harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\! \! \! \! \! \! \! \! \! \! -{\frac{1}{2}}\sin{\phi_{3}}\sin{\phi_{2}}\sin{\phi_{1}}\, t \, \hat{\rho}( - \theta_{3}- \theta_{2}) - \sin^{2}{\frac{\phi_{3}}{2}}\sin{\phi_{2}}\cos{\phi_{1}}\, t \, \hat{\rho}( 2 \theta_{3}- \theta_{2})
E ( \tau_{\small p h y s i c a l})_{\small d i e l e c t r i c}^{\small p h y s i c a l}={\frac{\hbar}{( 2 \pi )^{2}}}V_{\small p h y s i c a l}n^{3}{\frac{1 2}{( c \tau_{\small p h y s i c a l})^{4}}}.
|A-B|_2^2=\sum_{i\neq j}\int_{I(y_i, F_n(x_i))}dx\int_{I(y_j, F_n(x_j))}|K_n(x,y)|^2dy = \sum_{i\neq j}\int_{I(y_i, F_n(x_i))}dx\int_{I(y_j, F_n(x_j))}O(1)dy=\sum_{i\neq j}F_n(x_i)F_n(x_j)O(1) = \sum_{i\neq j}O\left(\frac{\ln n}{n^2}\right)O(1)=k(k-1)O\left(\frac{\ln n}{n^2}\right)=O\left(\frac{\ln n}{n^2}\right),
\big(\mathcal{W}_{n}\mathcal{F}_{n}\psi\big)_{j}(p)= \begin{cases}\zeta_{j}(p) \begin{pmatrix}1 \frac{p}{1+\sqrt{1+p^2}}\end{pmatrix}; \zeta_{j}(p) \begin{pmatrix}\frac{-p}{1+\sqrt{1+p^2}}1 \end{pmatrix}; \end{cases}
\gamma_1 = \frac{- \int_\Omega \psi_1^2 \left\vert \nabla \left( \frac{u_{2,\lambda}}{\psi_1}\right)\right\vert^2 + (2-q) E_\lambda (u_{2,\lambda}) - \lambda (p-q) A(u_{2,\lambda})}{\int_\Omega u_{2,\lambda}^2 + \int_{\partial \Omega}u_{2,\lambda}^2}.
0 \leq -2 \, ||^2 + 4n(12+N_0+N_1 r) (N_0+N_1 \sqrt{\rho}) r^2 (r^2-\rho)^{-3}+ 48n(12+N_0+N_1 r) \rho r^2 (r^2-\rho)^{-4}\leq - 8n(12+N_0+N_1 r)^2 r^4 (r^2-\rho)^{-4}+ 4n(12+N_0+N_1 r) (N_0+N_1 r) r^4 (r^2-\rho)^{-4}+ 48n(12+N_0+N_1 r) r^4 (r^2-\rho)^{-4}= -4n(12+N_0+N_1 r)^2 r^4 (r^2-\rho)^{-4}
L\left(s,\chi\right) =G\left(\chi\right)q^{-s}\left(2\pi\right)^{s-1}\Gamma\left(1-s\right)\sum_{\mu=\pm1}L\left(1-s,\overline{\chi}\right)\chi\left(-\mu\right)e^{i\mu\frac{\pi}{2}\left(1-s\right)}
h(x) \le \Bigl(g_{6+}(x)+\frac2{6^7}(3^7x-911)(54-2^7x) &\qquad\qquad+\frac{2}{6^8}(3^8 x - 2733)(107-2^8x)\Bigr)\Bigr|_{x=\frac{911}{3^7}}+ \frac1{10\cdot 6^8}- \eta &=-\frac{2903244137571860267233333557802343701}{11170068372714227709934762805538661558464000}<0,
{}_{2}F{}_{1}\left( \begin{array}{c}p-m,q-m-2\nu n+p+q\end{array}\big|-|z|^{2}\right) =\left( 1+|z|^{2}\right)^{m+\nu -\frac{p+q}{2}}R_{m-p,m-q+2\nu}^{n+p+q-1}\left( \left( 1+|z|^{2}\right)^{-\frac{1}{2}}\right) .
A\vec{\rho}=(\begin{matrix}\vec{E}_{1}^{\dagger}\vec{\rho}\\ \vec{E}_{2}^{\dagger}\vec{\rho}\\ \vec{E}_{3}^{\dagger}\vec{\rho}\\ \vdots\end{matrix})=(\begin{matrix}E_{1}\cdot\rho\\E_{2}\cdot\rho\\E_{3}\cdot\rho\\ \vdots\end{matrix})=(\begin{matrix}P(E_{1}|\rho)\\P(E_{2}|\rho)\\P(E_{3}|\rho)\\ \vdots\end{matrix})\approx(\begin{matrix}p_{1}\\p_{2}\\p_{3}\\ \vdots\end{matrix})=\vec{p}
\tilde{D}(\tilde{r}_i) =\left(1+\frac{1}{N}+\cdots+\frac{1}{N^i}\right)-\frac{1}{1+N+\cdots+N^i}\left((i+1)+\frac{i}{N}+\cdots+\frac{1}{N^i}\right) =\frac{1+N+\cdots+N^i}{N^i}-\frac{(i+1)N^i+iN^{i-1}+\cdots+1}{N^i(1+N+\cdots+N^i)}=\frac{(1+N+\cdots+N^i)^2-(1+2N+3N^2+\cdots+(i+1)N^i)}{N^i(1+N+\cdots+N^i)}
\begin{cases}\dot{u}_2 (t,x) = \Delta u_2 (t,x)+\displaystyle{\frac{1}{\varepsilon}}f(u_1 +u_2) ,\ \ \t > 0,\x\in \mathbb{R}, u_2 (0,x) = u_0^\varepsilon (x),\ \ \x\in \mathbb{R}, \end{cases}
u_{-n+k}(x) = 2^{n-k-1}\delta(x_{1})\cdots\delta(x_{n-k})h_1(x_{n-k+1})\cdots h_1(x_n) &= 2^{n-k-1}\delta(x_{1})\cdots\delta(x_{n-k})\frac{1}{x_{n-k+1}}\cdots \frac{1}{x_n}
\theta_i(k+1)= ~\theta_i(k)-\mu\!\!\!\sum_{j\in \mathcal{N}_i(k)\setminus \{i\}}\!\!\!\frac{\sin(\theta_i(k)-R_{ij}(k)\theta_j(k))}{\theta_i(k)-R_{ij}(k)\theta_j(k)}&~\times(\theta_i(k)-R_{ij}(k)\theta_j(k)).
