harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
e^{- 2 \hat{\phi}}= 1 +{\frac{2 \tilde{m}}{r}}\ , \qquad K = 1 + \sum_{1}^{s}{\frac{2 \hat{m}}{r}}\ , \qquad A_{n}={\frac{2 q_{n}}{r}}\ , \qquad n = 4 , \dots 8 , \qquad r^{2}\equiv \vec{x}^{2}.
\begin{cases}\widehat{u}_{ttt}+(\sigma+|\xi|^2)\widehat{u}_{tt}+2|\xi|^4\widehat{u}_t+(\sigma+|\xi|^2)|\xi|^4\widehat{u}=0, (\widehat{u},\widehat{u}_t,\widehat{u}_{tt})(0,\xi)=(\widehat{u}_0,\widehat{u}_1,\widehat{u}_2)(\xi),\end{cases}
p_{N{\mathrm{(int)}}}^{(1)}(\xi_1,\xi_2)=\frac{A_d}{2\xi_1^2}\int_{0}^{\infty}dk k^{d-2}\int_{0}^{\infty}d\omega \frac{K'_{\omega}(k\xi_2)(1+\omega^2/k^2\xi_1^2)}{K'_{\omega}(k\xi_1)N'_{\omega}(k\xi_1,k\xi_2)}.
C_{0 , B}= \frac 1 2 \dot{\phi}^{2}+ \frac{m_{B}^{2}}2 \phi^{2}+ \frac{\lambda_{B}}8 \phi^{4}+ \frac{\lambda_{B}}{3 2 \pi^{2}}\int_{0}^{\Lambda}\, \coth \frac{\beta W_{k}}2 \Biggl [ | \phi \dot{\varphi}_{k}- \dot{\phi}\varphi_{k}|^{2}+ k^{2}| \varphi_{k}|^{2}\phi^{2}\Biggr ] \, d k \ ,
f(x) = \log(x+2) + 2, \sigma(x,t) = \log(x-t+2) + \log(-x-t+2) + 4, \mu(x,t) =\log(x-t+2) - \log(-x-t+2) , \sigma(x,0) =\log (2-x)+\log (x+2)+4, \mu(x,0) =\log (x+2)-\log (2-x).
\delta S = \displaystyle \int d^{4}x \, v a r e p s i l o n^{\alpha}\displaystyle{\Biggl (}- 2 i ( \sigma^{\nu}{\bar{\sigma}}^{\mu})_{\alpha}{}^{\beta}D_{\beta}\partial_{\mu}{\widetilde T}_{\nu}+ \frac{i}{3}\sigma^{\mu}{}_{\alpha}{}^{\dot{\beta}}{\bar{D}}_{\dot{\beta}}\partial_{\mu}{\kern . 3 5 e m \bar{\kern - . 3 5 e m{\widetilde J}}}\displaystyle{\Biggr )}\displaystyle{\biggl \vert}\ .
{d\over{dt}}(L^n)_{\mu\nu}=i[{\cal H},(L^n)_{\mu\nu}] =[L^n,M]_{\mu\nu}=\sum_{\lambda\in{\cal R}}\left(\phantom{\mbox{\huge H}}\hspace{-15pt}(L^n)_{\mu\lambda}M_{\lambda\nu}- M_{\mu\lambda}(L^n)_{\lambda\nu}\right),\quad n=1,\ldots .
| h_{3}\rangle = ({\bf P}^{\bar{z}})^{k}\sum_{n = 0}^{k}\rho^{n}\frac{\Gamma ( \frac{D}{2}+ k - n - 2 )}{2^{n}n ! \Gamma ( \frac{D}{2}+ k - 2 )}({\bf M}^{\bar{z}j}{\bf M}^{\bar{z}j})^{n}V_{0}| 0 \rangle_{1}| 0 \rangle_{2}| 0 \rangle_{3},
{\cal A}= \left(- \frac{3}{4}(\nabla^2 X^\mu \nabla^2 X^\nu G_{\mu \nu}- 4 g^{\alpha \beta}P_{\alpha \beta}) - \frac{1}{2}R_{(2)}- \frac{1}{6}g^{\alpha \gamma}g^{\beta \delta}W_{\alpha \beta \gamma \delta}\right) \phi - \frac{5}{6}\nabla^2 X^\mu D_\mu \phi .
\sum_{v}\sum_{\substack{T_1\cup T_2\in \mathbb{F}^*_2 w=r(T_1), v=r(T_2)}}d_wd_v\omega(T_1\cup T_2) \geq \sum_{v}\sum_{\substack{T_1\cup T_2\in \mathbb{F}^*_2 w=r(T_1), v=r(T_2)}}\sum_{e\in E(r(T_2), T_1)}d_w\omega(T_1\cup T_2 \cup e) = (n-1)d_w\tau_w,
\Vert (a_i^{\Vert}, \phi_i^{\Vert}) \Vert_{L^{2,n}_{\nu + n}(C)}+ \Vert (a_i^{\perp}, \phi_i^{\perp}) \Vert_{L^{2,n}(C)}= \Vert (a_i , \phi_i) \Vert_{H_{n,\nu + n}(C)}\lesssim m_i^{-7/4},
P^{\beta,\epsilon}_{N,\omega}(d\varphi_{1},\dots,d\varphi_{N-2}):= \frac{e^{-H_{N,\omega}(\varphi)}}{Z^{\beta,\epsilon}_{N,\omega}}\prod_{i=1}^{N-2}\limits ( \epsilon \delta_{0}(d\varphi_{i}) +d\varphi_{i})
(L(v_{n+1}-v_{n}), v_{n+1})_{H}= (L^{1/2}(v_{n+1}-v_{n}), L^{1/2}v_{n+1})_{H}=\frac{1}{2}\|L^{1/2}v_{n+1}\|_{H}^2 - \frac{1}{2}\|L^{1/2}v_{n}\|_{H}^2 + \frac{1}{2}\|L^{1/2}(v_{n+1}-v_{n})\|_{H}^2
\int [ d \delta{\hat{g}}] e^{- | \delta{\hat{g}}|^{2}/ 2}\equiv \prod_{\sigma}\int [ d \delta{\hat{g}}]_{\sigma}e^{- | \delta{\hat{g}}|_{\sigma}^{2}/ 2}= 1 = J_{M}({\cal \phi},{\hat{g}}) \int [ d \delta{\cal \phi}]_{e^{\phi}{\hat{g}}}\int [ d \delta \eta ]_{\hat{g}}^{\prime}\int_{0}^{\infty}d l e^{- | \delta{\hat{g}}|^{2}/ 2}\quad ,
\mathcal{L}_k=\{\ell\ge 1: p_\ell<p_k\}\quad\textrm{and}G_k=\left[\sum_{\ell\in\mathcal{L}_k}2^{-\ell},2^{-k}+\sum_{\ell\in\mathcal{L}_k}2^{-\ell}\right].
