15 February 2014, Volume 30 Issue 2

    

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  Subordination in the Sense of Bochner of L^p-semigroups and Associated Markov Processes

  Oana LUPAŞCU

  Acta Mathematica Sinica. 2014, 30(2): 187-196. https://doi.org/10.1007/s10114-014-2751-1

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  We show that the subordination induced by a convolution semigroup (subordination in the sense of Bochner) of a C₀-semigroup of sub-Markovian operators on an L^p space is actually associated to the subordination of a right (Markov) process. As a consequence, we solve the martingale problem associate with the L^p-infinitesimal generator of the subordinate semigroup. We also prove quasi continuity properties for the elements of the domain of the L^p-generator of the subordinate semigroup. It turns out that an enlargement of the base space is necessary. A main step in the proof is the preservation under such a subordination of the property of a Markov process to be a Borel right process. We use several analytic and probabilistic potential theoretical tools.
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  Characterization of Gromov Hyperbolic Short Graphs

  José Manuel RODRÍGUEZ

  Acta Mathematica Sinica. 2014, 30(2): 197-212. https://doi.org/10.1007/s10114-013-2467-7

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  To decide when a graph is Gromov hyperbolic is, in general, a very hard problem. In this paper, we solve this problem for the set of short graphs (in an informal way, a graph G is r-short if the shortcuts in the cycles of G have length less than r): an r-short graph G is hyperbolic if and only if S_9r(G) is finite, where S_R(G):=sup{L(C) : C is an R-isometric cycle in G} and we say that a cycle C is R-isometric if d_C(x, y) ≤ d_G(x, y)| + R for every x, y ∈ C.
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  High-dimensional D. H. Lehmer Problem over Short Intervals

  Zhe Feng XU, Tian Ping ZHANG

  Acta Mathematica Sinica. 2014, 30(2): 213-228. https://doi.org/10.1007/s10114-014-3324-z

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  Let k be a positive integer and n a nonnegative integer, 0 < λ₁, . . . , λ_k+1 ≤ 1 be real numbers and w = (λ₁, λ₂, . . . , λ_k+1). Let q ≥ max{[ 1/λ_i] : 1 ≤ i ≤ k + 1} be a positive integer, and a an integer coprime to q. Denote by N(a, k,w, q, n) the 2n-th moment of (b₁ . . . b_k - c) with b₁ . . . b_kc ≡ a (mod q), 1 ≤ b_i ≤ λ_iq (i = 1, . . . , k), 1 ≤ c ≤ λ_k+1q and 2 |(b₁ +. . . + b_k + c). We first use the properties of trigonometric sum and the estimates of n-dimensional Kloosterman sum to give an interesting asymptotic formula for N(a, k,w, q, n), which generalized the result of Zhang. Then we use the properties of character sum and the estimates of Dirichlet L-function to sharpen the result of N(a, k,w, q, n) in the case of w = (1/2 , 1/2 , . . . , 1/2) and n = 0. In order to show our result is close to the best possible, the mean-square value of N(a, k, q) - φ^k(q)/2^k+2 and the mean value weighted by the high-dimensional Cochrane sum are studied too.
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  Landesman-Lazer Type Conditions and Multiplicity Results for Nonlinear Elliptic Problems with Neumann Boundary Values

  Edcarlos Domingos DA SILVA, Francisco Odair DE PAIVA

  Acta Mathematica Sinica. 2014, 30(2): 229-250. https://doi.org/10.1007/s10114-014-2750-2

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  We establish the existence and multiplicity of solutions for Steklov problems under nonresonance or resonance conditions using variational methods. In our main theorems, we consider a weighted eigenvalue problem of Steklov type.
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  Additive Hazards Regression under Generalized Case-Cohort Sampling

  Ji Chang YU, Yue Yong SHI, Qing Long YANG, Yan Yan LIU

  Acta Mathematica Sinica. 2014, 30(2): 251-260. https://doi.org/10.1007/s10114-014-3180-x