d\omega =\dfrac{1}{2}J_{p,t}^q\theta^t\wedge\theta^p\wedge\theta^q= &=\sum_{t<p<q}(J_{p,t}^q+J_{q,p}^t+J_{t,q}^p)\theta^t\wedge\theta^p\wedge\theta^q=:\sum_{t<p<q}d\omega(e_t,e_p,e_q)\theta^t\wedge\theta^p\wedge\theta^q.
\pi([x_1,\dots,x_n],[y_1,\dots,y_n])\tilde \eta(x_1,\dots,x_n) = \frac 1{n!}\sum_{\sigma\in\mathfrak S_n}\hat\pi((x_1,\dots,x_n),(y_1,\dots,y_n))\sigma \eta(x_1,\dots, x_n) &=\frac 1{n!}\sum_{\sigma\in\mathfrak S_n}\sigma\eta(y_1,\dots,y_n) &=\tilde \eta(y_1,\dots,y_n).
{V}_{i m p}\equiv \int d^{2}z \, \partial_{\alpha}( [ u_{i}X^{0}] \Theta ( X^{0}) \partial_{\alpha}X^{i}) = \int d \tau \, u_{i}\left( X^{0}\Theta ( X^{0}) \right) \partial_{n}X^{i}) ~ ; \qquad i = 1 , \ldots , 9 .
\sum_{k=0}^\infty\epsilon_k\le 2\frac{\hat{\beta}}{\delta}\sum_{k=0}^\infty\left(f(x^k)-f(x^{k+1})\right)= 2\frac{\hat{\beta}}{\delta}\left(f(x^0)-\lim_{k\to\infty}f(x^{k+1})\right)\leq 2\frac{\hat{\beta}}{\delta}\left(f(x^0)-f(x_*)\right)< \infty.
S_{eff}(A_{\mu})={\cal N}\int{\cal D}q ~ e^{- S(q,A_{\mu})}=\prod_{k=1}^{\infty}det \left ( \delta_{\mu\nu}-\frac{1}{\eta}F_{\mu\nu}\right )^{-1}= \sqrt{det \left ( \delta_{\mu\nu}-\frac{1}{\eta}F_{\mu\nu}\right )}.
dx=\sum_{i}\theta_{i}e_{i},de_{i}=\sum_{j}\theta_{ij}e_{j}+\sum_{k,j}h_{ij}^{k^{*}}\theta_{j}e_{k^{*}}, &de_{k^{*}}=-\sum_{i,j}h_{ij}^{k^{*}}\theta_{j}e_{i}+\sum_{l}\theta_{k^{*}l^{*}}e_{l^{*}}.
\theta(g_2, g_3)^{\alpha}\: + \: \theta(g_1, g_2 g_3)^{\alpha}\: = \:g_3^* \theta(g_1, g_2)^{\alpha}\: + \:\theta(g_1 g_2, g_3)^{\alpha}\: - \: d \log \omega^{g_1, g_2, g_3}_{\alpha}
(\frac{1}{2})^{10}[(\begin{matrix}10\\ 5\end{matrix})+(\begin{matrix}10\\ 6\end{matrix})+(\begin{matrix}10\\ 7\end{matrix})+(\begin{matrix}10\\ 8\end{matrix})+(\begin{matrix}10\\ 9\end{matrix})+(\begin{matrix}10\\ 10\end{matrix})]\approx0.623.
\theta(x_2) - \theta(x_1) = \sum_{k=0}^\infty \theta(x_2 + 2^{-k-1}(x_1-x_2) ) - \theta(x_2 + 2^{-k}(x_1-x_2)), &\leq 2 \|f\|_{L^\infty}^{1/2}\sum_{k=0}^\infty \left( \frac{2^{-k-1}(x_1-x_2)}{\log\left( 3 + 2^{k+2}\frac{x_2}{x_1-x_2}\right)}\right)^{1/2}, &\leq \frac{2 (1+\sqrt 2)}{\log 3}\|f\|_{L^\infty}^{1/2}(x_1-x_2)^{1/2}.
h ( \psi ; \tau , r ) \equiv \sum_{n = 1}^{\infty}\left( \frac 2{\pi n}\right)^{\frac 1 2}\cos [ n ( r - r_{n}) ] [^{A}\Psi_{n}\cos ( n \tau - \frac \pi 4 ) +^{B}\Psi_{n}\sin ( n \tau - \frac \pi 4 ) ]
\mathcal{Z}(\hat{\mu}_{M,D}) =\bigcup_{j=1}^{\infty}M^{*j}(\mathcal{Z}(m_D))=\bigcup_{j=1}^{\infty}M^{*j}(\mathcal{Z}_{D}^2+\mathbb{Z}^2) &\subset\left(\bigcup_{j=1}^{\infty}M^{*j}(\mathcal{Z}_{D}^2)\right)+\mathbb{Z}^2= \left(\bigcup_{j=1}^{p^2-1}M^{*j}(\mathcal{Z}_{D}^2)\right) +\mathbb{Z}^2\subset \mathring{E}_p^2+\mathbb{Z}^2.
