harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\frac{1}{2\pi}\,\int_{0}^{\infty}\,d\omega''\,\tilde{\beta}_{\omega''\omega}^{R *}\,\tilde{\beta}_{\omega''\omega}^{R}= \langle\, a_{\omega}^{R \dag}a_{\omega}^{R}\, \rangle \,_{\rm{in}}\,.
G_\lambda(x) \!=\! \left( \frac{\lambda}{c^2}I_4 \! + m \beta \! + \left( 1 \! - i \sqrt{\frac{\lambda^2}{c^2}- (m c)^2}|x| \right) \frac{i(\alpha \cdot x )}{c |x|^2}\right)\frac{e^{i \sqrt{\lambda^2/c^2\! - (m c)^2}|x|}}{4 \pi |x|};
\begin{aligned}|u(\bar x)-m^\gamma_u(2B_i)| \le |u(\bar x)-m^\gamma_u(B(\bar x,Cr_i))|+|m^\gamma_u(B(\bar x,Cr_i))-m^\gamma_u(2B_i)| &\le cr_i^{\beta}u_{\beta,Cr_i}^{\gamma/C,\#}(\bar x). \end{aligned}
(K_1 f) (x) := \int_0^1 \Bigl(G_{1}(x,t;\l) |Q^*(t)|^{1/2}\Bigr) f(t) dt, x \in [0,1], (K_2 f) (x) := \int_0^1 \Bigl(U(x) |Q(x)|^{1/2}G_{2}(x,t;\l)\Bigr) f(t) dt, x \in [0,1].
\mathbb{C}^N=\begin{cases}\mathbb{C}^{m}\otimes \Delta_r^+\oplus \overline{\mathbb{C}^{m}}\otimes \Delta_r^- , \overline{\mathbb{C}^{m}}\otimes \Delta_r^+\oplus \mathbb{C}^{m}\otimes \Delta_r^- ,\end{cases}
\begin{array}{lllllll}g_{rr}= \displaystyle \frac{l^{2}}{r^{2}}+O(r^{-4})\; g_{tt}= \displaystyle-\frac{r^{2}}{l^{2}}+O(1)\; begin{align*}2mm] g_{tr}= O(r^{-3})\; g_{\varphi \varphi}= r^{2}+O(1)\; begin{align*}1mm] g_{\varphi r}= O(r^{-3})\; g_{t\varphi}= O(1)\;\end{array}
\frac{\mathrm{K}_{0}(B)}{\mathrm{K}_{0}(\varphi) (\mathrm{K}_{0}(A))}=\frac{\mathrm{K}_0(C(Y))}{H}\cong \frac{C(Y,\mathbb{Z})}{\Delta(H)}= \frac{C(Y,\mathbb{Z})}{\{f \circ h: \ \f \in C(X, \mathbb{Z}) \}}.
K_{l}( u ) = \left( \begin{array}{l l}{2 x_{0}( q^{- 1}s^{- 1}u - q s u^{- 1})}&{x_{+}( q^{- 1}u^{2}- q u^{- 2})}\\{x_{-}( q^{- 1}u^{2}- q u^{- 2})}&{- 2 x_{0}( s u - s^{- 1}u^{- 1})}\\ \end{array}\right) ,
\lefteqn{(-\partial_1^2-|\partial_1|^{-1}\partial_2^2)u^\ell}&+P\Big(u^\ell\partial_2Ru^\ell+\partial_2\frac{1}{2}R(u^\ell)^2-u^\ell \partial_1\frac{1}{2}R(u^\ell)^2-\sigma\xi_\ell\Big)=0.
\times \left\langle \prod_{b = 1}^{N}\prod_{j = 1}^{n_{b}}\left[ : e^{+ i 2 \sqrt{\pi}\varphi^{( b )}( x_{j}^{( b )})}:_{Q^{0 , S}}: : e^{- i 2 \sqrt{\pi}\varphi^{( b )}( y_{j}^{( b )})}:_{Q^{0 , S}}\right] \right\rangle_{Q^{0 , S}}\; \leq
\int_{r_{1}}^{r_{2}}d r \sqrt{2 \mu ( E - V_{\it e f f}( r ) )}= \left( n + \frac{1}{2}+ \frac{1}{2}| l - \alpha | - \frac{1}{2}\sqrt{\eta}\right) \pi , \; \; ( n = 0 , 1 , 2 , \cdots )
\frac{1}{V(K_0)}V(z_0 h_{K_0};1)^2 = \frac{n-1}{n-p}V(z_0 h_{K_0}; 2) + \frac{1-p}{n-p}V(z^2_0 h_{K_0};1) ~, \frac{1}{V(K_0)}V(z_0 h_{K_0};1)^2 \geq \frac{n-1}{n-p_1}V(z_0 h_{K_0}; 2) + \frac{1-p_1}{n-p_1}V(z^2_0 h_{K_0};1) .
D\left(H_0,\delta\right)=\left\{H:\left|h_{j,j}-h_{j,j}^{(0)}\right|<\delta_{j,j},\left|u_{\ell,k}-u_{\ell,k}^{(0)}\right|<\delta_{\ell,k},\left|v_{\ell,k}-v_{\ell,k}^{(0)}\right|<\delta_{\ell,k}\right\},
\left( X^{\mu}\right) ( \tau , \sigma^{1}, \sigma^{2}) = \left( \begin{array}{c}{\tau}\\{X^{1}( \tau , \sigma^{1}, \sigma^{2})}\\{X^{2}( \tau , \sigma^{1}, \sigma^{2})}\\{X^{3}( \tau , \sigma^{1}, \sigma^{2})}\\ \end{array}\right) ,
\binom{n}{n^{0.7}}\binom{n}{n^{0.5 + 2\epsilon_0}}^r < n^{n^{0.7}}\Big(n^{n^{0.5 + 2\epsilon_0}}\Big)^r = e^{n^{0.7}\ln n}\Big(e^{(n^{0.5 + 4\epsilon_0})r \ln n}\Big) = e^{n^{0.7}\ln n + n^{(0.5 + 4\epsilon_0)}r\ln n}= e^{O(n^{0.7}\ln n)}.
