Math 115: College Algebra For Pre-Calculus, 2023 CUNY Queens College

#### Math 115: College Algebra For Pre-Calculus, Seth Lehman

*Open Educational Resources*

OER course syllabus for Math 115, College Algebra, at Queens College

Interpolation Problems And The Characterization Of The Hilbert Function, 2023 University of Arkansas, Fayetteville

#### Interpolation Problems And The Characterization Of The Hilbert Function, Bryant Xie

*Mathematical Sciences Undergraduate Honors Theses*

In mathematics, it is often useful to approximate the values of functions that are either too awkward and difficult to evaluate or not readily differentiable or integrable. To approximate its values, we attempt to replace such functions with more well-behaving examples such as polynomials or trigonometric functions. Over the algebraically closed field C, a polynomial passing through r distinct points with multiplicities m1, ..., mr on the affine complex line in one variable is determined by its zeros and the vanishing conditions up to its mi − 1 derivative for each point. A natural question would then be to consider …

Pairings In A Ring Spectrum-Based Bousfield-Kan Spectral Sequence, 2023 The Graduate Center, City University of New York

#### Pairings In A Ring Spectrum-Based Bousfield-Kan Spectral Sequence, Jonathan Toledo

*Dissertations, Theses, and Capstone Projects*

Bousfield and Kan traditionally formulated their homotopy spectral sequence over a simplicial set X resolved with respect to a ring R. By considering an adequate category of ring spectra, one can take a ring spectrum E, create from it a functor of a triple on the category of simplicial sets, and build a cosimplicial simplicial set **EX**. The homotopy spectral sequence can then be formed over such cosimplicial spaces by a similar construction to the original. Pairings can be established on these spectral sequences, and, for nice enough spaces, these pairings on the E_{2}-terms coincide with certain …

G-Coatomic Modules, 2023 Department of Mathematics, College of Education for Pure Sciences, University of Thi-Qar, Thi-Qar, Iraq

#### G-Coatomic Modules, Ahmed H. Alwan

*Al-Bahir Journal for Engineering and Pure Sciences*

Let *R* be a ring and *M* be a right *R*-module. A submodule of is said to be g-small in , if for every submodule , with implies that . Then is a g-small submodule of . We call g-coatomic module whenever and then . Also, is called right (left) g-coatomic ring if the right (left) -module (R) is g-coatomic. In this work, we study g-coatomic modules and ring. We investigate some properties of these modules. We prove is g-coatomic if and only if each is g-coatomic. It is proved that if is a g-semiperfect ring with , then …

On The Superabundance Of Singular Varieties In Positive Characteristic, 2023 University of Nebraska-Lincoln

#### On The Superabundance Of Singular Varieties In Positive Characteristic, Jake Kettinger

*Dissertations, Theses, and Student Research Papers in Mathematics*

The geproci property is a recent development in the world of geometry. We call a set of points Z\subseq\P_k^3 an (a,b)-geproci set (for GEneral PROjection is a Complete Intersection) if its projection from a general point P to a plane is a complete intersection of curves of degrees a and b. Examples known as grids have been known since 2011. Previously, the study of the geproci property has taken place within the characteristic 0 setting; prior to the work in this thesis, a procedure has been known for creating an (a,b)-geproci half-grid for 4\leq a\leq b, but it was not …

Automorphisms Of A Generalized Quadrangle Of Order 6, 2023 William & Mary

#### Automorphisms Of A Generalized Quadrangle Of Order 6, Ryan Pesak

*Undergraduate Honors Theses*

In this thesis, we study the symmetries of the putative generalized quadrangle of order 6. Although it is unknown whether such a quadrangle **Q** can exist, we show that if it does, that **Q** cannot be transitive on either points or lines. We first cover the background necessary for studying this problem. Namely, the theory of groups and group actions, the theory of generalized quadrangles, and automorphisms of GQs. We then prove that a generalized quadrangle **Q** of order 6 cannot have a point- or line-transitive automorphism group, and we also prove that if a group *G* acts faithfully on …

A Graphical User Interface Using Spatiotemporal Interpolation To Determine Fine Particulate Matter Values In The United States, 2023 Georgia Southern University

#### A Graphical User Interface Using Spatiotemporal Interpolation To Determine Fine Particulate Matter Values In The United States, Kelly M. Entrekin

*Honors College Theses*

Fine particulate matter or PM2.5 can be described as a pollution particle that has a diameter of 2.5 micrometers or smaller. These pollution particle values are measured by monitoring sites installed across the United States throughout the year. While these values are helpful, a lot of areas are not accounted for as scientists are not able to measure all of the United States. Some of these unmeasured regions could be reaching high PM2.5 values over time without being aware of it. These high values can be dangerous by causing or worsening health conditions, such as cardiovascular and lung diseases. Within …

