harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\exp \left( - i e \int_{x}^{y}A \right) h_{x}\Omega_{+}( x , y ) = \frac{1}{\pi ( x - y )^{2}}- \frac{\pi}{6 L^{2}}+ \frac{2 \pi}{L^{2}}( \phi - [ \phi ] - \frac{1}{2})^{2}+{\cal O}( x - y ) ,
\delta^{(m,n)}_{\texttt{a};\mu}(q):= \prod_{\substack{1\leq j<k\leq n \mu_j-\mu_k=0}}\frac{1-q^{k-j}}{1-q^{1+k-j}}\prod_{\substack{1\leq j<k\leq n \mu_j-\mu_k=m}}\frac{1-q^{n-k+j}}{1-q^{n+1-k+j}}.
\lambda^1 = (\xi + i \theta \gamma_5) \psi^1 + (\eta + i \varphi \gamma_5) \psi^2 + ...\ , lambda^2 = (\xi' + i \theta' \gamma_5) \psi^1 + (\eta' + i \varphi' \gamma_5) \psi^2 + ...\ .
{\cal O}= \exp \left( -\sum_{n=1}^\infty \alpha_{-n}A_n\right) :\exp \left[ \sum_{n,m=1}^\infty \alpha_{-n}\left( C_{nm}-\delta_{nm}\right) \alpha_m\right] : \exp \left( -\sum_{n=1}^\infty B_n\alpha_n\right){\rm e}^{-\phi}
C_1(z) = \frac{\partial}{\partial \nu'}\left[ \prod_p \left(1 + a(p)\frac{1-p^{\nu'-\nu}}{p^{3/2+3\nu'}-1}\right)\right]_{\nu'=\nu=z}&= \prod_p\left(1 + a(p)\frac{1-1}{p^{3/2+3z}-1}\right)\cdot \sum_q \left(1 + a(q)\frac{1-1}{q^{3/2+3z}-1}\right)^{-1}B(q,z) &= \sum_p \frac{-a(p) \log p}{p^{3/2+3z}-1},
f(\mu ,s,n) = \sum_{i = 1}^{\mu - 1}{\sum_{\nu = 1}^\infty{\frac{1}{{\nu^n (\nu + i )^s}}}}- \sum_{i = 1}^{\mu - 1}{\sum_{\nu = 1}^{\mu - i}{\frac{1}{{\nu^n (\nu + i )^s}}}}\,.
\varphi(S_i(S_j(v_0))) = \frac{\partial}{\partial x_i}\left( \varphi(S_j(v_0)) \right) =\frac{\partial f_j}{\partial x_i}\varphi(S_j(S_i(v_0))) = \frac{\partial}{\partial x_j}\left( \varphi(S_i(v_0)) \right) =\frac{\partial f_i}{\partial x_j}
t+\sum_{i=0}^{n-1}f(T^i x) \in [-B+C,B-C]\Rightarrow\, t+\sum_{i=0}^{n-1}g(T^i x) \in [-B,B] t+\sum_{i=0}^{n-1}g(T^i x) \in [-B,B]\Rightarrow\, t+\sum_{i=0}^{n-1}f(T^i x) \in [-B-C,B+C]
\bar{\phi}_{\bar{x}}^{\bar{i}}( \bar{y}) : = \bar{x}^{\bar{i}}+ \bar{y}^{\bar{i}}+ \frac 1 2 \bar{\phi}_{\bar{x}, \bar{j}\bar{k}}^{\bar{i}}\bar{y}^{\bar{j}}\bar{y}^{\bar{k}}+ \cdots
t \cdot [p_{\gamma_1, \alpha_1}(z_1), \; \ldots, \; p_{\gamma_n, \alpha_n}(z_n)] & =[p_{\gamma_1, \alpha_1}(t^{-\gamma_1(\alpha_1)}z_1), \; \gamma_1(t)p_{\gamma_2, \alpha_2}(z_2), \;p_{\gamma_3, \alpha_3}(z_3), \;\ldots, \; p_{\gamma_n, \alpha_n}(z_n)].
\hat{A}_{12}\hat{Q}^{-1}\hat{A}_{12}^{\top}=\left[\begin{array}{cc}\hat{A}_{1}\hat{Q}_{1}\hat{A}_{1}^{\top}\hat{A}_{1}\hat{Q}_{2}\hat{A}_{2}^{\top}\hat{A}_{2}\hat{Q}_{2}^{\top}\hat{A}_{1}^{\top}\hat{A}_{2}\hat{Q}_{3}\hat{A}_{2}^{\top}\end{array}\right]
\begin{aligned}\Psi_0=W_{abcd}k^am^bk^cm^d,\Psi_1=W_{abcd}k^al^bk^cm^d,\Psi_2=W_{abcd}k^am^bl^c\overline{m}^d, &\Psi_3=W_{abcd}l^ak^b\overline{m}^c l^d,\Psi_4=W_{abcd}l^a\overline{m}^bl^c\overline{m}^d.\end{aligned}
\frac{d\eta}{dt}=-\eta\nabla_{h}\cdot v_{h}-(\frac{\partial w}{\partial x}\frac{\partial v}{\partial z}-\frac{\partial w}{\partial y}\frac{\partial u}{\partial z})-\frac{1}{\rho^{2}}k\cdot(\nabla_{h}p\times\nabla_{h}\rho)
\sum_{k=0}^{n-1}q^{2k}[k+1]_{q^{2}}g_k(x;q^2) =\sum_{k=0}^{n-1}q^{2k}[k+1]_{q^{2}}\sum_{j=0}^k{k\brack j}_{q^2}^2{2j\brack j}_{q^2}x^j &=\sum_{j=0}^{n-1}[j+1]_{q^2}{2j\brack j}_{q^2}x^j\sum_{k=j}^{n-1}q^{2k}{k+1\brack j+1}_{q^2}{k\brack j}_{q^2}.
