Feb 2014, Volume 9 Issue 1
    

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  • RESEARCH ARTICLE
    Lei CHEN, Fuqing GAO, Shaochen WANG

    Let X? be a small perturbation Wishart process with values in the set of positive definite matrices of size m, i.e., the process X? is the solution of stochastic differential equation with non-Lipschitz diffusion coefficient: dXt?=?Xt?dBt+dBt'?Xt?+ρImdt, X0 = x, where B is an m × m matrix valued Brownian motion and B′denotes the transpose of the matrix B. In this paper, we prove that {Xt?-Xt0/?h2(?),?>0} satisfies a large deviation principle, and (Xt?-Xt0)/? converges to a Gaussian process, where h(?)+ and ?h(?)0 as ?0. A moderate deviation principle and a functional central limit theorem for the eigenvalue process of X? are also obtained by the delta method.

  • RESEARCH ARTICLE
    Xie-Bin CHEN

    The k-ary n-cube Qnk (n≥2 and k≥3) is one of the most popular interconnection networks. In this paper, we consider the problem of a faultfree Hamiltonian cycle passing through a prescribed linear forest (i.e., pairwise vertex-disjoint paths) in the 3-ary n-cube Qn3 with faulty edges. The following result is obtained. Let E0 (≠?) be a linear forest and F (≠?) be a set of faulty edges in Qn3 such that E0F = ?and |E0| + |F|≤2n - 2. Then all edges of E0 lie on a Hamiltonian cycle in Qn3-F,and the upper bound 2n-2 is sharp.

  • RESEARCH ARTICLE
    Yu CHEN, Weiping ZHANG, Chun SU

    We investigate the precise large deviations of random sums of negatively dependent random variables with consistently varying tails. We find out the asymptotic behavior of precise large deviations of random sums is insensitive to the negative dependence. We also consider the generalized dependent compound renewal risk model with consistent variation, which including premium process and claim process, and obtain the asymptotic behavior of the tail probabilities of the claim surplus process.

  • RESEARCH ARTICLE
    Yinghui DONG, Guojing WANG

    We consider a two-dimensional reduced form contagion model with regime-switching interacting default intensities. The model assumes the intensities of the default times are driven by macro-economy described by a homogeneous Markov chain as well as the other default. By using the idea of ‘change of measure’ and some closed-form formulas for the Laplace transforms of the integrated intensity processes, we derive the two-dimensional conditional and unconditional joint distributions of the default times. Based on these results, we give the explicit formulas for the fair spreads of the first-to-default and second-to-default credit default swaps (CDSs) on two underlyings.

  • RESEARCH ARTICLE
    Hui HE, Rugang MA

    We construct two kinds of stochastic flows of discrete Galton-Watson branching processes. Some scaling limit theorems for the flows are proved, which lead to local and nonlocal branching superprocesses over the positive half line.

  • RESEARCH ARTICLE
    Yutian LEI, Chao MA

    We obtain the optimal integrability for positive solutions of the Euler-Lagrange system of the weighted Hardy-Littlewood-Sobolev inequality in ?n:{u(x)=1|x|α?nυ(y)q|y|β|x-y|λdy,u(x)=1|x|β?nυ(y)p|y|α|x-y|λdy.C. Jin and C. Li [Calc. Var. Partial Differential Equations, 2006, 26: 447-457] developed some very interesting method for regularity lifting and obtained the optimal integrability for p, q>1. Here, based on some new observations, we overcome the difficulty there, and derive the optimal integrability for the case of p, q≥1 and pq 1. This integrability plays a key role in estimating the asymptotic behavior of positive solutions when |x| → 0 and when |x| → ∞.

  • RESEARCH ARTICLE
    Dongmei LI, Jinwang LIU, Weijun LIU

    We consider the following problem: for a collection of points in an n-dimensional space, find a linear projection mapping the points to the ground field such that different points are mapped to different values. Such projections are called normal and are useful for making algebraic varieties into normal positions. The points may be given explicitly or implicitly and the coefficients of the projection come from a subset S of the ground field. If the subset S is small, this problem may be hard. This paper deals with relatively large S, a deterministic algorithm is given when the points are given explicitly, and a lower bound for success probability is given for a probabilistic algorithm from in the literature.

