离散Leslie—Gower型捕食与被捕食系统的Neimark—Sacker分岔

本文利用映射的分岔理论讨论了离散Leslie—Gower型捕食与被捕食系统的Neimark—Sacker分岔,并通过数值模拟验证了所得结果的正确性。     

GMRES, L-Curves, and Discrete Ill-Posed Problems

The GMRES method is a popular iterative method for the solution of large linear systems of equations with a nonsymmetric nonsingular matrix. This paper discusses application of the GMRES method to the solution of large linear systems of equations that arise from the discretization of linear ill-posed problems. These linear systems are severely ill-conditioned and are referred to as discrete ill-posed problems. We are concerned with the situation when the right-hand side vector is contaminated by measurement errors, and we discuss how a meaningful approximate solution of the discrete ill-posed problem can be determined by early termination of the iterations with the GMRES method. We propose a termination criterion based on the condition number of the projected matrices defined by the GMRES method. Under certain conditions on the linear system, the termination index corresponds to the vertex of an L-shaped curve.     

Discrete Moving Frames and Discrete Integrable Systems

Group-based moving frames have a wide range of applications, from the classical equivalence problems in differential geometry to more modern applications such as computer vision. Here we describe what we call a discrete group-based moving frame, which is essentially a sequence of moving frames with overlapping domains. We demonstrate a small set of generators of the algebra of invariants, which we call the discrete Maurer–Cartan invariants, for which there are recursion formulas. We show that this offers significant computational advantages over a single moving frame for our study of discrete integrable systems. We demonstrate that the discrete analogues of some curvature flows lead naturally to Hamiltonian pairs, which generate integrable differential-difference systems. In particular, we show that in the centro-affine plane and the projective space, the Hamiltonian pairs obtained can be transformed into the known Hamiltonian pairs for the Toda and modified Volterra lattices, respectively, under Miura transformations. We also show that a specified invariant map of polygons in the centro-affine plane can be transformed to the integrable discretization of the Toda Lattice. Moreover, we describe in detail the case of discrete flows in the homogeneous 2-sphere and we obtain realizations of equations of Volterra type as evolutions of polygons on the sphere.     

Discrete,linear approximation problems in polyhedral norms

Summary We introduce a class of polyhedral norms and study discrete linear approximation problems under these norms. It is possible to give a uniform treatment, in particular, ofL ₁ and maximum norm problems, at least as regards notation; and we develop a general exchange algorithm in which we permit also linear inequality constraints.     

Discrete monodromy, pentagrams, and the method of condensation

This paper studies the pentagram map, a projectively natural iteration on the space of polygons. Inspired by a method from the theory of ordinary differential equations, the paper constructs roughly n algebraically independent invariants for the map, when it is defined on the space of n-gons. These invariants strongly suggest that the pentagram map is a discrete completely integrable system. The paper also relates the pentagram map to Dodgson’s method of condensation for computing determinants, also known as the octahedral recurrence. I dedicate this paper to Professor V. I. Arnold on the occasion of his 70th birthday     

Discrete strip-concave functions, Gelfand-Tsetlin patterns, and related polyhedra

Discrete strip-concave functions considered in this paper are, in fact, equivalent to an extension of Gelfand-Tsetlin patterns to the case when the pattern has a not necessarily triangular but convex configuration. They arise by releasing one of the three types of rhombus inequalities for discrete concave functions (or “hives”) on a “convex part” of a triangular grid. The paper is devoted to a combinatorial study of certain polyhedra related to such functions or patterns, and results on faces, integer points and volumes of these polyhedra are presented. Also some relationships and applications are discussed.In particular, we characterize, in terms of valid inequalities, the polyhedral cone formed by the boundary values of discrete strip-concave functions on a grid having trapezoidal configuration. As a consequence of this result, necessary and sufficient conditions on a pair of vectors to be the shape and content of a semi-standard skew Young tableau are obtained.     

Object-orientation,Discrete Simulation and the Three-Phase Approach

The three-phase approach provides a simple and robust way to develop discrete computer simulation programs. Object-orientation allows system developers to develop software which can be extended and also makes it impossible for important variables within a software system to be tampered with. This paper shows how the two approaches can be usefully combined and discusses the development of a simulation library, written in C++ and based on these ideas. Limitations of both approaches are also discussed.