harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\frac{1}{n}\sum_{t=1}^T\sum_{i=1}^n\sum_{j=1}^n\|x_{i,t}-x_{j,t}\| \le n\varepsilon_1+\tilde{\varepsilon}_2\sum_{t=1}^{T}\sum_{j=1}^n\|\epsilon^z_{j,t}\| &\le n\varepsilon_1+\frac{\tilde{\varepsilon}_2}{\sigma}\sum_{t=1}^{T}\sum_{j=1}^n(G_2\gamma_{0}\|[g_{j,t}(x_{j,t})]_+\|+G_1\alpha_{t}).
4D(v, w) = -8( w^3 - 32 w^2 + 65536 v^2 w -2097152 v^2 + 640 v w^2 - 20480 v w) &= -4\cdot 2((w^2 (w - 32) + 65536 v^2 (w - 32) + 640 wv (w - 32)),
\sum_{n\geq0}\rho^{n}\sin(n\alpha+\beta) =(\sin(\beta)-\rho\sin(\beta-\alpha))/(1-2\rho\cos(\alpha)+\rho^{2}), \sum_{n\geq0}\rho^{n}\cos(n\alpha+\beta) =(\cos(\beta)-\rho\cos(\beta-\alpha)/(1-2\rho\cos(\alpha)+\rho^{2}).
K_1< -\frac{1}{2\pi X(x_0,t)}\int_{-1}^{-X(x_0, t)}B_y(y,t)\, dy = -\frac{1}{2\pi X(x_0,t)}\left(B(-X(x_0,t),t)-B(-1,t)\right) = \ \frac{1}{2\pi X(x_0,t)}.
\Theta_0( \psi_k(g), \varphi_k(b)) &= \Theta_0((e,\dots,e,F_k(g),e,\dots,e), (e,\dots,e,F_k(b),F_k(h),\dots,F_k(h))) &= (e,\dots,e, F_k(g)F_k(b), \Upsilon(F_k(b))^{-1}F_k(h),e,\dots,e) &= (e,\dots,e, F_k(gb),e,e,\dots,e) &= \psi_k(g b).
\mathcal{B}_{n}^{\left( l\right)}\left( \lambda\right) =l!\binom{n}{l}\sum_{k=0}^{n-l}\genfrac{\{}{\}}{0pt}{}{n-l}{k}\binom{l+k-1}{k}\frac{\left( -\lambda\right)^{k}k!}{\left( \lambda-1\right)^{l+k}},
\int_{-\infty}^{\infty}\frac{\gamma(\frac{s}{2},z^{2}\pi)}{(z^{2}\pi)^{\frac{s}{2}}}e^{-2\pi ikz}dz=\frac{\Gamma(\frac{1-s}{2},k^{2}\pi)}{(k^{2}\pi)^{\frac{1-s}{2}}}.
\theta_{p+k,q+k}(\pi') =\pi_{1}(\lambda_{d},1,\Psi,\mu,\nu,(\varepsilon\mid(1,1,\dots,1)), \qquad(\kappa\mid(\frac{p+q}{2}-n,1+\frac{p+q}{2}-n,\dots,k-1+\frac{p+q}{2}-n)))
L \cdot \xi_n^{s+1}= f^s L_{1}\ldots L_r f^{-s-1}\cdot \xi_{P_1}^{s+1}\cdots \xi_{P_r}^{s+1}f^{s+1}= f^s \left( L_{1}\cdot \xi_{P_1}^{s+1}\right) \ldots \left( L_r \cdot \xi_{P_r}^{s+1}\right) = f^s b_{\xi_{\lambda_1}}(s) \xi_{P_1}^s \ldots b_{\xi_{\lambda_r}}(s) \xi_{P_r}^s &= b_{\xi_{\lambda_1}}(s) \ldots b_{\xi_{\lambda_r}}(s) \xi_n^s.
\begin{array}{l}{\partial_{0}x^{i}= - p^{i}, \qquad \partial_{0}p^{i}= - \partial_{1}\partial_{1}x^{i},}{( \partial_{0}+ \partial_{1}) \theta_{a}= 0 , \qquad ( \partial_{0}- \partial_{1}) \bar{\theta}_{\dot{a}}= 0 ,}\end{array}
n_j:=\theta(1_j) = \tilde\varphi([ -N_{\tau_j}\otimes \tau_j]) = - \tilde \varphi(N_{\tau_j}, \tau_j) = - \sum_{i=0}^{p-1}\varphi(\tau_j^i,\tau_j) , n_{j,k}:=\theta(1_{j,k}) = \tilde\varphi([\tau_k \otimes \tau_j] - [\tau_j \otimes \tau_k]) = \varphi(\tau_k,\tau_j)-\varphi(\tau_j,\tau_k),
\begin{aligned}\|S\mathcal{L}_{\mathcal{I}_l}W(x_j,x_k)u_\epsilon\|_{L_t^{\theta_q}(L^{q,2}_D)}= \lim_{n\rightarrow\infty}\left|\left<S\mathcal{L}_{\mathcal{I}_l}W(x_j,x_k)u_\epsilon,v_n\right>_{{L_t^{\theta_q}(L^{q,2}_D)},{L_t^{\theta_q'}(L^{q',2}_D)}}\right|.\end{aligned}
|\tau_{i}(\mathbf{V}^{T}_{11}) - \hat{\sigma}_{i}(\hat{\mathbf{V}}^{T}_{11})| \leq \|\mathbf{V}^{T}_{11}- \hat{\mathbf{V}}^{T}_{11}\|_{2}= \|\mathbf{V}_{11}- \hat{\mathbf{V}}_{11}\|_{2}\leq \|\mathbf{V}_{11}- \hat{\mathbf{V}}_{11}\|_F \leq \epsilon_v,
| 0 , 0 , - J , \pi > = \frac{1}{\sqrt{4}}( | \downarrow \downarrow \uparrow \uparrow > - | \downarrow \uparrow \uparrow \downarrow > - | \uparrow \downarrow \downarrow \uparrow > + | \uparrow \uparrow \downarrow \downarrow > ) \quad .
