harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\|e^{iv\cdot x}R(t)\|_{H^k}\leq\int_0^t \big\|e^{iv\cdot x}\big(|u|^{p-1}u-|\tilde{u}|^{p-1}\tilde{u}\big)(s)\big\|_{H^k}+\|\mathcal{E}\phi^{(\nu)}(s,\nu\cdot)\|_{H^k}ds &=\int_0^t I(s)+II(s)ds.
a \cong -\rho_r^4\vartheta^2(1-P^2), b \cong (1-P^2)(3(1-P^2)-1-3P^2)+\frac{1}{4}\rho_r^4\vartheta^2=(1-P^2)(2-6P^2)+\frac{1}{4}\rho_r^4\vartheta^2, c \cong -\frac{1}{4}\big(3(1-P^2)-(1-P^2)\big)=-\frac{1}{2}(1-P^2).
\mathcal{F}_{k}\circ\mathcal{F}_{k-1}\circ ...\circ\mathcal{F}_{2}\circ\mathcal{F}_1(A)=\bigcup_{i_1,i_2,...,i_k\in \{1,2\}}AP^{-1}S^{[1]}_{i_1},...,S^{[k-1]}_{i_{k-1}}S^{[k]}_{i_k}P.
\zeta_{2}^{D}( s ) = \frac{V \mu^{2 s}( S^{2}- M^{2})^{- s + \frac{m - 1}{2}}}{( 4 \pi )^{\frac{m - 1}{2}}\Gamma \left( \frac{m - 1}{2}\right)}\left[ ( - 1 )^{- s}\frac{\Gamma \left( \frac{m - 1}{2}\right) \Gamma \left( 1 - s \right)}{\Gamma \left( \frac{m + 1 - 2 s}{2}\right)}+ \right.
\mathbf{T}_t(\vartheta)=\sum_{\upsilon\subseteq\vartheta^t}\mathbf{L}^t(t_{|\upsilon|},\vartheta\setminus t_{|\upsilon|})\cdots\mathbf{L}^{t_3}(t_{2},\vartheta\setminus t_{2}) \mathbf{L}^{t_2}(t_{1},\vartheta\setminus t_{1})\mathbf{T}^{t_1}_0(\vartheta)
V_{z}(v) =\sum_{P<p\leq 2P}(\log p)\left(\sum_{|r|\leq Z(\log N)^{4}}g_{z}(r)e(rp^c)\right) e\left(v\left(N+j-p^{c}+\frac{z}{2Z}\right)^{\gamma}\right)+O\left(N^{-10}\right) &=\sum_{|r|\leq Z(\log N)^{4}}g_{z}(r)U\left(N+j+\frac{z}{2Z},r,v\right)+O\left(N^{-10}\right) &\ll N^{-10}+\frac{1}{Z}\sum_{|r|\leq R}\sup_{T\in[N,N+2]}|U(T,r,v)|,
\sum_{|\xi|\leq \lambda}\sum_{i,j=1}^{d_\xi}(A(d_\xi^{\frac{1}{2}}\xi_{ij}),d_\xi^{\frac{1}{2}}\xi_{ij})= (2\pi)^{-n}\left( \smallint\limits_{T^*\mathbb{S}(G)}\sigma_{\textnormal{loc}}(x,\xi')d\mu_L(x,\xi')\right)\lambda^n+O(\lambda^{n-1}),
\begin{array}{ccc}a^{(*)}a+c^{(*)}c=I, a a^{(*)}+ q^{2}c^{(*)}c=I, c^{(*)}c=c c^{(*)}, a c=q c a, a c^{(*)}=q c^{(*)}a. c^{(*)}a^{(*)}=q a^{(*)}c^{(*)}, c a^{(*)}=q a^{(*)}c. \end{array}
C_{\beta,2}:=\{\psi \in h_\alpha^\beta(\tau(\kappa_\alpha))(d_\beta^\tau): (H_\beta^2)^\tau(\psi(\kappa_\beta))=(H_\beta^2)^r(\psi (\kappa_\beta)) \}\in h_\alpha^\beta(\tau(\kappa_\alpha))(d_\beta^\tau).
\begin{aligned}\,\, \omega(x,y)=\omega(y,x) \,\,\, (x,y)\in G\times G; &\,\, \omega(x,x)=0 \quad\quad\quad\,\, \,\,\, x\in G; &\,\, \displaystyle \sum_{y\in G}\omega(x,y)<\infty \,\,\, x\in G\,.\end{aligned}
\frac{(n+\beta+1 )(n+N+2+\alpha+\beta)}{(2n+\alpha+\beta+2)}Q_n(x) +\frac{(N-n)(n+\alpha+1)}{(2n+\alpha+\beta+2)}Q_{n+1}(x) \qquad{}= (\beta+1+N-x) \hat Q_n(x), \frac{(n+1)}{(2n+\alpha+\beta+3)}\hat Q_n(x) + \frac{(n+\alpha+\beta+2)}{(2n+\alpha+\beta+3)}\hat Q_{n+1}(x)= Q_{n+1}(x) .
