School of Mathematics and Statistics  Theses
http://hdl.handle.net/11343/295
20190121T11:28:41Z

Selected problems in enumerative combinatorics: permutation classes, random walks and planar maps
http://hdl.handle.net/11343/219277
Selected problems in enumerative combinatorics: permutation classes, random walks and planar maps
Elvey Price, Andrew
In this thesis we consider a number of enumerative combinatorial problems. We solve the problems of enumerating Eulerian orientations by edges and quartic Eulerian orientations counted by vertices. We also find and prove an algebraic relationship between the counting functions for permutations sortable by a double ended queue (deque) and permutations sortable by two stacks in parallel (2sip). In each of these cases, our proof of the result uses an elaborate system of functional equations which is much more complicated than the result itself, leaving the door open for a more direct, combinatorial proof.
We find polynomial time algorithms for generating the counting sequence of dequesortable permutations and the cogrowth sequence of some groups, including the lamplighter group $L$ and the BrinNavas group $B$. For permutations sortable by two stacks in series and for the cogrowth sequence of Thompson's group $F$, we find exponential time algorithms which are significantly more efficient than the algorithms that previously existed in the literature. In each case an empirical analysis of the produced terms of the sequence leads to a prediction regarding its asymptotic form. In particular, this method leads us to conjecture that the growth rate of dequesortable permutations is equal to that of 2sipsortable permutations, a conjecture which we reduce to three conjectures of Albert and BousquetM\'elou about quarter plane walks. The analysis of the cogrowth sequence of Thompson's group $F$ leads us to conjecture that $F$ is not amenable.
We also study the enumeration of $1324$avoiding permutations, a notoriously difficult problem in the field of pattern avoiding permutations. Using a structural decomposition of these permutations, we improve the lower and upper bounds on the growth rate to $10.271$ and $13.5$ respectively.
Next we investigate the concept of combinatorial Stieltjes moment sequences. We prove that the counting sequence of returns in any undirected locallyfinite graph is a Stieltjes moment sequence. As a special case, this implies that any cogrowth sequence is a Stieltjes moment sequence. Based on empirical evidence, we conjecture that the counting sequence for $1324$avoiding permutations is a Stieltjes moment sequence, which would imply an improved lower bound of $10.302$ on its growth rate.
We then describe a general class of counting sequences of augmented perfect matchings, which we prove to be Stieltjes moment sequences. In fact, we prove the stronger result that these sequences are Hankel totallypositive as sequences of polynomials. As a special case, we show that the Ward polynomials are Hankel totallypositive.
In the final chapter we generalise an identity of DuminilCopin and Smirnov for the $O(n)$ loop model on the hexagonal lattice to the offcritical case. In the $n=0$ case, which corresponds to the enumeration of selfavoiding walks, we use our identity to prove a relationship between the halfplane surface critical exponents $\gamma_{1}$ and $\gamma_{11}$ and the exponent characterising the winding angle distribution of selfavoiding walks in the halfplane.
© 2018 Dr Andrew Elvey Price
20180101T00:00:00Z

