harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\nabla_{\alpha}R_{\beta\gamma}=\nabla_\beta R_{\alpha\gamma}+O_g'(r_q) &=\nabla_\beta R_{\mu\nu}J^\mu_\alpha J^\nu_\gamma+O_g'(r_q) &=\nabla_{\mu}R_{\beta\nu}J_{\alpha}^{\mu}J_{\gamma}^{\nu}+O_g'(r_q),
{\cal Z}_{\mathrm{m o n .}}= \sum_{N = 1}^{+ \infty}\sum_{q_{a}= \pm 1}^{}\frac{\zeta^{N}}{N !}\prod_{i = 1}^{N}\int d^{3}z_{i}\exp \left[ - \frac{\pi}{2 e^{2}}\int d^{3}x d^{3}y \rho_{\mathrm{g a s}}\left( \vec{x}{\,}\right) \frac{1}{\left| \vec{x}- \vec{y}{\,}\right|}\rho_{\mathrm{g a s}}\left( \vec{y}{\,}\right) \right] ,
\begin{array}{ll}\lambda_{\{1,2\}}(x,y) = \frac{(x,\, 9x)}{40}, \lambda_{\{1,3\}}(x,y) = \frac{(-x,\, 9x)}{41}, \lambda_{\{1,4\}}(x,y) = (-x,\, x), \lambda_{\{2,3\}}(x,y) = \frac{(x,\, x)}{9}, \lambda_{\{2,4\}}(x,y) = \frac{(x,\, 9x)}{41}, \lambda_{\{3,4\}}(x,y) = \frac{(9x,\, -x)}{40}.\end{array}
[\nabla_{\mu}, \nabla_{\nu}]\phi_{\mu_1\dots\mu_k}= - \sum_{i=1}^{k}R_{\mu_i\mu\nu}^{\lambda}\phi_{\mu_i\dots\mu_{i-1}\lambda\mu_{i+1}\dots\mu_k}+ W_{\mu\nu}\phi_{\mu_1\dots\mu_k},
\frac{d^2}{d\epsilon_1d\bar\epsilon_2}\vert_{\epsilon=0}P_{\epsilon\nu^{\epsilon\mu}}^{0,1}(f^{\epsilon\nu^{\epsilon\mu}})((\Phi_1^{\epsilon\mu})_*^{-1}(1-\vert \epsilon\mu\vert^2)\frac{d}{dt}\vert_{t=0}(\Phi_1^{\epsilon\mu+t{\tilde\mu}})P^{0,1}(\Phi_1^{\epsilon\mu+t{\tilde\mu}})^{-1}\epsilon\nu),
\frac{\overline{{{I}}}_{6}- \overline{{{I}}}_{4}}{\overline{{{I}}}_{4}- \overline{{{I}}}_{2}}={\frac{{. 9 9 5 1 2 2 6 1 4 6 \ldots - . 9 9 5 2 2 0 3 9 4 5 \ldots}}{{. 9 9 5 2 2 0 3 9 4 5 \ldots - . 9 9 5 5 7 5 2 2 1 2 \ldots}}}= . 2 7 5 5 7 \ldots
C_{x}^{\dagger}\, K_{x}= 0 \, \, \mathrm{w h i c h ~ ~ i m p l i e s}\, C_{x}^{\dagger}\, J ( x ) = 0 \qquad \mathrm{( ` ` s o u r c e ~ c o n s e r v a t i o n^{\prime \prime})}.
T ( k ) = \frac{\Gamma ( - 1 - i k / m ) \Gamma ( 2 - i k / m ) \Gamma ( 1 / 2 \mp i B - i k / m ) \Gamma ( 1 / 2 \pm i B - i k / m )}{\Gamma ( - i k / m ) \Gamma ( 1 - i k / m ) \Gamma^{2}( 1 / 2 - i k / m )}\; ,
p_{Z}(z)=\frac{b(z)\cdot d(z)}{a^{3}(z)}\frac{1}{\sqrt{2\pi}\sigma_{x}\sigma_{y}}[\Phi(\frac{b(z)}{a(z)})-\Phi(-\frac{b(z)}{a(z)})]+\frac{1}{a^{2}(z)\cdot\pi\sigma_{x}\sigma_{y}}e^{-\frac{c}{2}}
|[x_n(f_1)-x_n(f_2)](t)| =|a^{-n}(t)[f_1(t)-f_2(t)]| &\leq(1-r)^{-in}|f_1(t)-f_2(t)| &\leq 2(1-r)^{-in}f_0(t) &\leq 2(1-r)^{-in}e^{-i^2}&<\eta.
\mathrm{I m}m_{\mathrm{r e n}}^{2}= \pm i \pi \frac{\lambda}{3 2 \pi^{2}}( m^{2}- 2 M r^{- 2}- 4 M r^{- 3}- \frac{1}{2}M^{2}r^{- 2}( 1 - \frac{2 M}{r})^{- 1})
H(f)-H(\mathcal{M}_{\nu,\theta}) \geq H(f|\mathcal{M}_{0,1})+(1-\theta)\{H(\mathcal{M}_{0,1})-H(\mathcal{M}_{\Theta})\}\cr \geq H(f|\mathcal{M}_{0,1})+(1-\theta)\{H(\mathcal{M}_{0,1})-H(f)\}\cr = H(f|\mathcal{M}_{0,1})+(\theta-1) H(f|\mathcal{M}_{0,1}) \cr = \theta H(f|\mathcal{M}_{0,1}).