S_F(x, y) := \begin{cases}\max\left\{\frac{\mu_F}{2}\|x - y\|^2, \frac{1}{2L_F}\|\nabla F(x) - \nabla F(y)\|^2\right\}L_F < \infty; \frac{\mu_F}{2}\|x - y\|^2 \end{cases}
E_{z}=\frac{k_{o}^{2}\epsilon_{r}-k_{z}^{2}}{j\omega\epsilon_{o}\epsilon_{r}}(Ae^{jk_{x\epsilon}w}+Be^{-jk_{x\epsilon}w})e^{-jk_{xo}(x+w)}sin(\frac{m\pi}{a}y)e^{-jk_{z}z}(47)
\sigma = \Phi \! \! \left( \! \! \begin{array}{c c}{2}&{2}\\{1}&{1}\\ \end{array}\! \! \right) , \bar{\sigma}= \Phi \! \! \left( \! \! \begin{array}{c c}{1}&{1}\\{2}&{2}\\ \end{array}\! \! \right) ,
\nu_{\alpha \beta}^{g_{1}g_{2}}\: = \: \nu_{\alpha \beta}^{g_{2}}g_{2}^{*}\nu_{\alpha \beta}^{g_{1}}\left( h_{\alpha}^{g_{1}, g_{2}}\right) \left( h_{\beta}^{g_{1}, g_{2}}\right)^{- 1}
\begin{aligned}\mathcal{T}_{\alpha\beta}(t)=\xi*\biggl[ \tau_{ji}^{(\alpha)}*e_{ij}^{(\beta)}(t)+\mu_{kji}^{(\alpha)}*\kappa_{kji}^{(\beta)}(t) - \sigma_{i}^{(\alpha)}*E_{i}^{(\beta)}(t)-Q_{ji}^{(\alpha)}*V_{ij}^{(\beta)}(t)\biggr],\end{aligned}
\Bigl(\prod_{j=1}^q \cos t_j \cos t_j^\prime\Bigr)^l \cdot R_\lambda(k; t) R_\lambda(k; t^\prime)= &=\frac{1}{\kappa_p}\int_{B_q}\int_{SU(q)}\Delta(d(t,t^\prime;v,w))^l \cdot R_\lambda\bigl(k;\arccos(\sigma_{sing}(d(t,t^\prime;v,w)))\bigr)\cdot\Delta(I_q-w^*w)^{p-2q}dv dw.
\frac{\partial^{2}\tau}{\partial x_{i}\partial x_{j}}= 0 , ( \theta_{i}- \theta_{j}) \frac{\partial^{2}\alpha^{0}}{\partial x_{i}\partial x_{j}}= 2 [ \frac{\partial \tau}{\partial x_{i}}\frac{\partial \alpha^{0}}{\partial x_{j}}- \frac{\partial \tau}{\partial x_{j}}\frac{\partial \alpha^{0}}{\partial x_{i}}] .
C_{3,1}(9) = \{(6,1^3),\, (5,1^4),\, (4,2,1^3),\, (4,1^5),\, (3^2,1^3), \, (3,2^3), &\quad\ (3,2,1^4),\, (3,1^6),\, (2^4,1), \, (2^2,1^5),\, (2,1^7),\, (1^9)\}.
\nabla_{\!y}p_a(y)=\nabla_{\!y}\left[ p_a(x)+p_{a-1}(x)\,x_{n+1}^2\right]=\nabla_{\!x}p_a(x)+x_{n+1}^2\,\nabla_{\!x}p_{a-1}(x)+p_{a-1}(x)\,2\,x_{n+1}\,e_{n+1}\,,
\sum_{n = 1}^{N_{\nu}}\z_{\nu , n}^{- s}= \displaystyle{\frac{1}{2 \pi \imath}}\oint_{C}z^{- s}\displaystyle{\frac{{\Delta_{\nu}^{T E}}^{\prime}( z )}{\Delta_{\nu}^{T E}( z )}}\d z ,
\mathrm{R H S}= \displaystyle \frac{( 1 - \tau^{2}) \zeta_{N - 1}}{\prod_{j = 1}^{n}( z_{N - 1}- z_{j}\tau^{2})}\sum_{k = 1}^{n}\zeta_{k}\displaystyle \prod_{j = 1 \atop j \neq k}^{n}\frac{( z_{N - 1}- z_{j}) ( z_{k}- z_{j}\tau^{2})}{z_{k}- z_{j}}\displaystyle \overline{{{G}}}_{\varepsilon , k}^{( n - 1 , l - 1 )}( \zeta^{\prime}) .
{\rm Sym}^{r-1}(\widehat{\sigma}^1_1)\, :\,{\rm Sym}^{r-1}(J^1(\mathcal L)) \,\longrightarrow\,{\rm Sym}^{r-1}(\sigma^*_X\overline{J^1(\mathcal L)})\,=\, \sigma^*_X\overline{{\rm Sym}^{r-1}(J^1({\mathcal L}))}\, .
X_{( 0 )}^{\mu}( x , y ) = x^{\mu}+ \frac{1}{2}\alpha_{0}^{\mu}\left( ( x + y ) - ( x - y ) \right) + i \sum \frac{1}{m}\alpha_{m}^{\mu}e^{i m ( x - y )}+ i \sum \frac{1}{m}\alpha_{m}^{\mu}e^{- i m ( x + y )}
J_{\mu \nu}[ \xi | s ] = \tilde{g}\epsilon_{\mu \nu \rho \sigma}[ \bar{\psi}( \xi ( s ) ) \gamma^{\rho}t^{i}\psi ( \xi ( s ) ) ) ] \Omega_{\xi}^{- 1}( s , 0 ) t_{i}\Omega_{\xi}( s , 0 ) \dot{\xi}^{\sigma}( s ) ,
\sum_{| \vec{n}| = n}b^{\vec{n}}\left( \sum_{m = 0}\alpha_{m}^{\vec{n}^{'}, \vec{n}}( c ) ( \Delta + \theta )^{m}\right) \langle \Delta^{'}+ \theta | \Delta + \theta \rangle = 0 , \hspace{1 c m}\forall \: \: \: \vec{n}^{'}: | \vec{n}^{'}| = n ,
\left. \begin{array}{c c c}{{S_{\sigma \sigma^{\prime}}\nonumber}}\\{{S_{\Delta}\nonumber}}\\{{\tilde{S}_{\Delta}}}\\ \end{array}\right\}\varpi = ( \varpi + 2 ) \left\{\begin{array}{c c c}{{S_{\sigma \sigma^{\prime}}\nonumber}}\\{{S_{\Delta}\nonumber}}\\{{\tilde{S}_{\Delta}}}\\ \end{array}\right. .
