15 December 2010, Volume 27 Issue 1

    

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  Parabolic Marcinkiewicz Integrals along Surfaces on Product Domains

  Ahmad AL-SALMAN

  Acta Mathematica Sinica. 2011, 27(1): 1-18. https://doi.org/10.1007/s10114-010-9653-7

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  In this paper, parabolic Marcinkiewicz integral operators along surfaces on the product domain Rⁿ × R^m (n,m ≥ 2) are introduced. L^p bounds of such operators are obtained under weak conditions on the kernels.  
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  Diffeomorphisms with L^p-Shadowing Property

  Abbas FAKHARI, Keonhee LEE, Khosro TAJBAKHSH

  Acta Mathematica Sinica. 2011, 27(1): 19-28. https://doi.org/10.1007/s10114-011-0050-7

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  We give a characterization of structurally stable diffeomorphisms by making use of the notion of L^p-shadowing property. More precisely, we prove that the set of structurally stable diffeomorphisms coincides with the C¹-interior of the set of diffeomorphisms having L^p-shadowing property.    
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  Twists and Gromov Hyperbolicity of Riemann Surfaces

  Katsuhiko MATSUZAKI, José M. RODRÍGUEZ

  Acta Mathematica Sinica. 2011, 27(1): 29-44. https://doi.org/10.1007/s10114-011-9693-7

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  The main aim of this paper is to study whether the Gromov hyperbolicity is preserved under some transformations on Riemann surfaces (with their Poincaré metrics). We prove that quasiconformal maps between Riemann surfaces preserve hyperbolicity; however, we also show that arbitrary twists along simple closed geodesics do not preserve it, in general.  
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  Twisted Hamiltonian Lie Algebras and Their Multiplicity-Free Representations

  Ling CHEN

  Acta Mathematica Sinica. 2011, 27(1): 45-72. https://doi.org/10.1007/s10114-011-0204-7

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  We construct a class of new Lie algebras by generalizing the one-variable Lie algebras generated by the quadratic conformal algebras (or corresponding Hamiltonian operators) associated with Poisson algebras and a quasi-derivation found by Xu. These algebras can be viewed as certain twists of Xu’s generalized Hamiltonian Lie algebras. The simplicity of these algebras is completely determined. Moreover, we construct a family of multiplicity-free representations of these Lie algebras and prove their irreducibility.  
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  Partitioning a Graph into Defensive k-Alliances

  Ismael G. YERO, Sergio BERMUDO, Juan A. RODRíGUEZ-VELÁZQUEZ, José M. SIGARRETA

  Acta Mathematica Sinica. 2011, 27(1): 73-82. https://doi.org/10.1007/s10114-011-9075-1

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  A defensive k-alliance in a graph is a set S of vertices with the property that every vertex in S has at least k more neighbors in S than it has outside of S. A defensive k-alliance S is called global if it forms a dominating set. In this paper we study the problem of partitioning the vertex set of a graph into (global) defensive k-alliances. The (global) defensive k-alliance partition number of a graph Γ = (V,E), (ψ_k^gd (Γ)) ψ_k^d(Γ), is defined to be the maximum number of sets in a partition of V such that each set is a (global) defensive k-alliance. We obtain tight bounds on ψ_k^d(Γ) and ψ_k^gd(Γ) in terms of several parameters of the graph including the order, size, maximum and minimum degree, the algebraic connectivity and the isoperimetric number. Moreover, we study the close relationships that exist among partitions of Γ₁ × Γ₂ into (global) defensive (k₁ + k₂)-alliances and partitions of Γ_{em into (global) defensive kem}-alliances, i ∈ {1, 2}.  
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  An Oscillating Operator Related to Wave Equations in the Block Spaces

  Dashan FAN, Lijing SUN

  Acta Mathematica Sinica. 2011, 27(1): 83-96. https://doi.org/10.1007/s10114-011-9420-4

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  In this article, we study certain oscillating multipliers related to Cauchy problem for the wave equations on the Euclidean space and on the torus. We obtain that, at the end point, these operators are bounded from the L^p spaces to certain block spaces.  
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  L-fuzzy Roughness of n-ary Polygroups

  Yun Qiang YIN, Jian Ming ZHAN, P. CORSINI

  Acta Mathematica Sinica. 2011, 27(1): 97-118. https://doi.org/10.1007/s10114-011-9465-4

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  In this paper, we consider the relations among L-fuzzy sets, rough sets and n-ary polygroup theory. Some properties of (normal) TL-fuzzy n-ary subpolygroups of an n-ary polygroup are first obtained. Using the concept of L-fuzzy sets, the notion of υ-lower and T-upper L-fuzzy rough approximation operators with respect to an L-fuzzy set is introduced and some related properties are presented. Then a new algebraic structure called (normal) TL-fuzzy rough n-ary polygroup is defined and investigated. Also, the (strong) homomorphism of υ-lower and T-upper L-fuzzy rough approximation operators is studied.  
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  Interpolation of Vector Measures

  Ricardo del CAMPO, Antonio FERNÁNDEZ, Fernando MAYORAL, Francisco NARANJO, Enrique A. SÁNCHEZ-PÉREZ

  Acta Mathematica Sinica. 2011, 27(1): 119-134. https://doi.org/10.1007/s10114-011-9542-8

