A Stronger Strong Schottky Lemma For Euclidean Buildings, 2023 The Graduate Center, City University of New York

#### A Stronger Strong Schottky Lemma For Euclidean Buildings, Michael E. Ferguson

*Dissertations, Theses, and Capstone Projects*

We provide a criterion for two hyperbolic isometries of a Euclidean building to generate a free group of rank two. In particular, we extend the application of a Strong Schottky Lemma to buildings given by Alperin, Farb and Noskov. We then use this extension to obtain an infinite family of matrices that generate a free group of rank two. In doing so, we also introduce an algorithm that terminates in finite time if the lemma is applicable for pairs of certain kinds of matrices acting on the Euclidean building for the special linear group over certain discretely valued fields.

Translation Of: Familles De Surfaces Isoparamétriques Dans Les Espaces À Courbure Constante, Annali Di Mat. 17 (1938), 177–191, By Élie Cartan., 2023 College of the Holy Cross

#### Translation Of: Familles De Surfaces Isoparamétriques Dans Les Espaces À Courbure Constante, Annali Di Mat. 17 (1938), 177–191, By Élie Cartan., Thomas E. Cecil

*Mathematics Department Faculty Scholarship*

This is an English translation of the article "Familles de surfaces isoparamétriques dans les espaces à courbure constante" which was originally published in Annali di Matematica 17, 177–191 (1938), by Élie Cartan.

A note from Thomas E. Cecil, translator: This is an unofficial translation of the original paper which was written in French. All references should be made to the original paper.

**Mathematics Subject Classification Numbers: 53C40, 53C42, 53B25**

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 *APS(L)*. Our …

Slices Of C_2, Klein-4, And Quaternionic Eilenberg-Mac Lane Spectra, 2023 University of Kentucky

#### Slices Of C_2, Klein-4, And Quaternionic Eilenberg-Mac Lane Spectra, Carissa Slone

*Theses and Dissertations--Mathematics*

We provide the slice (co)towers of \(\Si{V} H_{C_2}\ul M\) for a variety of \(C_2\)-representations \(V\) and \(C_2\)-Mackey functors \(\ul M\). We also determine a characterization of all 2-slices of equivariant spectra over the Klein four-group \(C_2\times C_2\). We then describe all slices of integral suspensions of the equivariant Eilenberg-MacLane spectrum \(H\ulZ\) for the constant Mackey functor over \(C_2\times C_2\). Additionally, we compute the slices and slice spectral sequence of integral suspensions of $H\ulZ$ for the group of equivariance $Q_8$. Along the way, we compute the Mackey functors \(\mpi_{k\rho} H_{K_4}\ulZ\) and $\mpi_{k\rho} H_{Q_8}\ulZ$.

On The Uniqueness Of Continuation Of A Partially Defined Metric, 2023 Institute of Applied Mathematics and Mechanics of the NAS of Ukraine

#### On The Uniqueness Of Continuation Of A Partially Defined Metric, Evgeniy Petrov

*Theory and Applications of Graphs*

The problem of continuation of a partially defined metric can be efficiently studied using graph theory. Let $G=G(V,E)$ be an undirected graph with the set of vertices $V$ and the set of edges $E$. A necessary and sufficient condition under which the weight $w\colon E\to\mathbb R^+$ on the graph $G$ has a unique continuation to a metric $d\colon V\times V\to\mathbb R^+$ is found.

Multi-Trace Matrix Models From Noncommutative Geometry, 2022 The University of Western Ontario

#### Multi-Trace Matrix Models From Noncommutative Geometry, Hamed Hessam

*Electronic Thesis and Dissertation Repository*

Dirac ensembles are finite dimensional real spectral triples where the Dirac operator is allowed to vary within a suitable family of operators and is assumed to be random. The Dirac operator plays the role of a metric on a manifold in the noncommutative geometry context of spectral triples. Thus, integration over the set of Dirac operators within a Dirac ensemble, a crucial aspect of a theory of quantum gravity, is a noncommutative analog of integration over metrics.