\Big\|\sum_{k=1}^l\theta_k|a_k|u_k\Big\|\geq \langle\sum_{i=1}^n\frac{1}{\sqrt{\sharp s_i}}\sum_{j\in s_i}\theta_j u_j^*,\sum_{k=1}^l\theta_k|a_k|u_k\rangle= \sum_{i=1}^n\frac{1}{\sqrt{\sharp s_i}}\sum_{j\in s_i\cap\{1,\ldots,l\}}|a_j|\geq \geq \langle\phi,\sum_{k=1}^l a_ku_k\rangle.
\hat{\mathbf{h}}_{eff,2}^{(g)}=\left(\mathbf{X}^{(g)}\otimes \mathbf{SNR}^{total,(g)}_{mimo}\right)^H \left(\mathbf{R}^{(g)}_{code}\otimes\mathbf{SNR}^{total,(g)}_{mimo}+\mathbf{I}_{TD}\right)^{-1}\mathbf{y}^{(g)}
{\gamma_1'}^{\frac{n}{2}}{\gamma_2'}^{\frac{n}{2}-1}+{\gamma_1'}^{\frac{n}{2}-1}{\gamma_2'}^{\frac{n}{2}},\, \textrm{for}n\textrm{even}{\gamma_1'}^{\frac{n-1}{2}}{\gamma_2'}^{\frac{n-1}{2}},\, \textrm{for}n\textrm{odd.}
\pi{}^{(\alpha)}=( u_{i}^{(\alpha)},\theta{}^{(\alpha)},\varphi{}^{(\alpha)},e_{ij}^{(\alpha)},\kappa_{ijk}^{(\alpha)},\beta_{i}^{(\alpha)},E_{i}^{(\alpha)},V_{ij}^{(\alpha)}, \tau_{ij}^{(\alpha)},\mu_{ijk}^{(\alpha)},\sigma_{i}^{(\alpha)},Q_{ji}^{(\alpha)},\eta^{(\alpha)},q_{i}^{(\alpha)})\in\mathcal{K}.
\Big(\frac{1}{i}\partial_t-\mathcal{L}\Big)|\tilde{\psi}_1^\mathrm{r}\rangle=-\mathbb{H}|\tilde{\psi}_1^\mathrm{a}\rangle|\tilde{\psi}_1^\mathrm{r}(0)\rangle=0, &\Big(\frac{1}{i}\partial_t-\mathcal{L}\Big)|\tilde{\psi}_1^\mathrm{s}\rangle=-\tilde{\mathbb{H}}|\tilde{\psi}_1^\mathrm{a}\rangle|\tilde{\psi}_1^\mathrm{s}(0)\rangle=0
Q(n,u) =h_u^{s,(n-1)}&+ \sum_{i=1}^{n-2}\int_s^u \cdots \int_s^{v_{n-i+1}^*}\Phi(u-v_{n-1}) \prod_{k=n-i+1}^{n-1}\Phi(v_k-v_{k-1}) h_{v_{n-i}}^{s,(n-i-1)}dv_{n-i}\ldots dv_{n-1}&+ \mu \int_s^u \int_s^{v_{n-1}}\cdots \int_s^{v_2}\Phi(u-v_{n-1}) \prod_{k=2}^{n-1}\Phi(v_k-v_{k-1}) dv_1\ldots dv_{n-1},
F_{k,m}(z) = \int_{-1/2}^{1/2}\frac{(S_4^\ell \psi F_{k'})(z)}{(S_4^\ell \psi F_{k'})(\tau)}\frac{\psi(\tau)F_2(\tau)}{\psi(\tau)-\psi(z)}e^{2\pi i m \tau}du,
\Psi_{K_Y}(Y,T)=e^{iK_Y Y-i \sqrt{K_Y^2+m^2}T}=\int_{-\infty}^{\infty}\frac{dk_y}{2 \pi}e^{\pi k_y\over 2 H_5}H^{(2)}_{i k_y/H_5}(mt)e^{ik_y (y-H_5^{-1}\sinh^{-1}(\frac{K_Y}{m}))}
\|\phi\|_{Z}^2 \ =\ \sum_k \| P_k\phi \|^2_{Z_k}\ , \ \ \| \phi\|_{Z_k}^2 \ =\ \sup_{l<C}\sum_{\omega}2^{l}\|P^\omega_lQ_{k+2l}\phi \|_{L^1(L^\infty)}^2 \ .
\sigma_{0}:= \dfrac{4}{3(2-2\alpha+\beta)+\sqrt{(6-6\alpha+3\beta)^2+8(6\alpha+3\beta-4)}},\intertext{and}\tilde{\sigma_{0}}:= \dfrac{4}{(2-2\alpha+\beta)+\sqrt{(2-2\alpha+\beta)^2-8(2\alpha+\beta-4)}}.
