We study the rough bilinear fractional integral where 0<α<n, Ω is homogeneous of degree zero on the y variable and satisfies for some s≥1, and S^n−1 denotes the unit sphere of ℝn. By assuming size conditions on Ω, we obtain several boundedness properties of : where Our result extends a main theorem of Y. Ding and C. Lin [Math. Nachr., 2002, 246―247: 47―52].

This paper is a continuation of the study on the stability speed for Markov processes. It extends the previous study of the ergodic convergence speed to the non-ergodic one, in which the processes are even allowed to be explosive or to have general killings. At the beginning stage, this paper is concentrated on the birth-death processes. According to the classification of the boundaries, there are four cases plus one more having general killings. In each case, some dual variational formulas for the convergence rate are presented, from which, the criterion for the positivity of the rate and an approximating procedure of estimating the rate are deduced. As the first step of the approximation, the ratio of the resulting bounds is usually no more than 2. The criteria as well as basic estimates for more general types of stability are also presented. Even though the paper contributes mainly to the non-ergodic case, there are some improvements in the ergodic one. To illustrate the power of the results, a large number of examples are included.

In this paper, we investigate a renewal risk model in which the distribution of the interclaim times is a mixture of two Erlang distributions. First, the Laplace transform and the defective renewal equation for the Gerber-Shiu function are derived. Then, two asymptotic results for the Laplace transform of the time of ruin are given when the initial surplus tends to infinity for the light-tailed claims and heavy-tailed claims, respectively. Finally, an explicit expression for the Gerber-Shiu function is given.

The operator norms of weighted Hardy operators onMorrey spaces are worked out. The other purpose of this paper is to establish a sufficient and necessary condition on weight functions which ensures the boundedness of the commutators of weighted Hardy operators (with symbols in BMO()) on Morrey spaces.

The degree pattern of a finite group M has been introduced by A. R. Moghaddamfar et al. [Algebra Colloquium, 2005, 12(3): 431―442]. A group M is called k-fold OD-characterizable if there exist exactly k nonisomorphic finite groups having the same order and degree pattern as M. In particular, a 1-fold OD-characterizable group is simply called OD-characterizable. In this article, we will show that the alternating groups A_p+3 for p = 23, 31, 37, 43 and 47 are OD-characterizable. Moreover, we show that the automorphism groups of these groups are 3-fold OD-characterizable. It is worth mentioning that the prime graphs associated with all these groups are connected.

In this paper, we consider a system of two coupled wave equations with dispersive and viscosity dissipative terms under Dirichlet boundary conditions. The global existence of weak solutions as well as uniform decay rates (exponential one) of the solution energy are established.

l₁-regularized problems have a wide application in various areas such as signal processing. It minimizes a quadratic function combined with an l₁ norm term. Iterative soft-thresholding method (IST) is originally proposed to deal with these problems, and fixed point continuation algorithm (FPC) was proposed recently as an improved version of IST. This paper obtains a two-step version of FPC (TwFPC) by combining the new iterate of FPC with its previous two iterates. We also provide an analysis for the convergence of FPC and TwFPC. Various numerical experiments on image deconvolution and compressed sensing show that TwFPC improves IST significantly and is much faster than other competing codes. What is more important, it is very robust to the involved parameters and the regularization parameter.

Fast evaluation of the exact transparent boundary condition for the one-dimensional cubic nonlinear Schrödinger equation is considered in this paper. In [J. Comput.Math., 2007, 25(6): 730―745], the author proposed a fast evaluation method for the half-order time derivative operator. In this paper, we apply this method for the exact transparent boundary condition for the one-dimensional cubic nonlinear Schrödinger equation. Numerical tests demonstrate the effectiveness of the proposed method.