harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
(aq)_N\sum_{n=1}^{N}\left[\begin{array}{c}N n\end{array}\right]\frac{na^nq^{n^2}}{(aq)_n}=\displaystyle\sum_{n=1}^{N}\left[\begin{array}{c}N n\end{array}\right]\frac{(q)_n(-1)^{n-1}a^nq^\frac{n(n+1)}{2}}{1-q^n}.
\int_{\Omega_i}\nabla \cdot \Big( \varepsilon \nabla u^* \Big) \varphi_\epsilon\, dx+\eta_{e}\int_e u^* \bar{\varphi}\, ds =\int_{\Omega_i}f \varphi_\epsilon\, dx+\eta_{e}\int_e \hat{u}\bar{\varphi}\, ds.
\int_{\epsilon}^0 \exp\left( - ca_k K\tilde{H}(x) \right) dx \lesssim \sum_{\ell= N_0}^\infty \exp \left(-\ell \cdot\beta \log 2 \right) \lesssim 2^{-\beta N_0}\lesssim |\epsilon|^\beta < +\infty.
Ef=\int_{B_1^{d-n}}\sum_{m\in N^{-1}\mathbb{Z}^n \cap B_1^{n}}f(\xi,m) e^{i(x\cdot \xi+y\cdot m+t(\frac{\mathfrak{m}(N\xi)}{N^2 \mathfrak{m}(N)}+\frac{\mathfrak{m}(Nm)}{N^2 \mathfrak{m}(N)}))}\, d\xi,
\Big[ \sum_{l=M}^{L-M}{2^{-{t_l}d}\sum_{P\in\mathcal{Q}(l)}{\int_{\Omega}{\Big( \sum_{n=1}^{L-l}{2^{-2({s}+d){t_n}}\sum_{Q\in\mathcal{V}_n(l,P)}{\theta_Q(\omega)}}\Big)^{{q}/{2}}}d\lambda}}\Big]^{{1}/{q}}
\alpha(\mathbf{w},\mathbf{e}) =\frac{2}{(2N-1)^2}\cdot \Bigl \{(\lambda_0-\lambda_1)|E_1|^2 \left. + \sum\limits_{i=2}^{N-1}\left [ (\lambda_0-\lambda_1) + \frac{\lambda_1-\lambda_i}{2\cos^2 \frac{\pi}{2N-1}}\right ] |E_i|^2\right \}.
\int_{X_\epsilon}\partial_z \phi~ \partial_{\bar z}\phi~\frac{idz\wedge d\bar z}{2}=\int (\partial_z \phi_M \partial_{\bar z}\phi_M -2\alpha_1\phi_M\partial_z\partial_{\bar z}\phi_B)\frac{idz\wedge d\bar z}{2}+
\int_{- \infty}^{\infty}d \kappa \frac{e^{c \kappa}}{\frac{2}{\kappa}\sinh \left( \frac{\kappa \pi}{2}\right) 2 \cosh ( \frac{\kappa \pi}2 )}= \frac{1}{4 \cos ({\frac{c}{2}})^{2}}\, .
\qquad\Phi\otimes\overline{\Phi}((g,g')\cdot (x,x'))=\Phi(g\cdot x)\overline{\Phi}(g'\cdot x') &=e(k\cdot g-k\cdot g')\Phi(x)\overline{\Phi}(x')=e\Bigl((k,-k)\cdot(g,g')\Bigr)\Phi\otimes\overline{\Phi}(x,x')
\Omega_1\times\Omega_2\to\Omega_N,U_1\times U_2\mapsto R^{-1}\left[\begin{array}{cc}U_1 0 0 \mathfrak{J}\overline{U_2}\end{array}\right],\mathfrak{J}=\left[\begin{array}{cccc}J_{m}' 0 0 0 0 0 -1_r 0 0 1_r 0 0 0 0 0 J_{m}'\end{array}\right].
\ln \frac{f(x)}{f(x+1)}+\ln\frac{\Phi_0(x)}{\Phi_0(x+1)}=2\ln \frac{x+1}{x+\frac 12}+\ln\frac{x+a}{x+1+a}= \frac{-1/4 + a}{x^2}+O\left(\frac{1}{x^3}\right).
\begin{matrix}1&2&3&4\\ 2&1&4&3\\ 3&4&1&2\\ 4&3&2&1\end{matrix}\begin{matrix}1&2&3&4\\ 2&1&4&3\\ 3&4&2&1\\ 4&3&1&2\end{matrix}\begin{matrix}1&2&3&4\\ 2&3&4&1\\ 3&4&1&2\\ 4&1&2&3\end{matrix}\begin{matrix}1&2&3&4\\ 2&4&1&3\\ 3&1&4&2\\ 4&3&2&1\end{matrix}.
\left( \sum_{i=1}^2 \left( \frac{1}{r_i^2}\frac{\partial}{\partial r_i}r_i^2\frac{\partial}{\partial r_i}+ \frac{1}{r_i^2\, \sin\psi}\,\frac{\partial}{\partial \psi}\,\sin\psi\, \frac{\partial}{\partial \psi}- \frac{r_i^2}{g^2}\right)\! + \frac{2}{g^2}\, E \right) \Psi(r_1,r_2,\psi) = 0 \;.
\begin{aligned}\mu ({{\mathbf{q}}_w})\le{{c}_{\mu}}\mu ({{\mathbf{q}}_c}), \left( 1-\omega \right)\left( \mathbf{e}^T({{\mathbf{x}}_{o}}+{{\mathbf{s}}_{o}})/k+1 \right)={{c}_{xs}},{{c}_{\mu}}+{{c}_{xs}}\le1. \end{aligned}
W_{[n,0]}=a_{0}^{(0)}W_{[n-1,0]}+\sum_{j=1}^{p}\sum_{s=0}^{j}\sum_{\ell\in\mathbb{Z}}\sum_{k\in\mathbb{Z}}a_{-s}^{(j)}\, A_{[\ell-1,j-s-1]}^{(1)}B_{[k-\ell-1,s-1]}^{(1)}W_{[n-k-1,0]},
F_X(x) =F_X(0)+M\mathrm{sgn}(x)\int_0^x\mathrm{e}^{\beta t}|t|^\nu K_{\nu}(\alpha |t|)\,\mathrm{d}t &=F_X(0)+M\mathrm{sgn}(x)\sum_{k=0}^\infty\frac{\beta^k}{k!}\int_0^x (-1)^k|t|^{\nu+k}K_\nu(\alpha |t|)\,\mathrm{d}t &=F_X(0)+M\sum_{k=0}^\infty\frac{\beta^k}{k!}(\mathrm{sgn}(x))^{k+1}\int_0^{|x|}t^{\nu+k}K_\nu(\alpha t)\,\mathrm{d}t.
