harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\Phi_{t+1}^{\pm}(x_1,\dots, x_k) = \sum_{s=1}^k \Phi_t^{\pm}(x_1,\dots,x_s) x_s - q^{\mp 2}\sum_{s=1}^{k-1}\Phi_t^{\pm}(x_1,\dots,x_s) x_{s+1}&= x_1^{t+1}+ \sum_{s=2}^k \big( \Phi_t^{\pm}(x_1,\dots,x_s) x_s - q^{\mp 2}\Phi_t^{\pm}(x_1,\dots, x_{s-1})x_{s}\big)
\prod_{k=1}^{n-2}{\rm Cat}_{k}\prod_{1\le i < j \le n-1}\frac{2(m+1)+i+j-1}{i+j-1}=\prod_{j=m+1}^{m+n-2}\frac{1}{2j+1}{n+m+j \choose 2 j}\qquad{}=K_{A_{n+1}}\left(m+1,m+2,\ldots,n+m,-m n-{n \choose 2}\right) .
\begin{array}{rl}\hskip 0.3cm\nabla_{X_1^h}Y_2^{v}= \nabla_{Y_2^v}X_1^{h}=-cX_1(f_{_{1}})^hY_{2}(f_{_{2}})^v\big\{f_{_2}^vgrad(f_{_{1}}^h) +f_{_{1}}^hgrad(f_{_{2}}^v)\}&+\big(Y_2(\ln f_{_{2}})\big)^vX_1^h+\big(X_1(\ln f_{_{1}})\big)^hY_2^v, \end{array}
\rho_{134}(S_1)^{2}= (S_2S_1S_1^*+S_1S_2S_2^*) (S_2S_1S_1^*+S_1S_2S_2^*) & = S_2(S_1S_2)S_2^*+S_1(S_2S_1)S_1^*
\hat{\phi}(x,t)=\int\frac{d^{3}p}{(2\pi)^{3}}\frac{1}{\sqrt{2\omega_{p}}}(\hat{a}_{p}e^{-i\omega_{p}t+ip\cdot x}+\hat{a}_{p}^{\dagger}e^{i\omega_{p}t-ip\cdot x}).
V_{m}^{\prime}( \phi ) = \sum_{n = 1}^{\infty}v_{m m_{2}\ldots m_{n}}^{( n )}\phi_{m_{2}}\ldots \phi_{m_{n}}, \tilde{V}_{m}^{\prime}( \tilde{\phi}) = \sum_{n = 1}^{\infty}v_{m m_{2}\ldots m_{n}}^{( n )}\tilde{\phi}_{m_{n}}\ldots \tilde{\phi}_{m_{2}}
u(x,t_0+nT_1+t;t_0,u_0) =u(x,t_0+nT_1+t;t_0+nT_1,u(x,t_0+nT_1,t_0,u_0))\cr \geq \delta_2 e^{t(a_{\inf}-b_{\sup}M^+e^{T_1a_{\sup}})}\cr \geq \delta_2 e^{-T_1(a_{\inf}+b_{\sup}M^+e^{T_1a_{\sup}})}
\left(\frac{1}{\tilde{p}},\frac{1}{\tilde{q}}\right) = \frac{\sigma_2 -\sigma}{\sigma_2 -\sigma_1}\left(\frac{1}{p_1}, \frac{1}{q_1}\right) + \frac{\sigma -\sigma_1}{\sigma_2 -\sigma_1}\left(\frac{1}{p_2}, \frac{1}{q_2}\right),
\exists !\,f\in L^2({\mathbb{R}}^n)\mbox{such that, for all}t>0, \; u(t,\cdot)=\Gamma(t,0)f\mbox{in}L^2({\mathbb{R}}^n); &u \mbox{is a global weak solution of \eqref{eq1}in}L^\infty(L^2).
s = l + 2 , s^{2}= \nu^{2}+ \frac{h^{4}}{2 ( \nu^{2}- 1 )}+ \frac{( 5 \nu^{2}+ 7 ) h^{8}}{3 2 ( \nu^{2}- 1 )^{3}( \nu^{2}- 4 )}+ \cdot \cdot \cdot
\prod_{i=1}^{n}(1-a_{i}\overline{a}_{i}-b_{i}\overline{b}_{i}+a_{i}\overline{a}_{i}b_{i}\overline{b}_{i})=\prod_{i=1}^{n}(1-a_{i}\overline{a}_{i})\prod_{i=1}^{n}(1-b_{i}\overline{b}_{i})
\begin{aligned}-3F(2\pi/3)+\sum_{i=1}^{3}F(\theta_{i})\leq -3F(2\pi/3)+F(\pi/3)+\max(F(\pi/3),2F(\pi/6)) &\quad\leq \max\Big(-F(2\pi/3)-2[F(2\pi/3)-F(\pi/3)] \, , &\qquad\qquad\qquad\qquad\qquad\qquad-F(2\pi/3)+F(\pi/3)-2[F(2\pi/3) - F(\pi/6)]\Big).\end{aligned}
1-y_1\int\frac{\underline{m}^2(z)x^2dG_{y_2}(x)}{(1+x\underline{m}(z))^2}=\frac{h^2}{y_2}-\frac{y_1}{y_2}\cdot\frac{\left(1-\int\frac{y_2m_0}{t+m_0}dH(t)\right)^2}{1-\int\frac{y_2m_0^2}{(t+m_0)^2}dH(t)}.
[ \hat{x}_{1}^{\prime}, \hat{x}_{2}^{\prime}] = i \alpha^{2}, \hspace{0 . 4 c m}[ \hat{p}_{1}^{\prime}, \hat{p}_{2}^{\prime}] = i \alpha^{- 2}, \hspace{0 . 4 c m}[ \hat{x}_{i}^{\prime}, \hat{p}_{j}^{\prime}] = i \delta_{i j}.
