harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
d e t \vert A \vert = \prod_{\scriptscriptstyle n = 1}^{\infty}\vert \lambda_{\scriptscriptstyle n}\vert , \hspace{2 c m}d e t \vert B \vert = \prod_{\scriptscriptstyle n = 1}^{\infty}\vert 1 + \lambda_{\scriptscriptstyle n}\vert
{\cal L}^{\prime}= \bar{\psi}^{( i )}\mathrm{i}\partial \! \! \! / \psi^{( i )}+ \frac{g^{2}}{2}\left( \psi_{R}^{( i ) \dagger}\psi_{L}^{( i ) \dagger}+ \psi_{L}^{( i )}\psi_{R}^{( i )}\right)^{2}\ .
\widetilde{\phi}= e \Omega^{2}\delta ( \omega - \overrightarrow{k}\cdot \overrightarrow{v}) \left( \omega^{2}- v_{s}^{2}k^{2}\right) / 2 \pi \varepsilon k^{2}\left( v_{s}^{2}k^{2}- \omega^{2}- \Omega^{2}\right)
\limsup_{j\to\infty}\frac{1}{j}\log\|S(j)\mathfrak{r}_r(f)\|_{\sup}\leq -\lambda_1 r r>0 &\limsup_{j\to\infty}\frac{1}{j}\log\|S(j)\mathfrak{r}_r(f)\|_{L^1}\leq -\lambda_1 r r\leq 0.
KN^2/(KN+1) =N(KN)/(KN+1)\geq M_0\tau^{-i\theta}C[C+(2\tau^{\theta+\lambda})^iM_0^{\gamma}]^{-1}&=M_0^{1-\gamma}\tau^{-i(1-\gamma)\theta/2}[M_0^{-\gamma}\tau^{i(1+\gamma)\theta/2}+C^{-1}(2\tau^{(3+\gamma)\theta/2+\gamma})^i]^{-1}&\geq M_0^{1-\gamma}\tau^{-i(1-\gamma)\theta/2}(1+C^{-1})^{-1}&>N_0.
\mathcal{L}_{I_{1}}(s,d)=\exp\left(-\frac{2\pi}{\alpha_1}\lambda_{1}\left(\frac{P_1}{P_2}s\right)^{\frac{2}{\alpha_1}}B\left(\frac{2}{\alpha_1},1-\frac{2}{\alpha_1}\right)\right)\;.
\sigma_{p}i\infty = p \mathrm{and}\sigma_{p}^{-1}\Gamma_{X,p}\sigma_{p}= \langle\gamma_{\infty}\rangle,\quad\mathrm{where}\,\,\, \gamma_{\infty}=\left(\begin{array}{ccc}1 1 0 1 \end{array}\right)\quad \mathrm{and}\quad\Gamma_{X,p}=\langle\gamma_{p}\rangle
Q(\alpha,\alpha^{*})=\frac{1}{\pi}\langle\alpha|\hat{\rho}|\alpha\rangle=\frac{1}{\pi}|\langle n|\alpha\rangle|^{2}=\frac{1}{\pi n!}|\langle0|\hat{a}^{n}|\alpha\rangle|^{2}=\frac{|\alpha|^{2n}}{\pi n!}|\langle0|\alpha\rangle|^{2}.
\begin{aligned}\varphi'_m (x,t) = \sup_{y\in M}\varphi (y, m - 1) - \varphi(x, m - 1 + t), \varphi''_m(x,t) = \varphi (x, m - 1 + t) - \inf_{y\in M}\varphi(y, m - 1).\end{aligned}
4\pi e\left.\frac{\partial^2F}{\partial s^2}\right|_{s=0}=\langle V,(\Delta_L^{\perp}-1)V\rangle_{L^2}=2\langle f_1,(\Delta_L-1)f_1\rangle_{L^2}+2\langle f_2,(\Delta_L-1)f_2\rangle_{L^2}.
\int_\Omega \delta({\cal L}) \sqrt{g}\; d^4x =\int_\Omega J^{\mu\nu}{}_{\lambda\rho}\left\{-2 \nabla_{\mu}\nabla^{\lambda}(\delta g)_{\nu}{}^{\rho}+R_{\mu\nu}{}^{\sigma\lambda}(\delta g)_\sigma{}^{\rho}\right\}\sqrt{g}\; d^4x.
L_{g}= \left\{\eta:\,\mathbb{R}^{+}\rightarrow H_{0}^{1}(\Omega),\int_{\Omega}\int_{0}^{+\infty}g(s)\,|\nabla\boldsymbol{\eta}(\boldsymbol{x},\,s)|^{2}\ds\d\boldsymbol{x}<+\infty\right\},
R\left(\mathbb{1}_{P_{Q}^{n}\left(I_{1},\dots,I_{n-1},T\right)}\right)=\sum_{v\in\mathbb{Z}^{nd}}\mathbb{1}_{P_{Q}^{n}\left(I_{1},\dots,I_{n-1},T\right)}\left(v\right)-\int_{\mathbb{R}^{nd}}\mathbb{1}_{P_{Q}^{n}\left(I_{1},\dots,I_{n-1},T\right)}\left(v\right)dv,
\left|[u^c]\left(1- \frac{bq^b}{q^b-1}u(1-u)^{b-1}L(u,q^b)^{b}\right)^{N}\right| <\frac{3a_J}{8}.\frac{4}{3}\left(\frac{9q^b}{16}\right)^{-c}= \frac{a_J}{2}\left(\frac{9q^b}{16}\right)^{-c}.
\sum_{m\neq m'}^M \sum_{m\neq m'}^M \frac{z_{i_m}\, z_{i_{m'}}}{z_{i_m i_{m'}}\, z_{i_{m'}i_m}}= 2 \sum_{m < m'}^M \sum_{m\neq m'}^M \frac{z_{i_m}\, z_{i_{m'}}}{z_{i_m i_{m'}}\, z_{i_{m'}i_m}}\, .
