harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
W(a,b)(x_\cdot) = \prod_{i=1}^{m+n}\omega_{x_{i-1},x_i}^{a,b}=\prod_{i=1}^{t_1}H_i(a)\prod_{j=1}^{t_2}H_j(b) \prod_{i = (t_1\vee t_2)+1}^{m+n}\omega_{x_{i-1},x_i}^{a,b},
\frac{d\varphi_{k}}{du}=\frac{\ddot{\bar{h}}_{k}(u)}{(a_{k}+u|I_{k}|)^{H-\frac{1}{2}}}+\left(\frac{1}{2}-H\right)\cdot\frac{|I_{k}| \, \dot{\bar{h}}_{k}(u)}{(a_{k}+u|I_{k}|)^{H+\frac{1}{2}}}.
i_{V}\omega = \left. \int d x^{+}~ T r \left( s_{0}^{- 1}\Phi s_{0}\delta A + [ A , s_{0}^{- 1}\Phi s_{0}] s_{0}^{- 1}\delta s_{0}\right) \right\vert_{x^{-}= b}^{x^{-}= a}~ ~ .
d^c u = (\partial_r u) d^c r - (J\partial_r u)dr + d^c_b u, dd^c v = (\partial_r^2 v)dr \wedge d^c r + (\partial_r v)dd^c r - d(J\partial_r v) \wedge dr + dd^c_b v, \partial_r \,\lrcorner\,dd^c v = (\partial_r^2v + r^{-1}\partial_r v + (J\partial_r)^2v)d^c r+ d_b(J\partial_r v) + \mathcal{L}_{\partial_r}(d^c_b v),
\left\{A,B\right\}=-\widehat{\Sigma}_{\lambda}C^{\lambda}\!_{\mu\nu}\frac{\partial A}{\partial\widehat{\Sigma}_{\mu}}\frac{\partial B}{\partial\widehat{\Sigma}_{\nu}}-\frac{\partial A}{\partial\widehat{\Sigma}_{\mu}}\mathbf{R}_{\mu}B+\frac{\partial B}{\partial\widehat{\Sigma}_{\mu}}\mathbf{R}_{\mu}A.
II_n =\int_{\frac{\lambda_n}{|\xi|}\leq \frac{1}{C}}|\xi|^{2\sigma}\big(|1+\tfrac{\lambda_n}{|\xi|}|^{2\sigma}-(\tfrac{\lambda_n}{|\xi|})^{2\sigma}-2\sigma(\tfrac{\lambda_n}{|\xi|})^{2\sigma-1}-1\big)|\hat{v}_n(\xi)|^2d\xi &\lesssim\frac{1}{C^{2\sigma-1}}\||\nabla|^\sigma v_n\|_{L^2}^2\lesssim \frac{1}{C^{2\sigma-1}}\lesssim\epsilon.
T_R^* a(z^{-n-1}) \xi = R^{-n-1/2}(s^{1/2}a(z^{-1}(sz^{-1}+ \overline{w})^n) \otimes 1 + \Gamma \otimes a( \tilde W_{\phi_t}z^{-n-1}))T_R^*\xi,
\delta u_{st}^{\otimes 2}= \int_s^t 2 u_r \hat{\otimes}\Delta u_r + 2 u_r \hat{\otimes}\mathrm{div}(F(u_r)) dr + (\Gamma_{st}^1 + \Gamma_{st}^2) u_s^{\otimes 2}+ u_{st}^{\otimes 2, \natural}
\psi_\rho(\alpha) =\alpha+(1-\alpha)e^{(2-\rho)\ln{\frac{\alpha}{1-\alpha}}}+1-\rho &\geq\alpha+(1-\alpha)\left(1+(2-\rho)\ln{\frac{\alpha}{1-\alpha}}\right)+1-\rho &=(2-\rho)\left(1+(1-\alpha)\ln{\frac{\alpha}{1-\alpha}}\right)=(2-\rho)h(\alpha),
w[\mathbf{u},\mathbf{u}] = \int_{\mathbb{R}^d}\left\langle g_\wedge(\mathbf{x}) b_\wedge(\mathbf{D}) \mathbf{u}, b_\wedge(\mathbf{D}) \mathbf{u}\right\rangle_{\mathbb{C}^{m_\wedge}}d \mathbf{x}.
\pi(n)=\dfrac{\pi(1)}{\prod\limits_{i=1}^{n-1}\Bigl(1+\dfrac{b}{i}\Bigr)}=\pi(1) \dfrac{(n-1)!}{\prod\limits_{i=1}^{n-1}(b+i)}= \pi(1) \dfrac{\Gamma(n)\Gamma(b+1)}{\Gamma(b+n)}.
S = \left( \begin{array}{c c}{h}&{s}\\{- s^{\dagger}}&{- h^{T}}\\ \end{array}\right) , \quad V = \left( \begin{array}{c c}{h^{\prime}}&{a}\\{- a^{\dagger}}&{\left( h^{\prime}\right)^{T}}\\ \end{array}\right) .
P_n(\cos\theta_1\cos\theta_2\pm \sin\theta_1\sin\theta_2\cos\phi)=\sum_{k=-n}^n(\pm 1)^k\frac{(n-k)!}{(n+k)!}{\sf P}_n^k(\cos\theta_1){\sf P}_n^k(\cos\theta_2)\cos(k\phi),
\begin{aligned}\sum_{p = 1,2}|\alpha(g_p)f-f|_{\infty}^- \ge C \left| \sum_{\underset{p = 1,2}{g \in{\mathbb F}_2}}(\alpha(g_p)f-f)(g)H_p(g)\right| &= C \left| \sum_{\underset{p = 1,2}{g \in{\mathbb F}_2}}(f(g)(\alpha(g_p^{-1})H_p-H_p)(g)\right| &= C|f(e)| = C\end{aligned}
\frac{\partial^+ u}{\partial \nu_p^+}:= \nabla \mathcal{R}_p^+(u)\cdot \nu_p^+\bigg|_{\Gamma_p}, \frac{\partial^- u}{\partial \nu_p^-}:= \nabla \mathcal{R}_p^-(u)\cdot \nu_p^-\bigg|_{\Gamma_p}.