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  Case-cohort design usually requires the disease rate to be low in large cohort study, although it has been extensively used in practice. However, the disease with high rate is frequently observed in many clinical studies. Under such circumstances, it is desirable to consider a generalized case-cohort design, where only a fraction of cases are sampled. In this article, we propose the inference procedure for the additive hazards regression under the generalized case-cohort sampling. Asymptotic properties of the proposed estimators for the regression coefficients are established. To demonstrate the effectiveness of the generalized case-cohort sampling, we compare it with simple random sampling in terms of asymptotic relative efficiency. Furthermore, we derive the optimal allocation of the subsamples for the proposed design. The finite sample performance of the proposed method is evaluated through simulation studies.
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  Liouville Type Theorems for a System of Integral Equations on Upper Half Space

  Su Fang TANG, Jing Bo DOU

  Acta Mathematica Sinica. 2014, 30(2): 261-276. https://doi.org/10.1007/s10114-014-3071-1

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  In this paper, we consider the following system of integral equations on upper half space 
  where R₊ⁿ = {x = (x₁, x₂, . . . ,x_n) ∈ Rⁿ|x_n > 0}, x = (x1, x2, . . . , x_n-1,-x_n) is the reflection of the point x about the hyperplane x_n = 0, 0 < α < n, λ_i, μ_i, β_i ≥ 0 (i = 1, 2) are constants, p_i ≥ 0 and q_i ≥ 0 (i = 1, 2, 3, 4). We prove the nonexistence of positive solutions to the above system with critical and subcritical exponents via moving sphere method.
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  Differential Inequalities, Normality and Quasi-normality

  Xiao Jun LIU, Shahar NEVO, Xue Cheng PANG

  Acta Mathematica Sinica. 2014, 30(2): 277-282. https://doi.org/10.1007/s10114-014-2542-8

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  We prove that if D is a domain in C, α > 1 and C > 0, then the family F of functions f meromorphic in D such that |f'(z)|/(1+|f(z)|^α) > C for every z ∈ D is normal in D. For α = 1, the same assumptions imply quasi-normality but not necessarily normality.
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  Antimagic Labeling of Generalized Pyramid Graphs

  Subramanian ARUMUGAM, Mirka MILLER, Oudone PHANALASY, Joe RYAN

  Acta Mathematica Sinica. 2014, 30(2): 283-290. https://doi.org/10.1007/s10114-014-2381-7

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  An antimagic labeling of a graph with q edges is a bijection from the set of edges to the set of positive integers {1, 2, . . . , q} such that all vertex weights are pairwise distinct, where the vertex weight of a vertex is the sum of the labels of all edges incident with that vertex. A graph is antimagic if it has an antimagic labeling.
  In this paper, we provide antimagic labelings for a family of generalized pyramid graphs.
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  Supereulerian Graphs and the Petersen Graph

  Xiao Min LI, Lan LEI, Hong-Jian LAI, Meng ZHANG

  Acta Mathematica Sinica. 2014, 30(2): 291-304. https://doi.org/10.1007/s10114-014-2272-y

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  A graph G is supereulerian if G has a spanning eulerian subgraph. Boesch et al. [J. Graph Theory, 1, 79-84 (1977)] proposed the problem of characterizing supereulerian graphs. In this paper, we prove that any 3-edge-connected graph with at most 11 edge-cuts of size 3 is supereulerian if and only if it cannot be contractible to the Petersen graph. This extends a former result of Catlin and Lai [J. Combin. Theory, Ser. B, 66, 123-139 (1996)].
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  Depth in Bucket Recursive Trees with Variable Capacities of Buckets

  Ramin KAZEMI

  Acta Mathematica Sinica. 2014, 30(2): 305-310. https://doi.org/10.1007/s10114-013-2115-2

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  We consider bucket recursive trees of size n consisting of all buckets with variable capacities 1, 2, . . . , b and with a specific stochastic growth rule. This model can be considered as a generalization of random recursive trees like bucket recursive trees introduced by Mahmoud and Smythe where all buckets have the same capacities. In this work, we provide a combinatorial analysis of these trees where the generating function of the total weights satisfies an autonomous first order differential equation. We study the depth of the largest label (i.e., the number of edges from the root node to the node containing label n) and give a closed formula for the probability distribution. Also we prove a limit law for this quantity which is a direct application of quasi power theorem and compute its mean and variance. Our results for b = 1 reduce to the previous results for random recursive trees.
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  Almost Everywhere Convergence of Sequences of Cesàro and Riesz Means of Integrable Functions with Respect to the Multidimensional Walsh System