+ \int \! \! d^{D}\! z_{1}d^{D}\! z_{2}d^{D}\! z_{3}\left. \frac{\delta^{2}W}{\delta j ( x ) \delta j ( z_{1})}\, \frac{\delta^{2}W}{\delta j ( x ) \delta j ( z_{2})}\, \frac{\delta^{2}W}{\delta j ( x ) \delta j ( z_{3})}\, \frac{\delta^{3}\Gamma}{\delta \Phi ( z_{1}) \delta \Phi ( z_{2}) \delta \Phi ( z_{3})}\right] ,
\Phi_{p,q}(\lambda)-\Phi_{p-1,q+1}(\lambda)= \left( 3\,pq-6\,p-6\,q+12 \right){\lambda}^{3}+ \left( 18\,pq-34\,p-40\,q+74 \right){\lambda}^{2}&+ \left( 30\,pq-49\,p-67\,q+101 \right)\lambda+12\,pq-10\,p-22\,q+2.
e^{i \big( (y_1 - j_1K)\xi_1 + (s-j_{d+1}K^2) |\xi_1|^2 \big)}= \sum_{m,l=0}^\infty \frac{1}{m! l!}\big( \frac{10 i (y_1 - j_1 K)}{K}\big)^m \big( \frac{10i (s - j_{d+1}K^2)}{K^2}\big)^l \big( \frac{K \xi_1}{10}\big)^m \big( \frac{K^2|\xi_1|^2}{10}\big)^l.
\sum_{\substack{\xi_1, \xi_2, \xi_3 \xi_1', \xi_2', \xi_3 '}}\int_{\mathbb{T}^2}dx e(\langle \xi_1+ \xi_2+ \xi_3-\xi_1'- \xi_2'-\xi_3',x\rangle) &= |\{\xi_i \in \mathcal{E}(E) : \xi_1+ \xi_2+ \xi_3+\xi_4+ \xi_5+\xi_6=0\}|
|\nabla w|^2 =A^2\alpha^2d^{2\alpha-2}+4AB\alpha d^{\alpha-1}x'\cdot\nabla_{x'}d+4B^2|x'|^2, \Delta w =A\alpha(\alpha-1)d^{\alpha-2}+A\alpha d^{\alpha-1}\Delta d+2(n-1)B,
R_{\mu \nu}= R_{\mu \lambda \nu}^{\lambda}= \partial_{\rho}\gamma_{\mu \nu}^{\rho}- \partial_{\nu}\gamma_{\mu \rho}^{\rho}- \gamma_{\mu \lambda}^{\rho}\gamma_{\nu \rho}^{\lambda}+ \gamma_{\lambda \rho}^{\rho}\gamma_{\mu \nu}^{\lambda}.
\delta G_{\mu \nu}^{a b}\; = \; - i g^{- \frac{1}{2}}\, G_{\mu \lambda}^{a c}( \theta ) \epsilon_{\lambda \rho \sigma}\, f^{c d e}\, \delta \theta_{\rho}^{d}\, G_{\sigma \nu}^{e b}( \theta ) \; ,
E \int_0^T \left(e^{\beta t}|Y(t)|^{2}+ \int_t^T e^{\beta s}\|Z(t,s)\|^{2}ds\right) dt \leq Ce^{\beta T}E \int_0^T |\psi(t)|^{2}dt + \frac{C}{\beta}E \int_0^T \int_t^T e^{\beta s}|g(t,s)|^{2}dsdt.
\frac{1}{4 \pi^{2}}A ( \vec{x}) \wedge F ( \vec{x}) = - \frac{1}{8 \pi^{2}}\mathrm{t r}\left( J ( \vec{x}) \wedge d J ( \vec{x}) + \frac{2}{3}J ( \vec{x}) \wedge J ( \vec{x}) \wedge J ( \vec{x}) \right) .
\varepsilon_{k+1}=\frac{A_{k+1}}{\delta}\left(\frac{\tau_k^2}{2\mu}- \frac{\epsilon}{2}\right)\|\nabla f(x_k)\|^{2}+\frac{A_{k+1}}{\delta}\left(\frac{\tau_k}{2\epsilon}-\frac{\mu}{2\tau_k}\right)\|x_k - y_k\|^2,
p_{c}^{site}(d)=\sigma^{-1}+\frac{3}{2}\sigma^{-2}+\frac{15}{4}\sigma^{-3}+\frac{83}{4}\sigma^{-4}+\frac{6577}{48}\sigma^{-5}+\frac{119077}{96}\sigma^{-6}+O(\sigma^{-7})
F^{\#}\left( t\right) :=\sup\big\{\left\vert F(t,x)\right\vert :x\in\overline{\mathrm{Dom}\left( \varphi\right)}\big\}, G^{\#}\left( t\right) :=\sup\big\{\left\vert G(t,x)\right\vert :x\in\overline{\mathrm{Dom}\left( \varphi\right)}\big\}
\int \frac{d^{4}p^{\prime}}{\left( 2\pi \right)^{4}}\left[{D}\left(p,p^{\prime},P\right) +{K}\left( p,p^{\prime},P\right) \right]{G}^{\left(4\right)}\left( p^{\prime},p^{\prime \prime},P\right)=\delta^{\left( 4\right)}\left( p-p^{\prime \prime}\right) ,
\phi^4= \left( \begin{array}{cccc}k^1t \otimes 1_{N_1 \times N_1}0 \dots 0 0 k^2t\otimes 1_{N_2 \times N_2}\dots 0 \dots \dots \dots \dots 0 \dots \dots k^Kt \otimes 1_{N_K \times N_K}\end{array}\right)
M =\begin{bmatrix}M_{11}M_{12}\ldots M_{1k}M_{1,k+1}\cr \textbf{0}M_{22}\ddots \ddots M_{2,k+1}\cr \vdots \ddots \ddots \ddots \vdots\cr \textbf{0}\ldots \textbf{0}M_{kk}M_{k,k+1}\end{bmatrix},
{\bf A}_1 ={\bf G}\left[\begin{array}{cc}{\bf I}_r{\bf 0}{\bf 0}{\bf 0}\end{array}\right]{\bf G}' \mbox{and}{\bf A}_2 ={\bf G}\left[\begin{array}{cc}{\bf D}_r{\bf 0}{\bf 0}{\bf 0}\end{array}\right]{\bf G}'
\! \! \! \! - \frac{1}{2}\sum_{n_{1}, n_{2}}\log \left( \frac{\mid \tau \mid^{2}n_{1}^{2}- 2 \tau_{1}n_{1}n_{2}+ n_{2}^{2}+ \lambda^{2}}{\mid \tau \mid^{2}n_{1}^{2}- 2 \tau_{1}n_{1}n_{2}+ n_{2}^{2}}\right) .