L^{2}= - \left( \frac{d{\sf t}}{d \lambda}\right)^{2}+ e^{2{\sf t}/ l}\left( \frac{d{\sf r}}{d \lambda}\right)^{2}+ e^{2{\sf t}/ l}{\sf r}^{2}\sum_{b = 1}^{d - 2}\left( \prod_{a = 1}^{b - 1}\sin^{2}\theta_{a}\right) \left( \frac{d \theta_{b}}{d \lambda}\right)^{2},
\lim\limits_{\l\rightarrow0}L(\l)=\lim\limits_{\l\rightarrow0}\L(\l)\left[(p-1)\left(2-N+\sqrt{(N-2)^2-4\l}\right)+4\right]^2= &(16+o(1))\lim\limits_{\l\rightarrow0}\L(\l)>-16(N-1)\ge-16k(N-2+k)
{\cal L}_{G F}^{( 2 )}~ = ~ \frac{\kappa}{2 \pi}\epsilon^{m n}A_{n}{\dot{A}}_{m}+ \pi_{a}{\dot{z}}_{a}+ \pi_{a}^{*}\dot{z^{*}}_{a}+ \Omega_{1}^{( 0 )}\dot{\alpha}+ \Omega_{2}^{( 1 )}\dot{\beta}+ \Omega_{3}^{( 1 )}\dot{\gamma}+ \Omega_{4}^{( 2 )}\dot{\sigma}-{\cal H}^{( 2 )}
\iota : \mathbb{R}^{p,q}\hookrightarrow \mathbb{R}P^{p+q+1}\overrightarrow{x}=(x_1,\dots,x_{p+q}) \mapsto\iota(\overrightarrow{x}) =\left[ \frac{1-Q(\overrightarrow{x})}2:x_1:\dots:x_{p+q}:\frac{1+Q(\overrightarrow{x})}2 \right],
\sum_{n = 0}^{\infty}\frac{1}{y^{2}+ n^{2}}= \frac{\pi}{2 y}\left( \coth \pi y + \frac{1}{\pi y}\right) \simeq \frac{\pi}{2 y}+ \frac{1}{2 y^{2}}\equiv \bar{S}_{1}( y ){.}
u\in C([0,T];D_{A_{q}}^{1-\frac{1}{p},p})\cap L^{p}(0,T;W^{2,q}\cap W_{0}^{1,q})\,,\partial_{t}u\in L^{p}(0,T;L^{q}), p\in L^{p}(0,T;W^{1,q})\int_{\Omega}pdx=0, \theta\in C([0,T];W^{s,q}),\,\partial_{t}\theta \in L^{p}(0,T;W^{-1,q}).
\frac{\partial V ( \rho )}{\partial \rho}= \rho \left[ \frac{2}{G}- \frac{2 R}{( 4 \pi )^{2}}f ( \xi_{1}, \xi_{2}) \mathrm{E i}\left( - \frac{\rho^{2}}{\Lambda^{2}}\right) \right] = 0 .
\mu_{z}\approx z(\mu_{1},\mu_{2})+\frac{1}{2}\{\frac{\partial^{2}z}{\partial x_{1}^{2}}\sigma_{1}^{2}+\frac{\partial^{2}z}{\partial x_{2}^{2}}\sigma_{2}^{2}\}+\frac{\partial z^{2}}{\partial x_{1}\partial x_{2}}\sigma_{12}Eq(14)
\int f_{g,\delta,j_1}(z)\dots f_{g,\delta,j_{k-1}}(z)\nu_{g,j_k}(z)dz=\int(\hat{f}_{g,\delta,j_1}\overset{(k-1)-}{*\dots*}\hat{f}_{g,\delta,j_{k-1}})(-\omega) \hat{\nu}_{g,j_k}(\omega)d\omega=0.
\langle \varphi^{2}( x ) \rangle^{( a b )}\approx - \frac{( b - a )^{1 - D}}{( 4 \pi )^{D / 2}\Gamma ( D / 2 )}\int_{m_{0}}^{\infty}d y \, \frac{( y^{2}- m_{0}^{2})^{D / 2 - 1}}{c_{b}( y ) e^{2 y}/ c_{a}( y ) - 1}F ( y , r ) , \quad a , b \to \infty , b - a ={\mathrm{c o n s t}}.