\mathcal{\hat{B}}_{-}(\lambda )=\frac{(\zeta_{-}-1/\zeta_{-})}{\kappa_{-}e^{\tau_{-}}(\lambda^{2}/q-q/\lambda^{2})\prod_{n=1}^{\mathsf{N}}\alpha_{n}\beta_{n}}\mathcal{B}_{-}(\lambda ),
v_1 ( x, 0 ) = i \phi ''_{1}+ i(\lambda p/ 2) |\phi|^{p-2}\phi_1 + i(\lambda (p-2)/ 2 ) |\phi |^{p-4}\phi^2 \bar \phi_1 = \phi_2 (x) \in L^2(R^+)\, , &v_1 ( 0, t) = h'' (t) = h'_1 (t) = h_2 ( t ) \in{H^{\frac14}(0,T)}\, .
\int_A\int_{I_0^f}(F(x)-F(y))^2J^f(dxdy)=\int_A\int_{A^c}(F(x)-F(y))^2 J^f(dxdy) \leq \int_{A^c}\int_{I_0^f}(F(x)-F(y))^2 J^f(dxdy)
F ( a , b ; z ) = 1 +{\frac{a}{b}}z +{\frac{a ( a + 1 )}{b ( b + 1 )}}{\frac{z^{2}}{2 !}}+{\frac{a ( a + 1 ) ( a + 2 )}{b ( b + 1 ) ( b + 2 )}}{\frac{z^{3}}{3 !}}+ \cdots
\alpha_{4,\,5,\,6}=\alpha_{2,\,3,\,4}=\alpha_{1,\,2,\,6}=\sqrt{\tfrac{3}{2}},\,\alpha_{1,\,3,\,5}=-\sqrt{\tfrac{3}{2}},\,\alpha_{2,\,4,\,6}=-2\sqrt{\tfrac{3}{2}},
\begin{aligned}\tilde{\nabla}_i\tilde{h}_{jk}=\varphi f'f^2(\omega_jg_{ik}+\omega_kg_{ij}), \tilde{\nabla}_i\tilde{h}_{j0}=\tilde{\nabla}_i\tilde{h}_{0j}=f\nabla_i\omega_j\varphi, \tilde{\nabla}_i\tilde{h}_{00}=-2f'\omega_i\varphi, \tilde{\nabla}_0\tilde{h}_{i0}=\tilde{\nabla}_0\tilde{h}_{0i}=f\varphi'\omega_i,\end{aligned}
\frac{\Phi_{1a}(2,2;\tau)}{\Phi_{1b}(2,2;\tau)}=q^{-2/3}\prod_{n=1}^{\infty}\frac{(1-q^{9n-4})(1-q^{9n-5})}{(1-q^{9n-1})(1-q^{9n-8})}&=q^{-2/3}(1+q+q^2+q^3-q^5-q^6-q^7+\cdots)
\begin{aligned}H_{m,n}^{(1)}(k+1) - \frac{1}{c_{1,n}(k)}H_{m,n}^{(1)}(k) = -\frac{m}{a_{n+1}(k)c_{2,n}(k)}H_{m,n}^{(2)}(k) &H_{m,n}^{(2)}(k+1) - c_{2,n}(k)H_{m,n}^{(2)}(k) =- \frac{m}{a_{n}(k+1)c_{1,n}(k)}H_{m,n}^{(1)}(k) &+ \frac{m^2}{a_{n+1}(k)a_{n}(k+1)c_{1,n}(k)}H_{m,n}^{(2)}(k).\end{aligned}
S_{1}=\sum_{1<i<j<k}\left( \binom{n_{i}}{2}a_{i}^{2}+\binom{n_{j}}{2}a_{j}^{2}+n_{i}n_{j}a_{i}a_{j}\right) , S_{2}={\displaystyle\sum\limits_{i=2}^{k-1}}n_{i}a_{i}, c =n_{1}a_{1}+n_{k}a_{k}, l =\left( n_{1}+n_{k}\right) /2,
\pm \prod_{p \in{\cal P}_3}p^{a_p-b_p+\sum_{i=1}^n (\alpha_i-\beta_i) z_{ip}}\pm \prod_{p \in{\cal P}_2}p^{b_p-c_p+\sum_{i=1}^n (\beta_i-\gamma_i) z_{ip}}\pm \prod_{p \in{\cal P}_1}p^{c_p-a_p+\sum_{i=1}^n (\gamma_i-\alpha_i) z_{ip}}= 0.
\begin{aligned}(\Delta_R \otimes \mbox{id}) \circ \Delta_R = (\mbox{id}\otimes \Delta_{\mathcal{A}}) \circ \Delta_R, &\textsl{m}\circ(\mbox{id}\otimes \varepsilon_{\mathcal{A}}) \circ \Delta_R = \mbox{id}.\end{aligned}
e^{-{\frac{1}{2}}a \phi - ( p + 1 ) A}= 1 +{\frac{\lambda_{1}}{a \sqrt{2}}}{\frac{1}{r^{p + 1}}}, e^{+{\frac{1}{2}}a \phi - ( p + 1 ) A}= 1 +{\frac{\lambda_{2}}{a \sqrt{2}}}{\frac{1}{r^{p + 1}}},
\mathcal Q_k^n(\zeta)=\left(\zeta- (k+n)((k+n-1)a_{3,1}+a_{2,1})\right)\mathcal Q_{k-1}^n(\zeta)-\,(k-1)(k+n) a_{2,0}((k+n-1)a_{3,2}+a_{2,2})\mathcal Q_{k-2}^{n}(\zeta),k\geq 1,
I(\lambda, \tau, \eta) = \frac{2e^{-12\pi i \eta - 3\pi i \lambda}\theta_0(6\tau + 1/2;8\tau) \theta_0(\lambda; \tau)}{\theta_0(1/2; 2\tau)^3}\Big(\frac{\theta_0(2\lambda + 1/2; 2\tau)}{\theta_0(\tau+1/2; 2\tau)}I_1 - \frac{\theta_0(\lambda + 1/2; \tau)^2}{\theta_0(\tau; 2\tau)}I_2\Big).
