harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\begin{array}{ccc}\sup\limits_{N > 0}\inf\limits_{n > 0}P_N\left(\dfrac{\phi_n}{n}\right) = \lim\limits_{N \to +\infty}\lim\limits_{n \to +\infty}P_N\left(\dfrac{\phi_n}{n}\right) = \lim\limits_{n \to +\infty}\lim\limits_{N \to +\infty}P_N\left(\dfrac{\phi_n}{n}\right) = \inf\limits_{n > 0}P\left(\dfrac{\phi_n}{n}\right).\end{array}
\epsilon^{\alpha \mu \nu \beta}\gamma_{5}\gamma_{\mu}\partial_{\nu}\Psi_{\beta}^{i}-{\frac{\kappa \epsilon^{i j}}{\sqrt 2}}\big [ F^{\alpha \beta}+{\scriptstyle{\frac{1}{2}}}\gamma_{5}\epsilon^{\alpha \beta \rho \sigma}F_{\rho \sigma}\big ] \Psi_{\beta}^{j}= 0 \, .
(-1)^{\theta{m \choose 2}}\frac{q^{-\theta^2(N-1){m \choose 2}}}{\prod_{\substack{1\leq i<j\leq m 0\leq k<\theta}}{\big(x_i - q^kx_j\big)}}\prod_{i=1}^m{\frac{1}{\big(t^{N-i+1}; q\big)_{\theta(i-1)}}}.
Q(S)_n = \{m \in [n]: (\gamma_1 \gamma_2)^{\sigma_n}m = \gamma_1^{\sigma_n}\gamma_2^{\sigma_n}m \mbox{for all}\gamma_1,\gamma_2 \in S \}\cap \{m \in [n]: \gamma_1^{\sigma_n}m \neq \gamma_2^{\sigma_n}m \mbox{for all}\gamma_1 \neq \gamma_2 \in S \}
p(x) = \int_{\zeta}^x \exp \left[-2 \left(-\frac{3 \zeta^4}{4}+\frac{\zeta^3 \lambda}{3}+\frac{3 \zeta^2}{2}-\zeta \lambda +\frac{3 s^4}{4}-\frac{\lambda s^3}{3}-\frac{3 s^2}{2}+\lambda s\right)\right] \, ds,
{\cal L}_{\mathrm{b u l k}}( t , y ) = -{\frac{M_{3}^{2}\tilde{A}^{3}B}{\tilde{N}}}\left[ 3 \left({\frac{\dot{\tilde{A}}}{\tilde{A}}}\right)^{2}-{\frac{7}{1 2}}\dot{\phi}^{2}\right] .
\begin{array}{lcl}p^{-1}\star f(x,p) = fp^{-1}-\theta f'p^{-2}+{\theta}^{2}f'' p^{-3}-{\theta}^{3}f''' p^{-4}+... p^{-2}\star f(x,p) = fp^{-2}-2\theta f'p^{-3}+3{\theta}^{2}f'' p^{-4}-4{\theta}^{3}f''' p^{-5}+... p^{-3}\star f(x,p) = fp^{-3}-3\theta f'p^{-4}+6{\theta}^{2}f'' p^{-5}-10{\theta}^{3}f''' p^{-6}+... \end{array}
S = \int d^4x \sqrt{\hat g_4}\Bigl\{\hat R_4+ \frac{2 \hat \nabla_{\mu}{\cal A}\hat \nabla^{\mu}{\cal A}^{\dagger}}{({\cal A}-{\cal A}^{\dagger})^2}+ \frac{1}{8}Tr \bigl(l \hat \nabla m\bigr)^2- \frac{Q^2}{2}\frac{e^{\Phi_4}}{b^6}\Bigr\}
H_1(x; T):= \frac{1}{2 \pi i}\int_{\lambda + i T}^{c+i T}\frac{\Gamma(s)}{\zeta(k-2s)}x^{-s}{\rm d}s,{\rm and}H_2(x; T):= \frac{1}{2 \pi i}\int_{c - i T}^{\lambda - i T}\frac{\Gamma(s)}{\zeta(k-2s)}x^{-s}{\rm d}s
\frac{\beta(1+(1-2\alpha)r^m)}{(1-r^m)^{2(1-\alpha)+1}}+\frac{(1-\beta)r^m}{(1-r^m)^{2(1-\alpha)}}+\sum_{n=2}^{\infty}\prod_{k=0}^{n-2}\frac{k+2(1-\alpha)}{k+1}\phi_{n}(r) =\frac{1}{4^{1-\alpha}}-\phi_{1}(r).
\begin{array}{l}\lim_{n\rightarrow \infty}\frac{\ell_R(R/I(nn_1D)\cdots I(nn_rD_r))}{n^2}= \sum_{i=1}^t-\frac{1}{2}[S/m_i:R/m_R]((n_1\Delta(1)_i+\cdots+n_r\Delta(r)_i)^2) = \sum_{k_1+\ldots+k_r=2}\frac{1}{k_1!\cdots k_r!}\left(\sum_{i=1}^t-[S/m_i:R/m_R](\Delta(1)_i^{k_1}\cdot \ldots\cdot \Delta(r)_i^{k_r})\right)n_1^{k_1}\cdots n_r^{k_r}\end{array}
A_{(m+1)p_n-m}=A_{(m+1)p_n}-x^m A_{(m+1)p_n+m}-x^{m-1}A_{(m+1)p_n-1}&=(1-x^m)A_{(m+1)p_n+m}-x^{m-1}A_{(m+1)p_n-1}.
