Physical Sciences and Mathematics Commons^™

An Exploration In Ramsey Theory, Jake Weber

Dissertations and Theses @ UNI

We present several introductory results in the realm of Ramsey Theory, a subfield of Combinatorics and Graph Theory. The proofs in this thesis revolve around identifying substructure amidst chaos. After showing the existence of Ramsey numbers of two types, we exhibit how these two numbers are related. Shifting our focus to one of the Ramsey number types, we provide an argument that establishes the exact Ramsey number for h(k, 3) for k ≥ 3; this result is the highlight of this thesis. We conclude with facts that begin to establish lower bounds on these types of Ramsey …

On The Zeta Kirchhoff Index Of Several Graph Transformations, Danny Cheuk

Dissertations and Theses @ UNI

In this paper, we first derived the Ihara zeta function, complexity and zeta Kirchhoff index of the k-th semitotal point graph (of regular graphs), a construction by Cui and Hou [5], where we create triangles for every edge in the original graph. Then, we extend the construction to the creation of equilaterals and polygons.

We also derived the zeta Kirchhoff indices for numerous graph transformations on regular graphs, and some selected families of graphs.

At the end, a data summary is provided for enumeration computed on simple connected md2 graphs up to degree 10.

Properties Of Left-Separated Spaces And Their Unions, Eric Scheidecker

Dissertations and Theses @ UNI

Left-separated spaces are topological spaces which can be well ordered such that every initial segment is closed. In this paper, we examine what topological properties imply left-separation, and under what circumstances left-separation is preserved by unions. We also introduce several known theorems regarding elementary submodels as they are one of the primary tools that we use. We prove that for a topological space X;

1. If X has a point-countable base, then X is left-separated if and only if X has closed intersection with any elementary submodel M such that X ∈ M.

2. If every elementary submodel …

The Programmatic Manipulation Of Planar Diagram Codes To Find An Upper Bound On The Bridge Index Of Prime Knots, Genevieve R. Johnson

Dissertations and Theses @ UNI

The “bridge index” of a knot is the least number of maximal overpasses taken over all diagrams of the knot. A naïve method to determine the bridge index of a knot is to perform Reidemeister moves on diagrams of the knot, and this method quickly becomes tedious to implement by hand. In this paper, we introduce a sequence of Reidemeister moves which we call a “drag the underpass” move and prove how planar diagram codes change as Reidemeister moves are performed. We then use these results to programatically perform Reidemeister moves using Python 2.7 to calculate an upper bound on …

A Survey Of Butterfly Diagrams For Knots And Links, Mark Ronnenberg

Dissertations and Theses @ UNI

A “butterfly diagram” is a representation of a knot as a kind of graph on the sphere. This generalization of Thurston’s construction of the Borromean rings was introduced by Hilden, Montesinos, Tejada, and Toro to study the bridge number of knots. In this paper, we study various properties of butterfly diagrams for knots and links. We prove basic some combinatorial results about butterflies and explore properties of butterflies for classes of links, especially torus links. The Wirtinger presentation for the knot group will be adapted to butterfly diagrams, and we translate the Reidemeister moves for knot diagrams into so-called “butterfly …

Some Convergence Properties Of Minkowski Functionals Given By Polytopes, Jesse Moeller

Dissertations and Theses @ UNI

In this work we investigate the behavior of the Minkowski Functionals admitted by a sequence of sets which converge to the unit ball ‘from the inside’. We begin in R 2 and use this example to build intuition as we extend to the more general R n case. We prove, in the penultimate chapter, that convergence ‘from the inside’ in this setting is equivalent to two other characterizations of the convergence: a geometric characterization which has to do with the sizes of the faces of each polytope in the sequence converging to zero, and the convergence of the Minkowski functionals …

D-Spaces In Infinite Products Of Ordinals, Duncan Wright

Dissertations and Theses @ UNI

William Fleissner and Adrienne Stanley showed that, in finite products of ordinals, the following are equivalent: 1. X is a D-space. 2. X is metacompact. 3. X is metalindel¨of. 4. X does not contain a closed subset which is homeomorphic to a stationary subset of a regular, uncountable cardinal. In this paper we construct a counterexample that shows that this equivalence does not extend to infinite products of ordinals. We also introduce a new property, club-separable, which we show implies D for subsets of ωω1. We hope that club-separable will be able to replace property (4) above in order to …

Einstein Metrics On Piecewise-Linear Three-Spheres, Kyle Pitzen

Dissertations and Theses @ UNI

Einstein metrics on manifolds are in some ways the "best" or most symmetric metrics those manifolds will allow. There has been much work on these metrics in the realm of smooth manifolds, and many results have been published. These results are very difficult to compute directly, however, and so it is helpful to consider piecewise-linear approximations to those manifolds in order to more quickly compute and describe what these metrics actually look like. We will use discrete analogues to powerful preexisting tools to do analysis on two particular triangulations of the three dimensional sphere with the intent of finding Einstein …

Fuzzy Logic: An Analysis Of Logical Connectives And Their Characterizations, John F. Hamman

Dissertations and Theses @ UNI

The focus of this thesis is to determine exactly which functions serve as appropriate fuzzy negation, conjunction and disjunction functions. To this end, the first chapter serves as motivation for why fuzzy logic is needed, and includes an original demonstration of the inadequacy of many valued logics to resolve the sorites paradox. Chapter 2 serves as an introduction to fuzzy sets and logic. The canonical fuzzy set of tall men is examined as a motivating example, and the chapter concludes with a discussion of membership functions.

Four desirable conditions of the negation function are given in Chapter 3, but it …

A Partial "Squeezing Theorem" For A Particular Class Of Many-Valued Logics, Stephen Michael Walk

Dissertations and Theses @ UNI

The problem to be studied for this thesis was that of whether the usual statement calculus is a suitable formal system for every many-valued logic in a particular collection of logics. The logics in question are those that fall between the usual two-valued logic and a modified form of the Lukasiewicz-Tarski three-valued logic.

Since this betweenness relationship was an original concept and appeared nowhere in the literature, the first goal in the research plan was to define this relationship precisely. Preliminary concepts included truth value mapping and forgivingness of logics, concepts that, like betweenness, are original to this paper and …