N_{W_{\alpha}XY}(\hbar\omega)=(4\pi)^{-1}\psi_{W_{\alpha}XY}[1-\kappa]\int_{\Omega}\int_{0}^{\infty}\rho_{\alpha}(z)P_{W_{\alpha}}(\hbar\omega;z)exp[\frac{-z}{\lambda(W_{\alpha}XY)}cos\theta]dzd\Omega,
f'(z) = (2w+1)r - \frac{wr(1+z)^{r-1}}{1-w((1+z)^r-1)}> (2w+1)r - \frac{wr(1+z)^{r}}{1-w((1+z)^r-1)}>(2w+1)r - \frac{wr(1+1/(2w))}{1-w((1+1/(2w))-1)}=0
S^{I}_{p,q}\!\left(N,z^{\beta}\right) \; \equiv (-1)^N p^{-1}\, z_{2}^{\beta (N-1)}\int_C \frac{s^{N+q/p-1}\; e^{-s^{1/p}}}{s+z_{2}^{-\beta}}\; ds + \frac{2\pi i}{p}\; z_2^{-\beta q/p}& \times \sum_{j=1}^2 e^{i(2j-1)q \pi/p}\exp \!\left(-z_2^{-\beta/p}e^{i(2j-1) \pi/p}\right) \;\; ,
T_{7}+T_{8}+T_{9}=2\pi is\left( d,c;B_{b},A_{c},R_{2},-R_{1}\right)-\frac{2\pi i}{c\left( cz+d\right)}B_{0}(B)P_{2}\left( cR_{2}-dR_{1},A\right) & \quad-\frac{2\pi i}{c}(cz+d)B_{0}(A)P_{2}(R_{1},B_{b})+2\pi iP_{1}(R_{2},A_{c})f^{\ast}(bR_{1})\lambda_{R_{1}}.
\tilde{T}_{i}= p r_{2}^{!}\circ ( \int_{p r_{2}^{- 1}}) \circ ( \otimes s_{i}^{- \frac{1}{2}( \lambda + 2 \rho ) ( H_{i})}) \circ ( p r_{1}^{\ast}\otimes p r_{1}^{\ast}) .
\begin{aligned}\left(\frac{3 \boldsymbol{u}_h^n-3\widetilde{\boldsymbol{u}}_h^n}{2 \Delta t}+\nabla (p_h^{n}-p_h^{n-1}+\chi\nu\pi_{Q_h}(\nabla\cdot\widetilde{\boldsymbol{u}}_h^n)), \boldsymbol{y}_h \right) =0 (\nabla\cdot \boldsymbol{u}_h^n, q_h) =0 \boldsymbol{u}_h^n|_{\partial\Omega}=0\end{aligned}
\bar M_Z^2=m_1^2 \left( 1+\left(\frac{\Gamma_1}{m_1}\right)^2\right)^{-1}\quad{\rm and}\quad \bar\Gamma_Z^2=\Gamma_1^2 \left(1+\left(\frac{\Gamma_1}{m_1}\right)^2\right)^{-1},
\mathcal{H}= \left( a_{i}^{\dagger}\, , \, b_{j}\right) \left( \begin{matrix}{{\mathcal{E}_{i l}}}&{{\mathcal{F}_{j l}^{\dagger}}}\\{{\mathcal{F}_{i m}}}&{{- \mathcal{E}_{j m}^{T}}}\\ \end{matrix}\right) \left( \begin{matrix}{{a_{l}}}\\{{b_{m}^{\dagger}}}\\ \end{matrix}\right) .
\left(\mathcal{D}_0^\theta f\right)(x)=f(0) \left[\frac{1}{\theta-1}E_1\left(\frac{x}{\theta-1}\right)\right]+ \int_0^x \frac{1}{\theta-1}E_1\left(\frac{x-y}{\theta-1}\right)\frac{df}{dy}(y)\,dy,\mbox{a.e.}x\in [0,1].
\left(\ell \partial_{x_1}Y_\ell \pm \frac{\sqrt{\lambda_0}}{\sqrt{1-\ell^2}}Y_\ell\right) e^{\pm \frac{\ell \sqrt{\lambda_0}}{\sqrt{1-\ell^2}}x_1}-\ell \partial_{x_1}\left(Y_\ell e^{\pm \frac{\ell \sqrt{\lambda_0}}{\sqrt{1-\ell^2}}x_1}\right) =\pm \sqrt{\lambda_0}(1-\ell^2)^{\frac 12}Y_\ell e^{\pm \frac{\ell \sqrt{\lambda_0}}{\sqrt{1-\ell^2}}x_1}
\int_{\Sigma_{\varepsilon,k}}|H| = \int_{M}|H| + \int_{M'}|H| + \sum_{i=1}^N \int_{S_{\varepsilon,k,i}}|H| = 2\int_{M}|H| + \sum_{i=1}^N \int_{S_{\varepsilon,k,i}}|H|.
\prod_{\substack{n\geq 1 n\not\equiv 0,\pm (k_0 +1)}}(1-q^n)^{-1}= \sum_{n_1,n_2,\ldots,n_{[k/2]}\geq 0}\frac{q^{N_{1}^{2}+N_{2}^{2}+...+ N^{2}_{[k/2]}+N_{i}+N_{i+1}+\ldots+N_{[k/2]}}}{(q)_{n_1}(q)_{n_2}\cdots (q)_{n_{[k/2]}}}.
