harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
H_j(z)\in C^\infty(|z|<\delta),\ overline\partial H_j(z)=0,\ \ \forall \,j &\eta=\theta+\mathrm{arg}\,g(z)\quad\mbox{$(\mathrm{mod}\,2\pi)$\,\, where $\mathrm{arg}\,g(z)=\mathrm{Im}\log g(z)$}&\widetilde\varphi(H(z), \overline{H(z)})=\varphi(z, \overline z)+\log|g(z)|
\mathbf{e}^{j}(l_{1},m_{1})\mathbf{e}^{j}(l_{2},m_{2}) =\sum_{l=0}^{2j}(-1)^{2j+m}\sqrt{(2l_{1}+1)(2l_{2}+1)(2l+1)}& \cdot\left(\begin{array}[c]{ccc}l_{1}l_{2}l -m_{1}-m_{2}m\end{array}\right) \left\{\begin{array}[c]{ccc}l_{1}l_{2}l j j j\end{array}\right\}\mathbf{e}^{j}(l,m)
s^{\prime}= a \left( s + s_{o}- \frac{1}{2}\right) + b = a \left( s + s_{o}- \frac{1}{2}\right) + \frac{1}{2}[ 1 - a ( 2 \beta - 1 ) ] = a \left( s - \frac{\beta}{2}\right) + \frac{1}{2}.
\phi=\left(\begin{array}{cccccccccc}x_{1}x_{2}\cdots x_{k-1}x_{k}x_{k+1}\cdots x_{k+m-1}x_{k+m}x_{1}x_{2}\cdots x_{k-1}x_{k+1}x_{k+2}\cdots x_{k+m}ax_k \end{array}\right. \left.\begin{array}{ccc}x_{k+m+1}\cdots x_{r}x_{k+m+1}\cdots x_r\end{array}\right),
\widehat\theta\left[{\rm d}v^\varepsilon+v^\varepsilon_{xx}{\rm d}t+\left(x+\varepsilon\right)^{2\gamma}v^\varepsilon_{yy}{\rm d}t+\frac{\sigma}{(x+\varepsilon)^2}v^\varepsilon{\rm d}t\right]= P_1+(P_2+P){\rm d}t.
|( (-\Delta)^{1/2}(1 + \cdot^2)^{-q/2}) (x) | &\leq C\begin{cases}(1+x^2)^{-(q+1)/2}, \mathrm{if}0 < q < n, (1+x^2)^{-(n+1)/2}(1+\log (1+|x|)), \mathrm{if}q = n, (1+x^2)^{-(n+1)/2}, \mathrm{if}q > n.\end{cases}
N_{n m}^{w}= \frac 1{n m}\oint_{\infty}\frac{d z}{2 \pi i}z^{n}\oint_{\infty}\frac{d \zeta}{2 \pi i}\zeta^{m}\frac{w^{\prime}( z ) w^{\prime}( \zeta )}{( w ( z ) - w ( \zeta ) )^{2}}
\begin{aligned}\int_0^T\left<S\mathcal{K}_{\mathcal{I}_l}W_\epsilon(x_j,x_k)u_\epsilon,\mathcal{K}_{\mathcal{I}_l}v\right> = \left<SW_\epsilon(x_j,x_k)\mathcal{K}_{\mathcal{I}_l,j}u_\epsilon,\mathcal{K}_{\mathcal{I}_l,j}v\right> &+\left<S\nabla_jW_\epsilon(x_j,x_k)\mathcal{K}_{\mathcal{I}_l,j}u_\epsilon,\nabla_j\mathcal{K}_{\mathcal{I}_l,j}v\right>.\end{aligned}
d^2h(y)(w)=\lim_{\nu\to\infty}\frac{2[h_{\overline{k}}(y+\tau_{\!\nu}w^{\nu})-h_{\overline{k}}(y)-\langle\nabla h_{\overline{k}}(y),w^{\nu}\rangle]}{\tau_{\!\nu}^2}=\langle w,\nabla^2h_{\overline{k}}(y)w\rangle.
w=c_1v_1+\cdots+c_{a-1}v_{a-1}+c_ay_1+\cdots+ c_{a+b-2}y_{b-1}+c_{a+b-1}v_a+ &c_{a+b}u_1+\cdots+c_{a+2b-2}u_{b-1}+c_{a+2b-1}z,
\delta{\widetilde\varphi}= - \frac{{\rm e}^{\varphi_0}\,(m-\chi_0\,n)^2-{\rm e}^{-\varphi_0}n^2}{2\,{\widetilde\Delta}_{m,n}}\,\frac{{\widetilde Q}_1}{r^6}~~~,~~~\delta{\widetilde\chi}=\frac{{\rm e}^{-\varphi_0}\,(m-\chi_0\,n)\,n}{{\widetilde\Delta}_{m,n}}\,\frac{{\widetilde Q}_1}{r^6}~~,
y_t/ h_{t,1}= \underbrace{\sqrt{\rho}c_t}_{\rho}+\underbrace{\sqrt{\rho^{1- \alpha}}a_{t,3}}_{\rho^{1-\alpha}}+\underbrace{u_t /h_{t,1}}_{\rho^{0}}z_t/ g_{t,1}= \underbrace{\sqrt{\rho^{\alpha}}c_t}_{\rho^{\alpha}}+\underbrace{\sqrt{\rho^{0}}a_{t,3}}_{\rho^{0}}+\underbrace{v_t/ g_{t,1}}_{\rho^{0}}.