K_0(\|x\|)=\frac{1}{2}\int_{-\infty}^{+\infty}e^{-\|x\|\cosh t}dt =\frac{1}{2}\int_0^{+\infty}e^{-\|x\|\cosh\left(\ln(2y/\|x\|)\right)}\frac{dy}{y}&=\frac{1}{2}\int_0^{+\infty}e^{-y-\frac{\|x\|^2}{4y}}\frac{dy}{y}&=\frac{1}{2}\int_0^{+\infty}e^{-\frac{\|x\|^2}{4y}}e^{-y}\frac{dy}{y}.
\psi(x):= W_{\delta}(x)- PW_{\delta}(x) = \delta^{-\frac{n-4}{2}}\Big(\varphi_0\Big(\frac{e_n-x}{\delta}\Big)-\varphi_0\Big(\frac{e_n+x}{\delta}\Big)\Big)-\zeta_\delta(x),\qquad\zeta_\delta(x)=O(\delta^{\frac{6-n}{2}}).
r-\sum_{j\ne i}z_j^*(|v_j|-|v_i|) = r-\sum_{j=1}^{m_1}(|v_j|-|v_i|)-\sum_{j=m_1+1}^{m_0}z_j^* (|v_j|-|v_i|) &=r-\sum_{j=1}^{m_1}(|v_j|-|v_{m_1+1}|)=r-A(m_1)= 0,
\Delta E_{n}^{( 1 )}[ \chi \, ] = \frac{\langle V \rangle ( 2 - n )}{( 4 \pi )^{\frac{n}{2}}\Gamma \left( \frac{n}{2}\right)}\int_{0}^{\infty}( \omega ( k ) - m ) k^{n - 3}\, d k = \frac{\langle V \rangle m^{n - 1}}{( 4 \pi )^{\frac{n + 1}{2}}}\Gamma \Bigl ( \frac{1 - n}{2}\Bigr )
\sup_{f_1,f_2,f_3,F,G,H}\left(\mathcal{I}_2(W_0, f_1, f_2, f_3)+\mathcal{I}_1(W,F,G,H)\right) &=\sup_{f_1, f_2, f_3}\left(\mathcal{I}_2(W_0, f_1, f_2, f_3)\right)+\sup_{F, G, H}\left(\mathcal{I}_1(W,F,G,H)\right) &=I_{ini}(W_0)+I_{dyn}(W)
\begin{array}{lll}h_{b}(z,T,x^{10})=\tilde h_{b+N'/2}(\bar z,-1/\bar T,-x^{10}) , \tilde h_{b}(z,T,x^{10})=h_{b+N'/2}(\bar z,-1/\bar T,-x^{10}) h_{c}(z,T,x^{10})=\tilde h_{c}(\bar z,-1/\bar T,-x^{10}) , \end{array}
\sum_{\{j\,,\,\alpha_j<2m\}}(\lambda(\alpha)-\lambda_j)c_j^2 \ge C_2\sum_{\{j\,,\,\alpha_j<2m\}}c_j^2={C_2}\| Q\|_{L^2(\partial B_1)}^{2}\geq \frac{C_2}{C_1}M^2,
N= \bigg\|\int_{0}^{s}\int_{0}^{1}\bigg(\frac{\partial G}{\partial y}(t-r,\cdot,y)-\frac{\partial G}{\partial y}(s-r,\cdot,y)\bigg)v^{\delta+1}(r,\cdot)\d y\d r\bigg\|_{\L^p}.
