05 July 2007, Volume 23 Issue 7

    

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Articles
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  Selection Principles and Sierpinski Sets

  Marion SCHEEPERS

  Acta Mathematica Sinica. 2007, 23(7): 1153-1162. https://doi.org/10.1007/s10114-005-0810-3

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  We show that a set of real numbers is a Sierpiński set if, and only if, it satisfies a selection property similar to the familiar Menger property.
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  The Universal Kolyvagin Recursion Implies the Kolyvagin Recursion

  Yi OUYANG

  Acta Mathematica Sinica. 2007, 23(7): 1163-1172. https://doi.org/10.1007/s10114-005-0875-z

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  Let U_z be the universal norm distribution and M a fixed power of prime p. By using the double complex method employed by Anderson, we study the universal Kolyvagin recursion occurring in the canonical basis in the cohomology group H⁰(G_z,U_z/M U_z).We furthermore show that the universal Kolyvagin recursion implies the Kolyvagin recursion in the theory of Euler systems. One certainly hopes this could lead to a new way to find new Euler systems.
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  Estimate of the L^p-Fourier Transform Norm on Strong *-Regular Exponential Solvable Lie Groups

  A. BAKLOUTI, J. LUDWIG, L. SCUTO, K. SMAOUI

  Acta Mathematica Sinica. 2007, 23(7): 1173-1188. https://doi.org/10.1007/s10114-005-0845-5

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  We study the L^p-Fourier transform for a special class of exponential Lie groups, the strong *-regular exponential Lie groups. Moreover, we provide an estimate of its norm using the orbit method.
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  Oscillation Theorems for Quasilinear Elliptic Differential Equations

  Zhi Ting XU

  Acta Mathematica Sinica. 2007, 23(7): 1189-1198. https://doi.org/10.1007/s10114-005-0853-5

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  By using vector Riccati transformation and averaging technique, some oscillation criteria for the quasilinear elliptic di.erential equation of second order,
  
  are obtained. These theorems extend and include earlier results for the semilinear elliptic equation and PDE with p-Laplacian.
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  Normal Form for Families of Hamiltonian Systems

  Zhi Guo WANG

  Acta Mathematica Sinica. 2007, 23(7): 1199-1216. https://doi.org/10.1007/s10114-005-0844-6

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  We consider perturbations of integrable Hamiltonian systems in the neighborhood of normally parabolic invariant tori. Using the techniques of KAM-theory we prove that there exists a canonical transformation that puts the Hamiltonian in normal form up to a remainder of weighted order 2d + 1. And some dynamical consequences are obtained.
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  Log Structures on Generalized Semi-Stable Varieties

  Ting LI

  Acta Mathematica Sinica. 2007, 23(7): 1217-1232. https://doi.org/10.1007/s10114-005-0844-7

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  In this paper, a class of morphisms which have a kind of singularity weaker than normal crossing is considered. We construct the obstruction such that the so-called semi-stable log structures exists if and only if the obstruction vanishes. In the case of no power, if the obstruction vanishes, then the semi-stable log structure is unique up to a unique isomorphism. So we obtain a kind of canonical structure on this family of morphisms.
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  The Global Convergence of Self-Scaling BFGS Algorithm with Nonmonotone Line Search for Unconstrained Nonconvex Optimization Problems

  Hong Xia YIN, Dong Lei DU

  Acta Mathematica Sinica. 2007, 23(7): 1233-1240. https://doi.org/10.1007/s10114-005-0837-5

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  The self-scaling quasi-Newton method solves an unconstrained optimization problem by scaling the Hessian approximation matrix before it is updated at each iteration to avoid the possible large eigenvalues in the Hessian approximation matrices of the objective function. It has been proved in the literature that this method has the global and superlinear convergence when the objective function is convex (or even uniformly convex). We propose to solve unconstrained nonconvex optimization problems by a self-scaling BFGS algorithm with nonmonotone linear search. Nonmonotone line search has been recognized in numerical practices as a competitive approach for solving large-scale nonlinear problems. We consider two different nonmonotone line search forms and study the global convergence of these nonmonotone self-scale BFGS algorithms. We prove that, under some weaker condition than that in the literature, both forms of the self-scaling BFGS algorithm are globally convergent for unconstrained nonconvex optimization problems.
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  General Solutions of Matrix Extension Associated with Multiple Channel Biorthogonal Wavelet Filters