: \underset{\{T_s[i],T_u[j]\}}{\max}~ \underset{i=1}{\overset{N}{\sum}}\underset{j=i}{\overset{N}{\sum}}~w[i][j] ~f_{2,}\mathrm{s.t.}~ \underset{i=1}{\overset{N}{\sum}}\frac{1}{T_s[i]}\leq E_{s}, \underset{j=1}{\overset{N}{\sum}}\frac{1}{T_u[j]}\leq E_{u}, ~ T_{s}[i] \geq 0, T_{u}[j] \geq 0.
J_{r}={\frac{e n}{4}}( J_{2}+{\frac{r_{1}^{2}}{\sigma^{2}J_{2}}}-{\frac{2 r_{1}}{\sigma}}) ={\frac{e n}{4}}( J_{r}+ J_{1}+{\frac{r_{1}^{2}}{\sigma^{2}J_{2}}}-{\frac{2 r_{1}}{\sigma}}) ~ .
\mathbf{CK}_{x_3}^{\mu_3}={}_0F_{1}\left(\genfrac{}{}{0pt}{}{-}{\mu_3+1/2}\;\Big\rvert\;-\left(\frac{x_3\,\widetilde{\underline{D}}}{2}\right)^2\right) -\frac{\sigma_3\,x_3\,\widetilde{\underline{D}}}{2\mu_3+1}\;\;{}_0F_{1}\left(\genfrac{}{}{0pt}{}{-}{\mu_3+3/2}\;\Big\rvert\;-\left(\frac{x_3\,\widetilde{\underline{D}}}{2}\right)^2\right),
T_1 =(1-a_T)tu_T, T_2 =T(1-a_T)u_T\int_0^{t}\psi^T(T(t-s))s ds, T_3 =T^{1-\alpha/2}\sqrt{\frac{(1-a_T)}{\mu^*\delta^{-1}}}\int_0^t\psi^T(T(t-s)) Z^T_sds.
B_{n m}= \left( \begin{array}{c c}{{0}}&{{- 1}}\\{{1}}&{{h}}\\ \end{array}\right) , \qquad B^{n m}= \left( \begin{array}{c c}{{h}}&{{1}}\\{{- 1}}&{{0}}\\ \end{array}\right) , \quad B^{n m}B_{n m}= - 2 .
[L_n:k] = \prod_{i=0}^{n-1}[L_{i+1}:L_{i}] \leq\prod_{i=0}^{n-1}6^{8(g-g_i)}\cdot(8(g-g_i))! \leq\prod_{i=0}^g 6^{8(g-i)}\cdot(8(g-i))! = 6^{4g(g+1)}\prod_{r=0}^{g}(8r)! = 6^{4g(g+1)}\prod_{r=1}^{g}(8r)!
\sum_{n \in \mathbf B_N}\psi(n)^{-1}e^{2\pi i \xi n}=\sum_{n = 1}^N \psi(n)^{-1}\big(S_n(\xi) - S_{n-1}(\xi)\big) &=\psi(N+1)^{-1}S_N(\xi)+\sum_{n = 1}^N \big(\psi(n)^{-1}- \psi(n+1)^{-1}\big) S_n(\xi).
\psi_{P,\mathbf{a}}(\mathbf{Z};P)\equiv \frac{I_{\mathbf{a}}(\mathbf{A})}{\lambda_{\overline{\mathbf{a}}_{p}}(\mathbf{Z};P)}\lbrace Y - \chi_{\mathbf{a}}(P;\mathcal{G})\rbrace - \sum\limits_{k=0}^{p}g_{k}(\overline{\mathbf{A}}_{k}, \overline{\mathbf{Z}}_{k};P),
\times\bigg(\left(t_A\right)_{R_1r_1S_1s_1}\left(t_{C_1}\right)_{V_1v_1W_1w_1}\left(\left(t_{E_1\alpha_1}\right)_{S_1s_1V_1v_1}\delta_{W_1R_1}\delta_{w_1r_1}-\delta_{S_1V_1}\delta_{s_1v_1}\left(t_{E_1\alpha_1}\right)_{W_1w_1R_1r_1}\right)\bigg)\times
L_{1}^{\ast\ast}=\frac{\xi_{p}^{3}\phi(\xi_{p})(1+\sqrt{2\pi}\phi(\xi_{q}))-\xi_{q}^{3}\phi(\xi_{q})(1+\sqrt{2\pi}\phi(\xi_{p}))}{\Psi_{2}^{\ast}(-1,\frac{1}{2},1)F_{Y}(\xi_{p},\xi_{q})(1+\sqrt{2\pi}\phi(\xi_{p}))(1+\sqrt{2\pi}\phi(\xi_{q}))},
g(t,s,\mathfrak{X}) := \left( \varphi_t-\mathcal{L}_{\alpha(s,\mathfrak{X})}\varphi\right) \big(t,\mathfrak{X}(s)\big)\ge \widehat{\mathcal{G}}\big(t,\mathfrak{X}(s)\big)\,,\forall\ (t,s) \in (0, \infty)\times [0, \infty)\,.