Y_{k}(z) = (\mathbb{E}_{k}-\mathbb{E}_{k-1})\frac{1}{d_{N}}\frac{1+h_{k}^{\dagger}G_{k}(z)^{2}h_{k}}{\mathcal{H}_{kk}-z-h_{k}^{\dagger}G_{k}(z)h_{k}}
B(t,u,v) :=\sigma(\psi\circ \overline{\phi}'_t(u)-\overline{\mu}'_t\circ \psi(u),\overline{\mu}'_t(v)) &=\sigma(\psi\circ \overline{\phi}'_t(u),\overline{\mu}'_t(v))-\sigma(\overline{\mu}'_t\circ \psi(u),\overline{\mu}'_t(v)) &=\sigma(\psi\circ \overline{\phi}'_t(u),\overline{\mu}'_t(v))-\sigma(\psi(u),v),
\frac{Q_{r}(x)}{P_{r}(x)}= \sum_{n=1}^{r}\frac{Q_{r}(\lambda_{n})}{P_{r}^{\, \prime}(\lambda_{n}) (x-\lambda_{n})}= \sum_{m=0}^{\infty}\frac{1}{x^{m+1}}\sum_{n=1}^{r}\mu_{n}\lambda_{n}^{m}\ , \ \ \ \ |x| > \max_{1 \leq n \leq r}\, |\lambda_{n}| \ .
\sum_{l=1}^n \left(n \atop l \right)(1 - p)^{l(n-l)}\leq 2\sum_{l=1}^{\lceil \frac{n}{2}\rceil}\left(n \atop l \right)(1 - p)^{l(n-l)}\leq 2\sum_{l=1}^{\lceil \frac{n}{2}\rceil}n^l (1 - p)^{l \lfloor \frac{n}{2}\rfloor}
\sup_{|y|\le 1}\Big(\sum_{k=1}^d p|X_k(x+y)|^2+\langle x , X_0(x+y)\rangle \Big) &\le C(p)|x|^{2q_2}+\sup_{|y|\le 1}\Big(-|x+y|^{q_4}\langle x, (x+y)\rangle\Big) &\le C(p)|x|^{2q_2}-C(|x|^{q_4}-1)|x|^2+C\sup_{|y|\le 1}\big(|x|^{q_4+1}|y|\big) &\le -C(p)(1+|x|^{q_4+2}),
{g_n}^p=\prod_{i=1}^{6n}u_{n,i}=\prod_{i=1}^{6n}\left( v_{n, i,1}\cdot{v_{n,i,2}}^{f}\right)=\prod_{i=1}^{6n}\left( \prod_{k=1}^{m/2}{t_{n,i,1,k}}^{f^{2k}}\cdot \prod_{k=1}^{m/2}{t_{n,i,2,k}}^{f^{2k+1}}\right)
{\frac{1}{\sqrt{-g}}}\partial_{\mu}[\sqrt{-g}\partial^{\mu}\sigma] =-e^{-2 \sigma}{({\partial}_{\mu}b)}^2 +{\frac{2 \alpha'}{3}}\left({\frac{\delta \Delta}{\delta\sigma}}\right){\cal{R}}^2_{GB}+{\frac{2 \alpha'}{3}}\left({\frac{\delta \Theta}{\delta \sigma}}\right){\cal{R}}{\tilde{\cal{R}}}
\hat{\hat{T}}_{\hat{\hat{\mu}}\hat{\hat{\nu}}}{}^{\hat{\hat{\rho}}}= - \left( i_{\hat{\hat{k}}_{( n )}}\hat{\hat{C}}\right)_{\hat{\hat{\mu}}\hat{\hat{\nu}}}Q^{n m}\hat{\hat{k}}_{( m )}{}^{\hat{\hat{\rho}}}\, .
\tilde{C}(\Gamma_{4})=[\begin{matrix}\Gamma_{1}&D_{\Gamma_{1}}\Gamma_{3}^{*}\\ \Gamma_{3}D_{\Gamma_{1}}&-\Gamma_{3}\Gamma_{1}\Gamma_{3}^{*}+D_{\Gamma_{3}^{*}}\Gamma_{4}D_{\Gamma_{3}^{*}}\end{matrix}].
(-\frac{1}{u^{2}}\frac{\partial}{\partial u}(u^{2}\frac{\partial}{\partial u})+\frac{\hat{L}^{2}}{u^{2}}+\frac{1}{4}ku^{2}+\frac{1}{u})R_{l}(u)Y_{lm}(\hat{u})=E_{l}R_{l}(u)Y_{lm}(\hat{u}),
C_{a b}^{b}= E_{i}^{b}( \xi_{b}^{j}\xi_{a , j}^{i}- \xi_{a}^{j}\xi_{b , j}^{i}) = A_{~ c}^{b}( A_{a , j}^{~ c}\xi_{b}^{j}- A_{b , j}^{~ c}\xi_{a}^{j}) .
|u_{1}\overline{v}_{1}+\cdot\cdot\cdot+u_{n}\overline{v}_{n}|^{2}\le(|u_{1}|^{2}+\cdot\cdot\cdot+|u_{n}|^{2})(|v_{1}|^{2}+\cdot\cdot\cdot+|v_{n}|^{2}).
\frac{\Gamma(\frac{1}{5})\Gamma(\frac{4}{15})}{\Gamma(\frac{1}{3})\Gamma(\frac{2}{15})}=\frac{\sqrt{2}\sqrt[20]{3}}{\sqrt[6]{5}\sqrt[4]{5-\frac{7}{\sqrt{5}}+\sqrt{6-\frac{6}{\sqrt{5}}}}}
{\frac{1}{2}}\sum_{n = 0}^{\infty}{\frac{\epsilon^{n}t^{n}}{n !}}\left[ \left({\frac{3 i}{2}}\right)^{n}e^{i t}+ \left( -{\frac{3 i}{2}}\right)^{n}e^{- i t}\right] = \cos \left[ \left( 1 +{\frac{3}{2}}\epsilon \right) t \right] .
