harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\frac{1 6 \, \pi^{2}}{g_{i , \, \mathrm{b a r e}}^{2}}= \frac{1 6 \, \pi^{2}}{g_{i}^{2}( \mu )}- b_{i}\, ( 4 \pi )^{\epsilon}\int_{0}^{\infty}\frac{d t}{t^{1 - \epsilon}}\, e^{- t \frac{\mu^{2}}{M^{2}}}\, .
\begin{array}{r c l r c l}{{\chi_{1 2}}}&{{=}}&{{\chi_{3 4}}}&{{\equiv}}&{{\chi_{s},}}\\{{\chi_{1 4}}}&{{=}}&{{\chi_{2 3}}}&{{\equiv}}&{{\chi_{t},}}\\{{\chi_{1 3}}}&{{=}}&{{\chi_{2 4}}}&{{\equiv}}&{{\chi_{u}.}}\\ \end{array}
e^{t\Delta}\phi (x)= \bigg(\int_{|y| \le 1}\!\! +\int_{|y|\geq 1}\!\!\bigg) \frac{\exp\Big(-\frac{|x-y|^2}{4t}\Big)}{\big(4\pi t \big)^{n/2}}\phi(y) dy =: I_\alpha + I_\beta.
\langle x^{\prime \prime}, t^{\prime \prime}, \tau^{\prime \prime}| p_{x}^{\prime}, p_{t}^{\prime}, \tau^{\prime}\rangle = \exp \{i p_{x}^{\prime}x^{\prime \prime}+ i p_{t}^{\prime}t^{\prime \prime}\}/ [ ({p_{x}}^{\ 2}/ 2 m ) +{p_{t}}^{\prime}] ,
\frac1{\delta_1}+\frac1{\delta_2}+\frac1{\delta_3}+\frac1{\delta_4}=\delta\alpha, \ \ 0<\delta_1,\delta_2,\delta_3,\delta_4<1 \frac1{\theta_1}+\frac1{\theta_2}+\frac1{\theta_3}+\frac1{\theta_4}=1, \ \ \theta_1=\theta_3=\frac2{\delta}, \ \ 1<\theta_2, \theta_4<\infty.
V_{i}( k_{i}, \epsilon_{i}; z_{i}, \bar{z}_{i}) = ( \epsilon_{i}\cdot \partial X ( z_{i}) + i k_{i}\cdot \psi ( z_{i}) \epsilon_{i}\cdot \psi ( z_{i}) ) \, \mathrm{e}^{i k_{i}\cdot X ( z_{i}, \bar{z}_{i})}.
\frac{d^2}{ds^2}\Big\lvert_{s=0}g_H(T_*N,N) =-\frac{d}{ds}\Big\lvert_{s=0}\Bigl(g_H(T_*N,N^\perp)\bigl(-\kappa_\eta+g_H(T_*N,N)\kappa_1\bigr)\Bigr) &=\bigl(\kappa_\eta-g_H(T_*N,N)\kappa_1\bigr)\frac{d}{ds}\Big\lvert_{s=0}g_H(T_*N,N^\perp),
\displaystyle{\lim_{\tau \downarrow 0}}\, \displaystyle{\frac{f(\bar{x}+ \tau v) - f(\bar{x}) - \tau \, \nabla f(\bar{x})^Tv - \displaystyle{\frac{\tau^2}{2}}\, v^TF^{\, \prime}(\bar{x};v)}{\tau^2}}\, = \, 0,
\frac{a'_o}{a_o}\left( \frac{a'_o}{a_o}+\frac{n'_o}{n_o}\right)-\frac{b^2_o}{n^2_o}\left(\frac{\dot a_o}{a_o}\left( \frac{\dot a_o}{a_o}-\frac{\dot n_o}{n_o}\right)+\frac{\ddot a_o}{a_o}\right)-k\frac{b^2_o}{a^2_o}=-\frac{1}{6M^3}\Lambda b^2_o + \frac{1}{6M^3}T_{55},
K(L,r)=\int_{\frac{1-r^2}{1+r^2}}^1 \frac{s^{2L-n}}{1-s^{2L}}\int_0^{\alpha(s,r)}\left(2\cos\theta-2\cos \alpha(s,r)\right)^{n-2}\bigl(s+\frac 1s-2\cos\theta\bigr) d\theta\; ds
\mathbb{T}(z) := M(z) z \in \, , \mathbb{T}(z) := M(z) v_u(z) =M(z) \begin{pmatrix}1 0 e^{-iN \xi(z)}1 \end{pmatrix}z \in_u \, , \mathbb{T}(z) := M(z) v_l^{-1}(z) =M(z) \begin{pmatrix}1 0 e^{iN \xi(z)}1 \end{pmatrix}^{-1}z \in_l \, .
\gamma_{M}^{5}=i(\gamma^{0}\gamma^{1}\gamma^{2}\gamma^{3})_{M}=\frac{1}{i^{2}}(\gamma^{4}\gamma^{1}\gamma^{2}\gamma^{3})_{E}=(\gamma^{1}\gamma^{2}\gamma^{3}\gamma^{4})_{E}=\gamma_{E}^{5}.
\sum_{f \in \mathcal{F}_k(\mathcal{A},\mathcal{B})}m(f) L^{n(f)}= N^{|\mathcal{A}| + |\mathcal{B}|-k-1}L^k \sum_{S \subseteq \mathcal{A}\cup \mathcal{B}: \mathcal{B}\subseteq S}\left(\sum_{V \in S}|V| \right)
\lim_{n\to\infty}F_{s_{\tilde S^*_{0,\lfloor nt_1\rfloor}},s_{\tilde S^*_{\lfloor nt_1\rfloor+1,\lfloor nt_2\rfloor}}}(u_1,u_2)=\lim_{n\to\infty}F_{s_{\tilde S^*_{0,\lfloor nt_1\rfloor}}}(u_1)\lim_{n\to\infty}F_{s_{\tilde S^*_{{\lfloor nt_1\rfloor}+1,\lfloor nt_2\rfloor}}}(u_2)
\tilde{\phi}(T_{G,H}(x))=\tilde{\phi}(\prod_{i=1}^{n}g_{\pi_{x}(i)}^{-1}xg_{i}\cdot H^{\prime})=\prod_{i=1}^{n}\phi(g_{\pi_{x}(i)})^{-1}\phi(x)\phi(g_{i})\cdot\phi(H^{\prime}).
