harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
G ( E_{r}; \vec{x}_{b}, \vec{x}_{a}) = i \int_{0}^{\infty}d \tau_{f}\int{\cal D}\vec{p}{\cal D}\vec{x}e x p \{i \int_{0}^{\tau_{f}}d \tau [ \vec{p}\dot{\vec{x}}- f ( \vec{x}( \tau ) ) H ] \}
{\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}
\begin{array}{rl}r_1 u' - q^{{2}}r_3 v + q^{{2}}r_4 u - (q- 1)r_5 u = u'r_1- q v r_3 +ur_4, r_1 v' - q r_5 v + r_6 u = v' r_1 - q v r_5 +u r_6, r_2 u + r_3 v' -q r_5 u' = u r_2 + v' r_3 - qu'r_5, r_2 v+ r_4 v' -q r_6 u' = vr_2 + q^{{2}}v' r_4 - (q - 1) v' r_5 - q^{{2}}u'r_6.\end{array}
b_{\alpha, \beta}\left(\theta^\beta_\pm + \theta^\beta_\mp \right) = 2\mu c_\alpha a \left(| \alpha \cap (\alpha + \beta)| - |\alpha \cap \beta|\right) -2b_{\alpha, \beta}\xi_\beta = 2\mu c_\alpha a \left(|\alpha| -2 |\alpha \cap \beta| \right) &= \frac{2\mu^2b_{\alpha, \beta}c_\alpha}{\lambda}\xi_\beta &= 2(1-\lambda) b_{\alpha, \beta}\xi_\beta
\left[ \sum_{j = 1}^{n - 1}( \nu_{j}- \nu_{n}) \left( \partial_{j}\Delta_{n}\right) + ( d - \nu_{n}- \sum_{j = 1}^{n}\nu_{j}) G_{n - 1}\right] I_{n}^{( d )}= \sum_{j = 1}^{n}\left( \partial_{j}\Delta_{n}\right) I_{n , j}^{( d - 2 )}+ 2 \Delta_{n}I_{n}^{( d - 2 )}.
\phi_{n,n-2j}\left( 1+\mathrm{i},1-\mathrm{i},0,\ldots\right) =S\left(n,j;\kappa,1\right) , \psi_{n+1,n-2j}\left( 1+\mathrm{i},1-\mathrm{i},0,\ldots\right) =2\mathrm{i}S\left( n,j;\kappa,2\right) .
v_{+,a}(x)=2\phi_3(x)\frac{\int_{-\infty}^xdy\thinspace\phi^{-2}_3(y)\int_x^adt\phi^{-2}_3(t)\int_y^t\phi_3(z)dz}{\int_{-\infty}^a\phi^{-2}_3(y)dy},\x\le a;
g(x)-g(x-h) = e^{\frac{f(x)}{2}}- e^{-\frac{f(x)}{2}}-e^{\frac{f(x-h)}{2}}+ e^{-\frac{f(x-h)}{2}}& = \left( e^{\frac{f(x)}{2}}-e^{\frac{f(x-h)}{2}}\right) \left( 1 +e^{-\frac{f(x)}{2}-\frac{f(x-h)}{2}}\right) & \leq 2\left(e^{\frac{f(x)}{2}}-e^{\frac{f(x-h)}{2}}\right) \leq e^{\frac{f(x)}{2}}(f(x)-f(x-h)),
\frac{\partial^2}{\partial\sigma_i\partial\sigma_j}L(0)=2K(x^*)e_i\cdot e_j =\left(\frac{1}{\vert p-x^*\vert}+\frac{1}{\vert x-x^*\vert}\right)\left(I_3-\vartheta\otimes\vartheta\right)e_i\cdot e_j &=\left(\frac{1}{\vert p-x^*\vert}+\frac{1}{\vert x-x^*\vert}\right)\left(\delta_{ij}-(\vartheta\cdot e_i)(\vartheta\cdot e_j)\right)
(p_{n,x,u}-p_\cap) \biggl\|V^{-1/2}\begin{pmatrix}1-p_{n,x,u}-p_{n,y,v}\end{pmatrix}\biggr\|^{3}\leq (p_{n,x,u}-p_\cap)p_{n,y,v}^{3/2}|V|^{-3/2}\lesssim \Bigl(\frac{n}{k\|z\|}\Bigr)^{1/2},
\int_{1}^{\infty}\frac{ds}{\left(\int_{0}^{s}F(t)dt\right)^{\frac{p}{2p-q+1}}}<\infty\quad\mbox{and}\int_{1}^{\infty}\frac{s\ds}{\left(\int_{0}^{s}F(t)dt\right)^{\frac{p}{2p-q+1}}}=\infty.
\Biggl( \phi_1(x),{\phi_2(x)\choose\phi_3(x)},{\phi_4(x)\choose\phi_5(x)}\Biggr) \rightarrow \Biggl( c\, e\sp{\lambda x}\phi_1(x), U{e\sp{-\lambda\sp\prime x}\phi_2(x)\choose e\sp{-\lambda\sp\prime x}\phi_3(x)},V{e\sp{\lambda\sp{\prime\prime}x}\phi_4(x)\choose e\sp{\lambda\sp{\prime\prime}x}\phi_5(x)}\Biggr)
\left\|\frac{\partial}{\partial s}\left(\delta_h\nabla \psi_s\right)\right\|_{L^p(\mathbb{R}^N)}\le 2\,\left\|\frac{\partial}{\partial s}\nabla \psi_s\right\|_{L^p(\mathbb{R}^N)}\le C\, [\psi]_{\mathcal{B}^{\alpha,p}_\infty(\mathbb{R}^N)}\,s^\frac{\alpha-3}{2},
|Q(m,t)| = \frac{4^m\,2}{\pi \Gamma(m+1/2)}* \times \int_0^\infty e^{-t^2/(4v)}\int_0^v e^{-4w}w^{m-1}\Big( \frac{w}{v}\Big)^{3/2}\Big(1-\frac{w}{v}\Big)^{m-1/2}\,dw\, \frac{dv}{v^{1/2}}.
