School of Mathematics
http://hdl.handle.net/2262/1
School of Mathematics2016-12-03T02:32:33ZSolutions to dilation equations
http://hdl.handle.net/2262/77576
Solutions to dilation equations
Malone, David
This thesis aims to explore part of the wonderful world of dilation equations. Dilation equations have a convoluted history, having reared their heads in various mathematical ﬁelds. One of the early appearances was in the construction of continuous but nowhere differentiable functions. More recently dilation equations have played a signiﬁcant role in the study of subdivision schemes and in the construction of wavelets. The intention here is to study dilatione quations as entities of interest in their own right, just as the similar subjects of differential and difference equations are often studied. It will often be Lp(R) properties we are interested in and we will often use Fourier Analysis as a tool. This is probably due to the author’s original introduction to dilation equations through wavelets. A short introduction to the subject of dilation equations is given in Chapter 1. The introduction is ﬂeeting, but references to further material are given in the conclusion. Chapter 2 considers the problem of ﬁnding all solutions of the equation which arises when the Fourier transform is applied to a dilation equation. Applying this result to the Haar dilation equation allows us ﬁrst to catalogue the L2(R) solutions of this equation and then to produce some nice operator results regarding shift and dilation operators. We then consider the same problem in Rn where, unfortunately, techniques using dilation equations are not as easy to apply. However, the operator results are retrieved using traditional multiplier techniques. In Chapter 3 we attempt to do some hands-on calculations using the results of Chapter 2. We discover a simple ‘factorisation’ of the solutions of the Haar dilation equation. Using this factorisation we produce many solutions of the Haar dilation equation. We then examine how all these results might be applied to the solutions of other dilation equations. A technique which I have not seen exploited elsewhere is developed in Chapter 4. This technique examines a left-hand or right-hand ‘end’ of a dilation equation. It is initially developed to search for reﬁnable characteristic functions and leads to a characterisation of reﬁnable functions which are constant on intervals of the form [n, n +1). This left-hand end method is then applied successfully to the problem of 2- and 3- reﬁnable functions and used to obtain bounds on smoothness and boundedness. Chapter 5 is a collection of smaller results regarding dilation equations. The relatively simple problem of polynomial solutions of dilation equations is covered, as are some methods for producing new solutions and equations from known solutions and equations. Results regarding when self-similar tiles can be of a simple form are also presented.
2001-01-01T00:00:00ZFourier analysis, multiresolution analysis and dilation equations
http://hdl.handle.net/2262/77575
Fourier analysis, multiresolution analysis and dilation equations
Malone, David
This thesis has essentially two parts. The ﬁrst two chapters are an introduction to the related areas of Fourier analysis, multiresolution analysis and wavelets. Dilation equations arise in the context of multi resolution analysis. The mathematics of these two chapters is informal, and is intended to provide a feeling for the general subject. This work is loosely based on two talks which I gave, one during the 1997 Inter-varsity Mathematics competition and the other at the 1997 Dublin Institute for Advanced Studies Easter Symposium. The second part, Chapters 3 and 4, contain original work. Chapter 3 provides a new formal construction of the Fourier transform on Lp(Rn) (1 ≤ p ≤ 2) based on the ideas introduced in Chapter 2. The idea is to take some basic properties of the Fourier transform and show we can construct a bounded operator on L2(R) with these properties. I do this by constructing an operator on each level of the Haar multiresolution analysis, which I then show is well enough behaved to be extended by a limiting process to all of L2(R). Some of the important properties of the Fourier transform are also derived in terms of this construction, and the generalisations to Lp(Rn) are explored. Chapter 4 builds on the work of Chapters 3 and provides a uniqueness result for the Fourier transform. While searching for this result I also establish a related result for dilation equations (a subject also introduced in Chapter 2). Here the exact set of properties which were used to deﬁne the Fourier transform are varied in an effort to discover which are merely consistent with the Fourier transform and which strong enough to deﬁne it. I end up examining sets of dilation equations and determining when these will have a unique solution.
1998-01-01T00:00:00ZDiscrete exterior calculus with applications to flows and spinors
http://hdl.handle.net/2262/76932
Discrete exterior calculus with applications to flows and spinors
De Beaucé, Vivien
2005-01-01T00:00:00ZQuantum spectral curve as a tool for a perturbative quantum field theory
http://hdl.handle.net/2262/76887
Quantum spectral curve as a tool for a perturbative quantum field theory
VOLIN, DMYTRO
An iterative procedure perturbatively solving the quantum spectral curve of planar N=4N=4 SYM for any operator in the slsl(2) sector is presented. A Mathematica notebook executing this procedure is enclosed. The obtained results include 10-loop computations of the conformal dimensions of more than ten different operators.
We prove that the conformal dimensions are always expressed, at any loop order, in terms of multiple zeta-values with coefficients from an algebraic number field determined by the one-loop Baxter equation. We observe that all the perturbative results that were computed explicitly are given in terms of a smaller algebra: single-valued multiple zeta-values times the algebraic numbers.
PUBLISHED
2015-01-01T00:00:00Z