\int_{B_R}|\nabla u|^2\,dx+\Lambda|\{u>0\}\cap B_R| =\int_{B_R}|\nabla U_0|^2\,dx+\Lambda|\{|U_0|>0\}\cap B_R| &\le\int_{B_R}|\nabla (\xi \tilde u)|^2\,dx+\Lambda|\{|\xi\tilde u|>0\}\cap B_R| &=\int_{B_R}|\nabla \tilde u|^2\,dx+\Lambda|\{|\tilde u|>0\}\cap B_R|,
\int_{2\pi n-3\pi/2}^{+\infty}\frac{\sin^2(x)dx}{x^2(x-\pi n)(x+\pi n)}=\frac{1}{(\pi n)^3}\int_{2-3/(2n)}^{+\infty}\frac{\sin^2(\pi n t)dt}{t^2(t-1)(t+1)}&\leq \frac{1}{(\pi n)^3}\int_{2-3/(2n)}^{+\infty}\frac{dt}{(t-1)^4}=\frac{1}{(\pi n)^3}\frac{1}{3(1-\frac{3}{2n})^3}\leq \frac{4^3}{3(\pi n)^3}.
{\gamma_{sk}}= \frac{{{{\left\|{{{\bf{g}}_{a{p_0},s{k_0}}}}\right\|}^2}{{\left|{{X_{a{p_0},s{k_0}}}}\right|}^{- \beta}}}}{{I{n_{ap,sk}}+{{{\delta^2}}\mathord{\left/{\vphantom{{{\delta^2}}{{P_{ap}}}}}\right. \kern-\nulldelimiterspace}{{P_{ap}}}}}},
\vartheta_{C_1^*}.\vartheta_{C_{3}^*}= y, \vartheta_{C_i^*}.\vartheta_{C_{i}^*}= \vartheta_{2C_i^*}, \vartheta_{C_i^*}.\vartheta_{C_{i+1}^*}= \vartheta_{C_i^* + C_{i+1}^*},
\sum_{k}(\mathsf{M}_{m}^{n})_{k}^{i}(\mathsf{M}_{o}^{p})_{j}^{k}=u_{i}^{m*}\left(\sum_{k}u_{k}^{n}u_{k}^{o*}\right)u_{j}^{p}=\delta_{o}^{n}u_{i}^{m*}u_{j}^{p}=\delta_{o}^{n}(\mathsf{M}_{m}^{p})_{j}^{i}.
W_{\mathrm{q u}}= S_{\mathrm{q u}}+ T{\cal E}_{\mathrm{q u}}= \frac{4}{B}\left( \rho - \frac{\pi}{2}\right) + \frac{8 \pi}{\beta^{2}}\left( \pi a - \frac{\pi}{2}\right) + \pi = 2 N \pi .
y_{\omega_t}= \frac{1}{|G|}\sum_{g \in G}x\circ g(\{x,y\}) = \frac{t!(20-t)!}{|G|}|\{(u,v) \in C^2 \,\, : \,\, d_H(u,v)=t \}| &= \frac{|\{(u,v) \in C^2 \,\, : \,\, d_H(u,v)=t \}|}{2^{20}\binom{20}{t}}= \frac{|C| a_t}{2^{20}\binom{20}{t}},
2(2a)^{-s}\frac{1}{\Gamma(s)}\sum_{k=1}^{\infty}(-1)^{k}\int_{0}^{\infty}dt\, t^s\frac{e^{-(\frac32+\frac{c}{2a})t}}{1-e^{-t}}\frac{1}{(k\pi)^{2}+(\frac t2)^2}
\|uv\|_{\tilde L^q_t(B^s_{p,r})}\lesssim \|u\|_{\tilde L^{q_1}_t(B^s_{p,r})}\|v\|_{\tilde L^{q_2}(B^s_{p,r})},\, \|f(\rho) \|_{\tilde L^q_t(B^s_{p,r})}\leq C( \|\rho\|_{L^\infty_t(L^\infty)})\|\varrho\|_{\tilde L^q_t(B^s_{p,r})}\hbox{with}\frac 1q=\frac{1}{q_1}+\frac{1}{q_2}.
\mathcal{A}^{-1}=\begin{pmatrix}I_{d\times d}0_{d\times d}-A_{13}A_{11}-I_{d\times d}I_{d\times d}0_{d\times d}-A_{13}A_{11}0_{d\times d}I_{d\times d}A_{11}^T A_{21}0_{d\times d}-I_{d\times d}I_{d\times d}-A_{11}^T -A_{21}\end{pmatrix}.
\frac{d}{ds}A_{\alpha'}[\gamma_s]\Big\vert_{s=0}=\int_a^bd(\alpha+\beta)(\zeta\vert_{\gamma(t)},\dot\gamma(t))dt+\int_a^b d\big((\alpha+\beta)(\zeta\vert_{\gamma(t)})\big) &=(\alpha+\beta)(\zeta\vert_{\gamma(b)})-(\alpha+\beta)(\zeta\vert_{\gamma(a)}),
\lim_{x \rightarrow 0}x \frac{d \hat{G}( x )}{d x}= \lim_{x \rightarrow 0}\frac{1}{\pi^{2}N}\frac{\hat{G}^{3}( x ) x^{3}}{m^{2}\left( x^{2}+ m^{2}\right)}\rightarrow 0 ,
\left\langle \xi ( x ) \xi ( x^{\prime}) \right\rangle \equiv N ( x , x^{\prime}) = \left( \frac 1 8 \right) \left[ \left\langle \left\{\Phi^{2}( x ) , \Phi^{2}( x^{\prime}) \right\}\right\rangle_{0}- 2 < \Phi^{2}>_{0}^{2}\right]
\hat{S}\ [ \hat{X}^{\hat{\mu}}, \gamma ] = -{\textstyle \frac{p}{2}}\int d \xi \sqrt{| \gamma |}\ \gamma^{- 1}p a r t i a l_{\xi}\hat{X}^{\hat{\mu}}\partial_{\xi}\hat{X}^{\hat{\nu}}\hat{g}_{\hat{\mu}\hat{\nu}},
\times \, \exp \left\{- \sum_{- \infty}^{+ \infty}[{\bar{\chi}}_{n}^{( 1 )}( \not \! d + \omega_{n}+ e \varphi ) \chi_{n}^{( 1 )}+{\bar{\chi}}_{n}^{( 2 )}( \not \! d - \omega_{n}- e \varphi ) \chi_{n}^{( 2 )}] \right\}\; .
