Jerzy Ombach

Resource partitioning within a single species population and population stability: A theoretical model

Abstract Discrete population models which assume unequal resource partitioning among population members bring about population stability. These models also assume that individual resource share is independent of population density. The model presented here is an attempt to answer the question What does bring about population stability—the inequality of resource partitioning itself or the independence of resource share of population density? By developing a theoretical model with varying dependence of the resource share on the population size, it is shown that the inequality itself is not sufficient for population stability; rather it is the independence of the resource share from population size which brings about this property.

Shadowing, expansiveness and hyperbolic homeomorphisms

The purpose of this paper is to complete results concerning the class ℋ of expansive homeomorphisms having the pseudo orbits tracing property on a compact metric space. We show that hyperbolic homeomorphisms introduced by Mane in [8] are exactly those in the class ℋ then by the result of [12, 20] they form a class equal to the Smale space introduced by Ruelle in [18]. Next, assuming that the phase space is a smooth manifold, we show that a diffeomorphism is Anosov if and only if it is in the class ℋ and is a lower semi-continuity point of the map which assigns to any diffeomorphism the supremum of its expansive constants (possibly zero). Then we discuss the behavior of the dynamical systems generated by homeomorphisms from ℋ near their basic sets.

Consequences of the Pseudo Orbits Tracing Property and Expansiveness

Let f be an expansive homeomorphism with the pseudo orbits tracing property on a compact metric space. There are stable and unstable “manifolds” with similar properties as in the hyperbolic case, and similar behavior near periodic points is observed. Per ( f ) = Ω( f ) = CR(f) . Mappings Ω and CR are continuous at f .

Dynamical Properties of a Perceptron Learning Process: Structural Stability Under Numerics and Shadowing

In this paper two aspects of numerical dynamics are used for an artificial neural network (ANN) analysis. It is shown that topological conjugacy of gradient dynamical systems and both the shadowing and inverse shadowing properties have nontrivial implications in the analysis of a perceptron learning process. The main result is that, generically, any such process is stable under numerics and robust. Implementation aspects are discussed as well. The analysis is based on the theorem concerning global topological conjugacy of cascades generated by a gradient flow on a compact manifold without a boundary.

Shadowing property in analysis of neural networks dynamics

We show that the shadowing and the inverse shadowing, the notions considered in the theory of dynamical systems, may be successfully applied in the analysis of a multilayer neural networks learning process. Our main result is, that generically any such a process is robust. Implementation implications are discussed as well.

Nonautonomous Stochastic Search in Global Optimization

We present a general method how to prove convergence of a sequence of random variables generated by a nonautonomous scheme of the form Xt=Tt(Xt−1,Yt), where Yt represents randomness, used as an approximation of the set of solutions of the global optimization problem with a continuous cost function. We show some of its applications.

Numerical Methods for SDEs

To begin we briefly recall some background material on the numerical solution of deterministic ordinary differential equations (ODEs) to provide an introduction to the corresponding ideas for SDEs. We then discuss the stochastic Euler scheme and its implementation in some detail before turning to higher order numerical schemes for scalar Ito SDEs. We also indicate how MAPLE can be used to automatically determine the coefficients of these numerical schemes, in particular for vector SDEs.

A Tour of Important Distributions

This chapter reviews a selection of the best known probability distributions, which are important from both a theoretical and a practical point of view. Two other probability distributions, needed for statistical inference, will be mentioned in Chapter 6.

Measure and Integral

This Chapter provides a short introduction to measure theory and the theory of the Lebesgue integral. Avoiding as much as possible technical details, we present the basic ideas and also list the basic properties of measure and integral that will be used in this book. Let us note here that probability P(A) defined in Chapter 1 is nothing else but the measure of event A. Another basic probability concept, the mathematical expectation, established later in the book is nothing else but the integral with respect to the probability measure. To some extent, calculus of probability, theory of stochastic processes (including stochastic differential equations) and mathematical statistics can be thought of as parts of the theory of measure and integration.

Random Variables and Distributions

Most quantities that we deal with every day are more or less of a random nature. The height of the person we first meet on leaving home in the morning, the grade we will earn in the next exam, the cost of a bottle of wine we will buy in the nearest shopping centre, and many more, serve as examples of so called random variables. Any such variable has its own specifications or characteristics. For example, the height of an adult male may take all values from 150 cm to 230 cm or even beyond this interval. Besides, the interval (175, 180) is more likely than other intervals of the same length, say (152, 157) or (216, 221). On the other hand, the grade we will earn during the coming exam will take finite many values: excellent, good, pass, fail, and for the reader some of these values are more likely than others.