  • RESEARCH ARTICLE
    Hailong LI, Jing MA, Yuichi URAMATSU

    The aim of this paper is two folds. First, we shall prove a general reduction theorem to the Spannenintegral of products of (generalized) Kubert functions. Second, we apply the special case of Carlitz’s theorem to the elaboration of earlier results on the mean values of the product of Dirichlet L-functions at integer arguments. Carlitz’s theorem is a generalization of a classical result of Nielsen in 1923. Regarding the reduction theorem, we shall unify both the results of Carlitz (for sums) and Mordell (for integrals), both of which are generalizations of preceding results by Frasnel, Landau, Mikolás, and Romanoff et al. These not only generalize earlier results but also cover some recent results. For example, Beck’s lamma is the same as Carlitz’s result, while some results of Maier may be deduced from those of Romanoff. To this end, we shall consider the Stiletjes integral which incorporates both sums and integrals. Now, we have an expansion of the sum of products of Bernoulli polynomials that we may apply it to elaborate on the results of afore-mentioned papers and can supplement them by related results.

  • RESEARCH ARTICLE
    Dayong LU, Dengfeng LI

    An important tool for the construction of periodic wavelet frame with the help of extension principles was presented in the Fourier domain by Zhang and Saito [Appl. Comput. Harmon. Anal., 2008, 125: 68-186]. We extend their results to the dilation matrix cases in two aspects. We first show that the periodization of any wavelet frame constructed by the unitary extension principle formulated by Ron and Shen is still a periodic wavelet frame under weaker conditions than that given by Zhang and Saito, and then prove that the periodization of those generated by the mixed extension principle is also a periodic wavelet frame if the scaling functions have compact supports.

  • RESEARCH ARTICLE
    Chengxiu QIANG, Abdukadir OBUL

    We give a Gr?bner-Shirshov basis of quantum group of type F4 by using the Ringel-Hall algebra approach. We compute all skew-commutator relations between the isoclasses of indecomposable representations of Ringel-Hall algebras of type F4 by using an ‘inductive’ method. Precisely, we do not use the traditional way of computing the skew-commutative relations, that is first compute all Hall polynomials then compute the corresponding skewcommutator relations; instead, we compute the ‘easier’ skew-commutator relations which correspond to those exact sequences with middle term indecomposable or the split exact sequences first, then ‘deduce’ others from these ‘easier’ ones and this in turn gives Hall polynomials as a byproduct. Then using the composition-diamond lemma prove that the set of these relations constitute a minimal Gr?bner-Shirshov basis of the positive part of the quantum group of type F4. Dually, we get a Gr?bner-Shirshov basis of the negative part of the quantum group of type F4. And finally, we give a Gr?bner-Shirshov basis for the whole quantum group of type F4.

  • RESEARCH ARTICLE
    Guangjun SHEN, Litan YAN

    Let BH1,K1 and BH2,K2 be two independent bi-fractional Brownian motions. In this paper, as a natural extension to the fractional regression model, we consider the asymptotic behavior of the sequenceSn:=i=0n-1K(nαBiH1,K1)(Bi+1H2,K2-BiH2,K2),where K is a standard Gaussian kernel function and the bandwidth parameter αsatisfies certain hypotheses. We show that its limiting distribution is a mixed normal law involving the local time of the bi-fractional Brownian motion BH1,K1. We also give the stable convergence of the sequence Sn by using the techniques of the Malliavin calculus.

  • RESEARCH ARTICLE
    Yisheng SONG, Liqun QI

    We show that an (eventually) strongly increasing and positively homogeneous mapping T defined on a Banach space can be turned into an Edelstein contraction with respect to Hilbert’s projective metric. By applying the Edelstein contraction theorem, a nonlinear version of the famous Krein-Rutman theorem is presented, and a simple iteration process {Tkx/ ||Tkx||}(?xP+) is given for finding a positive eigenvector with positive eigenvalue of T. In particular, the eigenvalue problem of a nonnegative tensor A can be viewed as the fixed point problem of the Edelstein contraction with respect to Hilbert’s projective metric. As a result, the nonlinear Perron-Frobenius property of a nonnegative tensor A is reached easily.

  • RESEARCH ARTICLE
    Boping TIAN, Yongqiang FU

    The Perron method is used to establish the existence of viscosity solutions to the exterior Dirichlet problems for a class of Hessian type equations with prescribed behavior at infinity.

  • RESEARCH ARTICLE
    Xiaoming WANG

    The global crystal basis or canonical basis plays an important role in the theory of the quantized enveloping algebras and their representations. The tight monomials are the simplest elements in the canonical basis. We discuss the tight monomials in quantized enveloping algebra of type B3.