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Recognition by noncommuting graph of finite simple groups L4(q)
M. AKBARI, M. KHEIRABADI, A. R. MOGHADDAMFAR
Front Math Chin. 2011, 6 (1): 1-16.
https://doi.org/10.1007/s11464-010-0085-6
Let G be a nonabelian group. We define the noncommuting graph ?(G) of G as follows: its vertex set is G\Z(G), the set of non-central elements of G, and two different vertices x and y are joined by an edge if and only if x and y do not commute as elements of G, i.e., [x,y]≠1. We prove that if L ∈ {L4(7), L4(11), L4(13), L4(16), L4(17)} and G is a finite group such that ?(G)??(L), then G?L.
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Oscillatory integrals on unit square along surfaces
Jiecheng CHEN, Dashan FAN, Huoxiong WU, Xiangrong ZHU
Front Math Chin. 2011, 6 (1): 49-59.
https://doi.org/10.1007/s11464-010-0088-3
Let Q2 = [0, 1]2 be the unit square in two-dimensional Euclidean space ?2. We study the Lp boundedness of the oscillatory integral operator Tα,β defined on the set ?(?2+n) of Schwartz test functions byTα,βf(u,v,x)=∫Q2f(u-t,v-s,x-γ(t,s))t1+α1s1+α2eit-β1s-β2dtds,where x∈?n, (u,v)∈?2, (t,s,γ(t,s))=(t,s,tp1sq1,tp2sq2,?,tpnsqn) is a surface on ?n+2, and β1>α1, β2>α2. Our results extend some known results on ?3.
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Proximal alternating direction-based contraction methods for separable linearly constrained convex optimization
Bingsheng HE, Zheng PENG, Xiangfeng WANG
Front Math Chin. 2011, 6 (1): 79-114.
https://doi.org/10.1007/s11464-010-0092-7
Alternating direction method (ADM) has been well studied in the context of linearly constrained convex programming problems. Recently, because of its significant efficiency and easy implementation in novel applications, ADM is extended to the case where the number of separable parts is a finite number. The algorithmic framework of the extended method consists of two phases. At each iteration, it first produces a trial point by using the usual alternating direction scheme, and then the next iterate is updated by using a distance-descent direction offered by the trial point. The generated sequence approaches the solution set monotonically in the Fejér sense, and the method is called alternating direction-based contraction (ADBC) method. In this paper, in order to simplify the subproblems in the first phase, we add a proximal term to the objective function of the minimization subproblems. The resulted algorithm is called proximal alternating direction-based contraction (PADBC) methods. In addition, we present different linearized versions of the PADBC methods which substantially broaden the applicable scope of the ADBC method. All the presented algorithms are guided by a general framework of the contraction methods for monotone variational inequalities, and thus, the convergence follows directly.
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Stability of almost submetries
Xiaochun RONG, Shicheng XU
Front Math Chin. 2011, 6 (1): 137-154.
https://doi.org/10.1007/s11464-010-0076-7
In this paper, we consider a triple of Gromov-Hausdorff convergence: Ai→dGHA, Bi→dGHB and maps fi : Ai → Bi converge to a map f : A → B, where Ai are compact Alexandrov n-spaces and Bi are compact Riemannian m-manifolds such that the curvature, diameter and volume are suitably bounded (non-collapsing). When f is a submetry, we give a necessary and sufficient condition for the sequence to be stable, that is, for i large, there are homeomorphisms, Ψi : Ai → A, Φi : Bi → B such that f ? Ψi = Φi ? fi. When f is an ?-submetry with ?>0, we obtain a sufficient condition for the stability in the case that Ai are Riemannian manifolds. Our results generalize the stability/finiteness results on fiber bundles by Riemannian submersions and by submetries.
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13 articles
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