Continuous wavelet transforms on the unit sphere

Doutoramento em MatemáticaFCT - SFRH/BD/12744/200

Matricial approximations of higher dimensional master fields

We study matricial approximations of master fields constructed in [6]. These approximations (in non-commutative distribution) are obtained by extracting blocks of a Brownian unitary diffusion (with entries in R, C or K) and letting the dimension of these blocks to tend to infinity. We divide our study into two parts: in the first one, we extract square blocks while in the second one we allow rectangular blocks. In both cases, free probability theory appears as the natural framework in which the limiting distributions are most accurately described

Wavelets on Lie groups and homogeneous spaces

Within the past decades, wavelets and associated wavelet transforms have been intensively investigated in both applied and pure mathematics. They and the related multi-scale analysis provide essential tools to describe, analyse and modify signals, images or, in rather abstract concepts, functions, function spaces and associated operators. We introduce the concept of diffusive wavelets where the dilation operator is provided by an evolution like process that comes from an approximate identity. The translation operator is naturally defined by a regular representation of the Lie group where we want to construct wavelets. For compact Lie groups the theory can be formulated in a very elegant way and also for homogeneous spaces of those groups we formulate the theory in the theory of non-commutative harmonic analysis. Explicit realisation are given for the Rotation group SO(3), the k-Torus, the Spin group and the n-sphere as homogeneous space. As non compact example we discuss diffusive wavelets on the Heisenberg group, where the construction succeeds thanks to existence of the Plancherel measure for this group. The last chapter is devoted to the Radon transform on SO(3), where the application on diffusive wavelets can be used for its inversion. The discussion of a variational spline approach provides criteria for the choice of points for measurements in concrete applications

Universal Tutte characters via combinatorial coalgebras

The Tutte polynomial is the most general invariant of matroids and graphs that can be computed recursively by deleting and contracting edges. We generalize this invariant to any class of combinatorial objects with deletion and contraction operations, associating to each such class a universal Tutte character by a functorial procedure. We show that these invariants satisfy a universal property and convolution formulae similar to the Tutte polynomial. With this machinery we recover classical invariants for delta-matroids, matroid perspectives, relative and colored matroids, generalized permutohedra, and arithmetic matroids, and produce some new convolution formulae. Our principal tools are combinatorial coalgebras and their convolution algebras. Our results generalize in an intrinsic way the recent results of Krajewski--Moffatt--Tanasa.Comment: Accepted version, 51p

Quantum limits, counting and Landau-type formulae in hyperbolic space

In this thesis we explore a variety of topics in analytic number theory and automorphic forms. In the classical context, we look at the value distribution of two Dirichlet L-functions in the critical strip and prove that for a positive proportion these values are linearly independent over the real numbers. The main ingredient is the application of Landau's formula with Gonek's error term. The remainder of the thesis focuses on automorphic forms and their spectral theory. In this setting we explore three directions. First, we prove a Landau-type formula for an exponential sum over the eigenvalues of the Laplacian in PSL(2, Z)\H by using the Selberg Trace Formula. Next, we look at lattice point problems in three dimensions, namely, the number of points within a given distance from a totally geodesic hyperplane. We prove that the error term in this problem is O(X^{3/2}), where arccosh(X) is the hyperbolic distance to the hyperplane. An application of large sieve inequalities provides averages for the error term in the radial and spatial aspect. In particular, the spatial average is consistent with the conjecture that the pointwise error term is O(X^{1+\epsilon}). The radial average is an improvement on the pointwise bound by 1/6. Finally, we identify the quantum limit of scattering states for Bianchi groups of class number one. This follows as a consequence of studying the Quantum Unique Ergodicity of Eisenstein series at complex energies