\ddot{z}^{+}- \dot{z}^{+}\frac{\ddot{z}^{-}}{\dot{z}^{-}}- 2 \lambda \dot{z}^{+}( \dot{z}^{+}+ \dot{z}^{-}) + 4 \lambda \dot{z}^{+}\dot{z}^{-}= 0 \, ,
[y_1, y_1]=y_4, [y_1, y_2]=\frac{\alpha_1\alpha_4}{\gamma_1}y_5, [y_2, y_1]=\frac{\alpha_1\alpha_6}{\gamma_1}y_5, [y_2, y_2]=\frac{\alpha^2_1\beta_2}{\gamma^2_1}y_5, [y_1, y_3]=\beta_4y_5, [y_3, y_1]=\beta_6y_5, [y_2, y_3]=y_4, [y_3, y_2]=\frac{\alpha_1\gamma_4}{\gamma_1}y_5, [y_3, y_3]=\gamma_6y_5.
F(\mathbf{x}) = \int\limits_{\substack{\boldsymbol{\xi}\in \mathbb{R}^m \Vert \boldsymbol{\xi}\Vert_\infty \leqslant X}}c_X(\boldsymbol{\xi})e(\frac{\boldsymbol{\xi}\cdot \mathbf{x}}{N}) \, d\boldsymbol{\xi}+ O_{C}\left(MN\frac{\log X}{X}\right)
\sum_{k=0}^{\frac{p-3}2}\frac{\binom{\frac{p-1}2}k\binom{2k}k}{(2k+1)(-4)^k}\equiv-2q_p(2)+pq_p(2)^2+(-1)^{\frac{p-1}2}pE_{p-3}\pmod{p^2}.
\phi^\prime(x) \sim \frac{(2n)!}{(2m)!}e^{(2m+1)x-(2n+1)\phi(x)}=\frac{(2n)!}{(2m)!}e^{(2m+1)x-(2n+1)(kx+c+o(1))}&=\frac{(2n)!}{(2m)!}e^{-(2n+1)c+o(1)}=k+o(1).
\sum_{j_1+\cdots +j_{m+r}=n}\binom{n}{j_1,\cdots, j_{m+r}}B_{j_1}(x_1)\cdots B_{j_m}(x_m)E_{j_{m+1}}(\beta; x_{m+1})\cdots E_{j_{m+r}}(\beta; x_{m+r}) = \left ( B_n(m,\cdot)\times E(r,\beta;\cdot)\right )(x)=\sum_{k=0}^n v_k \Delta^k I_n(x),
I\left(f_{\omega_{\pi_\nu},s}^\circ,v^\circ_{\pi_{\nu}}\right)(I_m)=\int\limits_{Z_n(k_\nu)\backslash T^-}\mu(\underline{t}) \int\limits_{K_{n,\nu}^2}f_{\omega_{\pi_\nu},s}^\circ\left(\tilde{\varepsilon}t\left(I_m,A\underline{t}B\right)\right)\pi_{\nu}(A\underline{t}B)v^\circ_{\pi_{\nu}}dAdBd\underline{t},
\frac1p=\frac1{p_1}+\frac1{p_2}<\frac1{r_1}+\frac1{r_2}\le\frac1{\min\{r_1,2\}}+\frac1{\min\{r_2,2\}}<2-\frac1{\min\{r_3,2\}}\le\frac32.
A_i=\frac{(i+\alpha+\beta+1)(i+\alpha+1)(n-i)}{(2i+\alpha+\beta+1)(2i+\alpha+\beta+2)}\, ,C_i=\frac{i(i+\alpha+\beta+n+1)(i+\beta)}{(2i+\alpha+\beta)(2i+\alpha+\beta+1)}.
\mathbb{P}\left[\gamma\in F, ||\gamma-\gamma_{n}||_{\infty}< \delta \right] \le A\left(\kappa,n\right)e^{-\frac{I(F^{\delta})}{\kappa}}, A(\kappa, n) = \frac{4\left(\frac{2I(F^{\delta})}{\kappa}\right)^{\frac{n}{2}-1}}{2^{\frac{n}{2}}\Gamma(\frac{n}{2})}.
\mathrm{Cont}^{\ge m}(\mathfrak a)\cap\mathrm{Cont}^{\ge 1}((x_1,\dots,x_n)) = \mathrm{Spec}k[[x_i^{(j)}\ |\i=1,\dots,n,j\in\mathbb Z_{\ge 0}]]/(x_1^{(0)},\dots,x_n^{(0)},f_s^{(l)}\ |\ 1\le s\le r,\ 0\le l\le m-1).
q = \frac{1}{2 ( 2 \pi )^{D - 2}a^{3}}\int d^{N}{\bf k}\sum_{n = - \infty}^{+ \infty}\int_{0}^{\infty}\frac{d z}{\bar{J}_{n}^{2}( z ) + \bar{Y}_{n}^{2}( z )}
\langle g,T^{\ast}(K_a)\rangle = \langle Tg, K_a\rangle= \langle w\,(g\circ \varphi), K_a\rangle =w(a)\,g(\varphi(a))=w(a) \langle g, K_{\varphi(a)}\rangle =\langle g,\overline{w(a)}K_{\varphi(a)}\rangle.
\|u\|^p_{L^p_R}=\int_{R}^{\infty}(u(r))^pr^5dr,\|u\|^2_{\dot{H}^1_R}=\int_{R}^{\infty}(\partial_ru(r))^2r^5dr, \|\psi\|_{L^2(R,\infty)}=\int_{R}^{\infty}(\psi(r))^2dr.
