These notes are devoted to showing that the topological degree is a useful tool in the study of the properties of stability of periodic solutions of a scalar, time-dependent differential equation of Newtons type. Two different situations are considered depending on whether the equation has damping or not. When there is linear friction the asymptotic stability of a periodic solution can be characterized in terms of degree. When there is no friction the equation has a hamiltonian structure and some connections between Lyapunov stability and degree are discussed. These results are applied in two different directions: to prove that some classical methods in the theory of existence lead to instability (minimization of the action functional, upper and lower solutions) and to study the stability of the solutions of a concrete class of equations (equations of pendulum-type).

Some applications of the topological degree to stability theory

Edge detection in the presence of noise is a well-known problem. This paper examines an applications-motivated approach for solving the problem using novel techniques and presents a method developed by the authors that performs well on a large class of targets. ROC curves are used to compare this method with other well-known edge detection operators, with favorable results. A theoretical argument is presented that favors LMMSE filtering over median filtering in extremely noisy scenes. Simulated results of the research are presented.

Finding Edges in Noisy Scenes

Subharmonic solutions of second order nonlinear equations: a time-map approach

iii Preface The subject of Differential Equations is a well established part of mathematics and its systematic development goes back to the early days of the development of Calculus. Many recent advances in mathematics, paralleled by a renewed and flourishing interaction between mathematics, the sciences, and engineering, have again shown that many phenomena in the applied sciences, modelled by differential equations will yield some mathematical explanation of these phenomena (at least in some approximate sense). The intent of this set of notes is to present several of the important existence theorems for solutions of various types of problems associated with differential equations and provide qualitative and quantitative descriptions of solutions. At the same time, we develop methods of analysis which may be applied to carry out the above and which have applications in many other areas of mathematics, as well. As methods and theories are developed, we shall also pay particular attention to illustrate how these findings may be used and shall throughout consider examples from areas where the theory may be applied. As differential equations are equations which involve functions and their derivatives as unknowns, we shall adopt throughout the view that differential equations are equations in spaces of functions. We therefore shall, as we progress, develop existence theories for equations defined in various types of function spaces, which usually will be function spaces which are in some sense natural for the given problem.

/pdf/nonlinear-analysis-and-differential-equations-an-50qb7594ix.pdf

Nonlinear Analysis and Differential Equations An Introduction

Two linear feedback control algorithms for handling and preventing congestion in high-speed networks are proposed and analyzed. The fluid approximation model is described with a continuous-time system of delay-differential equations. The algorithms are asymptotically stable, and the transient processes are nonoscillatory. The control parameters are locally optimal (optimality is based on the asymptotic rate of convergence). The results of numerical experiments suggest that these parameters are globally optimal as well. The dependence of the quality of service on the duration of the control intervals is analyzed, and the performance of algorithms in a nonstationary environment is addressed.

Analysis and Optimization of Feedback Control Algorithms for Data Transfers in High-Speed Networks

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Some applications of the topological degree to stability theory

Finding Edges in Noisy Scenes

Subharmonic solutions of second order nonlinear equations: a time-map approach

Nonlinear Analysis and Differential Equations An Introduction

Analysis and Optimization of Feedback Control Algorithms for Data Transfers in High-Speed Networks

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