This dissertation has two principal components: the dimension of posets with planar cover graphs, and the cartesian product of posets whose cover graphs have hamiltonian cycles that parse into symmetric chains. Posets of height two can have arbitrarily large dimension. In 1981, Kelly provided an infinite sequence of planar posets that shows that the dimension of planar posets can also be arbitrarily large. However, the height of the posets in this sequence increases with the dimension. In 2009, Felsner, Li, and Trotter conjectured that for each integer h at least 2, there exists a least positive integer c(h) so that if P is a poset with a planar cover graph (the class of posets with planar cover graphs includes the class of planar posets) and the height of P is h, then the dimension of P is at most c(h). In the first principal component of this dissertation we prove this conjecture. We also give the best known lower bound for c(h), noting that this lower bound is far from the upper bound. In the second principal component, we consider posets with the Hamiltonian Cycle--Symmetric Chain Partition (HC-SCP) property. A poset of width w has this property if its cover graph has a hamiltonian cycle which parses into w symmetric chains. This definition is motivated by a proof of Sperner's theorem that uses symmetric chains, and was intended as a possible method of attack on the Middle Two Levels Conjecture. We show that the subset lattices have the HC-SCP property by showing that the class of posets with the strong HC-SCP property, a slight strengthening of the HC-SCP property, is closed under cartesian product with a two-element chain. Furthermore, we show that the cartesian product of any two posets from this strong class has the (weak) HC-SCP property.

In this paper we develop new preconditioned iterative methods for solving monotone nonlinear eigenvalue problems. We investigate the convergence and derive grid-independent error estimates for these methods. Numerical experiments demonstrate the practical effectiveness of the proposed methods for a model problem.

The biosynthesis of wax components containing secondary functional groups was investigated in the current study. Two fundamentally different pathways were proposed to introduce the secondary functional groups. One pathway involves hydroxylation of elongated substrates. Wax components characterized by two functional groups located on or near the centre of the carbon chain, nonacosane-14,15-diol, -14,16-diol and -13,15-diol as well as corresponding ketols were identified for the first time in Arabidopsis stem wax. The alkanediols and ketols were dominated by the C-14,15 isomers. The absence of alkanediols and ketols in Arabidopsis mah1 mutants that are deficient in secondary alcohol biosynthesis confirmed the biosynthetic relationship between secondary alcohols and alkanediols/ketols (Chapter 3). In pea (Pisum sativum) leaf wax, two novel compound classes were identified as primary/secondary alcohols dominated by octacosane-1,14-diol and secondary/secondary alkanediols hentriacontane-9,16-diol, -8,15-diol and -10,17-diol. Co-localization of the secondary/secondary alkanediols and hentriacontan-15-ol and -16-ol pointed to a biosynthetic relationship (Chapter 4). The diverse structures of compounds identified in the current study suggested that hydroxylases can use substrates other than alkanes. The predominance of isomers within homologues indicated a regiospecificity of the hydroxylases involved in wax biosynthesis. In addition to hydroxylation, secondary functional groups could also be introduced through elongation of carbon chains. Homologous series of 5-hydroxyaldehydes (C₂₄ and C₂₆-C₃₆) and 1,5-alkanediols (C₂₈-C₃₈) were identified in yew (Taxus baccata) needle wax. The relative position of both functional groups suggested that these two compound classes are biosynthetically related and their secondary functional groups are introduced during elongation (Chapter 5). The results of incubation of ¹⁴C-labeled malonyl-CoA and acyl-CoAs with different chain lengths in the presence of California poppy (Eschscholzia californica) microsomes provided the first evidence to support the elongation hypothesis. The results indicated that a carbonyl group rather than a hydroxyl group is introduced during elongation. To provide molecular tools for further investigations of the hypothetical pathway, three full length cDNAs encoding putative KCSs were cloned and one of them, PKCSI, was functionally characterized.