F_{t}(A_{2},A_{1}) =dA_{t}(A_{2},A_{1})+\frac{1}{2}\left[ A_{t}(A_{2},A_{1}),A_{t}(A_{2},A_{1})\right] , & =d\omega+tde+\frac{1}{2}\left[ \omega+te,\omega+te\right] , & =d\omega+tde+\frac{1}{2}\left[ \omega,\omega\right] +\frac{t}{2}\left[\omega,e\right] +\frac{t}{2}\left[ e,\omega\right] +\frac{t^{2}}{2}\left[e,e\right] ,
\begin{aligned}x_n^* - x_n^0 = \, x_n - \sum_{\beta=1}^\alpha x_\beta \rho_\beta (0, \tilde{x}) - \rho (0, \tilde{x}) > \, \rho (\hat{x}, \tilde{x}) - \sum_{\beta=1}^\alpha x_\beta \rho_\beta (0, \tilde{x}) - \rho (0, \tilde{x}) \geq 0\end{aligned}
\lambda_1(x) = -x_2-\frac{1}{2}\, x_3+1, \lambda_2(x) = x_2-\frac{1}{2}\, x_3, \lambda_3(x) = \frac{1}{2}\left(-2 x_1+x_3+1\right), \lambda_4(x) = \frac{1}{2}\left(2 x_1+x_3-1\right).
(i+i_1+\ldots i_n)\ldots(i+i_1+i_2)(i+i_1)\left([a^{i_1}q_1a^{i_2}q_2\ldots a^{i_n}q_n, a^i]-[a^{\sum_ki_k}q_1q_2\ldots q_n, a^i]\right)=0.
= \sum_{s=r-1}^{2a-r-1}\binom{s}{r-1}\sum_{b=0}^{a-s-1}\frac{(-1)^{b}}{a+b}\binom{a+b-s-1}{2b}\binom{2b}{b}= \sum_{s=r-1}^{a-1}\binom{s}{r-1}\sum_{b=0}^{a-s-1}\frac{(-1)^{b}}{a+b}\binom{a+b-s-1}{2b}\binom{2b}{b}.
\left|(y-x)-\frac{A_{x,y}(N)}N\right|\leq |x-x'|+|y-y'|+\left|(y'-x')-\frac{A_{x',y'}(N)}N\right| \leq 2\beta^n+(2L+S-2)\sum_{m=0}^{n+1}\left|\frac{R(1-(-S\beta)^m)+1}N \right|
\|L^{1/2}z_{1}\|_{H}^2 + (B_{1}v_{0}, z_{1})_{H}+ h(B_{1}z_{1}, z_{1})_{H}+ (A_{2}\varphi_{0}, z_{1})_{H}+ h(A_{2}v_{1}, z_{1})_{H}+ (\Phi\varphi_{1}, z_{1})_{H}+ ({\cal L}\varphi_{1}, z_{1})_{H}= (B_{2}\theta_{1}, z_{1})_{H}.
\begin{aligned}d_{x,r}(E,F) = r^{-1}\sup\big\{{\rm dist}(y,E) \, ; \, y\in F \cap B(x,r) \big\}\cr \hskip 1.2cm + r^{-1}\sup\big\{{\rm dist}(y,F) \, ; \, y\in E \cap B(x,r) \big\}.\end{aligned}
\begin{aligned}I\!I_{1}=\sum_{k=2}^{N-1}\|\nabla^k u\|_{H^1}^2\|\nabla^{N+1-k}n\|_{L^2}^2+\|\nabla u\|_{H^1}^2\|\nabla^{N}n\|_{L^2}^2 &\lesssim \sum_{k=2}^{N-1}(1+t)^{-(\frac{3}{2}+k)}(1+t)^{-(\frac{5}{2}+N-k)}+(1+t)^{-\frac{5}{2}}(1+t)^{-(\frac{1}{2}+N)}&\lesssim (1+t)^{-(N+4)}+(1+t)^{-(N+3)}&\lesssim (1+t)^{-(N+3)}.\end{aligned}
\frac{1}{2}\mathbb{E}{[\|x_t-x^*\|^2]}=\mathbb{E}[G_{\Gamma t}] \geq e^{-2\mu\gamma\Gamma t}G_{0}+\frac{m\tau_{N}^{2}\gamma}{4\mu}(1-e^{-2\mu\gamma\Gamma t}),
D_1 =D_2, \theta_{i^{(1)}}=\theta_{j^{(2)}}\ \mbox{for}\i^{(1)}\in \{1,\ldots,\ell^{(1)}\}\ \mbox{and}\j^{(2)}\in \{1,\ldots,\ell^{(2)}\}\ \mbox{when}\ \mathbf x_{i^{(1)}}=\mathbf x_{j^{(2)}},
\Delta^{(n,\sigma)}[a;c_1,c_{2}, \ldots ,c_n ](X) = \left( \begin{array}{cccc}a x_{11}+c_{1}x_{\sigma(1),\sigma(1)}0 \cdots 0 0 a x_{22}+c_{2}x_{\sigma(2),\sigma(2)}\cdots 0 \vdots \vdots \ddots \vdots 0 0 \cdots a x_{nn}+c_{n}x_{\sigma(n),\sigma(n)}\end{array}\right),
\delta_{\xi}g_{\mu \nu}= 2 \nabla_{( \mu}\xi_{\nu )}, \delta_{\xi}\Gamma_{\rho \sigma}^{\tau}= \frac{1}{2}g^{\tau \nu}\left( \nabla_{\rho}\delta_{\xi}g_{\sigma \nu}+ \nabla_{\sigma}\delta_{\xi}g_{\rho \nu}- \nabla_{\nu}\delta_{\xi}g_{\rho \sigma}\right) .