h(x_1,y_1,\ldots, x_{k'}, y_{k'}) =\Pi_{j=1}^{m'}(\epsilon_{j,1}x_1+\cdots +\epsilon_{j,k'}x_{k'})^d \Pi_{j=m'+1}^m (\epsilon_{j,1}y_1 +\cdots +\epsilon_{j,k'}y_{k'})^d
X(t) = X(0) + \sum_{k=1}^K \zeta_k \int_0^t 1_{\left[ \frac{q_{k-1}(s-)}{\lambda_0(s-,X(s-))}, \frac{q_{k}(s-)}{\lambda_0(s-,X(s-))}\right)}( \xi_{R_0(s-)})dR_0(s) R_0(s) = Y\big( \int_0^t \lambda_0(s,X(s)) ds\big),
u (x) \geq \left(\frac{1}{2}- C \delta\right) \sum_{\beta = \alpha + 1}^{n-1}b_\beta x_\beta^2 - C b_\alpha |\hat{x}|^2 \geq \frac{1}{4}\sum_{\beta = \alpha + 1}^{n-1}b_\beta x_\beta^2 - C b_\alpha |\hat{x}|^2
\left[-\frac{1}{2}, \frac{1}{2}\right]^d = \overset{\bullet}{\bigcup_{\lambda\in \{1, \dots, N\}^d}}I_\lambda I_\lambda \subset \overline{B}_{1/n}^{\|\cdot\|_{\ell^\infty}}(x) \subset \overline{B}_{d/N}^{|\cdot|}(x) x \in I_\lambda.
\chi_{A \, 1}~ = ~ - \, \frac{( 2 \alpha - 1 )}{8 ( \alpha - 1 )}\left[ y_{1}\left( P_{1}+ \frac{X_{1}}{2 ( 2 \alpha - 1 )}\right) + z_{1}\left( P_{2}+ \frac{X_{2}}{2 ( 2 \alpha - 1 )}\right) \right]
\sum_{e}\sum_{m,n=0}^{1}C_{e}(m,n)\theta^{4}[e] F_{R,tws}^{m,n}=E(z,0)^{-2}\sum_{i=2}^{3}C_{i}\theta_{i}^{12}(0|\tau)\frac{\theta^{2}_{i}(z|\tau)}{\theta_{i}^{2}(0|\tau)},
\begin{aligned}\|\nabla u_{0}(\mathbf x)\|_{L^{2}(\Lambda_h)}\leq \sqrt{2}\tau \|u_{0}\|_{L^{2}(\Lambda_h)}+\|e^{\rho \cdot \mathbf x}\|_{L^{4}(\Lambda_h)}\|\nabla \psi(\mathbf x)\|_{L^{4}(\Lambda_h)}\lesssim(1+\tau)(1+\tau^{-\frac{2}{3}})e^{-\zeta h\tau},\mbox{as $\tau \rightarrow \infty$.}\end{aligned}
\sum_i{|O_i|}\cdot (-1) \leq \sum_i{|O_i| \cdot \frac{1}{|O_i|}f\left(\sum_{\zeta \in O_i}{\zeta}\right)}= \sum_i{f\left(\sum_{\zeta \in O_i}{\zeta}\right)}= f\left(\sum_i{\sum_{\zeta \in O_i}{\zeta}}\right) &= f\left(M(-a - b\sqrt{2})\right) = -Mf(a + b\sqrt{c}) = -M(a + 2b) \Rightarrow \sum_i{|O_i|}\geq M(a + 2b).
\tilde{\lambda}(\tilde{\mathcal{C}}_n)=\prod_{\mathbf{u}_0\in\mathcal{U}_0^n}\prod_{\substack{(w,i) \in\mathcal{W}\times\mathcal{I}}}Q^n_{U_1|U_0}\big(\tilde{\mathbf{u}}_1(\mathbf{u}_0,w,i)\big|\mathbf{u}_0\big),
\mathrm{i}\,\frac{\mathrm{d}\psi_{1,B}}{\mathrm{d}\,t}+ \big( |\psi_{1,B}|^2 + e\,|\psi_{2,B}|^2 \big) \psi_{1,B}= 0, \mathrm{i}\,\frac{\mathrm{d}\psi_{2,B}}{\mathrm{d}\,t}+ \big( e\,|\psi_{1,B}|^2 + |\psi_{2,B}|^2 \big) \psi_{2,B}= 0,
w(t,x,\varkappa)=\int_{\mathbb R^d}e^{i \varkappa \cdot y}e^{-t\Lambda(b)}(x,y) dy - i\int_0^t \int_{\mathbb R^d}e^{-(t-s)\Lambda(b)}(x,z)(\varkappa \cdot b(z))w(s,z,\varkappa)dzds.
\widehat{\mu}_{(m)}=\frac{\mu_{(m)}^{G}+\sum_{k \neq 2}^{M}\varepsilon_{k}\mu_{(k+m)}^{G}}{1+\sum_{k \neq 2}^{M}\varepsilon_{k}\mu_{(k)}^{G}}=\frac{(m-1)!!\sigma^{m}+\sum_{k \neq 2}^{M}\varepsilon_{k}(k+m-1)!! \sigma^{k+m}}{1+\sum_{k \neq 2}^{M}\varepsilon_{k}(k-1)!! \sigma^{k}}.
\Big\|\sum_{\ell\in I}|\tilde{\chi}_\ell^\delta \tilde{g}|*|\chi_\ell^\delta g|\Big\|_{L_2(\mathbb{R}^2)}=\Big\|\Big(\sum_{\ell,\ell^\prime\in I}(|\tilde{\chi}_\ell^\delta \tilde{g}|*|\chi_{\ell^\prime}^\delta{g}|)^2\Big)^{1/2}\Big\|_{L_2(\mathbb{R}^2)}.
