harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
(-i\nabla-\epsilon A^0(x)-\kappa A(\epsilon x))^2\widetilde{\Lambda}^{\epsilon,\kappa}(x,\tilde{\beta}) =\widetilde{\Lambda}^{\epsilon,\kappa}(x,\tilde{\beta})\left\{(-i\nabla-\epsilon A^0(x))\,+\,\kappa a_\epsilon(x,\tilde{\beta})\right\}^2.
\uppercase\expandafter{\romannumeral2}\leq\sum\limits_{j=1}^{\infty}\vert \mu_{j}\vert\vert T(b_{j})(x)\vert\chi_{P_{j}^{*}}(x)+\sum\limits_{j=1}^{\infty}\vert\mu_{j}\vert\vert T(b_{j})(x)\vert\chi_{(P_{j}^{*})^{c}}(x)=:\uppercase\expandafter{\romannumeral1}_{1}+\uppercase\expandafter{\romannumeral1}_{2}
\renewcommand{\arraystretch}{2}\begin{array}{lcl}\displaystyle \Omega_1 = -\frac{1}{4}(F_2+F_3+F_4)+\cdots, ~ \displaystyle \Omega_2 = -\frac{1}{4}(-F_1+F_3+F_4)+\cdots, \displaystyle \Omega_3 = -\frac{1}{4}(-F_1-F_2+F_4)+\cdots, ~ \displaystyle \Omega_4 = -\frac{1}{4}(-F_1-F_2-F_3)+\cdots, \end{array}
\left[ \begin{array}{c c}{{\Psi_{1}( t )}}\\{{\Psi_{2}( t )}}\\ \end{array}\right] = \left[ \begin{array}{c c}{{\cos{\theta_{v}}}}&{{- \sin{\theta_{v}}}}\\{{\sin{\theta_{v}}}}&{{\cos{\theta_{v}}}}\\ \end{array}\right] \left[ \begin{array}{c c}{{\Psi_{e}( t )}}\\{{\Psi_{\mu}( t )}}\\ \end{array}\right] \, ,
\alpha = \begin{cases}\frac{1}{4}+\frac{1+\sqrt{8\gamma^2+1}}{16\gamma^2}, \mbox{if}~\gamma \ge \frac{1+\sqrt{2}}{2}, \frac{2\gamma+1}{8\gamma^2}+\frac{\sqrt{8\gamma+3}}{16\gamma^2}, \mbox{if}~\frac{1}{2}<\gamma< \frac{1+\sqrt{2}}{2}.\end{cases}
\epsilon_{xx}=\frac{\partial u_{x}}{\partial x}=-z\frac{\partial\varphi}{\partial x};\epsilon_{xz}=\frac{1}{2}(\frac{\partial u_{x}}{\partial z}+\frac{\partial u_{z}}{\partial x})=\frac{1}{2}(-\varphi+\frac{\partial w}{\partial x})
\lambda_n[t+1]=\left[\lambda_n[t]-\hat{\lambda}_n[t]\times\right. &\left.\left(T\log_2\left(1+\frac{a_n}{b_n+\frac{T}{1-T}}\right)-\mathcal{R}_\mathrm{eq}\right)\right]^+,\,\forall n \in \mathcal{N},
Z_{m,\theta}(\alpha,\beta,\gamma) = \frac{1}{2}\sum_{\theta \in \{0,1\}^2}\prod_{j \in \mathbb{T}_{m_1,m_2}}\left| \alpha + \beta \exp \left\{2 \pi i \frac{j_1 + \theta_1 /2}{m_1}\right\}+ \gamma \exp \left\{2 \pi i \frac{j_2 + \theta_2 /2}{m_2}\right\}\right|.
\rho_* \left( \rho^* \O(3,1))\otimes \O_\rho(1) \right) = \O(3,1) \otimes \rho_* \O_\rho(1) = \O(3,1) \otimes \big( \O(1,0) \oplus \O \big)^* = \O(2,1) \oplus \O(3,1).
[\begin{matrix}c_{3}c_{2}c_{1}-s_{3}s_{1}&-c_{2}s_{1}c_{3}-c_{1}s_{3}&c_{3}s_{2}\\c_{3}s_{1}+c_{1}c_{2}s_{3}&c_{3}c_{1}-c_{2}s_{3}s_{1}&s_{2}s_{3}\\ -c_{1}s_{2}&s_{1}s_{2}&c_{2}\end{matrix}]
\frac{q^2}{z_1^2}\log(1+\frac{z_1}{1+z_2})-\frac{q^2}{z_1(1+z_1+z_2)}=\xi(q)\Delta^2, \frac{(1-q)q}{(1+z_2)(1+z_1+z_2)}+\frac{(1-q)^2}{z_2^2}\log(1+z_2) -\frac{(1-q)^2}{z_2(1+z_2)}=[\xi(1)-\xi(q)]\Delta^2, \frac{q}{(1+z_2)(1+z_1+z_2)}+\frac{1-q}{1+z_2}=\xi'(1)\Delta^2.
\Delta C = -k_B\mathrm{Tr}\left[\rho\ \ln\left(\frac{\rho}{\rho^{eq}}\right)\right] -k_B\sum_{i=1}^k\lambda_i\left(\mathrm{Tr}\left[\rho F_i\right]-f_i\right) -k_B\mu\left(\mathrm{Tr}\left[\rho\right] - 1 \right).
