harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\left[ S_{\alpha},K_{\alpha}\right] =\frac{2}{r^{2}}\Delta -\frac{4\alpha^{2}}{r^{4}}\left\vert \frac{x}{r}+\varphi e_{1}\right\vert^{2}-\frac{1}{2}+\frac{2i\varphi^{\prime}}{r}\partial_{x_{1}}+\left[ A,K_{\alpha}\right] ,
\begin{cases}\frac{\partial^{\alpha}}{\partial t^{\alpha}}p_{k}(t)=-\lambda p_{k}(t)+\lambda p_{k-1}(t)k\geq 1 \frac{\partial^{\alpha}}{\partial t^{\alpha}}p_{0}(t)=-\lambda p_{0}(t) p_{k}(0)=\delta_{k,0},\end{cases}
f(x) = \left(\prod_{i = 0}^N \frac{1}{\log^{(i)}(x)}\right)\left(\frac{1}{\log^{(N + 1)}(x)}\right)^\alpha &= \frac{1}{x\log(x)\cdots \log^{(N)}(x) \left(\log^{(N + 1)}(x)\right)^\alpha}
\left|\det \begin{pmatrix}b_1 b_2 begin{align*}2pt] \partial_{\xi_d}b_1 \partial_{\xi_d}b_2 \end{pmatrix}\right|(x,\xi' + \zeta', \xi_d = i |\xi'|_x+ \delta) \geq C_1 |\xi'|_x^{k_1+k_2 -1},
|A_0|\geq \frac{1}{2}(1-\lambda)\alpha(1-\alpha) n - \delta(1-\alpha)n=\frac{1}{2}(1-\lambda)\alpha(1-\alpha) n \left (1-\frac{1}{2}(1-\lambda)(1-\alpha)\right ) \geq \frac{1}{4}(1-\lambda)\alpha(1-\alpha) n.
\left(\frac{d}{2}\right)^{-\frac{kp}{2}}\prod_{i=1}^{k}\frac{\Gamma \left( (d-i+1)/2\right)}{\Gamma \left((d-p-i+1)/2\right)}\det(S_{k})^{-\frac{p}{2}}\left(\prod_{i=1}^k c_i'\beta_i \right)^{d-p-k-1}e^{\frac{k}2 \|x\|^2},
(u_s^2)_t =(u_{s}^2)_{ss}-2u_{ss}^2 +2(u_s^2)_s\cdot u_s +6e^{2u}u_s^2 =(u_{s}^2)_{ss}-2u_{ss}^2 +4 u_s^2u_{ss}+6e^{2u}u_s^2
|\overline{\partial}_{w}v_{k}| = |\overline{\partial}_{w}(u_{k}\circ \Phi)| \leq |\partial_{z}u_{k}| |\overline{\partial}_{w}\Phi| + |\overline{\partial}_{z}u_{k}| |\partial_{w}\Phi| \lesssim |\partial_{z}u_{k}||\overline{\partial}_{w}\Phi| + k\big|u_{k}\overline{\partial}_{z}\xi\big|.
r^{n-1}\omega_1(r) =r^{n-1}+o(r^{n}), r^{n-2}\omega_2(r) =(n-1)r^{n-2}+o(r^{n-1}), r^{n-3}\omega_3(r) =(n-1)(n-2)r^{n-3}+(2dn-d-n)r^{n-1}+o(r^{n}),
\begin{array}{l}v_3 \colon y_1(1) = y_2(1) = y_3(0), y'_1(1) + y'_2(1) - y'_3(0) = 0, v_6 \colon y_3(1) = y_4(1) = y_5(1), y'_3(1) + y'_4(1) + y'_5(1) = 0.\end{array}
\left( g_{m n}\right)^{D}= \left( \begin{array}{c c}{g_{m n}+ g_{t s}{\Gamma^{t}}_{m}{\Gamma^{s}}_{n}}&{{\Gamma^{t}}_{n}g_{t m}}\\{{\Gamma^{t}}_{m}g_{t n}}&{g_{m n}}\\ \end{array}\right) .
L_{d}KLI(q\|p)=\sum_{j}p_{j}\sum_{i}\frac{\overline{q}_{i}}{a-b}\left[\left(\frac{\overline{q}_{i}}{\overline{p}_{i}}\right)^{a-1}-\left(\frac{\overline{q}_{i}}{\overline{p}_{i}}\right)^{b-1}\right]
H_{n}\left(\frac{B_{2k+1}\left(\frac{x+1}{2}\right)}{2k+1}\right)=\left(\frac{x}{2}\right)^{n+1}\prod_{\ell=1}^{n}\left(\frac{(2\ell)^{2}(2\ell-1)^{2}(x^{2}-(2\ell-1)^{2})(x^{2}-(2\ell)^{2})}{16(4\ell-3)(4\ell-1)^{2}(4\ell+1)}\right)^{n+1-\ell}.
\xi^*(\varphi_s^*\omega-\omega_T-\pi_T^*s^*\omega) = s^*\varphi(\xi)^*\omega-d\xi-s^*\omega &(d\xi\xi\Omega_\mathcal{F}^1 &P) = s^*(\omega+\pi^*d\xi)-d\xi-s^*\omega = 0
u_{m+1,n+1}=\cfrac{u_{m,n+1}\,u_{m+1,n+2}}{u_{m,n+2}(1+h\,k\,u_{m,n+1})},u_{m+1,n+1}=\cfrac{u_{m+1,n}\,u_{m+2,n+1}}{u_{m+2,n}(1+h\,k\,u_{m+1,n})}.
