harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\omega^t(z_1,z_2)=\frac{1}{2\pi i}d\log\!\left(\frac{z_1-z_2}{\overline z_1-z_2}\right)-\frac{t}{\pi i}d\log\!\left(\left|\frac{z_1-z_2}{\overline z_1-z_2}\right|\right)=\frac{1}{2\pi}d \mathrm{arg}\!\left(\frac{z_1-z_2}{\overline z_1-z_2}\right)+\frac{1-2t}{2\pi i}d\log\!\left(\left|\frac{z_1-z_2}{\overline z_1-z_2}\right|\right).
L^{n}_{p,q}(f;x)=L_{n}(f;p;q;x)= \sum\limits_{k=0}^{n}~~ \frac{f \bigg(\frac{p^{n-k}~[k]_{p,q}}{[n]_{p,q}}\bigg) ~\left[\begin{array}{c}n k\end{array}\right]_{p,q}p^{\frac{(n-k)(n-k-1)}{2}}q^{\frac{k(k-1)}{2}}~x^{k}~(1-x)^{n-k}}{\prod\limits_{j=1}^{n}\{p^{j-1}(1-x)+q^{j-1}x\}}.
\begin{aligned}c_{2,1}\left(\bigcap_{i=1}^{d}\left\{-2\eta<B_{s_{1}}^{i}-B_{t_{0}}^{i}<2\eta\right\}\right)\leq \left(1+d^{2}C_{H}\left(\frac{M}{\eta}\right)^{2}\right) \cdot P\left(\bigcap_{i=1}^{d}\left\{\left|B_{s_{1}}^{i}-B_{t_{0}}^{i}\right|<3\eta\right\}\right),\end{aligned}
R_{\mathrm{CPF}}^m = \max_{{\bf a}_m \in \mathbb{R}}\frac{1}{2}\log^+ \left( ||{\bf a}_m||^2 - \frac{P({\bf h}_m^{\mathrm{T}}{\bf a}_m)^2}{1+P||{\bf h}_m||^2}\right)^{-1}.
\varphi(\mathrm{v},\mathrm{y},t, \tau) := \displaystyle\int_{0}^{\tau}\dfrac{\alpha |\mathrm{y}-\mathrm{v}|^2}{4(\mathrm{s}-\tau)^2}\textbf{exp}\Big(-\dfrac{\alpha|\mathrm{y}-\mathrm{v}|^2}{4(\mathrm{s}-\tau)}\Big)\Phi^{\textbf{e}}(\xi,\mathrm{t};\mathrm{z},\mathrm{s})d\mathrm{s}.
|\Gamma_{\ell,m,0}|=(q-1)^3(q+1)q^{3\ell+m+6}|\Gamma_{\ell,m,m}| =(q-1)^3(q+1)q^{3\ell+6}|\Gamma_{\ell,\ell,m}|=(q-1)^3(q+1)q^{4\ell-m+6}.
{\rm d}Z_{m}^{k}={\rm d}z_{m}^{p}+\tau\sum_{j=1}^{r}a_{kj}{\rm d}(\delta_tZ_{m}^{j}), begin{align*}1mm]{\rm d}z_{m}^{p+1}={\rm d}z_{m}^{p}+\tau\sum_{k=1}^{r}b_{k}{\rm d}(\delta_tZ_{m}^{k}).
\mathcal{R}_{n,\alpha}(\varepsilon) = \mathcal{R}_{n,\alpha}(\theta(\mathcal{P}_k), (\cdot)^2, \varepsilon) = \inf_{Q \in \mathcal{Q}_\alpha}\inf_{\hat{\mu}}\sup_{P_{k, \varepsilon}\in \mathcal{P}_\varepsilon(\mathcal{P}_k)}\mathbb{E}_{P_{k, \varepsilon}, Q}\{(\hat{\mu}-\mu)^2\},
\Sigma_n =\lim_{k\to\infty}k^{-1}\int_{\Lambda_n}\Big(\sum_{j=0}^{k-1}v_n\circ T_n^j\Big)\Big(\sum_{j=0}^{k-1}v_n\circ T_n^j\Big)^T\,d\mu_n= \int_{\Delta_n}m_n\,m_n^T\,d\mu_{\Delta_n}.
\sum_{m = 1}^{\infty}\left\langle \frac{P_{n}( x_{1}) \tilde{Q}_{m}( y_{1})}{x_{1}- \phi_{a}}\right\rangle \left\langle \frac{P_{m}( x_{2}) \tilde{Q}_{n}( y_{2})}{y_{2}- \chi_{b}}\right\rangle = \left\langle \frac{P_{n}( x_{1}) \tilde{Q}_{n}( y_{1})}{( x_{1}- \phi_{a}) ( y_{1}- \chi_{b})}\right\rangle
2Ex= \theta x+(-1)^{p+q}x\theta=\xi_ia\otimes \xi^i\alpha+(-1)^{p+q}a\xi_i\otimes \alpha\xi^i= = \xi_ia\otimes \xi^i\alpha+(-1)^p a\xi_i\otimes \xi^i\alpha=\xi_i\wedge a\otimes \xi^i\alpha,
\int_{R^d}x\cdot\nabla \widetilde{\epsilon}_{0n}\,\widetilde{\epsilon}_{1n}= -\int_{R^d}\left[\frac{|\nabla \widetilde{\epsilon}_{0n}|^2}{2}+\frac{|\widetilde{\epsilon}_{1n}|^2}{2}\right]\,dx+o_n(1),\,\,{\rm as}\,\,n\to\infty.