\left\langle (\partial f)_x, (\partial y^{n_j})_x\right\rangle_{\mathcal{H}_x}=\sum_{i=1}^k\frac{\partial F}{\partial y^{n_i}}(y)Z^{n_i n_j}(x)=\left(Z_J(x)\nabla F(y)\right)_j,
{{Dp^{\mu}}\over{d\tilde{\lambda}}}\equiv{{dp^{\mu}}\over{d\tilde{\lambda}}}+\tilde{\Gamma}^{\mu}_{\rho\sigma}{{dx^{\rho}}\over{d\tilde{\lambda}}}p^{\sigma}=-\tilde{m}_0{{{\cal W}^{\prime}}\over{\cal W}}{{dy}\over{d\tilde{\lambda}}}{{dx^{\mu}}\over{d\tilde{\lambda}}},
\partial_{s}^2 \mathcal{N}(t,s,\xi,v,\overline v)= e^{(t-s) \mathcal{L}}\mathcal{C}^2\left[f, \mathcal{L}\right] \left(e^{s\mathcal{L}}v, e^{s\mathcal{L}}e^{\xi \mathcal{A}}\overline v \right)
\mathcal{I}-d^{-}(p,H) = 1-\mathcal{I}-d^{-}(p,H^{c}) = 1- \mathcal{I}-\limsup y_n = 1+ \mathcal{I}-\liminf (-y_n) = \mathcal{I}-\liminf (1- y_n) = \mathcal{I}-\liminf x_n = \mathcal{I}-d_{-}(p,H)
S_{\Psi}= \int d^{6}x \sqrt{^6 g}\bar{\Psi}i \Gamma^{A}D_{A}\Psi \simeq \epsilon^{2}\int_{0}^{\infty}d z \phi^{7}( \phi^{\prime})^{3 / 2}\int d^{4}x \sqrt{- \eta}\bar{\psi}i \gamma^{\nu}\partial_{\nu}\psi .
P_1^{[n]}(t; x, y) = \begin{cases}e^{-q_xt}\cdot\mathbf{1}_{\{x = y\}}\textrm{if}n=0, \int_0^t{e^{-q_xs}\cdot\sum_{z\neq x}{q_{x, z}P_1^{[n-1]}(t-s; z, y)}ds}\textrm{if}n\geq 1. \end{cases}
\exp(-\frac{s_0+1}{s_0}n^\frac{s_0}{s_0+1}\Re L_{p,q}(z))Q_n(z) = \sum_{\{(h,k) :\L_{h,k}(z)\in [L_{p,q}(z)]\}}\omega_{h,k,n}(z)I_{h,k,n}(z) &+ \sum_{\{(h,k) :\L_{h,k}(z)\not \in [L_{p,q}(z)]\}}\omega_{h,k,n}(z) I_{h,k,n}(z).
G_{E}^{+}\left( x , x^{\prime}\right) = \left\langle E \right| \phi \left( x \right) \phi \left( x^{\prime}\right) \left| E \right\rangle = \frac{\Gamma \left( h_{+}\right) \Gamma \left( h_{-}\right)}{1 6 \pi^{2}}F \left( h_{+}, h_{-}; 2 ; \frac{1 + P}{2}\right) ,
\sigma^{0 k}\equiv i \alpha^{k}= i \, \left( \begin{array}{c c}{{0}}&{{\sigma^{k}}}\\{{\sigma^{k}}}&{{0}}\\ \end{array}\right) \; ; \; \sigma^{i j}= \epsilon_{i j k}\, \left( \begin{array}{c c}{{\sigma^{k}}}&{{0}}\\{{0}}&{{\sigma^{k}}}\\ \end{array}\right) \; ,
\left| \left({1 + \frac{1}{2^{1+s}+ 1}}\right) / \left({1 + \frac{1}{2^s}}\right) \right| = \left|\frac{1 + \frac{1}{2c + 1}}{1 + \frac{1}{c}}\right| \ = \ \left|\frac{2c^2 +2c}{2c^2 + 3c +1}\right| \ = \ \left|\frac{2c}{2c + 1}\right|\ = \ \left| 1 - \frac{1}{2c + 1}\right| < 2.
\varepsilon_0=-\frac{a_3}{a_2},\varepsilon_1= \frac{a_1}{a_2}\ne0,\varepsilon_2= \frac{a_1^{3/2}}{a_2}>0,\varepsilon_3= \frac{\mathcal T^p}p>0,\varepsilon_4= \frac12a_{12}\varepsilon_3^\kappa, \varepsilon_5= \frac{\mathcal T^\omega}\omega\ne0,\varepsilon_6= a_{4}\varepsilon_3^\kappa.
z_s = e^{-\pi t + \pi \sigma i}, \cos\frac{\pi s}{2}= \frac{e^{\frac{\pi}{2}\left(t-\sigma i\right)}}{2}\left(1 + z_s\right), \quad \sin\frac{\pi s}{2}= \frac{e^{\frac{\pi}{2}\left(t-\sigma i\right)}i}{2}\left(1 - z_s\right).
\sum_{j=1}^n q^{\rho_j-2(n-j)}\Lambda_j^{1/2}\mu_j^{-1/2}{\cal X}^j{\cal X}^{-j}+\frac{q^{-2n+1}}{1+q}{\cal X}^0{\cal X}^0=-\frac{1}{a^2}~.
\frac{1}{4}\triangle(w_1+w_2+\cdots+w_{n-1})=|\gamma_1|^2e^{2w_1-w_2}+\sum_{i=2}^{n-2}|\gamma_i|^2e^{2w_i-w_{i-1}-w_{i+1}}+|\gamma_{n-1}|^2e^{2w_{n-1}-w_{n-2}}
\displaystyle\bigoplus_{0\le i_0<...< i_j\le n}(\displaystyle\sum_{0\le k^0_1,...,k^{j-1}_j \le 1}(-1)^{\triangle_{{i}^{k^0_1}_1...{i}^{k^{j-1}_j}_j}}g_{i_0{i}^{k^0_1}_1...{i}^{k^{j-1}_j}_j}).