\left\{w_h^{\wedge}(x), w_h^{\vee}(x)\right\}=\{T_{\Omega}w_h(x), T_{\Omega^*\setminus\overline{\Omega}}\,w_h(x)\}=\{T_{\Omega}u_h(x), T_{\Omega^*\setminus\overline{\Omega}}\,v_h(x)\}
\min_{(\mathbf{U},\mathbf{C})\in \Delta\times\mathbb{R}^{sN}}\sum_{i=1}^N\left\{\int_\Omega \left|I(x) - c_i\right|^2u_i^p(x) \dx\right. &\left.+\alpha\int_\Omega\int_{y\in\mathcal{N}(x)}(1-u_i(y))^p\left|I(y)-c_i\right|^2u^p_i(x)dydx\right\},
\int_0^1 u_N(x + iy_0) dx \leq \left(1 - \frac{2y_0}{\rho}\right) \int_0^1 u_N(x) dx + \frac{2y_0}{\rho}\frac{k}{N}\ln(C_v|\lambda|) + \frac{2y_0}{\rho}\frac{1}{N}\sum_{n = 1}^{N - k}\int_0^1\ln\left(|E - \lambda v(x + i\rho/2 + j_n\omega) - v_1(j_n)| + 1\right) dx.
g_{L^2}((\alpha_1,\dot\Phi_1),(\alpha_2,\dot\Phi_2)) = 2 \int_X \Re \langle\alpha_1,\alpha_2 \rangle + \Re \langle\dot\Phi_1,\dot\Phi_2\rangle \; dA, =\int_X \langle \dot A_1,\dot A_2 \rangle +2 \Re \langle\dot\Phi_1,\dot\Phi_2\rangle \; dA,
\begin{array}{l}\mathbb{E}|z(\cdot,t\wedge \tau_{n})|^2_{L^2(0,1)}=\mathbb{E}|z(\cdot,0)|^2_{L^2(0,1)}+2\mathbb{E}\int_0^{t\wedge \tau_{n}}z(1,s)\widetilde{w}(s)ds\cr -2\mathbb{E}\int_0^{t\wedge \tau_{n}}|z_x(\cdot,s)|^2_{L^2(0,1)}ds+\mathbb{E}\int_0^{t\wedge \tau_{n}}(\sigma^2-2c)|z(\cdot,s)|^2_{L^2(0,1)}ds.\end{array}
\vec U^T (I_n \otimes E^T P_{\partial\Omega}N_{x_i}E ){\bf A_i}\vec U = \sum_{j=1,N}\lbrack (E \vec U)^T (N_{i}{\bf A_i}) (E \vec U) \rbrack_j ds_j = \sum_{j=1,N}\lbrack \vec W^T \Lambda \vec W \rbrack_jds_j.
w(r, X^{s,x; \alpha,v}_r)=w(r, X^{s,x; \alpha[r,\xi],v}_r)\leq \mathbb{E}[w(t, X^{s,x; \alpha[r,\xi],v}_t)|\mathcal{F}_r] +(t-r)\varphi (t-r)\a.s.
+\sum_{i=1,2}\frac{1}{8}(\Phi^{\alpha}+ \overline{\Phi^\alpha})Q_{2}\big(\Gamma^\alpha((-1)^{i}2R_{3-i}\phi_0 + c_{+}R_i\Phi + c_{-}R_i\overline{\Phi}),(-1)^{i}2R_{3-i}\phi_0 + c_{+}R_i\Phi+ c_{-}R_i\overline{\Phi}) \Big).
\frac{\dim \Phi_{(p+1)qj}}{\dim \Phi_{pqj}}= \frac{p+q+n}{p+q+n-1}\cdot\frac{p+n}{p+n-j}\cdot\frac{p+n-j-1}{p+1}= \frac{\dim \Psi_{(p+1)q(j+1)}}{\dim \Psi_{pq(j+1)}}, \frac{\dim \Phi_{p(q+1)j}}{\dim \Phi_{pqj}}= \frac{p+q+n}{p+q+n-1}\cdot\frac{q+j}{q}\cdot\frac{q+n-1}{q+j+1}= \frac{\dim \Psi_{p(q+1)(j+1)}}{\dim \Psi_{pq(j+1)}};
\begin{aligned}0 = d_{(1,\ldots, n),(v_1, \ldots v_{w^{-1}(k)},\ldots,v_{h(w^{-1}(k))},e_{k-1},\ldots \widehat{v_{r}}, \ldots v_n)}&= d_{(1,\ldots, n),(v_1, \ldots v_{w^{-1}(k)},\ldots,v_{h(w^{-1}(k))}, ce_{j-1},\ldots \widehat{v_{r}}, \ldots v_n)}.\end{aligned}
\left( \frac{1}{n}\right)^{\frac{2k+1}{2}}\sum_{w~|~ ~ l(w)=2k+2}X_{w}= \left( \frac{1}{n}\right)^{\frac{2k+1}{2}}\sum_{r=3}^{2k+1}\sum_{w \in \mathfrak{W}_{2k+2,r,t}}X_{w}+ \left( \frac{1}{n}\right)^{\frac{2k+1}{2}}\sum_{w \in \mathcal{W}_{1,2k+1}}X_{w}.
\begin{array}{r l}{{\mathrm{( A )}}}&{{\pi^{*}\phi \cdot f ( x , t_{1}, \dots , t_{n}) ,}}\\{{\mathrm{~ ( B )}}}&{{\pi^{*}\phi \cdot f ( x , t_{1}, \dots , t_{n}) d t_{1}\wedge \dots \wedge d t_{n},}}\\ \end{array}
\gamma_{+}({\bf p}) = - \frac{e^{2}}{2}\int_{- \infty}^{+ \infty}d t ~ \left[ \frac{p_{3}}{{\bf p}^{4}}p_{k}\epsilon_{i j k}+ \pi \delta ( p_{1}, p_{2}) \Delta ( p_{3}) \epsilon_{i j 3}\right] a_{j}\dot{a}_{i}.
\delta_{-q}^* \hat{P}(p) &= -\sum_{\alpha, \beta=1}^3\lim_{\mu \searrow 0}\: \hat{P}_{m_\beta}\Big( p+\frac{q}{2}\Big)\: \big(g_{-q}^{\alpha,\beta}\big)^* \Big( p+\frac{q}{2}\Big)\:\frac{1}{2 p q + m_\alpha^2 - m_\beta^2 +i \mu \:\Big(p^0 - \frac{q^0}{2}\Big)}\:.
