harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\partial_t f+v\cdot \nabla_x f+{\rm{div}}_v((u-v)f)+{\rm{div}}_v(fL[f])-\Delta_vf=0, &\partial_t\rho+{\rm{div}}_x(\rho u)=0, &\partial_t(\rho u)+{\rm{div}}(\rho u\otimes u)+\nabla\rho^{\gamma}+\delta\nabla\rho^{\beta}-{\rm{div}}\mathbb{S}(\nabla u)=-\int_{\mathbb{R}^3}(u-v)f\,dv,
\frac{g_{n_s + \tau_s^+ - \tau_s^+}\big(q^{\lambda_1}t^{N-1}, \dots, q^{\lambda_N}; q, t\big)}{g_{n_s + \tau_s^+ - \tau_s^-}(t^{N-1}, \dots, t, 1; q, t)}{}= \frac{P_{\lambda}\big(q^{n_s + \tau_s^+ - \tau_s^-}t^{N-1}, t^{N-2}, \dots, t, 1; q, t\big)}{P_{\lambda}(t^{N-1}, \dots, t, 1; q, t)}\forall\, 1\leq s\leq m.
\displaystyle \int\limits_T| f(t,x_n(t)) - f(t,x(t))| d \mu(t) = \int\limits_{A^+_n}f(t,x_n(t)) - f(t,x(t)) d \mu(t) & + \int\limits_{A^-_n}f(t,x(t)) - f(t,x_n(t)) d \mu(t) &\leq \delta/2 + \delta/4< \delta;
D_{\zeta}(\zeta/\xi)^2 (1+\xi/\zeta)^{4}=2 (\zeta/\xi) (1+\xi/\zeta)^{2}(-\xi/\zeta + \zeta/\xi). &D_{\xi}(\zeta/\xi)^2 (1+\xi/\zeta)^{4}=2 (\zeta/\xi) (1+\xi/\zeta)^{2}(\xi/\zeta - \zeta/\xi).
B := B_{\varepsilon,\delta_1,\delta_2, \delta, K}:= A \cap \Big \{T_1 \leq K, \int_{T_1}^{T_1+\delta}e^{-\xi_s}\, ds \leq \delta_1, & \Delta \xi_s \not\in (c-\delta_2,c+\delta_2), \; \forall\, s\in (T_1,T_1+\delta] \Big\}
\begin{aligned}U = (9X-Y-104)/(11X-Y-236), V = (X^3-198X^2+916X+980Y+34336)/(11X-Y-236)^2, X = 2(14U^2-17U+V+4)/(U-1)^2, Y = 2(36U^3-25U^2+11UV-25U-9V+16)/(U-1)^3,\end{aligned}
\frac{L(u^{2},\chi^{2})L(u^{3},\chi^{3})}{L(u^{6},\chi^{6})}=\sum_{j=0}^{\deg Q-1}\sum_{l=0}^{\deg Q-1}\sum_{k=0}^{\infty}u^{2j+3l+6k}q^{\frac{j+k+l}{2}}\Lambda_{j}(\chi^{2})\Lambda_{l}(\chi^{3})Sym^{k}(\chi^{6})
a_{i,n+1}(z/w)= \begin{cases}q^{-1}(z/w-\xi)(z/w-1)+(\xi-1)(q^{-2}-1)z/w \ \i=n+1 (q^{-2}-1) q^{\bar{i}-\overline{n+1}}\xi \cdot (z/w-1) \ \i<n+1 (q^{-2}-1) q^{\bar{i}-\overline{n+1}}\cdot (z/w-1)z/w \ \i>n+1 \end{cases}.
P_{k,i}=\Pr\left(\gamma_{k,i}<\bar\gamma_{k,i}\right), &P_{k',i}=1-\Pr\left(\gamma_{k,i}>\bar\gamma_{k,i},\gamma_{k',i}>\bar\gamma_{k',i}\right), &P_{{},{k},i}=\Pr\left(\gamma_{{},{k},i}<\bar\gamma_{{},{k},i}\right),
Z[N,\{t_{\mathcal{B}}\}]= \int dT d\bar{T}\exp\Bigl(-N^{d-1}\sum_{\mathcal{B}}\frac{N^{-\frac{2}{(d-2)!}\omega(\mathcal{B})}}{|Aut(\mathcal{B})|}t_{\mathcal{B}}\mathcal{B}(T, \bar{T})\Bigr)
H_{3}= 2 \mu \delta_{J_{1}, J_{2}+ J_{3}}{\frac{\sqrt{J_{1}J_{2}J_{3}}}{N}}{\frac{J_{3}}{J_{1}}}{\frac{\sin^{2}( \pi n y )}{\pi^{2}( n y - m )^{2}}}\Pi_{2}^{\prime J_{1}}( q_{n}) \bar{\Pi}^{\prime J_{2}}O_{2}^{\prime J_{3}}( s_{m}) .
\mathcal{Q}\left({x^1},\xi^{1}\right) := \min_{{x^2}(\xi^{1})}\left\{c^{{2}^{\top}}\!\left(\xi^{1}\right){x^2}(\xi^{1}) T^1(\xi^{1}){x^1}+ W^2(\xi^{1}) x^2(\xi^{1})= h^2(\xi^{1}),\{x^2}(\xi^{1})\geq 0 \right\}.
