Spectral Sequences And Khovanov Homology, 2023 Dartmouth College

#### Spectral Sequences And Khovanov Homology, Zachary J. Winkeler

*Dartmouth College Ph.D Dissertations*

In this thesis, we will focus on two main topics; the common thread between both will be the existence of spectral sequences relating Khovanov homology to other knot invariants. Our first topic is an invariant *MKh(L)* for links in thickened disks with multiple punctures. This invariant is different from but inspired by both the Asaeda-Pryzytycki-Sikora (APS) homology and its specialization to links in the solid torus. Our theory will be constructed from a *Z^n*-filtration on the Khovanov complex, and as a result we will get various spectral sequences relating *MKh(L)* to *Kh(L)*, *AKh(L)*, and ...

Automorphism-Preserving Color Substitutions On Profinite Graphs, 2022 The University of Western Ontario

#### Automorphism-Preserving Color Substitutions On Profinite Graphs, Michal Cizek

*Electronic Thesis and Dissertation Repository*

Profinite groups are topological groups which are known to be Galois groups. Their free product was extensively studied by Luis Ribes and Pavel Zaleskii using the notion of a profinite graph and having profinite groups act freely on such graphs. This thesis explores a different approach to study profinite groups using profinite graphs and that is with the notion of automorphisms and colors. It contains a generalization to profinite graphs of the theorem of Frucht (1939) that shows that every finite group is a group of automorphisms of a finite connected graph, and establishes a profinite analog of the theorem ...

On The Thom Isomorphism For Groupoid-Equivariant Representable K-Theory, 2022 Dartmouth College

#### On The Thom Isomorphism For Groupoid-Equivariant Representable K-Theory, Zachary J. Garvey

*Dartmouth College Ph.D Dissertations*

This thesis proves a general Thom Isomorphism in groupoid-equivariant KK-theory. Through formalizing a certain pushforward functor, we contextualize the Thom isomorphism to groupoid-equivariant representable K-theory with various support conditions. Additionally, we explicitly verify that a Thom class, determined by pullback of the Bott element via a generalized groupoid homomorphism, coincides with a Thom class defined via equivariant spinor bundles and Clifford multiplication. The tools developed in this thesis are then used to generalize a particularly interesting equivalence of two Thom isomorphisms on TX, for a Riemannian G-manifold X.

Numerical Studies Of Correlated Topological Systems, 2022 University of Tennessee, Knoxville

#### Numerical Studies Of Correlated Topological Systems, Rahul Soni

*Doctoral Dissertations*

In this thesis, we study the interplay of Hubbard U correlation and topological effects in two different bipartite lattices: the dice and the Lieb lattices. Both these lattices are unique as they contain a flat energy band at E = 0, even in the absence of Coulombic interaction. When interactions are introduced both these lattices display an unexpected multitude of topological phases in our U -λ phase diagram, where λ is the spin-orbit coupling strength. We also study ribbons of the dice lattice and observed that they qualitative display all properties of their two-dimensional counterpart. This includes flat bands near the ...

Bbt Acoustic Alternative Top Bracing Cadd Data Set-Norev-2022jun28, 2022 East Tennessee State University

#### Bbt Acoustic Alternative Top Bracing Cadd Data Set-Norev-2022jun28, Bill Hemphill

*STEM Guitar Project’s BBT Acoustic Kit*

This electronic document file set consists of an overview presentation (PDF-formatted) file and companion video (MP4) and CADD files (DWG & DXF) for laser cutting the ETSU-developed alternate top bracing designs and marking templates for the STEM Guitar Project’s BBT (OM-sized) standard acoustic guitar kit. The three (3) alternative BBT top bracing designs in this release are

(a) a one-piece base for the standard kit's (Martin-style) bracing,

(b) 277 Ladder-style bracing, and

(c) an X-braced fan-style bracing similar to traditional European or so-called 'classical' acoustic guitars.