\bar{D}(r_{K-2}) = \frac{\sum_{i=0}^{1}\binom{K}{K-1+i}(N-1)^i N}{\binom{K-2}{K-3}+\sum_{i=0}^{1}\binom{K-1}{K-2+i}(N-1)^iN}&= \frac{N^2 +(K-1)N}{N^2+(K-2)N+(K-2)},
\partial_{a}\ln V_{N}\Gamma_{\hat{a}}\eta -{\frac{1}{V_{N}}}\partial_{a}E_{N}\Gamma_{\hat{a}}\Gamma_{N}\eta -{\frac{1}{2}}\sum_{N^{\prime}\neq N}\bar{q}_{N N^{\prime}}{\frac{1}{V_{N^{\prime}}}}\partial_{a}E_{N^{\prime}}\Gamma_{\hat{a}}\Gamma_{N^{\prime}}\eta \qquad \qquad
\mathfrak{X}_{{\rm ev}}'\coloneqq\bigoplus_{d=1}^{\infty}\bigoplus_{\substack{(l_{1},\dots,l_{d}):\,{\rm even}(l_{1},\dots,l_{d})\neq(1,\dots,1)}}\mathbb{Q}x_{l_{1}}\cdots x_{l_{d}}.
c_{f_{ww_0\lambda}, v_{w_0\lambda}}^{\lambda}.\ast(x) =c_{f_{ww_0\lambda}, v_{w_0\lambda}}^{\lambda}.\ast(\sum_{{\bf c}\in (\mathbb{Z}_{\geqq 0})^{l(w)}}{_{{\bf i}_w{\bf i}'_w}\zeta}_{({\bf c}, 0,\dots, 0)}^{x}E_{{\bf i}_w{\bf i}'_w}^{({\bf c}, 0,\dots, 0)}) &=c_{((\star),\cdot)_{\lambda}, v_{w_0\lambda}}^{\lambda},
M_{i j}\equiv \left( \begin{array}{c c c c c}{{2}}&{{1}}&{{\cdots}}&{{\cdots}}&{{1}}\\{{1}}&{{2}}&{{}}&{{}}&{{\vdots}}\\{{\vdots}}&{{}}&{{\ddots}}&{{}}&{{\vdots}}\\{{\vdots}}&{{}}&{{}}&{{\ddots}}&{{\vdots}}\\{{1}}&{{\cdots}}&{{\cdots}}&{{\cdots}}&{{2}}\\ \end{array}\right) .
\partial_i K(\xi -d\Gamma(m))\frac{1}{H_\mu(\xi)-\Sigma(\xi)+\omega(k)+1}-\partial_i K (\xi-d\Gamma(m))\frac{1}{H_\mu(\xi)-\Sigma(\xi)+1}&=\partial_i K(\xi-d\Gamma(m))\frac{1}{H_\mu(\xi)-\Sigma(\xi)+1}\frac{\omega(k)}{H_\mu(\xi)-\Sigma(\xi)+1+\omega(k)}
H_2(n,p) \ge \left(\frac{pk}{(k-1)(1+2/n)}\right)^n e^{-2/p}(1-o(1))c_k n^{1/2}E(n,{\textstyle \frac{k-1}{k}}) & = \left(\frac{1}{(1+2/n)}\right)^n e^{-2/p}(1-o(1))c_k n^{1/2}E(n,p) & = (1+o(1))c_p n^{1/2}E(n,p)
\overrightarrow{\pi}=(\pi_{1},\pi_{2},\cdots,\pi_{t})\in\mathcal{P}\times\mathcal{P}\cdots\times\mathcal{P}=\mathcal{P}^{t}(\textrm{resp}.~\mathcal{\overline{P}}^{t}, \mathcal{\hat{P}}^t),
Z_{1 / R}\to \int \frac{d^{2}\tau}{\tau_{2}}\tau_{2}^{- 5}| \eta ( \tau ) |^{- 1 6}\left( | Z_{0}^{1}( \tau ) |^{8}+ | Z_{1}^{0}( \tau ) |^{8}+ | Z_{0}^{0}( \tau ) |^{8}\right) \ ,
\vert n_1, n_2, \cdots, n_r \rangle \longrightarrow f_{n_1,n_2, \cdots, n_r}(\eta_1, \eta_2, \cdots, \eta_r) = c_{n_1,n_2,\cdots, n_r}{{\eta_1}}^{n_1}{{\eta_2}}^{n_2}\cdots{{\eta_r}}^{n_r}a_i^+ \longrightarrow{{\eta_i}}.
(23)(12)(23)[x_1,x_2,x_3] =(23)(12) Z^{d_3 d_2}[x_1,x_3,x_2] & = (23) Z^{d_3 d_2 + d_3 d_1}[x_3,x_1,x_2] & = Z^{d_3 d_2 + d_3 d_1 + d_2 d_1}[x_3,x_2,x_1] (12)(23)(12) [x_1,x_2,x_3] = (12)(23) Z^{d_2 d_1}[x_2,x_1,x_3] & = (12) Z^{d_2 d_1 + d_3 d_1}[x_2,x_3,x_1] & = Z^{d_2 d_1 + d_3 d_1 + d_3 d_2}[x_3,x_2,x_1]
\delta = \frac{Tf'(T)}{f(T)}&=T\frac{\Big(n(n-\tanh{T})-\frac{1}{\cosh^2{T}}\Big) e^{nT}\mp\Big(n(n+\tanh{T})-\frac{1}{\cosh^2{T}}\Big) e^{-nT}}{(n-\tanh{T})e^{nT}\pm(n+\tanh{T})e^{-nT}}.
