harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\frac{\sum_{|\beta| \leq N}(p_0)_{j,\beta}\prod_{(j',\alpha) \in S}(\partial^\alpha f_{j'}(x))^{\beta_{j',\alpha}}}{\sum_{|\beta| \leq N}(q_0)_{j,\beta}\prod_{(j',\alpha) \in S}(\partial^\alpha f_{j'}(x))^{\beta_{j',\alpha}}}= (y_0)_j, j = 1,\ldots,n,
p\Big(\Big(1+\frac{\Delta_h}{4\pi p}\Big)e^{-\frac{\Delta_h}{4\pi p}}-S_p\Big)f =p(1-\mu)f-p\Big(1-\Big(1+\frac{\Delta_h}{4\pi p}\Big)e^{-\frac{\Delta_h}{4\pi p}}\Big)f &=p(1-\mu)f-\frac{\Delta_h}{4\pi}\Psi(\Delta_h/4\pi p)\,f\;,
\sum_{I\in\mathcal D,\,I\subset J,\, I\cap E\not=\varnothing}|I|\ge \sum_{n\le s\le n+N}\sum_{I\in\mathcal D_s,\,I\subset J,\, I\cap E\not=\varnothing}|I| =\sum_{n\le s\le n+N}|J|=(N+1)|J|.
G^k_B\rho(x)-G^k_D\rho(x)=E^x\left[\int_{\tau_D}^{\tau_B}\rho(X_s)\,ds\right]=E^x\left[E^{X_{\tau_D}}\left[\int_0^{\tau_B}\rho(X_s)\,ds\right]\right]=H_DG^k_B\rho(x).
\mathbb{A}^{-1}_L=\mathbb{A}^{-1}_R=\mathbb{A}_\mathcal{G}^{-1}= \left(\begin{matrix}(a-bd^{-1}c)^{-1}-(db^{-1}a-c)^{-1}-(ac^{-1}d-b)^{-1}(d-ca^{-1}b)^{-1}\end{matrix}\right).
\mathcal{\overline{D}}_{fr}=\left\{\left(\omega_{1}^{l},\omega_{2}^{m}\right)\left|\omega_{1}=\frac{l}{r}C,\,\omega_{2}=\frac{m}{r}C;\,l,m\in\left\{0,1,\ldots,r\right\}\right.\right\}.
\lambda'_1 = \frac{1}{3}(4,5,6,3,4,2), \lambda'_2 = \frac{1}{3}(5,10,12,6,8,4), \lambda'_3 = (2,4,6,3,4,2) \lambda'_4 = (1,2,3,2,2,1), \lambda'_5 = \frac{1}{3}(4,8,12,6,10,5), \lambda'_6 = \frac{1}{3}(2,4,6,3,5,4).
\int_{x_0}^x \sin(2\pi \ell y)\frac{\sin (2\tilde{\theta}(y,{E})-2\tilde{\theta}(y,\hat{E}))}{(1+y-b)}dy= \int_{x_0}^x \frac{\cos(2\pi \ell y- (2\tilde{\theta}(y,{E})-2\tilde{\theta}(y,\hat{E})))}{2(1+y-b)}dy -\frac{\cos(2\pi \ell y+ (2\tilde{\theta}(y,{E})-2\tilde{\theta}(y,\hat{E})))}{2(1+y-b)}dy.
A^2_n:=\int_{\widehat{I}\cap (B_{2n+1}\setminus B_n)}\left(\widehat{f}'(\widehat{x})(\varphi_n(\widehat{x})-1)+\widehat{f}(\widehat{x})\varphi'_n(\widehat{x})\right)^2d\widehat{x}
{\rm OT}(\mu_0,\mu_1) = \inf_{\gamma \in \Gamma(\mu_0,\mu_1)}\Big\{\inf_{a> 0}\int_{X \times X}\frac{1}{a}c(x_0,x_1)d (a\gamma) \Big\}= \inf_{\gamma \in \Gamma(\mu_0,\mu_1)}\int_{X \times X}c(x_0,x_1)d\gamma,
\begin{gathered}A_n\circ A_m=A_{n+m},A_n\circ B_m=B_{n+m},B_n\circ B_m=A_{n+m}+4A_{n+m+1}, A_n\circ\varphi_m=\varphi_{n+m},A_n\circ\psi_m=\psi_{n+m},B_n\circ\varphi_m=\psi_{n+m},B_n\circ\psi_m=\varphi_{n+m}+4\varphi_{n+m},\end{gathered}
\lim\limits_{n\to \infty}n\P\left(X_{t_{2r+1}}^{\left(n\right)}\right. \left.>0,\,\ldots, X_{t_{[Tn]-1}}^{\left(n\right)}>0,\, X_{T}^{\left(n\right)}=j|X_{t_{2r}}^{\left(n\right)}=0\right) &=\sqrt{\frac{2}{\pi}}\frac{2pz}{(T-t)^{3/2}}e^{-\frac{z^2}{2(T-t)}}=4ph(T-t, z).
g_{2}= \frac{\sinh \xi}{4 \cosh^{5}\xi}- \frac{5 \sinh^{3}\xi}{4 \cosh^{7}\xi}+ C_{2}\left( \frac{\sinh^{2}\xi}{2 \cosh^{4}\xi}- \frac{3 \sinh^{4}\xi}{2 \cosh^{6}\xi}\right) + C_{3}\frac{\sinh^{3}\xi}{\cosh^{5}\xi}.
\left( \mathbf{G}_{k}\star\mathbf{f}\right) \left( x\right) =\int_{\mathbb{R}^{3}}g_{k}\left( \left\Vert x-y\right\Vert \right)\mathbf{f}\left( y\right) dy+\frac{1}{k^{2}}\nabla\int_{\mathbb{R}^{3}}g_{k}\left( \left\Vert \mathbf{x}-\mathbf{y}\right\Vert \right)\operatorname*{div}\mathbf{f}\left( \mathbf{y}\right) d\mathbf{y}.