H^1_{\rm et}(X',\, E_{G_c}(G)')\,\stackrel{\gamma'}{\longrightarrow}\, H^1_{\rm et}(X',\, (E_{G_c})')\,\stackrel{\beta'}{\longrightarrow}\,H^2_{\rm et}(X',\,{\mathbb Z}/2{\mathbb Z})\,=\,{\mathbb Z}/2{\mathbb Z}
\int_{-\infty}^{\infty}\frac{H(\epsilon)}{e^{\beta(\epsilon-\mu)}+1}d\epsilon=\int_{-\infty}^{\mu}H(\epsilon)d\epsilon+\frac{\pi^{2}}{6}(\frac{1}{\beta})^{2}H^{\prime}(\mu)+O(\frac{1}{\beta\mu})^{4}
{\rm (i)}\,\,\,\,\, \sigma_{A}(x, \xi)=u_{\xi}^{-1}(x)(Au_{\xi})(x);{\rm (ii)}\,\,\,\,\, \sigma_{A}(x,\xi)=u_{\xi}^{-1}(x)\int_{\Omega}K_{A}(x,y)u_{\xi}(y)dy.
Y_m = \Gamma \cdot Y^1_m \cdots Y^\ell_m = \Gamma \cdot (-1)^{s_m}\cdot q^{w_m}\cdot \prod_{1\leq j \leq \ell}^{\to}\left(\prod_{i\in Q_0^j}^{\to}y_{e_i}^{\gamma(i)}\right)
S_{\mathrm{main}}=\sum_{L=1}^{\infty}L\sum_{U,\, V <\frac{D}{L}}\tau (U)\tau(V)\,x_{UL}\bar{x}_{VL}\sum_{T|L}\frac{\mu(T)}{T}\sum_{A=1}^{\infty}\frac{G_k(A^2T^2UV\slash p)}{A},
S = \frac{r_{h}^{2}}{9 0}{\tilde{\epsilon}}^{- 2}+ \left[ \frac{\Delta_{h}^{\prime \prime}r_{h}^{2}}{1 8 0}+ \frac{r_{h}^{2}\mu^{2}}{6}\right] \log{\tilde{\epsilon}}+{\mathrm{t e r m s f i n i t e a s}}{\tilde{\epsilon}}\rightarrow 0 .
\left(-e^{\xi}\right)^{n}h_{n}\left(\sinh\xi\vert q\right)=\sum_{k=0}^{\infty}\frac{q^{\binom{k+1}{2}}e^{2k\xi}}{\left(q;q\right)_{k}}A_{k}\left(q^{\left(k-n\right)/2}e^{2\xi}\right),
\lVert \frac{1}{N}\sum_{n = 1}^N u_n^* T u_n \rVert_{\infty, 2}^2 \leq \frac{1}{N^2}\sum_{n, m = 1}^N \lVert u_m^* T u_m u_n^* T u_n \rVert_{\infty, 1}&\leq \frac{2}{N^2}\sum_{1 \leq m < n \leq N}2^{-n}+ \frac{1}{N^2}\sum_{n = 1}^N \lVert u_n^* T^2 u_n \rVert_{\infty, 1}&\leq \frac{2}{N}+ \frac{\| T \|^2}{N}\to 0.
E(x_{i}|t)|y_{1},\ldots ,y_{n}\rangle =\prod_{l=1}^{n}\left(1+x_{i}~y_{l}\right) ~|y_{1},\ldots ,y_{n}\rangle E(x_{i}|t)|z_{1},\ldots ,z_{k}\rangle =\left(\prod_{j=1}^{N}(1+x_{i}T_{j})+(-1)^{k}qx_{i}^{N}\right) \prod_{l=1}^{k}\frac{1}{1-x_{i}~z_{l}}~|z_{1},\ldots ,z_{k}\rangle \;.
\Vert \{a_{k}\}|~\mathcal{M}_{p}(l_{q_{1}}^{s_{1},\alpha},l_{q_{2}}^{s_{2},\alpha})\Vert =\Vert \{a_{k}\}\Vert_{l_{q_{1}}^{s_{1},\alpha}\rightarrow l_{q_{2}}^{s_{2},\alpha}}=\sup_{\Vert \{\lambda_{k}\}\Vert_{l_{q_{1}}^{s_{1},\alpha}=1}}\Vert\{a_{k}\lambda_{k}\}\Vert_{l_{q_{2}}^{s_{2},\alpha}}.
U=\frac{U_1+(U_2)^{\eta^{\ast}}}{2},\X=\frac{\tilde{X}+(\tilde{X})^{\eta^{\ast}}}{2}, Y=\frac{\tilde{Y}+(\tilde{Y})^{\eta^{\ast}}}{2},\Z=\frac{\tilde{Z}+(\tilde{Z})^{\eta^{\ast}}}{2}, &U_1=A_{1}^{\dagger}C_{1}+L_{A_{1}}S_1,\U_2=U_1^{\eta^{\ast}},
\left|\int\limits_{\partial Q_\ell}\frac{\Vec{n}x|u|^2}{|x|^2}\right|\,dx \leq \frac{2}{\ell}\int\limits_{\partial Q_\ell}|u|^2\,dx=\frac{2}{\ell}\left( A^{2,3}(-\frac{\ell}{2})+A^{2,3}(\frac{\ell}{2})+ A^{1,3}(-\frac{\ell}{2})+A^{1,3}(\frac{\ell}{2})+ A^{1,2}(-\frac{\ell}{2})+A^{1,2}(\frac{\ell}{2})\right).