\Phi(\kappa)=\beta_1C_n^2\left[1+\beta_2(\kappa/\kappa_l)-\beta_3(\kappa/\kappa_l)^{7/6}\right]\frac{\mathrm{exp}\left(-\kappa^2/\kappa_l^2\right)}{(\kappa_0^2+\kappa^2)^{11/6}},
\zeta_{N}^{c}( 2 , x^{2}) = \frac{\alpha R}{8 x^{3}}+ \frac{1 5 \alpha}{5 1 2 R x^{5}}+ \frac{4 0 3 5}{2^{1 7}}\frac{\alpha}{R^{3}x^{7}}+{\cal O}( x^{- 9}){.}
\int_{\mathbb{R}^n}\frac{A_0(x,y)}{|x-y|^{n+2s}}(\ell(x)-\ell(y)) dy = \int_{\mathbb{R}^n}\frac{a(x-y)}{|x-y|^{n+2s}}(\ell(x)-\ell(y)) dy & = \int_{\mathbb{R}^n}\frac{a(z)}{|z|^{n+2s}}(\ell(x)-\ell(x-z)) dz & = \int_{\mathbb{R}^n}\frac{a(z)}{|z|^{n+2s}}\mathcal{A}\cdot z dz =0,
= \lambda^{N}- \left( \sum_{a = 1}^{N}\lambda_{a}\right) \lambda^{N - 1}+ \left( \sum \sum_{b < a}^{N}\lambda_{a}\lambda_{b}\right) \lambda^{N - 2}+ \dots + ( - 1 )^{N}\prod_{a = 1}^{N}\lambda_{a}
\phi_1: U^1=1+Tk[[T]] \to \prod_{(i,p)=1}W(k) \prod_{(i,p)=1}^{\infty}\prod_{j=0}^{\infty}(1- a_{ij}T^i)^{p^j}\mapsto \left( \sum_{j=0}^{\infty}p^j \sum_{l|i}l [a_{lj}]^{i/l}\right)_i
\mathcal{L}_{\mathrm{C P - e v e n}}= i \, \bar{\psi}_{i}\gamma^{\mu}\partial_{\mu}\psi^{i}+ \bar{\psi}_{2}\partial_{5}\psi^{1}- \bar{\psi}_{1}\partial_{5}\psi^{2}- \omega ( \bar{\psi}_{1}\psi^{1}+ \bar{\psi}_{2}\psi^{2}) - 2 \Lambda \delta ( y ) \left[ \bar{\psi}_{1}\psi^{1}+ \rho \, \bar{\psi}_{2}\psi^{2}\right]
\int_{a_N}^u G_j(v) \: dv \geq -(s C_4 + \mu)(u - a_N) \geq -\frac{2 (s C_4 + \mu) a_N}{k_N}= -\frac{2(C_4 + \mu/s)}{k_N}\log \bigg( \frac{s}{\mu}\bigg) \geq -\log \bigg( \frac{s}{\mu}\bigg).
\langle f_{1}( M_{i_{1}}) f_{2}( M_{i_{2}}) \ldots f_{n}( M_{i_{n}}) \rangle = 0 \, m b o x{w h e n e v e r}\ \langle f_{j}( M_{i_{j}}) \rangle = 0 \ \mathrm{f o r ~ a l l}\j
\frac{d}{dx}\left[(1+x^2)\frac{d Q^k_\ell}{dx}\right] -\left[\ell(\ell+1)-\frac{k^2}{1+x^2}\right]Q^k_\ell =0 \frac{d}{dx}\left[(1+x^2)\frac{d Q^m_n}{dx}\right] -\left[n(n+1)-\frac{m^2}{1+x^2}\right]Q^m_n =0 .
\| U^{*}(\Phi(f\otimes a) \oplus \overbrace{\bigoplus_{k=1}^m f(z_k)\Sigma (a) \oplus \cdots \oplus\bigoplus_{k=1}^m f(z_k)\Sigma (a)}^m) U & - \Psi(f\otimes a)\oplus \overbrace{\bigoplus_{k=1}^m f(z_k)\Sigma (a) \oplus \cdots \oplus \bigoplus_{k=1}^m f(z_k)\Sigma (a)}^m\| < \varepsilon
n q = \mathsf{d}(r) \alpha_k r^k + \sum_{i \neq k}\mathsf{d}(r) \alpha_i r^i = \mathsf{n}(r) r^{k-1}+ (\alpha_k - 1) \mathsf{d}(r) r^k + \sum_{i \neq k}\mathsf{d}(r) \alpha_i r^i = \mathsf{n}(r) + \big( s + (\alpha_k - 1) \mathsf{d}(r) r^k + \sum_{i \neq k}\mathsf{d}(r) \alpha_i r^i \big),
\sum_{s=1}^{n}k_{rs}\theta^{s-1}\overset{\eqref{eq:a_i-k_ij-theta}}{{=}}a_{r}=\sum_{j=1}^{n}\Big(\sum_{h=1}^{n}p_{rjh}\theta^{h-1}\Big)a_{j}=\sum_{j=1}^{n}\Big(\sum_{h=1}^{n}p_{rjh}\theta^{h-1}\Big)\Big(\sum_{l=1}^{n}k_{jl}\theta^{l-1}\Big) \overset{\eqref{eq:theta-i-theta-j-c_ijk}}{=}\sum_{j,h,l=1}^{n}\sum_{s=1}^{n}p_{rjh}k_{jl}c_{hls}\theta^{s-1}.
\lambda_k = \int_0^1 \log 1/x \d\mu(x) + \int_0^1 \log \langle a_1(x);a_2(x),\dots,a_k(x) \rangle - \log 1/x \d\mu(x) = g + O\left( \int_0^1\left| \frac{1}{x}- \langle a_1(x);a_2(x), \dots, a_k(x) \rangle\right| \d\mu(x) \right) = g + O\left( \int_0^1 2^{-k}\d\mu(x) \right) &= g + O(2^{-k}).
