15 May 2008, Volume 24 Issue 5

    

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  Spectral Gap and Logarithmic Sobolev Constant for Continuous Spin Systems

  Mu Fa CHEN

  Acta Mathematica Sinica. 2008, 24(5): 705-736. https://doi.org/10.1007/s10114-007-7293-3

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  The aim of this paper is to study the spectral gap and the logarithmic Sobolev constant for continuous spin systems.A simple but general result for estimating the spectral gap of finite dimensional systems is given by Theorem 1.1,in terms of the spectral gap for one-dimensional marginals.The study of this topic provides us a chance,and it is indeed another aim of the paper,to justify the power of the results obtained previously.The exact order in dimension one (Proposition 1.4),and then the precise leading order and the explicit positive regions of the spectral gap and the logarithmic Sobolev constant for two typical infinite-dimensional models are presented (Theorems 6.2 and 6.3).Since we are interested in explicit estimates,the computations become quite involved.A long section (Section 4) is devoted to the study of the spectral gap in dimension one.
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  Statistically Convergent Difference Double Sequence Spaces

  Binod Chandra TRIPATHY, Bipul SARMA

  Acta Mathematica Sinica. 2008, 24(5): 737-742. https://doi.org/10.1007/s10114-007-6648-0

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  In this article we define the notion of statistically convergent difference double sequence spaces.We examine the spaces ₂e_∞(Δ,q),₂c(Δ,q),₂c^B(Δ,q),₂c^R(Δ,q),₂c^BR(Δ,q) etc.being symmetric,solid,monotone,etc.We prove some inclusion results too.
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  f-Colorings of Some Graphs of f-Class 1

  Xia ZHANG, Gui Zhen LIU

  Acta Mathematica Sinica. 2008, 24(5): 743-748. https://doi.org/10.1007/s10114-007-6194-9

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  An f-coloring of a graph G is an edge-coloring of G such that each color appears at each vertex v ∈ V(G) at most f(v) times.The minimum number of colors needed to f-color G is called the f-chromatic index of G and is denoted by x'_f(G).Any simple graph G has the f-chromatic index equal to Δ_f(G) or Δ_f(G)+1,where Δ_f(G)=max_v∈V(G){「d(v)/f(v)」}.If x'_f(G)=Δ_f(G),then G is of f-class 1; otherwise G is of f-class 2.In this paper,a class of graphs of f-class 1 are obtained by a constructive proof.As a result,f-colorings of these graphs with Δ_f(G) colors are given.
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  Sharp Upper Bounds for λ₁^Lr of Immersed Hypersurfaces and Their Stability in Space Forms

  Ru Shan WANG

  Acta Mathematica Sinica. 2008, 24(5): 749-760. https://doi.org/10.1007/s10114-007-6322-6

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  In this paper we derive new sharp upper bounds for the first positive eigenvalue λ₁^Lr of the linearized operator of the higher order mean curvature of a closed hypersurface immersed into a Riemannian space form Rⁿ⁺¹(c) (c>0).Our bounds are extrinsic in the sense that they are given in terms of the higher order mean curvatures.Under the assumption Hr+2>0,by establishing two valuable integral formulas,we obtain unified sharp upper bounds of λ₁^Lr.We also give an estimation of the upper bounds of the first eigenvalue of a Schrödinger-type operator,by which we prove those hypersurfaces with positive constant Hr+1 in any space forms are stable if and only if they are geodesic spheres,thus generalizing the previous result obtained only in the case c≤0.
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  Nonexistence of Ground States of -Δu=u^p-u^q

  Wei Min WANG, Li HONG, Kai Tai LI

  Acta Mathematica Sinica. 2008, 24(5): 761-770. https://doi.org/10.1007/s10114-007-4420-0

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  The authors use the method of moving spheres to prove the nonexistence of ground states of -Δu=u^p-u^q for n≥3,-∞<p<n+2/n-2 and q>max(1,p).In fact this conclusion is a special case of -Δu=f(u) for n≥2.
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  A Quadratically Approximate Framework for Constrained Optimization, Global and Local Convergence

  Jin Bao JIAN

  Acta Mathematica Sinica. 2008, 24(5): 771-788. https://doi.org/10.1007/s10114-007-4465-0