(\prescript{}{j-i_0+1}{\hat{s}})^*(i)= \prescript{}{j-i_0+1}{\hat{s}}(j-i) = \begin{cases}\hat{s}(j-i) j-i<j-i_0+1 \hat{s}(j-i+1) j-i_0+1\leq j- i\end{cases}= \begin{cases}\hat{s}(j-i) i_0\leq i \hat{s}(j-i+1) i<i_0\end{cases}
\partial_{t}Q-\Theta\sigma^{ij}_{k}\nabla_{ij}Q &=\frac{1}{P}(\partial_{t}P-\Theta\sigma^{ij}_{k}\nabla_{ij}P)-A\left[\partial_{t}\left(\frac{\rho^{2}}{2}\right)-\Theta\sigma^{ij}_{k}\nabla_{ij}\left(\frac{\rho^{2}}{2}\right)\right]+\frac{\Theta}{P^{2}}\sigma^{ij}_{k}\nabla_{i}P\nabla_{j}P.
\frac{d}{dt}\phi_\lambda \big( \eta(t,u) \big) \le - \frac{1}{2}\Psi \big( \eta(t,u) \big) \big\| \phi'_\lambda \big( \eta(t,u) \big) \big\| \le 0, \left\| \frac{d \eta}{dt}\right\|_\lambda = \big\| H(\eta) \big\|_\lambda \le 1
\phi_{\infty}=\left[\begin{array}{cccc}0 0 0 -1_{2m}0 0 -1_r 0 0 1_r 0 0 1_{2m}0 0 0\end{array}\right],\psi_{\infty}=\left[\begin{array}{cccc}0 0 0 J_m' 0 1_r 0 0 0 0 1_r 0 -J_m' 0 0 0\end{array}\right].
T_{12}:= \biggl|\int_{\mathcal{X}_n^c}\int_{\mathcal{X}_n}f(x)f(y) \log f(y) \int_{\tilde{u}_{n,x,y}}^\infty \log (uf(x)) \,d(\tilde{F}_{n,x}-F^-_{n,x})(u) \,dy \,dx\biggr|.
\int_{B(x_{0},r)}\frac{1}{|x|^{\beta}}\left(\exp(\alpha|f(x)|^{p'})-\sum_{0\leq k<p-1,\;k\in\mathbb{N}}\frac{1}{k!}(\alpha|f(x)|^{p'})^{k}\right)dx &\leq\sum_{p'k\geq p,\;k\in\mathbb{N}}\frac{(\alpha p'kC_{4}^{p'})^{k}}{k!}.
\begin{aligned}S_{N,M}=\sum\limits_{\boldsymbol\delta,\boldsymbol\Delta\eqref{bdelta,bDelta=}}\frac{(M!)^2}{\prod\limits_{i\in [N]}\delta_i!\,\Delta_i!}&=(M!)^2 [z_i^Mz_o^M] f_{k_1}(z_i)^N\,f_{k_2}(z_o)^N &=\Omega\left(N^{-1}(M!)^2\frac{f_{k_1}(z_i)^N\,f_{k_2}(z_o)^N}{z_i^M\,z_o^M}\right);\end{aligned}
p_{-k}=\overline{p_{k}}\mbox{, for all}k; p_{d}\in\Gamma(S^{\perp}); p_{k}\in\Gamma(S^{\perp})\mbox{if}k\mbox{and}d\mbox{have thesame parity;otherwise}p_{k}\in\Gamma(S); p(1)\neq0.
\partial_cT_0 dx^c \otimes dx^0 = \partial_c x^{-2}\hat{T}_0 dx^c \otimes dx^0 = x^{-2}\partial_c\hat{T}_0dx^c \otimes dx^0 = (x\partial_c\hat{T}_0) \frac{dx^c}{x}\otimes \frac{dx^0}{x^2}, \Gamma^d_{bc}T_d = \Gamma^d_{0c}T_d, & \Gamma^d_{0c}= \frac{1}{2}g^{dr}( \partial_cg_{r0}+\partial_xg_{rc}-\partial_rg_{0c}).
\begin{cases}f_1(x_1,x_2): =x_1^{p-1}+\nu\alpha x_1^{\frac{\alpha}{2}-1}x_2^{\frac{\beta}{2}}=1, f_2(x_1,x_2): =x_2^{p-1}+\nu\beta x_1^{\frac{\alpha}{2}}x_2^{\frac{\beta}{2}-1}=1,\end{cases}
x_{ij}x_{ik}=qx_{ik}x_{ij},x_{ij}x_{\ell j}=qx_{\ell j}x_{ij}, x_{ik}x_{\ell j}=x_{\ell j}x_{ik}\mbox{and}x_{ij}x_{\ell k}-x_{\ell k}x_{ij}=(q-q^{-1})x_{ik}x_{\ell j}.
