15 April 2005, Volume 21 Issue 2

    

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Articles
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  Automorphisms of Maps with a Given Underlying Graph and Their Application to Enumeration

  Lin Fan MAO, Yan Pei LIU, Feng TIAN

  Acta Mathematica Sinica. 2005, 21(2): 225-236. https://doi.org/10.1007/s10114-004-0494-0

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  A graph is called a semi-regular graph if its automorphism group action on its ordered pair of adjacent vertices is semi-regular. In this paper, a necessary and sufficient condition for an automorphism of the graph Γ to be an automorphism of a map with the underlying graph Γ is obtained. Using this result, all orientation-preserving automorphisms of maps on surfaces (orientable and non-orientable) or just orientable surfaces with a given underlying semi-regular graph Γ are determined. Formulas for the numbers of non-equivalent embeddings of this kind of graphs on surfaces (orientable, non-orientable or both) are established, and especially, the non-equivalent embeddings of circulant graphs of a prime order on orientable, non-orientable and general surfaces are enumerated.
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  The Relation of the Morse Index of Closed Geodesics with the Maslov-type Index of Symplectic Paths

  Chun Gen LIU

  Acta Mathematica Sinica. 2005, 21(2): 237-248. https://doi.org/10.1007/s10114-004-0406-3

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  In this paper, we consider the relation of the Morse index of a closed geodesic with the Maslov-type index of a path in a symplectic group. More precisely, for a closed geodesic c on a Riemannian manifold M with its linear Poincaré map P (a symplectic matrix), we construct a symplectic path γ(t) starting from identity I and ending at P, such that the Morse index of the closed geodesic c equals the Maslov-type index of γ. As an application of this result, we study the parity of the Morse index of any closed geodesic.
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  Rings Whose Modules Have Grade Zero

  Zhi Xiang WU

  Acta Mathematica Sinica. 2005, 21(2): 249-260. https://doi.org/10.1007/s10114-004-0504-2

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  In this paper, we prove that R is a two-sided Artinian ring and J is a right annihilator ideal if and only if (i) for any nonzero right module, there is a nonzero linear map from it to a projective module; (ii) every submodule of R _R is not a radical module for some right coherent rings. We call a ring a right X ring if Hom_R(M, R) = 0 for any right module M implies that M = 0. We can prove some left Goldie and right X rings are right Artinian rings. Moreover we characterize semisimple rings by using X rings. A famous Faith's conjecture is whether a semipimary PF ring is a QF ring. Similarly we study the relationship between X rings and QF and get many interesting results.
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  On Homogeneous Differential Polynomials of Meromorphic Functions

  Wei Chuan LIN, Hong Xun YI

  Acta Mathematica Sinica. 2005, 21(2): 261-266. https://doi.org/10.1007/s10114-004-0326-2

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  In this paper, we study one conjecture proposed by W. Bergweiler and show that any transcendental meromorphic functions f(z) have the form exp(αz+β) if f(z)f"(z)-a(f'(z))²≠0, where a ≠ 1, n±1/n, in n∈N. Moreover, an analogous normality criterion is obtained.
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  Extreme Points and Strongly Extreme Points of Musielak-Orlicz Sequences Spaces

  Xin Bo LIU, Ting Fu WANG, Fei Fei YU

  Acta Mathematica Sinica. 2005, 21(2): 267-278. https://doi.org/10.1007/s10114-004-0422-3

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  Criteria for extreme points and strongly extreme points in Musielak-Orlicz sequence spaces, equipped with both the Luxemburg norm and the Orlicz norm, are given.
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  Higher Level Orderings on Modules

  Min WU, Guang Xing ZENG

  Acta Mathematica Sinica. 2005, 21(2): 279-288. https://doi.org/10.1007/s10114-004-0495-z

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  The aim of this paper is to investigate higher level orderings on modules over commutative rings. On the basis of the theory of higher level orderings on fields and commutative rings, some results involving existence of higher level orderings are generalized to the category of modules over commutative rings. Moreover, a strict intersection theorem for higher level orderings on modules is established.
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  A Biordered Set Representation of Regular Semigroups

