Some Spectral Problems in Mathematical Physics

In this dissertation we deal with some spectral problems for periodic differential operators originating from mathematical physics. We begin by using quantum graphs to model a particular graphyne and related nanotubes. The dispersion relations, and thus spectra, of periodic Schrödinger operators on these structures are analyzed. We find highly directional Dirac cones, which makes some types of graphynes fascinating. Then, we study a conjecture that has been widely assumed in solid state physics. Namely, the extrema of the dispersion relation of a generic periodic difference operator on a class of discrete graphs are proven to be non-degenerate. Here, by non-degeneracy we mean extrema having non-degenerate Hessian. Finally, we present a technique of creating and manipulating spectral gaps for a (regular) periodic quantum graph by inserting appropriate internal structures into its vertices

Birefringent properties of the human cornea in vivo : towards a new model of corneal structure

The fundamental corneal properties of mechanical rigidity, maintenance of curvature and optical transparency result from the specific organisation of collagen fibrils in the corneal stroma. The exact arrangement of stromal collagen is currently unknown but several structural models have been proposed. The purpose of the present study is to investigate inconsistencies between current x‐ray derived structural models of the cornea and optically derived birefringence data. Firstly, the thesis reviews the current understanding of corneal structure, particularly in relation to corneal birefringence. It also reviews and develops the different analytical approaches used to model optical biaxial behaviour, particularly as applied to predict corneal optical phase retardation. The second part develops a novel technique of elliptic polarization biomicroscopy (EPB), enabling study of corneal birefringence in vivo. Using EPB, the pattern of corneal retardation is recorded for a range of human subjects. This dataset is then used to investigate both central and peripheral corneal birefringence as well as the corneal microstructure. A key finding is that the central parts of the cornea exhibit a retardation pattern compatible with a negative biaxial crystal, whereas the peripheral corneal regions do not. Furthermore, within the central regions of the cornea, orthogonal confocal conic fibrillar structures are identified which resemble the analytically derived contours of equal refractive index of an ideal negative biaxial crystal. The third part of this work presents a synthesis of previous published experimental, anatomical and theoretical findings and the experimental results presented in this thesis. Based on these findings, a novel corneal structural model is proposed that comprises overlapping spherical elliptic structural units. Finally, ensuing biomechanical and clinical consequences of the spherical elliptic structural model and of the EPB technique are discussed including their potential diagnostic and surgical applications

On Statistical Modelling and Hypothesis Testing by Information Theoretic Methods

The main objective of this thesis is to study various information theoretic methods and criteria in the context of statistical model selection. The focus in this research is on Rissanen’s Minimum Description Length (MDL) principle and its variants, with a special emphasis on the Normalized Maximum Likelihood (NML).We extend the Rissanen methodology for coping with infinite parametric complexity and discuss two particular cases. This is applied for deriving four NMLcriteria and investigate their performance. Furthermore, we find the connection between Stochastic Complexity (SC), defined as minus logarithm of NML, and other model selection criteria.We also study the use of information theoretic criteria (ITC) for selecting the order of autoregressive (AR) models in the presence of nonstationarity. In particular, we give a modified version of Sequentially NML (SNML) when the model parameters are estimated by forgetting factor LS algorithm.Another contribution of the thesis is in connection with the new approach for composite hypothesis testing using Optimally Distinguishable Distributions (ODD). The ODD-detector for subspace signals in Gaussian noise is introduced and its performance is evaluated.Additionally, we exploit the Kolmogorov Structure Function (KSF) to derive a new criterion for cepstral nulling, which has been recently applied to the problem of periodogram smoothing.Finally, the problem of fairness in multiaccess communication systems is investigated and a new method is proposed. The new approach is based on partitioning the network into subnetworks and employing two different multiple-access schemes within and across subnetworks. It is also introduced an algorithm for selecting optimally the subnetworks such that to achieve the max-min fairness