\sigma_{k}(n)=\prod_{i=1}^{\omega(n)}\frac{p_{i}^{(a_{i}+1)k}-1}{p_{i}^{k}-1}=\prod_{i=1}^{\omega(n)}(1+p_{i}^{k}+p_{i}^{2k}+\cdot\cdot\cdot+p_{i}^{a_{i}k}).
\alpha=\alpha(t)=\frac{P}{E}-\frac{\lambda P-|A^{3/2}u(t)|^2}{G^2-P}-\frac{e}{E}\frac{\lambda P-|A^{3/2}u(t)|^2}{\lambda e-E}=\lambda -\gamma(t)-\frac{e}{E}\beta(t).
\frac{1}{\prod_{(i,j)\in \lambda}(\lambda'_j+\mu_i-i-j+1+x)^2\prod_{(i,j)\in \mu}(\mu'_j+\lambda_i-i-j+1-x)^2}&=\prod_{i=1}^{K}\frac{\Gamma(m_i-l_i+x)^2\Gamma(l_i-m_i-x)^2(m_i-l_i+x)^2}{\Gamma(m_i+K+1+x)^2\Gamma(l_i+K+1-x)^2}\prod_{1\leq i ,j\leq K}(m_i-l_j+x)^2\ .
\sup_{n}\mathbb{E}_{\mathbb{P}}\left[\exp \left\{\kappa \sum_{i=1}^{n}\int_{T_{0}}^{T_{0}+\delta}\Delta K_{t}^{i,n}\cdot d W_{t}^{i}-\frac{\kappa}{2}\sum_{i=1}^n \int_{T_{0}}^{T_{0}+\delta}\left|\Delta K_{t}^{i, n}\right|^{2}d t\right\}\right]
I^{(m)}= \bigcap\limits_{i,j}(P_i+Q_j)^m = \bigcap\limits_{i,j}(P_i^m + P_i^{m-1}Q_j + \cdots + P_i Q_j^{m-1}+ Q_j^m) = \bigcap\limits_i P_i^m + \bigcap\limits_{i,j}P_i^{m-1}Q_j + \cdots + \bigcap\limits_{i,j}P_i Q_j^{m-1}+ \bigcap_j Q_j^m = I_1^{(m)}+ I_1^{(m-1)}I_2 + \cdots + I_1 I_2^{(m-1)}+ I_2^{(m)},
I(p_i,q_i,n_i,l_i,l_i') & = a_f(p_i^{2l_i})a_f(q_i^{2l_i'})(a_f(p_i^{2n_i}) - a_f(p_i^{2n_i-2}))(a_f(q_i^{2n_i}) - a_f(q_i^{2n_i-2}))
\partial_{\alpha}^n q=\sum_{k=1}^n\sum \frac{n!}{(k_1)!...(k_n)!}\Big(\sum_{j=1}^2\partial_{\alpha}^k f_j(\cdot,t)\circ g\Big)\prod_{l=1}^n \Big(\frac{\partial_{\alpha}^l g}{l!}\Big)^{k_l}
\begin{aligned}t(s) =\frac 12\int_0^s (x_V +x_W)\wedge (u_V+u_W) & = \frac 12\int_0^s x_V \wedge u_V + \frac 12\int_0^s x_V \wedge u_W + \frac 12\int_0^s x_W\wedge u_V +\frac 12 \int_0^s x_W\wedge u_W & =: t_V(s)+ t_{*}(s)+ t_W(s) \in\wedge^2 V \oplus (V\wedge W)\oplus \wedge^2 W,\end{aligned}
2\langle a, a-b\rangle = |a|^2+|a-b|^2-|b|^2 2\langle 3a-4b+c, a \rangle = |a|^2+(4|a|^2-4\langle a, b\rangle +|b|^2) +(|a|^2+4|b|^2+|c|^2-4\langle a, b\rangle-4\langle b, c\rangle+2\langle a, c\rangle) \quad-|b|^2-(4|b|^2-4\langle b, c\rangle+|c|^2) =|a|^2+|2a-b|^2+|a-2b+c|^2-|b|^2-|2b-c|^2
\psi_{1,\Gamma}\left( x\right) =\alpha_{0}x+\beta_{0}x\log x+\alpha_{1}+\beta_{1}\log x +F\left( \frac{1}{x}\right)+\frac{x^{2}}{2}+\sum_{\substack{\rho \left\vert \gamma \right\vert <T}}\frac{x^{\rho +1}}{\rho \left( \rho+1\right)}+O\left( \frac{x^{2}\log x}{T}\right) \ \left( x\rightarrow \infty \right)
{\cal L}_{0}^{\gamma f u l l}= \chi_{x}\upsilon_{t}+ \upsilon_{x}\chi_{t}- 2{\cal H}_{0}^{E}+ \lambda \left( u - \frac{\chi_{x}^{\gamma - 1}+ \upsilon_{x}^{\gamma - 1}}{\gamma - 1}\right) + \sigma \left( \rho - \chi_{x}\upsilon_{x}\right)
\max \left\{\lfloor \frac{1-T}{t}\rfloor,\lfloor \frac{1-T}{T-t}\rfloor\right\}+2 \leq \frac{1-T}{T-t}+2\leq \frac{1}{T-t}=\max \left\{\frac{1}{t},\frac{1}{T-t}\right\}.