S^{\rm grav.}= -{T_{KK11}\over 2}\int d^{7}\hat{\xi}\sqrt{|\hat{\gamma}|}\ \left[\hat{\hat{k}}^{4/7}\hat{\gamma}^{\hat{\imath}\hat{\jmath}}\hat{D}_{\hat{\imath}}\hat{\hat{X}}^{\hat{\hat{\mu}}}\hat{D}_{\hat{\jmath}}\hat{\hat{X}}^{\hat{\hat{\nu}}}\hat{\hat{g}}_{\hat{\hat{\mu}}\hat{\hat{\nu}}}-5 \right]\, ,
( H_{0}^{- ( 0 )})^{2}- ({\bf P}_{0}^{- ( 0 )})^{2}= 0 , \quad ( \widetilde{H}_{0}^{- ( 1 / 2 )})^{2}- ( \widetilde{\bf P}_{0}^{- ( 1 / 2 )})^{2}= 0 ,
\begin{aligned}\mathcal S_3(k,3)= \left\{F\in\binom{[n]}{k}:[2]\subseteq F,F\cap[3,k-1]\neq\emptyset\right\}\cup\left\{F\in\binom{[n]}{k}:[2]\cup[k,k+3]\subseteq F\right\}&\cup\{[2,k]\cup\{k+2\},\, [2,k+2]\setminus\{k\}, \, [2,k-1]\cup\{k+1,k+3\}\}&\cup\{[3,k+1]\cup\{1\},\, [3,k-1]\cup\{1,k+1,k+2\}, \, [3,k-1]\cup\{1,k+2,k+3\}\}.\end{aligned}
wt_{sp}(C)=\begin{cases}2 \mbox{if~}t_3=0, 3 \mbox{if~}t_3=1, 4 \mbox{if~}2 \le t_3 \le 25, 6 \mbox{if~}26 \le t_3 \le 50, 8 \mbox{if~}51 \le t_3 \le 62. \end{cases}
l^{2}= \left( 1 +{\frac{c_{p}\, N}{( r M_{P})^{7 - p}( R_{S}M_{P})^{2}( V_{T}M_{P}^{p})}}\right) R_{S}^{2}= R_{S}^{2}+{\frac{c_{p}\, N}{( r M_{P})^{7 - p}M_{P}^{2}( V_{T}M_{P}^{p})}}
\mathcal{J}= \mathbb{E}\{|e_k(i)|^2\}= \mathbb{E}\{|d_k(i)-y_k(i)|^2\}&=\mathbb{E}\{|d_k(i)|^2\}-\mathbb{E}\{d_k^*(i)y_k(i)\}-\mathbb{E}\{d_k(i)y_k^*(i)\}&\ \ \ +\mathbb{E}\{|y_k(i)|^2\},
\gamma_{\infty}(\kappa_1,\kappa_2) = \begin{cases}\beta, f^U = f^L = 0, \max\{1,\beta\}, \end{cases}\gamma_0(\kappa_1,\kappa_2) = \begin{cases}\alpha, f^U = f^L = 0, \min\{1,\alpha\}, \end{cases}c_1 := g^L c_2 := c_3 := f^U + g^U
\begin{aligned}\frac{\Sigma_{f,M_n}-Cd^{1/2}\log(d)G^{-1/2}}{\sqrt{\Sigma_{f,M_n}+Cd^{1/2}\log(d)G^{-1/2}}}\:\:\: \leq \:\:\: \limsup_{N\to\infty}\frac{|\sum_{n=1}^Np(M_nx)|}{\sqrt{2N\log(\log(N))}}\leq \:\:\: \frac{\Sigma_{f,M_n}+Cd^{1/2}\log(d)G^{-1/2}}{\sqrt{\Sigma_{f,M_n}-Cd^{1/2}\log(d)G^{-1/2}}}\textnormal{a.e.}\end{aligned}
F_0\leq 2 \Theta G = \frac{2 p}{q}\Big( 2^{\frac{n}{p}+1}\frac{q-p}{q}\Big)^{\frac{q-p}{q}}\big(\frac{4}{R}\big)^{\frac{n}{p}}\gamma^{\frac{q}{p}}G =\frac{\gamma'}{R^{\frac{n}{p}}}G.
\iint_Q{\rho}_5^2 |y_{tt}|^2 dxdt + \underset{[0,T]}{}\left({\rho}_5^2(t) \|y_{xt}(t)\|^2 \right) \leq C \left( \|y_{xt}(0)\|^2 + \iint_Q{\rho}_3^2 |G_{t}|^2 dxdt \right. &+ \iint_{\mathcal{O}\times (0,T)}{\rho}_1^2 |f|^2 dxdt + \iint_Q{\rho}_0^2 |y|^2 dxdt +\left. \sum_{i=1}^2 \iint_Q{\rho}_0^2 |p^i|^2 dxdt \right).
\tilde{\phi}\otimes \chi_{1}= \left( \begin{array}{c}{{\tilde{\phi}}}\\{{0}}\\{{0}}\\ \end{array}\right) \, , \quad \tilde{\phi}\otimes \chi_{2}= \left( \begin{array}{c}{{0}}\\{{0}}\\{{\tilde{\phi}}}\\ \end{array}\right) \, ,
\phi(aD)(z^2+r^2) = \sum_{k=0}^{\infty}\tfrac{a^k \phi^{(k)}(0)}{k!}\tfrac{d^k}{z^k}(z^2+r^2) & = \phi(0)(z^2+r^2) + a \phi'(0)(2z)+\tfrac{a^2}{2}\phi''(0) (2) & = z^2 + 2a\phi'(0) z +r^2 + a^2\phi''(0) & = (z+a\phi'(0))^2 + r^2 - \big([\phi'(0)]^2-\phi''(0)\big)a^2 & = (z-a\phi'(0))^2 + r^2-b_{\phi}a^2.
(\alpha -1)p_\theta({\bf{x}})^{\alpha -2}\partial_r [p_\theta ({\bf{x}})] = \partial_r[Z(\theta)^{\alpha - 1}][h({\bf{x}})+ w(\theta)^\top f({\bf{x}})] + Z(\theta)^{\alpha - 1}\partial_r[w(\theta)]^\top f({\bf{x}}).
\mathcal{L}_3:=\mathcal{T}^{-1}\mathcal{L}_2 \mathcal{T}=\Pi_S^{\perp}(\omega\cdot\partial_{\vartheta}+m_3\,\partial_{zzz}+D_S\,\partial_z+d_0)\Pi_S^{\perp}+\mathfrak{R}_3, &d_1:=(\mathcal{T}^{-1}c_1)+\omega\cdot\partial_{\vartheta}p, d_0:=\mathcal{T}^{-1}c_0, \mathfrak{R}_3:=\mathcal{T}^{-1}\mathfrak{R}_2 \mathcal{T}
R^{\rho}_{\phantom{1}\sigma\mu\nu}=\partial_{\mu}\Gamma^{\rho}_{\nu\sigma}-\partial_{\nu}\Gamma^{\rho}_{\mu\sigma}+\Gamma^{\rho}_{\mu\lambda}\Gamma^{\lambda}_{\nu\sigma}-\Gamma^{\rho}_{\nu\lambda}\Gamma^{\lambda}_{\mu\sigma}
\partial_{\Sigma^m G}\circ \theta[1^* ] =\partial_{\Sigma^m G}[\Sigma^{2m}\lambda_{1\cdots m}] &=\sum\limits_{i=1}^{m}(-1)^{m+m-i}\Sigma^{2m-1}\lambda_{1\cdots (i-1)(i+1)\cdots m}(1\otimes y_i-y_i\otimes 1) &=\sum\limits_{i=1}^{m}(-1)^{i}\Sigma^{2m-1}\lambda_{1\cdots (i-1)(i+1)\cdots m}(1\otimes y_i-y_i\otimes 1).