\begin{array}{rcl}\Delta_{(p_1, p_2)}= e^{-\phi_0}\left(P^{(1)}/P_0 \right)^2 + \left(P^{(2)}/P_0 + \chi_0 P^{(1)}/P_0 \right)^2 e^{\phi_0}= \left(P^{(1)}/P_0, P^{(2)}/P_0 \right){\cal M}_0 \left(\begin{array}{c}P^{(1)}/P_0 P^{(2)}/P_0\end{array}\right),\end{array}
\langle Y_i^{(m)}-X_{i-1}^{(m)},X_{i-1}^{(m)}-Z_{i-1}^{(m)}\rangle\mid_{X_{i-1}^{(m)}=z, X_i^{(m)}=z'}=~ -\langle Y_i^{(m)}-X_{i-1}^{(m)},X_{i-1}^{(m)}-Z_{i-1}^{(m)}\rangle\mid_{X_{i-1}^{(m)}=z, X_i^{(m)}=z''}
\begin{aligned}(\tilde{\Delta}_E\tilde{h}_1,\tilde{h}_2)_{L^2(\tilde{g})}= -16\lambda_i(\lambda_i-5)\int_0^{\infty}\varphi\psi\cosh \sinh^{3}dr, (\tilde{\Delta}_E\tilde{h}_2,\tilde{h}_3)_{L^2(\tilde{g})}= 24\lambda_i\int_0^{\infty}\psi\chi \cosh \sinh^{3}dr.\end{aligned}
\frac{dE(t)}{dt}=-\tau_{f}\sum_{i,j=1}^{N_{f}}\frac{dx_{i}}{dt}\frac{\partial^{2}L_{x}}{\partial x_{i}\partial x_{j}}\frac{dx_{j}}{dt}-\tau_{h}\sum_{\mu,\nu=1}^{N_{h}}\frac{dh_{\mu}}{dt}\frac{\partial^{2}L_{h}}{\partial h_{\mu}\partial h_{\nu}}\frac{dh_{\nu}}{dt}\le0
T_{h}= \left( \begin{matrix}{{n}}&{{0}}&{{\cdots}}&{{0}}&{{0}}\\{{0}}&{{n - 1}}&{{\cdots}}&{{0}}&{{0}}\\{{\vdots}}&{{\ddots}}&{{\ddots}}&{{\vdots}}&{{\vdots}}\\{{0}}&{{0}}&{{\cdots}}&{{- n + 2}}&{{0}}\\{{0}}&{{0}}&{{\cdots}}&{{0}}&{{- n}}\\ \end{matrix}\right) , \quad [ T_{a}, T_{b}] = f_{a b}^{c}T_{c}\, .
\left|G_j(\l)\right| = \left|\frac{o(e^{i \tau_k \l})}{(1+o(1))\Delta(\l)}\right| \leqslant \frac{C_1 e^{\Re(i \tau_k \l)}|\l|^s}{C e^{\Re(i \tau_k \l)}}= C_2 |\l|^s, \arg \l = \arg z_k, \ |\l|>R,
\tilde{E}_{n \delta}^{( i )}= \sum_{\begin{array}{c}{{k_{1}p_{1}+ . . . + k_{m}p_{m}= n}}\\{{0 < k_{1}< . . . < k_{m}}}\\ \end{array}}\frac{\left( q^{( \alpha_{i}, \alpha_{i})}- q^{- ( \alpha_{i}, \alpha_{i})}\right)^{\sum_{i}p_{i}- 1}}{p_{1}! \; . . . \; p_{m}!}( E_{k_{1}\delta}^{( i )})^{p_{1}}. . . ( E_{k_{m}\delta}^{( i )})^{p_{m}}
- \frac{\partial^2F}{\partial s^2}+ \frac{\partial}{\partial u}.(\mathcal{G}F)-\epsilon F\kappa : u\otimes u+\epsilon \frac{\partial}{\partial s}\Big( F\kappa:\lambda(F)\Big)=0 \textrm{on}S_2 \times ]0, 1[ & F(s=0) = F(s=1) = (1/4\pi)
\mathbb{P}[^{+}_{\omega \ge \omega_1}X^h(\omega) \in d\omega_2]=2\frac{j(\omega_2-h)}{j(\omega_1)}\int_{0}^{r^h(\omega_2)}\int_{-\infty}^{x^*}\Phi(\omega^{*}-h,y-x^*,\omega_2-h) \times \tilde{\Phi}(-\omega^*,y-x^*,-\omega_1)dydx^*
R l ( \alpha_{\mu}; \beta_{\nu}) \delta ( \alpha^{2}- p^{2}) \delta ( \beta^{2}- p^{2}) \delta \left( ( \alpha \beta ) - q \right) = l ( \alpha_{\mu}; P_{\rho}^{\sigma}( \alpha ) \beta_{\sigma}) \delta ( \alpha^{2}- p^{2}) \delta ( \beta^{2}- p^{2}) \delta \left( ( \alpha \beta ) - q \right)
F_m-F_{m-1}=\left( \tilde{U}_{m-1}\cdot \nabla\right)V_m+\left( U_{m-1}\cdot \nabla\right)\tilde{V}_{m-1}+\left( \tilde{U}_{m-1}\cdot \nabla\right)v_{}+\left( U_{}\cdot \nabla\right)\tilde{V}_{m-1}
A_k(t)=\frac{1}{n}\int (h_L-h_K)(u) dS_{M_k}(u) = \frac{1}{n}(h_L-h_K)(u_k) S_{M_k}(u_k)=\frac{\kappa_{n-1}}{n}(h_{L}(u_k)-h_K(u_k)).
\sum_{\boldsymbol{i}\in T, | \boldsymbol{i}| = n(k)}s_{\boldsymbol{i}}= \prod_{j=1}^k \Bigg( \sum_{| \boldsymbol{i}| = n(j) - n(j-1)}s^{(j-1)}_{\boldsymbol{i}}\Bigg) .
\underset{z\in(0,\infty)}\sup~\bigl(\frac{1}{(1+z)^n}-e^{-nz}\bigr)=\frac{1}{(1+z_n)^n}-e^{-nz_n}=\frac{z_n}{(1+z_n)^{n+1}}\le \frac{z_n}{(n+1)z_n}\le \frac{1}{n+1}.