Unsteady motion of small particles in fluid: from autonomous propulsion to the interrogation of liquidsolid interfaces
http://hdl.handle.net/11343/216669
Unsteady motion of small particles in fluid: from autonomous propulsion to the interrogation of liquidsolid interfaces
Collis, Jesse F.
Fluid flows at micro and nanometre length scales are governed by physical processes that are quite different to their macroscale counterparts. When these flows interact with suspended solid particles, a rich array of phenomena can occur. This thesis examines four such processes in which unsteadiness arises; due to both steady and oscillatory flow processes.
Axisymmetric particles immersed in shear flows undergo periodic rotational motions known as Jeffery orbits. While extensive measurements have been reported on these orbits, the axial rotation of the particles has been difficult to study. A new methodology and commensurate theory is developed to enable the observation of these axial rotations. Experimental measurements are performed to validate the proposed theory.
Hydrodynamic trapping of small particles in microvortices has been observed experimentally. These microvortices are generated by cylindrical nanorods rotating about their minor axis, driven by an external magnetic field. The physical mechanism underlying this phenomenon is currently unknown. Here, a hydrodynamic model based on a rotating disc is proposed in an attempt to explain these observations. An asymptotic expansion on the NavierStokes equations is used to include the effects of small fluid inertia in the system. Finite particle size and inertia effects are then included through use of a particle momentum equation. This model reveals a possible trapping mechanism due to the interaction of the particle with the surrounding nonuniform flow.
Nanorods suspended in acoustically actuated fluid chambers have been observed to undergo autonomous propulsion. These particles, which contain forandaft asymmetries due to both shape and density, create local streaming flows enabling their propulsion. A general theory is developed to describe their motion which is applied to a dumbbell model for analytic solution. This provides a significant advance on existing theory, by allowing axisymmetric particles of arbitrary shape and density distribution to be modelled. New physics is uncovered, demonstrating that these particles can change their propulsion direction at intermediate frequencies; this phenomena is yet to be observed experimentally.
The Navierslip condition is used widely to model noncontinuum flows. Its applicability to the gassolid interface is established from theory, while the liquidsolid interface does not enjoy such rigour. Here, a new experimental method for probing this boundary condition at the liquidsolid interface is developed, based on multimode measurements in Suspended Microchannel Resonators (SMRs). Analytic, numeric and statistic tools are formulated to enable interpretation of these measurements. A constant slip length (of 2.7 $\pm$ 0.6 nm) is measured for gold nanoparticles of varying radii, providing the first direct experimental evidence for Navier slip at the liquidsolid interface, i.e. the sliplength is a material parameter, independent of geometry. This array of measurements and theory advance our understanding of fluidstructure interactions of nanoparticles in fluid, and provide new information on the nature of the liquidsolid interface.
© 2018 Dr. Jesse F. Collis
20180101T00:00:00Z

The isoperimetric problem in block designs
http://hdl.handle.net/11343/214165
The isoperimetric problem in block designs
Surani, Muhammad Adib
This thesis focuses on the problem of determining the isoperimetric numbers of Levi graphs of balanced incomplete block designs. We study two versions of this: the vertexisoperimetric number and the edgeisoperimetric number. In the general case, we manage to obtain strong upper and lower bounds on these numbers for Levi graphs of arbitrary block designs. We also obtain stronger or even exact values for some families of graphs, particularly those arising from finite geometries or difference sets.
© 2018 Dr. Muhammad Adib Surani
20180101T00:00:00Z

Statistical models for the location of lightningcaused wildfire ignitions
http://hdl.handle.net/11343/214157
Statistical models for the location of lightningcaused wildfire ignitions
Read, Nicholas
Lightningcaused wildfire is a significant concern for fire management agencies worldwide. Unlike other ignition sources, lightning fires often occur in remote and inaccessible locations making detection and suppression particularly challenging. Furthermore, individual lightning storms result in a large number of fires clustered in space and time which can overwhelm suppression efforts. Victoria, Australia, is one of the most fire prone environments in the world and the increased frequency of largescale landscape fires over the last decade is of particular concern to local wildfire management authorities.
This thesis is concerned with modeling lightningcaused wildfire ignition locations in Victoria. Such models could be used for predicting daily lightningcaused ignition likelihood as well as simulating realistic point patterns for use in fire spread models for risk analyses.
The first half of this thesis looks at regression models. We review methods for the model selection, validation, approximation and interpretation of generalised additive models. A review of performance metrics, such as the AUC, shows the difficulties and subtleties involved in evaluating the predictive performance of models.
We apply this theory to construct a nonlinear logistic regression model for lightningcaused wildfires in Victoria. The model operates on a daily time scale, with a spatial resolution of 20 km and uses covariate data including fuel moisture indices, vegetation type, a lightning potential index and weather. We develop a simple method to deconstruct model output into contributions from each of the individual covariates, allowing predictions to be explained in terms of the weather conditions driving them. Using these ideas, we discuss ranking the relative 'importance' of covariates in the model, leading to an approximating model with similar performance to the full model.
The second half of this thesis looks at point process models for lightningcaused ignitions. We introduce general theory for point processes, focusing on the inhomogeneous Poisson process, cluster processes and replicated point patterns. The Kfunction is a useful summary function for describing the spatial correlation point patterns and for fitting models. We present a method for pooling multiple estimates of the Kfunction, such as those that arise when using replicated point patterns, intended to reduce bias.
We fit an inhomogeneous Poisson process model as well as a Thomas and Cauchy cluster process model to the Victorian lightningcaused ignition data set. The cluster process models prove to have significantly better fit than the Poisson process model, but still struggle to reproduce the complex behaviour of the physical process.
© 2018 Dr. Nicholas Read
20180101T00:00:00Z