B^\infty_{(m,\psi)}(|u|)(z) = \frac{|z|^m |u(z)|}{(1+|\psi(z)|)^m}e^{\frac{\alpha}{2}\big(|\psi(z)|^2-|z|^2\big)}= |u(z)| e^{\frac{\alpha}{2}\big(|\psi(z)|^2-|z|^2\big)},
\eta(\mathsf{I}_{\mathsf{F}-1},\mathsf{I}_{\mathsf{F}})=\eta(\mathsf{J}_{\mathsf{F}-1},\mathsf{B})=1-\kappa^2_{\mathsf{B}}\kappa^{-2}_{\mathsf{J}_{\mathsf{F}-1}}\geq 1-\kappa^2_{\mathsf{J}_{\mathsf{F}}}\kappa^{-2}_{\mathsf{J}_{\mathsf{F}-1}}=\eta(\mathsf{J}_{\mathsf{F}-1},\mathsf{J}_{\mathsf{F}})
\frac{q}{1+q}{}_2\phi_1 \left(\genfrac{}{}{0pt}{}{0,-q}{-q^2};q,q\right) = \frac{q}{1+q}\frac{\tiny 1}{(q;q)_\infty}{}_2\phi_2 \left(\genfrac{}{}{0pt}{}{0,\, q}{-q^2,\, 0};q,-q^2\right).
\begin{array}{l c l}{{T_{\scriptscriptstyle}}}&{{=}}&{{\frac{1}{4}\left( T_{1}+ T_{1 0}+ T_{1 9}+ T_{2 8}+ T_{3 7}+ T_{4 6}+ T_{5 5}+ T_{6 4}\right.}}\\{{}}&{{}}&{{\left. + i \left( T_{6 5}+ T_{7 4}+ T_{8 3}+ T_{9 2}+ T_{1 0 1}+ T_{1 1 0}+ T_{1 1 9}- T_{1 2 8}\right) \right) .}}\\ \end{array}
\int_{\{0,1\}^{\mathbb{Z}^d}}\left| \mathbb{E}_\eta(f(\eta_t)) - \mu(f) \right| \mathrm{d}\nu(\eta) = \|f - \mu(f)\|_\infty \int_{\{0,1\}^{\mathbb{Z}^d}}\left| \mathbb{E}_\eta(g(\eta_t)) \right| \mathrm{d}\nu(\eta)
\tau_{RS}(M)=\det(\phi_0^*\phi_0)^{-\frac{1}{2}}\det(\phi_3^*\phi_3)^{\frac{1}{2}}{\det}'(d^*_0d_0)^{\frac{1}{2}}{\det}'(d_1^*d_1)^{-\frac{1}{2}}{\det}'(d^*_2d_2) .
U_{2x,N-n}= U_{2x,N+n+1}= \frac{(-1)^x}{\sqrt{2}}\tilde Q_n(x;\alpha,\beta+1,N), U_{2x+1,N-n}= -U_{2x+1,N+n+1}= -\frac{(-1)^x}{\sqrt{2}}\tilde Q_n(x;\alpha+1,\beta,N),
i_{\alpha\beta}(t)=\frac{2}{3}[\begin{matrix}1&-\frac{1}{2}&-\frac{1}{2}\\ 0&\frac{\sqrt{3}}{2}&-\frac{\sqrt{3}}{2}\end{matrix}][\begin{matrix}i_{a}(t)\\i_{b}(t)\\i_{c}(t)\end{matrix}]
G_1 = G,G_{i+1}= \begin{bmatrix}P_G^i G P_G^{i-1}G \hdots G -P_G e_1 \hdots -P_G^i e_1 \end{bmatrix}B_1 = B, B_{i+1}= \begin{bmatrix}B P_B B \hdots P_B^i B P_B^i e_1 \hdots P_B e_1 \end{bmatrix},
\begin{cases}\widehat{\pi}(1,0) =\frac{\lambda}{1+\delta+\lambda}\widehat{\pi}(1,1)+\frac{\delta}{1+\delta+\lambda}\widehat{\pi}(0,1), \widehat{\pi}(0,1) =\frac{\gamma}{1+\gamma}\widehat{\pi}(1,0).\end{cases}
\frac{R_{31}}{i-\omega(1,1)}u_1+\frac{R_{32}}{i-\omega(2,2)}u_2=-i(iR_{31}-R_{32})=R_{31}+iR_{32}=0; \frac{R_{41}}{i-\omega(1,1)}u_1+\frac{R_{42}}{i-\omega(2,2)}u_2=R_{41}+iR_{42}=-1-1=-2=-2u_4.
w(l,\mu) \:=\; \int_{-\infty}^{+\infty}\!\! dp \int_{\mu}^{\infty}\!\! dx \;{\rm e}^{\textstyle -lp^2 -\sqrt{x}\,l}\;=\; \frac{2\sqrt{\pi}}{l^{5/2}}\;{\rm e}^{\textstyle -\sqrt{\mu}\,l}\; \left[ 1 + \sqrt{\mu}\,l \right],
\begin{array}{rcllll}f_{0,0}- \rho^* = \inf_{\varphi \in \mathbb{R}[\bar{z},z]}\langle \varphi , \delta_{0,0}\rangle A\varphi = b ~~ \& ~~ \varphi \in C, f_{0,0}- \rho\hphantom{^*}= \sup_{y \in \mathcal{H}}\langle b , y \rangle \delta_{0,0}- A^*y \in C^*.\end{array}
\sum_{\ell = i}^{j+K-1}\int_{\gamma_{\ell}\wedge u}^{\gamma_{\ell+1}\wedge u}s(j - \ell) \: dv \geq \sum_{\ell = j+1}^{j+K-1}s (j - \ell)(\gamma_{\ell+1}\wedge u - \gamma_{\ell}\wedge u) &\geq \frac{2 a_N}{k_N}\sum_{\ell = j+1}^{j+K-1}s(j - \ell) &= - \frac{K (K-1) s a_N}{k_N}.
