Dynamical systems can be used to model all sorts of statistical objects. In this thesis we model discrete time stochastic processes by the orbit of an initial condition under a transformation F...Show moreDynamical systems can be used to model all sorts of statistical objects. In this thesis we model discrete time stochastic processes by the orbit of an initial condition under a transformation F that maps from R to R. We can describe the divergence of the modelled stochastic process by an interval map defined on an interval I and an observable that maps from I to R and we can apply the central limit theorem to determine the distribution of this divergence. The expected value and variance of this divergence is called the drift and diffusion respectively. For complex systems determining the drift and diffusion coefficient explicitly can be very challenging. We show that for a family of transformations, the drift and diffusion coefficients admit a log-Lipschitz type continuity. When this family of transformations can be parametrized it is even shown a log-Lipschitz type continuity on the parameter values can be achieved. We extend the results found in Keller et al. (2008) to find explicit expressions for the constants involved in the log-Lipschitz continuity. We consider family of transformations for which the interval maps are asymmetric tent maps, check the assumptions and determine the log-Lipschitz constants.Show less
Edwards curves are curves of the form x^2 + y^2 = 1 + dx^2y^2. Edwards curves often find their use in cryptographic applications due to the computational advantages they provide but in this thesis...Show moreEdwards curves are curves of the form x^2 + y^2 = 1 + dx^2y^2. Edwards curves often find their use in cryptographic applications due to the computational advantages they provide but in this thesis we focus on their geometric and arithmetic properties.Show less
Expanders are sparse graphs that are highly connected. These two properties together make them prominent in both pure and applied mathematics, as well as computer science. Explicit constructions of...Show moreExpanders are sparse graphs that are highly connected. These two properties together make them prominent in both pure and applied mathematics, as well as computer science. Explicit constructions of these graphs are required for their use in many applications. But, although existence of expanders is rather easy to be proved, explicit constructions turn out to be surprisingly non-trivial. Ramanujan graphs are the optimal expanders, in the sense that they achieve asymptotically the largest expansion. In this thesis, we present an explicit construction of a family of constant degree Ramanujan graphs discovered by Pizer. These graphs are defined via the Brandt matrix of an Eichler order in quaternion algebras over $\mathbb{Q}$. We prove how these graphs attain the Ramanujan bound using the Ramanujan-Petersson Conjecture proved by Deligne. Furthermore, using the Deuring correspondence, we prove that the supersingular isogeny graphs is a subclass of these graphs and thus they are also Ramanujan.Show less
The construction of novel measures in measure theory is often done using Carathéodory’s Extension Theorem, though this can be a very tedious process. In a 1918 paper by Percy Daniell, he introduced...Show moreThe construction of novel measures in measure theory is often done using Carathéodory’s Extension Theorem, though this can be a very tedious process. In a 1918 paper by Percy Daniell, he introduced what are now known as Daniell integrals on vector lattices. The extension of this integral to a larger space naturally leads to a measure-constructing process. Given topological spaces X with some kind of projective structure, we introduce a strategy using projective systems for developing a Daniell integral on a vector lattice related to X, which in turn yields a measure on X. We apply this alternative strategy of constructing measures to some examples, where in particular we construct measures on infinite product spaces as well as infinite dimensional separable Hilbert spaces.Show less
Strong approximation is a property that some schemes have, which relates the geometry of their rational points to the geometry of their p-adic points. For S a subset of the places of the rational...Show moreStrong approximation is a property that some schemes have, which relates the geometry of their rational points to the geometry of their p-adic points. For S a subset of the places of the rational numbers, a scheme satisfies strong approximation away from S, if the rational points are dense in some product over all p not in S of the sets of p-adic points. In the last century, a couple of great, general results have been proven which give sufficient or necessary conditions for a scheme or a group scheme to satisfy strong approximation away from some set S. In 2019, Kok and Bright showed, using the Brauer-Manin obstruction, that the scheme representing primitive solutions to the equation X1^2 + 47 X2^2 - 103 X3^2 - 103 * 47 * 17 X4^2 = 0 does not satisfy strong approximation away from infinity. On the other hand, in 2020, Pagano and Bright proved a general result from which it follows that this scheme satisfies strong approximation away from infinity, 17, 47 and 103. This thesis shows that this scheme satisfies strong approximation away from infinity and 17, and it obtains a more general result for some equations of the form a X1^2 + b X2^2 + c X3^2 + d X4^2 = 0.Show less