\mathcal F(X \xleftarrow f M \xrightarrow g Y) :=f_*g^*:\mathcal F(Y) \to \mathcal F(X), \mathcal H^{Chern}_*(X \xleftarrow f M \xrightarrow g Y) := f_* \bigl(c(T_g)\cap g^* \bigr):\mathcal H^{Chern}_*(Y) \to \mathcal H^{Chern}_*(X).
\widetilde u(z) \coloneqq - \sum_{i=1}^N \frac{\dot{\lambda}_i}{z - z_i}dz + \sum_{i = 1}^\ell \Bigg( \sum_{j=1}^{m_i}\frac{1}{z - w^i_j}- \sum_{j=1}^{m_0}\frac{1}{z - w^0_j}\Bigg) \alpha_i dz.
\tilde t=T(t),\tilde x=(x+\delta_1)X^1+\delta_2,\tilde u=\frac{1}{T_t}\left( X^1 u +X^1_t(x+\delta_1)\right), begin{align*}1ex]\tilde \alpha(\tilde t)=\frac{1}{T_t}\left(\alpha(t)-2\frac{X^1_t}{X^1}+\frac{T_{tt}}{T_t}\right),\quad\tilde\beta(\tilde t)=\dfrac{(X^1)^5}{T_t}\beta(t),
\begin{aligned}\Phi(c(k)) = \Theta \, \Phi(k) k \in E_1 \cup E_6 &\Phi(t_{e_2}(k)) = \tau_{e_2}\Phi(k) k \in E_2 &\Phi(t_{e_1}c(k)) = \tau_{e_1}\Theta \, \Phi(k) k \in E_3 \cup E_4 &\Phi(t^{-1}_{e_2}(k)) = \tau^{-1}_{e_2}\Phi(k) k \in E_5.\end{aligned}
J_{\alpha}^{\mu{\mathbf i}}= 1 9 2 \left[ - 3 i \sigma_{\alpha \dot{\alpha}}^{\mu}{\bar{D}}_{\mathbf j}^{\dot{\alpha}}D^{{\mathbf i}{\mathbf j}}T - 3 i{\bar{\sigma}}^{\mu \dot{\alpha}\beta}{\bar{D}}_{\dot{\alpha}}^{\mathbf i}D_{\alpha \beta}T - 1 2 \partial^{\mu}D_{\alpha}^{\mathbf i}T + a \sigma_{\alpha}^{\mu \nu \, \beta}\partial_{\nu}D_{\beta}^{\mathbf i}T \right] | \ .
u_{R,r_1r_2 \ldots r_n}(t) := \left(u_{R,r_1}\circ u_{R,r_2}\circ \ldots \circ u_{R,r_n}\right)(t) = 1 - \frac{N}{\displaystyle 1+r_1 -\frac{N}{\displaystyle 1+ r_2 - \ddots -\frac{N}{r_n +t}}}.
(g_1,g_2,\dots,g_n)\cdot \big((z_1,w_1),(z_2,w_2),\dots, (z_n,w_n)\big) = \big((g_1z_1,g_1w_1), ((g_1^{-A^1_2})g_2z_2,g_2w_2),\dots, ((\prod_{k=1}^{n-1}g_k^{-A^k_n})g_nz_n,g_nw_n)\big)
\begin{aligned}\mbox{(i)}B=B_1\cup B_2, \widetilde B= E_1\cup E_2 \mbox{and}\omega(E_i)\ge \frac12\omega(\widetilde B), i=1,2; & \mbox{(ii)}b(x)-b(y) \mbox{does not change sign for all}(x,y)\in B_i\times E_i, i=1,2; &\mbox{(iii)}|b(x)-m_b(\tilde B)|\le |b(x)-b(y)| \mbox{for all}(x,y)\in B_i\times E_i, i=1,2.\end{aligned}
\psi_{k}^{(0)}(x) = e^{-i \theta_k + i \pi}\phi_{k}(x) \;, \psi_{k}^{(1)}(x) = \alpha_R \phi_k(0) e^{-i \theta_k + i \pi}\left( \sum_{n \neq k}\frac{\phi_n(x) \overline{\phi_n(0)}}{E_n-E_k}+ \int_{\sigma_c(H_0)}\frac{\chi_{\lambda}(x) \overline{\chi_{\lambda}(0)}}{\lambda- E_k}d \mu(\lambda) \right) \;,
I_{K_X}(q)=(dH)q^{H/z}\sum_{\substack{n\in \mathbb Q_{\geq 0}\exists j: nw_j\in \mathbb Z}}q^n\frac{\prod_{k=1}^{nd}(dH+kz)(-1)^{nd^\prime}\prod_{k=1}^{nd^\prime}(d^\prime H+kz)}{\prod_{i=1}^N\prod_{0<k\leq nw_i, \langle k\rangle=\langle nw_i\rangle}(w_iH+kz)}\textbf{1}_{\langle -n\rangle}.
\sum_{r=0}^{\ell}(-1)^r q^{\frac{r(r-1)}{2}}\frac{(q;q)_y}{(q;q)_r\, (q;q)_{y-r}}\frac{(\nu/\mu;q)_{\ell-r}}{(\nu;q)_{y-r}}= (-\nu/\mu)^\ell q^{\frac{\ell(\ell-1)}{2}}\frac{(\mu;q)_y}{(\nu;q)_y\, (\mu;q)_{y-\ell}}.
\bigg( \langle\langle F_{\mu\nu}(z_j) F_{\rho\sigma}(z_j^{\prime})\rangle\rangle - \langle\langle F_{\mu\nu}(z_j) \rangle\rangle \,\langle\langle F_{\rho\sigma}(z_j^{\prime}) \rangle\rangle\bigg)^{{\rm SR}}= 0 \> .
\omega ( z ) = \omega_{0}{\frac{\cal A}{\cal B}}\ , \quad \Delta \omega = \omega^{\prime}( z ) \chi ( z ) = \omega_{0}{\frac{\cal A}{\cal B}}\sum_{i = 1}^{\cal N}\left({\frac{1}{z - a_{i}}}-{\frac{1}{z - b_{i}}}\right) \chi ( z ) \ .
