Digital Commons Network^™

Regular Homotopy Of Closed Curves On Surfaces, Katherine Kylee Zebedeo

Boise State University Theses and Dissertations

The use of rotation numbers in the classification of regular closed curves in the plane up to regular homotopy sparked the investigation of winding numbers to classify regular closed curves on other surfaces. Chillingworth [1] defined winding numbers for regular closed curves on particular surfaces and used them to classify orientation preserving regular closed curves that are based at a fixed point and direction. We define geometrically a group structure of the set of equivalence classes of regular closed curves based at a fixed point and direction. We prove this group structure coincides with the one introduced by Smale [9] …

On The Spherical Symmetry Of Perfect-Fluid Stellar Models In General Relativity, Joshua M Brewer

Masters Theses

It is well known in Newtonian theory that static self-gravitating perfect fluids in a vacuum are necessarily spherically symmetric. The necessity of spherical symmetry of perfect-fluid static spacetimes with constant density in general relativity is shown.

A Convexity Theorem For Symplectomorphism Groups, Seyed Mehdi Mousavi

Electronic Thesis and Dissertation Repository

In this thesis we study the existence of an infinite-dimensional analog of maximal torus in the symplectomorphism groups of toric manifolds. We also prove an infinite-dimensional version of Schur-Horn-Kostant convexity theorem. These results are extensions of the results of Bao-Raiu, Elhadrami, Bloch-Flachka-Ratiu and Bloch-El Hadrami-Flaschka-Raiu.

A Homotopy Theory For Diffeological Spaces, Enxin Wu

Electronic Thesis and Dissertation Repository

Smooth manifolds are central objects in mathematics. However, the category of smooth manifolds is not closed under many useful operations. Since the 1970's, mathematicians have been trying to generalize the concept of smooth manifolds. J. Souriau's notion of diffeological spaces is one of them. P. Iglesias-Zemmour and others developed this theory, and used it to simplify and unify several important concepts and constructions in mathematics and physics.

We further develop the diffeological space theory from several aspects: categorical, topological and differential geometrical. Our main concern is to build a suitable homotopy theory (also called a model category structure) on the …

Contractible Theta Complexes Of Graphs, Chelsea Marian Mcamis

Masters Theses

We examine properties of graphs that result in the graph having a contractible theta complex. We classify such properties for tree graphs and graphs with one loop and we introduce examples of graphs with such properties for tree graphs and graphs with one or two loops. For more general graphs, we show that having a contractible theta complex is not an elusive property, and that any skeleton of a graph with at least three loops can be made to have a contractible theta complex by strategically adding vertices to its skeleton.

Degree Constrained Triangulation, Roshan Gyawali

UNLV Theses, Dissertations, Professional Papers, and Capstones

Triangulation of simple polygons or sets of points in two dimensions is a widely investigated problem in computational geometry. Some researchers have considered variations of triangulation problems that include minimum weight triangulation, de-launay triangulation and triangulation refinement. In this thesis we consider a constrained version of the triangulation problem that asks for triangulating a given domain (polygon or point sites) so that the resulting triangulation has an increased number of even degree vertices. This problem is called Degree Constrained Triangulation (DCT). We propose four algorithms to solve DCT problems. We also present experimental results based on the implementation of the …

On The Geometry Of Virtual Knots, Rachel Elizabeth Byrd

Boise State University Theses and Dissertations

The Dehn complex of prime, alternating virtual links has been shown to be non-positively curved in the paper "Generalized knot complements and some aspherical ribbon disc complements" by J. Harlander and S. Rosebrock (2003) [7]. This thesis investigates the geometry of an arbitrary alternating virtual link. A method is constructed for which the Dehn complex of any alternating virtual link may be decomposed into Dehn complexes with non-positive curvature. We further study the relationship between the Dehn space and Wirtinger space, and we relate their fundamental groups using generating curves on surfaces. We conclude with interesting examples of Dehn complexes …

Alexander And Conway Polynomials Of Torus Knots, Katherine Ellen Louise Agle

Masters Theses

We disprove the conjecture that if K is amphicheiral and K is concordant to K', then C_K'(z)C_K'(iz)C_K\(z²) is a perfect square inside the ring of power series with integer coefficients. The Alexander polynomial of (p,q)-torus knots are found to be of the form A_T(p,q)(t)= (f(t^q))/(f(t)) where f(t)=1+t+t²+...+t^p-1. Also, for (pⁿ,q)-torus knots, the Alexander polynomial factors into the form A_T(pⁿ ,q)=f(t)f(t^p)f(t^p2 )...f(t^pn-2 )f(t^pn-1 ). A new conversion from the Alexander polynomial to the …

Generalized Branching In Circle Packing, James Russell Ashe

Doctoral Dissertations

Circle packings are configurations of circle with prescribed patterns of tangency. They relate to a surprisingly diverse array of topics. Connections to Riemann surfaces, Apollonian packings, random walks, Brownian motion, and many other topics have been discovered. Of these none has garnered more interest than circle packings' relationship to analytical functions. With a high degree of faithfulness, maps between circle packings exhibit essentially the same geometric properties as seen in classical analytical functions. With this as motivation, an entire theory of discrete analytic function theory has been developed. However limitations in this theory due to the discreteness of circle packings …

Hyperbolic Structures From Link Diagrams, Anastasiia Tsvietkova

Doctoral Dissertations

As a result of Thurston's Hyperbolization Theorem, many 3-manifolds have a hyperbolic metric or can be decomposed into pieces with hyperbolic metric (W. Thurston, 1978). In particular, Thurston demonstrated that every link in a 3-sphere is a torus link, a satellite link or a hyperbolic link and these three categories are mutually exclusive. It also follows from work of Menasco that an alternating link represented by a prime diagram is either hyperbolic or a (2,n)-torus link.

A new method for computing the hyperbolic structure of the complement of a hyperbolic link, based on ideal polygons bounding the regions of a …

Orderly Ε-Homotopies Of Discrete Chains, Alexander Thomas Happ

Chancellor’s Honors Program Projects

No abstract provided.

On The Number Of Tilings Of A Square By Rectangles, Timothy Michaels

Chancellor’s Honors Program Projects

No abstract provided.

The Lamplighter Group, Scott Eckenthal

Senior Theses and Projects

The Lamplighter Group is an Algebraic Group whose behavior models the dynamics of a geometric system. In this thesis, a survey paper following a set of notes as written by Professor Jennifer Taback of Bowdoin College, we define this geometric system and the connection to the Lamplighter Group. Several subsequent results are then proven regarding word length of elements in the group and behavior within its Cayley Graph. A preliminary section is included to introduce the reader to several topics including Free Groups, Group Presentation, and the properties of the Cayley Graph.

Odd Or Even: Uncovering Parity Of Rank In A Family Of Rational Elliptic Curves, Anika Lindemann

Honors Theses

Puzzled by equations in multiple variables for centuries, mathematicians have made relatively few strides in solving these seemingly friendly, but unruly beasts. Currently, there is no systematic method for finding all rational values, that satisfy any equation with degree higher than a quadratic. This is bizarre. Solving these has preoccupied great minds since before the formal notion of an equation existed. Before any sort of mathematical formality, these questions were nested in plucky riddles and folded into folk tales. Because they are so simple to state, these equations are accessible to a very general audience. Yet an astounding amount of …