On a non-local perimeter-preserving curve evolution problem for convex plane curves

This paper deals with a non-local evolution problem for closed convex plane curves which preserves the perimeter of the evolving curve but enlarges the area it bounds and makes the evolving curve more and more circular during the evolution process. And the final shape of the evolving curve will be a circle in the C ^∞ metric as the time t goes to infinity. The first author is supported in part by the National Science Foundation of China (No.10671066) and the Shanghai Leading Academic Discipline Project (No. B407).     

An anisotropic area-preserving flow for convex plane curves

This paper will deal with an anisotropic area-preserving flow which keeps the convexity of the evolving curve and the limiting curve converges to a homothety of a symmetric smooth strictly convex plane curve.     

On the uniqueness of stable ultimate shapes for the anisotropic curve-shortening problem

The uniqueness of stable ultimate shapes for the generalized curve-shortening problem is established for a class of anisotropic factors. Received: 24 June 1999     

Positive diagonal scaling of a nonnegative tensor to one with prescribed slice sums

In this paper we give necessary and sufficient conditions on a nonnegative tensor to be diagonally equivalent to a tensor with prescribed slice sums. These conditions are variations of Bapat-Raghavan and Franklin-Lorenz conditions.     

On some applications of Wald's identity to dams

It is known that the main difficulty in applying the Markovian analogue of Wald's Identity is the presence, in the Identity, of the last state variable before the random walk is terminated. In this paper we show that this difficulty can be overcome if the underlying Markov chain has a finite state space. The absorption probabilities thus obtained are used, by employing a duality argument, to derive time-dependent and limiting probabilities for the depletion process of a dam with Markovian inputs.The second problem that is considered here is that of a non-homogeneous but cyclic Markov chain. An analogue of Wald's Identity is obtained for this case, and is used to derive time- dependent and limiting probabilities for the depletion process with inputs forming a non- homogeneous (cyclic) Markov chain.     

Green's function asymptotics of periodic elliptic operators on abelian coverings of compact manifolds

The main results of this article provide asymptotics at infinity of the Green's functions near and at the spectral gap edges for “generic” periodic second-order, self-adjoint, elliptic operators on noncompact Riemannian co-compact coverings with abelian deck groups. Previously, analogous results have been known for the case of ${\mathbb{R}}^n$ only. One of the interesting features discovered is that the rank of the deck group plays more important role than the dimension of the manifold.     

Boundary value problems of singular multi‐term fractional differential equations with impulse effects

We study a class of boundary value problems for nonlinear impulsive fractional differential equations. A weighted function Banach space and a completely continuous nonlinear operator are constructed firstly. Some existence results for solutions of these problems are established continuously. Our analysis relies on the well known Schauder's fixed point theorem. Examples are given to illustrate the main results finally.