0=\psi(\boldsymbol{x})= \sum_{i\in I_{\boldsymbol{x}}}e_{11}([a_i,b_i]-\overline{[a_i,b_i]})+\sum\limits_{j=1}^me_{jj}(c_j-\bar{c}_j)+\sum\limits_{j=1}^mu_{11}(\rho(c_j)) &+\sum\limits_{k=2}^ne_{11}(d_k-\bar{d}_k)+\sum\limits_{k=2}^n(-1)^{|d_k|}u_{kk}(d_k),
^{g}\Box_{(\underline{m},\underline{l})}:= \prod_{m_i > 0 \atop i \neq u}\hat{v}_{iu}^{m_i}\prod_{l_i > 0 \atop 0 \leq i \leq n}\mu_i^{l_i}- \prod_{m_i < 0 \atop i \neq u}\hat{v}_{iu}^{-k_i}\prod_{l_i > 0 \atop 0 \leq i \leq n}\mu_i^{-l_i}
\lambda_j^{\textrm{opt}}(t_j) = \frac{\|c_{1j}\|_{L^p\left(\Omega_j^{t_j}\right)}}{\|c_{1j}\|_{L^p\left(\Omega_j^{t_j}\right)}+ \|c_{2j}\|_{L^p\left(\Omega_j^{t_j}\right)}},
U_{1}(x_{1}, x_{2}, x_{3}) = -x_{3}w_{x_{1}}(x_{1}, x_{2}), U_{2}(x_{1}, x_{2}, x_{3}) = -x_{3}w_{x_{2}}(x_{1}, x_{2}), U_{3}(x_{1}, x_{2}, x_{3}) = \phantom{-x_{3}}w(x_{1}, x_{2}),
[z^n]G(z)= (-1)^n\sum_{k=0}^{p-1}\binom{k}{n-k}(-4)^k\cdot\frac{d}{dx}\bigg(\frac{(-\frac12(a+x))_k}{(1)_k}\cdot\frac{(\frac12+\frac12(a+x))_k}{k!}\bigg)\bigg|_{x=0}= (-1)^n\sum_{\frac 12n\leq k\leq p-1}\binom{k}{n-k}(-4)^k\cdot\frac{(\frac12+\frac12a)_k}{k!}\cdot\frac{d}{dx}\bigg(\frac{(-\frac12(a+x))_k}{(1)_k}\bigg)\bigg|_{x=0}.
\partial_x \left( \begin{array}{c}\phi_1 \phi_2 \end{array}\right) ={\bf U}\left( \begin{array}{c}\phi_1 \phi_2 \end{array}\right) , \partial_t \left( \begin{array}{c}\phi_1 \phi_2 \end{array}\right) ={\bf V}\left( \begin{array}{c}\phi_1 \phi_2 \end{array}\right),
M(X;\chi,u)=Res_{z=1}L(z, \chi) L(2z, \chi^{2}) G(z, \chi) F_{u}(z, \chi) \frac{X^{z}}{z}+\sum_{j=1}^4 \int_j-\int_5 + O \left(\frac{X^{1+ \varepsilon}\log X}{T}\right),
T_\phi u = T_\phi a + T_\phi b \in{\mathrm{L}}^{p}(\mathbb{R}^n_+) +{\mathrm{H}}^{1,p}(\mathbb{R}^n_+) \subset \mathrm{L}^p(\mathbb{R}^n_+) + \dot{\mathrm{H}}^{1,p}(\mathbb{R}^n_+)
\nabla^{\mu}\partial_{\mu}\Phi^{a}= -{\frac{\Phi^{b}( \nabla^{\mu}\partial_{\mu}\Phi^{b})}{\eta^{2}}}\Phi^{a}={\frac{( \partial^{\mu}\Phi^{b}) ( \partial_{\mu}\Phi^{b})}{\eta^{2}}}\Phi^{a}.
Z = \int \prod_{x}\{\mathrm{d}v_{\mu}\mathrm{d}j^{\mu}\mathrm{d}j_{5}^{\mu}\mathrm{d}| h |^{2}\mathrm{d}\alpha K [ j^{\mu}, j_{5}^{\mu}, | h |^{2}] \}e^{\mathrm{i}\int \mathrm{d}^{4}x{\cal L}[ v_{\mu}, j^{\mu}, j_{5}^{\mu}, | h | , \alpha ]},
\ \frac{1}{\pi}(1+r)^{-p}\int_0^1 \left(1 - \frac{4r}{(1+r)^2}x\right)^{-p/2}x^{-1/2}(1-x)^{-1/2}dx = (1+r)^{-p}{}_2F_1(p/2, 1/2;1; 4r/(1+r)^2)
P_b\left(\frac{1}{\rho_V}\right) = \overline{y}_S\left(\frac{1}{\rho_V}\right) + \overline{y}_{0,.}\left(\frac{1}{\rho_V}\right)-\frac{\rho_R(\rho_V)^K}{\rho_V^K Z(\frac{1}{\rho_V}, \rho_R(\frac{1}{\rho_V}))K!}= \overline{y}_{0,.}(\rho_V) + \overline{y}_S(\rho_V) -\frac{\rho_R(\rho_V)^K}{Z(\rho_V, \rho_R)K!}.
\begin{array}{c c c}{{\bf \mathrm{\large \Sigma_{3 6 0}}}}&{\mathrm{\normalsize{4 ( \delta_{i j})_{7 \times 7}}}}&{\mathrm{\normalsize{6 ( \delta_{i j})_{7 \times 7}}}}\\{}&{{\bf 1^{4}}}&{{\bf 1^{6}}}\\ \end{array}
\overline{{g}}_{\mu \nu}= \left[ \begin{array}{c c}{1 + l ( x^{0})^{2}\overline{{\lambda}}( x^{0}, x^{1})^{2}}{l ( x^{0})^{2}}{l ( x^{0})^{2}\overline{{\lambda}}( x^{0}, x^{1})^{2}}{l ( x^{0})^{2}}\end{array}\right] .
\int_{0}^{t}(t-s_{1})^{\theta-1}\int_{0}^{s_{1}}(s_{1}-s_{2})^{\theta-1}\cdots\int_{0}^{s_{n-1}}(s_{n-1}-s_{n})^{\theta-1}ds_{n}\cdots ds_{1}=\frac{\left\{\Gamma(\theta)\right\}^{n}}{\Gamma(n\theta+1)}t^{n\theta},
\langle \epsilon(g), \Gamma(\widehat{O})^*H_\mu(\xi)\Gamma(\widehat{O})\epsilon(f)\rangle =\langle \epsilon(\widehat{O}g), H_\mu(\xi)\epsilon(\widehat{O}f)\rangle &=\langle \epsilon(g), H_\mu(O\xi)\epsilon(f)\rangle.
(Y_{u})^{ab}=e^{\hat{K}/2}(\varepsilon_{lmn}+(1-e^{-2\pi i/M})s_{lmn})(t_{33}^{1/2})^{la}(t_{33}^{1/2})^{mb}(t_{33}^{1/2})^{nc}\langle h_{u}\rangle_{c}.