\begin{array}{l}{{\Phi^{\frac{d - 3}{2}}\partial^{\mu_{1}}\Phi \Phi^{s - 1}{\bf V}_{i_{2}}^{\mu_{2}}\cdots{\bf V}_{i_{s}}^{\mu_{s}}- \mathrm{t r a c e s}_{( i )}=}}\\{{={\frac{2}{2 s + d - 3}}\nabla^{\mu_{1}}\left( \Phi^{s +{\frac{d - 3}{2}}}{\bf V}_{i_{2}}^{\mu_{2}}\cdots{\bf V}_{i_{s}}^{\mu_{s}}- \mathrm{t r a c e s}_{( i )}\right)}}\\ \end{array}
e^{\psi}= \left| \frac{Q_{2}\sqrt{c_{1}}}{Q_{1}\sqrt{c_{2}}}\frac{\sinh \left[ \sqrt{c_{2}}( I ( A ) + \tilde{c}_{2}) \right]}{\sinh \left[ \sqrt{c_{1}}( I ( A ) + \tilde{c}_{1}) \right]}\right|^{1 / 2},
\rho_{n}\left( \underline{\theta}\right) ~\ddot{\theta}_{n}=\dot{f}_{n}-\sum_{m=1}^{N}\left\{\left[ \frac{\partial ~\gamma_{n}\left( \underline{\theta}\right)}{\partial ~\theta_{m}}+\frac{\partial ~\rho_{n}\left( \underline{\theta}\right)}{\partial ~\theta_{m}}~\dot{\theta}_{n}\right] ~\dot{\theta}_{m}\right\}~.
h^{\ast}(u,v)=\min\bigg\{\sum_{n\geq1}u_{n}(\ln u_{n}-1)\mid(u_{n})_{n\geq1}\in\mathbb{R}_{+},\ \sum_{n\geq1}u_{n}=u,\ \sum_{n\geq1}u_{n}\sigma_{n}=v\bigg\}
\begin{array}{ccl}\theta = \theta^i \mathbf{t}_i = (\theta{^i}_\mu(z)\hbox{d}x^\mu + \theta{^i}_j(z)\hbox{d}y^j)\mathbf{t}_i \pi = \pi_i\mathbf{t}^i = \frac{1}{(N-1)!}\pi_{iI_1\cdots I_{N-1}}(z)\hbox{d}z^{I_1}\wedge \cdots\wedge \hbox{d}z^{I_{N-1}}\mathbf{t}^i\end{array}
=q^{(a^2-b^2)/(2p)-(a-b)/2}\frac{(q^a;q^p)_{\infty}(q^{p-a};q^p)_{\infty}}{(q^b;q^p)_{\infty}(q^{p-b};q^p)_{\infty}}\textrm{,}|q|<1,
0 =(a_na_2)^{m_1}\sum_{i_1,\dotsc,i_s=0}^r\lambda_{i_1,\dotsc,i_s}(a_{n-1}a_1)^{i_1}\dotsm(a_{s+1}a_{s-1})^{i_{s-1}}a_s^{i_s}v &=(\lambda_1\lambda_2)^{m_1}\sum_{i_2,\dotsc,i_s=0}^r\lambda_{m_1,i_2,\dotsc,i_s}(a_{n-2}a_2)^{i_2}\dotsm(a_{s+1}a_{s-1})^{i_{s-1}}a_s^{i_s}v.
V_{i}\cdot{\cal N}_{\overline{{{\alpha V}}}}\equiv{\vec{U}}_{i}\cdot{\vec{P}}_{\overline{{{\alpha V}}}}- \sum_{r}V_{i}^{r}N_{\overline{{{\alpha V}}}}^{r}+ \sum_{\ell :{\mathrm{r i g h t}}}T_{i}^{\ell}J_{\overline{{{\alpha V}}}}^{\ell}- \sum_{\ell :{\mathrm{l e f t}}}T_{i}^{\ell}J_{\overline{{{\alpha V}}}}^{\ell}~ .
\lim_{n\rightarrow\infty}\vert e^{\pi H_{v_n}f(u_n)}-\cos(\pi P_{v_n}f(u_n))\vert=\lim_{n\rightarrow\infty}\vert e^{-\pi H_{v_n}f(u_n)}-\cos(\pi P_{v_n}f(u_n))\vert=0.
\{(k_1, \cdots, k_n)\in\mathbb{Z}^{\oplus n}|\ 0\leq k_r\leq l-1,\k_1+\cdots+k_n=lk-p\}\longrightarrow \{(k_1', \cdots, k_n')\in\mathbb{Z}^{\oplus n}|\ 0\leq k_r'\leq l-1,\k_1'+\cdots+k_n'=l(n-k)-(n-p)\}
\begin{aligned}u_{n}(c\tau_n, \tau_n)= \frac{\varepsilon_1(c)}{2}, &u_{n}(ct, t)\leq \frac{\varepsilon_1(c)}{2}, t\in(\tau_n, t_n), &u_{n}(c(\tau_n+t_n-\tau_n), \tau_n+t_n-\tau_n)\rightarrow 0, n\rightarrow \infty.\end{aligned}
\rho(p,r,d) +r(r+2) \geq \begin{cases}\frac{1}{2}r(r+1), \mbox{if}\; \; d = p-1, \gamma\Big(p-1-d\Big)\Big(r+1-\frac{1}{2}(p-1-d)(\gamma+1)\Big) \mbox{if}\; \; r \geq p-1-d >0.\end{cases}
\frac 1 4 \left( \langle D \bar F_2 - D F_2, D \bar F_1 +D F_1 \rangle -\langle D \bar F_2 + D F_2, D \bar F_1 -D F_1 \rangle\right) & = \frac 1 4 \left((1-\mu_2) ( 1+\mu_1) - (1+\mu_2) (1-\mu_1)\right) \langle D \bar F_2, D \bar F_1\rangle & = \frac{\mu_1 - \mu_2}{2}\langle D \bar F_2, D \bar F_1\rangle
\| \varphi(x) \widetilde{f}\|_{C^{k, \alpha}(B_a(0))}= \| f \|_{C^{k,\alpha}_{\phi, \varphi}(B_{a\phi(x)}(x))}\|\varphi (x)\widetilde{f}\|_{L^2 (B_a(0))}= \| f \|_{L^2_{\phi^{-n}\varphi^2}(B_{a\phi(x)}(x))}.