\texttt{maximize}&\int_{0}^{T}<p(t), \; S_i(I(t),p(t))> \sigma \; dt &\texttt{subject to inventory constraint}\frac{dI}{dt}\; = - \; S(I(t),p(t), t) \sigma & \texttt{and markdown boundary constraints}I(0) = I_0, \; I(T) = 0
p_{0 \mu}= p_{\mu}- \frac{1}{2}\tilde{S}^{a b}e_{a \mu b}= p_{\mu}- \frac{1}{2}\tilde{S}^{a b}\omega_{a b \mu}\; , \; \omega_{a b \mu}= \frac{1}{2}( e_{a \mu b}- e_{b \mu a}) ,
x_{k+1}^{\ell,j_{\ell}}= \frac{\sum_{i \in I_k}\big[m_k^{\ell}(x_k^i) \gamma_{\ell,k}(x_k^i) - \vartheta^{\ell}_{k}(x_k^i) \gamma_{\ell,k}'(x_k^i) \big] \, p_k^i}{\sum_{i \in I_k}\gamma_{\ell,k}(x_k^i) \, p_k^i}
\begin{array}{rl}\ \Phi(E,x)= \ \ E^2-\left( \omega^2 + \frac{\lambda^2}{4}+ \lambda E \right)\left(2x+2u-1\right)^2 + \frac{\lambda^2}{4}\left( 2 x +2 u -1 \right)^4 .\end{array}
\mathrm{TV}(P_{f_0},P_{f_1}) = \frac{1}{2}\int_{0}^1 |f_0(x) - f_1(x)|\,\mathrm{d}x = \frac{\gamma}{2}\int_{0}^1 |\psi_{jk}(x)|\,\mathrm{d}x = \frac{c}{2}\|\psi\|_12^{-j(\beta+1/2)}.
\sum_{\ell = 0}^\infty \sum_{\substack{2^\ell p < q < 2^{\ell+1}p q \leq N}}\frac{\psi(q)}{q}\sum_{\substack{r \in T_p, \s \in T_q |sp-rq| < 3 q \psi(p)}}1 \leq \sum_{\ell = 0}^\infty \frac{\psi(2^\ell p)}{2^\ell p}\sum_{\substack{Q_\ell < q < \min(2Q_\ell, N+1) r \in T_p, \s \in T_q |sp-rq| < L_\ell}}1,
\sum_{r=1}^{p-1}\frac{(p+t)!(r+1)!}{(r+1+t)!}= \sum_{r=p-t-1}^{p-2}(-1)^r\frac{(p+t)!}{(r+1+t)!(p-r-2)!}&= (p+t)\sum_{r=p-t-1}^{p-2}(-1)^r\binom{p+t-1}{r+t+1}&= t\sum_{i=0}^{t-1}(-1)^{i + p - t - 1}\binom{p + t - 1}{p+i}&= (-1)^tt\sum_{i=0}^{t-1}(-1)^{i}\binom{t - 1}{i}&= 0
\begin{aligned}z'_1 = z_1 + O( z \Xi ) + O(\Xi^2) + \sum_{J\neq K}O( z_J z_K ) , ..., z'_{2n}= z_{2n}+ O( z \Xi ) + O(\Xi^2) + \sum_{J\neq K}O( z_J z_K ), & \Xi ' =\Xi + O( z \Xi ) + O(\Xi^2) + \sum_{J\neq K}O( z_J z_K ) . \end{aligned}
\xi = \left( \begin{array}{c c}{0}&{1}\\{0}&{0}\\ \end{array}\right) = \frac{1}{2}( \sigma_{1}+ i \sigma_{2}) \ , \overline{{\xi}}= \left( \begin{array}{c c}{0}&{0}\\{1}&{0}\\ \end{array}\right) = \frac{1}{2}( \sigma_{1}- i \sigma_{2}) \ .
\theta(\theta(\rho_{j}, \rho_{i}), \theta(\rho_{j}, \rho_{\ell})) =\theta(1, \rho_n^{q(\ell-i)+B(j-\ell, n-\ell+i)})=\rho_n^{q(\ell-i)+B(j-\ell, n-\ell+i)}.
\left( 0 , a , 1 \right) = \left( \begin{array}{c c}{\mathrm{s g n}\left( a \right) \cdot \left( \begin{array}{c c}{\cosh \phi}&{\sinh \phi}\\{\sinh \phi}&{\cosh \phi}\\ \end{array}\right)}&{0}\\{0}&{{\bf 1}_{n - 2}}\\ \end{array}\right) \quad ,
\xi_q(-2m+j-1)=\sum_{i=0}^{2n-2}b_{2n-1,i}\sum_{p=0}^i \binom{i}{p}\frac{\beta_{n,q}(2m-j+1,i,p)}{p+2m-j+2}-\frac{\binom{n+q-1}{n}}{q}(n+1)^{2m-j}.
{\cal L}_{2}\equiv{\cal L}_{3}\left( \varphi, \partial_{\mu}\varphi, \partial_{\mu\nu}\varphi; B_{\rho}, F^{(1)}, F^{(2)}, C^{(1)}, C^{(2)}; B_{\rho\sigma}, D^{(1)}, D^{(2)}, G^{(2)}, G^{(3)}\right) .
\tan \mathrm{\boldmath \Large ~ \left( ~ \right. ~}\! \! \! \delta_{0}^{( D_{0})}( k ) - ( D / 2 - 1 ) \, \pi / 2 \mathrm{\boldmath \Large ~ \left. ~ \right) ~}\! \! = \frac{\pi}{\ln \left( k^{2}/ | E_{_\mathrm{( g s )}}| \right)}\; .
[\pi_1R(\lambda,{\cal A})e^{r{\cal A}}\Phi](\theta) = [\pi_1e^{r{\cal A}}R(\lambda,{\cal A})\Phi](\theta) &= [\pi_1 e^{(r+\theta){\cal A}}R(\lambda,{\cal A})\Phi](0) &= [\pi_1R(\lambda,{\cal A})e^{(r+\theta){\cal A}}\Phi](0) &= \pi_0 R(\lambda,{\cal A})e^{(r+\theta){\cal A}}\Phi.