\|b^{(\theta)}\|_{H^p}= \Big|\bigcup\mathcal{B}\Big|^{1/p}, \|b^{(\theta)}\|_{(H^p)^*}= \Big|\bigcup\mathcal{B}\Big|^{1/{p'}}\qquad \|b^{(\theta)}\|_{SL^\infty}= 1.
\mathbf{E}[(X^q)_{q\in J}\in A|X_0,Y_1,\ldots,Y_k, \{X_1^v,\ldots,X_k^v:|v|>m\}] = &\qquad\mathbf{E}[(X^q)_{q\in J}\in A|(X^q)_{q\in\partial J},(Y^{qr})_{q\in J,r\in J\cup\partial J,\|q-r\|=1}].
\phi^+_{i, \, n}= \begin{cases}(-1)^{n + 1}o_i^n \kappa_q \, q^{h_i}e'_{n \delta, \, \alpha_i}n > 0 q^{h_i}n = 0\end{cases}, \phi^-_{i, \, n}= \begin{cases}q^{-h_i}n = 0 (-1)^n o_i^n \kappa_q \, q^{-h_i}f'_{- n \delta, \, \alpha_i}n < 0\end{cases}.
S_3=\sum_{j=0}^t \binom{t}{t-g}\binom{-t-1}{j}\Big(\frac{k+1}{r}\Big)^j=\frac{1}{t!}\Big((1+x)^t\Big(1+\Big(\frac{k+1}{r}\Big) x\Big)^{-t-1}\Big)^{(t)}\big|_{x=0}.
\omega_{\xi}^{\ast}+ \frac{\alpha \xi^{\alpha}}{2(1 - \alpha)}= \frac{1 - \alpha \xi}{1 - \alpha}+ \frac{\frac{\alpha}{2}\xi^{\alpha}}{1 - \alpha}< \frac{1 - \alpha \xi + \frac{\alpha}{2}\xi}{1 - \alpha}= \frac{1 - \frac{\alpha}{2}\xi}{1 - \alpha}< 0,
\rho \ddot{u}_{i}= \big(A_{iJRs}u_{s, R}- \beta_{Ji}T - (C_{iJKSRl}u_{l, RS}+ M_{iJKR}\tau_{, R})_{, K}\big)_{, J}+ \rho f_{i}, a \ddot{\tau}=-\beta_{Ki}\dot{u}_{i, K}- M_{jLKI}u_{j, LKI}+ K_{IJ}\tau_{, IJ}+ \rho T_{0}^{-1}R
\left({}^{c}D_{-}^{\alpha,\beta,\gamma}f\right)(x)=(-1)^{[\operatorname{Re}(\alpha)]+1}\left(I_{-}^{-\alpha+[\operatorname{Re}(\alpha)]+1,-\beta-[\operatorname{Re}(\alpha)]-1,\alpha+\gamma}f^{\left([\operatorname{Re}(\alpha)]+1\right)}\right)(x).
\det T_{B}(C,D) = \det\begin{pmatrix}1 0 0 0 0 \alpha_1+h_1(\ell) 0 \beta_1 0 0 1 0 0 \beta_2 0 \alpha_2+h_2(\ell) \end{pmatrix}= (\alpha_1 + h_1(\ell))(\alpha_2 + h_2(\ell)) - \beta_1 \beta_2.
H_{1}= 1 + \sum_{i}{\frac{Q_{i}}{[ | \vec{x}- \vec{x}_{0 \, i}|^{2}+ 4 Q | \vec{z}- \vec{z}_{0}| ]^{2}}}, \ \ \H_{2}={\frac{Q}{| \vec{z}- \vec{z}_{0}|}}.
G(z) = \pi \frac{b-a}{2}\int_1^0 \sin(2\pi\theta) \frac{f\left ( \frac{b-a}{2}\cos (2\pi\theta) + \frac{b+a}{2}\right )}{z - \left ( \frac{b-a}{2}\cos (2\pi\theta) + \frac{b+a}{2}\right )}\mathrm{d}\theta.
E\left[e^{48c_{d,\sigma}k^2T\sup_{0\leq t\leq T}|\sigma_i\cdot B^i_t|^2}\right] = 1 + \sum_{p=1}^\infty \frac{(48c_{d,\sigma}\sigma_i^2k^2T)^p}{p!}E\left[\sup_{0\leq t\leq T}|B_t|^{2p}\right] \leq 1 + \sum_{p=1}^\infty \frac{(48k^2dT)^p}{p!}\left(\frac{2p}{2p-1}\right)^2\frac{(2p)!}{2^p\cdot p!}T^p.
\dot{f}( x^{2}) \equiv \frac{d}{d x^{2}}f ( x^{2}) = \frac{1}{2}\int_{0}^{1}d \alpha \Big [ d_{1}( \alpha^{2}x^{2}) + \alpha^{2}x^{2}d_{2}( \alpha^{2}x^{2}) \Big ] .
\Lambda_i = \frac{1}{\rho + \frac{1}{N}\sum_{l=1}^K \frac{r_{iil}^{-\alpha}}{1 + \Lambda_i r_{iil}^{-\alpha}}+ \frac{1}{N}\sum_{l=1}^{K'}\frac{r_{i\bar{i}l}^{-\alpha}}{1 + \Lambda_i r_{i\bar{i}l}^{-\alpha}}}.