E_{2n}=(2n)!\sum_{0\le k_{1},...,k_{n}\le n}(\begin{matrix}K\\k_{1},...,k_{n}\end{matrix})\delta_{n,\sum mk_{m}}(-\frac{1}{2!})^{k_{1}}(-\frac{1}{4!})^{k_{2}}\cdot\cdot\cdot(-\frac{1}{(2n)!})^{k_{n}},
w_0 =\theta(E_1\cdots E_r)=\psi(E_1\cdots E_r), w_1 =\theta(E_{r+1}\cdots E_{j-1})e_{j1}=\psi(E_{r+1}\cdots E_{j-1}E_{j1}), w_2 =e_{j2}\theta(E_{j+1}\cdots E_s)=\psi(E_{j2}E_{j+1}\cdots E_s)\textrm{and}w_3 =\theta(E_{s+1}\cdots E_t)=\psi(E_{s+1}\cdots E_t).
|1+z|^p \ge |1+u|^p (1+|1+u|^{-2}v^2)^{p/2}& \ge |1+u|^p (1+\tfrac{p}2 |1+u|^{-2}v^2 -C|1+u|^{-4}v^4) & = |1+u|^p +\tfrac{p}2 |1+u|^{p-2}v^2 -C|1+u|^{p-4}v^4.
\int_\Delta |H_\omega\circ \hat f_\omega^n|^p\,d\mu_\Delta =\int_\Delta |H_\omega|^p\,d\mu_\Delta=(1/\bar r)\int_Y\sum_{\ell=0}^{r(y)-1}|H_\omega(y,\varphi,\ell)|^p\,d\nu_Y(y,\varphi) \le \int_Y r \ |V^*_\omega|^p\,d\mu_Y\le |r|_a |{V^*_\omega}^p|_b = |r|_a |V^*_\omega|_{bp}^p.
\rho^T_t:=\frac{1}{T^{k/2}}\sum_{j_1 \neq \ldots \neq j_k}\sigma_{j_1}\ldots \sigma_{j_k}\langle \Lambda^{(k)}(x^{j_1}+\xi^{j_1},\ldots,x^{j_k}+\xi^{j_k};T),\mathbf{1}_{[0,t]}\rangle, t \geq 0.
F(u^k) :=S_0^*\left(\alpha (Su^k-z) + \beta_1 C^*_{\omega_R}H^+_\gamma(C_{\omega_R}Su^k) + \beta_2 C^*_{\omega_T}H^-_\gamma(C_{\omega_T}Su^k)\right), D_NF(u^k) = S_0^*\left(\alpha + \tfrac{\beta_1}\gamma C^*_{\omega_R}\chi^+(C_{\omega_R}S_0u^k) C_{\omega_R}+ \tfrac{\beta_2}{\gamma}C^*_{\omega_T}\chi^-(C_{\omega_T}S_0u^k)C_{\omega_T}\right)S_0,
\mathcal{L}= \beta^{2}\sqrt{1 + \frac{m^{2}}{\beta^{2}}A_{\mu}A^{\mu}}- \frac{1}{4}B_{\mu \nu}B^{\mu \nu}- \frac{\chi \theta}{2}\varepsilon^{\mu \nu \rho \lambda}\partial_{\mu}B_{\nu \rho}A_{\lambda}.
W_{1324}\equiv \langle F_{ab}(x_1) F_{cd}(x_3) \rangle\langle F_{cd}(x_3) F_{ef}(x_2) \rangle\langle F_{ef}(x_2) F_{gh}(x_4) \rangle\langle F_{gh}(x_4) F_{ab}(x_1) \rangle
\frac{[\ker(\Psi^0(\mathcal{F}))]}{\left[\left( \frac{\prod_{v\in S_{\infty}}H^0(K_v,\mathcal{F}_v)}{H^0_{et}(U,\mathcal{F})}\right)_{tor}\right]}= \frac{[H^0_{et}(U,\mathcal{F})_{tor}]}{\prod_{v\in S}[H^0(K_v,\mathcal{F}_v)_{tor}][\mathrm{cok}(\delta(\mathcal{F})_{tor})]}.