R_3' = \int_{\mathcal{X}_n}f(x) \int_0^\frac{a_n}{n-1}\log \biggl(\frac{V_d f(x) h_x^{-1}(s)^d}{s}\biggr) \mathrm{B}_{k,n-k}(s) \, ds \,dx &= O \biggl( \max \biggl\{\frac{k^{\frac{\alpha}{\alpha+d}- \epsilon}}{n^{\frac{\alpha}{\alpha+d}- \epsilon}}\, , \, \frac{k^{\beta/d}}{n^{\beta/d}}\biggr\}\biggr)
\frac{\psi^{\prime\prime}(r)}{\psi(r)}+\frac{(D-1)}{r}\frac{\psi^{\prime}(r)}{\psi(r)}=\frac{(Cg+D_1)g^{\prime2}}{1-g^2}+\frac{(Eg+F)g^{\prime2}}{(1-g^2)^2}+\frac{Gg^{\prime2}}{(\beta-\alpha)g-(\beta+\alpha)}+\frac{Kg^{\prime2}}{[(\beta-\alpha)g-(\beta+\alpha)]^2}-\frac{1}{2}\{g,r\}-\frac{(D-1)(D-3)}{4r^2}
S_{\mathrm{b r a n e}}= - T \int d^{4}x \sqrt{- g}e^{4 A}\left[ 1 + \frac{1}{2}e^{2 B - 2 A}g^{a b}\partial_{a}R \partial_{b}R \right] - q T \int d^{4}x \sqrt{- g}e^{\Lambda}.
E_q={1 \over 2\pi g^{1/2}(L)}\partial_{\bar\beta}\Gamma |_{\bar{\beta}=\beta_H}={1 \over 96\pi}\int_{x_+}^{L}{1 \over g}({4 \over \beta^2_H}-g'^2(x))dx-{1 \over 16\pi g^{1/2}(L)}g'(L) .
\tilde \pi_{12}^* \nabla_{\theta,\sigma}+ \tilde \pi_{23}^* \phi - \tilde m^* \tilde \pi_{13}^* \phi = (\tilde \pi_{23}^* \rho^* \pi_2^* - \tilde m^* \tilde \pi_{13}^*\rho^* \pi_2^*) \sigma( A)
V_{x_{{}_D}}\,\, U^{L, x_{{}_D}}\,\,{V_{x_{{}_D}}}^{-1}= \,\,{}^{\mu^{x_{{}_D}}}U^{L^{x_{{}_D}}}\,\,\, \textrm{and}\,\,\, V_{x_{{}_D}}\,\, \mathfrak{J}^{L,x_{{}_D}}\,\,{V_{x_{{}_D}}}^{-1}= \mathfrak{J}_{x_{{}_D}},
\theta ( g_{2}, g_{3})^{\alpha}\: + \: \theta ( g_{1}, g_{2}g_{3})^{\alpha}\: = \: g_{3}^{*}\theta ( g_{1}, g_{2})^{\alpha}\: + \: \theta ( g_{1}g_{2}, g_{3})^{\alpha}\: - \: d \log \omega_{\alpha}^{g_{1}, g_{2}, g_{3}}
\| E_{S}(\sum^k_{i,j=1}(a_i a_j^*) \otimes (b_i\sigma_g(x) b^*_j))\|_2 =\|\sum^k_{i,j=1}(a_i a^*_j) \otimes E_{S}(b_i \sigma_g (x) b^*_j)\|_2 =\|\sum^k_{i,j=1}\tau( \sigma_g (x) b^*_jb_i) a_i a^*_j \|_2 \leq\sum^k_{i,j=1}|\tau( x \sigma_{g^{-1}}(b^*_jb_i))| \| a_i a^*_j \|_2.
\tilde{H}{}_{a}^{0 f}= \frac{1}{2}\left[ - \frac{d^{2}}{d x^{2}}+ W_{- , \nu}^{2}- \nu - 3 - \frac{1}{2}( W_{- , \nu}^{\prime}- \nu + 1 ) \right] - \frac{1}{4}( W_{- , \nu}^{\prime}- \nu + 1 ) \cdot \sigma_{i}\otimes \sigma_{i}
\sum(-1)^{f_\sigma}\cdots E_{ij}E_{jk}\cdots-\sum (-1)^{f_{\sigma'}}\cdots E_{jk}E_{ij}\cdots = \sum (-1)^{\bar j +f_{\tilde\sigma}}\cdots E_{ik}\cdots-\sum(-1)^{f_{\tilde\sigma'}}\cdots E_{jj}\cdots = (m-n)\sum (-1)^{f_{\tilde\sigma}}\cdots E_{ik}\cdots-C_1\sum(-1)^{f_{\tilde\sigma'}}\cdots 1\cdots,
P^T\mathcal{L}_1(\lambda)P=\begin{bmatrix}\lambda P_7+P_6 -I_n 0 0 0 0 0 I_n 0 -\lambda I_n 0 0 0 0 0 \lambda I_n \lambda P_5+P_4 -I_n 0 0 0 0 0 I_n 0 -\lambda I_n 0 0 0 0 0 \lambda I_n \lambda P_3+P_2 -I_n 0 0 0 0 0 I_n 0 -\lambda I_n 0 0 0 0 0 \lambda I_n \lambda P_1+P_0\end{bmatrix},
D_Q\cap B_{\lambda^k}(x^k)\subset \bigcup_{j=1}^{J(k)}B_{\lambda^{k+1}}(x^{k+1}_j)\subset B_{2\lambda^k}(x^k_1) B_{\lambda^{k+1}}(x^{k+1}_j)\cap B_{\lambda^{k+1}}(x^{k+1}_i)=\emptyset\,j\neq i\,.