\sum_{k=0}^n \binom{a}{k}\binom{b}{n-k}\binom{k}{c}\binom{n-k}{d}k^2 = \binom{a+b-c-d}{n-c-d}\binom{a}{c}\binom{b}{d}\cdot \frac{n(a-c)[n(a-c-1)+2(bc-ad)+b+d]+(bc-ad)^2-bd(a-c)-ac(b-d)}{(a+b-c-d)(a+b-c-d-1)}
\tilde{R}(\bar{\kappa},\kappa_1) - \tilde{R}(\bar{\kappa},\kappa_2) = \tilde{R}(\bar{\kappa},\bar{0}) = \frac{1}{2}\left( \Phi(\bar{\kappa}) - \|\gamma(\bar{\kappa})\|_{L^2(m)}^2 +\Phi(\bar{0})- \|\gamma(\bar{0})\|_{L^2(m)}^2 - \Phi(\bar{\kappa}+\bar{0}) + \|\gamma(\bar{\kappa}+\bar{0})\|_{L^2(m)}^2 \right) &=0\ .
Pr[G\sim G(n,p)\mbox{is not Hamiltonian}]=\sum_{m=0}^N Pr[|E(G)|=m]\cdot Pr[G\mbox{is not Hamiltonian}|\, |E(G)=m] \le Pr[|E(G)|\not\in I] +\sum_{m\in I}Pr[|E(G)|=m]\,Pr[G\mbox{is not Hamiltonian}|\, |E(G)=m] = o(1) + \sum_{m\in I}Pr[|E(G)|=m]\,Pr[G\sim G(n,m)\mbox{is not Hamiltonian}|=o(1)\cdot Pr[ Bin(N,p)\in I] =o(1)\,.
L_a^4-\frac{2(\widetilde{F_-}^2+r^2\widetilde{F_+}^2)}{(1-r)^2}L_a^2+\frac{(\widetilde{F_-}^2-r^2\widetilde{F_+}^2)^2+(\widetilde{F_-}^2+r^2\widetilde{F_+}^2)^2\tan^2(\Delta\psi)}{(1-r)^4(1+\tan^2(\Delta\psi))}=0.
\frac{2}{2-t}(1-t)^{x}=\frac{1}{1-\frac{t}{2}}(1-t)^{x}=\sum_{l=0}^{\infty}\bigg(\frac{1}{2}\bigg)^{l}t^{l}\sum_{k=0}^{\infty}\binom{x}{k}(-1)^{k}t^{k}&=\sum_{n=0}^{\infty}\bigg(\sum_{k=0}^{n}\binom{x}{k}\bigg(\frac{1}{2}\bigg)^{n-k}(-1)^{k}\bigg)t^{n}.
\int_0^x t^\nu \mathbf{L}_{\nu+n}(t)\,\mathrm{d}t\sim\int_0^x \frac{t^{2\nu+n+1}}{\sqrt{\pi}2^{\nu+n}\Gamma(\nu+n+\frac{3}{2})}\,\mathrm{d}t=\frac{x^{2\nu+n+2}}{2^{\nu+n}(2\nu+n+2)\Gamma(\nu+n+\frac{3}{2})},
\theta = \left( \begin{array}{c c c c}{0}&{- i e_{1}}&{- i e_{2}}&{- i e_{3}}\\{i e_{1}}&{0}&{b_{3}}&{- b_{2}}\\{i e_{2}}&{- b_{3}}&{0}&{b_{1}}\\{i e_{3}}&{b_{2}}&{- b_{1}}&{0}\\ \end{array}\right) ,
e^{-b^{2}/2}\le Q_{n}(a,b)\le exp[-\frac{1}{2}(b^{2}+a^{2})]\sqrt{I_{0}(2ab)}\sqrt{\frac{2n-1}{2}+\frac{\zeta^{2(1-n)}}{2(1-\zeta^{2})}},\zeta<1,
\int_0^{\Delta}\left(p(1-\sigma(s))^{p-1}s\right)^{\frac{n-1}{p}}\mathrm{d}s\geq (1-\Delta)^{\frac{p-1}{p}(n-1)}\int_0^{\Delta}(ps)^{\frac{n-1}{p}}\mathrm{d}s=(1-\Delta)^{\frac{p-1}{p}(n-1)}\frac{1}{n-1+p}(p\Delta)^{\frac{n-1+p}{p}}\quad.
(P_{F_{n}}(\varphi,\varepsilon))^2= (2 \sin \pi( F_{n}\varphi + \varepsilon))^2 &\times \prod_{r=1}^{F_{n}-1}4 \left(\sin^2 \pi \left(r \varphi + \frac{(-\varphi)^{n}}{2}\right)- \sin^2 \pi \left(\frac{(-\varphi)^{n}}{2}- \varepsilon\right) \right).
\frac{|-1 + z_{1,n}|}{|z_{j,n}|}< \kappa(1,a) =\frac{\left|1 -\exp\bigl(\frac{\pi}{a}\bigr)\right|\exp\bigl(\frac{-\pi}{a}\bigr)}{\exp\bigl(\frac{\pi}{a}\bigr) +\left|1 -\exp\bigl(\frac{\pi}{a}\bigr)\right|},
[a_l,a_m,a_k,a_{\infty}] = \left( \frac{\theta[\eta_{S_l}-\tilde{\Delta}-2\eta_k](\Omega)\theta[\eta_{S_l}-\tilde{\Delta}](\Omega)}{\theta[\eta_{T_l}-\tilde{\Delta}](\Omega)^2}\right) \div \left(\frac{\theta[\eta_{S_m}-\tilde{\Delta}-2\eta_k](\Omega)\theta[\eta_{S_m}-\tilde{\Delta}](\Omega)}{\theta[\eta_{T_m}-\tilde{\Delta}](\Omega)^2}\right).