\left\{-\hbar^{2}\frac{\partial^{2}}{\partial x^{2}}+ \hbar^{2}\frac{\partial^{2}}{\partial y^{2}}+{(8\pi\chi_{0}^{3})}^{2}\left[ x\left(x-a_{0}- \frac{\alpha}{4\pi\chi_{0}^{3}}\right) - y^{2}\right] \right\}\Psi_{1}(x,y) = 0 ~.
\begin{bmatrix}I V U I\end{bmatrix}\begin{bmatrix}\frac{\phi}{\sigma_{-}}\frac{\psi}{\sigma_{+}}\end{bmatrix}= \begin{bmatrix}-P_{-}\left( \frac{q}{\sigma{-}}\right) -P_{+}\left( \frac{q}{z^{n}\sigma_{+}}\right)\end{bmatrix}.
o_n(1)=I'_\lambda(\omega_n)\Psi_n =(I'_\lambda(\omega_n)-I'_\lambda(\omega_\lambda^+))\Psi_n=\|\Psi_n\|_X^2-\displaystyle\int_{\mathbb{R}^N}b(x)|\Psi_n(x,0)|^{2_\alpha^*}dx,
{\rm I}(\mathcal{S}_R)\geq |V\cdot \coprod\limits_{i=1}^r b_i^{n_i-1}|+1=(|V|+1)+\sum\limits_{i=1}^r (n_i-1)={\rm D}({\rm U}(\mathcal{S}_R))+\Omega(K)-\omega(K).
\begin{pmatrix}1_t -A^{-1}B) 0 1_{n-t}\end{pmatrix}\begin{pmatrix}A^{-1}0 0 (C-{}^tBA^{-1}B)^{-1}\end{pmatrix}\begin{pmatrix}1_t 0 -{}^t(A^{-1}B) 1_{n-t}\end{pmatrix}=\begin{pmatrix}(C-{}^tBA^{-1}B)^{-1}\end{pmatrix},
D_{1}(L_{n})=nL_n, D_{1}(G_n)=nG_n, D_{1}(1)=0, &D_{2}(L_{n})=0, D_{2}(G_n)=G_n, D_{2}(1)=0, &D_{3}(L_{n})=nG_{n-1}, D_{3}(G_n)=0, D_{3}(1)=0, &D_{4}(L_{n})=0, D_{4}(G_n)=L_{n+1}, D_{4}(1)=0.
u_{j+1}-u_j =-{2j\choose j}{n-1\choose j}^2\frac{(9j+6)(j+1)n^2+(12j^2-8jn-4n+14j+4)n^3}{(j+1)^3(j+2)}begin{align*}5pt] =-{2j\choose j}{n-1\choose j}{n+1\choose j+1}\frac{(9j+6)n^2}{(j+1)(n+1)}begin{align*}5pt] \quad{}-{2j\choose j}{n\choose j+1}{n+1\choose j+1}\frac{(12j^2-8jn-4n+14j+4)n^2}{(j+1)(n+1)}.
\Omega^{\alpha\beta}=v^\alpha_{~\gamma}(y)v^\beta_{~\delta}(y)\left [(\gamma_a)^{\gamma\delta}e^a(x)+R(\gamma_{ab})^{\gamma\delta}\omega^{ab}(x) + \theta^{(\gamma}{\cal D}\theta^{\delta)}+(dvv^{-1})^{\gamma\delta}\right],
\Phi_{j}^{i}=\frac{1}{2}\frac{d}{dt}\frac{\partial f^{i}}{\partial v^{j}}-\frac{\partial f^{i}}{\partial u^{j}}-\frac{1}{4}\frac{\partial f^{i}}{\partial v^{k}}\frac{\partial f^{k}}{\partial v^{j}}.
\tilde{S}_{M}= N \frac{1}{1 - \mathrm{e}^{- 4 \tilde{\beta}}}\frac{1}{2}\mathrm{t r}_{N}\left[ \tilde{M}_{1}^{2}+ \tilde{M}_{2}^{2}- 2 \mathrm{e}^{- 2 \tilde{\beta}}\tilde{M}_{1}\tilde{M}_{2}\right] + N \sum_{a = 1}^{2}\mathrm{t r}_{N}\ln ( 1 - \tilde{g}\tilde{M}_{a}) ,
\begin{array}{ll}\partial_y (\frac{u_0^1-u_0^2}{\partial_y(u_0^1 + u_0^2)})|_{y=0}= (u_0^1-u_0^2)|_{y=0}\Big([\frac{1}{(u_0^1 + u_0^2)_y}]_y + \beta\frac{1}{(u_0^1 + u_0^2)_y}\Big) =0.\end{array}
\begin{array}{c c c c}{\bar{\varepsilon}_{i}\gamma^{A}\varepsilon^{\prime i}= 0}&{\bar{\rho}_{i}\tilde{\gamma}^{A}\rho^{\prime i}= 0}&{\bar{\varepsilon}_{i}\gamma^{[ A}\tilde{\gamma}^{\mu ]}\rho^{i}= 0}&{A = 4 , 5}\\ \end{array}
L\partial_{tt}\left(\frac1N\right)-2M\partial_{ts}\left (\frac1N\right)+N\partial_{ss}\left(\frac 1N\right) &\qquad+\frac{A_{2}}N\partial_t\left(\frac 1N\right)+\frac{A_{1}}N\partial_s\left(\frac1N\right)-\frac{A_0}{N^2}=0.
\left|N(f, r)-{q \choose r}q^{k-r}\displaystyle{\left(\sum_{j=0}^{d-r}(-1)^{j}{q-r \choose j}q^{-j}\right)}\right| &\leq \sum_{j=k+1}^d{j \choose r}{\frac{q}p+m\sqrt{q}+j \choose j}{m-1 \choose d-j}\sqrt{q}^{d-j}.
\Re\frac{d\Phi_n}{dx}= 2^{-1}(n+\alpha+\beta)P_{n-1}^{(\alpha+1,\beta)}+ \Re \mu\cdot 2^{-2}(n+\alpha+\beta)_2P_{n-2}^{(\alpha+2,\beta+1)}+ \big\{<n-2\big\}.