\sum_{0\leq n<m <\infty}\frac{\tau_A(p^{m}) \tau_B(p^{n}) \left( 1-\frac{p^w}{p^{2}}\right)}{p^{m(-\frac{1}{2}+s_1+w-z)}p^{n(\frac{3}{2}+s_2-w+z)}}= \frac{\tau_A(p)}{p^{-\frac{1}{2}+s_1+w-z}}+ O\bigg( \frac{1}{p^{1+\varepsilon}}\bigg).
\partial_{t}^{3}\hat n_{\mu}= - g^{kl}\partial_{k}\partial_{t}^2v^{\tau}\hat n_{\tau}\partial_{l}\eta_{\mu}- \left( \partial_{t}^2( g^{kl}\partial_{k}v^{\tau}\hat n_{\tau}\partial_{l}\eta_{\mu}) - g^{kl}\partial_{k}\partial_{t}^2v^{\tau}\hat n_{\tau}\partial_{l}\eta_{\mu}\right)
\zeta^{(p)}_0(\nu)={1 \over \pi \Gamma(\nu)}\int_0^\infty dx \int_0^1 dt ~t^{\nu-1}e^{-tx^2}{1 \over (4\pi t)^{(D-1)/2}}\sum_{n=0}^\infty \breve{a}_n(x)t^n.
-|u_1|_{d'}\cos \tau_1- |u_2|_3 \cos \tau_2 \ge -\frac{1}{3}\theta_{d'}- |u_2|_3 \ge -\frac{1}{3}\theta_{d'}+ \frac{1}{6}\theta_{d'}=-\frac{1}{6}\theta_d \mbox{.}
1 - \bar{S}_{F}\ast{\cal V}= \left[1 - S^{cov}_F\left(\gamma^{-}{A}_{-}+ \gamma^{j}{A}_{j}\right) \right]\ast \left[1 + \frac{e}{2}\gamma^{+}\frac{1}{i \partial_{-}}\ast{\cal W}_{-1}[\widehat{a}] \left(\gamma^{-}{A}_{-}+\gamma^{k}{A}_{k}\right)\right],
\begin{bmatrix}H_{1, a_{1, 1}}H_{1, a_{2, 1}}\dots H_{1, a_{M, 1}}\sigma_{1}H_{2, a_{1, 1}}\sigma_{2}H_{2, a_{2, 1}}\dots \sigma_{M}H_{2, a_{M, 1}}\vdots \vdots \vdots \vdots \sigma_{1}^{r - 1}H_{r, a_{1, 1}}\sigma_{2}^{r - 1}H_{r, a_{2, 1}}\dots \sigma_{M}^{r - 1}H_{r, a_{M, 1}}\end{bmatrix}.
(Q_{V(H)}x)(v) =\sum\limits_{e\in E_v(H)}w_H(e)\sum\limits_{u\in e}x(u) =\sum\limits_{\hat e\in E_{W_{E_v( H)}}(\hat H)}w_{\hat{H}}(\hat e)\sum\limits_{W_{E_0}\in \hat e}\hat x(W_{E_0}) =( Q_{V(\hat H)}(\hat x))(W_{E_v(H)}).
\exp \left( \sum_{g = 0}^{\infty}x^{2 g - 2}\mathcal{F}_{g}\right) = e^{- \frac{D ( 0 , 0 , 0 )}{2}\zeta ( 1 )}\prod_{( \ell , n , \gamma , j ) > 0}( 1 - p^{\ell}q^{n}\zeta^{\gamma}y^{j})^{D ( \ell n , \gamma , j )}\, .
e^{\rm trnc}(k;K_r,\omega)\,\le\,c_r\,\left(C(4,\omega)\,C(4\,r-6,\omega)\right)^{1/4}\left(\sum_{j=k+1}^\infty\xi_j\right) \left(\sum_{i=1}^\infty \xi_i\right)^{r-3/2},
I_{m,p}^{\star}(s,x)=\sqrt{\pi}2^{\frac{1}{2}-s}x^{s+\frac{1}{2}}\intop_{0}^{\infty}y^{s-\frac{1}{2}}J_{s-\frac{1}{2}}(xy)\,\left(\frac{2\pi p-y}{2\pi p+y}\right)^{m}\,e^{-my}dy
{\cal L}= -{\frac{1}{1 2}}[ 2 g^{\mu \nu}h_{i j}{\cal D}_{\mu}q^{i}{\cal D}_{\nu}q^{j}+ F_{\Lambda}^{a b}F_{\Lambda}^{a b}+ 2 g^{2}{\cal P}_{\Lambda}^{u}{\cal P}_{\Lambda}^{u}] \varepsilon_{c d e f}e^{c}e^{d}e^{e}e^{f}
\bigg\|\bigg(\ln\frac{u_{\alpha\beta}^2}{\bar{n}}\bigg)_+\bigg\|_{L^{k+1}(\Omega)}\leq\bigg(\frac{\alpha}{\sqrt{\bar{n}}}\bigg)^{\frac{1}{k+1}}\frac{4|D|_0}{\theta_L},k=1,2,3,\cdots.
L_{2r,\ell}(\tau) =\prod_{n=1}^{\infty}(1-q^n)^c(1-q^{\ell\cdot n})^d\sum_{m\geq \mu_{2r}}p_{[1^c\ell^d]}(\ell^{2r}m+n_{2r})q^m, L_{2r-1,\ell}(\tau) =\prod_{n=1}^{\infty}(1-q^{\ell\cdot n})^c(1-q^n)^d\sum_{m\geq \mu_{2r-1}}p_{[1^c\ell^d]}(\ell^{2r-1}m+n_{2r-1})q^m.
\lim_{N\rightarrow \infty}G\left(\frac{x}{\sqrt{N}},\frac{y}{\sqrt{N}}\right)=\sum_{m,n=0}^{\infty}\frac{x^{m}y^{n}}{\sqrt{m!n!}}\lim_{N\rightarrow \infty}P_{m,n}(i,k;N).