[w]_{A_{p,\mathcal{B}}}:=\sup_{B\in\mathcal{B}}\left(\frac{1}{\mu(B)}\int_B w\,d\mu \right)\left(\frac{1}{\mu(B)}\int_B w^{1-p'}\,d\mu \right)^{p-1},1<p<\infty.
\begin{gathered}\gamma_{-}(w_{1}) v_{1}\wedge v_{0}\wedge v_{-1}= q^{2}v_{0}\wedge v_{-1}, \gamma_{-}(w_{1}) v_{1}\wedge v_{0}\wedge v_{-1}= -q^{2}v_{1}\wedge v_{-1}, \gamma_{-}(w_{1}) v_{1}\wedge v_{0}\wedge v_{-1}= q^{4}v_{1}\wedge v_{0}.\end{gathered}
\frac{\log_p \lvert K P_i(G) \cap Z : P_i(G) \cap Z \rvert}{\log_p \lvert Z : P_i(G) \cap Z \rvert}= \frac{\log_p \lvert K \varrho_k \cap Z \varrho_k \rvert}{\log_p \lvert Z \varrho_k \rvert}.
\int_{X_{0}}\mathbb{E}_{\lambda}\left[\tilde{f}(g,\cdot)|\mathcal{F}^{s}\right]d\lambda=\int_{X_{0}}\tilde{f}\left(g,x_{0}\right)d\lambda(x_{0})=\int_{X}f\left(g.x_{0}\right)d\lambda(x_{0})=\int_{X}f\left(x\right)dg.\lambda(x).
V_{eff}= \int dx \Biggl( \frac{1}{2}\pi_c^2 + \frac{m^2}{2}\phi_c^2 \Biggr) +\frac{\hbar}{2}\sum_{n = 1}^{\infty}\Biggl(m^2 + \Bigl(\frac{n\pi}{L}\Bigr)^2 \Biggr)^{1/2}.
P_{0}= \varepsilon^{0 , 0}, \hspace{0 . 3 i n}\overline{{{P}}}= \varepsilon^{\mu , \mu}, \hspace{0 . 3 i n}\overline{{{\overline{{{P}}}}}}= \frac 1 2 \varepsilon^{[ \mu \nu ] , [ \mu \nu ]},
\alpha_P(x):=\begin{cases}x^p \mbox{if}P=\{1\}x^p(\prod_{k=0}^{p-1}\omega_P^k(x))^{-1}\mbox{if}P\neq\{1\}\mbox{and cyclic}x^p \mbox{if}P \mbox{is not cyclic}. \end{cases}
F_{2}\left( 1 , - m +{\frac{3}{2}}, i x \right) = \Gamma \left( m -{\frac{1}{2}}\right)^{- 1}\int_{- \infty}^{\infty}d \tau \, e^{- \tau^{2}}{\frac{\tau^{2 m}}{i x + \tau^{2}}}.
\sigma ( \omega ) = \lambda_{0}+ \sum_{i = 1}^{N}( \alpha_{i}- 1 ) \left\{\log \left| \frac{\vartheta_{1}( \omega - \omega_{i}| \tau )}{\eta ( \tau )}\right| - \frac{\pi}{\tau_{2}}( \omega_{y}- \omega_{i , y})^{2}\right\}.
X_{0}= \sum_{i,j=1}^{2}\delta_{ij}\frac{\partial}{\partial \Gamma_{ij}},\X_{1}= \sum_{i,j=1}^{2}( (B_{0}^{o})_{ij}- \Gamma_{ij}+ (B_{\infty})_{ik}\Gamma_{kj}- \Gamma_{ik}(B_{\infty})_{kj})\frac{\partial}{\partial \Gamma_{ij}}.
g_M(\nabla_{a'}\nabla_{10'}e_{10}',e_{d}')\eta^{ad}=e^{2\phi/3}\left(\frac{16}{9}|d\phi|^2-\frac{2}{3}\Delta\phi\right) g_M(\nabla_{10'}\nabla_{a'}e_{10}',e'_{d})\eta^{ad}=-\frac{1}{2}e^{8\phi/3}|G_2|^2 g_M(\nabla_{[a',10']}e_{10}',e_{d}')\eta^{ad}=\frac{4}{9}e^{2\phi/3}|d\phi|^2
\langle\underline t,\partial_{\underline x}\rangle M =\sum_{j=1}^pt_j\partial_{x_j}M=\sum_{j=1}^pt_j(\partial_{\theta}M)(\partial_{x_j}\theta)=\vert\underline t\vert^2\partial_{\theta}M=\partial_{\theta}M \partial_{\underline y}M =\sum_{j=1}^qe_{p+j}\partial_{x_{p+j}}M=\sum_{j=1}^qe_{p+j}(\partial_{\rho}M)(\partial_{x_{p+j}}\rho)=\underline\nu(\partial_{\rho}M).