H_{H P}^{( 1 )}= \frac{g^{2}\sqrt{N}}{4 \pi}\int_{0}^{\infty}\frac{d R}{\sqrt{R}}\int_{0}^{1}d z \sum_{n m l}\left( K_{n m l}( z ) - \tilde{K}_{n m l}( z ) \right) \left( b_{R}^{( n ) \dagger}b_{R z}^{( m )}b_{R ( 1 - z )}^{( l )}+ \mathrm{h . c .}\right) ,
H_{t,\mathbf{c}}:= H_{t,\mathbf{c}}(G)= TV^* \rtimes G\big/\langle [u,v]-t\omega_{V^*}(u,v)1_G+2\sum_{s\in S}\mathbf{c}(s)\omega_s(u,v)\cdot s: u,v\in V^*\rangle,
J^{(\mu)}\mathbf{\Psi}(t)=\frac{1}{\Gamma(\mu)}\int^t_0\frac{\mathbf{\Psi}(\xi)d\xi}{(t-\xi)^{1-\mu}}\cong\mathbf{\Psi}(t)\textbf{J}^{\mu,T}_{N_t\times N_t},
L_g \big( \Gamma_p^{\frac{2}{n-2}}G_p \big) = - \Delta_g \big( \Gamma_p^{\frac{2}{n-2}}G_p \big) + \frac{(n-2)}{4(n-1)}R_g \Gamma_p^{\frac{2}{n-2}}G_p &= 2(n-4)\alpha r^{-4}+ O(r^{4-n}) &= 2(n-4) \alpha r^{-4}+ O(r^{-3}), \ \ \mbox{if}5 \leq n \leq 7.
\mu_i(k)=\frac{1}{d(l-1)}\sqrt{\Big(\frac{1-\delta}{(d-1+\delta)(l-1)}\Big)^{k-1}}q_k\Big(\frac{\lambda_i-(l-2)(1-\delta)}{2\sqrt{(l-1)(1-\delta)(d-1+\delta)}}\Big)
\mathbb{E}\left[\left|\frac{t^{2Hl_{0}}}{\lambda_{t}^{(l)}}\right|^{2q}:\|U_{t}^{(l)}\|_{\textsc{HS}}<r\right] =\frac{1}{t^{4qH(l-l_{0})}}\mathbb{E}\left[\left|\frac{t^{2Hl}}{\lambda_{t}^{(l)}}\right|^{2q}:\|U_{t}^{(l)}\|_{\textsc{HS}}<r\right] \leq\frac{C_{1,q,l}}{t^{4qH(l-l_{0})}}.
D \left( \begin{array}{l}{f_{\pm}^{q}}\\{f_{\pm}^{- q}}\\ \end{array}\right) = d \left( \begin{array}{l}{f_{\pm}^{q}}\\{f_{\pm}^{- q}}\\ \end{array}\right) - i q A_{\pm}\sigma_{3}\left( \begin{array}{l}{f_{\pm}^{q}}\\{f_{\pm}^{- q}}\\ \end{array}\right) .
\mathrm{var}[\widehat{K}_{\mathrm{a-cpt-d}}^\mathbb{R}]=\frac{\psi}{2}\Big(\sum_{i\in\mathcal{K}'}\frac{\mathrm{var}[Z_i]}{N_1\varrho\vartheta_i}+\frac{1}{N_2\bar{\gamma}_c'}\Big).
\iota_{w}\mid_{q=1}\colon\mathbf{A}_{\mathbb{Q}[q^{\pm1}]}[N_{-}(w)\cap wG_{0}^{\min}]\otimes_{\mathcal{A}}\mathbb{C}\to\mathbf{A}_{\mathbb{Q}[q^{\pm1}]}[N_{-}^{w}]\otimes_{\mathcal{A}}\mathbb{C}
(a_{3,1}r^2+a_{3,2}r)\,\frac{d^2y}{dr^2}+(a_{2,0}r^2+a_{2,1}r+a_{2,2})\,\frac{dy}{dr}-(\tau_{1,0}r+\tau_{1,1})y=0,a_{3,1}^2+a_{3,2}^2\neq 0,~~a_{2,0}^2+\tau_{1,0}^2\neq 0,
G(t,s,p)=\begin{cases}k(t,s,v(s))+\displaystyle\frac{v(t)-p}{1+(v(t)-p)^2}, \p < v(t); k(t,s,p), v(t)\leq p \leq w(t); k(t,s,w(s))-\displaystyle\frac{w(t)-p}{1+(w(t)-p)^2}, \p > w(t). \end{cases}
\frac{\ddot{b}}{b}= \frac{\beta''}{\beta},~~~~\Biggl[\frac{d}{d t}\left(\frac{\dot{c}}{c}\right) + \frac{\dot{b}}{b}\frac{\dot{c}}{c}\Biggr] = \Biggl[\frac{d}{dx}\left(\frac{\kappa'}{\kappa}\right) +\frac{\beta'}{\beta}\frac{\kappa'}{\kappa}\Biggr].