\big\Vert C_{v^{(n)}_\Sigma}- C_{v^{(n+1)}_\Sigma}\big\Vert_{L^2 \to L^2}= \Vert C_\Sigma^- \Vert_{L^2\to L^2}\big\Vert v_\Sigma^{(n)}- v_{\Sigma}^{(n+1)}\big\Vert_{L^\infty(\Sigma)},
E_k \triangleright [E_\xi, E_{\xi^\prime}^*]_q = (E_k \triangleright E_\xi) E_{\xi^\prime}^* - q^{-(\xi + \alpha_k, \xi^\prime)}E_{\xi^\prime}^* (E_k \triangleright E_\xi) + q^{(\xi, \alpha_k)}E_\xi (E_k \triangleright E_{\xi^\prime}^*) - q^{-(\xi, \xi^\prime)}(E_k \triangleright E_{\xi^\prime}^*) E_\xi.
I_1 =-\frac{1}{2\pi i}\oint_{\partial U_{\delta}(\pm\eta)}\left(\sum_{j=1}^{\infty}\frac{T_{\pm\eta, j}(\xi,k)}{t^j}\right)ds =t^{-1}\textnormal{Res}_{k=\eta}T_{\eta,1}+t^{-1}\textnormal{Res}_{k=-\eta}T_{-\eta,1}+\mathcal{O}(t^{-2}),
\|\sigma_rf\|_k\le\sum_{l=0}^{\infty}r^{kl}\|\sigma_rP_l\|_k=\sum_{l=0}^{\infty}r^{kl}\|\sigma_rP_l\|_{H^2}\le\sum_{l=0}^{\infty}r^{kl}\|P_l\|_{H^2}\le\biggl(\sum_{l=0}^{\infty}r^{2kl}\biggr)^{1/2}\biggl(\sum_{l=0}^{\infty}\|P_l\|_{H^2}^2\biggr)^{1/2}=\frac{\|f\|_{H^2}}{(1-r^{2k})^{1/2}}.
F_N(1,\lambda)= \begin{pmatrix}1 \alpha_N(0,\lambda)\end{pmatrix}e^{-i\theta_N}+\frac{1}{(2i\lambda)^N}R_N(1,\lambda) G_N(1,\lambda)= \begin{pmatrix}\beta_N(0,\lambda) 1\end{pmatrix}e^{i\theta_N}+\frac{1}{(2i\lambda)^N}S_N(1,\lambda)
\frac{1}{2}\Delta|S|^2 =|\nabla S|^2+2\Big(c(1-|T|^2)+H^2-\frac{1}{2}|S|^2-\frac{c^2}{16H^2}|T|^4-\frac{c}{2H}\langle ST,T\rangle &+\sum_{\alpha>3}\det A_{\alpha}\Big)|S|^2.
\sum_{b=3}^{n-2}\sum_{\ell=0}^{n-1-b}(C_b-C_{b-1}-2^{b-2})=C_{n-2}-2^{n-3}+\sum_{\ell=1}^{n-4}\sum_{b=3}^{n-1-\ell}(C_b-C_{b-1}-2^{b-2}) &=C_{n-2}-2^{n-3}+\sum_{\ell=1}^{n-2}(C_{n-1-\ell}-2^{n-2-\ell})=C_{n-2}-3\cdot2^{n-3}+1+\sum_{\ell=1}^{n-2}C_\ell
J_{1 l i n}= \frac{m ( T_{o}) T_{o}}{8 \pi^{2}}\left( H e^{H t_{o}}\right)^{4}\int_{1}^{\infty}\frac{d z}{z}\frac{\sqrt{z^{2}- 1}}{\left[ \frac{\lambda_{R}T_{c}^{2}}{2 4}+ m^{2}( T ) z \right]^{2}}
\Delta (\lambda ):=\det \left(\begin{array}{cc}\int_{0}^{1}y_{1}(x,\lambda )\mathrm{d}\nu_{0}(x)-1 \int_{0}^{1}y_{2}(x,\lambda )\mathrm{d}\nu_{0}(x) \int_{0}^{1}y_{1}(x,\lambda )\mathrm{d}\nu_{1}(x)-y_{1}(1,\lambda ) \int_{0}^{1}y_{2}(x,\lambda )\mathrm{d}\nu_{1}(x)-y_{2}(1,\lambda )\end{array}\right) .
\left(\sum_{i=1}^n\left\|T(x_i)\right\|^p\right)^{\frac{1}{p}}\leq C \sup_{x^*\in B_{E^*}}\left(\sum_{i=1}^n\left|x^*(x_i)\right|^p\right)^{\frac{1}{p}}1\leq p<\infty, \max_{1\leq i\leq n}\left\|T(x_i)\right\| \leq C \sup_{x^*\in B_{E^*}}\left(\max_{1\leq i\leq n}\left|x^*(x_i)\right|\right) p=\infty.
\mathrm{Tr}(\Omega(c)|D|^{2-p}e^{-s^2D^2}) = \sum_{k=1}^p (-1)^{p-k}\mathrm{Tr}(\mathcal{W}_{k}(c)De^{-s^2D^2})+O(s^{-1}),s\downarrow 0.