I_1={\rm exp}\left\{\frac{1}{2\pi i}{\rm p.v.}\int_{-\infty}^{\infty}\frac{{\rm log}\left(1-b^2(s)\right)}{s}ds\right\}, I_2=\exp\left\{\frac{1}{2}{\rm log}\left(1-b^2\left(0\right)\right)\right\}.
\begin{aligned}x^{k+1}-{\bf 1}\otimes \bar{x}^{k+1}=(I+\gamma^k W\otimes I_d)x^k-{\bf 1}\otimes \bar{x}^{k}&\quad+\gamma^k\left(\zeta_w^k-\frac{1}{m}\sum_{i=1}^m{\bf 1}\otimes \zeta^k_{w,i}\right)-\lambda^k \left(g^k -\frac{1}{m}\sum_{i=1}^m{\bf 1}\otimes g_i^k\right)\end{aligned}
\partial_tf_1(J_t,L_t)|_{t=0^-}=\partial_t\phi(J_t,L_t)|_{t=0^-}&=\partial_t\left.\psi\left(\frac{L_t+u}{L_t-1},L_t\right)\right|_{t=0}&=\partial_1\psi(K,L)\partial_t\left. \left(\frac{L_t+u}{L_t-1}\right)\right|_{t=0}+\partial_2\psi(K,L)\partial_t L_t|_{t=0}&=\partial_2\psi(K,L)\partial_t L_t|_{t=0}
\begin{aligned}y(t_{n+1}) = y(0) + \frac{1}{\Gamma(1-\epsilon)}\int_{t_0}^{t_{n+1}}(t_{n+1}-\tau)^{-\epsilon}z(\tau)d\tau, z(t_{n+1}) = z(0)+J_0^{\alpha_2-1}f(t_{n+1})+\frac{1}{\Gamma(\alpha_2)}\int_{t_0}^{t_{n+1}}(t_{n+1}-\tau)^{\alpha_2-1}\Bigl{(}c(\tau)y(\tau)+b(\tau)z(\tau) \Bigr{)}d\tau.\end{aligned}
S = \frac{1}{2 \pi \alpha '}\int d \tau \int^{2 \pi \alpha ' p^+}_0 d \sigma \left[ \frac{1}{2}(\partial_{\tau}\vec{z})^2 - \frac{1}{2}(\partial_{\sigma}\vec{z})^2 - \frac{1}{2}\mu^2 \vec{z}^{\ 2}+ i \bar S (\slash \!\!\! \partial + \mu \Gamma^{1234}S )\right] ~,
\begin{cases}\displaystyle i\partial_t \Psi_1+\partial_\sigma^2\Psi_1-2\frac{\Psi_1-\Psi_2}{|\Psi_1-\Psi_2|^2}=0, \displaystyle i\partial_t \Psi_2-\partial_\sigma^2\Psi_2-2\frac{\Psi_1-\Psi_2}{|\Psi_1-\Psi_2|^2}=0. \end{cases}
\frac{d^2}{dt^2}R^2=2\frac{d}{dt}\bigg[\iint_{\mathbb T \times \mathbb R}\partial_{\theta}(\sigma\cdot J)\omega \sigma^{\perp}f ~ d\theta d\omega+K\iint_{\mathbb T \times \mathbb R}(\partial_{\theta}(\sigma\cdot J))^{2}f ~ d\theta d\omega\bigg].
\begin{cases}\lambda_{k}= (\lambda-\mu)\theta_{kl}c_{l}+(\lambda-\nu)\eta_{kl}b_{l}\mu_{k}= (\mu-\lambda)\theta_{kl}c_{l}+(\mu-\nu)\omega_{kl}a_{l}\nu_{k}= (\nu-\lambda)\eta_{kl}b_{l}+(\nu-\mu)\omega_{kl}a_{l}\,,\end{cases}
H_{F}= 1 + \sum_{i}{\frac{Q_{i}}{[ | \vec{x}- \vec{x}_{0 \, i}|^{2}+{\frac{4 Q_{p}}{( p - 4 )^{2}}}| \vec{z}- \vec{z}_{0}|^{p - 4}]^{\frac{( p - 3 )^{2}+ 3}{2 ( p - 4 )}}}}, H_{p}={\frac{Q_{p}}{| \vec{z}- \vec{z}_{0}|^{6 - p}}}.
\eta^a\xi_2^B\cdots \xi_{k+1}^E \nabla_a V_{B\cdots E}{}^P{}_Q =\nabla_\eta (\xi_2^B\cdots \xi_{k+1}^EV_{B\cdots E}{}^P{}_Q) -\sum_{s=2}^{k+1}\xi_2^B\cdots (\nabla_\eta\xi_s^R) \cdots\xi_{k+1}^E V_{B\cdots R\cdots E}{}^P{}_Q.
h_n'(x) = (-1)^n\g'(x) + \sum_{j=1}^n (-1)^{n-j}\ \frac{x^j}{j!}\g^{(j+1)}(x) + \sum_{j=1}^n (-1)^{n-j}\ \frac{x^{j-1}}{(j-1)!}\g^{(j)}(x) = \sum_{j=1}^{n+1}(-1)^{n+1-j}\ \frac{x^{j-1}}{(j-1)!}\g^{(j)}(x) + \sum_{j=1}^n (-1)^{n-j}\ \frac{x^{j-1}}{(j-1)!}\g^{(j)}(x) = \frac{x^n}{n!}\g^{(n+1)}(x)\,.
C =(1-\alpha)\max_{0\leq r\leq 1}\big[p(1-\alpha)h_2(r)+\alpha(h_2(pr)-rh_2(p))\big] C_{\mathrm{fb}}=(1-\alpha)\max_{0\leq r\leq 1}\big[p(1-\alpha)+\alpha(h_2(pr)-rh_2(p))\big].
\eta_{\pm}= \sqrt{-2 a_{20}^2 + 2 a_{20}b_{11}- b_{11}^2 - 8 i a_{20}b_{20}+ 2 i b_{11}b_{20}+ 2 b_{20}^2 + 2 r_{11}+ 4 i r_{20}\pm 2 a_{20}\xi \mp b_{11}\xi}.
\frac{S_{(a-1, 1)(r',1)}}{S_{(1, 1)(r', 1)}}= (-1)^{a}\frac{\sin\left(\frac{\pi b}{a}(a-1)r' \right)\sin\left(\frac{\pi a}{b}\right)}{\sin\left(\frac{\pi b}{a}r' \right)\sin\left(\frac{\pi a}{b}\right)}= (-1)^{a+br'+1}
\int_{G}^{}\phi_1 (g') \phi_2 (g'^{-1}g) dg' = \int_{G}^{}\int_{0}^{\infty}1_{L_1 (t_1)}(g') dt_1 \int_{0}^{\infty}1_{L_2 (t_2)}(g'^{-1}g) dt_2 dg'.
