Thesis (Ph.D.) - University of Queensland, 2005. / Includes bibliography.

Thesis (Ph. D.)--University of Washington, 2000. / Vita. Includes bibliographical references (p. 120-124).

A graph G is maximal nontraceable (MNT) (maximal nonhamiltonian (MNH)) if G is not traceable (hamiltonian), i.e. does not contain a hamiltonian path (cycle), but G+xy is traceable (hamiltonian) for all nonadjacent vertices x and y in G. A graph G is hypohamiltonian if G is not hamiltonian, but every vertex deleted subgraph G -u of G is hamiltonian. A graph which is maximal nonhamiltonian and hypohamiltonian is called maximal hypohamiltonian (MHH). Until recently, not much has appeared in the literature about MNT graphs, although there is an extensive literature on MNH graphs. In 1998 Zelinka constructed two classes of MNT graphs and made the conjecture, which he later retracted, that every MNT graph belongs to one of these classes. We show that there are many different types of MNT graphs that cannot be constructed by Zelinka's methods. Although we have not been able to characterize MNT graphs in general, our attempt at characterizing MNT graphs with a specified number of blocks and cut-vertices enabled us to construct infinite families of non-Zelinka MNT graphs which have either two or three blocks. We consider MNT graphs with toughness less than one, obtaining results leading to interesting constructions of MNT graphs, some based on MHH graphs. One result led us to discover a non-Zelinka MNT graph of smallest order, namely of order 8. We also present examples of MNTgraphs with toughness at least one, including an infinite family of 2-connected, claw-free graphs. We find a lower bound for the size of 2-connected MNT graphs of order n. We construct an infinite family of 2-connected cubic MNT graphs of order n, using MHH graphs as building blocks. We thus find the minimum size of 2-connected MNT graphs for infinitely many values of n. We also present a construction, based on MHH graphs, of an infinite family of MNT graphs that are almost cubic. We establish the minimum size of MNT graphs of order n, for all except 26 values of n, and we present a table of MNT graphs of possible smallest size for the excluded 26 values of n. / Mathematical Sciences / PHD (MATHEMATICS)

No description available.

Perovskite ferroelectrics and antiferroelectrics have attracted a lot of attention owing to their potential for device applications including THz sensors, solid state cooling, ultra high density computer memory, and electromechanical actuators to name a few. The discovery of ferroelectricity at the nanoscale provides not only new and exciting possibilities for device miniaturization, but also a way to study the fundamental physics of nanoscale phenomena in these materials. Ferroelectric nanowires show a rich variety of physical characteristics which are advantageous to the design of nanoscale ferroelectric devices such as exotic dipole patterns, a strong dependence of the polarization and phonon frequencies on the electrical and mechanical boundary conditions, as well as a dependence of the transition temperatures on the diameter of the nanowire. Antiferroelectricity also exists at the nanoscale and, due to the proximity in energy of the ferroelectric and antiferroelectric phases, a phase transition from the ferroelectric to the antiferroelectric phase can be facilitated through the application of the appropriate mechanical and electrical boundary conditions. While much progress has been made over the past several decades to understand the nature of ferroelectricity/antiferroelectricity in nanowires, many questions remain unanswered. In particular, little is known about how the truncated dimensions affect the soft mode frequency dynamics or how various electrical and mechanical boundary conditions might change the nature of the phase transitions in these ferroelectric nanowires. Could nanowires offer a distinct advantage for solid state cooling applications? Few studies have been done to elucidate the fundamental physics of antiferroelectric nanowires. How the polarization in ferroelectric nanowires responds to a THz electric field remains relatively underexplored as well. In this work, the aim is to to develop and use computational tools that allow first-principles-based modeling of electric-field-induced phenomena in ferroelectric/antiferroelectric nanowires in order to address the aforementioned questions. The effective Hamiltonian approach is a well validated model which reliably reproduces many static and dynamic properties of perovskite ferroelectric and antiferroelectrics. We begin by developing an effective Hamiltonian for the prototypical ferroelectric potassium niobate, a lead-free material which undergoes multiple structural phase transitions. Density functional theory calculations within the LDA and GGA are used to determine the effective Hamiltonian parameters for KNbO3 . By simulating an annealing within an NPT ensemble, we find that the KNbO3 parameters found from first principles underestimate the experimental transition temperatures. We apply a universal scaling technique to all of the first-principles derived parameters and are thus able to more accurately reproduce the transition temperatures predicted by experiment as well as a number of other static and dynamic properties of potassium niobate. Having determined the parameters of the effective Hamiltonian for KNbO3 , we use this as well as previously determined effective Hamiltonian parameters for PbTiO3 and BaTiO3 to study the electrocaloric effect in nanowires made of these materials. We determined that, in general, the electrocaloric effect in ferroelectric nanowires is diminished due to the reduced correlation length resulting from the finite lateral dimensions. However, certain temperature ranges were identified near ambient temperature where the electrocaloric response is enhanced with respect to bulk. The effective Hamiltonian model was also employed to study the response of the spontaneous polarization and temperature to tailored electric fields. We identified a novel means of reversing the polarization in ferroelectric nanowires which could potentially be used in the design of nanoscale THz sensors of ultra high density ferroelectric memory devices. While the soft mode frequency dynamics of bulk ferroelectrics under various mechanical boundary conditions have been studied extensively, the effects of different mechanical boundary conditions on the soft mode dynamics in ferroelectric nanowires remains relatively under-explored. We conduct a comprehensive study on PbTiO3 nanowires which explores the effects of hydrostatic pressure, applied uniaxial stress, and biaxial strain on the structural properties, transition temperatures, and soft mode dynamics. We found that depending on the particular type of mechanical boundary condition, the nanowire can exhibit either monodomain or polydomain vortex phases, drastically different from what is found for PbTiO3 bulk and originates from the critical role of the depolarizing field. We found a rich variety of dipole patterns, particularly for the polydomain states with the dipoles arranged in single and double polarization vortices depending on the type and strength of the mechanical boundary conditions. The soft mode frequency dynamics are also strongly affected by the mechanical boundary conditions. In particular we find that the frequency of the E mode in the P4mm phase is significantly larger than the A 1 mode which is in contrast with bulk PbTiO3 . This striking finding is attributed to the presence of the depolarizing field along the truncated directions which leads to mode hardening. In the last chapter, we identify the emergence of a ferroelectric state in antiferroelectric PbZrO3 nanowires and describe possible ways to stabilize the ferroelectric phase. Finally, we explore how our findings could potentially be used to improve existing technologies such as energy storage devices and electromechanical actuators as well as future technologies like solid state cooling devices.

