The presence of a small parameter can reduce the complexity of the stability analysis of pattern solutions. This reduction manifests itself through the complex-analytic Evans function, which... Show moreThe presence of a small parameter can reduce the complexity of the stability analysis of pattern solutions. This reduction manifests itself through the complex-analytic Evans function, which vanishes on the spectrum of the linearization about the pattern. For certain 'slowly linear' prototype models it has been shown, via geometric arguments, that the Evans function factorizes in accordance with the scale separation. This leads to asymptotic control over the spectrum through simpler, lower-dimensional eigenvalue problems. Recently, the geometric factorization procedure has been generalized to homoclinic pulse solutions in slowly nonlinear reaction-di ffusion systems. In this thesis we study periodic pulse solutions in the slowly nonlinear regime. This seems a straightforward extension. However, the geometric factorization method fails and due to translational invariance there is a curve of spectrum attached to the origin, whereas for homoclinic pulses there is only a simple eigenvalue residing at 0. We develop an alternative, analytic factorization method that works for periodic structures in the slowly nonlinear setting. We derive explicit formulas for the factors of the Evans function, which yields asymptotic spectral control. Moreover, we obtain a leading-order expression for the critical spectral curve attached to origin. Together these approximation results lead to explicit stability criteria. Show less
We begin by introducing the main ideas of the paper, and we give a brief description of the method proposed. Next, we discuss an alternative approach based on B-spline expansion, and lastly we make... Show moreWe begin by introducing the main ideas of the paper, and we give a brief description of the method proposed. Next, we discuss an alternative approach based on B-spline expansion, and lastly we make some comments on the method’s convergence rate. Show less
In this thesis, we studied the Hodge theory and deformation theory of nodal surfaces. We showed that nodal surfaces in the projective 3-space satisfy the infinitesimal Torelli property. We... Show moreIn this thesis, we studied the Hodge theory and deformation theory of nodal surfaces. We showed that nodal surfaces in the projective 3-space satisfy the infinitesimal Torelli property. We considered families of examples of even nodal surfaces, that is, those endowed with a double cover branched on the nodes. We gave a new geometrical construction of even 56-nodal sextic surfaces, while we proved, using existing constructions, that the sub-Hodge structure of type (1,26,1) on the double cover S of any even 40-nodal sextic surface cannot be simple. We also demonstrated ways to compute sheaves of differential forms on singular varieties using Saito's theory of mixed Hodge modules. Show less
Let A be a Banach algebra with a bounded left approximate identity {eλ}λ∈Λ" role="presentation">{eλ}λ∈Λ, let π" role="presentation">π be a continuous representation of A on a Banach space X, and let SShow moreLet A be a Banach algebra with a bounded left approximate identity {eλ}λ∈Λ" role="presentation">{eλ}λ∈Λ, let π" role="presentation">π be a continuous representation of A on a Banach space X, and let S be a non-empty subset of X such that limλπ(eλ)s=s" role="presentation">limλπ(eλ)s=s uniformly on S. If S is bounded, or if {eλ}λ∈Λ" role="presentation">{eλ}λ∈Λ is commutative, then we show that there exist a∈A" role="presentation">a∈A and maps xn:S→X" role="presentation">xn:S→X for n≥1" role="presentation">n≥1 such that s=π(an)xn(s)" role="presentation">s=π(an)xn(s) for all n≥1" role="presentation">n≥1 and s∈S" role="presentation">s∈S. The properties of a∈A" role="presentation">a∈A and the maps xn" role="presentation">xn, as produced by the constructive proof, are studied in some detail. The results generalize previous simultaneous factorization theorems as well as Allan and Sinclair’s power factorization theorem. In an ordered context, we also consider the existence of a positive factorization for a subset of the positive cone of an ordered Banach space that is a positive module over an ordered Banach algebra with a positive bounded left approximate identity. Such factorizations are not always possible. In certain cases, including those for positive modules over ordered Banach algebras of bounded functions, such positive factorizations exist, but the general picture is still unclear. Furthermore, simultaneous pointwise power factorizations for sets of bounded maps with values in a Banach module (such as sets of bounded convergent nets) are obtained. A worked example for the left regular representation of C0(R)" role="presentation">C0(R) and unbounded S is included. Show less
