CUNY Queensborough Community College
University of New Mexico
California State University, San Bernardino
Stephen F Austin State University
Missouri State University
Boise State University
Rhode Island College
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 E2-terms coincide with certain …
An Algebraic Characterization Of Total Input Strictly Local Functions,
2023
Universite Jean Monnet
An Algebraic Characterization Of Total Input Strictly Local Functions, Dakotah Lambert, Jeffrey Heinz
Proceedings of the Society for Computation in Linguistics
This paper provides an algebraic characteriza- tion of the total input strictly local functions. Simultaneous, noniterative rules of the form A→B/C D, common in phonology, are defin- able as functions in this class whenever CAD represents a finite set of strings. The algebraic characterization highlights a fundamental con- nection between input strictly local functions and the simple class of definite string languages, as well as connections to string functions stud- ied in the computer science literature, the def- inite functions and local functions. No effec- tive decision procedure for the input strictly local maps was previously available, but one arises …
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 …
Cohen-Macaulay Properties Of Closed Neighborhood Ideals,
2023
Clemson University
Cohen-Macaulay Properties Of Closed Neighborhood Ideals, Jackson Leaman
All Theses
This thesis investigates Cohen-Macaulay properties of squarefree monomial ideals, which is an important line of inquiry in the field of combinatorial commutative algebra. A famous example of this is Villareal’s edge ideal [11]: given a finite simple graph G with vertices x1, . . . , xn, the edge ideal of G is generated by all the monomials of the form xixj where xi and xj are adjacent in G. Villareal’s characterization of Cohen-Macaulay edge ideals associated to trees is an often-cited result in the literature. This was extended to chordal and bipartite graphs by Herzog, Hibi, and Zheng in …
Let’S Make Patterns!: Symmetric Rubik’S Cube Permutations,
2023
University of Nebraska at Omaha
Let’S Make Patterns!: Symmetric Rubik’S Cube Permutations, Danny Anderson
Theses/Capstones/Creative Projects
Rubik’s cubes are well-known for having several different patterns, or permutations, that can be made from them. Meanwhile, cubes generally display a wide variety of symmetries. Naturally, these ideas can be combined to form a notion of "symmetric Rubik's cube patterns." The goal of this paper is to find an algorithm that can produce all of the permutations that display symmetries.
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, …
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 …
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 …
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 Fq-rational solutions, Tate modules, supersingular curves, and applications to elliptic curves over Q. In particular, we approach the topic largely through the use …
