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Homogeneous Approximation Property for Wavelet Frames with Matrix Dilations,Ⅱ

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Acta Mathematica Sinica,English Series 2013年第1期29卷 183-192页

作者： Zhi Jing ZHAO Wen Chang SUN Department of Mathematics and lpmc Nankai University

The homogeneous approximation property （HAP） states that the number of building blocks involved in a reconstruction of a function up to some error is essentially invariant under time-scale shifts. In this paper, we show that every wavelet frame with nice wavelet function and arbitrary expansive dilation matrix possesses the HAP. Our results improve some known ones.

关键词： Wavelet frames homogeneous approximation property HAP matrix dilations

INVERTIBLE SEQUENCES OF BOUNDED LINEAR OPERATORS

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Acta Mathematica Scientia 2011年第5期31卷 1939-1944页

作者：臧丽丽孙文昌 Department of Mathematics and lpmc Nankai University

In this paper, we study the invertibility of sequences consisting of finitely many bounded linear operators from a Hilbert space to others. We show that a sequence of operators is left invertible if and only if it is ... 详细信息

关键词： frames g-frames Riesz bases g-Riesz bases

Minimal and Maximal Unconditional Bases with Respect to Framings

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Acta Mathematica Sinica,English Series 2013年第12期29卷 2295-2304页

作者： Rui LIU Department of Mathematics and lpmc Nankai University

This paper studies framings in Banach spaces, a concept raised by Casazza, Han and Larson, which is a natural generalization of traditional frames in Hilbert spaces and unconditional bases in Banach spaces. The minimal unconditional bases and the maximal unconditional bases with respect to framings are introduced. Our main result states that, if （xi, fi） is a framing of a Banach space X, and （eimin） and （eimax） are the minimal unconditional basis and the maximal unconditional basis with respect to （xi, fi）, respectively, then for any unconditional basis （ei） associated with （xi, fi）, there are A,B 〉 0 such that A｜｜i=1∑∞aieimin｜｜≤｜｜i=1∑∞aiei｜｜≤B｜｜i=1∑∞aieimax｜｜ for all （ai） ∈ c00.It means that for any framing, the corresponding associated unconditional bases have common upper and lower bounds.

关键词： Associated unconditional basis framing maximal unconditional basis minimal unconditional basis

SMALL INTO ISOMORPHISM FROM(c_0)INTO C(Ω)TYPE

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Acta Mathematica Scientia 2008年第2期28卷 307-314页

作者：定光桂 School of Mathematical Sciences and lpmc Nankai University

If T is an isomorphism of （c0） into C（Ω）（where Ω is a sequentially compact and paracompact space, or a compact metric space in particular）, which satisfies the condition ｜｜T｜｜·｜｜T^-1｜｜ ≤ 1 ＋ε for some ε ∈ （0,1/5）, then T/｜｜T｜｜ is close to an isometry with an error less than 9ε. The proof of this article is simple without using the dual space or adjoint operator.

关键词： Small into isomorphism isometry almost isometry

Graphs with vertex rainbow connection number two

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Science China Mathematics 2015年第8期58卷 1803-1810页

作者： LU ZaiPing MA YingBin Center for Combinatorics LPMC-TJKLC Nankai University

An edge colored graph G is rainbow connected if any two vertices are connected by a path whose edges have distinct colors. The rainbow connection number of a graph G, denoted by rc(G), is the smallest number of colors that are needed in order to make G rainbow connected. A vertex colored graph G is vertex rainbow connected if any two vertices are connected by a path whose internal vertices have distinct colors. The vertex rainbow connection number of G, denoted by rvc(G), is the smallest number of colors that are needed in order to make G vertex rainbow connected. In 2011, Kemnitz and Schiermeyer considered graphs with rc(G) = *** investigate graphs with rvc(G) = 2. First, we prove that rvc(G) 2 if |E(G)|≥n-22 + 2, and the bound is sharp. Denote by s(n, 2) the minimum number such that, for each graph G of order n, we have rvc(G) 2provided |E(G)|≥s(n, 2). It is proved that s(n, 2) = n-22 + 2. Next, we characterize the vertex rainbow connection numbers of graphs G with |V(G)| = n, diam(G)≥3 and clique number ω(G) = n- s for 1≤s≤4.