Dirac ensembles are closely related to random matrix ensembles. In order to determine properties of specific Dirac ensembles, we use techniques from random …

(R1518) The Dual Spherical Curves And Surfaces In Terms Of Vectorial Moments, 2022 Ordu University

#### (R1518) The Dual Spherical Curves And Surfaces In Terms Of Vectorial Moments, Süleyman Şenyurt, Abdussamet Çalışkan

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

In the article, the parametric expressions of the dual ruled surfaces are expressed in terms of the vectorial moments of the Frenet vectors. The integral invariants of these surfaces are calculated. It is seen that the dual parts of these invariants can be stated by the real terms. Finally, we present examples of the ruled surfaces with bases such as helix and Viviani’s curves.

Manufacturability And Analysis Of Topologically Optimized Continuous Fiber Reinforced Composites, 2022 Embry-Riddle Aeronautical University

#### Manufacturability And Analysis Of Topologically Optimized Continuous Fiber Reinforced Composites, Jesus A. Ferrand

*Doctoral Dissertations and Master's Theses*

Researchers are unlocking the potential of Continuous Fiber Reinforced Composites for producing components with greater strength-to-weight ratios than state of the art metal alloys and unidirectional composites. The key is the emerging technology of topology optimization and advances in additive manufacturing. Topology optimization can fine tune component geometry and fiber placement all while satisfying stress constraints. However, the technology cannot yet robustly guarantee manufacturability. For this reason, substantial post-processing of an optimized design consisting of manual fiber replacement and subsequent Finite Element Analysis (FEA) is still required.

To automate this post-processing in two dimensions, two (2) algorithms were developed. The …

Classifications Of Dupin Hypersurfaces In Lie Sphere Geometry, 2022 College of the Holy Cross

#### Classifications Of Dupin Hypersurfaces In Lie Sphere Geometry, Thomas E. Cecil

*Mathematics Department Faculty Scholarship*

This is a survey of local and global classification results concerning Dupin hypersurfaces in *S ^{n}* (or

**R**

*) that have been obtained in the context of Lie sphere geometry. The emphasis is on results that relate Dupin hypersurfaces to isoparametric hypersurfaces in spheres. Along with these classification results, many important concepts from Lie sphere geometry, such as curvature spheres, Lie curvatures, and Legendre lifts of submanifolds of*

^{n}*S*(or

^{n}**R**

*), are described in detail. The paper also contains several important constructions of Dupin hypersurfaces with certain special properties.*

^{n}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 …

Rendezvous Numbers Of Compact And Connected Spaces, 2022 University of Northern Iowa

#### Rendezvous Numbers Of Compact And Connected Spaces, Kevin Demler, Bill Wood Ph.D.

*Summer Undergraduate Research Program (SURP) Symposium*

The concept of a rendezvous number was originally developed by O. Gross in 1964, and was expanded upon greatly by J. Cleary, S. Morris, and D. Yost in 1986. This number exists for every metric space, yet very little is known about it, and it’s exact value for most spaces is not known. Furthermore, it’s exact value is difficult to calculate, and in most cases we can only find bounds for the value. We focused on their arguments using convexity and applied it to shapes in different metrics and graphs. Using sets of points that stood out (vertices, midpoints) as …

Left-Separation Of Ω1, 2022 University of Northern Iowa

#### Left-Separation Of Ω1, Lukas Stuelke, Adrienne Stanley Ph.D.

*Summer Undergraduate Research Program (SURP) Symposium*

A topological space is left-separated if it can be well-ordered so that every initial segment is closed. Here, we show that all countable ordinal numbers are left-separated. We then prove that a similar method could not work for ω_{1} , using the pressing-down lemma^{1} . We finish by showing that a left-separating well-ordering on ω_{1} necessarily leads to a contradiction.

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 …

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.

General Covariance With Stacks And The Batalin-Vilkovisky Formalism, 2022 University of Massachusetts Amherst

#### General Covariance With Stacks And The Batalin-Vilkovisky Formalism, Filip Dul

*Doctoral Dissertations*

In this thesis we develop a formulation of general covariance, an essential property for many field theories on curved spacetimes, using the language of stacks and the Batalin-Vilkovisky formalism. We survey the theory of stacks, both from a global and formal perspective, and consider the key example in our work: the moduli stack of metrics modulo diffeomorphism. This is then coupled to the Batalin-Vilkovisky formalism–a formulation of field theory motivated by developments in derived geometry–to describe the associated equivariant observables of a theory and to recover and generalize results regarding current conservation.

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

*UNO 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 to …

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.

(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.