S_0 =\sum'_{\substack{q'\sim Q_1 g_1'\sim G_1 g_2\sim G_2}}\sum'_{\substack{b_1'<d_0q' g_2 b_2'<d_0d_1 q' g_1'}}\sum'_{\substack{|\nu_1|\le E_0/X |\nu_2|\le E_0/X}}F_{X}\Bigl(\frac{b_1'}{d_0 q' g_2}+\nu_1\Bigr)F_{X}\Bigl(\frac{b_2'}{d_0 d_1 q' g_1'}+\nu_2\Bigr).
o(1)\cdot\|u_n\| =I'[u_n]u_n &= [u_n]_{s,p}^p+a\|u_n\|_p^p &\phantom{=}-b\int_{\mathbb{R}^N}(K\ast F(u_n))f(u_n)u_n dx-\varepsilon_g \|u_n\|_{p_g}^{p_g},
\int_{B_{kr}}|\overline{L}u(x)-Lu(x)|^{p}dx \leq c\sum_{i,j=1}^{q}\left((kr)^{Q}a_{R}^{\sharp}\right)^{1/\beta}\left( (kr)^{Q}(|X_{i}X_{j}u|^{p\alpha})_{B_{kr}}\right)^{1/\alpha}\ & =c(kr)^{Q}(a_{R}^{\sharp})^{1/\beta}\sum_{i,j=1}^{q}\left( (|X_{i}X_{j}u|^{p\alpha})_{B_{kr}}\right)^{1/\alpha}
= \frac 1{z^3}\int_{1-\delta}^1 \sigma^{(3)}(u) e^{uz}du + \frac{\sigma''(1) e^{z}}{z^3}- \frac{\sigma'(1) e^{z}}{z^2}+O\left( \frac{e^{x(1-\delta)}}{|z|}\right)
s*^{\alpha_p}s_{a_p}*^{\alpha_{p-1}}s_{a_{p-1}}*^{\alpha_{p-2}}\cdots*^{\alpha_1}s_{a_1}=t*^{\beta_q}s_{b_q}*^{\beta_{q-1}}s_{b_{q-1}}*^{\beta_{q-2}}\cdots*^{\beta_1}s_{b_1}
k^j \nabla \left[ \tilde{T}(\mathbf y) (\nabla k) \right] = k^j \tilde T(\mathbf y)(\nabla^2 k) + k^{j-1}\left[ \mathbf y_1^{-1}D(\tilde T) \right] (\mathbf y_1, \mathbf y_2) (\nabla k \cdot \nabla k) - k^{j-1}D(\tilde T) (\mathbf y_2, \mathbf y_1) (\nabla k \cdot \nabla k).
\lim_{n \to \infty}\frac{n - k_{n,x}D(k_{n,x})}{k_{n,x}}= \lim_{n \to \infty}\frac{n - \left[\Gamma(n) + \frac{xn}{(\log n)^2}\right] D(\Gamma(n))}{\Gamma(n) + \frac{xn}{(\log n)^2}}= \frac{-x}{a^2}.
(n!)^{3}\cdot H_{n}^{(3)}=[\begin{matrix}n+1\\ 2\end{matrix}]^{3}-3[\begin{matrix}n+1\\ 1\end{matrix}][\begin{matrix}n+1\\ 2\end{matrix}][\begin{matrix}n+1\\ 3\end{matrix}]+3[\begin{matrix}n+1\\ 1\end{matrix}]^{2}[\begin{matrix}n+1\\ 4\end{matrix}].
\partial_{-}h_{m +}= - \partial_{i}h_{i m}- e^{A}\left( \frac{D_{1}- 2}{2}A^{\prime}h_{r m}+ h_{m \underline{{l}};}^{\quad \underline{{l}}}- \kappa^{2}\left( \phi^{\dagger}D_{m}\Phi + ( D_{m}\Phi )^{\dagger}\phi + V_{\underline{{l}}}F_{n}^{\quad \underline{{l}}}\right) \right) ~ .
G_{\nu}^{\mu}( \{\}) = 8 \pi G ( 2 \alpha g^{\mu \alpha}\partial_{\alpha}\Lambda \partial_{\nu}\Lambda - \alpha \delta_{\nu}^{\mu}g^{\alpha \beta}\partial_{\alpha}\Lambda \partial_{\beta}\Lambda + T_{\nu}^{\mu}) = 8 \pi G \tilde{T}_{\nu}^{\mu}
K_i = \int \log(dP_{Kf(x_i)}/dP_{Kf_0(x_i)}) dP_{Kf(x_i)}V_i = \int\left( \log(dP_{Kf(x_i)}/dP_{Kf_0(x_i)}) - K_i \right)^2 dP_{Kf(x_i)}
\pi^{(i)}_*\left[ (1-\beta )^{-q_i}c(i)\cdot c(E_{p_i}/D_i) \right]_{q_i}= \left[ (1-\beta )^{-q_i}\cdot c(i) \cdot \pi^{(i)}_*c(E_{p_i}/D_i) \right]_{q_i-p_i+i}= \left[ (1-\beta )^{-q_i}\cdot c(i) \cdot (1-\beta)^{p_i-i}\right]_{q_i-p_i+i}= (1-\beta T_i)^{-q_i+p_i-i}\cdot c_{q_i-p_i+i}(i),
\mathsf{g}_1=\mathsf{x}_3+\mathsf{x}_6,\mathsf{g}_2=i(\mathsf{x}_6-\mathsf{x}_3),\qquad\mathsf{g}_3=\mathsf{x}_2-\mathsf{x}_5,\mathsf{g}_4=-(\mathsf{x}_2+\mathsf{x}_5),\mathsf{g}_5=\mathsf{x}_1,\qquad\mathsf{g}_6=-\mathsf{x}_4,
\mathcal{F}_{1 2}^{( 1 )}= \mathcal{F}_{1 2}^{( 2 )}= \ldots = \mathcal{F}_{1 2}^{( N )}, \quad \mathcal{F}_{3 4}^{( 1 )}= \mathcal{F}_{3 4}^{( 2 )}= \ldots = \mathcal{F}_{3 4}^{( N )}, \quad \mathcal{F}_{5 6}^{( 1 )}= \mathcal{F}_{5 6}^{( 2 )}= \ldots = \mathcal{F}_{5 6}^{( N )}.