  Bing Jun YU, Mang XU

  Acta Mathematica Sinica. 2005, 21(2): 289-302. https://doi.org/10.1007/s10114-004-0490-4

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  In this paper, for an arbitrary regular biordered set E, by using biorder-isomorphisms between the ω-ideals of E, we construct a fundamental regular semigroup W_E called NH-semigroup of E, whose idempotent biordered set is isomorphic to E. We prove further that W_E can be used to give a new representation of general regular semigroups in the sense that, for any regular semigroup S with the idempotent biordered set isomorphic to E, there exists a homomorphism from S to W_E whose kernel is the greatest idempotent-separating congruence on S and the image is a full symmetric subsemigroup of W_E. Moreover, when E is a biordered set of a semilattice E₀, W_E is isomorphic to the Munn-semigroup T_E0; and when E is the biordered set of a band B, W_E is isomorphic to the Hall-semigroup W_B.
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  Sharp Error Estimate for Maximum Likelihood Estimator of Nonstationary Diffusion Processes

  Zhi Yan XU, Wei An ZHENG

  Acta Mathematica Sinica. 2005, 21(2): 303-314. https://doi.org/10.1007/s10114-004-0371-x

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  We consider the maximum likelihood estimator of the unknown parameter in a class of nonstationary diffusion processes. We give further a precise estimate for the error of the estimator.
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  A Unified Convergence Theorem

  Jun De WU, Jian Wen LUO, Shi Jie LU

  Acta Mathematica Sinica. 2005, 21(2): 315-322. https://doi.org/10.1007/s10114-004-0481-5

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  We prove a unified convergence theorem, which presents, in four equivalent forms, the famous Antosik-Mikusinski theorems. In particular, we show that Swartz' three uniform convergence principles are all equivalent to the Antosik-Mikusinski theorems.
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  A System of Four Matrix Equations over von Neumann Regular Rings and Its Applications

  Qing Wen WANG

  Acta Mathematica Sinica. 2005, 21(2): 323-334. https://doi.org/10.1007/s10114-004-0493-1

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  We consider the system of four linear matrix equations A₁X = C₁, XB₂ = C₂, A₃XB₃ = C₃ and A₄XB₄ = C₄ over R, an arbitrary von Neumann regular ring with identity. A necessary and sufficient condition for the existence and the expression of the general solution to the system are derived. As applications, necessary and sufficient conditions are given for the system of matrix equations A₁X = C₁ and A₃X = C₃ to have a bisymmetric solution, the system of matrix equations A₁X = C₁ and A₃XB₃ = C₃ to have a perselfconjugate solution over R with an involution and char R≠2, respectively. The representations of such solutions are also presented. Moreover, some auxiliary results on other systems over R are obtained. The previous known results on some systems of matrix equations are special cases of the new results.
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  The Exceptional Set in Hua’s Theorem for Three Squares of Primes

  Jian Ya LIU, Tao ZHAN

  Acta Mathematica Sinica. 2005, 21(2): 335-350. https://doi.org/10.1007/s10114-004-0484-2

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  It is proved that with at most O(N^11/12+ε) exceptions, all positive integers n ≤ N satisfying some necessary congruence conditions are the sum of three squares of primes. This improves substantially the previous results in this direction.
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  Optimal Three Cylinder Inequality at the Boundary for Solutions to Parabolic Equations and Unique Continuation Properties

  Sergio VESSELLA

  Acta Mathematica Sinica. 2005, 21(2): 351-380. https://doi.org/10.1007/s10114-004-0498-9

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  Let Γ be a portion of a C ^1,α boundary of an n-dimensional domain D. Let u be a solution to a second order parabolic equation in D×(-T, T) and assume that u = 0 on Γ×(-T, T), 0∈Γ. We prove that u satis.es a three cylinder inequality near Γ×(-T, T) . As a consequence of the previous result we prove that if u (x, t) = O (|x|^k) for every t∈(-T, T) and every k∈N, then u is identically equal to zero.
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  On the Existence of Periodic Solutions for a Kind of Second Order Neutral Functional Differential Equation

