This thesis concerns representation growth of finitely generated torsion-free nilpotent groups. This involves counting equivalence classes of irreducible representations and embedding this counting into a zeta function. We call this the representation zeta function. We use a new, constructive method to calculate the representation zeta functions of two families of groups, namely the Heisenberg group over rings of quadratic integers and the maximal class groups. The advantage of this method is that it is able to be used to calculate the p-local representation zeta function for all primes p. The other commonly used method, known as the Kirillov orbit method, is unable to be applied to these exceptional cases. Specifically, we calculate some exceptional p-local representation zeta functions of the maximal class groups for some well behaved exceptional primes. Also, we describe the Kirillov orbit method and use it to calculate various examples of p-local representation zeta functions for almost all primes p.

Currently the degree sequence problem for 3-hypergraphs is still unsolved efficiently. This paper researches the 3-hypergraphic problem in terms of edge switching and exchanges in the sequence to implement Dewdney’s reduction. It proposes the idea of an irreducible decomposition and makes use of it to find some sufficient conditions for a 3-hypergraphic sequence. In addition, this paper explores a related problem: intersection preserving mappings. / May 2016

One of the most important tools to study the geometry of irreducible holomorphic symplectic manifolds is the monodromy group. The first part of this dissertation concerns the construction and studyof monodromy operators on irreducible holomorphic symplectic manifolds which are deformation equivalent to the 10-dimensional example constructed by O'Grady. The second part uses the knowledge of the monodromy group to compute the number of connected components of moduli spaces of bothmarked and polarised irreducible holomorphic symplectic manifolds which are deformationequivalent to generalised Kummer varieties.

As a hygroscopic salt, granular potash can easily absorb large quantities of water vapor from humid air during storage and transportation processes. Subsequent drying will result in potash particles sticking together to form clumps or cakes. In order to avoid or decrease caking, it is essential to know the local history of moisture content and moisture movement in a bed of potash. In this thesis, experimental measurements and numerical simulations are used to investigate moisture transport and redistribution by capillarity, gravity and diffusion effects within a potash bed. <p> The important properties required to model moisture transfer in granular porous potash (i.e. porosity, permeability, specific surface area and irreducible saturation) are investigated experimentally and theoretically. It is shown that for a mixture with a wide range of particle sizes the potash bed properties can be predicted knowing the properties for each narrow range of particle size in the mixture. <p> An experimental test facility was designed and constructed to test moisture transfer within a potash bed. The test procedures are presented along with an uncertainty analysis. The moisture content spatial distribution for different particle sizes under different initial conditions is investigated and data are presented. <p>A one-dimensional transient numerical model of moisture transport accounting for diffusion, capillarity and gravity effects within potash beds is developed. Two different moisture transport mechanisms are presented. In a wet region, where local moisture saturation level, S, is larger than an irreducible saturation, S0, liquid water exists as continuous liquid film on the particles; moisture is transferred by liquid film movement due to capillarity and gravity effects. In a dry region where S is less than S0, water vapor diffusion is the only mechanism of moisture transfer and water is adsorbed in layers on the surfaces. <p> From the experimental data and numerical simulation analysis, it is shown that the irreducible saturation, S0, is a strong function of particle size. It will decrease with a particle size increase. <p> The numerical model is validated by comparison with some typical experimental case studies. Agreement between the experimental data and simulation results is well within the experimental 95% uncertainty bounds. It is concluded from this research that the complex moisture transport process by diffusion, capillarity and gravity effects within a potash bed can be modeled and simulated. Experimental and simulation results indicate that direct water drainage will more readily occur for large particle sizes than for small particles for the same initial moisture content.

As a hygroscopic salt, granular potash can easily absorb large quantities of water vapor from humid air during storage and transportation processes. Subsequent drying will result in potash particles sticking together to form clumps or cakes. In order to avoid or decrease caking, it is essential to know the local history of moisture content and moisture movement in a bed of potash. In this thesis, experimental measurements and numerical simulations are used to investigate moisture transport and redistribution by capillarity, gravity and diffusion effects within a potash bed. <p> The important properties required to model moisture transfer in granular porous potash (i.e. porosity, permeability, specific surface area and irreducible saturation) are investigated experimentally and theoretically. It is shown that for a mixture with a wide range of particle sizes the potash bed properties can be predicted knowing the properties for each narrow range of particle size in the mixture. <p> An experimental test facility was designed and constructed to test moisture transfer within a potash bed. The test procedures are presented along with an uncertainty analysis. The moisture content spatial distribution for different particle sizes under different initial conditions is investigated and data are presented. <p>A one-dimensional transient numerical model of moisture transport accounting for diffusion, capillarity and gravity effects within potash beds is developed. Two different moisture transport mechanisms are presented. In a wet region, where local moisture saturation level, S, is larger than an irreducible saturation, S0, liquid water exists as continuous liquid film on the particles; moisture is transferred by liquid film movement due to capillarity and gravity effects. In a dry region where S is less than S0, water vapor diffusion is the only mechanism of moisture transfer and water is adsorbed in layers on the surfaces. <p> From the experimental data and numerical simulation analysis, it is shown that the irreducible saturation, S0, is a strong function of particle size. It will decrease with a particle size increase. <p> The numerical model is validated by comparison with some typical experimental case studies. Agreement between the experimental data and simulation results is well within the experimental 95% uncertainty bounds. It is concluded from this research that the complex moisture transport process by diffusion, capillarity and gravity effects within a potash bed can be modeled and simulated. Experimental and simulation results indicate that direct water drainage will more readily occur for large particle sizes than for small particles for the same initial moisture content.