\|u_t-v_t\| \leq\|u_t-w_t\|+\|w_t-v_t\|=\|w_tx_t-w_t\|+\|v_tp_t-v_t\| &\leq\|w_t(x_t-p_t)\|+\|v_tp_t-v_tq_t\|+\|v_tq_t-v_t\| &\leq \|x_t-p_t\|+\|p_t-q_t\|+\|v_tv_t^*v_t-v_t\|<10\eta.
D_{21}= \frac{(at_3)^{2-s_1-s_2-s_3}}{(s_1+s_2+s_3-2)(s_1+s_3-1)}&\quad+\frac{(at_3)^{2-s_1-s_2-s_3}}{2 \pi i (s_1+s_2+s_3-2) \Gamma(s_3)}\int_{(\frac{1}{2})}\frac{\Gamma(s_3+z) \Gamma(-z)}{s_1+s_3-1+z}dz.
\|e^{2\pi imx}f_{1,m}(y,z)\partial_jf_2\|_{L^2}^2=\sum\limits_{k\in\mathbb{Z}}\|2\pi ikf_{1,m}f_{2,k}\|_{L^2}^2\leq\sum\limits_{k\in\mathbb{Z}}(2\pi k)^2\|f_{1,m}\|_{L^2}^2\|f_{2,k}\|_{L^{\infty}}^2 \leq C\sum\limits_{k\in\mathbb{Z}}\|f_{1,m}\|_{L^2}^2(k^4\|f_{2,k}\|_{L^2}^2+\|\Delta f_{2,k}\|_{L^2}^2)\leq C\|f_{1,m}\|_{L^2}^2\|\Delta f_2\|_{L^2}^2,
\langle Av,v\rangle =d\langle (\cos{\psi}v_1-\sin{\psi}v_2,\sin{\psi}v_1+\cos{\psi}v_2),(v_1,v_2)\rangle &=d\cos{\psi}(v_1^2+v_2^2)=d\cos{\psi}\|v\|^2.
||[\phi_F,[\phi_G,\phi_{H}]||_l & \leq ||\phi_F[\phi_G,\phi_{H}](\phi_F)^{-1}||_l+||[\phi_G,\phi_{H}]^{-1}||_l &=2||[\phi_G,\phi_{H}]||_l \leq 2(||\phi_G\phi_{H}\phi_G^{-1}||_l+||(\phi_{H})^{-1}||_l) =4||\phi_{H}||_l\leq4||\phi_H||_{\partial_lQ_r}\leq 32r^2.
x_1\vee x_2\vee \cdots \vee x_n = \frac{1}{n!}\sum_{\sigma \in S_n}x_{\sigma(1)}\otimes x_{\sigma(2)}\otimes \cdots \otimes x_{\sigma(n)}; x_1\wedge x_2 \wedge \cdots \wedge x_n = \frac{1}{n!}\sum_{\sigma \in S_n}\epsilon(\sigma) x_{\sigma(1)}\otimes x_{\sigma(2)}\otimes \cdots \otimes x_{\sigma(n)},
d^{2}F_{x}=\sum_{i,j}\frac{de_{i}de_{j}^{\prime}}{r^{2}}[(1+\alpha_{1}\frac{u_{x}^{2}}{c^{2}}+\alpha_{2}\frac{u_{r}^{2}}{c^{2}})cos(rx)-\beta_{0}\frac{u_{x}u_{r}}{c^{2}}-\alpha_{0}\frac{ra_{r}^{\prime}}{2c^{2}}+\frac{ra_{x}^{\prime}}{2c^{2}}]
{\frac{\Lambda^{2}}{P^{+}}}> \mid{\frac{P_{\perp}^{2}}{P^{+}}}-{\frac{k_{1 \perp}^{2}}{k_{1}^{+}}}-{\frac{k_{2 \perp}^{2}}{k_{2}^{+}}}-{\frac{k_{3 \perp}^{2}}{k_{3}^{+}}}\mid .
E_{B}= \left( \begin{array}{c c}{{E_{1}}}&{{E_{2}}}\\ \end{array}\right) \;{\cal E}_{B}\; ; \;{\cal E}_{B}= \left( \begin{array}{c}{{1}}\\{{\nu_{B}^{c}}}\\ \end{array}\right) \;{\frac{1}{\sqrt{1 + \vert \nu_{B}\vert^{2}}}}\; .
\left( \frac{d^n}{dz^n}f(z) \right)^{\widehat{\,}}\, =\, (2\pi{\mathbf i}\xi)^n \widehat f(\xi) \left( z^n f(z) \right)^{\widehat{\,}}\, =\, \left(\frac{{\mathbf i}}{2\pi}\right)^n \frac{d^n \widehat f(\xi)}{d\xi^n}
{\Gamma^{0}}_{a}= \left\{\left( \begin{matrix}{{\sigma_{3}}}&{{0}}\\{{0}}&{{\sigma_{3}}}\\ \end{matrix}\right) , \quad \left( \begin{matrix}{{\sigma_{2}}}&{{0}}\\{{0}}&{{- \sigma_{2}}}\\ \end{matrix}\right) , \quad \left( \begin{matrix}{{0}}&{{\sigma_{2}}}\\{{\sigma_{2}}}&{{0}}\\ \end{matrix}\right) \right\}.
J^{(1,1)}_2 \leq 32C^2(\gamma)\max\{1,C^2_{\textsf{KMT}}\}T_*L^{-2}_*T^{2\gamma-1}_*+252\sqrt{C^{(2)}_{\textsf{KMT}}}T_*L^{-2}_*T^{-2}_*\,.
