In this thesis we present third-order corrections to heavy-quark production in photonexchange deep-inelastic scattering (DIS) and we study the resummation of leading and sub-leading contributions to DIS and other QCD hard processes at large values of the Bjorken variable x. We provide approximate next-to next-to-leading order (NNLO) corrections for the heavy-quark contribution to the structure function F2 in photon-exchange DIS. This is achieved by extending and combining the available NNLO information from different kinematic limits. In particular, we predict the full logarithmic behaviour of the coefficient functions near the threshold of heavy-quark production applying soft-gluon exponentiation (SGE). We utilize available all order-result in the high-energy limit to derive an analytic NNLO expression for the dominant contributions in this limit. Finally from known even integer Mellin moments of the heavy-quark operator matrix elements, as well as from the already mentioned high-energy behaviour, we construct an approximation for the heavy-quark coefficient functions at asymptotically large values of the exchanged momentum squared Q2. By combining these individual results we construct NNLO coefficient functions for heavy-quark DIS which, while still being approximate, represent the most comprehensive results possible at this point. The resulting improvement of the prediction, as well as the low-Q2 small-x limitations of the present NNLO results, are then illustrated in a phenomenological study. In the second part of this thesis we address the all-order resummation of QCD quantities in the large-x limit for which SGE is not applicable. After reviewing the resummation method that will be employed and the already available results for inclusive DIS (for which we will provide a closed analytic formula at next-to-next-to-leading logarithmic (NNLL) accuracy for the first time), we apply this method to electronpositron semi-inclusive annihilation (SIA). Also in this case, closed analytic formulae for the resummed time-like splitting and coefficient functions at NNLL accuracy are presented. This resummation method is then applied to the off-diagonal partonic cross sections in Drell-Yan lepton pair production and in Higgs hadro-production, for which we are able to resum only the leading logarithmic (LL) contributions.

The understanding of wave dynamics in interfacial microchannel flows is important for many technological applications in the micro-device industry. Here, a theoretical and numerical study is undertaken in order to understand the propagation of interfacial waves in a two-layer flow. The flow is considered to be driven by, in separate cases, the force of gravity and a pressure gradient. The results may provide steps towards more efficiently designed microfluidic products, and a better understanding of experimentally observed waves.

In this thesis we generalise the rigidity estimates of Friesecke et al. [2002] and Müller et al. [2014] to vector fields whose properties are constrained by both conditions on the support of their curl and the underlying discrete symmetries of the lattice Z2. These analytical estimates and other considerations are applied to a statistical model of a crystal containing defects based on work by Aumann [2015]. It is demonstrated in this thesis that we allow a finite density of defects. The main result is that regardless of crystal size, the ordering of the crystal, expressed via the L2-distance of a random vector field from the rotations, can be made arbitrarily small for sufficiently low temperature β-1.

We investigate the existence of phase transitions for a class of continuum multi-type particle systems. The interactions act on hyperedges between the particles, allowing us to define a class of models with geometry-dependent interactions. We establish the existence of stationary Gibbsian point processes for this class of models. A phase transition is defined with respect to the existence of multiple Gibbs measures, and we establish the existence of phase transitions in our models by proving that multiple Gibbs measures exist. Our approach involves introducing a random-cluster representation for continuum particle systems with geometry-dependent interactions. We then argue that percolation in the random-cluster model corresponds to the existence of a phase transition. The originality in this research is defining a random-cluster representation for continuum models with hyperedge interactions, and applying this representation in order to show the existence of a phase transition. We mainly focus on models where the interaction is defined in terms of the Delaunay hypergraph. We find that phase transitions exist for a class of models where the interaction between particles is via Delaunay edges or Delaunay triangles.