  Shi Ping LU, Wei Gao GE

  Acta Mathematica Sinica. 2005, 21(2): 381-392. https://doi.org/10.1007/s10114-004-0417-0

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  By means of the continuation theorem of coincidence degree theory, some new results on the non-existence, existence and unique existence of periodic solutions for a kind of second order neutral functional differential equation are obtained.
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  The Cauchy Problem of a Shallow Water Equation

  Xiao Feng LIU, Yong Yang JIN

  Acta Mathematica Sinica. 2005, 21(2): 393-408. https://doi.org/10.1007/s10114-004-0420-5

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  We consider the Cauchy problem of a shallow water equation and its local wellposedness.
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  (Z₂)^k-Actions with Fixed Point Set of Constant Codimension 2^k+2

  Zuo LIU, Zhen De WU

  Acta Mathematica Sinica. 2005, 21(2): 409-412. https://doi.org/10.1007/s10114-004-0320-8

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  Let J_*,k^r denote the ideal in MO_* of cobordism classes containing a representative that admits (Z₂)^k-actions with a fixed point set of constant codimension r. In this paper we determine J_*,k^{2k + 2} and J_*,3^{23 + 1}.
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  (g, f)-Factorizations Randomly Orthogonal to a Subgraph in Graphs

  Hao ZHAO, Gui Zhen LIU, Xiao Xia YAN

  Acta Mathematica Sinica. 2005, 21(2): 413-422. https://doi.org/10.1007/s10114-004-0482-4

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  Let G be a graph with vertex set V (G) and edge set E(G) and let g and f be two integervalued functions defined on V (G) such that 2k - 2 ≤ g(x) ≤ f(x) for all x ∈ V (G). Let H be a subgraph of G with mk edges. In this paper, it is proved that every (mg +m- 1,mf - m + 1)-graph G has (g, f)-factorizations randomly k-orthogonal to H under some special conditions.
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  On Some New Integral Inequalities for Functions in One and Two Variables

  Young Ho KIM

  Acta Mathematica Sinica. 2005, 21(2): 423-434. https://doi.org/10.1007/s10114-004-0463-7

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  In this paper, we consider a bound on a general version of the integral inequalities for functions and also study the qualitative behavior of the solutions of certain classes of the hyperbolic partial delay differential equations under the integral inequalities.
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  On Embeddings of Tori in Euclidean Spaces

  Matija CENCELJ, Dušan REPOVŠ

  Acta Mathematica Sinica. 2005, 21(2): 435-438. https://doi.org/10.1007/s10114-004-0462-8

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  Using the relation between the set of embeddings of tori into Euclidean spaces modulo ambient isotopies and the homotopy groups of Stiefel manifolds, we prove new results on embeddings of tori into Euclidean spaces.
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  Strong Approximation by Cesàro Means with Critical Index in the Hardy Spaces H^p(S^d-1) (0 < p ≤ 1)

  Feng DAI, Kun Yang WANG

  Acta Mathematica Sinica. 2005, 21(2): 439-448. https://doi.org/10.1007/s10114-004-0423-2

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  Let S^d-1 = {x:|x| = 1} be a unit sphere of the d-dimensional Euclidean space R^d and let H^p≡H^p(S^d-1) (0 < p ≤ 1) denote the real Hardy space on S^d-1. For 0 < p ≤ 1 and f∈H^p(S^d-1), let E_j(f,H^p) (j = 0, 1,…) be the best approximation of f by spherical polynomials of degree less than or equal to j, in the space H^p(S^d-1), its Cesàro mean of order δ > -1 is denoted by σ_k^δ(f). For 0 < p ≤ 1, it is known that δ(p):= d-1/p-d/2 is the critical index for the uniform summability of σ_k^δ in the metric H^p. In this paper, the following result is proved:   Theorem Let 0<p<1 and δ(p):= d-1/p-d/2. Then for f∈H^p(S^d-1),
  ,
  where A_N(f)≈B_N(f) means that there's a positive constant C, independent of N and f, such that C^-1 A_N(f)≤B_N(f)≤A_N(f).
  In the case d = 2, this result was proved by Belinskii in 1996.