(H_{\omega,\epsilon}-E)^{-1}- (H_{\omega^{(n)},\epsilon}-E)^{-1}= (H_{\omega,\epsilon}-E)^{-1}( V_\omega - V_{\omega^{(n)}}) (H_{\omega^{(n)},\epsilon}-E)^{-1}= (H_{\omega^{(n)},\epsilon}-E)^{-1}( V_\omega - V_{\omega^{(n)}}) (H_{\omega,\epsilon}-E)^{-1},
\sum_{n=0}^{\infty}\textup{PD}_\textup{t}(576n+72)q^n \equiv 2f_1^3+3\dfrac{f_1^3f_{4}^4f_{6}^6}{f_2^8f_{12}^3}&\equiv \left(2+3\dfrac{f_6^2}{f_{12}}\right)\cdot f_1^3 &\equiv\left(2+3\dfrac{f_6^2}{f_{12}}\right)\left(f_3a(q^3)-3qf_9^3\right)~(\textup{mod}~4),
\vert\vert S_{n_k}f\vert\vert=\vert\vert f+(S_{n_k}-I)f \vert\vert \le \vert\vert f \vert\vert+\vert\vert (S_{n_k}-I)f\vert\vert &\le \vert\vert f \vert\vert_{*}+4\cdot\frac{1}{4}\vert\vert (S_{n_k}-I)f \vert\vert &\le 5\vert\vert f \vert\vert_{*}
M=\begin{pmatrix}a_{0 0}\dots 1_{0\,r}0 0 \dots \dots \dots a_{1 0}a_{1 1}\dots 1 0 \dots \dots \dots \vdots \vdots \vdots \vdots \ddots \vdots \vdots \dots a_{q\,0}a_{q\,1}a_{q\,2}\dots \dots 1_{q\,q+r}0 \dots 0 a_{q+1\,1}a_{q+1\,2}a_{q+1\,3}\dots \dots 1 \dots 0 0 \ddots \vdots \vdots \vdots \vdots \ddots\end{pmatrix};
\Xi X = \left( \begin{array}{c c}{m}&{b}\\{\bar{b}}&{n}\\ \end{array}\right) \left( \begin{array}{c c}{M}&{B}\\{\bar{B}}&{N}\\ \end{array}\right) = \left( \begin{array}{c c}{m M + b \bar{B}}&{m B + b N}\\{\bar{b}M + n \bar{B}}&{\bar{b}B + n N}\\ \end{array}\right) ,
V_{\ell,k}^{\delta}:= A^{-T}( \delta k / \sqrt{d}) \left\langle \delta \cdot B_1 (\ell / \sqrt{d}) \right\rangle \times Q_{k}^{\delta}, Q_{k}^{\delta}:= Q_{\Phi,k/\sqrt{d}}^{(\delta,1)}= \Phi^{-1}\bigl(\delta \cdot B_1(k/\sqrt{d})\bigr),
F\Big(m_N, H - \sum\limits_{\ell=1}^{N-1}{U}_\ell\Big){\cal A}_k^\dagger{\cal A}_\ell =F\Big(m_N, H - \sum\limits_{\ell=1}^{N-1}{U}_\ell\, +1\Big){\cal A}_\ell{\cal A}_k^\dagger.
\frac{i}{2}\mathrm{T r}\left( \frac{1}{- \partial^{2}- m^{2}}\hat{T}_{x}^{( 1 )}\right) = \frac{i}{2}\sigma ( x ) \int \frac{d^{d}k}{( 2 \pi )^{d}}\frac{1}{k^{2}- m^{2}}.
\int_{\Omega_0}\bigl|w_{ij}^{(n)}(x)\bigr|^p\,d\mu(x) =\int_{\Omega_0}\bigl|\sum_{k=1}^d u_{ik}(\phi_n(x))v_{kj}^{(n)}(x)\bigr|^p\,d\mu(x) &=\int_{\Omega_0}\bigl|\sum_{k=1}^d u_{ik}(y)v_{kj}^{(n)}(\phi_n^{-1}(y))\bigr|^p J_{\phi_n^{-1}}(y)\,d\mu(y) \ ,
ds^{2}=(1+\frac{Gm}{2c^{2}r_{1}})^{4}(dx^{2}+dy^{2}+dz^{2})-c^{2}dt^{2}(1-\frac{Gm}{2c^{2}r_{1}})^{2}/(1+\frac{Gm}{2c^{2}r_{1}})^{2}
\langle \sigma_{1}( z_{1}) \sigma_{2}( z_{2}) \rangle = \left( \frac{1}{4 y_{1}y_{2}x}\right)^{1 / 8}\frac{1}{\vartheta_{3}( u ( x ) )}\vartheta_{2}( e^{2 i \varphi_{0}}, \sqrt{u ( x )}) .
\begin{array}{r c l}{{}}&{{}}&{{\partial G = 2 F^{1 \, m}H_{m}\, ,}}\\{{}}&{{}}&{{}}\\{{}}&{{}}&{{\partial H_{m}= \frac{3}{2}\epsilon_{m n p}F^{1 \, n}F^{2 \, p}\, ,}}\\ \end{array}
\mathfrak{Im}(\Omega)= -R_{-}^{3}\theta^{1567}+R_{+}^{2}R_{-}\left(\theta^{1237}+\theta^{1264}+\theta^{1534}\right) &+\frac{r}{2}dr\wedge\left(R_{-}\left(\theta^{267}+\theta^{537}+\theta^{564}\right)-\frac{R_{+}^{2}}{R_{-}}\theta^{234}\right).
J_{22}=-\frac{4}{mn\pi^{2}}\int_{0}^{\pi-h_{1}}\int_{0}^{h_{2}}H_{x,z_{1},y,z_{2}}(t_{1}+h_{1},t_{2})\frac{S(t_{1},t_{2})}{(t_{1}+h_{1})^{2}\left( 2\sin \frac{t_{2}}{2}\right)^{2}}dt_{1}dt_{2},
(I+\lambda C_n G^\omega_{n,n}(E+\iota 0)) \left[C_n\lim_{\epsilon\downarrow 0}\frac{1}{tr(G^{\omega,\lambda}_{n,n,n}(E+\iota\epsilon))}G^{\omega,\lambda}_{n,n,n}(E+\iota \epsilon)\right]=0,
t =\widetilde{\Phi}(s)\leq Ns^{1-\frac{1}{N}}e^{s^{\frac{1}{N}}}; \ln t \leq \ln N+\left( 1-\frac{1}{N}\right) \ln s+s^{\frac{1}{N}}\leq (2\ln(2N))s^{\frac{1}{N}},\ \ \ \ \ \ \s\geq \left( 2N\right)^{N}; \widetilde{\Phi}^{-1}(t) =s\geq \left( \frac{1}{2\ln(2N)}\ln t\right)^{N},\ \ \t\geq e^{2N}.