\begin{aligned}L^{\rho_+}_{-}(\gamma^2 n,n) \geq L^{\rho_+}(\xi_0 n^{2/3},0)+L^{\rho_+}_{(\xi_0 n^{2/3},0)\to (\gamma^2 n,n)}&\geq L^{\rho_+}(\xi_0 n^{2/3},0)+L_{(\xi_0 n^{2/3},0)\to (\gamma^2 n,n)},\end{aligned}
\Lambda ( \alpha , 0 ) = \int \frac{d^{D}k}{( 2 \pi )^{D}}\frac{1}{\left[ k^{2}- m_{0}^{2}\right]^{\alpha}}= \frac{i}{( 4 \pi )^{D / 2}}\left( - m_{0}^{2}\right)^{2 - \alpha}m_{0}^{2 \epsilon}\frac{\Gamma ( \alpha - 2 - \epsilon )}{\Gamma ( \alpha )}
c_3(t) = x_3 + \frac1{2\lambda^2 y}(e^{2\lambda t y}-1) (-\xi + \lambda z) + \frac1{\lambda^2}e^{\lambda t y}\sin(\lambda t \xi) & + \frac1{\lambda}\left(x_1 \, (e^{\lambda t y}\cos(\theta - \lambda t \xi) - \cos\theta) + x_2 \, (e^{\lambda t y}\sin(\theta - \lambda t \xi) - \sin\theta ) \right).
S(m) = \begin{bmatrix}A B^* B C \end{bmatrix}: \mathcal{H}_0 \oplus \mathcal{H}_1 \oplus \mathcal{H}_2 \oplus \cdots \oplus \mathcal{H}_m \rightarrow \mathcal{H}_0 \oplus \mathcal{H}_1 \oplus \mathcal{H}_2 \oplus \cdots\, \oplus\, \mathcal{H}_m.
M\le\frac{1}{4D^2}\left[\sum_{j=1}^k\left(\ell(I_j)R_j^2\right)^{1/3}\right]^{2}\left[\sum_{j=1}^k\left(\ell(I_j)R_j^2\right)^{1/3}\right]=\frac{1}{4D^2}\left[\sum_{j=1}^k\left(\ell(I_j)R_j^2\right)^{1/3}\right]^3.
D ( 1^{n_{1}}2^{n_{2}}\ldots l^{n_{1}}) = D ( \overbrace{j_{1}, \ldots , j_{1}}^{n_{1}}, \overbrace{j_{2}, \ldots , j_{2}}^{n_{2}}, \ldots , \overbrace{j_{l}, \ldots , j_{l}}^{n_{l}}) ,
x_i^*(y_i) = \frac{x_i^*(u_i + x_i - z_i)}{1+\varepsilon}= \frac{Px_i^*(u_i) + (x_i^*-Px_i^*)(x_i) + Px_i^*(x_i-z_i)}{1+\varepsilon}> \frac{(\|Px_i^*\|-\varepsilon)(1+\varepsilon) + (\|x_i^*-Px_i^*\|-\varepsilon)(1+\varepsilon) - \varepsilon}{1+\varepsilon}> \|x_i^*\| - 3\varepsilon=1-3\varepsilon.
\hat{\nabla}_{\hat{\mu}}\hat{F}_{( \hat{d})}{}^{\hat{\mu}\hat{\rho}_{1}\cdots \hat{\rho}_{d}}+{\textstyle \frac{2 \chi}{\sqrt{| \hat{g}|}}}\sum_{n}\mu_{n}{\displaystyle \int}d^{d}\xi_{n}\epsilon^{i_{1}\cdots i_{d}}\partial_{i_{1}}\hat{X}^{\hat{\mu}_{1}}\cdots \partial_{i_{d}}\hat{X}^{\hat{\mu}_{d}}\delta^{\hat{d}}( \hat{x}- \hat{X}_{n}) = 0 \, ,
{\mathcal{F}}( 0 ) \delta_{{\bf{k}}, 0}+ \sum_{n = 1}^{\infty}\frac{{\mathcal{F}}^{( n )}( 0 )}{n !}\mathrm{~}\sum_{\{{\bf{q}}_{i}\}}{\tilde{f}}_{{\bf{q}}_{1}}{\tilde{f}}_{{\bf{q}}_{2}}. . .{\tilde{f}}_{{\bf{q}}_{n}}\delta_{({\bf{k}}- \sum_{i = 1}^{n}{\bf{q}}_{i}) ,{\bf{0}}}
C\left\|\pi\right\|_{p'}\leq \left\|\nabla \pi\right\|_{-1,p'}=\sup_{\varphi\in \boldsymbol W^{1,p}_0(\Sigma,\mathbb{R}^2)}\tfrac{\left|\left(\pi,\nabla \cdot \varphi\right)\right|}{\left\|\nabla \varphi\right\|_p}.