= \sum_{j=1}^N \sum_{i=1}^{M-1}\Biggl\{\sum_{k=1}^{H-1}(a_{ik}b_{kj}- a^0_{ik}b^0_{kj}) + (a_{iH}- a^0_{iH})-\sum_{k=1}^{H-1}(a_{iH}b_{kj}- a^0_{iH}b^0_{kj}) \Biggr\}^2 &= \sum_{j=1}^N \sum_{i=1}^{M-1}\Biggl[(a_{iH}- a^0_{iH})+\sum_{k=1}^{H-1}\{(a_{ik}-a_{iH})b_{kj}- (a^0_{ik}-a^0_{iH}) b^0_{kj}\}\Biggr]^2 .
\begin{aligned}\min_{\mathbf{x}\in \mathbb{R}^{d_1}}\; \frac{1}{2}\mathbf{x}^TH\mathbf{x}+\mathbf{g}^T\mathbf{x}\hbox{s.t.}\; A\mathbf{x}= \mathbf{b}, \; \mathbf{x}- \mathbf{z}=0 & \mathbf{z}\geq 0 \end{aligned}
\int_{2^m}\chi_{Z(\omega)_\tau}(x) d \kappa(\omega) = \kappa (\{\omega: x \in \gamma^c Z(\omega)_{\tau(\gamma)}\mbox{for all}\gamma \in G \}) = \prod_{\gamma \in G}\kappa \left ( \left\{\omega: \omega(Y((\gamma^{-1})^cx)) = \tau(\gamma) \right\}\right) = 2^{-|G|}
T = \frac{d-2-2\kappa}{4 \pi r_H}\frac{r_0^{\frac{d-2}{2}}}{h^{\frac{d-2}{2}}}\sp \Omega_i =\frac{l_i}{(l_i^2 +r_H^2)}\frac{r_0^{\frac{d-2}{2}}}{h^{\frac{d-2}{2}}}
S_M = I - \hat{A}^{-1/2}A \hat{A}^{-1/2}+ \hat{A}^{-1/2}B^\top ( \hat{S}- C )^{-1}B \hat{A}^{-1/2}, S_N = I - \hat{S}^{-1/2}C \hat{S}^{-1/2}+ \hat{S}^{-1/2}B ( \hat{A}- A )^{-1}B^\top \hat{S}^{-1/2}.
P ( \ell , \Phi ) = \left[ \ell^{3}- \frac{\ell^{2}\, \Phi}{\pi}+ \ell \left( \frac{{\Phi}^{2}}{2 \,{\pi}^{2}}- 1 \right) + \frac{2 \,{\pi}^{2}\, \Phi -{\Phi}^{3}}{8 \,{\pi}^{3}}\right] ,
\mathbb{E}[|\mathbf{H}_{k,}\cdot \mathbf{H}_{l,}^H|^2] = \mathbb{E}_{\mathbf{H}_{k,}}\Bigl[\mathbf{H}_{k,}\cdot \mathbb{E}_{\mathbf{H}_{l,}}[\mathbf{H}^H_{l,}\cdot \mathbf{H}_{l,}] \cdot \mathbf{H}^H_{k,}\Bigr].
\sum_{\pi \in S_n}\frac{1}{n!}\big(Q_{X^n}\times Q_{Y^n}\big) W_{X^nY^n}^n[\pi] = \sum_{\pi \in S_n}\frac{1}{n!}Q_{X^n}W_{X^n}^n[\pi] \times Q_{Y^n}W_{Y^n}^n[\pi] = Q_{X^n}\times \sum_{\pi \in S_n}\frac{1}{n!}Q_{Y^n}W_{Y^n}^n[\pi] \,,
E_1(C; \tau) = \frac{C^2-1}{2}- C \sum_{\substack{1 \leq n < n_1(\tau) n}}\frac{(\log C)^{n}}{n!}, E_0(C; \tau) = \frac{C^2+1}{2}- C \sum_{\substack{0 \leq n < n_0(\tau) n}}\frac{(\log C)^{n}}{n!},
s_{\alpha}^{({\boldsymbol{q}})}({\boldsymbol{X}}_{n_1,\ldots,n_r})=\det \big|e_{j-i+ \alpha_i}^{({\boldsymbol{q}})}({\boldsymbol{X}}_{n_1,\ldots,n_r}) \big|_{1 \le i,j \le p}.
{\tilde{g}}_{1}(\zeta ) =\zeta +1+o(1){\tilde{g}}_{2}(\zeta ) =\zeta -\mu_{1}+o(1){\tilde{g}}_{3}(\zeta ) =\zeta +\frac{{\tilde{a}}}{\zeta^{p-1}}+o\left(\frac{1}{\zeta^{p-1}}\right){\tilde{g}}_{4}(\zeta ) =\zeta +\frac{{\tilde{b}}}{\zeta^p}+o\left(\frac{1}{\zeta^p}\right).
\begin{bmatrix}W - \epsilon \Lambda^{-1}T^T \tilde F \epsilon T^T \tilde F\end{bmatrix}^T\begin{bmatrix}\Lambda 0 0 - \Lambda^{-1}\end{bmatrix}\begin{bmatrix}W - \epsilon \Lambda^{-1}T^T \tilde F \epsilon T^T \tilde F\end{bmatrix}.
\frac{G^{\prime}(\frac{1}{2}+ \frac{1}{\sqrt{2}}-i{\beta},\frac{1}{2}- \frac{1}{\sqrt{2}}-i{\beta}; \frac{iz_0}{{\hbar}c})}{G(\frac{1}{2}+ \frac{1}{\sqrt{2}}-i{\beta},\frac{1}{2}- \frac{1}{\sqrt{2}}-i{\beta}; \frac{iz_0}{{\hbar}c})}= - \frac{i}{2{\hbar}c}- (\frac{1}{2}+i{\beta})\frac{1}{z_0}.