\max_{u\in U}\mathcal{H}(x^{(\bar{u},x_{0})}(t),u,\eta^{\bar{u}}(t),\mu^{\bar{u}}(t)) = \mathcal{H}(x^{(\bar{u},x_{0})}(t),\bar{u}(t),\eta^{\bar{u}}(t),\mu^{\bar{u}}(t)) = constant \mathrm{a.e.}~ t \leq t_1
\sum_{n=-\infty}^\infty \left[{a_1\atop b_1+\alpha n}\right]_{p_1}\left[{a_2\atop b_2+\alpha n}\right]_{p_2}\frac{1}{(-zq^n,-q^{1-n}/z;q)_\infty}&=\int_{-\infty}^\infty\left[{a_1\atop b_1+\alpha x}\right]_{p_1}\left[{a_2\atop b_2+\alpha x}\right]_{p_2}\frac{dx}{\left(-zq^x,-q^{1-x}/z;q\right)_\infty}.
\left. \begin{array}{l l l}{{a_{1}}}&{{=}}&{{( 0 . 7 7 , 1 . 9 5 ) , \quad a_{2}= ( 3 . 2 5 , 1 . 9 5 ) ;}}\\{{b_{1}}}&{{=}}&{{( 1 . 3 2 , 1 . 9 5 ) , \quad b_{2}= ( 2 . 7 0 , 1 . 9 5 ) .}}\\ \end{array}\right.
\frac{\bar{M}_{3}^{2}}{m_{2}G}\frac{1}{r}= 1 - \epsilon ( 2 \bar{E}\bar{M}_{3}^{2}+ m_{2}^{2}G^{2})^{1 / 2}( m_{2}G )^{- 1}\sin ( \phi - \phi_{0}) .
A^{-1}\big(R_y w\big) = A^{-1}\langle w_1,w_2,-w_3\rangle = w_1\cdot\partial_1\vec{\eta}+ w_2\cdot\partial_2\vec{\eta}- w_3\cdot\vec{n}= R_x\big(w_1\cdot\partial_1\vec{\eta}+ w_2\cdot\partial_2\vec{\eta}+ w_3\cdot\vec{n}\big) = R_x\big(A^{-1}w\big) ;
\frac{1}{2\pi}\varphi_{s,a,b}(0) := \sum_{j \in \mathbb{Z}}\frac{e^{2 \pi j i s}}{a^2 + (2 \pi j + b)^2}= \frac{1}{2\pi}\sum_{j \in \mathbb{Z}}\frac{\pi e^{- i (s-j)b}}{a}e^{- a |j-s|}= \frac{1}{2a}\left( \frac{e^{- s w}}{1 - e^{- w}}+ \frac{e^{- (1-s)\bar{w}}}{1 - e^{- \bar{w}}}\right),
\partial_{m}A^{m}= i \, e \int \frac{d^{3}k}{( 2 \pi )^{3}2 k^{0}}\left( e^{i k x}k^{m}a_{m}^{\dagger}( \vec{k}) - e^{- i k x}k^{m}a_{m}( \vec{k}) \right)_{\displaystyle |_{k^{0}= \sqrt{\vec{k}^{2}}}}
\Pi_\infty^{(s, v)}(z_1, z_2) : = \lim_{n \to \infty}\widetilde{\Pi}_n^{(n)}(z_1, z_2) = \frac{A_{II}^{(s,v)}(1, z_1) B_{II}^{(s,v)}(1, z_2) - A_{II}^{(s,v)}(1, z_2) B_{II}^{(s,v)}(1, z_1)}{2 \prod_{i=1}^m \sqrt{(v_i + z_1) ( v_i + z_2)}\cdot \big[ C_{II}^{(s,v)}(1)\big]^2 \cdot (z_1- z_2)}.
\ll \lambda, [\alpha] \gg =N^{n_0 - n_1}\prod_{\parbox{.6in}{\small faces $\sigma$}}tr(\alpha^\sigma_{\lambda(\sigma )})\prod_{\parbox{.7in}{\small tetrahedra $\tau $}}dim(\lambda(\tau))^{-1}\prod_{\parbox{.8in}{\small 4-simplices $\xi$}}\| \lambda, \xi \|
\frac{1}{n}\sum_{i=1}^n(\widetilde{f}_{m_0}(x_i)-f(x_i))^2 + \frac{1}{n}\sum_{i=1}^n(\widetilde{f}_{m_0}(x^{\prime}_i)-f(x^{\prime}_i))^2 \leq \frac{4vv_f}{m_0}+ 4v^2\epsilon^2_2.\stepcounter{equation}
\sigma(H^{\epsilon,\kappa})\cap \left [a_0,b_N\right ]\subset \bigcup_{k=0}^N [a_k,b_k]\,,\quad{\rm dim}\big({\rm Ran}E_{[a_k,b_k]}(H^{\epsilon,\kappa}) \big )= + \infty\,, & b_k-a_k\leq C_0\epsilon \big(\kappa + C_1\, \epsilon^{1/5}\big) \mbox{for}0\leq k\leq N\,, \mbox{and}a_{k+1}-b_k\geq\frac{1}{C_2}\epsilon \,, \mbox{for}0\leq k\leq N-1\,.
\begin{array}{r c l}{}&{\tilde{L}_{i}= \frac{( 2 \pi )^{2}{l_{p}}^{3}}{L_{i}R}}\\{}&{g^{2}= \frac{( 2 \pi )^{6}{l_{p}}^{6}}{L_{1}L_{2}L_{3}L_{4}R}}\\ \end{array}
C = e^{\pi \sqrt{- 1}[ ( n - 1 ) ( n + 2 ) + 2 ( n - 1 )^{2}\tau ] / 2 n}\displaystyle \frac{\prod_{i = 0}^{n - 1}( \sum_{j = 0}^{n - 1}\omega^{i j}\theta^{( j )}( 1 / 2 ) )}{h ( n / 2 - ( n - 1 ) \tau / 2 ) \prod_{k = 1}^{n - 1}h ( k \tau / n )^{n - k}},
H_2 = \Big|\Big(f_\epsilon^{'}(V)-\sum_{i=1}^k(-1)^if_0^{'}(PU_{\mu_i,\xi_i})\Big)\phi\Big|_{\frac{2n}{n+2}}\leq \Big|f_\epsilon^{'}(V)-\sum_{i=1}^k(-1)^if_0^{'}(PU_{\mu_i,\xi_i})\Big|_{\frac{n}{2}}|\phi |_{\frac{2n}{n-2}}\leq C \epsilon\ln\Big|\ln\frac{\epsilon}{|\ln\epsilon|^2}\Big| \|\phi\|.