This work explores the nature of the normal modes of vibration for harmonic lattices with the inclusion of disorder in one-dimension (1D) and three-dimensions (3D). The model systems can be visualised as a `ball' and `spring' model in simple cubic configuration, and the disorder is applied to the magnitudes of the masses, or the force constants of the interatomic `springs' in the system. With the analogous nature between the electronic tight binding Hamiltonian for potential disordered electronic systems and the isotropic Born model for phonons in mass disordered lattices we analyse in detail a transformation between the normal modes of vibration throughout a mass disordered harmonic lattice and the electron wave function of the tight-binding Hamiltonian. The transformation is applied to density of states (DOS) calculations and is also particularly useful for determining the phase diagrams for the phonon localisation-delocalisation transition (LDT). The LDT phase boundary for the spring constant disordered system is obtained with good resolution and the mass disordered phase boundary is verified with high precision transfer-matrix method (TMM) results. High accuracy critical parameters are obtained for three transitions for each type of disorder by finite size scaling (FSS), and consequently the critical exponent that characterises the transition is found as = 1:550+0:020 -0:017 which indicates that the transition is of the same orthogonal universality class as the electronic Anderson transition. With multifractal analysis of the generalised inverse participation ratio (gIPR) for the critical transition frequency states at spring constant disorder width k = 10 and mass disorder width m = 1:2 we confirm that the singularity spectrum is the same within error as the electronic singularity spectrum at criticality and can be considered to be universal. We further investigate the nature of the modes throughout the spectrum of the disordered systems with vibrational eigenstate statistics. We find deviations of the vibrational displacement uctuations away from the Porter-Thomas distribution (PTD) and show that the deviations are within the vicinity of the so called `bosonpeak' (BP) indicating the possible significance of the BP.

Surface partial differential equations model several natural phenomena; for example in uid mechanics, cell biology and material science. The domain of the equations can often have complex and changing morphology. This implies analytic techniques are unavailable, hence numerical methods are required. The aim of this thesis is to design and analyse three methods for solving different problems with surface partial differential equations at their core. First, we define a new finite element method for numerically approximating solutions of partial differential equations in a bulk region coupled to surface partial differential equations posed on the boundary of this domain. The key idea is to take a polyhedral approximation of the bulk region consisting of a union of simplices, and to use piecewise polynomial boundary faces as an approximation of the surface and solve using isoparametric finite element spaces. We study this method in the context of a model elliptic problem. The main result in this chapter is an optimal order error estimate which is confirmed in numerical experiments. Second, we use the evolving surface finite element method to solve a Cahn- Hilliard equation on an evolving surface with prescribed velocity. We start by deriving the equation using a conservation law and appropriate transport formulae and provide the necessary functional analytic setting. The finite element method relies on evolving an initial triangulation by moving the nodes according to the prescribed velocity. We go on to show a rigorous well-posedness result for the continuous equations by showing convergence, along a subsequence, of the finite element scheme. We conclude the chapter by deriving error estimates and present various numerical examples. Finally, we stray from surface finite element method to consider new unfitted finite element methods for surface partial differential equations. The idea is to use a fixed bulk triangulation and approximate the surface using a discrete approximation of the distance function. We describe and analyse two methods using a sharp interface and narrow band approximation of the surface for a Poisson equation. Error estimates are described and numerical computations indicate very good convergence and stability properties.

<p> We consider a classical lattice dipole gas with low activity in dimension <i>d</i> &ge; 3. We study long distance properties by a renormalization group analysis. We prove that various correlation functions have a infinite volume limit. We also get estimates on the decay of correlation functions.</p>

No description available.

<p> In this thesis we investigate two new Amplified Quantum Transforms. In particular we create and analyze the Amplified Quantum Fourier Transform (Amplified-QFT) and the Amplified-Haar Wavelet Transform. The Amplified-QFT algorithm is used to solve the Local Period Problem. We calculate the probabilities of success and compare this algorithm with the QFT and QHS algorithms. We also examine the Amplified-QFT algorithm for solving the Local Period Problem with Error Stream. We use the Amplified-Haar Wavelet Transform for solving the Local Constant or Balanced Signal Decision Problem which is a generalization of the Deutsch-Jozsa problem.</p>

No description available.