\frac{\hat{N}_2(L_2,S_2,U_2)}{D_2(L_2,S_2,U_2)}-\frac{N_0(L,S,U)}{D_0(L,S,U)}&=\frac{\hat{N}_2(L_2,S_2,U_2)D_0(L,S,U)-N_0(L,S,U)D_2(L_2,S_2,U_2)}{D_2(L_2,S_2,U_2)D_0(L,S,U)}
G_6+G_7 :=\frac{q(q-2)}{2}E\int_{t_0}^{u}|e_s^n|^{q-4}|\bar \sigma_s^{n*}e_s^n|^2ds +\frac{q}{2}E\int_{t_0}^{u}|e_s^n|^{q-2}|\bar \sigma_s^n|^2 ds & \leq K E\int_{t_0}^{u}|e_s^n|^{q-2}|\bar \sigma_s^n|^2 ds \leq K \int_{t_0}^{u}E\sup_{t_0 \leq r \leq s}|e_r^n|^{q}ds+ K E\int_{t_0}^{u}|\bar \sigma_s^n|^q ds
y^d=\nu(2(t-1)^3x(x-1)-6t^2(t-1)^2-4t(t-1)^3+12t(t-1)^2x(x-1)+3t^2(2t-20x(x-1)+ &4t(t-1)^3+6t^2(t-1)^2)+t^2(1-t)^3x(x-1).
\epsilon = 0 \hspace{0 . 7 c m}, \hspace{0 . 7 c m}x = \sqrt{\frac{7 - p}{8 ( 8 - p )}}\hspace{0 , 7 c m}, \hspace{0 . 7 c m}{\lambda}= \frac{7 - p}{1 6 x}\hspace{0 . 7 c m}, \hspace{0 . 7 c m}\nu = \frac{p - 3}{8 \sqrt{2}x}\quad .
I_+(r) =\int_0^\pi\frac{\sin(\theta_1)^{m-2}}{(r^2 + 1 - 2 r \cos(\theta_1) + \alpha^2 (r-1)^2)^{\frac{N+s}2}}d\theta_1 I_-(r) =\frac{\sin(\theta_1)^{m-2}}{(r^2 + 1 - 2 r \cos(\theta_1) + \alpha^2 (r+1)^2)^{\frac{N+s}2}}d\theta_1 ,
\Phi_t^{(1)}(\vec p ) = \, \frac{1}{2}\, e^{it \Delta E ( \vec p )}\delta(p_1 + p_2 - p_3 - p_4 ) \big( \hat V (p_1 - p_4 ) - \hat V (p_1 - p_3 ) \big) \big( \chi_{1234}+ \chi^\perp_{1234}\big)
|0-\frac{1}{3}|=\frac{1}{3}>\frac{1}{6}=|\frac{1}{3}-\frac{1}{2}|,d_{sup}(f(0),f(\frac{1}{3}))=\frac{4}{3}<\frac{3}{2}=d_{sup}(f(\frac{1}{3}),f(\frac{1}{2}))
\int_{t_{1}}^{t_{2}}\int_{\Gamma_{t}^{\lambda}}\Big|\kappa+\frac{\langle\gamma,N\rangle}{2(t_{0}-t)}\Big|^{2}\rho dsdt =-\Big[\int_{\Gamma_{t}^{\lambda}}\rho_{X_{0}}ds\Big]^{T-t_{2}/\lambda^{2}}_{T-t_{1}/\lambda^{2}}\rightarrow 0
\sum_i \frac{\left\Vert \sum_j\mu_j\langle \phi_i,\psi_j\rangle\psi_j\right\Vert^2}{(\zeta_i + \lambda)^{2}}=\sum_i\frac{\mu^2_i}{(\zeta_i+\lambda)^2}\sum_j\frac{\mu^2_j}{\mu^2_i}\langle \phi_i,\psi_j\rangle^2 \le \sum_i\frac{\mu^2_i}{(\zeta_i+\lambda)^2}\sup_i \frac{1}{\mu^2_i}\sum_j\mu^2_j\langle \phi_i,\psi_j\rangle^2.
B(v_1,v_2) + B(v_2,v_1) = \frac{k}{\ell}(-\alpha \sin \ell x_1 + \beta \cos \ell x_1) \cos k x_2 - \frac{\ell}{k}\cos k x_2 (-\alpha \sin \ell x_1 + \beta \cos \ell x_1) & = \left(\frac{k}{\ell}- \frac{\ell}{k}\right)(-\alpha \sin \ell x_1 + \beta \cos \ell x_1) \cos k x_2.
\sum_{k=1}^{r}\sum_{i=1}^d (\lambda_i - \widetilde \lambda_k)^2 \langle p_i, q_k\rangle^2 \leq 2 \sum_{k=1}^{r}\sum_{i=1}^d (\lambda_i - \widehat \lambda_k)^2 \langle p_i, q_k\rangle^2 +2 \sum_{k=1}^{r}\sum_{i=1}^d B(\widehat \lambda_k)^2 \langle p_i, q_k\rangle^2 ,
M = |z| = |q^{(e)}_\Lambda X^\Lambda -{i \over 2}q_{(m)}^\Lambda F_\Lambda|=e^{K/2}|q^{(e)}_0 + q^{(e)}_i Z^i +{i \over 2}(q_{(m)}^0 Z^i -q_{(m)}^i)f_i - iq^0_{(m)}f|\ ,
\gamma_{e}(e^*) + \beta_{e}(e) = 0, &\beta_{e^i}(e^*) + \beta_{e^{i+2}}(e) = 0, i \ge 1 &\alpha_v(e^*) + \beta_{e^2}(e) = 0, &\gamma_{e^{i+2}}(e^*) + \gamma_{e^i}(e) = 0, i\ge 1 &\alpha_v(e)+\gamma_{e^2}(e^*) =0.