(k-1)! \hat{D}(f_1, f_2, h; k-1) = \frac{1}{2 \pi i}\int_C \frac{F(\tau)}{q^{h+1}}d q &= \int_0^1 F\left(x + \frac{i}{h}\right) e^{- 2 \pi i h (x + (i/h))}d x.
(q;q)_\infty (-q^2;q^2)_\infty \sum_{n=0}^\infty a_k(n)q^n &=(-q^{2};q^{2})_\infty(q^{k};q^{k})_\infty\sum_{n=1}^{\infty}\frac{q^{kn}+ 2q^{2kn}+\dots+ (k-1)q^{(k-1)kn}}{1 + q^{kn}+ q^{2kn}+\dots+ q^{(k-1)kn}}.
(\int_{-\infty}^{\infty}x^{2}|f(x)|^{2}dx)(\int_{-\infty}^{\infty}\xi^{2}|\hat{f}(\xi)|^{2}d\xi)\ge(\frac{1}{4\pi}\int_{-\infty}^{\infty}|f(x)|^{2}dx)^{2}.
\hat{R}\, = \, \int \! d p_{0}\, \Lambda^{+ + +}\,{\frac{R}{1 - R}}\, \left[ G_{0 1}G_{0 2}K_{1 2}^{0}+ G_{0 2}G_{0 3}K_{2 3}^{0}+ G_{0 3}G_{0 1}K_{3 1}^{0}\right]
\int_{G}|z|^s|k_{l}(z)|dz =\int_{G}|z|^{s-\frac{n}{2}-\varepsilon}|z|^{\frac{n}{2}+\varepsilon}|k_{l}(z)|dz &\leq I_{s,n,\varepsilon}\Vert \psi_{l}\Vert_{{H}^{\frac{n}{2}+\varepsilon}(\widehat{G})},\,\textnormal{where,}\, I_{s,n,\varepsilon}:= \left( \int_{G}|z|^{2(s-\varepsilon)-n}dz\right)^{\frac{1}{2}}<\infty.
\prescript{L_i}{}{}x{}^{R_j}= \prescript{L_i}{}{}\left(\prescript L{}{}x{}^R\right){}^{R_j}\underset{}\leq\prescript{L_i}{}{}\left(\prescript L{}{}y{}^R\right){}^{R_j}= \prescript{L_i}{}{}y{}^{R_j}.
\Delta\left(|T|^2\right)+\left\langle X, \nabla\left(|T|^2\right)\right\rangle \geq-4 \sum_{j, l=1}^{n-1}R_{i j k l}T^{i k}T^{j l}-\frac{4}{n-1}R|T|^2 -O\left(r^{-\frac{5}{2}-16\delta}\right)|T|-O\left(r^{-\frac{9}{2}}\right).
T^{i}(x) T^{i+1}(x) \cdots T^{i+r_U( T^{i}(x) )-1}(x) = \varphi(y_i) T(\varphi(y_i)) \cdots T^{r_U (\varphi(y_i))-1}(\varphi(y_i)) & = \sigma(y_i).
\| H^i(z^{i})+(H^{i})'(z^{i})p^{i}\| \leq \frac{\eta^{i}-\tilde{\eta}}{1-\tilde{\eta}}\| H^i(z^{i})\|+\frac{1-\eta^{i}}{1-\tilde{\eta}}\| H^i(z^{i})+(H^{i})'(z^{i})\tilde{p}^{i}\| &=\frac{\eta^{i}-\tilde{\eta}}{1-\tilde{\eta}}\| H^i(z^{i})\|+\frac{1-\eta^{i}}{1-\tilde{\eta}}\tilde{\eta}\| H^i(z^{i})\| &=\eta^{i}\| H^i(z^{i})\|,
\partial=\partial^{\alpha}=\eta^{\alpha\beta}\partial_{\beta}=(\partial^{0},\partial^{1},\partial^{2},\partial^{3})=(\partial^{0},\partial^{i})=(\frac{1}{c}\frac{\partial}{\partial t},-\vec{\nabla})=(\frac{\partial_{t}}{c},-\vec{\nabla})=(\frac{\partial_{t}}{c},-\partial_{x},-\partial_{y},-\partial_{z})
J_{1}( \frac \phi v ) = \left( \begin{array}{c c c c}{{\phi_{2}^{1}}}&{{\phi_{3}^{1}}}&{{\cdots}}&{{\phi_{m}^{1}}}\\{{\phi_{2}^{2}}}&{{\phi_{3}^{2}}}&{{\cdots}}&{{\phi_{m}^{2}}}\\{{\vdots}}&{{\vdots}}&{{\ddots}}&{{\vdots}}\\{{\phi_{2}^{m - 1}}}&{{\phi_{3}^{m - 1}}}&{{\cdots}}&{{\phi_{m}^{m - 1}}}\\ \end{array}\right)
M_{k,t}=C_{n,r}\sum_{\substack{0\le i\le [T_k-S_k] 0\le j\le [T_t-S_t]}}\sum_{\substack{0\le u\le L_i^k 0\le v\le L_j^t}}\left(\hat y_i^k \hat y_j^t \right)^{-(n-r)}\left|\tilde A_{s_{i,u}^k s_{j,v}^t}^{(r)}\right| \exp\left(-\frac{\hat r\left( (\hat y_i^k)^2 + (\hat y_j^t)^2\right)}{2(1+|r(s_{i,u}^k-s_{j,v}^t)|)}\right),
\langle E_1^{\textbf{1},\alpha}E_1^{\mathbf{1}, \overline{\alpha_N}}(\cdot, s), f|B_{d'}\rangle = \frac{\Gamma(s + 1)}{d'^{s+1}(4 \pi)^{s + 1}}\sum_{n \geq 1}\frac{a_n \sigma_{1,\textbf{1},\alpha}(d'n)}{n^{s + 1}},
F_{ts}\left({t,s}\right) = \frac{1}{{ts}}H_{st}\left({\frac{1}{t},\frac{1}{s}}\right) - \frac{1}{t}H_t \left({\frac{1}{t},\frac{1}{s}}\right) - \frac{1}{s}H_s \left({\frac{1}{t},\frac{1}{s}}\right) + H\left({\frac{1}{t},\frac{1}{s}}\right) &= F_{st}\left({t,s}\right).