\|\overline X(0,n+m)\|_2^2\le \|\overline X(0,n)\|_2^2+\|\overline X(0,m)\|_2^2+4 \|\overline X(0,n)+\overline X(n,n+m)\|_2\|Z(n,m)\|_2 &+4\|Z(n,m)\|_2^2,
\det \left( ({\mathbf{I}}-{\mathbf{T}}_{b,a})|{\mathcal{B}}_{Q}^{[\underline{j}]_a^b}\right) = 1- \prod_{0\le i\le b-a}c_{i}= 1- \prod_{i\neq j\in T}q_{ij}= 1- \prod_{\{i,j\}\subset T}\sigma_{ij}
\lim\limits_{n\to+\infty}\dfrac{n G_n(x)}{(32\ln n)^{\frac{1}{2}}}=1,\ \lim\limits_{n\to+\infty}n G_n(x)=+\infty,\ \lim\limits_{n\to+\infty}n^{\gamma}G_n(x)=0,\ \forall\ \gamma<1.
\left\langle \varphi_{a}\right| \, \varphi_{b}( 1 ) \, \left| \varphi_{c}\right\rangle = \tilde{C}_{a b c}\ , \ \ \left\langle \varphi_{a}\right| \left. \varphi_{b}\right\rangle = \tilde{C}_{a b 1}
|\alpha_1|^2 \cos(\theta)e^{\alpha_1(x)\omega}\psi_1 +|\alpha_2|^2 \sin(\theta) e^{\alpha_2(x)\omega}\psi_2 = f(x)\big(\cos(\theta)e^{\alpha_1(x)\omega}\psi_1+\sin(\theta)e^{\alpha_2(x)\omega}\psi_2\big).
\frac{1}{\sigma_1(m)}T_m F_{n,\epsilon}(z_0)= \frac{1}{\sigma_1(m)}\left\{\sum_{\substack{ad=m, b\pmod{d}}}J_{N,n}\left(\frac{az_0+b}{d}\right) - \sum_{\substack{ad=m, b\pmod{d}}}P_{n,\epsilon}\left(\frac{az_0+b}{d}\right) \right\}.
k^{-1}b(k)=- \int_{-\infty}^{\infty}\bar{u}_y(y) \psi^-_{11}(y;k) e^{-2iz y}\mathrm{d}y, &\psi^-_{11}(x;k)=1-k \int_{-\infty}^{x}u_y(y) \psi^-_{21}(y;k) \mathrm{d}y, &\psi^-_{21}(x;k)=k \int_{-\infty}^{x}\bar{u}_y(y) \psi^-_{11}(y ; k) e^{2iz(x-y)}\mathrm{d}y.
c(u) \cdot c(\beta-u) = \frac{c(E^*;u)}{c(F;u)}\cdot \frac{c(F^*;\beta-u)}{c(E;\beta-u)}= \frac{c(E^*;u)}{c(F;u)}\cdot \frac{c(F;u)\,c(F^*;\beta)}{c(E^*;u)\,c(E;\beta)}= \frac{c(F^*;\beta)}{c(E;\beta)}= c(\beta),
\sum_{q=1}^{s}\sum_{\ell_{q-1}<j < \ell_q}(-1)^{p(\eta,j)-1}i_{\eta \cup j}= \sum_{q=1}^{s}\left(\prod_{r=s+1}^{d+1}{D_{\ell_{r}-1}}\right) \left(A(q-1)-A(q)\right) &=\left(\prod_{r=s+1}^{d+1}{D_{\ell_{r}-1}}\right) \sum_{q=1}^s \left(A(q-1)-A(q)\right) &=-A(s)\prod_{r=s+1}^{d+1}{D_{\ell_{r}-1}}=-\prod_{r=1}^{d+1}{D_{\ell_{r}-1}}.
D(Z_s^{(n_{k_j}+ 1)}, X_s^{(n_{k_j}+ 1)}, Z_s, X_s, \mu^{(n_{k_j})}, \mu) \le D(Z_s^{(n_{k_j}+ 1)}, X_s^{(n_{k_j}+ 1)}, Z_s, X_s, \mu^{(n_{k_j})}, \mu^{(n_{k_j})}) + D(Z_s, X_s, Z_s, X_s, \mu^{(n_{k_j})}, \mu) = I_1 + I_2
\gamma_{1}=\left\langle\sum_{X_i\in ({\it G}_1)_{X'_{i'}}}\sum_{\stackrel{i_{\alpha_{1}}< j_{\alpha_{1}}}{\left\{i_{\alpha_{1}},j_{\alpha_{1}}\right\}\in J_{X_{i}}}}t_{i_{\alpha_{1}}}t_{j_{\alpha_{1}}}{\bf 1}_{(w(t_{i_{\alpha_{1}}})=w(t_{j_{\alpha_{1}}}))}\chi\left(\left(i_{\alpha_{1}},j_{\alpha_{1}}\right)\in i'_{\alpha_{1}}\right)\right\rangle^{l.c.}_{w'}\;\mbox{and}
|G_{[0,M]}(E,T^{n_{0}-\frac{M}{2}}x)(m,n)|= |G_{[n_{0}-\frac{M}{2},n_{0}+\frac{M}{2}]}(E,x)(m+n_{0}-\frac{M}{2},n+n_{0}-\frac{M}{2})|<e^{M^{1-}-c_3|m-n|\chi_{|m-n|>\frac{M}{10}}}.