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  Let (Ω, Σ) be a measurable space and m0 : Σ → X₀ and m1 : Σ → X₁ be positive vector measures with values in the Banach Köthe function spaces X₀ and X₁. If 0 < α < 1, we define a new vector measure [m₀,m₁]α with values in the Calderón lattice interpolation space X₀^1-α X₁^α and we analyze the space of integrable functions with respect to measure [m₀,m₁]α in order to prove suitable extensions of the classical Stein-Weiss formulas that hold for the complex interpolation of L^p-spaces. Since each p-convex order continuous Köthe function space with weak order unit can be represented as a space of p-integrable functions with respect to a vector measure, we provide in this way a technique to obtain representations of the corresponding complex interpolation spaces. As applications, we provide a Riesz-Thorin theorem for spaces of p-integrable functions with respect to vector measures and a formula for representing the interpolation of the injective tensor product of such spaces.  
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  A Sufficient Degree Condition for a Graph to Contain All Trees of Size k

  Camino BALBUENA, Alberto MÁRQUEZ, José Ramón PORTILLO

  Acta Mathematica Sinica. 2011, 27(1): 135-140. https://doi.org/10.1007/s10114-011-9617-6

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  The Erdös-Sós conjecture says that a graph G on n vertices and number of edges e(G) > n(k - 1)/2 contains all trees of size k. In this paper we prove a sufficient condition for a graph to contain every tree of size k formulated in terms of the minimum edge degree ξ(G) of a graph G defined as ξ(G) = min{d(u) + d(v) - 2 : uv ∈ E(G)}. More precisely, we show that a connected graph G with maximum degree Δ(G) ≥ k and minimum edge degree ξ(G) ≥ 2k - 4 contains every tree of k edges if d_G(x) + d_G(y) ≥ 2k - 4 for all pairs x, y of nonadjacent neighbors of a vertex u of d_G(u) ≥ k.  
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  A Criterion of Normality Based on a Single Holomorphic Function

  Xiao Jun LIU, Shahar NEVO

  Acta Mathematica Sinica. 2011, 27(1): 141-154. https://doi.org/10.1007/s10114-011-9396-0

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  Let F be a family of functions holomorphic on a domain D ⊂ C. Let k ≥ 2 be an integer and let h be a holomorphic function on D, all of whose zeros have multiplicity at most k -1, such that h(z) has no common zeros with any f ∈ F. Assume also that the following two conditions hold for every f ∈ F :  Then F is normal on D.  
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  Hierarchical Management Scheme by Local Fields

  Zhi Hong YUE, Mao Zhi XU

  Acta Mathematica Sinica. 2011, 27(1): 155-168. https://doi.org/10.1007/s10114-011-9110-2

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  In the present paper we propose a new hierarchical management scheme within suitable extensions of 2 -adic numbers, in which the subfields are treated as classes, and the communications are realized by computation on elliptic curves. The security of our system is much stronger than that of the traditional public key cryptographic systems.  
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  The Critical Group of K_m× C_n

  Jian WANG, Yong Liang PAN, Jun Ming XU

  Acta Mathematica Sinica. 2011, 27(1): 169-184. https://doi.org/10.1007/s10114-011-9063-5

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  In this paper, the structure of the critical group of the graph K_m× C_n is determined,where m, n ≥ 3.  
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  A Characterization of Linearly Semisimple Groups

  Amelia ÁLVAREZ, Pedro SANCHO, Carlos SANCHO

  Acta Mathematica Sinica. 2011, 27(1): 185-192. https://doi.org/10.1007/s10114-011-8658-1

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  Let G = Spec A be an affine K-group scheme and à = {w ∈ A*: dimK A*·w < ∞, dimK w· A* < ∞}. Let <-, -> : A* × Ã → K, <w,w> := tr(ww), be the trace form. We prove that G is linearly reductive if and only if the trace form is non-degenerate on A*.  
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  Nonlinear Preserver Problems on B(H)

  Jian Lian CUI

  Acta Mathematica Sinica. 2011, 27(1): 193-202. https://doi.org/10.1007/s10114-011-9106-y

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  Let H be a complex Hilbert space of dimension greater than 2, and B(H) denote the Banach algebra of all bounded linear operators on H. For A, B ∈ B(H), define the binary relation A ≤_* B by A*A = A*B and AA* = AB*. Then (B(H), “≤_*”) is a partially ordered set and the relation “≤_*” is called the star order on B(H). Denote by B_s(H) the set of all self-adjoint operators in B(H). In this paper, we first characterize nonlinear continuous bijective maps on B_s(H) which preserve the star order in both directions. We characterize also additive maps (or linear maps) on B(H) (or nest algebras) which are multiplicative at some invertible operator.  
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  Local Analysis, Cardinality, and Split Trick of Quasi-biorthogonal Frame Wavelets

  Zhi Hua ZHANG

  Acta Mathematica Sinica. 2011, 27(1): 203-218. https://doi.org/10.1007/s10114-011-8172-5

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  The notion of quasi-biorthogonal frame wavelets is a generalization of the notion of orthogonal wavelets. A quasi-biorthogonal frame wavelet with the cardinality r consists of r pairs of functions. In this paper we first analyze the local property of the quasi-biorthogonal frame wavelet and show that its each pair of functions generates reconstruction formulas of the corresponding subspaces. Next we show that the lower bound of its cardinalities depends on a pair of dual frame multiresolution analyses deriving it. Finally, we present a split trick and show that any quasi-biorthogonal frame wavelet can be split into a new quasi-biorthogonal frame wavelet with an arbitrarily large cardinality. For generality, we work in the setting of matrix dilations.