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# Mathematics - Theses, Dissertations, and other Required Graduate Degree Essays

Receive updates for this collection## Mathematical Model of Thermoregulation in Honeybee Swarms

Honeybees are masters of regulating their temperatures collectively, even in the absence of a hive. A reproductive swarm consisting of a queen and about half a colony's workers will leave their hive to find a new home. Prior to settling on a permanent home, the honeybees form a stationary swarm, where the bees cling onto one another from a roof-type structure, completely exposed to the elements. Because bees are so sensitive to extremes of heat and cold, it is essential that the swarm has ways to control its temperature. We present a mathematical model to study how honeybees thermoregulate by adjusting their movement and metabolic heat output. We introduce a system of coupled partial differential equations and integral equations to describe the swarm temperature, density, and size, along with a corresponding numerical scheme. We then relax the assumption of spherical symmetry and extend the model by studying non-spherical swarms.

## Some Results on Structure in Graphs and Numbers

This thesis present three results. The first is a result in structural graph theory. It demonstrates a large family of complete graphs embedded (clique embeddings) as minors of the grid-like Chimera graph, which is an abstract graph representing the D-Wave quantum adiabatic processor where vertices of the Chimera graph represent qubits in the processor. These particular embeddings are uniform in the sense that each vertex of the complete graph is represented by an equal number of vertices in the Chimera graph, which is thought to improve performance of the D-Wave processor. We present a polynomial-time algorithm to find a largest clique in this family in arbitrary induced subgraphs. Then we use the output of our algorithm as a measure of quality of a particular induced subgraph and show that the size of a largest clique embedding grows logarithmically in the grid size when a fixed percentage of qubits have been deleted. The second is a result in design theory and combinatorial number theory; constructing a family of designs called Heffter arrays. A Heffter array is a (m x n) integer matrix whose entries' absolute values cover the interval [1,mn], where every row and column sums to zero modulo 2mn+1. Archdeacon uses Heffter arrays to construct embeddings of the complete graph K_{2mn+1} into orientable surfaces such that every edge appears in an m-cycle and an n-cycle, provided either m or n is odd. This chapter establishes that m \times n Heffter arrays exist if and only m>2 and n>2. The third is a result in combinatorial number theory and structural graph theory. For subsets A,B of a group G define the product set AB = {ab : a in A, b in B}. Presented in this chapter is a classification the sets A,B such that |AB| <= |A|+|B| when H

## Fast direct integral equation methods for the Laplace-Beltrami equation on the sphere

Integral equation methods for solving the Laplace-Beltrami equation on the unit sphere are presented and applied to the problem of point vortex motion. The Laplace-Beltrami equation is first posed on a simply connected domain on the sphere, then reformulated into an integral equation and discretized. The resulting linear system is solved by adapting current fast direct solvers from fully two and three dimensional problems to the surface of the sphere. The solution is achieved in O(N) operations, where N is the number of points lying on the contour of a single “island.” The performance of the solver is studied through several representative examples. To highlight the efficiency of the direct method for problems with multiple right hand sides, the solver is used to study point vortex motion. The relationship between the Laplace-Beltrami equation and the motion of a point vortex in the presence of coastlines is explained—both in terms of finding instantaneous streamlines of the fluid and the trajectory of a vortex over time. The solver is used to construct these instantaneous streamlines and trajectories, of which the latter requires the Laplace-Beltrami equation to be solved for each time step. In this case the performance of the direct solver is found to exceed previous iterative approaches using the fast multipole method. Lastly, the fast direct solver is adapted to the multiply connected case and several numerical examples are presented.

