The nonlinear Schrödinger equation coupling with stochastic weakly damped, forced KdV equation with additive noise can be solved pathwise, and the unique solution generates a random dynamical system. Then we prove that the system possesses a global weak random attractor.

Suppose that H is a subgroup of a finite group G. H is called ?-quasinormal in G if it permutes with every Sylow subgroup of G; H is called ?-quasinormally embedded in G provided every Sylow subgroup of H is a Sylow subgroup of some ?-quasinormal subgroup of G; H is called csupplemented in G if there exists a subgroup N of G such that G = HN and H ∩ N ≤ H_G = Core_G(H). In this paper, finite groups G satisfying the condition that some kinds of subgroups of G are either ?-quasinormally embedded or c-supplemented in G, are investigated, and theorems which unify some recent results are given.

This paper is a continuation and improvement over the results of Laszkiewicz and Zietak [BIT, 2006, 46: 345–366], studying perturbation analysis for polar decomposition. Some basic properties of best approximation subunitary matrices are investigated in detail. The perturbation bounds of the polar factor are also derived.

The classical SIS model with a constant transmission rate exhibits simple dynamic behaviors fully determined by the basic reproduction number. Behavioral changes and intervention measures influenced by the level of infection, likely with a time lag, require the transmission rate to be a nonlinear function of the total infectives. This nonlinear transmission, as shown in this paper via a combination of qualitative and numerical analysis, can generate interesting dynamical behaviors at the population level including backward and Hopf bifurcations. We conclude that sustained infections and periodic outbreaks can be consequences of delayed changes in behaviors or human intervention.

Let F_q be a finite field with q elements, where q is a prime power. Let G be a subgroup of the general linear group over F_q and F_q(x₁, ..., x_n) be the rational function field over F_q. We seek to understand the structure of the rational invariant subfield F_q(x₁, ..., x_n)^G. In this paper, we prove that F_q(x₁, ..., x_n)^G is rational (or, purely transcendental) by giving an explicit set of generators when G is the symplectic group. In particular, the set of generators we gave satisfies the Dickson property.

In this paper we construct an upwind finite volume element scheme based on the Crouzeix-Raviart nonconforming element for nonselfadjoint elliptic problems. These problems often appear in dealing with flow in porous media. We establish the optimal order H¹-norm error estimate. We also give the uniform convergence under minimal elliptic regularity assumption.

L. Addario-Berry et al. [Discrete Appl. Math., 2008, 156: 1168–1174] have shown that there exists a 16-edge-weighting such that the induced vertex coloring is proper. In this note, we improve their result and prove that there exists a 13-edge-weighting of a graph G, such that its induced vertex coloring of G is proper. This result is one step close to the original conjecture posed by M. Karónski et al. [J. Combin. Theory, Ser. B, 2004, 91: 151–157].

In this article, we provide a systematic method to construct examples of complete Ricci flat metrics on CP², with three lines in the general position deleted by Hsieh [Proc. AMS, 1995, 123: 1873–1877] and generalize them to higher dimensions. In addition, we further show that the examples of Hsieh are indeed all flat.

We study the ‘universal’ strong coercivity problem for variational integrals of degenerate p-Laplacian type by mixing finitely many homogenous systems. We establish the equivalence between universal p-coercivity and a generalized notion of p-quasiconvex extreme points. We then give sufficient conditions and counterexamples for universal coercivity. In the case of noncoercive systems we give examples showing that the corresponding variational integral may have infinitely many non-trivial minimizers in W₀^1,p which are nowhere C¹ on their supports. We also give examples of universally p-coercive variational integrals in W₀^1,p for p ≥ 2 with L^∞ coefficients for which unique minimizers under affine boundary conditions are nowhere C¹.