A $\Gamma$-convergence result for doubling metric measures and associated perimeters

Abstract. In this paper we study the notion of perimeter associated with doubling metric measures or strongly weights. We prove that the metric perimeter in the sense of L. Ambrosio and M. Miranda jr. coincides with the metric Minkowski content and can be obtained also as a -limit of Modica-Mortola type degenerate integral functionals. Received: 27 August 2001 / Accepted: 29 November 2001 / Published online: 10 June 2002 Investigation supported by University of Bologna, funds for selected research topics and by GNAMPA of INdAM, Italy. The authors are very grateful to Luigi Ambrosio and Francesco Serra Cassano for making their preprints available to them, for listening with patience and for many unvaluable suggestions.     

The Hopf Property for Subgroups of Hyperbolic Groups

A group is said to be Hopfian if every surjective endomorphism of the group is injective. We show that finitely generated subgroups of torsion-free hyperbolic groups are Hopfian. Our proof generalizes a theorem of Sela (Topology 35 (2) 1999, 301–321).     

试题质量评价研究

教学过程的规范化管理是保证教学质量的重要手段。考试是对教和学的效果的检验，是教学过程管理规范化必须要解决的问题，一直受到教学管理部门的重视，采取了多项措施解决有关问题，其中一个重要的手段是采用试题库。在采用试题库时，必须要有高质量的试题作保证。本探讨基于KDD技术的试题质量评价方法和指标，并通过实例来说明其有效性。     

The spline collocation method for parabolic boundary integral equations on smooth curves

Summary. We consider the spline collocation method for a class of parabolic pseudodifferential operators. We show optimal order convergence results in a large scale of anisotropic Sobolev spaces. The results cover the classical boundary integral equations for the heat equation in the general case where the spatial domain has a smooth boundary in the plane. Our proof is based on a localization technique for which we use our recent results proved for parabolic pseudodifferential operators. For the localization we need also some special spline approximation results in anisotropic Sobolev spaces. Received May 17, 2001 / Revised version received February 19, 2002 / Published online April 17, 2002     

Inner Radius of Univalence for a Strongly Starlike Domain

 The inner radius of univalence of a domain D with Poincaré density ρ _D is the possible largest number σ such that the condition ∥ S _f ∥ _D  = sup _{w∈ D} ρ _D (w) ⁻²∥ S _f (z) ∥ ≤ σ implies univalence of f for a nonconstant meromorphic function f on D, where S _f is the Schwarzian derivative of f. In this note, we give a lower bound of the inner radius of univalence for strongly starlike domains of order α in terms of the order α. The author was partially supported by the Ministry of Education, Grant-in-Aid for Encouragement of Young Scientists, 11740088. A part of this work was carried out during his visit to the University of Helsinki under the exchange programme of scientists between the Academy of Finland and the JSPS. Received November 26, 2001; in revised form September 24, 2002 Published online May 9, 2003     

An almost fourth order uniformly convergent difference scheme for a semilinear singularly perturbed reaction-diffusion problem

Summary. This paper is concerned with a high order convergent discretization for the semilinear reaction-diffusion problem: , for , subject to , where . We assume that on , which guarantees uniqueness of a solution to the problem. Asymptotic properties of this solution are discussed. We consider a polynomial-based three-point difference scheme on a simple piecewise equidistant mesh of Shishkin type. Existence and local uniqueness of a solution to the scheme are analysed. We prove that the scheme is almost fourth order accurate in the discrete maximum norm, uniformly in the perturbation parameter . We present numerical results in support of this result. Received February 25, 1994     

Semi-implicit complementary volume scheme for solving level set like equations in image processing and curve evolution

Summary. We introduce linear semi-implicit complementary volume numerical scheme for solving level set like nonlinear degenerate diffusion equations arising in image processing and curve evolution problems. We study discretization of image selective smoothing equation of mean curvature flow type given by Alvarez, Lions and Morel ([3]). Solution of the level set equation of Osher and Sethian ([26], \[30]) is also included in the study. We prove and estimates for the proposed scheme and give existence of its (generalized) solution in every discrete time-scale step. Efficiency of the scheme is given by its linearity and stability. Preconditioned iterative solvers are used for computing arising linear systems. We present computational results related to image processing and plane curve evolution. Received April 25, 2000 / Revised version received June 11, 2001 / Published online November 15, 2001     

Continuous modified Newton’s-type method for nonlinear operator equations

A nonlinear operator equation F(x)=0, F:H→H, in a Hilbert space is considered. Continuous Newton’s-type procedures based on a construction of a dynamical system with the trajectory starting at some initial point x ₀ and becoming asymptotically close to a solution of F(x)=0 as t→+∞ are discussed. Well-posed and ill-posed problems are investigated. Received: June 29, 2001; in final form: February 26, 2002?Published online: February 20, 2003 This paper was finished when AGR was visiting Institute for Theoretical Physics, University of Giessen. The author thanks DAAD for support