\left< W ( C ) \right> = \left< \exp \left[ \frac{i g}{4}\int_{\Sigma}^{}d \sigma_{\mu \nu}\left( f_{\mu \nu}+ h_{\mu \nu}\right) \right] \right>_{a_{\mu}, j_{\mu}}= \left< W ( C ) \right>_{a_{\mu}}\left< \exp \left( \frac{i g}{4}\int_{\Sigma}^{}d \sigma_{\mu \nu}h_{\mu \nu}\right) \right>_{j_{\mu}}.
a_{i+1}=\frac{a_i^{19k^4}}{2^{55k^6}}, b_{i+1}=\frac{a_i^{37k^2}}{2^{118k^4}}b_i^4, C_{i+1}=\frac{2^{122k^4}}{b_i^{4}a_i^{37k^2}}C_i, n_{i+1}=\frac{2^{14k^3}}{a_{i}^{4k}}n_{i}\,.
\begin{aligned}\Vert u(t) \Vert_{\infty}\leq K \left[ \left( t^{-\frac{N}{2q_0+N(m_1-1)}}\, \Vert u_0 \Vert_{q_0}^{\frac{2q_0}{2q_0+N(m_1-1)}}\right) \wedge \left( t^{-\frac{N}{2q_0+N(m_2-1)}}\, \Vert u_0 \Vert_{q_0}^{\frac{2q_0}{2q_0+N(m_2-1)}}\right) + \Vert u_0 \Vert_{q_0}\right]\end{aligned}
L(t)\, :=\, \exp\left(-\int_0^t\vartheta^\prime (s,\mathfrak{X})\, \mathrm{d}W(s) - \int_0^t \frac{1}{\,2\,}\,\big|\big|\vartheta(s,\mathfrak{X})\big|\big|^2 \mathrm{d}s\right), 0\leq t < \infty\,.
\left[ \sum_{j=1}^{n-1}(\nu_j-\nu_n) \left(\partial_j \Delta_n \right) +(d-\nu_n-\sum_{j=1}^{n}\nu_j)G_{n-1}\right]I_n^{(d)}=\sum_{j=1}^{n}\left(\partial_j \Delta_n\right)I_{n,j}^{(d-2)}+2 \Delta_n I_{n}^{(d-2)}.
\sum_{j=1}^\infty|q_j(z)|^2=\sum_{j=1}^L|q_j(z)|^2 \le \sum_{j=1}^L\|q_j\|_\infty^2\le C\sum_{j=1}^L\|q_j\|_{H^2}^2\le \frac{C}{\delta^{2\max m_j}}
\frac{1}{A_{1}^{\alpha_{1}}}=\frac{1}{(\alpha_{1}-1)!}\int_{0}^{\infty}ds_{1}s_{1}^{\alpha_{1}-1}e^{-s_{1}A_{1}}=\frac{1}{\Gamma(\alpha_{1})}\frac{\partial^{\alpha_{1}-1}}{\partial(-A_{1})^{\alpha_{1}-1}}(\int_{0}^{\infty}ds_{1}e^{-s_{1}A_{1}})
\big( (2m_{1}+m_{2}+2m_{3}) (\cot \langle \alpha_{2}, H\rangle )^{2}-(2n_{1}+m_{2}+2m_{3})\big) ((\cot \langle \alpha_{2}, H\rangle )^{2}-1) = (2m_{2}+4m_{3})(\cot \langle \alpha_{2}, H\rangle )^{2}.
\varphi\left(\begin{array}{c c}a b 0 d \end{array}\right) = \left(\begin{array}{c c c}a^{\ell_1}0 0 0 a^{\ell_1}c_1 \, a^{\ell_1}\, b^{p^{e_1}}\, d^{p^{e_1}}0 0 d^{- \ell_3}\end{array}\right) .
\begin{aligned}\int_{B_{r_{kk_0}}}G(x,y)|f|(y) dy\leq C_1Mr_{kk_0}^{s-n/p}\leq C_1Mr_{kk_0}^{\beta}\leq \hat{C}M r_{(k+1)k_0}^{\beta}\cdot\frac{C_1}{\hat{C}\tau_1^{\beta k_0}}.\end{aligned}
F_{j}(U^{(j)})\cdots F_3(U^{(3)})F_2(U^{(2)})\underline{F_{1}(U^{(1)})F(V_b)}&=F_{j}(U^{(j)})\cdots F_3(U^{(3)})\underline{F_2(U^{(2)})F_2(V_{b_1})}F_1(U^{(1)'}) &=F_{j}(U^{(j)})\cdots \underline{F_3(U^{(3)})F_3(V_{b_2})}F_2(U^{(2)'})F_1(U^{(1)'}) &=\cdots=F_{j+1}(U^{(j+1)'})\cdots F_3(U^{(3)'})F_2(U^{(2)'})F_1(U^{(1)'}),
{\bf g}\; :=\; \begin{pmatrix}g_1 g_2 g_3 \end{pmatrix}\; = \; \begin{pmatrix}-q^{3}KF+qFK q^{-3}KE-q^{-1}EK qEF-q^{-1}FE + \kappa K^2 -\kappa K'^2 \end{pmatrix}.
C_{pt}=\; (WC_{\xi}+UC_{\xi^2})_p =\; g(W_{\xi}C_{\xi}+WC_{\xi^2}+U_{\xi}C_{\xi^2}+UC_{\xi^3}) =\; g\Big(W_{\xi}-U\lambda\Big)C_{\xi}+g\Big(W+U_{\xi}-U\varphi\Big)C_{\xi^2},
M:= \lim_{x\to \infty}\frac1{x}\sum_{n\le x}f(n) = \prod_p \left(1-\frac1{p}\right) \sum_{\nu=0}^{\infty}\frac{f(p^\nu)}{p^\nu}= \prod_p \left(1-\frac1{p}\right) \sum_{\nu=0}^{\infty}\frac{(p^{\nu+1}-1)^2}{p^{2\nu}(p^{\nu+2}+p^{\nu+1}-(2\nu+3)p+2\nu+1)}.