This thesis concerns the mathematical analysis of certain random walks in a dynamic random environment. Such models are important in the understanding of various models in physics, chemistry and... Show moreThis thesis concerns the mathematical analysis of certain random walks in a dynamic random environment. Such models are important in the understanding of various models in physics, chemistry and biology. The interest is in questions such as how to determine the average velocity of the random walker and how to control fluctuations and deviations thereof. This is in general a very challenging problem due to the possibility of strong dependence both in space and time, and many problems are still wide open. After a general introduction in Chapter 1, we present several approaches for determining the asymptotic behaviour for random walks in a dynamic random environment in Chapter 2-5 of this thesis. Our work improves on the existing literature for general models with strongly mixing dynamics and provides new insight for certain models with poorly mixing dynamics. One particular model is analysed in more detailed, namely the so-called contact process. This model is a prototype of a dynamic random environment with poor mixing properties. In addition to results for certain random walks with the contact process as dynamic random environment, we also provide new insight for the contact process itself, given in Chapter 5. Show less
We consider the general properties of bounded approximate units in non-self-adjoint operator algebras. Such algebras arise naturally from the differential structure of spectral triples and... Show moreWe consider the general properties of bounded approximate units in non-self-adjoint operator algebras. Such algebras arise naturally from the differential structure of spectral triples and unbounded Kasparov modules. Our results allow us to present a unified approach to characterising completeness of spectral metric spaces, existence of connections on modules, self-adjointness and regularity of induced operators on tensor product C*-modules and the lifting of Kasparov products to the unbounded category. In particular, we prove novel existence results for quasicentral approximate units in non-self-adjoint operator algebras, allowing us to strengthen Kasparov's technical theorem and extend it to this realm. Finally, we show that given any two composable KK-classes, we can find unbounded representatives whose product can be constructed to yield an unbounded representative of the Kasparov product. (C) 2016 Elsevier Inc. All rights reserved. Show less
Kabboord, A.D.; Eijk, M. van; Fiocco, M.; Balen, R. van; Achterberg, W.P. 2016
In the thesis, `Patterns in natural systems’ the formation and evolution of patterns as solutions of several partial differential systems are studied. These mathematical systems model three... Show more In the thesis, `Patterns in natural systems’ the formation and evolution of patterns as solutions of several partial differential systems are studied. These mathematical systems model three different biological and ecological processes. First, the way that plankton concentrates in the water column, under the influence of light and nutrient availability. Second, how tumor cells invade their healthy surroundings when it is incorporated that tumor cells cannot survive in a very small concentration. Lastly, the phenomenon that vegetation in semi-deserts organizes in strikingly regular patterns is studied. The mathematical tools that are used in this thesis, mostly arise from asymptotic analysis and geometric singular perturbation theory. Show less
This thesis consists of two distinct topics. The first part of the thesis con- siders Gibbs-non-Gibbs transitions. Gibbs measures describe the macro- scopic state of a system of a large number of... Show moreThis thesis consists of two distinct topics. The first part of the thesis con- siders Gibbs-non-Gibbs transitions. Gibbs measures describe the macro- scopic state of a system of a large number of components that is in equilib- rium. It may happen that when the system is transformed, for example, by a stochastic dynamics that runs over a certain time interval, the evolved state is no longer a Gibbs measure. We study transitions from Gibbs tot non-Gibbs for mean-field systems and their relation to the large deviation rate function that is related to those systems. In the second part of the thesis we describe different notions of integrals for functions with values in a partially ordered vector space. We describe two extensions for integrals, called the vertical and the lateral extension. We compare combinations of them and compare them to other integrals. Another integral can be obtained for Archimedean directed ordered vector spaces, as they can be covered by Banach spaces in a natural way. This allows us to generalise the Bochner integral to function with values in such space. Show less