\|\vec{u}\|^2_{X(G_k^{j})}:=\sum_{0\leq i< j}2^{i-j}\sum_{G_{\alpha}^{i}\subset G_k^{j}}\|P_{\xi(G_{\alpha}^{i}), i-2\leq\cdot\leq i+2}\vec{u}\|^2_{U^2_{\triangle}(h^{1};G_{\alpha}^{i}\times\mathbb{R}^2)}+\sum_{i\geq j}\|P_{\xi(G_{k}^{j}),i-2\leq\cdot\leq i+2}\vec{u}\|^2_{U^2_{\triangle}(h^{1};G_{k}^{j}\times\mathbb{R}^2)}.
m\rq_{1,1}(x,y)+\phi_1(m\rq_{1,0}(x,y))=m_{1,1}(x,y)+m_{1,0}(\phi_1(x),y)+m_{1,0}(x,\phi_1(y)), &m\rq_{2,1}(x,y)+\phi_1(m\rq_{2,0}(x,y))=m_{2,1}(x,y)+m_{2,0}(\phi_1(x),y)+m_{2,0}(x,\phi_1(y)).
\bar{\Xi}_{\lambda_{2}}^{\lambda_{1}}\otimes f_{\dot{J}}^{\dot{I}}\otimes \Xi_{\lambda_{4}}^{\lambda_{3}}\left( \bar{\phi}^{\rho_{1}}\otimes s^{\dot{K}}\otimes \phi^{\rho_{2}}\right) = \delta_{\lambda_{2}}^{\rho_{1}}\delta_{\dot{J}}^{\dot{K}}\delta_{\lambda_{4}}^{\rho_{2}}\bar{\phi}^{\lambda_{1}}\otimes s^{\dot{I}}\otimes \phi^{\lambda_{3}}.
y_{\mathbf{r}}=\mu(r)\phi_\omega(r)\sum_{\mathbf{r}|\mathbf{f}}\frac{y_{\mathbf{f}}}{\phi_\omega(f)}\sum_{\substack{\mathbf{d}\mathbf{r}|\mathbf{d},\mathbf{d}|\mathbf{f}}}\mu(d)=\mu(r)\phi_\omega(r)\sum_{\mathbf{r}|\mathbf{d}}\frac{\lambda_{\mathbf{d}}}{d}.
\Psi_{h,k}(z,w) =\frac{\Gamma(s_0+1)}{s_0 k^{s_0+1}w^{s_0}}\sum_{r=1}^{k}z^r\Phi(z^k,s_0+1, \frac{r}{k}) A_{rh,k}\omega_{h,k,n}(z) = \exp(\sum_{r=1}^{k}\frac{z^r}{k}\Phi(z^k,1, \frac{r}{k})D_{rh,k}(0) - 2n\pi i \frac{h}{k}) g_{h,k}(z,w) =\frac{1}{k}\sum_{r=1}^{k}z^r \frac{1}{2\pi i}\int_{\sigma_0-i\infty}^{\sigma_0+i\infty}J(s;z,r,h,k,w)ds.
M_{d}^{2}| \Phi_{l}\rangle = ( l_{n}( l_{n}+ q ( d - 2 ) ) + l_{n - 1}( l_{n - 1}+ q ( d - 4 ) ) + \dots + l_{2}( l_{2}+ 2 q ) + l_{1}^{2}) | \Phi_{l}\rangle .
\begin{pmatrix}\sum_{k=1}^Ke_k^2(x_1)\eta_k\tau u_1^0 \vdots \sum_{k=1}^Ke_k^2(x_M)\eta_k\tau u_M^0\end{pmatrix}=\int_0^\tau\begin{pmatrix}\sum_{k=1}^Ke_k^2(x_1)\eta_ku_1^0 \vdots \sum_{k=1}^Ke_k^2(x_M)\eta_ku_M^0\end{pmatrix}ds.
r_{s-2}= \left( q_s + 3 \right) \left( r_{s-1}- r_{s+1}\right) + r_{s+1}- r_s + q_s r_{s+1}&= \left( q_s + 3 \right) r_s + r_{s-1}- 2r_s + q_s r_{s-1}- q_s r_s &= (q_s + 1) r_{s-1}+ r_s,
t_{\tau ,a}^{(h)}=\tau (\zeta_{a}^{(h-1)})t_{\tau ,a}^{(h-1)}-\frac{_{q}K_{+}(\xi_{a}^{(h-3/2)})_{q}\mathcal{U}_{-}(\xi_{a}^{(h-3/2)})}{(\xi_{a}^{(h-1/2)})^{2}-1/(\xi_{a}^{(h-1/2)})^{2}}t_{\tau,a}^{(h-2)}h\in \left\{1,...,p-1\right\}
log\Upsilon_{b}(w)=\int_{0}^{\infty}\frac{dt}{t}[(\frac{Q}{2}-w)^{2}e^{-t}-\frac{sinh^{2}\frac{1}{2}(\frac{Q}{2}-w)t}{sinh(\frac{1}{2}bt)sinh(\frac{1}{2}b^{-1}t)}].
\tilde{q}_{N_0}^1(\xi-\eta, \eta)= q_1^1(\xi-\eta, \eta) |\xi|^{2N_0}+ q_1^1(-\xi, \eta)|\xi-\eta|^{2N_0}+ 2 q_1^3(\xi-\eta,\eta) |\xi|^{2N_0},
\omega^{3}_{4}= - i \{\frac{1}{2}d\ln (\frac{d\overline{g}_{2}}{d\overline{z}_{2}}\frac{dz_{2}}{dg_{2}}) + \frac{(g_{2}d\overline{g}_{2}- \overline{g}_{2}dg_{2})}{(1-g_{2}\overline{g}_{2})}\},
\mathbf{s}_{m}^{a}C^{\alpha_{1}b c}= - \epsilon^{a c}B^{\alpha_{1}b}- \epsilon^{a b}B^{\alpha_{1}c}+ G_{\alpha_{0}\beta_{1}}^{\alpha_{1}}C^{\alpha_{0}a}C^{\beta_{1}b c}- \mathrm{~ \frac{1}{2}~}H_{\alpha_{0}\beta_{0}\gamma_{0}}^{\alpha_{1}}C^{\alpha_{0}a}C^{\beta_{0}b}C^{\gamma_{0}c}.