Sl(2,Z) Representations And 2-Semiregular Modular Categories, 2023 Louisiana State University and Agricultural and Mechanical College

#### Sl(2,Z) Representations And 2-Semiregular Modular Categories, Samuel Nathan Wilson

*LSU Doctoral Dissertations*

We address the open question of which representations of the modular group SL(2,Z) can be realized by a modular category. In order to investigate this problem, we introduce the concept of a symmetrizable representation of SL(2,Z) and show that this property is necessary for the representation to be realized. We then prove that all congruence representations of SL(2,Z) are symmetrizable. The proof involves constructing a symmetric basis, which greatly aids in further calculation. We apply this result to the reconstruction of modular category data from representations, as well as to the classification of semiregular categories, which are defined via an …

Unique Factorization In The Rings Of Integers Of Quadratic Fields: A Method Of Proof, 2023 Liberty University

#### Unique Factorization In The Rings Of Integers Of Quadratic Fields: A Method Of Proof, Zachary Warren

*Senior Honors Theses*

It is a well-known property of the integers, that given any nonzero *a* ∈ **Z**, where *a* is not a unit, we are able to write *a* as a unique product of prime numbers. This is because the Fundamental Theorem of Arithmetic (FTA) holds in the integers and guarantees (1) that such a factorization exists, and (2) that it is unique. As we look at other domains, however, specifically those of the form **O**(√*D*) = {*a* + *b*√*D* | *a*, *b* ∈ **Z**, *D* a negative, squarefree integer}, we find that …

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.

From Mirrors To Wallpapers: A Virtual Math Circle Module On Symmetry, 2023 Central New Mexico Community College

#### From Mirrors To Wallpapers: A Virtual Math Circle Module On Symmetry, Nicole A. Sullivant, Christina L. Duron, Douglas T. Pfeffer

*Journal of Math Circles*

Symmetry is a natural property that children see in their everyday lives; it also has deep mathematical connections to areas like tiling and objects like wallpaper groups. The Tucson Math Circle (TMC) presents a 7-part module on symmetry that starts with reflective symmetry and culminates in the deconstruction of wallpapers into their ‘generating tiles’. This module utilizes a scaffolded, hands-on approach to cover old and new mathematical topics with various interactive activities; all activities are made available through free web-based platforms. In this paper, we provide lesson plans for the various activities used, and discuss their online implementation with Zoom, …

Strong Homotopy Lie Algebras And Hypergraphs, 2023 Virginia Commonwealth University

#### Strong Homotopy Lie Algebras And Hypergraphs, Samuel J. Bevins, Marco Aldi

*Undergraduate Research Posters*

We study hypergraphs by attaching a nilpotent strong homotopy Lie algebra. We especially focus on hypergraph theoretic information that is encoded in the cohomology of the resulting strong homotopy Lie algebra.

Q-Polymatroids And Their Application To Rank-Metric Codes., 2023 University of Kentucky

#### Q-Polymatroids And Their Application To Rank-Metric Codes., Benjamin Jany

*Theses and Dissertations--Mathematics*

Matroid theory was first introduced to generalize the notion of linear independence. Since its introduction, the theory has found many applications in various areas of mathematics including coding theory. In recent years, q-matroids, the q-analogue of matroids, were reintroduced and found to be closely related to the theory of linear vector rank metric codes. This relation was then generalized to q-polymatroids and linear matrix rank metric codes. This dissertation aims at developing the theory of q-(poly)matroid and its relation to the theory of rank metric codes. In a first part, we recall and establish preliminary results for both q-polymatroids and …

Elliptic Curves Over Finite Fields, 2023 Colby College

#### Elliptic Curves Over Finite Fields, Christopher S. Calger

*Honors Theses*

The goal of this thesis is to give an expository report on elliptic curves over finite fields. We begin by giving an overview of the necessary background in algebraic geometry to understand the definition of an elliptic curve. We then explore the general theory of elliptic curves over arbitrary fields, such as the group structure, isogenies, and the endomorphism ring. We then study elliptic curves over finite fields. We focus on the number of **F**_{q}-rational solutions, Tate modules, supersingular curves, and applications to elliptic curves over **Q**. In particular, we approach the topic largely through the use …

Understanding And Advancing College Students' Mathematical Reasoning Using Collaborative Argumentation, 2023 Missouri State University