\widetilde{C}_{i,j}(1) = \left\{\begin{array}{cl}\frac{ij}{(i+j)}\binom{2i}{i}\binom{2j}{j}, i+j>0 0, i+j=0\end{array}\right\}, A_{i,j}= \frac{(2i+1)!(2j+1)!}{(i+j+1)(i!j!)^2}= \frac{(2i+1)(2j+1)}{(i+j+1)}\binom{2i}{i}\binom{2j}{j}.
M_{1}\lesssim\int_{\mathbb{D}}\Big|\int_{\mathbb{D}}K(w,z)g(z)dA(z)\Big|^{2}dA(w) &=\|B(g)\|_{L^{2}}^{2}\lesssim \|g\|_{L^{2}}^{2}\lesssim|I|^{p+2-2\beta}.
\left(\begin{bmatrix}0 A A^T 0\end{bmatrix},\begin{bmatrix}I 0 0 B^TB\end{bmatrix}\right),\qquad\left(\begin{bmatrix}0 B B^T 0\end{bmatrix},\begin{bmatrix}I 0 0 A^TA\end{bmatrix}\right),
\langle Y'_{W}(e^{xL_{-1}}a,z)w',w\rangle =\langle Y'_{W}(a,z+x)w',w\rangle =\langle w',Y_{W}(e^{(z+x)L_{1}}(z+x)^{-2\deg}(-1)^{\deg}a,(z+x)^{-1})w\rangle.
\mathfrak G^b_{n,j}(\theta) \equiv \frac{1}{\sqrt n}\sum_{i=1}^n \left(m_j(X_i^b,\theta)-\bar m_n(\theta)\right)/\sigma_{P,j}(\theta) &=\frac{1}{\sqrt n}\sum_{i=1}^n(M_{n,i}-1)m_j(X_i,\theta)/\sigma_{P,j}(\theta).
V_{1}( \mathrm{x ) = \frac{1}{3 2}\mathrm{x^{6}- \frac{\o m e g a}{1 6 \mathrm{g}}\mathrm{x^{4}+ ( \frac{\o m e g a^{2}}{3 2 \mathrm{g^{2}}}- \frac{3 j}{4}- \frac{3}{8}) \mathrm{x^{2}- \frac{\o m e g a}{4 \mathrm{g}}( 3 j - \frac{1}{2})}}}}
g_{\lambda, \rho}^2 (w_2) = \frac{n_0}{4 k_{\ell}}\log\left( 1 + \lambda (1 - \lambda \rho) w_2 P' \right) - \frac{(1 - \rho ) n_0}{2 k_{\ell}}\log \left( 1 + \lambda w_2 P' \right) - \frac{\rho \ell}{k_{\ell}}H_2 \left( \frac{w_2}{\ell}\right).
\mathbb{F}^+L^{E}_d(q,\bar q) = \lim_{\epsilon\rightarrow 0^+}\mathbb{F}^+L^{\epsilon,E}_d(q,\bar q) =(\bar q, \alpha(\bar q)), \mathbb{F}^-L^{E}_d(q,\bar q) = \lim_{\epsilon\rightarrow 0^+}\mathbb{F}^-L^{\epsilon,E}_d(q,\bar q)=(q, \alpha(q)),
\frac{\sqrt[8]{Q_{\{1,4\}}(x)^{16}+Q_{\{1,4\}}(x)^4\sqrt{Q_{\{1,4\}}(x)^{24}+64}}}{\sqrt[8]{2}}=Q_{\{1,4\}}\left(\frac{1-\sqrt{1-x^2}}{1+\sqrt{1-x^2}}\right),
\mathcal{F}^{i}_s=D(s((\chi_{i+1})_z\eta_{i+1}+(\chi_{i+1})_{\bar{z}}\bar{\eta}_{i+1})Id) &+D\Big(\left(\begin{array}{cc}0 si\chi_{i+1}(\eta_{i+1})_t -si\chi_{i+1}(\bar{\eta}_{i+1})_t 0\end{array}\right)\Big).
|p_k(M)|\leq \int_M|p_k(R)|\omega_g & \leq \frac{1}{(k!)^2(2\pi)^{2k}}\int_M\|R^k\|^2\omega_g &= \frac{1}{(k!)^2(2\pi)^{2k}}\int_M h_{4k}\omega_g &=\frac{(2k)!}{k!k!}\chi(M).
\mathcal{A}_{P}^{(1)}(F,G) = max_{i=1}^3 \frac{q_i \bigg\{|\langle f_i , g_i \rangle | + \| f_i\| \| g_i \| \bigg\}}{2}&= max_{i=1}^3 \bigg\{1,\frac{|1 + \beta| + \left|\sqrt{\alpha^2 + (1+ \beta)^2}\right|}{2}, \frac{|1 - \beta| + \left|\sqrt{\alpha^2 + (1 - \beta)^2}\right|}{2}\bigg\}.
\lambda^{\left( p\right)}\left( G\right) =2\sum_{\left\{i,j\right\}\in E\left( G\right)}x_{i}x_{j}\leq2e\left( G\right)^{1-1/p}\left(\sum_{\left\{i,j\right\}\in E\left( G\right)}x_{i}^{p}x_{j}^{p}\right)^{1/p}.