{\omega^\mu}_{\mu\mu}(\mu,\sigma) = 0,{\omega^\mu}_{\mu\sigma}(\mu,\sigma) = \frac{2\sigma^{-3}(\delta \mu) - 0}{-\sigma^{-2}(\delta \mu)}= -\frac{2}{\sigma},{\omega^\mu}_{\sigma\sigma}(\mu,\sigma) = 0.
\left\| \mu_Y - \mu_U \right\|_2 = \left\| \frac{F_p^{\otimes k}}{p^{k/2}}\left( \mu_Y - \mu_U \right) \right\|_2 & = \frac{1}{p^{k/2}}\left\| \lambda_Y - \lambda_U \right\|_2
d_2=\begin{pmatrix}p q 0 0 r 0 0 0 0 0 0 0 0 0 0\end{pmatrix}, b_2=\begin{pmatrix}0 0 0 0 0 0 0 0 a 0 0 0\end{pmatrix},\delta_2=\begin{pmatrix}0 x y -x 0 w -y -w 0\end{pmatrix}.
R_1(\lambda;x,y) = \tfrac{i}{2\sqrt{\lambda}}[e^{i|x-y|\sqrt{\lambda}}-e^{i(|x|+|y|)\sqrt{\lambda}}], R_2(\lambda;x,y) =\tfrac{1}{2(q-i\sqrt{\lambda})}e^{i(|x|+|y|)\sqrt{\lambda}}.
\langle \tilde{\Theta}_\pm \tilde{\Theta}_\pm D_a\varphi,\tilde{\Theta}_\pm B\varphi\rangle =\pm\langle D_a\varphi,\tilde{\Theta}_\pm B\varphi\rangle &=\mp\langle D_a\varphi,B\tilde{\Theta}_\pm \varphi\rangle &=\pm \langle B\varphi,D_a\tilde{\Theta}_\pm \varphi\rangle =\langle B\varphi,\tilde{\Theta}_\pm D_a\varphi\rangle,
L(^4g,s) = \prod_{q}(1-\alpha_q^4 q^{-s})^{-1}(1-\alpha_q^2 q^{-s})^{-1}(1-q^{-s})^{-1}(1-\beta_q^2 q^{-s})^{-1}(1-\beta_q^4 q^{-s})^{-1}
P_{1,n+1}(0)^* - P_{1,n+1}^*(a) = v_{n+1}^* \big[ I - R_{n+1}^*(a) \big] \Sigma_{1, n+1}= v_{n+1}^* a T_{n+1}^* R_{n+1}^*(a) \Sigma_{1, n+1}= -a v_{n}^* L_{2, n+1}^* T_{n+1}^* R_{n+1}^*(a) \Sigma_{1, n+1}= -a v_{n}^* R_{n}^*(a) L_{1, n+1}^* \Sigma_{1, n+1}
\int_{\mathbb{R}^{d}}K(x)\frac{f(x,u)}{\vert u\vert^{g^+-1}}\vert u\vert^{g^+-1}udx =\int_{\mathbb{R}^{d}}\int_{\mathbb{R}^{d}}g\left( \frac{u(x)-u(y)}{\vert x-y\vert^{\alpha}}\right)\frac{u(x)-u(y)}{\vert x-y\vert^{\alpha +d}}dxdy +\int_{\mathbb{R}^{d}}g(u)udx.
V^{\prime}_{Q}\,=\,\frac{{\hbar}^{2}}{8M}\left\{R + \frac{(\partial_{a}f)(\partial^{a}f)}{4 M}\left[ 3 g^{ij}_{\phantom{0},0}\, g_{ij,0}-{\left( g^{ij}\, g_{ij,0}\right)}^2\right] \right\}.
[c, z_1] [z_2, z_3, z_4] + [c, z_1] [z_3, z_4, z_2] + [c, z_1] [z_4, z_2, z_3] = [c, z_1] \bigl( [z_2, z_3, z_4] + [z_3, z_4, z_2] + [z_4, z_2, z_3] \bigr) = 0.
\begin{aligned}g(\alpha) =(r_i+\lambda_g)\alpha(1-\alpha)(\alpha-\gamma) &=\frac{(r_i+\lambda_g)(2-\omega)(\omega+1)(2-\omega-3\gamma)}{27}&=\frac{(2-\omega)(\omega+1)\left(-(\omega+1)r_i+(2-\omega)\lambda_g\right)}{27}.\end{aligned}
s_a(x,\xi) :=e^{-i\varphi_a(x,\xi)}H(e^{i\varphi_a(\cdot,\xi)})\left[x\right]-h_0(\xi) =\sum_{z\in\mathbb{Z}^d}f[z] e^{i(\varphi_a(x-z,\xi)-\varphi_a(x,\xi))}+V\left[x\right]-h_0(\xi)
X_i =\{x: c(g_0^{-1}, x)=h_i^{-1}\}, X_j =\{x: c(g_1^{-1}, x)=h_j^{-1}\}, X_k =\{x: c((g_0g_1)^{-1}, x)=h_k^{-1}\}, X_s =\{x: c((g_1g_2)^{-1}, x)=h_s^{-1}\}, X_l =\{x: c(g_2^{-1}, x)=h_l^{-1}\}.
\hat{H}_{1}^{( 1 )}= \displaystyle \frac{1}{\hat{q}_{1}^{( 0 )}}, \hat{H}_{i}^{( 1 )}= \displaystyle \frac{\hat{e}_{i - 1}^{( 0 )}\hat{H}_{i - 1}^{( 1 )}+ 1}{\hat{q}_{i}^{( 0 )}}, i = 2, \dots , N.
{\cal S}_{\rm probe}= - M_1 \int d^2\sigma\left[\frac{R_9}{\sqrt{2\alpha^\prime}}-\frac{R^2_9}{2\alpha^\prime}\frac{Q_1}{r^2}+\frac{2 V}{(4\pi^2\alpha^\prime)^2}\frac{Q_1}{r^2}\right].