## Unfolding of Diagramming and Gesturing between Mathematics Graduate Student and Supervisor during Research Meetings

This study has two main purposes. The first is to confirm and advance Gilles Châtelet’s account of the role of diagramming in mathematics invention by extending his results, which were based solely on historical, mathematical manuscripts, to the context of live mathematical activity. The second purpose is to elucidate the enculturation process of a graduate student into mathematical research, which has hitherto received limited research attention. These two purposes are related through the virtual and physical gestures that the graduate student engages in during diagramming thereby providing insights into mathematical invention and the enculturation process. This study adopts a qualitative methodology based on field notes, video-recordings and digital images from nine research meetings, which were held weekly over a period of three months. Current research theorises the role of mathematical diagrams in diametrically opposed ways: diagrams are either a visual representation of already existing mathematical objects and relations, or they are the means through which mathematical objects and relations emerge. The latter is due to Châtelet, who regards the diagram as a material site of engaging with and mobilizing the mathematics. His approach is employed in this thesis to create a window into the realm of mathematical thinking and invention by examining how a graduate student (as the less-expert mathematician) and his supervisor and two research colleagues (as the expert mathematicians) interact with diagrams. An embodied lens, based on the work of de Freitas, Roth, Rotman, Sinclair and Streeck, exposes the similarities and differences in the way that each class of mathematician gestures and diagrams. The analysis in this thesis reveals that gesturing and diagramming support and advance mathematical communication throughout the graduate student’s enculturation process. Furthermore, a collective study of the abundant diagrams produced during research meetings leads to a life-cycle of diagrams, whose phases disclose a variety of distinct relationships between mathematician and diagram. Lastly, a detailed examination of the evolution of a particular diagram uncovers how mathematical invention emerges through gesturing and diagramming. These findings have implications for the teaching and learning of mathematics at all levels.

## Parametric resonance in immersed elastic structures, with application to the cochlea

Examples of fluid motion driven by immersed flexible structures abound in nature. In many biological settings, for instance a beating heart, an active material generates a time-dependent internal forcing on the surrounding fluid. Motivated by such active biological structures, this thesis investigates parametric resonance in fluid-structure systems induced by an internal forcing via periodic modulation of the material stiffness. One particular application that we study is the cochlea which is the primary component for pitch selectivity in the mammalian hearing system. We present a 2D model of the cochlea in which a periodic internal forcing gives rise to amplification of basilar membrane (BM) oscillations. This forcing is inspired by experiments showing that outer hair cells within the cochlear partition change their lengths when stimulated, which can in turn distort the partition and modulate tension across the BM. We demonstrate the existence of resonant (unstable) solutions through a Floquet stability analysis of the linearized governing equations. Moreover, we show that an internal forcing is sufficient to produce travelling waves along the BM in the absence of any external stimulus. We next examine parametric instabilities in a 3D system by considering a closed spherical elastic membrane (or shell) immersed in a viscous, incompressible fluid. A Floquet analysis for both inviscid and viscous systems shows that parametric resonance is possible even in the presence of fluid viscosity. Numerical simulations are presented to verify the analysis and an application to cardiac fluid dynamics is discussed. Finally, we deviate from the topic of parametric instabilities to consider the natural oscillations of unforced spherical elastic membranes. We present a linear stability analysis to obtain a dispersion relation for immersed membrane oscillations for both inviscid and viscous fluids, as well as a nonlinear analysis of immersed membrane oscillations in an inviscid fluid. We then present an experiment where we measure oscillation frequencies of immersed water balloons in an attempt to corroborate the analytical results.

## Die Freude an der Gestalt: Methods, Figures, and Practices in Early Nineteenth Century Geometry

As recounted by later historians, modern geometry began with Jean Victor Poncelet, whose contributions then spread to Germany alongside an opposition between geometric methods that came to be exemplified by the antagonism of Julius Plücker, an analytic geometer, and Jakob Steiner, a synthetic geometer. To determine the participants, arguments, and qualities of this perceived divide, we drew upon historical accounts from the late nineteenth and early twentieth centuries. Several themes emerged from the historical perspective, which we investigated within the original sources. Our questions centred on how geometers distinguished methods, when opposition arose, in what ways geometry disseminated from Poncelet to Plücker and Steiner, and whether this geometry was "modern" as claimed. Our search for methodological debates led to Poncelet's proposal that within pure geometry the figure was never lost from view, while it could be obscured by the calculations of algebra. We examined his argument through a case study that revealed visual attention within constructive problem solving, regardless of method. Further, geometers manipulated and represented figures through textual descriptions and coordinate equations. In these same texts, Poncelet and Joseph-Diez Gergonne instigated a debate on the principle of duality. Rather than dismiss their priority dispute as external to mathematics, we consider the texts involved as a medium for communicating geometry in which Poncelet and Gergonne developed strategies for introducing new geometry to a conservative audience. This conservative audience did not include Plücker and Steiner, who adapted new vocabulary, techniques and objects. Through comparing their common research, we found they differentiated methods based on personal considerations. Plücker practiced a "pure analytic geometry" that avoided calculation. Steiner admired "synthetic geometry'' because of its organic unity. These qualities contradicted descriptions of analytic geometry as computational or synthetic geometry as ad-hoc. Finally, we turned to claims for novelty in the context of contemporary French books on geometry. Most of these books point to a pedagogical orientation, where the methodological divide was grounded in student prerequisites and "modern'' implied the use of algebra in geometry. By contrast, research publications exhibited evolving forms of geometry that evaded dichotomous categorization.

