This is joint work with Tomoyoshi Ibukiyama and Chia-Fu Yu.
Supervisor: Dr. Sjoerd Dirksen
Daily supervisor: Dr. Detlef Küchler (CERN)
Second reader: Dr. Palina Salanevich
Abstract: The GTS-LHC ion source provides heavy ions to theLarge Hadron Collider (LHC) ion injector chain situated at the EuropeanOrganisation for Nuclear Research (CERN) to conduct numerous physicsexperiments. The execution of such experiments relies on the stable operationof the ion source, which depends on frequent changes of the source’s settingsby a specialist. The objective of this research is to identify patterns in timeseries data from 2021 and forecast when the machine is going to fail, allowingthe source specialist to take preventive action. In this study, a classicalmachine learning algorithm and five different neural network architectures wereanalysed and implemented to predict a beam decay. The results of theforecasting methods were compared to a baseline model to decide whetherpatterns of a beam decay exist. The implemented models were able to reach theperformance of the baseline model but not surpass it. Given the currentmeasurements, it is not possible to predict a beam decay in a short-term orlong-term forecast. Additionally, two change point detection algorithms wereprovided to recognise abrupt changes in the status of the ion source onstreaming data. After the implementation of a high voltage breakdown filter, itis possible to identify quickly and efficiently a beam decay in the present byreducing the number of false alarms.
Second Reader: Dr. MikhailHlushchanka
Abstract: An endomorphism acting on a set of points on an ellipticcurve over a finite field defines a directed graph. Because of the underlyinggroup, one can expect these graphs to look regular. I will talk about how Iadapted theorems of Ugolini to describe the trees and cycles of these graphs.An important step is using a theorem of Lenstra to translate the problem intoan algebraic number-theoretic one. The next question in my thesis is, how thegraph changes for a fixed endomorphism when the number of points varies. Tostudy this one can vary the prime number of the finite field of the ellipticcurve. I will talk about the proportion of points in cycles and lower bounds onthe density of primes where this proportion is big. I will also discusspatterns found in a computational experiment where I analysed endomorphismswith norm <200 for p<1000.
This talk is about graph theory, elliptic curves,dynamical systems and algebraic number theory. The talk should be accessible toanyone with some most basic understanding of graphs, elliptic curves andalgebraic number theory.
The Fractional Laplacian: An adaptive finite difference approach in one and two dimensions with applications
Written under the supervision of Dr. Paul Zegeling, with Prof. Kees Oosterlee as secondreader.
Abstract: The fractional Laplacian can beviewed as a generalisation of the ordinary Laplacian and likewise can be usedto model systems in sectors such as engineering, hydrology and biology. In thisthesis we study super diffusion, a consequence of Lévy flights, which can bedescribed using the fractional Laplacian of order 0<alpha<2. Thefractional Laplacian has numerous definitions and here we will use the Rieszpotential definition in one and two dimensions (which is equal to the Rieszderivative in one dimension). Methods exist to approximate the fractionalLaplacian on fixed uniform grids. Here, we present new methods to approximatethe fractional Laplacian on non-uniform adaptive finite difference meshes inone and two dimensions. These methods are applied to space-fractional heat,advection-diffusion and Fisher’s PDEs and the results presented.
Keywords: Fractional calculus, non-uniformfinite differences, adaptive mesh
Supervisor: Prof. Dr. Gunther Cornelissen
SecondReader: Dr.Valentijn Karemaker
Abstract: One of thecentral objects in analytic number theory is the study of certain L-series. Forinstance, on $\mathbb{Q}$ we have the famous Riemann zeta function. Thisfunction has a generalization to an arbitrary number field K, which is calledthe Dedekind zeta function. This function has an analytic continuation to theentire complex plane with a simple pole at s=1. Its residue at s=1 involvesmany of the basic invariants of the number field. For instance the regulator,the class group, the finite torsion subgroup of the ring of integers, and thediscriminant of the number field appear in the residue. In other words, thisfunction encodes a lot of fundamental information about the number field.
The L-series ofan elliptic curve defined over a number field is the analogue of the Dedekindzeta function for a number field. This series is defined by an Euler product,i.e. a product indexed by the primes, on a part of the complex plane. It isconjectured that this function also has an analytic continuation to the entirecomplex plane. It is moreover conjectured by Bryan John Birch and PeterSwinnerton-Dyer that it has a zero of order equal to the rank of the ellipticcurve at s=1. The first non-zero coefficient of the corresponding Taylorexpansion at s=1 is conjectured to consist of multiple basic invariantsconcerning the set of global points on the elliptic curve. In particular theregulator, the Tate-Shafarevich group, the torsion subgroup of the globalpoints, and the period of the elliptic curve.
In this thesis wewill provide the required theory for understanding the two formulae, and toobtain as many similarities as possible between them. The obtained analogiescould function as a helping hand for a better understanding of the BSDconjecture.
The talk willgive a brief introduction to the BSD conjecture and the analytic class numberformula. Moreover, some of the obtained similarities between both formulae willbe discussed. It is convenient to have some background in algebraic numbertheory and elliptic curves.