\langle de^{i\delta_1}U(x_0)\,|\,de^{i\delta_1}V(x_0)\rangle_g= \langle de^{i(\delta_1-\frac{\sigma}{2})}U(x_0) \,|\, de^{i(\delta_1-\frac{\sigma}{2})}V(x_0)\rangle_g=\langle U(x_0)\,|\,V(x_0)\rangle_g.
[J^{x\bar{x}},Q^{\pm i}]=\pm\frac{1}{2}Q^{\pm i}\,,\quad[J^{x\bar{x}},Q^\pm_i]=\mp\frac{1}{2}Q^\pm_i\,,\quad[J^{x\bar{x}},S^\pm_i]=\pm\frac{1}{2}S^\pm_i\,,\quad[J^{x\bar{x}},S^{\pm i}]=\mp\frac{1}{2}S^{\pm i}\,.
\displaystyle - \left( \frac{6 5}{2 2 4}+ \frac{1 1}{1 4 0}\zeta ( 3 ) + \frac{1}{1 6}\zeta ( 4 ) + \frac{1}{1 2 0}\zeta ( 5 ) + \frac{1}{5 6 0}\zeta ( 3 )^{2}+ \frac{1}{1 1 2}\zeta ( 6 ) \right)
\limsup_{t \to \infty}\P_{s,\mu}(X_{t}\in B | \tau_{A}> t) = \limsup_{t \to \infty}\P_{s,\mu}(X_{t + u}\in B | \tau_{A}> t + u) = \limsup_{t \to \infty}\P_{s+u,\phi_{u+s,s}(\mu)}(X_{t}\in B | \tau_{A}> t) &= \limsup_{t \to \infty}\P_{s+u,\nu}(X_{t}\in B | \tau_{A}> t),
\lim_{n\longrightarrow\infty}{\bf P}_{X^n, Y^n}^{{\bf P}^*}\Big\{(X^n,Y^n)\in{\cal X}^n\times{\cal Y}^n:~\frac{1}{n+1}\left|{\bf E}^{{\bf P}^*}\{i^{{\bf P}^{*}}(X^n,Y^n)\}-i^{{\bf P}^{*}}(X^n,Y^n)\right|>\epsilon\Big\}=0
\frac{3}{2}U^{j}- 2 U^{j-1}+ \frac{1}{2}U^{j-2}= h f(U^j) + \frac{3}{2}g( U^{j-1}) \Delta_h W^j \quad- \frac{1}{2}g(U^{j-2}) \Delta_h W^{j-1}, j\in\{2,\dots,N_h\},
H_{J}:=-\alpha\gamma\sum_{d=1}^{m-1}\frac{[d]}{1+\beta\gamma[d]}{}+\sum_{r=1}^{m}\frac{(-\beta\delta)^{r-1}[r-1]!\,q^{-r(r-1)/2}}{\prod_{p=0}^{r-1}(1+\beta\gamma[m-1-p])}e_{r}(\widehat{X}_{j_{1}}, q\widehat{X}_{j_{2}}, \ldots , q^{m-1}\widehat{X}_{j_{m}}),
e_{\Psi}\left( \rho \right) =\rho \left( \mathfrak{e}_{\Psi}\right) \mathfrak{e}_{\Psi}\doteq \sum\limits_{\mathcal{Z}\in \mathcal{P}_{\mathrm{f}},\;\mathcal{Z}\ni 0}\frac{\Psi_{\mathcal{Z}}}{\left\vert \mathcal{Z}\right\vert}\in \mathcal{U}
\|T+{\mathcal K}(\tau;{\mathcal I}\|_{{\mathcal E}/{\mathcal K}}= \inf_{K \in{\mathcal K}(\tau;{\mathcal I})}|\|T-K\|| \ge \inf_{K \in{\mathcal K}(\tau;{\mathcal I})}\|T-K\| \ge \|p(T)\|.
\displaystyle - \frac{1}{4 0 3 2 0}( 4 2 7 6 8 1 + 1 0 4 8 8 1 2 \xi + 8 8 9 3 7 5 \xi^{2}+ 1 3 0 7 7 4 \xi^{3}- 1 4 5 6 3 0 \xi^{4})
\| U^k\|_{C^{t_k, \alpha}_{\phi, \varphi_k}(B_{\phi(x)/2}(x))}& \le C\left( \sum_{j=1}^{n+1}\|(L U)_j\|_{C^{-s_j, \alpha}_{\phi, \phi^{t_j+ s_j}\varphi_j}(B_{\phi(x)}(x))}+ \sum_{j=1}^{n+1}\| U^j \|_{L_{\phi^{-n}\varphi_j^2}^2(B_{\phi(x)}(x))}\right),
w_n(y,s) = \frac{1}{M_n}\,u_n\left(\bar x_n + r_n y, \bar t_n + r_n^2 s\right), (y,s) \in \frac{\Omega - \bar x_n}{r_n}\times \left(-\frac{\bar t_n}{r_n^2}, \frac{T-\bar t_n}{r_n^2}\right) = Q_n.
\begin{gathered}\begin{multlined}\Phi_\cdot^{a,b}(i,j): \{0,\ldots,a-1\}\times \{0,\ldots,b-1\}\rightarrow \{0,\ldots,ab-1\}\end{multlined}\Phi_\cdot^{a,b}(i,j) \coloneqq bi+j\;, \bar{\Phi}_\cdot^{a,b}(i) \coloneqq (\bar{\Phi}_{\cdot 0}^{a,b}(i), \bar{\Phi}_{\cdot 1}^{a,b}(i)) \coloneqq (\Phi_\cdot^{a,b})^{-1}(i)\;.\end{gathered}
\begin{array}{ll}ds_6^2 = e^{2\lambda}\left( dy^2 + dz^2\right) + e^{2\varphi}\left( - d\tau^2 + \tau^2 d\Omega_{3,-1}^2 \right) , H_e = id\left( e^{2\lambda}dy \wedge dz\right) .\end{array}
L_{0}^{\prime}^{( 0 )}={\L 1_{\eta}}_{0}^{( 0 )}-{\L 1_{u}}_{0}^{( 0 )}={\L 1_{\eta}}_{0}^{( 0 )}-{\L 1_{Q}}_{0}^{( 0 )}-{\L 1_{W}}_{0}^{( 0 )}.