<\Psi_1|\Psi_2>=N\int\limits_{p^2=-m^2}\frac{d{\bf p}}{p_0}\overline{\Theta}(p)_1^{A(l_1)B(l_2)...C(l_{r-1})a(l_r)}\Theta(p)_{2A(l_1)B(l_2)...C(l_{r-1})a(l_r)}
\left|\partial_t\rho_\eta^\varepsilon-\overline{\partial^\circ\eta}-k_d^{-1}\Bigl(\overline{V_\Gamma^2\eta}\Bigr)\right| \leq c\varepsilon\sum_{\xi=\eta,\zeta_0,\zeta_1}\left(|\bar{\xi}|+\Bigl|\overline{\partial^\circ\xi}\Bigr|+\Bigl|\overline{\nabla_\Gamma\xi}\Bigr|\right)
\Phi^{( \pm )}= e^{- i \omega t}Y_{\ell}( \cos \rho )^{2 h_{\pm}}( \sin \rho )^{\ell}P_{n}^{( \ell + \frac{d}{2}- 1 , 2 h_{\pm}- \frac{d}{2})}( \cos 2 \rho ) \ .
\Big|J_p(u_h(x)-u_h(y))-J_p(u(x)-u(y))\Big| \le (p-1)\,\left(|u_h(x)-u_h(y)|^{p-2}+|u(x)-u(y)|^{p-2}\right) &\times |(u_h(x)-u_h(y))-(u(x)-u(y))| &\le (p-1)\,\left(|u_h(x)-u_h(y)|^{p-2}+|u(x)-u(y)|^{p-2}\right) &\times \Big(|\delta_h u(x)|+|\delta_h u(y)|\Big).
\frac{\partial ~\gamma_{n}\left( \underline{\theta}\right)}{\partial~\theta_{m}}=\gamma_{n}\left( \underline{\theta}\right) ~\left\{\delta_{nm}~\sum_{\ell =1,~\ell \neq n}^{N}\left[ \cot \left( \theta_{n}-\theta_{\ell}\right) \right] -\left( 1-\delta_{nm}\right) ~\cot \left( \theta_{n}-\theta_{m}\right) \right\}~,
\psi \ll N(d)^{(n-2)(3/2-\Re(s)+\varepsilon)/4}N(b)^{3n/4-1-n \Re(s)/2+2\varepsilon}(1+|s|^2)^{\sum_{\infty}(k) \cdot (n/2-1/2)(3/2-\Re(s)+\varepsilon)},_{s=1+1/n}\psi \ll N(d)^{(n/4-1/2)(1/2-1/n+\varepsilon)}N(b)^{n/4-3/2+\varepsilon}.
\lim_{n \rightarrow \infty}\left( S^{\phi}_{q}(f_{\alpha}^{(n)}) - \frac{1}{q-1}\left(1-\frac{1}{\sqrt{q}}\left( \frac{2 \pi \alpha (1-\alpha)}{n}\right)^{\frac{1-q}{2}}\exp \left( \frac{(\alpha-\gamma)^2 (1-q)}{2\alpha (1-\alpha)q}\right) \right) \right) = 0
C_{2 m}( p_{i}; \Lambda ) = -{\frac{1}{2}}M^{3}\partial_{M}L_{2 m , 0}( p_{i}; \Lambda ) - \sum_{i}C_{i}( m , M , \Lambda_{R}, \lambda_{R}, k_{R}) O_{2 m}^{i}( p_{i}; 0 ; \Lambda )
\lim_{N\to \infty}\frac{\log \#\left(X, d^\sigma_N,\varepsilon\right)}{N}\leq \lim_{N\to \infty}\frac{\log \#\left(X, d^{\sigma, T_a}_{[0, N+L)\times [0,M)},\, \frac{1}{4a}\right)}{N}= \lim_{N\to \infty}\frac{\log \#\left(X, d^{\sigma, T_a}_{[0, N)\times [0,M)},\, \frac{1}{4a}\right)}{N}.
\lim_{T\rightarrow\infty}\overline{\Psi}_{f_{1}}\left(u,T\right)=\lim_{T\rightarrow\infty}J\left(u,T\right)\int_{\left(S_{p}\times S_{q}\right)^{n}}\lim_{T\rightarrow\infty}f_{1}\left(\left\Vert g_{0}^{-1}\left(u_{1}\eta_{1},\sqrt{u_{1}^{2}-u_{2}/T^{2}}\eta_{2},\dots,u_{2n}\eta_{2n}\right)\right\Vert \right)d\eta.
\|P_SVx\| =\left\|\sum_{j\in S}f_j(Vx)\tau_j\right\|=\left\|\sum_{j\in S}f_j\left(\sum_{k=1}^{n}g_k(x)\tau_k\right)\tau_j\right\|=\left\|\sum_{j\in S}\sum_{k=1}^{n}g_k(x)f_j(\tau_k)\tau_j\right\| &=\left\|\sum_{j\in S}g_j(x)\tau_j\right\|=\left(\sum_{j \in S}|g_j(x)|^p\right)^\frac{1}{p}=\|x\|_{S, g}, \forall x \in \mathcal{X}.
\mathbf{d}\,^j:=\pi_\mathcal{S}\,\mathbf{e}^j, \langle\boldsymbol{\nu}(x),\mathbf{d}\,^j(x)\rangle\equiv0, \sum_{j=1}^n\nu_k\mathbf{d}\,^k=0, j=1,\ldots,n,
\frac{p_{\nu_{1}\boxtimes\nu_{2}}(\xi)}{p_{\mu_{1}\boxtimes\mu_{2}}(\xi)}=\frac{|1-\eta_{\mu_{1}}(z)|^{2}}{\beta}\int_{\mathbb{T}}\frac{d\sigma(t)}{|t-\eta_{\mu_{1}}(z)|^{2}},\quad\xi\in\Gamma.