\exists\;\pmb{x}\; \forall\;\zeta \in \mathcal{Z}\; \exists\;\pmb{y}\colon A\pmb{x}+ B\pmb{y}\leq \pmb{d}(\pmb{\zeta}) & \pmb{x}= (\pmb{x}^c, \pmb{x}^d) \in (\mathbb{R}^{m_c}, \mathbb{Z}^{m_d}) & \pmb{y}= (\pmb{y}^c, \pmb{y}^d) \in (\mathbb{R}^{n_c}, \mathbb{Z}^{n_d}),
f'(x_1^*,\ldots,x_L^*;0,\ldots,d,\ldots,0) \ge\gamma_l\sum_{i\in\Lambda_1\cup\Lambda}\|(D_ld)_i\|_2 &=\gamma_l\sum_{i\in[m_l]}\|(D_ld)_i\|_2 &\ge\gamma_l\left(\sum_{i\in[m_l]}\|(D_ld)_i\|_2^2\right)^\frac{1}{2}&=\gamma_l\|D_ld\|_2 &\ge\gamma_l\sigma_{\min}(D_l)\|d\|_2 &=\gamma_l\sigma_{K_l,m_l,p_l}(D_l)\|d\|_2,
\|(\prod_{j=1}^{n-1}|E_{I_j}f|)^{\frac1{n-1}}\|_{L^{2(n+1)}(B_R)}\le C_{K,\epsilon}R^{\frac{1}{2(n+1)}+\epsilon}(\sum_{\Delta:\frac1{R^{1/n}}-\atop{\Delta\subset [0,1]}}\|E_{\Delta}f\|_{L^{2(n+1)}(w_{B_R})}^{2(n+1)})^{\frac{1}{2(n+1)}}
\begin{aligned}\dot u = \frac{1}{4}(u+v)^3 \left(u+\O((u+v)^3)\right), \dot v = - \frac{1}{4}(u+v)^3 \left(v+\O((u+v)^3)\right) , \dot{\tilde{z}}= \frac{1}{4}(u+v)^2(u-v) \tilde z + \O((u+v)^5), \dot \phi = \omega,\end{aligned}
\begin{array}{l l c l l}{{}}&{{V_{3}}}&{{\stackrel{C_{3}^{1 2}}{\longrightarrow}}}&{{V_{1}\otimes V_{2}}}&{{}}\\{{\Pi_{3}}}&{{\downarrow}}&{{}}&{{\downarrow}}&{{\Pi_{1}\otimes \Pi_{2}}}\\{{}}&{{V_{3}}}&{{\stackrel{C_{3}^{1 2}}{\longrightarrow}}}&{{V_{1}\otimes V_{2}}}&{{}}\\ \end{array}
\frac{1}{k^p (2k+m)(2k+n)}= \frac{1}{m n k^p}+ \frac{2}{m (m-n)}\left ( \sum_{j=0}^{p-2}\frac{(-2)^j}{m^{j+1}k^{p-1-j}}+ \Big ( \frac{2}{m}\Big )^{p-1}\frac{(-1)^{p-1}}{2k+m}\right ) & - \frac{2}{n (m-n)}\left ( \sum_{j=0}^{p-2}\frac{(-2)^j}{n^{j+1}k^{p-1-j}}+ \Big ( \frac{2}{n}\Big )^{p-1}\frac{(-1)^{p-1}}{2k+n}\right ).
\log\Gamma_M(w\,|\,a) = -\frac{1}{M!}B_{M, M}(w\,|\,a)\,\log(w) + \sum\limits_{k=0}^M\frac{B_{M,k}(0\,|\,a) (-w)^{M-k}}{k!(M-k)!}\sum\limits_{l=1}^{M-k}\frac{1}{l}+ + R_M(w\,|\,a), R_M(w\,|\,a) = O(w^{-1}), \,|w|\rightarrow\infty, \, |\arg(w)|<\pi.
\theta_s^2( a_pa_0\otimes a_1\otimes a_2\otimes\cdots\otimes a_{p-1}) ={\rm Tr}\left(\Gamma a_0\left(\prod_{l=1}^{m-2}b_l(a_l)\right)[F,\delta(a_{m-1})]\left(\prod_{l=m}^{p-1}b_{l+1}(a_l)\right)[T,a_p]\right)+Y_p^2.
\lim_{\delta \to 0}\frac{{\rm AoI}(S_{\mathrm{z}},Q_{\mathrm{uni}},F_\mathrm{s})}{\log_2 D}= \lim_{\delta \to 0}\frac{{\rm AoI}(S_{\mathrm{z}},Q_{\mathrm{uni}},F_\mathrm{s})}{H[Q_{\mathrm{uni}}(X)]}\frac{H[Q_{\mathrm{uni}}(X)]}{\log_2 D}= -\frac{3}{4}.
\partial_t\int_{\mathbb{R}}\frac{h}{h+1}=\int_{\mathbb{R}}\frac{-\partial_th}{(h+1)^2}=\int_{\mathbb{R}}\frac{1}{(h+1)^2}\bigg(\theta\partial_vh+\frac{\mathbb{P}_0f+h}{t}\bigg)=\int_{\mathbb{R}}\frac{\chi}{t(h+1)^2}=0,
\begin{array}{c c c c c c c}{}&{\circ_{0}}&{\left( \frac{1}{g}\right)}&{}&{}&{}&{}\\{}&{\mid}&{}&{}&{}&{}&{}\\{\left( \frac{R_{1}}{g}\right) \circ_{1}-}&{\circ_{2}}&{-}&{\circ_{3}}&{- \dots -}&{\circ_{d - 1}\left( \frac{1}{R_{d}}\right)}\\ \end{array}
{\sf M}^{a b}( x , y ) = \left. \frac{\delta \chi^{a}[ A^{\Omega}] ( x )}{\delta \varphi^{b}( y )}\right|_{\varphi = 0}\equiv{\sf M}[ A ] \qquad ; \qquad \Omega = \exp ( - g \varphi^{a}T^{a})
xa+yc = 1+u(bE_4^2-aE_6), xb+yd = u(bE_6-aE_4), x-u(xa+yc)E_6+u(xb+yd)E_4^2 = -u^2(bE_4^2-aE_6)E_6+u^2(bE_6-aE_4)E_4^2, y-6u(xa+yc)E_4+6u(xb+yd)E_6 = -6u^2(bE_4^2-aE_6)E_4+6u^2(bE_6-aE_4)E_6.