h^{(Q_N^1..Q_N^{k+1})}(u,\eta)= \big(\bar{f}_1^{Q_N^1}(u,\eta_1),\bar{f}^{Q_N^2}_2(u,\eta_2),\dots, \bar{f}^{Q_N^k}_k(u,\eta_k), \bar{f}^{Q_N^{k+1}}_{k+1}(u)\big)^{\top}
\left\langle -p_{0N}\right|h_{\mu\nu}(p_{0}) \alpha_{1}^{\mu}\tilde{\alpha}^{\nu}_{1}= \left. \frac{4}{\alpha'}h_{\mu\nu}(p_{0}) \frac{\partial^{2}}{\partial \bar{\psi}_{1\mu}\partial \psi_{1\nu}}\left\langle \Omega \left(\psi_{n},\bar{\psi}_{n};p_{0}\right) \right| \right|_{\psi,\bar{\psi}=0}~,
\Sigma_{n_{i}=0}^{\infty}\frac{H_{n_{i}}(\sqrt{\omega}x_{1i})H_{n_{i}}(\sqrt{\omega}x_{2i}) (\frac{z}{2})^{n_{i}}}{n_{i}!}= \frac{1}{\sqrt{1-z^{2}}}e^{\omega\left(\frac{2x_{1i}x_{2i}z -x_{1i}^{2}z^{2}-x_{2i}^{2}z^{2}}{1-z^{2}}\right)}
\lambda\left\{x \in \mathbb{I}:\frac{\log q_n(x)}{n}\geq b+ \varepsilon\right\}\geq \left(\frac{1}{3b_\varepsilon^2}\right)^{n-1}\cdot |I(b_\varepsilon)| &= \left(\frac{1}{3b_\varepsilon^2}\right)^{n-1}\cdot \frac{1}{b_\varepsilon(b_\varepsilon+1)}\geq \left(\frac{1}{3b_\varepsilon^2}\right)^{n}.
V(L,\Delta\Theta )~=~-\frac{2}{\pi}~\frac{(2g_{\mbox{\tiny YM}}^2N)^{\frac{1}{2}}}{L}(1-l^2)^{\frac{3}{2}}\left (\int_1^{\infty}\frac{dy}{y^2\sqrt{(y^2-1)(y^2+1-l^2)}}\right )^2~.
\sum_{B\subset S_8}E^{(S_4,B)}_{n,\ell}= \frac{\ell_1 \left( n-\ell_4 \right) \left( \sum_{j=0}^6 n^j p_j\left( \ell_1,\ldots,\ell_4 \right) \right)}{n^8 (n-1)(n-2)(n-3)(n-4)(n-5)(n-6)(n-7)},
\left(I_{-}^{\alpha,\alpha',\beta,\beta',\gamma}f\right)(x)=\frac{x^{-\alpha'}}{\Gamma(\gamma)}\int_x^{\infty}(t-x)^{\gamma-1}t^{-\alpha}F_3\left(\alpha,\alpha',\beta,\beta';\gamma;1-\frac{x}{t},1-\frac{t}{x}\right)f(t)\,dt,
0= G(x_1,x_2)= \pi^{-1}(v)(x_1,x_2)-\pi^{-1}(v)(x_2+x_1,x_2)+\pi^{-1}(v)(x_2+x_1,x_1) = \pi^{-1}(v)(x_1,x_2)-\pi^{-1}(v)(x_2-x_1,x_2)+\pi^{-1}(v)(x_2-x_1,x_1).
-(2 a_2 p_t+2a_4 q_t+2z_t)(A(Q_t)-A(\overline{Q}_t)) &\qquad\leq 2 a_2 C_A|q_t||p_t|+2a_4 C_A|q_t|^2+2 C_A|q_t||z_t| &\qquad\leq a_2 C_A (|q_t|^2+|p_t|^2)+2a_4 C_A|q_t|^2+C_A(|q_t|^2+|z_t|^2) &\qquad= (a_2+2a_4+1)C_A |q_t|^2+ a_2 C_A |p_t|^2+C_A |z_t|^2.
\textbf{BR}_{\nu,\mu}(a):=\left(\textbf{BR}_{\nu}(a)\right)^{\mu}=\lambda_1\sum^{\infty}_{n=0}\frac{(-1)^n}{k_1n+\lambda_1}\left(\begin{array}{cc}k_1n+\lambda_1 n\end{array}\right)a^n,
\begin{pmatrix}(n-2)(k-1)+1 (2-2k) F_{1,1}^{(1)}\dots (2-2k) F_{1,n-1}^{(1)}\end{pmatrix}=\begin{pmatrix}(n-2)(k-1)+1 0 \dots 0 12(k-1) \end{pmatrix}
u_t+bu_x+u_{xxx}+u_{xyy}=f_{1x},f_1\in L_2(Q_T), u\big|_{t=0}=u_0\in L_2,u\big|_{t=T}=u_T\in L_2, u\big|_{x=0}=u\big|_{x=R}=0,u_x\big|_{x=R}=\nu_1
|\begin{matrix}p_{v}-p_{u_{1}}&(p_{v}-p_{u_{1}})\cdot q_{u_{1}}\\p_{v}-p_{u_{2}}&(p_{v}-p_{u_{2}})\cdot q_{u_{2}}\\p_{v}-p_{u_{3}}&(p_{v}-p_{u_{3}})\cdot q_{u_{3}}\end{matrix}|=0.