\Vert f\Vert_{B^r_{p,q}}:=\Vert \{2^{jr}\Vert g_j \Vert_{L^p}\}_{j=0}^{\infty}\Vert_{l^q(\mathbb{N})}<\infty,\,\,\,g_0=f_0,\,0<p,q\leq \infty,\,r\in\mathbb{R},
V(R)\equiv\sum_{i=1}^{N}\frac{q_{i}}{4\pi\epsilon_{0}|r_{i}-R|}=\frac{1}{4\pi\epsilon_{0}}\sum_{l=0}^{\infty}\sum_{m=-l}^{l}(-1)^{m}I_{l}^{-m}(R)\sum_{i=1}^{N}q_{i}R_{l}^{m}(r_{i}),
\begin{array}{l}\Delta z^+ =0 \mbox{in}\;\; \Bbb E_s, z^+ =\xi \mbox{on}\;\; J_s, z^+ =0 \mbox{on}\;\; \Gamma_s, [0.3 cm] \Delta z^- =0 \mbox{in}\;\; \Bbb D_s, z^-=\xi \mbox{on}\;\; J_s, \partial_y z^- =0 \quad\mbox{on}\;\; \Gamma_0.\end{array}
\lim_{n}\left|\frac{1}{\phi(n)}\sum_{k=a_n+1}^{a_n+\phi(n)}\sum_{i=1}^b\mathbf{1}_{\{i\}}(\xi_{k-1})[p_k(i,j)-p(i,j)]\right| \leq \sum_{i=1}^b\lim_{n}\frac{1}{\phi(n)}\sum_{k=a_n+1}^{a_n+\phi(n)}|p_{k}(i,j)-p(i,j)|=0,\ \ \forall j\in \mathbf{X}.
u(x,t)=\frac{1}{2\pi^{2}}\left( \delta^{\prime}\left( t\right)u_{0}\left( x\right) +\delta \left( t\right) v_{0}\left( x\right) \right)\ast_{x,t}P\left( x,t\right) ,\;\;x\in\mathbb{R},\;t>0,
0=\sum_{i=1}^D\sum_{j=1}^D \alpha_i \bar{\alpha}_j \sum_{l,\, m \geq 1}G_k\left(\frac{lm}{p}\right)\frac{1}{\sqrt{lm}}\sum_{f\in \mathcal{B}_{2k}(p)}\omega_f \lambda_f(i) \lambda_f(j)\lambda_f(l) \lambda_f(m).
\frac{1}{n^2}\left|\sum\limits_{i=1}^{k^*}\sum\limits_{j=k^*+1}^n\left(1_{\left\{X_i\leq X_j\right\}}-\frac{1}{2}\right)\right| =\left|\frac{1}{n^2}\sum\limits_{i=1}^{k^*}\sum\limits_{j=k^*+1}^n1_{\left\{G(\xi_i)\leq G(\xi_j)+\Delta\right\}}-\frac{1}{n^2}\frac{k^*(n-k^*)}{2}\right|.
\mathcal{B}_\varepsilon =b(\mathbf{D})^*g^\varepsilon (\mathbf{x})b(\mathbf{D})+\sum_{j=1}^d \left( a_j^\varepsilon (\mathbf{x})D_j+D_ja_j^\varepsilon (\mathbf{x})^*\right) +Q^\varepsilon (\mathbf{x}).
\Sigma_2 : = p^{(w-1) h_p}\sum_{m= \max\{0,k_p-h_p\}}^{\infty}\sum_{n=m+h_p-k_p+1}^{\infty}\frac{\tau_A(p^{m}) \tau_B(p^{n}) \left( 1-\frac{p^w}{p^{2}}\right)}{p^{m(2-\alpha-w)}p^{n(1-\beta)}}.
V_{1}~=~\sqrt{\frac{8 \pi \eta_0}{3}}\,\frac{A_{1}}{k_0}\, \bigg(1 + \frac{1}{\textrm{j}k_0a}- \frac{1}{\textrm{j}(k_0a)^2}\bigg)\,e^{-\textrm{j}k_0a}~[], I_{1}~=~-\sqrt{\frac{8 \pi\eta_0}{3}}\,\frac{A_{1}}{k_0}\, \bigg(1 + \frac{1}{\textrm{j}k_0a}\bigg) \,e^{-\textrm{j}k_0a}~[].
\Phi(t,y)= \frac{i\beta(t) \int_{\mathbb{R}}F(t,x,y) x dx}{\int_{\mathbb{R}}F(t,x,y) dx}&\Psi(t,y,z) = \frac{\int_{\mathbb{R}}F(t,x,y) (e^{ix\psi(t,z)}-1)dx}{\int_{\mathbb{R}}F(t,x,y) dx}
\int_{\Sigma_r}f(V^2 + V^4|\nabla \tau|^2) +( div V^2 \nabla \tau) \sinh^{-1}( \frac{f div V^2 \nabla \tau}{|H||H_0|}) d\Sigma_r - \int_{\Sigma_r}\alpha_{H_0}(V^2 \nabla \tau) d\Sigma_r + \int_{\Sigma_r}\alpha_{H}(V^2 \nabla \tau)d\Sigma_r.