\Delta_0u_\epsilon = -(n-6)\frac{u_\epsilon}{(\epsilon^2+r^2)^2}(n\epsilon^2+4r^2), (\Delta_0u_\epsilon)' = (n-6)(n-4)\frac{u_\epsilon r}{(\epsilon^2+r^2)^3}\Big[(n+2)\epsilon^2+4r^2\Big].
\begin{gathered}a_{k^1,\ldots,k^{n-1}}(t^1,\ldots,t^{n-1},\cdot)= =\sum_{k^{n}\in\mathcal{K}_{n}}(\mathcal{J}^{u,\aleph_{t^2_{|k^2|},\ldots,t^{n-1}_{|k^{n-1}|}},\mathcal{S}^n}_{k^{n}}(a_{k^1,\ldots,k^{n}}(t^1,\ldots,t^{n-1},\cdot)))\circ\mathcal{G}^{t^2_{|k^2|},\ldots,t^{n-1}_{|k^{n-1}|}}.\end{gathered}
\Gamma_{\underline{{\alpha}}\underline{{\beta}}}^{i}= \left( \begin{array}{c c}{{0}}&{{\delta_{\beta}^{\alpha}( \gamma^{i})_{q}^{~ r}}}\\{{- \delta_{\alpha}^{\beta}( \gamma^{i})_{~ r}^{q}}}&{{0}}\\ \end{array}\right) \, , \quad i = 1 , . . . , 5 \, ,
x_{n}\rightarrow(\frac{2\pi}{a_{s}})x_{n},t\rightarrow(\frac{2\pi}{a_{s}})\sqrt{\frac{\epsilon_{s}}{2m_{a}}}t,a_{0}\rightarrow\frac{2\pi}{a_{s}},g\rightarrow g\frac{a_{s}/2\pi^{2}}{\epsilon_{s}/2}.
X=[\begin{matrix}1&x_{11}&\cdot\cdot\cdot&x_{1p}\\ 1&x_{21}&\cdot\cdot\cdot&x_{2p}\\ &&\vdots\\ 1&x_{n1}&\cdot\cdot\cdot&x_{np}\end{matrix}]\in\mathbb{R}^{n\times(p+1)};n\ge p+1
(\mathsf{N}_{m}^{n})^{*}=\mathsf{N}_{n}^{m},\quad\mathsf{N}_{m}^{n}\mathsf{N}_{o}^{p}=\delta_{o}^{n}\mathsf{N}_{m}^{p},\quad\mathrm{Tr}(\pi(K_{2\rho})\mathsf{N}_{m}^{n})=\delta_{m}^{n}q^{(2\rho,\lambda_{m})}.
v^k(x',x_n)=\rho_n(x_n) Z^k(x') (k=1,2,\cdots, n-1), &v^n (x',x_n)=\int_{a_n}^{x_n}u(x',s)ds -\tilde u(x') \int_{a_n}^{x_n}\rho_n(s) ds.
\limsup_{n \rightarrow + \infty}\sup_{u \in \left[ 0 , t \right]}\mathbb{E}\left( \left\Vert X_{u}\left( \xi \right) - X_{u}\left( \xi_n \right) \right\Vert_{N}^2 \right)\leq\lim_{n \rightarrow + \infty}K \left( t \right) \mathbb{E}\left( \left\Vert \xi - \xi_n \right\Vert_{N}^2 \right)= 0 .
\Delta_g(\bar u) =\bar u_{rr}+\frac{m-1}{r}\bar u_r+\frac{1}{r^2}\Delta_{\theta}\bar u-\frac{r}{2}\frac{\partial \bar u}{\partial r}&=\sum_{k=1}^{\infty}\Big((\bar f_k)_{rr}+(\frac{m-1}{r}-\frac{r}{2})(\bar f_k)_r-\frac{\lambda_k\bar f_k}{r^2}\Big)\varphi_k+(\bar f_0)_{rr}+(\frac{m-1}{r}-\frac{r}{2})(\bar f_0)_r &=0.
\left( \frac{\theta^n-\theta^{n-1}}{k_n}, \chi \right) = - i \left( \nabla \frac{\theta^n + \theta^{n - 1}}{2}, \nabla \chi \right)- \left( \frac{\rho^n-\rho^{n-1}}{k_n}, \chi \right) &+ i ( \omega^n, \chi) - ( r^n, \chi),
\mathbb{P}\left( J_0\nsubseteq \hat{J}_0 \right)\leq \mathbb{P}\left(\mathcal{E}_{\delta,q^*}^c\right)+\mathbb{P}\left(\mathcal{A}^c\right) &+\sum_{l=1}^{s}\sum_{m=0}^{q^*-(s-l)}\binom{s}{l}\binom{q-s}{m}4\exp\left( -c_1\frac{n^2(1-\rho_{q^*}^2)^2\kappa_l^2}{\sigma^4d_{q^*-s+l}+\sigma^2n(1-\rho_{q^*}^2)\kappa_l}\right).
\left\| \prod\limits_{i=1}^m f_i \right\|_{\mathcal{M}^{p}_{\phi}}\ge \frac{1}{\phi\left( K+K^{-\epsilon}\right)}\Biggl( \frac{1}{|B(0,K+K^{-\epsilon})|}\int\limits_{B(0,K+ K^{-\epsilon})}g_{\epsilon,K}(x)\dx\Biggr)^{\frac{1}{p}}&\ge \frac{C}{(K+K^{-\epsilon})^{\frac{\epsilon}{p}}\phi(K+K^{-\epsilon})}.
u_{s}(E_{2}-E_{1})=[\rho_{2}u_{2}(e_{2}+\frac{1}{2}u_{2}^{2}+\frac{p_{2}}{\rho_{2}})]-[\rho_{1}u_{1}(e_{1}+\frac{1}{2}u_{1}^{2}+\frac{p_{1}}{\rho_{1}})].
\inf_{c\in \mathbb{R}}\left(\frac{1}{\mu(E)}\int_E |f(x)-c|^pd\mu(x)\right)^{1/p}\leq\left(\frac{1}{\mu(E)}\int_E |f(x)-f_E|^pd\mu(x)\right)^{1/p}&\leq 2\inf_{c\in \mathbb{R}}\left(\frac{1}{\mu(E)}\int_E |f(x)-c|^pd\mu(x)\right)^{1/p}.
\xi(x) :=\frac{g(x)}{x}= \left[(1-\log x)^\tau-(-\log x)^\tau\right]^{-\alpha}, \eta(x) :=1 + \tau\alpha\left[(1-\log x)^\tau - (-\log x)^\tau \right]^{-1}\left[(1-\log x)^{\tau-1}- (-\log x)^{\tau-1}\right].
