15 March 2014, Volume 30 Issue 3

    

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  Explicit Stationary Distribution of the (L, 1)-reflecting Random Walk on the Half Line

  Wen Ming HONG, Ke ZHOU, Yi Qiang Q. ZHAO

  Acta Mathematica Sinica. 2014, 30(3): 371-388. https://doi.org/10.1007/s10114-014-3009-7

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  In this paper, we consider the (L, 1) state-dependent reflecting random walk (RW) on the half line, which is an RW allowing jumps to the left at a maximal size L. For this model, we provide an explicit criterion for (positive) recurrence and an explicit expression for the stationary distribution. As an application, we prove the geometric tail asymptotic behavior of the stationary distribution under certain conditions. The main tool employed in the paper is the intrinsic branching structure within the (L, 1)-random walk.
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  A Rao-type Characterization for a Sequence to Have a Realization Containing an Arbitrary Subgraph H

  Jian Hua YIN

  Acta Mathematica Sinica. 2014, 30(3): 389-394. https://doi.org/10.1007/s10114-014-2732-4

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  Let G be an arbitrary spanning subgraph of the complete graph K_r+1 on r+1 vertices and K_r+1-E(G) be the graph obtained from K_r+1 by deleting all edges of G. A non-increasing sequence π = (d₁, d₂,..., d_n) of nonnegative integers is said to be potentially K_r+1-E(G)-graphic if there is a graph on n vertices that has π as its degree sequence and contains K_r+1-E(G) as a subgraph. In this paper, a characterization of π that is potentially K_r+1-E(G)-graphic is given, which is analogous to the Erdős-Gallai characterization of graphic sequences using a system of inequalities. This is a solution to an open problem due to Lai and Hu. As a corollary, a characterization of π that is potentially K_s,t-graphic can also be obtained, where K_s,t is the complete bipartite graph with partite sets of size s and t. This is a solution to an open problem due to Li and Yin.
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  Large Deviations for Hitting Times of a Random Walk in Random Environment on a Strip

  Mei Juan ZHANG

  Acta Mathematica Sinica. 2014, 30(3): 395-410. https://doi.org/1007/s10114-014-2683-9

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  We consider a random walk in random environment on a strip, which is transient to the right. The random environment is stationary and ergodic. By the constructed enlarged random environment which was first introduced by Goldsheid (2008), we obtain the large deviations conditioned on the environment (in the quenched case) for the hitting times of the random walk.
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  Bifurcations of Limit Cycles from a Quintic Hamiltonian System with a Heteroclinic Cycle

  Li Qin ZHAO, De Ping LI

  Acta Mathematica Sinica. 2014, 30(3): 411-422. https://doi.org/10.1007/s10114-014-2615-8

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  In this paper, we consider Liénard systems of the form
  (dx)/(dt)= y,(dy)/(dt)= -(x + bx³-x⁵) +ε(α + βx³ + γx⁴)y,
  where b ∈ R, 0 < |ε|<<1, (α, β, γ) ∈ D ∈ R³ and D is bounded. We prove that for |b|>>1 (b < 0) the least upper bound of the number of isolated zeros of the related Abelian integrals I (h) =411(α + βx³ + γx⁴)ydx is 2 (counting the multiplicity) and this upper bound is a sharp one.
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  Sufficient Conditions for Error Bounds and Linear Regularity in Banach Spaces

  Zhou WEI, Qing Hai HE

  Acta Mathematica Sinica. 2014, 30(3): 423-436. https://doi.org/10.1007/s10114-014-2522-z

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  In this paper, we study error bounds for lower semicontinuous functions defined on Banach space and linear regularity for finitely many closed subset in Banach spaces. By using Clarke’s subdifferentials and Ekeland variational principle, we establish several sufficient conditions ensuring error bounds and linear regularity in Banach spaces.
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  Sharp Heat Kernel Estimates in the Fourier-Bessel Setting for a Continuous Range of the Type Parameter

  Adam NOWAK, Luz RONCAL

  Acta Mathematica Sinica. 2014, 30(3): 437-444. https://doi.org/10.1007/s10114-014-2512-1

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  The heat kernel in the setting of classical Fourier-Bessel expansions is defined by an oscillatory series which cannot be computed explicitly. We prove qualitatively sharp estimates of this kernel. Our method relies on establishing a connection with a situation of expansions based on Jacobi polynomials and then transferring known sharp bounds for the related Jacobi heat kernel.
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  The Product of the Restrained Domination Numbers of a Graph and Its Complement

  Johannes H. HATTINGH, Ernst J. JOUBERT

  Acta Mathematica Sinica. 2014, 30(3): 445-452. https://doi.org/10.1007/s10114-014-2492-1

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  Let G = (V,E) be a graph. A set S 445 V is a restrained dominating set if every vertex in V- S is adjacent to a vertex in S and to a vertex in V- S. The restrained domination number of G, denoted γ_r(G), is the smallest cardinality of a restrained dominating set of G. In this paper, we show that if G is a graph of order n ≥ 4, then γ_r(G)γ_r(G) ≤ 2n. We also characterize the graphs achieving the upper bound.
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  Liouvillian and Analytic Integrability of the Quadratic Vector Fields Having an Invariant Ellipse