\sum_{(d,2)=1}M_Z(d)\left( \frac{d}{p}\right) W\left( \frac{d}{X}\right)=\sum_{\substack{\alpha \leq Z (\alpha, 2p)=1}}\mu(\alpha) \sum_{(d,2)=1}\left( \frac{d}{p}\right) W\left( \frac{d\alpha^2}{X}\right).
\left( \mathrm{m}_{i}F_{p}\right) \left( \phi_{0}, \cdots , \hat{\phi}_{i - 1}, \phi_{i}, \cdots , \phi_{p}\right) = F_{p}\left( \phi_{0}, \cdots , \phi_{i - 2}, \phi_{i}, \phi_{i}, \cdots , \phi_{p}\right) + O ( \hbar )
J^{3,1,2}_\epsilon = \frac{1}{\epsilon}DF(s,X_s,X(s))(\beta)\cdot \int_{\beta-\epsilon}^{\beta}X_{s+\epsilon}(\alpha)d\alpha \Bigg|_{-r}^{0}- \int_{-r}^0 DF(s,X_s,X(s))(\beta)\cdot \frac{X_{s+\epsilon}(\beta) - X_s(\beta)}{\epsilon}d\beta.
\left\langle T_k, \tau ', \dots , \tau ', \frac{e^{-\tau_2/z}\alpha}{z+\psi}\right\rangle^\textnormal{coh}_{0,l+2,d}:= \sum_{n,m \geq 0}\frac{(-1)^{n+m}}{z^{n+m+1}}\left\langle T_k, \tau ', \dots , \tau ', \psi_{l+2}^n \tau_2^m \cup \alpha \right\rangle^\textnormal{coh}_{0,l+2,d}
'_b =VC\sum_{n\in\mathcal{A}_{m,b}}\mathbb{E}[S_{m,n,b}] &\leq C\left(2\sum_{n\neq a_b^{*}}\mathbb{E}[\theta_{n,b,K_b}] + 1 \right) &\leq C \left(2\sum_{n\neq a_b^{*}}\left[\frac{8\ln (K_b-u_n)}{\delta^2_{n,b}}+ 1+ \frac{\pi^2}{3}\right]+1\right ).
\frac{1}{t}d_K \big( x + t (Ax + v) \big) = \frac{1}{t}d_K \bigg( S_t x + tv + t \bigg( Ax - \frac{S_t x - x}{t}\bigg) \bigg) \leq \frac{1}{t}d_K ( S_t x + tv ) + \bigg\| Ax - \frac{S_t x - x}{t}\bigg\|,
b'_4(1,2,3,4) = b_4 + b_3 \cdot (-i\lambda_3)\left( \frac{i}{x_{1+2}}+ \right) &+b'_2 (1+2+3+4)\cdot (-i\lambda_3)^2 \left(\frac{i^2}{x_{1+2}x_{3+4}}+ \right) &+ b'_2(1+2+3+4) \cdot (-i\lambda_3)^2 \left( \frac{i^2}{x_{1+2}x_{1+2+3}}+ \right) .
\lim_{n \longrightarrow \infty}\frac{1}{n}{\bf E}\Big\{\sum_{t=1}^{n}||X_t||_{{\mathbb R}^{n_x}}^2\Big\}= \lim_{n \longrightarrow \infty}\frac{1}{n}\sum_{t=1}^{n}tr \Big(\Gamma P_t \Gamma^T +D K_{Z}D^T\Big) = tr \Big(\Gamma P \Gamma^T +D K_{Z}D^T\Big), \; \forall P_1\succeq 0.
\Delta_{-}( x^{\prime \prime}- x^{\prime}) = - i \int_{\Sigma ( \tilde{x}^{0})}d^{3}{\bf \tilde{x}}\hspace{0 . 2 5 e m}\Delta_{-}( x^{\prime \prime}- \tilde{x}) \stackrel{\leftrightarrow}{\partial_{0}}\Delta_{-}( \tilde{x}- x^{\prime}) .
( \mathbf{1}-\mathbf{Q})^{-1}= \frac{1}{1-3q^2-2q^3}\left[ \begin{array}{ccc}1-q^2 q+q^2 q+q^2 q+q^2 1-q^2 q+q^2 q+q^2 q+q^2 1-q^2\end{array}\right].
d\langle E_{AB},w^{(a)}_{{\cal M}_0}\rangle=\frac{g^2\lambda}{8\pi}d\langle E_{AB},w^{(a)}_{\rm flat}\rangle-\frac{\bar \mu}{2}d\left\langle E_{AB},\epsilon^{abc}\left(\sum_A dr^b_A\right)\wedge \left(\sum_B dr_B^c\right)\right\rangle.
\alpha_j = \frac{e_j}{q_j}+ z_j (e_j, q_j) =1, 0 < q_j \leq \tau_j := Y^{j/2}|z_j| \leq \frac{1}{q_j \tau_j}, z_j = \delta_j Y^{-j}(j=1,2, \dots, t).
\tan \theta = \frac{\tilde{b}}{\alpha^{\prime}}= \frac{( 2 \pi )^{2}\tilde{b}^{2}}{\alpha^{\prime}\tilde{V}_{2}}\frac{N_{p - 2}}{N_{p}}\Longrightarrow \ \ \frac{N_{p - 2}}{N_{p}}= \frac{\tilde{V}_{2}}{( 2 \pi )^{2}\tilde{b}}.
b_{i}({\bf{q}}) = \sum_{{\bf{k}}}[b_{i}({\bf{q}}), a^{\dagger}_{{\bf{k}}}({\bf{q}})]a_{{\bf{k}}}({\bf{q}}) - \sum_{{\bf{k}}}[b_{i}({\bf{q}}), a_{{\bf{k}}}(-{\bf{q}})]a^{\dagger}_{{\bf{k}}}(-{\bf{q}})
I_{W}= ( I_{C S G})_{| M \approx R \times \Sigma}= \frac{1}{\sqrt{\Lambda}}\int d t \int_{\Sigma}d^{2}x ( - 2 \epsilon^{i j}E_{i a}\dot{\omega}_{j}^{a}+ E_{t}^{a}\Psi_{a}+ \omega_{t a}{\cal G}^{a}) ,
\sum_{n=0}^\infty\sum_{|m|\leq n/2}(-1)^mq^{n^2-2m^2}(1-q^{2n+1}) = qg_{1,3,1}(q^6,q^6;q^4) + f_{1,3,1}(q^2,q^2;q^4) +q^4f_{1,3,1}(q^{10},q^{10};q^4)-q^9g_{1,3,1}(q^{14},q^{14};q^4).