\phi_v^H(s_\mu s^*_\nu) = \delta_{\mu,\nu}\Big(e^{-\beta|\mu|}\sum_{\lambda\in s(\mu)(E\backslash H)^*v}e^{-\beta|\lambda|}\Big)y_v^{-1}=\delta_{\mu,\nu}\Big(e^{-\beta|\mu|}\sum_{\lambda\in s(\mu)E^*v}e^{-\beta|\lambda|}\Big)y_v^{-1};
\begin{array}{l}\displaystyle\Theta = pd^mx + p_\alpha^i d u^\alpha \wedge d^{m-1}x_i + \frac{1}{n(ij)}\, p_\alpha^{1_i+1_j}d u_i^\alpha \wedge d^{m-1}x_j \, , begin{align*}10pt]\displaystyle\Omega = - d p \wedge d^mx - d p_\alpha^i \wedge d u^\alpha \wedge d^{m-1}x_i - \frac{1}{n(ij)}\, d p_\alpha^{1_i+1_j}\wedge d u_i^\alpha \wedge d^{m-1}x_j \, .\end{array}
\eta_{m\kappa}\left(0\right)+\eta_{-m,\kappa}\left(0\right)=\left(N_\kappa^++N_\kappa^-\right)\pi+\left(-1\right)^l\frac{\pi}{2}\left[\sin^2\eta_{m\kappa}\left(0\right)+\sin^2\eta_{-m,\kappa}\left(0\right)\right]\,,
\psi_{s}\psi_w=\begin{cases}\psi_{sw}, \ell(sw)=\ell(w)+1, 0, \ell(sw)=\ell(w)-1,\end{cases}\psi_{w}\psi_s=\begin{cases}\psi_{ws}, \ell(ws)=\ell(w)+1, 0, \ell(ws)=\ell(w)-1.\end{cases}
\ddot{\psi}_{n}= \frac{4 - h^{2}}{4 h^{2}}\cos \psi_{n}( \sin \psi_{n + 1}+ \sin \psi_{n - 1}) - \frac{4 + h^{2}}{4 h^{2}}\sin \psi_{n}( \cos \psi_{n + 1}+ \cos \psi_{n - 1})
m_{G}^{2}= - 2 e^{2}\int_{{\mathcal R}}\frac{d^{2}p}{( 2 \pi )^{2}}\frac{\partial}{\partial p_{\mu}}\left[ \frac{\Lambda^{2}}{p^{2}( p^{2}+ \Lambda^{2})}p_{\mu}\right] ,
\alpha(\delta(\mu))=\frac{a}{2}\rho_{j}[w_{(j,r_s)},w^*_{(j,r_s)}]+\frac{a}{2}\sum_{i\neq j}^{}\rho_{i}[w_{(i,r_s)},w^*_{(j,r_s)}]+\frac{a}{2}\,\,\sum_{t\neq j}[\alpha(w_{(t,r_s)}),w^*_{(t,r_s)}].
{\bf s}_{\textrm{dir}}(t) = [s_{\textrm{dir}_1}(t),\cdots,s_{\textrm{dir}_{N_r}}(t)]^T =\sum_{k=1}^{K}\widetilde{\alpha}_{1,k}{\bf a}_r(\theta_t,f_k){\bf a}_t^T(\theta_t,f_k){\bf F}_k{\bf s}_ke^{\mathrm{j}2\pi{f_k}t},
G_{4}^{\mu \nu \alpha \beta}( x ) = \int{\frac{d p^{4}}{( 2 \pi )^{4}}}\left({\frac{1}{2}}( g^{\mu \alpha}g^{\nu \beta}+ g^{\mu \beta}g^{\nu \alpha}) -{\frac{1}{3}}g^{\mu \nu}g^{\alpha \beta}+{\cal O}( p ) \right) D ( p^{2}, m_{0}^{2}, \Gamma ) e^{- i p ( x )},
B ( u ) \biggl |_{u \to \infty}= u \Bigl [ - 6 Q ( 1 ) + 4 \int_{0}^{1}d \gamma \frac{\gamma^{2}}{1 - \gamma^{2}}\Bigl ( Q ( \gamma ) - Q ( 1 ) \Bigr ) \Bigr ] + O ( \log u ) \quad .
J_8= 2\int \nabla (\overline{\phi}^{k+1}(\phi^{k+1}+\phi^k))\cdot Q(u^k) \cdot \overline{u}^{k+1}= -2 \int \sum_{i,j}\overline{\phi}^{k+1}(\phi^{k+1}+\phi^k)\partial_i a^{ij}_k\overline{u}^{k+1,j}-2\int \sum_{i,j}\overline{\phi}^{k+1}(\phi^{k+1}+\phi^k) a^{ij}_k\partial_i \overline{u}^{k+1,j}=: J_{81}+J_{82}+J_{83}+J_{84},
T=\begin{pmatrix}0 \frac{1}{4}\frac{1}{4}\frac{1}{2}\frac{1}{2}0 0 \frac{1}{2}0 \frac{1}{4}\frac{1}{4}\frac{1}{2}\frac{1}{2}0 0 \frac{1}{2}\end{pmatrix}
n^{2}( x^{\perp}, t ) = \frac{C^{2}}{4 t^{2}}\frac{\left( 4 t^{2}-{x^{\perp}}^{2}\right)^{2}}{C^{2}{x^{\perp}}^{2}+ 4 \sqrt{C^{4}- 4 I}x^{\perp}t + 4 C^{2}t^{2}}.