\frac{d}{dt}u_t(x) \geq - m u_t(x) - \langle a^{-}\rangle \psi^{-}_t u_t (x) begin{align*}.2cm] = - (m + \langle a^{-}\rangle q ) u_t (x) + \langle a^{-}\rangle (q - \psi^{-}_t)u_t (x) \geq - \langle a^{+}\rangle u_t (x),
K^{(1)}_{\bf a}(x,x',t)={{\bf a}\over 4}e^{{{\bf a}^2\over 4}t}e^{-{{\bf a}\over 2}(|x|+|x'|)}\left( 1-\Phi({|x|+|x'|\over 2\sqrt{t}}-{{\bf a}\over 2\sqrt{t}})\right)~~,
m(B_{t+\delta}(x_0)\setminus B_{t-\delta}(x_0))=m(B_{t+\delta}(x_0))-m(B_{t-\delta}(x_0)) \leq m(B_{t+\delta}(x_0))-\frac{F(t-\delta)}{F(t+\delta)}m(B_{t+\delta}(x_0)) =2\delta\cdot\frac{F'(t-\delta)}{F(t+\delta)}m(B_{t+\delta}(x_0))+o(\delta).
\,K\!\;\!(\alpha,m)\,=\;\mbox{${\displaystyle \min_{\delta \,>\,0}}$}\:\big\{\;\!\delta^{\mbox{}^{\scriptstyle \:\!-\,1/2}}\,\!\prod_{\:\!j\,=\,0}^{\;\!m}\:\!\bigl(\:\! \alpha + j/2 + \delta \;\!\bigr)^{\!\:\!1/2}\:\!\bigr\}.
\psi^{(+)}\stackrel{r \sim \infty}{\rightarrow}{{\Gamma(c)\Gamma(b-a)}\over{\Gamma(b)\Gamma(c-a)}}e^{(-Qr)}+{{\Gamma(c)\Gamma(a-b)}\over{\Gamma(a)\Gamma(c-b)}}e^{(+Qr)}
f(t)f(t+3k)-f(t+k)f(t+2k) &=\left(ar^t+\frac{b(-1)^{t+1}}{r^t}\right)\left(ar^{t+3k}+\frac{b(-1)^{t+k+1}}{r^{t+3k}}\right) &-\left(ar^{t+k}+\dfrac{b(-1)^{t+k+1}}{r^{t+k}}\right)\left(ar^{t+2k}+\frac{b(-1)^{t+1}}{r{t+2k}}\right) &=ab(-1)^{t+1}\left[\frac{(-1)^{k}}{r^{3k}}+r^{3k}-\frac{1}{r^k}-(-1)^{k}\right].
\lim_{N\rightarrow\infty}{\frac{P_{\lambda(N)}\big(x, t^{-1}, \dots, t^{1 - N}\big)}{P_{\lambda(N)}\big(1, t^{-1}, t^{-2}, \dots, t^{1 - N}\big)}}= \Phi^{\nu}(x; q, t),
\Omega=\{(\mathbf{x},y): \mathbf{x}\in\mathbb{R}^{n-1}, -\infty<y<\eta(\mathbf{x})\}\mbox{and}\Gamma=\{(\mathbf{x}, \eta(\mathbf{x})): \mathbf{x}\in\mathbb{R}^{n-1}\}
\gamma := \max \left(P, \left\lfloor \frac{n}{2}\right\rfloor + \iota\right)+1, \underline \kappa := m+1\bar \kappa:= \underline \kappa + 2 \gamma, \underline \iota := n+2 \iota+1\bar \iota:= \underline \iota + 2 \gamma.
W^2_2(N(\omega_1,\Sigma_1), N(\omega_2,\Sigma_2))= |\omega_1-\omega_2|^2+\mathrm{tr}\Sigma_1+\mathrm{tr}\Sigma_2- 2\mathrm{tr}\Big(\Sigma_1^{1/2}\Sigma_2\Sigma_1^{1/2}\Big)^{1/2},
\begin{aligned}|X_{t,n}- x_{t,n}| \le 2K\int_0^t |X_{s,n-1}- x_{s,n-1}|ds + \int_0^t|x_{s,n-1}||x_{0,1}- x_{s,1}|ds + |a_n|\int_0^t |x_{s,n}|ds + \int_0^t |x_{s,n-2}||x_{s,n-1}|ds.\end{aligned}
\mathbb{E}(P_{n}) = \frac{2p}{n-1}\sum_{k=1}^{n-1}\mathbb{E}(P_{k}) + \frac{1-p}{n-1}\sum_{k=1}^{n-1}\mathbb{E}(P_{k}) \mathbb{E}(P_{n-k}), n \ge 2, \mathbb{E}(P_{1})=1.
B^{ai}_b ={(\frac{\partial^2 A}{\partial \phi^i \partial \phi^j})}^{-1}(\frac{1}{4}\lambda^{a}_{bj}- \frac{1}{2}\frac{\partial^3 A}{\partial \phi^j \partial \phi^k \partial \phi^p}(\bar \psi^k_a \psi^{bp}+ \bar \psi^{bk}\psi_{a}^p)) ,
\delta G^{A B}\, \biggl ( \partial_{A}\partial_{B}\varphi - \overline{{{\Gamma}}}_{A B}^{C}\, \partial_{C}\varphi \biggr ) + \overline{{{G}}}^{A B}\, \biggl ( \partial_{A}\partial_{B}\chi - \overline{{{\Gamma}}}_{A B}^{C}\partial_{C}\chi - \delta \Gamma_{A B}^{C}\partial \varphi \biggr ) + \frac{\partial^{2}V}{\partial \varphi^{2}}\chi = 0 ,
\tilde{I}_4 \le \frac{c}{(\prod_{l = 1}^d h_l^*)^2}\frac{1}{T_n^2}\int_0^{T_n}\int_{D}^{T_n}e^{- \rho s'}e^{\rho \Delta_n}dt ds' & \le \frac{c}{T_n(\prod_{l = 1}^d h_l^*)^2}e^{- \rho D}
\gamma_{i}( \beta \sqrt{- \triangle}) = \int_{0}^{\infty}\! \!{\mathrm d}t g_{i}( t ) \left[ \frac{2 \pi}{\beta{\sqrt{- \triangle}}t}\frac{1}{{\mathrm s h}( 2 \pi t / ( \beta{\sqrt{- \triangle}}) )}- \frac{1}{t^{2}}\right] ,
-4 \int_{t}^{\infty}\frac{\lambda_{0,0}''(x) dx}{(\lambda_{0,0}(t)^{1-\alpha}+x-t)(1+x-t)^{3}}= -4 \int_{t}^{\infty}\frac{b}{x^{2}\log^{b+1}(x)}\frac{dx}{(\lambda_{0,0}(t)^{1-\alpha}+x-t)(1+x-t)^{3}}+ E_{v_{3,ip}}
c_{\Gamma''}[S]\in \Lambda,c_\Gamma[S] = S\, \frac{p_{m(e)}[S]}{p_{m(v)}[S]}p_g\left[\frac{c_{\Gamma''}[S]}{S}\right]=\frac{S\, p_{m(e)}[S]}{p_{m(v)}[S] p_g[S]}p_g[c_{\Gamma''}[S]].