W_{i j}^{( 0 )}( x , y ) = \left( \begin{array}{c c c c c}{{0}}&{{0}}&{{0}}&{{0}}&{{0}}\\{{0}}&{{0}}&{{0}}&{{0}}&{{0}}\\{{0}}&{{0}}&{{0}}&{{0}}&{{0}}\\{{0}}&{{0}}&{{0}}&{{0}}&{{0}}\\{{0}}&{{0}}&{{0}}&{{0}}&{{0}}\\ \end{array}\right) \delta^{2}( x - y ) \equiv \widetilde{W}_{i j}^{( 0 )}\delta^{2}( x - y ) ,
d s_{I}^{2}= - \left[ 1 - \frac{2 M}{r}\right] d t^{2}+ \left[ 1 - \frac{2 M}{r}\right]^{- 1}d r^{2}+ r^{2}\left( \sin^{2}\theta d \phi^{2}+ d \theta^{2}\right) ,
E=\frac{2e^{2}}{L_{B}}\int_{0}^{\infty}\frac{kdk}{k^{2}+k_{B}^{2}r_{ll^{\prime}}^{2}}J_{0}(cos\theta k)J_{0}(sin\theta k)J_{0}(k\frac{r_{12}}{r_{ll^{\prime}}})
|\phi^{N,i}(0,\boldsymbol{\xi})-\Psi^{N,i}(0,\boldsymbol{\xi})|\leq\frac{C}{N}\Big(1+\frac{1}{N}\underset{j=1}{\overset{N}\sum}|\xi^{i}-\xi^{j}|^2+\frac{1}{N^2}\underset{i,j=1}{\overset{N}{\sum}}|\xi^{i}-\xi^{j}|^2\Big)^{\frac{1}{2}},
\begin{array}{l}{{\vert 0 , z \rangle_{+ k}= \displaystyle \exp_{q^{- 2}}( - \frac{1}{2}| q^{2}z | ) \sum_{m = 0}^{\infty}\frac{( - q^{2}z )^{m}}{[ m ]_{q^{- 2}}!}({\cal D}_{+ k})^{m}\vert 0 \rangle_{+ k}~ , ~ ~{\cal X}^{+ k}\vert 0 \rangle_{+ k}= 0 ~ ,}}\\{{{\cal X}^{+ k}\vert 0 , z \rangle_{+ k}= z \vert 0 , z \rangle_{+ k}~ .}}\\ \end{array}
T_{n}^{\ast}(f;x)=\frac{n}{e_{\mu}(nx)}\sum_{k=0}^{\infty}\frac{(nx)^{k}}{\gamma_{\mu}(k)}\int_{\frac{k+2\mu \theta_{k}}{n}}^{\frac{k+1+2\mu \theta_{k}}{n}}f\left(\frac{nt+\alpha}{n+\beta}\right)\mathrm{d}t ,
\int_{-\infty}^\infty \int_{-\infty}^\infty \dots \int_{-\infty}^\infty \exp\left(-k\sum_{1\leq i \leq k}x_i^2\right) dx_1 dx_2 \dots dx_{k-1}= \pi^{\frac{k-1}{2}}k^{-\frac{k}{2}}.
\biggl|V_df(x)h_x^{-1}(s)^d - \sum_{l=0}^{\lceil \beta/2 \rceil -1}b_l(x) s^{1+2l/d}\biggr| \lesssim s \biggl\{\frac{a(f(x))^{d/(2 \wedge \beta)}s}{f(x)}\biggr\}^{\beta/d},
\int_{0}^{T}\int_{\mathbb{R}}\left( u_{t}^{\Delta x}\varphi v - \frac{\left(u^{\Delta x}\right)^{2}}{2}(\varphi v)_{x}- \left(\mathcal{H}u^{\Delta x}\right) (\varphi v)_{xx}\right) dx\,dt = \mathcal{O}(\Delta x),
\vec{y}= \left(x_1,\ldots,x_{i-1},(x_i+x_j)/2, 0,x_{i+1}, \ldots,x_{j-1}, 0,(x_i+x_j)/2, x_{j+1}, \ldots, x_m\right), \vec{z}= \left(x_1,\ldots,x_{i-1},0, (x_i+x_j)/2,x_{i+1}, \ldots,x_{j-1},(x_i+x_j)/2, 0,x_{j+1}, \ldots, x_m\right).
L \sigma_{\widetilde{X}}= \left( \mathcal{O}_{\widetilde{X}}, \alpha^\ast\mathcal{T}(-1),M, \alpha^\ast\mathcal{O}(1) , \mathcal{O}_{L_1}, \dots, \mathcal{O}_{L_7}, \mathcal{B}_1, \mathcal{B}_2,\mathcal{B}_3, \mathcal{B}_4 \right)
\hat{f}(s) = \Big[\prod_{i=1}^k \Lambda \big(\frac{s}{N^{i-1}}\big)\Big] \hat{f}\big(\frac{s}{N^k}\big)+ \sum_{i=0}^{k-1}\Big[\prod_{m=1}^i\Lambda \big(\frac{s}{N^{m-1}}\big)\Big] \hat{Q}\big(\frac{s}{N^i}\big),
\Pr_\theta [D(\theta) \leq \tfrac{\lambda n}{4}] \leq \Pr_\theta \Big [ (|G|-2D(\theta) )^{2\ell}\geq (\tfrac{\lambda n}{2})^{2\ell}\Big] \leq \frac{2^{-2\ell}r_\ell(A)}{(\lambda n/2)^{2\ell}}= \frac{r_\ell(A)}{(\lambda n)^{2\ell}}.
T_{1}g_{n}\,=\, g_n\,, \quad(T_{\tau}-id)(g_n)\,=\,\sum_{k=0}^{n-1}\frac{(2\pi \mathrm{i})^{n-k}}{(n-k)!}\,g_k\,, &g_n'\,=\,\sum_{k=0}^{n-1}(-1)^{n-k}\,E_{n-k+1}\,g_k\,.