\zeta^{(0)}(-m)=-\frac{1}{m+1}\sum_{k=0}^{m+1}(-1)^k \binom{m+1}{k}B_{2(m+1)-k}=-\frac{1}{m+1}\sum_{k=m+1}^{2m+2}(-1)^k \binom{m+1}{2m+2-k}B_k.
i{\cal M}_{f i}= ( i g )^{2}{\frac{i}{P^{2}- M^{2}+ i \varepsilon}}- ( i g )^{2}{\frac{i}{( P^{2}- M^{2}+ i \varepsilon )}}i \Pi ( P^{2}){\frac{i}{( P^{2}- M^{2}+ i \varepsilon )}}
\mbox{vol}_{n-2}(\Delta) = \frac{\det\Big( p_{i_1}(\xi,t) - q_j(\xi,t),\ldots, p_{i_{n-2}}(\xi,t) - q_j(\xi,t), n_j,\xi \Big)}{(n-2)! \sqrt{1 - \langle n_j,\xi\rangle^2}};
a_1J_{q,p}^t+a_2J_{p,q}^t+a_3J_p^rJ_{q,r}^t+a_4J_q^rJ_{p,r}^t+a_5J_p^rJ_q^sJ_{s,r}^t+a_6J_q^rJ_p^sJ_{s,r}^t+a_7J_q^sJ_{s,p}^t+a_8J_p^sJ_{s,q}^t=0
- \frac{1 9}{1 9 2}\{Q_{B R S T}, \partial b \partial e^{2 \phi}( z ) \}- \frac{5}{4 8}\{Q_{B R S T}, \partial^{2}b e^{2 \phi}( z ) \}- \frac{1}{9 6}\{Q_{B R S T}, b \partial^{2}e^{2 \phi}( z ) \},
\int_{\left[0,1\right]^d}\left(p(x) - \hat{p}\left(x\right) \right)^2dx \quad= \int_{\left[0,1\right]^d}\hat{p}\left(x\right)^2 dx - 2\int_{\left[0,1\right]^d}p(y)\hat{p}(y) dy + \int_{\left[0,1\right]^d}p(z)^2 dz.
-\mathfrak{s}\kappa \int_0^t \sum_{k\in \mathbb{N}}\operatorname{Re}\langle z\Phi e_k| z|^2, z \Phi e_k\rangle_{L_x^2}\d s+\mathfrak{s}\kappa \int_0^t \operatorname{Re}\langle z| z|^2, zF_{\Phi}\rangle_{L_x^2}\d s=0.
\begin{aligned}a^t_{\sigma,r}-a^t_{\sigma,r^j}=g_r-g_{r^j}&=j\left(\frac{q}{p}r_{\pi_{\hat{\beta}}(\hat{\gamma}-2)}+\frac{q}{p}r_{\pi_{\hat{\beta}}(\hat{\gamma})}+\lambda_{\pi_{\hat{\beta}}(\hat{\gamma}-1)}\right). \end{aligned}
|\chi_S \tilde{P}'_{\mathbf{d}, \mathcal{A}}(ab)| \leq \frac{d}{n}(24 d)^{-|S|}(4 |S|)! \cdot \left(C_8 + 2 C_8 |S| \binom{4 |S|}{4}^{-1}+ \frac{4 |S|^2}{70}\binom{4 |S|}{8}^{-1}+ o(1) \right) \leq \frac{d}{n}(24 d)^{-|S|}(4 |S|)! \cdot (2 C_8 + 1/2) ,
f=\epsilon_{0}[(\nabla\cdot E)E+(E\cdot\nabla)E]+\frac{1}{\mu_{0}}[(\nabla\cdot B)B+(B\cdot\nabla)B]-\frac{1}{2}\nabla(\epsilon_{0}E^{2}+\frac{1}{\mu_{0}}B^{2})-\epsilon_{0}\frac{\partial}{\partial t}(E\times B).
\lim_{b \uparrow\infty}\frac{\overline{Z}_a^{(q,r)}(b)}{W^{(q+r)}(b)}=\frac{r}{\Phi(q+r)}\overline{Z}^{(q)}(-a)+ \frac{\tilde{Z}^{(q,r)}(-a)}{\Phi(q+r)}.
J_k\rightarrow J_0= -\frac{\beta_+(x_0)m_+(x_0)+\beta_-(x_0)m_-(x_0)}{4\pi z^2}&-\frac{\beta_+(x_0)m_+(x_0)+\beta_-(x_0)m_-(x_0)}{4\pi z}\xi_z(0)+J_0'
Y(x,T)=u\tilde{Y}(u)=E_u[U_0\tilde{R}_{T-s}U_0^{-1}U_0U^{-1}_{T-s}U_{T-s}\tilde{Y}_{T-s}(U_{T-s})]=E_{(x,T)}[R_{T-s}P_{T-s}Z(X_{T-s})].
Y_s^{0,x;u_{\cdot}}= \xi + \int_s^{\tau}\hat{G}\bigl(r, X_r^{0,x;u_{\cdot}}, Y_r^{0,x;u_{\cdot}}, Z_r^{0,x;u_{\cdot}}\bigr) dr - \int_s^{\tau}Z_r^{0,x; u}dB_r,
\tilde \varphi_{mn}(\vec k) = \frac{1}{\sqrt{m! n!}}e^{\frac{k_c k_{\bar c}}{2}}\left( i \frac{\partial}{\partial k_c}\right)^m \left( i \frac{\partial}{\partial k_{\bar c}}\right)^n \tilde \chi_{00}(\vec k),
\Gamma_{F}[ \varphi_{0}, 0 ] \, = \, - \int d^{3}x \, \int \frac{d^{3}p}{( 2 \pi )^{3}}\left\{\ln [ 1 - e^{- L ( p + i e \varphi_{0})}] + \ln [ 1 - e^{- L ( p - i e \varphi_{0}}] \right\}\; ,
\begin{aligned}|N_{G}(h)\backslash V(T'\cup B)| \geq d_{G}(h)-|N_{G}(h)\cap V(B)| &-|N_{G}(h)\cap(V(T')\backslash\{u_{l+1},\cdots,u_{j}\})| &\geq (t+3)-2-(t-\lfloor\frac{j-l}{2}\rfloor) &\geq1+\lfloor\frac{j-l}{2}\rfloor,\end{aligned}
\partial_i T^{(r-1)}+\sum_{l=0}^r (A_i^{(l)}T^{(r-l)}-T^{(r-l)}\tilde{A}_i^{(l)})=0, (r-1){T}^{(r-1)}+\sum_{l=0}^r (B^{(l)}T^{(r-l)}-T^{(r-l)}\tilde{B}^{(l)})=0.