-\frac{i}{\pi}\int_0^\infty dt \int_0^{\pi/2}d \theta \frac{t^2 \left[ g(-i t) - g(i t) \right] \cos \theta}{(t^2 + z^2 \cos^2 \theta)^{3/2}}=-\frac{i}{\pi}\int_0^\infty dt \frac{t [ g(-it) - g(it)]}{(t^2 + z^2)},
\begin{array}{l}{\binom n{k_1k_2}}^{-1}B_{n,k_1}(1,\dots,i{\binom{i-1}k}^{-1}B_{i-1,k_2}(a_1,a_2,\dots),\dots)={\binom{n-k_1}{k_1k_2}}^{-1}B_{n-k_1,k_1k_2}(a_1,a_2,\dots). \end{array}
\sum_{m=2}^{\infty}\frac{\tau_{A_{s_1}\smallsetminus \{\alpha+s_1\}\cup \{-\beta-s_2\}}(p^m) \tau_{B_{s_2}\smallsetminus \{\beta+s_2\}\cup \{-\alpha-s_1\}}(p^m)}{p^m}\ll \sum_{m=2}^{\infty}\frac{p^{m(\frac{1}{2}- \epsilon+\varepsilon)}}{p^m}\ll \frac{1}{p^{1+\varepsilon}}
\tilde{T}_{a}\Psi^{(0)}= \left(\begin{array}{c}e^{i \chi(a; s^k)}\phi^{(0)}(s^i + a^i) e^{-i \chi(a; s^k)}\phi^{\ast (0)}(s^i + a^i) A^{(0)}_{k}(s^i + a^i) - \frac{1}{q}\partial_k \chi(a;s^i) \end{array}\right),
E^{\Pi^{D_T}}[e^{u\sqrt{T}\langle b_j-b_{0,j},a_\lambda \Phi_{\lambda,k}\rangle_{L^2}}|X^{T}] = e^{\frac{u}{\sqrt{T}}\int_0^T \tilde{\Phi}_{\lambda,k,j}(X_t).dW_t + \frac{u^2}{2}\|\tilde{\Phi}_{\lambda,k,j}\|_{\mu_0}^2 + ur_T + u^2\tilde{r}_T}& \times \frac{\int_{D_T}e^{S_T(b) + \ell_T(b_u)}d\Pi(b)}{\int_{D_T}e^{\ell_T(b)}d\Pi(b)},
e^{\widehat{\rho}}\widehat{R}= \sum_{\alpha\in A_N}\sum_{w\in W^\sharp}\sigma(w) w\left( t_{w^{-1}(\alpha)}\left(\frac{e^{\widehat{\rho}}}{(1+e^{\delta-\varepsilon_{N+1}})}\right)\right)= \sum_{\alpha\in A_N}\sum_{w\in W^\sharp}\sigma(w) w\left( t_{\alpha}\left(\frac{e^{\widehat{\rho}}}{(1+e^{\delta-\varepsilon_{N+1}})}\right)\right).
||Af||_{p}^p = \int_{\mathcal{D}}|f(x)|^p\sigma(x)^{p(1-p/2)}dx \leq \bigg(\int_{\mathcal{D}}|f(x)|^2dx\bigg)^{p/2}\bigg(\int_{\mathcal{D}}\sigma(x)^{bp(1-p/2)}dx\bigg)^{1/b}
c_2\le \P_x(T_{W_y}<\tau_U)=G^U\mu(x)=\int_U G^U(x,z)\mu (dz)\le \int_U G(x,z)\mu (dz) \le c_3 G(x,y) \mu(U)=c_3 G(x,y)\mathrm{Cap}^{Y^U}(W_y)\, .
\Omega_z(S_2^{h+k}(S_2^*)^k) = (S_2^{h+k}(S_2^*)^k v,v) = (S_2^{k}v,v) -\sum_{i=1}^{2^k-1}(S_2^hU^iS_2^{k}(S_2^*)^k U^{-i}v,v) & = -\sum_{i=1}^{2^k-1}\bar{z}^i(U^{2^hi}S_2^hS_2^{k}(S_2^*)^k v,v) = -\sum_{i=1}^{2^k-1}\bar{z}^i(S_2^hS_2^{k}(S_2^*)^k v,U^{-2^hi}v) & = -\sum_{i=1}^{2^k-1}z^{2^hi-i}(S_2^hS_2^{k}(S_2^*)^k v,v)
\begin{array}{c}\mbox{Ker}_{Z_1}\Omega_{l+1}=\Omega_l, \mbox{Ker}_{Z_2}\Omega(Z_1)_{l+1}=\Omega(Z_1)_l, \mbox{Ker}_{Z_3}\Omega(Z_1,Z_2)_{l+1}=\Omega(Z_1,Z_2)_l, \ldots \end{array}
\sum_{k=1}^{\infty}\sum_{n=1}^{\infty}\frac{1}{k^{\beta}+n^{\beta}}\int\int_{A_{k,n}}\frac{dxdy}{(|x_1|^{\alpha_1}+|x_2|^{\alpha_2})(|y_1|^{\alpha_1}+|y_2|^{\alpha_2})}<\infty.
M_{\mathfrak{g}_F}(v,w) = \begin{cases}d^{(i)}v = D^{(j)}, \, w=D^{(i)}, \, i \neq j 1 v=D^{(i)}, \, w = \nu_1^{(i)},\dots,\nu_{d^{(i)}-1}^{(i)}1 v=\nu_1^{(i)},\dots,\nu_{d^{(i)}-1}^{(i)}, \, w = D^{(i)}0 \end{cases}
{\widetilde H_{B F}}= H_{c}+{\displaystyle \int}d^{3}x \left( \frac{\displaystyle \pi_{\theta}^{2}}{\displaystyle 2 m^{2}}+ \frac{m^{2}}{2}( \partial_{i}\theta )^{2}- m^{2}\theta \partial_{i}A_{i}\right) ,
\sigma_{\Delta ,\mathbf{G},\mathbf{B}}^{2}\left( P\right) -\sigma_{\Delta ,\mathbf{B}}^{2}\left( P\right) =\sum_{a\in \mathcal{A}}c_{\mathbf{a}}^{2}E_{P}\left\{\pi_{\mathbf{a}}(\mathbf{G};P)var_{P}(Y\mid \mathbf{A}=\mathbf{a},\mathbf{G})var_{P}\left[ \frac{1}{\pi_{\mathbf{a}}(\mathbf{G,B};P)}\mid \mathbf{A=a},\mathbf{G}\right] \right\}\mathbf{\geq}0.
