b}]^{2}+ \theta^{T}( i \dot{\theta}- \Gamma_{a}[ X^{a}, \theta ] ) \right]
H_{\sigma,\alpha}\ :=\ \left\{\Psi \in{\mathcal H}_\sigma^{(0)}\bigm| Q_\alpha \ \Psi=0\right\}\ \big/ \ \left\{\Psi \mid \Psi = Q_\alpha \Phi, \Phi \in{\mathcal H}_{-\sigma}^{(0)}\right\},
S_{\rm eff}= S_{\rm clas}+ \frac{i}{2}{\rm{Tr}}\ln \hat{O}- i{\rm{Tr}}\ln \hat{G}_{\rm ghost}+\int{\rm d}^4x \sqrt{-g}F_{\mu\nu}(x) F_{\sigma\tau}(x)M^{\mu\nu\sigma\tau}(x,x)\; ,
f\left( y,r\right) -f\left( x,r\right) =\left( \frac{\psi\left(y+r\right) -\psi\left( r\right)}{y}-\frac{\psi\left( x+r\right)-\psi\left( r\right)}{x}\right) \frac{\left( 1-\frac{1}{r}\right)}{\Gamma\left( r\right)}& +\frac{1}{r\Gamma\left( r+1\right)}\frac{x-y}{\left( y+r\right)\left( x+r\right)}.
\frac{R_{12}}{i-\omega(2,2)}u_2+\frac{R_{13}}{i-\omega(3,3)}u_3=-i(R_{12}(-1)-iR_{13})=iR_{12}-R_{13}=i+i=2i=2u_1; \frac{R_{42}}{i-\omega(2,2)}u_2+\frac{R_{43}}{i-\omega(3,3)}u_3=iR_{42}-R_{43}=0-(-2)=2=2u_4.
\mathcal{L}_3=\Pi_S^{\perp}(\omega\cdot\partial_{\varphi}+m_3 \partial_{xxx}+\varepsilon \mathfrak{B}_1+\varepsilon^2 \mathfrak{B}_2+\tilde{d}_1 \partial_x+\tilde{d}_0)\Pi_S^{\perp}+\tilde{\mathcal{R}}_*
\begin{array}{r c l}{{F_{\sigma}^{( n )}( \zeta^{\prime \prime}, \zeta_{2 n - 1}, s x^{- 1}\zeta_{2 n - 1})^{k \cdots k^{\prime}k^{\prime}\pm 1 k_{2 n}}}}&{{=}}&{{\delta_{k^{\prime}, k_{2 n}}\frac{d_{s}}{\{k_{2 n}\}}F_{\sigma}^{( n - 1 )}( \zeta^{\prime \prime})^{k \cdots k^{\prime}}\; ( s = \pm ) ,}}\\ \end{array}
\mu_{t1}^{\pm}(x,\infty) := \left(\frac{t\,\P\left[\frac{X_{1}}{b(t)}\in (x,\infty) \right] -\nu_{2}(x,\infty)}{A(b(t))}\right)^{\pm}\to \left(x^{-2}(x^{-1}-1) \right)^{\pm}=:\chi_{1}^{\pm}(x,\infty) ,
\sum^{\infty}_{m=1}\sum_{j=1}^{\infty}\bigl\{E \bigl(\beta_{X_1,j}-\beta_{X_1,j}^{(m)}\bigr)^4 \bigr\}^{1/4}<\infty,\sum_{j=1}^{\infty}\bigl(E\beta_{X_1,j}^4 \bigr)^{1/4}<\infty,
\mathrm{rad}\left[ \frac{\mathcal{C}_1(-,?)}{I_1}\otimes \frac{\mathcal{C}_2(-,?)}{J_2}\right]= \mathrm{rad}\left[ \frac{\mathcal{C}_1(-,?)}{I_1}\right]\otimes \frac{\mathcal{C}_2(-,?)}{J_2}+ \frac{\mathcal{C}_1(-,?)}{I_1}\otimes \mathrm{rad}\left[ \frac{\mathcal{C}_2(-,?)}{J_2}\right].
\begin{pmatrix}0 \gamma''_{i-1}\end{pmatrix}= \begin{pmatrix}0 \rho_{i-2}\end{pmatrix}\begin{pmatrix}\psi_i -\beta_{i-1}0 -\phi_{i-1}\end{pmatrix}+ \phi_{i}\begin{pmatrix}0 \rho_{i-1}\end{pmatrix}.