S_t = \int_0^t \hat v_s\otimes(\hat v\circ F_s)\,ds=\int_0^t V_{N(s)}\otimes(\hat v\circ F_s)\,ds+\int_0^t (\psi\otimes\hat v)\circ F_s\,ds-\psi\otimes\hat v_t =Q_{N(t)}-\hat v\otimes\psi+V_{N(t)}\otimes(\psi\circ F_t)+\int_0^t (\psi\otimes \hat v)\circ F_s\,ds-\psi\otimes\hat v_t.
\left.\begin{array}{c}\max_{u_i \in \mathcal{U}_i}\Bigl \{\mathcal{L}_{\epsilon,i}^{u_i}\,\psi_{u_i}^{\ast}(x, u_i) + \lambda_{\epsilon,i}^{u_i^{\ast}}\,\psi_{u_i}^{\ast}(x) = 0 \Bigr \}, \forall x \in D \quad~ \psi_{u_i}^{\ast}(x) = 0, \forall x \in \partial D \end{array}\right.
CP_f(r)=\underset{i \in \psi}\prod\frac{1}{1+\max\{T, S_{th}\}r^\alpha d_i^{-\alpha}}e^{-\max\{T, S_{th}\}r^{\alpha}\frac{\sigma^2}{P}}+P[\hat{\eta}(r)>T,\eta(r)<S_{th}]
S_{N}( x ) ={\frac{1}{\sqrt{1 - \beta}}}\sum_{l = 0}^{N}\beta^{l}\sum_{n = 0}^{l}c_{n}{\frac{\Gamma ( n / 2 + l - 1 / 2 )}{\Gamma ( 3 n / 2 - 1 / 2 ) \Gamma ( l - n + 1 )}}\left[{\frac{g}{x ( x^{2}- 1 )}}\right]^{n}.
I_{1}( x , n ) = \int_{x}^{\infty}d \tau \frac{1}{e^{\tau}- 1}\tau^{n}= \sum_{q = 1}^{\infty}e^{- q x}( \frac{x^{n}}{q}+ \frac{n x^{n}}{q^{2}}+ . . . \frac{n !}{q^{n + 1}}) ,
\left[ A_{\kappa \rho}, A^{\lambda \sigma}\right] = \delta_{\kappa}^{\lambda}{M^{\sigma}}_{\rho}- \delta_{\kappa}^{\sigma}{M^{\lambda}}_{\rho}- \delta_{\rho}^{\lambda}{M^{\sigma}}_{\kappa}+ \delta_{\rho}^{\sigma}{M^{\lambda}}_{\kappa}\, .
(x^-_{\alpha}\otimes t)w_\lambda= 0,\ \ \alpha\in R^+_J,\ \ \alpha\ne \theta_J, \ \ (x^-_{\theta_J}\otimes t^2)w_\lambda=0, (x^-_{\theta_J}\otimes t)^r=0,\ \r >p=\max\{0, \lambda(h_{\theta_J})-\ell\}.
S ={\frac{1}{4}}\int T r ( B + F ) \wedge ( B + F ) ={\frac{1}{4}}\int d^{4}x \; \varepsilon^{\mu \nu \sigma \rho}( F_{\mu \nu}^{a}+ B_{\mu \nu}^{a}) ( F_{\sigma \rho}^{a}+ B_{\sigma \rho}^{a}) .
m_{00}=g_{1}\!-\!g_{20}\!-\!g_{24}\!-\!g_{32}\!-\!g_{29}\!+\!g_{9}\!+\!g_{4}\!+\!g_{13}, m_{11}=g_{24}\!-\!g_{9}\!+\!g_{20}\!-\!g_{13}\!-\!g_{32}\!+\!g_{1}\!+\!g_{4}\!-\!g_{29}, m_{22}=g_{1}\!-\!g_{13}\!-\!g_{32}\!-\!g_{4}\!+\!g_{29}\!+\!g_{9}\!-\!g_{24}\!+\!g_{20}, m_{33}=g_{24}\!+\!g_{13}\!+\!g_{29}\!+\!g_{1}\!-\!g_{4}\!-\!g_{9}\!-\!g_{20}\!-\!g_{32},
{\frac{1}{2}}\sqrt{- \eta}\; \eta^{\mu \nu}={\frac{1}{1 - \lambda^{2}}}\; \left( \begin{array}{c c}{{\lambda_{--}}}{{{\frac{1 + \lambda^{2}}{2}}}}{{{\frac{1 + \lambda^{2}}{2}}}}{{\lambda_{+ +}}}\end{array}\right)
\bigl<q| \hat{\eta}|\eta;t\bigr>=\Bigl[2i\hbar f(t)\tilde{\varphi}(t) \frac{\partial}{\partial q}+\Bigl(\frac{1}{f(t)}-2g(t)\tilde{\varphi}(t)\Bigr)q\Bigr]\bigl<q|\eta;t\bigr>=\eta\bigl<q|\eta;t\bigr>.