## Symmetric Differential Forms on the Barth Sextic Surface

This thesis concerns the existence of regular symmetric differential 2-forms on the Barth sextic surface, here denoted by X. This surface has 65 nodes, the maximum possible for a sextic hypersurface in P^3. This project is motivated by a recent work of Bogmolov and De Oliveira where it is shown that a hypersurface in P^3 with many nodes compared to its degree contains only finitely many genus zero and one curves. We find that there are symmetric differential 2-forms on X that are regular everywhere outside the nodes. We also find that none of these extend to a regular form on the minimal resolution of X. Using these forms we can prove that any genus 0 curve on X must pass through at least one node, and we determine the curves passing through just a select set of nodes.

## Solving variational problems and partial differential equations mapping between manifolds via the closest point method

Many applications from fields such as mathematical physics, image processing, computer vision and medical imaging require computation of maps between manifolds. A numerical framework is introduced for solving variational problems and partial differential equations that map from a source manifold M to a target manifold N . The numerics rely on the closest point representations of M and N. Using the closest point representation produces simple algorithms for handling complex surface geometries, since standard Cartesian numerical methods can be used. The framework is illustrated with harmonic maps and a straightforward algorithm is given for this case. Harmonic maps are important in applications such as texture mapping, brain image regularization and colour image denoising. Moreover, the harmonic mapping energy is part of numerous energy functionals. The algorithm is justified theoretically and shown to be first order accurate. It is implemented in two applications: removing noise from texture maps and colour image enhancement.

## On Laplace-Borel resummation of Dyson-Schwinger equations

On Laplace-Borel Resummation of Dyson-Schwinger Equations Abstract: In this work we conduct a complex analytic study of Dyson-Schwinger equations, the quantum equations of motion. Focusing on a particular family of those functional equations we consider the class of formal solutions G(x,L), series expansions in the coupling constant x and the energy scale L whose formal and real analytic aspects have already been studied. Taking the point of view of complex analysis we are able to shed some new light on the structure of these solutions and provide useful tools to consider asymptotic questions. This thesis is built around two functions. The anomalous dimension $\gamma_1$ which is closely tied to the energy scaling properties of quantum field theory and the Green function G(x,L), the actual solution to the Dyson-Schwinger equation. We study their dual aspects as formal power series and analytic functions in the variable x and L. Our tool of choice is the Laplace-Borel resummation method which proves suitable to take care of the divergent series occurring naturally in quantum field theory. Our main results consist in: i) conducting a Laplace-Borel analysis of the anomalous dimension $\gamma_1$; ii) constructing a Laplace-Borel solution to our Dyson-Schwinger equation by using the renormalization group equation.

## Simple eigenvalues of graphs and digraphs

The spectra of graphs and their relation to graph properties have been well-studied. For digraphs, in contrast, there are relatively few results. The adjacency matrix of a digraph is usually difficult to work with; it is not always diagonalizable and the interlacing theorem does not hold (in general) for adjacency matrices of digraphs. All acyclic digraphs have the same spectrum as the empty graph. This motivates the need to work with a different matrix which captures the adjacency of the digraph. To this end, we introduce the Hermitian adjacency matrix. Another way to extract more information out of the spectrum is by restricting to specific classes of digraphs. In this thesis, we look at vertex-transitive digraphs with simple eigenvalues. Intuitively, the property of having many simple eigenvalues tends to coincide with having few automorphisms. For example, the only vertex-transitive graph with all eigenvalues simple is K_2. In the case of graphs, we restrict to the cubic vertex-transitive case, where we find combinatorial properties of graphs with multiple simple eigenvalues. We also explore the eigenvectors of vertex-transitive digraphs with all eigenvalues distinct.