\partial_t u-\nabla\cdot( \mathsf{A}\nabla u)=-\nabla\cdot(u{\bf E}) \mbox{in}Q_T; (\mathsf{A}\nabla u-u{\bf E})\cdot{\bf n}=0 \mbox{on}(\partial\Omega\setminus \bar\Gamma)\times ]0,T[; (\mathsf{A}\nabla u-u{\bf E})\cdot{\bf n}+b(u)u=h \mbox{on}\Gamma\times ]0,T[,
{{\cal A}^{\prime}}_{( 2 )}^{I}= \frac{( 2 \pi \sqrt{\alpha^{\prime}})^{2}}{\sqrt{V}}{\cal A}_{( 2 )}^{I}, \qquad{b^{\prime}}^{I}= \frac{( 2 \pi \sqrt{\alpha^{\prime}})^{2}}{\sqrt{2 V}}{b}^{I}
(a)=\begin{pmatrix}1,0\end{pmatrix}, \;(a^*)=\begin{pmatrix}A_0^* A_1^*\end{pmatrix}, \;(\beta)=\begin{pmatrix}B_{00}B_{01}B_{10}B_{11}\end{pmatrix}, \;(\gamma)=\begin{pmatrix}0 1 C_{10}C_{11}\end{pmatrix}, \;(\delta)=\begin{pmatrix}D_{00}D_{01}D_{10}D_{11}\end{pmatrix}
F_{A_{K^{1},K^{2},K^{3}}}=\left(-\frac{9}{2}\frac{r(r^{2}+1)}{(r^{2}+2)^{\frac{7}{4}}(r+1)^{\frac{5}{2}}(r-1)^{\frac{5}{2}}}dr\wedge\theta^{1}+\frac{\theta^{25}+\theta^{36}+\theta^{47}}{(r^{2}+2)^{\frac{3}{4}}(r+1)^{\frac{3}{2}}(r-1)^{\frac{3}{2}}}\right)\otimes K^{i}e_{i}.
\Psi_{n l m}( r , \theta , \varphi ; \delta_{1}, \delta_{2}) = \sum_{n_{1}= 0}^{n - | m | - 1}\widetilde W_{n l m}^{n_{1}}( \delta_{1}, \delta_{2}) \Psi_{n_{1}n_{2}m}( \mu , \nu , \varphi ; \delta_{1}, \delta_{2}) ,
\|T_{k}^{\alpha}\|_{L^{p}(\mathbf{B},d\tau)\rightarrow L^{p}(\mathbf{B},dv_{pm-n})}> \frac{\pi^{n/2}\Gamma(p(m+\alpha)+\frac{n}{2}+1)\Gamma(m+\alpha+\frac{n}{p}+1)}{\Gamma(p(m+\alpha)+1)\Gamma(m+\alpha+\frac{n}{2}+\frac{n}{p}+1)}&\times\left(\frac{\Gamma(pm-n+1)}{\Gamma(pm-\frac{n}{2}+1)}\right)^{1/p}.
p_{A}= \bar{p}_{A}( \psi , \psi^{*}) \equiv m \frac{( \widetilde{\psi}\Gamma_{A B}\psi ) ( \widetilde{\psi}\Gamma^{B}\psi )^{*}+ c . c .}{2 \left| ( \widetilde{\psi}\Gamma_{C}\psi ) \right|^{2}}
\| \partial_{t}^{\ell}\nabla^{k}_{x}u(t) \|_{2}\le\begin{cases}C (1+t)^{-\frac{n}{4 (1-\sigma)}-\ell-\frac{k}{2(1-\sigma)}}(\sigma \in (0, \frac{1}{2}) ), & C (1+t)^{-\frac{n}{2}-(\ell+ k)}(\sigma =\frac{1}{2}), & C (1+t)^{-\frac{n}{4\sigma}- \frac{\ell+k}{2 \sigma}}(\sigma \in (\frac{1}{2}, 1] )\end{cases}
\mathcal{L}_{l_{1},l_{2},l_{3}}(\mathbf{n}_{1},\mathbf{n}_{2},\mathbf{n}_{3})=(-1)^{L}\mathcal{L}_{l_{2},l_{1},l_{3}}(\mathbf{n}_{2},\mathbf{n}_{1},\mathbf{n}_{3})=(-1)^{L}\mathcal{L}_{l_{1},l_{3},l_{2}}(\mathbf{n}_{1},\mathbf{n}_{3},\mathbf{n}_{2})
\omega_t(X^t,Y^t) = \mathrm{det}[X^t,Y^t,\eta_t] = (1+tg\lambda_1)(1+tg\lambda_2)\mathrm{det}[X,Y,\eta_t] + tX(g)(1+tg\lambda_2)\mathrm{det}[\eta,Y,\eta_t] + + tY(g)(1+tg\lambda_1)\mathrm{det}[\eta,X,\eta_t],
x_{k}=\frac{2}{N}[\frac{1}{2}\sum_{j=0}^{N-1}x_{j}+\sum_{n=1}^{N-1}\sum_{j=0}^{N-1}x_{j}cos[\frac{\pi}{N}(j+\frac{1}{2})n]cos[\frac{\pi}{N}(k+\frac{1}{2})n]]
\|b_{k,\ell}(\rho)\chi(\frac{\rho}M)\rho^{\frac{n-1}{2}}\|_{L_\rho^2}= \big\|\chi(\frac{\rho}M)\mathcal{H}_{\nu}\big[Y_{k,l}(\theta)a_{k,\ell}(r)\big](\xi)\big\|_{L_\xi^2}= \Big\|\mathcal{H}_{\nu}\Big[\chi(\frac{\rho}M)\mathcal{H}_{\nu}\big(Y_{k,l}(\theta)a_{k,\ell}(r)\big)(\xi)\Big]\Big\|_{L_x^2}.
E_{n}(x) = \sum_{k=0}^{n}\binom{n}{k}\frac{E_{k}}{2^{k}}\left( x - \tfrac{1}{2}\right)^{n-k}= \sum_{k=0}^{n}\binom{n}{k}E_{k}\left( \tfrac{1}{2}\right) \left( x - \tfrac{1}{2}\right)^{n-k},
S_{4}= ( 2 \pi )^{4}\int \left[ \prod_{j = 1}^{2}\frac{d^{4}P_{j}}{( 2 \pi )^{4}}\frac{d^{4}Q_{j}}{( 2 \pi )^{4}}\right] G_{4}H \delta^{4}\left( \sum_{l}P_{l}+ \sum_{j = 1}^{2}( Q_{j}- P_{j}) \right) .