|\theta| \frac{\max\{r,n^{1/2}\}\rho^r}{1-\rho}= \,O(n^{-\delta}hn^{1/2}\log n)=O\bigl(n^{1/2+\beta-\delta}\log n\bigr)\to 0, \frac{(r\theta)^2\rho^r}{1-\rho}= O(n^{1-2\delta}h\log^2n)=O\bigl(n^{1+\beta-2\delta}\log^2n)\to 0.
A_{2}^{( 1 )}( p_{1}, p_{2}) = \frac{N}{2}\frac{1}{( 4 \pi )^{d / 2}}\frac{1}{( 2 \alpha^{\prime})^{d / 2}}\left[ 2 g_{s}( 2 \alpha^{\prime})^{\frac{d - 2}{4}}\right]^{2}
\Big(\big(\psi\circ\chi\big)_\#(\mu)\Big)(x) =\mu\Big(\big(\psi\circ\chi\big)^{-1}(x)\Big)=\mu\Big(\chi^{-1}\big(\psi^{-1}(x)\big)\Big) &=\Big(\chi_\#(\mu)\Big)\big(\psi^{-1}(x)\big)=\Big(\psi_\#\big(\chi_\#(\mu)\big)\Big)(x).
R ( t ) = | a_{S}|^{2}e^{- \gamma_{S}t}+ | a_{L}\eta |^{2}e^{- \gamma_{L}t}+ 2 | a_{S}| | a_{L}| | \eta | e^{- ( \gamma_{L}+ \gamma_{S}){\frac{t}{2}}}\cos ( \Delta m t + \varphi )
\int e^{-2y^{2}}dy = \left(\frac{2}{\pi}\right)^{1/2}, \ \ \ \ \ \int y^{2}e^{-2y^{2}}dy =\left(\frac{2}{\pi}\right)^{1/2}\frac{1}{4}, \ \ \ \ \ \ \int y^{4}e^{-2y^{2}}dy =\left(\frac{2}{\pi}\right)^{1/2}\frac{3}{16},
\int_{\mathcal{Y}^d}(\Lambda_i+\widetilde{w}_{y_i})\widetilde{w}_{y_ix_j}\widetilde{m}_{x_j}dy+\int_{\mathcal{Y}^d}V_{x_j}\widetilde{m}_{x_j}dy=\int_{\mathcal{Y}^d}\frac{1}{\widetilde{m}}\widetilde{m}^2_{x_j}dy.
\langle I(r_*> r_0)\rangle\le\sum_{r\ge r_0}\sum_{n=1}^N\big\langle I\big(F_{r,n}^2\nabla(\phi,\sigma)\ge\frac{1}{C}(\frac{r_0}{r})^\beta\log(\frac{r}{r_0}+1)\big)\big\rangle.
\begin{array}{r c l}{{G_{\sigma}^{( n )}( \zeta^{\prime \prime}, \zeta_{2 n - 1}, \epsilon x^{- 1}\zeta_{2 n - 1})^{\varepsilon^{\prime \prime}\, s s^{\prime}}}}&{{=}}&{{s^{( 1 - \epsilon ) / 2}\delta_{s + s^{\prime}, 0}G_{\epsilon \sigma}^{( n - 1 )}( \zeta^{\prime \prime})^{\varepsilon^{\prime \prime}}~ ( s , s^{\prime}= \pm ) .}}\\ \end{array}
\{\theta: \|\theta-\theta_{0}\|^2 / B^2 \geq C (\varepsilon / B)^{4\alpha_{0}/(2\alpha_{0}+1)}\}=\{\theta: \|\theta-\theta_{0}\|^2 / \varepsilon^2 \geq C (B/\varepsilon)^{2/(2\alpha_{0}+1)}\}.
u_{n-1}- u_n \geq C \begin{cases}u_{n-1}a_{n-1}^{1-\kappa}= u_{n-1}^{-\kappa}\text{if $1 < \kappa < 2$}u_{n-1}a_{n-1}^{-1}\log a_{n-1}= u_{n-1}^2\log (1/u_{n-1}) \end{cases}
\sigma\rho_{u}=\tau +\eta\cdot\int_{t_{u}}^{T}\rho (s)ds=\tau +\eta\cdot\sum_{v=u}^{N-1}\int_{t_{v}}^{t_{v+1}}\rho (s)ds\geq\tau +\eta\cdot\sum_{v=u}^{N-1}\int_{t_{v}}^{t_{v+1}}\rho (t_{v+1})ds & \geq a_{j}^{(u)}+\sum_{v=u+1}^{N}(t_{v}-t_{v-1})K_{ij}^{(v,u)}\rho_{v}\mbox{(by (\ref{extclpeq213}) and (\ref{extclp214}))}.