F_{n}^{\mathcal{O}| \mu_{1}\ldots \mu_{n}}( \theta_{1}, \theta_{2}\ldots , \theta_{n}) : = \langle 0 | \mathcal{O}( 0 ) | Z_{\mu_{1}}^{\dagger}( \theta_{1}) Z_{\mu_{2}}^{\dagger}( \theta_{2}) \ldots Z_{\mu_{n}}^{\dagger}( \theta_{n}) \rangle .
f_{1,2,1}(x,y;q) =\Theta(y;q)m\Big (\frac{q^2x}{y^2},-1;q^3\Big )+\Theta(x;q)m\Big (\frac{q^2y}{x^2},-1;q^3\Big ) & \ \ \ \ \ \ \ \ \ \ - y\cdot \frac{(q^3;q^3)_{\infty}^3\Theta(-x/y;q)\Theta(q^2xy;q^3)}{\Theta(-1;q^3)\Theta(-qy^2/x;q^3)\Theta(-qx^2/y;q^3)}.
F'(z)\approx\frac{\partial}{\partial z}\Big(\kappa_1y^{\frac{1}{2}}+\kappa_2y^{\frac{1}{2}}\log(y)\Big)=\frac{i}{4}\Big((\kappa_1+2\kappa_2)y^{-\frac{1}{2}}+\kappa_2y^{-\frac{1}{2}}\log(y)\Big).
\Lambda(z^{(0)}) =\frac{e^{\frac{\sigma_{\mathrm{a}}^2}{\zeta}}\int_{\sigma_{\mathrm{w}}^2+\sigma_{\mathrm{a}}^2}^{\infty}\left( \frac{1}{v}\right)^{n}e^{-\frac{z^{(0)}}{v}}e^{-\frac{v}{\zeta}}dv}{\int_{\sigma_{\mathrm{w}}^2}^{\infty}\left( \frac{1}{v}\right)^{n}e^{-\frac{z^{(0)}}{v}}e^{-\frac{v}{\zeta}}dv}= \gamma,
\omega_{\psi}^{\mathbf{z}^{-1}}(\pi_{\mathbf{z}^{-1}}(t_{-\lambda})\Phi_{s_iw}^{\mathbf{z}^{-1}})= \mathcal{T}_i\,\omega_{\psi}^{\mathbf{z}^{-1}}(\pi_{\mathbf{z}^{-1}}(t_{-\lambda})\Phi_{w}^{\mathbf{z}^{-1}}).
\begin{array}{r c l}{{{\hat{S}}_{t}}}&{{=}}&{{{\hat{P}}_{x}\, ,}}\\{{{\hat{P}}_{t}}}&{{=}}&{{{\hat{P}}{\hat{S}}_{x}+{\hat{Q}}_{x}\, ,}}\\{{{\hat{Q}}_{t}}}&{{=}}&{{{2}{\hat{Q}}{\hat{S}}_{x}\, .}}\\ \end{array}
\mathbb{E}[x^{V_b};B_2]=\frac{1}{2}\Big{(}\sum_{k=0}^{\infty}\mathbb{P}(V_c=k)\sum_{j=0}^k\Big{(}\frac{1}{3}\Big{)}^k\binom{k}{j}x^j\Big{)}^2=\frac{1}{2}\Big{(}\sum_{k=0}^{\infty}\mathbb{P}(V_c=k)\Big{(}\frac{x+1}{3}\Big{)}^k\Big{)}^2=\frac{1}{2}f\Big{(}\frac{x+1}{3}\Big{)}^2
\nabla V_{\beta}(x) = \frac{\beta x}{(1+|x|^{2})^{\frac{2-\beta}{2}}}, \nabla^{2}V_{\beta}(x) = \frac{\beta I_d}{(1+|x|^{2})^{1-\frac{\beta}{2}}}+ \frac{\beta(\beta-2) xx^{\top}}{(1+|x|^{2})^{2-\frac{\beta}{2}}},
\frac{\partial}{\partial t}\big(\Phi''(P_tf)\Gamma(P_t f) \big)\Big|_{t=0}= \big[\Phi'''(P_t f)(LP_t f)\Gamma(P_t f)+\Phi''(P_t f)\cdot 2\,\Gamma(LP_t f,P_t f)\big]\big|_{t=0}\cr =\Phi'''(f)(L f)\Gamma( f)+ 2\,\Phi''( f)\Gamma(L f, f) =\Phi'''(f)(L f)\Gamma( f)+ 2\,\Gamma(L f, \Phi'( f)),
M\models \forall \; x \; \theta_{n}(x, \overline{a})\leftrightarrow \theta_{m}(x, \overline{b}) & M\models \forall \; x \; \theta_{n^{\prime}}(x, \overline{a}^{\prime})\leftrightarrow \theta_{m^{\prime}}(x, \overline{b}^{\prime}) & g_{m}(\overline{b})=c=c^{\prime}=g_{m^{\prime}}(\overline{b}^{\prime})
\begin{array}{lllll}\left|\nabla_v\left(f^{n+1}-H\right)\right| = \left|\int^t_0 \iint \nabla_v{\cal G}K^n \cdot \nabla_vf^{n+1}dwdy d \tau\right| \leq C \int^t_0 \iint \left| \nabla_w{\cal G}\right| w_0 \left|\nabla_vf^{n+1}\right| dwdy d \tau. \end{array}
- \frac{1}{4}\sqrt{- G}\, F_{M N}{}^{k}F^{M N}{}_{k}= - \frac{1}{4}\sqrt{- g}\, \Phi^{\frac{1}{D - 2}}[ B_{\mu \nu}{}^{k}B^{\mu \nu}{}_{k}+ \frac{2}{\Phi}g^{\mu \nu}D_{\mu}A^{k}\cdot D_{\nu}A_{k}] ,
w^{- 2 \delta}\langle P ( \chi , \chi ; - z^{\ast}) | V ( \psi , \rho ) | P \left( \overline{{\chi}}, \overline{{\chi}}; -{\frac{1}{w}}\right) \rangle = z^{- 2 \delta}\langle P ( \chi , \chi ; - w^{\ast}) | V ( \psi , \rho ) | P \left( \overline{{\chi}}, \overline{{\chi}}; -{\frac{1}{z}}\right) \rangle \, ,
\dot{\varrho}+ 3{\frac{\dot{a}_{0}}{a_{0}}}\left( \varrho +{\frac{\wp}{c^{2}}}\right) = -{\frac{\dot{G}_{4}}{G_{4}}}\varrho +{\frac{3 k c \dot{c}}{4 \pi G_{4}a_{0}^{2}}}-{\frac{3 C c \dot{c}}{4 \pi G_{4}a_{0}^{4}}}.