A = 2a(2a+1) + 2b_1(4a-2l-d+3), B_1 = 2b_1(4a+2b_1+3) - 4b_2(4a-2l-d+5), B_2 = - 4b_1^2 + 4b_2(4a+4b_1+5) - 6b_3(4a-2l-d+7), B_3 = - 16b_2(b_1-b_2) + 6b_3(4a+4b_1+7),
a_{m}(x)+\sum_{n=0}^{\infty}\frac{(-1)^{n+m+1}}{2\pi}a_{n}(x)\int_{-\infty}^{\infty}\frac{\varphi\left( \tau\right) z^{n+m+1}(\tau)d\tau}{\left(\frac{1}{2}-i\tau\right) \left( \frac{1}{2}+i\tau\right)}& =\frac{(-1)^{m}}{2\pi}\int_{-\infty}^{\infty}\frac{\varphi\left(\tau\right) z^{m+1}(\tau)d\tau}{\frac{1}{2}+i\tau},m=0,1,\ldots.
D_x^{\alpha=n}[f(x)\cdot g(x)] = \sum^{\infty}_{l= 0}C^{l}_{n}g^{(l)}(x) D_x^{(n-l)}[f(x)] = \sum^{n}_{l= 0}C^{l}_{n}g^{(l)}(x) f^{(n-l)}(x) = (f(x)\cdot g(x))^{(n)}.
\begin{aligned}u_a^{\varepsilon}(t,x,\xi) =t(\partial_tp_a^{\varepsilon}(t,x,\xi)+\partial_x p_a^{\varepsilon}(t,x,\xi)\cdot\xi) &=-2\phi'(|x/t-\xi_a|^2<{\varepsilon})\{|x/t-\xi_a|^2-(x^a\cdot\xi^a)/t\}\chi_a(\xi_a)^2\Phi_a(\xi).\end{aligned}
F(p, q) := \frac{1}{2}\sum_{k \in \mathbb{Z}}\int_{-1/2}^{1/2}\left\{\log\left( 1 + \left(\frac{p + k}{q}\right)^2 \right) - \log\left( 1 + \left( \frac{p + k + \phi}{q}\right)^2 \right) \right\}\mathrm{d}\phi.
\Delta^{\sigma(2+k)}_{h}\cdots \Delta^{\sigma(m-1+k)}_{h}H(x_0,\ldots,x_{m-1}) &=\Delta^{\sigma(2+k)}_{h}\cdots \Delta^{\sigma(m-1+k)}_{h}\Bigg[\sum\limits_{s=0}^{m-1}\prod\limits_{i=0}^{m-3}x_{i+s}-\sum\limits_{s=0}^{m-1}\prod\limits_{i=0}^{m-3}x_{\sigma(i+s)}\Bigg].
\lefteqn{\langle\xi(k')\xi(k'')\overline{\xi(l')}\,\overline{\xi(l'')}\rangle}&=\left\{\begin{array}{cl}\langle|\xi(k')|^2|\xi(k'')|^2\rangle \mbox{for}\;\{k',k''\}=\{l',l''\}\langle|\xi(k')|^2|\xi(l')|^2\rangle \mbox{for}\;k'+k''=l'+l''=0 0 \mbox{else}\end{array}\right\}.
c^{\ast}_{\varepsilon}:=\liminf_{|x|\rightarrow\infty}\inf_{t\geq T_{\varepsilon}}\left(2\sqrt{a_{\inf}-\frac{\chi\mu a_{\sup}}{b_{\inf}-\chi\mu}-\chi\mu\varepsilon}-\chi\|\nabla v(x,t+t_0;t_0,u_0)\|\right).
E(S_{\lfloor nt \rfloor}(r)(\overline{S_{\lfloor nu \rfloor}(s)-S_{\lfloor nt \rfloor}(s)}) = \sum_{k=1}^{m_{n}-1}k\gamma_{k}(r,s)+ m_{n}\sum_{k=m_{n}}^{\lfloor nu \rfloor -m_{n}}\gamma_{k}(r,s)+ \sum_{k=\lfloor nu \rfloor-m_{n}+1}^{\lfloor nu \rfloor-1}(\lfloor nu \rfloor-k)\gamma_{k}(r,s)
G \approx \frac{1}{2\pi(1+g^2/\pi)}\bigg[1-2^{(5\pi+3g^2)/(\pi+g^2)}\frac{(1+g^2/\pi)^2}{16\pi^2}\bigg(\frac{m^2}{T^2}\bigg)^{(\pi+2g^2)/(\pi+g^2)}\bigg].
0\leq \phi_{\nu}=\frac{(|\nabla u|^2)_{\nu}}{(b^2-\rho^2)^2}+\frac{2|\nabla u|^2(\rho^2\circ u)_{\nu}}{(b^2-\rho^2)^3}= -2H\frac{|\nabla u|^2}{(b^2-\rho^2)^2}+\frac{4\rho|\nabla u|^2\langle \nabla^N\rho,du(\nu) \rangle}{(b^2-\rho^2)^3}.
h_{(\sigma_{i})_{i},\mu}(g\alpha|\alpha:g\alpha \vee \alpha, F,\delta ) \leq \delta \log |A|+\limsup_{i\to\infty}\frac{1}{d_{i}}\log \binom{d_{i}}{\lfloor{\delta d_{i}\rfloor}}&= \delta \log|A|-\delta \log(\delta)-(1-\delta)\log(1-\delta),
\sum_{m\ge 1}m^{-\beta}\gamma_p(m,0)<\infty \, , \sum_{n\ge 1}n^{\gamma-\beta}\tilde \gamma_p(n)<\infty \, , \sum_{n\ge 1}\frac{1}{n^{1+\beta}}\sum_{m=1}^n \sum_{k=[m^\gamma]+1}^{n}\gamma_p(m,k)<\infty \, .