d h = \sum_{n = 0}^{\left[ \frac{p + 2}{2}\right]}\frac{1}{n !}d h^{( p + 2 - 2 n )}{\cal F}^{n}+ \sum_{n = 1}^{\left[ \frac{p + 2}{2}\right]}\frac{1}{( n - 1 ) !}( - 1 )^{p}h^{( p + 2 - 2 n )}( C \Gamma_{\mu}\Gamma_{1 1})_{\alpha \beta}\Pi^{\mu}\Pi^{\alpha}\Pi^{\beta}{\cal F}^{n - 1}\quad ,
D^{\boldsymbol{r}}f=\sum_{\boldsymbol{m}}\vartheta_{\boldsymbol{m}}D^{\boldsymbol{r}}\varphi_{\boldsymbol{N},\boldsymbol{m}}+\sum_{\boldsymbol{j},\boldsymbol{k}}\theta_{\boldsymbol{j},\boldsymbol{k}}D^{\boldsymbol{r}}\psi_{\boldsymbol{j},\boldsymbol{k}},
q_{c}( a , b , r ) = \frac{1}{8 \pi}\sum_{m = - \infty}^{+ \infty}\int_{- \infty}^{+ \infty}{d k \sum_{\gamma , \lambda}}\beta_{\lambda m}^{2}f_{\lambda m}^{( q )}( \gamma a , \gamma r ) , \quad q_{c}= \varepsilon , \, p_{i},
F_C^{(n)}(a,b; c_1, \ldots c_n; z_1, \ldots z_n) &= \sum_{k_1, \ldots k_n =0}^{\infty}\!\! \frac{(a)_{k_1 + \,\ldots \, k_n}(b)_{k_1 + \, \ldots \, k_n}}{(c_1)_{k_1}\ldots(c_n)_{k_n}}\frac{z_1^{k_1}\ldots z_n^{k_n}}{k_1! \ldots k_n!}
u(t,x) = (1+t)^{-\frac{\mu}{2}}w[u_0](t,x) +\frac{1}{2^{\sqrt{\delta}-1}}\int_0^t w[u_0](s,x) K_0(t,0;s; \mu,\nu^2) \, \mathrm{d}s +\frac{1}{2^{\sqrt{\delta}-1}}\int_0^t w[u_1+\mu\, u_0](s,x) K_1(t,0;s; \mu,\nu^2) \, \mathrm{d}s + \frac{1}{2^{\sqrt{\delta}-1}}\int_0^t \int_0^{t-b}w[f](s,x;b) E(t,0;b,s;\mu,\nu^2) \, \mathrm{d}s \, \mathrm{d}b,
A_i =(Z_{1i},{(Z_{1i}^2-1)}/{2}, (Z_{2i}^2-1)/2)^\tau, B_i =((Z_{1i}^2-1)(Z_{2i}^2-1)/2, Z_{1i}(Z_{2i}^2-1)/2, - (Z_{2i}^4-6Z_{2i}^2+3)/12)^\tau.
\eta^{D}(A\times[b,\infty))=e^{-\pi\lambda b}\langle\eta^{D},f_{b}\rangle =e^{-\pi\lambda b}\langle\eta^{D},f\rangle &=(\pi\lambda)^{-1}e^{-\pi\lambda b}Z_{\lambda}^{D}(A)=\int_{A\times[b,\infty)}Z_{\lambda}^{D}(\mathrm{d}x)e^{-\pi\lambda h}\mathrm{d}h
m'_{ij}= \big( c_{\lambda_i}(i) - (-1)^{k_s}e_{\lambda_i}(i) \big)\,\big( c_{\lambda_j}(j) + (-1)^{k_s}e_{\lambda_j}(j) \big)+ \sum_{m>0}\beta^m c_{\lambda_i+m}(i)\,c_{\lambda_j}(j) + \sum_{\ell>0}\sum_{m\geq 0}(-1)^\ell \left( \binom{\ell+m-1}{m}+ \binom{\ell+m}{m}\right) \beta^{m}\, c_{\lambda_i+\ell+m}(i)\, c_{\lambda_j-\ell}(j) .
{\cal H}_{T C S}( s , m , E_{c u t}) = \left\{\left| \Psi \right\rangle : \: P \left| \Psi \right\rangle = s \left| \Psi \right\rangle , Q \left| \Psi \right\rangle = m \left| \Psi \right\rangle , H_{C F T}\left| \Psi \right\rangle \leq E_{c u t}\left| \Psi \right\rangle \right\}\: .
T_{\lambda\nu}(f)(x) = f_{\lambda\nu}(\tau_\rho(y)) \cdot f(y) = \begin{cases}0, \rho \not= \lambda \nu f_{\lambda\nu}(x) \cdot f(y), \rho = \lambda\nu \end{cases}= f_\lambda(x) \cdot f_\nu(\tau_\nu(y) ) \cdot f(y)
\ell_4|_{\Lambda_0}= \,{-}(a_2^2 + a_2 b_2 + b_2^2)\widetilde{\ell}_0 - (a_2 + b_2)\widetilde{\ell}_2 \ell_6|_{\Lambda_0}= \, a_2 b_2 (a_2+b_2)\widetilde{\ell}_0 + a_2 b_2 \widetilde{\ell}_2.
\begin{cases}|\left< \hat{u}^n_t, \Psi(\cdot,y_1) \right>|\leq \left(1 + \frac{C}{\sqrt{n}}||\eta_t||_{-2,1}\right) ||\Psi||_{\mathcal{C}^2_b}, |\left< \hat{u}^n_t, \Psi(\cdot,y_1)-\Psi(\cdot,y_2) \right>|\leq \left(1 + \frac{C}{\sqrt{n}}||\eta_t||_{-2,1}\right) ||\Psi||_{\mathcal{C}^3_b}|y_1-y_2|.\end{cases}
R_{1}=(\underbrace{0,1,\cdots, 1}_{k+2},\underbrace{0, \cdots, 0}_{r}, \underbrace{0,\cdots, 0}_{t}) R_{2}=(\underbrace{1,0,\cdots, 0}_{k+2},\underbrace{1, \cdots, 1}_{r}, \underbrace{0,\cdots, 0}_{t}).