B( M_{1/\alpha}/t) \overset{\mathcal{L}}{=}B( S^{-1/\alpha}_{1/\alpha}/t) \overset{\mathcal{L}}{=}B(t S_{1/\alpha}^{1/\alpha}) \times S^{-1/{\alpha}}_{1/\alpha}/ t\overset{\mathcal{L}}{=}B(t S_{1/\alpha}^{1/\alpha}) \times M_{1/\alpha}/t.
{\cal{L}}( T \phi ) ={\cal{L}}( T \phi_{c l}+ T \eta ) ={\cal{L}}( \bar{\phi}_{c l}+ T \eta ) \neq{\cal{\tilde{L}}}( T \eta ) \Longrightarrow{\cal{\tilde{L}}}( T \eta ) \neq{\cal{\tilde{L}}}( \eta ) .
{x^{[2]}}({t_2}) = \frac{{{h^{[22]}}({t_1})}}{{{h^{[22]}}({t_2}- 2)}}u_1^{[2]},{x^{[3]}}({t_2}) = \frac{{{h^{[23]}}({t_1})}}{{{h^{[23]}}({t_2}- 2)}}u_1^{[3]}.
f_{\mu}^{a}( \psi_{, \mu}^{b}) - f_{\mu}^{a}( \psi^{a}, \lambda_{a}^{\mu}) - \lambda_{b}^{\nu}\frac{\partial f_{\nu}^{b}}{\partial \lambda_{a}^{\mu}}+ \frac{\partial L}{\partial \lambda_{a}^{\mu}}= 0 .
\psi_{_{W K B}}= C{\left( \pi \chi_{0}^{3}a_{0}^{2}( \eta - \frac{1}{2}s i n{2 \eta}) \right)}^{- 1 / 2}e x p \left( \frac{i}{\hbar}\pi \chi_{0}^{3}a_{0}^{2}( \eta - \frac{1}{2}s i n{2 \eta}) \right) .
0^p\cdot F_2 + 1^p\cdot F_1 + 2^p\cdot F_0 \ = \A_p F_2 + B_p F_3 - \sum_{k=0}^p \binom{p}{k}B_k 2^{p-k}, \intertext{and if we simplify this and pull out the $k=p$ term from the sum, we get}1 \ = \A_p + 2B_p - B_p - \sum_{k=0}^{p-1}\binom{p}{k}B_k 2^{p-k}.
\quad\quad\mathbf{V}_{\overline{\mathbf{x}}_{k}}^{post}= blkdiag\left(\left[ \mathbf{V}_{\mathbf{x}_{k-J}}^{post}\; \mathbf{V}_{\mathbf{x}_{k-J+1}}^{post}\; \ldots \; \mathbf{V}_{\mathbf{x}_{k}}^{post}\right]\right),
\{{\cal{Q}}_{a},{\overline{{\cal{Q}}}}_{b}\}={\cal{Z}}_{a b}= \left( \begin{matrix}{P_{0}+ P_{9}}&{P_{8}+ t_{r}P_{r}}\\{P_{8}- t_{r}P_{r}}&{P_{0}- P_{9}}\\ \end{matrix}\right) ,
f(t,y,z,u,\mu)=\hat f\big(t,y,z,u,\int\varphi d\mu\big),\,\,\,\,g(t,y,z,u,\mu)=\hat g\big(t,y,z,u,\int \phi d\mu\big), &h(t,y,z,u,\mu)=\hat h\big(t,y,z,u,\int\psi d\mu \big),\,\,\,\,\Phi(y,\mu_y)=\hat\Phi\big(y,\int \gamma d\mu_y\big),
{\cal A}_{\alpha_{j k}}={\frac{n}{4 \kappa}}\oint_{\alpha_{j k}}{\frac{\displaystyle u^{n}d u}{\displaystyle \sqrt{1 - e^{- ( 1 + u^{n}) / \kappa}}}}= e^{i \pi ( j + k + 1 ) / n}\sin \bigl ( \pi ( j - k ) / n \bigr ){\cal I}_{n}( \kappa )
\begin{array}{rclrcl}B_{t\underline{i}}= -\frac{1}{2\sqrt{2}}\frac{1+k^{2}}{k^{2}}\,,\hspace{1.5cm}B_{t}= 0\, , & B_{\underline{i}\underline{j}}= 0\, , B_{\underline{i}}= -\frac{1}{\sqrt{2}}{\cal A}_{\underline{i}}=k^{2}A_{\underline{i}}\, , & \phi = 0\, . \end{array}
F_{\gamma_E^{\infty}}(z) = \int_0^z f_{\gamma_E^{\infty}}(x) dx &\overset{c}{=}\frac{\sigma_n^2 (1-\rho)}{\sigma_d^2}\sum_{k=0}^{\infty}\frac{(2k+1)!\sigma_n^{2k}\rho^k}{(k!)^2\sigma_d^{2k}}& ~~~~\times \int_{\frac{\sigma_n^2}{\sigma_d^2}}^{z + \frac{\sigma_n^2}{\sigma_d^2}}y^{-(2k+2)}\left(y-\frac{\sigma_n^2}{\sigma_d^2}\right)^k dy
\lim_{n\to+\infty}\sum_{i=0}^{n-1}\int_{\{z\in M:-\Psi(z)\in I_{i,n}\backslash U_k\}}|\tilde{F}|^2_h \le \limsup_{n\to+\infty}\sum_{i=0}^{n-1}\frac{1}{\inf_{I_{i,n}\backslash U_k}c(t)}\int_{\{z\in M:-\Psi(z)\in I_{i,n}\backslash U_k\}}|\tilde F|^2_hc(-\Psi).
