School of Mathematics and Statistics  Theses
http://hdl.handle.net/11343/295
20180520T10:46:19Z

Chevalley groups and finite geometry
http://hdl.handle.net/11343/212228
Chevalley groups and finite geometry
Xu, Jon
This thesis explores the possible use of Schubert cells, Schubert varieties and flag varieties in finite geometry, particularly in regard to the question of these objects might be a source of understanding of ovoids or provide new examples. The main result provides a characterisation of those Schubert cells of finite Chevalley groups which satisfy the first property (thinness) of ovoids. Another result of the thesis is a realisation of key examples of ovoids as the points of a flag variety of a suitably chosen Chevalley group. The thesis is a result of an effort to create “interdisciplinary” communication and collaboration between the finite geometry community and the representation theory communities in Australia, and we hope that it will help to bridge the modern language barrier between these two fields.
© 2018 Dr. Jon Xu
20180101T00:00:00Z

Combinatorial enumeration and the Bethe Ansatz
http://hdl.handle.net/11343/210785
Combinatorial enumeration and the Bethe Ansatz
Foxcroft, John Spencer Hugh
This thesis focuses on combinatorial enumeration of lattice paths using the Constant Term method. In particular we make use of the Bethe Ansatz, an Ansatz given by a linear combination of solutions to the free ``no boundaries'' problem (these solutions are called ``bulk'' solutions).
We begin with a short review of these enumeration methods using the simple example of Dyck paths, and provide an example as to how the Constant Term method lends itself to combinatorial understanding with a new bijection to weighted Dyck paths.
We then move to more complicated lattice paths called Osculating Paths. Previous solutions to this problem gave ambiguous results due to the ill defined nature of the Constant Term of a rational function. The Constant Term has a simple definition if the argument is a Laurent series, but if the argument is a rational function then the value of the Constant Term depends on how the rational function is embedded into a ring of Laurent series (of which there are many possible). A careful choice of which embeddings to choose leads to a valid solution to the initial condition. A proof that the embedding satisfies the initial condition is provided, along with a combinatorial interpretation of why.
This Rational Constant Term method is then used to solve the Alternating Sign Matrix problem as a lattice path problem. Interestingly this method produces a summation form for the enumeration of Alternating Sign Matrices, in contrast to the more familiar product form. It remains an open problem to find a connection between these summation and product forms, similar to that which has been found for many nonintersecting path problems.
The latter half of this thesis focuses on the broader application and limits of the Constant Term method. We consider nearest neighbour walks in the quarterplane and eighthplane, considering which step sets are readily solvable and why. These chapters consider the connection between the Constant Term and the method of images, using reflection groups to satisfy the boundary conditions. These step sets can be split into three types of reflection groups, orthogonal reflections across lines that pass through the origin, oblique reflections across lines that pass through the origin, and affine reflections (reflections across lines parallel to lines through the origin). We can further split the step sets into those with lines of reflection that align with the quarterplane boundaries, and those with lines of reflection that cut through the quarterplane.
As the affine reflection group is infinite, it is not clear what form the Bethe Ansatz should take. For many problems an algebraic solution leads to a combinatorial understanding, but in solving these affine step sets the combinatorics of the problem leads to an algebraic solution. This method requires an infinite number of images to solve boundary conditions for finite lattice paths.
The eighthplane step set problems illustrate that care must be taken in writing down the partial difference equations. These paths are only satisfied if we can ensure that this diagonal boundary cannot be ``jumped'' by a single step in these paths.
© 2017 Dr. John Spencer Hugh Foxcroft
20170101T00:00:00Z