\mathcal{I}_{h}=\begin{cases}\h \mathcal{I}_{1}h>0, \ -h \mathcal{I}_{-1}h<0, \ 0 h=0,\end{cases}S_{h}=\begin{cases}\h^{1/2}S_{1}h>0, \ (-h)^{1/2}S_{-1}h<0, \ \emptyset h=0.\end{cases}
d s^{2}= \left( \prod_{i = 1}^{i = n}H_{i}^{- 2 r_{i}}\right) d t^{2}- \left( \prod_{i = 1}^{i = n}H_{i}^{- 2 r_{i}}\right)^{- \frac{1}{d - 3}}d \vec{x}^{\ 2},
\mathrm{Sub}(D)=\sum_{k=n}\star+\sum_{k\neq n, j=n}\star+\sum_{k\neq n, j\neq n, k=j}\star+\sum_{k\neq n, j\neq n, k\neq j}\star\triangleq\mathrm{I}+\mathrm{II}+\mathrm{III}+\mathrm{IV},
\int_{\mathbb{R}_{\geqslant0}}x^{\kappa}m_{\alpha,\beta}(\mathrm{d}x)\leqslant\left(\int_{\mathbb{R}_{\geqslant0}}xm_{\alpha,\beta}(\mathrm{d}x)\right)^{\kappa}=\left(\frac{\alpha}{\beta}\right)^{\kappa},
\left({\cal X}{\bar{H}}- E_{i}\right) | i \rangle = \left[ \Xi^{\dagger}{\cal A}^{\dagger}H_{\mathrm{\scriptsize I}}+ \left({\cal X}{\bar{H}}- E_{i}\right) \left( 1 - \Xi^{\dagger}{\cal A}^{\dagger}\right) \right] | i \rangle .
\begin{aligned}Y = f\left({\textbf{X}}\right) \overset{(a)}{=}{\theta^{- 1}}\left({\sum\limits_{i = 1}^p{{\phi_i}({X_i})}}\right)= \psi \left({\varphi \left({{\textbf{X}}}\right)}\right) \end{aligned}
\varphi^*\left(\begin{array}{c c}a b 0 d \end{array}\right) =\left(\begin{array}{c c c}a^{2 \, p^{e_1}}a^{p^{e_1}}\, b^{p^{e_1}}\frac{1}{2}\, b^{2 \, p^{e_1}}0 1 b^{p^{e_1}}\, d^{p^{e_1}}0 0 d^{2 \, p^{e_1}}\end{array}\right) (\, e_1 \geq 0 \, ) .
S_{a,b,r}^{\alpha_{s},\beta_{s},r_{s},p_{s}}\left( p,k\right) =\frac{\left( r!\right)^{p}\prod_{s=2}^{l}\left( r_{s}!\right)^{p_{s}}}{k!}\sum_{j=0}^{k}\left( -1\right)^{j}\binom{k}{j}\binom{\left( k-j\right) a+b}{r}^{p}\prod_{s=2}^{L}\binom{\left( k-j\right) \alpha_{s}+\beta_{s}}{r_{s}}^{p_{s}}.
|x_{i+1}(t)-x_{i}(t)| \leq \int_a^t|[k(t,s,x_i(s))-k(t,s,x_{i-1}(s))]|\Delta s &\leq L\int_a^t |x_i(s)-x_{i-1}(s)|\Delta s &\leq L\int_a^tL^i M\frac{(s-a)^i}{i!}\Delta s &\leq M\frac{L^{i+1}}{i!}\int_a^t(s-a)^i\ds &= M\frac{L^{i+1}(t-a)^{i+1}}{(i+1)!}\cdot
J^{( N )}( N \! - \! 1 ; 1 , \ldots , 1 ) = \frac{2 \mathrm{i}^{1 - 2 N}\pi^{( N - 1 ) / 2}}{\prod m_{l}}\; \Gamma \left( \frac{N \! + \! 1}{2}\right) \int_{0}^{\infty}\ldots \int_{0}^{\infty}\prod \mathrm{d}\alpha_{l}\left( \sum_{i = 1}^{N}\frac{\alpha_{i}}{m_{i}}\right) \delta \left( \alpha^{T}\| c \| \alpha \! - \! 1 \right) .