#### Understanding And Advancing College Students' Mathematical Reasoning Using Collaborative Argumentation, Rachel Kay Heili

*MSU Graduate Theses*

This study explored students’ mathematical reasoning skills and offered supports to advance them through a collaborative argumentation framework in a college intermediate algebra class. The goals of this study were to make observations about student reasoning, identify specific actions to address those observations, and document student growth in reasoning as a result of those actions. An iterative analysis, mixed method study was conducted in which the researcher engaged students in responding to questions that required conceptual understandings using a collaborative argumentation framework as a tool to identify and code components of their responses—claim, evidence, and reasoning. After coding and analyzing …

Higher Spanier Groups, 2023 West Chester University

#### Higher Spanier Groups, Johnny Aceti

*West Chester University Master’s Theses*

When non-trivial local structures are present in a topological space X, a common ap- proach to characterizing the isomorphism type of the n-th homotopy group πn(X, x0) is to consider the image of πn(X, x0) in the n-th ˇCech homotopy group ˇπn(X, x0) under the canonical homomorphism Ψn : πn(X, x0) → ˇπn(X, x0). The subgroup ker Ψn is the obstruc- tion to this tactic as it consists of precisely those elements of πn(X, x0), which cannont be detected by polyhedral approximations to X. In this paper we present a definition of higher dimensional analouges of Thick Spanier groups use …

The Lie Algebra Sl2(C) And Krawtchouk Polynomials, 2023 University of North Florida

#### The Lie Algebra Sl2(C) And Krawtchouk Polynomials, Nkosi Alexander

*UNF Graduate Theses and Dissertations*

The Lie algebra L = sl2(C) consists of the 2 × 2 complex matrices that have trace zero, together with the Lie bracket [y, z] = yz − zy. In this thesis we study a relationship between L and Krawtchouk polynomials. We consider a type of element in L said to be normalized semisimple. Let a, a^∗ be normalized semisimple elements that generate L. We show that a, a^∗ satisfy a pair of relations, called the Askey-Wilson relations. For a positive integer N, we consider an (N + 1)-dimensional irreducible L-module V consisting of the homogeneous polynomials in two variables …

Explorations In Well-Rounded Lattices, 2023 Claremont Colleges

#### Explorations In Well-Rounded Lattices, Tanis Nielsen

*HMC Senior Theses*

Lattices are discrete subgroups of Euclidean spaces. Analogously to vector spaces, they can be described as spans of collections of linearly independent vectors, but with integer (instead of real) coefficients. Lattices have many fascinating geometric properties and numerous applications, and lattice theory is a rich and active field of theoretical work. In this thesis, we present an introduction to the theory of Euclidean lattices, along with an overview of some major unsolved problems, such as sphere packing. We then describe several more specialized topics, including prior work on well-rounded ideal lattices and some preliminary results on the study of planar …

An Inquiry Into Lorentzian Polynomials, 2023 Harvey Mudd College

#### An Inquiry Into Lorentzian Polynomials, Tomás Aguilar-Fraga

*HMC Senior Theses*

In combinatorics, it is often desirable to show that a sequence is unimodal. One method of establishing this is by proving the stronger yet easier-to-prove condition of being log-concave, or even ultra-log-concave. In 2019, Petter Brändén and June Huh introduced the concept of Lorentzian polynomials, an exciting new tool which can help show that ultra-log-concavity holds in specific cases. My thesis investigates these Lorentzian polynomials, asking in which situations they are broadly useful. It covers topics such as matroid theory, discrete convexity, and Mason’s conjecture, a long-standing open problem in matroid theory. In addition, we discuss interesting applications to known …

(R1509) Topsis And Vikor Methods For Spherical Fuzzy Soft Set Aggregating Operator Framework, 2022 Saveetha Institute of Medical and Technical Sciences

#### (R1509) Topsis And Vikor Methods For Spherical Fuzzy Soft Set Aggregating Operator Framework, M. Palanikumar, K. Arulmozhi, Lejo J. Manavalan

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

The Spherical Fuzzy Soft (SFS) set is a generalization of the Pythagorean fuzzy soft set and the intuitionistic fuzzy soft set. We introduce the concept of aggregating SFS decision matrices based on aggregated operations. The techniques for order of preference by similarity to ideal solution (TOPSIS) and viekriterijumsko kompromisno rangiranje (VIKOR) for the SFS approaches are the strong points of multi criteria group decision making (MCGDM), which is various extensions of fuzzy soft sets. We define a score function based on aggregating TOPSIS and VIKOR methods to the SFS-positive and SFS-negative ideal solutions. The TOPSIS and VIKOR methods provide decision-making …