\sum\nolimits^{\mathbb{C}}Q\times \sum\nolimits^{\mathbb{C}}Q\rightarrow \mathbb{C}l_{(n+m)}(\varphi_{1},\varphi_{2})=([p,[\varphi_{1}]],[p,[\varphi_{2}]])\mapsto\left\langle \left\langle \lbrack \varphi_{1}],[\varphi_{2}]\right\rangle\right\rangle =\tau ([\varphi_{2}])[\varphi_{1}],
H_{2,1}(F_{f}/2) =\frac{1}{48}\left(a^4_2 - 12 a^2_3 + 12 a_2 a_4\right) &=\frac{1}{3072}\left(-3c^4_{1}-8c^2_{1}c_2 -48c^2_{2}+64c_1 c_3\right).
\eta_{X, Y}^\lambda(\mathsf{P}) := \sum_{i,j,k}q^{(\lambda,\lambda_{i})}(2c_{j}^{i}-\delta_{j}^{i})\varepsilon(X\triangleright\mathsf{P}_{k}^{j})\varepsilon(Y\triangleright\mathsf{P}_{i}^{k}).
\frac{\hat c_{n}}{\hat c_{n-1}}= \frac{1}{\Gamma(b/a)}\int_0^1 \theta^{\frac{nb+\nu}{a}-1}(1-\theta)^{\frac{b}{a}-1}\theta^{-\frac{b}{a}}F_3\left(\frac{\nu}{a}-1, \frac{b}{a}, 1, \mu,\frac{b}{a}, 1-\theta,1- \frac1\theta \right) d\theta.
C_{1}= 2 i , \hspace{0 . 2 c m}C_{2}= 8 ( n + 2 ) , \hspace{0 . 2 c m}C_{3}= - 4 8 i ( n + 2 ) ( n + 4 ) , \hspace{0 . 2 c m}C_{4}= - 3 8 4 ( n + 2 ) ( n + 4 ) ( n + 6 ) .
\widetilde{\nu}=-(\eta^{-\frac{2}{n-2}}r^2+o(r^2))\partial_r \Phi_u\partial_{\widetilde{\nu}}\Phi_u =A(n-2)\eta^{-\frac{2}{n-2}}r^{n-1}+o(r^{n-1}) ds^{\widetilde{g}}=G^{\frac{2(n-1)}{n-2}}ds^g=(\eta^{\frac{2(n-1)}{n-2}}r^{-2(n-1)}+o(r^{-2(n-1)}))ds^g.
\frac{\partial G}{\partial z_1}(z_1,z_2,\bar{z}_1 z_2 + \bar{z}_1^2 )=\overline{F_1}(\bar{z}_1)\frac{\partial F_2}{\partial z_1}(z_1,z_2,\bar{z}_1 z_2 + \bar{z}_1^2) .
{\gamma_{{\rm{MRT}}}}= \frac{{P{{\left\|{{\bf{\bar h}}}\right\|}^2}}}{{P\sum\nolimits_{i = -{n_{{\rm{span}}}},i \ne 0}^{{n_{{\rm{span}}}}}{{{\left|{{{{\bf{\bar g}}}^H}\left[ i \right]{\bf{\bar h}}/\left\|{{\bf{\bar h}}}\right\|}\right|}^2}}+{\sigma^2}}}.
\min_{\emptyset \neq J \subset \{1,\hdots,k\}}\inf_{\substack{V\subset U \lambda_t^k(V)\geq \lambda_t^k(U)/2^{k+1}}}\lambda^{|J|}_t(\Pi_J(V)) \geq \min_{j=1,\hdots,k}(2^dt\kappa_d \tau^d)^{-(k-j)}2^{-k-1}(t\kappa_d \tau^d)^{k-1}t\ell_d(A) & \geq 2^{-k-1}\min_{j=1,\hdots,k}2^{-d(k-j)}(t\kappa_d \tau^d)^{j-1}t \ell_d(A),
\gamma_k^\mathrm{C}[\iota] = \begin{cases}\frac{M\sigma_{\mathrm{s}k}^2[\iota]}{\sum_{i=1}^{K}\beta_{\mathrm{s}i}+ 1/\rho_\mathrm{s}}, \frac{M\sigma_{\mathrm{s}k}^2[\iota]}{\sum_{i=1}^{K}\beta_{\mathrm{s}i}+ \left(\rho_\mathrm{d}\beta_\mathrm{LI}+1\right)/\rho_\mathrm{s}}, .\end{cases}
\nabla^2 u(z) = 0 \mbox{if}z\in \Omega, u(z) = 1 \mbox{if}z\in \partial\Omega\cap\overline{B(z_0,r)}, u(z) = 0 \mbox{if}z\in \partial\Omega\backslash \overline{B(z_0,r)}.
L_{1}=\frac{\xi_{p}(1+|\xi_{p}|)\exp(-|\xi_{p}|)-\xi_{q}(1+|\xi_{q}|)\exp(-|\xi_{q}|)}{2F_{Y}(\xi_{p},\xi_{q})},~ L_{2}=\frac{F_{Y_{(1)}}(\xi_{p},\xi_{q})}{F_{Y}(\xi_{p},\xi_{q})},
{\cal F}^{S^1}:=\{\phi\in L^2(S^1, d\theta/2\pi):\H\phi\in W^{1,2}({\cal D})\},{\cal E}^{S^1}(\phi,\psi) :={\cal E}^{\cal D}(H^{\cal D}\phi,H^{\cal D}\psi)\ \ \ \phi,\psi\in{\cal F}^{S^1}.
a_{2,2}=\int_{{\cal M}_3}h^{1/2}d^3x~\mbox{Tr}\left[-{1 \over 32}F^2-{1\over 12}F^{jk}F_{jk}\right]=-{1 \over 24}\int_{{\cal M}_3}h^{1/2}d^3x~F^{jk}F_{jk}
2\frac{(q^2;q^2)_\infty^4[-1/q;q^2]_\infty}{[x, -xq, -x/q;q^2]_\infty}=2 \frac{(q^2;q^2)_\infty^2}{[-q;q^2]_\infty}\sum_{n=-\infty}^{\infty}\frac{(-1)^{n}q^{3n(n+1)}}{1-xq^{2n}}-\sum_{n=-\infty}^{\infty}\frac{q^{3n^2-1}[1+(3n-2)(1+xq^{2n-1})]}{(1+xq^{2n-1})^2}&-\sum_{n=-\infty}^{\infty}\frac{q^{3n^2-1}[1+3n(1+xq^{2n-1})]}{(1+xq^{2n-1})^2}.
