Applications Of Equivariant Topology In Cascading Makeev Problems, 2018 Bard College

#### Applications Of Equivariant Topology In Cascading Makeev Problems, Andres N. Mejia

*Senior Projects Spring 2018*

Many solutions to problems arising in discrete geometry have come from insights in equivariant topology. Configuration-Space/Test Map (CS/TM) type setups, pioneered by Zivaljevic, offer reductions of combinatorial or geometric facts to showing the nonexistence of certain $G$-equivariant maps $f:X \to V\setminus Z$. In particular, partitions of objects by arcs, planes, and convex sets, and Tverberg theorems have been particularly amenable to topological methods , since their solutions affect the global structure of the relevant topological objects. However, there have been limits to the method as demonstrated by a failure to solve of the celeberated and now ...

I’M Being Framed: Phase Retrieval And Frame Dilation In Finite-Dimensional Real Hilbert Spaces, 2018 University of Central Florida

#### I’M Being Framed: Phase Retrieval And Frame Dilation In Finite-Dimensional Real Hilbert Spaces, Jason L. Greuling

*Honors Undergraduate Theses*

Research has shown that a frame for an n-dimensional real Hilbert space oﬀers phase retrieval if and only if it has the complement property. There is a geometric characterization of general frames, the Han-Larson-Naimark Dilation Theorem, which gives us the necessary and suﬃcient conditions required to dilate a frame for an n-dimensional Hilbert space to a frame for a Hilbert space of higher dimension k. However, a frame having the complement property in an n-dimensional real Hilbert space does not ensure that its dilation will oﬀer phase retrieval. In this thesis, we will explore and provide what necessary and suﬃcient ...

3-Maps And Their Generalizations, 2018 Virginia Commonwealth University

#### 3-Maps And Their Generalizations, Kevin J. Mccall

*Theses and Dissertations*

A 3-map is a 3-region colorable map. They have been studied by Craft and White in their paper 3-maps. This thesis introduces topological graph theory and then investigates 3-maps in detail, including examples, special types of 3-maps, the use of 3-maps to find the genus of special graphs, and a generalization known as n-maps.

Extensions Of The Morse-Hedlund Theorem, 2018 Bucknell University

#### Extensions Of The Morse-Hedlund Theorem, Eben Blaisdell

*Honors Theses*

Bi-infinite words are sequences of characters that are infinite forwards and backwards; for example "...*ababababab*...". The Morse-Hedlund theorem says that a bi-infinite word *f* repeats itself, in at most *n* letters, if and only if the number of distinct subwords of length *n* is at most *n*. Using the example, "...*ababababab*...", there are 2 subwords of length 3, namely "*aba*" and "*bab*". Since 2 is less than 3, we must have that "...*ababababab*..." repeats itself after at most 3 letters. In fact it does repeat itself every two letters. Interestingly, there are many extensions of this theorem to multiple dimensions ...

Geometric Serendipity, 2018 Virginia Commonwealth University

#### Geometric Serendipity, Dakota Becker

*Auctus: The Journal of Undergraduate Research and Creative Scholarship*

The central focus of my practice is the serendipitous exploration into geometry, symmetry, design, and color. I have found more and more that the affinity I have for hard-edge geometric abstraction is a deeper reflection of the way in which I process my thoughts and surroundings. In the past year, I have sought to challenge myself by questioning the core of my practice and pushing it to go beyond its individual elements. In this way, I seek to create work that is more than its parts. As a result, I have become more purposeful with my designs and push both ...

The Role Of Topology In Magnetic Solitary Wave Dynamics, 2018 University of Colorado at Boulder

#### The Role Of Topology In Magnetic Solitary Wave Dynamics, Maximilian Emil Ruth

*Applied Mathematics Graduate Theses & Dissertations*

Topological solitary waves have recently attracted attention from the applied mathematics and physics communities because of both their perceived robustness and technological applications, e.g. storage and logic. In the field of magnetism, topological structures include the one-dimensional domain wall and the two-dimensional magnetic skyrmion. Topology in these structures is the result of a quantized winding number, as the magnetization vector is restricted to the unit sphere. The winding number provides a notion of “topological protection”, meaning that topological wave structures cannot be continuously deformed into other structures with different winding numbers. This thesis presents two problems in magnetic solitary ...