\sum_{r \in \mathbb{Z}}c^F_r(\widetilde{W_v},\widetilde{U_v})^lq_F^{-lr/2}X^{-lr}= \omega_\sigma(-1)^{n-2}\gamma(X,\pi,\sigma,\psi)^l \sum_{r \in\mathbb{Z}}c^F_r(W_v,U_v)^lq_F^{lr/2}X^{lr}
f_{n}(z)=\sum_{k\ge0}(\frac{k!}{|G|}[z^{k}]g(z)^{n})\frac{z^{k}}{k!}=\frac{1}{|G|}\sum_{k\ge0}z^{k}[z^{k}]g(z)^{n}=\frac{g(z)^{n}}{|G|}.
\begin{array}{l}\Phi^{d-3\over 2}\partial^{\mu_1}\Phi \Phi^{s-1}{\bf V}^{\mu_2}_{i_2}\cdots{\bf V}^{\mu_s}_{i_s}-\mbox{traces}_{(i)}= begin{align*}5pt]={2\over 2s+d-3}\nabla^{\mu_1}\left( \Phi^{s+{d-3\over 2}}{\bf V}^{\mu_2}_{i_2}\cdots{\bf V}^{\mu_s}_{i_s}-\mbox{traces}_{(i)}\right)\end{array}
\int_{e}\overline{R}_e^2 \tau_e ds = \int_{e}\overline{R}_e (\overline{R}_e-R_e) \tau_e ds +a_{\omega_e}(u_h-u,\overline{R}_e \tau_e) &{}- \int_{\omega_e}R_K\overline{R}_e \tau_e dx- \int_{e}g(\lambda- \lambda_h)\overline{R}_e \tau_e ds.
\sup_{\{p(x_t|s_{t-1},q_{t-1})\}_{t\geq 1}}\liminf_{N\rightarrow \infty}\frac{1}{N}\sum_{i=1}^N I(X_i,S_{i-1};Y_i|Q_{i-1}) = \sup_{P_{X|S,Q}}\liminf_{N\rightarrow \infty}\frac{1}{N}\sum_{i=1}^N I(X_i,S_{i-1};Y_i|Q_{i-1}),
\left|z+\dfrac{a_{n-1}}{a_n}-(t_1-t_2)\right|> \left(t_2+\left|\dfrac{a_{n-1}}{a_n}\right|\right)(\sin\alpha-\cos\alpha) &+\dfrac{2t_2|a_{k+1}|+2|a_k|}{|a_n|t_1^{n-k-1}}\cos\alpha+2\sin\alpha\sum\limits_{\nu=0}^{n-2}\left|\dfrac{a_{\nu}}{a_n}\right|\dfrac{1}{t_1^{n-\nu-1}}.
J(u,v)= \alpha\big(\|\nabla u\|_{2}^{2}+\|\nabla v\|_{2}^{2}\big)+(\alpha-1)\big(\|u\|_{2}^{2}+\omega\|v\|_{2}^{2}\big) &+(3\alpha-2)\Big(B(u)+B(v)\Big)-\frac{p\alpha-1}{p}F(u,v).
(x_1^{\prime},x_2^{\prime},...,x_k^{\prime})\preceq (x_1,x_2,...,x_k) \Longleftrightarrow x_i^{\prime}=x_i\ \ \ \i>0\ \ & x_j^{\prime}<x_j, x_i^{\prime}=x_i\ \ \ \j>0\ \ \ \i>j.
\beta_{\rm max}=1,\,\alpha=10^{-5},\,\eta_2=0.1,\, \tau_{\rm min}={10^{-3}}/{[2(\alpha\!+\!\delta)\!+\!L_{\!f}]},\,\tau_{\rm max}=10^{6},\ \tau_{0,0}={10}/{\|\widetilde{A}\|}.
\delta_+(k,\xi) = \delta_-(k,\xi) (1+|r(k)|^2), \k\in(-\infty, k_0=-\xi), &\lim\limits_{k\to\infty}\delta(k) = 1, &\delta(k) \mbox{is analytic in}k\in\mathbb{C}\setminus(-\infty,k_0].
\widetilde{\mathbf{A}}\left(x,t,y,\zeta\right):=-\bar{y}^{2(p-1)}\mathbf{A}\left(x,t,\bar{y}^{-1},-\bar{y}^{-2}\zeta\right) \quad{\textrm{with}}\bar{y}:=\min\left\{y,\frac{1}{m}\right\}
\varpi_{ABC}{}^{C}=\widetilde{\varpi}_{ABC}{}^{C}+\varpi_{ABC}^{{\footnotesize (T)}}{}^{C},\varpi_{A^{\prime}B^{\prime}C}{}^{C}=\widetilde{\varpi}_{A^{\prime}B^{\prime}C}{}^{C}+\varpi_{A^{\prime}B^{\prime}C}^{{\footnotesize (T)}}{}^{C}.