R =\frac{1}{NB}\left(|\mathcal{M}_2|\times|\mathcal{A}| +|\mathcal{M}_1|\times|\mathcal{A}^c|+ B\times|\mathcal{I}|\right) =[H(p_2)-H(p_1)]\times q_1 +[H(p_2^*)-H(p_1^*)]\times q_2+[H(p_1^*)-H(p_2)] =[H(p_1^*)-H(p_1)]\times q_1 +[H(p_2^*)-H(p_2)]\times q_2,
A_L^N:= \prod_{k=N+1}^\infty \frac{\left|\mu_k-\lambda_1\right|\left|\lambda_k-\mu_1\right|}{\left|\lambda_k-\lambda_1\right|\left|\mu_k-\mu_1\right|}= \prod_{k=N+1}^\infty \Big|1+ \frac{(\mu_k-\lambda_k)(\lambda_1-\mu_1)}{(\lambda_k-\lambda_1)(\mu_k-\mu_1)}\Big|
h_{(j,N)}^{-}=1_{A_{j}^{(N)}}g_{-}-1_{A_{j}^{(N)}}\mathbb{E}\left(g_{+}\;|\;A_{j}^{(N)}\right) & \leqslant\phi_{(j,N)}f\leqslant1_{A_{j}^{(N)}}g_{+}-1_{A_{j}^{(N)}}\mathbb{E}\left( g_{-}\;|\;A_{j}^{(N)}\right) =h_{(j,N)}^{+},
\begin{cases}\Psi''-\Delta \Psi+ \overline A_1^T\Psi=0 \hbox{in}(0, T)\times \Omega, \Psi=0 \hbox{on}(0, T)\times \Gamma_0, \partial_\nu \Psi+\overline B_1^T\Psi=0 \hbox{on}(0, T)\times\Gamma_1\end{cases}
F(t) =\frac{1}{2}H(e^{-2t})=\frac{2\cosh(it)[\sinh((n-i)t) + \sinh(it)\cosh((n-2k)t)]}{\sinh(nt)}&=\frac{2\cosh(it)\sinh((n-i)t) + \sinh(2it)\cosh((n-2k)t)}{\sinh(nt)}
\delta \Psi_{M}= 0 \qquad \Longrightarrow \qquad \hat{\nabla}_{M}\epsilon ( x , y ) = 0 \Longrightarrow \left( \begin{matrix}{\hat{\nabla}_{\mu}\eta ( x ) = 0}\\{\hat{\nabla}_{\alpha}\varsigma ( y ) = 0}\\ \end{matrix}\right) \ .
D_m=\omega_Q^{\frac{1}{\bar{p}_2}-\frac{1}{\bar{p}_1}}\omega_Q^m\int_0^\infty...\int_0^\infty\frac{\prod_{i=1}^m|r|^{Q-\frac{Q}{p_i}-1}}{(1+\prod_{i=1}^m|s|)^m}dr_1...dr_m.
\psi_3^\star = \psi_3 -\frac32 \psi_2 + \frac35 \psi_1-\frac{1}{20}\psi_0 =\begin{cases*}8x^3-6x^2+\frac65 x -\frac{1}{20}, $x \in [0,1/2),$ -8x^3+18x^2-\frac{66}{5}x+\frac{63}{20}, $x \in [1/2,1). $ \end{cases*}
g(x_0, x_1, \ldots , x_{\ell k_0 + m}) := (g_0(x_0, \ldots, x_{k_0-1}), g_0(x_{k_0}, \ldots, x_{2k_0 - 1}), \ldots, g_0(x_{(\ell-1)k_0}, \ldots, x_{\ell k_0 - 1}), x_{\ell k_0}, \ldots, x_{\ell k_0 + m-1}).
\begin{array}{c c}{w = \textstyle{\frac{1}{4}}w_{\mu \nu}\tilde{\sigma}^{\mu}\sigma^{\nu}\, , ~ ~ ~ ~}{~ ~ ~ ~ \tilde{w}= \textstyle{\frac{1}{4}}w_{\mu \nu}\sigma^{\mu}\tilde{\sigma}^{\nu}\, .}\end{array}
m(\xi,\eta,\rho) = O\big( f(|\xi|,|\eta|,|\rho-\xi|,|\rho+\eta|) \big) & \, \Longleftrightarrow \,\big\| \mathcal{F}^{-1}\big(m^{k_1,k_2,k_3,k_4}\big)\big\|_{L^1}\lesssim f(2^{k_1},2^{k_2},2^{k_3},2^{k_4}) \mathbf{1}_{\mathcal{Y}}(k_1,k_2,k_3,k_4).
S=-\sum_{I=1}^K\,\int d\tau_I\,(m^I_0\,\sqrt{v_\mu^I\,v_I^\mu}+{\rm tr}\,Z_I\lambda^{-1}_I{\dot\lambda}_I)-\int d^4x\,{\rm tr}\,\bigl(j_\mu\,A^\mu+{1\over 16\pi}\,F_{\mu\nu}\,F^{\mu\nu}\bigr).
\tilde{\boldsymbol{K}}_{j-1}=\sum_{r=1}^n \frac{\tilde{S}_{jr}}{\tilde{S}_{1r}}\pi_r\big(\tilde{\boldsymbol{K}}\big)=\sum_{i =0}^{n-1}a_i^{(j-1)}(\boldsymbol{q}) \tilde{\boldsymbol{K}}^i, j=1,\ldots,n .
\sum_{h_1}f'(h_1)(\xi) =\sum_{h_1}\sum_jf(g_j)(\pi(\xi)|_{X_j})=\sum_g\sum_{h_1, h_1\in c(g, X)}f(g)(\pi(\xi)|_{X_{g, h_1}}) &=\sum_gf(g)(\pi(\xi))=\sum_{g}\pi^*(f(g))(\xi).