\left\lbrace\begin{array}{ll}\widehat U''-{\Delta}\widehat U+A\widehat U=0 \hbox{in}(0, +\infty)\times \Omega, \widehat U= 0 \hbox{on}(0, +\infty)\times\Gamma_0, \partial_\nu \widehat U + B\widehat U = D\widehat H \hbox{on}(0, +\infty)\times\Gamma_1, t=0:\widehat U =\widehat U'=0 \hbox{in}\Omega.\end{array}\right.
\frac{1}{\phi(n)}\log\Lambda_{a_n,\phi(n)}(s,\omega) = \frac{1}{\phi(n)}\sum_{k=a_n+1}^{a_n+\phi(n)}\{sg_k(\xi_{k-1},\xi_{k})-\log{E[e^{sg_k(\xi_{k-1},\xi_k)}|\xi_{k-1}]}\}.
D_{a}= \frac{\partial}{\partial \theta^{a}}- \frac{i}{2}\overline{{{\theta}}}_{a}\frac{\partial}{\partial \tau}, \quad \overline{{{D}}}^{a}= \frac{\partial}{\partial \overline{{{\theta}}}_{a}}- \frac{i}{2}\theta^{a}\frac{\partial}{\partial \tau},
\sigma_{m,k;n,\ell}:= \left(1+\frac{m}{k}\right)\left(1+\frac{n}{\ell}\right)\sigma_{m+k-1,n+\ell-1}-\left(1+\frac{m}{k}\right)\frac{n}{\ell}\sigma_{m+k-1,n-1}& -\frac{m}{k}\left(1+\frac{n}{\ell}\right)\sigma_{m-1,n+\ell-1}+\frac{mn}{k\ell}\sigma_{m-1,n-1},
\sum_{m=0}^\infty \frac{M_p(m)}{p-1}t^m = \sum_{m = 0}^\infty \frac{(2m)!}{m! (m+1)!}p^{m+1}t^{2m}- \sum_{m = 0}^\infty \sum_{k = 1}^m \frac{(2k+1)(2m)!}{(m-k)!(m+k+1)!}p^{m-k}t^{2m}&= pC(pt^2) - \sum_{k = 1}^\infty C(pt^2)^{2k + 1}t^{2k}&= pC(pt^2) - C(pt^2)^3\cdot \frac{t^2}{1 - t^2C (pt^2)^2}.
\sigma^{\infty}( \tilde{x_1}, \tilde{x_2}, 1 , \tilde{m}) =\tilde{\sigma}^*_1 ( \tilde{x_1}, \tilde{x_2},\tilde{m}) \forall ( \tilde{x_1},\tilde{x_2}, \tilde{m}) \in \mathbb{R}^2 \times \mathbb{R}_+,
s =\sum_{i=0}^{r-1}\left\{\langle\frac{-2np^i}{q-1}\rangle+\langle\frac{ndp^i}{q-1}\rangle+\langle\frac{-n(d-1)p^i}{q-1}\rangle-\langle\frac{-np^i}{q-1}\rangle\right\}&=\sum_{i=0}^{r-1}\left\{\lfloor\frac{-np^i}{q-1}\rfloor-\lfloor\frac{-2np^i}{q-1}\rfloor-\lfloor\frac{ndp^i}{q-1}\rfloor-\lfloor\frac{-n(d-1)p^i}{q-1}\rfloor\right\},
\partial_t \phi - \Delta \phi + \bar{u}\cdot \nabla \phi + \phi \cdot \nabla \bar{u}+ \phi \cdot \nabla (\phi \eta) & + \phi \cdot \nabla \psi + \psi \cdot \nabla \phi + \bar{u}\cdot \nabla \psi + \psi \cdot \nabla \bar{u}+ \nabla q=0\, ,
\mathcal{C}= \left\{(r_1,r_2): 0\leq \sum_{i\in \Omega}r_i\leq \gamma(\sum_{i\in \Omega}P_i),\ \Omega\subseteq \{1,2\}\right\}=\left\{\mathbf{r}:\mathbf{r}\in \mathbb{R}_2^+, f(\mathbf{r})\leq 0 \right\},
S_{(0,0,0,0,0)}\left( \frac{1}{x_0^2}, \frac{1}{x_1^2}, \frac{1}{x_2^2}\right) = & \left( x_0 x_1 x_2 \right)^{-4}\cdot (-x_0 + x_1 - x_2 ) (-x_0 + x_1 + x_2) (x_0 + x_1 - x_2) (x_0 + x_1 + x_2).
V_{Selb}(z)=\frac{1}{t}\log\big(z^{-t}(1+z)^{2t}\big),\Delta^{(Selb)}_m= \sum_{i=0}^{\big[\frac{N-m}{2}\big]}\frac{(-1)^{(2i+m)-1}}{2^{(2i+m)}(2i+m)}\binom{2i+m}{i}.
\widetilde{\mathcal{E}}=\{\Psi\in(\mathcal{L}^{\beta})^*|\, H\Psi=0\}, \mathcal{E}=\widetilde{\mathcal{E}}\cap(\mathbb{C}\mathbf{1}\oplus\mathcal{L}^{\beta-2}), \mathsf{E}=\widetilde{\mathcal{E}}\cap\mathcal{L}^{\beta-2}.
