En 中n维单形二面角的角平分面的性质

利用距离几何的理论与方法研究了En中n维单形内二面角与外二面角的角平分面的性质,获得了n维单形内二面角与外二面角平分面的2个定理.     

Singularly perturbed reaction diffusion equations with time delay

Abstract A class of initial boundary value problems of differential-difference equations for reaction diffusion with a small time delay is considered. Under suitable conditions and by using the stretched variable method, a formal asymptotic solution is constructed. Then, by use of the theory of differential inequalities, the uniform validity of the solution is proved.     

Stefan problems for reflected SPDEs driven by space–time white noise

We prove the existence and uniqueness of solutions to a one-dimensional Stefan Problem for reflected SPDEs which are driven by space–time white noise. The solutions are shown to exist until almost surely positive blow-up times. Such equations can model the evolution of phases driven by competition at an interface, with the dynamics of the shared boundary depending on the derivatives of two competing profiles at this point. The novel features here are the presence of space–time white noise; the reflection measures, which maintain positivity for the competing profiles; and a sufficient condition to make sense of the Stefan condition at the boundary. We illustrate the behaviour of the solution numerically to show that this sufficient condition is close to necessary.     

On the computation of a very large number of eigenvalues for selfadjoint elliptic operators by means of multigrid methods

Recent results in the study of quantum manifestations in classical chaos raise the problem of computing a very large number of eigenvalues of selfadjoint elliptic operators. The standard numerical methods for large eigenvalue problems cover the range of applications where a few of the leading eigenvalues are needed. They are not appropriate and generally fail to solve problems involving a number of eigenvalues exceeding a few hundreds. Further, the accurate computation of a large number of eigenvalues leads to much larger problem dimension in comparison with the usual case dealing with only a few eigenvalues. A new method is presented which combines multigrid techniques with the Lanczos process. The resulting scheme requires O(mn) arithmetic operations and O(n) storage requirement, where n is the number of unknowns and m, the number of needed eigenvalues. The discretization of the considered differential operators is realized by means of p-finite elements and is applicable on general geometries. Numerical experiments validate the proposed approach and demonstrate that it allows to tackle problems considered to be beyond the range of standard iterative methods, at least on current workstations. The ability to compute more than 9000 eigenvalues of an operator of dimension exceeding 8 million on a PC shows the potential of this method. Practical applications are found, e.g. in the numerical simulation of quantum billiards.     

Global stability analysis of epidemiological models based on Volterra–Lyapunov stable matrices

In this paper, we study the global dynamics of a class of mathematical epidemiological models formulated by systems of differential equations. These models involve both human population and environmental component(s) and constitute high-dimensional nonlinear autonomous systems, for which the global asymptotic stability of the endemic equilibria has been a major challenge in analyzing the dynamics. By incorporating the theory of Volterra–Lyapunov stable matrices into the classical method of Lyapunov functions, we present an approach for global stability analysis and obtain new results on some three- and four-dimensional model systems. In addition, we conduct numerical simulation to verify the analytical results.     

A New Lower Bound for the Kirchhoff Index using a numerical procedure based on Majorization Techniques

In this note, we use a procedure, proposed in [Bianchi, M., and A. Torriero, Some localization theorems using a majorization technique, Journal of Inequalities and Applications 5 (2000), 433–446], based on a majorization technique, which localizes real eigenvalues of a matrix of order n. Through this information, we compute a lower bound for the Kirchhoff index (see [Bianchi M., A. Cornaro, J.L. Palacios and A. Torriero, Bounds for the Kirkhhoff index via majorization techniques, Journal of Mathematical Chemistry, (2012) online first]) that takes advantage of additional eigenvalues bounds. An algorithm has been developed with MATLAB software to evaluate the above mentioned bound. Finally, numerical examples are provided showing how tighter results can be obtained.     

Periodic solutions to a delayed predator–prey model with Hassell–Varley type functional response

In this paper, a delayed predator–prey model with Hassell–Varley type functional responses is studied. Some sufficient conditions are obtained for the existence of positive periodic solutions to it by applying the coincidence degree theorem. It is interesting that the result is based on the delay, which is different from the previous work (the results are delay-independent). Furthermore, the simulation shows that some conditions are sharp.     

The estimated number of irreducible binomials

Let ${\mathbb{F}}_q$ be the finite field with q elements, and T a positive integer. In this article, we find an asymptotic formula for the total number of monic irreducible binomials in ${\mathbb{F}}_q[x]$ of degree less or equal to T, when T is large enough. We also show explicit lower and upper bounds for the number of binomials in the case when T is small.