\lVert f+g\rVert_S =\lVert \sum_{v\in I\cup J}(\alpha(v)+\beta(v))\#v\rVert_S &=\max\{\lvert v\rvert_S\mid v\in I\cup J, \alpha(v)+\beta(v)\neq 0\}&\leq \max\{\lvert v\rvert_S\mid v\in I\cup J, \alpha(v)\neq 0 or \beta(v)\neq 0\}&=\max(\lVert f\rVert_S,\lVert g\rVert_S).
S_{I,\theta}= -\frac{\sigma}{2}\int \frac{d\nu}{2\pi}\int_{\bar R_1}^{\bar R_2}dr \tilde\theta_I^* \left[-\frac{\partial}{\partial r}\gamma^{-1}r^2 \frac{\partial}{\partial r}- (\nu^2 - \omega^2) \gamma r^2 \right] \tilde\theta_I \,,
\Phi^\prime_\epsilon(r) = \frac{1}{\epsilon r}\sum^2_{j=0}\alpha_j\int_{\partial^*\!E_j\cap(B_r \setminus B_{r(1-\epsilon)})}\frac{|x|}{r}\, d\mathcal{H}^{n-1}(x), a.e.\, r\in(0,d),
{}^{i}\hat{O}^{r_{1}r_{2}}_{\lambda,\mu}={}^{i}\hat{O}^{r_{1}}_{\lambda}\otimes{}^{i}\hat{O}^{r_{2}}_{\mu}\equiv{}^{i}\hat{O}^{r}_{\lambda,\mu}={}^{i}O^{r}_{\lambda,\mu}(\alpha_{\lambda}\otimes \alpha_{\mu}),
\mu_j(K_{H/2}M_gK_{H/2}) \geq\mu_j\left[(M_{g}K_{H/2}M_{g})^2\right] &=\left(\mu_j(M_{g}K_{H/2}M_{g})\right)^2 &\geq c_{H}\max_{1\leq i\leq2n-1}|u_i||\ell_{i+1}-\ell_i|^{H}\widetilde{\mu}_j^2,
ds^{2}= -\frac{2Q^{2}}{\gamma^{4}}\left ( 1 - \frac{4M}{r}\right ) dt^{2}+ \frac{2Q^{2}}{r^{2}}\left ( 1 - \frac{4M}{r}\right )^{-1}dr^{2}+ 2Q^{2}d\Omega^{2}
F(u)=\int_{1}^{u}(a^{n}+x^{n})^{\frac{1}{n}}dx, F^{\prime}(u)=\frac{(a^{n}+u^{n})^{\frac{1}{n}}}{u}, F^{\prime\prime}(u)=\frac{u^{n-1}(a^{n}+u^{n})^{\frac{1-n}{n}}}{u}-\frac{(a^n+u^{n})^{\frac{1}{n}}}{u^{2}}.
\frac{1}{n^2}\sum\limits_{i=1}^{\lfloor n\tau\rfloor}\sum\limits_{j=\lfloor n\tau\rfloor +1}^n1_{\left\{G(\xi_i)\leq G(\xi_j)+\Delta\right\}}\overset{P}{\longrightarrow}\tau(1-\tau)\int_{\mathbb{R}}F(x+\Delta)dF(x)
\sum_{k=0}^n\frac{q^{r\binom{k+1}{2}+(1-r)k}[r]_q^k\,[k]_{q^r}!S_{r}[n,k]\,t^k}{\prod_{i=0}^k(1-tq^{ri})}= \sum_{m=0}^{\infty}[rm+1]_q^{n}\,t^{m}.
\begin{aligned}d(\Theta(\cdot,t),\bar{\Theta}(\cdot,t)) \leqslant C \Bigl( 1+|F|^p + |Z|^2+|w|^2+|V|^2\Bigr) &\leqslant C \Bigl( 1+G(\Xi^{j-1}) + G(\Xi^{j})+|v^{j-1}|^2 +|v^j|^2\Bigr)\end{aligned}
\mathcal{R}_2^+= \sum_{m \geq 1}\sum_{n \geq 1}\frac{A(m,n)}{\sqrt{m^2 n}}g\Big(\frac{m^2 n}{N}\Big)\sum_{C_2/m \leq c \leq C_1/m}\frac{S(n,1;c)}{c}H_{m,n}^+ \Big( \frac{4\pi \sqrt{n}}{c}\Big),
[t_1v_1,t_2v_2]={}^{t_1}[v_1,t_2]\cdot{}^{t_1t_2}[v_1,v_2]\cdot{}^{t_2}[t_1,v_2]\in [^{t_1}v_1,t_2]\cdot[t_1,{}^{t_2}v_2]\,[N,N].
\int_{-T_0}^{T_0}\frac{|\zeta(\b(\l+it-z_1))|^4}{|\zeta(\b(2(\l+it)-z_1-z_2))(\l+it)|}dt &=\int_{-T_0}^{T_0}|\chi(\b(\l+it-z_1))|\frac{|\zeta(1-\b(\l+it-z_1))|^4}{|\zeta(\b(2(\l+it)-z_1-z_2))(\l+it)|}dt &\ll T_0^{(a_1-a_2)\b/2}\int_{-T_0}^{T_0}\frac{|\zeta(1/2+(a_1-a_2)\b/2+i(t+t'))|^4}{|\zeta(1+2it))(\l+i(t+t''))|}dt,
x^5(t,s)= \,x^5(t)+\frac{\cos(sb(t))-1}{2b(t)^2}\Big(-(b_1(t)a^2(t)+b_3(t)a^4(t))x^1(t) &+(b_1(t)a^1(t)+b_3(t)a^3(t))x^2(t)-(b_3(t)a^2(t)-b_1(t)a^4(t))x^3(t) &-(b_3(t)a^1(t)+b_1(t)a^3(t))x^4(t)\Big)+\frac{\sin(sb(t))}{2b(t)}\Big(a^3(t)x^1(t)+a^4(t)x^2(t) &-a^1(t)x^3(t)-a^2(t)x^4(t)\Big).