\beta_k \!=\! \rho^2_{\rm r}\!\left( \! E \bigg[ \left\{\frac{(\boldsymbol{w}^T_k)}{\epsilon_{\rm r}\!+\! |\boldsymbol{w}^T_{k-1}|}\left( \boldsymbol{I}- \mu\boldsymbol{R}\right)^{-1}\frac{(\boldsymbol{w}_k)}{\epsilon_{\rm r}\!+\! | \boldsymbol{w}_{k-1}|}\right\}\!\bigg]\! \right) .
v(r)= (\gamma-1)^{\frac{1}{q-(p-1)}}\int_r^R\left( \frac{-\Phi '(s)}{\Phi (s)+1}\right)^{\frac{1}{q-(p-1)}}ds = \theta\left[ r^{-\frac{p-q}{q-(p-1)}}-R^{-\frac{p-q}{q-(p-1)}}\right],
\phi_1 = \frac{1}{<\phi_1 | \phi_1 >}\left( e^{-i \omega t}e^{-\frac{\omega x^2}{2}}x^{l+\frac{1}{2}}L_n^l (\omega x^2) - \frac{i \sqrt{n+1}}{\sqrt{n+l+1}}e^{i \omega t}e^{-\frac{\omega x^2}{2}}x^{l + \frac{1}{2}}L_{n+1}^l (\omega x^2) \right) ,
||\T_f(r) \le\dfrac{\sharp J}{q-(2n-k+2)+\sharp J}\biggl (\sum_{j\in J}N^{[k]}_{(f,a_{i_j})}(r)+\sum_{j=2n-k+3}^qN^{[k]}_{(f,a_{i_j})}(r)\biggl ) &+o(T_f(r)) + O(\max_{1\le i \le q}T_{a_i}(r)).
|\mathcal{J}^*_{A,j,i}| \le \sum_{i' \le K}\left(\frac{2|\mathcal J_{i,i'}|}{K-1}+1 \right) \overset{{\rm (c)}}{=}\frac{2|\mathcal J|}{K(K-1)}+K \overset{{\rm (b)}}{\le}\frac{(1/2 - 2\mu - 2\rho) n}{K(K-1)}+ K & \le (1-4 \mu - 3 \rho) m/K
\frac{d}{dz}\left\lbrace{}^{\Psi}F_{p}^{(\alpha,\beta)}(a,b;c;z)\right\rbrace =\frac{d}{dz}\left\lbrace \sum_{n=0}^{\infty}(a)_{n}\frac{{}^{\Psi}B_{p}^{(\alpha,\beta)}(b+n,c-b)}{B(b,c-b)}\frac{z^{n}}{n!}\right\rbrace &=\sum_{n=1}^{\infty}(a)_{n}\frac{{}^{\Psi}B_{p}^{(\alpha,\beta)}(b+n,c-b)}{B(b,c-b)}\frac{z^{n-1}}{(n-1)!}.
\left\|\begin{bmatrix}A X X^* B\end{bmatrix}\right\|_{(k)}\le \left\| \frac{A+B}{2}+ (r+\omega)I\right\|_{(k)}+ \left\| \frac{A+B}{2}- rI\right\|_{(k)}& =\left\| A+B + \omega I\right\|_{(k)}.
(\alpha \circ v) (L_{x_i^{k-2}}) = a_{1i}^{k-2}x^{k-2}+ (k-2)a_{1i}^{k-3}a_{i2}x^{k-1}(\alpha \circ v) (L_{x_i^{k-1}}) = a_{1i}^{k-1}x^{k-1}
= \int_{N_L \backslash B_{L,n}K_{n,L}}{|\nu(g_2)|^{s-1}\ell(\pi(g_2)v_1)\,dg}= \int_{T_{L,n}}{\delta_{B_L}^{-1}(t)|\nu(t)|^{s-1}\ell(\pi(t)v_1)\,dt}
\frac{c |mh\pm nk|}{gx Q}W\left( \frac{c |mh\pm nk|}{gx Q}\right) = \frac{1}{2\delta}\int_{-\delta}^{\delta}\frac{c |mh\pm e^{\xi}nk|}{gx Q}W\left( \frac{c |mh\pm e^{\xi}nk|}{gx Q}\right)\,d\xi +O\left(\frac{c nk \delta}{gx Q}\right).
\frac{d^{n}}{dx^{n}}f(g(x))=\sum\frac{n!}{m_{1}!1!^{m_{1}}m_{2}!2!^{m_{2}}\cdot\cdot\cdot m_{n}!n!^{m_{n}}}\cdot f^{(m_{1}+\cdot\cdot\cdot+m_{n})}(g(x))\cdot\prod_{j=1}^{n}(g^{(j)}(x))^{m_{j}},
T_g(\omega,h^E)-T_g(\omega',h^E)= - \widetilde{{\rm ch}}_g(f_*E,h^{f_*E},h'^{f_*E}) &+ \int_{W_{\mu_n}/{V_{\mu_n}}}\widetilde{{\rm Td}}_g(Tf,h^{Tf},h'^{Tf}){\rm ch}_g(E,h^E)
\big(\tau_{k, \ell}^{(\alpha, \beta)}\big)^{2}=\tau_{k, \ell}^{(\alpha, \beta + 1)}\tau_{k, \ell}^{(\alpha, \beta -1)}- \tau_{k, \ell + 1}^{(\alpha, \beta - 1)}\tau_{k, \ell - 1}^{(\alpha, \beta +1)}-\tau_{k-1, \ell}^{(\alpha,\beta-1)}\tau_{k + 1, \ell}^{(\alpha, \beta + 1)},
\lambda^1 T = Y^1_T-Y^1_0 + \int_0^T [H_1(X_{s}, Z^1_s,u^*_s,v^*_s) - Z^1_sR(u_t,v^*_s) - L_1(X_s,u_s,v^*_s)]ds &- \int_0^T Z^1_s dW^{u,v^*}_s + \int_0^T L_1(X_s,u_s,v^*_s)ds.