  Wen Bin WEI, Min HUANG

  Acta Mathematica Sinica. 2007, 23(7): 1241-1250. https://doi.org/10.1007/s10114-005-0860-6

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  This paper is concerned with seeking the general solutions of matrix equation (ξ)M^*(ξ)=I_s for the construction of multiple channel biorthogonal wavelets, provided that some special solution of its is known.
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  C⁰-Sufficiency, Kuiper-Kuo and Thom Conditions for Non-isolated Singularity

  Xu XU

  Acta Mathematica Sinica. 2007, 23(7): 1251-1256. https://doi.org/10.1007/s10114-005-0879-8

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  In this paper, a criterion on the C⁰-sufficiency for a function germ with non-isolated singularity is obtained analogously to that of Kuiper-Kuo for the case of isolated singularities. Moreover, the Kuiper-Kuo condition and the Thom condition for an analytic function germ with non-isolated singularity are proved to be equivalent.
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  A Continuation Algorithm for Max-Cut Problem

  Feng Min XU, Cheng Xian XU, Xing Si LI

  Acta Mathematica Sinica. 2007, 23(7): 1257-1264. https://doi.org/10.1007/s10114-005-0881-1

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  A continuation algorithm for the solution of max-cut problems is proposed in this paper. Unlike the available semi-definite relaxation, a max-cut problem is converted into a continuous nonlinear programming by employing NCP functions, and the resulting nonlinear programming problem is then solved by using the augmented Lagrange penalty function method. The convergence property of the proposed algorithm is studied. Numerical experiments and comparisons with the Geomeans and Williamson randomized algorithm made on some max-cut test problems show that the algorithm generates satisfactory solutions for all the test problems with much less computation costs.
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  Precise Asymptotics for Lévy Processes

  Zhi Shui HU, Chun SU

  Acta Mathematica Sinica. 2007, 23(7): 1265-1270. https://doi.org/10.1007/s10114-005-0868-y

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  Let {X(t), t ≥ 0} be a Lévy process with EX(1) = 0 and EX²(1) < ∞. In this paper, we shall give two precise asymptotic theorems for {X(t), t ≥ 0}. By the way, we prove the corresponding conclusions for strictly stable processes and a general precise asymptotic proposition for sums of i.i.d. random variables.
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  Global Attractors for a Nonclassical Diffusion Equation

  Chun You SUN, Su Yun WANG, Cheng Kui ZHONG

  Acta Mathematica Sinica. 2007, 23(7): 1271-1280. https://doi.org/10.1007/s10114-005-0909-6

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  We prove the existence of global attractors in H₀¹(Ω) for a nonclassical diffusion equation. Two types of nonlinearity f are considered: one is the critical exponent, and the other is the polynomial growth of arbitrary order.
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  Ruin Probabilities in Cox Risk Models with Two Dependent Classes of Business

  Jun Yi GUO, Kam C. GUO, Ming ZHOU

  Acta Mathematica Sinica. 2007, 23(7): 1281-1288. https://doi.org/10.1007/s10114-005-0819-7

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  In this paper we consider risk processes with two classes of business in which the two claim-number processes are dependent Cox processes. We first assume that the two claim-number processes have a two-dimensional Markovian intensity. Under this assumption, we not only study the sum of the two individual risk processes but also investigate the two-dimensional risk process formed by considering the two individual processes separately. For each of the two risk processes we derive an expression for the ruin probability, and then construct an upper bound for the ruin probability. We next assume that the intensity of the two claim-number processes follows a Markov chain. In this case, we examine the ruin probability of the sum of the two individual risk processes. Specifically, a differential system for the ruin probability is derived and numerical results are obtained for exponential claim sizes.
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  Explicit Convergence Rates of the Embedded M/G/1 Queue

  Yuan Yuan LIU, Zhen Ting HOU

  Acta Mathematica Sinica. 2007, 23(7): 1289-1296. https://doi.org/10.1007/s10114-005-0917-6

