Department of Mathematics
http://hdl.handle.net/1957/13737
2016-10-01T17:12:49ZNonuniform Sampling Of Band-limited Functions
http://hdl.handle.net/1957/59862
Nonuniform Sampling Of Band-limited Functions
Al-Hammali, Hussain Y.
In this thesis, we will study certain generalizations of the classical Shannon Sampling Theorem, which allows for the reconstruction of a pi-band-limited, square-integrable function from its samples on the integers. J. R. Higgins provided a generalization where the integers can be perturbed by less than 1/4, which includes nonuniform and nonperiodic sampling sets. We generalize Higgins’ theorem by allowing for sampling sets that are perturbations of the set of zeros of a π-sine-type function.
A second type of generalization allows for functions f that, while still band-limited, need not be square-integrable but may have polynomial growth when restricted to the real line. We investigate two ways to achieve this goal, again using nonuniform sampling sets. The first is an approximate method that uses the multiplication of f by a smooth and rapidly decaying auxiliary function. The second method is exact and uses oversampling by finitely many additional points. It is also shown that oversampling by finitely many points is not only economical and may lead to faster convergence of the series, but also enables the perturbed sampling points to go beyond a quarter from the integers. Furthermore, oversampling by finitely many points is applied to control the error stemming from a quantization of the sampled function values.
The final topic considered is the so-called peak value problem, where one seeks to find an upper bound for the infinity norm of a function from knowledge of the supremum of its sampled values. We generalize an existing approach by first proving and then applying a nonuniform version of the Valiron-Tschakaloff sampling theorem.
Graduation date: 2017
2016-08-22T00:00:00ZPersistence of Populations in Environments with an Interface
http://hdl.handle.net/1957/59861
Persistence of Populations in Environments with an Interface
Rekow, James A.
We model a fish population in a spatial region comprising a marine protected area and a fishing ground separated by an interface. The model assumes conservation of biomass density and takes the form of a reaction diffusion equation with a logistic reaction term. At the interface, in addition to continuity of biomass density flux, two possible matching conditions are considered: continuity of population density and continuity of biomass density. Neumann conditions are imposed at the physical boundaries.
The eigenvalues of the elliptic problem resulting from the linearization of the model are computed. Necessary and sufficient conditions for the largest eigenvalue to be positive are determined. We show that there exist positive eigenfunctions corresponding to this eigenvalue. If the largest eigenvalue is positive, the population persists, whereas if this eigenvalue is negative, the population goes extinct. A simple sufficient condition for persistence when biomass density is continuous is that the spatially averaged net growth rate is positive. Similarly, if the spatially averaged net body mass growth rate is positive, the population persists when population is continuous.
A brief introduction is given on connections between parabolic partial differential equations and stochastic processes. Questions relating branching stochastic processes and properties of population models that incorporate interfaces are identified.
Graduation date: 2017
2016-09-09T00:00:00ZTopology of non-negatively curved manifolds
http://hdl.handle.net/1957/59713
Topology of non-negatively curved manifolds
Escher, Christine; Ziller, Wolfgang
An important question in the study of Riemannian manifolds of positive sectional curvature
is how to distinguish manifolds that admit a metric with non-negative sectional
curvature from those that admit one of positive curvature. Surprisingly, if the manifolds
are compact and simply connected, all known obstructions to positive curvature are already
obstructions to non-negative curvature. On the other hand, there are very few
known examples of manifolds with positive curvature. They consist, apart from the rank
one symmetric spaces, of certain homogeneous spaces G/H in dimensions 6, 7, 12, 13 and
24 due to Berger [Be], Wallach [Wa], and Aloff-Wallach [AW], and of biquotients K\G/H
in dimensions 6, 7 and 13 due to Eschenburg [E1],[E2] and Bazaikin [Ba], see [Zi] for a
survey. Recently, a new example of a positively curved 7-manifold was found which is
homeomorphic but not diffeomorphic to the unit tangent bundle of S⁴, see [GVZ, De].
And in [PW] a method was proposed to construct a metric of positive curvature on the
Gromoll-Meyer exotic 7-sphere.
This is an author's peer-reviewed final manuscript, as accepted by the publisher. The published article is copyrighted by Springer and can be found at: http://link.springer.com/article/10.1007%2Fs10455-013-9407-8
2014-06-01T00:00:00ZA Polynomial Chaos Method for Dispersive Electromagnetics
http://hdl.handle.net/1957/59535
A Polynomial Chaos Method for Dispersive Electromagnetics
Gibson, Nathan L.
Electromagnetic wave propagation in complex dispersive media is governed by the time dependent
Maxwell’s equations coupled to equations that describe the evolution of the induced macroscopic polarization.
We consider “polydispersive” materials represented by distributions of dielectric parameters in a
polarization model. The work focuses on a novel computational framework for such problems involving
Polynomial Chaos Expansions as a method to improve the modeling accuracy of the Debye model and allow
for easy simulation using the Finite Difference Time Domain (FDTD) method. Stability and dispersion
analyzes are performed for the approach utilizing the second order Yee scheme in two spatial dimensions.
This is an author's peer-reviewed final manuscript, as accepted by the publisher. The published article is copyrighted by Global-Science Press and published by Cambridge University Press. It can be found at for CiCP: http://www.global-sci.com/ and also at for Cambridge University Press: http://journals.cambridge.org/action/displayJournal?jid=CPH
2015-11-01T00:00:00Z