15 April 2012, Volume 28 Issue 4

    

• Select all
  |

Articles
• Select
  On a Variant of Giuga Numbers

  José María GRAU, Florian LUCA

  Acta Mathematica Sinica. 2012, 28(4): 653-660. https://doi.org/10.1007/s10114-011-1148-7

  Abstract ( ) Download PDF ( )   Knowledge map   Save

  In this paper, we characterize the odd positive integers n satisfying the congruence Σ_j=1^n-1 j^n-1/2≡0 (mod n). We show that the set of such positive integers has an asymptotic density which turns out to be slightly larger than 3/8.
• Select
  On Product-Cordial Index Sets and Friendly Index Sets of 2-Regular Graphs and Generalized Wheels

  Harris KWONG, Sin Min LEE, Ho Kuen NG

  Acta Mathematica Sinica. 2012, 28(4): 661-674. https://doi.org/10.1007/s10114-011-0515-8

  Abstract ( ) Download PDF ( )   Knowledge map   Save

  A vertex labeling f:V→Z₂ of a simple graph G=(V,E)induces two edge labelings f⁺, f^*:E→Z₂ defined by f⁺(uv)=f(u)+f(v)andf^*(uv)=f(u)f(v). For each em∈Z₂, let vf(i)= |{v∈V:f(v)=i}|, e_f⁺(em)=|{e∈E:f⁺(e)=em}| and e_f^*(em)=|{e∈E:f^*(e)=i}|. We call f friendly if |vf(0)-vf(1)|≤1. The friendly index set and the product-cordial index set of G are defined as the sets{|e_f⁺(0)-e_f⁺(1)|:f is friendly} and {|e_f^* (0)-e_f^*(1)|:f is friendly}. In this paper we study and determine the connection between the friendly index sets and product-cordial index sets of 2-regular graphs and generalized wheel graphs.
• Select
  Edge-Fault-Tolerant Edge-Bipancyclicity of Bubble-Sort Graphs

  Xin Ping XU, Min XU, Jin JING

  Acta Mathematica Sinica. 2012, 28(4): 675-686. https://doi.org/10.1007/s10114-011-0511-z

  Abstract ( ) Download PDF ( )   Knowledge map   Save

  The bubble-sort graph B_n is a bipartite graph. Kikuchi and Araki [Edge-bipancyclicity and edge-fault-tolerant bipancyclicity of bubble-sort graphs. Information Processing Letters, 100, 52-59 (2006)] have proved that B_n is edge-bipancyclic for >n≥ 5 and B_n-F is bipancyclic when n≥ 4 and |F| ≤ n-3. In this paper, we improve this result by showing that for any edge set F of B_n with |F| ≤>n-3, every edge of B_n-F lies on a cycle of every even length from 6 to n, for n≥ 5 and every edge of B_n-F lies on a cycle of every even length from 8 to n, for >n= 4.
• Select
  On the Random Conjugate Spaces of a Random Locally Convex Module

  Tie Xin GUO, Shi En ZHAO

  Acta Mathematica Sinica. 2012, 28(4): 687-696. https://doi.org/10.1007/s10114-011-0408-x

  Abstract ( ) Download PDF ( )   Knowledge map   Save

  Theoretically speaking, there are four kinds of possibilities to define the random conjugate space of a random locally convex module. The purpose of this paper is to prove that among the four kinds there are only two which are universally suitable for the current development of the theory of random conjugate spaces. In this process, we also obtain a somewhat surprising and crucial result: if the base (Ω, F, P) of a random normed module is nonatomic then the random normed module is a totally disconnected topological space when it is endowed with the locally L⁰-convex topology.
• Select
  Structure of Independent Sets in Direct Products of Some Vertex-transitive Graphs

  Xing Bo GENG, Jun WANG, Hua Jun ZHANG

  Acta Mathematica Sinica. 2012, 28(4): 697-706. https://doi.org/10.1007/s10114-011-0311-5

  Abstract ( ) Download PDF ( )   Knowledge map   Save

  Let Circ(r, n) be a circular graph. It is well known that its independence number α(Circ(r, n))=r. In this paper we prove that α(Circ(r, n)×H)=max{r|H|, nα(H)} for every vertex transitive graph H, and describe the structure of maximum independent sets in Circ (r, n)×H. As consequences, we prove α(G×H)=max{α(G)V(H)|, α(H)|V(G)|} for G being Kneser graphs, and the graphs defined by permutations and partial permutations, respectively. The structure of maximum independent sets in these direct products is also described.
• Select
  Normality and Quasinormality of Zero-free Meromorphic Functions

  Jian Ming CHANG

  Acta Mathematica Sinica. 2012, 28(4): 707-716. https://doi.org/10.1007/s10114-011-0297-z