\frac{d}{d t}\langle F ( x ) \rangle_{t}\sim \frac{1}{Z_{t}}\int_{D_{t}}d x \frac{\partial}{\partial x}e^{S ( x )}F^{\prime}( x ) = \left. \frac{1}{Z_{t}}e^{S ( x )}F^{\prime}( x ) \right|_{x_{-}( t )}^{x_{+}( t )}.
v_{0}=\begin{bmatrix}1 0\end{bmatrix}\wedge\begin{bmatrix}0 1\end{bmatrix}\wedge\begin{bmatrix}z 0\end{bmatrix}\wedge\begin{bmatrix}0 z\end{bmatrix}\wedge\begin{bmatrix}z^{2}0\end{bmatrix}\wedge\begin{bmatrix}0 z^{2}\end{bmatrix}\wedge\cdots,
\begin{aligned}0 \leq \langle x_1-\bar{x}_1,(z_1-x_1)-(\bar{z}_1-\bar{x}_1)\rangle = \langle x_2-\bar{x}_1,(z_1-x_1)-(\bar{z}_1-\bar{x}_1)\rangle + \langle x_1-x_2,(z_1-x_1)-(\bar{z}_1-\bar{x}_1)\rangle.\end{aligned}
\left[ - \frac{4 \kappa \xi}{\left( 1 + \kappa \xi \Phi^{2}\right)}\left( \frac{1}{\kappa}\Lambda - \frac{1}{2}\mu^{2}\Phi^{2}+ \frac{1}{4}\lambda \Phi^{4}\right) - \mu^{2}+ \lambda \Phi^{2}\right] \Phi = 0 .
R_{\alpha_{0}, j}^{i}R_{\beta_{0}}^{j}- ( - 1 )^{\epsilon_{\alpha_{0}}\epsilon_{\beta_{0}}}R_{\beta_{0}, j}^{i}R_{\alpha_{0}}^{j}= - R_{\gamma_{0}}^{i}F_{\alpha_{0}\beta_{0}}^{\gamma_{0}}- M_{\alpha_{0}\beta_{0}}^{i j}S_{\mathrm{c l}, j},
\int_{S^2}W_0 \tilde{\nabla}\tilde{X}^i \cdot \tilde{\nabla}P_j dS^2 =- \int_{S^2}\tilde{\nabla}(\rho \tilde{\nabla}\tilde{X}^i ) P_j dS^2 = 2\int_{S^2}W_0 \tilde{X}^i P_j dS^2- \int_{S^2}P_j \tilde{\nabla}\tilde{X}^i \cdot \tilde{\nabla}W_0 dS^2.
\left[ e^{i \gamma \varphi}\right]_{n}( \underline{{\theta}}( \lambda ) ) = \left[ e^{i \gamma \varphi}\right]_{m}( \underline{{\theta}}^{\prime}) \left[ e^{i \gamma \varphi}\right]_{n - m}( \underline{{\theta}}^{\prime \prime}) + O ( e^{- \lambda}) .
\hat{\sigma}_{ABM|Z=\mathbf{z}}&= \hat{D}_A\left(\frac{\lambda\mathbf{z}}{\sqrt{\eta}}\right)\hat{D}_B\left(\frac{(1-\lambda)T\mathbf{z}}{\sqrt{\eta-1}}\right)\hat{\rho}_{ABM}{\hat{D}_A\left(\frac{\lambda\mathbf{z}}{\sqrt{\eta}}\right)}^\dag{\hat{D}_B\left(\frac{(1-\lambda)T\mathbf{z}}{\sqrt{\eta-1}}\right)}^\dag
I_1(B) = - \frac{eB}{{\pi}^{\frac{3}{2}}\beta}\sum_{k=1}^{\infty}\frac{(-1)^{k+1}}{k}\int_0^{\infty}dx \;e^{- \frac{x^2}{4}- \frac{1}{x^2}\beta^2 k^2 m^2}\frac{1}{e^{\frac{2 \beta^2 eB k^2}{x^2}}-1}\; .
W[J]=-iln(\langle0|0\rangle_{J})=\frac{1}{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})].
\widetilde{\mathbb{B}}(\bar t)=\frac{T_{N-1,N}(\bar t^{N-1})T_{N-2,N-1}(\bar t^{N-2}) \cdots T_{23}(\bar t^2) T_{12}(\bar t^1) |0\rangle}{\prod_{i=1}^{N-1}\lambda_{i+1}(\bar t^{i})\prod_{i=1}^{N-2}f(\bar t^{i+1},\bar t^i)}.
\frac{1}{r}\Big(\widetilde{\Omega}_{r+t}[u_0](x)+\widetilde{\Omega}_{r-t}[u_0](x)\Big) = \frac{1}{r}\Big(\Omega_{r+t}[u_0](x)-\Omega_{t-r}[u_0](x)\Big) \xrightarrow [r\to 0]{}2\frac{\partial}{\partial t}\, \Omega_{t}[u_0](x).
I ( s ) \equiv \int_{1}^{\infty}{\frac{( t^{2}- 1 )^{- s}d t}{\mathrm{e}^{2 \pi a t}- 1}}= \int_{1}^{t_{0}}{\frac{( t^{2}- 1 )^{- s}d t}{\mathrm{e}^{2 \pi a t}- 1}}+ \int_{t_{0}}^{\infty}{\frac{( t^{2}- 1 )^{- s}d t}{\mathrm{e}^{2 \pi a t}- 1}}\, ,
-i\hbar\frac{\partial\delta\psi^{*}}{\partial t}=-\frac{\hbar^{2}}{2m}\nabla^{2}\delta\psi^{*}+V\delta\psi^{*}+g(2|\psi_{0}|^{2}\delta\psi^{*}+(\psi_{0}^{*})^{2}\delta\psi).