\theta^{(d)}_M(t)=\Vert M\Vert K^{(d)}_M(o,t) =\Vert M\Vert\sum_{(z_1,\ldots,z_d)\in\prod^{d}_{j=1}m_j\mathbb{Z}}e^{-2dt}\prod^{d}_{j=1}I_{z_j}(2t) &=\Vert M\Vert\sum^{\infty}_{h=0}\sum_{z=(z_1,\ldots,z_d)\in R^{(d)}_{M}(h)}m^{(d)}_{M}(z)e^{-2dt}\prod^{d}_{j=1}I_{z_j}(2t).
\mathsf b(z;\eta) = \bigl(\exp \eta \partial_{\tilde e_0}\bigr) \, \sum_{r\geq0}(-z)^r \tilde{e}_r, \mathsf c(z;\eta) = \exp\bigl( \eta \sum_{r\geq 0}(-z)^r \partial_{\tilde e_r}^\perp \bigr)
(T_k1)(x) = -\frac{1}{2\pi k}\int_0^\infty e^{ix\xi}\hat u(\xi)~d\xi + \frac1{2\pi k}\int_0^\infty\frac{\xi e^{ix\xi}}{\xi-k}\hat u(\xi)~d\xi &= -\frac{C_+u(x)}{k}+ \frac1{2\pi k}\int_0^{\infty}\frac{\xi^{1-\epsilon}e^{ix\xi}}{\xi-k}[\xi^{\epsilon}\hat u(\xi)]~d\xi.
N \epsilon_{Y_{j, v}}(A) \lambda_{Y_{j, v}, v}\left(\pi_{j, v}(P)\right) \leq N\left(\epsilon_{j, v}+\delta'\right) \lambda_{Y_{j, v}, v}\left(\pi_{j, v}(P)\right) &=N\left(\epsilon_{j, v}+\delta'\right) \lambda_{E_{j, v}, v}(P) & \leq \lambda_{\pi_{j, v}^{*}F_{j, v}, v}(P)=\lambda_{F_{j, v}, v}\left(\pi_{j, v}(P)\right)
\hat{\boldsymbol{\theta}}=\arg\underset{\boldsymbol{\theta}}{\min}\{-\log(p(\boldsymbol{\Upsilon}_{\mathrm{c}}\vert\mathbf{X},\boldsymbol{\theta})\}=\arg\underset{\boldsymbol{\theta}}{\min}\Bigl\{\log|\mathbf{K}|+\boldsymbol{\Upsilon}_{\mathrm{c}}^{\mathrm{T}}\,\mathbf{K}{}^{-1}\,\boldsymbol{\Upsilon}_{\mathrm{c}}\Bigr\}.
\delta_{a}( l ) = - \int \frac{k^{\prime}}{2 l}d l - \frac{1}{1 6 \pi^{2}}( C_{G}^{a}- C_{M}^{a}) k ( l ) - \frac{2}{1 6 \pi^{2}}\sum_{A}C_{A}^{a}\ln ( 1 - p_{A}l ) .
d(\bar{y}(t),S(\bar{u}(t))) \leq \left\lVert \frac{1}{m}\sum_{k=1}^m y(t+\tau_{n_k}) - \bar{y}(t) \right\rVert + d\left(\frac{1}{m}\sum_{k=1}^m y(t+\tau_{n_k}), S(\bar{u}(t))\right) &\leq \left\lVert \frac{1}{m}\sum_{k=1}^m y(t+\tau_{n_k}) - \bar{y}(t) \right\rVert + d\left(\frac{1}{m}\sum_{k=1}^m S_{\ell_k^t}(u(s_{\ell_k^t})), S(\bar{u}(t))\right),
\varpi_k=\eta_k+\sum_{\ell=0}^{k-1}\eta_\ell\sum_{j=1}^{k-\ell}\frac{C_{k\ell j}}{\lambda_{2}^j}\sum_{{\mbox{\tiny$\begin{array}{c}3\leq n_1,\ldots,n_j\leq 2n+3 n_1+\cdots+n_j=k-\ell+2j \end{array}$}}}\lambda_{n_1}\cdots\lambda_{n_j},
\begin{array}{c}\Psi_i[{\bf r}(\beta^{\prime});\beta_n,\beta_1]=\psi_i({\bf r}_n,\beta_n)\psi_i({\bf r}_{n-1},\beta_{n-1})...\psi_i({\bf r}_1,\beta_1) \int D{\bf r}(\beta^{\prime})\cdot \Psi_i^{*}[{\bf r}(\beta^{\prime});\beta ,\beta_0]\cdot \Psi_j[{\bf r}(\beta^{\prime});\beta ,\beta_0]=\delta_{ij}\end{array}
S^{(1)}_{E_{ij}}= \Phi_1(p_{i}\varphi^{-1}(\varphi(S^{(1)}_{E_{ij}})) p_{j}) = \Phi_1(\varphi^{-1}(p_{i'}\varphi(S^{(1)}_{E_{ij}})p_{j'})) = \Phi_1(\varphi^{-1}(\sum^{\infty}_{n=k}a_nS^{(n)}_{E_{i'j'}})) & = \sum^{\infty}_{n=k}a_n\Phi_1(\varphi^{-1}(S^{(n)}_{E_{i'j'}})) = 0
\frac{\delta S}{\delta A_{0}}= - \frac{\sqrt{3}}{8 \pi}\partial_{i}\, \mathrm{I m}\left[ \sigma_{\alpha \beta}^{0 i}\left( D^{\alpha \beta}{\cal F}^{\prime}+{\cal F}^{\prime \prime}D^{\alpha \beta}{\cal A}\right) \right] | .