In this master thesis, we study a stochastic model for genetic evolution. In particular, we add random resampling rates to the standard Moran model. Before the process starts, we let every...Show moreIn this master thesis, we study a stochastic model for genetic evolution. In particular, we add random resampling rates to the standard Moran model. Before the process starts, we let every individual in the population of size N choose at random a resampling rate from a finite set of size K of possible rates. We look at the K-vector of fractions of individuals with a given resampling rate and one of the two possible types. We show that, as N goes to infinity, the scaled process converges in distribution in the Meyer-Zheng topology, which is a specific topology on the space of càdlàg paths. The limiting process lives on the K-dimensional simplex and its components are deterministic fractions of the total sum of components. The total sum performs a Wright-Fisher diffusion, with a diffusion constant that is the weighted average of the resampling rates. If the resampling rates scale with N and all converge to the same constant r > 0 as N goes to infinity, then we obtain a similar result. In that case, the limiting process has diffusion constant r. If the resampling rates scale with N and converge to 0, then the random process converges in distribution in the Skorohod topology to a deterministic process.Show less
This thesis is about rational points on so-called del Pezzo surfaces, which are a certain type of surfaces with relatively simple geometry. The lower the degree of a del Pezzo surface, the more...Show moreThis thesis is about rational points on so-called del Pezzo surfaces, which are a certain type of surfaces with relatively simple geometry. The lower the degree of a del Pezzo surface, the more intricate is its geometry. Let S be a del Pezzo surface over a field k. It is known that if the degree of S is not 1 and S(k) is non-empty (with a mild extra condition for degree 2), the surface S is k-unirational, meaning that there is a dominant rational map from some projective space to S. If k is an infinite field, this implies that the set S(k) of k-rational points lies Zariski dense in S. But in general, we do not know whether unirationality holds when the degree is 1, and the answer to this question seems way out of reach. So if we want to prove the density of the k-rational points of del Pezzo surfaces of degree 1, we have to search for alternative methods. In this thesis, a result is proven that gives sufficient and necessary conditions for the Zariski density of the rational points on a certain family of del Pezzo surfaces of degree 1.Show less
Invariant theory is the study of invariants of homogeneous polynomials. Perhaps the best known example of an invariant is the discriminant b^2 - 4ac of a quadratic polynomial. In this thesis, we...Show moreInvariant theory is the study of invariants of homogeneous polynomials. Perhaps the best known example of an invariant is the discriminant b^2 - 4ac of a quadratic polynomial. In this thesis, we will consider invariants of binary quintics, which are homogeneous polynomials in two variables of degree five. Given values for the invariants, we (re)construct a binary quintic that attains this tuple of values. We will also discuss the implementation of these methods in SageMath, a free open-source mathematics software system.Show less
Adèles and idèles are nowadays frequently used in theoretical algebraic number theory, for example in class field theory. For explicit computations however, people still use the classical ideals....Show moreAdèles and idèles are nowadays frequently used in theoretical algebraic number theory, for example in class field theory. For explicit computations however, people still use the classical ideals. In this thesis we define representations of adèles and idèles, enabling us to perform explicit computations in adèle rings and idèle groups. We also discuss two applications of our representations: computing the profinite Fibonacci graph and computing Hilbert class fields of imaginary quadratic number fields using Shimura’s reciprocity law. We implemented these representations as well as the applications in the computer algebra package SageMath.Show less
We consider the family of skew tent maps Tα,β : [0, 1] → [0, 1] defined by Tα,β(x) = ( αx + α+β−αβ β for x ∈ [0, 1 − 1 β ], β − βx for x ∈ [1 − 1 β , 1] with α, β > 1 and α + β ≥ αβ. By A....Show moreWe consider the family of skew tent maps Tα,β : [0, 1] → [0, 1] defined by Tα,β(x) = ( αx + α+β−αβ β for x ∈ [0, 1 − 1 β ], β − βx for x ∈ [1 − 1 β , 1] with α, β > 1 and α + β ≥ αβ. By A. Lasota and J.A. Yorke [LY73] we know that each skew tent map has a unique acim. We fix the parameter β and show that the measure-theoretic entropy of the skew tent maps, with respect to the unique acim, depends continuously on α on a part of the parameter domain. The stability of the acim under small perturbations plays an important role in showing this result. We also investigate the relation between the measure theoretic entropy and the topological entropy for skew tent maps.Show less
