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Full-Text Articles in Physical Sciences and Mathematics
The Axiom Of Choice In Topology, Ruoxuan Jia
The Axiom Of Choice In Topology, Ruoxuan Jia
Honors Theses
Cantor believed that properties holding for finite sets might also hold for infinite sets. One such property involves choices; the Axiom of Choice states that we can always form a set by choosing one element from each set in a collection of pairwise disjoint non-empty sets. Since its introduction in 1904, this seemingly simple statement has been somewhat controversial because it is magically powerful in mathematics in general and topology in particular. In this paper, we will discuss some essential concepts in topology such as compactness and continuity, how special topologies such as the product topology and compactification are defined, …
An Investigation Of The Four Vertex Theorem And Its Converse, Rebeka Kelmar
An Investigation Of The Four Vertex Theorem And Its Converse, Rebeka Kelmar
Honors Theses
In the study of curves there are many interesting theorems. One such theorem is the four vertex theorem and its converse. The four vertex theorem says that any simple closed curve, other than a circle, must have four vertices. This means that the curvature of the curve must have at least four local maxima/minima. In my project I explore different proofs of the four vertex theorem and its history. I also look at a modified converse of the four vertex theorem which says that any continuous real- valued function on the circle that has at least two local maxima and …
Normal Surfaces And 3-Manifold Algorithms, Josh D. Hews
Normal Surfaces And 3-Manifold Algorithms, Josh D. Hews
Honors Theses
This survey will develop the theory of normal surfaces as they apply to the S3 recognition algorithm. Sections 2 and 3 provide necessary background on manifold theory. Section 4 presents the theory of normal surfaces in triangulations of 3-manifolds. Section 6 discusses issues related to implementing algorithms based on normal surfaces, as well as an overview of the Regina, a program that implements many 3-manifold algorithms. Finally section 7 presents the proof of the 3-sphere recognition algorithm and discusses how Regina implements the algorithm.
Some Examples Of The Interplay Between Algebra And Topology, Joseph D. Malionek
Some Examples Of The Interplay Between Algebra And Topology, Joseph D. Malionek
Honors Theses
This thesis presents several undergraduate and graduate level concepts in the fields of algebraic topology and topological group theory in a manner which requires very little mathematical background of the reader. It uses non-rigorous interpretations of concepts while introducing the reader to the rigorous ideas with which they are associated. In order to give the reader an idea of how the fields of algebra and topology are closely affiliated, the paper goes over five main concepts, the fundamental group, homology, cohomology, Eilenberg-Maclane spaces, and group dimension.
Tying The Knot: Applications Of Topology To Chemistry, Tarini S. Hardikar
Tying The Knot: Applications Of Topology To Chemistry, Tarini S. Hardikar
Honors Theses
Chirality (or handedness) is the property that a structure is “different” from its mirror image. Topology can be used to provide a rigorous framework for the notion of chirality. This project examines various types of chirality and discusses tools to detect chirality in graphs and knots. Notable theorems that are discussed in this work include ones that identify chirality using properties of link polynomials (HOMFLY polynomials), rigid vertex graphs, and knot linking numbers. Various other issues of chirality are explored, and some specially unique structures are discussed. This paper is borne out of reading Dr. Erica Flapan’s book, When Topology …