\begin{array}{r c l}{{\delta_{\epsilon}A^{a}}}&{{=}}&{{\{A^{a}, Q_{\epsilon}\}= g f^{a b c}\epsilon^{b}A^{c}\ \ \ ,}}\\{{\delta_{\epsilon}\pi^{a}}}&{{=}}&{{\{\pi^{a}, Q_{\epsilon}\}= g f^{a b c}\epsilon^{b}\pi^{c}\ \ \ ,}}\\{{\delta_{\epsilon}\phi^{a}}}&{{=}}&{{- \dot{\epsilon}^{a}+ g f^{a b c}\epsilon^{b}\phi^{c}\ \ \ ,}}\\ \end{array}
\mathbf U(t) = \mathbf S(t) \mathbf u_0 + i \int_0^{t \wedge \mu}\mathbf S(t-s) \mathbf N(\mathbf U(s)) \, ds + i \int_0^t \mathbf S(t-s) \mathbf M(\mathbf U(s)) \, dW(s), \mathbf V(t) = \mathbf S(t) \mathbf u_0 + i \int_0^{t \wedge \mu}\mathbf S(t-s) \mathbf N(\mathbf V(s)) \, ds + i \int_0^t \mathbf S(t-s) \mathbf M(\mathbf V(s)) \, dW(s).
V_L = \frac{1}{2}J_{-N - 1}( \sigma_{-N - 1}^{(1)}\sigma_{-N}^{(1)}+ \sigma_{-N-1}^{(2)}\sigma_{-N}^{(2)}), V_R = \frac{1}{2}J_{N}( \sigma_{N}^{(1)}\sigma_{N+1}^{(1)}+ \sigma_{N}^{(2)}\sigma_{N+1}^{(2)}),
\nu_{+}- \nu_{-}= \frac{N}{2 4 \cdot 8 \pi^{2}}\int_{\cal M}\mathrm{t r}( R \wedge R ) - \frac{g^{2}N_{f}}{3 2 \pi^{2}}\int_{\cal M}F_{\mu \nu}^{a}\tilde{F}^{\mu \nu a}d^{4}x ,
M_{\tau}(t):=\mathbb{E}\Big(e^{t\tau}\Big)=e^{-\lambda b'}=e^{-\frac{\lambda}{\sigma}\ln\left(\frac{K k_2}{k_2-1}\right)^{\frac{1}{k_2}}}=\left(\frac{Kk_2}{k_2-1}\right)^{-\frac{\lambda_+(t)}{k_2\sigma}}
|w_1^\beta-w_2^\beta| \leq \int_0^1\beta\left(tw_1+(1-t)w_2\right)^{\beta-1}|w_1-w_2|dt \leq \int_0^1\beta\left(tw_1^{\beta-1}+(1-t)w_2^{\beta-1}\right) |w_1-w_2|dt \leq \frac{\beta}{2}\left(w_1^{\beta-1}+w_2^{\beta-1}\right) |w_1-w_2|.
[z^N]\log \frac{z}{f(z)}=\frac{1}{2\pi N i}\int_{|z|=\delta}\frac{1}{z^{N}}d\log\frac{z}{f(z)}&=-\frac{1}{2\pi N^2i}\int_{|w|=\delta_1}\frac{d\Phi(w)^N}{w^N}&=-\frac{1}{N}[w^N] \Phi(w)^N.
S_{cl}(e^i_\mu,\omega_\mu) = \int{d^2}x\;e\bigg(\frac{1}{16\alpha}R_{\mu\nu}{}^{ij}R^{\mu\nu}{}_{ij}-\frac{1}{8\beta}T_{\mu\nu}{}^{i}T^{\mu\nu}{}_{i}-\gamma\bigg),
\left( \left( T_{1}+T_{2}\right)^{2}+\left( T_{1}-T_{2}\right)^{2}\right) \psi_{nj}& =\left( j+1\right) \left( j+2\right) \psi_{n-2,j+2}-\left(2N\kappa+n+j+1\right) \left( 2N\kappa-2\kappa+n+3j+1\right) \psi_{n-2,j}& +2\left( 2N\kappa+n+j+1\right) \left( 2N\kappa+n-j+1\right)\psi_{n-2,j-2}.
\mathfrak{G}_{\sigma_{\lambda}}^{(\beta)}(X_n)={\operatorname{DET}\big| x_i^{\lambda_j +n-j}(1+\beta x_i)^{j-1}\big|_{1 \le i,j \le n}\over \prod\limits_{1 \le i < j \le n}(x_i-x_j)},
\widehat{Q}(x) := \Theta^{-1}(x) \begin{pmatrix}p_1+h_1' p_1-h_1' h_2 -h_2 p_1+h_1' p_1-h_1' h_2 -h_2 -h_2 -h_2 p_2+h_2' p_2-h_2' h_2 h_2 p_2+h_2' p_2-h_2' \end{pmatrix}.
\delta(s(e)) = \tilde{\mathbf Q}_\alpha( g_p ) \cdot \begin{cases}: \delta\big(e^{-\pi i ( \frac{k_L}{3}-\frac{k_R}{3}+\frac{1}{2})}\big) : \delta \big( e^{-\pi i ( -\frac{k_R}{3}- \frac{1}{6}+ \frac{1}{2})}\big) \ .\end{cases}
\exp{[ T ]}( a_{1}^{\dagger})^{M}| 0 \rangle = \prod_{j = 2}^{r}\exp ( \xi_{j}X_{( - 1 , j )}) \prod_{j = 2}^{r}\exp ( \xi_{- j}X_{( - 1 , - j )}) ( a_{1}^{\dagger})^{M}| 0 \rangle .
T_{\eta,1}=\tilde{P}f^{-\frac{\sigma_3}{4}}\Psi_0^{Ai}\frac{\mathcal{K}_1}{f^{3/2}}(\Psi_0^{Ai})^{-1}f^{\frac{\sigma_3}{4}}{\tilde{P}}^{-1}, f=f_{\eta}=-\left(\frac{3}{2}g\right)^{\frac{2}{3}}.