Stainless steel has garnered attention as an alternative structural material to conventional carbon steel due to its corrosion resistance properties and aesthetic appearance. Of interest are single angles, which are frequently used in trusses, transmission towers, and as bracing diaphragms. When subjected to compression, knowledge concerning the behavior, analysis, and design of stainless steel single angles is very limited. This thesis addresses the behavior of duplex stainless steel single equal-leg angles subject to concentric compressive loading. Two complementary approaches are used in this study, the first of which was experimental and consisted of conducting 33 full-scale buckling tests on S32003 duplex stainless steel single equal-leg angle components. Angles specimens had slenderness ratios ranging from 35 to 350 and leg width-to-thickness ratios of 7.5 to 12.3. In the second approach, computational models that accounted for material nonlinearity, material anisotropy, and geometric out-of-straightness were developed and validated using the experimentally obtained test results. These models were subsequently used to perform numerical buckling experiments to shed light on the behavior of axially loaded compression duplex stainless steel single angles for a wide range of practical leg width-to-thickness ratios. Results from the full-scale tests and from the numerical models are shown to correlate well with the classical mechanics-based formulae, which considers nonlinear stress-strain relationships, for predicting flexural and flexural-torsional buckling strengths of singly symmetric stainless steel members. Finally, design criteria in the form of load and resistance factor design (LRFD) with a reliability index of 3 for buckling limit states are proposed for possible adoption in future US national standards.

Darbo tikslas: • Išanalizuoti šviesos spindulio diferencialinių lygčių sferiškai simetriškus sprendinius. Darbo uždaviniai: • Užrašyti ir išspęsti konkrečioje koordinačių sistemoje šviesos spindulio sklidimo diferencialines lygtis. • Iliustruoti sprendinius su Mathcad programa. / Work is to analyze the differential equations spherically symmetric solutions. Recorded and resolved in a particular coordinate system the light specific propagation in differential equations.Using the original beam equations found in specific cases spread radius of the sphere in which the refractive index depends only on the distance on the center. Derived formulas we framed when the beam trajectory equations given different refractive index dependence on the distance to the center of spherical symmetry.

In this thesis, we seek exact solutions for the interior of a radiating relativistic star undergoing gravitational collapse. The spherically symmetric interior spacetime, when matched with the exterior radiating Vaidya spacetime, at the boundary of the star, yields the governing equation describing the gravitational behaviour of the collapsing star. The investigation of the model hinges on the solution of the governing equation at the boundary. We first examine shear-free models which are conformally flat. The boundary condition is transformed to an Abel equation and several new solutions are generated. We then study collapse with shear in geodesic motion. Two classes of solutions are generated which are regular at the stellar centre. Our treatment extends the results of Naidu et al (2006) which had the undesirable feature of a singularity at the centre of the star. In an attempt to find more general models, we transform the fundamental equation to a Riccati equation. Two general classes of solution are found and are used to study the thermal evolution in the causal theory of thermodynamics. These solutions are shown to reduce to the Friedmann dust solution in the absence of heat flow. Furthermore, we obtain new categories of solutions for the case of gravitational collapse with expansion, shear and acceleration of the stellar fluid. This is achieved by transforming the boundary condition into a Riccati equation. In special cases the Bernoulli equation is regained. The solutions are given in terms of elementary functions and they permit the investigation of the physical features of radiative stellar collapse. / Thesis (Ph.D.)-University of KwaZulu-Natal, Durban, 2007.

In this thesis we study spherically symmetric spacetimes related to the Einstein field equations. We consider only neutral matter and apply the Einstein field equations with isotropic pressures. Our object is to model relativistic stellar systems. We express the Einstein field equations and the condition of pressure isotropy in terms of Schwarzschild coordinates and isotropic coordinates. For Schwarzschild coordinates we consider the transformations due to Buchdahl (1959), Durgapal and Bannerji (1983), Fodor (2000) and Tewari and Pant (2010). The condition of pressure isotropy is integrated and new exact solutions of the field equations are obtained utilizing the transformations of Buchdahl (1959) and Tewari and Pant (2010). These exact solutions are given in terms of elementary functions. For isotropic coordinates we can express the condition of pressure isotropy as a Riccati equation or a linear equation. An algorithm is developed that produces a new solution if a particular solution is known. The transformations reduce to a nonlinear Bernoulli equation in most instances. There are fundamentally three new classes of solutions to the condition of pressure isotropy. / Thesis (M.Sc.)-University of KwaZulu-Natal, Westville, 2011.