{\mathfrak{A}}_L^{dis}:=\Big\{A=\bigl(a_{rs}^{\alpha\beta}\bigr)_{\substack{1\leq r,s\leq n 1\leq\alpha,\beta\leq M}}\in{\mathfrak{A}}_L:\, \mbox{conditions \eqref{Ea4-fCii-n3}-\eqref{Ea4-fCii-n2B}hold for each}\,\, begin{align*}0pt] s,s'\in\{1,...,n\}\,\,\mbox{and}\,\,\alpha,\gamma\in\{1,...,M\}\Big\}.
d s^{2}= \left( \frac{r^{2}}{R^{2}}\right) ( - d t^{2}+ d x_{1}^{2}+ d x_{2}^{2}+ d x_{3}^{2}) + \frac{R^{2}d r^{2}}{r^{2}}+ R^{2}d \Omega_{5}\
M_N(a, b, \lambda) \triangleq_{\{t_1\cdots t_N\}}\prod\limits_{j=1}^N (1-t_j)^a(1-1/t_j)^b \prod\limits_{k<l}^N \Bigl(1-\frac{t_k}{t_l}\Bigr)^\lambda\Bigl(1-\frac{t_l}{t_k}\Bigr)^\lambda, = \prod_{j=0}^{N-1}\frac{\Gamma(1+(j+1)\lambda)\Gamma(1+a+b+j\lambda)}{\Gamma(1+\lambda)\Gamma(1+a+j\lambda)\Gamma(1+b+j\lambda)}.
{\cal D}_{\cal P}(U) \: = \:_{\cal P}Mod \Big\{\sum \limits_{\iota \in I}\, d_{\iota}f_{\iota}\: : \: \sum \limits_{\iota}\, f_{\iota}\in{\cal P}(U) \Big\}\quad \subset \:{\cal D}_{\cal P}^0 (U),
\frac{q^{n^{2}/2}}{\left(a;q\right)_{n}}=\frac{1}{\sqrt{\pi\log q^{-2}}}\int_{-\infty}^{\infty}\frac{\exp\left(\frac{x^{2}}{\log q^{2}}+inx\right)}{\left(a,-aq^{-1/2}e^{ix};q\right)_{\infty}}dx
\gamma^{m}(A^m_{\varepsilon})=\int_{(1- \varepsilon) \sqrt{m-1}}^{(1+ \varepsilon) \sqrt{m-1}}g_{m}(r) dr\leq \frac{e^{-\rho(m/2)}}{\sqrt{\pi}}e^{1/2}\left(1-\frac1m\right)^{(m-1)/2}\cdot 2\varepsilon\sqrt{m-1}, 1-\gamma^{m}(A^m_{\varepsilon})=\int_{0}^{(1- \varepsilon_m) \sqrt{m-1}}g_{m}(r) dr+\int_{(1+ \varepsilon_m)\sqrt{m-1}}^\infty g_{m}(r) dr.
\lambda_{m}( \alpha ) = \sum_{l = 0}^{m}\left({\frac{i \alpha}{2}}\right)^{l}{\frac{( m + l ) !}{l ! ( m - l ) !}}\times \left\{\begin{array}{c l}{{- \lambda_{2 N}( \alpha )}}&{{\mathrm{~ f o r ~ o d d ~ m + l ~}}}\\{{\lambda_{0}( \alpha )}}&{{\mathrm{~ f o r ~ e v e n ~ m + l ~}}}\\ \end{array}\right\}
{\cal L}= \sqrt{-\hat{G}}\,[{\cal R}_{\hat{G}}+ \hat{G}^{MN}\partial_M \Phi \partial_N \Phi -{\textstyle{1\over{12}}}\hat{H}_{MNP}\hat{H}^{MNP}-{\textstyle{1\over 4}}\hat{F}^I_{MN}\hat{F}^{I\,MN}],
Z_k=\begin{cases}X_{-(\epsilon_i-\epsilon_{i+1})}^{p-1}X_{\epsilon_{i+1}+\epsilon_j}\beta_k=(\epsilon_i+\epsilon_j)\in \Phi_{1}\cr X_{-(\epsilon_i-\epsilon_{i+1})}X^{p-1}_{\epsilon_{i+1}-\epsilon_j}\beta_k=\epsilon_i-\epsilon_j\in \Phi_0^+\backslash\Pi_0 \end{cases}i<j.
Q_2(x):= P_2(x)/x = f'(0) + \frac{a}{12}x^3 + \frac{b}{6}x^2 + \frac{c_2}{2}x \ge 0 , x\in [0,1], Q_1(x) := P_1(x)/x = f'(0) + \frac{a}{12}x^3 + \frac{b}{6}x^2 + \frac{c_1}{2}x \ge 0 , x\in [-1,0].
\prod_{i=1}^k (1+y X_i) \times \prod_{j=1}^{n-k}(1+y \tilde{X}_i) = \prod_{i=1}^n (1+y T_i) -\frac{q}{1-q}y^{n-k}\tilde{X}_1 \cdot \ldots \cdot \tilde{X}_{n-k}\bigl( \prod_{i=1}^k (1+y X_i) -1 \bigr) \/.