K_{(0,\lambda)}[\psi_\lambda](T) =\frac{1}{\Gamma(1-\alpha)}\left(\int_{T-\lambda}^T\frac{\partial_t\psi_\lambda(s)}{(t-s)^\alpha}ds-\frac{\psi_\lambda(T)-\psi_\lambda(T-\lambda)}{\lambda^\alpha}\right) &=\frac{a}{2\lambda^\alpha}\left(\Gamma(1+\alpha)-\frac{1}{\Gamma(1-\alpha)}\right).
\frac{B_{n}^{\left(q\right)}}{n!}=\sum_{p=1}^{n}\left(\sum_{m=p}^{n}\binom{m}{p}\binom{m+q-1}{q-1}\right)\sum_{k_{1}+\dots+k_{p}=n}\frac{\left(-1\right)^{p}}{\left(k_{1}+1\right)!\dots\left(k_{p}+1\right)!}.
R_\ell (r) ={1\over \sqrt v}\;{\Gamma (\ell + 1+i\eta)\over \Gamma (2\ell +1)}\;e^{-\pi\eta/2}\; (2ikr)^\ell \; e^{ikr}F(\ell + 1+i\eta\;,\; 2\ell +2\; ;\; -2ikr)
h(x,\xi)= h_0(\xi)+V(x), V_\rho(t,x)= V(x)\chi(\rho x)\chi\left(\frac{\langle\log\langle t \rangle\rangle x}{\langle t \rangle}\right), h_\rho(t,x,\xi)= h_0(\xi)+V_\rho(t,x), \nabla_x^2 V_\rho(t,x)={}^t\nabla_x\nabla_x V_\rho(t,x),
\int_1^\infty t^{-\mu}|f(t)|\, dt \leq \int_1^\infty t^{-\mu}\int_t^\infty |df(s)|\, dt=\int_1^\infty |df(s)|\int_1^s t^{-\mu}\, dt \lesssim \int_1^\infty t^{1-\mu}|df(t)|
T^{eff}_{\mu\nu}(\phi) = \phi_{,\mu}\phi_{,\nu}- \frac{1}{2}\overline{g}_{\mu\nu}\phi_{,\alpha}\phi_{,\beta}\overline{g}^{\alpha\beta}+ \overline{g}_{\mu\nu}V_{eff}(\phi), V_{eff}(\phi) = \frac{1}{4U(\phi)}(V+M)^{2}.
\phi^{(\epsilon)}_{ij}(\xi)=\begin{cases}\displaystyle{\phi_{ij}(\epsilon)+\int_{\epsilon}^0\phi_{ij}'(\rho_\epsilon(\eta))\,d\eta+\phi_{ij}'(\rho_\epsilon(0))\xi}, \xi\leq 0, \phi_{ij}(\xi), \epsilon\leq \xi.\end{cases}
\Psi_n(x)y =d^{2(n+k)}\sum_{a,b}(a e_1 e_2\cdots e_{n}x e_{n+1}e_n \cdots e_{k+2})(z_{a,b}e_{k+1})(e_k \cdots e_2e_1 b^*) &=d^{2(n+k)}\sum_{a,b}a e_1 e_2\cdots e_{n}xz_{a,b}e_{n+1}e_n \cdots e_2e_1 b^*
r^{\mu}\bigg|\int_M^N (rt)^\nu f(t) j_\alpha(rt)\, dt\bigg| =r^{\mu+\nu}\bigg| g_{\alpha,r}^\nu(N) f(N)- g_{\alpha,r}^\nu(M)f(M) -\int_M^N g_{\alpha,r}^\nu(t)\, df(t)\bigg| \lesssim r^{\mu+\nu-\alpha-3/2}\bigg( N^{\nu-\alpha-1/2}|f(N)|+M^{\nu-\alpha-1/2}|f(M)| \phantom{=}+ \int_M^N t^{\nu-\alpha-1/2}\, |df(t)|\bigg).
R^0\widehat{S}(D_{A_2}(a_{2,*}(\omega_X\otimes a_1^*P)))=R\widehat{S}(D_{A_2}(a_{2,*}(\omega_X\otimes a_1^*P))) = R\widehat{S}(D_{A_2}\circ Ra_{2,*}(\omega_X\otimes a_1^*P))=R\widehat{S}(Ra_{2,*}\circ D_X(\omega_X\otimes a_1^*P)) = R\widehat{S}(Ra_{2,*}R\mathcal{H}om(\omega_X\otimes a_1^*P, \omega_X[2]))=R\hat{S}Ra_{2,*}(a_1^*P^{\vee})[2],
N_{R,p,q}^{\prime}(v)=\frac{1}{[p+q]!}\sup_{R/2\leq r<R}(R-r)^{p+q+2}\Vert\partial_{y}^{q+2}\nabla_{x}^{p}v\Vert_{L^{2}(B_{r}^{+})},p\geq0,q\geq-2.
D_{V^{*}}^2=I-VV^* =I-V_1V_1^*+V_1V_1^*-V_1V_2V_2^*V_1^*= D_{V_1^*}^2+V_1D_{V_2^*}^2V_1^* &=I-V_2V_2^*+V_2V_2^*-V_2V_1V_1^*V_2^*=V_2D_{V_1^*}^2V_2^*+D_{V_2^*}^2.