Extinction in branching processes with countably many types
http://hdl.handle.net/11343/210538
Extinction in branching processes with countably many types
Braunsteins, Peter Timothy
Multitype branching processes describe the evolution of populations in which individuals give birth independently according to a probability distribution that depends on their type. In this thesis, we consider the extinction of branching processes with countably infinitely many types.
We begin by developing a class of iterative methods to compute the global extinction probability vector $\bm{q}$. In particular, we construct a sequence of truncated and augmented branching processes with finite but increasing sets of types. A pathwise approach is then used to show that, under some sufficient conditions, the corresponding sequence of extinction probability vectors converges to the infinite vector $\bm{q}$. Besides giving rise to a family of algorithmic methods, our approach leads to new sufficient conditions for $\bm{q}=\bm{1}$ and $\bm{q}<\bm{1}$.
When then turn our attention to a specific class of branching processes which we refer to as lower Hessenberg branching processes. These are branching processes with typeset $\mathcal{X}=\{0,1,2,\dots\}$, in which individuals of type $i$ may give birth to offspring of type $j \leq i+1$ only. For this class of processes, we study the set of fixed points of the progeny generating vector. In particular, we highlight the existence of a continuum of fixed points whose minimum is the global extinction probability vector $\bm{q}$ and whose maximum is the partial extinction probability vector $\bm{\tilde{q}}$. We derive a computationally efficient partial extinction criterion. In the case where $\bm{\tilde q}=\bm{1}$, we derive a global extinction criterion, and in the case where $\bm{\tilde{q}}<\bm{1}$, we develop necessary and sufficient conditions for $\bm{q}=\bm{\tilde{q}}$.
Finally, we consider a more general class of structured branching processes which we refer to as block lower Hessenberg branching processes. For these processes, we derive partial and global extinction criteria, and we study the probability of extinction $\bm{q}(A)$ in any set of types $A$. In particular, we develop conditions for $\bm{q}(A)$ to be different from the global and partial extinction probability vectors, we present an iterative method to compute the vectors $\bm{q}(A)$, and we investigate their location in the set of fixed points of the progeny generating vector.
© 2018 Dr. Peter Timothy Braunsteins
20180101T00:00:00Z

Networks of interacting stochastic fluid models
http://hdl.handle.net/11343/208775
Networks of interacting stochastic fluid models
Sonenberg, Nikki
The focus of this thesis is the study of Markov modulated fluid models in the performance modelling of ad hoc telecommunications networks – that is networks where nodes communicate via multihop routes as there is no central administration.
Nodes rely on energy sources, for example batteries, to transmit to each other within a given radius, from which the multihop routes are defined. The energy consumption of a node is a function of its communication activity, and consumption rates may vary according to whether a node is a source, transit, or destination of a call. Latouche and Taylor introduced an approach to analysing such networks of fluid models, each modulated by the same background process, that incorporates feedback from the fluid level at a node to the background phase process. Assuming that the battery at the node can be recharged, they study the energy resource required to ensure the network can remain active without having to bring down calls. The network behaves as a loss network, that is networks of links each of which is capacity constrained and where calls unable to be accepted are lost.
A contribution of this thesis is to extend the Latouche and Taylor analysis of a network with infinite buffers at each node to the situation where the buffers are finite. The stationary density at a node with a finite buffer is derived using a Markov regenerative approach, allowing for a mixture of behaviour at each boundary.
Of significance to the analysis of the finite model at a network node is the understanding that the stationary density can be expressed in a way that effectively separates the behaviour of the process at the boundaries from its behaviour in the interior. So the description of the stationary density of the finite buffer model rests heavily on the expression for the stationary density of the infinite buffer model, which we derive using matrix analytic methods in a way that sets out all the steps of the fluid model analysis, and integrates ideas from many papers from the literature in a consistent form.
For the network model, a reduced load method is used to develop approximations accounting for the interactions between nodes, as an empty buffer at a node can lead to the dropping of calls in the network. Expressions for dropout probabilities across the network are derived using an analysis that requires key quantities associated with the stationary density of the finite buffer model.
Since ad hoc networks rely on transit nodes, and because data forwarding consumes valuable battery energy, each node along the path has an inherent disincentive to cooperate. We investigate a mechanism introduced by Latouche and Taylor to promote collaboration throughout the network that involves a network with two fluid models at each node, one with energy and another with credit. A contribution of this thesis is to consider call blocking behaviour in such networks, obtained by drawing on the results derived for the stationary density of finite buffer models with varied behaviour at the boundaries.
© 2017 Dr. Nikki Sonenberg
20170101T00:00:00Z