\Re h_1= -C_1/(8x) + \frac{\sqrt{4+C_1^2}( C_2 F_{0}(C_1/2,1/(8x)) + C_3 G_{0}(C_1/2,1/(8x)) )}{8x( C_2 F_{1}(C_1/2,1/(8x)) + C_3 G_{1}(C_1/2,1/(8x)) )}
\alpha_{1,t,C}(x) =\int_0^{tC^2}C\alpha_{1,tC^2-s,1}(0)\rho_{\theta_0^{Cx}}(s)ds\quad \beta_{1,t,C}(x) =\int_0^{tC^2}C\beta_{1,tC^2 -s,1}(0) \rho_{\tau_0^{Cx}}(s)ds
(\kappa_{k-1},\lambda_{k-1})=\max_{\kappa,\lambda}\mathrm{R}\left(f(x)-\left(\begin{array}{ccccccc}c \kappa_0 \kappa_{k-2}\kappa \cline{1-1}\cline{3-3}\cline{5-5}\cline{7-7}\Phi_0(\nu;x) + x+\lambda_0 +\cdots+ x+\lambda_{k-2}+ x+\lambda\end{array}\right)\right).
E_{\exp}^{\left( 1 \right)}= \left. - \frac \mu 2 \frac d{d t}\left( \sum_{n = - \infty}^{\infty}e^{- t \left( \left( \frac{2 n \pi}{L \mu}\right)^{2}+ \left( \frac m \mu \right)^{2}\right)^{\frac 1 2}}\right) \right\rfloor_{t = 0}
\begin{aligned}|r^{(k)}_j|_p \leq\max\left(|\beta_j^k r_j^{(0)}|_p,\max_{1\leq i\leq k}(|d_i\beta_j^{k-i}|_p)\right) &\leq\max\left(|r_j^{(0)}|_p,|\beta|_p\right)<\infty.\end{aligned}
\delta_{i_1 i_2}[\delta_{i_1 i_2}D_{\pi(i_3 \cdots i_n) k}] = \delta_{1 1}D_{\pi(i_3 \cdots i_n) k}+ \delta_{2 2}D_{\pi(i_3 \cdots i_n) k}+ \delta_{3 3}D_{\pi(i_3 \cdots i_n) k}= 3D_{\pi(i_3 \cdots i_n) k}.
L^* L=\begin{bmatrix}L_1^* L_1 L_1^*L_2 L_1^*L_{2T-1}L_2^*L_1 L_1^*L_1 L_1^*L_{2T-2}& L_{2T-1}^*L_1 L_{2T-2}^*L_1 L_1^*L_1\end{bmatrix}.
\frac{T_{p + 2}}{\kappa}\int_{\mathcal{M}_{p + 3}}C_{p + 3}= \frac{T_{p + 2}}{\kappa}\int_{\mathcal{M}_{p + 3}}A_{p + 1}\wedge \omega_{2}= \frac{T_{p}}{2 \kappa}\int_{\mathcal{M}_{p + 1}}\frac{A_{p + 1}}{2 \pi^{2}\alpha^{\prime}},
S\left( \begin{array}{cc}a d b c \end{array}\right)(\theta_{12})( \pi_{a,b}(\theta_1) \otimes \pi_{b,c}(\theta_2) )\Delta (g) =( \pi_{a,d}(\theta_2) \otimes \pi_{d,c}(\theta_1) )\Delta (g)S\left( \begin{array}{cc}a d b c \end{array}\right)(\theta_{12})
\max_{v \in \{v_x,v_y\}}|vf(x)-1- 3 k^{-1/2}\log^{1/2}n| \lesssim a(f(x) \wedge f(y)) \Bigl( \frac{k}{nf(x)}\Bigr)^{\beta/d}= o(k^{-1/2}),
A\tilde{w}_r(x) = \frac{c_2}{C_3}AV(\Psi)(x) + A\eta_r(x) \ge - \frac{c_2}{C_3}|LV(\Psi)(x)| + A\eta_r(x) \ge -\frac{c_2}{V(4r)}+ \frac{c_2}{V(4r)}= 0
\bigg(1 - \frac{1}{p^{\alpha + 1}}\bigg)^{-k}\bigg(1 - \frac{k - 1}{p}- \frac{k}{p^{\alpha + 1}}\bigg) = 1 - \frac{k - 1}{p}+ O_k\bigg(\frac{1}{p^{\alpha + 2}}\bigg);
[m_1, m_2] &= m_1\circ m_2 +m_2\circ m_1 &= m_1(x_1, m_2(x_2, x_3)) + m_2(x_1, m_1(x_2, x_3)) + m_1(m_2(x_1, x_3), x_2) + m_2(m_1(x_1, x_3), x_2) &- m_1(m_2(x_1, x_2), x_3) - m_2(m_1(x_1, x_2), x_3).
\sum_{k=0}^{\infty}\frac{q^k(1+q^{2k+1})}{(1-q^{2k+1})^{2}}= \frac{(q^{2}; q^{2})_{\infty}^{4}}{(q;q^{2})_{\infty}^{4}}, \sum_{k=0}^\infty\dfrac{q^k(1+q^{2k+1})}{(1-q^{2k+1})^2}=\dfrac{(q^2;q^2)_\infty^4}{(q;q^2)_\infty^4}.
\big|a_1(T){\mathrm{e}}^{\omega_1/T}+\cdots+a_n(T){\mathrm{e}}^{\omega_n/T}\big|_\infty \prod_{i=2}^n |a_i(T)|_\infty \ge C(n)^{-1}, |a_1(T)|_\infty \prod_{i=2}^n \big|a_1(T){\mathrm{e}}^{\omega_i/T}-a_i(T){\mathrm{e}}^{\omega_1/T}\big|_\infty \ge C(n)^{-(n-1)}.
|x_1 +\dots + x_n|^2 =(x_1 +\dots + x_n)^*(x_1 +\dots + x_n)=x_1^*x_1+x_2^*x_2+\dots+x_n^*x_n &=|x_1|^2+|x_2|^2+\dots+|x_n|^2.
\sigma_n(x',x_3) = \big(\left( \sigma \circ d_n \right)(\cdot,x_3) \ast \rho_{\epsilon_n}\big)(x'), \bar{\sigma}_n(x') = \big(\left( \bar{\sigma}\circ d_n \right) \ast \rho_{\epsilon_n}\big)(x'), \hat{\sigma}_n(x') = \big(\left( \hat{\sigma}\circ d_n \right) \ast \rho_{\epsilon_n}\big)(x').