\partial'_{N}P(n) = \partial'_{N_0N_1}P \big( (Q'! - 1)N_0N_1 + n \big) \cdots \partial'_{N_0N_1}P (n) = \rho \big( (Q'! - 1)N_0N_1 + n \big) \cdots \rho (n) \cdot q \big( (Q'! - 1)N_0N_1 + n \big) \cdots q (n) = \rho \big( (Q'! - 1)N_0N_1 + n \big) \cdots \rho (n) \cdot q(n)^{Q'!}\gamma.
s(F^{[k]}) = \frac{1}{k}\left(\sum_{i=0}^{2k}\sum_{j=0}^{\min\{i,2\}}(-1)^{i-j}\binom{r-1+i}{r-1+j}q_1^{(i-j)}(s_j(F))\right)\left(s(F^{[k-1]})\right).
\displaystyle{\frac{2}{\kappa}}V^{^1{scalar}}_{\mu\nu}(k_1;p_1,p_2)=\,p_{_1\mu}\,p_{_2\nu}+ \,p_{_1\nu}\,p_{_2\mu}- p_1 \cdot p_2\,\eta_{\mu \nu}-{m^2}\,\eta_{\mu \nu}+2\, \xi \,\left( \,k_{_1\mu}\,k_{_1\nu}- \,k_1^{2}\,\eta_{\mu \nu}\right)
\mathbf{Y}^{0,x, \varpi, \mathbf{u}}_t = \big( X_t^{\#_{I^\varpi_1}, [(x,\mathbf{u})]_1^{\varpi}}, X_t^{\#_{I^\varpi_2}, [(x,\mathbf{u})]_2^{\varpi}}, \dots , X_t^{\#_{I^\varpi_{\#_\varpi}}, [(x,\mathbf{u})]_{\#_{\varpi}}^\varpi}, X^{0,x}_t \big)
e^{w_{\lambda}(\Pi^{-1}_{\xi_k}(\lambda z))}=e^{U_{\lambda,\xi_k}}\big[1+4\pi\lambda^2 \sum\limits_{j,j\neq k}z\nabla^2_x G(\Pi^{-1}_{\xi_k}(x),\xi_j)\vert_{x=0}z^T
\sum_{i}e^{i}(R(Z, e_{i})Z + R(Y, e_{i})Y) = - \sum_{i}(f^{i}(R(\phi e_{i}, Y)Y) + e^{i}(R(e_{i},Y)Y))+ (n-1) g(Y,Y) &+ (-n+1) g(Y,Y) + n \eta (Y)^{2}- n f^{2}= -{\rm Ric}(Y, Y) + g(Y,Y) + n \eta (Y)^{2}- n f^{2}.
C_1(0) F(1,1,6;1,3,4;p)= F(2,1,5;1,3,4;p) , C_2(0) F(2,1,5;1,3,4;p)= F(2,2,4;1,3,4;p) , C_2(1) F(2,2,4;1,3,4;p)= F(2,3,3;1,3,4;p) .
B = ( - t_{1}+ c +\alpha z) C_{1}+ t_{2}(\lambda +\gamma t_{2}z^{\lambda})C_{2}-\frac{z}{2}( \lambda + 1 +2\gamma t_{2}z^{\lambda})D -\gamma z^{\lambda +2}E,\ \lambda \in \mathbb{Z}_{\geq 1}B= ( -t_{1}+ c+\alpha z) C_{1}+(\lambda t_{2}+\gamma z^{-\lambda}) C_{2}-\frac{z}{2}(\lambda +1)D,\ \lambda \in{\mathbb Z}_{\leq -1},
\prod_{i = 1}^{n - 1}\left[ \int d^{D}x | f_{i}( x ) |^{q q_{1}/ q}\right]^{\frac{1}{q q_{1}/ q}}\; \left[ \int d^{D}x | f_{n}( x ) |^{p}\right]^{\frac{1}{p}}\; = \; \prod_{i = 1}^{n}\| f_{i}\|_{q_{i}}\; .
\sigma_k(\lambda)=\sigma_k(\lambda|i)+\lambda_i\sigma_{k-1}(\lambda|i), \forall \,1\leq i\leq n, &\sum_i \lambda_i\sigma_{k-1}(\lambda|i)=k\sigma_{k}(\lambda), &\sum_i\sigma_{k}(\lambda|i)=(n-k)\sigma_{k}(\lambda).
H^{\prime}\equiv - i \frac{3 \hbar c}{2}\gamma^{0}\gamma^{a}( - \sigma^{- 1})_{, a}= - i \frac{3 \hbar c}{2}\left( \begin{matrix}{{\epsilon_{0}}}&{{\vec{\sigma}\cdot \vec{\epsilon}}}\\{{\vec{\sigma}\cdot \vec{\epsilon}}}&{{\epsilon_{0}}}\\ \end{matrix}\right) \,{,}
\|\bar{u}_\ell\|_{C^{k+2,\alpha}_{\nu_{\ell,j}+2}(U_\ell)}+ \sum_{i=0}^{I_{\ell,j}}|A_{\ell,i}| \leq c c_j\left(\|\bar{f}\|_{C^{k,\alpha}_\nu(M)}+ |f_\ell|\right).
\Gamma^{kln}_{\mu \nu \lambda}(p,q,r)= -(2 \pi)^{4}\left( \frac{2}{g'}C^{n}_{kl}\right)\left[\delta_{\mu \lambda}(r + q - p)_{\nu}+\delta_{\nu \lambda}(q - r)_{\mu}+ \delta_{\mu \nu}(p - q -r)_{\lambda}\right] \, .
f(\omega)= \begin{cases}\left(\frac{j_M^0}{j_M^*}\right)^{1/3}\omega^{-7/6}\left(1+O(\epsilon \omega^{\delta})\right) \mbox{for}\omega\leq 1, \left(\frac{j_M^\infty}{j_M^*}\right)^{1/3}\omega^{-7/6}\left(1+O(\epsilon \omega^{-\delta})\right) \mbox{for}\omega> 1, \end{cases}
\partial_{-}( \partial_{+}\Gamma^{( 1 )}\, \Gamma^{( 1 ) - 1}) = - \Gamma^{( 2 )}\, \Gamma^{( 1 ) - 1}, \qquad \partial_{-}( \partial_{+}\Gamma^{( 2 )}\, \Gamma^{( 2 ) - 1}) = \Gamma^{( 2 )}\, \Gamma^{( 1 ) - 1}.