\|{\mathbb{T}}\|^\ell_{\dot{B}_{p,1}^{\gamma_3}}\le C(1+t)^{-\frac{(\frac n2-\sigma-1)(1-\theta_{4})}{2}}=C(1+t)^{-\frac{n}{2}(\frac 12-\frac 1p)-\frac{\gamma_3-\sigma-1}{2}}.
\underline C \coloneqq \frac{\sigma}{1+\delta_s}, c_0 = \frac{1+\gamma + \sqrt 3}{\gamma}, \bar C \coloneqq \frac{\sigma}{\kappa(c_0, s)}\bigg( 1+ \frac{\sigma \kappa(c_0,s) ( \sqrt s + 2\sqrt{\log3})}{\lambda \sqrt s}+ \frac{\sqrt 3}{\sqrt{\log(9ep/s)}}\bigg) .
\bar{\Gamma}(\bar x,\bar v)={\displaystyle\sum_{j=1}^n\bar{v}^{j}\frac{\partial}{\partial \bar{x}^{j}}}+{\displaystyle\sum_{i=1}^n}\left({\displaystyle\sum_{j=1}^n}\frac{1}{f}\bar{v}^{j}\frac{\partial f}{\partial \bar x^{j}}\bar{v}^{i}+f^{2}X^{i}\left (\bar x, \dfrac 1f \,\bar v\right)\right)\frac{\partial}{\partial \bar{v}^{i}},
f_\tau(u) = \prod_{\nu\in \Phi_\tau}h_{\nu}(u), \quad\Phi_\tau = \{\nu\in \Phi:\nu|_{E^{\mathrm{ur}}}= \tau\}v_\tau(u) = \prod_{\nu\in \Phi_\tau^c}h_{\nu}(u), \quad\quad\Phi_\tau^c = \{\nu\in \Phi^c:\nu|_{E^{\mathrm{ur}}}= \tau\}
\left|\frac{1}{4\pi}\int\limits_{\beta_{kn}}\,\frac{\Delta f_0(M)}{\rho_{M_0}(M)}\, dm_3(M)\right| &\leq C \int\limits_{B_{2\Lambda_{n-2}(M_{k,n-2})}}\,\frac{\omega(d(M))}{d^{2}(M)\cdot 2^{-n+1}}\, dm_3(M) &\leq C\omega(2^{-n+2})\leq C\omega(2^{-n})
\begin{aligned}\Vert (u_0,h_0)\Vert_{H^m_l}-C(u_e,u_b,h_e)\leq \Vert (u^0_p,h^0_p)(0,y)\Vert_{H^m_l}\leq \Vert (u_0,h_0)\Vert_{H^m_l}+C(u_e,u_b,h_e).\end{aligned}
\begin{aligned}\| D_x^{\beta}(\int \frac{e^{it(\varphi(\xi + \tau,\eta)- \varphi(\xi,\eta))}-1}{|\tau|^{1+r_1}}\,\widehat u_0(\xi+\tau, \eta)\, d\tau)^{\lor}\|_2 \leq c_{\alpha,\beta}(1+|t|) (\|u_0\|_2+\|D_x^{\beta+2\alpha}u_0\|_2 + \|D_y^{\beta+2\alpha}u_0\|_2). \end{aligned}
Q_{n}[ \overline{{\psi}}, \psi , A , h ] : = P [ \overline{{\psi}}, \psi , A , h ] \Big ( S_{M}[ \overline{{\psi}}, \psi ] \Big )^{n}\longrightarrow \tilde{Q_{n}}[ G , \frac{\delta}{\delta a}, A , h ] .
V=\frac{5+5\sqrt{5}}{6\sqrt{3}}\sqrt{18+6\sqrt{5}+a(3+3\sqrt{5}+a)}+\frac{5+3\sqrt{5}}{24\sqrt{2}}\sqrt{72+(5+\sqrt{5})a(3+3\sqrt{5}+a)}\approx37.61664996273336,
\beta_1(z) =\psi_k(1)z+\psi_k(2)z^2+\dots+\psi_k(s_1)z^{s_1};{\mathcal E}(\beta_1(z)) = \frac{1}{(1-z)^{\psi_k(1)}\cdots(1-z^{s_1})^{\psi_k(s_1)}}\approx\frac{\mu}{(1-t)^M}, t\to 1{-}0; M =\psi_k(1)+\dots+\psi_k(s_1)=\dim_K F({\bf{N}}_{s_1},k), \mu=\prod_{q=2}^{s_1}q^{-\psi_k(q)}.
\sum_{l=-N/2}^{N/2-1}(1+\mu_l^2+\mu_l^4)\left|\widehat{(\gamma^{n+1}\psi^n)}_l(0)-\widetilde{(R^{n+1}\Psi^n)}_l\right|^2 &\lesssim\|I_N(\gamma^{n+1}\psi^n(\cdot,0))-I_N(R^{n+1}\psi^n))\|_{H^2}^2+h^{2m_0}&\lesssim\|e_\psi^n\|_{H^2}^2+\|\gamma^{n+1}-I_NR^{n+1}\|_{H^2}^2+h^{2m_0}.
d_i(h\otimes (x \otimes v_1 \wedge\ldots \wedge v_i)) = \sum_{j=0}^i (-1)^j(h v_j \otimes x \otimes v_1 \wedge \ldots \wedge \hat{v}_j \wedge \ldots \wedge v_i -h \otimes v_j. x \otimes v_1 \wedge \ldots \wedge \hat{v}_j \wedge \ldots \wedge v_i ) .