\partial^\alpha t^\beta (M_\omega T_x g(t)) = \sum_{\gamma_1 \leq \alpha}\sum_{\gamma_2 \leq \beta}\begin{pmatrix}\alpha \gamma_1\end{pmatrix}\begin{pmatrix}\beta \gamma_2\end{pmatrix}x^{\gamma_2}(2 \pi i \omega)^{\gamma_1}M_\omega T_x ( \partial^{\alpha-\gamma_1}t^{\beta-\gamma_2}g(t)),
\mathcal{C}\sim a^{4-h}\, \sum_{n}\, \frac{1}{\left( 1 + \eta \, \lambda_{n}^{(3)}\right)}\, \langle \overset{1}{\mathbb{P}}\left(N\left(e_{n}^{(3)}\right)\right) ; e^{(1)}_{n_{0}}\rangle \langle \mathcal{P}(\cdot,0); e^{(1)}_{n_0}\rangle \otimes \int_{B}e^{(3)}_{n}(x) \, dx = \mathcal{O}\left( a^{6-h}\right),
\sum_{n=0}^{\infty}(\frac{(-1)^{n}}{2n+1})^{5}=\frac{1}{1^{5}}-\frac{1}{3^{5}}+\frac{1}{5^{5}}-\frac{1}{7^{5}}+\cdot\cdot\cdot=\frac{5\pi^{5}}{1536}
\hat{q}_\tau^2(\xi-\eta,\eta)= |\xi|^{2\tau}\big(q_1(\eta, \xi-\eta)+\eta(\xi-\eta)|\eta|^{-1/2}\theta(\xi-\eta, \eta)\big) + |\xi-\eta|^{2\tau}\big(q_2(\eta, -\xi) + q_2(-\xi,\eta) \big),
zH_k^{(s,t)}H_{k-1}^{(s+1,t)}(z)=H_k^{(s+1,t)}H_{k-1}^{(s,t)}(z)+H_{k-1}^{(s+1,t)}H_k^{(s,t)}(z), &H_{k}^{(s+1,t)}H_{k}^{(s,t)}(z)=H_{k+1}^{(s,t)}H_{k-1}^{(s+1,t)}(z)+H_{k}^{(s,t)}H_{k}^{(s+1,t)}(z).
\Psi_{\hat u',{\bf x}}= (\Phi_{\overline{\bf y}(\hat u',{\bf x}),\delta})^{-1}\circ (\varphi(\hat u',{\bf x}))^{-1}\circ \tilde{\psi}\vert_{\Sigma_1({\bf x}))}: \Sigma_1({\bf x})(\delta') \to \Sigma(\delta).
J(d_1 ,\ldots ,d_n ;e_1 ,\ldots ,e_n ) \leq \begin{cases}\frac{\beta^{n+1}r^{n+2}}{(r-2e\alpha )(\beta -1)(r-1)}+\frac{(2e\alpha )^{n+2}\beta^{n+1}}{(\beta -1)(r-2e\alpha )(2e\alpha \beta -1)}, r >2e\alpha \frac{\beta^{n+1}(2e\alpha )^{n+2}}{(2e\alpha -r)(\beta -1)(2e\alpha -1)}+\frac{\beta^{n+1}r^{n+2}}{(\beta -1)(2e\alpha -r)(\beta r -1)}., 2e\alpha >r . \end{cases}
p_{i+1}= p_1 - \sum_{j=1}^i 3g_i(1-\ln g_i) > p_1 - \sum_{j=1}^i 3x_i(1-\ln x_i) \ge\intertext{(because $x\mapsto 3x(1-\ln x)$ is increasing for $x\in(0;1]$)}\ge p_1 - s(x_1) \ge p.
R_{\mu \nu}= Tr [\frac{1}{2}J^p_{\mu}J^p_{\nu}- 2p(F_{\mu \lambda}{F^T}^{\nu \lambda}- \frac{1}{4}g_{\mu \nu}FF^T) +pH_{\mu \lambda \sigma}pH_{\nu}^{\lambda \sigma}- \frac{1}{3}g_{\mu \nu}(pH)^2]
\left\| \partial_{t}^{\ell}\nabla_{x}^{k}E_{3}(t)g\right\|_{2}\le Ct^{1- \ell}(1+t)^{-\frac{n}{2}-k}\| g \|_{1}+ C e^{-ct}\| \nabla^{(k+\ell-1)_{+}}_{x}g\|_{2},
\begin{aligned}{H}_{i}(t)= H_{i}(t,x,y,z,z_1,z_2,u_1,u_2;q_i,k_i,k_{1i},k_{2i},p_i) = b(t,x,u_1,u_2)q_i(t)+\sigma(t,x,u_1,u_2)k_i(t)+\sum_{j=1}^2\sigma_j(t,x,u_1,u_2)k_{ji}(t) - [f(t,x,y,z,z_1,z_2,u_1,u_2)-\sum_{j=1}^2h_j(t)z_j(t)]p_i(t)+l_i(t,x,y,z,z_1,z_2,u_1,u_2), \end{aligned}
D_{ij}=\frac{\partial^2S_N}{\partial \lambda_i \partial \lambda_j}=\sum_{k=1}^N \frac{\partial c_k}{\partial \lambda_i}\frac{\partial}{\partial c_k}\frac{\partial S_N}{\partial \lambda_j}=\sum_{k=1}^N \frac{\partial c_k}{\partial \lambda_i}\frac{\partial H_j}{\partial c_k},
[e_1, e_1]=\alpha_1e_4+\alpha_2e_5, [e_1, e_2]=\alpha_3e_3+ \alpha_4e_4+ \alpha_5e_5, [e_2, e_1]=-\alpha_3e_3+\beta_1e_4+ \beta_2e_5, [e_2, e_2]=\beta_3e_4+\beta_4e_5, [e_1, e_3]=\beta_5e_4+\beta_6e_5, [e_2, e_3]=\gamma_1e_4+\gamma_2e_5, [e_3, e_1]=\gamma_3e_4+\gamma_4e_5, [e_3, e_2]=\gamma_5e_4+\gamma_6e_5, [e_3, e_3]=\gamma_7e_4+\gamma_8e_5.
\mathbf{D}_1(v) = \frac{1}{8 \sqrt{\pi}\, \gamma^{3/2}}\, \left\{\mathrm{C}\left( \gamma^{1/2}|a| \right)\,{\bf I}+\mathrm{T}\left( \gamma^{1/2}|a| \right) \frac{|a|^2{\bf I}- a \otimes a}{|a|^2}\right\}
\mathcal{A}^\alpha(\omega, \delta) = \begin{pmatrix}\mathcal{S}_D^{k_b}-\mathcal{S}_D^{\alpha,k}-\frac{1}{2}+ \mathcal{K}_D^{k_b, *}-\delta( \frac{1}{2}+ (\mathcal{K}_D^{-\alpha,k})^*)\end{pmatrix}, \,\, \Phi^\alpha= \begin{pmatrix}\varphi^\alpha \psi^\alpha\end{pmatrix}.