\sum^r_{i=0}\bigg[\bigg(-\frac{1}{2}\frac{d}{dt}-\frac{i+s}{2}\tanh t +i \coth t\bigg)c_i(t)+(r-i+1)(\sinh t)^{-1}c_{i-1}(t)\bigg]f^{r+1,s-2}_i\otimes f^{\Lambda^\vee}_{r-i}=0,
\lim_{t\rightarrow x^+}\chi_{\mathcal{E}}(t)=\lim_{t\rightarrow x^+}t+\frac{1-e^{-C(t)}}{C'(t)}=x &\lim_{t\rightarrow x^+}\eta_{\mathcal{E}}(t)=\lim_{t\rightarrow x}1+\frac{e^{C(t)}+e^{-C(t)}-2}{C'(t)}=1.
\frac{(p+r+q+s)!}{(p+r)!(q+s)!}{\rm Alt}(\omega_1\omega_2)=(-1)^{qr}\Big(\frac{(p+q)!}{p!q!}{\rm Alt}\omega_1\Big)\wedge \Big(\frac{(r+s)!}{r!s!}{\rm Alt}\omega_2\Big).
\Big(\frac D2\Big)^p(1-r^{-p})\Big[\frac1{r^{d/2-p}-1}+\frac1{1-r^{-p}}\Big]\Big(\frac{\sqrt A r^{d}}{2(r^{d/2}-1)}\Big)^{2p/d}=\frac{D^pA^{p/d}r^p(r^{d/2}-1)^{1-2p/d}}{2^{p+2p/d}(r^{d/2-p}-1)}=\kappa_{d,p}.
R_{us_j,vs_j}^{J,q}(q)= (-1)^{\ell(vs_j)-\ell(us_j)}(1-q)^2\prod_{k\in D(us_j,vs_j)}\left(1-q^{a_k(us_j,vs_j)}\right) begin{align*}5pt]= (-1)^{\ell(v)-\ell(u)}(1-q)^2\frac{1-q^{a_j(us_j,vs_j)}}{1-q^{a_{j+1}(u,v)}}\prod_{k\in D(u,v)}\left(1-q^{a_k(u,v)}\right).
{\bf E}^{\tau p}={\frac{1}{2}}\left( \begin{array}{c c}{{a_{1}^{\tau p}}^{\dagger}}{{a_{2}^{\tau p}}^{\dagger}}\end{array}\right){\bf \sigma}\left( \begin{array}{c}{a_{1}^{\tau p}}{a_{2}^{\tau p}}\end{array}\right)
\pi^n_k(a)(\eta_1 \otimes \cdots \otimes \eta_k) = a(n - k) \eta_1 \otimes \cdots \otimes \eta_k \intertext{and}\tau^n_k(\xi)(\eta_1 \otimes \cdots \otimes \eta_k) = \xi(n - k) \otimes \eta_1 \otimes \cdots \otimes \eta_k.
\dot{\tilde{R}}~~ = \tilde{R}(- k_\beta\psi_\omega +\hat{R}\tilde{b}_\omega )^\times , \dot{\tilde{p}}~~ = \tilde{R}(-k_\beta\psi_v+\hat{p}^\times \hat{R}\tilde{b}_\omega + \hat{R}\tilde{b}_v ), \dot{\psi}_{\tilde{R}}= E(\tilde{R}) (- k_\beta\psi_\omega +\hat{R}\tilde{b}_\omega),
c_0 = e^{i \phi_0}\left[ 1 + a_{-1}a_1 +\frac{1}{2\sqrt{2}}a_{-1}^2a_{2}+\frac{1}{2\sqrt{2}}a_{-2}a_{1}^2 +\frac{1}{2}a_{-2}a_{2}+\cdots \right].
E_{z}^{a}= \frac{N ( z , t )}{f^{\prime}( z , t )}\left( \begin{array}{c}{{f}}\\{{1}}\\{{f^{2}}}\\ \end{array}\right) \; , \; \; E_{u}^{a}= \frac{n ( u , t )}{\dot{f}( u , t )}\left( \begin{array}{c}{{f}}\\{{1}}\\{{f^{2}}}\\ \end{array}\right)
\mathcal{A}^T_{1,n}=P_n\mathcal{A}^T_1 P_n\quad\qquad\qquad\qquad\mathcal{A}^T_{2,n}=Q_n\mathcal{A}_2^T Q_n \mathcal{B}^T_n=P_n\mathcal{B}^T Q_n \mathcal{C}^T_n=P_n\mathcal{C}^T\qquad\mathcal{D}^T_n=Q_n\mathcal{D}^T.