The CADD data set for each of the three (3) top bracing designs includes

(a) a nominal 24" x 18" x 3mm (0.118") Baltic birch plywood laser layout of

(1) the one-piece base with slots,

(2) pre-radiused and pre-scalloped vertical braces with tabs to ensure proper orientation and alignment, and

(3) various gages and jigs and

(b) a nominal 15" x 20" marking template.

The 'provided as is" CADD data is formatted for use on a Universal Laser Systems (ULS) laser cutter digital (CNC) device. Each CADD drawing is also provided in two (2) formats: Autodesk AutoCAD 2007 .DWG and .DXF R12. Users should modify and adapt the CADD data as required to fit their equipment. This CADD data set is released and distributed under a Creative Commons license; users are also encouraged to make changes o the data and share (with attribution) their designs with the worldwide acoustic guitar building community.

On A Relation Between Ado And Links-Gould Invariants, 2022 Louisiana State University at Baton Rouge

#### On A Relation Between Ado And Links-Gould Invariants, Nurdin Takenov

*LSU Doctoral Dissertations*

In this thesis we consider two knot invariants: Akutsu-Deguchi-Ohtsuki(ADO) invariant and Links-Gould invariant. They both are based on Reshetikhin-Turaev construction and as such share a lot of similarities. Moreover, they are both related to the Alexander polynomial and may be considered generalizations of it. By experimentation we found that for many knots, the third order ADO invariant is a specialization of the Links-Gould invariant. The main result of the thesis is a proof of this relation for a large class of knots, specifically closures of braids with five strands.

Unomaha Problem Of The Week (2021-2022 Edition), 2022 University of Nebraska at Omaha

#### Unomaha Problem Of The Week (2021-2022 Edition), Brad Horner, Jordan M. Sahs

*Student Research and Creative Activity Fair*

The University of Omaha math department's Problem of the Week was taken over in Fall 2019 from faculty by the authors. The structure: each semester (Fall and Spring), three problems are given per week for twelve weeks, with each problem worth ten points - mimicking the structure of arguably the most well-regarded university math competition around, the Putnam Competition, with prizes awarded to top-scorers at semester's end. The weekly competition was halted midway through Spring 2020 due to COVID-19, but relaunched again in Fall 2021, with massive changes.

Now there are three difficulty tiers to POW problems, roughly corresponding ...

Introduction To Classical Field Theory, 2022 Department of Physics, Utah State University

#### Introduction To Classical Field Theory, Charles G. Torre

*All Complete Monographs*

This is an introduction to classical field theory. Topics treated include: Klein-Gordon field, electromagnetic field, scalar electrodynamics, Dirac field, Yang-Mills field, gravitational field, Noether theorems relating symmetries and conservation laws, spontaneous symmetry breaking, Lagrangian and Hamiltonian formalisms.

(R1961) On Fuzzy Upper And Lower Theta Star Semicontinuous Multifunctions, 2022 J. J. College of Arts and Science (Autonomous)

#### (R1961) On Fuzzy Upper And Lower Theta Star Semicontinuous Multifunctions, A. Mughil, A. Vadivel, O. Uma Maheswari

*Applications and Applied Mathematics: An International Journal (AAM)*

This work introduces the concepts of fuzzy upper and lower theta star (respectively theta)- semicontinuous multifunction on fuzzy topological spaces in the Sostak sense. In L-fuzzy topological spaces, the mutual relationships of these fuzzy upper (resp. fuzzy lower) theta star (resp. theta)-semicontinuous multifunctions are established, as well as several characterizations and properties. Later, researchers looked at the composition and union of these multifunctions.

(R1898) A Study On Inextensible Flows Of Polynomial Curves With Flc Frame, 2022 Ordu University

#### (R1898) A Study On Inextensible Flows Of Polynomial Curves With Flc Frame, Süleyman Şenyurt, Kemal Eren, Kebire Hilal Ayvacı

*Applications and Applied Mathematics: An International Journal (AAM)*

In this paper, we investigate the inextensible flows of polynomial space curves in R3. We calculate that the necessary and sufficient conditions for an inextensible curve flow are represented as a partial differential equation involving the curvatures. Also, we expressed the time evolution of the Frenet like curve (Flc) frame. Finally, an example of the evolution of the polynomial curve with Flc frame is given and graphed.