\frac{1}{\pi}=8\sqrt{15}\sum_{k=0}^{\infty}s_{6B}(k)(\frac{1}{2}-\frac{3\sqrt{5}}{20}+k)(\frac{1}{\phi^{12}})^{k+\frac{1}{2}},j_{6B}(\sqrt{\frac{-5}{6}})=(\frac{1+\sqrt{5}}{2})^{12}=\phi^{12}
\int_{\mathbb{R}^d}\varepsilon L^\varepsilon \nabla \tilde\rho \cdot\nabla \varphi^\varepsilon \, dx= \int_{\mathbb{R}^d}-\varepsilon (\nabla \varphi^\varepsilon\otimes \nabla \varphi^\varepsilon) : \nabla^2\tilde \rho + e_{\varepsilon}\Delta \tilde\rho \, dx.
\sup_{k,\tau}\sum_{\substack{|k_2|\gtrsim |k| k_2^*\ll |k| |\Phi|\gtrsim k^2}}\frac{\langle k\rangle^{2s+2\varepsilon-2+}}{\prod_{i=1}^p\langle k_i\rangle^{2s}}\lesssim \langle k\rangle^{2\varepsilon -2+(p-1)\max(1-2s, 0)+},
\int_{B^n(x,r_i)}\Psi(\vert\nabla u\vert)\, d\mu_{\alpha}\geq\begin{cases}CR_i\log^{\lambda-(n+\alpha-1)}\left(\frac{1}{R_i}\right) ~~p=n+\alpha-1,~\lambda>n+\alpha-1, CR_i^{n+\alpha-p}\log^{\lambda}\left(\frac{1}{R_i}\right) ~.\end{cases}
\sum_{m\geq 0}B_{m}(1-z)\frac{u^{m}}{m!}= \frac{u}{e^{u}-1}e^{(1-z)u}= \frac{ue^{u}}{e^{u}-1}e^{-zu}= \frac{-u}{e^{-u}-1}e^{-zu}= \sum_{m\geq 0}B_{m}(z)\frac{(-u)^{m}}{m!}= \sum_{m\geq 0}(-1)^{m}B_{m}(z)\frac{u^{m}}{m!}.
(gh)(\sqrt{\alpha_i}) =g(h(\sqrt{\alpha_i})) &=g(x_{h,\sigma_h(i)}\sqrt{\alpha_{\sigma_h(i)}}) &=x_{h,\sigma_h(i)}g(\sqrt{\alpha_{\sigma_h(i)}}) &=x_{h,\sigma_h(i)}x_{g,\sigma_g(\sigma_{h}(i))}\sqrt{\alpha_{\sigma_g(\sigma_{h}(i))}}&=x_{h,\sigma_h(i)}x_{g,\sigma_{gh}(i)}\sqrt{\alpha_{\sigma_{gh}(i)}},
|(w_{[t^{-2},1]},r_{[t^{-2},1]})(\alpha)| \lesssim \ \sum_{2^\ell = t^{-2}}^{\alpha^{-2}t^2}|t|^{-1}+ |\alpha|^{-\frac12}+ \sum_{2^\ell=\alpha^2 t^{-2}}^12^{-\frac{\ell}2}|\alpha|^{-1}\approx \ |t|^{-1}|\log (1+t^2 |\alpha|^{-1})| + |\alpha|^{-\frac12}+ |\alpha|^{-1}\lesssim |\alpha|^{-\frac12}.
E^{T, \Upsilon, Y}\cup Z^a\big[ T_1, ..., T_n \big]= \bigg( \bigcup_{i=1}^n E^{T_i, \Upsilon_i, Y_i}\bigg) \cup \Big\{\tilde{\phi}^{T, \Upsilon, Y}\big[ \tilde{h}_j^{Y}\big] : \tilde{h}_j^{Y}\in Z^a\big[ Y_1, ..., Y_n \big] \Big\}.
\frac{(-1)^{k-1}}{(q;q)_\infty}\sum_{n=1-k}^{k}(-1)^{n}q^{n(3n-1)/2}= (-1)^{k-1}+ \sum_{n=k}^\infty \frac{q^{{k\choose 2}+(k+1)n}}{(q;q)_n}\begin{bmatrix}n-1 k-1\end{bmatrix},
(x^-_{i_p,i_{p+1}}\otimes t^{\nu(h_{i_p,i_{p+1}})+2})\tilde v_\mu= 0 \ \{\rm and}(x^-_{i_p, i}\otimes t^{\nu(h_{i_p,i})+1})\tilde v_\mu = (x^-_{j, i_{p+1}}\otimes t^{\nu(h_{j,i_{p+1}})+1})\tilde v_\mu= (x^-_{i,j}\otimes t^{\nu(h_{i,j})})\tilde v_\mu=0,
\partial_{B}^{2}\left( f \circ \theta \right) = ( \partial_{_{w}}^{2}f ) ( x ) + \partial_{\int_{\lbrack 0 , \cdot ]^{2}}1_{r \leq \tau}[ A d_{x ( r )}B ( r ) , A d_{x ( \tau )}B ( \tau ) ] \, d r d \tau}f ( x ) .
\underline{{{h s ( 2 , 2 )}}}: f ( a , \bar{a}) = \alpha + \alpha_{i \bar{j}}a^{i}\bar{a}^{\bar{j}}+ \alpha_{i_{1}i_{2}\bar{j}_{1}\bar{j}_{2}}a^{i_{1}}a^{i_{2}}\bar{a}^{\bar{j}_{1}}\bar{a}^{\bar{j}_{2}}+ \ldots
\epsilon \simeq \frac{( 2 \pi )^{2}g_{s}^{2}F^{2}}{2 ( M_{s}^{2}A_{T})^{2}}\left( \frac{M_{P}}{\Psi}\right)^{2}, \qquad \eta \simeq - \frac{2 \pi g_{s}F}{( M_{s}^{2}A_{T})}\left( \frac{M_{P}}{\Psi}\right)^{2}.