\int_{2^-}^{u}\frac{G(v)}{(v-1)^2}\,dv \ge\int_{2}^{u}\frac{1}{(v-1)^2}\left(\log\frac{v}{2}-\frac{c_1}{\log^2y}\right)\,dv &=-\frac{1}{u-1}\log\frac{u}{2}+\int_{2}^{u}\frac{1}{v(v-1)}\,dv-\frac{c_1}{\log^2y}\left(1-\frac{1}{u-1}\right) &=-\frac{u}{u-1}\log\frac{u}{2}+\log(u-1)-\frac{c_1}{\log^2y}\left(1-\frac{1}{u-1}\right).
\left\langle\begin{array}{cc}r t\end{array}\right\rangle_{n-1}\left\langle\begin{array}{cc}r q\end{array}\right\rangle_{n-1}=\left\langle\begin{array}{cc}r q\end{array}\right\rangle_{n-1}\left\langle\begin{array}{cc}r t\end{array}\right\rangle_{n-1}
\|g\|_{\ell_1(L_\infty)}:= \|\{\|g\|_{L_{\infty}(m+K)}\}_{m \in \mathbb{Z}^p}\|_{\ell_1(\mathbb{Z}^p)}, \|g\|_{\ell_{1,\infty}(L_\infty)}:= \|\{\|g\|_{L_{\infty}(m+K)}\}_{m \in \mathbb{Z}^p}\|_{\ell_{1,\infty}(\mathbb{Z}^p)}.
\begin{cases}\rho_t + \left(\rho \, v\right)_x = 0, t> 0, ~ x\in{\mathbb{R}}, \left[\rho \, (v+p(\rho))\right]_t + \left[\rho \, v \, (v+p(\rho))\right]_x = 0, t> 0, ~ x\in{\mathbb{R}}.\end{cases}
f(x) = (x - 2)^{-r}\sum\limits_{i=0}^{\lfloor -r/2 \rfloor}\delta_i \left(\frac{x}{x-2}\right)^{2i}= x^{-r}\sum\limits_{i=0}^{\lfloor -r/2 \rfloor}\delta_i \left(\frac{x-2}{x}\right)^{-r-2i}.
{w}_\beta^{~\alpha}= -2 d\theta^\alpha D_\beta \Phi -2 d\theta_\beta D^\alpha \Phi + \Pi^{\gamma\dot{\gamma}}w_{\gamma\dot{\gamma}\beta}^{~~~~\alpha}~, \qquad w_{\dot{\beta}}^{~\dot{\alpha}}= (w_\beta^{~\alpha})^*~.
\big<\nabla v, u\big>_{W^{-1,q}(\Omega)\times W^{1,p}_0(\Omega)}= -\big<v, (u)\big>_{(L^q(\Omega)/\mathbb{R})\times (L^p(\Omega)/\mathbb{R})}\forall~ v\in L^q(\Omega)/\mathbb{R},\forall~ u\in W^{1,p}_0(\Omega;\mathbb{R}^d).
S ={N \over g}tr \left({1 \over 2 \cosh (\beta) (1 + c^*)}S^2 +{1 \over2 \cosh(\beta) (1 - c^*)}D^2 +{1 \over4}S^4 +{1\over 4}D^4 +{1 \over 2}( S D S D + 2 S^2 D^2 ) \right),
d s^{2}= - \frac{f^{2}}{l^{2}}\left( r_{+}d \phi - r_{-}\frac{d t}{l}\right)^{2}- \frac{f{' \; 2}d r^{2}}{( \frac{f^{2}}{l^{2}}+ 1 )}+ \left( \frac{f^{2}}{l^{2}}- 1 \right) \left( r_{-}d \phi - r_{+}\frac{d t}{l}\right)^{2}
\mathcal{V}_Q \, \rho(x,\omega) \, \mathcal{V}_Q^{-1}f(t) = e^{\pi i Q t \cdot t}e^{\pi i x \cdot \omega}e^{2 \pi i \omega \cdot(t-x)}e^{- \pi i Q(t-x) \cdot (t-x)}f(t-x) & = e^{\pi i (\omega + Q x) \cdot x}e^{2 \pi i (\omega + Qx) \cdot (t-x)}f(t-x) & =\rho(x, Qx + \omega, \tau) f(t) = \rho(V_Q (x,\omega), \tau) f(t),
n \: C \alpha_{1}^{1 - p - \gamma}\: \beta_{1}^{1 - q}\alpha_{i_{0}}^{\gamma}\left( \prod_{j = 2}^{p}\alpha_{1}^{a_{j}- \alpha_{j}}\: \alpha_{j}^{- a_{j}}\right) \left( \prod_{k = 2}^{q}\beta_{1}^{b_{k}- \beta_{k}}\: \beta_{k}^{- b_{k}}\right) ,
\left .\sum_{\substack{n \leq x n\in \mathcal{N}_2}}\frac{1}{n}=\int_{x_0}^{x}\frac{1}{t}d N_2(t) =\frac{N_2(t)}{t}\right |_{x_0}^{x}+\int_{x_0}^{x}\frac{N_2(t)}{t^2}d t,
\rho_{p\rightarrow 1+2+3}=\frac{(32\pi)^{2-2\ell}}{(\Gamma(\ell-1/2))^2(p^2)^{\ell-1}}\int_{(m_1+m_2)^2}^{(\sqrt{p^2}-m_3)^2}s^{1-\ell}{\cal D}^{\ell-3/2}\,ds,
O(\frac{Qmn}{\delta}\cdot(Q+m^{2}ln\frac{Q}{m}))=O(\frac{Q^{2}mn+Qm^{3}nln\frac{Q}{m}}{\delta})\approx O(m^{8}lnmln^{2}(\frac{mn}{gh})+\frac{m^{4}nlnm}{h}ln(\frac{mn}{gh}))
\begin{cases}-\phi''-\frac{z}{2}\phi'+\frac{z}{2}e_\sigma\psi'=f_1, -\psi''-\frac{z}{2}\psi'+\frac{z}{2}e_\sigma\phi'=f_2, \phi(0)=\phi(r)=0,\psi(0)=\psi(r)=0.\end{cases}
\Delta_{j,\ell}= \delta_{\ell x_j}+\sum_{k=1}^{\ell-1}\binom{\ell}{k}\delta_{(\ell-k)x_j}*\sigma_{j-1}^{*k}\, &=\delta_{\ell x_j}+\sum_{k=1}^{\ell-1}\binom{\ell}{k}\sum_{1\le \nu_1,\dots,\nu_k\le j-1}\delta_{(\ell-k)x_j+x_{\nu_1}+\dots+x_{\nu_k}}\,.