\mathrm{T r}e^{i F_{9}}- \mathrm{t r}e^{i F_{9}}\, \mathrm{t r}e^{i F_{\bar{9}}}+ \mathrm{T r}_{S}e^{i F_{\bar{9}}}={\frac{1}{2}}\, ( \mathrm{S t r}e^{i F_{9}})^{2}-{\frac{1}{2}}\, \mathrm{S t r}e^{2 i F_{9}}.
S^-_{k,j}:=\left\{\omega\in S_k\Bigl|\left|w_{\psi_a}(\omega)\right|< q\|\psi_a\|\,\forall\, 1\leq a<j, \mbox{and}w_{\psi_j}(\omega)\leq -q\|\psi_j\|\Bigr.\right\}.
\begin{aligned}|\mathcal B^{(d)}|\geq \left\lceil k-d-\frac{1}{2}-C\frac{\binom{n-d-1}{k-d-1}}{\binom{n-d}{k-d}}\right\rceil = \left\lceil k-d-\frac{1}{2}-C\frac{(k-d)}{n-d}\right\rceil \geq k-d,\end{aligned}
g_\alpha(p) = p^{3n -k-\ell-9}+ (n-\ell-1) (p-1) (p+1)p^{2n -k-7}&+(n-k-1)(p-1) p^{2n-\ell-6}- (n-\ell-1)p^{n-4}(p-1) &+(n-k-2)(n-\ell-1) (p-1)^2p^{n-4}& + \left((n-\ell-2) + \binom{n-\ell-2}{2}\right)(p-1)(p-2)(p-3)p^{n-4}.
\frac{\lambda_2^{K_{\ell}}}{\lambda_2^L}=\Bigl\{2^{(\dim M_2^{K_{\ell}}-m)/2}\cdot|M_2^{K_{\ell}}|^{-1}\cdot\prod_{i=1}^{m}(2^{2i}-1)\Bigr\}\Bigl\{2^{(\dim M_2^L-m)/2}\cdot|M_2^L|^{-1}\cdot\prod_{i=1}^{m}(2^{2i}-1)\Bigr\}^{-1}\leq 3\cdot 2(2^{n_{2,\nu_2}}-1).
\mathcal{C}' = \left\{\left.\left(\mathsf{a},u,\Phi \in \mathcal{C}^{1,p}_{\ast}\right)~~ \right\vert ~~ \left(\begin{array}{c}\partial_{\mathsf{a}}u - (X_{\Phi})^{1,0}\ast\overline{\partial}\eta^{1,0}- \mu_{c}\circ u\end{array}\right) = 0 \right\}
Z_{g h}= \int D [ C , \tilde{\cal P};{\cal P}, \tilde{C}] \cdot \exp \{i \int_{\tau_{i}}^{\tau_{f}}( \tilde{\cal P}_{a}\dot{C}^{a}- \dot{\tilde{C}}_{a}{\cal P}^{a}- \tilde{\cal P}_{a}{\cal P}^{a}) d \tau \}.
( 2 \pi )^{d}\delta^{( d )}\! \left( \sum_{i = 1}^{M}p_{i}\right) A_{P}( p_{1}\dots p_{M}) = \frac{N}{( 2 \alpha^{\prime})^{d / 2}}\frac{\langle B_{0}| D | B_{M}\rangle}{\pi}.
\left\{j_{\alpha}^{0}, j_{\beta}^{0}\right\}\approx \left[ 2 \Gamma_{\alpha \beta}^{m}\cdot j_{m}^{0}\; + \left( e m b^{*}S_{\alpha \beta}\left( \Delta \right) \right)_{r}\cdot \Lambda^{r}\right] \cdot \delta \left( \sigma - \sigma^{\prime}\right) \quad ,
f(x_{0}+h)=f(x_{0})+\frac{f^{\prime}(x_{0})}{1!}h+\frac{f^{(2)}(x_{0})}{2!}h^{2}+\cdot\cdot\cdot+\frac{f^{(n)}(x_{0})}{n!}h^{n}+R_{n}(x),
\mathcal{I}_{\chi}(\alpha){=}\left. s_{\chi}(t) .* h_{\chi}(t-t_o) \right|_{t=t_o}{=}\int\nolimits_{-\frac{B_{\chi}+\alpha B}{2}}^{\frac{B_{\chi}+\alpha B}{2}}S_{\chi}(f) S_{\chi}^{*}(f) df,
\|\mathcal{T}[F](t)\|_{L^1}\leq \iint_{\Omega\times\mathbb{R}^3}\big|\bar{f_0}\big(X(t_0),V(t_0)\big)\big|dxdv + \iint_{\Omega\times\mathbb{R}^3}\int_{t_0}^t \big|Q^\varepsilon[F]\big(s,X(s),V(s)\big)\big| ds dxdv begin{align*}3pt] =: I + II ,
\limsup_{n\to\infty}\frac{1}{a_c^{(n)}}\log P(B_1^{(n)'}) \leq\limsup_{n\to\infty}\frac{1}{a_c^{(n)}}\log\left(\sum_{t=a_n}^{\lfloor\kappa_n(x_0)\rfloor}P(S_n(t)\leq t-a_n)\right) &\leq\limsup_{n\to\infty}\frac{1}{a_c^{(n)}}\log((\lfloor\kappa_n(x_0)\rfloor-a_n+1)P(S_n(t_n)\leq t_n-a_n)) &\leq-J(x_0).