\lambda^3-\frac{3\lambda}{4}- \frac{1}{4}\cos{3\alpha}-\left(3\lambda^3-\frac{3}{4}\cos3\alpha\right)\cdot\Delta^2= &2\sqrt{\lambda^6+\frac{3\lambda^4}{4}-\frac{\lambda^3}{2}\cos3\alpha+\frac{9\lambda^2}{16}-\frac{3\lambda}{16}\cos3\alpha+\frac{1}{16}\cos^23\alpha}\cdot\Delta^3.
a_0 =\frac12\log\left(\frac{2\pi}{A^4}\right), a_n =\frac{1}{4n}-\frac{C}{\pi^2n^2}-\frac{\log n}{4\pi^2n^2}-\frac{1}{\pi^2}T_n\quad(n\ge1), b_n =\frac{\gamma+\log(n)-H_n}{2\pi n}+\frac{1}{2\pi n^2}\quad(n\ge1)
\varphi((T^R_{\lambda_R^+})^{-(n + k)})e_G(\sigma)(E_{o_-}(w(\lambda_R^+)^k)) = \varphi((T^R_{\lambda_R^+})^{-(n + k)})\sigma(E^P_{o_{-,P}}(w(\lambda_R^+)^k)) & = \psi((T^{P\cap R}_{\lambda_{R}^+})^{-(n + k)})\sigma(E^P_{o_{-,P}}(w(\lambda_R^+)^k)).
\begin{array}[c]{c}\displaystyle\frac{n}{q\sqrt{1-v^{2}/c^{2}}}\rho_{t}+\frac{n}{c^{2}\left(\sqrt{1-v^{2}/c^{2}}\right)^{3}}vv_{t}+\frac{n}{q\sqrt{1-v^{2}/c^{2}}}v\rho_{r}+\frac{n}{\left( \sqrt{1-v^{2}/c^{2}}\right)^{3}}v_{r}\displaystyle+\frac{2n}{r\sqrt{1-v^{2}/c^{2}}}v=0\end{array}
W [{\psi}, A ] = \int_{- \infty}^{\infty}d t \int_{0}^{\mathrm{L}}d x \sum_{k = 1}^{2}\overline{{{\psi}}}_{k}({\gamma}^{\mu}i{\partial}_{\mu}- m_{k}){\psi}_{k}+ \frac{1}{2}\int_{- \infty}^{\infty}d t \int_{0}^{\mathrm{L}}d x \int_{0}^{\mathrm{L}}d y J^{0}( x , t ) D ( x , y | \mathrm{L}) J^{0}( y , t )
\begin{aligned}K^\dagger \left(t,x_1,x_2,z_1,z_2;\lambda\right)= \frac{\lambda^2}{2\nu}G_{\nu/2}(t,\bar{x}-\bar{z}) \exp\left(-\frac{\lambda^2(|\Delta x|+|\Delta z|)}{2\nu}+\frac{\lambda^4t}{4\nu}\right) &\times \Phi\left(\frac{\lambda^2\sqrt{t}}{\sqrt{2\nu}}-\frac{|\Delta x|+|\Delta z|}{\sqrt{2\nu t}}\right),\end{aligned}
\{z_i^\prime, \; z_k^\prime\}= \{(t^{-1})^* z_i, \, (t^{-1})^* z_k\}= (t^{-1})^*\{z_i, \, z_k\}= ((t^{-1})^* f_{i,k})(z_1, \ldots, z_n) &= f_{i,k}(z_1^\prime, \ldots, z_n^\prime).
\max_{\boldsymbol{P}}~~ \sum_{i=1}^K \ln\left(1+\frac{P_i^\prime}{\sigma^2+\sum_{j=1,j\neq i}^K P_j^\prime}\right), \mbox{s.t.}~~ P_i^\prime\ge 0, ~\forall i, &P_i^\prime\le h_iP^{pk},~\forall i, &\sum_{i=1}^K \frac{g_i}{h_i}P_i^\prime \le I.
u_0^{r,(h)}(r,z) = \sum_{M\le j\le M+\sqrt M}2^{-3j}\frac 1 r (-\partial_z a_0)(2^j(r,z)), &u_0^{z,(h)}(r,z) = \sum_{M\le j\le M+\sqrt M}2^{-3j}\frac 1 r (\partial_r a_0)(2^j (r,z)), &\omega_0^{\theta,(h)}(r,z) = \sum_{M\le j\le M+\sqrt M}2^{-j}b_0(2^j(r,z)),
\hspace{1 0 m m}\tilde{B}_{0 \mu}= G_{0 \mu}/{G_{0 0}}\hspace{3 m m}, \hspace{3 m m}\tilde{B}_{\mu \nu}= B_{\mu \nu}- \frac{G_{0 \mu}B_{0 \nu}- G_{0 \nu}B_{0 \mu}}{G_{0 0}}.
\|1_A\|\ = \ \|z-1_{B\backslash E}- P_E(z)\| \ \leqslant\ \mathbf C_\ell \|z-P_E(z)\| \ \leqslant\ \mathbf C_\ell \mathbf C_{\lambda, rp}\sigma^{\mathcal{U}_X, R}_m(z) \ \leqslant\ \mathbf C_\ell \mathbf C_{\lambda, rp}\|1_B\|.
\hat{u}_{2}\circ p_{12}\circ u_{2}=\hat{u}_{2}\circ u'_{2}\circ p_{12}~~~~~~(\because p_{12}\circ u_{2}=u'_{2}\circ p_{12}) &=\hat{u}_{2}\circ p_{12}~~~(\because \hat{u}=(\hat{u}_{1},\hat{u}_{2})\in U^{G}(X_{1},X_{2})).