\left(\prod^{k-1}_{j=0}\dfrac{1}{P(D_{j})}\right)^{\frac{1}{k(k-1)}}= \prod^{k-1}_{j=0}\left(n_{0,j}\frac{n_{1,j}}{3}\frac{n_{2,j}}{3}{n_{3,j}}\ldots n_{k-2,j}\frac{n_{k-1,j}}{3}\right)^{\frac{1}{k(k-1)}}
\begin{array}{rcl}\partial_i\partial_j = q^{-1}\partial_j\partial_i~,~~~~~{\rm for}~i<j~ ~{\rm and}~~i\not=-j~, \partial_{-i}\partial_i = \partial_i\partial_{-i}+\lambda q^{i-2}\Delta_{i-1}= q^{-2}\partial_i\partial_{-i}+\lambda q^{i-2}\Delta_i~,~~~~~{\rm for}~i>0~.\end{array}
{\cal S}[ \psi ] = \int \, d^{5}x \sqrt{g}\left\{\frac{i}{2}\, \left[ \overline{{{\psi}}}\tilde{\gamma}^{\hat{\mu}}\tilde{\nabla}_{\hat{\mu}}\psi - ( \overline{{{\tilde{\nabla}_{\hat{\mu}}\psi}}}) \tilde{\gamma}^{\hat{\mu}}\psi \right] - M \overline{{{\psi}}}\psi \right\}
\begin{aligned}|b_h(u,v,w)| \leq \|\nabla\times u\|_{2}\|v\|_{6}\|w\|_{3}\leq C\|\nabla u\|_{2}\|\nabla v\|_{2}\|w\|_{2}^{1/2}\|\nabla w\|_{2}^{1/2}, \end{aligned}
\displaystyle{\raisebox{0 . 9 e x}{\scriptsize{\diamond}}}\mathrm{}\rho_{\tau}( Q + P ) \simeq \frac{1}{\pi}\frac{\gamma_{q}}{[ q_{0}- \tau q + p_{0}- \tau{\bf p}\cdot \hat{{\bf q}}]^{2}+ \gamma_{q}^{2}},
\Pr(\phi_{2,j}= 1) = \sum_{j'=1}^n \Pr(\{\phi_{2,j}=1\}\cap \{\phi_{1,j'}= 1\}) = \sum_{j' \neq j}\Pr(\{\phi_{2,j}= 1\}| \{\phi_{1,j'}=1 \}) \Pr(\phi_{1,j'}=1) = \sum_{j' \neq j}\frac{1}{n-1}\cdot \frac{1}{n}= \frac{1}{n}
\Lambda_{ni}^{(2)}= \sum_{k \neq n}\frac{P_{ni}^{(1)}H^{(1)}P_{k}^{(0)}H^{(1)}P_{ni}^{(1)}}{\lambda_{n}^{(0)}- \lambda_{k}^{(0)}}+ P_{ni}^{(1)}H^{(2)}P_{ni}^{(1)}
\begin{array}{c}{\pi_{x}= p_{x}+ \frac{\lambda}{2}\left( x \left( x p_{x}+ y p_{y}\right) + \left( x p_{x}+ y p_{y}\right) x \right) ,}\\{\pi_{y}= p_{y}+ \frac{\lambda}{2}\left( y \left( x p_{x}+ y p_{y}\right) + \left( x p_{x}+ y p_{y}\right) y \right) .}\\ \end{array}
g^2R^{(\pm)}_2(\nu;0)=g^2\int_{1}^{\infty}d\alpha(\sum_{\ell=1}^{7}\frac{\Phi_{\ell}^{(3)}(\alpha)}{\alpha^{\ell}})[\alpha^{\frac{\nu}{2}}+\alpha^{\frac{-\nu}{2}}]=2\xi(\frac{1}{2}+\nu).
\epsilon \frac{dc}{dt}=\mu h\frac{b+c}{1+c}-\frac{\Gamma}{K_1}\frac{c}{K+c}=\frac{F(c,h)}{K_1}=\epsilon F=f(c,h), \frac{dh}{dt}=\frac{1}{1+c^2}-h=G(c,h).
0 \geq \lambda u_1^+ (\hat h) - f(\hat x) + g(\hat x) + \frac{1}{T_1 (\hat h)}\int_0^{T_1(\hat h)}f(X(t, \hat x)) \, dt > \lambda u_1^+ (\hat h) - \frac{\eta}{2}+ \frac{1}{T_1 (\hat h)}\int_0^{T_1(\hat h)}\Big( g(X(t, \hat x)) - \frac{\eta}{2}\Big) \, dt = \lambda u_1^+ (\hat h) + \overline G_1 (\hat h, \phi' (\hat h)) - \eta.
\sum_{k > L}\frac{R^k}{k}\min\{q^{k/2}, d(\chi)\}\le d(\chi) \sum_{k>L}\frac{R^k}{k}\le \frac{d(\chi)}{L+1}\sum_{k>L}R^k = \frac{d(\chi)}{L+1}\frac{R^{L+1}}{1-R},
\sum_{j=1}^{k}\frac{t^2}{(\epsilon n)^2}\frac{C_u^j}{1-C_u}\leq \sum_{j\geq 1}\frac{t^2}{(\epsilon n)^2}\frac{C_u^j}{1-C_u}= \frac{t^2}{\epsilon^2n^2}\frac{C_u}{(1-C_u)^2}\leq D_1 \frac{t^2}{n^2},
\left<L(\gamma )\frac{1}{2e^2}\int tr F^2 dx \right> = -\frac{\epsilon}{e^2}\left<tr(\nabla_\mu F_{\mu \nu}(\gamma (0))\dot{\gamma}_\nu (0)U(\gamma ))\right> = -\frac{\epsilon}{e^2}\frac{\partial^2}{\partial \gamma^2_\mu (0)}W(\gamma )
\Lambda(n) =\sum_{\genfrac{}{}{0pt}{}{kl=n}{l\le v}}\log k\ \mu(l) -\sum_{l\le v^2}\sum_{kl=n}\Pi_{v}(l)+\sum_{\genfrac{}{}{0pt}{}{kl=n}{k>v, l>v}}\Lambda(k)\Xi_v(l),
(n-2)\mu_i[\Gamma_{[n]\setminus \{i\}}, \Gamma_{\{i,k\}}]=\mu_i\smashoperator{\sum_{\substack{j\in [n] j\neq i\neq k}}}[\Gamma_{[n]\setminus \{i,j\}}, \Gamma_{\{i,j,k\}}]+[\Gamma_{\{j,k\}},\Gamma_{[n]\setminus \{j\}}].
0=A(u)+5B(u)+3C(u)+\frac{1}{4}\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}e^{-|x-y|}u^{2}(x)u^{2}(y)dxdy+\frac{3\omega}{2}\int_{\mathbb{R}^{3}}u^{2}dx.