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  This paper investigates the explicit convergence rates to the stationary distribution π of the embedded M/G/1 queue; specifically, for suitable rate functions r(n) which may be polynomial with r(n) = nl, l > 0 or geometric with r(n) = αn, α > 1 and "moments" f ≥ 1, we find the conditions under which for all #em/em# ∈ E. For the polynomial case, the explicit bounds on M(#em/em#) are given in terms of both "drift functions" and behavior of the first hitting time on the state 0; and for the geometric case, the largest geometric convergence rate α^* is obtained.
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  Asymptotic Formulae for Brillouin Index on Riemannian Manifolds

  Wen Xiang SUN

  Acta Mathematica Sinica. 2007, 23(7): 1297-1302. https://doi.org/10.1007/s10114-005-0904-y

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  We establish several asymptotic formulae for Brillouin index on flat tori. As an application of these formulae it is proved that the topological entropy of a geodesic flow on a flat torus is zero.
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  Approximation Properties of Julia Polynomials

  Daniyal M. ISRAFILOV, Burcin ISRAFILOV

  Acta Mathematica Sinica. 2007, 23(7): 1303-1310. https://doi.org/10.1007/s10114-005-0730-2

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  Let G be a finite simply connected domain in the complex plane C, bounded by a rectifiable Jordan curve L, and let w = φ₀ (z) be the Riemann conformal mapping of G onto D(0, r₀):={w:|w| < r₀}, normalized by the conditions φ₀ (z₀) = 0, φ'₀ (z₀) = 1.In this work, the rate of approximation of φ₀ by the polynomials, defined with the help of the solutions of some extremal problem, in a closed domain G is studied. This rate depends on the geometric properties of the boundary L.
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  On the Norm of a Self-Adjoint Operator and a New Bilinear Integral Inequality

  Bi Cheng YANG

  Acta Mathematica Sinica. 2007, 23(7): 1311-1316. https://doi.org/10.1007/s10114-005-0895-8

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  In this paper, the expression of the norm of a self-adjoint integral operator T:L²(0,∞) → L²(0,∞) is obtained. As applications, a new bilinear integral inequality with a best constant factor is established and some particular cases are considered.
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  Local Isoperimetric Inequalities for Sectors on Surfaces and Cones

  Fernando GIMÉNEZ, Javier Orengo VALVERDE

  Acta Mathematica Sinica. 2007, 23(7): 1317-1326. https://doi.org/10.1007/s10114-005-0826-8

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  On surfaces we give conditions under which the solution of a restricted local isoperimetric problem for sectors with small solid angle is the circular sector and we characterize these surfaces. Also we study this problem for general spherical cones on hypersurfaces in higher dimensional Riemannian manifolds.
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  A Covering Lemma on the Unit Sphere and Application to the Fourier-Laplace Convergence

  Rong HUANG, Kun Yang WANG

  Acta Mathematica Sinica. 2007, 23(7): 1327-1332. https://doi.org/10.1007/s10114-005-0890-0

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  Suppose f ∈ L(Σ_n-1), n ≥ 3. If f satisfies the condition at every point x in a set E of positive measure in Σ_n-1, then the Cesàro means of critical order (n-2)/2 of the Fourier-Laplace series of f converge to f at almost every point x in E.
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  A Theorem on the Structure of the Ring A_cris

  Hao LIN

  Acta Mathematica Sinica. 2007, 23(7): 1333-1336. https://doi.org/10.1007/s10114-005-0898-5

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  In this paper, we give a new description for the structure of the p-adic period subring A_cris using formal groups.
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  K-theory of Continuous Deformations of C^*-algebras

  Takahiro SUDO

  Acta Mathematica Sinica. 2007, 23(7): 1337-1340. https://doi.org/10.1007/s10114-005-0779-y

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  We study K-theory of continuous deformations of C^*-algebras to obtain that their K-theory is the same as that of the fiber at zero. We also consider continuous or discontinuous deformations of Cuntz and Toeplitz algebras.
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  On the Mean Value of the Complete Trigonometric Sums with Dirichlet Characters

  Zhe Feng XU

  Acta Mathematica Sinica. 2007, 23(7): 1341-1344. https://doi.org/10.1007/s10114-007-0916-x

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  The main purpose of this paper is, using the analytic method, to study the mean value properties of the complete trigonometric sums with Dirichlet characters, and give an exact calculating formula for its fourth power mean.