R_3\leq{I({X_3;Y_2})}=h(Y_2)-h(Y_2|X_3) &=\frac{1}{2}\log\frac{|{\hat{\bf{C}}}_{y_2}|}{|{\hat{\bf{C}}}_{s_2}|}&=\frac{1}{2}\log\frac{{C^{2}_{y_2}}-{|\tilde{C}_{y_2}|^{2}}}{{C^{2}_{s_2}}-{|\tilde{C}_{s_2}|^{2}}}=L_3,
\begin{aligned}x^0(u,v) =\frac{1}{2}\int(1+q^2)fdu-(1+r^2)gdv, x^1(u,v) =-\frac{1}{2}e^{\mu_1x^0(u,v)}\int(1-q^2)fdu+(1-r^2)gdv, x^2(u,v) =-e^{\mu_2x^0(u,v)}\int qfdu+rgdv. \end{aligned}
H_0(b_1,\cdots,b_d) = \sum_{j=1}^{d}\Bigg( \left( -i \partial_{x_j}+ \frac{b_{j}y_{j}}{2}\right)^{2}+ \left( -i\partial_{y_j}- \frac{b_{j}x_{j}}{2}\right)^{2}\Bigg),
\prod_{\substack{m,n \geq 1 m \leq n ; \, (m,n)=1}}\left( \frac{1}{1- y^m z^n}\right)^{\frac{m^3}{n^4}}= \exp\left\{\sum_{n=1}^{\infty}\left( \frac{n^3 y^{n+4}+(3n^3+6n^2-4) y^{n+2}}{(1-y)^4}\right) \frac{z^n}{n^4}\right\}
\lim_{t \to \infty}t^{\frac 12 - \frac 1p}\|v(t) - \alpha \Theta(t,\cdot)\|_{L^p(\mathcal F_0)}= 0 \forall \,p \in (2,\infty) & \sup_{t >t_0}(t-t_0)^{\frac 1q}|\ell(t)| < \infty
T=\displaystyle\frac{2\left( \int_{0}^{R}f(r)v_{0}dr\right) \left( \int_{0}^{R}\frac{f^{2}(r)}{{f}^{\prime}(r)}dr\right)}{\left( \int_{0}^{R}f(r)v_{0}dr\right)^{2}-2C\left( \int_{0}^{R}\frac{f^{2}(r)}{{f}^{\prime}(r)}dr\right) \left( \int_{0}^{R}f(r)dr\right)}>0
\mathcal{A}(a, b) = \sum_i \mathcal{C}(i, \mathcal{F}_a(x - 1))\mathcal{C}(i, \mathcal{F}_b) + \sum_i \mathcal{C}(i, \mathcal{F}_a)\mathcal{C}(i, \mathcal{F}_b(x - 1)) = [\mathcal{F}_a(x - 1) \bigcirc \mathcal{F}_b(x)] + [\mathcal{F}_a(x) \bigcirc \mathcal{F}_b(x - 1)],
H^{(orb)}_{\lambda=1}= \frac{1}{sin \gamma}\sum^L_{j=1}\left( \tau^+_j \tau^-_{j+1}+ \tau^-_j \tau^+_{j+1}+\rho^+_j \rho^-_{j+1}+ \rho^-_j \rho^+_{j+1}- \sigma^0_j -\sigma^0_{j+1}\right)
g_{\mathrm{\footnotesize \boldmath a}}^{( n )}( z , v ) = \sum_{\mathrm{\footnotesize \boldmath \alpha}\in G_{m}}\omega^{2 \langle \mathrm{\footnotesize \boldmath \alpha}, \mathrm{\footnotesize \boldmath a}\rangle}u_{2 \mathrm{\footnotesize \boldmath \alpha}}^{( n )}( z , v ) u_{2 ( \mathrm{\footnotesize \boldmath a}+ \mathrm{\footnotesize \boldmath \alpha})}^{( n )}( - z , v ) .
\mathcal{E}(\mathbf{x},s)=-\dfrac{\Gamma \left( \dfrac{s}{2}\right)}{\pi^{s}}\sum_{\mathbf{k}\in \mathbb{Z}^{n}\setminus \mathbf{\lbrace 0\rbrace}}\dfrac{e^{i\pi\mathbf{x}.\mathbf{k}}}{\Vert\mathbf{k}\Vert^{s}}.
\mathfrak{A}(n;q_1,q_2,q_2')=\sum_{\alpha_1\bmod{q_1}}K_{q_1}(\alpha_1\ell\bar{q}_2^3)K_{q_1}(-\alpha_1\ell\bar{q_2'}^3)e_{q_1}(\alpha_1 \bar{q}_2\bar{q}_2' n)
f_k(x):=\begin{cases}-\frac{1}{2^k}\left(1-\frac{d_X(x,\omega_j)}{\varepsilon_k}\right) \textrm{if}x\in I_{j,k}\textrm{for}j=1,\dots, a_k; +\frac{1}{2^k}\left(1-\frac{d_X(x,\omega_j)}{\varepsilon_k}\right) \textrm{if}x\in I_{j,k}\textrm{for}j=a_k+1,\dots, 2a_k; 0 \textrm{elsewhere}.\end{cases}
\tilde Q_{jk}=\big(\tilde UX\tilde U^\dag\big)_{jk}=\phi_j\delta_{jk}-\mathrm{i}g\frac{1-\delta_{jk}}{\lambda_j-\lambda_k}, \tilde L_{jk}=\big(\tilde UP\tilde U^\dag\big)_{jk}=\lambda_j\delta_{jk},j,k=1,\dots,n.
\Phi\left(\cap_{i=1}^k A_i \right) (\leqq \times \cdots \times \leqq) \Phi\left(\cap_{i=1}^k A'_i \right) \Longleftrightarrow \pi(\ell_i(A_i)) \leqq \pi(\ell_i(A'_i)) (\forall i) & \Longleftrightarrow \cap_{i=1}^k A_i \subset \overline{\cap_{i=1}^k A'_i}.