\lim_{\delta \to 0}\frac{II}{\delta}= \frac{X_{\beta,2}}{f_{\beta,2}'(g_{\beta,2}(x))}\cdot \frac{f_{\beta,2}''(g_{\beta,2}(x))}{(f_{\beta,2}' ( g_{\beta,2}(x) ) )^2}.
\begin{cases}\Delta^2 u^\prime(\lambda) + V_1 u^\prime(\lambda) - \lambda u^\prime(\lambda) = u(\lambda) + (V_2 - V_1)u^\prime_2(\lambda) \, B_R, u^\prime(\lambda) = 0 \,\partial B_R, \Delta u^\prime(\lambda) = 0 \,\partial B_R,\end{cases}
G_k(q) =\sum_{\mathfrak J\in S_2}\tilde \chi^k(\mathfrak J) q^{N(\mathfrak J)}+3 \sum_{\mathfrak J\in \Sigma}\tilde \chi^k((-2)\mathfrak J) q^{N((-2)\mathfrak J)}& =\sum_{\mathfrak J\in S_2}\tilde \chi^k(\mathfrak J) q^{N(\mathfrak J)}+3(-2)^k \sum_{\mathfrak J\in \Sigma}\tilde \chi^k(\mathfrak J) q^{4N(\mathfrak J)}.
a(z^{-n-1})^*T(\Omega \otimes \Omega) = T(a(s^{1/2}z^{-1}(sz^{-1}+ \overline{w})^n)^*\Omega \otimes \Omega) + T(\Omega \otimes a(\overline{z W_{\phi_t}z^n})^*) = 0.
\int_\Omega |\nabla u_{n,k}|^2 + \int_\Omega \lambda V |u_{n,k}|^2 \leq b_\nabla 2^{-(d-1) m n}\cdot 2^{\gamma m n}\left(1- 2 \lambda 2^{- \gamma m n \tfrac{2}{d}(-1 + \epsilon)}\cdot 2^{-2 \gamma m n}\right) <0
\left\|x + t\frac{y_k-x}{\|y_k-x\|}- y_k \right\| - \|b-y_k\| = \| x - y_k \| \left( 1 - \frac{t}{\|x-y_k\|}\right) - \|b-y_k\| = \|x - y_k \| - \|b-y_k \| - t
\nabla_i\nabla_j\big(\frac{\partial R}{\partial t}\big)=\nabla_i\nabla_j\big(\big(1+\frac{\alpha'}{2}R\big)\Delta R\big)+\frac{\alpha'}{2}\nabla_i\nabla_j|\nabla R|^2+\nabla_i\nabla_jR^2+\frac{\alpha'}{4}\nabla_i\nabla_jR^3-r\nabla_i\nabla_jR.
\frac{\left| u'_{\theta,\vartheta_1 \ldots \vartheta_n}(t) \right|}{\left| u'_{\theta,\vartheta_1 \ldots \vartheta_n}(r) \right|}=\left(\frac{q_{\theta,n}+r q_{\theta,n-1}}{q_{\theta,n}+t q_{\theta,n-1}}\right)^2 \leq\left(\frac{q_{\theta,n}+\theta q_{\theta,n-1}}{q_{\theta,n}}\right)^2 \leq \left( 1+\theta^2 \right)^2.
P=P_{g}+P_{b}=\frac{-1}{9\times 960\pi^{2}}(\Lambda_{betw}^{2}-\Lambda_{out}^{2})-\frac{\eta^{4}\Lambda_{betw}^{2}J_3(B_1,B_2)}{36\pi^{3/2}a^{4}\Gamma (3/2)}.
({\gamma_{\mu}}^{\prime}) ({\gamma_{\mu - 1}}^{\prime}) \cdots ({\gamma_{2}}^{\prime}) ( \gamma_{g + 1}) ( D_{g})^{- 1}= \{S^{g + 1};{S_{2}}^{\prime},{S_{3}}^{\prime}, \cdots ,{S_{\mu}}^{\prime}\},
- 2 \, G_{\mu \nu}\,{\frac{\delta \Gamma}{\delta G_{\mu \nu}}}+ \beta^{i}\,{\frac{\delta \Gamma}{\delta \phi^{i}}}= - 2 \, \sqrt{G}\, ({\cal W}_{d}+ \nabla_{\mu}{\cal J}_{d}^{\mu}) - \beta^{i}\,{\frac{\delta S_{\mathrm{l o c}; \, 0}}{\delta \phi^{i}}}.
div_{\sigma}\alpha_{H_0}= -(\widehat{H}\hat{\sigma}^{ab}-\hat{\sigma}^{ac}\hat{\sigma}^{bd}\hat{h}_{cd})\frac{\nabla_b\nabla_a \tau'}{\sqrt{1+|\nabla\tau'|^2}}+ div_\sigma (\frac{\nabla\tau'}{\sqrt{1+|\nabla\tau'|^2}}\cosh\theta_0|{H_0}|-\nabla\theta_0)
R_{rn+p}^{(t)}(x)=\underset{j=0}{\overset{}{\sum}}\binom{rn+p-j-1}{j}_{r}((r-1)(rn+p-1)-rj)...((r-1)(rn+p-1)-rj-t+1)x^{(r-1)(rn+p-1)-rj-t}.
\left((2p-1)x+\frac{4pK}{f(x)}\right)\frac{f^{(1)}(x)}{f(x)^2}=(2p-1)x+\frac{6pK+1}{f(x)}-2(p+1)\frac{\displaystyle \int_0^x f(t)^3 \;dt}{f(x)^3}+\frac{4e^{3F(0)}}{f(x)^3}.
\Phi^{\iota_1\iota_2\iota_3}(\xi,x,y) := -\Lambda(\xi) + \iota_1\Lambda(\xi+x) + \iota_2\Lambda(\xi+y) + \iota_3\Lambda(\xi+x+y), & c^{\ast,\iota_1\iota_2\iota_3}_{\xi;k_1,k_2,k_3}(x,y) := c^{\iota_1\iota_2\iota_3}(\xi,-x,-\xi-x-y) \cdot \varphi'_{k_1}(\xi+x) \varphi'_{k_2}(\xi+y)\varphi'_{k_3}(\xi+x+y).