(R1956) Neutrosophic Soft E-Compact Spaces And Application Using Entropy Measure, 2022 Government Polytechnic College for Women, Annamalai University

#### (R1956) Neutrosophic Soft E-Compact Spaces And Application Using Entropy Measure, P. Revathi, K. Chitirakala, A. Vadivel

*Applications and Applied Mathematics: An International Journal (AAM)*

In this paper, the concept of neutrosophic soft e-compactness is presented on neutrosophic soft topological spaces using the definition of e-open cover and its types. In addition, neutrosophic soft e-compactness and neutrosophic soft e-separation axioms are associated. Also, the concept of neutrosophic soft locally e-compactness is introduced in neutrosophic soft topological spaces and some of its properties are discussed. Added to that, an application in decision making problem is given using entropy.

Sangaku In Multiple Geometries: Examining Japanese Temple Geometry Beyond Euclid, 2022 Murray State University

#### Sangaku In Multiple Geometries: Examining Japanese Temple Geometry Beyond Euclid, Nathan Hartmann

*Honors College Theses*

When the country of Japan was closed from the rest of the world from 1603 until

1867 during the Edo period, the field of mathematics developed in a different way

from how it developed in the rest of the world. One way we see this development

is through the *sangaku*, the thousands of geometric problems hung in various Shinto and Buddhist temples throughout the country. Written on wooden tablets by people from numerous walks of life, all these problems hold true within Euclidean geometry. During the 1800s, while Japan was still closed, non-Euclidean geometries began to develop across the globe ...

Finite Dimensional Approximation And Pin(2)-Equivariant Property For Rarita-Schwinger-Seiberg-Witten Equations, 2022 University of Arkansas, Fayetteville

#### Finite Dimensional Approximation And Pin(2)-Equivariant Property For Rarita-Schwinger-Seiberg-Witten Equations, Minh Lam Nguyen

*Graduate Theses and Dissertations*

The Rarita-Schwinger operator Q was initially proposed in the 1941 paper by Rarita and Schwinger to study wave functions of particles of spin 3/2, and there is a vast amount of physics literature on its properties. Roughly speaking, 3/2−spinors are spinor-valued 1-forms that also happen to be in the kernel of the Clifford multiplication. Let X be a simply connected Riemannian spin 4−manifold. Associated to a fixed spin structure on X, we define a Seiberg-Witten-like system of non-linear PDEs using Q and the Hodge-Dirac operator d∗ + d+ after suitable gauge-fixing. The moduli space of solutions M ...

Slope Conjecture And Normal Surface Theory, 2022 University of South Alabama

#### Slope Conjecture And Normal Surface Theory, Helene Swanepoel

*Theses and Dissertations*

In this thesis we explore the relationship between quantum link invariants and the geometric and topological properties for a family of pretzel knots P(−2r, 2r + 1, 2r + s), r ≥ 1 and s ≥ 3, as predicted by the Slope conjecture. This conjecture asserts a connection between the degree of the colored Jones polynomial and boundary slopes of these knots. We introduce fundamentals of knot theory and prove that the minimum degree of the Jones polynomial for the family of pretzel knots P(−2r, 2r + 1, 2r + s), r ≥ 1 and s ≥ 3, is −6r − 3r + 3. In addition, we consider ...

The Examination Of The Arithmetic Surface (3, 5) Over Q, 2022 California State University - San Bernardino

#### The Examination Of The Arithmetic Surface (3, 5) Over Q, Rachel J. Arguelles

*Electronic Theses, Projects, and Dissertations*

This thesis is centered around the construction and analysis of the principal arithmetic surface (3, 5) over Q. By adjoining the two symbols i,j, where i^{2} = 3, j^{2} = 5, such that ij = -ji, I can produce a quaternion algebra over Q. I use this quaternion algebra to find a discrete subgroup of SL_{2}(R), which I identify with isometries of the hyperbolic plane. From this quaternion algebra, I produce a large list of matrices and apply them via Mobius transformations to the point (0, 2), which is the center of my Dirichlet domain. This list of ...