\partial_tQ =-\varrho\det(a_{ij})(p-n-1+e^{2w})|\nabla w|^2 &+\varrho\det(a_{ij})\sum a^{ij}(Q_{ij}-w_{ik}w_{kj}+w_iw_j-\delta_{ij}|\nabla w|^2) &\leq-\varrho\det(a_{ij})(p-n-1+e^{2w})|\nabla w|^2+\varrho\det(a_{ij})(\max_ia^{ii}-\sum a^{ii})|\nabla w|^2 &\leq-C_0Q
|v_\pm (\lambda)\rangle \,=\, \sqrt{\frac{\omega (\lambda)\mp M}{2 \, \omega (\lambda)}}\, \left(\begin{array}{c}|v_1 (\lambda)\rangle \displaystyle{\frac{C}{\omega (\lambda) \mp M}}\, |v_1 (\lambda) \rangle\end{array}\right)\;,
L_{M}= - \frac{1}{4}F_{\mu \nu}^{a}F^{a \mu \nu}- \frac{1}{2}D_{\mu}\phi^{a}D^{\mu}\phi^{a}- \frac{\beta^{2}g^{2}}{4}( \phi^{a}\phi^{a}- v^{2})^{2}\ ,
A_{n_1,\ldots,n_4}^{(p_1,p_2)}= \delta\tbinom{n_1,n_2}{p_1,p_2}a_{n_3,n_4}+ \delta\tbinom{n_1,n_3}{p_1,p_2}a_{n_2,n_4}+\delta\tbinom{n_1,n_4}{p_1,p_2}a_{n_2,n_3}&+\delta\tbinom{n_2,n_3}{p_1,p_2}a_{n_1,n_4}+\delta\tbinom{n_2,n_4}{p_1,p_2}a_{n_1,n_3}+\delta\tbinom{n_3,n_4}{p_1,p_2}a_{n_1,n_2}.
\hat{Z}_l=\sum_{1\leq i<j}^{n_l}L_{ij}^2-\bigg(\sum_{a=1}^{l}r_a^2 \bigg)\bigg(\sum_{a=1}^{l}\frac{1}{r_a^2}g_a(\Omega_a) \bigg),~~~l=2,\cdots,N-1.
(\lambda + A)^{-1}y = \lim\limits_{n \rightarrow \infty}\lim\limits_{t \rightarrow 0+}-t^{1 - 2 \alpha}( \lambda + A)^{-1}U'(t)x_n = \lim\limits_{n \rightarrow \infty}\lim\limits_{t \rightarrow 0+}-t^{1 - 2 \alpha}U'(t) ( \lambda + A)^{-1}x_n = \lim\limits_{n \rightarrow \infty}c_{\alpha}J^{\alpha}_A (\lambda + A)^{-1}x_n = 0.
\int\int\frac{(x^{2}+y^{2})^{2}y^{2}-4x^{2}y^{2}(x^{2}+y^{2})+3y^{2}(x^{2}+y^{2})^{2}-4y^{4}(x^{2}+y^{2})}{(x^{2}+y^{2})^{4}}dxdy
\begin{array}{r l}{{\delta e}}&{{= - \partial_{\tau}( a e ) -{i}\alpha ( \tau ) \psi ,}}\\{{\delta \psi}}&{{= - a ( \tau ) \dot{\psi}-{\frac{3}{2}}\dot{a}\psi -{\frac{1}{4}}\alpha ( \tau ) \dot{e}-{\frac{1}{2}}\dot{\alpha}( \tau ) e .}}\\ \end{array}
\mu \equiv \frac{\sqrt 3 h}{\xi}, \quad \chi \equiv \frac{\psi}{\xi}, \quad \nu \equiv \frac{N}{\xi^{2}}, \quad \zeta \equiv \frac{\pm 3 K}{\xi^{2}}, \quad \lambda \equiv \frac{\pm \Lambda_{\mathrm{M}}}{\xi^{2}}, \quad \frac{d}{d T}\equiv \xi \frac{d}{d \tau},
F_t(g) = t^\alpha \int_0^\infty g\left(e^{-t\lambda}\right)e^{-t\lambda}\mathrm d\nu(\lambda),G(g) = \frac{c}{\Gamma(\alpha)}\int_0^\infty g\left(e^{-t}\right)t^{\alpha-1}e^{-t}\mathrm dt.
\tau_j(\psi(pap)) =\sum_{i,\ell=1}^{n}\langle a\xi_\ell, \xi_i\rangle\tau_j(b_i^*b_\ell) &=\sum_{i,\ell=1}^{n}\langle a\xi_\ell, \xi_i\rangle\langle\pi_\tau(b_i^*b_\ell)c_j\hat 1_B, \hat 1_B\rangle &=\sum_{i,\ell=1}^{n}\langle a\xi_\ell, \xi_i\rangle\langle c_j\hat b_\ell, b_i^*\rangle &=\langle a\otimes c_j(\xi),\xi\rangle,
0= \int\limits_X \bar\partial\xi \wedge \overline{\rho_\omega^{2,\,0}}\wedge \rho_\omega^{2,\,0}\wedge \omega_{n-3}-\int\limits_X \xi \wedge \overline{\rho_\omega^{2,\,0}}\wedge \bar\partial\rho_\omega^{2,\,0}\wedge \omega_{n-3}-\int\limits_X \xi \wedge \overline{\rho_\omega^{2,\,0}}\wedge \rho_\omega^{2,\,0}\wedge \bar\partial\omega_{n-3}.
