15 July 2012, Volume 28 Issue 7

    

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  Exponential Polynomials as Solutions of Certain Nonlinear Difference Equations

  Zhi Tao, WEN Janne, HEITTOKANGAS, Ilpo LAINE

  Acta Mathematica Sinica. 2012, 28(7): 1295-1306. https://doi.org/10.1007/s10114-012-1484-2

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  Recently, C.-C. Yang and I. Laine have investigated finite order entire solutions f of nonlinear differential-difference equations of the form fⁿ + L(z, f) = h, where n ≥ 2 is an integer. In particular, it is known that the equation f(z)² +q(z)f(z +1) = p(z), where p(z), q(z) are polynomials, has no transcendental entire solutions of finite order. Assuming that Q(z) is also a polynomial and c ∈ C, equations of the form f(z)ⁿ +q(z)e^Q(z)f(z +c) = p(z) do posses finite order entire solutions. A classification of these solutions in terms of growth and zero distribution will be given. In particular, it is shown that any exponential polynomial solution must reduce to a rather specific form. This reasoning relies on an earlier paper due to N. Steinmetz.
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  Continuous Embeddings of Besov-Morrey Function Spaces

  Dorothee D. HAROSKE, Leszek SKRZYPCZAK

  Acta Mathematica Sinica. 2012, 28(7): 1307-1328. https://doi.org/10.1007/s10114-012-1119-7

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  We study embeddings of spaces of Besov-Morrey type, MB_p1,q1^s1,r1 (R^d) → MB_p2,q2^s2,r2 (R^d), and obtain necessary and sufficient conditions for this. Moreover, we can also characterise the special weighted situation B_p1,r1^s1 (R^d,w) → MB_p2,q2^s2,r2 (R^d) for a Muckenhoupt A_∞ weight w, with w_α(x) = |x|^α, α > -d, as a typical example.
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  Atomic Decomposition and Boundedness Criterion of Operators on Multi-parameter Hardy Spaces of Homogeneous Type

  Guo Zhen LU, Ya Yuan XIAO

  Acta Mathematica Sinica. 2012, 28(7): 1329-1346. https://doi.org/10.1007/s10114-012-0630-1

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  The main purpose of this paper is to derive a new (p, q)-atomic decomposition on the multi-parameter Hardy space H^p(X₁ × X₂) for 0 < p₀ < p ≤ 1 for some p₀ and all 1 < q < ∞, where X₁ × X₂ is the product of two spaces of homogeneous type in the sense of Coifman and Weiss. This decomposition converges in both L^q(X₁ × X₂) (for 1 < q < ∞) and Hardy space H^p(X₁ × X₂) (for 0 < p ≤ 1). As an application, we prove that an operator T, which is bounded on L^q(X₁ × X₂) for some 1 < q < ∞, is bounded from H^p(X₁ × X₂) to L^p(X₁ × X₂) if and only if T is bounded uniformly on all (p, q)-product atoms in L^p(X₁ × X₂). The similar boundedness criterion from H^p(X₁ × X₂) to H^p(X₁ × X₂) is also obtained.
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  New Construction of Complex Manifold via Conifold Transition

  Jin Xing XU

  Acta Mathematica Sinica. 2012, 28(7): 1347-1368. https://doi.org/10.1007/s10114-012-0666-2

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  For a generic anti-canonical hypersurface in each smooth toric Fano 4-fold with rank 2 Picard group, we prove there exist three isolated rational curves in it. Moreover, for all these 4-folds except one, the contractions of generic anti-canonical hypersurfaces along the three rational curves can be deformed to smooth threefolds which is diffeomorphic to connected sums of S³?S³. In this manner, we obtain complex structures with trivial canonical bundles on some connected sums of S³?S³. This construction is an analogue of that made by Friedman [On threefolds with trivial canonical bundle. In: Complex Geometry and Lie Theory, volume 53 of Proc. Sympos. Pure Math., Amer. Math. Soc., Providence, RI, 1991, 103-134], Lu and Tian [Complex structures on connected sums of S³?S³. In: Manifolds and Geometry, Pisa, 1993, 284-293] who used only quintics in P4.
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  Characterizations of Universal Finite Representability and B-convexity of Banach Spaces via Ball Coverings