D_kR_N\Big(\sum_{k'}T_N^{-1}D^N_{k'}D_{k'}(f)\Big)(x) =D_kR_N \sum_{k'}\sum_{m=0}^\infty (R_N)^m D^N_{k'}D_{k'}(f)(x) &=\sum_{k'}\sum_{m=0}^\infty D_k(R_N)^{m+1}D^N_{k'}D_{k'}(f)(x)
\phi_i \wedge \rho^{(1)}- \sum_{p=1}^n\tau_p \wedge{\theta ''}_{pi}= \sum_{j<k}\sum_{l=1}^n \phi_j \wedge \phi_k \wedge \bar \phi_l \left( \delta_{ij}\rho^{(1)}_{k\bar l}- \delta_{ik}\rho^{(1)}_{i \bar l}- T^{p}_{jk}A_{pi, \bar l}\right) \, .
\int_0^1 P_l(x) Q_k(x) dx = \frac{1}{2\pi i}\oint_{\Sigma_l}\frac{(t+1-k)_k}{(t-l)_{l+1}}dt= \frac{1}{2\pi i}\oint_{\Sigma_l}\frac{\Gamma[t-l]}{\Gamma[t+1-k]}dt.
\int_{\mathbb R}J_1(\xi-y)\underline{\phi}(y)dy \geq\int_{\mathbb R}J_1(\xi-y)\left(1-\displaystyle\frac{1-b}{2}e^{\eta y}\right)dy =1-\frac{1-b}{2}e^{\eta \xi}\int_{\mathbb R}J_1(y)e^{-\eta y}dy.
{\cal A}_{\mu}~ = ~ \left( \begin{array}{l}{V^{A}{}_{\mu}}{W_{\mu A}}\end{array}\right){}~ ~ ~ ~ ~ ~ ~ ~ ~ ~{\cal F}_{\mu \nu}~ = ~ \left( \begin{array}{l}{V^{A}{}_{\mu \nu}}{W_{\mu \nu A}}\end{array}\right)
\mathcal{C}_{\rm{i}}=\left\{\left({\mathcal{R}}^{\rm{s}},{\mathcal{R}}^{\rm{c}}\right)|{\mathcal{R}}^{\rm{s}}\!\in\!\left[0,\mathcal{R}_{\rm{s}}^{\alpha}\right],{\mathcal{R}}^{\rm{c}}\!\in\!\left[0,\mathcal{R}_{\rm{c}}^{\alpha}\right],\alpha\!\in\!\left[0,\!1\right]\right\}.
1-\alpha\lambda_\mathrm{n_0}^{(3)}= \mathcal{O}(\delta^\mathrm{h}) \iff \dfrac{\omega_\mathrm{p}^2}{\omega_0^2-\omega^2-i\gamma\omega}= -1 + \varepsilon_\infty^{-1}\lambda_\mathrm{n_0}^{(3)}- \dfrac{\varepsilon_\infty^{-1}(\lambda_\mathrm{n_0}^{(3)})^2}{\varepsilon_\mathrm{m}+ \lambda_\mathrm{n_0}^{(3)}}+ \mathcal{O}(\delta^\mathrm{h}).
\begin{tabular}{ll}$\mathcal{A}_{0;1;2}^2=-\frac 16\frac{\cos \theta}{\sin \theta},$ $\mathcal{A}_{0;1;2}^3=\frac 1{p(y)},$ $\mathcal{A}_{0;1;3}^2=\frac 1{3\sqrt{3}}\frac 1{\sin \theta},$ $\mathcal{A}_{0;1;3}^3=0$ \,.\end{tabular}
\left( \begin{matrix}{{\sigma_{0}^{( \mathrm{o p )}}}}\\{{\pi_{0}^{( \mathrm{o p )}}}}\\ \end{matrix}\right) = -{\frac{\mu^{2}}{N}}\left( m_{\mathrm{Z M}}^{2}- \partial_{\bot}^{2}\right)^{- 1}\left[ : \bar{\Psi}_{M}^{a}\left( \begin{matrix}{{1}}\\{{i \gamma_{5}}}\\ \end{matrix}\right) \Psi_{M}^{a}: \right]_{0},
F(x) =\Phi((xA+H)^p)^{1/2}\Psi((xB+K)^q)\Phi((xA+H)^p)^{1/2}&=x^\gamma\Phi((A+x^{-1}H)^p)^{1/2}\Psi((B+x^{-1}K)^q)\Phi((A+x^{-1}H)^p)^{1/2},x\in(R,\infty).
x_{n+1}-y_{n+1}=\big[(1-\alpha_n)x_n +\alpha_n s_n\big]-\big[(1-\alpha_n)P_Lx_n +\alpha_n P_Ls_n\big] &=(1-\alpha_n)(x_n -P_Lx_n) +\alpha_n(s_n -P_Ls_n) &=\big((1-\alpha_n) +\alpha_n(1-\lambda_n)(1-\mu_n)\big)(x_n -y_n).
L^p_{d\mu}:= \left\{f\in L^p_{\rm loc}(\Omega) \;;\; \|f\|_{L^p_{d\mu}}:= \left(\int_{\Omega}|f(x)|^p a(x) \,dx\right)^{\frac{1}{p}}<\infty \right\}, 1\leq p<\infty
\langle T ( z ) \Phi_{1}( z_{1}) \cdots \Phi_{N}( z_{N}) \rangle = \sum_{j = 1}^{N}\left( \frac{h_{i}}{( z - z_{j})^{2}}+ \frac{1}{z - z_{j}}\frac{\partial}{\partial z_{j}}\right) \langle \Phi_{1}( z_{1}) \cdots \Phi_{N}( z_{N}) \rangle ~ .