A_{ij}=A_{(ij)}+A_{[ij]},\,\,\,\,\,\,\,\,\,\,A_{(ij)}=\frac{1}{2}\left(A_{ij}+A_{ji}\right)\,\,\,\,\,\,\,\,\,\&\,\,\,\,\,\,\,\,\,A_{[ij]}=\frac{1}{2}\left(A_{ij}-A_{ji}\right)
T_{1+s}(\pi_{XY}) \quad:=\inf\left\{R:\;\lim_{n\to\infty}D_{1+s}(P_{X^{n}Y^{n}}\|\pi_{X^{n}Y^{n}})=0\right\}, \widetilde{T}_{1+s}(\pi_{XY}) \quad:=\inf\Big\{R:\;\lim_{n\to\infty}\frac{1}{n}D_{1+s}(P_{X^{n}Y^{n}}\|\pi_{X^{n}Y^{n}})=0\Big\}.
\left( \begin{array}{cc}0 B 0 0\end{array}\right) \cdot aAv = tr(\psi(aA).Q.B^{T})v &= tr(\psi(a).\psi(A).Q.B^{T})v = tr(\psi(a).(A \otimes I -1\otimes A^{T}).Q.B^{T})v &=tr(Q.B^{T}.\psi(a).(A \otimes I))v- tr(\psi(a).A^{T}.Q.B^{T})v &= tr(Q.B^{T}.\psi(a))Av-tr(\psi(a).(Q^{T}.A)^{T}.B^{T})v &= tr(Q.B^{T}.\psi(a))Av.
{}_pF_q\left[\begin{matrix}a_1\;,\;\ldots\;,\;a_p b_1\;,\;\ldots\;,\;b_q\end{matrix};z\right] :=\sum_{l=0}^\infty\frac{(a_1)_l\ldots (a_p)_l}{(b_1)_l\ldots (b_q)_l}\frac{z^l}{l!},
\intop_{0}^{\infty}\frac{x^{s+\frac{1}{2}}J_{s-\frac{1}{2}}(bx)}{x^{2}+a^{2}}\,dx=a^{s-\frac{1}{2}}\,K_{s-\frac{1}{2}}(ab),\,\,\,\,a,b>0.
\int_{|z|\geq R}\left(|z|^{\frac{\sigma-l-1}{2}}e^{-\langle \rho, z\rangle}e^{|\rho|\frac{|z|}{2}}e^{-\frac{\xi}{2}|z|}\right)^{p'}|z|^{-\alpha p'}dz \leq C <\infty.
E\sum_{k=0}^{[T/\epsilon ]}\int_{k\epsilon}^{(k+1)\epsilon}\mathcal\mathbb 1_{[0,\tau_N^{n,m}]}(r)|b(r,X^m_{k\epsilon \wedge \tau^m},\mu^m_r,\alpha^m_r)- b(r,X^m_{r\wedge \tau^m},\mu^m_r,\alpha^m_r)|^2 dr\leq C(N,T)\epsilon.
K [ t , p , q , x ] = \frac{\pi^{2}}{2 t^{2}}\big ( \mathrm{e}^{- t ( p^{2}x - q^{2}x^{2})}- \mathrm{e}^{- t ( p^{2}x - q^{2}x^{2}) - p \circ p / t}\cos 2 x \theta ( p , q ) \big ) ~ .
P_2 \left( n \right) = \delta^{4n - 8}\left( \frac{1}{n - 3}\bar B_1 \left( \frac{1}{{\delta^2}},n - 1,n - 2 \right) + \frac{1}{{n - 1}}\bar B_1 \left( \frac{1}{{\delta^2}},n - 3,n \right) \right. \left. - \frac{2}{{n - 2}}\bar B_1 \left( \frac{1}{{\delta^2}},n - 1,n - 2 \right) \right).
- \beta \int d \tau \sqrt{\dot{x}^{2}}\, \frac{\varepsilon_{\mu \nu \rho}\dot{x}^{\mu}\, \ddot{x}^{\nu}\stackrel{\ldots}{x}^{\rho}}{{( \dot{x}\ddot{x})^{2}\, - \, \dot{x}^{2}\, \ddot{x}^{2}}},
Rm^+(X,Y,Z,W)= Rm(X,Y,Z,W)+\frac{1}{2}\nabla_XH(Y,Z,W)-\frac{1}{2}\nabla_YH(X,Z,W) &-\frac{1}{4}\langle H(X,W),H(Y,Z)\rangle+\frac{1}{4}\langle H(Y,W),H(X,Z)\rangle,
\widehat{|h_0|^2}(\xi) =\lim_{A\to\infty}\int_1^A \frac{e^{-2\pi i \xi s}}{s}ds=\int_1^\infty \frac{e^{-2\pi i \xi s}}{s}ds= \int_{2\pi \xi}^{\mathrm{sgn}(\xi)\times\infty}\frac{e^{-is}}{s}ds,
E(u(t)):=\frac{1}{2}\int_{\mathbb R^3}|\nabla u(t)|^2\,dx+\frac{1}{q+1}\int_{\mathbb R^3}|u(t)|^{q+1}\,dx-\frac{1}{p+1}\int_{\mathbb R^3}|u(t)|^{p+1}\,dx,
I_1 := \int_{M\setminus B_R}V^{-\beta+\frac{t}{\sigma-p+1}}|\nabla\varphi|^{\frac{p(\sigma-t)}{\sigma-p+1}}d\mu, I_2 := \int_{B_{2nR}\setminus B_{nR}}\varphi^{\frac{p(\sigma-t)}{\sigma-p+1}}|\nabla\eta_n|^{\frac{p(\sigma-t)}{\sigma-p+1}}V^{-\beta+\frac{t}{\sigma-p+1}}d\mu.
