This thesis provides explicit expressions for the density functions of absolutely continuous invariant measures for general families of interval maps, that include randommaps and infinite measure... Show moreThis thesis provides explicit expressions for the density functions of absolutely continuous invariant measures for general families of interval maps, that include randommaps and infinite measure transformations, not necessarily number systems. Natural extensions, the Perron-Frobeniusoperator and the dynamical phenomenon of matching are some of the techniques exploited to obtain such results. In particular, in this thesis the notion of matching is for the first time recognised in an infinite measure system and the definition, known so far for deterministic transformations only, is extended to cover random interval maps as well. The thesis also presents new developments in the area of number expansions Show less
In recent years HLA epitope matching is becoming a hot topic in transplant community to prevent donor-specific antibody formation after transplantation, as such antibodies are associated with... Show moreIn recent years HLA epitope matching is becoming a hot topic in transplant community to prevent donor-specific antibody formation after transplantation, as such antibodies are associated with inferior graft survival. The number of HLA epitope mismatches between donor and recipient correlates with donor-specific antibody formation, but not every epitope mismatch will trigger an antibody response. Therefore, it is pivotal to define the immunogenic epitopes. In my thesis we describe the development of the software HLA-EMMA to determine HLA amino acid mismatches between donor and recipient to identify immunogenic amino acids. In addition, the generation of recombinant human HLA-DR monoclonal antibodies for antibody-verification of eplets/epitopes is described. These tools will contribute to the definition of immunogenic epitopes, which is required before introducing HLA epitope matching in the clinic. Show less
In this dissertation, matching, entropy, holes and expansions come together. The first chapter is an introduction to ergodic theory and dynamical systems. The second chapter is on, what we called... Show moreIn this dissertation, matching, entropy, holes and expansions come together. The first chapter is an introduction to ergodic theory and dynamical systems. The second chapter is on, what we called Flipped $\alpha$-expansions. For this family we have an invariant measure that is $\sigma$-finite infinite. We calculate the Krengel entropy for a large part of the parameter space and find an explicit expression for the density by using the natural extension. In Chapter 3 Ito Tanaka's $\alpha$-continued fractions are studied. We prove that matching holds almost everywhere and that the non-matching set has full Hausdorff dimension. In the fourth chapter we study $N$-expansions with flips. We use a Gauss-Kuzmin-Levy method to approximate the density for a large family and use this to give an estimation for the entropy. In the last Chapter we look at greedy $\beta$-expansions. We show that for almost every $\beta\in(1,2]$ the set of points $t$ for which the forward orbit avoids the hole $[0,t)$ has infinitely many isolated and infinitely many accumulation points in any neighborhood of zero. Furthermore, we characterize the set of $\beta$ for which there are no accumulation points and show that this set has Hausdorff dimension zero. Show less