I_{{\rm bl}}^{\mathfrak{m}}(a,1,b)=-I_{{\rm bl}}^{\mathfrak{m}}(a+b+1)-\sum_{j=2}^{a-1}I_{{\rm bl}}^{\mathfrak{m}}(j,a+1-j,b)-\sum_{j=2}^{b-1}I_{{\rm bl}}^{\mathfrak{m}}(a,b+1-j,j)
\sum\limits_{\substack{\beta \in \Pi \beta \ne \alpha}}\frac{\mu_\beta \bigl(e(\beta)+e(\alpha)\bigr)^{2k-1}}{\bigl(e(\beta)-e(\alpha)\bigr)^{2k-1}}= 0 \mbox{\rm and}\sum\limits_{\substack{\beta \in \Pi \beta \ne \alpha}}\frac{\mu_\beta (\mu_\beta +1)e(\beta) \bigl(e(\beta)+e(\alpha)\bigr)^{2k-1}}{\bigl(e(\beta)-e(\alpha)\bigr)^{2k+1}}= 0;
d l^2 = 4M^2 \frac{r - M}{r + M}\big( d \psi + \cos \theta \d \varphi \big)^2 + \frac{r + M}{r - M}dr^2 + \big( r^2 - M^2 \big) \big( d \theta^2 + \sin^2 \theta \d \varphi^2 \big).
c_2(2n,k)=\sum_{i=k}^{2n}{i-1\choose k-1}c_1(2n,i)=\sum_{j=\lceil\frac k2\rceil}^{n}{2j-1\choose k-1}c_1(2n,2j) =\sum_{j=\lceil\frac k2\rceil}^{n}a^{n-j}{2j-1\choose k-1}{n+j-1\choose n-j}.
\frac{i \theta^{i m}}{2 \pi}\frac{i \theta^{j n}}{2 \pi}\int_{0}^{1}d \tau^{\prime}~ e^{- 2 \pi i \tau^{\prime}}l^{m^{\prime}}F_{m m^{\prime}}( x + l \tau^{\prime}) \int_{0}^{1}d \tau^{\prime \prime}~ e^{2 \pi i \tau^{\prime \prime}}l^{n^{\prime}}F_{n n^{\prime}}( x + l \tau^{\prime \prime}) .
\pi_{\mu \nu}= - \frac{1}{4}T_{\mu \alpha}T_{\nu}^{\alpha}+ \frac{1}{1 2}T T_{\mu \nu}+ \frac{1}{8}g_{\mu \nu}T_{\beta}^{\alpha}T_{\alpha}^{\beta}- \frac{1}{2 4}g_{\mu \nu}T^{2}
\lim_{R\to \infty}\lim_{k\to \infty}\int_{\mathbb{R}^{N}}(1-\Phi_{R}(x))\big[ U_k(x)+V_k(x)\big]\,dx = \lim_{R\to \infty}\int_{\mathbb{R}^{N}}(1-\Phi_{R}(x))\,d\mu.
\left(\int_{\varphi_n(E_M)}|Dh_n|^p\,d\mu\right)^{\frac{1}{p}}=\left(\int_{\varphi_n(E_M)}|D\phi_n^0\circ f_n|^p |Df_n|^p\,d\mu\right)^{\frac{1}{p}}&\leq\|D\phi_n^{0}\|_{L^\infty}\left(\int_{\varphi_n(E_M)}|Df_n|^p\,d\mu\right)^{\frac{1}{p}}\ .
F_-(t)u[x] =(2\pi)^{-d}\int_{\mathbb{T}^d}\sum_{y\in\mathbb{Z}^d}e^{i\Phi(x,y,\xi;t)}({}^tL_2)^\ell (s_-(x,\xi)p_+(y,\xi)) u[y] d\xi &=(2\pi)^{-d}\int_{\mathbb{T}^d}\sum_{y\in\mathbb{Z}^d}e^{i(\varphi_a(x,\xi)-\varphi_a(y,\xi))}e^{-it h_0(\xi)}({}^tL_2)^\ell (s_- p_+) u[y] d\xi
O(\epsilon^2 m^3) = \left\| \prod_{m<k< n}P_{n-k}(\epsilon) \left(\prod_{0\le k \le m}P_{n-k}(\epsilon) - \tilde{\Pi}+ \epsilon\tilde{\Pi}\tilde{P}^{(1)}(I-\tilde{P}+\tilde{\Pi})^{-2}\tilde{P}\right) \right\| &= \left\| \prod_{k=1}^{n}P_k(\epsilon) - \tilde{\Pi}+ \epsilon\tilde{\Pi}\tilde{P}^{(1)}(I-\tilde{P}+\tilde{\Pi})^{-2}\tilde{P}\right\|,
F_{2}( n , k ; \Delta^{2}) = - 2 \pi^{2}i \{\frac{1}{1 6 k \mu^{2}}\frac{[ \sqrt{k^{2}- \Delta^{2}}+ k - i \mu ]}{[ k + i \frac{( \Delta^{2}- \mu^{2})}{2 \mu}]}+ \frac{1}{1 6 k \mu^{2}}\frac{[ \sqrt{k^{2}- \Delta^{2}}- k - i \mu ]}{[ k - i \frac{( \Delta^{2}- \mu^{2})}{2 \mu}]}\},
\frac{d}{d\tau}(\frac{g_{\lambda\nu}\dot{x}^{\nu}+g_{\mu\lambda}\dot{x}^{\mu}}{\sqrt{-g_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu}}})=\frac{g_{\mu\nu,\lambda}\dot{x}^{\mu}\dot{x}^{\nu}}{\sqrt{-g_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu}}}(3)
s_1t_1(1-s_2t_2) = \sqrt{g(s_1)g(s_2)g(t_1)g(t_2)}\sqrt{\frac{f(s_1)}{f(s_2)}\frac{f(t_1)}{f(t_2)}}=:F(s_1)G(s_2)F(t_1)G(t_2),
\left\{Q_{r}, Q_{s}\right\}= \left( C^{( 1 3 )}\Gamma_{M N}^{( 1 3 )}\right)_{r s}Z_{\bf 0}^{M N}+ \left( C^{( 1 3 )}\Gamma_{M_{1}M_{2}M_{3}}\right)_{r s}Z_{\bf 0}^{M_{1}M_{2}M_{3}},
S_{I I}+ S_{I I I}~ \sim ~ M^{2}\tau_{f}\left\{1 - \frac{2}{\pi r^{2}}\left( \frac{1}{\sigma_{0}}+ \frac{1}{2 \pi - \sigma_{0}}\right){\sin}^{2}\, \left( \frac{r \sigma_{0}}{2}\right) \right\}