Enriched structures on stable curves over fields were defined by Mainò in the late 1990s, and have played an important role in the study of limit linear series and degenerating jacobians. In this... Show moreEnriched structures on stable curves over fields were defined by Mainò in the late 1990s, and have played an important role in the study of limit linear series and degenerating jacobians. In this paper we solve three main problems: we give a definition of enriched structures on stable curves over arbitrary base schemes, and show that the resulting fine moduli problem is representable; we show that the resulting object has a universal property in terms of Néron models; and we construct a compactification of our stack of enriched structures. Show less
Arntzen, V.H.; Fiocco, M.; Leitzinger, N.; Geskus, R.B. 2023
We consider the problem of explicitly computing Beilinson–Bloch heights of homologically trivial cycles on varieties defined over number fields. Recent results have established a congruence, up to... Show moreWe consider the problem of explicitly computing Beilinson–Bloch heights of homologically trivial cycles on varieties defined over number fields. Recent results have established a congruence, up to the rational span of logarithms of primes, between the height of certain limit mixed Hodge structures and certain Beilinson–Bloch heights obtained from odd-dimensional hypersurfaces with a node. This congruence suggests a new method to compute Beilinson–Bloch heights. Here we explain how to compute the relevant limit mixed Hodge structures in practice, then apply our computational method to a nodal quartic curve and a nodal cubic threefold. In both cases we explain the nature of the primes occurring in the congruence. Show less
Explaining treatment response variability between and within patients can support treatment and dosing optimization, to improve treatment of individual patients. This thesis discussed multiple... Show moreExplaining treatment response variability between and within patients can support treatment and dosing optimization, to improve treatment of individual patients. This thesis discussed multiple aspects of treatment variability and the associated statistical learning techniques which can be used to explain and/or predict part of that variability. Even though in recent times the availability of several high-throughput measurement technologies has created many new opportunities to develop improved treatment strategies, deriving actionable insights from such data remains a challenge. To this end, the use of longitudinal and high-dimensional data analysis techniques is needed to explore omics data for explaining treatment response and clinical course, and to answer clinical questions from routine healthcare data from hospitals and research institutes. Show less
In this work we consider a generalization of graph flows. A graph flow is, in its simplest formulation, a labelling of the directed edges with real numbers subject to various constraints. A common... Show moreIn this work we consider a generalization of graph flows. A graph flow is, in its simplest formulation, a labelling of the directed edges with real numbers subject to various constraints. A common constraint is conservation in a vertex, meaning that the sum of the labels on the incoming edges of this vertex equals the sum of those on the outgoing edges. One easy fact is that if a flow is conserving in all but one vertex, then it is also conserving in the remaining one. In our generalization we do not label the edges with real numbers, but with elements from an arbitrary group, where this fact becomes false in general. As we will show, graphs with the property that conservation of a flow in all but one vertex implies conservation in all vertices are precisely the planar graphs. Show less
Signorelli, M.; Tsonka, R.; Aartsma-Rus, A.; Spitali, P. 2023
We consider the spatially inhomogeneous Moran model with seed-banks introduced in den Hollander and Nandan (2021). Populations comprising active and dormant individuals are structured in colonies... Show moreWe consider the spatially inhomogeneous Moran model with seed-banks introduced in den Hollander and Nandan (2021). Populations comprising active and dormant individuals are structured in colonies labelled by Zd, d≥1. The population sizes are drawn from an ergodic, translation-invariant, uniformly elliptic field that form a random environment. Individuals carry one of two types: ♡, ♠. Dormant individual resides in what is called a seed-bank. Active individuals exchange type from seed-bank of their own colony and resample type by choosing parent from the active populations according to a symmetric migration kernel. In den Hollander and Nandan (2021) by using a dual (an interacting coalescing particle system), we showed that the spatial system exhibits a dichotomy between clustering (mono-type equilibrium) and coexistence (multi-type equilibrium). In this paper we identify the domain of attraction for each mono-type