Department of Mathematics and Statistics Graduate Theses
http://hdl.handle.net/1974/758
2016-12-09T05:48:06ZAn impulsive differential equation model for Marek's disease
http://hdl.handle.net/1974/14944
An impulsive differential equation model for Marek's disease
Rozins, Carly
Many dynamical processes are subject to abrupt changes in state. Often these perturbations can be periodic and of short duration relative to the evolving process. These types of phenomena are described well by what are referred to as impulsive differential equations, systems of differential equations coupled with discrete mappings in state space. In this thesis we employ impulsive differential equations to model disease transmission within an industrial livestock barn.
In particular we focus on the poultry industry and a viral disease of poultry called Marek's disease. This system lends itself well to impulsive differential equations. Entire cohorts of poultry are introduced and removed from a barn concurrently. Additionally, Marek's disease is transmitted indirectly and the viral particles can survive outside the host for weeks. Therefore, depopulating, cleaning, and restocking of the barn are integral factors in modelling disease transmission and can be completely captured by the impulsive component of the model.
Our model allows us to investigate how modern broiler farm practices can make disease elimination difficult or impossible to achieve. It also enables us to investigate factors that may contribute to virulence evolution.
Our model suggests that by decrease the cohort duration or by decreasing the flock density, Marek's disease can be eliminated from a barn with no increase in cleaning effort. Unfortunately our model also suggests that these practices will lead to disease evolution towards greater virulence. Additionally, our model suggests that if intensive cleaning between cohorts does not rid the barn of disease, it may drive evolution and cause the disease to become more virulent.
Thesis (Ph.D, Mathematics & Statistics) -- Queen's University, 2016-09-22 10:30:23.342
2016-09-22T00:00:00ZIrreducibility of Random Hilbert Schemes
http://hdl.handle.net/1974/14882
Irreducibility of Random Hilbert Schemes
Staal, Andrew P
We prove that a random Hilbert scheme that parametrizes the closed
subschemes with a fixed Hilbert polynomial in some projective space is
irreducible and nonsingular with probability greater than $0.5$. To
consider the set of nonempty Hilbert schemes as a probability space,
we transform this set into a disjoint union of infinite binary trees,
reinterpreting Macaulay's classification of admissible Hilbert
polynomials. Choosing discrete probability distributions with
infinite support on the trees establishes our notion of random Hilbert
schemes. To bound the probability that random Hilbert schemes are
irreducible and nonsingular, we show that at least half of the
vertices in the binary trees correspond to Hilbert schemes with unique
Borel-fixed points.
Thesis (Ph.D, Mathematics & Statistics) -- Queen's University, 2016-09-11 13:52:03.771
2016-09-13T00:00:00ZCodomain Rigidity of the Dirichlet to Neumann Operator for the Riemannian Wave Equation
http://hdl.handle.net/1974/14738
Codomain Rigidity of the Dirichlet to Neumann Operator for the Riemannian Wave Equation
Milne, Tristan
We study the Dirichlet to Neumann operator for the Riemannian wave equation on a compact Riemannian manifold. If the Riemannian manifold is modelled as an elastic medium, this operator represents the data available to an observer on the boundary of the manifold when the manifold is set into motion through boundary vibrations. We study the Dirichlet to Neumann operator when vibrations are imposed and data recorded on disjoint sets, a useful setting for applications. We prove that this operator determines the Dirichlet to Neumann operator where sources and observations are on the same set, provided a spectral condition on the Laplace-Beltrami operator for the manifold is satisfied. We prove this by providing an implementable procedure for determining a portion of the Riemannian manifold near the area where sources are applied. Drawing on established results, an immediate corollary is that a compact Riemannian manifold can be reconstructed from the Dirichlet to Neumann operator where sources and observations are on disjoint sets.
Thesis (Master, Mathematics & Statistics) -- Queen's University, 2016-08-23 15:00:13.283
2016-08-23T00:00:00ZGeometry of Dirac Operators
http://hdl.handle.net/1974/14633
Geometry of Dirac Operators
Beheshti Vadeqan, Babak
Let $M$ be a compact, oriented, even dimensional Riemannian manifold and let $S$ be a Clifford bundle over $M$ with Dirac operator $D$.
Then
\[
\textsc{Atiyah Singer: } \quad
\text{Ind } \mathsf{D}= \int_M \hat{\mathcal{A}}(TM)\wedge \text{ch}(\mathcal{V})
\]
where $\mathcal{V} =\text{Hom}_{\mathbb{C}l(TM)}(\slashed{\mathsf{S}},S)$.
We prove the above statement with the means of the heat kernel of the heat semigroup $e^{-tD^2}$.
The first outstanding result is the McKean-Singer theorem that describes the index in terms of the supertrace of the heat kernel.
The trace of heat kernel is obtained from local geometric information. Moreover, if we use the asymptotic expansion of the
kernel we will see that in the computation of the index only one term matters.
The Berezin formula tells us that the supertrace is nothing but the coefficient of the Clifford top part, and at the end, Getzler calculus enables us to find the integral of these top parts in terms of characteristic classes.
Thesis (Master, Mathematics & Statistics) -- Queen's University, 2016-07-04 20:27:20.386
2016-07-05T00:00:00Z