\psi_{\mathrm{A B}}( r , \theta ) ={\frac{i}{4 \pi}}e^{( N +{\frac{1}{2}}) \theta}\int_{\cal C}d t \; e^{- i k r \cos t}\left[{\frac{e^{i \gamma t}}{\sin{\frac{1}{2}}( t - \theta )}}+{\frac{e^{- i \gamma t}}{\sin{\frac{1}{2}}( t + \theta )}}\right] .
C=\frac{A_{x}}{2}[\frac{y^{\prime}(0)+\sqrt{{y^{\prime}(0)}^{2}+1}}{1-\frac{V_{t}}{V_{d}}}+\frac{y^{\prime}(0)-\sqrt{{y^{\prime}(0)}^{2}+1}}{1+\frac{V_{t}}{V_{d}}}]
F_{4,\{1,3\}}=\sum_{i_1<i_2\leq i_3<i_4}x_{i_1}x_{i_2}x_{i_3}x_{i_4}=\sum\limits_{i_1<i_2< i_3<i_4}x_{i_1}x_{i_2}x_{i_3}x_{i_4}+\sum_{i_1<i_2<i_4}x_{i_1}x_{i_2}^2x_{i_4}.
y^2=x^3+\sum_{k=-4}^{4}\sum_{l=0}^{8-nk}f_{kl}z^{\prime l}z^{4+k}p^{8-l-qk}x+\sum_{k=-6}^{6}\sum_{l=0}^{12-nk}g_{kl}z^{\prime l}z^{6+k}p^{12-l-qk}
\int_{\tau_{1}}^{\tau_{2}}F_{rad}\cdot vdt=\int_{\tau_{1}}^{\tau_{2}}-Pdt=-\int_{\tau_{1}}^{\tau_{2}}\frac{\mu_{0}q^{2}}{6\pi c}a^{2}dt=-\int_{\tau_{1}}^{\tau_{2}}\frac{\mu_{0}q^{2}}{6\pi c}\frac{dv}{dt}\cdot\frac{dv}{dt}dt.
erf_{\kappa}(x)=(2+\kappa)\sqrt{\frac{2\kappa}{\pi}}\frac{\Gamma(\frac{1}{2\kappa}+\frac{1}{4})}{\Gamma(\frac{1}{2\kappa}-\frac{1}{4})}\int_{0}^{x}exp_{\kappa}(-t^{2})dt
H_{l , c}= - p_{u}= \frac{1}{2}( E - p_{y}) = \frac{p_{A}^{2}}{4 p_{v}}+ \frac{1}{4 \alpha^{\prime}p_{v}}( \hat{N}_{L}+ \hat{N}_{R}) + f ( \hat{J}_{L}+ \hat{J}_{R}) .
{\cal A}( x ) = \lim_{M \rightarrow \infty}\sum_{l = 0}^{\infty}e^{- \frac{l ( l + 1 )}{M^{2}}}( 2 l + 1 ) ( \frac{- 2}{M^{2}}) [ i \epsilon_{i j k}x_{k}( \tilde{\cal L}_{i}a_{j}) + 2 x_{i}a_{i}] .
G^{J J^{\prime}}( x ) = \frac{\delta^{J J^{\prime}}}{| x |^{2 J}}\left( \frac{g_{\mathrm{Y M}}^{2}N}{8 \pi^{2}}\right)^{J}\cdot J \cdot \frac{\sinh \left( \frac{1}{2}\frac{J^{2}}{N}\right)}{\frac{1}{2}\frac{J^{2}}{N}}
\partial (q_1 \ast q_2) = (1, q_1 \ast q_2) &=(1, q_1) \ast (\phi(e_{q_1}), q_2) &=(1,q_1) \ast \big(\phi(e_{q_1}) \cdot(1,q_2) \big) &= \partial(q_1) \ast \big(\phi(e_{q_1}) \cdot \partial(q_2) \big),
\begin{gathered}X_{m,n}= f(x_{m,n}),Y= e(x_{m,n},y_{m,n}) = y_{m,n}\: f_x(x_{m,n}) + g(x_{m,n}), U_{m,n}= u_{m,n}+ \frac{e_x(x_{m,n},y_{m,n})}{f_x(x_{m,n})}\end{gathered}
\sum_{k = 0}^{\infty}\left( \frac{1}{32}\right)^{k}\binom{2k}{k}^2 \frac{H_{2k}}{k+1}= 4-\frac{3 \Gamma^2 \left(\frac{1}{4}\right)}{2 \pi^{3/2}}- \frac{2 \sqrt{\pi}(\pi +3 \ln (2)-4)}{\Gamma^2 \left(\frac{1}{4}\right)},
\widetilde{P}_{n,m}(r)=\left[{L}_m^{(\ell+\frac{1}{2})}(-y)L_n^{(\ell-\frac{1}{2})}(y)-L_m^{(\ell-\frac{1}{2})}(-y){\partial_yL_n^{(\ell-\frac{1}{2})}(y)}\right]_{y=\frac{1}{2}\omega r^2}.
\mathrm{Ai}(x) = \frac{x^{-1/4}}{2\pi}\exp\left( -\frac{2}{3}x^{3/2}\right) \sum\limits_{n=0}^{\infty}\frac{(-1)^n \Gamma\left( \frac{3n+1}{2}\right)}{3^{2n}(2n)!}x^{-3n/2}, x\to +\infty,