\bar{\Delta}\Delta = f^{- 1}1_{[ 2 ] \times [ 2 ]}= \left( \! \begin{array}{c c}{{f_{1}^{- 1}}}&{{0_{[ 2 ] \times [ 2 ]}}}\\{{0_{[ 2 ] \times [ 2 ]}}}&{{f_{2}^{- 1}}}\\ \end{array}\! \right) ,
\sum_{k=1}^\infty \int_{D(z_k,r)}\Bigg( \frac{\mu(D(z, 2r))}{(1+|z|)^s}\Bigg)^p dV(z) \geq \sum_{k=1}^\infty \int_{D(z_k,r)}\Bigg( \frac{\mu(D(z_k, r))}{(1+|z_k|)^s}\Bigg)^p dV(z) &\gtrsim \sum_{k=1}^\infty \Bigg( \frac{\mu(D(z_k, r))}{(1+|z_k|)^s}\Bigg)^p
\rho(\frac{\partial v_{i}}{\partial t}+v_{j}\frac{\partial v_{i}}{\partial x_{j}})=-\frac{\partial p}{\partial x_{i}}+\frac{\partial}{\partial x_{j}}[\mu(\frac{\partial v_{i}}{\partial x_{j}}+\frac{\partial v_{j}}{\partial x_{i}})+\lambda\frac{\partial v_{k}}{\partial x_{k}}]+\rho f_{i}
u_k (x) = \int\limits_{\mathbb{R}^3}\sum\limits^3_{\ell=1}\Psi_{k\ell}(x -{y}) f_\ell ({y}) \mbox{d}{y}, p(x) = \int\limits_{\mathbb{R}^3}\langle \Theta (x -{y}),{f}) ({y})\rangle \mbox{d}{y}
\pi ={\cal P}+ \frac{i}{W}\frac{\partial W}{\partial z}( \psi \overline{{{\psi}}}- \frac{3}{2}) , \overline{{{\pi}}}= \overline{{{{\cal P}}}}- \frac{i}{W}\frac{\partial W}{\partial \overline{{{z}}}}( \psi \overline{{{\psi}}}- \frac{1}{2}) ,
E_\mu(u) = \, -\frac1\pi \int \mu(\xi) \ln |T(-\xi/2)| d\xi = \, \frac1\pi \real \int_1^\infty \overline{[\mu](\tau)}\Big( \ln T(i\tau/2) + \sum_{j=0}^{N-1}i^j H_{j}\tau^{-j-1}\Big) dt + \sum_{j=0}^{N-1}\frac1{j!}\overline{\mu^{(j)}(0)}H_{j}
|\lambda|^{n}f^0(\lambda)\big|A_N\bigl(e^{i\lambda}\bigr)\bigl(1-e^{i\lambda\mu}\bigr)^{n}g^0(\lambda)+C^{\mu,0}_{N}\bigl(e^{i\lambda}\bigr)\big| &=\alpha_1\big|1-e^{i\lambda\mu}\big|^{n}\bigl(f^0(\lambda)+\lambda^{2n}g^0(\lambda)\bigr),
\frac{n}{a_n^2}\Gamma_n(\frac{a_n^2}{n}t)=\frac{n}{a_n^2}\log W_n(\frac{a_n}{n}t)+\frac{n}{a_n^2}\lambda_n(\frac{a_n^2}{n}t)-\frac{n}{a_n^2}\log W_n(0).
\int_{G}t_{m n}^{( \rho )}( g ) \, d g = \sum_{j = 1}^{\ell}P^{( j ) m}P_{n}^{( j )}, \qquad P^{( j ) m}= \eta^{m}( w^{( j )}) , \quad P_{n}^{( j )}= \theta^{( j )}( v_{n}) .
R(t) \geq-\sin(\frac{\pi}{3})(1-{\mathcal M}(L^+_{\frac{\pi}{3}}(t)) -{\mathcal M}(L^-_{\frac{\pi}{3}}(t))+\sin\frac{\pi}{3}{\mathcal M}(L^+_{\frac{\pi}{3}}(t))-{\mathcal M}(L^-_{\frac{\pi}{3}}(t)) &=\sqrt{3}{\mathcal M}(L^+_{\frac{\pi}{3}}(t)) - \frac{\sqrt{3}}{2}-{\mathcal M}(L^-_{\frac{\pi}{3}}(t)) \Big (1-\frac{\sqrt{3}}{2}\Big ).
e T_{\mu \nu}={\tilde{\overline{\Psi}}_{L}}{\gamma_{\mu}}i \left( D_{\nu}- \frac{1}{2}\Gamma_{\alpha \nu}^{\alpha}\right){\tilde{\Psi}}_{L}- \frac{i}{2}g_{\mu \nu}\partial_{\alpha}({\tilde{\overline{\Psi}}}_{L}\gamma^{\alpha}{\tilde{\Psi}_{L}}) . \nonumber \,
C^{t_1,\ldots,t_p}= C^{t_1,t_2}\stackrel{C^{t_2}(.)}{\star}C^{t_2,t_3}\stackrel{C^{t_3}(.)}{\star}\ldots\stackrel{C^{t_{p-1}}(.)}{\star}C^{t_{p-1},t_p}.
A_{K,M}^{(1,1)}\buildrel \Delta \over = \max \left\{{\frac{1}{{\left|{1 - \frac{1}{{MN + K{N_{\cal S}}}}\left({{\lambda_{\cal M}}+{\lambda_{\cal S}}}\right)}\right|}},\frac{1}{{\left|{1 - \frac{1}{{MN + K{N_{\cal S}}}}\left({\hat \lambda_{{\cal M},j}^{(K)}+ \hat \lambda_{{\cal S},j}^{(K)}}\right)}\right|}}}\right\}.