  Wen ZHANG

  Acta Mathematica Sinica. 2012, 28(7): 1369-1374. https://doi.org/10.1007/s10114-012-0721-z

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  By a ball-covering B of a Banach space X, wemean that B is a collection of open (or closed) balls off the origin whose union contains the unit sphere of X; and X is said to have the ball-covering property provided it admits a ball-covering of countably many balls. This paper shows that universal finite representability and B-convexity of X can be characterized by properties of ball-coverings of its finite dimensional subspaces.
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  Multilinear Fractional Integrals on Morrey Spaces

  Takeshi IIDA, Enji SATO, Yoshihiro SAWANO, Hitoshi TANAKA

  Acta Mathematica Sinica. 2012, 28(7): 1375-1384. https://doi.org/10.1007/s10114-012-0617-y

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  In the present paper we obtain and extend the boundedness property of the Adams type for multilinear fractional integral operators. Also, we deal with the Olsen type inequality.
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  Some Strong Laws of Large Numbers for Blockwise Martingale Difference Sequences in Martingale Type p Banach Spaces

  Andrew ROSALSKY, Le Van THANH

  Acta Mathematica Sinica. 2012, 28(7): 1385-1400. https://doi.org/10.1007/s10114-012-0378-7

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  For a blockwise martingale difference sequence of random elements {V_n, n ≥ 1} taking values in a real separable martingale type p (1 ≤ p ≤ 2) Banach space, conditions are provided for strong laws of large numbers of the form lim_n→∞∑_i=1ⁿ V_i/g_n = 0 almost surely to hold where the constants g_n ↑ ∞. A result of Hall and Heyde [Martingale Limit Theory and Its Application, Academic Press, New York, 1980, p. 36] which was obtained for sequences of random variables is extended to a martingale type p (1 < p ≤ 2) Banach space setting and to hold with a Marcinkiewicz-Zygmund type normalization. Illustrative examples and counterexamples are provided.
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  Probabilistic Normed Riesz Spaces

  Celaleddin SENCIMEN, Serpil PEHLIVAN

  Acta Mathematica Sinica. 2012, 28(7): 1401-1410. https://doi.org/10.1007/s10114-012-0305-y

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  In this paper, the concepts of probabilistic normed Riesz space and probabilistic Banach lattice are introduced, and their basic properties are studied. In this context, some continuity and convergence theorems are proved.
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  Some Sufficient Conditions for Convergent Multivariate Subdivision Schemes with Nonnegative Finite Masks

  Min WU, Jia Li ZHOU

  Acta Mathematica Sinica. 2012, 28(7): 1411-1420. https://doi.org/10.1007/s10114-012-0267-0

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  It is well known that the convergence of multivariate subdivision schemes with finite masks can be characterized via joint spectral radius. For nonnegative masks, we will present in this paper some computable simply sufficient conditions for the convergence, which will cover a substantially large class of schemes.
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  Upper Bounds for Ruin Probabilities under Stochastic Interest Rate and Optimal Investment Strategies

  Jin Zhu, LI Rong WU

  Acta Mathematica Sinica. 2012, 28(7): 1421-1430. https://doi.org/10.1007/s10114-012-0153-9

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  In this paper, we study the upper bounds for ruin probabilities of an insurance company which invests its wealth in a stock and a bond. We assume that the interest rate of the bond is stochastic and it is described by a Cox-Ingersoll-Ross (CIR) model. For the stock price process, we consider both the case of constant volatility (driven by an O-U process) and the case of stochastic volatility (driven by a CIR model). In each case, under certain conditions, we obtain the minimal upper bound for ruin probability as well as the corresponding optimal investment strategy by a pure probabilistic method.
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  Complementarity Properties of the Lyapunov Transformation over Symmetric Cones