John Horton Conway: The Man And His Knot Theory, 2022 East Tennessee State University

#### John Horton Conway: The Man And His Knot Theory, Dillon Ketron

*Electronic Theses and Dissertations*

John Horton Conway was a British mathematician in the twentieth century. He made notable achievements in fields such as algebra, number theory, and knot theory. He was a renowned professor at Cambridge University and later Princeton. His contributions to algebra include his discovery of the Conway group, a group in twenty-four dimensions, and the Conway Constellation. He contributed to number theory with his development of the surreal numbers. His Game of Life earned him long-lasting fame. He contributed to knot theory with his developments of the Conway polynomial, Conway sphere, and Conway notation.

De Rham Cohomology, Homotopy Invariance And The Mayer-Vietoris Sequence, 2022 California State University - San Bernardino

#### De Rham Cohomology, Homotopy Invariance And The Mayer-Vietoris Sequence, Stacey Elizabeth Cox

*Electronic Theses, Projects, and Dissertations*

This thesis will discuss the de Rham cohomology, homotopy invariance and the Mayer-Vietoris sequence. First the necessary information for this thesis is discussed such as differential *p*-forms, the exterior derivative as well as pull back of a map. The de Rham cohomology is defined explicitly, some properties of the de Rham cohomology will also be discussed. It will be shown that the de Rham cohomology is in fact a homotopy invariant as well as some examples using homotopy invariance are provided. Finally the Mayer-Vietoris sequence will be established, an example of using the Mayer-Vietoris sequence to compute the de ...

A Coarse Approach To The Freudenthal Compactification And Ends Of Groups, 2022 University of Tennessee, Knoxville

#### A Coarse Approach To The Freudenthal Compactification And Ends Of Groups, Hussain S. Rashed

*Doctoral Dissertations*

The main purpose of this work is to present a coarse counterpart to the Freudenthal compactification and its corona (the space of ends) that generalizes the Freudenthal compactification of a Freudenthal topological space X (connected, locally connected, locally compact and σ-compact) and its corona; then applying it to groups as coarse space to obtain generalizations to many well-known results in the theory of ends of groups. To this end, we present two constructions:

1. The Coarse Freudenthal compactification of a proper metric space which is a coarse compactification that coincides with the Freudenthal compactification when the metric space is geodesic ...

How To Guard An Art Gallery: A Simple Mathematical Problem, 2022 St. John Fisher College

#### How To Guard An Art Gallery: A Simple Mathematical Problem, Natalie Petruzelli

*The Review: A Journal of Undergraduate Student Research*

The art gallery problem is a geometry question that seeks to find the minimum number of guards necessary to guard an art gallery based on the qualities of the museum’s shape, specifically the number of walls. Solved by Václav Chvátal in 1975, the resulting Art Gallery Theorem dictates that ⌊*n*/3⌋ guards are always sufficient and sometimes necessary to guard an art gallery with *n* walls. This theorem, along with the argument that proves it, are accessible and interesting results even to one with little to no mathematical knowledge, introducing readers to common concepts in both geometry and graph ...

A Fractal Geometry For Hydrodynamics, 2022 Ouachita Baptist University

#### A Fractal Geometry For Hydrodynamics, Jonah Mears

*Honors Theses*

Experiments have shown that objects with uneven surfaces, such as golf balls, can have less drag than those with smooth surfaces. Since fractal surfaces appear naturally in other areas, it must be asked if they can produce less drag than a traditional surface and save energy. Little or no research has been conducted so far on this question. The purpose of this project is to see if fractal geometry can improve boat hull design by producing a hull with low friction.