No description available.

For distinct vertices u and ν of a nontrivial connected graph G, the detour distance D(u, ν) between u and ν is the length of a longest u-ν path in C. For a vertex ν in G, define D+(ν) = max{D(u, ν) : u ∈ V(G) - {ν}}. A vertex u is called a hamiltonian neighbor of ν if D(u, ν) -D+(ν). A vertex v is said to hamiltonian dominate a vertex u if u = ν or u is a hamiltonian neighbor of ν. A set 5 of vertices of G is called a hamiltonian dominating set if every vertex of G is hamiltonian dominated by some vertex in S. A hamiltonian dominating set of minimum cardinality is a minimum hamiltonian dominating set and this cardinality is the hamiltonian domination number γH(G) of G. It is shown that if T is a tree of order n ≥ 3 and p is the order of the periphery of T, then γH(T) = n - p. It is also shown that if G is a connected graph of order n ≥ 3, then γH(G) ≤ n - 2. Moreover, for every pair k, n of integers with 1 ≤ k ≤ n - 2, there exists a connected graph G of order n such that γH(G) = k. For a vertex ν in G, define D- (ν) -min{D(u, ν) : u ∈ V(G) -{ν}}. A vertex u is called a detour neighbor of ν if D(u, ν) = D- (ν). The detour domination number γD(G) of G is defined analogously to γH(G)- It is shown that every pair a, b of positive integers is realizable as the domination number and hamiltonian domination number, respectively, of some graph. For integers a, b ≥ 2, the corresponding result is shown for the detour domination number and hamiltonian domination number. The problem of determining those rational numbers r and s with 0 < r, s < 1 for which there exists a graph G of order n such that γD(G)/n = r and γH/(G)/n = s is discussed.

No description available.

We begin this dissertation by presenting a brief introduction to the theory of solitons and integrability (plus some classical methods applied in this field) in Chapter 1, mainly using the Korteweg-de Vries equation as a typical model. At the end of this Chapter a mathematical framework of notations and terminologies is established for the whole dissertation. In Chapter 2, we first introduce two specific matrix spectral problems (with 3 potentials) associated with matrix Lie algebras $\mbox{sl}(2;\mathbb{R})$ and $\mbox{so}(3;\mathbb{R})$, respectively; and then we engender two soliton hierarchies. The computation and analysis of their Hamiltonian structures based on the trace identity affirms that the obtained hierarchies are Liouville integrable. This chapter shows the entire process of how a soliton hierarchy is engendered by starting from a proper matrix spectral problem. In Chapter 3, at first we elucidate the Gauge equivalence among three types $u$-linear Hamiltonian operators, and construct then the corresponding B\"acklund transformations among them explicitly. Next we derive the if-and-only-if conditions under which the linear coupling of the discussed u-linear operators and matrix differential operators with constant coefficients is still Hamiltonian. Very amazingly, the derived conditions show that the resulting Hamiltonian operators is truncated only up to the 3rd differential order. Finally, a few relevant examples of integrable hierarchies are illustrated. In Chapter, 4 we first present a generalized modified Korteweg-de Vries hierarchy. Then for one of the equations in this hierarchy, we build the associated Riemann-Hilbert problems with some equivalent spectral problems. Next, computation of soliton solutions is performed by reducing the Riemann-Hilbert problems to those with identity jump matrix, i.e., those correspond to reflectionless inverse scattering problems. Finally a special reduction of the original matrix spectral problem will be briefly discussed.

We prove the existence and linear stability of quasi-periodic breather solutions in a 1d Hamiltonian lattice of identical, weakly-coupled, anharmonic oscillators with general on-site potentials and under the effect of long-ranged interaction, via de KAM technique. We prove the persistence of finite-dimensional tori which correspond in the uncoupled limit to N arbitrary lattice sites initially excited. The frequencies of the invariant tori of the perturbed system are only slightly deformed from the frequencies of the unperturbed tori.