\overline{Q}_{k,a}(x;q) =\sum_{i=1}^a (xq)^i \left[\sum_{h=1}^{k-i}(xq^2)^h\overline{Q}_{k,k-h}(xq^2;q) +\sum_{h=0}^{k-i-1}(xq^2)^h\overline{Q}_{k,k-h}(xq^2;q)\right] &\quad+ \sum_{i=0}^{a-1}(xq)^i\left[\sum_{h=1}^{k-i}(xq^2)^h\overline{Q}_{k,k-h}(xq^2;q)+\sum_{h=0}^{k-i-1}(xq^2)^h\overline{Q}_{k,k-h}(xq^2;q)\right].
\mathbb{E}[\zeta | d(x, \hat{x}) = k] = s(n-s) \left( 1 - 4\frac{k}{n}+ 4\frac{k^2}{n^2}- 4\frac{k(n-k)}{n^2(n-1)}\right) = s(n-s) \left( \left(1 - 2\frac{k}{n}\right)^2 - 4\frac{k(n-k)}{n^2(n-1)}\right)
v^{\mu}( x_{+}, \theta ) = a^{\mu}+ 2 i \theta \sigma^{\mu}\bar{\xi}+ w_{~ \nu}^{\mu}x_{+}^{\nu}+ \lambda x_{+}^{\mu}- 2 \theta \sigma^{\mu}( x_{+}\cdot \tilde{\sigma}) \tilde{\zeta}+ b^{\mu}x_{+}^{2}- 2 x_{+}^{\mu}b \cdot x_{+}
\int_{a_2}^{\widetilde r_3}(x-a_2)^{p-1}\Psi(x)dx =\int_{\widetilde a_2}^{\widetilde r_3}(x-a_2)^{p-1}\Psi(x)dx=\int_{\widetilde r_2}^{\widetilde a_2}(a_2-x)^{p-1}\Psi(x)dx &=\int_{r_2}^{a_2}(a_2-x)^{p-1}\Psi(x)dx=\int_{a_2}^{r_3}(x-a_2)^{p-1}\Psi(x)dx.
\tilde{F}=F_{1}(\delta\tilde{v},u_{L}+\tilde{u}_{1})+F_{1}(u_{L}+\tilde{v}_{2},\delta\tilde{u}) & +F_{2}\left( \delta\tilde{v},\nabla P_{L}+\nabla\tilde{P}_{1}\right)+F_{2}(u_{L}+\tilde{v}_{2},\nabla\delta\tilde{P}), \tilde{G}=-\left( A_{(u_{L}+\tilde{v}_{1})}-A_{(u_{L}+\tilde{v}_{2})}\right) \left( u_{L}+\tilde{u}_{2}\right) ,
\widehat{\theta}_{a b}^{\alpha}= \frac{1}{\alpha !}\frac{i^{\alpha - 1}}{2^{\alpha}}\sum_{i = 1}^{\alpha}\{2 \theta_{a b}\prod_{k \ne i}( \theta^{\mu_{k}\nu_{k}}\partial_{\mu_{k}}\partial_{\nu_{k}}) + ( x_{a}\theta^{\mu_{i}}\,_{b}- x_{b}\theta^{\mu_{i}}\,_{a}) \prod_{j \ne i}( \theta^{\mu_{j}\nu_{j}}\partial_{\nu_{j}}) \prod_{k = 1}^{\alpha}\partial_{\mu_{k}}\}.
S_I(T_0,T_1,\dots,T_r):=\sum_{(n,\alpha)\in \mathbb N^{r+1}}\sum_{\begin{smallmatrix}\beta\in\mathbb N^m \theta'(\beta,\alpha)\end{smallmatrix}}\sum_{\begin{smallmatrix}(k_j)_{j\in I}\in \mathbb N_{>0}^I (\ref{eq3.12})_{n,\beta}\end{smallmatrix}}\L^{-\sum_{j\in I}k_j\nu_j}T_0^nT_1^{\alpha_1}\cdots T_r^{\alpha_r}.
f_{D 0}^{A}= 2 \pi V_{1 1}\int_{0}^{\infty}\frac{d t}{t^{5 / 2}}\sum_{n^{I}\ne 0}e^{- \frac{\pi}{t}{\cal M}^{2}}= V_{1 1}\sum_{n^{I}\ne 0}\left({\cal M}^{2}\right)^{- 3 / 2}, \quad{\cal M}^{2}= n^{I}g_{I J}n^{J},
\lbrack v]_{x_{N},x^{\prime},\overline{Q}}^{(1,(1-\omega)\gamma)}=\sup_{\substack{\theta>0,\overline{h}\in R^{N-1}(x,t)\in\overline{Q}}}\frac{|\Delta_{\theta,x_{N}}^{2}\Delta_{\overline{h},x^{\prime}}v(x,t)|}{\theta|\overline{h}|^{(1-\omega)\gamma}},
\frac{1}{8}\,W^{+}_{ijkl}W^{+}_{ijpq,t}W^{+}_{klpq,t}= \,\, \lambda |d\lambda|^{2}+\mu|d\mu|^{2}+\nu|d\nu|^{2}&-\lambda(\mu-\nu)^{2}|a|^{2}-\mu(\nu-\lambda)^{2}|b|^{2}-\nu(\lambda-\mu)^{2}|c|^{2}\,.
\kappa_{i[jk]}=\mathbf{A}_{[jk],i}=\frac{1}{2}\left(A_{jk,i}-A_{kj,i}\right)=\frac{1}{2}\left(-\epsilon_{jkl}\vartheta_{l,i}+\epsilon_{kjl}\vartheta_{l,i}\right)=\frac{1}{2}\left(\epsilon_{kjl}+\epsilon_{kjl}\right)\vartheta_{l,i}=\epsilon_{kjl}\,\vartheta_{l,i}.