0 \leq P_s*(\chi_{B_0}\mu)(\bar x) \lesssim \int_{\bar y\in B_0}\frac1{|\bar x- \bar y|_p^N}\,d\mu(\bar y) &\leq \int_{|\bar x-\bar y|_p\leq 4d}\frac1{|\bar x- \bar y|_p^N}\,d\mu(\bar y)\lesssim d^{2s-1}=|t-u|^{1-\frac1{2s}}.
\times \Bigg [ \int_{\Lambda}\prod_{c = 1}^{N}\left[ \prod_{j = 1}^{n_{c}}d^{2}x_{j}^{( c )}d^{2}y_{j}^{( c )}\right] \prod_{d < e}^{N}\Big | B ( d , e ) \Big |^{\frac{g}{\pi + g N}q}\Bigg ]^{\frac{1}{q}}
a(n) R_n(\lambda ( x);\gamma,\delta,N) + b(n) R_{n+1}(\lambda ( x);\gamma,\delta,N) ={\hat{d}}(x) R_{n}(\lambda (\hat x);\hat{\gamma},\hat{\delta},\hat{N}) \hat{a}(n) R_{n}(\lambda (\hat x);\hat{\gamma},\hat{\delta},\hat{N}) + \hat{b}(n) R_{n+1}(\lambda (\hat x);\hat{\gamma},\hat{\delta},\hat{N}) = d(x) R_{n+1}(\lambda ( x);\gamma,\delta,N) .
\big(\Phi^\delta(nT, 0)u^r\big)(x) \geq \big(\Phi(nT, 0)u^r\big)(x)-C(nT,\delta) &=e^{\lambda_2^rnT}u^r(x)-C(nT,\delta) &=e^{(\lambda_2^r-\epsilon)nT}e^{\epsilon nT}u^r(x)-C(nT,\delta),
\begin{pmatrix}h\left(0\right)-h'\left(0\right) h\left(1\right)+h'\left(1\right)\end{pmatrix}\longmapsto\varphi\left(0\right)\begin{pmatrix}h\left(0\right)-h'\left(0\right) h\left(1\right)+h'\left(1\right)\end{pmatrix}
A = \frac{1}{P_r(\vec{1})}\left( \nabla P_r(\vec{1})\cdot (x+i\tilde{y})\left(\frac{1}{2m}- \frac{P_{r+1}(\vec{1})}{P_r(\vec{1})}\right) + \nabla P_{r+1}(\vec{1})\cdot (x+i\tilde{y})\right)
H(t,k,\xi)=\sum_{q\neq 0}e^{-2|k-q|}\frac{\varepsilon_0\eta_0}{(\xi-tq)^2+q^2},H(t,\xi)=\varepsilon_0+\sum_{|k-k_0|\leq 2\sigma k_0}H(t,k,\xi),
\mathcal{H}^s(\Omega_n(f,\psi)) \le \mathcal{H}^s(B\cup(\cup_{i=1}^\infty\Omega_n(f,\psi)\cap I_i)) &\le \mathcal{H}^s(B)+\sum_{i=1}^\infty\mathcal{H}^s(\Omega_n(f,\psi)\cap I_i)=0
\sum_{i,k=1}^d (\lambda_i - \widetilde \lambda_k)^2 \langle p_i, q_k\rangle^2 \leq \left(\sum_{i=1}^{r}\sum_{k=1}^d+ \sum_{k=1}^{r}\sum_{i=1}^d+ \sum_{i,k=r+1}^{d}\right)(\lambda_i - \widetilde \lambda_k)^2 \langle p_i, q_k\rangle^2.
\sum_{j=a}^\infty\sum_{l=a}^\infty e^{-{\frac{1}{2}}|j-l|}\left(\int_{j}^{j+1}\theta\right)\left( \int_{l}^{l+1}\theta\right) \le \left(\sum_{j=a}^\infty \int_j^{j+1}\theta\right)^2= \left(\int_a^\infty \theta\right)^2
\|V_{in}^R\|^{(2)}_{q,B_{R,T}}\leq C\bigg(\|f\|_{q,B_{R,T}},\|h_i\|_{q,B_{R,T}}, \|g_i\|^{2-1/q}_{q,\partial_pB_{R,T}}\bigg),\i=1,2,
\big(D^{(n)}_q(t;p)f\big)(z_1,\dots,z_n)=\sum_{\vec{\sigma}\in \{\pm 1\}^n}\frac{\prod\limits_{1\le i<j\le n}\theta_p\big(t z_i^{\sigma_i}z_j^{\sigma_j}\big)}{\prod\limits_{1\le i\le j\le n}\theta_p\big(z_i^{\sigma_i}z_j^{\sigma_j}\big)}f\big(q^{\sigma_1/2}z_1,\dots,q^{\sigma_n/2}z_n\big).
\|D^r\tilde{f}_P\|_p = \frac{\|D^r g\|_p}{h^{d+r}}\left(\frac{1}{S}\sum_{i=1}^Sp_i^p\right)^{\frac{1}{p}}= \frac{\|D^r g\|_p}{h^{r-s}}\cdot d_0Ln\ln n\left(\frac{1}{S}\sum_{i=1}^Sp_i^p\right)^{\frac{1}{p}}
H_{mirror}=i\hbar\zeta(\hat{a}_{c}\hat{a}_{s1}^{\dagger}-\hat{a}_{c}^{\dagger}\hat{a}_{s1}+\hat{a}_{s2}\hat{a}_{c}^{\dagger}e^{-ik_{c}z}-\hat{a}_{s2}^{\dagger}\hat{a}_{c}e^{ik_{c}z})
\|u^h_t-u^h_s\|^2_{m-1}=2\int_s^t \left(u^h_r-u^h_s, (L^h_r+J^h)u_r+ f_r \right)_{m-1}dr & \leq N \int_s^t \|u^h_r-u^h_s\|^2_{H^m}+\|(L^h_r+J^h)u^h_r\|^2_{H^{m-2}}+ \|f_r\|^2_{H^{m-2}}dr & \leq N \int_s^t \sup_{t' \leq T}\|u^h_{t'}\|^2_{H^m}+\|f_r\|^2_{H^{m-2}}dr & \leq N(\mathcal{K}_m^2+K')(t-s),
\mathbf{I}_2 =\left( \begin{array}{cc}1 0 0 1 \end{array}\right), \mathbf{\sigma}_x =\left( \begin{array}{cc}0 1 1 0 \end{array}\right), \mathbf{\sigma}_y =\left( \begin{array}{cc}0 -i i 0 \end{array}\right), \mathbf{\sigma}_z =\left( \begin{array}{cc}1 0 0 -1 \end{array}\right).