\langle{\cal O}\rangle_{\sigma}= G_{\sigma}^{( n )}( \overbrace{x^{- 1}\zeta , \cdots , x^{- 1}\zeta}^{n}, \overbrace{\zeta , \cdots , \zeta}^{n})^{- \varepsilon_{n}\cdots - \varepsilon_{1}\varepsilon_{1}^{\prime}\cdots \varepsilon_{n}^{\prime}}.
F_{r}( \kappa ) = \left\{\prod_{A = 1}^{9}m_{\pm}( \beta_{A}f_{A}^{\mathrm{i n}}) \right\}\times \left\{\prod_{i = 2 , 3}p ( \beta_{i}^{\mathrm{v a c}}f_{i}^{\mathrm{v a c}}) \right\}\ .
f_1 = \frac{(2 n - 1) \sqrt{x}- n}{2 (n - 1) \sqrt{x}}\; \; < 1 \; , \; \; \; f_2 = \frac{(2 n - 1) \sqrt{x}+ n}{2 (n - 1) \sqrt{x}}\; \; > 1 \; .
\delta_{n}^{(1)}= \delta_{n}^{(1)}(v,A,b,\sigma,q,B,\eta,\gamma) = \frac{b K_n}{\gamma^{1/q}n^{1/2-1/q}}+\frac{(b \sigma^2 K_n^2)^{1/3}}{\gamma^{1/3}n^{1/6}}.
d s_{5}^{2}\approx e^{- 2 | y | / \ell}\left[ - \rho^{2}\frac{\Delta}{\Sigma^{2}}d t^{2}+ \frac{\Sigma^{2}}{\rho^{2}}\left( d \varphi + \frac{2 G_{4}J r}{\Sigma^{2}}d t \right)^{2}\sin^{2}{\theta}+ \frac{\rho^{2}}{\Delta}d r^{2}+ \rho^{2}d \theta^{2}\right] + d y^{2},
\Omega_{2}^{( 1 )}( x ) = \int d y \ \lambda_{i_{1}}^{( 1 )}( x , y ) L_{i_{1}}^{( 1 )}( y ) = - \partial_{1}\alpha_{2}^{( 1 )}+ e \alpha_{3}^{( 1 )}+ e \alpha_{4}^{( 1 )}+ \alpha_{5}^{( 1 )}= 0 .
\mathcal{L}^{\mathrm{optimal}}(\overline{\pmb{\theta}}) = \max\{\mathcal{L}^{\mathrm{NCP}}(\overline{\pmb{\theta}}), \mathcal{L}^{\mathrm{CLIP}}(\overline{\pmb{\theta}}), \mathcal{L}^{\mathrm{KL}}(\overline{\pmb{\theta}})\}.
\left( \begin{array}{c}\phi \psi \end{array}\right) (\theta )|\frac{1}{2}\rangle =\left( \begin{array}{ll}\mathcal{A}_{+}\mathcal{B}\tilde{\mathcal{B}}\mathcal{A}_{-}\end{array}\right) \left( \begin{array}{c}\phi \psi \end{array}\right) (-\theta )|\frac{1}{2}\rangle
\widetilde{F}\left( \beta \right) = - \frac{1}{\beta}\log \widetilde{Z}\left( \beta \right) = - \frac{1}{\beta}\sum_{\ell = 1}^{\infty}\ell^{2}\log \left( 1 - e^{- \beta \sqrt{\ell^{2}+ \mu_{e f f}^{2}}/ r}\right) .
v_{\mu}^{(1)}=-\frac{1}{2\pi\kappa}H_{\mu}^{(2)}+G_{\mu}^{(1)}\;\;\;, \;\;\; v_{\mu}^{(2)}=-\frac{1}{2\pi\kappa}H_{\mu}^{(1)}+G_{\mu}^{(2)}\;\;,
\left.\mathbb{R}^{N}\ni y\longmapsto p\left( y\right) ,p^{2}\left( y\right):=\inf_{v\in\mathcal{L}^{2}\left( q;\mathbb{R}^{N}\right)}\mathbb{E}^{0,\gamma_{0}}\left[ \int_{0}^{T}\left\vert\Pi_{\left( \ker B^{\ast}\left( \Gamma_{t}^{\gamma_{0}}\right) \right)^{\bot}}\left( Y_{t}^{y,v}\right) \right\vert^{2}dt\right] \right.
S_{e f f}= \int{d^{4}x \left({\Pi^{i}\stackrel{\bf{\cdot}}{A_{i}}+{\cal P}\stackrel{\bf{\cdot}}{C}+{\overline{{{\cal P}}}}\stackrel{\bf{\cdot}}{\overline{{C}}}+ \Pi^{0}\stackrel{\bf{\cdot}}{A_{0}}- H_{0}- \left[{\Omega , \Psi}\right]}\right)},
\nabla_\mu e_\nu^{\!~~A a}\equiv \partial_\mu e_\nu^{\!~~A a}-\Gamma_{\mu\nu}^{~~~\rho}e_\rho^{\!~~A a}+\omega_{\mu~~B}^{\!~~A}e_\nu^{\!~~B a}+\omega_{\mu\!~~b}^{\!~~a}e_\nu^{\!~~A b}= 0,
|\psi (t)-\psi (u)| =\psi (t)=\dfrac{t-a_{2n-1}}{b_{2n-1}-a_{2n-1}}\leq \dfrac{t-u}{b_{2n-1}-a_{2n-1}}\leq \dfrac{t^{1/k}}{b_{2n-1}-a_{2n-1}}< \dfrac{(b_{2n-1})^{1/k}}{b_{2n-1}-a_{2n-1}}<\dfrac{(b_{2n-1})^{1/k}}{2^{2n-1}(b_{2n-1})^{1/k}}<\dfrac{1}{2^{2n-1}}.