\hat\sigma^M_{n,j}(\theta_n) =\hat \mu_{n,j}(\theta_n)\hat\sigma_{n,j}(\theta_n)+(1-\hat \mu_{n,j}(\theta_n))\hat\sigma_{n,j+R_1}(\theta_n) &=(1+o_{P_n}(1))\hat\sigma_{n,j}(\theta_n)+(1-\hat \mu_{n,j}(\theta_n))O_{P_n}(\hat\sigma_{n,j}(\theta_n)),
F_{4}\, \stackrel{\eta \rightarrow \infty}{\sim}\, \left( F_{\infty}- 9 \gamma^{2}b ( 6 \kappa )^{- 1 4 / 3}[ \frac{3}{4}( \eta - \eta_{0}) ]^{- 8 / 3}\right) v o l_{1 , 3}\, ,
\frac{\lambda_{R}}{2 m_{f , R}^{2}}( \langle \Phi^{2}( \tau ) \rangle - \langle \Phi^{2}( 0 ) \rangle ) = 3 ({\cal{D}}^{H F}( 0 , \tau ) -{\cal{D}}^{H F}( 0 , 0 ) )
(\begin{matrix}d[X^{i},X^{j}]&d[X^{i},Y^{j}]\\d[Y^{i},X^{j}]&d[Y^{i},Y^{j}]\end{matrix})=\frac{|\alpha|\hbar}{2m}\delta^{ij}(\begin{matrix}1+cos\phi&sin\phi\\sin\phi&1-cos\phi\end{matrix})dt
N \cdot \tilde{D}\cdot \tilde{S}_1 = (a \varphi^*A - \frac{e}{r}E) \cdot (\varphi^*A - \mu E) \cdot (a_1 \varphi^*A - \frac{e_1}{r}E) &= a a_1 (A^3) - \mu \frac{e e_1}{r^2}(E^3) & < a a_1 (A^3) - \frac{e e_1}{r^3}(E^3) \le 0.
\alpha = \sqrt{G}v = \frac{M_{W}}{e M_{\mathrm{P l}}}\ , \ \, b e t a = \frac{\sqrt{\lambda}}{e}= \frac{M_{H}}{\sqrt{2}M_{W}}\ , \ \, g a m m a = \kappa v = \frac{\kappa M_{W}}{e}\ ,
u_{\bf n}^{( + )}({\bf x}) = \sqrt{2 / l^{3}}\cos \left( 2 \pi{\bf n}\cdot{\bf x}/ l \right) , u_{\bf n}^{( - )}({\bf x}) = \sqrt{2 / l^{3}}\sin \left( 2 \pi{\bf n}\cdot{\bf x}/ l \right) .
A_Z(\xi) =\d{}{t}\bigg\vert_{t=0}(\tau^{-1}_t X_{\zeta_t(x)}) =(\nabla_ZX)_x &= (\nabla_X Z)_x + T(Z,X)_x + (\mathcal{L}_ZX)_x = \nabla_\xi Z + T(Z_x,\xi)
a_k=a_{[g/2]+1-k}^{-1}\ \ \mathrm{and}\ \b_k=a_{[g/2]+1-k}b_{[g/2]+1-k}^{-1}a_{[g/2]+1-k}^{-1}, \ \ 1\leq k\leq t; & a_{[g/2]+1-k}= a_k^{-1}\ \ \mathrm{and}\ \b_{[g/2]+1-k}=a_{[g/2]+1-k}^{-1}b_k^{-1}a_{[g/2]+1-k}, \ \ 1\leq k\leq t; a_1=a_2=\cdots=a_{t-1}=a_t^n=b_{t+2}=b_{t+3}=\cdots=b_{2t}=c_{2t}=c_t=1.
\Phi\left(x,x^{1/3}\right)\ge\frac{x}{\log x^{1/3}}\left(\omega\left(\frac{\log x}{\log x^{1/3}}\right)+\frac{\Delta_3^-}{\log x^{1/3}}\right)=\frac{3x}{\log y}\left(\frac{\omega(3)}{u}+\frac{3\Delta_3^-}{u^2\log y}\right).