\begin{cases}\dot{h}_1= A{h_1}+BR^{-1}B^T{h_4}+G{h_2}+\gamma{h_3}, \dot{h}_2 = (A + G){h_2}+\gamma{h_3}, \dot{h}_3 = -(A+G)^T{h_3}-(I-\Gamma)^T Q ({h_1}- \Gamma{h_2}) , \dot{h}_4=-A^T{h_4}+Q({h_1}-\Gamma{h_2}),\end{cases}
S_{i \; j}^{k \; \ell}\left( \begin{array}{c c}{{a}}&{{d}}\\{{b}}&{{c}}\\ \end{array}\right) :{\cal V}_{a , b}( \theta_{1}) \otimes{\cal V}_{b , c}( \theta_{2}) \rightarrow{\cal V}_{a , d}( \theta_{2}) \otimes{\cal V}_{d , c}( \theta_{1})
\left ( \sum^{m+1}_{i=1}\lambda_i\left({\bf x}_0 \right)_i\right )^{p-1}-\left ( \sum^m_{i=1}\lambda_i\left({\bf x}_0 \right)_i\right )^{p-1}= (p-1)\lambda_{m+1}\left({\bf x}_0 \right)_{m+1}\xi^{p-2}<(p-1)\lambda_{m}\left({\bf x}_0 \right)_m\left ( \sum^m_{i=1}\lambda_i\left({\bf x}_0 \right)_i\right )^{p-2},
d s_{( \mu < 0 )}^{2}= l^{2}\left[ - d \tau^{2}+ \frac{1 + m}{4}\frac{d z^{2}}{z^{2}}+ \frac{1 + m}{4}\frac{d \bar{z}^{2}}{\bar{z}^{2}}+ \left( e^{- 2 \tau}+ \frac{( 1 + m )^{2}}{1 6}\frac{1}{z^{2}\bar{z}^{2}}e^{2 \tau}\right) d z d \bar{z}\right] ,
\bar{\vartheta}(t_2)-\bar{\vartheta}(t_1) =\int_{(t_1,t_2]}\, d\mu(s) \int_{\mathbb R}{p}(t_2-s, x-y)\sigma(s, y)\,dy &\quad{}+\int_{(0,t_1]}\, d\mu(s) \int_{\mathbb R}\bigl({p}(t_2-s, x-y)-{p}(t_1-s, x-y) \bigr)\sigma(s, y)\,dy &:=F_1+F_2.
\alpha_2=\frac{\beta_{X^2}^2-\beta_{X^4}}{2(-1+2\beta_{X^2}-\beta_{X^4})},\alpha_3=\frac{-\beta_{XY}^2+\beta_{X^2}-\beta_{X^4}}{2(\beta_{X^2}-\beta_{X^4})},\alpha_4=\frac{-2\beta_{XY}^2+\beta_{XYXY}+\beta_{X^2}-\beta_{X^4}}{2(\beta_{XYXY}+\beta_{X^2}-\beta_{X^4})}.
\pi(g^{[k](\alpha)}_{-})=\begin{bmatrix}S^{+}(z)\tau^{(\alpha)}_{k}S^{+}(z)\tau^{(\alpha)}_{k-1}/z S^{-}(z)\tau^{(\alpha)}_{k+1}/z S^{-}(z)\tau^{(\alpha)}_{k}\end{bmatrix}/\tau^{(\alpha)}_{k}.
\lim_{N \to \infty}\Big(\frac{1}{\varphi(x)}\Big)^k R_{N}^{(k)}\Big(Nx+\frac{u_1}{\varphi(x)}, \ldots, Nx+\frac{u_k}{\varphi(x)}\Big)= \det \Big[\frac{\sin \pi(u_{i}-u_{j})}{\pi(u_{i}-u_{j})}\Big]_{i,j = 1}^{k}
|u(t)-\overline{u}(t)|_{p}=|\phi_x(t)-\overline{\phi}_x(t)|_{p}\indent \overset{p}{\lesssim}|\phi(t)-\overline{\phi}(t)-R|^{1/2p}_{1}|u_x(t)-\overline{u}_x(t)|^{1-1/2p}_{1}\indent \overset{p}{\lesssim}|\phi(t)-\overline{\phi}(t)-R|^{1/2p}_{1}. \indent \overset{p}{\lesssim}|\phi(t)-\overline{\phi}(t)-R|^{1/2p}_{\infty}.
\begin{array}{ll}\frac{\partial \hat{\rho}(t,0)}{\partial t}= \frac{i}{\hbar}[\hat{\rho}(t,0),\hat{H}(t)] = \left. \frac{i}{\hbar}\partial_s^2{\rm Log}\left( e^{\int \hat{\rho}(t,s) ds}e^{\int \hat{H}(t) ds}\right) \right|_{s=0},\end{array}
\begin{array}{r c l}{{{\cal L}_{G G \phi}}}&{{=}}&{{\displaystyle + \frac{g^{2}v}{2 \sqrt{2}}\left( A^{\mu}( \pi_{+}W_{\mu}^{-}+ W_{\mu}^{+}\pi_{-}) + Z^{\mu}( W_{\mu}^{-}\pi_{+}+ \pi_{-}W_{\mu}^{+}) \right)}}\\{{}}&{{}}&{{\displaystyle + \frac{g^{2}v}{2}( W_{\mu}^{-}W^{+ \mu}+ Z_{\mu}Z^{\mu}) \sigma ,}}\\ \end{array}
\prod^{N-1}_{i=3}(\int dz_{i}\oint dz'_{i}) \langle{\cal O}^{1}(z_{1}){\cal O}^{2}(z_{2}) b(z'_{3}){\cal O}^{3}(z_3) \cdots b(z'_{N-1}){\cal O}^{N-1}(z_{n-1}){\cal O}^{N}(z_{N})\rangle,
D^i_sD^j_tp\vert_{R}(w)=D^i_sD^j_tp\vert_{\tilde R}(w), i=0,...,n_1-r_1-2\,j=0,...,r_2, &D^i_sD^j_tp\vert_{R}(\hat w)=D^i_sD^j_tp\vert_{\tilde R}(\hat w), i=0,...,r_1\,j=0,...,r_2,
V_{4}=-\varepsilon\int_{\mathbb{T}^{d}}(\mu^{1}-\mu^{2})(\nabla\mu^{2})\cdot\left(\Theta_{p}(t,x,\mu^{1},Dw^{1})-\Theta_{p}(t,x,\mu^{2},Dw^{2})\right)\dx,
\begin{aligned}\alpha \omega_1 + \beta \omega_2 + \gamma \omega_3 + \delta \omega_4 + \mu \omega_5 + \nu \omega_6 = n(n-1)-1 -\alpha \omega_1 - \gamma \omega_3 - \delta \omega_4 = -(n-1) -\beta \omega_2 +\gamma \omega_3 - \nu \omega_6 = - \frac{n(n-3)}{2}\alpha \omega_1 + \delta \omega_4 - \mu \omega_5 = - \binom{n-1}{2}.\end{aligned}