In this paper, a new method is used to calculate the chromatic polynomials of graphs. The chromatic polynomials of the complements of a wheel and a fan are determined. Furthermore, the adjoint polynomials of \footnotesize$F_n$ with $n$ vertices are obtained. This supports a conjecture put forward by R.Y. Liu et al.

This paper deals with the asymptotic behavior of solutions for the nonlinear Sobolev-Galpern equations. We first show the existence of the global weak attractor in $H^2(\Omega )\cap H_0^1(\Omega )$ for the equations. And then by an energy equation we prove that the global weak attractor is actually the global strong attractor. The finite-dimensionality of the global attractor is also established.

In [6], Guo and Tan have shown that $2$ is a Laplacian eigenvalue of any tree with perfect matchings. For trees without perfect matchings, we study whether $2$ is one of its Laplacian eigenvalues. If the matching number is $1$ or $2$, the answer is negative; otherwise, there exists a tree with that matching number which has (has not) the eigenvalue $2$. In particular, we determine all trees with matching number $3$ which has the eigenvalue $2$.

Geometric process was first introduced by Lam$^{[10,11]}$. A stochastic process $\{X_{i}, \ i = 1, 2,\cdots \}$ is called a geometric process (GP) if, for some $a > 0, \{a^{i-1}X_{i}, \ i = 1, 2,\cdots \}$ forms a renewal process. In this paper, the GP is used to analyze the data from a series of events. A nonparametric method is introduced for the estimation of the three parameters in the GP. The limiting distributions of the three estimators are studied. Through the analysis of some real data sets, the GP model is compared with other three homogeneous and nonhomogeneous Poisson models. It seems that on average the GP model is the best model among these four models in analyzing the data from a series of events.

This paper deals with the connectedness of the cone-efficient solution set for vector optimization in locally convex Hausdorff topological vector spaces.The connectedness of the cone-efficient solution set is proved for multiobjective programming defined by a continuous cone-quasiconvex mapping on a compact convex set of alternatives. The generalized saddle theorem plays a key role in the proof.

A periodic predator-prey model with stage structure for prey and time delays due to negative feedback and gestation of predator is proposed. By using Gaines and Mawhin's continuation theorem of coincidence degree theory, sufficient conditions are derived for the existence of positive periodic solutions to the proposed model. Numerical simulations are presented to illustrate the feasibility of our main result.

In this note we present a perturbation bound of unitary polar factors. We add the orthogonal projectors into the bound; as a result, a uniform perturbation bound of $Q$-factors is obtained, and the previous bounds are improved.

The strong approximations of a class of $\Bbb R^d$-valued martingales are considered. The conditions used in this paper are easier to check than those used in [3] and [9]. As an application, the strong approximation of a class of non-homogenous Markov chains is established, and the asymptotic properties are established for the multi-treatment Markov chain adaptive designs in clinical trials.

Subject to the assumption that the common distribution of claim sizes belongs to the extended regular variation class, the present work obtains a simple asymptotic formula for the ruin probability within a random or nonrandom horizon in the renewal model.