School of Mathematics and Statistics - Theses
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ItemMathematical approaches to pattern formation in dermatology(University of Melbourne, 2005)
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ItemAspects of mixed longitudinal growth analysis(University of Melbourne, 2010)This thesis presents practical approaches to the analysis of mixed longitudinal growth data. Longitudinal studies of the human population are specifically designed to investigate changes over a limited age range in a characteristic which is measured repeatedly for each study participant. This type of data poses several methodological challenges. First, models for the analysis of longitudinal data must recognize the relationship between the observations taken from each study participant. The mixed nature of the data calls for the use of random effects and variance and correlation structures for the within group errors. Secondly, the models must be flexible enough so that they can be easily differentiated for the timing of the population growth spurts. And thirdly, longitudinal growth data of human subjects is more often than not affected by the missing data problem. In practice, the missing data mechanism needs to be understood and taken into consideration when fitting the models. These aspects of mixed longitudinal growth analysis are covered in detail in this thesis using a comprehensive data set of repeated measures of human height of hundreds of Melbourne school children ranging form the ages of 5 to 18 years.
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ItemQuaternionic modular forms mod p( 2023-08)In a 1987 letter, Serre proves that the systems of Hecke eigenvalues arising from mod p modular forms are the same as those arising from certain functions on the adelic points of D^*, where D is the unique quaternion algebra over Q ramified at p and infinity. We give a detailed account of this proof, the key idea of which is to restrict our study of mod p modular forms to the supersingular locus of the modular curve using the Hasse invariant, and then we extend the result to other level structures. We then incorporate additional ramification into the quaternion algebra D, and this correlates with the study of modular forms on Shimura curves.
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ItemA shape theorem for the half-orthant model and the features of its shape( 2022)We study degenerate random environments which are site-based models of random media involving a parameter p in [0,1]. We are focused on a particular case, the half-orthant model, and will prove a shape theorem for this model when pp_c(2), which is not addressed in this thesis. At last, we prove the general case when p
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ItemAn explained sum of squares approach to nonparametric regression with measurement error( 2019)We introduce a new method in nonparametric regression problems in the presence of measurement error, known as the explained sum of squares. We discuss its theoretical properties and provide evidence of practical application in both univariate and multivariate settings. We show that our estimator is theoretically consistent, and is a novel approach to typical deconvolution methods popular in the measurement error literature.
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ItemTriangulating Cappell-Shaneson homotopy 4-spheres( 2017)The smooth 4-dimensional Poincare conjecture states that if a smooth 4-manifold is homeomorphic to the 4-sphere then it is diffeomorphic to the standard 4-sphere. Historically, one of the most promising families of potential counterexamples to this conjecture is the family of Cappell-Shaneson homotopy 4-spheres. Over time, with difficult Kirby calculus computations, an infinite subfamily of Cappell-Shaneson homotopy 4-spheres was shown to be standard, that is, diffeomorphic tothe 4-sphere and hence are not counterexamples. More recently, Gompf showed that a strictly larger subfamily is standard using the fishtail surgery trick. In another direction, Budney-Burton-Hillman discovered a fascinating ideal triangulation of a smooth 4-manifold which they show is homeomorphic to a certain Cappell-Shaneson 2-knot complement X^4. They pose as a problem to show that it is in fact diffeomorphic to X. We solve this problem by directly triangulating X and observing that the triangulation we obtain is combinatorially isomorphic to theirs. In fact, we show that this triangulation is a layered triangulation. We then generalise our construction to show how to construct layered triangulations of once-punctured 3-torus bundles, generalising the well-known Floyd-Hatcher triangulations of once-punctured torus bundles. Next, we describe how we can use these triangulations of once-punctured 3-torus bundles to construct triangulations of Cappell-Shaneson homotopy 4-spheres. This involves understanding the Gluck twisting operation via triangulations. For some particular examples, we simplify the triangulations using Pachner moves to a standard triangulation of the 4-sphere. For these examples, this provides a new computational proof that the corresponding Cappell-Shaneson homotopy 4-spheres are diffeomorphic to the 4-sphere.
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ItemIntrinsic damping in metal nanoparticles( 2015)Micro-scale mechanical resonators typically exhibit quality factors in the tens of thousands, while those of nanoscale devices are considerably smaller. This thesis is a study of the damping measured using ultrafast laser measurements of metal nanoparticles. Continuum, quantum, and molecular approaches are used to investigate the origins of observed damping. Together these approaches provide evidence that acoustic damping in metal nanoparticles is dominated by phonon-phonon interactions.
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ItemGalois representations and theta operators for Siegel modular forms( 2015)Modular forms are powerful number theoretic objects, having attracted much study and attention for the last 200 years. In the modern area, one of their primary points of interest is their role in the Langlands program. The work of Deligne (see [Del71]) and Serre (see [S+87]) provided a connection between modular forms and Galois representations. An integral piece of this connection is the theta operator, which allows tight manipulation of the modular forms and Galois representations. There is a larger picture, in which modular forms are merely a special instance of objects known as Siegel modular forms. In this thesis, we describe generalisations of the above concepts and theories to the Siegel case. We first demonstrate some generalisations of the theta operator, and subsequently describe the connection between Siegel modular forms and Galois representations. Finally we give a description of the effect of the theta operator on the Galois representations which are conjecturally arising from these Siegel modular forms.
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ItemThe Steiner ratio conjecture for eight points( 2014)Given a set X of n points in the euclidean plane R2, a Steiner minimal tree is a shortest network interconnecting X. Note that a Steiner minimal tree may contain extra vertices, referred to as Steiner points, not already in X. The problem of finding a Steiner minimal tree is NP-hard. A Steiner minimal tree can, however, be approximated by a minimum spanning tree, which can be found in polynomial time in n. A measure for the performance of a Steiner minimal tree compared with that of a minimum spanning tree is given by the Steiner ratio ρ, the ratio of lengths between a Steiner minimal tree and a minimum spanning tree. In 1968, Gilbert and Pollak conjectured that ρ ≥ √3/2 for any X. The conjecture was subsequently verified for several small values of n. For example, Rubinstein and Thomas applied some ideas from the calculus of variations to verify the conjecture for n = 6. Then, in 1992, Du and Hwang published a paper concerning the general problem, with convexity and flexibility as the major ideas. It has since been pointed out, however, that the argument of Du and Hwang contains a gap. Nevertheless, their argument contains many elegant ideas which may still be retained. In this thesis, we use a combination of techniques drawn from both the work of Rubinstein and Thomas as well as that of Du and Hwang to verify the conjecture for n = 8. Whereas Du and Hwang chose to work with a restricted set of spanning trees, we opt to work with the full set of spanning trees. We introduce the notion of symmetry breaking into the conception of the problem in order to fully exploit the idea of flexibility. We also adapt some material from rigidity theory, the study of bar-and-joint frameworks, in order to ensure that our arguments apply to both generic and nongeneric situations.
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ItemCores of vertex-transitive graphs( 2013)The core of a graph $\Gamma$ is the smallest graph $\Gamma^\ast$ for which there exist graph homomorphisms $\Gamma\rightarrow\Gamma^\ast$ and $\Gamma^\ast\rightarrow\Gamma$. Thus cores are fundamental to our understanding of general graph homomorphisms. It is known that for a vertex-transitive graph $\Gamma$, $\Gamma^\ast$ is vertex-transitive, and that $\left|V(\Gamma^\ast)\right|$ divides $\left|V(\Gamma)\right|$. The purpose of this thesis is to determine the cores of various families of vertex-transitive and symmetric graphs. We focus primarily on finding the cores of imprimitive symmetric graphs of order $pq$, where $p