In this dissertation we study the excedance permutation statistic. We start by extending the classical excedance statistic of the symmetric group to the affine symmetric group eSn and determine the generating function of its distribution. The proof involves enumerating lattice points in a skew version of the root polytope of type A. Next we study the excedance set statistic on the symmetric group by defining a related algebra which we call the excedance algebra. A combinatorial interpretation of expansions from this algebra is provided. The second half of this dissertation deals with the topology of the Frobenius complex, that is the order complex of a poset whose definition was motivated by the classical Frobenius problem. We determine the homotopy type of the Frobenius complex in certain cases using discrete Morse theory. We end with an enumeration of Q-factorial posets. Open questions and directions for future research are located at the end of each chapter.

Inorganic based membranes which have a symmetric/asymmetric structure have been produced using an immersion induced phase inversion and sintering method. An organic binder solution (dope) containing yttria-stabilised zirconium (YSZ) particles is spun through a triple orifice spinneret to form a hollow fibre precursor, which is then sintered at elevated temperatures to form a ceramic support. The phase inversion process for the formation of hollow fibre membranes was studied in order to produce the best morphological structure/support for palladium based membranes. The spinning parameters, particle size, non-solvent concentration, internal coagulant as well as the calcination temperature were investigated in order to determine the optimum values. Sintering temperature was also investigated, which would yield a sponge-like structure with an optimized permeability, while retaining a smooth outer surface. The supports produced by phase inversion were characterized in terms of dimension by mercury porosimetry, compressed air permeability, Surface Electron Microscopy (SEM) and Atomic Force Microscopy (AFM). The morphology of the produced ceramic support showed either dense or porous characteristics governed by the dynamics of the phase inversion process. The particle size of YSZ was examined in order to decrease the amount of agglomerates in the spinning suspension. Zetasizer tests indicated that at 15 minutes, the ultrasonic bath effectively homogenised the YSZ particles and prohibited soft agglomerates from reforming in the spinning suspension. In this study, an increase in air gap had no noticeable effect on the finger like voids but it had a considerable effect on both the inner diameter (ID) and outer diameter (OD) of the green fibres, while an increase in bore liquid flow rate and extrusion pressure promoted viscous fingering and significant effect on the ID and OD of the fibres, respectively. There was a decrease in porosity and permeability with increasing sintering temperature, addition of water concentration in the spinning suspension and varying Nmethylpyrrolidone (NMP) aqueous solution of the internal coagulant. The amount of YSZ added to the starting suspension influenced the properties of the support structure. Viscous deformation was observed for dope with lower particle loading thus resulted in the formation of cracks and defects during sintering. / >Magister Scientiae - MSc

Most of the cryptosystems currently used are based on number theoretic problems. We focus on cryptosystems based on finite automata (FA) which are lightweight in nature and have relatively small key sizes. The security of these systems relies on the difficulties in inverting non-linear finite automata and factoring matrix polynomials. In symmetric or single key encryption, the secret key consists of two finite automata and their inverses. By applying the inverses of the automata to the cipher text, the plain text can be effectively calculated. In case of asymmetric or public key encryption, the public key consists of another automaton, which is the combination of the two finite automata while the private key consists of the inverse of the two individual automata. It is hard to invert the combined automaton without the knowledge of the private key automata. We propose a third variant which is based on a 128-bit key and uses a DES-based key generation algorithm. We implement and test all three cryptosystems - the standard single key and public key cryptosystems as well as our novel DES-based FA cryptosystem. We also extensively test the finite automata cryptosystems on a standard desktop machine as well as the Nokia N900 smartphone. All statistical tests carried out on the ciphertext are satisfactory. / Graduate