2|E(G)| = \sum_{f\in{F(G)}}\ell(f) = \sum_{f \in X}\ell(f) + \sum_{f \in Y}\ell(f) + \sum_{f \in Z}\ell(f) + \sum_{f \in F(G)\setminus{X \cup Y \cup Z}}\ell(f) \geq 3|X| + 4|Y| + 5|Z| + 6(|F(G)| - |X| - |Y| - |Z|) = 6|F(G)| - 3|X| - 2|Y| - |Z|.
\int_{M}\left[ a|\nabla_{g}u_{k}|_{g}^{2}+b|u_{k}|^{2}\right] dV_{g}=2\left( J_{g}(u_{k})+\frac{1}{2^{\ast}}|u_{k}|_{g,c,2^{\ast}}^{2^{\ast}}\right) \leq C+o(\Vert u_{k}\Vert_{g}).
\quad\ 2h^{ij}\left(\nabla_j R_{\nu li}^{\ \ \l}+\nabla_l R_{\nu ij}^{\ \ \l}\right) =2S(\nabla_2 R_{\nu 2}-\nabla_1R_{\nu 2}) &=2S\left(\nu(S|\nabla f|)-2Ric(\nabla_{\nu}e_2,e_2)+2Ric(\nabla_{\nu}e_1,e_1)-S|\nabla f|^2\right).
\mathrm{GL}(2,\mathbb{R}) =\left\{A\in\mathrm{Mat}_{2\times 2}(\mathbb{R}) : \det(A)=\pm 1\right\}\mathrm{SL}(2,\mathbb{R}) =\left\{A\in\mathrm{Mat}_{2\times 2}(\mathbb{R}) : \det(A)=1\right\}
\hat{h}_t(p,\xi) = \hat{h}_0(p,\xi)e^{-2\pi i \left(\xi \cdot v(p)\right)t}\;\;\;\;\;\;\;\;- \int_0^t \widehat{S_h}(\tau,p, \xi )e^{-2\pi i \left(\xi \cdot v(p)\right)(t-\tau)}d\tau.
\alpha_{n}^{\mu}| 0 \rangle_{\alpha}| 0 \rangle_{\widetilde \alpha}| p \rangle = \tilde{\alpha}_{n}^{\mu}| 0 \rangle_{\alpha}| 0 \rangle_{\widetilde \alpha}| p \rangle = 0 ~ ~ \forall n > 0 ~ ~ , ~ ~
R_{\tilde \Sigma}(\eta) = R_{\Sigma+ E}(\eta) = (\Sigma+ E-\eta I)^{-1}&= \left[ (\Sigma - \eta I)(I +(\Sigma - \eta I)^{-1}E) \right]^{-1}&= (I +R_{\Sigma}(\eta)E)^{-1}R_{\Sigma}(\eta) &= \sum_{k\geq 0}(-1)^{k}[R_{\Sigma}(\eta)E]^k R_{\Sigma}(\eta),\eta\in\gamma_r.
\frac{z^n p_n(x)}{\beta_n}=\frac{1-z^{2n+2}}{1-z^2}+\sum_{i=0}^{n-1}\left\{(1-4a_{i+1}^2)z^2(\frac{1-z^{2n-2i-2}}{1-z^2})-2b_i z(\frac{1-z^{2n-2i}}{1-z^2})\right\}\frac{z^i p_i(x)}{\beta_i}.
\mathrm{G}( z , \bar{z}, \hat{z}, \hat{\bar{z}}) = \frac{1}{4 \pi^{2}}\int_{- \infty}^{+ \infty}\int_{- \infty}^{+ \infty}\exp \{\dot{\imath}[ q ( \hat{z}- z ) + \bar{q}( \hat{\bar{z}}- \bar{z}) ] \}\widetilde{\mathrm{G}}( z , \bar{z}, q , \bar{q}) \, \mathrm{d}q \, \mathrm{d}\bar{q}\, .
\int_0^{\delta(x)/2}\psi(t^s)\;dt\lessgtr\int_0^{\delta(x)/2}t^s\psi'(t^s)\;dt=\frac{\delta(x)}{2s}\psi\left(\frac{\delta(x)^s}{2^s}\right)-\frac1s\int_0^{\delta(x)/2}\psi(t^s)\;dt
[{\cal P}_{0}, X_{0}] = i , \quad [{\cal P}_{i}, X_{j}] = - i \, \delta_{i j}\left( 1 - \frac{\vec{{\cal P}}\,{}^{2}}{\kappa^{2}}\right) \, e^{{\cal P}_{0}/ \kappa}, \quad [{\cal P}_{0}, X_{i}] = - \frac{2 i}\kappa \,{\cal P}_{i}\, e^{{\cal P}_{0}/ \kappa}
\int_0^1 f'(x)^2 dx = \sum_{1 \leq |j| \leq \frac{n-1}{2}}(2 \pi j)^2 |a_j|^2 \leq (n-1)^2 \pi^2 \sum_{1 \leq |j| \leq \frac{n-1}{2}}|a_j|^2 = (n-1)^2 \pi^2 \int_0^1 f(x)^2 \,dx.
w z\Phi =Uv^*\Bigl[\|v^*\mathbf n\|^{2\dagger}\langle M_0v^*\mathbf n,\Phi\rangle \mathbf n-G_{0,0}v\Phi\Bigr] &= U(1-S)M_0\Phi-UM_0\Phi+\Phi &=\Phi.