\mathrm{T r}\left[ \gamma^{0}\langle \psi ( \vec{x}_{0}, t ) \bar{\psi}( \vec{x}_{0}, t ) \rangle_{\beta}\right] = \frac{\Delta_{n}}{2 \pi}\int_{- \infty}^{\infty}d \omega F_{1 / 2 , n}( \omega ) .
\int_{0}^{t/2}\frac{\sin(u)}{u \log^{a}(t/u)}du = \lim_{\epsilon \rightarrow 0}\left(\int_{\epsilon}^{t/2}dy \frac{e^{-y}}{y\log^{a}(\frac{t}{i y})}\right)-\left(\int_{0}^{\frac{\pi}{2}}i d\theta \frac{e^{\frac{i t}{2}\cos(\theta)}e^{-\frac{t}{2}\sin(\theta)}}{\log^{a}(2 e^{-i \theta})}\right)
\varphi_0(w) = \varphi_0\left(v_1 l_m^{g_2}r_k^{g_1}v_2\right) + \varphi_0(v_1 P_\Omega v_2) = \varphi_0\left(v_1 l_m^{g_2}r_k^{g_1}v_2\right) + \varphi_0(v_1) \varphi_0(v_2)
\sum_{\substack{q\ge1 N\mid q}}\frac{r_q(n)}{q^{2s}}= \begin{cases}\frac{\sigma_{1-2s}(n; N)}{\zeta^{(N)}(2s)}n\neq 0, N^{1-2s}\prod_{p\mid N}(1-p^{-1}) \frac{\zeta(2s-1)}{\zeta^{(N)}(2s)}n=0. \end{cases}
A(a,x) = 1 + \frac{x}{a}(A(a,x) - A(0,x)) + aA(a,x) + xB(0,a,x), B(a,b,x) = \frac{bx}{a(1-b)(1-x)}(A(a,x) - A(0,x)) + \frac{bx}{(1-b)(1-x)}B(a,b,x) &\qquad+ \frac{x}{a(1-x)}(B(a,b,x) - B(0,b,x)).
\Gamma(\phi_{0})=\int d^{3}{\bf x}\left\{\frac{1}{2}(\partial_{i}\phi_{0})^{2}+ \frac{1}{2}\mu_{R}^{2}\phi_{0}^{2}+ \frac{\lambda_{R}T}{4}\phi_{0}^{4}+ \frac{(N+26)}{9}\frac{\lambda_{R}^{3}\zeta(3)}{2^{9}\pi^{4}}\phi_{0}^{6},\right\}.
\widehat{\Theta}^+_1 =\widehat{\Theta}_1 + \frac{2\eta_1}{|\widehat{\mathcal{C}}_1|}\sum_{i \in \widehat{\mathcal{C}}_1 \cap \mathcal{C}_1}(X^{(i)}- \widehat{\Theta}_{1}Z^{(i)})Z^{(i),\top}+ \frac{2\eta_1}{|\widehat{\mathcal{C}}_1|}\sum_{i \in \widehat{\mathcal{C}}_1 \cap \overline{\mathcal{C}_1}}(X^{(i)}- \widehat{\Theta}_{1}Z^{(i)})Z^{(i),\top},
D(t,t) \ =\ 1 \ \ 1\leq t \leq (k+1)n D\big((k+i)n+t,\, (k+i)n+t)\big)\ =\u_i(t)\ 1\leq i \leq k \ \ 1\leq t \leq n \ =\ 1 \ \t=(2k+1)n +1 \ .
(\rho(s_{ij}(w)))_{1\leq i,j \leq 2}= \begin{pmatrix}(q-q^{-1}w) E_{11}+ (1-w) E_{22}(q-q^{-1}) E_{12}(q-q^{-1})w E_{21}(1-w)E_{11}+ (q^{-1}-qw) E_{22}\end{pmatrix}.
