Let X^ϵ be a small perturbation Wishart process with values in the set of positive definite matrices of size m, i.e., the process X^ϵ is the solution of stochastic differential equation with non-Lipschitz diffusion coefficient: \mathrm{d}{X}_t^{ϵ}=\sqrt{ϵX_t^{ϵ}}\mathrm{d}{B}_t+\mathrm{d}{B}_{\mathit{t}}^{\mathit{\text{'}}}\sqrt{ϵX_t^{ϵ}+\rho I_m\mathrm{d}t}, 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 \{X_t^{ϵ}-X_t^{\mathrm{0}}/\sqrt{ϵh^{\mathrm{2}}(ϵ)},ϵ>\mathrm{0}\} satisfies a large deviation principle, and (X_t^{ϵ}-X_t^{\mathrm{0}})/\sqrt{ϵ} converges to a Gaussian process, where h(ϵ)\to +\infty and \sqrt{ϵ}h(ϵ)\to \mathrm{0} as ϵ\to \mathrm{0}. A moderate deviation principle and a functional central limit theorem for the eigenvalue process of X^ϵ are also obtained by the delta method.

The k-ary n-cube Q_n^k (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 Q_n^{\mathrm{3}} with faulty edges. The following result is obtained. Let E₀ (≠Ø) be a linear forest and F (≠Ø) be a set of faulty edges in Q_n^{\mathrm{3}} such that E₀ ∩ F = Øand |E₀| + |F|≤2n - 2. Then all edges of E₀ lie on a Hamiltonian cycle in Q_n^{\mathrm{3}}-F,and the upper bound 2n-2 is sharp.

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.

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.

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.

We obtain the optimal integrability for positive solutions of the Euler-Lagrange system of the weighted Hardy-Littlewood-Sobolev inequality in {ℝ}^n:

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| → ∞.

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.

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.

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.

We give a Gröbner-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 Gröbner-Shirshov basis of the positive part of the quantum group of type F₄. Dually, we get a Gröbner-Shirshov basis of the negative part of the quantum group of type F₄. And finally, we give a Gröbner-Shirshov basis for the whole quantum group of type F₄.

Let B^H1,K1 and B^H2,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 sequence

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 B^H1,K1. We also give the stable convergence of the sequence S_n by using the techniques of the Malliavin calculus.

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 {T^kx/ ||T^kx||}(\forall x\in P^{+}) 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.

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.

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.