Theses (Mathematics and Statistics)
http://hdl.handle.net/11122/4613
Fri, 23 Oct 2020 10:46:24 GMT2020-10-23T10:46:24ZAn exposition on the Kronecker-Weber theorem
http://hdl.handle.net/11122/11349
An exposition on the Kronecker-Weber theorem
Baggett, Jason A.
The Kronecker-Weber Theorem is a, classification result from Algebraic Number Theory. Theorem (Kronecker-Weber). Every finite, abelian extension of Q is contained in a cyclotomic field. This result was originally proven by Leopold Kronecker in 1853. However, his proof had some gaps that were later filled by Heinrich Martin Weber in 1886 and David Hilbert in 1896. Hilbert's strategy for the proof eventually led to the creation of the field of mathematics called Class Field Theory, which is the study of finite, abelian extensions of arbitrary fields and is still an area of active research. Not only is the Kronecker-Weber Theorem surprising, its proof is truly amazing. The idea of the proof is that for a finite, Galois extension K of Q, there is a connection between the Galois group Gal(K/Q) and how primes of Z split in a certain subring R of K corresponding to Z in Q. When Gal(K/Q) is abelian, this connection is so stringent that the only possibility is that K is contained in a cyclotomic field. In this paper, we give an overview of field/Galois theory and what the Kronecker-Weber Theorem means. We also talk about the ring of integers R of K, how primes split in R, how splitting of primes is related to the Galois group Gal(K/Q), and finally give a proof of the Kronecker-Weber Theorem using these ideas.
Thesis (M.S.) University of Alaska Fairbanks, 2011
Sun, 01 May 2011 00:00:00 GMThttp://hdl.handle.net/11122/113492011-05-01T00:00:00ZInvestigations in phylogenetics: tree inference and model identifiability
http://hdl.handle.net/11122/11303
Investigations in phylogenetics: tree inference and model identifiability
Yourdkhani, Samaneh
This thesis presents two projects in mathematical phylogenetics. The first presents a new, statistically consistent, fast method for inferring species trees from topological gene trees under the multispecies coalescent model. The algorithm of this method takes a collection of unrooted topological gene trees, computes a novel intertaxon distance from them, and outputs a metric species tree. The second establishes that numerical and non-numerical parameters of a specic Prole Mixture Model of protein sequence evolution are generically identifiable. Algebraic techniques are used, especially a theorem of Kruskal on tensor decomposition.
Thesis (Ph.D.) University of Alaska Fairbanks, 2020
Fri, 01 May 2020 00:00:00 GMThttp://hdl.handle.net/11122/113032020-05-01T00:00:00ZAn exploration of two infinite families of snarks
http://hdl.handle.net/11122/10547
An exploration of two infinite families of snarks
Ver Hoef, Lander
In this paper, we generalize a single example of a snark that admits a drawing with even rotational symmetry into two infinite families using a voltage graph construction techniques derived from cyclic Pseudo-Loupekine snarks. We expose an enforced chirality in coloring the underlying 5-pole that generated the known example, and use this fact to show that the infinite families are in fact snarks. We explore the construction of these families in terms of the blowup construction. We show that a graph in either family with rotational symmetry of order m has automorphism group of order m2m⁺¹. The oddness of graphs in both families is determined exactly, and shown to increase linearly with the order of rotational symmetry.
Thesis (M.S.) University of Alaska Fairbanks, 2019
Wed, 01 May 2019 00:00:00 GMThttp://hdl.handle.net/11122/105472019-05-01T00:00:00ZOn the Klein-Gordon equation originating on a curve and applications to the tsunami run-up problem
http://hdl.handle.net/11122/10490
On the Klein-Gordon equation originating on a curve and applications to the tsunami run-up problem
Gaines, Jody
Our goal is to study the linear Klein-Gordon equation in matrix form, with initial conditions originating on a curve. This equation has applications to the Cross-Sectionally Averaged Shallow Water equations, i.e. a system of nonlinear partial differential equations used for modeling tsunami waves within narrow bays, because the general Carrier-Greenspan transform can turn the Cross-Sectionally Averaged Shallow Water equations (for shorelines of constant slope) into a particular form of the matrix Klein-Gordon equation. Thus the matrix Klein-Gordon equation governs the run-up of tsunami waves along shorelines of constant slope. If the narrow bay is U-shaped, the Cross-Sectionally Averaged Shallow Water equations have a known general solution via solving the transformed matrix Klein-Gordon equation. However, the initial conditions for our Klein-Gordon equation are given on a curve. Thus our goal is to solve the matrix Klein-Gordon equation with known conditions given along a curve. Therefore we present a method to extrapolate values on a line from conditions on a curve, via the Taylor formula. Finally, to apply our solution to the Cross-Sectionally Averaged Shallow Water equations, our numerical simulations demonstrate how Gaussian and N-wave profiles affect the run-up of tsunami waves within various U-shaped bays.
Thesis (M.S.) University of Alaska Fairbanks, 2019
Wed, 01 May 2019 00:00:00 GMThttp://hdl.handle.net/11122/104902019-05-01T00:00:00Z