中国科学院数学与系统科学研究院期刊网

15 March 2015, Volume 31 Issue 3
    

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  • Choonkil PARK
    Acta Mathematica Sinica. 2015, 31(3): 353-366. https://doi.org/10.1007/s10114-015-4278-5
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    In this paper, we introduce an additive functional inequality and a quadratic functional inequality in normed spaces, and prove the Hyers-Ulam stability of the functional inequalities in Banach spaces. Furthermore, we introduce an additive functional inequality and a quadratic functional inequality in non-Archimedean normed spaces, and prove the Hyers-Ulam stability of the functional inequalities in non-Archimedean Banach spaces.
  • Eunjeong YI
    Acta Mathematica Sinica. 2015, 31(3): 367-382. https://doi.org/10.1007/s10114-015-4160-5
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    Let G= (V (G),E(G)) be a graph with vertex set V (G) and edge set E(G). For two distinct vertices x and y of a graph G, let RG{x, y} denote the set of vertices z such that the distance from x to z is not equal to the distance from y to z in G. For a function g defined on V (G) and for UV (G), let g(U) =∑ s∈U g(s). A real-valued function g: V (G) → [0, 1] is a resolving function of G if g(RG{x, y}) ≥ 1 for any two distinct vertices x, yV (G). The fractional metric dimension dimf (G) of a graph G is min{g(V (G)): g is a resolving function of G}. Let G1 and G2 be disjoint copies of a graph G, and let σ: V (G1) → V (G2) be a bijection. Then, a permutation graph Gσ = (V,E) has the vertex set V = V (G1) ∪ V (G2) and the edge set E = E(G1) ∪ E(G2) ∪ {uv | v = σ(u)}. First, we determine dimf (T) for any tree T. We show that 1 < dimf (Gσ) ≤ 1/ 2 (|V (G)| + |S(G)|) for any connected graph G of order at least 3, where S(G) denotes the set of support vertices of G. We also show that, for any ε > 0, there exists a permutation graph Gσ such that dimf (Gσ) - 1 < ε. We give examples showing that neither is there a function h1 such that dimf (G) < h1 (f (Gσ)) for all pairs (G, σ), nor is there a function h2 such that h2(dimf (G)) > dimf (Gσ) for all pairs (G, σ). Furthermore, we investigate dimf (Gσ) when G is a complete k-partite graph or a cycle.
  • Gao Cheng YUE, Cheng Kui ZHONG
    Acta Mathematica Sinica. 2015, 31(3): 383-410. https://doi.org/10.1007/s10114-015-4178-8
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    Longtime behavior of degenerate equations with the nonlinearity of polynomial growth of arbitrary order on the whole space RN is considered. By using -trajectories methods, we proved that weak solutions generated by degenerate equations possess an (LU2 (RN), Lloc2 (RN))-global attractor. Moreover, the upper bounds of the Kolmogorov ε-entropy for such global attractor are also obtained.
  • Shi Hui ZHU
    Acta Mathematica Sinica. 2015, 31(3): 411-429. https://doi.org/10.1007/s10114-015-4349-7
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    We study the blow-up solutions for the Davey-Stewartson system (D-S system, for short) in Lx2 (R2). First, we give the nonlinear profile decomposition of solutions for the D-S system. Then, we prove the existence of minimal mass blow-up solutions. Finally, by using the characteristic of minimal mass blow-up solutions, we obtain the limiting profile and a precisely mass concentration of L2 blow-up solutions for the D-S system.
  • Yan LIN, Mei XU
    Acta Mathematica Sinica. 2015, 31(3): 430-444. https://doi.org/10.1007/s10114-015-4277-6
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    In this paper, we obtain the (WHω1,WLω1) type estimate for the Marcinkiewicz integral and the (WHb,ω1,WLω1) type estimate for the commutator generated by a BMO function and the Marcinkiewicz integral, where the kernel satisfies a certain logarithmic type Lipschitz condition.
  • Robert X. J. HAO, Larry X. W. WANG, Harold R. L. YANG
    Acta Mathematica Sinica. 2015, 31(3): 445-455. https://doi.org/10.1007/s10114-015-4209-5
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    We consider context-free grammars of the form G = {ffb1+b2+1ga1+a2, g → fb1ga1+1}, where ai and bi are integers subject to certain positivity conditions. Such a grammar G gives rise to triangular arrays {T(n, k)}0≤kn satisfying a three-term recurrence relation. Many combinatorial sequences can be generated in this way. Let Tn(x) =∑k=0n T(n, k)xk. Based on the differential operator with respect to G, we define a sequence of linear operators Pn such that Tn+1(x) = Pn(Tn(x)). Applying the characterization of real stability preserving linear operators on the multivariate polynomials due to Borcea and Bräandén, we obtain a necessary and sufficient condition for the operator Pn to be real stability preserving for any n. As a consequence, we are led to a sufficient condition for the real-rootedness of the polynomials defined by certain triangular arrays, obtained by Wang and Yeh. Moreover, as special cases we obtain grammars that lead to identities involving the Whitney numbers and the Bessel numbers.