\prod_{p|a}\left(1+\frac{g(p)}{p}\right) \leq \prod_{p|a}\left(1+ p^{-1}\right)^H \leq \prod_{w(N) < p < p_m}(1 + p^{-1})^H \leq \left( \frac{\log (w(N)+ E \log \log N) + O(1)}{\log w(N) + O(1)}\right)^H = \left( 1 + O_E\left(\frac{1}{\log w(N)}\right) \right)^H = 1 + O_{E,H}\Big(\frac{1}{\log w(N)}\Big),
\zeta \left( \begin{array}{r c r}{s_{1}}&{\ldots}&{s_{k}}\\{\sigma_{1}}&{\ldots}&{\sigma_{k}}\\ \end{array}\right) : = \sum_{n_{j}> n_{j + 1}> 0}\quad \prod_{j = 1}^{k}\frac{\sigma_{j}^{n_{j}}}{n_{j}^{s_{j}}}
m_{m-i+1, n-m-j+1}^\circ=\sum_{\substack{\gamma^\circ:m-i+1 \rightarrow n-m-j+1}}\prod_{f\in \hat{\gamma}^\circ}X_f^\circ=\sum_{\gamma:i\rightarrow j}\prod_{f\notin \hat{\gamma}}X_f^{-1}=\frac{1}{c}\sum_{\gamma:i\rightarrow j}\prod_{f\in \hat{\gamma}}X_f=\frac{1}{c}m_{ij}.
S^{\star}=\sum_{C}\left\{-\frac{1}{\oint_{C}dx_{t}}\oint_{C}dx_t\biggl| \left( \frac{\delta}{\delta\sigma^{\mu t}}- i \frac{2\pi}{e}W_{\mu t}\right) \Phi[C] \biggr|^{2}- M_{0}^{2}\biggl| \Phi[C]\biggr|^{2}\right\},
\frac{(h^t)^{n-p}}{(n-p)!}\circ(\ast h^p)= \widehat{(h^t)}_R\circ i_{h^p}\frac{g^n}{n!}= \widehat{(h^t)}_R\circ i_{\widehat{h}_R(g^p)}\frac{g^n}{n!}=i_{g^p}\circ \widehat{(h^t)}_R(\frac{g^n}{n!}) &=i_{g^p}\circ \frac{(h^t)^n}{n!}=\det (h^t) i_{g^p}\circ \frac{g^n}{n!}=(\det h)\frac{p!}{(n-p)!}g^{n-p}.
E_{m}= \frac{1}{i L}\frac{\partial}{\partial A_{m}}- \frac{e}{2 \pi i m}\left( j_{+}{}^{m}+ ( j_{-}{}^{m})^{\dagger}\right) , \ \m \neq 0 , \quad E_{0}= \frac{1}{i L}\frac{\partial}{\partial c},
\left[ r_{\dot{J}}^{\dot{I}}, f_{\dot{L}}^{\dot{K}}\right] = \sum_{\dot{K}_{1}\dot{K}_{2}= \dot{K}}\delta_{\dot{J}}^{\dot{K}_{2}}f_{\dot{L}}^{\dot{K}_{1}\dot{I}}- \sum_{\dot{L}_{1}\dot{L}_{2}= \dot{L}}\delta_{\dot{L}_{2}}^{\dot{I}}f_{\dot{L}_{1}\dot{J}}^{\dot{K}}.
e^{2t}| T_u v(t) - u(t)| = \left| \int_t^\infty e^{2t}\Phi(t, s) f(v(s)) ds \right| & \le C_1K_1C'^2 e^{-2t}+ C_1 K_2 C'^3 e^{-4t}& \le C_1 K_1 C'^2 + C_1 K_2 C'^3 & = 4C_1 K_1 r^2 + 8C_1 K_2 r^3
\frac{|\delta_2|}{|z-1|^{k+1}}\quad= \zeta^{-p/2-(k+1)}\frac{(1/2)^{k+1}}{(k+1)!}\prod_{i=0}^{k}(p+2i) \le \left( 1- \frac{1}{2p}\right)^{-p/2}\left( 1- \frac{1}{2p}\right)^{-(k+1)}p^{k+1}\quad\le (2e)^{1/4}2^{k+1}p^{k+1},
c=\frac{2}{3}\sqrt{-m_{c}^{2}+2m_{b}^{2}+2m_{a}^{2}}=\sqrt{2(b^{2}+a^{2})-4m_{c}^{2}}=\sqrt{\frac{b^{2}}{2}-a^{2}+2m_{b}^{2}}=\sqrt{\frac{a^{2}}{2}-b^{2}+2m_{a}^{2}}.
x\left(t\right) = \left(k-1\right) \Delta x \quad\mbox{for}\left( j-1 \right) \Delta \tau < t \leq j \Delta \tau , k \in \left\lbrace 1,2,\ldots,n \right\rbrace, j \in \left\lbrace 1,2,\ldots,Q \right\rbrace, Q = \operatorname{round}\left(t_\mathrm{max}/ \Delta \tau\right) .
b=\frac{2}{3}\sqrt{-m_{b}^{2}+2m_{a}^{2}+2m_{c}^{2}}=\sqrt{2(a^{2}+c^{2})-4m_{b}^{2}}=\sqrt{\frac{a^{2}}{2}-c^{2}+2m_{a}^{2}}=\sqrt{\frac{c^{2}}{2}-a^{2}+2m_{c}^{2}}
\begin{cases}u_t - a(x)u_{xx}- \displaystyle \frac{\lambda}{b(x)}u=h(t,x) \chi_{\omega}(x), (t,x) \in Q_T, Bu(0)=Bu(1)=0, t \in (0,T), u(0,x)=u_0(x), x \in (0,1),\end{cases}
\int_0^1\frac{B^2(t)}{t(1-t)}\mathrm{d}t =\sum_{k,l=1}^{\infty}\sqrt{\lambda_k\lambda_l}\int_{0}^{1}Z_kZ_l f_k(t)f_l(t)\mathrm{d}t &=\sum_{k=1}^{\infty}\frac{1}{k(k+1)}Z_k^2\ .