M^{-1}.x_{ij}=\sum_{\ell=1}^n m_{i\ell}x_{\ell j}M^{-1}.x_{i_1j_1}\dots x_{i_rj_r}=\left(\sum_{\ell_1=1}^n m_{i_1\ell_1}x_{\ell_1 j_1}\right)\cdots\left(\sum_{\ell_r=1}^n m_{i_r\ell_r}x_{\ell_r j_r}\right) =\sum_{\ell_1,\dots,\ell_r=1}^n m_{i_1\ell_1}\cdots m_{i_r\ell_r}(x_{\ell_1j_1}\dots x_{\ell_rj_r})
\mathcal{L}= - \frac{1}{2}(a^{\dagger}b+b^{\dagger}a), \mathcal{A}= \frac{i}{2}(a^{\dagger}b - b^{\dagger}a), \widetilde{\mathcal{L}}= -\frac{1}{2}(ba^{\dagger}+ab^{\dagger}), \widetilde{\mathcal{A}}= \frac{i}{2}(ba^{\dagger}- ab^{\dagger}).
\left(\begin{matrix}x=0 \,\left( \frac12 \right) x=\infty \, \left( \frac12 \right) \overbrace{\begin{matrix}\sqrt{t_2}0 -\sqrt{t_2}0\end{matrix}}\overbrace{\begin{matrix}1 0 -t_1/2 0 -1 0 t_1/2 0\end{matrix}}\end{matrix}\right).
\gamma_{\sigma, k}:=\begin{cases}\frac{n}{4(1-\sigma)}-\frac{\sigma}{1-\sigma}+\frac{k}{2(1- \sigma)}\ \ 0 \le \sigma < \frac{1}{2}, \frac{n}{2}+1-k \ \ \sigma = \frac{1}{2}, & \frac{n}{4\sigma}-\frac{1}{2\sigma}+\frac{k}{2\sigma}\ \ \frac{1}{2}< \sigma \le 1, \end{cases}
{\cal Z}( x ) = 1 + \frac{\alpha}{4 \pi}\frac{( 4 \pi )^{\epsilon}}{\Gamma ( 2 - \epsilon )}\xi \left[{\frac{\epsilon}{2 - \epsilon}}\int_{0}^{x}d y{\frac{y^{1 - \epsilon}}{x^{2}}}{\cal Z}( y ) - \int_{x}^{\infty}d y y^{- \epsilon - 1}{\cal Z}( y ) \right] .
M^{2}\equiv \left. \frac{\langle \psi | H | \psi \rangle}{\langle \psi | \psi \rangle}\right|_{m i n}= 1 + \frac{2 \pi}{\sqrt{3}}\sqrt{m^{2}- \frac{3}{4 \pi^{2}}\theta_{e f f}^{2}},
\aligned \sum_{j_r}\,\frac{a_{j_r}}{a_1^\bullet\overline a_1^\bullet}+p_i\,t_1+q_i\,\overline t_2=0 \ \ \ \{\rm and}\ \ \ \ \sum_{j'_r}\,\frac{a_{j'_r}}{a_1^\bullet\overline a_1^\bullet}+q_i\,\overline t_1+p_i\,t_2=0, \endaligned
\int\varphi(h_{\gamma}(A))d\mu_{T}(A)=\frac{\int_{G}\varphi(x)Q_{Ta}(x)Q_{Tb}(x^{-1}wzw^{-1}z^{-1})dxdwdz}{\int_{G}Q_{Tc}(wzw^{-1}z^{-1})dwdz},
\prod_{i=1}^{2k-\ell-1}\frac{f\Bigl( \|(R_{n}+a(R_{n})\rho)\theta + y_i\|e_1 \Bigr)}{f(R_ne_1)}= \prod_{i=1}^{2k-\ell-1}\frac{L \Bigl( R_{n}+ a(R_{n}) \bigl( \rho + \xi_n(\rho,\theta,y_i) \bigr) \Bigr)}{L(R_n)}&\times \exp \left\{-\int_0^{\rho + \xi_n(\rho,\theta,y_i)}\frac{a(R_{n})}{a \bigl( R_{n}+ a(R_{n})r \bigr)}\, dr \right\},
\alpha \,<\, \int_X \,\int_G\, \psi(t x)\,d\mu^{mk}(t) \,d\rho(x) \,=\, \int_X \,\int_G\, \psi(t x)\,d\mu^{n_{j_0}+s_0}(x) \,d\rho(x) \,<\,{\alpha}/{2}\,.
{\frac{t_{\mathrm{d e c a y}}^{\mathrm{H M}}}{t_{\mathrm{d e c a y}}^{\mathrm{C D L}}}}= \operatorname{e x p}\left[ 8 \pi^{2}\left({\frac{8}{T^{2}}}-{\frac{3}{V_{1}}}\right) \right] \ .
\left[ B_{\vec{k}}^{b}, \exp \sum_{a}C_{3}\left( \left| \vec{p}\right| \right) B_{\vec{p}}^{a +}A_{\vec{p}}^{L , a +}\right] = - C_{3}\left( \left| \vec{p}\right| \right) B_{\vec{p}}^{b +}\delta_{\vec{k}, \vec{p}}\exp \sum_{a}C_{3}\left( \left| \vec{p}\right| \right) B_{\vec{p}}^{a +}A_{\vec{p}}^{L , a +}.