\delta_k(A\wr_I B) = \sum_{\ell=0}^n \delta_\ell(B) \sum_{j_1+\dots+j_\ell=k}\prod_{r=1}^{\ell}\delta_{j_r}(A) \quad\textup{and hence}\delta(A\wr_I B) = \sum_{\ell=0}^n \delta_\ell(B) \delta(A)^\ell.
\mu=m_{z}\{4[\frac{\partial}{\partial r}(\frac{1}{r}\frac{\partial\psi}{\partial\theta})]^{2}+[\frac{1}{r^{2}}\frac{\partial^{2}\psi}{\partial\theta^{2}}-r\frac{\partial}{\partial r}(\frac{1}{r}\frac{\partial}{\partial r})]^{2}\}^{(n-1)/2}
\lim_{N \rightarrow \infty}\max_{l \leq N_k^{\delta}}\max_{\mu, \nu}\left \vert \mathbb{E}^V \theta(N \zeta_{\alpha^{\prime}}(\mu)\zeta_{\alpha^{\prime}}(\nu))-\mathbb{E}^V \theta[\frac{N}{\pi}\int_{I}\tilde{G}_{\mu \nu}(z) \mathcal{X}(E) dE] \right\vert =0,
\int_{M_j}\int_{\partial D_j}\left|\frac{f^*}{g_l}\right|^2|dz_j|d\mu_j(\hat w_j) \le C_0\int_{M_j}\int_{\partial D_j}\left|f^*\right|^2e^{-2\log|g_l|}\frac{\rho}{e^{-2\log|g_l|}}|dz_j|d\mu_j(\hat w_j) \le C_0\|f\|^2_{\partial M,\rho}< +\infty
\|F(g)\|_S < \left\{\frac{1}{40}+ \frac{1}{32}+\frac{1}{8}\left(\frac8{7}+ 1\right)\right\}+ \frac{1}{2}= \frac{1}{40}+ \frac{1}{32}+ \frac{1}{7}+\frac{1}{8}+\frac{1}{2}<1.
\eta = \bigoplus_{\substack{I \subseteq [n] |I| = t}}\eta_{I}= \bigoplus_{\substack{I \subseteq [n] |I| = t}}\sum_{p \in \mathbb{Z}}\eta_{I,p}\in \bigoplus_{\substack{I \subseteq [n] |I| = t}}\Gamma(D(x_{I}), \Omega_{X}^{j}(\log f))
\Phi_k \left(G_{N,S}\right)=\max_{0\leq d_2\leq \dots \leq d_k}\ \sideset{}{^*}\max_{s_1,\dots, s_k}\max_{1\leq m_1\leq m_2\leq N-d_k}\left|\sum_{n=m_1}^{m_2}e_{n+d_1}(s_1)\dots e_{n+d_{k}}(s_k)\right|,
\lim_{T\rightarrow\infty}\frac{\mathbb{E}_x\left[Z_T^{b^\ast}\right]}{T}= \lim_{T\rightarrow\infty}\frac{1}{T}\mathbb{E}_x\int_0^T \mu\left(X_t^{Z^{b^\ast}}\right)X_t^{Z^{b^\ast}}\,dt.
a(v, \xi) = <v, \overrightarrow{z\xi}> - <DF_p(v), \overrightarrow{F_p(z)f(\xi)}> , b(v, \xi) = \frac{1}{2}\left(d^2 B^{f(\xi)}(DF_p(v), DF_p(v) ) - d^2 B^{\xi}(v, v)\right).
\begin{aligned}-\left(e^{-s}C_pq\left| \dot{\xi}(s)\right|^{q-2}\dot{\xi}(s)\right)^\prime +e^{-s}Df\left( \xi (s)\right) =0 \Longrightarrow -\left(e^{-s}C_pq\left| \dot{\xi}(s)\right|^{q-2}\dot{\xi}(s)\right)^\prime \cdot \dot{\xi}(s) e^s+ Df\left( \xi (s)\right)\cdot \dot{\xi}(s) =0.\end{aligned}
\frac{\mathrm{d}}{\mathrm{d}\ln s}\, \, \ln \left| \frac{F ( s )}{s}\right| \, \, \rightarrow \, \, \frac{2}{\pi}\; \operatorname{a r c t a n}\left( \frac{\mathrm{R e}\, F ( s )}{\mathrm{I m}\, F ( s )}\right) \, \, \, .
\left(\tilde\zeta^{(i)}_{k},\tilde\zeta^{(j)}_{k}\right)=\left((\tilde\zeta^{(i)}_{k-1}+1){\bf1}\left\{U_k{\le}q_{\tilde\zeta^{(i)}_{k-1}}\right\},(\tilde\zeta^{(j)}_{k-1}+1){\bf1}\left\{U_k{\le}q_{\tilde\zeta^{(j)}_{k-1}}\right\}\right),
V^{\tau}( R , l , E ) = E - \frac{1}{2}\frac{( f_{+})^{( 3 k + c_{1}- 1 ) / 2}}{( f_{-})^{( 3 k + c_{1}+ 1 ) / 2}}\left[ E^{2}- \left( \frac{f_{-}}{f_{+}}\right)^{2 k}- \frac{l^{2}}{r^{2}}\frac{( f_{-})^{k - 1 / 2}}{( f_{+})^{k + 1 / 2}}\right] .
\Phi_{v,ij,{\bf 0}}(\lambda) = \sum_{l\in L_{v,\hat{\imath}\hat{\jmath}}}f_{v_i}^{(l_i)}(\lambda_i)f_{v_j}^{(l_j)}(\lambda_j)\prod_{\substack{m=1 m\neq i,j}}^N [v_m]_{l_m}\lambda_m^{v_m+l_m}.