\int_{\Omega}|\nabla u_{0}|^{2}q_{0}(q_{0}- 1) u_{0}^{q_{0}-2}\phi_{1}= \int_{\Omega}(-\Delta u_{0}) q_{0}u_{0}^{q_{0}-1}\phi_{1}+ u_{0}^{q_{0}}(\Delta\phi_{1}) & = \int_{\Omega}(a u_{0}^{q_{0}}) q_{0}u_{0}^{q_{0}-1}\phi_{1}+u_{0}^{q_{0}}(-q_{0}au_{0}^{q_{0}-1}\phi_{1}- \sigma_{1}\phi_{1}) & = -\sigma_{1}\int_{\Omega}u_{0}^{q_{0}}\phi_{1},
\overrightarrow{C}_{(mzw:n)}= \left(\oplus_{i=1}^{t-1}\overrightarrow{C}_{(m:n)}\otimes Z_i\right)\oplus \left(\overrightarrow{C}_{(m:n)}\otimes Z_{t}\right)\oplus &\left(\overrightarrow{C}_{(m:n)}\otimes Z_{t+1}\right)\oplus\left(\oplus_{i=t+2}^{zw}\overrightarrow{C}_{(m:n)}\otimes Z_i\right)
\begin{array}{rcl}u_t = \displaystyle\left(u-\frac34 a u^2-\frac16 b u_{xx}+\frac18 a^2 u^3 -\frac{13}{48}a b u_x^2-\frac{5}{12}a b u u_{xx}+\frac{d}{2}h u\right. begin{align*}5mm] \left.+\displaystyle\frac14 b d (-h''u+ h u_{xx})-\frac{19}{360}b^2 u_{xxxx}\right)_x \end{array}
f(\gamma) \triangleq \frac{\gamma \chi(\gamma) e^{- \gamma}}{e^{-\gamma}+\tau(1-e^{-\gamma})}= \frac{\gamma \left( q_1 + \frac{q_2}{1!}\gamma + \frac{q_3}{2!}\gamma^2 + \cdots + \frac{q_M}{(M-1)!}\gamma^{M-1}\right) e^{- \gamma}}{e^{-\gamma}+\tau(1-e^{-\gamma})}
\mathbb{E}_{wo}[\|x_k - x_* \|^2 ] \leq \mathbb{E}_{wo}\Big [ \| A_k^*\cdots A_1^*A_1 \cdots A_k \|\Big ] \| x_0 - x_* \|^2 + \rho^2 \gamma^2 \sum_{l=1}^k \mathbb{E}_{wo}\Big [ \| A_k^*\cdots A_{l+1}^*A_{l+1}\cdots A_k \| \Big ] \|a_{i_l}\|^2
K_{n}^{(3)}K_{m}^{(3)}+K_{n+1}^{(3)}K_{m+1}^{(3)}+K_{n+2}^{(3)}K_{m+2}^{(3)}&=\left\lbrace \begin{array}{c}\left(2^{n}+M_{n}^{(2)}\right)\left(2^{m}+M_{m}^{(2)}\right) +\left(2^{n+1}+M_{n+1}^{(2)}\right)\left(2^{m+1}+M_{m+1}^{(2)}\right) +\left(2^{n+2}+M_{n+2}^{(2)}\right)\left(2^{m+2}+M_{m+2}^{(2)}\right)\end{array}\right\rbrace .
(\mathcal{D}V)(x) =(a-bx)V'(x)+\frac{\sigma^{2}x}{2}V''(x) =-b\kappa x^{\kappa}+\kappa x^{\kappa-1}\left(a+\frac{\sigma^{2}(\kappa-1)}{2}\right)\leqslant-b\kappa x^{\kappa}+c_{3}
\vartheta((\mathrm{W}, \mathrm{X})+\mu)=e(\mu, (\mathrm{W},\mathrm{X}))\vartheta(\mathrm{W},\mathrm{X})=\frac{1}{a(\mu)b((\mathrm{W},\mathrm{X}),\phi(\mu))}\vartheta(\mathrm{W},\mathrm{X}).
\Phi(x)=\frac{1}{2}+\frac{1}{\sqrt{2\pi}}\cdot e^{-x^{2}/2}[x+\frac{x^{3}}{3}+\frac{x^{5}}{3\cdot5}+\cdot\cdot\cdot+\frac{x^{2n+1}}{(2n+1)!!}+\cdot\cdot\cdot]
(K^{-1})^{V_i}_{V_j}= \frac{2}{B(T+T^*)^{2\bar{n}}\|\Pi\|}\left(\delta_{ij}\|\Pi\|^2 + \Pi^*_i\Pi_j \frac{1-\frac{\bar{n}}{3}(1-2\bar{n})(1-A|z|^2)B\|\Pi\|}{1-\frac{\bar{n}}{3}(1-A|z|^2)B\|\Pi\|}\right),
\begin{aligned}\left (\tau \circ (S\otimes S)\circ \Delta\right)(x_i) =\left(\tau \circ (S\otimes S)\right )(x_i\otimes x_1+x_1\otimes x_i), &= \tau\left (-x_1^{-1}x_ix_1^{-1}\otimes x_1^{-1}-x_1^{-1}\otimes x_1^{-1}x_ix_1^{-1}\right ) &= -x_1^{-1}\otimes x_1^{-1}x_ix_1^{-1}-x_1^{-1}x_ix_1^{-1}\otimes x_1^{-1}\end{aligned}
J_{1}\big(v_{1}(\cdot), u_{2}(\cdot)\big)- J_{1}\big(u_{1}(\cdot), u_{2}(\cdot)\big) = \frac{1}{2}\mathbb{E}\int_{0}^{T}\left[l_{1}(t)\big (y^{v_{1}, u_{2}}(t)-y(t)\big)^{2}+m_{1}(t)\big (v_{1}(t)-u_{1}(t))^{2}\right]dt & +\frac{1}{2}\mathbb{E}\left[r_{1}\big (y^{v_{1}, u_{2}}(0)-y(0)\big)^{2}\right]+\Theta,
f^{( n , 0 )}( r_{1}. . . r_{n - 2}, r_{n - 1}+ 2 , r_{n}) \left( \frac{( r_{n}+ 1 )^{2}}{( 2 r_{n}+ 3 ) ( 2 r_{n}+ 1 )}- \frac{( r_{n - 1}+ 1 )^{2}}{( 2 r_{n - 1}+ 3 ) ( 2 r_{n - 1}+ 1 )}\right) = 0 .