\frac{1}{\Phi(Q)^{2}}\sum_{\substack{\chi\neq\chi_{0}\chi \in \Gamma_{6-prim}^{6-odd}(Q)}}|\sum_{\substack{2j+3l+6k=n 0\leq j \leq N 0\leq l \leq N 0\leq k}}q^{\frac{j+k+l}{2}}\Lambda_{j}(\chi^{2})\Lambda_{l}(\chi^{3})Sym^{k}(\chi^{6})|^2
A_{5}:=\sup_{R>0}\left(\int_{|x|\geq R}\phi_{5}(x)dx\right)^{1/q}\left(\int_{0}^{R}\left(\int_{\wp}r^{Q-1}\psi_{5}(ry)d\sigma(y)\right)^{1-p^{\prime}}dr\right)^{1/p^{\prime}}<\infty,
\log \rho_{g}^{\left( p,q\right)}\left( f\right) =\underset{\sigma\rightarrow +\infty}{\overline{\lim}}\left[ \log^{\left[ p+1\right]}M_{g}^{-1}\left( \sigma \right) -\log^{\left[ q+1\right]}M_{f}^{-1}(\sigma )\right] ,
w(x,y)=\frac{16q_{0}}{\pi^{6}D}\sum_{m=1,3,5,...}^{\infty}\sum_{n=1,3,5,...}^{\infty}\frac{1}{mn(\frac{m^{2}}{a^{2}}+\frac{n^{2}}{b^{2}})^{2}}sin\frac{m\pi x}{a}sin\frac{n\pi y}{b}
\left\Vert \left (M_0-\hat M\right )_{\Omega}\right \Vert_2^2\leq \frac{2\lambda}{3}\,\left \Vert \hat M-M_0\right \Vert_{*}+\lambda \left (\left\Vert\mathbf{P}_{M_0}\left (M_0-\hat M\right )\right \Vert_{*}-\left \Vert \mathbf P^{\bot}_{M_0}\left (\hat M\right )\right \Vert_{*}\right ).
(F^s E_{1}^{n}F_{2}^{m}\otimes F_1) \cdot w_{p_1 - p_2}= \left( \sum_{i=0}^{s}\binom{s}{i}((F)^{s-i}(E_{1}^{n}F_{2}^{m})) F^i \otimes F_1 \right) \cdot w_{p_1 - p_2}& = ((F)^{s}(E_{1}^{n}F_{2}^{m}) \otimes F_1) \cdot w_{p_1 - p_2},
P_{\widetilde{\varphi_{\ell}^{H_1}}}(T) =\frac{1}{(-1)^{\ell/4}+2^{(\ell-4)/2}}&\sum_{i=1}^{\ell/4-1}(-1)^{i}(4^{i-1}T^{4(i-1)}+2\times 4^{i-1}T^{4(i-1)+1}+2\times 4^{i-1}T^{4(i-1)+2}).
\int_{\Omega}f(x)\, dx = \int_{B_H}f(\nabla \widetilde u (\xi)) \Phi(\xi)\, d\xi = \int_{B_{H_0}}f(\nabla \widetilde u (\nabla V_0(y)) \det(\nabla_{\xi}^2V(\nabla V_0(y)))\,{\rm det}(\nabla^2 V_0(y))\, dy = \int_{B_{H_0}}f(\nabla \widetilde u (\nabla V_0(y))) \, dy
\max_{S(t), t=1,.., N}\frac{V_{\rm{tot}}(t)}{N}=\max_{S(t), t=1,.., N}\rm{E}[\frac{1}{N}\sum_{t=1}^{N}\sum_{s_i\in S_m}1(s_i\in S(t)) 1(B_i(t)\geq1)]
\sum_{d=0}^{\kappa}(d+2)^uF_{d+2}(n)=\left(\sum_{d=0}^\kappa (d+2)^ue^{-d^2-d}\right)+O\left(\sum_{d=0}^\kappa (d+2)^u\frac{(\log n)^6}{n}e^{d^2+d}\right),
\varphi (x,t)=-2\int_0^t\int_\Sigma \frac{\partial p}{\partial \nu_x}(x,y,s)\varphi (y,t-s)dA(y)ds &-2\int_\Omega \frac{\partial p}{\partial \nu_x}(x,y,t)\psi (y)dV(y),\;\; (x,t)\in \Sigma \times (0,+\infty ).
D_{s\eta_1}=(1+\rho^1)D^{(1)}+s((\chi_1)_z\eta_1+(\chi_1)_{\bar{z}}\bar{\eta}_1)(e_1\partial_t)+\Theta_s^0+\mathcal{R}_s^0+\mathcal{H}_s^0+\mathcal{F}_s^0+\delta^{(1)}.
g_k^{(m)}:= \begin{cases}g_k,\k = 1, \ldots, m; b_m,\k = m + 1, \ldots, m^2,\end{cases}b_m := -\frac{m^2}{2(m^2 - m)}a - \frac{g_1 + \ldots + g_m}{m^2 - m}.
{\tilde b}= \frac{2}{3}\Pi_{\mu\nu}\Pi^\mu\Pi^\nu+\frac{4}{3}\Pi_\mu^{(z)}\Pi^\mu \Pi-\frac{2}{15}\Pi_{\alpha\beta}\Pi^\alpha\Pi^\beta -\frac{3}{5}\Pi_{\mu\alpha}\Pi^\mu\Pi^\alpha-\frac{3}{5}\Pi_\alpha^{(z)}\Pi\Pi^\alpha-2\Pi{\cal F}\ .
\|y-z\|^2 \leq \sum_{i\in I}\omega_i\|T_ix-z\|^2\leq\sum_{i\in I}\omega_i\big(\|x-z\|^2 -\sigma_i\|x-T_ix\|^2\big) &=\|x-z|^2-\sum_{i\in I}\omega_i\sigma_i\|x-T_ix\|^2\leq\|x-z\|^2-\sigma_+\|x-y\|^2,
\int_{-\frac{\widetilde{T}}{2}}^{\frac{\widetilde{T}}{2}}\Theta_m(t)e^{\lambda_n t}dt =\int_{-\frac{\widetilde{T}}{2}}^{\frac{\widetilde{T}}{2}}\Theta_m(t)e^{-i(i\lambda_n) t}dt=\Psi_m(i\lambda_n) =P_m(i\lambda_n)\frac{\sin(\delta i(\lambda_n-\lambda_m))}{\delta i(\lambda_n-\lambda_m)}=\delta_{nm}.