  Yuan Min LI, Xing Tao WANG, De Yun WEI

  Acta Mathematica Sinica. 2012, 28(7): 1431-1442. https://doi.org/10.1007/s10114-012-0138-8

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  The well-known Lyapunov's theorem in matrix theory/continuous dynamical systems asserts that a square matrix A is positive stable if and only if there exists a positive definite matrix X such that AX+XA* is positive definite. In this paper, we extend this theorem to the setting of any Euclidean Jordan algebra V . Given any element a ∈ V, we consider the corresponding Lyapunov transformation L^a and show that the P and S-properties are both equivalent to a being positive. Then we characterize the R₀-property for L^a and show that L^a has the R₀-property if and only if a is invertible. Finally, we provide L^a with some characterizations of the E₀-property and the nondegeneracy property.
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  Strong Connections and Invertible Weak Entwining Structures

  J. N. ALONSO ÁLVAREZ, J. M. FERNÁNDEZ VILABOA, R. GONZÁLEZ RODRÍGUEZ

  Acta Mathematica Sinica. 2012, 28(7): 1443-1460. https://doi.org/10.1007/s10114-012-0088-1

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  In this paper we obtain a criterion under which the bijectivity of the canonical morphism of a weak Galois extension associated to a weak invertible entwining structure is equivalent to the existence of a strong connection form. Also we obtain an explicit formula for a strong connection under equivariant projective conditions or under coseparability conditions.
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  On Semilattice Decomposition of an Abel-Grassmann's Groupoid

  Madad KHAN, Saima ANIS

  Acta Mathematica Sinica. 2012, 28(7): 1461-1468. https://doi.org/10.1007/s10114-012-0014-6

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  In this paper, we have decomposed an AG-groupoid. Let S be an AG-groupoid with left identity and a relation γ be defined on S as: aγb if and only if there exist positive integers m and n such that b^m ∈ (Sa)S and aⁿ ∈ (Sb)S for all a and b in S. We have proved that S/γ is a maximal separative semilattice homomorphic image of S. Every AG-groupoid S is uniquely expressible as a semilattice Y of archimedean AG-groupoids S_α (α ∈ Y ). The semilattice Y is isomorphic to S/γ and the S_α (α ∈ Y) are the equivalence classes of S mod γ.
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  A Note on the Uniqueness of Koebe-Andreev-Thurston Theorem

  Xiao Jun HUANG, Zi Peng WANG

  Acta Mathematica Sinica. 2012, 28(7): 1469-1474. https://doi.org/10.1007/s10114-012-9584-6

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  In this paper, we present a new proof of the uniqueness of Koebe-Andreev-Thurston theorem. Our method is based on the argument principle in complex analysis and reviews the connection between the circle packing theorem and complex analysis.
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  ε₀-Regularity for Mean Curvature Flow from Surface to Flat Riemannian Manifold

  Xiao Li HAN, Jun SUN

  Acta Mathematica Sinica. 2012, 28(7): 1475-1490. https://doi.org/10.1007/s10114-012-9271-7

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  In this paper we prove an ε₀-regularity theorem for mean curvature flow from surface to a flat Riemannian manifold. More precisely, we prove that if the initial energy ∫_Σ0 |A|² ≤ ε₀ and the initial area μ₀(Σ₀) is not large, then along the mean curvature flow, we have ∫_Σt |A|² ≤ ε₀. As an application, we obtain the long time existence and convergence result of the mean curvature flow.
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  Some Dichotomy Results for the Quenching Problem

  Gao Feng ZHENG

  Acta Mathematica Sinica. 2012, 28(7): 1491-1506. https://doi.org/10.1007/s10114-012-9143-1

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  It is shown that any solution to the semilinear problem
   
  either touches 1 in finite time or converges smoothly to a steady state as t→∞. Some extensions of this result to higher dimensions are also discussed.