\begin{aligned}\Omega_n\big(t_\lambda^\Lambda{t_\mu^\Lambda}^*\big) =t_{\lambda(0,e_i)}^\Lambda\otimes_{\mathcal{T}C^*(\Lambda^i)}\Omega_{n-1}\big(t_{\lambda(e_i,d(\lambda))}^\Lambda{t_\mu^\Lambda}^*\big) &\end{aligned}
={\frac{1}{( d - 2 ) !}}\sum_{k = 0}^{\infty}C_{k}( z ) \sigma^{k}\sum_{n = 0}^{\infty}\left( n +{\frac{d + 1}{2}}\right)^{- 2 z - 2 k}\left[ d^{2}+ d ( n - 3 ) + 4 - n \right] \sum_{p = 0}^{d - 3}n^{p}w_{p}^{\prime},
\begin{aligned}\|x^{k+1}- q\| = \|\beta_k v^k + (1 - \beta_k) Tu^k - q \| &\leq \beta_k \|v^k - q \| + (1 - \beta_k) \|Tu^k - q \| &\leq \beta_k \|v^k - q \| + (1 - \beta_k) \|u^k - q \|.\end{aligned}
E(u_0)< (1-\mu)\mathcal{E}(\phi)M(\phi)^{\sigma_c}M(u_0)^{-\sigma_c}=(1-\mu)\frac{N(p-2)-4s_1}{2N(p-2)}x_1^2 &<(1-\mu)\frac{N(p-2)-4s_1}{2N(p-2)}\|(-\Delta)^{\frac{s_1}{2}}u(t)\|_2^2,
T=\frac{(\sum_{i}p_{i})^{2}}{2M_{N}}+\frac{\sum_{i}p_{i}^{2}}{2m_{e}}=\frac{\sum_{i}p_{i}^{2}}{2M_{N}}+\frac{1}{M_{N}}\sum_{i>j}p_{i}\cdot p_{j}+\frac{\sum_{i}p_{i}^{2}}{2m_{e}}
P_{\hat{\underline S}| \underline X^L Y^L}({\underline s}| \underline x^L, y^L) = P_{{\underline S}| \underline X^L Y^L}({\underline s}| \underline x^L, y^L) & = \frac{P_{{\underline S}, \underline X^L Y^L}({\underline s}, \underline x^L, y^L)}{P_{\underline X^L Y^L}(\underline x^L, y^L)}.
(-1)^{k}(\begin{matrix}z\\k\end{matrix})=(\begin{matrix}-z+k-1\\k\end{matrix})=\frac{1}{\Gamma(-z)}\frac{1}{(k+1)^{z+1}}\prod_{j=k+1}\frac{(1+\frac{1}{j})^{-z-1}}{1-\frac{z+1}{j}}
\delta{\cal L}= \frac{- 1}{1 6{\pi}^{2}}\sum_{\alpha}\int d^{2}\theta \frac{W^{\alpha}W_{\alpha}}{4}\left( [ c ( G_{\alpha}) - \sum_{r_{\omega}}T ( r_{\omega}) ] g ( \phi ) + 2 \sum_{r_{\omega}}T ( r_{\omega}) \log d e t h_{\alpha \beta}( \phi ,{\bar{\phi}}) \right) + h . c
\mu(\{x \in P: \, \Phi_P(x) > 1\}) \le \int_P \Phi_P^s\,d\mu = C_2^{-s}\Big\| \sup_{\epsilon > 0}\Big| \mathop{\mathop{\sum_{Q \in \mathcal{D}:\, Q \subset P}}_{Q^a = F}}_{\ell(Q) > \epsilon}\epsilon_Q D_Q f \Big|\, \Big\|_{L^s(\mu)}^s &\le C_2^{-s}C_1\mu(P) \le \mu(P)/2,
\beta_{-1}\cdot \alpha_{-1}= (a_1 \cdot v_{(2,3)}) \cdot (a_1 \cdot v_{(2,3)}) - \frac{(11t -1)t}{2^4}a_1 - \frac{(2t -1)t}{2^4}a_{-1}+ \frac{t}{2^4}v_{(2,3)}+ \frac{3t + 2}{2^4}a_1 \cdot v_{(2,3)}.
\{Y_l^m(\theta,\varphi)\}_{l,m}:=\{\sqrt{\frac{2l+1}{4\pi}\frac{(l-|m|)!}{(l+|m|)!}}P_l^{|m|}(\cos\theta)e^{im\varphi}\}_{l,m},\,m=-l,\ldots,l;\,l=0,1,2,\ldots,
\prod_{x\in B_i}\bigl(1-f(x,y)\bigr)^{\eta_x}\leq\prod_{x\in B_i}\left(1-\frac{c_1}{(y_1-x_1)^{\frac{d-1}{2}}}\right)^{\eta_x}&\leq\left(1-c_1\alpha^{-\frac{2(k+i+1)(d-1)}{2}}\right)^{\alpha^{(d+1)(k+i-1)}}&=\left(1-c_1\alpha^{-(k+i+1)(d-1)}\right)^{\alpha^{(d+1)(k+i-1)}}\,.
I_i \cap \Big(I_0 +\sum_{\substack{j=1 j \neq i}}^k I_j\Big) =(f_iRf_i+f_iRf_0+f_0Rf_i+f_0Rf_iRf_0) \cap & \cap(f_i'Rf_i'+f_i'Rf_0+f_0Rf_i'+f_0Rf_i'Rf_0+I_0),
\tilde{\Gamma}_{I J}^{i}= \Gamma_{I J}^{i}- ( T^{i}+ \bar{T^{i}}) G_{I J}-{\frac{2}{3}}( T^{i}+ \bar{T^{i}}) ( T^{k}+ \bar{T^{k}}) K_{T k j}\Gamma_{I J}^{j}
\begin{aligned}\frac{1}{2}\|A(u+v(u))-y\|_2^2 = \frac{1}{2}\langle A(u+v(u))-y,Au-y\rangle \qquad\qquad{}+ \frac{1}{2}\langle A^*(A(u+v(u))-y),v(u)\rangle = \frac{1}{2}\langle A(u+v(u))-y,Au-y\rangle - \frac{\beta}{2}\|v(u)\|_2^2. \end{aligned}
\sum_{j=1}^3\sum_k\int_{\mathbb{R}}\big(\mathbf{1}_{|k|\geq D T_0^{21/20}}+\mathbf{1}_{|\xi|\geq DT_0^{21/20}}\big)e^{(|k|+|\eta|)/4}\big|\widehat{g_j'}(t,k,\xi)\big|^2\,\mathrm{d}\xi\lesssim e^{-DT_0^{21/20}/4}(Z(t))^2.