\lambda \left\{x \in \mathbb{I}:q_m^{-2}(x) \geq \frac{\delta}{\beta^{n+1}}\right\}= \lambda \left\{x \in \mathbb{I}:q_m^{-2t}(x) \geq \left(\frac{\delta}{\beta^{n+1}}\right)^t\right\}\leq \frac{\beta^{t(n+1)}\mathrm{E}\left(q_m^{-2t}\right)}{\delta^t}.
\begin{aligned}\hat{p}_{n+1}(x) =\hat{a}_{n+1}(x)\hat{p}_n(x)+\beta_{n+1}\hat{p}_{n-1}(x), \hat{q}_{n+1}(x) =\hat{a}_{n+1}(x)\hat{q}_n(x)+\beta_{n+1}\hat{q}_{n-1}(x).\end{aligned}
\frac{p_{\lambda}(u(0)) \ldots p_{\lambda}(u(\ell - 2)) p_{\lambda}( \ell \rho - \sum_{i = 0}^{\ell - 2}u(i) )}{p_{\lambda}^{\ast \ell}(\ell \rho)}= \frac{p_0 (u(0)) \ldots p_0 (u(\ell - 2)) p_0 ( \ell \rho - \sum_{i = 0}^{\ell - 2}u(i) )}{p_0^{\ast \ell}(\ell \rho)}
\left(J_1',J_2',J_3'\right) = \left( \alpha, -\frac{H_{\mu,2}^{(0)}(J_1,J_2,J_3,\alpha)}{H_{\mu,2}^{(1)}(J_1,J_2,J_3,\alpha)}, -\frac{H_{\mu,3}^{(0)}(J_1,J_2,J_3,\alpha)}{H_{\mu,3}^{(1)}(J_1,J_2,J_3,\alpha)}\right) \ .
\mathbf P_t^{(1)}=\mathbf e_{k+n}\mathbf e_{j+n}^T\mathbf M_t +\mathbf M_t^*\mathbf e_{j+n}\mathbf e_{k+n}^T, \mathbf P_t^{(2)}=\mathbf e_{k}\mathbf e_{j}^T\mathbf M_t +\mathbf M_t^*\mathbf e_{j}\mathbf e_{k}^T.
\bigl( \max_{x \in \tau}\partial_\mu v_\delta(t,x) \bigr) - \bigl( \min_{x \in \tau}\partial_\mu v_\delta(t,x) \bigr) = \partial_\mu v_\delta(t,x') - \partial_\mu v_\delta(t,x'') & = \sum_k \mu_k \bigl( \partial_k v_\delta(t,x') - \partial_k v_\delta(t,x'') \bigr) \le \sum_k |\mu_k| \delta.
{}^C\left(t^{1-2H}\frac{\partial}{\partial t}\right)^{\alpha}P(x,t) = \left(t^{1-2H}\frac{\partial}{\partial t}\right)^{\alpha}\left(P(x,t)-P(x,0)\right) =(2H)^{\alpha}t^{-2H\alpha}I^{0,-\alpha}_{2H}\left(P(x,t)-P(x,0)\right) &=H^{\alpha}\frac{\partial^2}{\partial x^2}P(x,t),
\widehat{w}:=\Big(\prod_{i=1}^{m-1}w_i^{\frac1{p_i}}\Big)^{\varrho}\mbox{and}W:=w^{\frac{r_m}{p}}\widehat{w}^{- \frac{r_m}{\delta_{m+1}}}=w_m^{\frac{r_m}{p_m}}\widehat{w}^{\frac{r_m}{\delta_m}}
\mathcal Q(\ell, \omega) = \frac1\ell \mathcal Q_0 + \frac{4 | \Gamma| \cdot |\partial_2 \psi_{\varkappa, 2}(0)|^2}{\sqrt{\Lambda -\omega}\cdot \sqrt{2 \zeta_1 ''(\varkappa)}}\; T_\ell^* \bigr( P_+ + P_- \bigr) T_\ell + R(\ell,\omega)
\|u\|_{L^1}= 2\pi \int_0^\infty \bigl|\int_1^r \frac 1t \, e^{f(t)}\, dt\bigr|\; r\,e^{-f(r)}dr &=2\pi \int_0^1 \int_r^1 \frac rt \, e^{f(t)-f(r)}\, dt \, dr + 2\pi \int_1^\infty \int_1^r \frac rt \, e^{f(t)-f(r)}\, dt\, dr.
\psi^+_{k_y}=\frac{i}{2}\sqrt{\frac{1}{\pi H_5}}\int_{-\infty}^{\infty}\frac{dK_Y}{\sqrt{K_Y^2+m^2}}e^{i{k_y\over H_5}\sinh^{-1}(K_Y/m)}\exp(iK_Y Y -i \sqrt{K_Y^2+m^2}T),
\mathcal{I}:= \big\{(\iota_1,\iota_2,\iota_3,\iota_4) \in \{+,-,0\}^4 \, \mbox{s.t.}\, \exists \, a,b \in \{1,2,3,4\}\,\, \mbox{with}\,\, \iota_a = +, \iota_b = - \big\}.
S(\delta t_{\perp})=\sum_{n=0}^{m}|c_{n}|^{2}e^{-i\pi\frac{E_{n}}{E_{max}}}=\sum_{n=0}^{m}|c_{n}|^{2}(cos(\pi\frac{E_{n}}{E_{max}})-isin(\pi\frac{E_{n}}{E_{max}}))=0
\chi_{1 2}( p_{3 0}) \, = \, \beta_{1}\beta_{2}\, T_{1 2}^{0}( p_{3 0}) \, \int d \, p_{1 2 0}\, G_{0 1}G_{0 2}\, \beta_{1}\beta_{2}\, [ \, \, \chi_{2 3}( p_{1 0}) \, + \, \chi_{3 1}( p_{2 0}) \, ]
p_1 = \sum \beta^2 - \sum \gamma^2 - \sum \delta^2 = 4(x_1^2 + x_2^2 + x_3^2) + (x_1\pm x_2)^2 + (x_1\pm x_3)^2+ (x_2\pm x_3)^2 -4y_1^2 - 4y_2^2 = 8(x_1^2 + x_2^2 + x_3^2) - 4(y_1^2 + y_2^2).