  György GÁT

  Acta Mathematica Sinica. 2014, 30(2): 311-322. https://doi.org/10.1007/s10114-013-1766-3

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  The aim of this paper is to prove the a.e. convergence of sequences of the Cesàro and Riesz means of the Walsh-Fourier series of d variable integrable functions. That is, let a= (a₁, . . . ,a_d) : N → N^d (i>d ∈ P) such that a_j(n + 1) ≥ δ sup_k≤n a_j(k) (j = 1, . . . , d, n ∈ N) for some δ > 0 and a₁(+∞) = . . . = a_d(+∞) = +∞. Then, for each integrable function f ∈ L¹(I^d), we have the a.e. relation for the Cesàro means lim_n→∞ σ_a(n)^αf = f and for the Riesz means lim_n→∞ σ_a(n)^α,γf = f for any 0 < α_j ≤ 1 ≤ γ_j (j = 1, . . . , d). A straightforward consequence of our result is the so-called cone restricted a.e. convergence of the multidimensional Cesàro and Riesz means of integrable functions, which was proved earlier by Weisz.
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  The Role of BMOA in the Boundedness of Weighted Composition Operators on the Unit Ball

  Chao ZHANG, Guang Fu CAO

  Acta Mathematica Sinica. 2014, 30(2): 323-330. https://doi.org/10.1007/s10114-013-1742-y

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  We study the weighted composition operators W_h,ø on Hardy space H²(B) whenever H ∈ BMOA (resp. H ∈ VMOA). Analogous results are given for H^p(B) spaces and the scale of weighted Bergman spaces. In the latter case, BMOA is replaced by the Bloch space (resp. VMOA by the little Bloch space).
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  Hausdorff Dimension of Quadratic Rational Julia Sets

  Jia ZHOU, Liang Wen LIAO

  Acta Mathematica Sinica. 2014, 30(2): 331-342. https://doi.org/10.1007/s10114-013-1640-3

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  Suppose a quadratic rational map has a Siegel disk and a parabolic fixed point. If the rotation number of the Siegel disk is an irrational of bounded type, then the Julia set of the map is shallow. This implies that its Hausdorff dimension is strictly less than two.
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  Every Toroidal Graph Is Acyclically 8-Choosable

  Jian Feng HOU, Gui Zhen LIU

  Acta Mathematica Sinica. 2014, 30(2): 343-352. https://doi.org/10.1007/s10114-013-1497-5

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  A proper coloring of a graph G is acyclic if G contains no 2-colored cycle. A graph G is acyclically L-list colorable if for a given list assignment L = {L(v) : v ∈ V (G)}, there exists a proper acyclic coloring φ of G such that φ(v) ∈ L(v) for all v ∈V (G). If G is acyclically L-list colorable for any list assignment L with |L(v)| ≥ k for all v ∈ V (G), then G is acyclically k-choosable. In this article, we prove that every toroidal graph is acyclically 8-choosable.
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  A Minimizing Property of Homographic Solutions

  Abdalla M. MANSUR, Daniel C. OFFIN

  Acta Mathematica Sinica. 2014, 30(2): 353-360. https://doi.org/10.1007/s10114-013-1299-9

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  In this paper, we prove that the homographic solutions to the rhombus four body problem are the variational minimizers of the Lagrangian action restricted on a holonomically constrained rhombus loop space.
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  Triple Positive Periodic Solutions of Nonlinear Singular Second-order Boundary Value Problems

  Qing Liu YAO

  Acta Mathematica Sinica. 2014, 30(2): 361-370. https://doi.org/10.1007/s10114-013-1291-4

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  This paper studies the positive solutions of the nonlinear second-order periodic boundary value problem u''(t) + λ(t)u(t)=f(t, u(t)), a.e. t ∈ [0, 2π], u(0) = u(2π), u' (0) = u' (2π), where f(t, u) is a local Carathéodory function. This shows that the problem is singular with respect to both the time variable t and space variable u. By applying the Leggett-Williams and Krasnosel'skii fixed point theorems on cones, an existence theorem of triple positive solutions is established. In order to use these theorems, the exact a priori estimations for the bound of solution are given, and some proper height functions are introduced by the estimations.