F_{2}(a,b_{1},b_{2};c_{1},c_{2};x,y)=\sum_{m,n=0}^{\infty}\frac{(a)_{m+n}(b_{1})_{m}(b_{2})_{n}}{(c_{1})_{m}(c_{2})_{n}m!n!}x^{m}y^{n}
O_{\tilde{L}}( \{p , \zeta \}, \tilde{p}_{1}, \{t \}, \{q \}) = \exp \biggl [ \sum_{j = 1}^{2}\frac{( 2 \eta_{j}\zeta_{( j )}+ i p_{j}\vartheta_{j})}{8 \sqrt{z_{j}- u}}\hat{\Psi}\biggl ] \, ,
\tilde{v}(x) = c_1^{-1}2^{j\kappa}v(x_0 + 2^{-j}x),\tilde{w}_i(x) = c_1^{-1}2^{j\kappa}w_i(x_0 + 2^{-j}x), \tilde{f}_1(x) = 2^{j(\kappa-2)}\frac{f(x_0 + 2^{-j}x) - a_jg(x_0 + 2^{-j}x)}{c_1(b_j - a_j)}, \tilde{f}_2(x) = 2^{j(\kappa-2)}\frac{b_jg(x_0 + 2^{-j}x) - f(x_0 + 2^{-j}x)}{c_1(b_j - a_j)}.
g_{\mu \nu}\equiv{\frac{\rho_{0}}{c_{0}}}\left[ \begin{matrix}{{- ( c_{0}^{2}- v_{0}^{2})}}&{{\vdots}}&{{- v_{0}^{j}}}\\{{\cdots \cdots \cdots \cdots}}&{{\cdot}}&{{\cdots \cdots}}\\{{- v_{0}^{i}}}&{{\vdots}}&{{\delta_{i j}}}\\ \end{matrix}\right] .
\int_{0}^{\infty}J_{0}(r \xi) \left(\partial_{r}v_{4,c}(t,r) + \frac{v_{4,c}(t,r)}{r}\right) r dr = -\int_{0}^{\infty}v_{4,c}(t,r) J_{0}'(r\xi) \xi r dr = \xi \int_{0}^{\infty}v_{4,c}(t,r) J_{1}(r\xi) r dr &=\xi \widehat{v_{4,c}}(t,\xi)
\|y_{k+1}-y_k\|= \| T_{y_k}^{-1}[-F(y_k)-r_k]\|, \max_{w\in \{-r_k,~ r_k \}}\left\|T_{x_0}^{-1}w\right\| \leq \theta \left\|T_{x_0}^{-1}[-F(y_k)]\right\|.
\delta^+(k)\stackrel{k\rightarrow\infty}{\cong}\frac{1}{k}\int_0^\infty dxV(x) \cos^2(k x)\hspace{1cm}\delta^-(k)\stackrel{k\rightarrow\infty}{\cong}\frac{1}{k}\int_0^\infty dxV(x) \sin^2(k x)
j^1(x, t)\Bigm|_{x=0}= \frac{1}{2}(\psi_1^\dagger (0)\psi_1(0^-) - \psi_2^\dagger (0)\psi_2(0^-) + \psi_1^\dagger (0^-)\psi_1(0) - \psi_2^\dagger (0^-)\psi_2(0)) = 0,
N=q^{2n+1}-q^{n+2}+2q^{n+1}-1, &k=q^{2n+1}-q^{n+2}+\frac{5q^{n+2}+q^n-q^3+q^2-2q+2}{2(q+1)}-a(2q^2-2q-1), &d\geq 2a(q^2-q-1)-\frac{q^2(q^n-2q^{n-1}-q^{n-2}-q+1)}{q+1}.
\hat{F}\circ \hat{G}(S) = \bigcap_{0 \in U \in \hat{G}(S)}{\rm cl}_{T^{\max}(\hat{X})}[I(U)] = \bigcap_{0 \in V \in \hat{T}}{\rm cl}_{T^{\max}(\hat{X})}[I(X) \cap V] .
\frac{1}{2}\frac{\partial}{\partial t}(v_i^{-})^2+\frac{1}{2}\frac{\partial}{\partial t}(u_j^{-})^2 = -\tilde d_i v_i^{-}\Delta_M v_i -v_i^{-}F_i(u^{+}, v^{+}) &\quad- d_j u_j^{-}\Delta u_j-u_j^{-}H_j(u^{+})
\rho_{s}( \lambda_{1}, \cdots , \lambda_{p}) = \left\langle \prod_{a = 1}^{p}\frac{1}{N}\mathrm{t r}\, \delta ( \lambda_{a}- M ) \right\rangle = \det_{1 \leq a , b \leq p}K_{s}( \lambda_{a}, \lambda_{b}) ,
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 M^{ra_3(l+1)}c_0^2\alpha^4\|J_4^{l+1}\|_{l+1,r}+c\alpha^{-2}.
\begin{aligned}u_{\tau\tau t}- F^{ij}u_{\tau\tau ij}- F^{ij}\chi_{\tau\tau, ij}&= -(\psi)_{\tau\tau}+ F^{ij}\chi_{ij,\tau\tau}- F^{ij}\chi_{\tau\tau, ij}+ F^{ij,kl}U_{ij,\tau}U_{kl,\tau}& -2F^{ij}u_{im}R^m_{\tau\tau j}- 2F^{ij}u_{m\tau}R^m_{i\tau j}- F^{ij}u_m \big( R^m_{\tau\tau i,j}+ R^m_{i\tau j,\tau}\big)\end{aligned}
\psi_\omega(T_xT_y^*) =\psi_\omega(T_b^sT_b^{*t}) &=\big(\pi_{U,V}(T_b^sT_b^{*t})(e_{0,e}\otimes \xi_\omega)\,|\,e_{0,e}\otimes \xi_\omega\big) &=\big(U^sU^{*t}(e_{0,e}\otimes \xi_\omega)\,|\,e_{0,e}\otimes \xi_\omega\big) &=\big(U^{*t}(e_{0,e}\otimes \xi_\omega)\,|\,U^{*s}(e_{0,e}\otimes \xi_\omega)\big).