\Omega := A^{-N-1,N}(r_0,R_0)\times [\lambda_0,\lambda_0+1] := \big \{(y,z)\in E^{-N,N}\times E_{-N-1}\ \big |\r_0\leq \|y\|^2 + \|z\|^2 \leq R_0\big \}\times [\lambda_0,\lambda_0+1]
\begin{array}{lll}2g(H(x,y),z) = g(T(x,y),z)+g(T(z,x),y)+g(T(z,y),x) = 2g(\alpha(g(x,y)\xi-\eta(y)x)+\beta(g(\varphi x,y)\xi-\eta(y)\varphi x),z).\end{array}
cos\frac{2\pi}{21}+cos(2\cdot\frac{2\pi}{21})+cos(4\cdot\frac{2\pi}{21})+cos(5\cdot\frac{2\pi}{21})+cos(8\cdot\frac{2\pi}{21})+cos(10\cdot\frac{2\pi}{21})=\frac{1}{2}.
c_m:=\frac{e^{-2\beta\|J\|_{\infty}}}{e^{2\beta\|J\|_{\infty}}+e^{-2\beta\|J\|_{\infty}}}\leq c(x,\sigma)\leq \frac{e^{2\beta\|J\|_{\infty}}}{e^{2\beta\|J\|_{\infty}}+e^{-2\beta\|J\|_{\infty}}}=:c_M.
\begin{cases}\lim_{n\to \infty}\underbar r_1^n=\underbar r_1,\lim_{n\to \infty}\bar r_1^n=\bar r_1, \cr \lim_{n\to \infty}\underbar r_2^n=\underbar r_2, \lim_{n\to \infty}\bar r_2^n=\bar r_2. \end{cases}
C_2(\nu)\|e^{\nu t\Delta}u_0\|_{\widetilde{L}^1_{T'}(\dot{B}^{s+1}_{2,2})}+T'\big(M_0^{\frac{1}2}+C_2(\nu)M_0\big) &\quad+C_2(\nu)(T')^{\frac{r-2}{2r}}(2C_1(\nu)M_0+32C_2M^4_0)<\frac{1}{2C_3}.
\begin{cases}\bar{\mu}_\varepsilon^{\delta_n}\to \bar{\mu}_\varepsilon \mbox{weakly-$ * $ in $ L^\infty(0, T; H) $,}\bar{\mu}_\varepsilon^{\delta_n}(t) \to \bar{\mu}_\varepsilon(t) \mbox{in $ H $,}\mbox{for a.e. $ t \in (0, T) $,}\end{cases}\mbox{as $ n \to \infty $,}
g_S(s) = \int_{\mathbb{R}}\cdots \int_{\mathbb{R}}\left[ \prod_{j=2}^{d}h_j(x_{j-1}) \right] h_1 \left( s - \sum_{j=1}^{d-1}x_j \right) \, \d x_1 \cdots \, \d x_{r-1},
Q_{\alpha}^{( 1 )}= - \frac{i}{2}\, [ A_{\mu}, A_{\nu}]^{i j}\Gamma_{\alpha \beta}^{\mu \nu}\, \frac{\partial}{\partial \bar{\psi}_{\beta}^{i j}}+ i \Gamma_{\mu \alpha \beta}\psi_{\beta}^{i j}\, \frac{\partial}{\partial A_{\mu}^{i j}}
\mathcal{V}_2\left( \frac{m}{p_2 (p_1q)^2}\right) = \frac{p_2^{1/4}(p_1q)^{1/2}}{m^{1/4}}\frac{N^{3/4}}{N^{i(t+\nu)}}\,\sum_{\pm}\mathcal{I}^{\pm}(m,u,q,p_1,p_2),
\frac12=\int_0^{\infty}\varphi^G_{\zeta}(u){\rm d}u \ge \varphi^G_{\zeta}(0)\int_{0}^{u_0}e^{-u\varphi^G_{\zeta}(0)}{\rm d}u =\int_{0}^{u_0 \varphi^G_{\zeta}(0)}e^{-u}{\rm d}u=1-e^{-u_0\varphi^G_{\zeta}(0)},
(\dfrac{f+W^{2}}{W^{2}})_{k}=\dfrac{f_{k}}{W^{2}}-\dfrac{f}{W^{4}}(f_{k}+2\hat{u}^{l}\hat{u}_{l;k})=\dfrac{1}{W^{2}}(f_{k}+2\dfrac{f\eta_{k}}{\eta}).
I_2(x,t):=\dfrac{1}{(2\pi)^d\, t^{d/\ell}}\int\limits_{R_{\pi t^{1/\ell}}\setminus R_{\pi t^{1/L}}}(-1)^J \left(A(t^{-1/\ell}\xi)\right)^J e^{-tA(t^{-1/\ell}\xi)}\cdot e^{ix t^{-1/\ell}\xi}\, d\xi,
\frac{d^{2}a}{dt^{2}}_{|_{t=t_{}}}=\frac{3\; a\left(t_{}\right)}{2t_{}^{2}\left(2t_{}^{2}+3\right)}\left[\left(4\; a\left(t_{}\right)^{2}-2\right)t_{}^{2}+3\right].
I_{N}( p , \{\nu \}) = c_{N \ell}\Delta^{( N - 1 ) ( 2 \ell + 1 ) - 2 \sum_{i}\nu_{i}}( \prod_{i}m_{i}^{\ell - \nu_{i}- 1}) / ( 4 \pi )^{\ell ( N - 1 )}\sigma^{( 2 - N ) \ell - N + \sum_{i}\nu_{i}},
\hat{H}_{AF}=\hbar[(g_{c}\hat{\sigma}_{+}\hat{a}_{c}-g_{c}^{*}\hat{\sigma}_{-}\hat{a}_{c}^{\dagger})+(-g_{c}\hat{\sigma}_{+}\hat{a}_{c}^{\dagger}+g_{c}^{*}\hat{\sigma}_{-}\hat{a}_{c})]
\Vert e^{-\mathcal{A}_{D,\varepsilon}t}-e^{-\mathcal{A}_D^0t}\Vert_{L_2(\mathcal{O})\rightarrow L_2(\mathcal{O})}\leqslant4\pi^{-1/2}\widehat{C}_{12}\varepsilon t^{-1/2}e^{-c_*t/2}&\leqslant \check{C}_{12}\varepsilon (t+\varepsilon^2)^{-1/2}e^{-c_*t/2},t\geqslant \varepsilon^2,
({\gamma}^{\mu}i{\partial}_{\mu}- m_1){\psi}_1(x,t) -\frac{1}{2}e_1 b(t){\gamma}^1{\psi}_1(x,t) +e_1 \int_{0}^{\rm L}dy D(x,y|{\rm L}) J^0(y,t){\gamma}^0{\psi}_1(x,t) =0,
4\Big( \frac32u_3 - 2 u_2 + \frac12 u_1 , u_3 \Big) = |u_3|^2-|u_2|^2 + |2u_3-u_2|^2 - |2u_2-u_1|^2 + |u_3-2u_2+u_1|^2,
{\rm res}(\tilde{r},a)=\frac{1}{z-\bar{a}}u(z)|_{z=a}=\frac1{a-\bar{a}}u(a)=\overline{\frac1{\bar{a}-a}\overline{u(a)}}=\overline{\frac{1}{\bar{a}-a}{u(\bar{a})}}=\overline{{\rm res}(\tilde{r},\overline{a})}.