This thesis is an examination of the structure of irreducible continua, with a particular emphasis on local connectedness and monotone maps. A continuum is irreducible if there exist a pair of points such that no proper subcontinuum contains both, with the arc being the most basic example. Being irreducible has a number of interesting implications for a continuum, both locally and globally, and it is these consequences we shall focus on. As mentioned above, the arc is the most straightforward example of an irreducible continuum. Indeed, an intuitive understanding of an irreducible continuum would be that it is structured like an arc, with the points of irreducibility at either end joined by a subspace with no loops or offshoots. In Chapter 2 we will see that for a certain class of continua this intuition is well founded by constructing a monotone map from an irreducible continuum onto an arc. This monotone map will preserve much of the structure of our continuum and as such will provide an insight into that structure. We will next examine a generalisation of irreducibility which considers finite sets of points rather than just pairs. A number of classical results will be re-examined in this light in Chapter 3. While the majority of these theorems will be shown to have close parallels in higher finite and infinite irreducibility there will be several which do not hold without further conditions on the continuum. Such anomalies will be particularly prevalent in continua which have indecomposable subcontinua dominating their structure. In Chapter 4 monotone maps will be constructed for finitely irreducible continua similar to the map to an arc mentioned previously. Chapters 7 and 8 will generalise irreducibility further to the infinite case and we will again construct monotone maps preserving the structure of our continuum. Along with the arc, another highly significant irreducible continuum is the sin 1 x continuum. Chapter 5 will focus on this continuum, which will be the basis for a nested sequence of continua. A number of results concerning continuous images of these continua will be presented before using the sequence of continua to define an indecomposable continuum. This continuum will be investigated, and it will be shown that the union of our nested continua form a composant of the indecomposable continuum. In Chapter 6 we will turn to the question of compactifications. If a space X is connected then any metric compactification of X will be a continuum. This chapter will answer the question of when a compactification is an irreducible continuum, with the remainder of the compactification consisting of all of the irreducible points. A list of properties will given such that a continuum has such a compactification if and only if it has each property on the list. It will also be demonstrated that each of these properties is independent of the others. Finally, in Chapter 9 we will revisit the idea of structure-preserving monotone maps, but this time in continua which are not irreducible. Motivated by the fibres of the maps in previous chapters, we will introduce two categories of subcontinua of a continuum X. The first will be nowhere dense subcontinua which are maximal with this property and the second will be subcontinua about which X is locally connected and which are minimal with this property. Continua in which every point lies in a maximal nowhere dense subcontinuum will be examined, as well as spaces in which every point lies in a unique minimal subcontinuum about which X is locally connected. We will also look at the properties of monotone maps arising from partitions of X into such subcontinua, and will prove that if every point of X lies in a maximal nowhere dense subcontinuum then the resulting quotient space will be one dimensional.

222 pages / Efforts have been made in this study to trace the developments of the Irreducible School Fund and the factors responsible for depriving the schools of the legacy bequeathed by far-seeing statesmen during the formative period of American development. An attempt has been made to analyse the errors of the past and to estimate the resultant losses to the public schools of Oregon. It is now apparent that what was once intended as substantial school aid actually has been a paltry "drop-in-the-bucket" relative to needs.

This thesis deals with k-irreducible triangulations of closed, compact 2-manifolds without boundary. A triangulation is k-irreducible, if all its closed cycles of length less than k are nullhomotopic and no edge can be contracted without losing this property. k-irreducibility is a generalization of the well-known concept of irreducibility, and can be regarded as a measure of how closely the triangulation approximates a smooth version of the underlying surface. Research follows three main questions: What are lower and upper bounds for the minimum and maximum size of a k-irreducible triangulation? What are the smallest and biggest explicitly constructible examples? Can one achieve complete classifications for specific 2-manifolds, and fixed k?

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The main results of this dissertation show evidence supporting the Successive Surface Scaffolding Conjecture. This is a new conjecture that, if true, guarantees the existence of all the wye-delta-order minimal obstructions of a surface S as subgraphs of the irreducible triangulations of the surface S with a crosscap added. A new data structure, i.e. an augmented rotation system, is presented and used to create an exponential-time algorithm for embedding graphs in any surface with a constant-time check of the change in genus when inserting an edge. A depiction is a new formal definition for representing an embedding graphically, and it is shown that more than one depiction can be given for nonplanar embeddings, and that sometimes two depictions for the same embedding can be drastically different from each other. An algorithm for finding the essential cycles of an embedding is given, and is used to confirm for the projective-plane obstructions, a theorem that shows any embedding of an obstruction must have every edge in an essential cycle. Obstructions of a general surface S that are minor-minimal and not double-wye-delta-minimal are shown to each have an embedding on the surface S with a crosscap added. Finally, open questions for further research are presented. / Graduate