Advanced Enrichment Topics In An Honors Geometry Course, 2018 John Carroll University

#### Advanced Enrichment Topics In An Honors Geometry Course, Kayla Woods

*Masters Essays*

No abstract provided.

Self-Assembly Of Dna Graphs And Postman Tours, 2018 University of North Florida

#### Self-Assembly Of Dna Graphs And Postman Tours, Katie Bakewell

*UNF Graduate Theses and Dissertations*

DNA graph structures can self-assemble from branched junction molecules to yield solutions to computational problems. Self-assembly of graphs have previously been shown to give polynomial time solutions to hard computational problems such as 3-SAT and k-colorability problems. Jonoska et al. have proposed studying self-assembly of graphs topologically, considering the boundary components of their thickened graphs, which allows for reading the solutions to computational problems through reporter strands. We discuss weighting algorithms and consider applications of self-assembly of graphs and the boundary components of their thickened graphs to problems involving minimal weight Eulerian walks such as the Chinese Postman Problem and ...

Uses Of Mathematics In Computer Animation And 3d Rendering Software, 2018 University of Colorado, Boulder

#### Uses Of Mathematics In Computer Animation And 3d Rendering Software, Peter Rock

*Undergraduate Honors Theses*

As the title suggests, this paper discusses the applications of several mathematical concepts to computer animation software generally used in the creation of movies and video games. Topics covered will include differential forms, conformal maps, surface texturing, and lighting techniques. It is not the goal of this paper to present anything particularly novel to the mathematical community, but rather to present something that is entertaining to read that will hopefully engage both mathematicians and sane people alike. This paper has been carefully crafted so that it should be accessible to most people with a Calc. I background. That being said ...

A Journey To The Adic World, 2018 Georgia Southern University

#### A Journey To The Adic World, Fayadh Kadhem

*Electronic Theses and Dissertations*

The first idea of this research was to study a topic that is related to both Algebra and Topology and explore a tool that connects them together. That was the entrance for me to the “adic world”. What was needed were some important concepts from Algebra and Topology, and so they are treated in the first two chapters.

The reader is assumed to be familiar with Abstract Algebra and Topology, especially with Ring theory and basics of Point-set Topology.

The thesis consists of a motivation and four chapters, the third and the fourth being the main ones. In the third ...

Using Geogebra To Explore Properties Of Circles In Euclidean Geometry, 2018 John Carroll University

#### Using Geogebra To Explore Properties Of Circles In Euclidean Geometry, Erin Hanna

*Masters Essays*

No abstract provided.

From Convergence In Measure To Convergence Of Matrix-Sequences Through Concave Functions And Singular Values, 2017 Scuola Normale Superiore, Pisa, Italy

#### From Convergence In Measure To Convergence Of Matrix-Sequences Through Concave Functions And Singular Values, Giovanni Barbarino, Carlo Garoni

*Electronic Journal of Linear Algebra*

Sequences of matrices with increasing size naturally arise in several areas of science, such as, for example, the numerical discretization of differential and integral equations. An approximation theory for sequences of this kind has recently been developed, with the aim of providing tools for computing their asymptotic singular value and eigenvalue distributions. The cornerstone of this theory is the notion of approximating classes of sequences (a.c.s.), which is also fundamental to the theory of generalized locally Toeplitz (GLT) sequences, and hence to the spectral analysis of PDE discretization matrices. Drawing inspiration from measure theory, here it is introduced ...

Introduction To The Usu Library Of Solutions To The Einstein Field Equations, 2017 ian.anderson@usu.edu

#### Introduction To The Usu Library Of Solutions To The Einstein Field Equations, Ian M. Anderson, Charles G. Torre

*Tutorials on... in 1 hour or less*

This is a Maple worksheet providing an introduction to the USU Library of Solutions to the Einstein Field Equations. The library is part of the DifferentialGeometry software project and is a collection of symbolic data and metadata describing solutions to the Einstein equations.