关键词： vertex-coloring vertex rainbow connection number clique number

Context-free Grammars for Triangular Arrays

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Acta Mathematica Sinica,English Series 2015年第3期31卷 445-455页

作者： Robert X.J.HAO Larry X.W.WANG Harold R.L.YANG Center for Combinatorics LPMC-TJKLC Nankai University

We consider context-free grammars of the form G = {f → f^b1＋b2＋1g^a1＋a2, g → f^b1 g^a1＋1},where ai and bi are integers sub ject to certain positivity conditions. Such a grammar G gives rise to triangular arrays {T（n, k）}0≤k≤n satisfying a three-term recurrence relation. Many combinatorial sequences can be generated in this way. Let Tn （x） =∑k=0^n T（n, k）x^k. Based on the differential operator with respect to G, we define a sequence of linear operators Pn such that Tn＋1（x） = Pn（Tn（x））. Applying the characterization of real stability preserving linear operators on the multivariate polynomials due to Borcea and Br？ndén, we obtain a necessary and sufficient condition for the operator Pn to be real stability preserving for any n. As a consequence, we are led to a sufficient condition for the real-rootedness of the polynomials defined by certain triangular arrays, obtained by Wang and ***, as special cases we obtain grammars that lead to identities involving the Whitney numbers and the Bessel numbers.

关键词： Context-free grammar stable polynomials real-rootedness Bessel numbers Whitneynumbers

The Gerber-Shiu Discounted Penalty Function for a Compound Binomial Risk Model with By-claims

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Acta Mathematicae Applicatae Sinica 2015年第1期31卷 181-190页

作者： Jin-zhu LI Rong WU School of Mathematical Sciences and lpmc Nankai University

A recursive formula of the Gerber-Shiu discounted penalty function for a compound binomial risk model with by-claims is obtained. In the discount-free case, an explicit formula is given. Utilizing such an explicit expression, we derive some useful insurance quantities, including the ruin probability, the density of the deficit at ruin, the joint density of the surplus immediately before ruin and the deficit at ruin, and the density of the claim causing ruin.

关键词： compound binomial risk model Gerber-Shiu function by-claims

On a family involving R.L. Cohen's ζ-element(II)

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Science China Mathematics 2015年第8期58卷 1745-1752页

作者： HONG JianGuo LIU XiuGui ZHENG Da School of Mathematical Sciences and lpmc Nankai University

In 1981, Cohen constructed an infinite family of homotopy elements ζk∈ π＊（S） represented by h0bk ∈ ExtA3,2（p-1）（pk＋1＋1）（z/p,Z/p） in the Adams spectral sequence, where p 〉 2 and k ≥ 1. In this paper, we make use of the Adams spectral sequence and the May spectral sequence to prove that the composite map ζn-1β2γs＋3 is nontrivial in the stable homotopy groups of spheres πt（s,n）-s-8（S）, where p ≥7, n 〉 3, 0≤s 〈p-5 andt（s,n） =2（p-1）[pn＋（s＋3）p2＋（s＋4）p＋（s＋3）]＋s.

关键词： stable homotopy groups of spheres Adams spectral sequence May spectral sequence

The dividend function in the jump-diffusion dual model withbarrier dividend strategy

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Applied Mathematics and Mechanics(English Edition) 2008年第9期29卷 1239-1249页

作者：李波吴荣 School of Mathematical Sciences and lpmc Nankai University

A dual model of the perturbed classical compound Poisson risk model is considered under a constant dividend barrier. A new method is used in deriving the boundary condition of the equation for the expectation function by studying the local time of a related process. We obtain the expression for the expected discount dividend function in terms of those in the corresponding perturbed compound Poisson risk model without barriers. A special case in which the gain size is phase-type distributed is illustrated. We also consider the existence of the optimal dividend level.

关键词： compound Poisson process diffusion process Gerber-Shiu function integro-differential equation time of ruin surplus before ruin deficit at ruin