\sum\limits_{i=1}^{\lfloor nr\rfloor}\sum\limits_{j=\lfloor nr\rfloor+1}^n\left(1_{\{X_i\leq X_j\}}-\frac{1}{2}\right)-\frac{n-\lfloor nr\rfloor}{n-\lfloor n\lambda\rfloor}\sum\limits_{i=1}^{\lfloor n\lambda\rfloor}\sum\limits_{j=\lfloor n\lambda\rfloor+1}^n\left(1_{\{X_i\leq X_j\}}-\frac{1}{2}\right)=0
\Phi_{\rm{st},\beta}(t) = \frac{S_\beta}{\pi E_{\rm{b}}}\bigg(\frac{P_{\rm{cu}}}{R^2_{\beta}}+ \lambda_{\beta}\left( 2^{C/W}- 1 \right)P_{\rm{tr},1}(h_{\beta,1}^*)R^2_{\beta}\bigg).
a=\left[\begin{array}{cc}a_{11}a_{12}0 a_{22}\end{array}\right], b=\left[\begin{array}{cc}b_{11}b_{12}0 b_{22}\end{array}\right], i=\left[\begin{array}{c}i_{1}i_{1}\end{array}\right], f=\left[\begin{array}{c}f_{1}f_{2}\end{array}\right], a'=f_2+a_{11}, b'=B_{12}f_2+b_{11}.
M_n=(-1)^n\sum_{k=0}^n(-2)^{n-k}\binom{n}{k}M_k, &C_{n+1}=(-1)^n\sum_{k=0}^n(-4)^{n-k}\binom{n}{k}C_{k+1}, &S_{n+1}=(-1)^n\sum_{k=0}^n(-6)^{n-k}\binom{n}{k}S_{k+1}.
d \Omega_{d - 1}^{2}= \sum_{i = 1}^{d}( d \omega^{i})^{2}= d \theta_{1}^{2}+ \sin^{2}\theta_{1}d \theta_{2}^{2}+ \cdots + \sin^{2}\theta_{1}\cdots \sin^{2}\theta_{d - 2}d \theta_{d - 1}^{2}.
\sum_{m_{1}}R_{U}( J , J^{\prime}; \varpi )_{m_{\phantom{2}}m^{\prime}}^{m_{2}m_{1}}\prod_{t = 1}^{J^{\prime}+ m_{2}}( y^{\prime}q^{2 J}+ y q^{\varpi + 4 J - 2 J^{\prime}+ 2 t - 1}) \prod_{s = 1}^{J + m_{1}}( y^{\prime}+ y q^{- ( \varpi - 2 J + 2 m_{2}+ 2 s - 1 )})
V_{( \infty^{p}, - 1 )}^{( M )}( \psi ) = \oint_{0}\frac{d \zeta}{\zeta^{M + 1}}V \left[ \left( \frac{( \zeta + 1 )}{\zeta^{p - 1}}\right)^{L_{0}}\psi , \zeta \right] \, .
|Dw|^2= w_1^2+\sum_{2\leq i\leq n}w_i^2, w_i= -G_{x_i}=-\psi_{x_i}d-\psi\gamma^i,i=2,\ldots,n, w_1= (1-G_u)u_1-G_{x_1}=(1-G_u)u_1 -\psi_{x_1}d-\psi\gamma^1.
\left (\sum_{k=0}^\infty c_{ijk}p^k\sum_{\substack{(i,j)\in\mathbb{J}_S \frac{m}{i}=\frac{n}{j}\in\mathbb{Z}_{\geq 0}}}(-j)[-a_{ijk}]^{m/i}\right )_{(m,n)\in\mathbb{J}_S},
\sum\limits_{k=0}^K (2k+1)\frac{1}{2\mu}\frac{\left( -1\right)^{k}}{\Gamma \left( \frac 12\right)\Gamma \left( k+1\right)}\left( \frac{t}{2\mu}\right)^{2k}\left[r_{d+2k+1}(-k-\frac 12) +\right.
\left[ \hat{G}\left( \mathbf{k}\right) \right]_{lm,l^{\prime}m^{\prime}}=\left( 4\pi\right)^{2}\left( -i\right)^{-l+l^{\prime}}a^{l+l^{\prime}+2}\frac{j_{l}\left( ka\right) j_{l^{\prime}}\left( ka\right)}{k^{2}}Y_{lm}^{\ast}\left( \mathbf{\hat{k}}\right) Y_{l^{\prime}m^{\prime}}\left(\mathbf{\hat{k}}\right) .
\omega(q)=\sum_{n\ge0}\frac{q^{2n(n+1)}}{(q;q^{2})_{n+1}^{2}}=\frac{1}{\prod_{n>0}(1-q^{2n})}\sum_{n\ge0}(-1)^{n}q^{3n(n+1)}\frac{1+q^{2n+1}}{1-q^{2n+1}}
\frac{d^{l-m}}{dx^{l-m}}(x^{2}-1)^{l}=\sum_{k=m}^{l}\frac{(l-m)!}{(k-m)!(l-k)!}\frac{l!}{k!}(x+1)^{k}\frac{l!}{(l-k+m)!}(x-1)^{l-k+m}