\langle A'_{\lambda_1,\lambda_2}(v), h\rangle= \langle A'_{\lambda_1,\lambda_2}(v_n), h\rangle +o(1) = \langle A'_{\lambda_1,\lambda_2}(u_n), h_n\rangle +o(1) = o(1)\|h_n\|+o(1) = o(1)\|h\|+o(1),
A |_{I_{\mu}}\; = \; \left( \begin{array}{c c c c}{{\mu}}&{{1}}&{{\cdots}}&{{0}}\\{{0}}&{{\mu}}&{{\cdots}}&{{0}}\\{{\vdots}}&{{\vdots}}&{{\ddots}}&{{1}}\\{{0}}&{{0}}&{{\cdots}}&{{\mu}}\\ \end{array}\right) \; .
\|f-f_n\|^2 = \sum_{u\in V}|\left< f-f_n,e_u\right>|^2= \sum_{u\in V}|f(u)-f_n(u)|^2 = = \sum_{u\in U}|f(u)-f_n(u)|^2 = \sum_{j=n+1}^\infty |f(u_j)|^2 \stackrel{n\to\infty}{\longrightarrow}0,
r_+(z)=p (z)+ \sum_{|z_{j}|>1}\sum_{\ell=1}^{L_{j}}c_{j,\ell}(z-z_{j})^{-\ell},r_-(z)=\sum_{|z_{j}|<1}\sum_{\ell=1}^{L_{j}}c_{j,\ell}(z-z_{j})^{-\ell}.
\begin{aligned}(i)\R^{l}_{ijk}+R^{l}_{jki}+R^{l}_{kij}=0, &(ii) R^{l}_{ijk}=-R^{l}_{jik}\end{aligned}\begin{aligned}(iii)\R^{l}_{ijk}=-R^{k}_{ijl}&(iv)\R^{l}_{ijk}=R^{j}_{kli}\end{aligned}
\frac{1}{2}\partial_{t}\lVert \Lambda^{r_{2}}b\rVert_{L^{2}}^{2}+ \lVert \Lambda^{2r_{2}}b\rVert_{L^{2}}^{2}= -\int (u\cdot\nabla) b \cdot \Lambda^{2 r_{2}}b - (b\cdot\nabla) u \cdot \Lambda^{2r_{2}}b \leq (\lVert (u\otimes b) \rVert_{L^{2}}+ \lVert (b\otimes u) \rVert_{L^{2}}) \lVert \Lambda^{2r_{2}}b\rVert_{L^{2}}
\lambda_{k}= \min_{\stackrel{u \in \mathbb{P}_{k}}{\|u\|_{L^{2}(\Omega)}=1}}\left\{\int_{\Omega}|\nabla u|^2 \, dx +\alpha \iint_{\mathbb{R}^{2n}}\dfrac{|u(x)-u(y)|^2}{|x-y|^{n+2s}}\, dxdy \right\},
Z_{0}^{K 3}( \tau ) = \frac{N_{1}}{4}G ( 2 \tau ){\tilde{Z}}_{a}( \tau ) + N \left( Z_{e v e n}^{S_{0}}( \tau ){\tilde{Z}}_{2}( \tau ){\tilde{Z}}_{-}( \tau ) + Z_{o d d}^{S_{0}}( \tau ){\tilde{Z}}_{2}( \tau ){\tilde{Z}}_{+}( \tau ) \right) ,
\det\left(M_{n\mathbf{m}}(\lambda,b,\Omega_{\mathbf{m}}^{\pm}(\lambda,b))\right)\underset{n\rightarrow \infty}{=}d_{\infty}(\lambda,b,\mathbf{m})+\frac{\widetilde{d}_{\infty}(\lambda,b,\mathbf{m})}{n}+O_{\lambda,b,\mathbf{m}}\left(\frac{1}{n^{3}}\right),
{\cal G}_{\phi \phi}\left(\rho,z\right)={\displaystyle\frac{1}{4\pi A{\cal C}_{66}}\sum_{l=1}^{4}{\frac{\rho^2\Lambda_{\rm b}\left(a=-A_l\right)+z^2\Gamma_{\rm b}\left(a=-A_l\right)}{\rho^2\sqrt{A_l\rho^2+z^2}\prod_{j=1,(j\neq l)}^{4}{\left(A_j-A_l\right)}}}}
F(\mathfrak A)(\mathsf U^{\mathsf V}) \cap F(\mathfrak A)(\mathsf U^{\mathsf W}) = \mathfrak A(\mathsf V) \cap \mathfrak A(\mathsf W) = \mathfrak A(\mathsf V \cap \mathsf W) = F(\mathfrak A)(\mathsf U^{\mathsf V \cap \mathsf W}) = \sum_{\mathsf Z \in \mathsf U^{\mathsf V \cap \mathsf W}}F(\mathfrak A)(\mathsf U^{\mathsf Z}).