\Delta_f H l_{b_j}=H\Delta_f l_{b_j}+2\langle\nabla H, \nabla l_{b_j}\rangle+l_{b_j}\Delta_f H &=-H l_{b_j}-2 \langle AX^T,b\rangle+H l_{b_j}(1-\|A\|^2) &=-\|A\|^2H l_{b_j},
\begin{array}{c}p_\varepsilon=2c_0\frac{u_\varepsilon^2+z_\varepsilon^2}{u_\varepsilon^4+c_0^2}-2c_0+O(\varepsilon)e^{-\left(\sqrt{2}\sqrt{1-4c_0^2}+O(\varepsilon)\right)|x|}, q_\varepsilon=O(\varepsilon)e^{-\left(\sqrt{2}\sqrt{1-4c_0^2}+O(\varepsilon)\right)|x|}, \end{array}
\mu\otimes\mu\left([0,1]^2\setminus \Delta\right)=[a^2(\lambda\otimes\lambda)+a(1-a)(\lambda\otimes \varepsilon_{\frac{1}{2}}) &+a(1-a)(\varepsilon_{\frac{1}{2}}\otimes\lambda)+(1-a)^2 (\varepsilon_{\frac{1}{2}}\otimes \varepsilon_{\frac{1}{2}})]\left([0,1]^2\setminus \Delta\right) &=a^2+2a(1-a)=a(2-a).
A \equiv \frac{( - K^{\prime}( 0 ) ) ( -{\textstyle \int}p^{2}K^{\prime})}{( -{\textstyle \int}K^{\prime}) K ( 0 )}, \B \equiv \frac{K^{\prime \prime}( 0 ) K ( 0 )}{( - K^{\prime}( 0 ) )^{2}}.
\mathcal{Z}_{m,n}^{\gamma}(z,\overline{z})={(-1)^{m}(\gamma+m+1)_{n}}\left(1-|{z}|^{2}\right)^{-\gamma}\sum_{j=0}^{+\infty}\dfrac{(-\gamma-m)_{j}}{j!}\dfrac{\partial^{m}}{\partial z^{m}}(z^{j+n}){\overline{z}^{j}}.
w=[\begin{matrix}\sigma_{a}^{2}&-\sigma_{b}^{2}\end{matrix}],k=[\begin{matrix}1&1\end{matrix}],\lambda=\frac{\mu_{a}-\mu_{b}}{\sigma_{a}^{2}-\sigma_{b}^{2}}[\begin{matrix}\sigma_{a}^{2}&\sigma_{b}^{2}\end{matrix}].
\sum_{d<x}\frac{\phi(d)}d \sum_{\substack{{p\leq x}{p\equiv 1 (d)}}}\frac 1p = \int_2^x\frac1{u^2}\sum_{k\leq u}\frac{\phi(k)}k\pi(x;k,1)du +O(1) =\int_2^x \frac{1}{u^2}\left( u + C \frac{u}{\log u}+ O\left(\frac{u}{\log^2 u}\right)\right) du +O(1) =\log x + C \log\log x + O(1)
x_1 = \left(\begin{matrix}1 \alpha + \alpha^2 \alpha\end{matrix}\right), x_2 = \left(\begin{matrix}1 \sigma(\alpha) + \sigma(\alpha)^2 \sigma(\alpha)\end{matrix}\right), x_3 = \left(\begin{matrix}1 \sigma^2(\alpha) + \sigma^2(\alpha)^2 \sigma^2(\alpha)\end{matrix}\right),
\sum\limits_{j=0}^{2}\varepsilon^{\frac{j}{2}}\left\vert \lambda\right\vert^{1-\frac{j}{2}}\left\Vert u_{2}^{\left( j\right)}\right\Vert_{X}+\left\Vert Au_{2}\right\Vert_{X}\leq C\sum\limits_{k=1}^{2}\left[\left\Vert f_{k}\right\Vert_{E_{k}}+\left\vert \lambda \right\vert^{1-\theta_{k}}\left\Vert f_{k}\right\Vert_{E}+\right.
\int_{{[0,2\pi]}^m}\frac{dx_1dx_2\cdots dx_m}{\displaystyle\sum_{j=1}^{m}\left|\cos\left(\left(\frac{\alpha-\beta}{2}\right)x_j\right)-\cos \left(\left(n-\frac{\alpha+\beta}{2}\right)x_j\right)\right|^{2s}}&=2^{2m-2s}\int_{{[0,\pi/2]}^m}\frac{dx_1dx_2\cdots dx_m}{\displaystyle\sum_{j=1}^m\sin^{2s}((n-\alpha) x_j)\sin^{2s}((n-\beta) x_j)},
\|u-u_N\|_{H^s(\mathbb R)}\le c \big(\|\tilde e_N\|_{H^s(\mathbb R)}+ \|I_N^\lambda f-f\|_{L^2(\mathbb R)}\big) &\le cN^{s-m}|u|_{{\mathbb B}^m_{\lambda}(\mathbb R)}+ cN^{-k}|f|_{{\mathbb B}^{k}_{\lambda}(\mathbb R)}.
\,\varphi_{\underline{d},{\underline{d}}''}(K_{il})= \,V_{X_1}(-(d-i)A)+\,V_{X_2}(-iA-lB)+\,V_{X_3}(-(d-l)B) &-\,(ev^{il})-\,V_{X_1}(-(d-i+1)A).
- \int_U D^2v Dv \cdot \big[vDv \psi+\frac12 v^2 D\psi\big]\,dx =- \frac12 \int_U (D|Dv|^2\cdot Dv) v \psi\,dx-\frac14 \int_U (D|Dv|^2\cdot D\psi)v^2 \,dx,
DQ^{2}-EP^{2}=\frac{[(P_{x}m+P_{y})\alpha^{3}+(P_{x}-2P_{y}m)\alpha^{2}-P_{y}m][(P_{x}m-P_{y})\alpha^{3}+P_{x}\alpha^{2}-2P_{y}\alpha+P_{y}m]}{\alpha^{2}(1+\alpha^{2})(a-m)^{2}}
I_{A/E}= \frac{1}{2}\sum_{x=-1,1}\int_{-\infty}^{+\infty}p_{A/E}(\xi \mid X=x) &\times \log_{2}\frac{2p_{A/E}(\xi \mid X=x)}{p_{A/E}(\xi \mid X=-1)+p_{A/E}(\xi \mid X=1)}d\xi,\raisetag{36pt}
\int_{0}^{1}\cdot\cdot\cdot\int_{0}^{1}(\prod_{i=1}^{k}t_{i})\prod_{i=1}^{n}t_{i}^{\alpha-1}(1-t_{i})^{\beta-1}\prod_{1\le i<j\le n}|t_{i}-t_{j}|^{2\gamma}dt_{1}\cdot\cdot\cdot dt_{n}
(\widetilde{V}_1 \widetilde{V}_2 \widetilde{V}_c)^2 \widetilde{C}_3 \widetilde{C}\widetilde{B}_2 \widetilde{W}_1 \widetilde{W}_2 \widetilde{W}_c \widetilde{W_1^\prime}\widetilde{W_2^\prime}\widetilde{W_c^\prime}= \partial^2.