\Big|\sum_{x}e(\alpha' x^2+x\beta')\Big|^2= \sum_{x_1}\sum_{x_2}e\big(\alpha' (x_1^2-x_2^2)+(x_1-x_2)\beta'\big) = \sum_{h}\sum_{x}e(2\alpha'hx+h\beta')\ \le\sum_{h}\Big|\sum_{x}e(2\alpha'hx)\Big|.
\liminf_{n\to\infty}\frac{1}{(1-r^{-1})^{-1}h(x)a_c^{(n)}}\log P\left(\frac{S_n(\kappa_n(x))}{(1-r^{-1})^{-1}h(x)a_c^{(n)}}<\frac{\varepsilon}{h(x)}\right) \geq-\inf_{y\in\left(-\infty,\frac{\varepsilon}{h(x)}\right)}H(y) &=-H\left(\frac{\varepsilon}{h(x)}\right),
\begin{array}{rl}\Lambda \sqrt{\sigma_{r,3}}= \omega_{r,0}- i \gamma_{r,0}= \Lambda\sqrt{\sigma_{r,1}^{*}}\, , \varrho_{r,2}i^{-\frac{3}{2}}(\sigma_{r,3})^{\frac{1}{4}}= \chi_{r,0}\, e^{-i \varphi_{r,0}}= \left[\varrho_{r,0}i^{\frac{3}{2}}(\sigma_{r,1})^{\frac{1}{4}}\right]^{*}\, .\end{array}
\frac{\partial T}{\partial t}+u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}+w\frac{\partial T}{\partial z}=\frac{\partial}{\partial x}\kappa\frac{\partial T}{\partial x}+\frac{\partial}{\partial y}\kappa\frac{\partial T}{\partial y}+\frac{\partial}{\partial z}\kappa\frac{\partial T}{\partial z}+A
\frac{(q^n-1)(q^n-q)\cdots(q^n-q^{k-1})}{(q^k-1)(q^k-q)\cdots(q^k-q^{k-1})}=\frac{(q^n-1)(q^{n-1}-1)\cdots(q^{n-k+1}-1)}{(q^k-1)(q^{k-1}-1)\cdots(q-1)}.
\frac{[x^{n-\Sigma_{\mathrm{k}}}]S_{\neq \mathrm{k}}(x)}{[x^n]S(x)}=(1+o(1)) \cdot z_n^{\Sigma_{\mathrm{k}}},s_n<k_1<\cdots<k_\ell<s_n+o(s_n) n\to\infty,
\sum_{i=1}^n \sqrt{(x_i-x_{i-1})^2+(f(x_i)-f(x_{i-1}))^2}&\qquad\le \sum_{i=1}^n\left((x_i-x_{i-1})+(f(x_i)-f(x_{i-1}))\right) &\qquad=(b-a)+(f(b)-f(a)) &\qquad\le (b-a)+(d-c).
\sup_{\mathbb{B}_{R}}G \leq \sup_{\mathbb{B}_{2R}}(\phi G)\leq \,\, n\bigg(\frac{2\lambda}{2-\lambda}K+\frac{\lambda}{2-\lambda}A+\lambda (2n(p-1))^{p-1}a+\frac{12\lambda \left\vert b \right\vert}{(2-\lambda)n}& +\frac{2n\lambda^{2}C_{1}^{2}}{(2-\lambda)^{2}R^{2}}\bigg).
G^{J_1 J_2,K_1 K_2}_{x_1 x_2,y_1 y_2}\Big|^{q=1}_{{\rm free}}=D^{J_1 J_2,K_1 K_2;0}_{x_1 x_2,y_1 y_2}\,\frac{4 J}{\sqrt{J_1 J_2 K_1 K_2}}\,\frac{\sinh \frac{J J_1}{2 N}\,\sinh \frac{J_2 K_1}{2 N}\,\sinh \frac{J_2 K_2}{2 N}}{\frac{J^2}{2 N}}.
{\rm K}_{n,{2(m+1)\over 2m+1-n}}={n! \over \pi}\left \{{(2m-1)!!\over (2m)!!}B \left ({m+1 \over n+1}+{1\over 2}\;,{1\over 2}\right )\right \}^{\!\! n+1 \over 2(m+1)},
S_{n}^{(2m)}=(S_{n}^{(2)})^{m}+\sum_{k=1}^{\lfloor\frac{m}{2}\rfloor}(\begin{matrix}m\\ 2k\end{matrix})(\begin{matrix}2k\\k\end{matrix})R^{2k}(S_{n}^{(2)}-R^{2})^{k}(S_{n}^{(2)})^{m-2k}
\displaystyle\int_{M}(P_{0}^{k}\lambda^{\perp})(P_{0}^{k+1}\lambda^{\perp})\,d\mu_{0}=\displaystyle\int_{M}(P_{0}^{k}\lambda^{\perp})(P_{0}P_{0}^{k}\lambda^{\perp})d\mu_{0}\geq \Upsilon \displaystyle\int_{M}(P_{0}^{k}\lambda^{\perp})^{2}d\mu_{0}.
|\nabla^{2}W|^{2}= \sum_{ijklst}W_{ijkl,st}^{2}= \frac{1}{4}\sum_{ijklst}\Big(W_{ijkl,st}-W_{ijkl,ts}\Big)^{2}+\frac{1}{4}\sum_{ijklst}\Big(W_{ijkl,st}+W_{ijkl,ts}\Big)^{2}&\geq \frac{1}{4}\sum_{ijklst}\Big(W_{ijkl,st}-W_{ijkl,ts}\Big)^{2}+\frac{1}{4}|\Delta W|^{2}
\theta_1(s) = \int\limits_0^s \frac{-a'_1 \, e^{-\pi i/3}\left[(3\cos(s'/R_0)-2)^{\frac 23}- 1\right]}{2e^{-\pi i/4}R_0^{\frac 12}\sqrt{1-\cos(s'/R_0)}}\, ds' & = \frac12 |a'_1| \, e^{-\pi i/12}R_0^{\frac 12}\int\limits_0^{s/R_0}\frac{(3\cos x -2)^{\frac 23}- 1}{\sqrt{1-\cos x}}\, dx \,.