t + 1 = \frac{R_*^{\frac 32}}{L_*\beta_0^{\frac 32}}\Big[\frac 7{2}\Big( \arctan \Big( \frac{7}{2\sqrt{\alpha_0}}\Big) -\arctan \Big( \frac{7 - 2\beta_0R}{2 \sqrt{g(R)}}\Big)\Big) + \left( \sqrt{\alpha_0}- \sqrt{g(R)}\right)\Big].
v_{k}^{(\alpha)}=\frac{h_{k+1}^{(\alpha)}}{h_{k}^{(\alpha+1)}},w_{k}^{(\alpha)}=\frac{h_{k}^{(\alpha+1)}}{h_{k}^{(\alpha)}}, v_{-1}^{(\alpha)}=\frac1{h_{-1}^{(\alpha)}}=0.
\left( \begin{array}{c}3n n\end{array}\right) =\sum_{j=n}^{2n}\left( -1\right)^{n+j}\left( \begin{array}{c}2n j\end{array}\right) \frac{2n+1}{j+1}+\sum_{j=0}^{n-1}\left( \begin{array}{c}n+j j\end{array}\right) \left( \begin{array}{c}2n-j-1 n-j\end{array}\right) .
f(b_1,\cdots,b_{d-1},b_d)=\sum_{\alpha\in\mathbb{Z}^{d-1}}A_{\alpha}(b_d) e^{\alpha_1\frac{2\pi ib_1}{T_1}}\cdots e^{\alpha_{d-1}\frac{2\pi ib_{d-1}}{T_d}}
\int_{-\infty}^{\infty}q^{\beta^{2}}\left(q^{-\beta+n+1};q\right)_{\infty}L_{n}^{(-\beta)}\left(-1;q\right)A_{q}\left(-q^{\beta+n}\right)d\beta=\frac{\sqrt{\pi/\log q^{-1}}}{\left(q;q\right)_{n}},
\sum_{m=1}^{R}\left| \sum_{|t_j| \leq T}a_j u_j(z_m) + \frac{1}{4 \pi}\sum_{\mathfrak{a}}\int_{-T}^{T}a_{\mathfrak{a}}(t) E_{\mathfrak{a}}(z_m, 1/2+it) dt \right|^4 \ll (T^4 +T^2 \delta^{-2}) \|a\|_{*}^4.
g_{\varphi_{\epsilon}}^{p \bar{q}}g_{\varphi_{\epsilon}j \bar{m}}R_{\omega_{\epsilon}}^{\bar{m}j}{}_{p \bar{q}}=\frac{1+\varphi_{\epsilon i \bar{i}}}{1+\varphi_{\epsilon j \bar{j}}}R_{\omega_{\epsilon}}^{i\bar{i}}{}_{j\bar{j}},
U \;=\;\int \frac{d^4 p_1}{(2 \pi)^4}\cdots \int \frac{d^4 p_r}{(2 \pi)^4}\; \delta(a_1 - p_1^2) \:\cdots\: \delta(a_r - p_r^2) \;f^{\{\alpha\}}(p_1,\ldots,p_k) \;\;\;\; .
\mathfrak{K}_{\mathcal{L}}= \{(M,(R^{\mathcal{M}})_{R \in \mathcal{L}}) \in K(\mathcal{Q}) \times \prod_{R \in \mathcal{L}}K(\mathcal{Q}^{\alpha(R)}) \mid R^{\mathcal{M}}\subseteq M^{\alpha(R)}\}.
\int_{A'\cap D_{k}''}f(\widetilde{y}, \widetilde{w}, t_i)\d\widetilde{y}d\widetilde{w}\leq \frac{C}{\alpha_k^2}\int_{A'\cap D_{k}''}\widetilde{w}^2 f(\widetilde{y}, \widetilde{w}, t_i)\d\widetilde{y}d\widetilde{w}&
G_g(m^\prime, m_d, l)= \frac{1}{q^{d+1}}\sum_{x^\prime\in \mathbb F_q^{d-1}, x_d, s\in \mathbb F_q}\chi(m^\prime \cdot x^\prime+ m_dx_d+ls) ~\widehat{G_g}(x^\prime, x_d, s) = \frac{1}{q^{d+1}}\sum_{s\neq 0}\sum_{(x^\prime, x_d)\in \mathbb F_q^d}\chi(m^\prime \cdot x^\prime+ m_dx_d+ls) ~\widehat{g}(x^\prime, x_d s).
\|A\| - \|I^\sigma I_\sigma(A)\| = \sum_{\substack{\tau \in A \sigma \not \subset \tau}}w(\tau) = \sum_{\substack{\tau \in A+I^\sigma(B) \sigma \not \subset \tau}}w(\tau)= \|A+I^\sigma(B)\| - \|I^\sigma I_\sigma(A+I^\sigma(B))\|.
\mathbf{J}^{-1}= \left( \begin{array}{ccc}e^{-\mathcal{F}+\alpha}e^{-\mathcal{F}+ \beta}0 e^{-\mathcal{G}+\alpha}-e^{-\mathcal{G}+\beta}0 0 0 1 \end{array}\right).
2\pi=\sum_{n\geq 4}v_{n}^{\mathrm{int}}\left(2\pi-\frac{\pi}{2}n\right)+\sum_{n_1,n_2}v_{n_1,n_2}^{\mathrm{ex}}\left(\frac{3\pi}{2}-\frac{\pi}{4}(n+n_1)\right)+\sum_{n_1,n_2}v_{n_1,n_2}^{\mathrm{cl}}\left(\frac{3\pi}{2}-\frac{\pi}{4}(n+n_1)\right), \sum_{n}v_n^{\mathrm{ex}}=3,