  Abstract ( ) Download PDF ( )   Knowledge map   Save

  Let k, K∈N and F be a family of zero-free meromorphic functions in a domain D such that for each f∈F, f^(k)-1 has at most K zeros, ignoring multiplicity. Then F is quasinormal of order at most v=K/k+1, where v is equal to the largest integer not exceeding K/k+1. In particular, if K=k, then F is normal. The results are sharp.
• Select
  On Ha's Version of Set-valued Ekeland's Variational Principle

  Jing Hui QIU

  Acta Mathematica Sinica. 2012, 28(4): 717-726. https://doi.org/10.1007/s10114-011-0294-2

  Abstract ( ) Download PDF ( )   Knowledge map   Save

  By using the concept of cone extensions and Dancs-Hegedus-Medvegyev theorem, Ha [Some variants of the Ekeland variational principle for a set-valued map. J. Optim. Theory Appl., 124, 187-206 (2005)] established a new version of Ekeland's variational principle for set-valued maps, which is expressed by the existence of strict approximate minimizer for a set-valued optimization problem. In this paper, we give an improvement of Ha's version of set-valued Ekeland's variational principle. Our proof is direct and it need not use Dancs-Hegedus-Medvegyev theorem. From the improved Ha's version, we deduce a Caristi-Kirk's fixed point theorem and a Takahashi's nonconvex minimization theorem for set-valued maps. Moreover, we prove that the above three theorems are equivalent to each other.
• Select
  Strongly Irreducible Operators on Banach Spaces

  Yun Nan ZHANG, Huai Jie ZHONG

  Acta Mathematica Sinica. 2012, 28(4): 727-740. https://doi.org/10.1007/s10114-011-0157-x

  Abstract ( ) Download PDF ( )   Knowledge map   Save

  This paper firstly discusses the existence of strongly irreducible operators on Banach spaces. It shows that there exist strongly irreducible operators on Banach spaces with w^*-separable dual. It also gives some properties of strongly irreducible operators on Banach spaces. In particular, if T is a strongly irreducible operator on an infinite-dimensional Banach space, then T is not of finite rank and T is not an algebraic operator. On Banach spaces with subsymmetric bases, including infinite-dimensional separable Hilbert spaces, it shows that quasisimilarity does not preserve strong irreducibility. In addition, we show that the strong irreducibility of an operator does not imply the strong irreducibility of its conjugate operator, which is not the same as the property in Hilbert spaces.
• Select
  Modified Block Iterative Method for Solving Convex Feasibility Problem, Equilibrium Problems and Variational Inequality Problems

  Shi Sheng ZHANG, Chi Kin CHAN, H. W. JOSEPH LEE

  Acta Mathematica Sinica. 2012, 28(4): 741-758. https://doi.org/10.1007/s10114-011-0099-3

  Abstract ( ) Download PDF ( )   Knowledge map   Save

  The purpose of this paper is by using the modified block iterative method to propose an algorithm for finding a common element in the intersection of the set of common fixed points of an infinite family of quasi-φ-asymptotically nonexpansive and the set of solutions to an equilibrium problem and the set of solutions to a variational inequality. Under suitable conditions some strong convergence theorems are established in 2-uniformly convex and uniformly smooth Banach spaces. As applications we utilize the results presented in the paper to solving the convex feasibility problem (CFP) and zero point problem of maximal monotone mappings in Banach spaces. The results presented in the paper improve and extend the corresponding results announced by many authors.
• Select
  Quantum Hyperbolic Invariants for Diffeomorphisms of Small Surfaces

  Xiaobo LIU

  Acta Mathematica Sinica. 2012, 28(4): 759-770. https://doi.org/10.1007/s10114-011-0094-8

  Abstract ( ) Download PDF ( )   Knowledge map   Save

  An earlier article [Bonahon, F., Liu, X. B.: Representations of the quantum Teichmüller space and invariants of surface diffeomorphisms. Geom. Topol., 11, 889-937 (2007)] introduced new invariants for pseudo-Anosov diffeomorphisms of surface, based on the representation theory of the quantum Teichmüller space. We explicitly compute these quantum hyperbolic invariants in the case of the 1-puncture torus and the 4-puncture sphere.
• Select
  Tame Kernels of Pure Cubic Fields

  Xiao Yun CHENG

  Acta Mathematica Sinica. 2012, 28(4): 771-780. https://doi.org/10.1007/s10114-011-0089-5

  Abstract ( ) Download PDF ( )   Knowledge map   Save

  In this paper, we study the p-rank of the tame kernels of pure cubic fields. In particular, we prove that for a fixed positive integer m, there exist infinitely many pure cubic fields whose 3-rank of the tame kernel equal to m. As an application, we determine the 3-rank of their tame kernels for some special pure cubic fields.
• Select
  On the Rates of the Other Law of the Logarithm