\langle0|0\rangle_{J}=exp(\frac{i}{2}\int dxdx^{\prime}[J_{\mu}(x)\Delta(x-x^{\prime})J^{\mu}(x^{\prime})+\frac{1}{m^{2}}\partial_{\mu}J^{\mu}(x)\Delta(x-x^{\prime})\partial_{\nu}^{\prime}J^{\nu}(x^{\prime})])
\int_{\mathbb{D}}|\hat f(z)|^{\alpha}(1-|z|^2)^{\alpha-2}{dxdy}= \int_{-1}^{1}dx\int_{-\sqrt{1-x^2}}^{\sqrt{1-x^2}}(1-x^2-y^2)^{\alpha-2}f^{\alpha}(x)dy &=\int_{-1}^1\frac{\sqrt{\pi}\left(1-x^2\right)^{\alpha-\frac{3}{2}}\Gamma[\alpha-1]}{\Gamma\left[\alpha-\frac{1}{2}\right]}f^{\alpha}(x)dx.
{\cal A}_{\mathrm{c l o s e d}}\: = \: \frac{M_{\min}^{2}L_{P L}^{6}}{1 2 6 0 \pi m_{0}^{2}}\: \left( \frac{\frac{c^{2}\beta_{S}}{7}\: + \: \frac{1}{9 m_{0}}}{\frac{c^{2}\beta_{S}}{3}\: + \: \frac{1}{5 m_{0}}}\right)
\delta(\mu)=\frac{1}{2}\sum_{r_s,t}^{}[w_{(t,r_s)},w^*_{(t,r_s)}],\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,1\leq s\leq m ,\,\,\,\,\,\, \,\,\,\,\,\,\,\,\,\,1\leq t\leq n_{r_s}.
J_{2,3}=6T_{xy^2}T_{xy}-3T_{x^2y}T_{y^2}-2T_{xy^2}T_{x}T_{y}+T_{x^2y}T_{y}^2-4T_{xy}^2T_{y}& +2T_{xy}T_{y}^2T_{x}-3T_{y^3}T_{x^2}+4T_{y^2}T_{y}T_{x^2}-T_{y}^3T_{x^2}+T_{y^3}T_{x}^2-T_{y^2}T_{y}T_{x}^2.
\widehat{\gamma}_{W, c}= \sum_{w \in W}(-1)^w\mathcal{F}_\zeta[\delta_w] - \mathcal{F}_\zeta[c \delta_0] = \mathcal{F}_\zeta \left[ \sum_{w \in W}(-1)^w\delta_w - c \delta_0\right] = \mathcal{F}_\zeta \left[ \sum_{w \in W}(-1)^w\delta_w - c \delta_0\right] = \mathcal{F}_\zeta \left[ \gamma_{W,c}\right].
\left\{3^a(u_1^b u_2^c)^2\beta^d:a,c\in\mathbb{Z}_{\geq0},b\in\mathbb{Z},d=0,1,2,\textnormal{and}cx_1^{-1}+y_{1,d}x_1^{-1}\geq b\geq cx_2-y_{2,d}\right\}.
\frac{1}{g_{YM}^{\ 2}}=\frac{\left(\alpha^{\prime}\right)^{\frac{3-p^{\prime}}{2}}}{\left(2\pi\right)^{p^{\prime}-2}G_{s}}=\frac{\left(\alpha^{\prime}\right)^{\frac{3-p^{\prime}}{2}}}{\left(2\pi\right)^{p^{\prime}-2}g_{s}}\left(\frac{\det (g+2\pi\alpha^{\prime}B)}{\det G}\right)^{\frac{1}{2}}~.
Q(t^{\frac{a+\epsilon}{2}},t^{\frac{1-a-2\epsilon}{2}})=P(|\theta(T_{t^{\frac{a+\epsilon}{2}}})|>|\tilde{\theta}(\tilde{T}_{t^{\frac{1-a-2\epsilon}{2}}})|)=V(\frac{a+\epsilon}{1-a-2\epsilon}).
\widetilde{B}(\zeta) = \sum_{j=0}^{\infty}B_{j}\zeta^{j}\mbox{with}\widetilde{B}(\zeta) = 1 + \zeta\big ( \tau^{-\alpha}\delta (\zeta)^{\alpha}+A_h \big )^{-1}\tau^{\gamma-1}\delta(\zeta)^{-\gamma}.
\sum_{\{\Delta:\Omega\}=(1,p)}c(\Omega^{p^{r-2}m}) =(\widetilde\lambda_1(p^2)\kappa(p^{r-2})-\chi(p^2)p^{2k-3}\eta(p^{r-2}))c(\Delta^m) &\quad-\chi(p)p^{k-2}\alpha(\Delta^{p^{r-2}};p)\kappa(p^{r-2})c(\Delta^m) &=\eta(p^r)c(\Delta^m).
_{p}F_{q}\left[ \begin{array}{r}\left( a_{p}\right) ; \left( b_{q}\right) ;\end{array}z\right] =\sum\limits_{n=0}^{\infty}\frac{\Pi_{j=1}^{p}\left( a_{j}\right)_{n}}{\Pi_{j=1}^{q}\left( b_{j}\right)_{n}}\frac{z^{n}}{n!}
\Pi_{m=0}^{\infty}\left( 1-\nu^{2m}r^2\right )\frac{a_{i}}{a_{j}}< \frac{a_{I,i}}{a_{I,j}}< \Pi_{m=1}^{\infty}\left( 1+\nu^{2m}r^2\right )\frac{a_{i}}{a_{j}}.