T_{m,\lambda}(a_1, a_2, \ldots, a_n) = T_m(a_1 a_2 \cdots a_{\lambda_1})T_m(a_{\lambda_1 + 1}a_{\lambda_1 + 2}\cdots a_{\lambda_1 + \lambda_2}) \cdots T_m(a_{n - \lambda_{l(\lambda)}+1}a_{n - \lambda_{l(\lambda)}+1}\cdots a_{n})
R(r)=Ce^{\eta{r}}r^{(\frac{1-\tau}{2})}\Big[e^{-2\eta{r}}r^{-(1-\tau+\mathrm{c})}\Big]_{-(1+\mathrm{c}E^{-1})}\quad\Bigg(\mathrm{c}=-\Big(\frac{\beta+\eta(1-\tau)}{2\eta}\Big)\Bigg),
v(x)= \begin{cases}O \left( \left(\frac{1}{|x|}\right)^{n-2}\right), \frac{b}{1-a}<n-2 o\left( \left(\frac{1}{|x|}\right)^{\frac{b}{1-a}+\varepsilon}\right), \frac{b}{1-a}\geq n-2 \end{cases}
G=(\begin{matrix}g_{11}&g_{12}\\g_{21}&g_{22}\end{matrix})=(\begin{matrix}a_{1}\cdot a_{1}&a_{1}\cdot a_{2}\\a_{2}\cdot a_{1}&a_{2}\cdot a_{2}\end{matrix})=(\begin{matrix}a_{1}^{2}&a_{1}a_{2}cos(\alpha_{1})\\a_{1}a_{2}cos(\alpha_{1})&a_{2}^{2}\end{matrix})
|\mathbb{P}(a\le c_{1}X_{1}+\cdot\cdot\cdot+c_{n}X_{n}\le b)-\frac{1}{\sqrt{2\pi}}\int_{a}^{b}e^{-\frac{1}{2}t^{2}}dt|\le C(c_{1}^{4}+...+c_{n}^{4}).
Q_{z,t}(h_1,h_2): =8\pi k ix h_1h_2+4\pi k yt^2h_1^2 +4\pi k y t^{-2}h_2^2= 4\pi \Big( (\sqrt{ky}th_1 + \frac{ix\sqrt{k}}{t\sqrt{y}}h_2)^2+\frac{k|z|^2h_2^2}{yt^2}\Big).
E^{2}_{p,q}(n) = H_{p}(O_{n,n}, H_{q}(C_{*}(n))) \Rightarrow H_{p+q}(O_{n,n}, C_{*}(n)) E^{1}_{p,q}(n) = H_{q}(O_{n,n}, C_{p}(n)) \Rightarrow H_{p+q}(O_{n,n}, C_{*}(n)).
Y_{l,m}=\frac{(-1)^{l}}{\lambda_{l,m}}J_{-}^{l-m}(x+iy)^{l}\lambda_{l,m}=\frac{l! \ 2^{l}}{\sqrt{2l+1}}\sqrt{\frac{(l-m)!}{(l+m)!}}0\leq m\leq l
\langle \Psi ( z^{i}, d z^{j}) | \Xi ( z^{k}, d z^{l}) \rangle : = \Psi^{+}( \partial_{d z^{j}}, \partial_{z^{i}}) \Xi ( z^{k}, d z^{l}) |_{z^{1}= z^{2}= . . . z^{D}= d z^{1}= d z^{2}= . . . = d z^{D}= 0},
\psi_{n,c}(1) \geq \left(\chi_n(c)\right)^{1/4}\sqrt{\frac{\pi}{2 \mathbf K(\sqrt{q_n})}}\left(1-\frac{3}{(1-q_n) \sqrt{\chi_n(c)}}\right),q_n= \frac{c^2}{\chi_n(c)},
\mathbb{L}''(\hat{\mathbf U},\hat{\Psi})(({\mathbf V},\Psi),(\tilde{{\mathbf V}},\tilde{\Psi})):=\frac{d}{d\varepsilon}\mathbb{L}'(\hat{\mathbf U}+\varepsilon \tilde{{\mathbf V}},\hat{\Psi}+\varepsilon\tilde{\Psi})({\mathbf V},\Psi)\Big|_{\varepsilon=0},
d\left(f_{\lambda,\phi}\circ\pi_2|_{\Gamma_{\phi}}\right)=(\pi_2|_{\Gamma_{\phi}})^{*}(\phi^*\lambda-\lambda)=(\pi_{1}^{*}\lambda-\pi_{2}^{*}\lambda)|_{\Gamma_{\phi}}=\Lambda|_{\Gamma_{\phi}}.
i \hbar \frac{\partial}{\partial t}\left[ \begin{array}{c c}{\Psi_{e}( t )}\\{\Psi_{\mu}( t )}\\ \end{array}\right] = H_{\mathrm{f l a v o r}}\left[ \begin{array}{c c}{\Psi_{e}( t )}\\{\Psi_{\mu}( t )}\\ \end{array}\right] ,
\pi \circ_1 \pi_{N+1}- \pi \circ_2 \pi_{N+1}+ \pi_{N+1}\circ_1 \pi - \pi_{N+1}\circ_2 \pi = - \sum_{i+j =N+1, i, j \geq 1}\big( \pi_i \circ_1 \pi_j - \pi_i \circ_2 \pi_j \big).
< O_{n}O_{m}>_{q}= \frac{1}{2^{n + m}}\sum_{k = 0}^{n}\sum_{l = 0}^{m}\left( \begin{array}{c}{n}\\{k}\\ \end{array}\right) \left( \begin{array}{c}{m}\\{l}\\ \end{array}\right) < W_{q}^{n - k , k}W_{- q}^{m - l , l}> .
\hat h_{\mu\nu}^{\rm odd}= \left (\begin{array}{cccc}0 0 0 h_0(r) 0 0 0 h_1(r) 0 0 0 0 h_0(r) h_1(r) 0 0\end{array}\right ) e^{-i\omega t}\sin\theta{{d P_l(\theta)}\over{d\theta}}
\hat{L}\hat{u}(x_s,t) = \hat{L}\tilde{u}^l_s(x_s,t) - \frac{\partial^2 \tilde{u}^l_s}{\partial \nu^2}(x_s,t) + \frac{\partial^2 \hat{u}}{\partial \nu^2}(x_s,t) &= L\tilde{u}_s(a(x_s),t) + \frac{\partial^2 \tilde{u}_s}{\partial s^2}(a(x_s),t),\forall x_s \in \mathcal{M}_s, \forall s \in (-\delta,\delta),
K_{x}=K_{1}=i\frac{\partial\hat{B}(\varphi,\hat{e}_{x})}{\partial\varphi}|_{\varphi=0}=i(\begin{matrix}0&1&0&0\\ 1&0&0&0\\ 0&0&0&0\\ 0&0&0&0\\ \end{matrix}),
B^{\mu}( p ) = \frac{1}{\sqrt 2}\left[ \left( \begin{array}{c}{0}\\{1}\\{i}\\ \end{array}\right) + \frac{p^{x}+ i p^{y}}{M ( E + M )}\left( \begin{array}{c}{E + M}\\{p^{x}}\\{p^{y}}\\ \end{array}\right) \right] \phi ( p ) .
