Algebra Commons^™

College Algebra With Review, Lanee Young Ph.D., Jayme Goetz

All Open Educational Resources

This text is disseminated via the Open Education Resource (OER) LibreTexts Project (https://LibreTexts.org) and like the thousands of other texts available within this powerful platform, it is freely available for reading, printing, and "consuming." The LibreTexts mission is to bring together students, faculty, and scholars in a collaborative effort to provide an accessible, and comprehensive platform that empowers our community to develop, curate, adapt, and adopt openly licensed resources and technologies; through these efforts we can reduce the financial burden born from traditional educational resource costs, ensuring education is more accessible for students and communities worldwide. Most, but …

Representations Of Finite Groups And Diagrammatic Algebras, Hudson Yeend

CMC Senior Theses

Representation theory allows mathematicians to study abstract mathematical objects using the powerful and concrete tools of linear algebra. This thesis aims to present some foundational concepts in representation theory and apply these concepts to specific groups and algebras. We begin by examining representations of finite groups, culminating with a proof of Maschke's theorem. We then use the correspondence between a group and its group algebra to segue into a study of representations of diagrammatic algebras, where we introduce analogous notions of decomposition. We end with a study of quiver representations, noting that Gabriel's theorem and the kQ-modular structure transcend …

Lie-Galois Theory, Giovanni Reed

Honors Undergraduate Theses

Differential equations are a much-studied topic in the field of mathematics, as well as other sciences, such as engineering, economics, and biology. While much is known concerning these, there is still a large gap in our knowledge about such equations. It is important therefore, both to mathematics and other sciences, that we gain a more complete knowledge of differential equations, in particular the nature of their solutions. In this research, we investigate the solution space of linear ordinary differential equations (ODEs) from the standpoint of differential algebra. Differential algebra allows an ODE to be treated similarly to a polynomial, allowing …

Principal Quandles, Jesse Parrish

Electronic Theses and Dissertations

This thesis concerns principal quandles and, as a special case, Alexander quandles. A principal quandle is a coset quandle, Q(G,H, f), in which the subgroup H is trivial. If in addition the group G is abelian, then the coset quandle is called an Alexander quandle. The isomorphism types of Alexander quandles were classified [16], and some progress was made on describing the isomorphism types of principal quandles in [14] and [13]. The first part of this thesis applies the ideas used in Holmes’ paper to give a classification of the isomorphism types of principal quandles. The second …

Dimension Of Subalgebras Of Fomin-Kirillov Algebras, Taro Ikeda, Sirous Homayouni

Annual Student Research Poster Session

One of the central open problems concerning the Fomin-Kirillov algebras was whether their dimension is finite or infinite. This question was recently resolved in one research by C. Barligea, where it was shown that FK(n) is infinite-dimensional for all n ≥ 6. In our work, we investigate the dimension of a certain subalgebra of FK(6).

Universal Centralizers, Morita Abelianization, And Wonderful Models In Lie Theory, Peter Crooks

Funded Research Records

No abstract provided.

Career: Algebra And Representation Theory For Non-Semisimple Topological Field Theory, Matthew Young

Funded Research Records

No abstract provided.

Self-Tor Persistence Of Modules Over Determinantal Rings, Tatheer F. Ajani

Mathematics Dissertations - Archive

Tor-persistence is the claim that Tor of a module with itself is only zero if the module has finite projective dimension. Work done by Avramov, Iyengar, Nasseh, Sather-Wagstaff, and various other authors have proved Tor-persistence of modules over certain rings. In this work, we will prove Tor-persistence for certain modules over determinantal rings, specifically for the hypersurface defined by the determinant of a generic matrix. We will then give an explicit proof that Tor^R_2(M,M) is never zero, that Tor^R_1(M,M)=0, and due to the periodicity of the given free resolution, our result can be extended to the entire complex, showing that …

The Algebra Behind Magic, Lois Carpenter

Undergraduate Research Awards

Card Tricks have long been a staple in the common magician’s repertoire, and while many tricks can be explained through sleight of hand alone, others rely on seemingly random shuffling methods that leave the magician with significant control over the deck. Applied card magic (frequently referred to as ‘cheating’) makes significant use of this ability. Thus, the utility of this topic is clear- anyone with basic mastery of perfect shuffles has complete control over the arrangement of cards in a deck, and with it a fundamental advantage against other players in any game of cards. While most perfect shuffles are …

A Comparative Analysis Of Early Algebraic Thinking Activities From U.S. And Singapore Primary Textbooks, Christian Joel Hernandez

Theses and Dissertations

Many studies highlight the challenges students face when transitioning to algebra at the secondary level. Introducing algebraic concepts and fostering algebraic thinking at the primary level can help mitigate these difficulties. Prior to formal algebra instruction, early algebra can be cultivated as a mode of thinking known as algebraic thinking. Several international curricula, such as Singapore Math, incorporate early algebraic thinking into the early stages of schooling. Singapore Math, renowned for its high performance in international assessments, has been widely adopted by schools seeking to replicate its success.

