Ph.D. Theses (Mathematics)
http://hdl.handle.net/1957/15739
2017-09-09T14:54:59ZA New Algorithm for Computing the Veech Group of a Translation Surface
http://hdl.handle.net/1957/60633
A New Algorithm for Computing the Veech Group of a Translation Surface
Edwards, Brandon (Brandon Gary)
We give a new characterization of elements in the Veech group of a translation surface. This provides a computational test for Veech group membership. We use this computational test in an algorithm that detects when the Veech group is a lattice (has co-finite area), and in this case computes a fundamental polygon for the action of the Veech group on the hyperbolic plane. A standard result, essentially due to Poincaré, provides that a complete set of generators for the Veech group can then be obtained from the side pairings associated to this fundamental polygon.
Our approach introduces a new computational framework used to formulate a membership criterion for the Veech group of a compact translation surface (X,ω). We represent (X,ω) on a certain non-compact translation surface O that can be used to represent any translation surface within the SL(2,ℝ) orbit of the translation equivalence class of (X,ω). The surface O has an easily computed SL(2,ℝ)-action. When this action is restricted to the translation surface representations mentioned above, it corresponds to the usual SL(2,ℝ)-action on the set of equivalence classes of translation surfaces. The Veech group of a compact translation surface is therefore the stabilizer of its representation on O.
Graduation date: 2017
2017-03-17T00:00:00ZNonuniform 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
2016-08-22T00:00:00ZNew Families of pseudo-Anosov Homeomorphisms with Vanishing Sah-Arnoux-Fathi Invariant
http://hdl.handle.net/1957/59294
New Families of pseudo-Anosov Homeomorphisms with Vanishing Sah-Arnoux-Fathi Invariant
Do, Hieu Trung
Translation surfaces can be viewed as polygons with parallel and equal sides
identified. An affine homeomorphism φ from a translation surface to itself is called
pseudo-Anosov when its derivative is a constant matrix in SL₂(R) whose trace is larger
than 2 in absolute value. In this setting, the eigendirections of this matrix defines the
stable and unstable flow on the translation surface. Taking a transversal to the stable
flows, the first return map of the flow induces an interval exchange transformation T.
The Sah-Arnoux-Fathi invariant of φ is the sum of the wedge product between the
lengths of the subintervals of T and their translations. This wedge product does not
depend on the choice of transversal. We apply Veech’s construction of pseudo-Anosov
homeomorphisms to produce infinite families of pseudo-Anosov maps in the stratum
H(2, 2) with vanishing Sah-Arnoux-Fathi invariant, as well as sporadic examples in
other strata.
Graduation date: 2016
2016-05-27T00:00:00ZCompatible Discretizations for Maxwell's Equations with General Constitutive Laws
http://hdl.handle.net/1957/59160
Compatible Discretizations for Maxwell's Equations with General Constitutive Laws
McGregor, D. A. (Duncan A. O.)
In this thesis we construct compatible discretizations of Maxwell's equations. We use the term compatible to describe numerical methods for Maxwell's equations which obey many properties of vector Calculus in a discrete setting. Compatible discretizations preserve the exterior Calculus ensuring that the divergence of the curl and the curl of a gradient are zero in a discrete setting. This compatibility of discretizations with the continuum Maxwell's equations guarantees that the numerical solutions are physically meaningful.
We focus on the construction of a class of discretizations called Mimetic Finite Differences (MFD). The MFD method is a generalization of both staggered finite differences and mixed finite elements. We construct a parameterized family of MFD methods with equivalent formal order of accuracy. For time-dependent problems, we exploit this non-uniqueness by finding parameters which are optimal with respect to a certain criteria, for example, minimizing dispersion error. Dispersion error is a numerical artifact in which individual frequencies in a wave propagate at incorrect speeds; dominating the error in wave problems over long time propagation.
The novelty of this work is the construction of an MFD discretization for Maxwell's equations which reduces dispersion error for transient wave propagation in materials that are modeled by a general class of linear constitutive laws. We provide theoretical analysis of these new discretizations including an analysis of stability and discrete divergence. We also provide numerical demonstrations to illustrate the theory.
In addition to applications in the time domain we consider equilibrium Magnetohydrodynamic (MHD) generators. MHD generators extract power directly from a plasma by passing it through a strong magnetic field. Used as a topping cycle for traditional steam turbine generator, MHD offers a theoretical thermal efficiency of 60% compared to 40% of traditional systems. However, this technology has high life cycle costs due to equipment failure. One source of failure is arcing: the formation of high density currents which damage the generator. In this work we develop, analyze, and simulate a model of these generators. We use these simulations to show the viability of detecting electrical arcs by measurements of their magnetic fields outside of the generator.
Graduation date: 2016
2016-05-26T00:00:00Z