a_{0}=\int_{\cap_{i=1}^{k}D_{\Sigma_{i}}}^{}\sqrt{-1}\partial\bar{\partial}f(s)=\int_{\mathbb{C}\setminus{0}}^{}\sqrt{-1}f''(s)\frac{dw\wedge d\bar{w}}{\lvert w\rvert^{2}}=2\pi\big(\lim_{s\rightarrow\infty}f'(s)-\lim_{s\rightarrow -\infty}f'(s)\big)=2\pi m
l_R(b)l_R(-c) =l_R(b)l_R(-abd)=l_R(b)l_R(ad)=l_R(b)l_R(a)+l_R(b)l_R(d) =l_R(a)l_R(b)+l_R(b)l_R(d)=l_R(c)l_R(d)+l_R(b)l_R(d) =l_R(bc)l_R(d)=l_R(ad)l_R(d).
-12q^7-136q^7s-48q^7s^2 + q^8 + 32q^8s + 16 q^8s^2 + 16q^9 = \left(-12q^7+ q^8\right) + \left(-136q^7s-48q^7s^2 \right)+ \left(32q^8s + 16 q^8s^2 + 16q^9 \right) \le -184q^7s^2 + 64q^8s \le 0 \, ;
\frac{1}{2\pi i}\oint_{C}\frac{p^{\prime}(z)}{p(z)}z^{m}dz=\sum_{z\in G:p(z)=0}\frac{p^{\prime}(z)z^{m}}{p^{\prime}(z)}=\sum_{z\in G:p(z)=0}z^{m}.
g(\mathsf{x},t,\mathsf{y})=\sum_{\mathsf{n}\in \mathbb{N}^{d}}\exp \left[-tE_{\mathsf{n}}\right] \mathsf{h}_{\mathsf{n}}\left( \mathsf{x}\right) \mathsf{h}_{\mathsf{n}}\left( \mathsf{y}\right)
E\left[T_2\right] = \frac{\nu_2}{\alpha_2}\, M H_{\nu_2 M}\, + \, O\left(e^{-\varepsilon M}\right)= \big(\lambda^{-1}\nu_1 + \nu_2\big) M H_{\nu_2 M}\, + \, O\left(e^{-\varepsilon M}\right),M \to \infty.
Q_0(x) =\dfrac{1}{2}\ln\left(\dfrac{1+x}{1-x}\right), Q_1(x) =\dfrac{x}{2}\ln\left(\dfrac{1+x}{1-x}\right)-1, Q_2(x) =\dfrac{3x^2-1}{4}\ln\left(\dfrac{1+x}{1-x}\right)-\dfrac{3x}{2}, Q_3(x) =\dfrac{5x^3-3x}{4}\ln\left(\dfrac{1+x}{1-x}\right)-\dfrac{5x^2}{2}+\dfrac{2}{3}.
\begin{aligned}\boldsymbol{K}_1( \widehat{\gamma}_{1,3}, \mu_1) \leq \left( \frac{\eta}{\eta_0}\right) \boldsymbol{K}_1(\gamma^{\eta_1, \delta}_{1,3}, \mu_1) + \left( 1- \frac{\eta}{\eta_0}\right) \boldsymbol{K}_1(\widetilde{\gamma}_{1,3}, \mu_1) \leq \eta. \end{aligned}
\Big(\frac{\tau z - 1}{z - \tau}\frac{w - \tau}{\tau w - 1}\Big)^{N-n}= \Big(1 - \frac{\frac{1}{\tau}- \tau}{z - \tau}\Big)^{N-n}\Big(1- \frac{\frac{1}{\tau}-\tau}{w - \tau}\Big)^{-N+n}\to e^{- \frac{2 \kappa}{z - 1}+ \frac{2 \kappa}{w - 1}}.
\psi(\{u,v,w\}_{T})=\psi(\theta(T(v),T(w))u)=\theta(\phi(T(v)),\phi(T(w)))\psi(u) \;\;\; =\theta(T^{'}(\psi(v)),T^{'}(\psi(w)))\psi(u)=\{\psi(u),\psi(v),\psi(w)\}_{T^{'}}.
D(t_{s+1}) \leq D(t_{s})\prod_{\ell=t_{s}+1}^{t_{s+1}}a(\ell)+\sum_{\tau=t_s}^{t_{s+1}-1}b(\tau)\prod_{\ell=\tau+2}^{t_{s+1}}a(\ell)\cr \leq D(t_{s})\lambda+c(s),
s_0^2 \quad=\frac{M_d^2}{2\sigma^2}- \sqrt{p\left(\frac{M_d^2}{2\sigma^2}- \frac{p}{4}\right)}\quad\ge\frac{M_d^2}{4\sigma^2}\quad= (\log{d}) \frac{3}{4}\frac{\xi_1(1-\tau)}{\sigma^2 \gamma_1},
\begin{array}{ll}e = r^{m_{j}-2}\left(e_1({\theta}) + re_{2}(r, \theta) \right), f = r^{m_{j}-2}\left(f_1({\theta}) + rf_{2}(r, \theta) \right), g = r^{m_{j}-2}\left(g_1({\theta}) + rg_{2}(r, \theta) \right).\end{array}
{\rm Tr}=-\frac{\Gamma K}{(K+c^{\star})^2}-1<0,\,\,{\rm Det}=\frac{\Gamma K}{(K+c^{\star})^2}-\lambda T'(c^{\star}),\,\,{\rm Discr}=\left(\frac{\Gamma K}{(K+c^{\star})^2}-1 \right)^2+4\lambda T'(c^{\star})>0.
r_{2}=r_{1}(1+\frac{b}{r_{1}}sin\theta+\frac{b^{2}}{2r_{1}^{2}}cos^{2}\theta+\cdot\cdot\cdot)=r_{1}+bsin\theta+\frac{b^{2}}{2r_{1}}cos^{2}\theta+\cdot\cdot\cdot.