\left( \Gamma_{1}\otimes \Gamma_{2}\right) \left( \Gamma_{3}\otimes \Gamma_{4}\right) = ( - 1 )^{\mathrm{d e g}\left( \Gamma_{2}\right) \mathrm{d e g}\left( \Gamma_{3}\right)}\Gamma_{1}\Gamma_{3}\otimes \Gamma_{2}\Gamma_{4},
\begin{aligned}{\underline{y}_d^c}_k = \Big\lbrack \int_{\Omega}{y_d^c}_k \varphi_j \,d\boldsymbol{x}\Big\rbrack_{j=1,\dots,n}\mbox{and}{\underline{y}_d^s}_k = \Big\lbrack \int_{\Omega}{y_d^s}_k \varphi_j \,d\boldsymbol{x}\Big\rbrack_{j=1,\dots,n}.\end{aligned}
T = \tilde{O}\left( \frac{B^{\frac{(1+\kappa)}{\kappa}}}{R_0^{\frac{(r-1)(1+\kappa)}{\kappa}}}\cdot \left[\frac{\mu_r R_0}{2\varepsilon}\right]^{\left(\frac{(r-1)(1+\kappa)}{r\kappa}\right)}\right), B = \frac{2^{(1+r)}\left(2M_2 + \sqrt{d}a_{2,q^*}\mathcal{D}+ 2\sigma_{q,\kappa}\right)}{\mu_r}
\begin{bmatrix}\ast 1 \ast \hdots \ast 0 0 0 \hdots \ast 0 \ast \hdots \ast 1 0 0 \hdots \end{bmatrix}\sqcup\begin{bmatrix}1 0 \ast \hdots \ast 0 0 0 \hdots 0 \ast \ast \hdots \ast 1 0 0 \hdots \end{bmatrix}
- \log \vert \det(1-A)\vert \leqslant \sum_{k=1}^\infty \frac{1}{k}\vert \mathrm{Re}(\mathrm{tr}(A^k)) \vert \leqslant \sum_{k=1}^\infty \frac{1}{k}\vert \mathrm{tr}(A^k) \vert.
\{Q_{\alpha},Q_{\beta}\}= (\Gamma^{\mu}{\cal C})_{\alpha\beta}P_{\mu}+(\Gamma^{\mu_1\mu_2}{\cal C})_{\alpha\beta}Z_{\mu_1\mu_2}+ (\Gamma^{\mu_1\cdots\mu_5}{\cal C})_{\alpha\beta}X_{\mu_1\cdots\mu_5},
\frac{15+1}{4^3\cdot 6}aN^{-7/4}(x-N)^3 =\frac{1}{24}aN^{-7/4}(x-N)^3 &\le \frac18 aN^{-3/4}(x-N)^2+\frac{80}{4^4\cdot 24}a N^{-11/4}(x-N)^4.
{\mathcal{R}}_{}=\frac{N}{L}\max_{{\mathsf{tr}}\left({\textbf{S}}{\textbf{S}}^{\mathsf{H}}\right)\leq p_{}}\log_2\det\left({\textbf{I}}_L+{\textbf{S}}^{\mathsf{H}}{\textbf{R}}_{}{\textbf{S}}\right).
M(x,t,k)=\left( \begin{array}{cc}\frac{k+v_1(x,t)}{k-\frac{iA}{2}}\frac{v_1(x,t)}{k}\frac{-\overline{v_1(-x,-t)}}{k-\frac{iA}{2}}\frac{k-\overline{v_1(-x,-t)}}{k}. \end{array}\right)
\sim \sum_{r_2,\dots,r_{m-k+1}}L_{r_2,\dots,r_{m-k+1}}(t) t^{(2-\alpha) r_2}\cdots t^{(m-k+1-\alpha) r_{m-k+1}}&= \sum_{r_2,\dots,r_{m-k+1}}L_{r_2,\dots,r_{m-k+1}}(t) t^{2 r_2 + \cdots + (m-k+1) r_{m-k+1}- \alpha \left( r_2 + \cdots + r_{m-k+1}\right)}&= \sum_{r_2,\dots,r_{m-k+1}}L_{r_2,\dots,r_{m-k+1}}(t) t^{m - \alpha k},
\left. \begin{array}{l}{\left[ q_{\mu}, q_{\nu}\right] = i Q_{\mu \nu}( g )}\\{{}}\\{R_{\mu \nu}- ( 1 / 2 ) R g_{\mu \nu}= 8 \pi T_{\mu \nu}( \psi )}\\{{}}\\{F ( \psi ) = 0 ,}\\ \end{array}\right.
\delta \left( \begin{matrix}{g_{i j}}\\{\pi^{i j}}\\ \end{matrix}\right) = \{\left( \begin{matrix}{g_{i j}}\\{\pi^{i j}}\\ \end{matrix}\right) , T_{\mu}\}F^{\mu}, \: \: \delta N^{\mu}= \dot{F}^{\mu}- U_{\beta \gamma}^{\mu}N^{\beta}F^{\gamma}.
\hat{T}= \sum_{\{\Delta_{0}S_{\gamma}\}}\prod_{p \in \Delta_{0}S_{\gamma}}\hat{W}_{p}^{n_{p}}\exp{[ \, - \frac{1}{2 \beta_{0}}\sum_{\ell}\hat{E}_{\ell}^{2}- \frac{1}{2 \beta_{s}}\sum_{p_{s}\in ( S_{t + a_{0}}- S_{t})}n_{p}^{2}\, ] \,},
\textbf{1}= \sum_{n=0}^{\infty}a^{\dagger}_{m_{1}}\ldots a^{\dagger}_{m_{n}}\mid 0 \rangle \langle 0 \mid a_{m_{n}}\ldots a_{m_{1}}= \sum_{n=0}^{\infty}\tilde{a}^{\dagger}_{m_{1}}\ldots \tilde{a}^{\dagger}_{m_{n}}\mid 0 \rangle \langle 0 \mid \tilde{a}_{m_{n}}\ldots \tilde{a}_{m_{1}}