equilibrium in the clustering regime for a fixed environment. We also show that when the migration kernel is recurrent, for a.e. realization of the environment, the system with an initially consistent type distribution converges weakly to a mono-type equilibrium in which the fixation probability to type-♡ configuration does not depend on the environment. A formula for the fixation probability is given in terms of an annealed average of type-♡ densities in dormant and active population biased by ratio of the two population sizes at the target colony.Primary techniques employed in the proofs include stochastic duality and the environment process viewed from particle, introduced in Dolgopyat and Goldsheid (2019) for random walk in random environment on a strip. A spectral analysis of Markov operator yields quenched weak convergence of the environment process associated with the single-particle dual process to a reversible ergodic distribution, which we transfer to the spatial system of populations by using duality. Show less
This dissertation consists of two parts, each of which considers a different research area related to random interval maps. In the first part we are interested in random interval maps that are... Show moreThis dissertation consists of two parts, each of which considers a different research area related to random interval maps. In the first part we are interested in random interval maps that are critically intermittent. In Chapter 2 we consider a large class of such systems and demonstrate the existence of a phase transition, where the absolutely continuous invariant measure changes between finite and infinite. For a closely related class we derive in Chapter 3 statistical properties like decay of correlations and the Central Limit Theorem. In Chapter 4 we investigate whether a similar phase transition remains to exist when the critical behaviour is toned down. Random interval maps can also be used to generate number expansions, which will be the main object of study in the second part. In Chapter 5 we generalize Lochs’ Theorem, which compares the efficiency between representing real numbers in decimal expansions and regular continued fraction expansions, to a wide class of pairs of random interval maps that produce number expansions. Closely related to this result, we study in Chapter 6 the efficiency of beta-encoders as a potential source for pseudo-random number generation by comparing the output of a beta-encoder with its corresponding binary expansion. Show less
A Chung–Lu random graph is an inhomogeneous Erdős–Rényi random graph in which vertices are assigned average degrees, and pairs of vertices are connected by an edge with a probability that is... Show moreA Chung–Lu random graph is an inhomogeneous Erdős–Rényi random graph in which vertices are assigned average degrees, and pairs of vertices are connected by an edge with a probability that is proportional to the product of their average degrees, independently for different edges. We derive a central limit theorem for the principal eigenvalue and the components of the principal eigenvector of the adjacency matrix of a Chung–Lu random graph. Our derivation requires certain assumptions on the average degrees that guarantee connectivity, sparsity and bounded inhomogeneity of the graph. Show less
The main theme of this thesis is the theoretical study of Gaussian processes as a tool in Bayesian nonparametric statistics. We are interested in the frequentist properties of Bayesian... Show moreThe main theme of this thesis is the theoretical study of Gaussian processes as a tool in Bayesian nonparametric statistics. We are interested in the frequentist properties of Bayesian nonparametric techniques in an asymptotic regime. We will be focusing specifically on consistency, convergence rates, uncertainty quantification and adaptation. These properties will be studied in the context of non-parametric problems, that is to say models with few modeling constraints. Moroever, the thesis will cover the topic of scalability of Bayesian techniques. Indeed, Bayesian methods are computation-hungry and rapidly become intractable as the number of observations grows. This issue led to the introduction of distributed Bayesian methods in order to decrease the computational complexity of the techniques. Show less
The CM class number one problem for elliptic curves asked to find all elliptic curves defined over the rationals with non-trivial endomorphism ring. For genus-2 curves it is the problem of... Show moreThe CM class number one problem for elliptic curves asked to find all elliptic curves defined over the rationals with non-trivial endomorphism ring. For genus-2 curves it is the problem of determining all CM curves of genus 2 defined over the reflex field. We solve the problem by showing that the list given in Bouyer and Streng [LMS J. Comput. Math., 18(1):507{538, 2015] is complete. Show less
Arpin, S.A.; Bozlee, S.; Herr, L.S.; Smith, H. 2023
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