Existence of a dense orbit and topological transitivity: When are they equivalent?

Topological transitivity and existence of a dense orbit are two notions which play an important role in every definition of chaos [2], [4]. Unfortunately, in the literature there are often misunderstandings (or even false statements) about their relationship. In this note we show that generally neither condition implies the other and then we give two propositions (Propositions1 and 2) for implication in either direction. This revised version was published online in June 2006 with corrections to the Cover Date.     

Sufficient conditions for robust stability of large-scale dynamical systems including delayed states and parameter perturbations in interconnections

The problem of the robust stability of large-scale dynamical systems including delayed states and parameter perturbations in interconnections is considered. By using algebraic Riccati equations and some analytical methods, some sufficient conditions on linear decentralized state feedback controllers are derived so that the systems remain stable in the presence of delayed states and parameter perturbations. Such conditions give some bounds on the perturbations of interconnections with delayed states and uncertain parameters, and result in a quantitative measures of robustness for large-scale dynamical systems including delayed states and uncertain parameters in interconnections. The results obtained in this paper are applicable not only to large-scale systems with multiple time-varying delays, but also to large-scale systems without exact knowledge of the delays, i.e., large-scale systems with uncertain delays.     

The stability of boundary conditions for an angled-derivative difference scheme

Tidal forcing of the shallow water equations is typical of a class of problems where an approximate equilibrium solution is sought by long time integration of a differential equation system. A combination of the angled-derivative scheme with a staggered leap-frog scheme is sometimes used to discretise this problem. It is shown here why great care then needs to be taken with the boundary conditions to ensure that spurious solution modes do not lead to numerical instabilities. Various techniques are employed to analyse two simple model problems and display instabilities met in practical computations; these are then used to deduce a set of stable boundary conditions.Dedicated to Professor J. Crank on the occasion of his 80th birthday     

Remote two-time boundary conditions and special states in quantum mechanics

For an oscillating universe, the assumptions of wave-function localization and approximate symmetry for the initial and final states impose restrictions on the intermediate motion. We argue that one of these restrictions is the avoidance of superpositions of macroscopically distinct states.     

Heat Content Asymptotics for Operators of Laplace Type with Spectral Boundary Conditions

Let P be an operator of Dirac type and let D=P ² be the associated operator of Laplace type. We impose spectral boundary conditions and study the leading heat content coefficients for D.     

A Modified Relaxation Scheme for Mathematical Programs with Complementarity Constraints

In this paper, we consider a mathematical program with complementarity constraints. We present a modified relaxed program for this problem, which involves less constraints than the relaxation scheme studied by Scholtes (2000). We show that the linear independence constraint qualification holds for the new relaxed problem under some mild conditions. We also consider a limiting behavior of the relaxed problem. We prove that any accumulation point of stationary points of the relaxed problems is C-stationary to the original problem under the MPEC linear independence constraint qualification and, if the Hessian matrices of the Lagrangian functions of the relaxed problems are uniformly bounded below on the corresponding tangent space, it is M-stationary. We also obtain some sufficient conditions of B-stationarity for a feasible point of the original problem. In particular, some conditions described by the eigenvalues of the Hessian matrices mentioned above are new and can be verified easily. This work was supported in part by the Scientific Research Grant-in-Aid from the Ministry of Education, Science, Sports, and Culture of Japan. The authors are grateful to an anonymous referee for critical comments.     

Optimality conditions in multiobjective differentiable programming

A theorem stated in Ref. 1 is corrected.     

On sufficient optimality conditions for mathematical programs with equaiity constraints

A number of sufficiency theorems in the mathematical programming literature, concerning problems with equality constraints, are shown to be trivial consequences of the corresponding results for inequality constraints.This work was supported by NSF Grant No. ECS-8214081. Research by the first author was done while a visitor at La Trobe University.     

THE SOLVABILITY CONDITIONS FOR AN INVERSE EIGEN-PAIR PROBLEM

This paper discusses problem IEP:Given n×m matrix X and m×m diagonal matrix A, find an n×n matrix A such that AX=XA.The new solvablily conditions for the problem IEP are obtained. The eigenvalue dislribulaion of the solutions for the problem IEP are described in detail.     

Semidifferentiable functions and necessary optimality conditions

We consider Lyapunov's equationPA+A ^T P+Q=0, whereQ is symmetric positive definite andA is in controllable companion form. We prove that a necessary and sufficient condition thatA be stable is that the first rowP ₁ of theP-matrix be a stablen–1 coefficient vector. This result is related to the minimum phase property of linear systems and is useful in designing robust controllers.