The Discrete Nodal Domain Theorem states that an eigenfunction of the k-th largest eigenvalue of a generalized graph Laplacian has at most k (weak) nodal domains. We show that the number of strong nodal domains cannot exceed the size of a maximal induced bipartite subgraph and that this bound is sharp for generalized graph Laplacians. Similarly, the number of weak nodal domains is bounded by the size of a maximal bipartite minor.

In this theses, an iterativ linear solver for the non-linear p-Poisson problem is introduced. After the theoretical convergence results some numerical examples of a fully adaptive solver are presented.

In this work we deal with conversion of 3D triagonal polygonal meshes to the 3D spline patches using spectral analysis. The converted mesh is divided into quadrilaterals using eigenvectors of Laplacian operator. These quadrilaterals will be converted into spline patches. We will present some interesting results of this method. The assets and imperfections of this method will be briefly discussed.

In this dissertation we study positive solutions to a singular p-Laplacian elliptic boundary value problem on a bounded domain with smooth boundary when a positive parameter varies. Our main focus is the analysis of a challenging class of singular p-Laplacian problems. We establish the existence of a positive solution for all positive values of the parameter and the existence of at least two positive solutions for a certain explicit range of the parameter. In the Laplacian case, we also prove the uniqueness of the positive solution for large values of the parameter. We extend our existence and multiplicity results to classes of singular systems and to the case when a domain is an exterior domain. We prove our existence and multiplicity results by the method of sub and supersolutions and our uniqueness result by establishing apriori and boundary estimates. Such results are well known in the literature for the nonsingular case. In this study, we extend these results to the more difficult singular case.

A graph G is a (t, r)-regular graph if every collection of t independent vertices is collectively adjacent to exactly r vertices. If a graph G is (2, r)-regular where p, s, and m are positive integers, and m ≥ 2, then when n is sufficiently large, then G is isomorphic to G = Ks+mKp, where 2(p-1)+s = r. A nested (2,r)-regular graph is constructed by replacing selected cliques with a (2,r)-regular graph and joining the vertices of the peripheral cliques. For example, in a nested 's' graph when n = s + mp, we obtain n = s1+m1p1+mp. The nested 's' graph is now of the form Gs = Ks1+m1Kp1+mKp. We examine the network properties such as the average path length, clustering coefficient, and the spectrum of these nested graphs.

This thesis extends an existing cell population modelling framework to investigate two different hypotheses for what drives the phenomenon of cell sorting, which is the spontaneous self-reorganization of cell populations. This behaviour cause cells to find their way back into their original configuration after they have been scrambled. Original tissue function may also be regained. The modelling framework is called discrete Laplacian cell mechanics (DLCM), and models cell movement on a lattice as a result of pressure differences. The first hypothesis suggests that cells exhibit type-specific adhesion properties which cause cells of the same type to adhere more to each other than to cells of a different kind. The other, more recent, hypothesis explains cell sorting behaviour as a consequence of interfacial tension, where cells of different types exhibit larger tension between them compared to cells of the same type. Adhesion is implemented as a passive force between cells of the same type, which counteract the pressure-driven events, while interfacial tension is implemented as pressure sources arising due to contact with cells of a different type. This thesis investigates whether these additions on the scale of individual cells can be sufficient to induce sorting behaviour on the cell population scale. Subsequently the suitability of implementing these effects in the DLCM framework can be evaluated. Starting from a scrambled cell configuration of two types, the results show that differential adhesion can result in the cell population sorting into smaller clusters, with the addition of Brownian motion improving the sorting ability significantly. Differential interfacial tension as it is implemented here demonstrates the effect of dissociation between cells of different type, but this is not sufficient to achieve sorting. The behaviour can be likened to a form of localized Brownian motion where more unsorted areas are prone to more movement events. Therefore, differential tension is not deemed suitable within the DLCM framework on its own. The cohesive effect of differential adhesion together with the dissociative effect of differential interfacial tension proved to work well together, acheiving a high degree of sorting both overall and compared to the case of only differential adhesion with some Brownian motion. Full separation into one distinct cell mass for each cell type present could not be achieved.

In this dissertation, we study the existence and nonexistence of positive solutions to some classes of singular p-Laplacian boundary value problems with a parameter. In the first study, we discuss positive solutions for a class of sublinear Dirichlet p- Laplacian equations and systems with sign-changing coefficients on a bounded domain of Rn via Schauder Fixed Point Theorem and the method of sub- and supersolutions. Under certain conditions, we show the existence of positive solutions when the parameter is large and nonexistence when the parameter is small. In the second study, we discuss positive radial solutions for a class of superlinear p- Laplacian problems with nonlinear boundary conditions on an exterior domain via degree theory and fixed point approach. Under certain conditions, we show the existence of positive solutions when the paprameter is small and nonexistence when the paramter is large. Our results provide extensions of corresponding ones in the literature from the Laplacian to the p-Laplacian, and can be applied to the challenging infinite semipositone case

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"We cannot afford the luxury any longer of ignoring the nature of the bonding in these interesting compounds .... " P.E.D. Morgan, (1974). An understanding of bonding is paramount to furthering our understanding of materials (Morgan, 1974). The properties of materials are governed by the interactions between atoms. These interactions are governed by the nature of the bonds. In this study, two methods are explored which provide insight into chemical interactions. First, promolecule radii, calculated for nitride, oxide, and sulfide coordinated polyhedra with bond lengths fixed at the sums of effective ionic and crystal radii, are analyzed. Radii calculated for transition and non-transition cations for the first four rows of the periodic table are highly correlated with crystal radii derived for oxide and sulfide crystals and with ionic radii derived for nitride crystals. Promolecule radii calculated for the coordination polyhedra match experimentally determined bonded radii to within ~0.02Å, on average. Calculated radii anions tend to match ionic radii when bonded to highly electropositive cations and atomic radii when bonded to highly electronegative cations. In the second study, molecular orbital calculations were completed on a series of small molecules containing the nitride anion. Bond type can be characterized by studying the systematics of parameters derived from the bond critical point properties of the electron density distributions. A set of criteria is established to suggest how covalent or ionic a bond is. This criteria is based on bond critical point properties such as the Laplacian of the electron density distribution evaluated at the bond critical point, the electron density distribution at the critical point, the local energy density at the critical point, the relative electronegativity of the cation, the curvatures of the electron density distribution, and the distance from the nucleus of the nitride anion to the bond critical point, (the bonded radius of the nitrogen atom). Parameters computed for promolecule data indicate that these easily obtained results offer a method of calculating bond critical properties which are close in value to the more extensive results derived from molecular orbital calculations. / Ph. D.