A vast number of questions and problems concerning probability theory need conditional probability for providing answers and solutions. From traditional games of dice to modern statistical...Show moreA vast number of questions and problems concerning probability theory need conditional probability for providing answers and solutions. From traditional games of dice to modern statistical applications and machine learning, all use conditional probability in some sense to obtain more insight in the problem. As fundamental conditional probability is, it is not without controversy. Problems and paradoxes like the Borel-Kolmogorov paradox, Monty Hall’s three door problem and the two envelope problem have puzzled mathematicians, statisticians and psychologists for centuries, resulting into much debate and a vast amount of literature. This thesis concerns some of the most well-known paradoxes in conditional probability. In all paradoxes, the paradoxical result arises from wrongly stating the probability space concerning the problem or wrongly applying conditional probability, like not giving the accompanying σ-algebra or not conditioning on a partition. All problems can be easily avoided by always stating the probability space with the σ-algebra when applying conditional probability. The two most obvious examples are the Borel-Kolmogorov paradox and Monty Hall’s problem. The Borel-Kolmogorov paradox is a good example of why conditioning on sets with zero measure is only possible with much care and why it is necessary to provide the accompanying σ-algebra with your solution. Monty Hall’s three door problem is a prime example of wrongly conditioning on a set of subsets that cannot form a partition of the sample space. The original problem asks for a single probability, however correctly applying conditional probability reveals that the probability of the car being behind the other door is dilated between the two values 1 2 to 1. In both cases the paradoxical results vanish when the whole probability space is considered and when conditional probability is applied correctly. The dilation of the conditional probability like in Monty Hall’s problem is investigated further in this thesis. Problems like Monty Hall and the boy or girl problem resemble each other in such a fundamental fashion that a generalization exists, encompassing them both. Furthermore, safe probability introduced by Grünwald [Grff18b] can be applied to answer the following question: if one should pin a single probability on for example in Monty Hall’s game the car being behind the other door, which probability should it be? This generalization can be applied to all problems with a countable space of outcomes with fixed probability measure and finite set of possible observations with sufficiently many possible probability measures, resolving several paradoxes in probability at once.Show less
Metastability is a phenomenon where a dynamical system can move between different states that are not its global equilibrium state. On short time scales the system can find itself equilibrized in a...Show moreMetastability is a phenomenon where a dynamical system can move between different states that are not its global equilibrium state. On short time scales the system can find itself equilibrized in a certain region of its state space (a local equilibrium), whereas on a long time scale it will make quick transitions between new, different regions of its state space. These local equilibria are referred to as the metastable states. One of the uses of metastability is for model reduction. In this thesis we will restrict ourselves to Markovian processes and consider the networks associated to the transitions of the Markov chains. Instead of considering a Markov process on a very large state space, one can look at the process on a reduced state space representing these metastable states. The idea is that this coarse-grained network "mimics" the behaviour of the original network. We shall give two different mathematical definitions for metastability of Markov chains. In most cases where metastability is studied, limiting asymptotics are wielded. One must think of taking limits of large volume or low temperature. However in the paper [1] by Avena, Castell, Gaudillière and Mélot a new framework is introduced by which to describe "metastability" without the use of these limits. The network of transitions of a given Markov process is coarse-grained to a state space that represents probability measures which focus on different regions of the original finite state space (the local equilibria). It does so through the use of intertwining dualities. We say that a n × n-matrix A is intertwined with a m × m-matrix C with respect to a m × n-matrix B if BA = CB. For our discussion we are given a Markov process on a finite state space with an associated transition matrix P in order to find another Markov process on a smaller state space with transition matrix P and a matrix Λ such that ΛP = PΛ where the rows of Λ are probability measures on the original state space (representing the local equilibria). In this thesis we will explore this framework based on intertwining on a toy model consisting of three nodes that we want to reduce to a network of two nodes. The goal is to illustrate the method in [1] in this explicit model and explore which evolutions among the local equilibria can be described; how this relates to the spectrum of the transition matrix P of the Markov chain in this model; and its implications on the mixing time.Show less