This study compares two primary-level mathematics curricula—CCSSM-aligned textbooks and Singapore Math—specifically focusing …

Making Sandwiches: A Novel Invariant In D-Module Theory, David Lieberman

Department of Mathematics: Dissertations, Theses, and Student Research

Say I hand you a shape, any shape. It could be a line, it could be a crinkled sheet, it could even be a the intersection of a cone with a 6-dimensional hypersurface embedded in a 7-dimensional space. Your job is to tell me about the pointy bits. This task is easier when you can draw the shape; you can you just point at them. When things get more complicated, we need a bigger hammer.

In a sense, that “bigger hammer” is what the ring of differential operators is to an algebraist. Then we will say some things and stuff …

Math 75: Introduction To Linear Algebra, Sarah K. Merz

Pacific Open Texts

This text is intended to use in a first course of Linear Algebra with a prerequisite of Calculus 1. Topics covered include systems of linear equations, matrix operations and inverses, linear transformations, Markov chains, determinants, eigenvalues and eigenvectors, diagonalization, vector geometry, projections and planes, homogeneous coordinates, subspaces, spanning sets, linear independence, orthogonality, fundamental subspaces, and least squares.

On Properties Of Pair Operations, Sarah Jane Poiani

Mathematics & Statistics ETDs

For any closure operation $\cl$ and interior operation $\ri$ on a class of $R$-modules, we develop the theory of $\cl$-prereductions and $\ri$-postexpansions. A pair operation is a generalization of closure and interior operations. Using Epstein, R.G. and Vassilev's duality \cite{ERGV-nonres}, we show that these notions are in fact dual to each other. We discuss the relationship between the core and hull and prereductions and postexpansions. We further the thematic notion of duality and seek to understand how it arises in the context of properties pair operations can be endowed with and focus on inner product spaces and properties demonstrated by …

College Algebra, Leslie Bain

ATU Faculty OER Book Reviews

Review of OER College Algebra textbook by Carl Stitz, available at https://open.umn.edu/opentextbooks/textbooks/college-algebra

Pairs Of Quadratic Forms Over P-Adic Fields, John Hall

Theses and Dissertations--Mathematics

Given two quadratic forms $Q_1, Q_2$ over a $p$-adic field $K$ in $n$ variables, we consider the pencil $\mathcal{P}_K(Q_1, Q_2)$, which contains all nontrivial $K$-linear combinations of $Q_1$ and $Q_2$. We define $D$ to be the maximal dimension of a subspace in $K^n$ on which $Q_1$ and $Q_2$ both vanish. We define $H$ to be the maximal number of hyperbolic planes that a form in $\mathcal{P}_K(Q_1, Q_2)$ splits off over $K$. We will determine which values for $(D, H)$ are possible for a nonsingular pair of quadratic forms over a $p$-adic field $K$.

Slₖ-Tilings And Paths In ℤᵏ, Zachery T. Peterson

Theses and Dissertations--Mathematics

An SLₖ-frieze is a bi-infinite array of integers where adjacent entries satisfy a certain diamond rule. SL₂-friezes were introduced and studied by Conway and Coxeter. Later, these were generalized to infinite matrix-like structures called tilings as well as higher values of k. A recent paper by Short showed a bijection between bi-infinite paths of reduced rationals in the Farey graph and SL₂-tilings. We extend this result to higher k by constructing a bijection between SLₖ-tilings and certain pairs of bi-infinite strips of vectors in ℤᵏ called paths. The key ingredient in the proof is the relation to Plucker friezes and …

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

Department of Mathematics: Dissertations, Theses, and Student Research

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 …

Mth 125 - Modeling With Exponential Functions, Stivi Manoku

Open Educational Resources

The file includes a variety of problems that emphasize the importance of modeling exponential growth and/or radioactive decay. Through different exercises and problems, the assignment goal is to improve their comprehension of exponential functions and hone their problem-solving abilities.

The Mceliece Cryptosystem As A Solution To The Post-Quantum Cryptographic Problem, Isaac Hanna

Senior Honors Theses

The ability to communicate securely across the internet is owing to the security of the RSA cryptosystem, among others. This cryptosystem relies on the difficulty of integer factorization to provide secure communication. Peter Shor’s quantum integer factorization algorithm threatens to upend this. A special case of the hidden subgroup problem, the algorithm provides an exponential speedup in the integer factorization problem, destroying RSA’s security. Robert McEliece’s cryptosystem has been proposed as an alternative. Based upon binary Goppa codes instead of integer factorization, his cryptosystem uses code scrambling and error introduction to hinder decrypting a message without the private key. This …

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.

