The study of Banach algebras began in the twentieth century and originated from the observation that some Banach spaces show interesting properties when they can be supplied with an extra...Show moreThe study of Banach algebras began in the twentieth century and originated from the observation that some Banach spaces show interesting properties when they can be supplied with an extra multiplication operation. A standard example was the space of bounded linear operators on a Banach space, but another important one was function spaces (of continuous, bounded, vanishing at infinity etc. functions as well as functions with absolutely convergent Fourier series). Nowadays Banach algebras is a wide discipline with a variety of specializations and applications. This particular paper focuses on Gelfand theory — the relation between multiplicative linear functionals on a commutative Banach algebra and its maximal ideals, as well as with the spectra of its elements. Most of the content of chapters 1 thorough 3 is meant, in one way or another, to lead towards this theory. The central ingredient of Gelfand theory is the well-known Gelfand-Mazur theorem which says that if a Banach algebra is a division algebra then it is isomorphic to C. The first chapter is a purely algebraic one and provides us with all the necessary algebraic techniques, particularly concerning algebras without identity. The second and third chapters introduce normed algebras and Banach algebra and other concepts like the spectrum, and prove several important results among which the Gelfand-Mazur theorem. The fourth chapter is the pivotal one — where Gelfand theory is developed. In the fifth chapter several examples of Banach algebras are discussed in detail, together with their Gelfand representations. Some practical applications of the theory are also mentioned, among which Wiener’s famous theorem about zeroes of functions with absolutely Fourier series, proven entirely from the context of Banach algebras.Show less
Consider the release into the air of a contaminated particle that could be harmful to nearby wildlife and agriculture. To understand the effect of this particle on the environment, it becomes...Show moreConsider the release into the air of a contaminated particle that could be harmful to nearby wildlife and agriculture. To understand the effect of this particle on the environment, it becomes important to find out whether it hits the ground and how long it takes to do so. Mathematically speaking, we are searching for the the exit probability and the expected exit time. Observations show that the particle doesn’t just move along with the wind. It also performs some random motion which can cause it to move against the wind. This has to be taken into account when choosing a model to describe the movement of the particle. In this thesis we look at the situation described above and also at other practical examples. Our main focus will be on finding the exit probability and the expected exit time. We start with an introduction to stochastic differential equations (SDE’s), because the models we consider will be in that form. Connected with each SDE is a partial differential equation called the Fokker-Planck (FP) equation. This FP-equation will then lead us to our first means to obtain the expected exit time. Next, we introduce the numerical simulation of SDE’s and this will provide us with a second way to obtain the expected exit time. Finally, four examples are chosen from non-mathematical research areas. Both approaches to finding the expected exit time will be applied and the results will be compared. The differential equations that arise in the second approach are approximated using singular perturbations. All results can be verified with the MATLAB code from the appendix. The examples we look at are a population model from biology, a membrane voltage model from neurology, a particle movement model from physics and a model for groundwater pollution from hydrology.Show less
With the steady increase in air traffic, nowadays most of the major hub airports experience severe traffic congestion during certain portions of the day. The main limitation at the moment is found...Show moreWith the steady increase in air traffic, nowadays most of the major hub airports experience severe traffic congestion during certain portions of the day. The main limitation at the moment is found to be the amount of landing aircraft an airport can accommodate during a period of time. A specific runway capacity constraint is the required minimum wake turbulence separation distance between consecutive landing aircraft. Currently research is being done to determine safe reduced wake vortex separation constraints that dynamically alter with the occurring local weather type. This document describes the work undertaken in the final part of the author’s studies at the mathematics department of the Universiteit Leiden, carried out at the Nationaal Lucht- en Ruimtevaartlaboratorium (NLR). The objective was to develop an airport runway capacity model capable of investigating the direct benefits in the application of weather dependent reduced distance separation requirements. The effort resulted in a further development and implementation of existing analytical runway capacity modelsShow less
We present a variant of Newton’s Method for computing travelling wave solutions to bistable lattice differential equations. We prove that the method converges to a solution, obtain existence and...Show moreWe present a variant of Newton’s Method for computing travelling wave solutions to bistable lattice differential equations. We prove that the method converges to a solution, obtain existence and uniqueness of solutions to such equations with a small second order term and study the limiting behaviour of such solutions as this second order term tends to zero. The robustness of the algorithm will be discussed using numerical examples. These results will also be used to illustrate phenomena like propagation failure, which are encountered when studying lattice differential equations. We finish by outlining some properties of higher dimensional systems, including a period two bifurcation.Show less