\prod_{\mathit{v\in S}}\prod_{j=1}^{d}|L_{\mathit{v},j}(\mathbf{X})|_\mathit{v}\leq \prod_{v\in M_\infty}\prod_{j=1}^{d}|q|_\mathit{v}=\prod_{j=1}^{d}|q|^{\sum_{\mathit{v}\in M_\infty}d(\rho_\mathit{v})/[K:\mathbb{Q}]}=|q|^d.
f(x)= \widetilde V\left(x,\lambda_{t+1}, B_{k+1}\backslash \{\lambda_{t+1}\},J_k+tT_{k+1},J_k+(t+1)T_{k+1}, tT_{k+1},\frac{w_{\lambda_{t+1}}^\prime(J_k+tT_{k+1})}{w_{\lambda_{t+1}}(J_k+tT_{k+1})}\right),
\int_0^\infty\sigma_r(A\times B)^2 \,r^{2d-1}\,dr \lesssim \int_0^\rho r^{2d-1}\,dr+ \int_\rho^\infty (r^{-d}|A|)^{1/2}\sigma_r(A\times B)\,r^{2d-1}\,dr \lesssim \rho^{2d}+ \rho^{-d/2}|A|^{1/2}\int_0^\infty \sigma_r(A\times B)\,r^{2d-1}\,dr \lesssim \rho^{2d}+ \rho^{-d/2}|A|^{1/2}\cdot|A|\cdot|B| = 2|A|^{6/5}|B|^{4/5}.
\nabla\cdot\vec{E}=\frac{\rho}{\epsilon_{0}}\nabla\times\vec{B}-\epsilon_{0}\mu_{0}\frac{\partial\vec{E}}{\partial t}=\mu_{0}\vec{J}\nabla\times\vec{E}+\frac{\partial\vec{B}}{\partial t}=0\nabla\cdot\vec{B}=0
\Big(\bar{x}^{(1)}_\nu(\theta_t\omega)e^{O_t^{(1)}(\omega)},\bar{x}^{(2)}_\nu(\theta_t\omega)e^{O_t^{(2)}(\omega)},\dots,\bar{x}^{(N)}_\nu(\theta_t\omega)e^{O_t^{(N)}(\omega)}\Big)^{\top},
i\sqrt{2}\partial^{x}_{+}G(x;y)+g^{2}\int dw^{-}G(x^{+},x^{-};x^{+},w^{-})|x^{-}-w^{-}|G(x^{+},w^{-};y^{+},y^{-})=i\delta(x^{-}-y^{-}).
\log\|s\| = d_1 \log|u| + d_2 \log|v| + d_3 \frac{\log|u|\log|v|}{\log|t_1|}+ \log|\nu|+\sigma + \frac{1}{\log|t_1|}\gamma_1+ \frac{\log|u|}{\log|t_1|}\gamma_2 + \left(\frac{\log|u|}{\log|t_1|}\right)^2 \gamma_3
g_{11}=\langle X_1(s),X_1(s)\rangle =2(1+sf)^2+4s^2\big(\textstyle\frac{\partial f}{\partial\theta_1}\big)^2, g_{12}=\langle X_1(s),X_2(s)\rangle=4s^2\textstyle\frac{\partial f}{\partial\theta_1}\frac{\partial f}{\partial \theta_2}, g_{22}=\langle X_2(s),X_2(s)\rangle =2(1+sf)^2+4s^2\big(\textstyle\frac{\partial f}{\partial\theta_2}\big)^2.
A = \left( \begin{array}{c c c c c c c}{{1}}&{{0}}&{{1}}&{{0}}&{{1}}&{{0}}&{{1}}\\{{1}}&{{1}}&{{0}}&{{0}}&{{0}}&{{1}}&{{1}}\\{{0}}&{{1}}&{{1}}&{{1}}&{{0}}&{{0}}&{{1}}\\ \end{array}\right) ,
[2]P_0 = (x_1(t), y_1(t)) = \left (\frac 1{64}t^2 - \frac{41}2t + 315, -\frac{55}{512}t^2 + \frac{2637}{8}t - 5670\right ) P_1\dot{-}P_2 = (x_2(t), y_2(t)) = \left (t^2 + 192t + 8640, - t^3 -301t^2 - 27936t - 803520 \right )
\left|\frac{1}{m+1}\sum_{i=0}^mw_{n+i}-w\right| <C\left|\frac{1}{m+1}\sum_{i=0}^mx_{n+i}-x+\frac{1}{m+1}\sum_{i=0}^my_{n+i}-y\right. &+\left.\frac{1}{m+1}\sum_{i=0}^mz_{n+i}-z\right|,
B^{\mu\nu}(s)=-\frac{1}{32\pi^{2}}\Lambda^{s/4\pi}(g^{\mu\nu}-\frac{q^{\mu}q^{\nu}}{q^{2}})\int_{0}^{1}dx\int_{0}^{\Lambda}dm\; m^{-1-s/4\pi}\ln{(1+\frac{x(1-x)q^{2}}{m})}
\int_{r_{1}}^{r_{2}}dr \sqrt{2\mu \left( E - \tilde{V}(r) -\frac{\eta+\frac{1}{4}}{2\mu r^{2}}\right)}= \left( n +\frac{1}{2}\right) \pi. \;\;\;\;\; ( n = 0, 1, 2, \cdots )
F_{q}^{(k)}(s;q^{\prime}) = \displaystyle \sum_{n=0}^{\infty}\frac{\left(-\frac{1}{1-q^{\prime}}\right)_{n}}{Q_{2n+1}(2-q)}\frac{((1-q^{\prime}) \alpha)^{n}}{n!}(-1)^{k}\frac{\Gamma(k+2n+1)}{s^{2n+k+1}}.
