Johan F. Kaashoek

Potentialised Partial Differential Equations in Economic Geography and Spatial Economics: Multiple Dimensions and Control

The present study is a follow-up to previous recent publications in the field of theoretical economic geography and spatial economics. Earlier results are generalised and simulated in higher dimensions (in terms of variables and topological dimensions), and given possible undesirable outcomes of the process (which can behave chaotically), application of control methods to it is being studied.

A neural' network applied to tlie calculation of lyapunov exponents 1

Chaotic deterministics systems are characterised by the instability of orbits on an attractor. The largest Lyapunov exponent measures on average the exponential growth rate of small deviations along an orbit and gives as such an indication whether or not the dynamic generating process is unstable. The direct method for calculation of the Lyapunov exponent, based on finite differences as formulated by the so-called Wolf-algorithm,fails on medium sized data sets. Alternatively, one can use a neural network with backpropagation to estimate a data generating function. This so-calletl indirect method enables us to recover the theoretical value of the largest Lyapunov exponent in several examples.

On potentialized partial differential equations in theoretical spatial economics

Abstract For use in theoretical spatial economics, classical partial differential equations should be generalized by potentializing the spatial variables: this has been undertaken for the one-dimensional wave equation. Three potentials (linear, exponential, Tanner) have been investigated; the exponential one especially has led to meaningful analytical solutions in terms of spatial economic patterns. Problems of stability and ‘strangeness’ of the dynamical solutions are discussed, and lines for further research set out.

A Note on the Detection of Chaos in Medium Sized Time Series

The length of macro-economic time series is often restricted, e.g., to 200 observations. Either no more data are available, or economic intervention policy makes a restriction to a limited size necessary. We have examined the presence of “chaotic properties” for medium sized time series by using numerical procedures for calculating the largest Lyapunov exponent and the correlation dimension.

Pattern Formation for a One Dimensional Evolution Equation Based on Thom’s River Basin Model.

A one component, one dimensional diffusion model is presented in which spatial structure is generated by means of a density dependent diffusive mechanism such that for some density values mass flow is proportional to the mass density gradient. Although stability and attractivity properties of a set of analytic periodic stationary solutions are not strong enough, the analytical and numerical work reported here, show that this evolution equation in one space variable with zero-flux boundary conditions will have stationary attracting periodic limit distributions.

Counting and Summation

The previous chapters should have given you some idea of the possibilities of computer algebra for Analysis and Linear Algebra. In this chapter, we shall focus on another field of application. In contrast to the continuous (or limit) processes we have encountered so far, in the present chapter we shall work with finite processes such as ordinary counting. We shall have occasion to look into the ways Maple can assist us with counting and summation processes, both numerical and symbolic. The examples we have chosen are mainly taken from the fields of combinatorics and discrete probability theory.

Matrices and Vectors

Just like Mathematical Analysis, Linear Algebra too offers a number of techniques of great importance for many areas of mathematical application. We leave Analysis for the moment; in a later chapter we shall return to it with topics like differentiation and integration of mathematical functions. Although the present chapter is not of an analytic nature, what we have learned so far, especially about functions, will be rather useful to us in this chapter as well.

Vector Spaces and Linear Mappings

In this closing chapter we will be engaged again with notions and techniques typical for the field of Linear Algebra, but here our attention will also be focused on the abstract structure of vector spaces and linear mappings (or linear transformations) between them. Vectors—the elements of a vector space—do not need to be ordered lists of numbers, polynomials can also be viewed as vectors. Of course, matrices continue to play a prominent role, namely as representations of linear mappings between vector spaces with predetermined bases.

A Tour of Maple V

In this chapter you will learn by example how to get on with the Maple computer algebra system (CA system or CAS for short). In getting to know Maple, we shall discover how to deal with basic concepts and techniques such as giving instructions to the computer, correcting errors in input lines, reading input and writing output files, and we shall get acquainted with first principles of Maple’s programming language. For the time being we shall merely focus on getting familiar with the computer and the Maple CA system, and not on mathematics as yet. Later, in the chapters to follow, this practical knowledge will be supplemented in order to let Maple play an active role in helping you to improve your understanding of difficult mathematical concepts, and to generally enhance your overall knowledge of mathematics. These chapters introduce the use of the Maple system in branches of mathematics such as Mathematical Analysis, Linear Algebra, Probability Theory, and Discrete Mathematics. Also, in passing, we shall briefly pay attention to the graphical possibilities a CA system has to offer.

Functions and Sequences

After the first chapter’s tour of Maple, in which you may have gained some insight into the workings of the Maple system, it is time to concentrate on mathematics. Naturally, you won’t be an accomplished Maple user just yet, but, provided you carefully read the first chapter and worked through both worksheets, you should have some idea as to how Maple could assist in extending and deepening your knowledge of mathematics. Moreover, in this chapter and the ones to come you will gradually learn a great deal about Maple, so that at the end of this book there is a good chance that you will be a reasonably proficient user of the Maple system with an improved insight in mathematics to boot. Anyhow, that is what we aim for. But before reaching that state of affairs you need quite a bit of practice, so let us continue to explore the Maple avenues without further delay.