q_{1}= \theta_{1}\beta+\beta K + \theta_{1}^{1-\beta}|b(0)|^{\beta}+ \frac{C_{d,\alpha}\beta(3-\beta)\sqrt{d}\sigma_{d-1}}{2(2-\alpha)}+ \frac{C_{d,\alpha}\beta\sigma_{d-1}}{\alpha-\beta}+ \left(\frac{\theta_{1}}{4}\right)^{1-\beta}\left(\frac{C_{d,\alpha}\sigma_{d-1}}{\alpha-1}\right)^{\beta},
\left\lbrace\begin{aligned}\partial_tu^\epsilon(t,q,p) =\frac{1}{\epsilon}\Bigl(\nabla_qu^\epsilon(t,q,p)\cdot p+\nabla_pu^\epsilon(t,q,p)\cdot f(q)\Bigr) &+\frac{1}{\epsilon^2}\Bigl(-\nabla_pu^\epsilon(t,q,p)\cdot p+\nabla_{p}^2u^\epsilon(t,q,p):a(q)\Bigr), u^\epsilon(0,q,p) =\varphi(q),\end{aligned}\right.
\sum_{n=M_-}^{M_+-1}A_2(n)|C_n| \leq{\frac{c_1}{(\gamma t)^{(d+1)/2}}}\sum_{n=0}^{\lfloor t^{1/2+\varepsilon}\rfloor}n^{d-1}+ \frac{c_2}{M (\gamma t)^{d/2}}\sum_{n=\lfloor t^{1/2+\varepsilon}\rfloor+1}^{M_+}n^{d-1}\leq c_3 (\gamma t)^{-1/2+\varepsilon d}.
{\cal L}= \bar{\psi}\gamma^{\mu}( i \nabla_{\mu}+ A \omega_{\mu}+ a_{\mu}) \psi + \frac{g^{2}}{4}\bar{\psi}\gamma^{\mu}\psi \bar{\psi}\gamma_{\mu}\psi - \epsilon^{\mu \nu}\rho ( \partial_{\mu}a_{\nu}- \frac{B}{2}\epsilon_{a b}e_{\mu}^{a}e_{\nu}^{b}) ,
\int_{0^-}^{\infty}e^{-s x}dG_3(x)=\frac{\mathbb E\left[e^{-s \overline{X^1}_{e(q)}}\right]}{\mathbb E\left[e^{-s \overline{X^1}_{e(p+q)}}\right]}=e^{\int_{0}^{\infty}(e^{-s x}-1) \Pi_3(dx)}, \ \s\geq 0,
\limsup_{i \rightarrow \infty}\frac{\phi(z_i(T))-\phi(y_k(T))}{s_i}\geq \lim_{i \rightarrow \infty}\frac{V(t_k,z_i(t_k))-V(t_k,y_k(t_k))}{s_i}&=\lim_{i \rightarrow \infty}\frac{V(t_k,y_k(t_k)+s_i \theta)-V(t_k,y_k(t_k))}{s_i}=\langle \nabla_x V (t_k,x_k),\theta \rangle > 0.
\tilde{\pi}_1(a): =[\tilde{f}_{11}(a),\tilde{g}_{12}(1)]-[\tilde{f}_{11}(1),\tilde{g}_{12}(a)], \tilde{\pi}_2(a): =[\tilde{f}_{11}(a),\tilde{g}_{12}(e^{-1})]-[\tilde{f}_{11}(1),\tilde{g}_{12}(ae^{-1}),
\frac{1}{s_d^{2+\delta}}\sum_{i=1}^d \mathrm{E}[|Q_i-\mu_i|^{2+\delta}] \leq \frac{5^{2+\delta}}{s_d^{2+\delta}}\sum_{j=1}^5 C_j(\delta) \sum_{i=1}^d |T_{ji}|^{2+\delta}&\leq 5^{2+\delta}\sum_{j=1}^5 C_j(\delta) \frac{\sum_{i=1}^d |T_{ji}|^{2+\delta}}{\left( \sum_{i=1}^d T_{ji}^2 \right)^{1+\delta/2}}
\sum_{j = 1}^{\infty}j a_{ij}r^j = c_i r + \sum_{j= 1}^n c_{ij}\left( \sum_{q = 1}^{\infty}a_{jq}r^q \right) + \sum_{k_1 + \cdots + k_{n+1}\geq 2}c_{i k_1 \cdots k_{n+1}}\left(\sum_{j = 1}^{\infty}a_{k_1 j}r^j\right)^{k_1}\cdots \left(\sum_{j = 1}^{\infty}a_{k_n j}r^j\right)^{k_n}r^{k_{n+1}}
\left(\frac{33 r}{\sqrt{d}}\right)^d \gamma_1 \log^{d/2}\left(\frac{\gamma_2}{\gamma_1}\right) = \gamma'_d < \|f^{=d}\|_2 \leq \left(\frac{33\sqrt{d}}{d \log^{1/2}\left(\frac{\gamma_2}{\gamma_1}\right)}\right)^{d}\gamma_1 \log^{d}\left(\frac{\gamma_2}{\gamma_1}\right),