\begin{aligned}R_{\mathrm{c},k}=\log_{2}\left(1+\frac{\left|h_{k}\right|^{2}P_{\mathrm{c}}}{\left|h_{k}\right|^{2}P_1+\left|h_{k}\right|^{2}P_2+1}\right), R_{k}=\log_{2}\left(1+\frac{\left|h_{k}\right|^{2}P_k}{\left|h_{k}\right|^{2}P_j+1}\right),\end{aligned}
\begin{array}{l}\Psi(u)=R({\bf B}_{f})\Psi_{f}(u_{f}), \quad\bar{\Psi}(u)=\bar{\Psi}_{f}(u_{f})\widetilde{R}^{+}({\bf B}_{f}), g(u)\nabla\Psi(u)=S(B_{f})R({\bf B}_{f})\gamma_{f}D\Psi_{f}(u_{f}), \left( \nabla\bar{\Psi}(u)\right)g(u)=S(B_{f})\left( D\bar{\Psi}_{f}(u_{f})\right)\gamma_{f}\widetilde{R}^{+}({\bf B}_{f}).\end{array}
\begin{aligned}\dot x(t) =Ax(t)+B_1w(t)+B_2K[C_2x(t_k-\eta_k)+D_2v(t_k-\eta_k)], z(t) =C_1x(t)+D_1K[C_2x(t_k-\eta_k)+D_2v(t_k-\eta_k)].\end{aligned}
\overset{1a}{\Delta}_{m+1}B =T^{tt}(\alpha_{+,m+1},\beta_{+,m+1})-T^{tt}(\alpha_{+,m},\beta_{+,m+1}), \overset{1b}{\Delta}_{m+1}B =T^{tt}(\alpha_{+,m},\beta_{+,m+1})-T^{tt}(\alpha_{+,m},\beta_{+,m}).
\mathcal{R}^n_t (\varphi) = \frac{1}{\ell}\int_0^t \sum_{j \in \mathbb{Z}}\bigg[ - \sum_{i=1}^{\ell-1}(W^2_{j+i}-W^2_{j+i-1})\psi_i - W_j (W_j - W_{j-1}) + \sigma_n^2(\rho) \bigg] \varphi^n_j (s )ds .
(-\underbar{q}, -\underbar{p}) \cdot ( \dot{p}, \dot{q}) =-\underbar{p}\underbar{q}(1-\underbar{q}) - p \frac{\underbar{q}}{m}\Big( \frac{2m}{1+m}- \frac{m}{\lambda}\big( 1-\frac{2}{1+m}\big) - \underbar{q}\Big) \ge -\underbar{p}\underbar{q}(1-\underbar{q}) =: \delta >0.
d_{\beta}d^c_{\beta}F = -\frac{1}{2}\Big(\Delta_{g_0}u + \big\langle 2t_0\alpha_0-\theta_{g_0}, du \big\rangle_{g_0}+ t_0(1-t_0)|\alpha_0|_{g_0}^2 u \Big) e^{t_0f}\, F_0 \wedge F_0.
s_{2i+1}-s_i = \frac{\omega^{a_{h(2i+1)}(2i+1)}}{2^{h(2i+1)}}\Biggl[z^{h(2i+1)}+ \frac{1}{2}\left(\sum_{j=0}^{k-h(2i+1)-1}z^j\right) (z^{h(2i+1)+1}-1) \quad\qquad\qquad\qquad\qquad-\omega^{-1}\left(\sum_{j=0}^{k-h(2i+1)}z^j\right) (z^{h(2i+1)}-1)\biggr].
(f_i\circ f_1)\star f_j+(-1)^{(m_1-1)m_j}f_2\star(f_3\circ f_1)-(-1)^{(m_1-1)(m_j-1)}(f_i\star f_j)\circ f_1 = (-1)^{(m_1+m_i)(m_j-1)+(m_1-1)(m_i-1)}f_2\star_0(f_1\star_0 f_j)-f_i\star_1(f_1\star_1 f_j).
\begin{bmatrix}0 -C_1 C_2 \end{bmatrix}^t \,\,=\,\, \bigl[\,0 \,,\,-c_{001},-c_{010},-c_{100},\,\,c_{002},\,c_{011},\,c_{020},\,c_{101},\,c_{110},\,c_{200}\, \bigr]^t .
A_{d}(p^{2}+\omega_{n}^{2})^{\frac{1}{2}d-2}\left[ \left( 1+v^{2}\right)^{\frac{1}{2}d-\frac{3}{2}}+\frac{\Gamma (\frac{1}{2}d-\frac{1}{2})}{\sqrt{\pi}\Gamma (d)}\frac{(v^{2})^{\frac{d}{2}-1}}{1+v^{2}}{}_{2}F_{1}\left(\textstyle{{\frac{1}{2}d-\frac{1}{2},1;\frac{1}{2}d;\frac{v^{2}}{1+v^{2}}}}\right) \right] \,\ \cdot
V = \frac{1}{\sqrt{q^{10}+q^6+q^4+1}}\Biggl( \begin{matrix}0 0 0 q 0 0 q^4+q^2-1 0 0 q^5-q^3-q 0 0 - q^4 0 0 0 \end{matrix}\, \Biggr) \,.
