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Reduced-order extrapolation spectral-finite difference scheme based on POD method and error estimation for three-dimensional parabolic equation
Jing AN,Zhendong LUO,Hong LI,Ping SUN
Front. Math. China. 2015, 10 (5): 1025-1040.
https://doi.org/10.1007/s11464-015-0469-8
In this study, a classical spectral-finite difference scheme (SFDS) for the three-dimensional (3D) parabolic equation is reduced by using proper orthogonal decomposition (POD) and singular value decomposition (SVD). First, the 3D parabolic equation is discretized in spatial variables by using spectral collocation method and the discrete scheme is transformed into matrix formulation by tensor product. Second, the classical SFDS is obtained by difference discretization in time-direction. The ensemble of data are comprised with the first few transient solutions of the classical SFDS for the 3D parabolic equation and the POD bases of ensemble of data are generated by using POD technique and SVD. The unknown quantities of the classical SFDS are replaced with the linear combination of POD bases and a reducedorder extrapolation SFDS with lower dimensions and sufficiently high accuracy for the 3D parabolic equation is established. Third, the error estimates between the classical SFDS solutions and the reduced-order extrapolation SFDS solutions and the implementation for solving the reduced-order extrapolation SFDS are provided. Finally, a numerical example shows that the errors of numerical computations are consistent with the theoretical results. Moreover, it is shown that the reduced-order extrapolation SFDS is effective and feasible to find the numerical solutions for the 3D parabolic equation.
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Maximal function characterizations of Musielak-Orlicz-Hardy spaces associated with magnetic Schr?dinger operators
Dachun YANG,Dongyong YANG
Front. Math. China. 2015, 10 (5): 1203-1232.
https://doi.org/10.1007/s11464-015-0432-8
Let φ be a growth function, and let A:=-(?-ia)?(?-ia)+V be a magnetic Schr?dinger operator on L2(?n),n≥2, where α:=(α1,α2,?,αn)∈Lloc2(?n,?n) and 0≤V∈Lloc1(?n). We establish the equivalent characterizations of the Musielak-Orlicz-Hardy space HA,φ(?n), defined by the Lusin area function associated with {e-t2A}t>0, in terms of the Lusin area function associated with {e-tA}t>0, the radial maximal functions and the nontangential maximal functions associated with {e-t2A}t>0 and {e-tA}t>0, respectively. The boundedness of the Riesz transforms LkA-1/2,k∈{1,2,?,n}, from HA,φ(?n) to Lφ(?n) is also presented, where Lk is the closure of ??xk-iαk in L2(?n). These results are new even when φ(x,t):=ω(x)tp for all x∈?nand t ∈(0,+∞) with p ∈(0, 1] and ω∈A∞(?n) (the class of Muckenhoupt weights on ?n).
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Flag-transitive 2-(v, k, λ) symmetric designs with (k, λ) = 1 and alternating socle
Yan ZHU, Haiyan GUAN, Shenglin ZHOU
Front. Math. China. 2015, 10 (5): 1483-1496.
https://doi.org/10.1007/s11464-015-0480-0
Consider the flag-transitive 2-(v, k, λ) symmetric designs with (k, λ) = 1. We prove that if is a nontrivial 2-(v, k, λ) symmetric design with (k, λ) = 1 and G≤Aut() is flag-transitive with Soc(G) = An for n≥5, then is the projective space PG2(3,2) and G = A7.
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14 articles
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