The Zariski-Riemann Space As A Universal Model For The Birational Geometry Of A Function Field, Giovan Battista Pignatti Morano Di Custoza

Dissertations, Theses, and Capstone Projects

Given a function field $K$ over an algebraically closed field $k$, we propose to use the Zariski-Riemann space $\ZR (K/k)$ of valuation rings as a universal model that governs the birational geometry of the field extension $K/k$. More specifically, we find an exact correspondence between ad-hoc collections of open subsets of $\ZR (K/k)$ ordered by quasi-refinements and the category of normal models of $K/k$ with morphisms the birational maps. We then introduce suitable Grothendieck topologies and we develop a sheaf theory on $\ZR (K/k)$ which induces, locally at once, the sheaf theory of each normal model. Conversely, given a sheaf …

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 i² = 3, j² = 5, such that ij = -ji, I can produce a quaternion algebra over Q. I use this quaternion algebra to find a discrete subgroup of SL₂(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 …

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.

Varieties Of Nonassociative Rings Of Bol-Moufang Type, Ronald E. White

All NMU Master's Theses

In this paper we investigate Bol-Moufang identities in a more general and very natural setting, \textit{nonassociative rings}.

We first introduce and define common algebras. We then explore the varieties of nonassociative rings of Bol-Moufang type. We explore two separate cases, the first where we consider binary rings, rings in which we make no assumption of it's structure. The second case we explore are rings in which, $2x=0$ implies $x=0$.

Counting The Moduli Space Of Pentagons On Finite Projective Planes, Maxwell Hosler

Senior Independent Study Theses

Finite projective planes are finite incidence structures which generalize the concept of the real projective plane. In this paper, we consider structures of points embedded in these planes. In particular, we investigate pentagons in general position, meaning no three vertices are colinear. We are interested in properties of these pentagons that are preserved by collineation of the plane, and so can be conceived as properties of the equivalence class of polygons up to collineation as a whole. Amongst these are the symmetries of a pentagon and the periodicity of the pentagon under the pentagram map, and a generalization of …

College Algebra Through Problem Solving (2021 Edition), Danielle Cifone, Karan Puri, Debra Maslanko, Ewa Stelmach

Open Educational Resources

This is a self-contained, open educational resource (OER) textbook for college algebra. Students can use the book to learn concepts and work in the book themselves. Instructors can adapt the book for use in any college algebra course to facilitate active learning through problem solving. Additional resources such as classroom assessments and online/printable homework is available from the authors.

Factoring: Difference Of Squares, Thomas Lauria

Open Educational Resources

This lesson plan will explain how to factor basic difference of squares problems

Zn Orbifolds Of Vertex Operator Algebras, Daniel Graybill

Electronic Theses and Dissertations

Given a vertex algebra V and a group of automorphisms of V, the invariant subalgebra V^G is called an orbifold of V. This construction appeared first in physics and was also fundamental to the construction of the Moonshine module in the work of Borcherds. It is expected that nice properties of V such as C2-cofiniteness and rationality will be inherited by V^G if G is a finite group. It is also expected that under reasonable hypotheses, if V is strongly finitely generated and G is reductive, V^G will also be strongly finitely generated. This is an analogue …

On Properties Of Positive Semigroups In Lattices And Totally Real Number Fields, Siki Wang

CMC Senior Theses

In this thesis, we give estimates on the successive minima of positive semigroups in lattices and ideals in totally real number fields. In Chapter 1 we give a brief overview of the thesis, while Chapters 2 – 4 provide expository material on some fundamental theorems about lattices, number fields and height functions, hence setting the necessary background for the original results presented in Chapter 5. The results in Chapter 5 can be summarized as follows. For a full-rank lattice L ⊂ Rd, we are concerned with the semigroup L+ ⊆ L, which denotes the set of all vectors with nonnegative …

Abstract Algebra: Theory And Applications, Thomas W. Judson

eBooks

Tom Judson's Abstract Algebra: Theory and Applications is an open source textbook designed to teach the principles and theory of abstract algebra to college juniors and seniors in a rigorous manner. Its strengths include a wide range of exercises, both computational and theoretical, plus many nontrivial applications. Rob Beezer has contributed complementary material using the open source system, Sage.

An HTML version on the PreText platform is available here.

The first half of the book presents group theory, through the Sylow theorems, with enough material for a semester-long course. The second-half is suitable for a second semester and …