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  This paper presents a quadratically approximate algorithm framework (QAAF) for solving general constrained optimization problems,which solves,at each iteration,a subproblem with quadratic objective function and quadratic equality together with inequality constraints.The global convergence of the algorithm framework is presented under the Mangasarian-Fromovitz constraint qualification (MFCQ),and the conditions for superlinear and quadratic convergence of the algorithm framework are given under the MFCQ,the constant rank constraint qualification (CRCQ) as well as the strong second-order sufficiency conditions (SSOSC).As an incidental result,the definition of an approximate KKT point is brought forward,and the global convergence of a sequence of approximate KKT points is analysed.
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  On Randers Metrics with Isotropic S-Curvature

  Zhong Min SHEN, Hao XING

  Acta Mathematica Sinica. 2008, 24(5): 789-796. https://doi.org/10.1007/s10114-007-5194-0

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  In this paper,we study a class of Finsler metrics defined by a vector field on a Riemannian space form.We give an explicit formula for those with isotropic S-curvature.This class contains all Randers metrics of constant flag curvature.
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  Normal Forms for Periodic Orbits of Real Vector Fields

  Ping WANG, Hao WU, Wei Gu LI

  Acta Mathematica Sinica. 2008, 24(5): 797-808. https://doi.org/10.1007/s10114-007-0982-0

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  We consider normal forms of real vector fields near periodic orbits and provide sufficient conditions for their smooth linearization.In addition,the main results also assert the existence of vertical local foliations,whose leaves are all transversal to the periodic orbit.
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  Exponential Attractor of the 3D Derivative Ginzburg-Landau Equation

  Shu Juan LÜ, Qi Shao LU

  Acta Mathematica Sinica. 2008, 24(5): 809-828. https://doi.org/10.1007/s10114-007-5324-8

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  In this paper,we consider a derivative Ginzburg-Landau-type equation with periodic initial-value condition in three-dimensional spaces.Sufficient conditions for existence and uniqueness of a global solution are obtained by uniform a priori estimates of the solution.Furthermore,the existence of a global attractor and an exponential attractor with finite dimensions are proved.
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  Averaging Technique for the Oscillation of Second Order Damped Elliptic Equations

  Zhi Ting XU

  Acta Mathematica Sinica. 2008, 24(5): 829-842. https://doi.org/10.1007/s10114-007-5397-4

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  By using the averaging technique,we establish some oscillation theorems for the second order damped elliptic differential equation 
   D_i[a_ij(x)D_jy]+ b_i(x)D_iy+c(x)f(y)=0,
  which extend and improve some known results in the literature.
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  An Alternating Iterative Method and Its Application in Statistical Inference

  Ning Zhong SHI, Guo Rong HU, Qing CUI

  Acta Mathematica Sinica. 2008, 24(5): 843-856. https://doi.org/10.1007/s10114-007-1017-6

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  This paper studies non-convex programming problems.It is known that,in statistical inference,many constrained estimation problems may be expressed as convex programming problems.However,in many practical problems,the objective functions are not convex.In this paper,we give a definition of a semi-convex objective function and discuss the corresponding non-convex programming problems.A two-step iterative algorithm called the alternating iterative method is proposed for finding solutions for such problems.The method is illustrated by three examples in constrained estimation problems given in Sasabuchi et al.(Biometrika,72,465-472 (1983)),Shi N.Z.(J.Multivariate Anal.,50,282-293 (1994)) and El Barmi H.and Dykstra R.(Ann.Statist.,26,1878-1893 (1998)).
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  Global Well-posedness for gKdV-3 in Sobolev Spaces of Negative Index

  Zhi Fei ZHANG

  Acta Mathematica Sinica. 2008, 24(5): 857-866. https://doi.org/10.1007/s10114-007-5597-y

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  The initial value problem for the generalized Korteweg-deVries equation of order three on the line is shown to be globally well posed for rough data.Our proof is based on the multilinear estimate and the I-method introduced by Colliander,Keel,Staffilani,Takaoka,and Tao.
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  Generalized I-Nonexpansive Selfmaps and Invariant Approximations

  M. A. AL-THAGAFI, Naseer SHAHZAD

  Acta Mathematica Sinica. 2008, 24(5): 867-876. https://doi.org/10.1007/s10114-007-5598-x

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  Common fixed point results for new classes of noncommuting selfmaps satisfying generalized I-contraction or I-nonexpansive type conditions are established.We apply them to obtain several invariant approximation results which unify,extend,and complement the well-known results.
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  Extensions of 2-Homogeneous Polynomials on c₀

  Pei-Kee LIN

  Acta Mathematica Sinica. 2008, 24(5): 877-880. https://doi.org/10.1007/s10114-007-5636-8

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  We prove that every 2-homogeneous polynomial on the complex c0 has a unique normpreserving extension to its bidual e_∞.