\nu_{d,d}\left( \mathbb U^c(\Lambda)\right) \quad\le[\nu_{d,p}(\mathbb G^c)]^{-1/2}\int_{\mathcal O_d}\nu_{d,p}(\mathbb J^c(U\Lambda U'))\;\nu_{d,d}(dU) &\quad=[\nu_{d,p}(\mathbb G^c)]^{-1/2}\int_{\mathcal O_d\times \mathcal O_p}\nu_{d,p}(\mathbb J^c(U\Lambda U'))\;(\nu_{d,d}\otimes \nu_{p,p}) (dU,dV) &\quad=[\nu_{d,p}(\mathbb G^c)]^{1/2},
\begin{array}{l}{\frac{M}{A_{8 - p}}= \frac{8 - p}{2}r_{H}^{7 - p}+ \frac{7 - p}{2}\alpha ,}\\{\left( \frac{Q_{0}}{( 7 - p ) A_{8 - p}}\right)^{2}= \alpha \left( \alpha + r_{H}^{7 - p}\right) ,}\\ \end{array}
\frac{n^{4d}}{m^4}\sum_{l = 2}^m \mathbb{E}_0\left[(D^S_{nl})^4 \right], \frac{n^{4d}}{m^4}\sum_{l = 2}^m \mathbb{E}_0\left[(D^T_{nl})^4 \right], \frac{n^{2d}}{m^4}\sum_{l = 2}^m \mathbb{E}_0\left[(D^Z_{nl})^4 \middle| \mathcal{F}_{n, l- 1}\right] \longrightarrow 0.
y^\delta \tilde{\Gamma}^\alpha{}_{\beta \delta}\tilde{B}^\beta|_{1+3}=y^\delta dx^\sigma [\Gamma^\alpha{}_{\beta \delta}B^\beta{}_{,\sigma}+\Gamma^\alpha{}_{\beta \delta}\Gamma^\beta{}_{\sigma \tau}B^\tau +\Gamma^\alpha{}_{\beta \delta ,\sigma}B^\beta ].
F^{2}= E_{j}\rho ( E_{i}^{( n )}) - \frac{\sin ( n - 1 ) \alpha}{2 \cos \alpha \sin \alpha}\rho^{2}( E_{i}^{( n - 1 )}) E_{j}\rho ( E_{i}^{( n )}) = \frac{\sin ( n + 1 ) \alpha}{2 \cos \alpha \sin n \alpha}F
\frac{\partial}{\partial x_{\mu}}( \left| x - a \right| + \left| x - b \right| + \left| x - c \right| ) = \frac{( x - a )_{\mu}}{\left| x - a \right|}+ \frac{( x - b )_{\mu}}{\left| x - b \right|}+ \frac{( x - c )_{\mu}}{\left| x - c \right|}
R(x_1, x_2) x_2 = - (1 + (3/4) \lambda^2) x_1, \ R(x_1, x_n) x_n = (1/4) \lambda^2 x_1, R(x_2, x_1) x_1 = - (1 + (3/4) \lambda^2) x_2, \ R(x_2, x_n) x_n = (1/4) \lambda^2 x_2, R(x_n, x_1) x_1 = (1/4) \lambda^2 x_n, \ R(x_n, x_2) x_2 = (1/4) \lambda^2 x_n.
f(5^2+\underbrace{1^2+\dotsb+1^2}_{5}+ \underbrace{1^2+\dotsb+1^2}_{k-6}) = f(4^2 + a_1^2 + \dotsb + a_5^2 + \underbrace{1^2+\dotsb+1^2}_{k-6})
\frac{\partial}{\partial\eta_i}~ f_{n_1, \cdots, n_i, \cdots, n_r}(\eta_1,\cdots, \eta_i, \cdots, \eta_r) = \sqrt{n_i(k+1-(n_1+\cdots+n_i+\cdots+n_r))}f_{n_1, \cdots, n_i-1, \cdots, n_r}(\eta_1,\cdots, \eta_i, \cdots. \eta_r)
A_0 \left( \overrightarrow{\eta}( \overrightarrow{X},t), t \right) = \hat{A}_0 ( \overrightarrow{X},t)- \frac{e \theta}{2(1+e\theta \widehat{B})}\widehat{A}_{l}( \overrightarrow{X},t)\varepsilon_{kj}\partial_t \widehat{A}_j ( \overrightarrow{X},t)e_{kl}( \overrightarrow{X},t)\, ,
Q_{\beta}= \vec{\Theta}_{\alpha}\cdot \bigl ( - \i \gamma_{\alpha \beta}^{t}\vec{\nabla}_{t}+ \frac{1}{2}\vec{q}_{s}\times \vec{q}_{t}\gamma_{\beta \alpha}^{s t}\bigr ) , \qquad ( \beta = 1 , \ldots , s_{d}) , \nonumber
\frac{e_{t+1}(L_{t}\cup M \cup (V(e)\setminus \{w\}), F_{t+1})}{|L_{t}\cup M \cup V(e) \setminus\{w\}|}= \frac{e_{t+1}(L_{t}\cup M, F_{t+1})+\binom{m-1}{2}+2(m-1)}{|L_{t}\cup M|+m-1}& = \frac{e_{t+1}(L_{t}\cup M, F_{t+1})}{|L_{t}\cup M|}\cdot \frac{|L_{t}\cup M|}{|L_{t}\cup M|+m-1}& + \frac{m+2}{2}\frac{m-1}{|L_{t}\cup M| + m-1}.