\begin{bmatrix}A X X^* B\end{bmatrix}= U\begin{bmatrix}\frac{A+B}{2}+{\mathrm{Re}\,}X 0 0 0\end{bmatrix}U^* +V\begin{bmatrix}0 0 0 \frac{A+B}{2}-{\mathrm{Re}\,}X\end{bmatrix}V^*
\int_{-\infty}^{-\sigma}p(x)dx+\int_{+\sigma}^{+\infty}p(x)dx=2\int_{+\sigma}^{+\infty}p(x)dx=2\int_{+\sigma}^{+\infty}\frac{1}{\sigma\sqrt{2\pi}}e^{-x^{2}/2\sigma^{2}}dx
\int_{a_{k}}^{b_{k}}P_{n,k}(\tau)\,\tau^{s}\,\frac{|H_{n,k}(\tau)|\,d|\sigma_{n,k}|(\tau)}{|P_{n,k-1}(\tau)\,P_{n,k+1}(\tau)|}=0,s=0,\ldots,Z(n,k)-1,k=0,\ldots,p-1.
\phi(T_{b^t})=(1-e^{-\beta}d)\Big(\phi_P(T_{b^t})+\sum_{\{k\geq 1\,:d\,\mid\, c^jd^{-j}t0\leq j<k\}}e^{-\beta k}\phi_P(T_{b^{c^kd^{-k}t}})\Big),
[Q]= \frac{[\mathbb{III}^1(T)]\prod_p\frac{[H^0(\hat{\mathbb{Z}},H^1(I_p,\hat{T}))^{D}]}{[S_p]}[R]}{[\mathrm{cok}\Delta]}= \frac{[\mathbb{III}^1(T)][\mathbb{III}^2(T)][R]\prod_p[H^0(\hat{\mathbb{Z}},H^1(I_p,\hat{T}))^{D}]}{[H^1(K,\hat{T})]\prod_p[S_p]}.
k_3(\langle H\rangle,1)=\frac{1}{128}\left[2(q^2-20q+81)+2 u(-p)^t+3q^2\cdot{_{3}}F_{2}\left(\begin{array}{ccc}\chi_4, \chi_4, \overline{\chi_4}\varepsilon, \varepsilon\end{array}| 1\right) \right].
{}_2F_1\bigg[\begin{matrix}\alpha \frac12+\alpha &\frac32-2\alpha\end{matrix}\bigg|\,-\frac13\bigg]=\frac{8^{-2\alpha}}{9^{-2\alpha}}\cdot\frac{\Gamma(\frac43)\Gamma(\frac32-2\alpha)}{\Gamma(\frac32)\Gamma(\frac43-2\alpha)},
\frac{\bar{D}}{Dt}\left( \frac{p'}{\bar{\rho}c^2}\right) +\nabla \cdot u^a =0, \frac{\bar{D}}{Dt}\nabla \cdot u^a+\frac{1}{\bar{\rho}}\Delta p' = 0, \nabla \times \frac{\bar{D}}{Dt}u^v =-\nabla \times \nabla \cdot ( u^a \otimes \bar{u}).
V(y_i,\theta) = - V_{p+1}(i)^p \int_0^\infty \frac{dt}{t}(8 \pi^2\alpha'\,t)^{-\frac{p+1}{2}}e^{-\frac{y^2}{2 \pi \alpha'}t}16 t^4\simeq y^{p-7}\,.
S_1=2\,\frac{\sqrt{(|y|^2+|x_0|^2)^3+(|y|^2+|x_0|^2)^2I_2+(|y|^2+|x_0|^2)(I_2^2-I_4)/2+|I_3|^2}}{|y|^2+|x_0|^2}\, ,
\hat{D}_{\mu\nu\rho\underline{x}}={\textstyle\frac{3}{8}}\left(C_{\mu\nu\rho}- A_{[\mu}^{(i)}B_{\nu\rho]}^{(i)}-\epsilon^{ij}A_{[\mu}^{(i)}A_{\nu}^{(j)}B_{\rho]}\right)\, ,\hspace{.5cm}\hat{\rho}_{\mu\nu\underline{x}}={\textstyle\frac{1}{2}}\chi_{\mu\nu}\, .
\left(\mu\frac{\partial}{\partial\mu}+\beta\left(g^{2}\right)\frac{\partial}{\partial g^{2}}-\gamma_{2}\left(g^{2}\right)\int d^{2}xJ\frac{\delta}{\delta J}+\eta\left(g^{2},\zeta\right)\frac{\partial}{\partial\zeta}\right)E = 0
p_{\mu_{1}\boxtimes\mu_{2}}(f(t)) =\frac{1}{2\pi}\Re\frac{1+\eta_{\mu_{1}}(z(t))}{1-\eta_{\mu_{1}}(z(t))}=\frac{1}{2\pi}\int_{\mathbb{T}}\frac{1-|z(t)|^{2}}{|\xi-z(t)|^{2}}\,d\mu_{1}(\overline{\xi}) =\frac{1}{2\pi\beta}\beta\int_{\mathbb{T}}\frac{1-|z(t)|^{2}}{|\xi-z(t)|^{2}}\,d\mu_{1}(\overline{\xi})=\frac{-\log R(t)}{2\pi\beta}.