\|u_{0,n}\|_{L^2}= \mu_n \|Q\|_{L^2}, \|\nabla u_{0,n}\|_{L^2}= \mu_n \lambda_n \|\nabla Q\|_{L^2}, \|u_{0,n}\|_{\dot{H}^1_c}= \mu_n \lambda_n \|Q\|_{\dot{H}^1_c},
g_{\mu \nu}\left( x\right) =\eta_{mn}\,e_\mu^m\left( x\right) e_\nu^n\left( x\right) \,\,,\quad g^{\mu \nu}\left( x\right) =\eta^{mn}\,e_m^\mu \left( x\right) e_n^\nu \left( x\right) \,\,,
\mathrm{Pr}^{(\alpha,n)}V=V+\eta^{\alpha,t}\partial_{\mathcal{T}_t^\alpha u}+\phi^{\alpha,t}\partial_{\mathcal{T}_t^\alpha v}+\eta^x\partial_{u_x}+\phi^x\partial_{v_x}+\eta^{xx}\partial_{u_{xx}}+\phi^{xx}\partial_{v_{xx}}+\cdots,
\Vert P_{S}M y\Vert_{\infty}= \Vert P_{S}M x + P_{S}M (y-x)\Vert_{\infty}= \Vert P_{S}M x + P_{S}M P_{A}(y-x)\Vert_{\infty}&\leq \Vert P_{S}M x\Vert_{\infty}+ \Vert P_{S}M P_{A}(y-x)\Vert_{\infty}<\|Mx\|_{\infty}+ \Vert M\, : \, \ell_{\infty}\rightarrow \ell_{\infty}\Vert \, \eta &\leq 1/(8d) +\eta d < a,
(\mu+1)\int_0^{\infty}e^{-(\mu+1)t}(\sinh t)^{-s}\,dt =\int_0^{\infty}\frac{d}{dt}(1-e^{-(\mu+1)t})(\sinh t)^{-s}\,dt &=s\int_0^{\infty}(1-e^{-(\mu+1)t})(\sinh t)^{-s-1}(\cosh t)\,dt.
w(y,t)= 2^{\alpha k+1}(v(2^{-k}y+x_0,2^{-k\gamma}t+t_0)-m_k) \leq 2^{\alpha k+1}(a_{k-\ell-1}-m_k) &\leq 2^{\alpha k+1}(a_{k-\ell-1}-b_{k-\ell-1}+b_{k}-m_k) &\leq 2^{\alpha k+1}(2^{-\alpha(k-\ell-1)}-\frac12 2^{-k\alpha}) &\leq 2^{1+\alpha(\ell+1)}-1\leq 2|2y|^\alpha-1 &\leq 2|2y|^\eta-1.
Z_{\left\lfloor Tr\right\rfloor}^{\ast}(d) =T^{1/2-d}(\log T)^{-1}\mathsf{D}_{d}\Delta_{+}^{-d}\xi_{\left\lfloor Tr\right\rfloor}& =T^{1/2-d}(\log T)^{-1}\sum_{n=1}^{\left\lfloor Tr\right\rfloor -1}\pi_{n}(d)\sum_{k=0}^{n-1}(k+d)^{-1}\xi_{\left\lfloor Tr\right\rfloor-n}\Rightarrow W(r;d).
{\mathbf O}\, \,{\mathbf R}\, \,{\mathbf O}^{- 1}\: = \: - 8 m +{\cal M}_{\nu}^{\mu}{\mathbf a}_{\nu}^{\dagger}{\mathbf a}^{\mu}+{\cal N}_{\nu}^{\mu}{\mathbf b}^{\dagger \nu}{\mathbf b}_{\mu}.
\boldsymbol{\nu}^{b, \pm}:=(\underbrace{1, \ldots , 1}_{m_{1}}, \ldots , \underbrace{b, \ldots , b}_{m_{b}\pm 1}, \ldots \underbrace{N, \ldots , N}_{m_{N}}) \in I_{N, c\pm 1}^{+}.
\epsilon_{0}^{>}= - \frac{8 k^{2}}{\ell_{0}^{2}}B_{1}^{<}, \quad \epsilon_{2}^{>}= k^{2}B_{2}^{<}- \frac{3 w^{( 2 ) >}k^{2}}{\ell_{0}^{2}}B_{1}^{<}\, .
[{\rm ad}x,D(d)]=[\sum[{\rm ad}x_{i1},{\rm ad}x_{i2}],D(d)] =\sum((-1)^{\left|D\right|\left|x\right|}D([[{\rm ad}x_{i1},{\rm ad}x_{i2}],d])-(-1)^{\left|D\right|\left|x\right|}[[D({\rm ad}x_{i1}),{\rm ad}x_{i2}],d] -(-1)^{\left|D\right|\left|x\right|}(-1)^{\left|D\right|\left|x_{i1}\right|}[[{\rm ad}x_{i1},D({\rm ad}x_{i2})],d]).
u_{2/k}(x,t)=\frac{1}{2^{1-\frac{1}{k}}t^{2H/k}}\sum_{n=0}^{\infty}\frac{(-1)^n}{n!\Gamma(-\frac{n}{k}+1-\frac{1}{k})}\frac{2^{n/k}|x|^n}{t^{\frac{2}{k}Hn}}.
\mathcal{Z}\left(\hat{\mu}\right)=\bigcup_{i=1}^{r}\mathcal{Z}\left(\hat{\nu}_{i}\right),\ \mathcal{Z}\left(\hat{\nu}_{i}\right)=\rho^{-i}\bigcup_{j=0}^{\infty}\left(\frac{q}{p}\right)^{j}\frac{a_{j}}{N_{j}},\a_{j}\in\mathbb{Z}\setminus N_{j}\mathbb{Z}.
M^{ra_3l}c_0^2\alpha^4\Bigg(\|F_4^l\|_{l,r}+\sum_{n=2}^{\infty}\|T_4^{l,(n)}\|_{l,r}\Bigg)\le c_0^2\alpha^4\|J_4^0\|_{0,r}+c|l|\alpha^{-2}.