  • Yuan Qing ZHANG, Guang Ren YANG
    Acta Mathematica Sinica. 2015, 31(3): 456-478. https://doi.org/10.1007/s10114-015-3569-1
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    In this article, we study estimation of a partially specified spatial panel data linear regression with random-effects. Under the conditions of exogenous spatial weighting matrix and exogenous regressors, we give an instrumental variable estimation. Under certain sufficient assumptions, we show that the proposed estimator for the finite dimensional parameter is root-N consistent and asymptotically normally distributed and the proposed estimator for the unknown function is consistent and asymptotically distributed. Consistent estimators for the asymptotic variance-covariance matrices of both the parametric and unknown components are provided. The Monte Carlo simulation results verify our theory and suggest that the approach has some practical value.
  • Xiao Zhi WANG
    Acta Mathematica Sinica. 2015, 31(3): 479-500. https://doi.org/10.1007/s10114-015-4130-y
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    This paper deals with the existence of a positive solution for two classes of critical quasilinear system. We prove these results by a variant of mountain pass lemma, combining two convergence theorems and two estimate results. Here we avoid the usual compactness arguments (e.g., Palais-Smale condition or Cerami condition) and reveal the potential of some energy level estimates for the existence of nontrivial solutions.
  • Jian Bo FANG
    Acta Mathematica Sinica. 2015, 31(3): 501-510. https://doi.org/10.1007/s10114-015-3573-5
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    Let x: Mn-1 → Rn be an umbilical free hypersurface with non-zero principal curvatures. M is called Laguerre isoparametric if it satisfies two conditions, namely, it has vanishing Laguerre form and has constant Lauerre principal curvatures. In this paper, under the condition of having constant Laguerre principal curvatures, we show that the hypersurface is of vanishing Laguerre form if and only if its Laguerre form is parallel with respect to the Levi-Civita connection of its Laguerre metric.
  • Deng PAN, Yan Yan LIU, Yuan Shan WU
    Acta Mathematica Sinica. 2015, 31(3): 511-525. https://doi.org/10.1007/s10114-015-3628-7
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    Additive hazards model with random effects is proposed for modelling the correlated failure time data when focus is on comparing the failure times within clusters and on estimating the correlation between failure times from the same cluster, as well as the marginal regression parameters. Our model features that, when marginalized over the random effect variable, it still enjoys the structure of the additive hazards model. We develop the estimating equations for inferring the regression parameters. The proposed estimators are shown to be consistent and asymptotically normal under appropriate regularity conditions. Furthermore, the estimator of the baseline hazards function is proposed and its asymptotic properties are also established. We propose a class of diagnostic methods to assess the overall fitting adequacy of the additive hazards model with random effects. We conduct simulation studies to evaluate the finite sample behaviors of the proposed estimators in various scenarios. Analysis of the Diabetic Retinopathy Study is provided as an illustration for the proposed method.
  • Christopher S. WITHERS, Saralees NADARAJAH
    Acta Mathematica Sinica. 2015, 31(3): 526-542. https://doi.org/10.1007/s10114-015-2003-z
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    We give expansions about the Gumbel distribution in inverse powers of n and log n for Mn, the maximum of a sample size n or n+1 when the j-th observation is μ(j/ n)+ej, μ is any smooth trend function and the residuals {ej} are independent and identically distributed with 

    as x→∞. We illustrate practical value of the expansions using simulated data sets.
  • Mohamed ACHACHE
    Acta Mathematica Sinica. 2015, 31(3): 543-556. https://doi.org/10.1007/s10114-015-1314-4
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    In this paper, we establish the polynomial complexity of a primal-dual path-following interior point algorithm for solving semidefinite optimization (SDO) problems. The proposed algorithm is based on a new kernel function which differs from the existing kernel functions in which it has a double barrier term. With this function we define a new search direction and also a new proximity function for analyzing its complexity. We show that if q1 > q2 > 1, the algorithm has O((q1+1) n q1+1 /2(q1-q2) log n/∈) and O((q1+1) 3q1-2q2+1 2(q1-q2) √n log n/∈) complexity results for large- and small-update methods, respectively.