INVERSE METHOD OF NANOINDENTATION BY LASER INTERFEROMETRY

Nanoindentation is a process characterized by the continuous measurement of the force applied to and the resulting displacement of a sample. Commercial indenters are cost-prohibitive; therefore, a low-cost module using a unique inverse method of nanoindentation was developed. Using this method, the force and displacement data are obtained indirectly from the interferometer images. A final design was selected, analyzed, and constructed. Lastly, a series of indentation events were performed, validating both the mechanism and the approach to nanoindentation

Deterministic Abelian Sandpile Models and Patterns

In this thesis we want to study the ASM in connection with its capability to produce interesting patterns. it is a surprising example of model that shows the emergence of patterns but maintains the property of being analytically tractable. Then it is qualitatively different from other typical growth models --like Eden model, the diffusion limit aggregation, or the surface deposition -- indeed while in these models the growth of the patterns is confined on the surfaces and the inner structures, once formed, are frozen and do not evolve anymore, in the ASM the patterns formed grow in size but at the same time the internal structures aquire structure, as it has been noted in several papers. There have been several earlier studies of the spatial patterns in sandpile models. The first of them was by Liu et.al. The asymptotic shape of the boundaries of the patterns produced in centrally seeded sandpile model on different periodic backgrounds was discussed in a work of Dhar of 1999. Borgne et.al. obtained bounds on the rate of growth of these boundaries, and later these bounds were improved by Fey et.al. and Levine et.al. An analysis of different periodic structures found in the patterns were first carried out by Ostojic who also first noted the exact quadratic nature of the toppling function within a patch. Wilson et.al. have developed a very efficient algorithm to generate patterns for a large numbers of particles added, which allows them to generate pictures of patterns with N up to 2^26. There are other models, which are related to the Abelian Sandpile Model,e.g., the Internal Diffusion-Limited Aggregation (IDLA), Eulerian walkers (also called the rotor-router model), and the infinitely-divisible sandpile, which also show similar structure. For the IDLA, Gravner and Quastel showed that the asymptotic shape of the growth pattern is related to the classical Stefan problem in hydrodynamics, and determined the exact radius of the pattern with a single point source. Levine and Peres have studied patterns with multiple sources in these models, and proved the existence of a limit shape. Limiting shapes for the non-Abelian sandpile has recently been studied by Fey et.al. The results of our investigation toward a comprehension of the patterns emerging in the ASM are reported along the thesis. In chapter 3 we will introduce some new algebraic operators, $a^\dagger_i$ and $\Pi_i$ in addition to $a_i$, over the space of the sandpile configurations, that will be in the following basic ingredients in the creation of patterns in the sandpile. We derive some Temperley-Lieb like relations they satisfy. At the end of the chapter we show how do they are closely related to multitopplings and which consequences has that relation on the action of $\Pi_i$ on recurrent configurations. In chapter 4 we search for a closed formula to characterize the Identity configuration of the ASM. At this scope we study the ASM on the square lattice, in different geometries, and in a variant with directed edges, the F-lattice or pseudo-Manhattan lattice. Cylinders, through their extra symmetry, allow an easy characterization of the identity which is a homogeneous function. In the directed version, the pseudo-Manhattan lattice, we see a remarkable exact self-similar structure at different sizes, which results in the possibility to give a closed formula for the identity, this work has been published. In chapter 5 we reach the cardinal point of our study, here we present the theory of strings and patches. The regions of a configuration periodic in space, called patches, are the ingredients of pattern formation. In a last paper of Dhar, a condition on the shape of patch interfaces has been established, and proven at a coarse-grained level. We discuss how this result is strengthened by avoiding the coarsening, and describe the emerging fine-level structures, including linear interfaces and rigid domain walls with a residual one-dimensional translational invariance. These structures, that we shall call strings, are macroscopically extended in their periodic direction, while showing thickness in a full range of scales between the microscopic lattice spacing and the macroscopic volume size. We first explore the relations among these objects and then we present full classification of them, which leads to the construction and explanation of a Sierpinski triangular structure, which displays patterns of all the possible patches