T_{*}= \frac{4 \pi m_{W}^{2}}{g^{2}\left[ \left( 5 + 8 \, \frac{m_{W}^{2}}{m_{H}^{2}}\right) \ln \frac{m_{W}}{g^{2}}+ O \left( \ln \ln \frac{m_{W}}{g^{2}}\right) \right]},
1+\frac{cos\frac{1}{2}\beta cos\frac{1}{2}\gamma}{cos\frac{1}{2}\alpha}:1+\frac{cos\frac{1}{2}\gamma cos\frac{1}{2}\alpha}{cos\frac{1}{2}\beta}:1+\frac{cos\frac{1}{2}\alpha cos\frac{1}{2}\beta}{cos\frac{1}{2}\gamma}
\Delta^{\prime}= \frac{\Delta}{( R^{2 n},{\cal Z})_{E}}s d e t \Bigl ( [ \kappa^{( l )}\tilde{W_{0}}D^{( 2 l + 1 )}( R^{s t}) + \kappa^{( l ) \prime}W_{0}D^{( 2 l )}( R^{s t}) ] W_{0}^{- 1}\Bigr ) .
A=(\begin{matrix}29&2&1\\ 2&6&1\\ 1&1&\frac{1}{5}\end{matrix})\Rightarrow D^{-1}(L+U)=(\begin{matrix}0&\frac{2}{29}&\frac{1}{29}\\ \frac{1}{3}&0&\frac{1}{6}\\ 5&5&0\end{matrix})\Rightarrow\rho(D^{-1}(L+U))\approx1.0661.
d \Sigma_{p q}^{2}= \lambda^{2}( d \psi + p \cos \theta_{1}d \phi_{1}+ q \cos \theta_{2}d \phi_{2})^{2}+ \Lambda_{1}^{- 1}( d \theta_{1}^{2}+ \sin^{2}\theta_{1}d \phi_{1}^{2})
\begin{aligned}(X_u-X_s)\cdot \mu(X_u) + \frac{1}{2}(p-1) \cdot\sigma^2(X_u) = X_u\mu(X_u) + \frac{1}{2}(p-1)\cdot \sigma^2(X_u) - X_s\mu(X_u) & \le X_u\mu(X_u) + \frac{1}{2}(p_0-1)\cdot \sigma^2(X_u) + |X_s|\cdot |\mu(X_u)| & \le c\cdot \bigl( 1+X_u^2 + |X_s|\cdot (1 + |X_u|^{\ell_\mu+1})\bigr).\end{aligned}
\begin{array}{lll}t=\tau t^{\prime},u(x,t) = \frac{\hat{u}}{\tau}(x,t^{\prime}),p(x,t) = \hat{p}(x,t^{\prime}),\varrho(x,t) = \hat{\varrho}(x,t^{\prime}),S(x,t) = \hat{S}(x,t^{\prime}),\end{array}
(\log f(z))' =\frac{f'(z)}{f(z)}=\frac{e^{i(\Theta -\theta )}(R'_r+iR\Theta '_r)}{Re^{i\Theta}}=e^{-i\theta}((\ln R)'_r+i\Theta '_r)=(\cos \theta -i\sin \theta )((\ln R)'_r+i\Theta '_r) =((\ln R)'_r\cos \theta +\Theta '_r\sin \theta )+i(\Theta '_r\cos \theta -(\ln R)'_r\sin \theta ) ,
\iint_D f(a_1, a_2, b, c) \cdot \phi_{n-2}(-a)\phi_{n-1}(-b) \, dabc = \iint_D f(a_1,a_2,b,c) \cdot \left(1 + \frac{2\alpha_1}{n}+ \frac{\alpha_2(a+b)}{n}- \frac{a^2+b^2}{2n}\right)\,dabc + o(n^{-1}) \; .
\begin{aligned}\int_{Q_T}\left(s\Theta (z_x)^2 + s^3 \Theta^3 z^2\right)e^{2s\Phi}dxdt\le C\left(\int_{Q_T}h^{2}e^{2s\Phi}dxdt -(\mbox{B.T.})\right),\end{aligned}
\widetilde{\bigtriangleup}^{A}_{\mu \nu}(p)= \bigtriangleup^{A}_{\mu \nu}+ \bigtriangleup^{A}_{\mu \mu'}V^{\lambda \sigma, \mu'}\bigtriangleup^{B}_{\lambda \sigma, \lambda' \sigma'}V^{\lambda' \sigma', \nu'}\bigtriangleup^{A}_{\nu' \nu}+ \cdots .
{C}_{\theta}(\tau_i,\tau_{-i})=\theta \underbrace{e^{-\beta\tau_i}x_i(\tau_i)}_{}+(1-\theta)\underbrace{\frac{e^{-\beta\tau_i}x_i(\tau_i)}{l_1+l_2\left(\frac{1}{N}\sum\limits_{j=1}^Ne^{-\beta\tau_j}x_j(\tau_j)\right)}}_{}.
\left( L_{+}, L_{-}\, | \, L_{+}, A , L_{-}\right) = 2 \hbar^{2}\left[ A , I \right] - 2 \hbar^{3}\left[ L_{0}, A \right] - \hbar \left[ L_{-}, \left[ L_{+}, \left[ A , L_{0}\right] \right] \right] - \hbar \left[ L_{+}, \left[ L_{-}, \left[ A , L_{0}\right] \right] \right] \; ,
q^{-1}L_{6}(q^2) = 2\sum_{\substack{n \geq 1 -n \leq j \leq n-1}}q^{3(2n)^2 - (2j+1)^2}+ 2\sum_{\substack{n \geq 0 -n \leq j \leq n}}q^{3(2n+1)^2 - (2j)^2}.