\pi^n(a) \cdot b = \iota_n(\pi^n(a))(b) = \pi^{n+1}(a)(b) = a(n+1)(b) = 0, \tau^n(\xi) \cdot b = \iota_n(\tau^n(\xi)) \tau^{n+1}(\xi)(b) = \xi(n+1)b = 0, \tau^n(\xi)^* \cdot b = \iota_n(\tau^n(\xi))^*(b) = \tau^{n+1}(\xi)^*(b) = 0.
\Psi(s) = \frac{\int_0^{L_m^*}\frac{|\phi'|^2 V_{\gamma_m}}{\sigma_s}+ \frac{k^2 |\phi|^2 \sigma_s}{V_{\gamma_m}}\,dt}{\int_0^{L_m^*}|\phi|^2 V_{\gamma_m}\sigma_s \,dt}
\Phi(s,x,t)=\frac{-x}{\sqrt{2\pi(t-s)^3}}e^{-\frac{x^2}{2(t-s)}}\left( \phi'(s)x-\frac{1}{2}\int_{s}^{t}(\phi'(u))^2 du \right) \times \mathbb{E}^{(s,-x) \rightarrow (t,0)}\left[ \left (-\int_{s}^{t}\phi''(u)B^{br}(u)du \right)\right]
\frac{1}{2}\| u (\cdot, \cdot, t) \|_{L^2}^2 + \alpha \int_0^t \| u_y (\cdot, \cdot, s) \|_{L^2}^2 ds= \frac{1}{2}\| u_0 \|_{L^2}^2 + \beta \int_0^t \| u (\cdot, \cdot, s) \|_{L^2}^2 d\tau
[m]^2\sum_{n=1}^{\frac{m-1}{2}}[3n]\frac{(1-q)^2(q;q^2)_{n}(q^{m+2};q^2)_{n-1}^2 q^{m-{n+1\choose 2}-\frac{(2n+1)(m-1)}{2}}}{(q;q)_{n}^2 (q^2;q^2)_{n}},
\left(\frac{\partial \phi^*(y^k)}{\partial x^i}\Big|_{m}\right)_{i,k=1, \ldots,r}\mbox{of the diagonal block $B_0$}\mbox{and}\left(\frac{\partial \phi^*(\theta_{\mu}^a)}{\partial \xi_{\mu}^b}\Big|_{m}\right)_{a,b=1, \ldots,s_{\mu}}\mbox{of the diagonal blocks $B_{\mu}$}\;,
\lambda^{*}(\alpha) := \inf \left \{\frac{\int_{\Omega}|\nabla u|^{p}+|u|^{p}dx}{\int_{\partial \Omega}|u|^{p}\, d\mu^*}\colon u\in W^{1,p}(\Omega),\ \mu^*(\{u=0\}\cap \partial\Omega) \ge \alpha \mu^*(\partial\Omega)\right\},
[e_1, e_2]=\alpha_2e_4+ \alpha_3e_5, [e_2, e_1]=-\alpha_2e_4+\alpha_4e_5, [e_3, e_1]=\beta_3e_5, [e_3, e_2]=\beta_6e_5, [e_3, e_3]=\beta_7e_5.
L_{1}^{\pm}(v)^{-1}=\begin{pmatrix}* * y^{\pm}I * * -e_{2}^{\pm}(v)k_{3}^{\pm}(v)^{-1}I x^{\pm}I -k_{3}^{\pm}(v)^{-1}f_{2}^{\pm}(v)I k_{3}^{\pm}(v)^{-1}I\end{pmatrix}
0 = [[\beta_{-1}, \beta_{-m+1}],\beta_n] +(m+n)(m-2) \beta_{n-m}+ \sum_{j=1}^{m-2}j \alpha_{n-j}\alpha_{-m+1+j}\beta_{-1}-\sum_{j=1}^{m-2}j \alpha_{n-j}\beta_{-1}\alpha_{-m+1+j}&-\sum_{j=1}^{m-2}j(n-j) \alpha_{n-j-1}\alpha_{-m+1+j}+ \sum_{j=1}^{m-2}(n+1)j \alpha_{n-1-j}\alpha_{-m+1+j}.
-\frac12\Delta_B|\phi|^2= \sum_a\{|\nabla_{\bar V_a}\phi|^2 +|\nabla_{V_a}\phi|^2\}+\langle \phi,\sum_aR^Q(\bar V_a,V_a)\phi\rangle +\frac12\sum_a\{\langle\omega^a\wedge (\nabla_{V_a}H^{1,0})\lrcorner\,\phi,\phi\rangle +\langle\phi,\omega^a\wedge (\nabla_{V_a}H^{1,0})\lrcorner\,\phi\rangle\}.
\Sigma'(f,g):=E \left\{\frac{Ew(X_{2})-\theta w_{\theta}(X_{1})}{\theta w_{\theta}(X_{1})}\left[f(Y_{1})-R(f)\frac{\theta w_{\theta}(X_{1})}{Ew(X_{2})}\right]\times\right. &\quad\left.\times\left[g(Y_{1})-R(g)\frac{\theta w_{\theta}(X_{1})}{Ew(X_{2})}\right]\right\},f,g\in\mathcal{F},
P_A(z)= \sum_{i=0}^{C}\binom{A+B+C+1}{i}\binom{A+C-i}{A}(-z)^i, Q_A(z)= (-1)^{C}\sum_{i=0}^{A}\binom{A+C-i}{C}\binom{B+i}{i}(z)^i, E_A(z)= \sum_{i=0}^{B}\binom{A+i}{i}\binom{A+B+C+1}{A+C+i+1}(-z)^i.
\wp(p|\tau)=\frac{2g_{3}(C-\tau)^{3}-4(C\eta_{1}-\eta_{2})^{3}-g_{2}(C\eta_{1}-\eta_{2})(C-\tau)^{2}}{(C-\tau)[12(C\eta_{1}-\eta_{2})^{2}-g_{2}(C-\tau)^{2}]},