\begin{aligned}\Lambda :=\sum_{i=1}^n\|a_{ii}\|_{C(\bar U)}+ \sum_{i,j=1}^n\|\tilde a_{ij}\|_{C(\bar U)}+\sum_{i=1}^n\|b_i\|_{C(\bar U)}&\quad+ \sum_{i=1}^n\sum_{l=1}^m\|c_{il}\|_{C(\bar U)}+\sum_{l,p=1}^m\|d_{lp}\|_{C(\bar U)}+\sum_{l=1}^m\|e_l\|_{C(\bar U)}. \end{aligned}
(\eta\delta)_*\beta_\alpha(U)s = (\eta\delta)(U)\beta_\alpha(U)s &=(\eta\delta)(U)\alpha(U)[U\rightarrow U,\bar s] &=\delta(U)\alpha(U)[U\rightarrow U,\bar s] &=(\delta\alpha)(U)[U\rightarrow U,\bar s] &=\beta_{\delta\alpha}(U)s &=\beta_{\delta_*\alpha}(U)s.
\int\hat{\Delta}_{\ell}(w)^{2}F_{0}(dw) =\int[\hat{\alpha}_{\ell}(x)-\alpha_{0}(x)]^{2}[\lambda(w,\hat{\gamma}_{\ell})-\lambda(w,\gamma_{0})]^{2}F_{0}(dw) & \leq O_{p}(1)\int[\lambda(w,\hat{\gamma}_{\ell})-\lambda(w,\gamma_{0})]^{2}F_{0}(dw)\overset{p}{\longrightarrow}0.
\operatorname{Tr}(\mathcal{X}_E*\vartheta_{\tilde{\eta}}(Q_1))=\frac{N}{\pi}\int_{E^-}^{E_U}\operatorname{Im}m_2(w+i\tilde{\eta})dw, \ \ \sum_{\mu \nu}\vert G_{\mu \nu}(w+i\tilde{\eta})\vert^2=\frac{N \operatorname{Im}m_2(w+i\tilde{\eta})}{\tilde{\eta}}.
g ( t , \eta , \bar{\eta}) = e^{- i t H}e^{i ( \eta_{i}Q^{i}+ \bar{\eta}^{i}\bar{Q}_{i})}e^{i ( \lambda_{i}S^{i}+ \bar{\lambda}^{i}\bar{S}_{i})}e^{i z D}e^{i \omega K}e^{i \phi J_{1}}e^{i \theta J_{2}}e^{i \psi J_{3}}\quad ,
g([X,Y], Z) =-g(h(X,tY), \varphi Z)-g(\widetilde\nabla_{X}Z, t'nY+n'nY) -g(h(Y,tX), \varphi Z)-g(\widetilde\nabla_{Y}Z, t'nX+n'nX)-2g(\widetilde\nabla_{Y}X, Z).
\frac{\|\partial_{y_n}^k u\|_{H_0^1}}{k!}\leq \sum_{l=1}^k \left(\frac{\theta_n}{4}\right)^{l}\frac{\|\partial_{y_n}^{k-l}u\|_{H_0^1}}{(k-l)!}+ \left(\frac{\theta_n}{4}\right)^k (1+\|f(y)\|_{L^2}) &\leq \left(\frac{\theta_n}{2}\right)^k \frac{1}{2}(\|u\|_{H^1_0}+ 1 + \|f(y)\|_{L^2})
m_{G}^{2}= - 2 e^{2}\int_{{\mathcal R}}\frac{d^{2}p}{( 2 \pi )^{2}}\frac{\partial}{\partial p_{\mu}}( \frac{p_{\mu}}{p^{2}}) + 2 e^{2}\int_{{\mathcal R}}\frac{d^{2}p}{( 2 \pi )^{2}}\frac{\partial}{\partial p_{\mu}}( \frac{p_{\mu}}{p^{2}+ \Lambda^{2}}) ,
\begin{array}{c c l}{{\xi_{k}^{\alpha}( x )}}&{{=}}&{{\left( \left( \frac 1{R \left( A ( x ) \right) R \left( A ( x ) \right)}\right)^{1 / 4}\right)_{k k^{^{\prime}}}^{\alpha \alpha^{^{\prime}}}B_{k^{^{\prime}}}^{\alpha^{^{\prime}}}( x )}}\\{{}}&{{=}}&{{\frac 1{\pi \sqrt{2}}\int_{0}^{\infty}d \lambda \lambda^{- 1 / 4}\frac 1{\lambda + R \bullet R}\bullet B ( x )}}\\ \end{array}
\frac{1}{2iT}\int_{\frac{1}{2}-iT}^{\frac{1}{2}+iT}\chi(1-s)\zeta(s+ia)\zeta(s)ds =\frac{2\pi}{T}D_{ia}(T/2\pi) \;+\; \mathcal{O}\Big(T^{-1/2+\varepsilon}\big( 1+ |a|^{1/6}\big)\Big) &= \zeta(1+ia)+\zeta(1-ia)\Big(\frac{T}{2\pi}\Big)^{-ia}\frac{1}{1-ia}& \;+\; \mathcal{O}\big(T^{-1/3}(1+|a|^{1/6})\big),
\tilde Z^{(1)}(q_{N_r};l_{1r},l_{2r})=\frac{k_r^{\sigma_rl_{1r}}H_{(1)}(q_{N_r})[1+(-1)^{l_{2r}}k_r]^{2l_{1r}}}{k_r^{(3-2l_{1_r})/2}[1+(-1)^{l_{2r}}k_r^{1/2}]^{4l_{1r}}}Z^{(1)}(k_r;l_{1r},l_{2r})
f_1= \sum_{i,j=1}^2 \frac{\partial \chi_1}{\partial X_{ij}}(I_2)(\delta(g^{-1})g+g^{-1}A g)_{ij}=b+\eta f_2= \sum_{i,j=1}^2 \frac{\partial \chi_2}{\partial X_{ij}}(I_2)(\delta(g^{-1})g+g^{-1}A g)_{ij}=b-\eta.