  Jaume LLIBRE, Claudia VALLS

  Acta Mathematica Sinica. 2014, 30(3): 453-466. https://doi.org/10.1007/s10114-014-2484-1

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  We characterize the Liouvillian and analytic integrability of the quadratic polynomial vector fields in R² having an invariant ellipse. More precisely, a quadratic system having an invariant ellipse can be written into the form  = x² + y²-1 + y(ax + by + c), = -x(ax + by + c), and the ellipse becomes x² + y² = 1. We prove that
  (ⅰ) this quadratic system is analytic integrable if and only if a = 0;
  (ⅱ) if x² + y² = 1 is a periodic orbit, then this quadratic system is Liouvillian integrable if and only if x² + y² = 1 is not a limit cycle; and
  (ⅲ) if x² + y² = 1 is not a periodic orbit, then this quadratic system is Liouvilian integrable if and only if a = 0.
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  On an Entropy of Z₊^k-Actions

  Yu Jun ZHU, Wen Da ZHANG

  Acta Mathematica Sinica. 2014, 30(3): 467-480. https://doi.org/10.1007/s10114-014-2357-7

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  In this paper, a definition of entropy for Z₊^k(k ≥ 2)-actions due to Friedland is studied. Unlike the traditional definition, it may take a nonzero value for actions whose generators have finite (even zero) entropy as single transformations. Some basic properties are investigated and its value for the Z₊^k-actions on circles generated by expanding endomorphisms is given. Moreover, an upper bound of this entropy for the Z₊^k-actions on tori generated by expanding endomorphisms is obtained via the preimage entropies, which are entropy-like invariants depending on the “inverse orbits” structure of the system.
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  Entire Functions Sharing an Entire Function of Smaller Order with Their Difference Operators

  Xiao Min LI, Hong Xun YI

  Acta Mathematica Sinica. 2014, 30(3): 481-498. https://doi.org/10.1007/s10114-014-2042-x

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  We study a uniqueness question of entire functions sharing an entire function of smaller order with their difference operators, and deal with a question posed by Liu and Yang. The results in this paper extend the corresponding results obtained by Liu-Yang and by Liu-Laine respectively. Examples are provided to show that the results in this paper, in a sense, are the best possible.
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  A Note on Faber Operator

  Hua Ying WEI, Mei Li WANG, Yun HU

  Acta Mathematica Sinica. 2014, 30(3): 499-504. https://doi.org/10.1007/s10114-013-2423-6

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  For a rectifiable Jordan curve Γ with complementary domains D and D^*, Anderson conjectured that the Faber operator is a bounded isomorphism between the Besov spaces B_p (1 < p < ∞) of analytic functions in the unit disk and in the inner domain D, respectively. We point out that the conjecture is not true in the special case p = 2, and show that in this case the Faber operator is a bounded isomorphism if and only if Γ is a quasi-circle.
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  On the Clique-Transversal Number in (Claw, K₄)-Free 4-Regular Graphs

  Ding Guo WANG, Er Fang SHAN, Zuo Song LIANG

  Acta Mathematica Sinica. 2014, 30(3): 505-516. https://doi.org/10.1007/s10114-013-2083-6

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  A clique-transversal set D of a graph G is a set of vertices of G such that D meets all cliques of G. The clique-transversal number, denoted by τ_C(G), is the minimum cardinality of a clique-transversal set in G. In this paper, we first present a lower bound on τ_C(G) and characterize the extremal graphs achieving the lower bound for a connected (claw, K4)-free 4-regular graph G. Furthermore, we show that for any 2-connected (claw, K4)-free 4-regular graph G of order n, its clique-transversal number equals to 「n/3」.
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  Some Remarks on One-dimensional Functions and Their Riemann-Liouville Fractional Calculus

  Qi ZHANG

  Acta Mathematica Sinica. 2014, 30(3): 517-524. https://doi.org/10.1007/s10114-013-2044-0

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  A one-dimensional continuous function of unbounded variation on [0, 1] has been constructed. The length of its graph is infinite, while part of this function displays fractal features. The Box dimension of its Riemann-Liouville fractional integral has been calculated.
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  A Class of Stochastic Differential Equations with the Time Average

  Rong Cheng YIN, Fu Ke WU, Shi Geng HU

  Acta Mathematica Sinica. 2014, 30(3): 525-538. https://doi.org/10.1007/s10114-013-1268-3

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  Stochastic differential equations with the time average have received increasing attentions in recent years since they can offer better explanations for some financial models. Since the time average is involved in this class of stochastic differential equations, in this paper, the linear growth condition and the Lipschitz condition are different from the classical conditions. Under the special linear growth condition and the special Lipschitz condition, this paper establishes the existence and uniqueness of the solution. By using the Lyapunov function, this paper also establishes the existence and uniqueness under the local Lipschitz condition and gives the p-th moment estimate. Finally, a scalar example is given to illustrate the applications of our results.
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  On Graphs with Cut Vertices and Cut Edges

  Kun Fu FANG, Jin Long SHU

  Acta Mathematica Sinica. 2014, 30(3): 539-546. https://doi.org/10.1007/s10114-014-1230-z

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  Let G(n, k, t) be a set of graphs with n vertices, k cut edges and t cut vertices. In this paper, we classify these graphs in G(n, k, t) according to cut vertices, and characterize the extremal graphs with the largest spectral radius in G(n, k, t).