R^{1}_{2}= - 2i \frac{dg_{1}\wedge d\overline{g}_{1}}{(1- g_{1}\overline{g}_{1})^{2}},~~~R^{3}_{4}=+2i \frac{dg_{2}\wedge d\overline{g}_{2}}{(1+ g_{2}\overline{g}_{2})^{2}},
g_{ij}=\begin{pmatrix}a_3(u^2)^2+a_2(u^3)^2 -a_3 u^1u^2+\alpha u^3 -a_2 u^1 u^3+\alpha u^2 -a_3 u^1u^2+\alpha u^3 a_2+a_3(u^1)^2 -2\alpha u^1 -a_2 u^1 u^3+\alpha u^2 -2\alpha u^1 a_3+a_2(u^1)^2\end{pmatrix}, w_{12}=\sqrt{\frac{(a_2^2-\alpha^2)(a_3^2-\alpha^2)}{\det g}}u^3,w_{23}=0,w_{31}=\sqrt{\frac{(a_2^2-\alpha^2)(a_3^2-\alpha^2)}{\det g}}u^2,
\Gamma =(\alpha'(\rho/\alpha)^{2/3},(\alpha/\rho)^{1/3},(\alpha/\rho)^{1/3}) \, , \Gamma^0 =(\alpha_0'(\rho/\alpha_0)^{2/3},(\alpha_0/\rho)^{1/3},(\alpha_0/\rho)^{1/3}).
\begin{cases}\frac{|\Lambda_i+ (\widetilde{w}_i (x,y_i) )_{y_i}|^2}{2}+V_i(x,y_i )=\ln \widetilde{m}_i (x,y_i )+\widetilde{H}_i (x,\Lambda_i), \big(\widetilde{m}_i (x,y_i) (\Lambda_i+ (\widetilde{w}_i (x,y_i))_{y_i})\big )_{y_i}=0, \int_0^1 \widetilde{m}_i (x,y_i)dy=1.\end{cases}
\quad\left((2n+3)^{2}-(2n+5)^{2}\right)\frac{2(n+2)!^{2}(2n+3)}{4^{2n+5}(4n+9)!!}\binom{4n+8}{2n+4}\binom{2}{1}=\frac{(2n+4)^{2}((2n+3)^{2}-(2n+4)^{2})}{4(4n+7)(4n+9)}\frac{2(n+1)!^{2}2n+3}{4^{2n+3}(4n+5)!!}\binom{4n+6}{2n+3}\binom{0}{0},
\int^{i\infty}_{-i\infty}dz \prod_{i=1}^3 s_b(\tfrac{iQ}{2}+a_i+z) s_b(\tfrac{iQ}{2}+b_i-z)= \prod_{i,j=1}^3s_b(\tfrac{iQ}{2}+a_i+b_j) \; ,
1+b_1+\cdots b_{\lfloor\frac{k-1}{2}\rfloor}=\left(1+\left(a_1-f_1\right)+\cdots+\left(a_{\lfloor\frac{m+k-1}{2}\rfloor}-f_{\lfloor\frac{m+k-1}{2}\rfloor}\right) \right)\left(1+{p}_1+\cdots\right)^{-1}
\int_{\tilde\Gamma}2\mbox{Re}((\nu_1+i\nu_2){(i\Psi_0)\Phi'(\mathcal P_1{\mbox{\bf a}},\mathcal P_2{\mbox{\bf b}})})+2\mbox{Re}((\nu_1-i\nu_2){}{\bar\Phi'(\mathcal C_1\overline{(-i \Psi_0)\mbox{\bf a}},\mathcal C_2\bar{\mbox{\bf b}})})d\sigma=0.
\liminf_{\epsilon\to0}\lambda^2(\epsilon)\log\mathbb P\big(X^{\epsilon}\in B(\varphi,\gamma)\big) \geq \lim_{\delta\to0}\liminf_{\epsilon\to0}\lambda^2(\epsilon)\log\mathbb P\Big(\hat{X}^{\epsilon}\in B(\varphi,\delta)\Big) & \geq -\mathcal{I}^{\varphi}(\varphi)=-\mathcal{I}(\varphi),
\mathcal{G}_{D,0}(z,s) := -2^{s-2}\pi^{-\frac{s}{2}-1}|D|^{\frac{s}{2}}L_D(s) \Gamma\biggl(\frac{s}{2}+1\biggr) P_{\frac{3}{2}, 0}\biggl(z, \frac{s}{2}+ \frac{1}{4}\biggr).
0=\int(g_{\mu\nu}\frac{d^{2}x^{\nu}}{d\tau^{2}}+\frac{1}{2}\frac{dx^{\alpha}}{d\tau}\frac{dx^{\nu}}{d\tau}(\partial_{\alpha}g_{\mu\nu}+\partial_{\nu}g_{\mu\alpha}-\partial_{\mu}g_{\alpha\nu}))\delta x^{\mu}d\tau
[\begin{matrix}0&0&0&1\\ 0&0&1&0\\ 0&1&0&0\\ 1&0&0&0\end{matrix}]^{*}=[\begin{matrix}0&0&0&1\\ 0&0&1&0\\ 0&1&0&0\\ 1&0&0&0\end{matrix}]
\int dx \phi^n_\star=\int dp_1 dp_2 \ldots dp_n \exp(-\frac{i}{2}\sum_{i<j}(p_i)_\mu\theta^{\mu \nu}(p_j)_\nu) \phi(p_1)\phi(p_2)\ldots\phi(p_n) \delta(p_1+p_2+\ldots+p_n)
c_{1}+d_{1}(1-|v|^2)=0, &{-c_{0}v{\bf i}}+c_{1}\frac{v(1-|v|^2)}{2|v|^2}-2d_{1}v=0, &{c_{0}\overline{v}{\bf i}}+c_{1}\frac{\overline{v}(1-|v|^2)}{2|v|^2}-2d_{1}\overline{v}=0.
|H\cap W'|q^{d^2-d}\theta_{d-1}\le\begin{cases}|W'\cap \tau_i|q^{d^2-d}\theta_{d-1}H\cap U=\tau_ii\in\{1,2\}, (q^{d-1}+\theta_{d-1})q^{d^2-d}\theta_{d-1}\end{cases}