( \bar{\psi}^{T}\zeta ) ( \zeta^{T}\Gamma_{1 2}\psi ) = \left( \sum_{A = 1}^{8}\bar{\psi}_{A}\zeta_{A}\right) \cdot \sum_{a = 1}^{4}( - 1 )^{a}\left( \zeta_{2 a - 1}\psi_{2 a}- \zeta_{2 a}\psi_{2 a - 1}\right) .
\sum_{\pi\in \Sigma_2}\ [\bar{e}_{i,k_{\pi(1)}}, [\bar{e}_{i,k_{\pi(2)}}, \bar{e}_{i\pm 1,l}]]=0 \ \mathrm{and}\ [\bar{e}_{i,k}, \bar{e}_{j,l}]=0\ \mathrm{for}\j\ne i, i\pm 1,
|v(x)-v(y)| =|v(x)-v(\bar x)|+|v(\bar x)-v(\bar y)|+|v(\bar y)-v(y)| &\le c\lambda \big(d(x,\bar x)^\beta+d(\bar x,\bar y)^\beta+d(y,\bar y)^\beta\big)\le c\lambda d(x,y)^\beta.
\sum_{k=0}^a \binom{a}{k}\binom{b}{m+k}\binom{k}{c}\binom{m+k}{d}H_{b-m-k}={}\binom{a+b-c-d}{m+a-d}\binom{a}{c}\binom{b}{d}(H_{b-m-c}-H_{a+b-c-d}+H_{b-d})
(z^{k})^{(\nu)}=\begin{cases}0 \mbox{if}\k<[\nu-1]+1; \frac{\Gamma(k+1)\Gamma(b+k+\nu-1-[\nu-1])}{\Gamma(b+k)\Gamma(k-[\nu-1])}z^{k-1-[\nu-1]}\mbox{if}\k\geq[\nu-1]+1.\end{cases}
\widehat \zeta=(\widehat y,\widehat p,\widehat z,\widehat q)=(y- y',p-p',z-z',q-q'), &\widehat{\overline \zeta}=(\widehat{\overline y},\widehat{\overline p},\widehat{\overline z},\widehat{\overline q})=(\overline y-\overline y',\overline p-\overline p',\overline z-\overline z',\overline q-\overline q').
I_{N}^{\lambda}u(x)=\sum_{n=0}^{N}\tilde{u}_{n}^{\lambda}R_{n}^{\lambda}(x),{\rm where}\tilde{u}_{n}^{\lambda}=\frac{1}{\gamma_{n}^{\lambda}}\sum_{j=0}^{N}u(x_{j}^\lambda)R_{n}^{\lambda}(x_j^\lambda)\omega_{j}^\lambda.
\tilde{f}^{[i]}_{\ell,m-j}= \sum_{r=\ell+1}^{m-j+1}\sum_{h=1}^{m-j+1}b^{[i]}_{h,r,m-j+1}D B^{[i]}_{h,m-j+1}= \sum_{r=\ell+1}^{m-j+1}\sum_{h=1}^{m-j+1}b^{[i]}_{h,r,m-j+1}\left(D g^{[i]}_{h,m-j+1}-Dg^{[i]}_{h+1,m-j+1}\right),
\ldots =\frac{v - \delta(v-d-1)}{v-\lfloor d/2 \rfloor}\sum_{j=0}^{\lfloor d/2 \rfloor}\binom{v-j}{v-2j-\delta}\sum_{k=0}^{2j}\binom{j+\delta}{k-j+1}, &=\frac{v - \delta(v-d-1)}{v-\lfloor d/2 \rfloor}\sum_{j=0}^{\lfloor d/2 \rfloor}\binom{v-j}{v-2j-\delta}2^{j+\delta}.
S_M(X; \pi, \hat{\phi}, W) = \sum_{p}\frac{a_{\pi}(p)\log p}{\sqrt{p}}\hat{\phi}\left( \frac{\log p}{M\log X}\right) \left( \frac{2}{p}\right)\sum_{(d,2)=1}M_Z(d)\left( \frac{d}{p}\right) W\left( \frac{d}{X}\right).
\widetilde\phi (x) = K^{\partial_x^2}\phi (x) = t_0 + t_1 K^{- \frac{1}{{R^2}}}\cos \left({\frac{x}{R}}\right) + t_2 K^{- \frac{4}{{R^2}}}\cos \left({\frac{{2x}}{R}}\right) +\cdots,
\bar\partial_Mf=\sum_{j=1}^{m-1}\sideset{}{'}\sum_{\vert J\vert=q}(\overline{L_j}f_J)\overline{\omega_j}\wedge \overline{\omega_J}+\sideset{}{'}\sum_{\vert J\vert=q}f_J\bar\partial_M\overline{\omega_J}.