\begin{aligned}|\dot{\phi}| <\frac{M}{R}+K\iint_{\mathbb T \times \mathbb R}|\sin(\theta-\phi)|f(\theta,\omega,t) ~ d\theta d\omega. <\frac{M}{R}+\sqrt{\frac{K}{R}}\sqrt{KR\iint_{\mathbb T \times \mathbb R}\sin^{2}(\theta-\phi)f(\theta,\omega,t) ~ d\theta d\omega}&<\frac{M}{R}+\sqrt{\frac{K}{R}}\sqrt{M+\dot{R}}.\end{aligned}
(i_f X\wedge Y)(g_2,\dots, g_{k+l})+(-1)^{kl}(i_f Y\wedge X)(g_2,\dots, g_{k+l}) &=(i_f X\wedge Y)(g_2,\dots, g_{k+l})+(-1)^{kl+k(l-1)}(X\wedge i_f Y)(g_2,\dots, g_{k+l}) &=(i_f X\wedge Y)(g_2,\dots, g_{k+l})+(-1)^{k}(X\wedge i_f Y)(g_2,\dots, g_{k+l})
\mathbb{P}\left(\frac{\log Z_{n}-n \mu}{\sigma \sqrt{n}}\leq x\right)-\Phi(x) =-\left[\mathbb{P}\left(\frac{\log Z_{n}-n \mu}{\sigma \sqrt{n}}> x\right)-(1-\Phi(x))\right] &\leq-\left[(1-\Phi(x))\exp\left\{-C\frac{1+x^{3}}{\sqrt{n}}\right\}-(1-\Phi(x))\right] &=-(1-\Phi(x))\left(\exp\left\{-C\frac{1+x^{3}}{\sqrt{n}}\right\}-1\right).
\exp{(2\pi i (x\tau(t)+y\delta)(c\tau(t)+d\delta)^{-1})}=\exp{\Bigg(2\pi i (x \y)\begin{pmatrix}\tau(t) \delta\end{pmatrix}\begin{pmatrix}c\tau(t)+d\delta\end{pmatrix}^{-1}\Bigg)}.
\left(\frac{f_1}{f_k}\right)^p + \frac{a_{k-1}}{a_k}\left(\frac{f_2}{f_k}\right)^p+\cdots + \frac{a_1}{a_k}= \frac{a_1}{a_k}\left(\frac{f_1}{f_k}\right)^{p-1}\;\; 1\le k\le h
a_{1}^{1234}=a_{1}^{1243}=a_{1}^{1324}=a_{1}^{1342}=a_{1}^{1423}=a_{1}^{1432}= a_{1}^{2134}=a_{1}^{2143}=a_{1}^{2314}=a_{1}^{2341}=a_{1}^{2413}=a_{1}^{2431}= a_{1}^{3124}=a_{1}^{3142}=a_{1}^{3214}=a_{1}^{3241}=a_{1}^{3412}=a_{1}^{3421}= a_{1}^{4123}=a_{1}^{4132}=a_{1}^{4213}=a_{1}^{4231}=a_{1}^{4312}=a_{1}^{4321}
\rho\left(M_\alpha(z), M_\beta(w)\right) = \det\left(\mu_{\alpha}(z)\right)^{k+\nu}\det\left(\lambda_\alpha(z)\right)^{-\nu}\det\left(\mu_{\beta}(w)\right)^{k'+\nu'}\det\left(\lambda_\beta(w)\right)^{-\nu'},
n_{\rm eff}^2(n,\eta) \, = \, n^2-n^2\frac{\vert b_m \vert}{\ell_{_{\rm C}}^{-2m}}\biggl(\frac{n}{2\pi}\biggr)^{2m}\biggl[\ell_0-\frac{\ell_0-\ell_{\rm b}}{1+( \eta /\eta_0)^2}\biggr]^{-2m},\quad \frac{a''}{a}= \frac{2(\ell_0-\ell_{\rm b})}{\eta_0^2\ell_{\rm b}}\frac{1-3(\eta /\eta_0)^2}{[1+(\eta /\eta_0)^2]^3},
\int_0^r \left[(-\phi''-\frac{z}{2}\phi'+\frac{z}{2}e_\sigma\psi')\cdot\phi+(-\psi''-\frac{z}{2}\psi'+\frac{z}{2}e_\sigma\phi')\cdot\psi\right]{\rm{d}}z =\int_0^r |(\phi',\psi')|^2+\frac{1}{4}\int_0^r |(\phi,\psi)|^2-\frac{1}{2}\int_0^r (e_\sigma+ze_\sigma')\psi\phi.
\langle 0 \vert B \vert \theta_{2},\dots,\theta_{k},\theta_{1},\theta_{k+1},\dots,\theta_{n}\rangle =S_{\rm gs}\langle0 \vert B \vert \theta_{1},\dots,\theta_{n}\rangle_{\rm in}, S_{\rm gs}=\prod\limits_{l=2}^{k}S(\theta_{l}-\theta_{1}), \theta_{1}>\cdots >\theta_{n}.
-{\frac{1}{r^{2}}}\left( 1 -{\frac{r_{0}}{r}}\right){\frac{d}{d r}}\left( r ( r - r_{0}){\frac{d R_{\ell}}{d r}}\right) +{\frac{\ell ( \ell + 1 )}{r^{2}}}\left( 1 -{\frac{r_{0}}{r}}\right) R_{\ell}= k^{2}R_{\ell}
\partial_t \zeta_t(x) = \frac{(x-\gamma(t,\zeta_t(x))\cdot(\partial_s\gamma(t,\zeta_t(x))\times \partial_s^3\gamma(t,\zeta_t(x)))}{1-(x-\gamma(t,\zeta_t(x))\cdot\partial_s^2\gamma(t,\zeta_t(x))}.
\min_{r > 0}\left[ v \frac{P_{n \ell}^{2}( 1 )}{r^{2}}+ \beta \sqrt{m^{2}+ r^{2}}\right] \leq{\cal E}_{n \ell}( v ) \leq \min_{r > 0}\left[ v \frac{P_{n \ell}^{2}( 2 )}{r^{2}}+ \beta \sqrt{m^{2}+ r^{2}}\right] ,