{\cal T}(y)=\left(\begin{array}{cccc}{\rm e}^{i\varphi_{1}}& \ddots & \ddots &{\rm e}^{i\varphi_{N}}\end{array}\right),\quad{\rm with}\quad \varphi_{i}\equiv \sum_{b=1}^n{h_{b, i}\over L^b}y^b, \quad \sum_{i=1}^N h_{b, i}=0.
\liminf \frac{1}{M}\underset{i = 1}{\stackrel{M}{\sum}}\varphi_{N,K,i,a}(\eta_i(t)) \geq \frac{1}{2^L}\underset{j = 1}{\stackrel{2^L}{\sum}}\tilde{\varphi}\left(\int_{(j-1)/2^L}^{j/2^L}{\eta_*(\theta)d\theta}\right)
M_2 = \frac{\left(1+2\rho^2\right)^2}{\sqrt{1+\rho^2+\rho^4}\sqrt{1+5\rho^2+9\rho^4}}\le \left(\frac{1+2\rho^2}{1+\frac{3}{2}\rho^2}\right)^2 \le 1+ \frac{2\rho^2}{2+3\rho^2}\, ,
\Delta_{\kappa}( \vec{x}, x_{0}) \sim{\frac{- i \kappa \cosh \left({\frac{\pi \kappa x^{0}}{2}}\right) \Gamma \left( 1 + i \kappa x^{0}\right)}{\left( 2 \pi \right)^{2}\sqrt{1 +{\frac{m^{2}}{\kappa^{2}}}}\, | \vec{x}| \left( \kappa | \vec{x}| \right)^{1 + i \kappa x^{0}}}}
\sup\mathcal{A}(x,y,\omega)\leq\sup\mathcal{A}(x,z,\omega)+\sup\mathcal{A}(z,y,\omega)\;\;\textrm{and}\;\;\mathcal{A}(x,y,\tau_z\omega)=\mathcal{A}(x+z,y+z,\omega).
u_i =-\Psi_i\eta_i-k_i\rho_i(e_{vi})e_{vi}-\theta_i{\rm sgn}(e_{vi}) \dot{\eta}_i =F_i\eta_i+G_iu_i \dot{k}_i =-\lambda k_{i}+\rho_i(e_{vi})e_{vi}^2, \dot{\theta}_i=|e_{vi}|.
\begin{array}{ll}\int_{_{B(x_{i_{l_{k}}}, R_{i_{l_k}}) / \bigcup_{k^{'}=k+N}^{K}B (x_{i_{l_{k^{'}}}},R_{i_{l_{k'}}})}}\ |u(\bar{t})|^{1_{2}^{*}}\, dx \geq(c')^{\delta^{-\frac{1}{2}}},\end{array}
[\tilde{M}_Q(t_1,t_2,\ldots,t_{n-1})]_{jk}= \begin{cases}0, k < j, d_j, k = j, g_j h_k, k = j + 1, g_j b_{j+1}b_{j+2}\cdots b_{k-1}h_k, k > j + 1.\end{cases}
\langle [\xi,\eta]_{\ast},[X,Y]\rangle-\langle{\mathrm{ad}_{X}^{\ast}}^{(l)}\xi,{\mathrm{ad}_{\eta}^{\ast}}^{(r)}Y\rangle+\langle{\mathrm{ad}_{Y}^{\ast}}^{(r)}\xi,{\mathrm{ad}_{\eta}^{\ast}}^{(r)}X\rangle+\langle{\mathrm{ad}_{Y}^{\ast}}^{(l)}\xi,{\mathrm{ad}_{\eta}^{\ast}}^{(r)}X\rangle=0,
\lambda_{d_1, \ldots, d_k}:= \left(\prod_{i = 1}^k \mu(d_i)|d_i|\right) \sum_{\substack{r_1, \ldots, r_k d_i \mid r_i \forall i (r_i, W) = 1 \, \forall i}}\frac{\mu(r_1, \ldots, r_k)^2}{\prod_{i=1}^k \Phi(r_i)}F\left(\frac{\log|r_1|}{\log R}, \ldots, \frac{\log|r_k|}{\log R}\right)
BS=\frac{1}{N}\sum_{k=1}^{K}{n_{k}(f_{k}-\overline{o}_{k})}^{2}-\frac{1}{N}\sum_{k=1}^{K}{n_{k}(\overline{o}_{k}-\overline{o})}^{2}+\overline{o}(1-\overline{o})
\Phi(\omega_{1},\omega_{2})=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}p(y_{1},y_{2})e^{j(\omega_{1}y_{1}+\omega_{2}y_{2})}dy_{1}dy_{2}=exp\{-\frac{\omega_{1}^{2}+\omega_{2}^{2}+2\rho\omega_{1}\omega_{2}}{2}\}
\langle F_\mu^i[\xi|s] F_{\mu'}^{i'}[\xi'|s']\rangle=-{3i\over{4a_{\xi'}(s')}}\delta^{ii'}g_{\mu\mu'}\,\delta(s-s')\prod_{\bar s = 0}^{2\pi}\, \delta^4(\xi(\bar s)-\xi'(\bar s)),
G_{tt}= e^{2(\alpha -\beta)}\beta_r r^{-1}, G_{tr}= \beta_t r^{-1}, G_{rr}= \alpha_r r^{-1}, G_{\theta \theta}= r^2 \bigl(e^{-2\beta}(-\beta_r\alpha_r + \alpha^2_r + \alpha_{rr}) -e^{-2\alpha}(\beta^2_t -\beta_t\alpha_t + \beta_{tt}) \bigr), G_{t \theta}= 0, G_{r \theta}=0.