\left\{\begin{pmatrix}\alpha 0 0 \beta\end{pmatrix}\, \vert \, \alpha, \beta \in \mathbb C^{\times}\right\}\bigcup \left\{\begin{pmatrix}0 \gamma \delta 0\end{pmatrix}\, \vert \, \gamma, \delta \in \mathbb C^{\times}\right\}.
[ \bar{\epsilon}Q \ , \bar{Q}\epsilon^{\prime}]^{*}= \bar{\epsilon}( /{\hspace{- 0 . 2 7 c m}{\bf P}}+ \Gamma^{1 1}Z ){\frac{/{\hspace{- 0 . 2 7 c m}{\bf P}}}{2{\bf P}^{2}}}( /{\hspace{- 0 . 2 7 c m}{\bf P}}+ \Gamma^{1 1}Z ) \epsilon^{\prime}=
f \left( T_{a}( z ) , \zeta \right) = T_{a}^{\prime}( z ) \left[ f ( z , \zeta ) + \sum_{b}\frac{1}{2}( \zeta - k ) ( a_{b}z + b_{b})^{2}\ln \left( \frac{a_{b}( d_{b}a_{a}- c_{b}b_{a})}{c_{b}( b_{b}a_{a}- a_{b}b_{a})}\right) \right] \ .
\frac{\partial}{\partial \lambda}\mathrm{B}_{z^{2}}\left( \lambda,0\right) =2\ln z\,\mathrm{B}_{z^{2}}\left( \lambda ,0\right) -\frac{z^{2\lambda}}{\lambda^{2}}\,_{3}F_{2}\left( \left.\begin{array}{c}1,\lambda ,\lambda \lambda +1,\lambda +1\end{array}\right\vert z^{2}\right) .
\int_{M\setminus\Sigma}|\nabla(\mathfrak{u}_n-\mathfrak{u})|^2 =\int_{M\setminus\Sigma}|(\nabla\chi_n)\mathfrak{u}+\chi_n\nabla\mathfrak{u}|^2 &\leq 2\int_{M\setminus\Sigma}|(\nabla\chi_n)\mathfrak{u}|^2+|\chi_n\nabla\mathfrak{u}|^2.
\sum_{\substack{m,n<x^2 m\neq n}}\frac{1}{|m-n|}= 2\sum_{m<x^2}\sum_{n<m}\frac{1}{m-n}< 2\sum_{m<x^2}\sum_{k<x^2}\frac{1}{k}< 4x^2\log x + 2\gamma x^2 + 1,
K_{\eta_{n}}(x,dy)=P(\overline{X}_{n+1}\in dy|\overline{X}_{n}=x)=\frac{1}{\sqrt{2\pi}\sigma}exp\{-\frac{1}{2\sigma^{2}}(y-[b(x)\int_{R}a(z)\eta_{n}(dz)+c(x)])^{2}\}dy
m(r'+ ap) = m(r') + \sum_{1\leq i < u}\frac{(a p)^i}{i!}\frac{d^i m}{(dx)^i}(r') \pmod{p^u}= \sum_{1\leq i < u}\frac{(a p)^i}{i!}\frac{d^i m}{(dx)^i}(r')\pmod{p^u}= a p \sum_{1\leq i < u}\frac{(a p)^{i-1}}{i!}\frac{d^i m}{(dx)^i}(r')\pmod{p^u}
I_3(p) \sim \frac{1}{(\sigma \omega)^{(3-D)/2}}\Gamma\left(\frac{3-D}{2}\right)\;F\left( \frac{3-D}{4},\; \frac{5-D}{4}; \; \frac{D}{2}; \frac{p^2}{\sigma^2}\right) \; ,
V(u)=E\left \{1-\frac{1}{2}\left(\frac{u}{u_*}\right)^{7-p}\left [1-\frac{1}{\left(1+(u_*/u)^{7-p}\right)^2}\right] \right\}+\frac{Eu_{**}^2}{2u^2 \left (1+(u_*/u)^{7-p}\right)^2},
\hat{J}^{(r)}_a(z)\hat{J}^{(s)}_b(w)=\frac{\hat{G}_{ab}\delta_{r+s,0\hspace{.05in}mod\hspace{.04in}\lambda}}{(z-w)^2}+\frac{if_{ab}^{\ \c}\hat{J}_c^{(r+s)}(w)}{(z-w)}+ O((z-w)^0)
f=\sum_i\frac{h_i(\ell_{i(n_i+1)},\dots,\ell_{ip})}{L_{i1}^{s_{i1}}\dots L_{in_i}^{s_{in_i}}}+\phi_i(L_{i1},\dots,L_{in_i},\ell_{i(n_i+1)},\dots,\ell_{ip})
{\rm{Var}}\left({w_1^i\left({{t_0}}\right)}\right) = \int_{- \infty}^{+ \infty}{{S_{w_1^i}}\left( f \right)df}={{{N_0}}\mathord{\left/{\vphantom{{{N_0}}2}}\right.\kern-\nulldelimiterspace}2}
\left<{<\!\!<\!\tilde{W}^{n}\!>\!\!>}_{t}(\varphi_{1}), \varphi_{2}\right> = \frac{1}{n}\int_{0}^{t}\left<\tilde{u}^n_{z},\varphi_{1}\varphi_{2}\Psi(\cdot,\tilde{\gamma}^n_z)\right>dz,