Classification Of Minimal Separating Sets In Low Genus Surfaces, 2017 Portland State University

#### Classification Of Minimal Separating Sets In Low Genus Surfaces, J. J. P. Veerman, William Maxwell, Victor Rielly, Austin K. Williams

*Mathematics and Statistics Faculty Publications and Presentations*

Consider a surface *S* and let *M* ⊂ *S*. If *S* \ *M* is not connected, then we say *M* separates *S*, and we refer to *M* as a separating set of *S*. If *M* separates *S*, and no proper subset of *M* separates *S*, then we say *M* is a minimal separating set of *S*. In this paper we use computational methods of combinatorial topology to classify the minimal separating sets of the orientable surfaces of genus *g* = 2 and *g* = 3. The classification for genus 0 and 1 was done in earlier work, using methods of algebraic topology.

Constructing A Square An Ancient Indian Way Activity, 2017 Pittsburg State University

#### Constructing A Square An Ancient Indian Way Activity, Cynthia J. Huffman Ph.D.

*Open Educational Resources - Math*

In this activity students use string to model one of the ways that was used in ancient India for constructing a square. The construction was used in building a temporary fire altar. The activity is based on a translation by Sen and Bag of the Baudhāyana-śulba-sūtra.

Constructing A Square Indian Fire Altar Activity, 2017 Pittsburg State University

#### Constructing A Square Indian Fire Altar Activity, Cynthia J. Huffman Ph.D.

*Open Educational Resources - Math*

In this activity, we will model constructing a square fire altar with a method similar to one used by people in ancient India. The fire altars, which were made of bricks, had various shapes. Instructions for building the altars were in Vedic texts called *Śulba-sūtras*. We will follow instructions for constructing a square *gārhapatya *fire altar* *from the *Baudhāyana-śulba-sūtra*, which was written during the Middle Vedic period, about 800-500 BC.

Euler Construction Activity, 2017 Pittsburg State University

#### Euler Construction Activity, Cynthia J. Huffman Ph.D.

*Open Educational Resources - Math*

Original sources of mathematics provide many opportunities for students to both do mathematics and to improve their problem solving skills. It is also interesting to explore original sources in new ways with the use of technology. In this activity, students can gain experience with dynamic geometry software and enhance their geometric intuition by working through a construction given by Euler in 1783.

Statistical Computational Topology And Geometry For Understanding Data, 2017 University of Tennessee, Knoxville

#### Statistical Computational Topology And Geometry For Understanding Data, Joshua Lee Mike

*Doctoral Dissertations*

Here we describe three projects involving data analysis which focus on engaging statistics with the geometry and/or topology of the data.

The first project involves the development and implementation of kernel density estimation for persistence diagrams. These kernel densities consider neighborhoods for every feature in the center diagram and gives to each feature an independent, orthogonal direction. The creation of kernel densities in this realm yields a (previously unavailable) full characterization of the (random) geometry of a dataspace or data distribution.

In the second project, cohomology is used to guide a search for kidney exchange cycles within a kidney ...

Localization Of Large Scale Structures, 2017 University of Tennessee, Knoxville

#### Localization Of Large Scale Structures, Ryan James Jensen

*Doctoral Dissertations*

We begin by giving the definition of coarse structures by John Roe, but quickly move to the equivalent concept of large scale geometry given by Jerzy Dydak. Next we present some basic but often used concepts and results in large scale geometry. We then state and prove the equivalence of various definitions of asymptotic dimension for arbitrary large scale spaces. Some of these are generalizations of asymptotic dimension for metric spaces, and many of the proofs are new. Particularly useful in proving the equivalences of the various definitions is the notion of partitions of unity, originally set forth by Jerzy ...

Generalizations Of Coarse Properties In Large Scale Spaces, 2017 University of Tennessee, Knoxville

#### Generalizations Of Coarse Properties In Large Scale Spaces, Kevin Michael Sinclair

*Doctoral Dissertations*

Many results in large scale geometry are proven for a metric space. However, there exists many large scale spaces that are not metrizable. We generalize several concepts to general large scale spaces and prove relationships between them. First we look into the concept of coarse amenability and other variations of amenability on large scale spaces. This leads into the definition of coarse sparsification and connections with coarse amenability. From there, we look into an equivalence of Sako's definition of property A on uniformly locally finite spaces and prove that finite coarse asymptotic definition implies it. As well, we define ...