I_\varepsilon (u) = \frac{1}{2}\displaystyle\int_{\mathbb{R}^N}|(-\Delta)^{\frac{\gamma}{2}}u(x)|^2dx - \frac{N - 2\gamma}{2N}\displaystyle\int_{\mathbb{R}^N}(1 + \varepsilon K)|u|^{\frac{2N}{N - 2\gamma}}dx
\widehat{A}(TM,\nabla^{TM}) ={\det}^{1/2}\Big(\frac{\sqrt{-1}R^{TM}/(4\pi)}{\sinh\big(\sqrt{-1}R^{TM}/(4\pi)\big)}\Big), \widehat{L}(TM,\nabla^{TM}) ={\det}^{1/2}\Big(\frac{\sqrt{-1}R^{TM}/(4\pi)}{\tanh\big(\sqrt{-1}R^{TM}/(4\pi)\big)}\Big).
\dfrac{M_{1}(\mathcal{C}(G))}{|v(\mathcal{C}(G))|}=\dfrac{2^{5k}-4 \cdot 2^{4k}+ 4 \cdot 2^{3k}+4 \cdot 2^{2k}-5 \cdot 2^{k}-4}{2^{3k}-2^{k}-1}
P_{mk}=\begin{bmatrix}c_{1n}c_{2n}\ldots c_{kn}c_{1n}c_{2n}\ldots c_{kn}\hdotsfor{4}c_{1n}c_{2n}\ldots c_{kn}\end{bmatrix},Q_{kk}=\begin{bmatrix}1 c_{1n}\ldots c_{k-2,n}c_{k-1,n}0 1 \ldots c_{k-3,n}c_{k-2,n}\hdotsfor{5}0 0 \ldots 1 c_{1n}0 0 \ldots 0 1\end{bmatrix}.
I_{\epsilon}(z) \> =\> \sum_{n=0}^{\infty}f^{(n)}(0) \, \frac{z^{n+1}}{(n+1)!}\> \Gamma \left [ 1 + (n+1) \epsilon \right ] \, \cos \left [ (n+1) \epsilon \frac{\pi}{2}\right ] \;\;\;.
\frac{d}{dz}(\theta_{b_1}\circ \theta_{b_{2}}\circ \cdots \circ \theta_{b_n})(z)= \frac{(-1)^n}{(B_{n-1}z + B_n)^2}= \frac{(-1)^n}{B_{n-1}^2(z + B_n/B_{n-1})^2}.
\bar{\theta}_{R}^{a}\theta_{L}^{b}| 0 , p^{+}\rangle_{U}\quad \longleftrightarrow \quad{\frac{1}{\sqrt{2 J N^{J + 2}}}}( \sum_{l = 0}^{J}T r [ \chi^{a}\tilde{Z}^{l}\psi^{b}Z^{J - l}] + \sum_{l = 0}^{J}T r [ \psi^{a}Z^{l}\chi^{b}\tilde{Z}^{J - l}] ) ,
\sum_{\mu}s_{\kappa/\mu}(\rho_2) s_{\nu/\mu}(\rho_1)\mathcal{U}^{\llcorner}_{\rho_1, \rho_2}(\pi \vert \nu, \mu, \kappa) = \frac{s_{\pi/\kappa}(\rho_1)s_{\pi/\nu}(\rho_2)}{H(\rho_2 ; \rho_1)}.
j_{p}= d \zeta_{\tilde{m}}^{\wedge p}\int_{\partial{\cal M}^{p + 1}}d{\hat{\zeta}}^{\tilde{m}}( \tau ) \delta^{p + 1}\left( \zeta -{\hat{\zeta}}( \tau ) \right) = ( - )^{p}d \zeta_{\tilde{n}}^{\wedge p}j^{\tilde{n}}. \qquad
ES_{\alpha}(X)=-\xi-\frac{\lambda}{2\alpha}[exp(\frac{1-2\gamma\delta}{2\delta^{2}})\Phi(\Phi^{-1}(\alpha)-\frac{1}{\delta})-exp(\frac{1+2\gamma\delta}{2\delta^{2}})\Phi(\Phi^{-1}(\alpha)+\frac{1}{\delta})]
\sum_{r_\beta\leq s}s^{-k}\int_{B_s(x)}D_\mu(y,r_\beta)\,d\mu(y) \leq C\sum_{r_\beta\leq s}s^{-k}\int_{B_s(x)}W_{r_\beta}(y)\,d\mu(y) &=C s^{-k}\int_{B_s(x)}\sum_{r_\beta\leq s}W_{r_\beta}(y)\,d\mu(y) &\leq C\, s^{-k}\int_{B_s(x)}\big|\theta_s(y)-\theta_0(y)\big|\,d\mu(y)\leq C(n,K_N,\Lambda,\epsilon)\eta\, ,
\sum_{k_{1},k_{2},..,k_{N}=1}^{\infty}\Big(\frac{1}{\sum\limits_{i=1}^{N}d_{\ell_{i}}\lambda_{k_{i}}}\mathop{\textstyle \prod}\limits_{i=1}^{N}\lambda_{k_{i}}^{2\gamma_1-1}\emph{q}_{k_{i},k_{i}}^{\ell_{i},\ell_{i}}\Big)^{\frac{1}{2}}<\infty .
\sum\limits_{i= 0}^\infty \gamma_i x^i = \frac{1}{x}\cdot \frac{d^{2k-1}}{dx^{2k-1}}\left[x^{2k-2}\left(\frac{x^2}{1-x}\right)\psi\left( \frac{x^2}{1 - x}\right) \right],