  Li-Xin ZHANG, You-You CHEN

  Acta Mathematica Sinica. 2012, 28(4): 781-792. https://doi.org/10.1007/s10114-011-0082-z

  Abstract ( ) Download PDF ( )   Knowledge map   Save

  Let X,X₁,X₂, . . . be i.i.d. random variables, and set S_n=X₁+· · ·+X_n, M_n=max_k≤n|S_k|, n≥1. Let a_n=o(√n/logn). By using the strong approximation, we prove that, if EX=0, VarX=σ² > 0 and E|X|^2+ε < ∞ for some ε > 0, then for any r > 1,   We also show that the widest a_n=o(√n/logn).
• Select
  A Note on Lower Bound of Centered L₂-discrepancy on Combined Designs

  Yi Ju LEI, Zu Jun OU, Hong QIN, Na ZOU

  Acta Mathematica Sinica. 2012, 28(4): 793-800. https://doi.org/10.1007/s10114-011-0009-8

  Abstract ( ) Download PDF ( )   Knowledge map   Save

  This note provides a theoretical justification of optimal foldover plans in terms of uniformity. A new lower bound of the centered L₂-discrepancy values of combined designs is obtained, which can be used as a benchmark for searching optimal foldover plans. Our numerical results show that this lower bound is sharper than existing results when more factors reverse the signs in the initial design.
• Select
  The Stability of the Elliptic Equilibrium of Planar Quasi-periodic Hamiltonian Systems

  Yun Chao WU, Yi Qian WANG

  Acta Mathematica Sinica. 2012, 28(4): 801-816. https://doi.org/10.1007/s10114-011-0006-y

  Abstract ( ) Download PDF ( )   Knowledge map   Save

  In this paper, we study the planar Hamiltonian system  where f is real analytic in x and θ, A(θ) is a 2×2 real analytic symmetric matrix, J=(1^-1) and ω is a Diophantine vector. Under the assumption that the unperturbed system x= JA(θ)x, θ=ω is reducible and stable, we obtain a series of criteria for the stability and instability of the equilibrium of the perturbed system.
• Select
  The Structure on Invariant Measures of C¹ Generic Diffeomorphisms

  Wen Xiang SUN, Xue Ting TIAN

  Acta Mathematica Sinica. 2012, 28(4): 817-824. https://doi.org/10.1007/s10114-011-9723-5

  Abstract ( ) Download PDF ( )   Knowledge map   Save

  Let Λ be an isolated non-trivial transitive set of a C¹ generic diffeomorphism f∈Diff(M). We show that the space of invariant measures supported on Λ coincides with the space of accumulation measures of time averages on one orbit. Moreover, the set of points having this property is residual in Λ (which implies that the set of irregular⁺ points is also residual in Λ). As an application, we show that the non-uniform hyperbolicity of irregular⁺ points in Λ with totally 0 measure (resp., the non-uniform hyperbolicity of a generic subset in Λ) determines the uniform hyperbolicity of Λ.
• Select
  Equivariant Normal Forms for Parameterized Delay Differential Equations with Applications to Bifurcation Theory

  Shang Jiang GUO, Yu Ming CHEN, Jian Hong WU

  Acta Mathematica Sinica. 2012, 28(4): 825-856. https://doi.org/10.1007/s10114-011-9718-2

  Abstract ( ) Download PDF ( )   Knowledge map   Save

  In this paper, we develop an efficient approach to compute the equivariant normal form of delay differential equations with parameters in the presence of symmetry. We present and justify a process that involves center manifold reduction and normalization preserving the symmetry, and that yields normal forms explicitly in terms of the coefficients of the original system. We observe that the form of the reduced vector field relies only on the information of the linearized system at the critical point and on the inherent symmetry, and the normal forms give critical information about not only the existence but also the stability and direction of bifurcated spatiotemporal patterns. We illustrate our general results by some applications to fold bifurcation, equivariant Hopf bifurcation and Hopf-Hopf interaction, with a detailed case study of additive neurons with delayed feedback.
• Select
  L^r Convergence for B-valued Random Elements

  Ping Yan CHEN, Ding Cheng WANG

  Acta Mathematica Sinica. 2012, 28(4): 857-868. https://doi.org/10.1007/s10114-011-9475-2

  Abstract ( ) Download PDF ( )   Knowledge map   Save

  The paper investigates L^p convergence and Marcinkiewicz-Zygmund strong laws of large numbers for random elements in a Banach space under the condition that the Banach space is of Rademacher type p, 1 < p < 2. The paper also discusses L^r convergence and L^r bound for random elements without any geometric restriction condition on the Banach space.