[\begin{matrix}t-i&0\\ \frac{2t}{t+i}&\frac{t^{2}-1}{t+i}\\ \end{matrix}]U=\frac{1}{\sqrt{2}}[\begin{matrix}t-i&t-i\\i\frac{(t-i)^{2}}{t+i}&-i(t+i)\\ \end{matrix}]
\sum_{2< p\le 23}\frac{1}{m_p^2}{\rm d}({\rm L}_{p}) = \frac{1}{\mu_7^2}\sum_{2< p\le 7}{\rm d}({\rm L}_{p}) + \frac{1}{\mu_{19}^2}\sum_{7< p\le 19}{\rm d}({\rm L}_{p}) + \frac{1}{\mu_{23}^2}{\rm d}({\rm L}_{23}) = 0.390126\cdots,
\Psi_{p,q}^{\rm e}(\gamma) = \begin{cases}-nq \gamma \sim \pm S_p^{\,n}\ (1 \leq n \leq p-1), -np \gamma \sim \pm U_q^{\,n}\ (1 \leq n \leq q-1), \Psi_{p,q}(\gamma) = \Psi_{p,q}^\mathrm{h}(\gamma) .\end{cases}
\phi(y)(z) =-{z}^{-1}-{\frac{1}{2}}-{\frac{1}{12}}z+{\frac{1}{720}}{z}^{3}+ O\!\left({z}^{4}\right), \phi(dy)(z) ={z}^{-2}-{\frac{1}{12}}+{\frac{1}{240}}{z}^{2}+O\!\left({z}^{4}\right), \phi(d^2y)(z) =-2\,{z}^{-3}+{\frac{1}{120}}z-{\frac{1}{1512}}{z}^{3}+ O\!\left({z}^{4}\right)
\prod_{j=1}^N \frac{1}{1-q^j}=\sum_{\substack{0\leqslant h<k \leqslant N (h,k)=1}}\sum_{\ell=1}^{\lfloor N/k \rfloor}\frac{C_{hk\ell}(N)}{(q-e^{2\pi ih/k})^\ell}.
E[ | \beta_t-x |^{-2}]=2\pi^2 E[ G_\beta( \beta_t, x )]=2 \pi^2 \int_t^\infty \phi_s(x) ds = t^{-1}\varphi_1(|x|^2/t) = |x|^{-2}\varphi_2(t/|x|^2) \,,
\lim_{t_0\to-\infty}\underline{U}(t,t_0)=\frac{1}{b+\chi_2\mu_2-\chi_1\mu_1}\Big\{a+\frac{1}{\lambda_2}\big(\left[(\chi_2\mu_2\lambda_2-\chi_1\mu_1\lambda_1)_{+}+\chi_1\mu_1(\lambda_1-\lambda_2)_{+}\right]\underline{u}_{0}-\left[(\chi_2\mu_2\lambda_2-\chi_1\mu_1\lambda_1)_{-}+\chi_1\mu_1(\lambda_1-\chi_2)_{-}\right]\overline{u}_0\big)\Big\}.
f_{1}= \frac{( 2 n - 1 ) \sqrt{x}- n}{2 ( n - 1 ) \sqrt{x}}\; \; < 1 \; , f_{2}= \frac{( 2 n - 1 ) \sqrt{x}+ n}{2 ( n - 1 ) \sqrt{x}}\; \; > 1 \; .
\frac{1}{|{\bf x}|}\equiv \sum_{n = - \infty}^{+ \infty}\frac{1}{\sqrt{\vec{x}^{2}+ ( \beta n )^{2}}}= 2 T \sum_{n = - \infty}^{+ \infty}K_{0}( 2 \pi T | \vec{x}{\,}| n ) \simeq - 2 T \ln ( \mu | \vec{x}{\,}| ) .
\Gamma(\theta_s,\theta_x) := \begin{cases}\displaystyle \frac{2\left[(2-\theta_s)m + \theta_sn\right]}{\theta_s\theta_x mn}m \geq n begin{align*}3mm] \displaystyle \frac{2\left[(2-\theta_x)n + \theta_xm\right]}{\theta_s\theta_x mn}m \leq n. \end{cases}
b_{11}^{(k+1)}-b_{33}^{(k+1)}= b_{11}^{(k)}-b_{44}^{(k)}+ (b_{13}^{(k)}+b_{24}^{(k)})\tan\phi_{k+1}+b_{24}^{(k)}(\tan\psi_{k+1}-\tan\phi_{k+1}),
S = \int d^{D}x \sqrt{- g}\left[ - \sum_{k = 1}^{N}\overline{{\psi}}_{k}\gamma^{\mu}\nabla_{\mu}\psi_{k}+ \frac{\lambda_{0}}{2 N}( \sum_{k = 1}^{N}\overline{{\psi}}_{k}\psi_{k})^{2}\right] ,
\prod_{1 \leq i <j \leq k}d(U_i',U_j') \geq \prod_{1 \leq i<j \leq k}d(U_i,U_j)-\eta \binom k2 \geq \prod_{1 \leq i<j \leq k}d(U_i,U_j)-\frac \eta \delta \binom k2.
f =q - 4096\, q^2 + 97956\, q^3 + 16777216\, q^4 + 341005350\, q^5 - 401227776\, q^6 +\cdots g =q + 4096\, q^2 + (2048-a/2)\, q^3 + 16777216\, q^4 + (431848374+162\, a)\, q^5 +\cdots g' =q + 4096\, q^2 + (2048+a/2)\, q^3 + 16777216\, q^4 + (431848374-162\, a)\, q^5 +\cdots \, ,