Wavelets on Lie groups and homogeneous spaces

Within the past decades, wavelets and associated wavelet transforms have been intensively investigated in both applied and pure mathematics. They and the related multi-scale analysis provide essential tools to describe, analyse and modify signals, images or, in rather abstract concepts, functions, function spaces and associated operators. We introduce the concept of diffusive wavelets where the dilation operator is provided by an evolution like process that comes from an approximate identity. The translation operator is naturally defined by a regular representation of the Lie group where we want to construct wavelets. For compact Lie groups the theory can be formulated in a very elegant way and also for homogeneous spaces of those groups we formulate the theory in the theory of non-commutative harmonic analysis. Explicit realisation are given for the Rotation group SO(3), the k-Torus, the Spin group and the n-sphere as homogeneous space. As non compact example we discuss diffusive wavelets on the Heisenberg group, where the construction succeeds thanks to existence of the Plancherel measure for this group. The last chapter is devoted to the Radon transform on SO(3), where the application on diffusive wavelets can be used for its inversion. The discussion of a variational spline approach provides criteria for the choice of points for measurements in concrete applications

Micromagnetic Simulations of High-Speed Magnonic Devices

An emerging field of research in recent years has been magnonics, the manipulation of coherent spin excitations, spin-waves, in magnetically ordered materials. Recent advances in experimental techniques for high-frequency magnetisation dynamics and the advent of micromagnetic simulations has led to the propositions of functional magnetic devices based upon the control of spin-waves. This thesis presents work for characterisation and future development of high-speed magnonic devices derived from micromagnetic simulations, and numerical techniques for the solution of the Landau-Lifshitz equation for micromagnetic simulations in the finite-difference time-domain approach. In chapter 3, spin-waves were controlled in the propagation along a thin film magnonic waveguide via resonant scattering from a mesoscale chiral magnetic resonator, in the backwards volume, forwards volume and Damon-Eshbach geometries. The scattering interaction demonstrated non-reciprocity associated with devices acting as spin-wave diodes. Additionally, such devices demonstrated the possibility of phase-shifting. The results obtained were numerically fit and interpreted in terms of a phenomenological model of resonant chiral scattering. The origin of the chiral coupling was discussed in terms of the stray field. In chapter 4, the phenomenon of spin-wave confinement, wavelength conversion and Möbius mode formation was demonstrated in the backwards volume configuration of thin-film magnetic waveguides. The presence of magnetic field gradients or thickness gradients modified the position of the Γ-point of the dispersion relation for Backwards Volume Dipolar-Exchange Spin-Waves (BVDESW), such that back-scattering and wavelength conversion occurred from the field/thickness gradients due to the “valleys” of the spin-wave dispersion. This work highlights a basis for not only experimental observation of such phenomena, but the potential for devices based upon valleytronics, an exploitation of the valley degree-of-freedom due to the spin-wave dispersion. In chapter 5, motivated by numerical error encountered in previous work in the thesis, the validity of implicit methods formulated for the numerical solution of the Landau-Lifshitz equation for finite-difference time-domain micromagnetic simulations were demonstrated. The implicit methods were tested for single spin precession in an external field, the μMAG standard problems and additional test cases. A source of numerical instability in explicit integration methods, numerical stiffness in systems of differential equations, was demonstrated to occur in existing explicit numerical methods, applied to the Landau-Lifshitz equation, common to popular micromagnetic software. The stability of implicit methods was demonstrated to be advantageous over explicit methods in micromagnetic scenarios where numerical stiffness could occur. Additionally, it was demonstrated that the quality of the numerical results was improved compared to explicit methods when the implicit method possessed L-stability, a damping of stiff, high wave number spin waves in the simulation.Engineering and Physical Sciences Research Council (EPSRC

A Generalised Gangolli-Levy-Khintchine Formula for Infinitely Divisible Measures and Levy Processes on Semi-Simple Lie Groups and Symmetric Spaces

In 1964 R.Gangolli published a Levy-Khintchine type formula which characterised K bi-invariant infinitely divisible probability measures on a symmetric space G=K. His main tool was Harish-Chandra's spherical functions which he used to construct a generalisation of the Fourier transform of a measure. In this paper we use generalised spherical functions (or Eisenstein integrals) and extensions of these which we construct using representation theory to obtain such a characterisation for arbitrary infinitely divisible probability measures on a non-compact symmetric space. We consider the example of hyperbolic space in some detail