\begin{aligned}A(W_{2n};x) = x + (2n-1)x^{2n-3}+ (2n-1)(2n-3)x^{2n-1}+ x^{2n+1}+ + \displaystyle\sum_{k=2}^{n-1}{2n-1 \choose k-1}x^{2k-1}+{2n-1 \choose n-1}x^{2n-1}+ + \displaystyle\sum_{k=n+1}^{2n-1}a_{2n,k}x^{2n-1}+ \sum_{k=n+1}^{2n-1}b_{2n,k}x^{2n+1}+ x^{2n+3}. \end{aligned}
\zeta_{AV,2}(s_1,s_2,s_3) = \sum_{m \leq at_3}\sum_{n<m}\frac{1}{m^{s_1}n^{s_2}(m+n)^{s_3}}+ \begin{cases}O (t_3^{-\sigma_1-\sigma_3}) (\sigma_2>2) O (t_3^{-\sigma_1-\sigma_3}\log t_3 ) (\sigma_2=2) O (t_3^{\frac{3}{2}-\sigma_1-\sigma_2-\sigma_3}) (\sigma_2<2). \end{cases}
|f_0(W_\delta) - f_0(U_{\delta,e_n})+f_0(U_{\delta,-e_n})|_{\frac{(p+1)(1+\gamma)}{p}}=\begin{cases}O(\delta^{n-2}) 4\leq n<6,\ \gamma\geq 0, O(\delta^4|\log \delta|^\frac{2}{3}) n=6,\ \gamma=0, O(\delta^{\frac{n+2}{2}}) n=6,\ \gamma>0 n>6,\ \gamma\geq 0.\end{cases}
p_{L,N+1}^{\ell_2}(x) = \sum\limits_{\ell=0}^N\dfrac{\sum_{j=0}^N\omega_j\tilde{\Phi}_{\ell}(x_j)f(x_j)}{1+\lambda\mu_{\ell}^2}\tilde{\Phi}_{\ell}(x) &=\sum_{j=0}^N\omega_jf(x_j)\sum_{\ell=0}^N\dfrac{\tilde{\Phi}_{\ell}(x_j)\tilde{\Phi}_{\ell}(x)}{1+\lambda\mu_{\ell}^2}.
\begin{array}{ccc}\delta \langle X\rangle \equiv \delta \langle A_{1}(x_{1}),...,A_{N}(x_{N})\rangle =\delta \int{\cal D}\varphi \, A_{1}...A_{N}\, e^{-S[\phi ]}= \delta \int{\cal D}\varphi \, \left[ \sum_{k}A_{1}...\delta A_{k}...A_{N}\, e^{-S}-A_{1}...A_{N}\, e^{-S}\delta S\right] \end{array}
Q_{\Lambda_{d,m}}\negmedspace\left(\sum_{l=1}^{L}a_{ml}\sum_{\mu=1}^{j}\mathbf{t}_{l,\mu}\right)=Q_{\Lambda_{d,m}}\left(\sum_{l\in\mathcal{S}^{\left\langle j\right\rangle}}\negmedspace a_{ml}\negmedspace\sum_{\mu=\pi_{c}\left(l\right)}^{j}\mathbf{t}_{l,\mu}\right)
{\frac{\delta{\cal H}_{2 n + 1}}{\delta u_{- j}}}= \sum_{k = 0}^{\infty}( - 1 )^{( j + 1 ) k +{\frac{1}{2}}k ( k + 1 )}\left( D^{k}{\frac{\partial{\cal H}_{2 n + 1}}{\partial u_{- j}^{[ k ]}}}\right) ,
P(y) ={}\int_{\mathbb R^n}P(y;x) \Delta_n(x)\det \left[ f_{k-1}(x_j) \right]_{j,k=1}^n dx_1 \dotsm dx_n ={}\frac{\Delta_n(y)}{Z_n\widehat Z_n}\int_{\mathbb R^n}\det \left[ f_{k - 1}(x_j) \right]^n_{j, k = 1}\det \left[ e^{-\frac{1}{2}(y_j - x_k)^2}\right]^n_{j, k = 1}dx_1 \dotsm dx_n.
b^n_l=(-\partial_{a_l})(U^n_l)Str\left(\ldots \phi^{j_l}\ldots \right)-k(A_1+A_2)-\frac{n}{l}[ k(B_1+B_2)+b^n_l+D^n_l|_{i_n\leftrightarrow a_{l+1}}+\ldots +D^n_l|_{i_n\leftrightarrow a_p}]
R^{(i)}_{}(D)=\min\limits_{p(w|u_i)}\min\limits_{{g_j: \mathcal{U}_j\times\mathcal{W}\rightarrow\mathcal{U}_i}\atop{\mathbb{E}[d(U_i, g_j(U_j, W))]\le D}}I(U_i; W|U_j)
\left[ - \frac 1 2 \frac{\partial^{2}}{\partial x^{2}}+ \frac 1 2 \frac{\omega^{2}}{4 z_{0}}x^{2}\right] \varphi_{0}( z ( x ) ) = \frac{\epsilon_{1}}{4 z_{0}}\varphi_{0}( z ( x ) )
C_{I J K}\, h^{I}h^{J}h^{K}= 3 \left\{h^{1}\, \big ( h^{2}\big )^{2}- h^{1}\, \big ( h^{\mu}\big )^{2}- h^{2}\, \big ( h^{i}\big )^{2}+ \gamma_{\mu i j}\, h^{\mu}\, h^{i}\, h^{j}\right\}.
1=C(\varepsilon) \left(1-\varepsilon \mu_{(p)}^{G}+\frac{\varepsilon^{2}}{2}\mu_{(2p)}^{G}-...\right)=C(\varepsilon) \sum_{n=0}^{\infty}\frac{(-\varepsilon)^{n}}{n!}\mu_{(np)}^{G}.
\sigma_{\rm a}^2(x)=\left(\psi_{1}x+\psi_{2}\right)T_{\rm s}\,e^{-\frac{\left(\psi_1x+\psi_2\right)\tau_{\rm d}}{N_{\rm a}}}-\frac{\left(\psi_{1}x+\psi_{2}\right)^2T_{\rm s}\tau_{\rm d}}{N_{\rm a}}\left(2-\frac{\tau_{\rm d}}{T_{\rm s}}\right)e^{-\frac{2\left(\psi_1x+\psi_2\right)\tau_{\rm d}}{N_{\rm a}}}.